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Laser Beam Shaping Techniques 9m
Q%Q
Fred M. Dickey,* Louis S. Weichrnan, Richard N. Shagam ~~~
+&
Sandia National Laboratories, MS 0328, Albuquerque, NM 87185-0328 -~m Cl
ABSTRACT
Industrial, military, medical, and research and development applications of lasers frequently require abeam with aspecified
irradiance distribution in some plane. Acommon requirement is alaser profile that is uniform over some cross-section.
Such applications include laser/material processing, laser material interaction studies, fiber injection systems, optical
datdimage processing, lithography, medical applications, and military applications. Laser beam shaping techniques can be
divided in to three areas: apertured beams, field mappers, and multi-aperture beam integrators. An uncertainty relation
exists for laser beam shaping that puts constraints on system design. In this paper we review the basics of laser beam
shaping and present applications and limitations of various techniques.
Keywords: Laser beam shaping, beam profiles, field mapping, beam integrators
1. INTRODUCTION
Beam shaping is the process of redistributing the irradiance and phase of abeam of optical radiation. The beam shape is
defined by the irradiance distribution and the phase of the shaped beam is amajor factor in determining the propagation
properties of the beam profile. Applications of beam shaping include laser/material processing, laser/material interaction
studies, laser weapons, optical datdimage processing, lithography, printing, and laser art patterns. In this paper we provide
an overview of the techniques for producing shaped beams that are uniform over aspecified cross-section. The shaped
beam cross section may be arbitrary, including rectangular, circular, triangular, hexagonal, and ring shaped. In-depth
infromation is provided by the references. The theory and techniques of laser beam shaping are addressed in the book
edited by Dickey and Holswade.l
Laser beam shaping techniques can be divided into three broad classes. The first is the trivial, but usefid, aperturing of the
beam illustrated in Fig. 1. In this case the beam is expanded and an aperture is used to select asuitably flat portion of the
beam. The resulting irradiance can be imaged with magnification to control the size of the output beam. The major
Fig. 1Uniformirradianceobtainedby aperturing inputbeam.
“Correspondence:Email:fmdicke@sandia. ~ov; Telephonti 505844 9660;Fax 5058449554
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DISCLAIMER
This report was prepared as an account of work sponsored
by an agency of the United States Government. Neither
the United States Government nor any agency thereof, nor
any of their employees, make any warranty, express or
implied, or assumes any legal liability or responsibility for
the accuracy, completeness, or usefulness of any
information, apparatus, product, or process disclosed, or
represents that its use would not infringe privately owned
rights. Reference herein to any specific commercial
product, process, or service by trade name, trademark,
manufacturer, or otherwise does not necessarily constitute
or imply its endorsement, recommendation, or favoring by
the United States Government or any agency thereof. The
views and opinions of authors expressed herein do not
necessarily state or reflect those of the United States
Government or any agency thereof.
DISCLAIMER
Portions of this document may be illegible
in electronic image products. Images are
produced from the best available original
document.
.—.-’
.,
disadvantage of this technique is that it is not lossless. In most cases it is desirable, for obvious reasons, that the beam
shaping operation conserve energy. Further, if the input beam irradiance is not suitably smooth, it might not be possible to
find an aperture size and position that gives the desired result. In that case, some form of input beam homogenization
might be required. This type of beam shaping will not be treated further.
The second major technique for beam shaping is what might be called field mapping. Field mappers transform the input
field into the desired field in acontrolled manner. The basic field mapper concept is illustrated in Fig. 2for the case of
mapping asingle mode Gaussian beam into abeam with auniform irradiance. In the figure, Gaussian distributed rays are
bent in aplane so
II
——I
Fig.2Schematicof thefieldmappingconcept(FromRef. 1).
so that they are uniformly distributed in the output plane. The ray bending described in the figure defines awavetlont that
can be associated with an optical phase element. Field mappers can be made effectively lossless. The field mapping
approach to beam shaping is applicable to well defined single mode laser beams.
The remaining class of beam shapers is beam integrators, also known as beam homogenizers. Arepresentative example of
abeam integrator is shown in Flg.3. In this configuration, the input beam is broken up into beamlets by aIenslet array and
T’
D
J_ &
Ts
Fig. 3Amulti-aperturebeam integrator(FromRef. 1).
superimposed in the output plane by the primary lens. The term integrator comes horn the fact that the output pattern is a
sum of diffraction patterns determined by the Ienslet apertures. Beam integrators are especially suited to multimode lasers
with arelatively low degree of spatial coherence. They can also be designed to be effectively Iossless.
In Section 2we discuss aconstraint on beam shaping that results from the application of diffraction theory to the beam
shaping problem. The result is an uncertainty principle type relation that involves the input and output beam sizes. The
theory and design of field mappers is discussed in Section 3. Beam integrators are treated in Section 4. In Section 5we
briefly discuss beam shaping using diffractive diffisers, which are technically field mappers but exhibit speckle properties
t
associated with beam integrators. Finally, in Section 6we discuss the importance of validating and refining the beam
shaping system design using sophisticated optical software.
2. THE UNCERTAINTY PRINCIPLE AND 13
The concept of field mapping is applicable to beam integrators as well as field mappers. The basic field mapping problem
can be expressed in terms of the Fresnel integral as,
where k=2z/~,
U(XI, yl )is the complex representation of the input beam,
V(xl, yl )is the phase function representing the lossless beam shaping element,
u(xo, y. )is the shaped complex field in the output plane at distance Z.
By expanding the last exponential in the integrand and including the remaining quadratic phase function in the beam
shaping element, ~, one can express the beam shaping problem as aFourier transform (Fraunhofer integral),
exp(ikz) [
U(xo, Yo) =ik ‘Xp ’02+ YO 2JJu(~1>Y1)exPv(x1 jY1)exP[-i~(xoxl +YoY1)]~1~Y17 (z)
where ~in this equation differs from that of Eq. (1) by aquadratic phase factor. In terms of either of these two equations,
the beam shaping problem is to determine the phase function, V, when U(XI .Y1)and the magnitude of U(xo. y. )are
specified, This is equivalent to simultaneously specifying the magnitude of afunction and the magnitude of its Fourier
transform.
The uncertainty principle of quantum mechanics, or equivalently the time-bandwith produce inequality associated with
signal processing can be applied to the beam shaping problem. The uncertainty principle is aconstraint on the lower limit
of the product of the root-mean-square width of afunction and its root-mean-square bandwith,23
1
AXAV>— (3)
4Z7“
Applying the uncertainty principle to the beam shaping problem of Eq. (1) or Eq. (2) one obtains aparameter ~of the
form,l
f?=c++, (4)
‘+)
where r. is the input beam half-width, Y. is tie outputbeamhalf-width, and Cis aconstant that depends on the exact
definition of beam widths. As will be discussed in the following sections, agood field mapping solution to Eq. (1) or Eq.
(2) is obtainable if~ is suitably large. Also, and not unrelated, Pis the parameter involved in the stationary phase method
of solution of Eq. (1) or Eq. (2).1
3. FIELD MAPPERS
The beam shaping problem described by Eq. (2) can be directly implemented by the system shown in Fig. 4.1’4’5’6In the
figure, the last two elements comprise the beam shaping system; the first two elements are abeam expanding telescope.
The beam shaping system consists of ashaping element (phase function 1#)and aFourier transform (focusing) lens. The
beam expanding telescope, which may or may not be necessary, provides amean of increasing Pby increasing the input
beam diameter. Using the method of stationary phase, Romero and DickeyGhave obtained solutions for converting
,
I
\u
P/’=7
Fig.4BeamshapingsystemimplementingEq.2 (FromRef. 5).
Gaussian beams into uniform profiles with both rectangular and circular cross sections. In these solutions, the phase ~in
Eq. (1) and Eq. (2) is given by ~=fl~. For acircular Gaussian beam input, the problem of turning aGaussian beam into
aflat-top beam with rectangular cross section is separable. That is, the solution is the product of two one-dimensional
solutions. ~and $(~) are thus calculated for each dimension. The phase element will then produce the sum of these
phases (@X~(X)+flY@Y(Y)].The corresponding one dimensional solution for $is
where
and r.= l/e2 radius of the incoming Gaussian beam.
The sohrtion for the problem of turning acircular Gaussian beam into aflat-top beam with circular cross section is
where
&fi”’,r.
and r=radial distance from the optical axis.
(5)
(6)
As previously mentioned, the quality of these solutions depend strongly on the parameter ~. For the two solutions given
in Eq. (3) and Eq, (4), ~is given by
where: ro=
Yo=
p= 26 r. y. ,(7)
fA
l/e2 radius of incoming Gaussian beam,
half-width of desired spot size (the radius for acircular spot, or half the width of asquare or rectangular
spot).
The effects of ~on the quality of the solution for the problem of mapping aGaussian beam into aflat-top beam with a
rectangular cross section is illustrated in Fig. 5. In the figure we give simulation results for ashaped beam profile with a
rectangular cross section with Pvalues of 4,8, and 16.
1
,
(a) (b) (c)
Fig. 5Simulatedshapedbeamwith squarecrosssection.(a) P=Q. (b) ~=8. (c) ~=16.
The beam shaping configuration just discussed is very general. Solutions for different profiles and cross sections can be
obtained using the method of stationary phase. Also, the phase element can be designed using genetic algorithmsl’7and the
Gerchberg-Saxton algorithms There are several properties associated with the Iossless beam shaping configuration shown
in Fig, 5that are important to system design considerations. We will list them here, noting that the details are provided in
the references.
●
●
●
●
●
●
Element S~acin~: Assuming the validity of the Fresnel integral, the spacing between the phase element and
the Fourier transform lens is not critical.
Sinde Element Desire: The phase element and the Fourier transform lens can be combined as one element.
_The Fourier transform lens focal length may be changed to scale the output spot size without
changing ~.
Positive/Ne~ative Phase: The sign of the beam shaping element phase, v, can be changed without changing
the output beam profile (irradiance). It does, however, change properties of the beam before and after the
output plane. In one case the beam goes through afocus before the output (focal) planq in the other case the
beam goes through afocus after the output plane.
Quadratic Phase Correction: The solutions given in Eq. (5) and Eq. (6) were derived assuming aplane wave
(uniform phase) input beam. Small quadratic phase deviations associated with adiverging input correspond to
asmall shift in the output plane with aproportional scaling of the shaped beam size.
Collimation: The output beam can be collimated using aconjugate phase plate in the output plane that cancels
any non-uniform phase component.
Another interesting field mapping configuration has been suggested by Rhodes and Shealyg that is especially applicable to
the production of relatively large collimated beams with auniform irradiance profile. The basic concept is illustrated in
Fig. 6. In the figure, the second surface of the fust lens directs the incident rays so that they are uniformly distributed at the
b- <
=‘D
Fig. 6Twolenssystemfor large collimatedbeams.
first surface of the second lens. That surface then redirects the rays so that they are collimated. Ageneral theory of this
two lens beam shaping system is detailed by Shealy (Chapter 4) and, Evans and Shealy (Chapter 5) in Reference 1. Their
approach is geometrical optics, which assumes a large P. The design approach starts with the eikonal and invokes
conservation of energy along aray bundle between the two surfaces. Special attention is given to constant optical path
length designs that give acollimated (minimum divergence) output. The result is adifferential equation for the lens
surface. They also develop parallel methods for two lens systems using gradient-index (GRIN) glasses.
,.
It is interesting to note that the fust lens in Fig. 6effectively accomplishes the same function as the combination of the
shaping element and transform lens in Fig. 4. The main difference is that the output surface for the system in Fig. 4is
assumed planar, while the output surface of the fust lens in Fig. 6is the f~st surface of the second lens.
4. BEAM INTEGRATORS
Amulti-aperture integrator system basically consist of two components; 1) asubaperture array consisting of one or more
Ienslet arrays which segments the entrance pupil or cross section of the beam into an array of beamlets and applies aphase
aberration to each bearnlet, and 2) abeam integrator or focusing element which overlaps the beamlets from each
subaperture at the target plane. The target is located at the focrd point of the primary focusing element, where the chief rays
of each subaperture intersect. Brown et all provide adetailed treatment of beam integrators.
Beam integrators can be loosely divided into two categories; diffracting and imaging. Asimple diffracting beam integrator
(also called anon-imaging integrator) is illustrated in Fig. 3, consisting of asingle Ienslet array and apositive primary lens.
The target irradiance is the sum of defocused diffraction spots (point spread functions) of an on-axis object point at infinity
(assuming acollimated input wavefront). The diffracting beam integrator is based on the assumption that the output is the
superposition of the diftlaction fields of the beamlet apertures. The diffraction field is obtained using the Fresnel integral.
If the beam is not spatially coherent over each beamlet aperture amore complicated integral is required and, generally, one
would not be able to obtain areasonable replica of the Ienslet aperture. For example, aspatially incoherent field is
approximated by aLambertian source that radiates over ahuge angle and would not produce alocalized irradiance
distribution at the output plane.
Figure 7illustrates an imaging multi-aperture beam integrator. This type of integrator is especially appropriate for spatially
incoherent sources. From aray optics perspective, these sources produce awavefront incident over arange of field angles
on the lenslet apertures. The first lenslet array segments the beam as before and focuses the bearnlets onto asecond lenslet
array, That is, each Ienslet in the first array is designed to confine the incident optical radiation within the corresponding
aperture in the second array. Asecond lenslet array, separated from the fwst by adistance equal to the focal length of the
secondary lenslets, together with the primary focusing lens forms areal image of the subapertures of the first lenslet array
on the target plane. The primary lens overlaps these subaperture images at the target to form one integrated image of the
subapertures of the first array element. Re-imaging the lenslet apertures mitigates the dil%action effects of the integrator in
Fig. 3. Imaging integrators are more complicated than diftiacting integrators in that they require asecond lenslet array and
an associated alignment sensitivity. Diffracting integrators are more frequently the integrators of choice.
ff
1 2
Fig.7Basicconfigurationfor the imagingintegrator.
There are four major assumptions in the development of dit%acting beam integrators. They are as follows:
1) The input beam amplitude (or equivalently irradiance) is approximately uniform over each subaperture. This
allows for the output to be the superposition of the difliaction patterns of the beamlet defining apertures. It is
expected that small deviations will average out in the output plane. That is, the errors associated with a
particular aperture will not dominate.
2) The phase across each subaperture is uniform. The discussion in 1) applies in this case also. In addition, a
linear phase across asubaperture results in aredirection of the beamlets, causing amisalignment in the output.
3) The input beam divergence does not vary significantly with time. Generally, an input beam divergence will
result in anon-overlapping of the beamlets in the target plane. This can be corrected in many cases with
correction optics in the input beam. However, atime varying divergence would negate the possibility of
correction.
1
,>,
4) The input beam field should be spatially coherent over each subaperture. This is inherent in assumption 1)
since the diffraction patterns are assumed to be described by aFresnel integral.
The imaging integrator does not require assumption 4) since it does not necessarily require that the output pattern be
described by adiffraction integral.
The basic problem for the diffracting integrator is that each lenslet then maps auniform input intensity into auniform
output intensity via the Fourier transform. It can be shown that the desired Ienslet phase function is
(8)
This quadratic phase factor describes athin lens. Again the solution includes the parameter ~that is ameasure of the
quality of the solution. This parameter, for this case, is given by
p=%!, (9)
where d, S, and Fare defined in Fig. 3.
Note that /3 is adimensionless constant and, as previously discussed, is related to the mathematical uncertainty principle.
Increasing ~decreases the effects of difiiaction in the ou~-ut.
Using paraxial geometrical optics it can be shown that the spot size
primary lens divided by the f-number of the subaperture lens,
F
‘=j7Z”
Son the target is equal to the focal length Fof the
(lo)
This result is also obtained using diffraction theory and Fourier optics.
In addition to the diffraction effects discussed above with respect to ~, multi-aperture beam integrators generally exhibit
interference effects. They are effectively multiple beam interferometers. The coherence theory of multi-aperture beam
integrators is developed in Reference 1. Depending on the degree of spatial coherence of the source, the output irradiance
will contain an interference or speckle component. For these conditions, the integrated irradiance of the coherent
component is adequately described by
i
2
I(X> y) =‘~NAmnexp{i[k(amx+ flny)+em IF(X,Y]2,
0,0
(11)
where %and flnare the direction cosines associated with each beamlet, d~ is the phase of the beamlet, Am the amplitude of
the beamlet field, and the function. F(x,y) is the diffraction integral of the beamlet-limiting aperture. F(x,y) is the Fourier
transform of the aperture function for the optical configuration in Fig. 3.
The first factor in Eq. (11) describes the averaging and interference effects of the integrator. The interference effect is a
result of the sum of linear (in xand y) phase terms, which can be viewed as aFourier series. The spatial period for the
resulting interference pattern is given by
Period=?. d(12)
“,
It should be noted that when the coherence between the beamlets is negligible Eq. 11 reduces to
1(x,y)= ‘~NlAmn121F(x,y]’,
0,0 (13)
which isjust the diffraction pattern of asingle Ienslet aperture. Simulations of beam integrator outputs are presented in
Section 6.
5. DIFFRACTIVE DIFFUSERS
The characteristics of diffractive diffusers, also called diffuser beam shapers, are discussed by Brown (Chapter 6) in
Reference 1, Diffuser beam shapers are essentially field mappers. They are designed to diffract the incident beam into the
desired irradiance distribution with abuilt in speckle (or random) pattern. The basic design procedure is;* 1) multiply the
desired irradiance pattern (magnitude) by arandom function (speckle pattern), 2) inverse Fourier transform this result to get
the input field, 3) binarize the phase of inverse transform function to define the beam shaping diffuser. The technique is
illustrated in the following fig~e.
(a)
Fig. 8Simulateddiffuserbeam shaping.(a) Aportion of the binary (2 levels) phase structure
for the ring diffuser.(b) Simulatedintensityplot of the ring diffuser. This simulationused a
Gaussianinput beam of diameter 0.5 mm. Asphericalphase curvaturewas then applied to
simulate alens with afocal length of 10.0 mm. Using scalar wave theory the field was
propagated10.0mmto the focalplaneof the lens.
Diffuser beam shapers generally offer the advantage of being much more tolerant to alignment errors than conventional
field mappers. Although we class them as field mappers, their speckle and alignment tolerance properties are more like
beam integrators. Perhaps they can be viewed as beam integrators with Ienslet aperture size approaching zero with the
phase varying from Ienslet to Ienslet.
6. DESIGN AND ANALYSIS CONSIDEIWTIONS
An important and necessary step in the application of alaser beam shaping technique is the modeling and simulation of the
beam shaper element using acomputer-based high fidelity opticrd design and analysis program. Two types of analysis
programs are available. General geometric optical design programs like ZEMAXW and OSLOw10 establish and verify the
basic geometrical properties of an optical system using beam shaping technology. They also usually have some capability
for the calculation of beam propagation and diffraction properties with some fidelity. These programs can, of course, be
used to perform optimization of specific geometrical and system parameters. In contrast, more detailed, realistic wavefiont
*1These programs also incorporate
analysis can be performed with optical propagation codes like GLADW and ASAIY.
effects like high order or multimode laser inputs and can account for other factors, including beam polarization and spatial
coherence effects. Additionally, ASAP, being araytrace-based propagation code, can permit the simultaneous visualization
of both geometric and diffraction performance of an optical system.
*v
As brief examples, we illustrate how one propagation analysis code, ASAP, can be used to visualize and fine tune the
performance of both afield mapping-based and aIenslet-based beam integrator system. For the field mapper example, we
use the system illustrated in Fig. 4, and choose alaser beam radius, r. =6.75 mm (after afocal magnification), and adesired
spot half-width, y. =2mm. The transform lens f=400 mm, and the laser operates at A=10.6 pm. The coefficients
describing the aspheric profile of the beam shaper are extracted from Ref. 4. The value of ~for this system is 16.
Figure 9(a) is an isometric view of the intensity distribution, and Fig. 9(b) superimposes the intensity contours over the
spot diagram at the irradiance plane. In this view we may visualize the spherical aberration responsible for the beam
shaping effect.
-1.00
(a)
Ffm I m-m for2=606
,21
,a5E-09
.00
(b)
Fig. 9Beamirradianceperformancefor afieldmappersystemwith P=16.a)
Isometricview.b) Geometricspotdiagramsuperimposedover irradiancecontours.
The ASAP-based irradiance calculation shows that the 98 percent radius is 2.10 mm, a5percent error. Axial adjustment of
the spot focal plane by 6 mm yields the nearly desired spot, although the beam profile has been altered asmall amount.
(Fig.” 10 (a)),
If the mode structure, including the amplitude and phase distribution of aspecific laser are known then the effect on
performance can also be calculated. The program permits inclusion of known phase and amplitude distributions. The
sensitivity to manufacturing tolerance and misalignment on the design can also be studied. As an example, Fig. 10 (b)
illustrates the effect on the beam after a10 degree tilt in the transform lens.
t’lm /SQ+X for 2=606
,
1
\
I
I
I
,
1
)
1
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I
I
)
I
1
~
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~
~
i
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i
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yfi ,-~
,,, ,-,--?7, ,.— -T- :;:.~,::J~.,-:T -.7 $7;;,:;.: ,,.......--m 7— . ,; ~,...-.,!++ ,.,.,.,~; ,~+ .
-. —.
,. ,, . ,,,.,,-, ..0’,’.,$ 1 . . .... r.
.-.. —. .
,... . . ,. ,,. --’; 7: “7s7--”- -‘--- “:.:’ f.. >..’”- -“--’--’
,:.,,,.. -.,
.4,00
Fwx Ise-m for Moo
-4.00
1.08
L8UE-10
1.00
(a) (b)
Fig. 10 Effectof geometricmodificationsto field mapperbeamshapersystem
(a)A6mm axial focalplaneshiftchangesbeamscaleby 5 percent.
(b) Effectof 10degreerotationin transformlens.
In the second example, abeam integrator system using amultiple Ienslet array, as illustrated in Figure 3, would be diftlcult
to design and analyze purely analytically. However, three-dimensional propagation codes like ASAP have been used to
simulate and verify the design fairly readily. The example system uses an array of hexagonally packed Ienslets, each with a
, .
1,5 mm aperture, to reduce acollimated h=1.06 pm, 5mm diameter beam to a220 pm diameter for injection into an
optical fiber. Fiber injection systems are treated in detail by Weichman et a112. ASAP was used to analyze and optimize
performance as afunction of axial focus spot position. Figure 11 shows the geometric spot diagram and through-focus
diffraction effect on spot size.
L6D28
.180
.170
.160
.134
.Ia
.130
J
.11022 zPOs
23 ‘M
Fig. 11 (a) Focalplanegeometricspot diagram.
(b) Diffractionspotsizevs focalplaneposition for central98 percentofbeamenergy.
Figure 12 shows the irradiance distribution at the minimum spot position for auniform illumination, perfectly coherent
laser beam. Note how the irradiance pattern is dominated by interference between the Ienslet beams, demonstrating that for
the given parameters, the resultant spot is not uniform.
o-b 0.00
$-0.00
@32120.68
(a)
FWX I sq-w forz=22.9
/’h3212E+0
2.40tE-04
-0.250 .250
(b)
Fig. 12 Lensletarraybeamintegratoranalysis-fully coherentlaserbeam input
(a) Imagesimulationand(x,y)profiles. (b) Isometricview. (at fiberinjectionplane)
If we use asource laser with limited spatial coherence, we cti reduce the interference effects on the spot irradiance
distribution. Figure 13 shows the calculation of aspatial coherence width small compared to each lenslet aperture. Note
how the contrast of the interference fringes is reduced significantly.
7. SUMMARY
In this paper we have discussed what we believe to be the three major approaches to laser beam shaping, outlined the basic
characteristics of each, and suggested the importance of the use of high fidelity optical software to the design problem.
, t. , *
Although we have included basic design equations, space limits require that the interested reader consult
more detailed treatment.
t. Cob. GIidth=O. 6!3 mm I. —m.+
ll_2
FW2 IS!l-~ for 2=22.9
XN4(6E+03
o
.250
the literature for a
(a) (b)
Fig. 13 Lensletarray beamintegratoranalysis-input laser beam with coherencewidth of 0.65 mm.
(a) Imagesimulationand(x,y)profiles. (b) Isometricview. (at fiber injectionplane).
ACKNOWLEDGEMENT
Sandia is amultiprogram laboratory operated by Lockheed Martin for the
Contract DE-AC04-94AL85000.
REFERENCES
United States Department of Energy under
1F, M.Dickeyand ScottC. Holswade,Luser Beam Shaping: Theory and Techniques, MarcelDekker,Inc., NewYork,in print.
2LEFranks,Signal Theory, Prentice-Hall,Inc.,NewJersey,1969.#.
3R, N. Bracewell,The Fourier Transform and its Applications, McGraw-Hill,NewYork, 1978.
4F, M.Dickeyand S. C. Holswade,“Gaussianlaserbeam profile shaping: Opt. Eng. 35, pp. 3285-3295,1996.
5SCHolswadeandF. M. Dickey,“Gaussianlaserbeamprofileshaping:test and evaluation; Proc.SPZE 2863, pp. 237-245,1996.
. .
6L, A.Romeroand F. M,Dickey,“LosslessLaserBeamShaping,”1Opt. Sot. Am.A 13, pp. 751-760,1996.
7DRBrownand A. Kathman,“Multi-elementdiffractiveopticaldesignsusingevolutionaryprograming; Proc. SPIE 2863,1996.
,.
sH. Stark,Image Recovery: Theoty and Application, AcademicPress,NewYork, 1987.
9PWRhodes and D. L. Shealy, “Refractive optical systems for irradiance redistribution of collimated radiation: their
de~ign”andanalysisj’ Appl. Opt. 19, pp. 3545-3553,1980.
10
ZEMAX is aproductof Focus Softswxe,Inc.Tucson,AZ. OSLOis aproductof SinclairOpticS,Jnc.Fairport,NY.
1*GLADis aproductof AppliedOpticsResearch. ASAPis aproductof BreaultResearchOrganization.
12LSWeichrnan,F. M, Dickey,and R. N. Shagam,“BeamShapingElementfor CompactFiber InjectionSystems,”Proc. SPIE 3929,
2000.“
