U.S. patent application number 10/909108 was filed with the patent office on 2005-01-13 for passive optical resonator with mirror structure suppressing higher order transverse spatial modes.
This patent application is currently assigned to Axsun Technologies, Inc.. Invention is credited to Kuznetsov, Mark E..
Application Number | 20050007666 10/909108 |
Document ID | / |
Family ID | 25263542 |
Filed Date | 2005-01-13 |
United States Patent
Application |
20050007666 |
Kind Code |
A1 |
Kuznetsov, Mark E. |
January 13, 2005 |
Passive optical resonator with mirror structure suppressing higher
order transverse spatial modes
Abstract
An optical resonator is designed to suppress higher order
transverse spatial modes. Higher order transverse modes in the
inventive optical resonator are forced to be unstable, and
ultimately achieving single transverse mode resonator operation.
Specifically, the mirror shape or intracavity lens profile is
tailored to bound the lower order modes while rendering the higher
order modes unstable. This has application in MEMS/MOEMS devices by
reducing side mode suppression ratio (SMSR) dependence on alignment
tolerances, for example.
Inventors: |
Kuznetsov, Mark E.;
(Lexington, MA) |
Correspondence
Address: |
J GRANT HOUSTON
AXSUN TECHNOLOGIES INC
1 FORTUNE DRIVE
BILLERICA
MA
01821
US
|
Assignee: |
Axsun Technologies, Inc.
Billerica
MA
|
Family ID: |
25263542 |
Appl. No.: |
10/909108 |
Filed: |
July 30, 2004 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
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10909108 |
Jul 30, 2004 |
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09833139 |
Apr 11, 2001 |
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6810062 |
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Current U.S.
Class: |
359/578 |
Current CPC
Class: |
H01S 2301/166 20130101;
H01S 5/18388 20130101; H01S 3/08072 20130101; H01S 5/141 20130101;
H01S 3/08059 20130101; H01S 3/08045 20130101 |
Class at
Publication: |
359/578 |
International
Class: |
G02B 027/00 |
Claims
What is claimed is:
1. A passive optical resonator comprising at least one optical
cavity defined by at least two mirror structures in which at least
one of the mirror structures has a mirror profile having a diameter
and sag that are selected in combination with a length of the
cavity to degrade a stability of transverse modes with mode numbers
4 and greater.
2. A resonator as claimed in claim 1, wherein the length of the
optical cavity is less than about 50 micrometers, the sag of the
mirror profile is less than about 200 nanometers, and a full width
at half maximum diameter of the mirror profile is less than 30
micrometers.
3. A resonator as claimed in claim 1, wherein the length of the
optical cavity is less than about 30 micrometers, the sag of the
mirror profile is less than about 150 nanometers, and a full width
at half maximum diameter of the mirror profile is less than 20
micrometers.
4. A resonator as claimed in claim 1, wherein the length of the
optical cavity is less than about 20 micrometers, the sag of the
mirror profile is less than about 100 nanometers, and a full width
at half maximum diameter of the mirror profile is less than 15
micrometers.
5. A resonator as claimed in claim 1, wherein the sag of the mirror
profile is less than about 150 nanometers.
6. A resonator as claimed in claim 1, wherein the sag of the mirror
profile is less than about 100 nanometers.
7. A resonator as claimed in claim 1, wherein an optical distance
between the mirror structures is tunable.
8. A resonator as claimed in claim 1, wherein an optical distance
between the mirror structures is tunable by out-of-plane deflection
of one of the mirror structures.
9. An optical resonator comprising at least one optical cavity
defined by at least two mirror structures wherein a net profile of
the mirror structures is concave in a center region surrounding an
optical axis and flat and/or convex in an annular region
surrounding the center region, and wherein a diameter of the center
regions is selected in response to a mode field diameter of a
lowest order mode of the resonator.
Description
RELATED APPLICATIONS
[0001] This application is a Continuation of application Ser. No.
09/833,139 filed on Apr. 11, 2001 which is incorporated herein by
reference in its entirety.
BACKGROUND OF THE INVENTION
[0002] Optical resonators include two or more mirror structures
that define the resonator cavity. Optical resonators can be passive
cavity devices as used, for example, in tunable Fabry-Perot (FP)
filters. Active cavity devices also include a gain medium, such as
a semiconductor or a solid-state material, inside the cavity
between the mirror structures. The most common example of an active
cavity optical resonator is the laser.
[0003] A reoccurring issue in optical resonator design, both in
macroscopic and micro optical systems, is transverse spatial mode
control. At scales associated with micro optical systems, which
include single mode optical fiber, semiconductor gain media, and/or
micro-optoelectromechanical system (MOEMS) devices, spatial mode
control can dictate many system design variables.
[0004] Typically, fundamental transverse mode operation is desired
in laser devices because of the optical beam spatial profile
requirements for long distance beam propagation, focusing of beams
into small spots, and beam coupling into single mode transmission
fibers. In addition, different spatial modes of an optical
resonator typically have different resonant optical frequencies,
which characteristic is detrimental for active and passive cavity
applications requiring spectral purity. A typical application
requiring spectral purity of the resonator operation is optical
spectral monitoring using tunable Fabry-Perot filters of optical
signals such as wavelength-division-multiplexed (WDM) optical
communication signals.
[0005] In active cavity devices, such as edge-emitting
semiconductor lasers, the transverse spatial mode problem is
addressed by the judicious design of the laser waveguide to ensure
that it supports only a single transverse mode. In vertical cavity
surface emitting lasers (VCSEL's), oxide confining layers and other
aperturing techniques are used to achieve single transverse mode
operation in small aperture devices. Problems begin to arise as
higher output power devices are designed, however. There is
contention between the desire to increase modal volume and beam
diameter while at the same time suppressing oscillation of the
higher-order transverse modes.
[0006] In passive cavity devices, the transverse mode problems can
be more intractable, since the design degree of freedom associated
with the gain medium is not present. One solution is to incorporate
single mode fiber into the design. The inclusion of fiber, however,
tends to complicate increased device integration, creates
fiber-coupling requirements, and does not resolve all of the
spatial mode problems. For example, a detector can be responsive to
the input TEM20 mode even with spatial mode filtering offered by a
single mode fiber; this is due to the substantial amount of power
propagating in the leaky and cladding modes of the fiber.
[0007] A related solution to controlling the transverse side mode
suppression ratio (SMSR) contemplates the use of intracavity
apertures or spatial filters. Higher order spatial modes generally
have larger mode field diameters than the fundamental TEM00 mode.
As a result, apertures in the optical train can induce loss for
higher order transverse modes and may be used to improve the side
mode suppression. These spatial filters, however, can introduce
some loss to the fundamental mode as well as to the higher order
modes; they also require precise alignment.
[0008] Still another solution concerns cavity design. In a confocal
Fabry-Perot cavity, where cavity length is equal to the mirror
radius of curvature, all transverse modes are degenerate, i.e., all
the transverse modes coexist on the same frequencies, or
wavelengths, as the longitudinal mode frequencies or the
longitudinal mode frequencies shifted by a half spectral period.
MOEMS micro optical cavities typically have large free spectral
ranges, or spectral periods, corresponding to small cavity lengths
of only tens of micrometers, however. Therefore the confocal MOEMS
micro cavity configuration would require mirrors with
correspondingly small radii of curvature, i.e., tens of
micrometers, which are difficult to fabricate, and have small mode
sizes, which are difficult to align.
[0009] A more typical configuration for MOEMS tunable filter
Fabry-Perot cavity is termed a hemispherical cavity. In such
cavities, one of the reflectors is near planar and the other
reflector is a spherical reflector. The advantage is reduced
alignment criticalities because of the general radial homogenatiy
of the flat reflector. In such configurations, spatial mode
spectral degeneracy is not present and higher order transverse
modes present a problem--spurious peaks are observed in the filter
transmission spectrum.
[0010] These problems have led to solutions that focus on
minimizing the excitation of higher order modes by precise control
of how light is launched into the cavity. For example, U.S. patent
application Ser. No. 09/666,194, filed on 21 Sep. 2000 by Jeffrey
A. Korn, and Ser. No. 09/747,580, filed 22 Dec. 2000 by Walid A.
Atia, et al., concern, in part, alignment of a tunable filter
relative to the surrounding optical train. U.S. patent application
Ser. No. 09/809,667, filed on 15 Mar. 2001 by Jeffrey A. Korn,
concerns mode field matching between the launch light mode and the
lowest order spatial mode of the filter. Such approaches minimize
excitation of higher order spatial modes and thus yield systems
with high side mode suppression ratios.
SUMMARY OF THE INVENTION
[0011] The present invention is directed to the design of the
optical resonator cavity mirrors and/or intracavity lenses. The
design intention for these mirrors/lenses is to degrade the ability
of the resonator to support higher order transverse spatial modes.
Higher order transverse modes are forced to be unstable in the
inventive optical resonator, ultimately achieving improved
transverse mode operation to single transverse mode resonator
operation.
[0012] Generally, previous approaches to transverse mode control in
optical resonators spatially varied only the magnitude of optical
beam transmission or reflection inside the resonator in order to
introduce differential loss for higher order stable transverse
modes. In contrast, the present invention includes spatially
varying the phase of optical beam transmission or reflection in the
resonator in order to make higher order transverse modes of the
resonator fundamentally unstable or unbound.
[0013] The invention can be analogized to optical fibers, which
achieve single transverse mode operation by using spatially varying
refractive indices, and hence spatially varying optical phase
delay, to make higher order transverse modes of the fiber unbound
and hence to suppress their propagation. Unfolded optical
resonators are represented by discrete lens waveguides, with
certain analogies to the continuously guiding waveguides, such as
optical fibers. In the present invention, analogously with
controlling the fiber refractive index profile, the mirror shape or
lens profile is tailored to produce the desired spatial
distribution of the intracavity optical phase delay, which
selectively suppresses resonance of higher order transverse modes
in the cavity. As a result, single transverse mode operation of
optical resonators is achieved.
[0014] The path to implementing the invention is in the context of
micro optical systems, which include single mode optical fiber,
semiconductor gain media, and/or MOEMS devices. In such systems,
the maximum beam diameters are typically less than a few
millimeters. Specifically, in the present implementations, the
maximum beam diameters in the system are less than 500 micrometers
(.mu.m). Small beams are consistent with the small form factors
required in most optical communications applications, and also
reduce the beam diameter translation requirements when coupling
into and out of single mode fibers, which typically have a 5 to 10
.mu.m mode diameter. Further, the beams as launched into the
optical cavities are small, typically less than 200 .mu.m. In
current implementations, the launched beams are actually
considerably smaller, less than 50 .mu.m, or between 5 and 30
.mu.m.
[0015] In more detail, according to one aspect, the invention
features a passive optical resonator comprising at least one
optical cavity defined by at least two mirror structures in which
at least one of the mirror structures has a mirror profile having a
diameter and sag that are selected in combination with a length of
the cavity to degrade a stability of transverse modes with mode
numbers 4 and greater.
[0016] Generally, in such implementations, the optical cavity
length is less than about 50 micrometers, a sag of the optical
surface is less than about 200 nanometers, and the full width half
maximum of the mirror diameter is less than about 30 micrometers.
More commonly, the optical cavity length is less than about 30
micrometers, a sag of the optical surface is less than about 150
nanometers, and the full width half maximum of the mirror diameter
is less than about 20 micrometers. As described in more detail
below, optical cavity lengths of less than about 20 micrometers,
with optical surface sags of less than about 100 nanometers, and
the full width half maximum mirror diameters of less than 15
micrometers have been produced. In these cases, the sag is an
important parameter, since the relationship between the mirror sag
and the effective mode deflections for the higher order transverse
modes leads to the inability of the cavity to support these modes,
according to one line of analysis.
[0017] The above and other features of the invention including
various novel details of construction and combinations of parts,
and other advantages, will now be more particularly described with
reference to the accompanying drawings and pointed out in the
claims. It will be understood that the particular method and device
embodying the invention are shown by way of illustration and not as
a limitation of the invention. The principles and features of this
invention may be employed in various and numerous embodiments
without departing from the scope of the invention.
BRIEF DESCRIPTION OF THE DRAWINGS
[0018] In the accompanying drawings, reference characters refer to
the same parts throughout the different views. The drawings are not
necessarily to scale; emphasis has instead been placed upon
illustrating the principles of the invention. Of the drawings:
[0019] FIG. 1 is a schematic view of a conventional hemispherical
resonator;
[0020] FIG. 2 is a plot of an exemplary mirror profile in
nanometers (nm) as a function of radial distance in micrometers for
a curved spherical mirror in the hemispherical resonator
design;
[0021] FIGS. 3A through 3F are plots in the x-y plane of the mode
intensity profiles in arbitrary units for some exemplary
Hermite-Gaussian transverse modes of spherical mirror
resonators;
[0022] FIG. 4 is a plot of transverse mode frequencies in terahertz
(THz) as a function of mode number, illustrating the modal
structure of a conventional spherical resonator cavity;
[0023] FIG. 5 is a plot of normalized mirror height as a function
of radial distance for a family of super-secant hyperbolic mirror
profiles, which may be used in the implementation of the present
invention;
[0024] FIG. 6 is a plot of a mirror profile as a function of radial
distance for an exemplary super-secant mirror having a diameter of
15 micrometers (.mu.m) and a sag of 2 nm, also shown are effective
mode lengths of the mirror in a 20 .mu.m long cavity for the three
stable transverse radial modes;
[0025] FIG. 7 is a plot of the spatial mode profiles of the three
stable radial modes as a function of radial distance for a
curved-flat resonator constructed with the super-secant mirror of
FIG. 6 in which the cavity length is L.sub.c=20 .mu.m, mirror
diameter is 15 .mu.m, and sag is 2 .mu.m;
[0026] FIG. 8 is a plot of transverse mode frequencies in terahertz
(THz) as a function of mode number illustrating the modal structure
of a curved-flat resonator constructed with the super-secant mirror
of FIG. 6 in which the cavity length is L.sub.c=20 .mu.m, mirror
diameter is 15 .mu.m, and sag is 260 nm;
[0027] FIG. 9 is a plot of a mirror profile as a function of radial
distance for an exemplary super-secant mirror having a diameter of
15 .mu.m and a sag of 115 nm, also shown are effective mode lengths
for the mirror in a 20 .mu.m long cavity for the three stable
transverse modes with the highest order mode approaching
cutoff;
[0028] FIG. 10 is a plot of the spatial mode profiles of the three
stable radial modes, with the highest order mode spatially
broadened near its cutoff, as a function of radial distance for a
curved-flat resonator constructed with the super-secant mirror of
FIG. 9 in which the cavity length is L.sub.c=20 .mu.m, mirror
diameter is 15 .mu.m, and sag is 115 nm;
[0029] FIG. 11 is a plot of transverse mode frequencies in
terahertz (THz) as a function of mode number illustrating the modal
structure of a curved-flat resonator constructed with the
super-secant mirror of FIG. 9 in which the cavity length is
L.sub.c=20 .mu.m, mirror diameter is 15 .mu.m, and sag is 115
nm;
[0030] FIG. 12 is a plot of a mirror profile as a function of
radial distance for an exemplary super-secant mirror having a
diameter of 15 .mu.m and a sag of 100 nm, also shown are effective
mode lengths for the mirror in a 20 .mu.m long cavity;
[0031] FIG. 13 is a plot of the spatial mode profiles as a function
of radial distance of a curved-flat resonator constructed with the
super-secant mirror of FIG. 12 in which the cavity length is
L.sub.c20 .mu.m, mirror diameter is 15 .mu.m, and sag is 100
nm;
[0032] FIG. 14 is a plot of transverse mode frequencies in
terahertz (THz) as a function of mode number illustrating the modal
structure of a curved-flat resonator constructed with the
super-secant mirror of FIG. 12 in which the cavity length is
L.sub.c=20 .mu.m, mirror diameter is 15 .mu.m. and sag is 100
nm;
[0033] FIG. 15 is a plot of a mirror profile as a function of
radial distance for an exemplary super-secant mirror having a
diameter of 15 .mu.m and a sag of 25 nm, also shown is the
effective mode length for the mirror in a 20 .mu.m long cavity for
the single stable transverse mode;
[0034] FIG. 16 is a plot of the spatial mode profile of the single
stable transverse mode as a function of radial distance of a
curved-flat resonator constructed with the super-secant mirror of
FIG. 15 in which the cavity length is L.sub.c20 .mu.m, mirror
diameter is 15 .mu.m, and sag is 25 nm;
[0035] FIG. 17 is a plot of transverse mode frequencies in
terahertz (THz) as a function of mode number illustrating the
single modal structure of a curved-flat resonator constructed with
the super-secant mirror of FIG. 15 in which the cavity length is
L.sub.c=20 .mu.m, mirror diameter is 15 .mu.m, and sag is 25
nm;
[0036] FIG. 18 is a schematic view of a general curved-flat
resonator illustrating the relationship between the curved mirror
structure profile, including mirror diameter and depth, and the
resonator mode profiles and mode effective length of some exemplary
modes;
[0037] FIG. 19 is a schematic view of a curved-flat resonator,
contrasting operation of the conventional spherical curved mirror
with a multitude of transverse modes and the inventive finite-depth
mirror with a only a few stable transverse modes;
[0038] FIG. 20 is the .LAMBDA.-V transverse mode stability diagram
for optical resonators using mirrors with a finite deflection,
specifically secant hyperbolic surface profile;
[0039] FIG. 21 is the .LAMBDA.-V transverse mode stability diagram
for optical resonators illustrating the universal nature of the
resonator V parameter;
[0040] FIG. 22 illustrates modal optical k-vector propagation
directions and the effective mode lengths in the plane-plane and
curved mirror optical resonators, from which the fundamental mode
size of these resonators is derived, as well as the behavior of the
resonator unstable modes;
[0041] FIG. 23 shows the mode diameter to mirror diameter ratio as
a function of the resonator V-parameter, as calculated for optical
resonators with the secant hyperbolic profile mirrors;
[0042] FIG. 24 shows the spectral positions of the stable bound and
unstable unbound transverse cavity modes in the Fabry-Perot
resonator spectrum;
[0043] FIG. 25 shows the regimes of single mode and multi
transverse mode optical resonator operation as dependent on the
mirror diameter, mirror height, and the cavity length;
[0044] FIG. 26 is a cross-sectional view illustrating a mass
transport process for making mirror structures for carrying out the
present invention;
[0045] FIG. 27A is a three-dimensional plot showing the measured
profile of a mirror fabricated using a mass transport process from
a 12 .mu.m mesa precursor structure;
[0046] FIG. 27B is a cross-sectional view showing a measured mirror
profile for a mass transported 12 .mu.m mesa precursor
structure;
[0047] FIG. 28 is the .LAMBDA.-V transverse mode stability diagram
comparing optical resonators using mirrors with secant hyperbolic
and mass-transported surface profiles;
[0048] FIG. 29A is a plot of measured spectral power in decibels
(dB) as a function of wavelength in nanometers (nm) illustrating
the transmission spectrum of a Fabry-Perot filter configured
according to the present invention, where the resonator V parameter
is V.sub.r=3.6;
[0049] FIG. 29B is a plot of measured spectral power in decibels
(dB) as a function of wavelength in nanometers (nm) illustrating
the transmission spectrum of a Fabry-Perot filter configured
according to the present invention, where the resonator V parameter
is V.sub.r=2.5;
[0050] FIG. 29C is a plot of measured spectral power in decibels
(dB) as a function of wavelength in nanometers (nm) illustrating
the transmission spectrum of a Fabry-Perot filter configured
according to the present invention, where the resonator V parameter
is V.sub.r=1.6 and the resonator operates essentially with a single
transverse mode;
[0051] FIG. 30 is a plot of measured transverse mode finesse as a
function of pre-transport mesa diameter illustrating the mode
discrimination of Fabry-Perot resonators configured according to
the present invention;
[0052] FIG. 31 is a plot of measured transverse mode finesse as a
function of the resonator V parameter illustrating the mode
discrimination of Fabry-Perot resonators configured according to
the present invention;
[0053] FIG. 32A is an image and contour plot of the measured
Fabry-Perot filter side mode suppression ratio for the conventional
spherical mirror resonator, illustrating filter SMSR sensitivity to
the lateral x-y displacements of the filter input beam;
[0054] FIG. 32B is an image and contour plot of the measured
Fabry-Perot filter side mode suppression ratio for the inventive
resonator with mass-transported mirrors, illustrating much lower
filter SMSR sensitivity to the lateral x-y displacements of the
filter input beam;
[0055] FIG. 33 is a perspective view of a MOEMS Fabry-Perot filter
to which the present invention is applied;
[0056] FIG. 34 is a plot of spectral power in decibels (dB) as a
function of wavelength in nanometers (nm) illustrating the
transmission spectrum of a MOEMS Fabry-Perot tunable filter
configured according to the present invention;
[0057] FIG. 35 is a schematic view of curved-flat resonator,
according to the present invention, illustrating the parasitic
modes that are created by membrane bow;
[0058] FIG. 36A shows the net effective mirror profile and the
effective single mode deflection for an optical cavity with a
secant-hyperbolic-shape finite deflection mirror paired with a flat
membrane second mirror;
[0059] FIG. 36B shows the net effective mirror profile and the
effective multiple mode deflections for an optical cavity with a
secant-hyperbolic-shape finite deflection mirror paired with a
positively-bowed-membrane second mirror, illustrating parasitic
modes of the cavity;
[0060] FIG. 36C shows the net effective mirror profile and the
effective single mode deflection for an optical cavity with a
deeper sag secant-hyperbolic-shape finite deflection mirror paired
with a negatively-bowed-membrane second mirror; and
[0061] FIG. 37 is a plot of spectral power in decibels (dB) as a
function of wavelength in nanometers (nm) illustrating the
transmission spectrum of a MOEMS Fabry-Perot filter with a
mass-transported inventive mirror when parasitic modes due to
positive membrane bow are present.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
[0062] Transverse Mode Structure For Conventional Hemispherical
(Spherical/Parabolic Mirror) Resonator
[0063] To understand the present invention, it is helpful to
understand the transverse mode structure for a conventional
hemispherical resonator. A hemispherical resonator is a two mirror
resonator that is defined by a flat or relatively flat mirror and a
concave mirror having a spherical/parabolic surface profile.
[0064] FIG. 1 illustrates such a hemispherical resonator. Optical
radiation in the cavity 50 oscillates between the flat mirror 52
and the concave spherical/parabolic mirror 54. FIG. 2 is a plot of
the mirror profile cross section for the curved optical surface of
the mirror 54. In the present example, the radius of curvature for
mirror 54 is R.sub.c=1000 micrometers and the length of the cavity
50 is L.sub.c=20 micrometers.
[0065] FIGS. 3A through 3F illustrate the transverse mode intensity
profiles for some exemplary Hermite-Gaussian resonator modes for
this defined resonator. For the lowest order mode illustrated in
FIG. 3A, at the flat mirror, intensity 1/e.sup.2 diameter is 15.6
micrometers in the cavity 50 of the resonator defined relative to
FIGS. 1 and 2. However, in addition to the lowest order mode, there
are many other stable transverse modes as illustrated in FIGS. 3B
through 3F for the resonator.
[0066] FIG. 4 is a plot of the transverse mode frequencies in
terahertz of the hemispherical resonator of FIG. 1. The cavity
length of L.sub.c20 .mu.m corresponds to the m=26 longitudinal mode
frequency of f.sub.00=194.87 THz. For this cavity the transverse
mode spacing is 0.339 THz and the longitudinal mode spacing is 7.49
THz. Each one of the stable transverse modes of the hemispherical
resonator cavity resonates at a slightly different frequency, i.e.,
the mode frequencies are not "degenerate" as in the confocal
cavity.
[0067] Transverse Mode Structure For Resonators With Mirrors Having
A Bounded Mirror Deflection
[0068] One embodiment of the present invention surrounds the use of
mirrors in which the mirror surface profile, including its diameter
and height and profile or curvature, is tailored to produce a
stable resonance for the fundamental mode, while making some or all
of the higher order transverse modes of the resonator unstable.
[0069] A properly chosen mirror profile in the case of a curve-flat
resonator or mirror profiles in the case of a curved-curved
resonator, according to the present invention, increase the
diameter of the higher order transverse modes, spreading these
modes strongly outside the central region of the mirror(s), making
these modes eventually unbound and unstable. In contrast, the
lowest order, or fundamental, transverse mode is impacted primarily
by the central region of the mirror and its behavior and stability
are similar to that found in a conventional hemispherical
cavity.
[0070] In one example of the inventive curved mirror, moving from
the center to the edge of the mirror generally within the spatial
extent of the transverse modes of interest, the mirror profile
deflection is bounded to less than a certain maximum deflection or
sag. This maximum in mirror deflection is achieved, for example, by
varying the mirror curvature from positive near the mirror center
to negative away from the mirror center. Contrast this with the
conventional spherical, and also aspheric mirrors, where mirror
deflection increases, at least from the perspective of many of the
higher order modes, without bound. FIG. 2 illustrates the unbounded
parabolic deflection of a spherical mirror. Such conventional
spherical mirrors have only positive profile curvature within the
range of the modes.
[0071] FIG. 5 illustrates some exemplary mirror profiles that have
a finite maximum mirror deflection or sag; these mirrors have
regions of both positive and negative curvature about the mirror
center. The full two-dimensional mirror profile is generated from
the cross section by making a surface of revolution, for example.
The specific illustrated profiles are super-secant hyperbolic.
These super-secant mirror profiles are defined by the equation
d(x)=d.sub.0(1-sec h((2x/x.sub.0).sup.n)). Among other possible
mirror profiles are super-Gaussian profiles, as well as many
others.
[0072] To control the transverse modes in optical resonator
cavities, a number of parameters must be selected so that the
Fabry-Perot cavity will support only the lower order modes, and
typically only the lowest order mode. Specifically, these
parameters include the maximum mirror deflection height or sag
d.sub.0, the step diameter x.sub.0, cavity length L.sub.c, and the
specific mirror profile.
[0073] Modeling the inventive cavity to illustrate the transverse
mode discrimination is performed in the context of curved mirror
structures with secant hyperbolic mirror profiles having diameters
of x.sub.0=15 micrometers and sags of d.sub.0=260, 115, 100, and 25
nanometers; here the mirror super-secant order is n=1. The full
width at half maximum (FWHM) diameter of the mirror is
w=x.sub.fwhm=1.317x.sub.0=19.8.mu.m. The curved mirror structures
are paired with flat or relatively flat mirror structures defining
Fabry-Perot (FP) cavities of length L.sub.c=20 micrometers.
[0074] Modes of optical resonators with a mirror surface given by a
surface of revolution are best described in cylindrical
coordinates; such modes have a radial mode number n.sub.radial that
characterizes radial variation of the mode intensity and an
azimuthal mode number n.sub.azim that characterizes azimuthal
variation of the mode intensity.
[0075] FIG. 6 shows the mirror profile cross section associated
with the first modeled super-secant mirror FP cavity. The sag of
the curved mirror structure is d.sub.0=260 nm. The figure also
shows positions of the mode effective deflections corresponding to
the stable radial modes of the mirror in a 20 .mu.m long cavity;
V.sub.r=4.60.
[0076] FIG. 7 shows the radial field profiles of the three stable
radial modes of the resonator using mirror of FIG. 6 in a 20 .mu.m
long cavity; vertical lines indicate the curvature inflection
points of the secant hyperbolic mirror profile. The azimuthal mode
number is zero for these modes, n.sub.azim=0, i.e., these modes are
circularly symmetric.
[0077] FIG. 8 is a plot of the stable transverse mode frequency
offset as a function of the radial mode number for a cavity with
super-secant mirror of FIG. 6. This shows the result of the use of
mirrors with a bounded mirror deflection, in this case the
super-secant mirrors. Specifically, the resonator that implements
the principles of the present invention does not support some of
the higher order transverse modes, such as modes with radial mode
numbers 4 and greater.
[0078] FIG. 9 shows the mirror profile associated with the second
modeled secant hyperbolic mirror cavity; the maximum mirror
deflection of sag is d.sub.0=115 nm. Also shown are positions of
the mode effective deflections corresponding to the stable radial
modes of the mirror in a 20 .mu.m long cavity; V.sub.r=3.06. This
figure illustrates the condition when a mode, in this case the
third order radial mode, is approaching cutoff.
[0079] FIG. 10 shows the mode field profiles for the three stable
transverse modes of the cavity using mirror of FIG. 9. Here the
azimuthal mode number is zero and the vertical lines indicate the
mirror curvature inflection points. The third radial mode is
approaching cutoff, as evidenced by large spreading of the mode
intensity beyond the mirror diameter.
[0080] FIG. 11 is a plot of the stable transverse mode frequency
offset as a function of the radial mode number for the cavity using
the mirror of FIG. 9. Only three lowest order radial modes are
stable.
[0081] FIG. 12 shows a mirror profile cross section for a secant
hyperbolic mirror with a further compressed sag; specifically, sag
is now reduced to d.sub.0=100 nanometers. FIG. 12 also shows
positions of the mode effective deflections corresponding to the
stable radial modes of the mirror in a 20 .mu.m long cavity;
V.sub.r=2.85. For this mirror height, there are only two stable
radial modes remaining.
[0082] FIG. 13 is a plot of the mode field profiles for the mirror
of FIG. 12 in a 20 .mu.m long cavity; again, vertical lines
indicate the curvature inflection points of the secant hyperbolic
mirror.
[0083] FIG. 14 is a plot of the stable transverse radial mode
frequency offset as a function of radial mode number for the mirror
of FIG. 12.
[0084] FIG. 15 shows a mirror profile cross section for a secant
hyperbolic mirror with a sag or maximum deflection compressed to
d.sub.0=25 nanometers; V.sub.r=1.43. This 20 .mu.m long cavity
supports a single radial mode. Also shown is the effective
deflection of the mode.
[0085] FIG. 16 is a plot of the mode field profile for a cavity
with the small sag mirror of FIG. 15; again, vertical lines
indicate the mirror curvature inflection points. This modeling
suggests that the cavity supports only a single transverse radial
mode. Further modeling indicates that transverse modes with
azimuthal mode number 1 are all unstable; therefore the resonator
indeed supports only a single transverse mode.
[0086] FIG. 17 is a plot of the stable transverse radial mode
frequency offset as a function of radial mode number. This shows
the result of the use of the secant hyperbolic mirrors with the
small maximum deflection or sag. The cavity essentially supports a
single transverse mode with the mode frequency shifted by 0.156 THz
from the flat-flat cavity resonant frequency.
[0087] Modal Analysis Of Conventional And Inventive Optical
Resonator Cavities
[0088] FIG. 18 illustrates operation of a general curved-flat
optical resonator configuration. Specifically, for resonator 200,
one end of the resonator is defined by a substantially or
relatively flat mirror structure 52. The second mirror structure 54
has a curved profile; the spatial variation of this mirror profile
deflection d(x) defines transverse modes of the resonator.
Resonator cavity length L.sub.c is defined as the distance between
the relatively flat mirror and the apex of the curved mirror. If
the curved mirror deflection d(x) has a maximum or an upper bound,
we term this maximum deflection or sag d.sub.0.
[0089] Optical resonator of FIG. 18 has a set of longitudinal mode
orders with longitudinal mode frequencies: 1 f m = m c 2 n _ L c (
1 )
[0090] where m is the longitudinal mode number, c is the speed of
light, and {overscore (n)} is the refractive index of the resonator
medium. Each longitudinal mode order m has a corresponding set of
transverse modes with transverse mode numbers t and transverse mode
frequencies 2 f m , t m c 2 n _ L m , t = f m + f m , t . ( 2 )
[0091] For each transverse mode (m,t), the mode effective length is
defined as: 3 L m , t m c 2 n _ f m , t ( 3 )
[0092] and the mode effective deflection
.DELTA.L.sub.m,t.ident.L.sub.c-L.sub.m,t (4)
[0093] Generally, transverse mode of an optical resonator is stable
only when its modal effective length falls within the effective
span of the cavity lengths:
(L.sub.c-d.sub.0)<L.sub.m,t<L.sub.c (5)
[0094] In other words, a transverse mode of an optical resonator is
stable when the mode effective deflection is within the range of
the mirror deflections:
0<.DELTA.L.sub.m,t<d.sub.0 (6)
[0095] Conversely, a transverse mode is unstable if its mode
effective deflection falls outside this range. This stability
condition can be also written in terms of the transverse mode
frequency shifts .DELTA.f.sub.m,t: 4 0 < f m , t < f m d 0 L
c - d 0 ( 7 )
[0096] The stability criterion (5,6) is in analogy with optical
waveguide or fiber propagation, where an optical mode is guided by
the waveguide only if the mode effective index falls between the
waveguide core and cladding refractive indices. When a transverse
mode is stable or bound, its power is confined in the vicinity of
the fiber core or the mirror diameter region, in the present
situation. When a mode approaches cutoff, its power spreads far
outside the fiber or mirror core, eventually making the mode
unbound or unstable. For the optical fiber, mode cutoff condition
corresponds to the mode effective index approaching the cladding
index. For the optical resonator, the mode cutoff condition
corresponds to the mode effective deflection approaching the
largest deflection of the bounded deflection mirror, i.e.,
.DELTA.L.sub.m,t.apprxeq.d.sub.0. Generally, the mirror center
corresponds to the fiber core, while the mirror edges correspond to
the fiber cladding.
[0097] FIGS. 6, 9, 12, and 15 illustrate positions of the mode
effective deflections of the stable transverse cavity modes within
the deflection span of the bounded deflection secant hyperbolic
mirrors.
[0098] For the conventional curved mirrors, such as spherical
mirrors, the mirror deflection d(x) increases monotonically with
distance x from the mirror apex throughout the region of x spanned
by the optical modes of interest. Therefore, conventional, e.g.,
spherical, mirrors have no mirror deflection maximum or bound; all
transverse modes are stable for resonator cavities with such
unbounded deflection mirrors. In practice, maximum deflection for
conventional mirrors occurs at the physical edge of the mirror;
this makes unstable only the very high order transverse modes that
spill beyond the edge of the mirror aperture.
[0099] For the inventive mirrors, the mirror profile deflection
reaches a substantially maximum deflection d.sub.0, or the
deflection is bounded to less than d.sub.0, within the spatial
range of the transverse modes of interest, such as the fundamental
transverse mode. By adjusting the mirror profile width and the
maximum deflection, which is the mirror sag, we can limit the
number of stable modes of the resonator. For example, the resonator
can be forced to have only a single stable transverse mode.
[0100] FIG. 19 illustrates the difference between a conventional,
e.g., spherical, mirror (dotted line) 54 and the inventive curved
mirror 212. The inventive mirror 212 has a maximum deflection or
sag d.sub.0 and a diameter w defined by the mirror profile full
width at half maximum (FWHM) deflection. We fit approximately a
parabolic mirror 54 to the inventive mirror by passing the surface
through the inventive mirror apex and the half sag points. Such
parabolic/spherical mirror 54 has a radius of curvature 5 R c = w 2
4 d 0 .
[0101] The spherical mirror here has no maximum deflection, it
supports transverse modes (m,0), (m,1), (m,2) . . . that have
effective modal lengths falling within the unbounded spherical
mirror deflection. The inventive mirror 212 has a maximum
deflection do; here only two transverse modes (m,0) and (m,1) are
stable, while the (m,2) and higher order transverse modes are
unstable.
[0102] We obtain here an approximate condition for the single
transverse mode operation of the inventive optical resonator. Such
resonator supports only the fundamental transverse mode when the
first higher order mode (m,1) becomes unstable, i.e., it violates
the stability condition (6) or (7). Therefore the single transverse
mode condition can be written as 6 f m , 1 = ( f m , 1 - f m ) >
f m d 0 L c - d 0 = c d 0 L c - d 0 ( 8 )
[0103] where .lambda.=c/f.sub.m is the wavelength, in vacuum, of
light in the vicinity of the optical modes of interest.
[0104] The fitted spherical mirror resonator has Hermite-Gaussian
transverse modes with transverse mode frequencies given by 7 f mts
= ( m + ( t + s + 1 ) 1 L c R c ) c 2 n _ L c = ( m + ( t + s + 1 )
1 4 L c d 0 w 2 ) c 2 n _ L c ( 9 )
[0105] where t and s are the integer transverse mode numbers in
Cartesian coordinates. For the spherical mirror, which is
approximately parabolic near its apex, the transverse modes (9) are
equally spaced in frequency, just as the equally spaced propagation
constants in an optical waveguide with parabolically graded
refractive index, or equally spaced energy levels of the harmonic
oscillator with a parabolic potential.
[0106] If we assume that the first higher order mode (m,1) of the
inventive resonator has its mode frequency given approximately by
the fitted spherical mirror, the (m,1) mode frequency is given by
(9) with (t+s)=1. Therefore 8 f m , 1 = f m01 = ( m + 2 4 L c d 0 w
2 ) c 2 n _ L c
[0107] and the single mode condition (8) for optical resonators
becomes approximately 9 w n _ d 0 / L c ( 1 - d 0 / L c ) w n _ d 0
/ L c < 2 ( 10 )
[0108] since d.sub.0/L.sub.c<<1. Compare this with the single
mode condition for a step index fiber with the core diameter w,
core and cladding indices n.sub.core and n.sub.clad, and the index
difference .DELTA.n=(n.sub.core-n.sub.clad): 10 V f = w n core 2 -
n clad 2 w 2 n clad n < 2.405 ( 11 )
[0109] Here V.sub.f is the dimensionless V-parameter for the fiber.
The two conditions for the resonators and fibers are qualitatively
similar; in case of optical resonators, the role of core-cladding
index difference .DELTA.n is played by the normalized mirror
deflection d.sub.0/L.sub.c.
[0110] In analogy with optical fibers, we define a dimensionless
V-parameter for optical resonators: 11 V r w n _ d 0 / L c ( 12
)
[0111] Furthermore, to characterize the resonator transverse modes,
we define a dimensionless mode parameter .LAMBDA.:
.LAMBDA..ident.1-.DELTA.L.sub.m,t/d.sub.0 (13)
[0112] which is simply related to the ratio of the mode effective
deflection .DELTA.L.sub.m,t and the mirror maximum deflection
d.sub.0. The A parameter ranges between 0 and 1 and characterizes
the strength of the mode confinement by the mirror:
.LAMBDA..about.1 corresponds to the strongly confined or strongly
bound modes, .LAMBDA..about.0 corresponds to the mode cutoff
condition when the modes become unbound.
[0113] For different values of the optical resonator parameters w,
d.sub.0, L.sub.c, and .lambda., we calculate the resonator V
parameter and the transverse mode frequencies with the
corresponding mode effective deflection and the mode A parameter. A
.LAMBDA.-V diagram is a plot of the modal .LAMBDA. values as a
function of the resonator V parameter.
[0114] Generally, in the discussions, a curved-flat optical cavity
is considered, in which the curved optical surface is formed on one
of the mirror structures with the other mirror structure being
relatively flat, possibly only having a bow. These cavities have
advantages in implementation due to the reduced assembly tolerance
in the alignment of the mirror structures. The analysis can be
generalized to curved-curved optical cavities, however, by
introducing the concept of net optical curvature or a net mirror
profile. The net mirror profile is the total round-trip profile
that the optical wave sees in the cavity, as if an equivalent
curved-flat cavity were being analyzed.
[0115] FIG. 20 shows the .LAMBDA.-V diagram for optical resonators
made using the secant hyperbolic profile mirrors paired with a flat
mirror; the mode labels (n.sub.radial, n.sub.azim) correspond to
the radial and azimuthal transverse mode numbers. For large cavity
V numbers, V.sub.r>4, the resonator supports multiple modes:
(0,0), (0,1), (1,0), (1,1), (2,0), etc. As the cavity V number
decreases, the modal A parameters also decrease, as the modes
become more weakly bound. The higher order transverse modes reach
cutoff one-by-one as their A parameter decreases to zero. The (2,0)
mode gets cutoff near V.about.3.5 with only four remaining stable
modes, the (1,1) mode near V.about.2.8 with three remaining stable
modes, the (1,0) near V.about.2.2 with two remaining stable modes.
The (0,1) first higher order mode gets cutoff near
V=V.sub.0.about.1.5. For values of the resonator V parameter less
than V.sub.0.about.1.5, the optical resonator supports only a
single transverse mode, namely the fundamental mode. The
fundamental mode apparently becomes cutoff and unbound for
V<0.6. Therefore, the resonator single mode condition is:
V.sub.r<1.5 (14)
[0116] The approximate single mode condition we gave in (10) by
comparing inventive mirrors with spherical mirrors is V.sub.r<2,
which is not too far from the more accurate condition (14) above
for the secant hyperbolic shaped mirrors.
[0117] To confirm that the resonator V.sub.r parameter (12) is
indeed a universal cavity parameter and the .LAMBDA.-V diagram is a
universal diagram, we plot in FIG. 21 the .LAMBDA.-V plot for
optical cavities with secant hyperbolic mirrors where we scan
separately the mirror deflection height, the mirror diameter, and
the cavity length. For all these cavities the modal .LAMBDA.-V
plots essentially correspond, confirming the universality of the
.LAMBDA.-V plot. There is only a small deviation in the plots for
the cavity length scan at the larger V parameters, V>4, which is
far from the single mode condition of interest. Therefore, all the
different optical cavities with the same V.sub.r parameter, while
having different mirror diameters, mirror deflections and cavity
lengths, will have the same modal .LAMBDA. parameter.
[0118] From the A parameter of the fundamental mode we can
determine the 1/e.sup.2 diameter x.sub.w of the mode intensity
distribution. Referring to FIG. 22, for a plane-plane cavity of
length L.sub.c all modes have the resonance condition
{overscore (n)}k.sub.mL.sub.c=m.pi. (15)
[0119] where k.sub.m=2.pi.f.sub.m/c=2.pi./.lambda..sub.m is the
optical wavevector and f.sub.m is the longitudinal mode frequency
(1). The mode effective lengths are all equal to the cavity length
and the modes are transversely unbound plane waves. When one or
both mirrors of the cavity become curved, the fundamental mode has
a mode effective length L.sub.m,t that is shorter than the cavity
length, L.sub.m,t<L.sub.c, with the resulting new resonance
condition for the curved mirror cavity that follows from the
effective length definition in (3):
{overscore (n)}k.sub.m,tL.sub.m,t=m.pi. (16)
[0120] As a result, we now have {overscore
(n)}k.sub.m,tL.sub.c>m.pi. for the curved mirror resonator,
i.e., the optical k-vector is too long to resonate at the cavity
length. The reason is that the mode now has a finite transverse
extent and contains plane wave components with k-vector tilted
relative to the z axis at angle .theta..sub.d. Angle .theta..sub.d
gives the divergence half angle of the resonator mode. The k.sub.z
component of the k-vector along the z-axis now satisfies the curved
mirror cavity resonance condition:
{overscore (n)}k.sub.zL.sub.c=m.pi. (17)
[0121] where k.sub.z=k.sub.m,t cos.theta..sub.d. From (16) and (17)
we obtain expression for the k-vector tilt angle, or the beam
divergence half angle, .theta..sub.d: 12 cos d = L m , t L c and
tan d = ( L c L m , t ) 2 - 1 2 L L c ( 18 )
[0122] since .DELTA.L<<L.sub.c. Angle .theta..sub.d gives the
divergence half angle of the resonator mode. From the standard
Gaussian beam expressions, the modal beam waist diameter x.sub.w at
1/e.sup.2 intensity point is obtained directly from this divergence
angle: 13 x w = 2 tan d 2 L c L ( 19 )
[0123] The ratio between the fundamental mode 1/e.sup.2 waist
diameter x.sub.w and the mirror full width half max diameter w
becomes 14 x w w = w 2 L c L = 1 V 2 d 0 L = 1 V 2 1 - ( 20 )
[0124] FIG. 23 shows the plot, for the fundamental transverse
cavity mode, of this mode to mirror diameter ratio as a function of
the cavity V-parameter, as calculated for the secant hyperbolic
profile mirror resonators. The mode diameter is smaller than the
mirror diameter for the larger V numbers (V>3). Generally, the
ratio of the mode to the mirror diameter is less than 0.7 for V
numbers greater than 3. The mode diameter increases relative to the
mirror diameter as the cavity V numbers get smaller, with the ratio
at unity for V.sub.r=2, and for still smaller values of V the mode
diameter becoming greater than the mirror diameter and finally
diverging near the V.about.0.5 fundamental mode cutoff. For the
single mode condition of V at about 1.5, the ratio is about 1.2
[0125] We now consider in more detail the mode cutoff condition and
the unstable or unbound transverse mode regime. The mode stability
condition (5) can also be written as:
{overscore (n)}k.sub.m,t(L.sub.c-d.sub.0)<{overscore
(n)}k.sub.m,tL.sub.m,t<{overscore (n)}k.sub.m,tL.sub.c (21)
[0126] or
{overscore (n)}k.sub.m,t(L.sub.c-d.sub.0)<m.pi.<{overscore
(n)}k.sub.m,tL.sub.c (22)
[0127] since by definition in (16) we have {overscore
(n)}k.sub.m,tL.sub.m,t=m.pi.. We have seen that the inequality in
(22) on the right,
m.pi.<{overscore (n)}k.sub.m,tL.sub.c (23)
[0128] implies a confined or bound mode with a diverging beam that
has k-vectors tilted from the z axis by an angle .theta..sub.d, the
beam divergence half angle, such that
m.pi.={overscore (n)}k.sub.m,t cos .theta..sub.dL.sub.c (24)
[0129] with the angle .theta..sub.d given in the stable regime by
15 cos d = L m , t L c ( 18 )
[0130] In turn, when a mode becomes unstable, the stability
inequality in (22) on the left is violated, becoming the mode
instability condition:
{overscore (n)}k.sub.m,t(L.sub.c-d.sub.0)>m.pi. (25)
[0131] This implies that in the unstable regime the mode consists
of rays with k-vectors tilted from the z axis by an angle
.theta..sub.r, such that
{overscore (n)}k.sub.m,t cos .theta..sub.r(L.sub.c-d.sub.0)=m.pi.
(26)
[0132] The ray angle .theta..sub.r is given in this unstable mode
regime by 16 cos r = L m , t L c - d 0 = 1 - L / L c 1 - d 0 / L c
= 1 - ( 1 - ) d 0 / L c 1 - d 0 / L c ( 27 )
[0133] and for .DELTA.L<<L.sub.c we have
tan .theta..sub.r.apprxeq.{square root}{square root over
(-2.LAMBDA.d.sub.0/L.sub.c)} (28)
[0134] Note that in the unstable regime .DELTA.L>d.sub.0 and
.LAMBDA.<0.
[0135] In the unstable regime, the resonator transverse modes are
rays, tilted by angle .theta..sub.r relative to the z axis, that
bounce between the two cavity mirrors in the plane-plane region of
the cavity outside the curved mirror diameter, or the cavity core.
These rays are only slightly perturbed by the curved mirror
deflection near the cavity core. These unstable ray modes of the
resonator correspond to the unbound radiation modes of the optical
fiber or waveguide. For a fiber, radiation modes have modal
effective indices that fall below the cladding refractive index;
these modes are plane wave like and are only slightly refracted by
propagation through the waveguide core.
[0136] We now illustrate the spectral positions of the stable bound
and unstable unbound transverse modes of optical resonators. The
transverse mode wavelength can be written as 17 m , t = 2 k m , t =
2 n _ m ( L c - L m , t ) = 2 n _ m L c ( 1 - ( 1 - m , t ) d 0 / L
c ) ( 29 )
[0137] Further, for the m-th longitudinal mode we define the
longitudinal mode wavelength 18 m = 2 n _ m L c ( 30 )
[0138] and the mode cutoff wavelength 19 m , c = 2 n _ m ( L c - d
0 ) ( 31 )
[0139] We therefore obtain a simple relation between the modal
wavelength .lambda..sub.m,t and the modal .LAMBDA..sub.m,t
parameter:
(.lambda..sub.m,t-.lambda..sub.m,c).apprxeq..lambda..sub.m,c.LAMBDA..sub.m-
,t(d.sub.0/L.sub.c) (32)
[0140] where we have assumed that (d.sub.0/L.sub.c)<<1. Thus
for .LAMBDA..sub.m,t=0 at mode cutoff we have
.lambda..sub.m,t=.lambda..sub.m- ,c and for .LAMBDA..sub.m,t=1 for
fully confined modes, we have .LAMBDA..sub.m,t=.lambda..sub.m. Note
that the mode cutoff wavelength .lambda..sub.m,c depends only on
the mirror sag d.sub.0 and the cavity length L.sub.c; it is
independent of the mirror diameter. For bound modes, which satisfy
0<.DELTA.L.sub.m,t<d.sub.0 from (6), we then have
.lambda..sub.m,c<.lambda..sub.m,t<.lambda..sub.m or
0<.LAMBDA.<1 (33)
[0141] For the unbound modes, which satisfy
.DELTA.L.sub.m,t>d.sub.0, we have
.lambda..sub.m,t<.lambda..sub.m,c or .LAMBDA.<0 (34)
[0142] FIG. 24 illustrates the positions of the bound and unbound
cavity modes in the Fabry-Perot resonator spectrum. This is
analogous to the spectrum of the guided and unguided modes in an
optical fiber. The spectral width of the bound mode region is
determined solely the mirror sag:
(.lambda..sub.m-.lambda..sub.m,c)=(2{overscore (n)}/m)d.sub.0; the
spacing of the bound modes inside the bound mode region is also
determined by the mirror diameter and the cavity length.
[0143] Optical resonator analysis in this section can also be
applied to the conventional spherical mirror resonators. For
example, expression (19) applies also to the spherical mirror
resonators. For a spherical mirror the transverse mode frequencies
are given by (9) and the effective mode deflections are 20 L = L c
( t + s + 1 m ) 1 L c R c ( 35 )
[0144] for resonators with a short cavity length with
L.sub.c<<R.sub.c. From (19) we then obtain the following
expression for the fundamental t=s=0 mode beam waist diameter at
1/e.sup.2 point: 21 x w , spherical 2 = 4 R c L c ( 36 )
[0145] which is in agreement with conventional expressions for
spherical mirror resonators. The .LAMBDA.-V diagram can also be
constructed and applied to the spherical mirror resonators. For a
spherical mirror with an aperture diameter of w.sub.a and a radius
of curvature R.sub.c, the mirror sag is
d.sub.0=w.sub.a.sup.2/(8R.sub.c) and the resonator V parameter
becomes 22 V spherical = w a 2 4 R c L c ( 37 )
[0146] For a spherical micro mirror with an aperture of w.sub.a=100
.mu.m and a radius of curvature of R.sub.c=1500 .mu.m, the 20 .mu.m
long cavity at 1.55 .mu.m wavelength has a V parameter of
V.sub.spherical=29.3. The .LAMBDA.-V diagram of FIG. 20, which
looks somewhat different for spherical rather than secant
hyperbolic shaped mirrors, indicates that such a spherical mirror
cavity indeed supports a multitude of transverse modes.
[0147] The single mode condition (14) for optical resonators
indicates that the product of the mirror full width half max
diameter w and the square root of the normalized mirror sag
d.sub.0/L.sub.c has to be sufficiently small in order to suppress
higher order transverse modes.
[0148] FIG. 25 illustrates the single mode condition (14) of the
inventive optical resonators for a set of resonator parameters
applicable to the tunable Fabry-Perot MOEMS filters. We plot here
the maximum mirror deflection or sag d.sub.0 versus the mirror FWHM
diameter that allows the single transverse mode operation; the plot
is repeated for several values of the cavity length L.sub.c. For
each cavity length, the region below the corresponding curve gives
single-mode operation, while in the region above this curve first
one and then more higher-order modes become stable. Note the small
required mirror diameters, 5 to 30 micrometers, and mirror sags, 20
to 120 nanometers, for the cavity lengths of 10 to 25
micrometers.
[0149] FIGS. 6, 9, 12 and 15 show the secant hyperbolic mirror
profiles together with the calculated effective deflection
positions of the stable transverse modes. As the secant hyperbolic
mirror height is reduced from 2 to 25 nm for the fixed mirror
diameter of x.sub.0=15 .mu.m (w=x.sub.fwhm=19.8.mu.m), fewer
transverse modes fit their effective deflections within the mirror
deflection range, eventually only a single fundamental transverse
mode remains within the range and is the only stable mode. For the
mirror FWHM diameter of 20 .mu.m, the single mode condition in FIG.
25 predicts transition from multi mode to single transverse mode
operation at the mirror height of about 30 nm, in agreement with
secant hyperbolic mirror resonator calculations in FIGS. 6-17.
[0150] We have described the use of controlled profile curved
mirrors in order to control the transverse modes of the optical
resonator. One can accomplish the same goal by using an intracavity
lens or a graded index lens to provide the required optical phase
profile distribution inside an optical resonator.
[0151] Experimental Results: Optical Resonators With Finite
Deflection Mirrors
[0152] In order to test the operation of the inventive finite
deflection mirrors in Fabry-Perot resonators, we have fabricated
arrays of such micro mirrors on semiconductor substrates using
photolithographic techniques. The mirrors were formed by etching
cylindrical blind holes, or inverted mesas, into silicon, Si, and
gallium phosphide, GaP, substrates and then executing a mass
transport process in order to smooth out the contour. This yields a
mirror profile of a given diameter and depth that has both negative
and positive curvature regions.
[0153] FIG. 26 illustrates the mass transport operation. Initially,
a cylindrical blind hole or inverted mesa 120 is etched into the
substrate 122. This hole is produced by reactive ion etching (RIE),
in one example. The etched substrate 122 is then exposed to
elevated temperature and possibly a controlled atmosphere to
initiate the mass transport process thereby yielding a mirror with
a positive curvature region 112 that smoothly transitions to a
negative curvature region 110.
[0154] Such mirror profiles are alternatively produced with a
direct etching process such as by reflowing a deposited resist
followed by a non or partially selective etch process.
[0155] FIGS. 27A and 27B are profile plots of an exemplary 12
micrometer mesa after mass transport, which plots are based on
actually metrology data. After mass transport, the mirror diameter,
full width at half max, is slightly larger than the starting mesa
diameter. As the starting mesa diameters get smaller, the final
transported mirror depth or sag also gets slightly smaller. The
initial mesa diameters varied between 30 and 12 .mu.m;
correspondingly, the final transported mesa diameters, full width
at half maximum, varied between 31 and 17 .mu.m and the final
mirror sags varied between 80 and 50 nm. For comparison, FIG. 27B
also shows calculated secant hyperbolic mirror profile with the
same full width half max as the fabricated mass-transported
mirror.
[0156] FIG. 28 shows the calculated .LAMBDA.-V diagram for the
resonators using the mass-transported and the secant hyperbolic
mirror profiles. The .LAMBDA.-V curves for the two mirror profiles
are very similar for the fundamental and the lowest order modes,
which is the region of most interest, typically. The curves for the
two profiles increasingly differ for higher order transverse modes.
More generally, FIG. 28 illustrates that the inventive modal
operating principles, as well as the inventive single transverse
mode operation, apply to resonators with a variety of mirror
profiles and are not restricted to the secant hyperbolic shaped
mirrors.
[0157] FIGS. 29A, 29B, and 29C show the measured spectra of the
optical resonators constructed using the mass-transported GaP
mirrors paired with flat mirrors in 23 .mu.m long cavities.
[0158] For the wider mirror, shown in FIG. 29A, with mesa diameter
of 30 m and V.sub.r=3.6, the resonator supports three stable
transverse modes; the fourth mode 310 is approaching cutoff and has
a broad spectral line corresponding to low mode finesse.
[0159] For the narrower mirror, shown in FIG. 29B, with mesa
diameter of 20 .mu.m and V.sub.r=2.5, the resonator supports only
two stable transverse modes.
[0160] For the narrowest tested mirror, shown in FIG. 29C, with
mesa diameter of 12 .mu.m and V.sub.r=1.6, the resonator supports
only one stable transverse mode, with the first higher order mode
just at the cutoff condition. These modal spectra and modal
stabilities are in agreement with the .LAMBDA.-V mode stability
diagram in FIG. 28.
[0161] Generally, the remaining stable higher order modes can be
preferentially excited through input beam misalignment and through
mode field size mismatching. This has been used to measure the
finesse, and thus the intracavity loss, of the individual resonator
transverse modes.
[0162] FIG. 30 is a plot of the measured transverse mode finesse as
a function of the starting mesa, i.e., prior to mass transport,
diameter, although the data represents resonant cavities made using
post transport mirrors. The cavity length was approximately 23
.mu.m for all filters. The lowest order, i.e., fundamental,
transverse mode (mode 0) exhibits a stable finesse of approximately
2,500 without regard to the mesa diameter. A second order mode.
(mode 2) finesse, in contrast, starts deteriorating at an
approximately 30 micrometer mesa diameter and becomes fully
degraded at approximately 24 micrometers. Similarly, the first
order mode (mode 1) finesse starts exhibiting deterioration at a
mesa diameter of 21 micrometers and is completely degraded at about
a 12 to 14 micrometer mesa diameter.
[0163] FIG. 31 is a plot of the measured mode finesse as a function
of the resonator V parameter. This plot generally illustrates that
cavities with V parameters below 3.5 generally exhibit improved
mode control in which the number of modes is limited to 3 or less.
V parameters below 2.5 begin to exhibit single mode or near single
mode operation, with a V parameter of less than about 1.7 typically
resulting in single mode operation.
[0164] More specifically, mode 2 finesse degrades and the mode
disappears near V.sub.r.about.2.7, which agrees well with the mode
(1,0) cutoff near V.sub.r.about.2.6 for mass-transported mirrors in
FIG. 28. Mode 1 in FIG. 31 degrades finesse and disappears near
V.sub.r1.6, in agreement with the mode (0,1) cutoff near V.sub.r1.7
for mass-transported mirrors in FIG. 28.
[0165] These experimental measurements confirm the modal analysis
of the optical resonators. FIG. 25 superimposes our experimental
results on top of the theoretical stability diagram: the two
circles connected by a dashed line indicate the range of
experimental mirror filters with starting mesa diameters from 22 to
12 .mu.m that showed the transition from low loss first higher
order mode (mode 1) to highest loss for this mode. Thus, this
dashed line with two end points indicates the measured transition
from two mode to single mode operation; this line approaches the
theoretical single mode regime boundary for these filters with a 23
.mu.m long cavity.
[0166] Mode finesse degradation as the mode approaches cutoff,
which is observed in FIGS. 30 and 31, likely occurs because of the
large spatial spread of the loosely bound modes near modal cutoff.
Such widely spread modes are more sensitive to mirror and cavity
imperfections which can induce loss. For example, angular
misalignment of the finite deflection mirror and the essentially
flat mirror causes diffraction loss and reduced finesse, with a
much larger effect on the loosely bound modes near cutoff. This
mirror tilt loss of the resonators is analogous to the bend
radiation loss in optical waveguides, which is also much stronger
for weakly guided modes. Angular misalignment in spherical mirror
resonators causes only a lateral shift of the mode and a tilt of
the optic axis. Tilt misalignment in finite deflection mirror
resonators also causes a mode shift inside the mirror, but the
small diameter mirrors can tolerate only a small mode shift and
thus only small mirror tilts.
[0167] When the inventive optical resonator is constructed such
that the first higher order is not fully cutoff, the achievable
side mode suppression ratio, the SMSR, is still better than for
conventional spherical mirror resonators. In addition, the SMSR for
inventive resonators is more tolerant of lateral misalignment of
the resonator input excitation beam.
[0168] FIGS. 32A and 32B show the measured contour plots of the
optical filter SMSR for the conventional hemispheric tunable
resonator, FIG. 32A, and the inventive tunable resonator, FIG. 32B,
as functions of the lateral x and y displacement of the input beam
in micrometers. Here the cavity length is .about.16 .mu.m and the
inventive mass transported mirror resonator has a V number of V=1.9
(FIG. 32B). The inventive resonator achieves a finesse of .about.37
dB as compared with a finesse of .about.31 dB for the spherical
mirror resonator. Also, to keep the SMSR above 25 dB, the spherical
mirror resonator can tolerate input beam x-y displacements of only
1.0 .mu.m, or +/-0.5 .mu.m; in contrast, for the same SMSR range
the inventive resonator can tolerate displacements of 6 .mu.m, or
+/-3 .mu.m. This implies relaxed assembly tolerances for the
inventive optical filters.
[0169] Exemplary Tunable Fabry-Perot MOEMS Filter
Implementation
[0170] One embodiment of the present invention is as a tunable
Fabry-Perot filter.
[0171] In some implementations, the filter is fabricated by shaping
the ends of a solid material with the desired mirror curvatures and
then HR coating the ends. In such case, the net mirror curvature
can be distributed between both mirrors. The optical distance
between the ends is adjusted mechanically or thermally, for
example.
[0172] In an other implementation, the flat first mirror structure
210 and/or the inflection mirror structure 212 is deflectable or
movable in a Z-axis direction using MOEMS technology, for example,
to thereby provide for a tunable filter with a tunable pass
band.
[0173] Generally, the mode field diameter for the lowest order mode
as defined by the intensity 1/e.sup.2 diameter of the mode 214,
generally fits within the central portion of the mirror (see FIG.
19). Typically, the ratio of the mode field 1/e.sup.2 diameter to
the diameter w of the mirror FWHM is slightly greater than about
0.5, usually greater than 0.7. For fully single mode resonators,
the ratio is typically greater than about 0.9 to greater than 1.2,
or more.
[0174] In contrast, the mode field diameter of a higher order mode,
when stable, extends into the negative curvature portion 110, and
possibly the flat portions 111 surrounding the regions with the
optical curvature. This eventually makes the cavity unstable for
that mode. In this way, the invention utilizes phase profiling or a
phase aperture. A variation in phase is introduced across
transverse plane to preferentially preserve the lowest order mode
while making the higher order modes unstable.
[0175] FIG. 33 shows an exploded, exemplary
micro-opto-electro-mechanical system (MOEMS) Fabry-Perot tunable
filter 100 having a resonant cavity 200 that is constructed
according to the principles of the present invention.
[0176] Generally, in the FP filter 100, a spacer device 414
separates the mirror device 412 from the membrane device 410 to
thereby define the Fabry-Perot (FP) cavity 200.
[0177] The optical membrane device 410 comprises handle material
215 that functions as a support. An optical membrane or device
layer 211 is added to the support material 215. The membrane
structure 214 is formed in this optical membrane layer 211. An
insulating layer 216 separates the optical membrane layer 211 from
the support material 215. During manufacture, this insulating layer
216 functions as a sacrificial/release layer, which is partially
removed to release the membrane structure 214 from the support
material 215. This insulating layer defines the electrostatic
cavity between the membrane structure 214 and the handle wafer in
the illustrated implementation. Electrical fields are established
across this cavity to provide the forces necessary to deflect the
membrane out-of-plane and therefore tune the filter 100 by
modulating the size of the cavity 200.
[0178] In the current implementation, the membrane structure 214
comprises a body portion 218. The optical axis 10 of the device 100
passes concentrically through this body portion 218 and orthogonal
to a plane defined by the membrane layer 211. A diameter of this
body portion 218 is preferably 300 to 600 micrometers; currently it
is about 500 micrometers.
[0179] Tethers 220 extend radially from the body portion 218 to an
outer portion 222, which comprises the ring where the tethers 220
terminate. In the current implementation, a spiral tether pattern
is used.
[0180] An optical coating dot 230 is typically deposited on the
body portion 218 of the membrane structure 214. A second optical
coating is deposited on the mirror device 412 to thereby define the
other end of the FP cavity. The optical coatings are preferably a
highly reflecting (HR) dielectric mirror stacks, comprising 6 or
more layers of alternating high and low index material. This yields
a highly reflecting, but low absorption, structure that is
desirable in, for example, the manufacture of high finesse
Fabry-Perot filters.
[0181] According to the invention, the curved-flat resonator
incorporating the principles of the present invention is used in
the filter 100. Specifically, in one embodiment, one end of the
resonator 200 is defined by a substantially flat mirror structure
210. The second mirror structure 212 has a profile that provides
the spatial mode selectivity of the present invention.
[0182] Depending on the FP filter implementation, either the mirror
device 412 or center region 250 of the membrane 214 is patterned to
have the second mirror structure's mode selective profile, with the
other end of the cavity 200 being defined by the relatively flat
mirror structure 210. That is, in one implementation, the flat
first reflector 210 is on the mirror device 412 with the second
mirror structure 212 being etched or otherwise formed in region
250. In the other implementation, the flat first reflector 210 is
on the membrane 214, with the mirror device 412 having the second
mirror structure 212.
[0183] FIG. 34 is a plot of spectral power in decibels as a
function of wavelength illustrating the filter spectrum for the
tunable filter of FIG. 33. The plot shows a well defined
fundamental mode; only one higher order mode is observable, it is
suppressed to .about.38 dB below the fundamental mode.
[0184] Parasitic Modes In Fabry-Perot MOEMS Tunable Filters
[0185] FIG. 35 is a schematic diagram illustrating an
implementation issue sometimes associated with deploying the
present invention in a tunable MOEMS-type device, where controlled
deflection of a movable membrane mirror, for example, is used to
tune the Fabry-Perot resonator. Thin membranes become bowed due to
stress in the deposited dielectric reflective mirror coating, for
example. The resulting bow affects the transverse mode structure of
the resonator.
[0186] Specifically, as a result of the bow in a previously flat
membrane mirror 210, 214, previously unstable transverse modes of
the resonator can become stable and the resonator can acquire new
stable parasitic modes, as compared with the case where the
membrane mirror is unbowed, or substantially flat. Similar results
arise when the curved finite deflection mirror is on the movable
membrane or the fixed mirror side of the resonator.
[0187] Observing FIG. 35, we see that both mirrors 210, 212 now can
have some curvature. We can consider the effect of the two mirrors
on the optical mode as one net effective mirror profile that the
optical wave sees on one roundtrip in the cavity; the second
effective mirror of the effective cavity is then flat.
[0188] FIG. 36A shows the net mirror profile and the effective mode
deflection for a 20 .mu.m long cavity where the finite deflection
mirror has a secant hyperbolic shape and the membrane is flat. The
secant hyperbolic mirror has a sag of d.sub.0=40 nm and a full
width half max diameter of w=15 .mu.m. The cavity has a single
stable transverse mode.
[0189] FIG. 36B shows the net mirror profile and the effective
modal deflections of the same secant hyperbolic shape mirror paired
in a cavity with a positively bowed membrane. We assume the
membrane bow is approximately spherical. The membrane radius of
curvature is R.sub.mems=30 mm, while the center of the secant
hyperbolic mirror has a radius of curvature of R.sub.sech=0.81 mm .
The net mirror profile now has effectively unbounded deflection,
with the resulting multitude of stable transverse modes. Just the
radial modes are shown here in which n.sub.azim=0. The fundamental
mode is due to the secant hyperbolic mirror, while the higher order
modes are similar to the modes of the conventional flat-spherical
mirror cavity formed by the spherically bowed membrane and the flat
annular regions outside the center of the secant hyperbolic mirror.
These higher order modes have the smaller mode and effective
displacement spacing approximately corresponding to this parasitic
flat-spherical cavity with a large spherical radius of curvature.
These higher order modes can also have the larger mode diameters
associated with the large radius of curvature of the parasitic
cavity, as compared with the smaller mode diameter of the
fundamental mode of the secant hyperbolic mirror cavity.
[0190] FIG. 36C shows the net mirror profile and the effective
modal deflection of a secant hyperbolic mirror paired with a
negatively bowed membrane. Here the secant hyperbolic mirror has
the same diameter as before, namely w=15 .mu.m, but a bigger sag,
d.sub.0=120 nm.
[0191] In this case of a negative bow only a single stable
transverse mode is observed, but a bigger secant hyperbolic mirror
sag was required to achieve the desired single mode operation.
[0192] FIG. 37 shows the measured transmission spectrum of a MOEMS
Fabry-Perot filter with a mass-transported inventive mirror when
parasitic modes due to membrane bow are present. There is a
well-defined fundamental mode, labeled A, due to the
mass-transported curved mirror. We also observe at shorter
wavelengths a set of parasitic modes, labeled B, C that correspond
to the parasitic cavity due to the membrane bow. The parasitic mode
spacing is consistent with the membrane bow radius of curvature of
approximately 30-60 mm. Observations of filter mode profiles with
an infrared camera show much larger parasitic mode diameters, of
the order of 50 .mu.m, as compared with the approximately 17 .mu.m
diameter of the fundamental mode. These mode profile, as well as
spectral spacing, observations confirm that the observed higher
order modes are indeed due to the parasitic cavity formed by the
bowed down membrane and the flat regions the curved mirror
structure.
[0193] These parasitic modes due to membrane bow are also addressed
using other techniques. For example, options include: 1) HR coating
stress control to induce a relatively flat or negatively bowed
membrane; 2) fabricate an additional negative curvature section in
the annular region 111 outside the center of the curved mirror; 3)
remove HR coating in the annular region 111 or on the opposed
portions of the membrane 210 to suppress the finesse of the
parasitic modes; 4) induce loss in the annular region 111 or the
opposed portions of membrane 210 via surface roughening, surface
blackening, for example, again in order to suppress the finesse of
the parasitic modes; 5) laterally shift the curved mirror relative
to the bowed membrane, such that the parasitic modes are centered
away from the curved mirror center, and are thus more weakly
excited and more easily suppressed.
[0194] While this invention has been particularly shown and
described with references to preferred embodiments thereof, it will
be understood by those skilled in the art that various changes in
form and details may be made therein without departing from the
scope of the invention encompassed by the appended claims.
* * * * *