U.S. patent application number 14/580492 was filed with the patent office on 2015-06-25 for tunable acoustic gradient index of refraction lens and system.
The applicant listed for this patent is TRUSTEES OF PRINCETON UNIVERSITY. Invention is credited to Craig B. Arnold, Euan McLeod, Alexandre Mermillod-Blondin.
Application Number | 20150177592 14/580492 |
Document ID | / |
Family ID | 39721818 |
Filed Date | 2015-06-25 |
United States Patent
Application |
20150177592 |
Kind Code |
A1 |
Arnold; Craig B. ; et
al. |
June 25, 2015 |
TUNABLE ACOUSTIC GRADIENT INDEX OF REFRACTION LENS AND SYSTEM
Abstract
A tunable acoustic gradient index of refraction (TAG) lens and
system are provided that permit, in one aspect, dynamic selection
of the lens output, including dynamic focusing and imaging. The
system may include a TAG lens and at least one of a source and a
detector of electromagnetic radiation. A controller may be provided
in electrical communication with the lens and at least one of the
source and detector and may be configured to provide a driving
signal to control the index of refraction and to provide a
synchronizing signal to time at least one of the source and the
detector relative to the driving signal. Thus, the controller is
able to specify that the source irradiates the lens (or detector
detects the lens output) when a desired refractive index
distribution is present within the lens, e.g. when a desired lens
output is present.
Inventors: |
Arnold; Craig B.;
(Princeton, NJ) ; McLeod; Euan; (Alhambra, CA)
; Mermillod-Blondin; Alexandre; (Berlin, DE) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
TRUSTEES OF PRINCETON UNIVERSITY |
Princeton |
NJ |
US |
|
|
Family ID: |
39721818 |
Appl. No.: |
14/580492 |
Filed: |
December 23, 2014 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
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14039741 |
Sep 27, 2013 |
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14580492 |
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13473364 |
May 16, 2012 |
8576478 |
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14039741 |
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12528347 |
Mar 11, 2010 |
8194307 |
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PCT/US08/54892 |
Feb 25, 2008 |
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13473364 |
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60998427 |
Oct 10, 2007 |
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60903492 |
Feb 26, 2007 |
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Current U.S.
Class: |
359/305 |
Current CPC
Class: |
G02B 3/0087 20130101;
G02B 7/023 20130101; G02F 1/332 20130101; G02B 3/12 20130101; G02B
3/0081 20130101; G02B 3/14 20130101; G02F 2201/58 20130101; G02F
2001/294 20130101; G02F 1/113 20130101; G02F 2203/18 20130101; G02B
27/0927 20130101; G02B 27/0025 20130101; G02F 1/33 20130101 |
International
Class: |
G02F 1/33 20060101
G02F001/33; G02B 27/09 20060101 G02B027/09; G02B 27/00 20060101
G02B027/00 |
Claims
1. A method for driving a tunable acoustic gradient index of
refraction lens to produce a desired refractive index distribution
within the lens, comprising: selecting a desired refractive index
distribution to be produced within the lens; determining the
frequency response of the lens; using the frequency response to
determine a transfer function of the lens to relate the index
response to voltage input; decomposing the desired refractive index
distribution into its spatial frequencies; \ converting the spatial
frequencies into temporal frequencies; representing the voltage
input as an expansion having voltage coefficients; determining the
voltage coefficients from the representation of the decomposed
refractive index distribution; using the determined voltage
coefficients to determine the voltage input in the time domain; and
driving a tunable acoustic gradient index of refraction lens with
the determined voltage input responsively to a control signal.
2. The method according to claim 1, wherein the step of converting
the spatial frequencies comprises converting the decomposed
refractive index distribution into discrete spatial frequencies to
provide a discretized representation of the decomposed refractive
index distribution.
3. The method according to claim 1, wherein the step of determining
the voltage coefficients is preformed before the step of
representing the voltage input.
4. The method according to claim 1, wherein the step of decomposing
a desired refractive index distribution comprises using a
Fourier-Bessel transform to decompose the refractive index into its
spatial frequencies.
5. The method according to claim 1, wherein the step of determining
the voltage coefficients comprises using an inverse Fourier-Bessel
transform.
6. The method according to claim 1, wherein the step of decomposing
a desired refractive index distribution comprises using a Fourier
transform to decompose the refractive index into its spatial
frequencies.
7. The method according to claim 1, wherein the step of determining
the voltage coefficients comprises using an inverse Fourier
transform.
8. The method according to claim 1, wherein the step of converting
the spatial frequencies, comprises converting into discrete
temporal frequencies that are integer multiples of the ratio of a
frequency with which the desired refractive index distribution
repeats in time within the lens to the speed of sound within the
lens.
9. The method according to claim 1, wherein the desired refractive
index distribution comprises a parabolic refractive index
distribution, where the refractive index in the lens varies as the
square of the radius of the lens.
Description
CROSS-REFERENCE TO RELATED APPLICATION
[0001] This application is a continuation of U.S. patent
application Ser. No. 14/039,741, filed Sep. 27, 2013, which is a
continuation of U.S. patent application Ser. No. 13/473,364, filed
May 16, 2012, which is a continuation of U.S. patent application
Ser. No. 12/528,347, filed Mar. 11, 2010, now issued as U.S. Pat.
No. 8,194,307, which is a national stage entry of International
Application No. PCT/US08/54892, filed Feb. 25, 2008, which claims
the benefit of priority of U.S. Provisional Application Nos.
60/903,492 and 60/998,427, filed on Feb. 26, 2007 and Oct. 10,
2007, respectively, the entire contents of which application(s) are
incorporated herein by reference.
FIELD OF THE INVENTION
[0002] The present invention relates generally to a tunable
acoustic gradient index of refraction (TAG) lens, and more
particularly, but not exclusively, to a TAG lens that is configured
to permit dynamic focusing and imaging.
BACKGROUND OF THE INVENTION
[0003] When it comes to shaping the intensity patterns, wavefronts
of light, or position of an image plane or focus, fixed lenses are
convenient, but often the need for frequent reshaping requires
adaptive optical elements. Nonetheless, people typically settle for
whatever comes out of their laser, be it Gaussian or top hat, and
use fixed lenses to produce a beam with the desired
characteristics. In laser micromachining, for instance, a
microscope objective will provide a sharply focused region of given
area that provides sufficient power density to ablate the
materials.
[0004] However, in a variety of applications, it is useful or even
necessary to have feedback between the beam properties of the
incident light and the materials processes that are induced. A
classical example is using a telescope to image distant objects
through the atmosphere. In this case, the motion of the atmosphere
causes constant perturbations in the wavefront of the light. One
can measure the fluctuations and, using adaptive elements, adjust
the wavefront to cancel out these effects. Still other,
laboratory-based, imaging applications such as ophthalmologic
scanning, confocal microscopy or multiphoton microscopy on living
cells or tissue, would benefit greatly from the use of direct
feedback to correct for wavefront aberrations induced by the sample
under investigation, or to provide rapid scanning through focal
planes.
[0005] Advanced materials processing applications also require
precise beam intensity or wavefront profiles. In these cases,
unlike imaging, one is modifying the properties of a material using
the laser. For instance, laser forward-transfer techniques such as
direct-write printing can deposit complex patterns of
materials--such as metal oxides for energy storage or even living
cells for tissue engineering--onto substrates. In this technique, a
focused laser irradiates and propels a droplet composed of a
mixture of a liquid and the material of interest toward a nearby
substrate. The shape of the intensity profile of the incident laser
plays a critical role in determining the properties of the
deposited materials or the health of a transferred cell. In cases
such as these, the ability to modify the shape of the incident beam
is important, and with the ability to rapidly change the shape, one
adds increased functionality by varying the laser-induced changes
in a material from one spot to another.
[0006] Even traditional laser processes like welding or cutting can
benefit from adaptive optical elements. In welding, a
continuous-wave laser moves over a surface to create a weld bead
between the two materials. Industrial reliability requires uniform
weld beads, but slight fluctuations in the laser, the material, or
the thermal profile can diminish uniformity. Therefore, with
feedback to an adaptive optical element, more consistent and
regular features are possible.
[0007] Whether the purpose is to process material, or simply to
create an image, the applications for adaptive optics are quite
varied. Some require continuous-wave light, others need pulsed
light, but the unifying requirement of all applications is to have
detailed control over the properties of the light, and to be able
to change those properties rapidly so that the overall process can
be optimized.
[0008] Fixed optical elements give great choice in selecting the
wavefront properties of a beam of light, but there exist few
techniques for modifying the beam temporally. The simplest approach
is to mount a lens or a series of lenses on motion control stages.
Then one can physically translate the elements to deflect or
defocus the beam. For instance, this technique is useful for
changing the focus of a beam in order to maintain imaging over a
rough surface, or changing the spot size of a focused beam on a
surface for laser micromachining. However, this approach suffers
from a drawbacks related to large scale motion such as vibrations,
repeatability and resolution. Moreover, it can be slow and
inconvenient for many industrial applications where high
reliability and speed are needed. Nonetheless, for certain
applications such as zoom lenses on security cameras, this is a
satisfactory technique. Recently, more advanced methods of inducing
mechanical changes to lenses involve electric fields or pressure
gradients on fluids and liquid crystals to slowly vary the shape of
an element, thereby affecting its focal length.
[0009] When most people think of adaptive optical elements, they
think of two categories, digital micromirror arrays and spatial
light modulators. A digital mirror array is an array of small
moveable mirrors that can be individually addressed, usually
fabricated with conventional MEMS techniques. The category also
includes large, single-surface mirrors whose surfaces can be
modified with an array of actuators beneath the surface. In either
case, by controlling the angle of the reflecting surfaces, these
devices modulate the wavefront and shape of light reflected from
them. Originally digital mirror arrays had only two positions for
each mirror, but newer designs deliver a range of motion and
angles.
[0010] Spatial light modulators also modify the wavefront of light
incident on them, but they typically rely on an addressable array
of liquid crystal material whose transmission or phase shift varies
with electric field on each pixel.
[0011] Both digital mirror arrays and spatial light modulators have
broad capabilities for modulating a beam of light and thereby
providing adaptive optical control. These are digitally
technologies and can therefore faithfully reproduce arbitrary
computer generated patterns, subject only to the pixilation
limitations. These devices have gained widespread use in many
commercial imaging and projecting technologies. For instance,
digital mirror arrays are commonly used in astronomical
applications, and spatial light modulators have made a great impact
on projection television and other display technology. On the
research front, these devices have enabled a myriad of new
experiments relying on a shaped or changeable spatial pattern such
as in optical manipulation, or holography.
[0012] Although current adaptive optical technologies have been
successful in many applications, they suffer from limitations that
prevent their use under more extreme conditions. For instance, one
of the major limitations of spatial light modulators is the slow
switching speed, typically on the order of only 50-100 Hz. Digital
mirror arrays can be faster, but their cost can be prohibitive.
Also, while these devices are good for small scale applications,
larger scale devices require either larger pixels, leading to
pixilation errors, or they require an untenable number of pixels to
cover the area, decreasing the overall speed and significantly
increasing the cost. Finally, these devices tend to have relatively
low damage thresholds, making them suitable for imaging
applications, but less suitable for high energy/high power laser
processing. Accordingly, there is a need the in the field of
adaptive optics for devices which can overcome current device
limitations, such as speed and energy throughput for materials
processing applications.
SUMMARY OF THE INVENTION
[0013] To overcome some of the aforementioned limitations, the
present invention provides an adaptive-optical element termed by
the inventors as a "tunable-acoustic-gradient index-of-refraction
lens", or simply a "TAG lens." In one exemplary configuration, the
present invention provides a tunable acoustic gradient index of
refraction lens comprising a casing having a cavity disposed
therein for receiving a refractive material capable of changing its
refractive index in response to application of an acoustic wave
thereto. To permit electrical communication with the interior of
the casing, the casing may have an electrical feedthrough port in
the casing wall that communicates with the cavity. A piezoelectric
element may be provided within the casing in acoustic communication
with the cavity for delivering an acoustic wave to the cavity to
alter the refractive index of the refractive material. In the case
where the refractive material is a fluid, the casing may include a
fluid port in the casing wall in fluid communication with the
cavity to permit introduction of a refractive fluid into the
cavity. Additionally, the casing may comprise an outer casing
having a chamber disposed therein and an inner casing disposed
within the chamber of the outer casing, with the cavity disposed
within the inner casing and with the piezoelectric element is
disposed within the cavity.
[0014] In one exemplary configuration the piezoelectric element may
comprise a cylindrical piezoelectric tube for receiving the
refractive material therein. The piezoelectric tube may include an
inner cylindrical surface and an outer cylindrical surface. An
inner electrode may be disposed on the inner cylindrical surface,
and the inner electrode may be wrapped from the inner cylindrical
surface to the outer cylindrical surface to provide an annular
electric contact region for the inner electrode on the outer
cylindrical surface. In another exemplary configuration, the
piezoelectric element may comprise a first and a second planar
piezoelectric element. The first and second planar piezoelectric
elements may be disposed orthogonal to one another in an
orientation for providing the cavity with a rectangular
cross-sectional shape.
[0015] The casing may comprise an optically transparent window
disposed at opposing ends of the casing. At least one of the
windows may include a curved surface and may have optical power.
One or more of the windows may also operate as a filter or
diffracting element or may be partially mirrored.
[0016] In another of its aspects, the present invention provides a
tunable acoustic gradient index of refraction optical system. The
optical system may include a tunable acoustic gradient index of
refraction lens and at least one of a source of electromagnetic
radiation and a detector of electromagnetic radiation. A controller
may be provided in electrical communication with the tunable
acoustic gradient index of refraction lens and at least one of the
source and the detector. The controller may be configured to
provide a driving signal to control the index of refraction of the
lens. The controller may also be configured to provide a
synchronizing signal to time at least one of the emission of
electromagnetic radiation from the source or the detection of
electromagnetic radiation by the detector relative to the
electrical signal controlling the lens. In so doing, the controller
is able to specify that the source irradiates the lens (or detector
detects the lens output) when a desired refractive index
distribution is present within the lens. In this regard, the source
may include a shutter electrically connected to the controller (or
detector) for receiving the synchronizing signal to time the
emission of radiation from the source (or detector).
[0017] The controller may be configured to provide a driving signal
that causes the focal length of the lens to vary with time to
produce a lens with a plurality of focal lengths. In addition, the
controller may be configured to provide a synchronizing signal to
time at least one of the emission of electromagnetic radiation from
the source or the detection of electromagnetic radiation by the
detector to coincide with a specific focal length of the lens. In
another exemplary configuration, the controller may be configured
to provide a driving signal that causes the lens to operate as at
least one of a converging lens and a diverging lens. Likewise the
controller may be configured to provide a driving signal that
causes the lens to operate to produce a Bessel beam output or a
multiscale Bessel beam output. Still further, the controller may be
configured to provide a driving signal that causes the optical
output of the lens to vary with time to produce an output that
comprises a spot at one instance in time and an annular ring at
another instance in time. In such a case, the controller may be
configured to provide a synchronizing signal to time at least one
of the emission of electromagnetic radiation from the source or the
detection of electromagnetic radiation by the detector to coincide
with either the spot or the annular ring output from the lens. As a
further example, the controller may be configured to provide a
driving signal that causes the optical output of the lens to vary
with time to produce an output that comprises a phase mask or an
array of spots. To facilitate the latter, the lens may comprise a
rectangular or square cross-sectional shape.
[0018] As a still further exemplary configuration the controller
may be configured to provide a driving signal that creates a
substantially parabolic refractive index distribution, where the
refractive index in the lens varies as the square of the radius of
the lens. The substantially parabolic refractive index distribution
may exist substantially over the clear aperture of the lens or a
portion of the aperture. In turn, the source of electromagnetic
radiation may emit a beam of electromagnetic radiation having a
width substantially matched to the portion of the clear aperture
over which the refractive index distribution is substantially
parabolic. In this regard the source may include an aperture to
define the width of the emitted beam. Alternatively, the controller
may be configured to provide a driving signal that creates a
plurality of substantially parabolic refractive index distributions
within the lens. The driving signal may comprise a sinusoid, the
sum of at least two sinusoidal driving signals of differing
frequency and/or phase, or may comprise a waveform other than a
single frequency sinusoid.
[0019] In another of its aspects, the present invention provides a
method for driving a tunable acoustic gradient index of refraction
lens to produce a desired refractive index distribution within the
lens. The method includes selecting a desired refractive index
distribution to be produced within the lens, determining the
frequency response of the lens, and using the frequency response to
determine a transfer function of the lens to relate the index
response to voltage input. In addition the method includes
decomposing the desired refractive index distribution into its
spatial frequencies, and converting the spatial frequencies into
temporal frequencies representing the voltage input as an expansion
having voltage coefficients. The method further includes
determining the voltage coefficients from the representation of the
decomposed refractive index distribution, and using the determined
voltage coefficients to determine the voltage input in the time
domain. The method then includes driving a tunable acoustic
gradient index of refraction lens with the determined voltage
input. In this method, the decomposed refractive index distribution
may be converted into discrete spatial frequencies to provide a
discretized representation of the decomposed refractive index
distribution.
[0020] In yet another its aspects, the invention provides a method
for controlling the output of a tunable acoustic gradient index of
refraction optical lens. The method includes providing a tunable
acoustic gradient index of refraction lens having a refractive
index that varies in response to an applied electrical driving
signal, and irradiating the optical input of the lens with a source
of electromagnetic radiation. In addition the method includes
driving the lens with a driving signal to control the index of
refraction within the lens, and detecting the electromagnetic
radiation output from the driven lens with a detector. The method
then includes providing a synchronizing signal to the detector to
select a time to detect the electromagnetic radiation output from
the driven lens when a desired refractive index distribution is
present within the lens.
[0021] In still a further aspect of the invention, a method is
provided for controlling the output of a tunable acoustic gradient
index of refraction optical lens. The method includes providing a
tunable acoustic gradient index of refraction lens having a
refractive index that varies in response to an applied electrical
driving signal, and irradiating the optical input of the lens with
a source of electromagnetic radiation. In addition the method
includes driving the lens with a driving signal to control the
index of refraction within the lens, and detecting the
electromagnetic radiation output from the driven lens with a
detector. The method then includes providing a synchronizing signal
to the detector to select a time to detect the electromagnetic
radiation output from the driven lens when a desired refractive
index distribution is present within the lens.
BRIEF DESCRIPTION OF THE DRAWINGS
[0022] The foregoing summary and the following detailed description
of the preferred embodiments of the present invention will be best
understood when read in conjunction with the appended drawings, in
which:
[0023] FIG. 1A schematically illustrates an exploded view of an
exemplary configuration of a TAG lens in accordance with the
present invention having a cylindrical shape;
[0024] FIG. 1B schematically illustrates an isometric view of
assembled the TAG lens of FIG. 1A;
[0025] FIG. 2 schematically illustrates an exploded view of an
exemplary configuration of a TAG lens similar to that of FIG. 1A
but having a piezoelectric tube that is segmented along the
longitudinal axis;
[0026] FIGS. 3A and 3B schematically illustrate an exploded
side-elevational view and isometric view, respectively, of another
exemplary configuration of a TAG lens in accordance with the
present invention;
[0027] FIG. 4A schematically illustrates an isometric view of
another exemplary configuration of a TAG lens in accordance with
the present invention having a rectangular shape;
[0028] FIG. 4B schematically illustrates the rectangular center
casing of the lens of FIG. 4A;
[0029] FIG. 5 illustrates a characteristic pattern created by
illuminating a circular TAG lens with a wide Gaussian collimated
laser beam;
[0030] FIG. 6 illustrates the dependence of the peak refractive
power of the lens, RP.sub.A, on the inner radius of the lens,
r.sub.0, assuming resonant driving conditions;
[0031] FIG. 7 illustrates the dependence of the peak refractive
power of the lens, RP.sub.A, on the static refractive index,
n.sub.0, assuming resonant driving conditions;
[0032] FIG. 8 illustrates the dependence of the peak refractive
power, RP.sub.A, on fluid sound speed, c.sub.s, assuming resonant
driving conditions;
[0033] FIG. 9 illustrates the dependence of refractive power on
driving frequency f=.omega./(2.pi.), assuming resonant driving
conditions;
[0034] FIG. 10 illustrates the nonresonant dependence of the
refractive power of the lens, RP.sub.A, on the inner radius of the
lens, r.sub.0;
[0035] FIG. 11 illustrates the nonresonant dependence of refractive
power, RP.sub.A, on fluid sound speed, c.sub.s;
[0036] FIG. 12 illustrates the nonresonant dependence of refractive
power on driving frequency f=.omega./(2.pi.);
[0037] FIG. 13 schematically illustrates a flow chart of a process
for solving the inverse problem of specifying the driving waveform
required to produce a desired refractive index profile;
[0038] FIG. 14 schematically illustrates a ray diagram showing a
TAG lens acting as a simple converging lens;
[0039] FIG. 15 illustrates both the goal and the actual deflection
angle as a function of radius at time t=0;
[0040] FIG. 16 illustrates both the goal and the actual refractive
index as a function of radius at time t=0;
[0041] FIG. 17 illustrates both the continuous and discretized
spatial frequencies of n.sub.goal(r);
[0042] FIG. 18 illustrates one period of the time domain voltage
signal required to generate the actual lensing effects portrayed in
FIGS. 15 and 16;
[0043] FIGS. 19A and 19B illustrate characteristic TAG-generated
multiscale Bessel beams, with each figure showing two major rings
plus the central major spot, FIG. 19A showing the pattern at a low
driving amplitude (30 V) without minor rings and FIG. 19B showing
the pattern at a higher driving amplitude (65 V) with many minor
rings;
[0044] FIGS. 20A and 20B schematically illustrate an experimental
setup used to study the TAG beam characteristics and the coordinate
system utilized, respectively;
[0045] FIG. 21A illustrates the predicted index profile at one
instant in time, with a linear approximation to the central peak
(dashed line);
[0046] FIG. 21B illustrates the predicted index profile one
half-period later in time than that shown in FIG. 21A, with linear
approximations made to the two central peaks (dashed line), with
the scale of the spatial axis set by the driving frequency, in this
case, 497.5 kHz;
[0047] FIGS. 21C and 21D illustrate theoretical predictions for the
instantaneous intensity patterns corresponding to a and b,
respectively, observed with 355 nm laser light 50 cm behind the TAG
lens with n.sub.A=1.5.times.10.sup.-5 and scale bars set at 2 mm
long;
[0048] FIGS. 21E and 21F show stroboscopic experimental images
obtained in conditions identical to those of FIGS. 21C and 21D with
the laser repetition rate synchronized to the TAG driving frequency
and TAG lens driving amplitude of 5 V;
[0049] FIG. 22 illustrates the experimentally determined
time-average intensity enhancement and propagation of the TAG
central spot and first major ring with the lens driven at 257.0 kHz
with an amplitude of 37.2 V, and with the x and z axes having
significantly different scales--note the characteristic fringe
patterns emanating from each peak in the index profile (cf. FIG.
21);
[0050] FIG. 23 illustrates the experimental and theoretical
intensity profile of the TAG beam imaged 70 cm behind the lens
showing that the fringe patterns extend similar to what one would
expect from an axicon, with the lens driven at 257.0 kHz with an
amplitude of 37.2 V, and for the theory, the value of n.sub.A is
4.times.10.sup.-5;
[0051] FIG. 24 illustrates the beam divergence of the theoretical
TAG, experimental TAG, Gaussian, and exact Bessel beams, with the
TAG and Gaussian beams achieve their maximum intensity
approximately 58 cm behind the lens, all beams having the same beam
width at this location, and with the TAG lens driven at 257.0 kHz
with an amplitude of 37.2 V, and for the theory, the value of
n.sub.A is 4.times.10.sup.-5;
[0052] FIG. 25 illustrates propagation similar to FIG. 22 with a
1.25 mm diameter circular obstruction placed 27 cm behind the lens,
with the TAG lens driven at 332.1 kHz with an amplitude of 5 V;
[0053] FIG. 26 illustrates experimental and theoretical locations
of the first major ring as a function of driving frequency, with
the solid line representing the theory given by Eq. 69, the squares
representing this theory, but also account for deflection in
optical propagation due to the asymmetry of the refractive index on
either side of the major ring, and with the remaining symbols
representing experimental results from various trials;
[0054] FIG. 27A illustrates experimental variation in the intensity
enhancement 50 cm behind the lens as a function of driving
amplitude, with the TAG lens driven at 257.0 kHz;
[0055] FIG. 27B illustrates theoretical variation in the intensity
enhancement 50 cm behind the lens as a function of driving
amplitude when driving the lens at 257.0 kHz, showing good
agreement with FIG. 27A;
[0056] FIG. 28 illustrates an experimental and theoretical central
spot size as a function of driving amplitude when the lens is
driven at 257.0 kHz and the beam is imaged 50 cm behind the lens,
with error bars representing the size of a camera pixel;
[0057] FIG. 29 schematically illustrates an experimental setup with
a pair of lenses L.sub.1 and L.sub.2 forming a telescope to reduce
the size of the beam for micromachining, and with the delay between
the AC signal and the laser pulse set by a pulse delay generator,
the inset image on the left showing the spatial profile of the
incident Gaussian beam, and the inset image on the right showing
the resulting annular beam after passing through the TAG lens;
[0058] FIG. 30 illustrates intensity images of instantaneous
patterns obtained by changing the TAG lens driving frequency, with
pictures taken at a distance of 50 cm away from the lens, and
frequencies from left to right of 719 (bright spot), 980, 730, 457,
367, 337 kHz and amplitude of the driving signal fixed at 9.8 V
peak to peak; the ring diameter listed at the bottom, and there is
a half a period phase shift between the spot and the ring
pattern;
[0059] FIGS. 31A and 31B illustrate a micromachined ring on the
surface of a polyimide sample, with FIG. 31A showing an optical
micrograph of micromachined ring structure and FIG. 31B showing the
profilometry analysis through the dashed line in FIG. 31A
demonstrating that material is removed over a depth of
approximately 0.9 .mu.m with little recast material;
[0060] FIG. 32 illustrates a different intensity distribution at
each laser spot, with a two pattern basis and demagnification of
50.times. used, and with the lens is driven at 989 kHz in the upper
image and driven at 531 kHz in the lower image;
[0061] FIGS. 33A and 33B illustrate the relation between driving
voltage, and ring radius, FIG. 33A, and number of rings, FIG.
33B;
[0062] FIG. 34 schematically illustrates the index of refraction in
a cylindrical TAG lens, which is a zeroth order Bessel function due
to the acoustic wave in a cylindrical geometry and showing that as
light enters this modulated-index field, it will be bent according
to the local gradients;
[0063] FIG. 35 schematically illustrates an experimental setup of a
TAG system in accordance with the present invention for dynamic
focusing and imaging of an object of interest;
[0064] FIG. 36 illustrates images of the object of interest at
three object locations as a function of TAG driving signal and
laser pulse timing for the TAG system of FIG. 35;
[0065] FIG. 37 schematically illustrates another experimental setup
of a TAG system in accordance with the present invention for
dynamic focusing and imaging of an object of interest;
[0066] FIGS. 38A-38C illustrate images of the object of interest at
three object locations, respectively, as a function of TAG driving
signal and laser pulse timing for the TAG system of FIG. 37;
[0067] FIG. 39 illustrates images of a Bessel beam taken 70 cm
behind the lens of FIGS. 3A, 3B to illustrate the time from which
the driving frequency is first changed to which the beam reaches a
steady state, with the driving frequency being 300 kHz and
amplitude 60 V.sub.p-p, and each image exposed for 0.5 ms;
[0068] FIG. 40 illustrates the intensity of a TAG lens generated
beam with respect to time after the driving voltage is switched off
at t=0;
[0069] FIG. 41 illustrates three plots with differing viscosities
of switching speed with respect to the driving frequency;
[0070] FIG. 42 illustrates the time-average output pattern from the
lens of FIGS. 4A, 4B with the periodicity of the spots on the order
of 0.1 mm;
[0071] FIG. 43 illustrates a theoretical plot of an instantaneous
pattern from a rectangular TAG lens driven at a frequency of 250
kHz, with the amplitude of the refractive index wave (both
horizontal and vertical) being 3.65.times.10.sup.-5; and
[0072] FIG. 44 illustrates the time-averaged pattern of the
theoretical plot of FIG. 31.
DETAILED DESCRIPTION OF THE INVENTION
[0073] Referring now to the figures, wherein like elements are
numbered alike throughout, FIGS. 1B, 1B schematically illustrates
an exemplary configuration of a TAG lens 100 in accordance with the
present invention having a cylindrical shape. The TAG lens 100 is a
piezoelectrically driven device that uses sound waves to modulate
the wavefront of an incident light beam. The lens 100 is composed
of a hollow piezoelectric tube 10 that is constrained by two
transparent windows 30 on either end for optical access and filled
with a refractive material, such as a gas, solid, liquid, plasma,
or optical gain medium, for example. The TAG lens 100 works by
creating a standing acoustic wave in the refractive liquid. The
acoustic standing wave is created by applying an alternating
voltage, typically in the radio-frequency range, to the
piezoelectric tube 10 by a controller 90. The controller 90 may
include a function generator passed through an RF amplifier and
impedance matching circuit.
[0074] Turning to FIGS. 1A, 1B in more detail, the TAG lens 100
includes a piezoelectric tube 10 having a generally cylindrical
shape, though other shapes such as a square, triangular, hexagonal
cross-section, etc. may be used by utilizing multiple piezoelectric
elements as described below. The piezoelectric tube 10 includes an
outer electrode 14, and an inner electrode which may be wrapped
from the inside surface of the piezoelectric tube 10 to the outer
surface of the tube 10 (using a conductive copper tape adhered to
the inside and wrapped around to the outside, for example) to
provide an annular electrode contact region 12 for the inner
electrode. The annular electrode contact region 12 and the outer
electrode 14 may be electrically separated from one another by an
annular gap 15 disposed therebetween. While a single piezoelectric
tube 10 is shown, multiple tubes can be used end-to-end or a single
piezoelectric tube 10a can be segmented along the longitudinal axis
into different zones 14a, 14b which can be separately electrically
addressed and driven to permit separate electrical signals to be
delivered to each tube or zone 14a, 14b, FIG. 2. For example, the
piezoelectric tube 10a may include an annular gap 15b disposed
between the two ends of the tube 10a to electrically isolate two
outer electrode zones 14a, 14b from one another along the axis of
the tube 10a. A similar electrical gap may be provided internally
to the tube 10a to electric isolate to inner electrode zones. Each
of the separated inner electrode zones may be wrapped to a
respective end of the tube 10a to provide a respective annular
electrode contact region 12a, 12b disposed at opposing ends of the
tube 10a. Each of the separate longitudinal zones 14a, 14b may then
be driven by a separate signal. Such a configuration can, for
example, permit a single TAG lens to be operated as if it were a
compound lens system, with each tube or longitudinal zone 14a, 14b
corresponding to a separate optical element or lens. Additionally,
the piezoelectric tube 10 may be segmented circumferentially so
that multiple electrically addressable zones may exist at a given
longitudinal location. Likewise, electrically addressable zones
that have an arbitrary shape and size may be provided on the inner
and outer cylindrical surfaces of the piezoelectric tube 10, where
a given zone on the inner cylindrical surface may (or may not)
coincide with an identical zone on the outer surface.
[0075] A cylindrical gasket 18 having an inner diameter larger than
the outer diameter of the piezoelectric tube 10 may be provided to
slide over the piezoelectric tube 10 to center and cushion the
piezoelectric tube 10 within the rest of the structure. An opening,
such as slot 19, may be provided in the cylindrical spacer gasket
18 to permit access to the piezoelectric tube 10 for purpose of
making electrical contact with the piezoelectric tube 10 and
filling the interior of the piezoelectric tube 10 with a suitable
material, e.g., a fluid (liquid or gas). The spacer gasket 18 may
be housed within a generally cylindrical inner casing 20 which may
include one or more fluid ports 22 in the sidewall through which
fluid may be introduced into or removed from the inner casing 20
and the interior of the piezoelectric tube 10 disposed therein. One
or more outlet/inlet ports 44 having barbed protrusions may be
provided in the fluid ports 22 to permit tubing to be connected to
the outlet/inlet ports 44 to facilitate the introduction or removal
of fluid from the TAG lens 100. In this regard the inner casing 20,
spacer gasket 18, and piezoelectric tube 10 are configured so that
fluid introduced through the inlet port 44 may travel past the
spacer gasket 18 and into the interior of the piezoelectric tube
10. In addition, one or more electrical feedthrough ports 24 may be
provided in a sidewall of the inner casing 20 to permit electrical
contact to be made with the piezoelectric tube 10. For instance,
wires may be extended through the electrical feedthrough ports 24
to allow electrical connection to the piezoelectric tube 10.
[0076] At either end of the inner casing 20 transparent windows 30
may be provided and sealed into place to provide a sealed enclosure
for retaining a refractive fluid introduced through the inlet port
44 within the inner casing 20. To assist in creating a seal, an
O-ring 26 may provided between the ends of the inner casing 20 and
the transparent windows 30, and the end of the inner casing 20 may
include an annular groove into which the O-rings 26 may seat.
Likewise spacer O-rings 16 may be provided between the ends of the
piezoelectric tube 10 and the transparent windows 30. The windows
30 may comprise glass or any other optical material that is
sufficiently transparent to the electromagnetic wavelengths at
which the lens 100 is to be used. For instance, the windows 30 may
be partially mirrored, such to be 50% transparent, for example. In
addition, the windows 30 may comprise flat slabs or may include
curved surfaces so that the windows 30 function as a lens. For
example, one or both of the surfaces of either of the windows 30
may have a concave or convex shape or other configuration, such as
a Fresnel surface, to introduce optical power. Further, the windows
30 may be configured to manipulate the incident optical radiation
in other manners, such as filtering or diffracting.
[0077] The inner casing 20 and transparent windows 30 may be
dimensioned to fit within an outer casing 40 which may conveniently
be provided in the form of a 2 inch optical tube which is a
standard dimension that can be readily mounted to existing optical
components. To secure the inner casing 20 within the outer casing
40, the outer casing 40 may include an internal shoulder against
which one end of the inner casing 20 seats. In addition, the outer
casing 40 may be internally threaded at the end opposite to the
shoulder end. A retaining ring 50 may provided that screws into the
outer casing 40 to abut against the end of the inner casing 20 to
secure the inner casing 20 with and the outer casing 40. The outer
casing 40 may include an access port 42, which may be provided in
the form of a slot, through which the inlet/outlet ports 44 and
electrical connections, such as a BNC connector 46, may pass. In
order to supply the driving voltage to lens the 100, a controller
90 may be provided in electrical communication with the connector
46, which in turn is electrically connected to the piezoelectric
tube 10, via the annular electrode contact region 12 and outer
electrode 14, for example.
[0078] Turning next to FIGS. 3A and 3B, an additional exemplary
configuration of a cylindrical TAG lens 300 in accordance with the
present invention is illustrated. Among the differences of note
between the TAG lens 300 of FIGS. 3A and 3B in the TAG lens 100 of
FIGS. 1A and 1B are the manner in which electrical contact is made
with the inner surface of the piezoelectric tube 310 and the
relatively fewer number of parts. The TAG lens 300 includes a
piezoelectric tube 310 which may be similar in configuration to the
piezoelectric tube 10 of the TAG lens 100. In order to make contact
with the inner electrode surface 312, an inner electrode contact
ring 320 may be provided that includes an inner electrode contact
tab 322 which may extend into the cavity of the piezoelectric tube
310 to make electrical contact with the inner electrode surface
312. To prevent electrical communication between the inner
electrode contact ring 320 and the outer electrode 314, an annular
insulating gasket 352 may be provided between the piezoelectric
tube 310 and the inner electrode contact ring 320.
[0079] To create a sealed enclosure internal to the piezoelectric
tube 310 in which a refractive fluid may be contained, two housing
end plates 340 may be provided to be sealed over the ends of the
piezoelectric tube 310. In this regard, annular sealing gaskets 350
may be provided between the ends of the piezoelectric tube 310 and
the housing end plates 340 to help promote a fluid-tight seal. The
housing end plates 340 may include a cylindrical opening 332
through which electromagnetic radiation may pass. In addition, the
housing end plates 340 may include windows 330 disposed within the
opening 332, which may include a shoulder against which the windows
330 seat. A refractive fluid may be introduced and withdrawn from
the lens 300 through optional fill ports 344, or by injecting the
refractive fluid between the sealing gasket 350 and the housing end
plates 340 using a needle. An electrical driving signal may be
provided by a controller 390 which is electrically connected to the
outer electrode 314 and the inner electrode contact ring 320 by
wires 316 to drive the piezoelectric tube 310. Like the controller
90 of FIG. 1A, the controller 390 may include a function generator
passed through an RF amplifier and impedance matching circuit.
Though the lens 300 of FIG. 3 contains fewer parts than the lens
100 of FIG. 1A, the lens 100 may be more convenient to use due to
the increased ease with which the lens 100 may be filled and
sealed. In addition, the electrode configuration of the lens 100,
specifically the inclusion of the annular electrode contact region
12 for making electrical contact with the inner electrode of the
piezoelectric tube 10, may lead to the creation of more
axisymmetric acoustic waves (i.e., about the longitudinal axis of
the piezoelectric tube 10) than would be possible with the point
contact provided by contact tab 322 of the lens 300.
[0080] Turning next to FIGS. 4A and 4B, an alternative exemplary
configuration of a rectangular TAG lens 200 in accordance with the
present invention is illustrated. The lens 200 may include two
piezoelectric plates 210 oriented 90.degree.. with respect to one
another to provide two sides of the square cross-section of the
rectangular enclosure, FIG. 3B. To complete the square
cross-sectional shape of the lens cavity two planar walls 212 may
be provided opposite the two piezoelectric plates 210. Providing
two piezoelectric plates 210 can be useful for generating arbitrary
patterns by combining several input signals that generate two
independent orthogonal wavefronts, and as such is not limited to
circularly symmetric patterns. Electrical wires 216, 218 may be
connected to opposing sides of the piezoelectric plates 210, with
the "hot" wires 216 electrically connected to the surface of the
piezoelectric plate 210 closest to the interior of the lens 200,
FIG. 3B. The wires 216, 218 may in turn be electrically connected
with a controller that provides the driving voltage for the
piezoelectric plates 210.
[0081] The piezoelectric plates 210 and planar walls 212 are
enclosed within a center casing 240 which may have threaded holes
through which adjustment screws 214 may pass to permit adjustment
of the location of the walls 212. The piezoelectric plates 210 in
turn may be secured with an adhesive 213 to the center casing 240
to secure them in place. Sealing washers 215 may be provided
internally to the center casing 240 on the adjustment screws 214 to
help seal a refractive fluid, F, within the center casing 240. The
center casing 240 may be provided in the form of an open-ended
rectangular tube, to which two end plates 220 may be attached to
provide a sealed enclosure 250 in which the refractive fluid, F,
may be retained. Attachment may be effected through the means of
bolts 242, or other suitable means. The bolts 242 pass through the
end plates 220 and center casing 240. To aid in providing a
fluid-tight seal between the center casing 240 and the end plates
220, sealing gaskets 226 may be provided between each end face of
the center casing 240 and the adjoining end plate 220. The
electrical wires 216, 218 may pass between the sealing gaskets 226
and the center casing 240 or end plates 220. The end plates 220 may
also include a central square opening 232 in which transparent
windows 230 may be mounted (e.g., with an fluid-tight adhesive or
other suitable method) to permit optical radiation to pass through
the lens 200 and the refractive fluid, F, in the central enclosure
250. The refractive fluid, F, may be introduced into the sealed
enclosure 250 via fluid ports or by injecting the refractive fluid,
F, into the sealed enclosure 250 by inserting a needle between the
sealing gasket 226 and the center casing 240 or end plate 220. The
particular exemplary lens 200 fabricated in tested (results in FIG.
42) was 0.5 inches thick, and 2 by 2 inches in the other two
dimensions, and used PZT5A3, poled with silver electrodes, Morgan
Electro Ceramics as the piezoelectric plates 210. Each of the
piezoelectric plates 210 were driven at the same frequency and
amplitude, and were in phase. Silicone oils of 0.65 and 5 cS have
been used successfully. The patterns seen in FIG. 42 may be seen
over a range frequencies (200-1000 kHz) and driving amplitudes
(5-100 V.sub.p-p), e.g. 400 kHz and 30 V.sub.p-p.
[0082] Having provided various exemplary configurations of TAG
lenses 100, 200, 300 in accordance with the present invention,
discussion of their operation follows.
I. Operation of Tag Lens
[0083] A predictive model for the steady-state fluid mechanics
behind TAG lenses 100, 200, 300 driven with a sinusoidal voltage
signal is presented in this section. The model covers inviscid and
viscous regimes in both the resonant and off-resonant cases. The
density fluctuations from the fluidic model are related to
refractive index fluctuations. The entire model is then analyzed to
determine the optimal values of lens design parameters for greatest
lens refractive power. These design parameters include lens length,
radius, static refractive index, fluid viscosity, sound speed, and
driving frequency and amplitude. It is found that long lenses 100,
200, 300 filled with a fluid of high refractive index and driven
with large amplitude signals form the most effective lenses 100,
200, 300. When dealing with resonant driving conditions, low
driving frequencies, smaller lens radii, and fluids with larger
sound speeds are optimal. At nonresonant driving conditions, the
opposite is true: high driving frequencies, larger radius lenses,
and fluids with low sound speeds are beneficial. The ease of
tunability of the TAG lens 100, 200, 300 through modifying the
driving signal is discussed, as are limitations of the model
including cavitation and nonlinearities within the lens 100, 200,
300.
[0084] The TAG lens 100, 200, 300 uses acoustic waves to modulate
the density of an optically transparent fluid, thereby producing a
spatially and temporally varying index of refraction--effectively a
time-varying gradient index lens 100, 200, 300. Because the TAG
lens 100, 200, 300 operates at frequencies in the order of 10.sup.5
Hz, the patterns observed (FIG. 5) by passing a CW collimated laser
beam through the TAG lens 100, 200, 300 are time-average images of
a temporally periodic pattern. The minor rings around each bright
major ring approximate nondiffracting axicon-generated Bessel
beams. The mechanics behind these patterns is explained below, and
the optic of pattern formation are discussed in section III below.
The square TAG lens 200 has been seen to produces patterns such as
those in FIG. 42.
[0085] An exemplary TAG lens 300 used in the analyses of this
section is illustrated in FIGS. 3A, 3B, and may comprise a
cylindrical cavity formed by a hollow piezoelectric tube 310 with
two flat transparent windows 330 on either side for optical access.
The cavity may be filled with a refractive fluid and the
piezoelectric tube 310 driven with an AC signal generating
vibration in several directions, the important of which is the
radial direction. This establishes standing-wave density and
refractive index oscillations within the fluid, which are used to
shape an incident laser beam.
[0086] Based on the TAG lens 300 configuration of FIGS. 3A, 3B, a
predictive model is developed for the fluid mechanics and local
refractive index throughout the lens 300 under steady-state
operation. This model is also expected to generally apply to the
cylindrical lens 100 of FIGS. 1A, 1B, 2. (A previous model for an
acoustically-driven lens had been proposed, however this model had
invoked time-invariant nonlinear acoustic theories which ignore the
more significant linear effects occurring in these lenses. Cf.
Higginson, et al., Applied Physics Letters, 843 (2004).)
Experimentation has shown that TAG beams are strongly time-varying,
and the following linear acoustic model better explains all
characteristics of the TAG lens 300. The results of the present
model will be provided using "base case" TAG lens parameters.
Optimizing the refractive capabilities of the lens 300 relative to
this base case for desired applications is also discussed. The
effects of modifying the lens dimensions, filling fluid, and
driving signal are all examined, as well as how modifying the
driving signal can be used to tune the index of refraction within
the lens 300.
Base Case Parameters
[0087] FIG. 5 shows the pattern generated by a "base case" TAG lens
300 (except for a driving frequency shifted to 299.7 kHz) as
observed 80 cm behind the lens 300. The lens 300 itself is
diagrammed in FIGS. 3A, 3B. The base case parameters for this lens
300 are listed in Table I.
TABLE-US-00001 TABLE I Base case parameters for the TAG lens,
divided into geometric, fluid, and driving signal parameters,
respectively. Parameter Symbol Base Case Value Lens inner radius
r.sub.s 3.5 cm Lens length L 4.06 cm Fluid Viscosity V 100 cS
Static refractive Index n.sub.s 1.4030 Speed of sound 1.00 Fluid
Density p 964 kg/m Voltage Amplitude V.sub.A 10 V Peak inner wall
velocity v.sub.A 1 cm/s Resonant Frequency f 246.397 kHz
Off-Resonant Frequency f 253.5 kHz indicates data missing or
illegible when filed
[0088] The piezoelectric material used for the tube 310 is lead
zirconate titanate, PZT-8, and the filling fluid for the lens 300
is a Dow Corning 200 Fluid, a silicone oil. The piezoelectric tube
310 is driven by the controller 390 which includes a function
generator (Stanford Research Systems, DS345) passed through an RF
amplifier (T&C Power Conversion, AG 1006) and impedance
matching circuit, which can produce AC voltages up to 300 V.sub.pp
at frequencies between 100 kHz and 500 kHz. Other impedance
matching circuits could be used to facilitate different frequency
ranges. Two different driving frequencies are used, corresponding
to resonant and off-resonant cases, listed in Table I.
Mechanics
Piezoelectric Transduction
[0089] As indicated above, the piezoelectric transducer used to
drive the TAG lens 300 comes in the form of a hollow cylinder or
tube 310. The electrodes 312, 314 are placed on the inner and outer
circumferences of the tube 310. The driving voltage frequency and
amplitude is applied to the piezoelectric tube 310 so that
V = V ? sin ( ? ) . ? indicates text missing or illegible when
filed ( 1 ) ##EQU00001##
[0090] The theory behind how a hollow piezoelectric tube 310 will
respond to such a driving voltage has already been published
(Adelman, et al., Journal of Sound and Vibration, 245 (1975)),
which leads to inner wall velocities on the order of V.sub.A=1
cm/s, assuming driving voltage amplitudes on the order of 10 V. It
is important to note that the wall velocity is always proportional
to the driving voltage amplitude.
Fluid Mechanics
[0091] The mechanics of the fluid within the lens 300 is described
by three equations: conservation of mass, conservation of momentum,
and an acoustic equation of state. Stated symbolically, these
equations are:
.differential. .rho. .differential. + .gradient. ( .rho. v ) = 0 ,
( 2 ) .differential. .differential. t ( .rho. v ) + .gradient.
.rho. + .gradient. ( .rho. v v ) + V D = 0 , ( 3 ) p = p 0 = ? ( p
- p 0 ) , ? indicates text missing or illegible when filed ( 4 )
##EQU00002##
where {circle around (x)} represents the tensor product and D is
the viscous stress tensor whose elements are given by
? = - ( .eta. - 2 .mu. / 3 ) ( .gradient. v ) ? - .mu. (
.differential. ? .differential. ? + .differential. ? .differential.
? ) . ? indicates text missing or illegible when filed ( 6 )
##EQU00003##
Here, .rho. is the local density, v is the local fluid velocity, p
is the local pressure, .mu. is the dynamic shear viscosity, and
.eta. is the dynamic bulk viscosity. Bulk viscosities are not
generally tabulated and are difficult to measure. For most fluids,
.eta. is the same order of magnitude as .mu.. For the base case, it
is assumed that .eta.=.mu.. Equation 4 assumes small amplitude
waves where c.sub.s is the speed of sound within the fluid at the
quiescent density and pressure, .rho..sub.0 and .rho..sub.0. This
equation represents the linearized form of all fluid equations of
state.
[0092] Substituting the equation of state (Eq. 4) into the momentum
conservation equation (Eq. 3) yields two coupled differential
equations for the dependent variables .rho. and v. Applying no-slip
conditions at the boundaries of the cell translates to these
boundary conditions:
v ? = v ? cos ( ? ) ? , ( 6 ) v ? = v ? = 0. ? indicates text
missing or illegible when filed ( 7 ) ##EQU00004##
[0093] The radial boundary condition is determined from the
velocity of the inner wall 312 of the piezoelectric tube 310. This
assumes that the piezoelectric tube 310 is stiff compared to the
fluid and that acoustic waves within the fluid do not couple back
into the piezoelectric motion. Impedance spectroscopy conducted on
the TAG lens 300 shows that except near resonances, the TAG lens
300 impedance is the same regardless of the filling fluid chosen.
Hence, this assumption is generally true, however some corrections
may be needed when near resonance. The presence of the dead
space-created by the sealing gaskets 350 between the piezoelectric
tube ends and windows 330, especially in the configuration of FIGS.
3A, 3B, will also modify the boundary condition in Eq. 6, however
this effect is neglected, because it is expected to only be
significant near the sealing gaskets 350 themselves.
[0094] Typically, a unique solution for .rho. and v at all times
would require two initial conditions as well as the above boundary
conditions. However for the steady-state response to the vibrating
wall, the initial conditions do not affect the steady-state
response.
[0095] The following assumptions reduce the dimensionality of the
problem, making it more tractable. First, the azimuthal dependence
can be eliminated because of the lack of angular dependence within
the boundary conditions (Eqs. 6 and 7). Second, the z-dependence of
the boundary conditions only appears in the no-slip conditions at
the transparent windows 330. Physically, this effect is expected to
be localized to a boundary layer of approximate thickness
.delta. = 2 .mu. .rho. 0 .omega. . ( 8 ) ##EQU00005##
[0096] For the base case parameters, this thickness comes out to
approximately 10 .mu.m. Thus, the lens 300 is operating in the
limit .delta.<<L, and solving the problem outside the
boundary layer will account for virtually all the fluid within the
lens 300. Furthermore, because radial gradients are expected to be
reduced within the boundary layer, the boundary layer effect can be
approximated by simply using a reduced effective lens length.
Gradients in the z-direction are expected to be much larger within
the boundary layer because the fluid velocity transitions to zero
at the wall. However for a normally incident beam of light, all
that is significant is the transverse gradient in total optical
path length through the lens 300. Optical path length differences
due to density gradients in the z-direction within the thin
boundary layer are insignificant compared to the optical path
length differences within the bulk. The result of these
considerations is that an approximate solution can be found by
solving the one dimensional problem, assuming .rho. is only a
function of r and then applying that solution to all values of z
within the lens 300.
[0097] The problem can be further simplified by linearization. This
assumes that the acoustic waves have a small amplitude relative to
static conditions. Each variable is expanded in terms of an
arbitrary amplitude parameter, .lamda.:
.rho. ( r , t ) = .rho. ? + .lamda. .rho. ? ( r , ? ) + .lamda. ?
.rho. ? ( r , t ) + ( 9 ) ? ( r , t ) = 0 + .lamda. v ? ( r , t ) +
.lamda. ? v ? ( r , t ) + ? indicates text missing or illegible
when filed ( 10 ) ##EQU00006##
[0098] Furthermore, the wave amplitudes are assumed small and
therefore any second order or higher term (.lamda..sup.2,
.lamda..sup.3, etc.) is much less than the zeroth or first order
terms, so the higher order terms can be dropped from the equations.
Keeping only the zeroth and first order terms results in .rho.(r,
t)=.rho..sub.0+.lamda.p.sub.1(r, t) and v(r, t)=.lamda.v.sub.1(r,
t), and Eqs. 2 and 3 can be rewritten as:
.lamda. [ .differential. ? .differential. t + ? ( ? ) ] = 0. ( 11 )
.lamda. [ .differential. .differential. t ( ? ) + ? ? ? + ? ? ] = 0
, ? indicates text missing or illegible when filed ( 12 )
##EQU00007##
where D.sub.1 is defined in the same way as D in Eq. 5, except with
v replaced by v.sub.1.
Inviscid Solution
[0099] One solution of interest is the inviscid solution because it
reasonably accurately predicts the lens output patterns for low
viscosities in off-resonant conditions while retaining a simple
analytic form. This solution is found by setting .mu.=.eta.=0. In
the one dimensional case, the problem becomes:
? .differential. t + 1 r .differential. .differential. r ( r ? v )
= 0 , ( 13 ) .differential. .differential. t ( .rho. o v ) + ?
.differential. .rho. 1 .differential. r = 0 , ( 14 ) ? = ? cos ( ?
) . ? indicates text missing or illegible when filed ( 15 )
##EQU00008##
It can be directly verified by substitution that the solution to
this problem is
? ( r , t ) = ? ( ? ) sin ( .omega. ? ) , ( 16 ) v ( r , ? ) = - ?
( .omega. r / ? ) cos ( .omega. ? ) , ? indicates text missing or
illegible when filed ( 17 ) ##EQU00009##
where
.rho..sub.A=-(.rho..sub.o.nu..sub.A)/(c.sub.sJ.sub.1(.omega.r.sub.0-
/c.sub.s)). For the base case off-resonant frequency, .rho..sub.A
is expected to be 0.090 kg/m.sup.3.
Viscous Solution
[0100] An effective kinematic viscosity is defined as
.nu.'=(.eta.+4.mu./3). In cases where this viscosity is large
compared to c.sub.s.sup.2/.omega. or when the lens 300 is driven
near a resonant frequency of the cavity, viscosity becomes
significant and the solution is somewhat more complex. To put the
viscosity threshold in context, the base case fluid, 100 cS
silicone oil, is considered low viscosity for frequencies
f<<c.sub.s.sup.2/(2.pi..nu.')=700 MHz.
[0101] Differentiating Eq. 11 with respect to time and taking the
divergence of Eq. 12, the equations can be decoupled and all
dependence on v eliminated to yield the damped wave equation,
? ( ? + v ' .differential. p i .differential. t ) - .differential.
2 p 1 .differential. t 2 = 0. ? indicates text missing or illegible
when filed ( 18 ) ##EQU00010##
By evaluating Eqs. 11 and 12 at r=r.sub.0 and assuming a curl-free
velocity field there, Eq. 6 can be converted from a boundary
condition in velocity to the following Neumann boundary condition
in density,
.differential. p i .differential. r ? = ? sin ( .omega. t ) - ? cos
( .omega. t ) . ? indicates text missing or illegible when filed (
19 ) ##EQU00011##
The steady-state one-dimensional solution to the above wave
equation and boundary condition can be expanded as a sum of
eigenfunctions:
.rho. 1 ( r , t ) = r ( A sin ( .omega. t ) + B cos ( .omega. t ) )
+ ? J 0 ( ? r ) [ ? sin ( .omega. t ) + D m cos ( .omega. t ) ] , ?
indicates text missing or illegible when filed ( 20 )
##EQU00012##
where k.sub.m=x.sub.m/r.sub.0 with x.sub.m being the location of
the m.sup.th zero of J.sub.1(x) and x.sub.0=0. A and B can be found
by substituting this solution into Eq. 19. C.sub.m and D.sub.m can
be found by substituting the solution into Eq. 18 and integrating
against the orthogonal eigenfunction J.sub.0(k.sub.nr) over the
entire circular domain. The resulting expressions are:
A = .rho. 0 v A .omega. ? v '2 .omega. 2 + ? ( 21 ) B = .rho. 0 v A
.omega. 2 v ' v '2 .omega. 2 + ? , ( 22 ) ? = ( ? ) ? , ( 23 ) D m
= ( ? ) .omega. v ' ? ? indicates text missing or illegible when
filed ( 24 ) ##EQU00013##
In the expressions above, E.sub.m and F.sub.m are the
nondimensional integrals,
E ? = .intg. 0 ? ? J ? ( x ? x ) x , ( 25 ) V ? = .intg. 0 ? J 0 (
x ? x ) x . ? indicates text missing or illegible when filed ( 26 )
##EQU00014##
By taking the limit .nu.'.fwdarw.0 and using the same trick of
integrating against an orthogonal eigenfunction, the inviscid
solution in Eq. 16 can be recovered.
Resonant Driving Conditions
[0102] Another important limit is that of operating near a
resonance of the cavity using a relatively low viscosity fluid.
Operating at the n.sup.th (>0) resonance means that
.omega.=c.sub.sk.sub.n. Note that at resonant frequencies, the
inviscid solution in Eq. 16 diverges because
J.sub.1(kr.sub.o).fwdarw.0 in the denominator of .rho..sub.A.
Consequently, in order to get a valid solution near resonance, the
full viscous solution is necessary--even at low viscosities. As
discussed in the previous section, low viscosity means that
.nu.'<<c.sub.s.sup.2/.omega.. In this limit, the coefficients
of the viscous solution look as follows:
A .fwdarw. ? , ( 27 ) B .fwdarw. - v ' ? , ( 28 ) ? .fwdarw. ? , (
29 ) ? .fwdarw. - ? , ( 30 ) ? .fwdarw. - v ' ? ? , ( 31 ) ?
.fwdarw. - 1 v ' ? . ? indicates text missing or illegible when
filed ( 32 ) ##EQU00015##
[0103] Note that as the viscosity vanishes, the only term that
diverges is the D.sub.m=n term. All the other terms either vanish
or do not change. This means that when driving on resonance with a
low viscosity fluid only the D.sub.m=n, term is significant, and
the solution for the density becomes,
? ( r , t ) .fwdarw. - 1 v ' ( ? ) J 0 ( ? ) cos ( .omega. t ) . ?
indicates text missing or illegible when filed ( 33 )
##EQU00016##
At the resonant base case frequency, the amplitude of .rho..sub.1
takes the value 9.1 kg/m.sup.3.
From Density to Refractive Index
[0104] The Lorentz-Lorenz equation can be used to determine the
local index of refraction from the fluid density. This relationship
is
n = 2 Q .rho. + 1 1 - Q .rho. , ( 34 ) ##EQU00017##
where Q is the molar refractivity, which can be determined from
n.sub.o and .rho..sub.o. For small .rho..sub.1, this equation can
be linearized by a Taylor expansion about the static density and
refractive index. Substituting for Q, this takes the form,
n = n o + n o 4 + n o 2 - 2 6 n o ( .rho. 1 .rho. 0 ) . ( 35 )
##EQU00018##
[0105] In the resonant base case, the amplitude of oscillation of
the density standing wave is less than 1% of the static density.
Comparing the true Lorentz-Lorenz equation with the linearized
version, one finds that the error in refractive index due to
linearization is less than 0.2%.
[0106] In the inviscid linearized acoustic case, the refractive
index given by Eq. 35 assuming the density distribution in Eq. 16
or 33, depending on resonance, reduces to an expression of the
form,
n = n ? + n ? J ? ( kr ) sin ( ? ) , ? indicates text missing or
illegible when filed ( 36 ) ##EQU00019## in-the off-resonant case,
or
n = n ? + n ? J ? ( kr ) cos ( ? ) , ? indicates text missing or
illegible when filed ( 37 ) ##EQU00020##
at resonance. The full expression for n.sub.A in the low-viscosity
off-resonant case is:
n A = ( ? ) , ? indicates text missing or illegible when filed ( 38
) ##EQU00021##
and in the resonant case n.sub.A is given by:
n A = ( ? ) ? indicates text missing or illegible when filed ( 39 )
##EQU00022##
[0107] For the base case, n.sub.A is expected to have an
off-resonant value of 4.3.times.10.sup.-5. On resonance, it is
expected to have a base case value of 4.3.times.10.sup.-3. Similar
solutions can be obtained for the viscous case.
Optimizing the Figure of Merit: Refractive Power
[0108] In order to get the most out of a TAG lens 300 under steady
state operation, one wishes to maximize the peak refractive power.
The lower bound is always zero, given by the static lens 300
without any input driving signal. Higher refractive powers increase
the range of achievable working distances and Bessel beam ring
spacings. The refractive power, RP, is defined here to be the
magnitude of the transverse gradient in optical path length. This
is given by the product of the transverse gradient in refractive
index and the length of the lens 300. Under thin lens and small
angle approximations, the maximum angle that an incoming collimated
ray can be diverted by the TAG lens 300 is equal to its refractive
power. For a simple converging lens, its RP is also equal to its
numerical aperture.
[0109] Maximizing the refractive power can be accomplished by
altering the dimensions of the lens 300, the filling fluid, or the
driving signal. Because the base case TAG lens 300 is well within
the low viscosity range of the parameter space, discussion in this
section will be limited to only low-viscosity fluids in the
resonant and off-resonant cases so that Eq. 36 or 37 applies with
n.sub.A given by Eq. 38 or 39.
[0110] The first step is to calculate the TAG lens peak refractive
power, RP.sub.A, using Eq. 36 and assuming azimuthal symmetry
within the lens 300:
RP A = ? .gradient. OPL r ^ = ? L .gradient. n r ^ = ? Lkn A ? ( kr
) . ? indicates text missing or illegible when filed ( 40 )
##EQU00023##
Therefore, RP.sub.A is maximized by maximizing |Lkn.sub.A|, while
the term J.sub.1(kr) only determines at what radial location this
maximum is achieved. In order to maximize |Lkn.sub.A|, each of the
parameters in Eqs. 38 and 39 is considered. Because the dependence
on these parameters can vary between resonant and off-resonant
driving conditions, the analysis has been divided into the two
subsections below.
Optimizing Resonant Conditions
[0111] It is first assumed that the lens 300 will be driven under
resonant conditions. This will yield the highest refractive,
powers. At the resonant base case frequency, RP.sub.A takes the
value 0.16. The model for this section uses the refractive index
given by Eq. 37 with n.sub.A given by Eq. 39.
Optimizing Lens Dimensions
[0112] The size of the TAG lens 300 is considered first. This is
determined by the piezoelectric tube length L and inner radius
r.sub.0. The refractive power of the TAG lens 300 is proportional
to L, so longer lenses are desirable. With increasing length, thin
lens approximations will become increasingly erroneous, and
eventually the TAG lens 300 will function as a waveguide.
[0113] The dependence on transverse lens size is not a simple
relationship because of the Bessel functions in the denominator of
Eq. 39 and the fact that the value of n in E.sub.n and F.sub.n
depends on r.sub.0. The relationship between the refractive power
and the inner lens radius is plotted in FIG. 6. This figure shows
that on resonance, higher refractive powers can be achieved with
lenses having a smaller radius. Discrete points are plotted because
resonance is only achieved at discrete inner radii. This effect can
be attributed to increased viscous losses due to increased acoustic
wave propagation distance.
Optimizing the Refractive Fluid
[0114] The relevant properties of the refractive fluid include its
static index of refraction n.sub.0, its effective kinematic
viscosity V', and the speed of sound within the material,
c.sub.s.
[0115] Increasing the value of n.sub.0 affects only the first term
of Eq. 39 and increases the TAG lens refractive power. Due to the
nature of the Lorentz-Lorenz equation, the same fractional
variation in density will have a greater effect on the refractive
index of a material with a naturally high refractive index than it
will on a material with a lower refractive index. This effect is
plotted in FIG. 7. It is clear that higher static indices of
refraction improve lens performance.
[0116] In the resonant case, the viscosity of the fluid is
significant and lower viscosities are more desirable because the
refractive index amplitude is inversely proportional to the
effective kinematic viscosity. In symbols,
n.sub.A.varies..nu.'.sup.1. This result is expected because lower
viscosities will mean less viscous loss of energy within the lens
300.
[0117] As with the inner radius, the effect of the sound speed on
the refractive power cannot be easily analytically represented
because of the Bessel functions in the denominator of Eq. 39 and
the dependence of E.sub.n and F.sub.n on c.sub.s. These effects are
plotted in FIG. 8. This shows that higher sound speeds are
preferable.
[0118] Listed in Table II are a variety of filling materials and
their relevant properties. For resonant driving conditions, water
and 0.65 cS silicone oil are best because of their low viscosities.
Nitrogen would make a poor choice because of its very low value of
static index of refraction. Because of their high viscosities,
Glycerol and 100 cS silicone oil are less desirable for resonant
operation.
TABLE-US-00002 TABLE II Properties of potential fluids. All values
for temperatures in the 20-30.degree. C. range. V c p.sub.a Fluid
n.sub.s (cS) (m/s) (kg/m.sup.3) Silicone Oil 1.4030 100 985 964
Silicone Oil 1.375 0.65 873.2 761 Glycerol 1.4746 740 1904 1260
Water 1.33 1.00 1493 1000 Nitrogen 1.0003 16.1 355 1.12 indicates
data missing or illegible when filed
Optimizing the Driving Signal
[0119] While only sinusoidal driving signals are discussed at
present, the controller 390 (or controller 90) can provide more
complicated signals to produce arbitrary index profiles that repeat
periodically in time as discussed below in section II. There are
two variable parameters of the sinusoidal driving signal: its
amplitude, V.sub.A, and its frequency, .omega.. These two
parameters will determine the inner wall velocity, which are
treated herein as a given parameter.
[0120] It has been noted that voltage amplitude, V.sub.A, is
proportional to inner wall velocity, .nu..sub.A. These amplitudes
have a very simple effect on the refractive index. From Eqs. 38 and
39, it can be seen that lens refractive power is directly
proportional to .nu..sub.A, and hence, V.sub.A, and that larger
wall velocities and driving voltages are desirable.
[0121] Similar to the lens radius and sound speed, the driving
frequency .omega. has an effect on the refractive power of the lens
300 that cannot be given in a simple analytic form. This effect is
plotted in FIG. 9 and illustrates that lower frequencies yield
greater refractive powers. This is because higher frequencies
exhibit greater viscous damping.
Optimizing Nonresonant Conditions
[0122] There are conditions where driving on resonance is
impractical. For example, due to the sharpness of the resonant
peaks, a small error in lens properties or driving frequency can
result in a large error in refractive index. Operating off
resonance can be more forgiving in terms of error, however this
comes at the expense of reduced refractive powers. In this section
the off-resonant base case frequency is used.
[0123] Since the lens 300 is operating off resonance, the
refractive index is given by Eq. 36 with n.sub.A, given by Eq. 38,
which yields an RP.sub.A of 0.0016. The dependencies of RP.sub.A on
lens length L, static refractive index n.sub.0, and driving
amplitude V.sub.A(.nu..sub.A) are all identical to what was found
for resonant driving conditions. This is because these variables
only appear in the common prefactors in Eqs. 38 and 39. Hence these
parameters will not be reexamined in this section. Note that if
referring to FIG. 7 (RP.sub.A vs. n.sub.0), the RP axis will have
to be scaled appropriately because the values of RP.sub.A differ
between the resonant and off-resonant cases.
[0124] The difference between the resonant and off-resonant driving
conditions is found in the lens radius r.sub.0, the sound speed
c.sub.s, and the driving frequency .omega.. These dependencies are
plotted in FIGS. 10-12. These parameters all exhibit opposite
trends from resonant driving conditions. For best off-resonant
performance, large radius lenses filled with low speed of sound
fluids driven at high frequencies are desirable. This occurs
because viscous damping no longer affects the refractive power.
These results are expected because larger lenses vibrating at the
same wall speed cause more acoustic power to be focused at the
center of the lens 300, increasing refractive powers. Also, higher
driving frequencies condenses the spatial oscillations in density,
producing higher gradients in refractive index.
[0125] Looking at the values in Table II, it is evident that for
off-resonant driving, both silicone oils, glycerol, and water all
become viable fluid choices now that viscosity is unimportant.
These fluids all have appreciable static refractive indices
compared to nitrogen. The silicone oils are expected to have
somewhat better performance over glycerol and water because of
their low sound speeds.
Other Considerations
[0126] Preventing cavitation is another consideration involved in
selecting a filling material other than simply maximizing the
refractive power. If the pressure within the lens 300 drops below
the vapor pressure of the fluid, then cavitation can occur,
producing bubbles within the lens 300 that disrupt its optical
capability. Specifically, this can happen when
? > ? - ? / ? ? indicates text missing or illegible when filed (
41 ) ##EQU00024##
where p.sub.v is the vapor pressure of the fluid. There are a
couple ways that cavitation can be avoided. First, one can choose a
fluid with a low vapor pressure. Second, the lens 300 can be filled
to a high static pressure.
[0127] Another danger in blindly maximizing the refractive power is
that at high RP values, the model may break down. This is because
the linearization performed above is only valid at relatively small
amplitudes. Once the order of .rho..sub.A or n.sub.A becomes
comparable to the order of .rho..sub.o or n.sub.0, the
linearization loses accuracy. It is likely that the general trends
observed in this section will hold to some degree in the nonlinear
regime, although the specific form of the dependence of refractive
power on all the variables requires further analysis. It is
possible to increase the domain of the linear regime by selecting
fluids of large density. One should also note that the selection of
n.sub.0 does not affect the linearization of the fluid mechanics.
Therefore, increasing the refractive power via increasing the
fluid's refractive index will not endanger the fluid linearization,
although it may endanger the Lorentz-Lorenz linearization. However,
when the linear models no longer apply, it is still possible to
obtain solutions via full numerical simulations.
[0128] The results of the predictive model are useful for
optimizing the TAG lens design in terms of maximizing its ability
to refract light in steady-state operation. A TAG lens 300 is most
effective when it is long, filled with a fluid of high refractive
index, and driven with large voltage amplitudes. If driving on
resonance, lower frequencies, smaller lens radii, and fluids with
larger sound speeds and lower viscosities enhance refractive power.
Off resonance, higher frequencies, larger lenses, and lower sound
speeds are preferred. Viscosity is irrelevant for nonresonant
driving.
[0129] It is important to note that these choices are only best for
optimizing the steady-state refractive power where the linear model
is applicable. If wave amplitudes become too great, then a
nonlinear model will be required, which could be implemented
numerically. Also, different optimization parameters will occur if,
for example, one wishes to optimize the TAG lens 300 for pattern
switching speed or high damage thresholds--two of the potential
advantages of TAG lenses over spatial light modulators.
[0130] The above modeling has been done in a circular cross-section
geometry so as to model a TAG lens 300 capable of generating Bessel
beams. Other geometries are also possible for creating complicated
beam patterns. The natural example is that of a rectangular cavity,
e.g., FIGS. 4A, 4B, in which the Bessel eigenfunctions would be
replaced by sines and cosines. With other geometries that break the
circular symmetry, it may also be possible to create
Laguerre-Gaussian and higher-order Bessel beams. It has been shown
that passing a Laguerre-Gaussian beam through an axicon creates
higher-order Bessel beams. This same method can be implemented with
the TAG lens 300 replacing the axicon to produce tunable higher
order Bessel modes.
II. Determination of Voltage Signal to Create Specific Refractive
Index Profile
[0131] In the linear regime, the cylindrical TAG lens 100, 300 has
the potential to create arbitrary (non-Bessel) axisymmetric beams.
By driving the lens 100, 300 with a Fourier series of signals at
different frequencies, interesting refractive index distributions
within the lens 100, 300 can be generated. This is because the lens
100, 300 effectively performs a Fourier-Bessel transform of the
electrical signal into the index pattern. As this pattern will vary
periodically in both space and time, it will be best resolved with
a pulsed laser synchronized to the TAG lens 100, 300.
[0132] This section solves the inverse problem: determining what
voltage signal is necessary to generate a desired refractive index
profile. However, before directly tackling this question, it is
easier first to find the response of the lens 100, 300 to a single
frequency, and then to solve the forward problem addressed in the
next section: determining the index profile generated by a given
voltage input.
[0133] The first step of the procedure is to find the frequency
response of the TAG lens 100, 300. A linear model of the TAG lens
100, 300 is assumed. That is, the oscillating refractive index
created within the lens 100, 300 is assumed linear with respect to
the driving frequency. Listed below are the single-frequency input
signal and resulting output refractive index within the lens 100,
300.
V ( ? , f ) = ? [ V ^ ( f ) ? ] ( 42 ) ? = ? + ? [ ? ( f ) ? ] ?
indicates text missing or illegible when filed ( 43 )
##EQU00025##
[0134] Here, f is the electrical driving frequency of the lens 100,
300, n.sub.0 is the static refractive index of the lens 100, 300,
V(f) is the driving voltage complex amplitude, and k is the spatial
frequency given by f/c.sub.s where c.sub.s is the speed of sound
within the fluid.
[0135] From this frequency response, a transfer function can be
defined to relate the index response to the voltage input:
.PHI. ( f ) = n ^ ( f ) V ^ ( f ) .di-elect cons. C . ( 44 )
##EQU00026##
This transfer function can either be determined empirically, or
through modeling, and accounts for both variations in amplitude and
phase.
Forward Problem
[0136] For the forward problem, it is assumed that the lens 100,
300 is driven with a discrete set of frequencies at varying
amplitudes and phase shifts. One could also phrase the problem in
terms of a continuous set of input frequencies, however as is seen
later, the solution to the discrete set will be more useful when
dealing with the inverse problem.
[0137] An input signal of the form,
V ( t ) = Re [ ? ] , ? indicates text missing or illegible when
filed ( 45 ) ##EQU00027##
is assumed where each V.sub.m is a given complex amplitude.
[0138] By linearity and the results of the frequency response in
Eq. 43, the corresponding refractive index in the lens is known to
be,
? = ? [ ? ] . ? indicates text missing or illegible when filed ( 46
) ##EQU00028##
The coefficients n.sub.m can be determined from the frequency
response to be,
? = .PHI. ( ? ) ? ? indicates text missing or illegible when filed
( 47 ) ##EQU00029##
Eqs. 46 and 47 are the solution to the forward problem.
Inverse Problem
[0139] The inverse problem is to determine what input voltage
signal, V(t), is required to produce a desired refractive index
profile, n.sub.goal(r). From Eqs. 42 and 43, it is evident that the
actual refractive index is a function of both space and time, while
the input electrical driving signal is only a function of time. As
a result, it is not possible to create any arbitrary index of
refraction profile defined in both space and time, however it is
possible to approximate an arbitrary spatial profile that repeats
periodically in time. It will be assumed that the arbitrary profile
is centered around the static index of refraction, n.sub.0. The
deviation of the goal from n.sub.0 is denoted as n.sub.goal(r), and
the frequency with which it repeats in time as f.sub.rep.
[0140] The procedure is depicted as a flow chart in FIG. 13. First,
n.sub.goal(r) is decomposed into its spatial frequencies using a
Fourier-Bessel transform. This result is then discretized so that
only spatial frequencies that are integer multiples of
f.sub.rep/c.sub.s are included. Then, in order to write the index
response in the form of Eq. 46, an inverse Fourier-Bessel transform
is applied to the discrete series. This gives the coefficients
n.sub.m, which are used in conjunction with the frequency response
to yield the required voltage signal in the frequency domain.
Summing over all modes provides the final answer, V(t).
[0141] The first step is to decompose the desired index profile
into its spatial frequencies using a windowed Fourier-Bessel
transform because of the circular geometry of the lens 100, 300.
Given n.sub.goal(r), n.sub.goal(k) can be computed as
? ( k ) = ? ( r ) ? ( 2 .pi. rk ) 2 .pi. r r , ? indicates text
missing or illegible when filed ( 48 ) ##EQU00030##
where r.sub.0 is the inner radius of the lens 100, 300. Depending
on the desired index goal, n.sub.goal(r) this windowing may
introduce undesirable Gibbs phenomenon effects near the edge of the
lens 100, 300 if n.sub.goal(r) does not smoothly transition to zero
at r=r.sub.0. However, the significance of these effects can be
reduced by either modifying the goal signal, extending the limit of
integration beyond r.sub.0, or simply using an optical aperture to
obscure the outer region of the lens 100, 300.
[0142] From inverse-transforming it is known that,
? ( r ) ? ? indicates text missing or illegible when filed ( 49 )
##EQU00031##
However, each of the spatial frequencies will oscillate in time at
its own frequency given by f=c.sub.sk. As a result, the goal
pattern can only generated at one point in time. If this time is
t=0, then the time dependent index of refraction will be given
by,
? ( r , t ) ? [ ? ] ? indicates text missing or illegible when
filed ( 50 ) ##EQU00032##
[0143] In practice, one would wish the goal index pattern to repeat
periodically in time, as opposed to achieving it only at one
instant in time. Therefore, the second step of the procedure is to
discretize the spatial frequencies used so that the n.sub.goal(r)
can be guaranteed to repeat, with temporal frequency f.sub.rep.
This is achieved by only selecting spatial frequencies that are
multiples of f.sub.rep/c.sub.s. That is, it is assumed that,
k .di-elect cons. ? = ? . ? indicates text missing or illegible
when filed ( 51 ) ##EQU00033##
The upper limit, M, is set sufficiently large so that the
contribution to n.sub.goal(r) is negligible from spatial
frequencies higher than Mf.sub.rep/c.sub.s. It is also required
that n.sub.goal(r) and f.sub.rep are chosen so that the
contribution is negligible from spatial frequencies lower than
f.sub.rep/c.sub.s and so that the discretization accurately
approximates the continuous function. The lower f.sub.rep, the more
accurately the discretization will reflect the continuous solution,
however it also means that there will be longer intervals between
pattern repetition.
[0144] This discretization changes the integral in Eq. 50 into a
sum:
? = ? [ ? ] , ? indicates text missing or illegible when filed ( 52
) ##EQU00034##
where .DELTA.k is the spacing between spatial frequencies, in this
case given by f.sub.rep/c.sub.s. The third step of the procedure is
to compute this sum. At this point, one should compare
n.sub.goal(r) with n(r,0) to ensure good agreement. If the
agreement is poor, lowering f.sub.rep, raising M, smoothing
n.sub.goal(r), and continuing the integration in Eq. 48 beyond
r.sub.0 can all improve the approximation.
[0145] By comparing Eq. 52 with Eq. 46, it is evident that,
? = ? .DELTA. k ? ? indicates text missing or illegible when filed
( 53 ) ##EQU00035##
Using the frequency response in Eq. 47 and rewriting k.sub.m and
.DELTA.k in terms of f.sub.rep, this expression can be used to find
the voltage signal coefficients:
? = ? ? indicates text missing or illegible when filed ( 54 )
##EQU00036##
The time domain signal is given by the same expression as Eq.
45,
V ( t ) = Re [ ? ] . ? indicates text missing or illegible when
filed ( 55 ) ##EQU00037##
Eqs. 54 and 55 represent the last step and solution to the inverse
problem.
[0146] In fact, discretization is not absolutely necessary to
provide this transformation. One can analytically perform the same
functions resulting in a temporal spectrum of the voltage function.
In this case, the equations presented above would replace
summations and series with definite integrals. However, the
resulting voltage function would not necessarily repeat
periodically in time.
[0147] As a theoretical, exemplary problem, a simple converging
lens with a specific focal length, l, is created, as shown in FIG.
14. Given this criteria, the goal is to determine the voltage
signal to be supplied to the TAG lens 100, 300 so that it adopts
this converging lens configuration with temporal period t.sub.rep.
The parameters for the TAG lens 100, 300 in this example are given
in Table 3. For the sake of simplicity in illustrating the above
procedure, a constant frequency response, .PHI.(f), is assumed.
Note that in reality, .PHI.(f) would vary greatly with frequency
near resonances within the lens 100, 300. However, since this
affects only the last step of the procedure, the exact form of the
function, while important for practical implementation, is
unimportant here in demonstrating the solution process.
TABLE-US-00003 TABLE 3 Parameters used in the example inverse
problem. Name Symbol Value Focal Length l 1 m Temporal Period t 1
ms Inner Radius r.sub.s 5 cm Lens Length L 5 cm Sound Speed c 1000
.sup.-1 Transfer Func. .PHI.(f) 10.sup.-3 V.sup.-1 Largest Mode M
300 indicates data missing or illegible when filed
[0148] The first step of this example is to determine the
refractive index profile for a simple converging lens with this
focal length. Using small angle approximations, the angle by which
normally incident incoming rays should be deflected is given
by,
.theta. ? ( r ) = - r ? . ? indicates text missing or illegible
when filed ( 56 ) ##EQU00038##
This goal angular deflection is shown in FIG. 15.
[0149] The corresponding refractive index profile required to
deflect rays by this angle is given by,
? = ? = - ? ? indicates text missing or illegible when filed ( 57 )
##EQU00039##
This function is plotted in FIG. 16. If the effective converging
lens were to have a high numerical aperture, were optically thick,
or were used in a situation where the small angle approximations
were not accurate enough, then the goal refractive index would no
longer be parabolic. However, the TAG lens could still be used to
emulate the new goal function.
[0150] The procedure described above is now followed to obtain
V(t). Equation 48 is used to decompose the goal refractive index
into its spatial frequencies. These spatial frequencies are then
discretized with the lower bound and spacing between frequencies
given by f.sub.rep=1/t.sub.rep=1 kHz. The upper bound is chosen to
be 300 kHz, which corresponds to M=300. Both the continuous and
discretized spatial frequencies are plotted in FIG. 17. The value
of M was chosen large enough so that the actual angular deflection
reasonably approximates the goal, as shown in FIG. 15.
[0151] The actual index of refraction at time t=0 is obtained from
the discretized frequencies using Eq. 52 and is plotted in FIG. 16.
Note the very good agreement between the goal and actual refractive
index profiles. This good agreement is due to f.sub.rep being
relatively small, and M being relatively large.
[0152] The angular deflection is obtained by differentiating the
refractive index profile. This deflection is plotted in FIG. 15.
Note that the discrepancy between the goal and actual angular
deflections is amplified relative to the refractive index profile.
This is because of the amplifying properties of the derivative.
However, there is still good agreement between the goal and actual
deflections, except at the edge of the lens 100, 300 where the
Gibbs phenomenon can be observed because the refractive index does
not smoothly transition to zero there. Better agreement could be
achieved by choosing a larger M or smoothing out the desired index
goal. In some situations, choosing a larger t.sub.rep would also be
helpful.
[0153] Using Eqs. 54 and 55, the actual voltage signal to be
generated by the controller 90, 390 can be computed. It is periodic
with period t.sub.rep. One period is plotted in FIG. 18. It is
important to note that changes in .PHI.(f) can change the form of
this function. This signal can be output from the controller 90,
390 to drive the lens 100, 300.
[0154] The above method to approximately generate arbitrary
axisymmetric index patterns which repeat at regular intervals
allows a cylindrical TAG lens 100, 300 to act as an axisymmetric
spatial light modulator. If instead of a cylindrical geometry, a
rectangular, triangular, hexagonal or other geometry were used for
the TAG lens 200 with two or more piezoelectric actuators 210, then
arbitrary two dimensional patterns may be approximated without the
axisymmetric limitation. The only mathematical difference will be
the use of Fourier transforms instead of using Fourier-Bessel
transforms to determine the spatial frequencies. As such, the TAG
lenses 100, 200, 300 may be used as a tunable phase mask (or
hologram generator) or the adaptive element in a wavefront
correction scheme.
[0155] Compared to nematic liquid crystal SLMs, TAG lenses 100, 300
can have much faster frame rates limited only by the liquid
viscosity and sound speed. The frame rate of the theoretical
example presented above was 1 kHz. If the voltage signal was not
precisely periodic in time, but varied slightly with each
repetition, then pattern variations could be achieved at this rate.
This method used only steady state modeling, however with fully
transient modeling even higher frame rates would be possible.
Because of the simplicity and flexibility in the optical materials
used in a TAG lens 100, 300, it is possible to design one to
withstand extremely large incident laser energies. Due to its
analog nature, TAG lenses also avoid pixilation issues.
In addition to their advantages, TAG lenses do have some
limitations that may make SLMs more suitable in certain
applications. Specifically, TAG lenses may work best when
illuminated periodically with a small duty cycle, whereas SLMs are
"always-on" devices. In some special cases, continuous wave
illumination of TAG lenses may be acceptable if the index pattern
in the middle of the cycle is not disruptive. Moreover, TAG lenses
may be operated in modes other than that of a simple positive lens
with fixed focal length, for example, in modes where multiscale
Bessel beams are created.
III. Optical Analysis of Multiscale Bessel Beams
[0156] In this section multiscale Bessel beams are analyzed which
are created using the TAG lens 300 of FIGS. 3A, 3B as a rapidly
switchable device. The shape of the beams and their nondiffracting
and self-healing characteristics are studied experimentally and
explained theoretically using both geometric and Fourier optics.
The spatially and temporally varying refractive index within the
TAG lens 300 leading to the observed tunable Bessel beams are
explained. As discussed below, experiments demonstrate the
existence of rings, and the physical theory (geometric and
diffractive) accurately predicts their locations. By adjusting the
electrical driving signal, one can tune the ring spacings, the size
of the central spot, and the working distance of the lens 300. The
results presented here will enable researchers to employ dynamic
Bessel beams generated by TAG lenses. In addition, this section
discusses in detail how to tune the electrical driving signal to
alter the observed patterns.
[0157] Experimentally, it has been observed that the TAG beam has
bright major rings that may each be surrounded by multiple minor
rings, depending on driving conditions (see FIG. 19). On the large
scale, the periodicity of the major rings is that of the square of
a Bessel function. When driven at a sufficiently high amplitude,
minor rings become evident. The periodicity of the minor rings
around the central spot is also given by the square of a Bessel
function. Therefore, the TAG lens 300 beam can be thought of as a
multiscale Bessel beam, i.e., one having at least one bright major
ring surrounded by at least one minor ring. The images of FIGS. 19A
and 19B are both taken 50 cm behind the lens 300 with driving
frequency 257.0 kHz.
[0158] As shown in FIG. 34, the density variations in the standing
wave result in refractive-index variations that focus the light
passing through the lens 300. The acoustic standing wave is created
by an alternating voltage, typically in the radio-frequency range,
applied to the piezoelectric element 310 by the controller 390. The
relations between the drive frequency and amplitude, and the
resulting refractive-index modulation, are nonlinear and complex,
but are predictable as shown in section I.
[0159] Although the refractive-index variation induced by the
voltage is small, the lens 100, 200, 300 is thick enough to allow
significant focusing. Because the index variation is periodic, the
TAG lens 100, 300 is able not only to shape a single beam of light,
but in a rectangular configuration the TAG lens 220 can also take a
single beam and create an array of smaller beams as do other
adaptive optical devices.
[0160] The optical properties of the lens 300 are determined by a
number of experimentally controllable variables. First and
foremost, the geometry and symmetry of the lens 300 determine the
symmetries of the patterns that can be established. For instance, a
square shaped lens 200 can produce a square array of beamlets,
while a cylindrical geometry can produce Bessel-like patterns of
light. The density and viscosity of the refractive fluid, the
static filling pressure, as well as the type of piezoelectric
material, will all play a role in determining the static and
dynamic optical properties of the TAG lens 300. These properties
can be designed and optimized for different applications.
[0161] The two "knobs" that control the TAG lens 300 effect on a
continuous (CW) beam are the amplitude and frequency of the
electrical signal applied to the piezoelectric transducer 310. For
a pulsed beam or detector, extra control is provided by the timing
of pulse, as discussed in section V. The amplitude determines the
volume of the sound wave and the corresponding amplitude of the
index variation. Of course, there are fundamental limitations to
how much index amplitude is possible due to the relatively small
compressibility of liquids. However, the lens 300 does not need to
be filled with a liquid at all. The physics of this device should
work equally well on a gas, solid, plasma, or a multicomponent,
complex material. The frequency of the drive signal determines the
location of the maxima and minima in the index function. Multiple
driving frequencies and segmented piezoelectric elements can be
used to give full functionality to this device and enable creation
of arbitrary patterns.
[0162] FIG. 20A depicts the experimental setup. The TAG lens 300 is
primarily studied by illuminating it with a wide Gaussian beam of
collimated 532 nm CW laser light. (Although in the next section the
effects of pulsed illumination are discussed.) The intensity
pattern produced by the lens 300 is then sampled at various
distances using a 1/2'' CCD camera 370 (Cohu 2622). In order to
achieve intensity profiles as a function of radius, azimuthal
averaging was used to filter out CCD noise. In order to observe the
time dependence of the beam, a pulsed 355 nm laser with 20 ns pulse
length is also used to strobe the pattern.
[0163] The TAG lens 300 has an inner diameter of 7.1 cm and a
length of 4.1 cm including the piezoelectric element 310, contact
ring 320, and gaskets 350, 352. The fluid used is 0.65 cS Dow
Corning 200 Fluid (silicone oil), which has an index of refraction
of n.sub.0=1.375, and speed of sound of 873 ms.sup.-1 under
standard conditions. The TAG lens 300 is driven by a controller 390
comprising a function generator (Stanford Research Systems, DS345)
passed through an RF amplifier (T&C Power Conversion, AG 1006).
An impedance-matching circuit of the controller 390 is used to
match the impedance of the TAG lens 300 at its operating
frequencies with the 50.OMEGA. output impedance of the RF
amplifier. A fixed component impedance matching circuit is used,
which works well over the range 100 kHz-500 kHz. Most of the data
presented here is acquired at a frequency of 257.0 kHz. If driven
near an acoustic resonance of the lens 300, then the amplifier and
impedance matching circuit are unnecessary, and this modified setup
has been used to acquire data over larger frequency ranges. The
data presented herein cover the range 250 kHz-500 kHz at amplitudes
from 0-100 V peak-to-peak.
[0164] The driving parameters were chosen to best illustrate the
multiscale nature of the Bessel beam. The TAG lens frequencies are
chosen so that the lens 300 appears to be operating close to a
single-mode resonance. Driving amplitude was chosen to provide
well-defined major and minor rings for this example. The imaging
distance for fixed-z figures was chosen to be approximately at the
midpoint of the multi-scale Bessel beam.
[0165] The coordinate system used for presenting the theoretical
calculations and experimental results is defined with z in the
direction of the propagation of the light, x and y being transverse
coordinates at the image plane, and .xi. and .eta. being transverse
coordinates at the lens plane, as shown in FIG. 20B. The radial
coordinates are given by
? = ? + ? and .rho. = ? + ? . ? indicates text missing or illegible
when filed ##EQU00040##
Theory and Numerical Methods
[0166] The ultimate goal of the following theory is to describe the
physics of light propagation through the lens 300, particularly in
the case when coherent, collimated light is shone through it.
Refractive Index Profile
[0167] The first step in modeling the TAG lens 300 is to determine
the index of refraction profile. It has been calculated in section
I above that the refractive index within the TAG lens 300 is of the
form,
? = ? ? indicates text missing or illegible when filed ( 58 )
##EQU00041##
assuming a low viscosity filling fluid (kinematic viscosity much
less than the speed of sound squared divided by the driving
frequency) with linearized fluid mechanics, where n.sub.0 is the
static index of refraction of the filling fluid, .omega. is the
driving frequency of the lens 300, c.sub.s is the speed of sound of
the filling fluid, and .rho. is the radial coordinate in the lens
plane. This function is plotted in FIGS. 21A and 21B, at two
different times. If driven on resonance, then the sine changes to a
cosine, however as this section focuses on the standing waves
within the cavity, and not on transient effects, the temporal phase
shift is irrelevant.
[0168] The only parameter in Eq. 58 with some uncertainty is
n.sub.A. The modeling in section I estimates its value, however
n.sub.A is very sensitive to a number of experimental parameters,
most notably how close the driving frequency is to a resonance.
Because of this high sensitivity and experimental uncertainty in
some of the modeling parameters, here n.sub.A is treated as a
fitting parameter, adjusting its value in order to achieve the best
agreement between the theory and the experiments. This results in
values for n.sub.A on the order of 10.sup.-5 to 10.sup.-4, in good
agreement with modeling predictions of section I above.
[0169] It is important to note that the refractive index is a
standing wave that oscillates in time. This time-dependent index is
illustrated in FIGS. 21A and 21B. The theoretical predictions for
the corresponding time-dependent patterns are presented in FIGS.
21C and 21D. Experimental images are also presented for comparison
in FIGS. 21E and 21F. The experimental images were acquired by
operating a pulsed 20 ns laser at a repetition rate synchronized to
the TAG lens driving frequency. This synchronization was
implemented so that the laser fired at the same relative phase in
each period of the TAG lens oscillation, however the laser did not
fire on every TAG period, resulting in effective laser repetition
rates below 1 kHz. By adjusting the relative phase between the TAG
lens 300 and the laser, one can shift the pattern from FIG. 21E to
FIG. 21F and back to FIG. 21E. The phase delay between these two
patterns corresponds to half the period of the driving signal of
the TAG lens 300. As expected, one sees bright regions at local
maxima in the refractive index, and dark regions at local minima.
This time-varying nature of the beam presents interesting
opportunities for generating annular patterns in pulsed-laser
applications. The theoretical and experimental CW patterns
presented in the remainder of this section are a time-average of
the intensity patterns resulting from both upward and downward
pointing index profiles (and the continuously varying intermediate
profiles). As a result, major scale rings are observed with
Bessel-squared periodicity.
Geometric Optics
[0170] The TAG lens 300 is modeled using the thin lens
approximation, that is, a light ray exits the lens 300 at the same
transverse location where it entered the lens 300. The conditions
where this approximation is valid are examined below. Under the
thin lens approximation, the phase transformation for light passing
through a lens 300 is given by:
t ? ( .xi. , .eta. ) = exp ( ? ( nL + L ? - L ) ) . ? indicates
text missing or illegible when filed ( 59 ) ##EQU00042##
where k.sub.0 is the free-space propagation constant
(2.pi./.lamda.), L.sub.0 is the maximum thickness of the lens 300,
L is the thickness at any given point in the lens 300, and n is the
index of refraction at any given point in the lens 300. Since the
lens 300 is a gradient index lens, L=L.sub.0 throughout the lens
300, and it is only n that varies transversally.
[0171] In some cases, it may be useful to express the phase
transformation in Eq. 59 as the angle at which a collimated light
ray would leave the lens 300. After having traveled through the
bulk of the lens 300, but just before exiting, the equation for the
wavefront is k.sub.0n(.rho.)(L.sub.0+z)=const. At this point, an
incident light ray would have been deflected by an angle, .theta.,
that is perpendicular to the wavefront and is hence given by:
tan ( ? ) = - ? ? n ( .rho. ) .rho. , ? indicates text missing or
illegible when filed ( 60 ) ##EQU00043##
assuming the thin lens approximation. To get the angle of a light
ray leaving the lens 300, Snell's law is applied to the fluid-air
interface (since the transparent window 330 of the lens 300 is
flat, it has no effect on the angle of an exiting ray). This yields
.theta., the angle that a ray will propagate after leaving the lens
300.
sin ( .theta. ( .rho. ) ) = n ( .rho. ) sin ( ? ( .rho. ) ) , ?
indicates text missing or illegible when filed ( 61 )
##EQU00044##
assuming that n(.rho.)/n.sub.air.apprxeq.n(.rho.) . . . . Applying
small angle approximations to Eqs. 60 and 61 yields
.theta. ( .rho. ) = - L o n .rho. . ( 62 ) ##EQU00045##
[0172] In order to illustrate the physics behind the minor ring
interference patterns created by the TAG lens 300, it is useful to
consider a linear approximation to one of the peaks, as is shown in
FIG. 21A. In this case, linearization about the inflection point of
the central peak has been performed. A gradient index of refraction
lens 300 with this linear profile is fundamentally equivalent to a
uniform-index conical axicon. As long as the input beam completely
covers the central peak, this approximation is valid and reproduces
the key elements of the observed beam. However, if one apertured
the lens 300 so that the input beam only covered the rounded tip of
the central peak, then the linear/axicon approximation would not be
sufficient, and a second-order parabolic approximation would be
required.
[0173] The equivalence between the TAG lens 300 and an axicon can
be shown quantitatively. To determine the angle that a light ray
leaves an axicon, Snell's law is used:
sin ( .PHI. + .theta. ? ) = n ? sin ? . ? indicates text missing or
illegible when filed ( 63 ) ##EQU00046##
Here, .theta..sub.ax is the angle from the z-axis that a light ray
leaves the axicon, and .phi. is the angle between the z-axis and
the normal to the output face of the axicon. The cone angle of this
axicon is given by .alpha.=.pi.-2.phi.. Substituting in for .phi.
from Eq. 63, setting .theta..sub.ax=.theta. from Eq. 62, and
applying small angle approximations, it is possible to express the
cone angle of the corresponding axicon in terms of the parameters
of the linear gradient in index of refraction lens:
? = .pi. - 2 .PHI. = .pi. - 2 ? n .rho. . ? indicates text missing
or illegible when filed ( 64 ) ##EQU00047##
[0174] This equation forms the basis for the effects of tuning the
lens 300 by changing the driving amplitude. Increasing the driving
amplitude increases dn/d.rho. and is therefore identical to
increasing the cone angle of the equivalent axicon.
[0175] Even though this model of the TAG lens 300 ignores the
curvature of the refractive index, it does a good job of
qualitatively explaining the visible features, and furthermore the
experimental pattern around the central spot does closely resemble
Bessel beams generated by axicons, as shown below in the section
labeled "Beam characteristics". This model has so far neglected the
time dependence of the refractive index. However, simulations show
that the periodicity of the time-average pattern surrounding the
central spot is closely approximated by the instantaneous pattern
produced when the refractive index at the center of the lens 300 is
at its peak. While not significantly shifting their positions, the
time-averaging does decrease the contrast of the minor rings.
[0176] The major rings and their surrounding minor rings can be
explained in a similar way. The only difference between these minor
rings and those surrounding the central spot is that these rings
are derived from circular ridges in the refractive index as opposed
to a single peak. For example, the first major ring is established
from the peaks highlighted in FIG. 21B, a half-period later in time
with respect to the central spot from FIG. 21A. Due to the
time-averaging CW nature of the imaging method, images such as FIG.
19 exhibit major rings at the locations of the peaks in FIG. 21A,
as well as the peaks in FIG. 21B. The result is that the major ring
locations have a Bessel-squared periodicity--the same periodicity
as conventional Bessel beams.
Fourier Optics
[0177] Using the phase transformation for light passing through the
lens 300 given by Eq. 59, the electric field of the light upon
exiting the lens 300 is given by:
U ? ( .xi. , .eta. ) = t ? ( .xi. , .eta. ) U ? ( .xi. , .eta. ) ,
? indicates text missing or illegible when filed ( 65 )
##EQU00048##
[0178] where U.sub.0(.xi.,.eta.) is the electric field of the light
entering the lens 300.
[0179] In order to find the intensity profile at the image plane,
the field U.sub.TAG(.xi.,.eta.) must be propagated using a
diffraction integral. The Rayleigh-Sommerfeld diffraction integral
is used in this simulation. The assumptions involved in this
integral are that the observation point is many wavelengths away
from the lens 300 (r>>.lamda.) and the commonly accepted
assumptions of all scalar diffraction theories. The field at a
distance z from the lens plane is given by:
? ( x , y , z ) = ? ( .xi. , .eta. ) ? ? ? , ? indicates text
missing or illegible when filed ( 66 ) ##EQU00049##
where the integration is performed over the entire aperture of the
lens 300 and s(x, y, .xi., .eta.) is the distance between a point
(, .eta.) on the lens plane and a point (x, y) on the image plane,
given by:
? = ? + ( x - .xi. ) ? + ( y - .eta. ) ? ? indicates text missing
or illegible when filed ( 67 ) ##EQU00050##
[0180] Computationally, the integral in Eq. 66 can be difficult to
evaluate because the magnitude of k.sub.0 (on the order of 10.sup.7
m.sup.-1) leads to a phase factor in the integrand that varies
rapidly in .xi. and .eta.. Numerical approximations therefore
require a sufficient number of points in the transverse directions
to accurately represent the variation of this phase factor in the
domain of interest. The closer the image plane to the lens plane,
the more quickly that phase factor will vary (because s becomes
more strongly dependent on .xi. and .eta. now that z is small), and
the more points are required to accurately compute the integral.
Note that the integral is a convolution as the integrand is a
product of two functions, one of (.xi., .eta.) and another of
(x-.xi., y-.eta.). In this study the convolution integral is
computed using fast Fourier transforms (FFTs).
[0181] Finally, the intensity profile at the image plane is found
from the electric field as follows:
? ( x , y , z ) = 1 2 ? ? ( x , y , z ) ? ? indicates text missing
or illegible when filed ( 68 ) ##EQU00051##
[0182] In the following section, all theoretical figures are
obtained using this Fourier method, assuming a refractive index of
the form of Eq. 58 and averaging many images corresponding to
instantaneous patterns generated at different times within one
period of oscillation.
Beam Characteristics
[0183] This section describes the characteristics of the
TAG-generated time-average multiscale Bessel beam. The TAG beam
characteristics are divided into two categories: the nature of the
beam propagation and the ability to tune the beam. In each
category, theoretical predictions, experimental results, and
comparisons between the two are presented. The specific propagation
characteristics are the beam profile, the axial intensity
variation, the beam's nondiffracting nature, and the beam's
self-healing nature. The parameters of the beam that are tunable
include the major ring locations, the minor rings locations, the
central spot size, and the working distance. Intensity values in
all plots (except on-axis intensity) have been normalized so that
the wide Gaussian beam (full width at half-maximum greater than 1
cm) incident to the TAG lens 300 has a peak intensity of 1.
Beam Propagation
[0184] The first important demonstration is that the lens 300 does
produce a multiscale Bessel beam. FIG. 19 shows the multi-scale
nature of the intensity pattern, while FIG. 22 shows how the minor
scale and first major ring of the experimental TAG beam propagate.
Note that the central lobe of the beam propagates over a meter
without significant diffraction--one of the key properties of a
Bessel beam. No experimental method of creating an approximate
Bessel beam creates a truly nondiffracting beam because all
experimental methods limit beams with finite apertures, whereas a
true Bessel beam is infinite in transverse extent.
[0185] An intensity profile of the multiscale Bessel beam is
plotted in FIG. 23. This is a slice of FIG. 22, 70 cm behind the
lens 300. One can observe four minor rings with radius less than 1
mm. The location of these rings is consistent between the
experimental and theoretical curves, however the peak intensity is
significantly lower in the experiment, most likely due to wavefront
errors either from the incident beam or from scattering within the
TAG lens fluid or windows. Another possible explanation is a
discrepancy between Eq. 58 and the real system, perhaps because of
nonlinear acoustic effects within the lens 300, or because of the
excitation of higher order acoustic Bessel modes (J.sub.n(
),n>0) because of slight nonuniformities in the piezoelectric.
Despite this error in peak intensity, good theoretical-experimental
agreement was found in the minor ring location by fitting n.sub.A
from Eq. 58 to the experimental data. The resulting value of
n.sub.A is 4.times.10.sup.-5. Neglecting the intensity difference,
the following figures will show that this single parameter fit
accurately describes all the spatial characteristics of the
observed TAG beam. Note that the location of the first major ring
is accurately predicted by the model.
[0186] The minor ring fringes in FIG. 23 are similar to the fringes
one would expect from an axicon. Performing the linear
approximation illustrated in FIG. 21A, and using Eq. 64, the cone
angle of the corresponding axicon is 179.5.degree.. For many
applications, cone angles close to 180.degree. are used because of
their long working distance and large ring spacing. Experimentally
so far, equivalent cone angles as sharp as 178.degree. have been
achieved, however sharper cone angles can be achieved by increasing
the thickness of the lens 300 (L.sub.0) or further increasing the
driving amplitude, which are both linearly proportional to the
range of cone angles (Eq. 58). In addition, optimizing the driving
frequency can lead to sharper effective cone angles by taking
advantage of the acoustic resonances at certain frequencies.
[0187] The essential characteristic of nondiffracting beams is that
the transverse dimensions of the central lobe remain relatively
constant in z. From FIG. 22, one can see that this is the case for
the TAG beam. This property is plotted quantitatively in FIG. 24.
Although the TAG beam does diverge in both theory and experiment
because it is not an exact Bessel beam, this divergence is small
compared to that of a Gaussian beam, and similar to that of an
axicon-generated beam.
[0188] If a conventional collimated Gaussian beam is focused so
that it has a minimum spot size (radius at which the electric field
amplitude falls to 1/e of its peak value) of 150 .mu.m at z=58 cm,
then by the time the light reaches z=100 cm, the spot size will be
more than 500 .mu.m. In contrast, a TAG beam with this beam waist
would only diverge to a size of 175 .mu.m after this distance. This
Gaussian beam waist is chosen to match the experimental width of
the TAG beam at the location where the theoretical TAG beam reaches
its peak intensity.
[0189] Apart from being nondiffracting, the other major feature of
Bessel beams is their ability to self-heal. Because the wavevectors
of the beam are conical and not parallel to the apparent
propagation direction of the central lobe of the beam, the
intensity pattern of Bessel beams is capable of reforming behind
obstacles placed on-axis. This feature is experimentally
demonstrated for a TAG beam in FIG. 25. An obstruction is placed
slightly after the initial formation of the TAG minor scale Bessel
beam. This obscures the beam for a short distance, however the
Bessel beam eventually heals itself and reforms approximately 30 cm
beyond the obstruction.
Tunability
[0190] One of the most innovative features about the TAG lens 300
is the ability to control the shape of the emitted beam. One can
directly tune the major and minor ring sizes and spacings without
physically moving any optical components. The major and minor
scales of the Bessel beam are both adjustable because of the two
degrees of freedom in the time-average pattern: the driving
frequency and the driving amplitude. Changing the driving amplitude
modifies only the minor scale, while changing the driving frequency
excites different cavity modes and will affect both scales of the
beam. Independent tunability of the major and minor rings is useful
in applications such as optical tweezing for manipulating trapped
particles relative to each other, laser materials processing for
fabricating features of different size, and scanning beam
microscopy for switching between high-speed coarse images and
slow-speed high-resolution images.
[0191] Tuning the major ring spacing can be achieved by modulating
the driving frequency as shown in section I above. This is because
the major rings occur near the extrema of
J.sub.0(.omega..rho./c.sub.s) from Eq. 58. Increasing the driving
frequency compresses the major rings, while decreasing the driving
frequency increases their spacing. The radial coordinate of the
first major ring, .rho.*, is approximately given by
? = ? .omega. , ? indicates text missing or illegible when filed (
69 ) ##EQU00052##
where 3.832 is the radial coordinate of the first minimum of
J.sub.0(.rho.). This function is plotted in FIG. 26, along with
experimental measurements. Optical propagation slightly shifts the
theoretical position of the intensity maximum relative to the index
maximum because the refractive index profile is not locally
symmetric between the inside and the outside of the first major
ring. The experimental results agree closely to the predictions,
however they do exhibit some variability between trials. This is
attributed to different filling conditions of the prototype lens
300. When refilling the lens 300 between trials, small changes in
the pattern are noticed, and also some azimuthal asymmetries. Any
asymmetries represent the contribution of non-J.sub.0 modes within
the lens 300. These modes exhibit different radial index
distributions and therefore result in slight shifting of the first
major ring. The effects of changing the driving frequency on the
minor scale pattern are complicated due to resonant amplitude
enhancement at certain driving frequencies.
[0192] The continuous tunability of the minor Bessel rings is
demonstrated experimentally (FIGS. 19A, 19B, 27A, 33A, 33B) and
theoretically (FIG. 27B) by varying the driving amplitude of the
TAG lens 300. The refractive index amplitude, n.sub.A, is directly
proportional to the driving voltage amplitude. This conclusion is
qualitatively supported by the similarity between FIGS. 27A and
27B. The theoretical change in the pattern with increasing index
amplitude closely resembles the experimentally observed change in
the pattern with increased voltage amplitude.
[0193] Increasing the driving amplitude increases the number of
discernable minor rings (FIGS. 27A, 27B, 33B) and decreases the
spacing between those rings (FIGS. 28, 33A) because it alters the
angle that light rays surrounding a peak are deflected, similar to
sharpening the cone angle of an axicon and increasing the spatial
frequency of the resulting interference fringes. (Note in FIGS. 27A
and 27B the gray-scale map has been scaled down for clarity. The
actual peak intensity is 51.) The range of plotted refractive index
amplitudes corresponds (at this frequency) to equivalent cone
angles from 180.degree. (lens off) to 179.degree.. Eventually, a
large enough amplitude results in interference between the minor
rings surrounding adjacent major rings. Driving amplitude does not
affect the major scale beam because it does not alter which
acoustic modes are excited within the lens 300. Such changes
directly correspond to changing the axicon's cone angle.
[0194] The driving amplitude can also be used to tune the working
distance. Here, the working distance is defined as the smallest
distance behind the TAG lens 300 where a minor ring is observed
surrounding the central peak (the distance until the start of the
Bessel beam). This can be inferred from FIGS. 27A and 27B. At low
amplitudes, the Bessel interference pattern is not apparent,
implying that the working distance at these amplitudes is greater
than 50 cm, the distance behind the lens 300 that these plots were
obtained. As the driving amplitude increases, the working distance
decreases, and the multiscale Bessel beam forms nearer the lens
300. This ability to tune working distance should be especially
useful for dynamic metrological applications, such as those
involved in long range straightness measurements. The TAG lens 300
would be helpful for using a single optical setup to very quickly
switch between straightness measurements on the order of
centimeters to straightness measurements on the order of hundreds
of meters. In addition, being able to tune the long working
distance of the Bessel beam should prove useful for scanning beam
microscopy.
[0195] This section has modeled and experimentally characterized
TAG-generated multiscale Bessel beams. This characterization has
verified the refractive index model for the TAG lens 300 presented
earlier. In addition, the connection between the minor scale of the
TAG beam and refractive axicons has been established. The
nondiffracting and self-healing characteristics of the TAG lens
beam have been experimentally proven and theoretically justified.
The ability to independently tune the major and minor scales of the
beam through driving frequency and amplitude has also been
presented, along with the tunability of the central spot size and
working distance.
[0196] Because of this tunability, TAG lenses may be used in
applications where dynamic Bessel beam shaping is required. In
particular, applications include optical micromanipulation,
laser-materials processing, scanning beam microscopy, and metrology
and others.
[0197] If a similar analysis is performed for a TAG lens 200 with a
square chamber, then instantaneous patterns such as those in FIG.
43 would be predicted, while time average patterns would look like
those shown in FIG. 44. Indeed similar time-average patterns are
seen experimentally (FIG. 42), if without the fine detail, due to
system aberrations. One sees that this pattern is similar to that
created by a square array of lenslets.
Justification for Approximation that TAG Lens is a Thin Lens
[0198] The TAG lens 300 can be approximated as a thin lens. This
approximation is valid if a ray of light does not significantly
deflect while passing through the lens 300. A corollary to this is
that a ray experiences a constant transverse gradient in index of
refraction throughout its travel within the lens 300. A specific
definition for what it means for a deflection to be "significant"
is provided below.
[0199] A bounding argument is used to justify the thin lens
approximation. It can be shown that the actual transverse
deflection of a ray passing through the lens 300 is less than the
deflection predicted by a thin lens model, which is in turn much
less than the characteristic transverse length scale: the spatial
wavelength of the acoustic Bessel modes within the lens 300. This
length scale is chosen because it implies that the deflected ray
experiences a relatively constant gradient in refractive index
while passing through the lens 300:
[0200] Let |.delta.|.sub.max be the maximum deflection experienced
by a ray. If the lens 300 is thin then the corresponding exit angle
of the ray (before the fluid-window interface) is given by
.theta.(.rho.) from Eq. 60, reproduced here:
tan ( ? ( .rho. ) ) = - ? n ( .rho. ) n ( .rho. ) .rho. . ?
indicates text missing or illegible when filed ( A1 )
##EQU00053##
In the thin lens case, there is a one-to-one correspondence between
the ray with the largest exit angle, .theta..sub.max and the ray
with the largest deflection, |.delta.|.sub.thin,max. Furthermore,
from Eq. A1, one can see that this ray passes through the region of
the lens 300 with the greatest gradient in refractive index,
(dn/d.rho.).sub.max.
[0201] If the lens 300 was not a thin lens, light rays passing
through it would deflect and therefore a single ray could not
experience a refractive gradient of (dn/d.rho.).sub.max during its
entire trip. In fact, in some regions it must experience smaller
gradients, and hence the deflection of a ray passing through the
same lens 300 with thick lens modeling would actually be smaller
than calculated with thin lens modeling. The deflection predicted
under the thin lens model, |.delta.|.sub.hrmthin,max, therefore
serves as an upper bound for the true deflection,
|.delta.|.sub.max.
[0202] Going back to the thin lens model, one can bound the
transverse deflection based on the exit angle. The angle of the
light ray monotonically increases as it passes through the lens 300
until it reaches the value of the exit angle. Hence, the total
deflection, |.delta.|, must be less than the length of the lens 300
times the tangent of the exit angle.
[0203] Putting all of the above inequalities together, one can now
bound the total deflection in the lens 300 assuming a thick lens
model:
? = ? ? indicates text missing or illegible when filed ( A2 )
##EQU00054##
One can easily compute the value of the term on the right hand side
of Eq. A2. If this turns out to be much less than the spatial
wavelength of the acoustic fluid mode, then one can conclude that
the TAG lens 300 is a thin lens. For the patterns studied here, the
spatial wavelength of the acoustic mode is approximately 3 mm. For
n.sub.A=4.times.10.sup.-5 (best fit value from experimental data),
the maximum gradient in index of refraction is 0.043 m.sup.-1. Eq.
A2 then gives |.delta.|.sub.max<50 .mu.m, which is much less
than 3 mm. Therefore the thin lens approximations made throughout
this section are justified, and one can build on the results of
this section to exploit TAG lenses to effect dynamic pulsed-beam
shaping.
IV. Dynamic Pulsed-Beam Shaping
[0204] The ability to dynamically shape the spatial intensity
profile of an incident laser beam enables new ways to modify and
structure surfaces through pulsed laser processing. In one of its
aspects, the present invention provides a device and method for
generating doughnut-shaped Bessel beams from an input Gaussian
source. The TAG lens 100, 300 is capable of modulating between
focused beams and annular rings of variable size, using sinusoidal
driving frequencies. Laser micromachining may be accomplished by
synchronizing the TAG lens to a 355 nm pulsed nanosecond laser.
Results in polyimide demonstrate the ability to generate adjacent
surface features with different shapes and sizes.
[0205] The experimental setup used for dynamic pulsed-beam shaping
is shown in FIG. 29. The light source is a Nd:YVO.sub.4 laser 362
(Coherent AVIA) delivering 15 ns duration pulses at a 355 nm
wavelength and maximum repetition rate of 250 kHz. The output beam
has a Gaussian profile, with a measured diameter of about 3.5 mm at
1/e.sup.2 and an M.sup.2 approximately 1.3. The beam is directed
into the 3.5 cm radius TAG lens 300 of FIGS. 3A, 3B filled with
silicone oil (0.65 cS Dow Corning 200 Fluids). The controller 390
comprising a wavefunction generator (Stanford Research Systems
Model DS 345) provides a radio frequency (RF) sinusoidal signal
between 0.33 and 1.20 MHz to drive the piezoelectric crystal,
generating vibrations inside the silicone oil, though other signals
than a single frequency sinusoid may be used, such as a sum of two
or more sinusoids of differing frequency, or a Fourier series per
Eqs. 54 and 55, for example. The refractive index of the TAG lens
cavity continuously changes with the instantaneous value of the AC
signal.
[0206] Synchronization of the laser 362 and TAG lens 300 is
accomplished using a pulse delay generator triggered off the same
AC signal. A pulse delay generator (Stanford Research Systems Model
DG 535), which may be provided as part of the controller 390, is
programmed to provide a specific phase shift from the trigger
signal that can be much greater than 2.rho.. In this way, it is
possible to synchronize the laser pulses with the TAG lens 300 so
that each pulse meets the lens 300 in the same state of vibration.
Because the phase shift is greater than one period, the effective
repetition rate of the laser pulse can be arbitrarily controlled
within the specifications of the laser source.
[0207] For micromachining, the size of the shaped laser beam is
reduced by a pair of lenses, L.sub.1, L.sub.2, with focal lengths
of 500 mm and 6 mm respectively. The demagnified laser beam
illuminates the surface of a thin layer (about 4.7 .mu.m) of
polyimide coated on a glass plate 380, which is mounted on an x-y-z
translation system. Photomodified samples are then observed under
an optical microscope and characterized by profilometry.
Results, Instantaneous Patterns (Basis)
[0208] There are three main parameters of the TAG lens 300 that
affect the dimensions and shapes of the patterns that can be
generated: the driving amplitude, the driving frequency, and the
phase shift between the driving signal and the laser trigger. For
this section, attention focuses on simple shapes including annuli
and single spots, although the TAG lens 300 is capable of more
complex patterns. These instantaneous patterns are denoted herein
as the "basis". In general, the frequency affects the diameter of
the ring, the amplitude affects the sharpness and width of the
rings. The phase selects the nature of the instantaneous pattern.
For instance, when the index of refraction is at a global maximum
in the center, the instantaneous pattern is a spot, but at half a
period later when it becomes a global minimum, the instantaneous
pattern is doughnut or annular shaped.
[0209] In FIG. 30, a few elements of the basis are presented. These
patterns are acquired 50 cm away from the TAG lens 300 but without
the reducing telescope lenses L.sub.1, L.sub.2. Rings with various
diameters, ranging over an order of magnitude from a single bright
spot to about 4 mm dimension are obtained. Different driving
frequencies ranging from 0.33 to 1.29 MHz are used to generate the
annular shapes shown in the figure with fixed amplitude (9.8 V peak
to peak) and phase angle chosen to optimize the ring shape. For
high throughput micromachining, one would use this basis to define
a lookup table that establishes the correspondence between driving
amplitude, frequency, phase shift and the observed instantaneous
intensity distribution. A slight eccentricity in the micromachined
rings is noted due to minor imperfections and nonuniformities in
fabricating the piezoelectric tube 310. Furthermore, these
asymmetries depend on the driving frequency due to resonance
behavior in the tube 310. The configuration of the TAG lens 100 of
FIGS. 1A, 1B may cancel this unwanted effect.
Sample Micromachining
[0210] The TAG lens 300 is capable of high energy throughput
without damage and can therefore be used for pulsed laser
micromachining FIGS. 31A, 31B demonstrates this point for a
polyimide film. In this case, the incident laser energy on the
sample is 8.2 .mu.J and the driving frequency is 700 kHz resulting
in a 15 .mu.m diameter ring in the film. Calculating the actual
fluence is difficult because the background of the beam, although
subthreshold, still carries significant energy as expected for a
Bessel-like beam. A profilometry analysis of the irradiated
polyimide thin film shows a well defined annular structure with
depth of 0.9 .mu.m and a width of approximately 3.5 .mu.m.
Additional studies could be performed to assess the heat affected
zone surrounding the laser-induced structures using these
non-traditional intensity profiles.
[0211] In contrast to many other methods of producing annular
beams, the TAG lens 300 gives the added ability to rapidly change
the pattern according to the structure or pattern required. To
demonstrate this effect, FIG. 32 gives an example of adjacent
laser-induced surface structures that alternate between a central
spot and an annular beam. For this figure, the two basis elements
are selected by varying the phase shift such that the index
function is at either a maximum or minimum while the substrate is
manually translated. In the upper image of the figure, a driving
frequency of 989 kHz is used. The bright central spot shows
indication of a second ring that causes damage to the polyimide
film. However, when driven at lower frequencies as in the lower
image (531 kHz), one is able to obtain single spots in the image.
This effect can be ameliorated by better optimization of the phase
shift and amplitude in order to maximize the energy difference
between the central spot and the outer rings. Furthermore,
aperturing or reducing the size of the beam incident on the TAG
lens 300 can be used to remove unwanted outer rings.
Switching Time of the TAG Lens
[0212] The ability to switch rapidly between two distinct intensity
distributions is a key parameter in evaluating the relevance of a
beam shaping strategy for micromachining or laser marking purposes.
When using a TAG lens 100, 200, 300, two situations have to be
considered. Either the elements of the basis can be reached by
driving the TAG lens 300 at a single frequency (FIG. 32), or the
lens 300 may be operated at different frequencies, as illustrated
in FIG. 30.
[0213] When all the desired shapes can be generated by using the
same lens driving frequency, the theoretical minimum switching time
is given by half of the driving signal period. In FIG. 32, the TAG
lens 300 was driven at a frequency of 989 kHz (upper picture) and
531 kHz (lower picture) implying that toggling times between two
adjacent patterns are theoretically as short as 0.5 .mu.s and 0.9
.mu.s, respectively, or twice the driving frequency. This converts
to a switching frequency of approximately 1-2 MHz. Although these
values are too low for optical communication and switching
requirements, these rates are more than sufficient for pulsed laser
processing.
[0214] In the case that frequency changes are needed, the minimum
amount of time required to switch the pattern is equal to the
amount of time it takes to propagate the sound wave from the
piezoelectric to the center of the lens 300. This is denoted as the
TAG lens 300 response time. The instantaneous pattern is
established at this time, followed by a transient to reach steady
state. In the context of pulsed laser processing, it is the
response time that is the relevant test of lens speed. As an
example, considering the sound velocity in the silicone oil to be
about 900 ms.sup.-1 and a radius of 3.5 cm for the lens 300, the
response time is as short as 40 .mu.s. However, by changing
temperature or the refractive filling fluid, the speed of sound can
be increased and the response time can be significantly
decreased.
[0215] The effects of transients in the output of the TAG lens 300
to reach steady state using the silicone oil with a viscosity of
0.65 cS are relatively fast ranging from 2-3 ms is shown in FIG.
39. The first picture in the upper left hand corner shows the TAG
output when the driving frequency is first turned on. Within the
first 0.5 ms, the pattern begins to establish a bright central spot
with emerging major rings. These rings continue to sharpen over the
next 0.5 ms as does the central spot. After 1 ins, the minor rings
begin to form and continue developing for the next 0.5 ms. Finally
after 2.0 ms, the pattern remains steady.
[0216] To gain more precise information about the time needed to
reach steady state, a high speed photodiode was used to measure the
intensity of the central, spot. FIG. 40 illustrates this
measurement as the lens 300 is turned off and the acoustic wave
dissipates in glycerol. The TAG lens's driving voltage is set to
zero at t=0. The liquid in the TAG lens 300 was replaced with
glycerol although results with the silicone oils were also
obtained. The plots are similar but, due to viscosity differences,
have different time scales. At negative times, the lens 300 is
operating in the steady state producing the oscillatory behavior
discussed in Section I. Once the driving frequency is shut off, the
oscillations rapidly decay toward the constant value. The decay
time is quantified by extracting the time for the voltage to
decrease to 1/e of the initial steady state. FIG. 41 shows these
results for different filling liquids and different initial driving
frequencies.
[0217] The steady state time is expected to be dependent on the
viscosity of the fluid and the driving frequency. The data in FIG.
41 show a modest dependence on driving frequency that is within the
measurement noise. A higher viscosity fluid damps out transients
and reaches steady state more quickly as shown in the figure. The
average time constant is 2.1 ms, 650 .mu.s, and 320 .mu.s for 0.65
cS silicone oil, 100 cS silicone oil, and glycerol respectively.
The images of FIG. 39. showing silicone oil at 0.65 cS agree with
the decay time measured in this manner. Therefore one can expect
switching rates of 1/decay time or 500-3000 Hz.
V. Dynamic Focusing and Imaging
[0218] In another of its aspects, the present invention provides a
TAG lens 100, 300 and method for a rapidly changing the focal
length. The TAG lens 100, 300 is capable of tuning the focal length
of converging or diverging beams by using an aperture to isolate
portions of the index profile and synchronizing the TAG lens 100,
300 with either a pulsed illumination source, or a pulsed imaging
device (camera).
[0219] The experimental setup is similar to that described earlier
in FIG. 29. In this experiment, a pulsed UV laser 560 is used at
355 nm with 15 ns pulse durations and a maximum of 250 kHz
repetition rate to illuminate an object of interest 510, FIG. 35.
In order to improve imaging and reduce interference, a diffuser or
scattering plate, SP, is used. The light may be directed using a
mirror 512 and beam splitter 514 through the object of interest 510
which for this example is a US Air Force calibration standard (USAF
1951) that is located at a position, d, away from the entrance to
the TAG lens 300. An aperture 520, or iris, with approximately 1.5
mm interior diameter, is co-axially located on either the input or
output side of the TAG lens 300 to restrict optical access to the
portion of the TAG lens 300 at which the desired index of
refraction is located (cf. FIGS. 21A, 21B, 34 showing refractive
index versus lens radius). Finally a lens, L1, may be disposed in
the output path of the TAG lens 300 after the aperture 520 to
produce an image at an image plane 530 where a detector 540, e.g.
CCD camera, is located to record the image. The filling materials,
details of the electronic circuitry, and the typical driving
frequencies for the TAG lens 300 are given in paragraph[00204]
above.
[0220] In order to successfully use the TAG lens 300 as a dynamic
focusing and imaging device, it is necessary to synchronize the
incident laser pulse (or camera shutter) to trigger at the
appropriate temporal phase location of the TAG lens driving signal.
This is accomplished and described in detail above by using a pulse
delay generator that is triggered from the RF signal driving the
TAG lens 300. It is possible to accurately control the exact phase
difference between the laser and the TAG driving signals and
therefore, the instantaneous state of the index of refraction
profile when light is passing through it.
[0221] The detailed physics of the lens operation is described
earlier in section I and with the driving voltage at 334 kHz and
9.8 V.sub.p-p. However, in order to understand this implementation
of the TAG lens 300, it is instructive to refer to FIG. 21. As can
be seen, at a time t=0 (FIG. 21A), the index of refraction profile
is at a global maximum in the center of the lens 300 (.rho. or
.xi.=0) and at half a period later, t=T/2 (FIG. 21B), the index of
refraction is at a global minimum in the center. Based on the
sinusoidal temporal dependence of equation for the index of
refraction profile Eq. 58, one can easily see that the
instantaneous profile will oscillate continuously between this
global maximum and global minimum at .rho.=0. At times in between,
the index profile will be either a local maximum or minimum at
.rho.=0.
[0222] As noted in FIGS. 21A, 21B, if one were to look at the
inflection points in the index profile, one can approximate the
index as linear and produce a Bessel beam. However, if instead, one
were to look only in the small region near .rho.=0 this region of
the index profile is more accurately represented as a parabolic
function. This result can be seen by taking a Taylor expansion of a
Bessel function about .rho.=0 and keeping the lowest order term in
.rho. which for a Bessel function is .rho..sup.2. Therefore, if one
were to use an aperture to filter out the region of the lens 300
that is not within this parabolic region, one is left with an index
of refraction profile that resembles a simple converging (or
diverging) lens 300 with a focal length given by the amplitude and
width of this region.
[0223] What is notable about this interpretation of the TAG lens
index of refraction profile is that since the curvature, and
therefore the effective focal length, depends on the instantaneous
amplitude and driving frequency of the acoustic wave within the
lens 300, the effective focal length will change continuously in
time. Thus in the same manner that one can synchronize individual
patterns of the TAG lens 300, one can synchronize the light source,
e.g. laser 560, or imaging device to select any focal length that
is needed subject to the limitations of the driving signal. For
example, synchronizing the light source to the pattern in FIG. 21A
would give the shortest converging focal length while synchronizing
to the pattern in FIG. 21B would give the shortest diverging
(negative) focal length.
[0224] In order to demonstrate this point, the described apparatus
of FIG. 35 has been used to image a calibration standard 510 at
various locations along the optical path. FIG. 36 shows three
images of the resolution standard 510 located at positions, d=7.5,
26, and 59 cm, from the entrance to the TAG lens 300. As can be
seen from the images, at each location, it is possible to
accurately generate an image at the image plane 350 with the CCD
array of the camera is located, without changing the position of
the lens 300 relative to the image plane 530. Each of the three
images in FIG. 35 is taken with a different temporal phase delay
between the TAG and the light source, as illustrated in the graph
of FIG. 36 showing the laser pulse timing against TAG driving
signal. This result indicates that it is possible to synchronize
the lens 300 so that an object 510 located an arbitrary distance
away can be imaged.
[0225] In carefully looking at the images in FIG. 36, one notices
that not only are the images brought into focus at the different
locations, but that the magnification of the image changes as a
function of the position along the optical axis as -would be
expected. Such an effect is consistent with the fact that the focal
length of a single lens 300 is being changed and is a benefit for
those desiring not only the ability to image an object 510 at an
arbitrary distance, but also to change the size of the image.
However, one can remove the change in magnification with the
appropriate combination of standard optical elements and
additional, synchronized TAG lenses in the optical path, if
desired.
[0226] The speed at which one can move between the different object
locations is exceedingly fast compared to any other adaptive
optical element since one only needs to change the phase difference
between the TAG lens 300 and laser driving signal. Therefore, times
that are mere fractions of the oscillation period can accommodate
large changes in the location of the object plane. For instance, in
FIG. 21A, the amount of time needed to switch between the first
image at d=0 and the other two images is 0.28 and 0.48
microseconds, respectively. A more complete discussion on the
switching time is presented in section IV above.
[0227] In addition to being able to move an object and image it at
a different location, it is possible to rapidly switch between
existing objects located at different places on the optical axis.
In FIG. 37, an experimental setup similar to that of FIG. 35 is
shown, but instead of a calibration standard 510, there is a series
of wires A, B, C positioned at different locations in front of the
TAG lens 300 with different offset angles to render each wire A, B,
C identifiable in the image plane 530. In this setup, the lens 300
can be synchronized with the laser pulse to bring any one of the
three wires A, B, C, into focus on the image plane 530 without the
need to remove the other two non-imaged wires. This also indicates
that the apertured TAG lens 300 is not just acting as a pinhole
camera because only one wire A, B, C is in focus in any given
image.
[0228] The other notable aspect of this experiment is that the TAG
lens 300 can be synchronized to be either a converging or diverging
lens 300 depending upon the phase difference. In the experimental
setup with the three wires A, B, C, the focal length of the lens L1
and the distances on the optical axis are configured so that it is
necessary for the TAG lens 300 to be either converging, diverging,
or planar in order to bring one of the wires A, B, C into focus. As
can be seen from the image in FIG. 38A, when the laser 560 is
synchronized so that it illuminates the TAG lens 300 while the
instantaneous index of refraction shows negative curvature, a
converging lens 300 is encountered and the wire A closest to the
lens 300 is brought into focus. When the laser is synchronized to
pass through the TAG lens 300 when the index function has zero
amplitude, the TAG lens 300 acts as a flat plate and therefore does
not disrupt the wavefront curvature. Thus, the middle wire B is in
focus by operation of the lens L1 alone, FIG. 38B. Finally, when
the synchronization is done such that the instantaneous index of
refraction exhibits positive curvature and the laser pulse passes
through the lens 300, it behaves as a diverging lens 300 and the
furthest wire C is brought into focus, FIG. 38C. Thus, ability to
dynamically position the focal position of a converging or
diverging TAG lens 300, or to rapidly change the location of and
magnification in an image plane, enables fundamentally new methods
of imaging with great potential in the fields of industrial
controls, homeland security, biological imaging and many other
important areas.
[0229] These and other advantages of the present invention will be
apparent to those skilled in the art from the foregoing
specification. Accordingly, it will be recognized by those skilled
in the art that changes or modifications may be made to the
above-described embodiments without departing from the broad
inventive concepts of the invention. It should therefore be
understood that this invention is not limited to the particular
embodiments described herein, but is intended to include all
changes and modifications that are within the scope and spirit of
the invention as set forth in the claims.
* * * * *