U.S. patent application number 09/930606 was filed with the patent office on 2002-11-07 for double chirped mirror.
Invention is credited to Kartner, Franz X., Keller, Ursula, Matuschek, Nicolai.
Application Number | 20020163727 09/930606 |
Document ID | / |
Family ID | 22160396 |
Filed Date | 2002-11-07 |
United States Patent
Application |
20020163727 |
Kind Code |
A1 |
Kartner, Franz X. ; et
al. |
November 7, 2002 |
DOUBLE CHIRPED MIRROR
Abstract
The invention is a double chirped mirror and a method of
constructing a double chirped mirror for a frequency range of
electromagnetic radiation, comprising specifying a design including
a plurality of layers, the plurality of layers being transparent to
the electromagnetic radiation and having refractive indices which
vary between layers in the plurality of layers, and wherein for a
first set of layers the optical thickness of alternate layers in
the set of layers varies monotonically and the total optical
thickness of a layer and the two adjacent half layers in the set of
layers varies monotonically. The design is optimized by adjusting
the optical thickness of layers in the plurality of layers.
Inventors: |
Kartner, Franz X.;
(Cambridge, MA) ; Matuschek, Nicolai;
(Lottstetten, DE) ; Keller, Ursula; (Zurich,
CH) |
Correspondence
Address: |
WILSON SONSINI GOODRICH & ROSATI
650 PAGE MILL ROAD
PALO ALTO
CA
943041050
|
Family ID: |
22160396 |
Appl. No.: |
09/930606 |
Filed: |
August 15, 2001 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
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09930606 |
Aug 15, 2001 |
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09080904 |
May 18, 1998 |
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6301049 |
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Current U.S.
Class: |
359/588 |
Current CPC
Class: |
H01S 3/08059 20130101;
H01S 3/08004 20130101 |
Class at
Publication: |
359/588 |
International
Class: |
G02B 001/10 |
Claims
What is claimed is:
1. A method of constructing a double chirped mirror for a frequency
range of electromagnetic radiation, comprising: specifying a design
including a plurality of layers, the plurality of layers being
transparent to the electromagnetic radiation and having refractive
indices which vary between layers in the plurality of layers and
wherein for a first set of layers the optical thickness of
alternate layers in the set of layers varies monotonically and the
total optical thickness of a layer and the two adjacent half layers
in the set of layers varies monotonically; and optimizing the
design by adjusting the optical thickness of layers in the
plurality of layers.
2. The method of claim 1, wherein: optimizing the design includes
reducing the difference between the group delay dispersion for a
plurality of frequencies in the frequency range of interest and a
design goal group delay dispersion for the plurality of
frequencies.
3. The method of claim 1, wherein: optimizing the design includes
reducing the difference between the reflectance for a plurality of
frequencies in the frequency range of interest and a design goal
reflectance for the plurality of frequencies.
4. The method of claim 1, wherein: specifying the design includes
specifying a second set of layers in the plurality of layers,
wherein the optical thickness of a layer is the same as the optical
thickness of two adjacent half layers in the set of layers, and the
total optical thickness of the layer and the two adjacent half
layers in the second set of layers varies monotonically.
5. The method of claim 1, wherein: specifying the design includes
specifying a third set of layers in the plurality of layers,
wherein the optical thickness of each layer in the third set of
layers is substantially a quarter-wavelength thick for a frequency
in the frequency range of electromagnetic radiation.
6. The method of claim 1, wherein: specifying the design includes
specifying a fourth set of layers in the plurality of layers,
wherein the fourth set of layers comprises an antireflection
coating.
7. The method of claim 1, wherein: optimizing the design by adjust
ing the optical thickness of layers in the plurality of layers
includes constructing a function having as its input a design goal
and a current design parameter, wherein the out put of the function
is related to a difference between the design goal and the current
design parameter.
8. The method of claim 7, wherein: optimizing the design by
adjusting the optical thickness of layers in the plurality of
layers includes adjusting the optical thickness of layers in the
plurality of layers until the output of the function reaches a
predetermined value.
9. The method of claim 1, wherein: optimizing the design by
adjusting the optical thickness of layers in the plurality of
layers includes using a merit function.
10. The method of claim 1, wherein: optimizing the design includes
adjusting the optical thickness of layers in the plurality of
layers such that the design has a reflectivity of at least 99% for
incident electromagnetic radiation in the frequency range of
electromagnetic radiation and a reflectivity of less than 99% in
each of two contiguous frequency ranges adjacent to the frequency
range of electromagnetic radiation, and such that the group delay
dispersion versus frequency characteristic of the mirror is smooth
and varies between 100% and 20% from its average value over at
least a continuous half-part of the frequency range of the
electromagnetic spectrum.
11. A method of constructing a double chirped mirror for a
frequency range of electromagnetic radiation, comprising:
specifying a design including a plurality of layers, the plurality
of layers being transparent to the electromagnetic radiation and
having refractive indices which vary between layers in the
plurality of layers specifying a first set of layers in the
plurality of layers wherein layers in the first set of layers
include a double chirp; specifying a second set of layers in the
plurality of layers adjacent to the first set of layers wherein
layers in the second set of layers include a simple chirp;
optimizing the design by adjusting the optical thickness of layers
in the plurality of layers.
12. The method of claim 11, wherein: specifying a third set of
layers in the plurality of layers adjacent to the second set of
layers wherein layers in the third set of layers includes a
quarter-wave stack.
13. The method of claim 11, wherein: specifying a fourth set of
layers in the plurality of layers adjacent to the first set of
layers wherein layers in the fourth set of layers includes an
antireflection coating.
14. The method of claim 11, wherein: optimizing the design includes
constructing a finction having as its input a design goal and a
current design parameter, wherein an output of the function is
related to a difference between the design goal and the current
design parameter.
15. The method of claim 14, wherein: optimizing the design includes
adjusting the optical thickness of layers in the plurality of
layers such that the output of the function reaches a predetermined
value.
16. The method of claim 15, wherein: optimizing the design includes
optimizing a reflectance of the mirror, optimizing a group delay of
the mirror, and optimizing a group delay dispersion of the
mirror.
17. A double chirped mirror for a frequency range of
electromagnetic radiation, comprising: a design including a
plurality of layers, the plurality of layers being transparent to
the electromagnetic radiation and having refractive indices which
vary between layers in the plurality of layers and wherein for a
first set of layers the optical thickness of alternate layers in
the set of layers varies monotonically and the total optical
thickness of a layer and the two adjacent half layers in the set of
layers varies monotonically; and optimizing the design by adjusting
the optical thickness of layers in the plurality of layers.
18. The double chirped mirror of claim 17, wherein: optimizing the
design includes reducing the difference between the group delay
dispersion for a plurality of frequencies in the frequency range of
interest and a design goal group delay dispersion for the plurality
of frequencies.
19. The double chirped mirror of claim 17, wherein: optimizing the
design includes reducing the difference between the reflectance for
a plurality of frequencies in the frequency range of interest and a
design goal reflectance for the plurality of frequencies.
20. The double chirped mirror of claim 17, wherein: the design
includes a second set of layers in the plurality of layers, wherein
the optical thickness of a layer is the same as the optical
thickness of two adjacent half layers in the set of layers, and the
total optical thickness of the layer and the two adjacent half
layers in the second set of layers varies monotonically.
21. The double chirped mirror of claim 17, wherein: the design
includes a third set of layers in the plurality of layers, wherein
the optical thickness of each layer in the third set of layers is
substantially a quarter-wavelength thick for a frequency in the
frequency range of electromagnetic radiation.
22. The double chirped mirror of claim 17, wherein: the design
includes a fourth set of layers in the plurality of layers, wherein
the fourth set of layers comprises an antireflection coating.
23. The double chirped mirror of claim 17, wherein: optimizing the
design by adjusting the optical thickness of layers in the
plurality of layers includes constructing a finction having as its
input a design goal and a current design parameter, wherein the
output of the function is related to a difference between the
design goal and the current design parameter.
24. The double chirped mirror of claim 23, wherein: optimizing the
design by adjusting the optical thickness of layers in the
plurality of layers includes adjusting the optical thickness of
layers in the plurality of layers until the output of the function
reaches a predetermined value.
25. The double chirped mirror of claim 17, wherein: optimizing the
design by adjusting the optical thickness of layers in the
plurality of layers includes using a merit function.
26. The double chirped mirror of claim 17, wherein: optimizing the
design includes adjusting the optical thickness of layers in the
plurality of layers such that the design has a reflectivity of at
least 99% for incident electromagnetic radiation in the frequency
range of electromagnetic radiation and a reflectivity of less than
99% in each of two contiguous frequency ranges adjacent to the
frequency range of electromagnetic radiation, and such that the
group delay dispersion versus frequency characteristic of the
mirror is smooth and varies between 100% and 20% from its average
value over at least a continuous half-part of the frequency range
of the electromagnetic spectrum.
27. A method of constructing a double chirped mirror for a
frequency range of electromagnetic radiation, comprising:
specifying a design including a plurality of layers, the plurality
of layers being transparent to the electromagnetic radiation and
having refractive indices which vary between layers in the
plurality of layers and wherein for a first set of layers in the
plurality of layers the optical thickness of a layer is the same as
the optical thickness of two adjacent half layers in the set of
layers, and the total optical thickness of the layer and the two
adjacent half layers in the first set of layers varies
monotonically and wherein for a second set of layers in the
plurality of layers the second set of layers comprises an
antireflection coating; and optimizing the design by adjusting the
optical thickness of layers in the plurality of layers.
28. The method of claim 27, wherein: optimizing the design includes
reducing the difference between the group delay dispersion for a
plurality of frequencies in the frequency range of interest and a
design goal group delay dispersion for the plurality of
frequencies.
29. The method of claim 27, wherein: optimizing the design includes
reducing the difference between the reflectance for a plurality of
frequencies in the frequency range of interest and a design goal
reflectance for the plurality of frequencies.
30. The method of claim 27, wherein: specifying the design includes
specifying a third set of layers in the plurality of layers,
wherein the optical thickness of each layer in the third set of
layers is substantially a quarter-wavelength thick for a frequency
in the frequency range of electromagnetic radiation.
Description
BACKGROUND OF THE INVENTION
[0001] 1. Field of the Invention
[0002] The invention relates to the field of reflective dielectric
structures, and more particularly to broadband reflective
dielectric structures used as mirrors in laser systems.
[0003] 2. Description of Related Art
[0004] Ultra short-pulse generation has advanced to a level where
the bandwidth of standard Bragg mirrors, e.g. composed of TiO.sub.2
and SiO.sub.2 quarter-wave layers, limits the pulse width or
tunability of the generated laser pulses. The limitation is two
fold. First, due to the limited difference in refractive index of
both materials, e.g., n.sub.TiO.sub..sub.2.apprxeq.2.4 and
n.sub.SiO.sub..sub.2.apprxeq.1.45 the high reflectivity bandwidth
of a standard quarter-wave Bragg mirror centered at 800 nm is only
about 200 nm. Second, the higher order group delay dispersion (GDD)
produced by quarter-wave Bragg mirrors further limits the useful
bandwidth to about 100 nm which is just enough bandwidth for 10 fs
pulses.
[0005] In a chirped mirror, the Bragg wavelength, .lambda..sub.B,
of the individual layer pairs is varied from layer pair to layer
pair (e.g. linearly), so that longer wavelengths penetrate deeper
into the mirror structure than shorter wavelengths before being
reflected. Such mirrors show an enlarged high reflectivity range
and show a negative dispersion. However, the dispersion properties
of these mirrors may be inadequate for ultra short pulse
generation.
[0006] Chirped mirrors are also beneficial for the compression of
high energy pulses, because they produce high dispersion with
little material in the beam path, thereby avoiding nonlinear
effects in the compressor. Thus, the design of these mirrors is
extremely important for the further development of ultra fast laser
sources.
[0007] It turns out that the design of a chirped mirror does not
necessarily lead to a smooth and controlled GDD of the mirror.
Using standard transfer matrix analysis of the multilayer structure
as discussed in "Exact coupled mode theories for multilayer
interference coating with arbitrary strong index modulations," IEEE
J. Quant. Elec., vol. 33, March 1997, which is hereby incorporated
by reference, one observes that the group delay produced by such a
chirped mirror does not vary linearly with wavelength, as one would
expect for a mirror with linearly chirped Bragg wavelength. The
local average of the group delay shows the expected tendency to
increase linearly with increasing wavelength. However, it also
exhibits strong oscillations. The cause of these oscillations is
the following. Longer wavelengths have to pass the first section of
the Bragg mirror, which acts as a transmission grating for these
wavelengths. The slight reflection in the front section interferes
with the strong reflections from the deeper layers, as in a
Gires-Tournouis Interferometer (GTI). The oscillations in the group
delay have an amplitude of several tens of femtoseconds, which make
these simple-chirped mirrors less useful for ultra short pulse
generation.
[0008] What is needed is a mirror design which reduces the
oscillations in the group delay, allowing control of the group
delay dispersion while maintaining broad band reflectivity and low
group delay dispersion.
SUMMARY OF THE INVENTION
[0009] One embodiment of the present invention is an optimized
double-chirped mirror in which an initial design of a double
chirped mirror is generated and then optimized to improve the group
delay dispersion and reflectance characteristics. In the initial
design, the oscillations in the group delay are avoided by varying
not only the Bragg period or Bragg wavelength in the mirror, but
also by tailoring the coupling between the forward and the backward
propagating waves inside the mirror such that spurious reflections
leading to GTI-effects are consistently avoided. In this embodiment
of the invention both quantities, the Bragg wavelength and the
coupling coefficient, are chirped. The oscillations in the group
delay are reduced or eliminated by a sufficiently slow increase in
the coupling of the wave incident onto the mirror and the wave
reflected from the mirror as the incident wave moves through the
first sections of the mirror. This slow increase in the coupling of
the waves, to avoid spurious reflection in the front section of the
mirror, is provided by the matching sections.
[0010] In this embodiment of the invention, the mismatch and
therefore the oscillations in the group delay are reduced by
addressing two of the matching problems encountered in a standard
chirped mirror. First, the medium from which the radiation is
incident on the mirror is matched to the first layer of a second
matching section mirror by a first matching section, which is, for
example, provided by a high quality broadband antireflection
coating. The antireflection coating can be designed with
commercially available dielectric coating design programs. The
second matching section has to match from the antireflection
coating section to a simple chirped section. In this way, the
double-chirped mirror is generated with a controlled group delay
and an extended high reflectivity range when compared to standard
dielectric Bragg mirrors. This analytic starting structure helps to
avoid internal resonances in the multilayer structure.
[0011] In this embodiment, the material parameters used to
construct the initial design are assumed to be constant in
frequency over the frequency range of interest. Once the initial
design has been constructed, a subsequent computer based
optimization of the initial design can take into account the
wavelength dependence of the refractive indices of the materials
used to construct the mirror, and other design goals, such as a
highly transmitting wavelength range close to the high reflection
wavelength range of the mirror which may be useful for coupling a
pump laser beam into a laser cavity.
[0012] In one embodiment of the invention, each of the sections of
the mirror comprises sets of layers in a plurality of layers.
Layers in the plurality of layers are composed of materials with a
high or low index of refraction for a frequency range of
electromagnetic radiation.
[0013] In one embodiment of the invention, the optimization of the
initial design is achieved by constructing a merit function
including the reflectance, group delay, and/or group delay
dispersion of the mirror, or any combination of these parameters.
The merit function is then optimized by adjusting the thicknesses
of layers in sections of the mirror until acceptable reflectance,
group delay, and/or group delay dispersion are achieved. In other
embodiments, the reflectance portion of the merit function includes
minimizing the reflectance of the mirror in a wavelength range
which may be used to transmit a pump laser beam through the mirror.
In another embodiment of the invention, the merit function is
alternately optimized for reflectivity, group delay and/or group
delay dispersion by varying weightings of the optimization
function.
[0014] In still another embodiment, the merit function is optimized
to achieve a final design such that the GDD varies by less than
100% from its average value, but by more than 20% from its average
value over more than a continuous half part of the high
reflectivity band of the mirror. In yet another embodiment of the
invention, the merit function is optimized to achieve a final
design such that the GDD varies by less than 200% from its average
value, but by more than 20% from its average value over more than a
continuous half part of the high reflectivity band of the mirror.
The high reflectivity band of the mirror is defined to mean a
continuous frequency range over which the reflectivity of the
mirror is higher than 99%.
[0015] In still other embodiments the merit function is optimized
to achieve a final design such that the GDD varies by less than
200% from its average value over the high reflectivity band of the
mirror, less than 100% from its average value over the high
reflectivity band of the mirror, less than 50% from its average
value over the high reflectivity band of the mirror, or less than
20% from its average value over the high reflectivity band of the
mirror.
[0016] Yet another embodiment of the invention is a method of
constructing a double chirped mirror for a frequency range of
electromagnetic radiation, comprising specifying a design including
a plurality of layers, the plurality of layers being transparent to
the electromagnetic radiation and having refractive indices which
vary between layers in the plurality of layers, and wherein for a
first set of layers the optical thickness of alternate layers in
the set of layers varies monotonically and the total optical
thickness of a layer and the two adjacent half layers in the set of
layers varies monotonically. The design is optimized by adjusting
the optical thickness of layers in the plurality of layers.
[0017] In another embodiment, optimizing the design includes
reducing the difference between the group delay dispersion for a
plurality of frequencies in the frequency range of interest and a
design goal group delay dispersion for the plurality of
frequencies. Optimizing the design may also include reducing the
difference between the reflectance for a plurality of frequencies
in the frequency range of interest and a design goal reflectance
for the plurality of frequencies. Furthermore, optimizing the
design may also include constructing a function having as its input
a design goal and a current design parameter, wherein the output of
the function is related to a difference between the design goal and
the current design parameter
[0018] In yet another embodiment, specifying the design includes
specifying a second set of layers in the plurality of layers,
wherein the optical thickness of a layer is the same as the optical
thickness of two adjacent half layers in the set of layers, and the
total optical thickness of the layer and the two adjacent half
layers in the second set of layers varies monotonically. Specifying
the design may also include specifying a third set of layers in the
plurality of layers, wherein the optical thickness of each layer in
the third set of layers is substantially a quarter-wavelength thick
for a frequency in the frequency range of electromagnetic
radiation. Additionally, specifying the design includes specifying
a fourth set of layers in the plurality of layers, wherein the
fourth set of layers comprises an antireflection coating.
[0019] Still another embodiment of the invention includes a double
chirped mirror as part of a laser cavity. In this embodiment the
laser may be a short pulse laser, or an ultrashort pulse laser,
including a femtosecond laser. Yet another embodiment of the
invention includes a double chirped mirror as part of an amplifier.
In this embodiment the amplifier may be a short pulse amplifier, or
an ultrashort pulse amplifier, including a femtosecond pulse
amplifier. In another aspect of this embodiment, the amplifier is a
high power amplifier.
BRIEF DESCRIPTION OF FIGURES
[0020] FIG. 1 depicts a simplified diagram showing an embodiment of
the double chirped mirror. The number of layers represented is not
intended to depict the actual number of layers for an embodiment of
the device, but rather it shows the basic differences in the
layering of the different sections.
[0021] FIGS. 2A-C depict the optical and physical thicknesses of
the layers for an embodiment of the invention at the initial design
phase, and after optimization.
[0022] FIGS. 3 depict the reflectivity, group delay and group delay
dispersion for an embodiment of the invention at the initial design
stage.
[0023] FIGS. 4 depict the reflectivity, group delay and group delay
dispersion for an embodiment of the invention after the group delay
has been optimized with all other weights at zero.
[0024] FIG. 5 depict the reflectivity, group delay and group delay
dispersion for an embodiment of the invention after the group delay
and reflectivity have been optimized with all other weights at
zero.
[0025] FIG. 6 depict the reflectivity, group delay and group delay
dispersion for an embodiment of the invention after the group
delay, reflectivity, group delay dispersion and pump reflectivity
have all been optimized.
[0026] FIGS. 7A-C depict the optimized results for the layers of
the corresponding optimized results of FIGS. 2A-C.
DETAILED DESCRIPTION
[0027] FIG. 1 depicts an initial design for an embodiment of the
present invention. This embodiment of the invention provides high
reflectivity over a wavelength range of interest with a group delay
dispersion which is suitable for use for dispersion compensation in
ultrashort pulse lasers. The initial design for double chirped
mirror 100 includes up to 4 sections: antireflection coating
section 102, double chirp section 104, simple chirp section 106 and
optional quarter wave stack 108. Each section is comprised of a
plurality of alternate layers of a high refractive index material
and a low refractive index material both of which are transparent
over the frequency range of interest.
[0028] The layer can be deposited by techniques well known to those
skilled in the art including but not limited to PVD, sputtering,
ion plating, ion beam sputtering, CVD, MOCVD, molecular beam
epitaxy, and chemical depth (sol gel). The low refractive index
layer may be any material with a refractive index lower than that
of the material chosen for the high refractive index material, and
may be chosen from, but is not limited to the following materials:
SiO.sub.2, MgF.sub.2, Al.sub.2O.sub.3, and AIF.sub.9. The high
refractive index layer may be any material with a refractive index
higher than that of the material chosen for the low refractive
index material, and may be chosen from, but is not limited to the
following materials: TiO.sub.2, HfO.sub.2, NbO.sub.2, ZrO.sub.2,
Y.sub.2O.sub.2, AlO.sub.2, and Gd.sub.2O.sub.3.
[0029] Antireflection coating section 102 is a broadband
antireflection coating that matches medium 110 to the first layer
of double chirp section 104. Medium 110 is any medium from which
radiation is incident upon double chirped mirror 100 and includes
but is not limited to air, glass, or a laser crystal.
Antireflection coating section 102 can be any general broad band
antireflection coating designed for use over the wavelength range
of interest. See for example "Thin Film Optical Filters," by H. A.
Macleod, published by Bristol, Adam Hilges, 1985, which is hereby
incorporated by reference. In another embodiment of the invention,
the antireflection coating is designed to be anti-reflecting for
both the wavelength range of interest and a pump wavelength
range.
[0030] Adjacent to antireflection coating section 102 is double
chirp section 104. Double chirp section 104 initially acts to
impedance match the radiation from antireflection coating 102 to
simple chirp section 106.
[0031] In double chirp section 104, the total optical thickness of
a layer and the two adjacent half layers in the set of layers 104A
increases monotonically from matching structure 102 to simple chirp
section 106. This configuration causes the mirror to have a
negative dispersion since the longer wavelength radiation is
reflected from deeper in the mirror. Additionally, the optical
thickness of a set of alternate layers 104B changes monotonically
from matching structure 102 to simple chirp section 106. As
depicted in FIG. 1, the optical thicknesses of layers 104A
decreases monotonically from matching structure 102 to simple chirp
section 106. In another embodiment if positive dispersion is
desired then the total optical thicknesses of a layer and the two
adjacent half layers in the set of layers 104A should decrease in
thickness from matching structure 102 to simple chirp section 106.
Set of alternating layers 104B is chosen based on what material
matching structure 102 is designed to match to. For example, in one
embodiment if medium 1 10 has a low index of refraction, then
matching layer 102 should be chosen to match medium 110 to a low
index layer of double chirp section 104. In this embodiment, set of
alternate layers 104B will then be made of a low index of
refraction material whose optical thickness will increase
monotonically from matching layer 102.
[0032] Adjacent to double chirp section 104 is simple chirp section
106. Simple chirp section 106 comprises a set of layers wherein the
optical thickness of a layer, e.g., 106A, is the same as the
optical thickness of two adjacent half layers 106 B in the set of
layers, and the total optical thickness of a unit 106C composed of
a layer and the two adjacent half layers varies monotonically from
double chirp section 104 to optional quarter wave stack 108. The
thicknesses of layers in the simple chirp section is discussed in
"Design and fabrication of double-chirped mirrors," by F. X.
Kartner, M. Matuschek, T. Schibli, U. Keller, H. A. Haus, C. Heine,
R. Morf, V. Scheuer, M. Tilsch, T. Tschudi, Optics Lett. 22, 831
(1997), which is hereby fully incorporated by reference. In the
embodiment of FIG. 1, the thicker units 106C are deeper in the
mirror causing the mirror to have a negative dispersion, but
without departing from the invention, the thickness of the units
could be monotonically decreased rather than monotonically
increased from section 102 to section 108 to give a positive
dispersion.
[0033] Adjacent to simple chirp section 106 is optional quarter
wave stack 108. Optional quarter wave stack 108 comprises
alternating layers each of which are a quarter wavelength optical
thickness at a wavelength in the frequency range of interest. In
one embodiment, the quarter wave stack is designed to provide high
reflectance at the longer wavelength end of the frequency range of
interest and thus the layers are a quarter wavelength optical
thickness at the selected longer wavelength. In this embodiment,
the shorter wavelength radiation does not penetrate as deeply into
the mirror, and it may therefore experience less loss than the
longer wavelengths. In another embodiment of the invention, if the
chirp of sections 104 and 106 were reversed then quarter wave stack
108 should be designed to reflect radiation at the shorter
wavelength end of the frequency range of interest.
[0034] It should be noted that although the embodiment of the
mirror depicted in FIG. 1 includes the quarter wave stack, this
section is not required in order to use the present invention and
it may be omitted.
[0035] In this embodiment of the invention, the optimization of the
initial design is achieved by constructing a merit function that
includes terms for the reflectivity, the group delay, and the group
delay dispersion over the reflectivity range of interest, and
optionally, terms for the reflectivity in a range that includes the
pump wavelength. The merit function is then optimized for group
delay, then for group delay and reflectivity, then for group delay,
reflectivity and group delay dispersion, and finally for the
inclusion of all terms including, optionally, the pump wavelength
reflectivity by adjusting the thicknesses of layers in sections of
the mirror until acceptable values of reflectivity, group delay,
group delay dispersion, and optionally pump wavelength transmission
over the desired reflectivity range are achieved.
[0036] FIGS. 2A-C are graphs depicting another embodiment of the
invention. The embodiment depicted in the graphs of FIG. 2 is for a
mirror designed to operate over a wavelength range from 680 nm to
980 nm, and for a pump wavelength of 500 nm +/-20 nm. In this
embodiment, 56 layers have been used to construct the initial
design. These 56 layers are comprised of alternating layers of a
high index material TiO.sub.2 of index n.sub.h.apprxeq.2.4 and
thickness d.sub.h,m for the high index layer in the m.sup.th layer
pair, and a low index material SiO.sub.2 of index
n.sub.1.apprxeq.1.45 and thickness d.sub.1,m for the low index
layer in the m.sub.th layer pair.
[0037] The total number of layers used to construct the initial
design depends on the final result desired and can be determined
through simple trial and error. Typically, 14 layers should be used
for the antireflection coating matching section. For a discussion
of antireflection coatings see "Optimal single band
normal-incidence antireflection coatings," by J. A. Dobrowlski, A.
V. Tikhonrurov, M. K. Trubetskoc, J. T. Sullivan, and P. G. Verly,
Applied Optics, 35, pp. 644-658, (1996) which is hereby fully
incorporated by reference. Approximately 40 layers should be used
for the rest of the mirror in order to achieve high reflectivity
over a 300 to 400 nm reflectivity range. More layers may be used,
but as described below, this may result in layers being removed
from the final optimized design.
[0038] FIG. 2A represent the optical thickness of the layers for
the initial design of the mirror. The optical thickness is the
physical thickness "d" of a layer multiplied by the index of
refraction "n" of the layer. Layers 0 to 14 comprise the
antireflection coating section of the mirror. Layers 15 to 38
comprise the double chirp section of the mirror. The double chirp
of this section is apparent in the upwardly sloping line comprised
of the high index layer material, and the downwardly sloping line
comprised of the low index material. In this embodiment the high
index layers increase monotonically in optical thickness going
deeper into the mirror. This results in a gradual increase in the
coupling coefficient giving a stronger coupling of the incident
wave to the reflected wave, increasing local reflectance, as the
incident wave passes through this section. This gradual increase of
the coupling coefficient produces an impedance matching effect that
reduces the GTI effect mentioned above.
[0039] Layers 39 to 54 comprise the simple chirp section. In this
section, the optical thickness of a high index layer is equal to
the optical thickness of the two adjacent low index half layers,
and the optical thickness of adjacent units of the high index layer
and the two adjacent low index half layers increases monotonically.
The last few layers comprise the quarter wave stack in which each
layer is a quarter wave optical thickness for a frequency in the
long wavelength end of the frequency range of interest.
[0040] FIG. 2B depict the optical thickness of an adjacent pair of
high and low index layers of the initial design depicted in FIG.
2A. In FIG. 2B, the 14 layer antireflection coating extends up to
layer pair 7, the double chirp section extends from layer pair 8
through layer pair 19, and the simple chirp extends from layer pair
20 through layer pair 27. The quarter-wave section is layer pair
28. As is clear from FIG. 2B, the optical thickness of adjacent
layer pairs maintains the same linear chirp through both the double
chirp section and the simple chirp section. FIG. 2C depict the
physical thickness of the high and low index layers of the
mirror.
[0041] In another embodiment of the invention, the chirp in the
simple chirp and double chirp sections need not be linear. A
non-linear chirp can be used to shape higher order components of
the GDD. Additionally, a non-linear chirp can be used to trade off
reflectivity range and smoothness of the group delay.
[0042] FIGS. 3A, 3B and 3C show the reflectivity, group delay, and
group delay dispersion, respectively, of the initial design. The
reflectivity plotted in FIG. 3A is the power reflectivity of the
initial design mirror. The group delay, GD, of the mirror and the
group delay dispersion, GDD, are related to the phase, .o
slashed.(.omega.), by building the first and second derivatives
with respect to the circular frequency (.omega.=2.pi.c/.lambda.). 1
GD ( ) = .0. ( ) GDD ( ) = 2 .0. ( ) 2
[0043] As shown in FIG. 3A, the reflectivity of the initial design
is high over more than a 200 nm bandwidth, and the GDD fluctuates
approximately +/-50% on the short wavelength side and approximately
+/-200% on the long wavelength side of the high reflectivity band
of the mirror.
[0044] This initial design is used as the starting point for an
optimization routine. In the optimization routine, the actual
variations of the index of refraction over wavelength for the high
and low index materials is taken into account.
[0045] In one embodiment, the optimization proceeds as follows. A
merit function F({d.sub.m}) is defined which is to be minimized.
This is a function of all layer thicknesses d.sub.m of the low and
high index layers where "m" denotes the layer number.
[0046] A suitable merit function should measure the difference
between the current mirror characteristics and the desired
characteristics.
[0047] In this embodiment the following merit function is used: 2 F
( { d m } ) = w p i = 1 N 1 R ( i ) - R ref ( i ) + w R i = N 2 N R
( i ) - R ref ( i ) + w GD i = N 2 N GD ( i ) - GD ref ( i ) + w
GDD i = N 2 N GDD ( i ) - GDD ref ( i )
[0048] The merit function splits the frequency range into three
intervals:
[0049] .left brkt-bot..omega..sub.1, .omega..sub.N.right brkt-bot.,
.left brkt-bot..omega..sub.N, .omega..sub.N.right brkt-bot. and
.left brkt-bot..omega..sub.N, .omega..sub.N.right brkt-bot..
[0050] The first interval covers a frequency range for high
transmission for a laser pump beam. The second interval does not
enter the merit function and third interval is the high
reflectivity range of the mirror for which the other properties are
also optimized.
[0051] In this embodiment, 49 points were selected for the merit
function, and this set of points were uniformly spaced throughout
the interval [.omega..sub.1, .omega..sub.N]. Without departing from
the present invention, the set of points selected for optimization
in the merit function can be as large as is computationally
feasible, and they need not be non-uniformly spaced in the
interval(s) of interest. Fewer points are needed if the initial
design is close to the design goals, and more points may be needed
if the initial design is far from the design goals. In other
embodiments of the invention, the number of points used throughout
the optimization process is varied. For example, fewer points are
chosen at the beginning of the optimization when the initial design
is far from the design goals. Then as the design approaches the
goals, more points are used. In still other embodiments of the
invention, non-uniformly spaced points are used at the ends of the
frequency range of interest in order to achieve better results at
these end points.
[0052] In the merit function above, the desired mirror properties
in reflectivity (R), group delay (GD), and group delay dispersion
(GDD) have the index ref. The weights, w.sub.R w.sub.GD and
w.sub.GDD set how strong the different mirror characteristics will
contribute to the total merit function and can be adjusted
interactively during the optimization process to ensure the
usefulness of the final design. The exponent .alpha. is most often
chosen as 1. If it is chosen large, it weights more strongly those
frequency ranges where the largest deviation from the desired
behavior occurs.
[0053] Without departing from the present invention, other merit
functions can be used. A suitable merit function need only have as
its inputs the design goals and the current design parameters. The
merit function should also output one or more values related to the
difference between the goals and the current parameters so that the
output of the merit function can be used as feedback for the
optimization process.
[0054] In this embodiment, the Broyden-Fletscher-Goldfarb-Shanno
Algorithm was used to optimize the design by minimizing the merit
function. This algorithm is described in detail in the book
"Numerical Recipes in Fortran" by William H. Press, S. A.
Technology, W. T. Vetterling, and B. P. Flannery in Cambridge,
University Press (1986), which is hereby fully incorporated by
reference. Without departing from the present invention, other
algorithms can be used, including but not limited to statistical
optimization routines, Monte-Carlo simulations, or any of the
optimization routines in the "Numerical Recipes in Fortran" book
cited above.
[0055] Initially, w.sub.R is set equal to 1, and all of the other
weights are set to 0 and the merit function is optimized. The
results of this optimization are shown in FIG. 4. Note the
reduction in the group delay difference from the goal over the high
reflectivity region from 680-980 nm, and the slight improvement in
the reflectivity. A typical goal for optimization of the group
delay at this step is a +/-1 fs variation in the group delay from
the design goal. If this cannot be achieved, then more layers may
have to be added to the initial design.
[0056] Once the group delay has been optimized, the next step is to
slowly start to increase the weight factor for reflectivity,
w.sub.R. One method that can be used is to increase w.sub.R until F
has just about doubled and then optimize F again. Now double the
value of w.sub.R and optimize again. Repeat this step until there
is a noticeable change in the group delay. The results of this
optimization are depicted in FIG. 5. In one embodiment, a
noticeable change in the group delay is a change of about +/-1 fs
at any of the points in the set of points.
[0057] Next, the group delay dispersion is optimized by slowly
increasing w.sub.GDD. In one embodiment, w.sub.GDD is increased
until it contributes about 10% to the total value of F. The merit
finction is then re-optimized. Then, continue increasing w.sub.GDD
until the reflectivity starts to degrade. Next, w.sub.P is
increased slowly and F is re-optimized until degradation of other
features is noted. The results are depicted in FIG. 6. Note the
degradation in reflectivity at approximately 730 nm.
[0058] If the optimization procedure has resulted in layers which
are thinner than is practical to fabricate, then these layers are
removed. A thin layer can be removed from the design by switching
its index to the index of an adjacent layer, while maintaining the
optical thickness of the layer. The merit function is then
re-optimized. The removal of very thin layers may not even affect
the merit flnction. Thick layers can be thinned by removing a
half-wave of optical thickness for a wavelength near the center of
the frequency range of interest for the mirror. The merit fUnction
is then re-optimizing. If the removal of these layers has too large
of an effect on F, then they may have to be kept in the design.
[0059] In other embodiments F can be split up further so that
wavelength dependent weighting factors can be used. For example, a
gaussian reflectivity weighting over a small wavelength region
(e.g., containing a reflectivity spike) can be used to help reduce
the size of the spike.
[0060] Without departing from the present invention, The weights in
the merit function may be increased in any order or one or more at
a time. What is important in the present invention is that during
the optimization procedure, progress toward achieving one of the
design goals does not cause the optimization routine to settle in
another local minimum of the merit function from which the best
design for achieving the other design goals cannot be achieved.
[0061] FIGS. 4-6 depict the reflectivity, group delay and group
delay dispersion for at various stages of the optimization process.
In FIG. 4, only the group delay has been optimized. In FIG. 5 the
group delay and reflectivity have been optimized with all other
weights at zero. In FIG. 6 the group delay, reflectivity, group
delay dispersion and pump reflectivity have all been optimized.
[0062] FIGS. 7A-C depict the optimized results for the layers of
the mirror. These figures are the corresponding optimized results
of FIGS. 2A-C.
[0063] The foregoing description of embodiments of the invention
has been provided for the purposes of illustration and description.
It is not intended to be exhaustive or to limit the invention to
the precise form disclosed. Many modifications and variations will
be apparent. The embodiments were chosen and described in order to
best explain the principles of the invention and its practical
application, thereby enabling others to understand the invention
for various embodiments and with various modifications as are
suited to the particular use contemplated. It is intended that the
scope of the invention be defined by the following claims.
* * * * *