Design and performance ofdiffractive optics for custom laser resonators

James R. Leger, Diana Chen, and Greg Mowry

Diffractive optical elements are used as end mirrors and internal phase plates in an optical resonator. Asingle diffractive end mirror is used to produce an arbitrary real-mode profile, and two diffractive mirrorsare used to produce complex profiles. Diffractive mirror feature size and phase quantization are shownto affect the shape of the fundamental mode, the fundamental-mode loss, and the discrimination againsthigher-order modes. Additional transparent phase plates are shown to enhance the modal discrimina-tion of the resonator at the cost of reduced fabrication tolerances of the diffractive optics. A 10-cm-longdiffractive resonator design is shown that supports an 8.5-mm-wide fundamental mode with a theoreticalsecond-order mode discrimination of 25% and a negligible loss to the fundamental mode.

1. Introduction

Conventional spherical-mirror laser resonators differin design, depending on the characteristics of the gainmedium. Low-gain systems, such as He–Ne gaslasers, require a resonator with a very low fundamen-tal-mode loss. Higher-gain lasers can tolerate someloss to the fundamental mode in exchange for moredesirable mode discrimination. Establishing a com-mon mode across an array of lasers 1such as a diodelaser array2 requires special optics for producing afundamental mode with peaks that match the indi-vidual waveguides.The majority of low-gain lasers utilize a stable

Fabry–Perot resonator to establish the laser mode.Although this resonator design has a low fundamental-mode loss, it has several inherent disadvantages:112 To achieve adequate modal discrimination, longresonator lengths and small-diameter modes andmode-selecting apertures must be used; 122 the smallmodal diameter results in limited interaction with thegain medium, reducing the amount of power that canbe extracted; 132 the Gaussian shape of the fundamen-tal mode promotes spatial hole burning, resulting inexcitation of higher-order modes at high-power lev-els; and 142 although the Gaussian profile of thefundamental mode has many desirable properties,there are many applications that could benefit from

The authors are with the Department of Electrical Engineering,University of Minnesota, Minneapolis, Minnesota 55455.Received 8 August 1994; revised manuscript received 23 Decem-

ber 1994.0003-6935@95@142498-12$06.00@0.

r 1995 Optical Society of America.

2498 APPLIED OPTICS @ Vol. 34, No. 14 @ 10 May 1995

an alternative mode shape. In particular, a funda-mental mode with uniform intensitymay find applica-tion in a variety of laser systems.Many of the problems with stable Fabry–Perot

resonators can be circumvented by the use of unstableresonator geometries.1,2 Unstable resonators cansupport a large-diameter fundamental mode whilesimultaneously preserving adequate higher-ordermode discrimination. These resonators have beenused with great success in high-gain laser systemssuch as high-power Nd:YAG, CO2, and excimer.However, the unstable geometry has an inherentlylossy fundamental mode and is not suitable for low-and medium-gain laser systems. In addition, un-stable resonators sometimes have an obstructed out-put aperture that produces an undesirable near-fieldpattern.Resonators that use unconventional mirrors have

been employed to improve the modal properties oflasers. Variable reflectivity mirrors have been usedin unstable resonators to reduce the diffraction fromthe hard edges of a conventional unstable resonatorand to increase the uniformity of the near-field inten-sity.3–9 This idea has been extended to variablephase mirrors to provide flat near-field patterns10–12in CO2 lasers. Finally, a spatial filter in the form of agrid of opaque strips has been used to flatten thebeam profile from a Nd:YAG laser.13External resonators have also been applied to

arrays of semiconductor lasers. Fourier spatial fil-tering,14–17 as well as Talbot cavities and Talbotfilters,18–22 has been demonstrated to generatesubstantial powers from both one-dimensional21,23and two-dimensional arrays.24

Many of the characteristics of the above lasercavities can be greatly enhanced if the conventionalmirrors are replaced with diffractive optical elements.A single diffractive element has been used to producea square flattop Nd:YAG laser beam and to provideenhanced discrimination against higher-order spatialmodes.25 This configuration has also been applied toarrays of semiconductor lasers26,27 and to large-areasemiconductor amplifiers.28,29 Additional diffractiveelements placed inside the resonator can providefurther modal discrimination, allowing the cavitylength to be reduced and the mode size increased.30In this paper we discuss some of the design aspects ofthis new type of resonator. In the first section, wereview the technique of diffractive mode-selectingmirrors 1DMSM’s2 in a laser cavity. In Section 2 thedesign of the DMSM for optimizing mode shape andmode discrimination is described. In particular, weare concerned with the feature-size requirements andthe number of phase levels needed to achieve thedesired performance. Finally, we analyze the effectof a second diffractive element in the cavity anddescribe some of the design considerations for optimiz-ing performance. We explore several possible selec-tions for the phase mask and report on the impact ofthis selection on the feature-size requirements of theDMSM.

2. Concept of Diffractive Optic Resonators

A diffractive optic resonator that consists of tworeflective diffractive optical elements with transpar-ent diffractive elements placed between the mirors isshown in Fig. 1. In addition, various apertures maybe included inside the cavity to provide selectivefiltering of the higher-order modes. Because diffrac-tive optics can be fabricated to approximate anyrequired phase reflectance, the resulting optical reso-nator can have any desired fundamental-mode shape.The designer starts by specifying the desired geomet-ric shape and complex-mode profile of the fundamen-tal mode just to the left of the output mirror. Thediffraction pattern of this mode at the diffractive endmirror is then calculated numerically with the scalarnonparaxial diffraction formula andwith the propaga-tion through intracavity phase plates and aperturestaken into account. An end-mirror phase reflec-tance that will return the complex conjugate of thisspecific wave front is then calculated. It is easy toshow11,25 that the return wave that is incident uponthe output mirror is the complex conjugate of theoriginal desired distribution. If we design this sec-

Fig. 1. General design of a diffractive optical resonator.

ond mirror to reflect the complex conjugate of thiswave front and if both mirrors and apertures aresufficiently large, the resulting field is identical to theoriginal desired field. This, by definition, is a modeof the cavity.For a mirror to produce the complex-conjugate

wave front, it need only modify the phase of the wavefront. Thus both mirrors are phase only and can befabricated as diffractive optical elements if integermultiples of a wavelength are subtracted from thephase-conjugating surface and if the remaining sur-face-relief structure is represented as a series ofquantized phase steps. By careful choice of othersystem parameters, these mirrors can also be used todiscriminate against all higher-order modes. Forthis reason, we have called these mirrors DMSM’sand the resonator based on them a DMSM resonator.The theoretical performance of this resonator can

be assessed by the solution of the eigenvalue equa-tion,

eK1x, x82Un1x82d2x8 5 gnUn1x2, 112

where the integral kernel K1x, x82 describes the round-trip propagation in the cavity, Un1x2 are the eigenfunc-tions of the equation and gn their correspondingeigenvalues. The squared magnitude of an eigen-value associated with a particular mode gives theround-trip attenuation of that mode, which is due todiffraction. The gain required for overcoming thisattenuation is called the modal threshold gain and isdefined as

Gn 5 1@gn2. 122

The modal threshold gain is used exclusively in thispaper as a measure of modal discrimination andfundamental-mode loss, in which a threshold gain ofunity corresponds to a lossless cavity.The eigenfunctions of Eq. 112 describe the amplitude

and the phase of the various modes. The kernelK1x, x82 contains all the information regarding thespecific implementation of the diffractive optic.Hence solutions to this equation can be used toanalyze the effect of feature-size limits, phase quanti-zation, etc., on the resonator performance.

3. Design of a Single-Element DiffractiveMode-Selecting Mirror Resonator

The simplest form of the DMSM resonator consists ofa single DMSM for an end reflector and a flat mirrorfor an output mirror. This requires the mode at theoutput mirror to be a real function 1so that a conven-tional mirror can return the complex conjugate2.There are several goals in the design of a DMSM

resonator. First, the fundamental mode must beacceptably close to the desired mode shape and musthave a sufficiently low loss for the particular lasersystem. Second, the separation in threshold gainbetween the fundamental mode and the higher-order

10 May 1995 @ Vol. 34, No. 14 @ APPLIED OPTICS 2499

modesmust be sufficient to preventmultimode oscilla-tion. Third, the fabrication of the DMSM must bewithin the state of the art, and losses introduced byquantization and fabrication errors must be mini-mized. The design parameters consist of cavitylength, aperture size and placement, number of DMSMphase-quantization levels, and feature-size limita-tions of the DMSM. In this section, we address eachof these issues by modeling a particular DMSM cavityand solving for the eigenfunctions and the eigenval-ues of Eq. 112.Two different mode shapes are studied in this

paper. The first is a single beam with a super-Gaussian profile, chosen to be very close to a uni-formly illuminated square. Its amplitude is given by

A11x, y2 5 exp321x@v02204exp321y@v02

204, 132

where v0 is the half-width of the square. The secondis a mode from a laser array consisting of M 1 1Gaussian beams, expressed by

A21x, y2 5 om52M@2

M@2

exp32 1x 2 ma22 1 y2

v082 4, 142

where v08 is the width of a single Gaussian beam anda is the separation between the beams.

A. Optimum Cavity Length and Aperture Sizes

It is well known that the cavity length has a greateffect on the mode discrimination in conventionalFabry–Perot optical resonators. Because the lengthand the mode size are dependent on one another, thecavity is often desribed by a dimensionless parameter.The approximate parameter is different for a singlebeam and for an array of beams. We describe thecavity characteristics of each of these fundamentalmodes below.

1. Super-Gaussian Fundamental ModeFor the single super-Gaussian mode of Eq. 132, thecavity Fresnel numbers can be used to describethe effects of cavity length and aperture sizes. TheFresnel numbers N1 and N2 that correspond to thetwo apertures are defined as

Ni 5di2

4lz, 152

where d1 is the width of the aperture covering theoutput mirror, d2 is the width of the DMSM aperture,z is the cavity length, and l is the wavelength of light.The modal threshold gain required for overcomingthe losses to the fundamental and the second-ordercavity modes as a function of these two Fresnelnumbers is shown in Fig. 2. For every point on thiscurve, the DMSM has been adjusted to produce asuper-Gaussian whose width v0 is slightly smallerthan the output aperture, such that d1@2 5 1.08 v0.This value was chosen to produce negligible clipping

2500 APPLIED OPTICS @ Vol. 34, No. 14 @ 10 May 1995

of the super-Gaussian. It is clear from the graphthat a sufficiently large value ofN2 ensures a low-lossfundamental mode for all cavity lengths. This re-sults simply from using a DMSM with sufficient sizeto collect and phase conjugate virtually all the dif-fracted light from the output aperture. In addition,using a smaller value of N2 does not appreciablyincrease the discrimination between the fundamentaland the second-order modes, whereas the resultingcavity mode shape can differ significantly from thedesired one because of clipping at the mirror. Hencethere is no advantage to using a limiting aperture inthe DMSM plane.The Fresnel number governed by the output aper-

tureN1 has a dramatic effect onmodal discrimination.The discrimination increases with decreasingN1 untilN1 reaches a value of approximately 1@p, whereuponit tends to saturate 1for low N22 or decrease 1for highN22. This implies that the optimum cavity length isroughly given by the Rayleigh range z0 5 pv0

2@ldefined by a conventional Gaussian with beam waistv0. Thus, for large beam widths, the optimal cavitylength can be quite large. This issue is addressed inSection 4.Figure 2 was based on the fabrication of a DMSM

with a mode size approximately equal to the output-aperture size. We now consider the effect of usingan output aperture that is not matched to the funda-mental mode of the cavity. Figure 3 shows the losses

Fig. 2. Threshold gain of fundamental and second-order modes asa function of two cavity Fresnel numbers N1 and N2. The DMSMis adjusted to be phase conjugate to a super-Gaussian that isslightly smaller than the output aperture.

Fig. 3. Threshold gains of the fundamental and the second-ordermodes as a function of normalized output-mirror-aperture size.Normalization is with respect to the super-Gaussian full width.

to the first two modes as functions of the normalizedaperture width d8 5 d1@2v0. The cavity length ischosen to be one Rayleigh range of a conventionalGaussian 1pv0

2@l2, and the Fresnel number of theDMSM aperture 1N22 is infinite. From this result, itis apparent that the output aperture is very impor-tant for modal discrimination. Normalized aperturesizes of d8 : 1 reduce the modal discriminationsignificantly, whereas sizes of d8 9 1 result in asubstantial fundamental-mode loss. When d8 5 1,the super-Gaussian fundamental uniformly illumi-nates the aperture with very little clipping, whereasthe higher-order modes 1with substantal light outsidethe aperture2 are efficiently filtered out.The apertures also affect the shape of the fundamen-

tal mode, and care must be taken to avoid significantmodal distortion. Figure 41a2 shows the effect ofchanging the Fresnel number N2 by the use of differ-ent aperture sizes at the DMSM. For N2 $ 20, theeffect on the mode is minimal, and the modal shape isvery close to the expected super-Gaussian. As theFresnel number is decreased, the clipping of thefundamental mode becomes significant and the modeis distorted. The increased loss is also evident in theslightly smaller mode height. From this result, it isapparent that the DMSM should be fabricated with asize sufficient to result in a Fresnel number of N2 $20.The effect of the output-mirror-aperture size on

modal shape is shown in Fig. 41b2. For maximummodal discrimination, it is desirable to have theaperture approximately equal to the mode size.However, the slight clipping that results distorts themode. The solid curve corresponds to an aperturethat is 1.08 times the mode size. There is essentiallyno clipping, and the mode shape is almost a perfectsuper-Gaussian of the 20th order. The dashed curvecorresponds to an aperture that is equal to the modesize. A small amount of ripple that is due to theclipping of the fundamental mode by the outputaperture can be observed. Although the distortion issmall in this example, lower-order super-Gaussianswill have more clipping, and the distortion maybecome significant. In these cases, the aperturemust be enlarged to preserve the modal shape at theexpense of the modal discrimination.

2. Gaussian Array Fundamental ModeThe DMSM cavity has been studied for arrays ofGaussian beams suitable for application with diodelaser arrays. In this case, the output aperture isformed by the waveguides on the laser array itself.The dimensionless design parameters are the fillfactor of the array 3 f 5 2v08@a from Eq. 1424 and thecavity length normalized to a Talbot distance Zt 52a2@l. The modal threshold gain of the second-ordermode as a function of normalized cavity length Z@Zt isshown in Fig. 51a2. As the DMSM size was chosen tobe large, the fundamental-mode loss was negligibleand is not shown in the figure. It is apparent thatthe cavity length affects themodal behavior in amuchmore complex manner than the single beam result ofFig. 2. The lengths corresponding to large modaldiscrimination are at one-half Talbot distance andone-quarter Talbot distance. The actual heights ofthe peaks and the optimum choice between these twocavity lengths are functions of the fill factor of thearray.27 Figure 51b2 shows the effect of the fill factoron the second-order mode at a constant cavity lengthof one-half Talbot distance. A larger fill factor re-duces the coupling between the individual Gaussianbeams and hence reduces the discrimination of thesecond-order mode. However, unlike a simple Talbotcavity,31 very small fill factors also reduce the modaldiscrimination.27 Thus there is an optimal fill factorfor an array of lasers that maximizes the modaldiscrimination.

B. Fabrication Tolerances and Requirements

The above resonator models assumed perfect diffrac-tive optics. We have shown that, for sufficientlylarge DMSM’s, the loss to the fundamental mode isnegligible. However, because of fabrication limita-tions, the effects of finite feature size and phasequantization must be taken into account. In mostother diffractive optics applications, we are interestedprimarily in the efficiency and the possible distortionof the designed response that are due to these quanti-zations. In a resonator application, however, theefficiency of both the fundamental and the second-order modes must be considered 1where it is desirableto minimize the efficiency of the latter2. In this

Fig. 4. Fundamental-mode shape as a function of aperture size: 1a2 aperture is applied to DMSM, and curves correspond to FresnelnumbersN2 5 4, 8, and 20; 1b2 aperture is applied to output mirror, normalized to the super-Gaussian full width.

10 May 1995 @ Vol. 34, No. 14 @ APPLIED OPTICS 2501

Fig. 5. Modal threshold gain of second-order mode for an array of eight Gaussian sources: 1a2 threshold gain as a function of normalizedcavity length for an array fill factor of 0.12, 1b2 threshold gain as a function of the fill factor for a cavity length of zt@2.

section, we consider the effects of these fabricationtolerances on the modal behavior of the resonator.

1. Feature-Size RequirementsThe feature-size effects were studied by the applica-tion of a size quantization across the phase pattern atthe DMSM. The phase was held at a constant aver-age value across the width of the feature. Figure 6shows the modal threshold gain of the fundamentaland the second-order modes as functions of thisfeature width quantization for the single super-Gaussian beam of Eq. 132. The cavity is assumed tohave a Fresnel number N1 5 1@p to maximize themodal discrimination and an output aperture slightlylarger than themode size d1@25 1.08 v0. The secondFresnel number N2 is sufficientlylarge so that theclipping at the DMSM is negligible. The feature sizeDl is expressed in dimensionless units Dy, where

Dy 5Dl

Œlz162

and z is the cavity length. Because N1 5 1@p1corresponding to a cavity length z 5 pv0

2@l2, thereduced units are given simply by Dy 5 Dl@Œpv0.Thus, for a beam half-width of 0.6 mm, the reducedfeature size corresponds to the approximate actualfeature size in millimeters. It is apparent that the

Fig. 6. Threshold gains of super-Gaussian fundamental andsecond-order modes as functions of minimum DMSM feature size.

2502 APPLIED OPTICS @ Vol. 34, No. 14 @ 10 May 1995

feature-size requirements are very lax for this applica-tion. For example, a square beamwith a width of 1.2mm 1v0 5 0.6 mm2 can be fabricated with a mode lossof less than 1% by the use of a minimum feature sizeof 40 µm. This is easily achievable with simplelithography and wet chemical etching.A feeling for the effect of finite feature size on the

loss to the fundamental mode can be obtained whenthe requirements of a spherical diffractive mirrorused to establish a simple Gaussian beam of the samebeam waist v0 are considered. We first calculate theradius of curvature R1z2 at the mirror by propagatingthe beam the length of the cavity 1one Rayleigh range2.We then have

R1z 5 z02 5 z11 1z02

z 2 2 5 2z0. 172

The focal length of the mirror used to reflect this wavemust be equal to half the radius of curvature of thebeam, or z0. The mirror diameter must be chosen tocollect the majority of the light. We choose a mirrorwith Fresnel numberN2. Thus, at a Rayleigh range,we have a mirror diameter d of

d 5 2v0ŒpN2 182

and an f-number for this mirror of

f@# 5z0

2v0ŒpN2

. 192

It is well known that the minimum feature size atthe edge of a two-level diffractive optic mirror is givenby l1 f@#2. Thus the minimum feature size Dl at theedge of ourM-level mirror is

Dl 52l f@#

M5

v0

MΠp

N2

, 1102

or, in dimensionless units,

Dy 51

MŒN2

, 1112

where the Rayleigh range z0 5 pv02@l has been

substituted. This indicates that, for a given effi-ciency 1corresponding to a certain number of levels atthe edge of the mirror2 and a given diffractive loss atthe DMSM 1corresponding to a specific Fresnel num-ber N22, the required feature size is directly propor-tional to the beam size v0. So, for example, if a1.2-mm beam is desired 1v0 5 0.6 mm2 and a Fresnelnumber N2 5 20 is sufficient to keep diffractive lossesto an acceptable level, the required 16-phase-levelmirror would have a feature size at the edge of themirror or Dl 5 15 µm.The local diffraction efficiency of the mirror at its

edge hedge can be approximated by the relationship

hedge 5 0sin1p@M2

p@M 02

< 1 21

3 1p

M22, 1122

where the approximation is true for large M. Solv-ing Eq. 1102 forM, substituting this value into Eq. 1122,and defining the loss at the edge of the mirror Ledge

that is due to the finite feature size as Ledge 5 1 2 hedge

results in

Ledge <pN2

3 1Dlv022. 1132

In terms of reduced coordinates, the loss can besimply expressed as

Ledge <p2N2

3Dy2. 1142

Finally, approximation 1142 can be expressed in termsof the fundamental modal threshold gainG1 as

G1 5 11 2p2N2

3Dy22

21

< 1 1p2N2

3Dy2, 1152

where the approximation holds for Dy 9 11@p2 Œ3@N2.The actual threshold gain of the element shown inFig. 6 consists of an average across the entire mirrorand is thus expected to be considerably smaller. Theeffect of feature size on the second-order mode issimilar to the fundamental mode, so that the modediscrimination stays approximately constant, indepen-dent of feature size.The feature-size constraints of the DMSM fabri-

cated for the array of Gaussians from Eq. 142 can befar more demanding. Figure 7 shows the modalthreshold gain of the first two modes for an array ofeight lasers with a fill factor of f 5 2v08@a 5 0.12,where a is the spacing between Gaussian beams.The mirror is placed at a distance of one-half Talbotlength to maximize the modal discrimination. The

normalized feature size Dy now becomes

Dy 5Dl

Œlz5

Dl

a5

fDl

2v08. 1162

From Eq. 1162, we see that the smaller the Gaussianbeam waist v0, the smaller the required feature sizeDl. This is as expected, as beams with smallerwaists have larger divergences, and thus the angularplane-wave spectrum incident upon the DMSM islarger. This, in turn, requires finer pitch gratings forphase conjugating the higher-frequency plane waves.As an example, a standard single-mode waveguide ina laser array has a lateral beam waist v0 of 3 µm.From Fig. 7, we see that the fundamental-modethreshold gain becomes substantial for values of Dy .0.06. This corresponds to a minimum feature size Dlof 3 µm.

2. Phase-Quantization RequirementsBecause diffractive optics are often fabricated withdiscrete phase levels, it is of interest to determine theeffect of this phase quantization on modal thresholdgain and mode shape. Figure 81a2 shows the result ofquantizing the phase of a diffractive optic tovarious numbers of phase levels. As before, thediffractive optic is designed to produce a 20th-ordersuper-Gaussian mode with a cavity length of oneRayleigh range 1based on a conventional Gaussianbeam2. No feature-size restrictions are used in thismodel, and the aperture size d1 at the output mirror isadjusted to be 1.08 times the fundamental-modewidth 2v0. The modal threshold gain of the funda-mental mode is seen to be similar to that expectedfrom a conventionalmultiphase lens or grating 1shownas a dotted curve in the figure2. The modal thresholdgain of the second-order mode also increases with areduced number of phase levels. Hence the overallmodal discrimination is large even with a low numberof phase levels.The mode shape is also influenced by the number of

phase levels. Figure 81b2 shows the resultant funda-

Fig. 7. Threshold gains of Gaussian-array fundamental andsecond-order modes as functions of minimum DMSM feature size.

10 May 1995 @ Vol. 34, No. 14 @ APPLIED OPTICS 2503

Fig. 8. Effect of phase quantization on laser resonator performance: 1a2 threshold gain of fundamental and second-order modes asfunctions of number of phase-quantization levels; 1b2mode shapes for 4, 8, and 32 phase-quantization levels.

mental mode when 32, 8, and 4 phase levels are usedto produce the 20th-order super-Gaussian. It is clearthat eight phase levels produce significant distortionof the fundamental mode. In addition, there is asignificant cavity loss, as seen by the smaller modeheights. However, as the feature sizes are verylarge, even when many phase levels are used, it isrelatively easy to fabricate mirrors with many phaselevels, resulting in low-loss fundamental modes andaccurate mode shapes. Figure 9 shows the fourmask patterns required for fabricating a 16-levelDMSM for a 20th-order square super-Gaussian with abeam width of 1.2 mm.

4. Laser Resonators With Internal Phase Plates

A simple two-mirror cavity design can establish anarbitrary real-mode profile with a single DMSM andan arbitrary complex profile with two mode-selectingmirrors. When simple mode profiles such as super-Gaussians are desired, however, large discriminationbetween spatial modes occurs only when the cavitylength is approximately one Rayleigh range. Thus,for large beam diameters, these methods can result invery large cavity lengths, compromising mechanicalstability and increasing the pulse length for Q-switched laser operation.In this section we describe a diffractive laser cavity

that contains a phase plate inside a single DMSMresonator. The setup is as shown in Fig. 1, with asimple flat mirror used as the output mirror. Thedesign of the cavity proceeds in the same way asabove. The mode at the output mirror can be se-lected to be any desired real function, but the diffrac-tion pattern incident upon the DMSM is modified bythe phase plate. As in the above, the DMSM isdesigned to return the complex conjugate of thispattern. As the light passes through the phase plateon the return trip, it recreates the desired mode.Thus, if a laser crystal is placed between the phaseplate and the output mirror, the mode in the crystalcan be simple 1e.g., super-Gaussian2 and can extractpower from the medium in an optimum fashion.The mode to the left of the phase plate is far morecomplex, however. Below we show that this in-creased complexity greatly increases the modal dis-crimination of the cavity. The following sections

2504 APPLIED OPTICS @ Vol. 34, No. 14 @ 10 May 1995

describe the performance enhancement offered byboth periodic-grating phase plates and pseudorandomphase plates.

A. DMSM with a Sine-Wave Phase Plate

We first describe the performance of a sine-wavephase plate placed 20 cm from the output mirror and30 cm from the DMSM, for a total cavity length of 50cm. The same fundamental mode and output aper-ture were chosen as in the above simulation 12v0 5 1.2mm, d1 5 1.3 mm2. However, the total cavity lengthof 50 cm corresponds to less than one-half of aconventional Gaussian–Rayleigh range at the operat-ing wavelength of 1.06 µm.The phase plate studied was a simple phase grating

with amplitude transmittance t1x, y2 given by

t1x, y2 5 exp3 jm sin12p fgx 1 f24, 1172

where m is the modulation index, fg is the frequency,and f is the shift of the phase grating.The choice of the aperture size ensured that the loss

to the fundamental mode was negligible 1,0.1%2.Figure 10 shows the loss to the second-order TEM10mode as a function of grating frequency. The modu-lation depthmwas set to unity, and the grating phasef was set to zero. A grating frequency of fg 5 0corresponds to a simple DMSM cavity and has asecond-order modal threshold gain g2 5 1.34. Thisvalue can be improved remarkably if the gratingfrequency is increased to 3.75 mm21, where thesecond-order modal threshold gain is 3.7 1correspond-ing to a loss of 73%2. Increasing the frequency pastthis point leads to a rapid decrease in modal discrimi-nation, followed by a more gradual reduction. Forsufficiently high grating frequencies, the modal dis-crimination reduces to the value obtained with nophase plate present 1g2 5 1.342. This is to be ex-pected, as at these high spatial frequencies there isvery little overlap between the grating orders. TheDMSM is then illuminated by mulitiple copies of thesame distribution from a simple DMSM resonator,and there is no improvement in performance.The high modal discrimination exhibited by the

diffractive resonator can be utilized to reduce the

Fig. 9. Four mask patterns for fabricating 16-level DMSM. The smallest feature size is 50 µm.

cavity length or to increase the mode size. For thisexample, if a modal threshold gain of 1.15 is adequateto discriminate against higher-order modes, the modediameter of the diffractive resonator can be increased

Fig. 10. Second-order TEM10 modal threshold gain as a functionof sinusoidal-grating frequency. The loss to the fundamentalmode is less than 0.1%.

to 5 mm 1from 1.2 mm2 while still ensuring single-spatial-mode operation.The optimum phase of the sinusoidal grating was

studied next. Figure 11 shows the modal thresholdgain of the second-order TEM10 mode as a function ofgrating shift f. For each grating shift, a new DMSMwas calculated to establish a low-loss fundamentalmode. The loss to the second-order mode was thendetermined. For our cavity parameters, a sinusoidalpattern 1f 5 02 has a considerably larger mode-discrimination ability than the corresponding cosinu-soidal pattern 1f 5 90°2. However, the actual opti-mum value for f appears to be a function of thespecific cavity parameters, such as modulation depthm and grating frequency fg.Figure 12 shows the second-order TEM10 modal

threshold gain as a function of grating modulationdepth m. The solid curve corresponds to a gratingfrequency of 2 mm21, and the dashed curve corre-

10 May 1995 @ Vol. 34, No. 14 @ APPLIED OPTICS 2505

sponds to a frequency of 3.75 mm21. For the lowerfrequency, an increase in discrimination is observedwith a higher modulation depth of up to approxi-mately m 5 12. This increase is expected, as phasegratings with larger modulation indices contain morediffraction orders. As above, these higher ordersinterfere and increase the complexity of the mode-selecting mirror, thereby increasing the modal dis-crimination. The higher-frequency grating 1corre-sponding to the optimum point in Fig. 102 achieves amaximum at a much lower modulation index. Amodulation depth greater than unity does not appearto increase the modal discrimination. One possibleexplanation for this is that the overlap of the diffrac-tion orders at higher grating frequencies is reduced.Consequently the higher orders generated by a largermodulation index do not interfere with the lowerorders and so do not improve the mode-discriminationability of the DMSM.

B. Selection of Phase-Plate Pattern

The phase function encoded on the phase plate has animpact on the mode-discrimination ability of thecavity as well as on the fabrication and the alignmenttolerances of the DMSM. We have investigated bothperiodic gratings and pseudorandom phase arrays forthis element. A periodic phase grating can haveseveral practical advantages. First, it may be easierto fabricate, as sine-wave and square-wave gratingscan be fabricated by conventional means. In addi-

Fig. 11. Second-order TEM10 modal threshold gain for sinusoidalphase grating with different phase shifts.

Fig. 12. Second-order TEM10 modal threshold gain for a sinusoi-dal phase grating with different modulation depths.

2506 APPLIED OPTICS @ Vol. 34, No. 14 @ 10 May 1995

tion, the initial gross alignment is easier, as theabsolute center of the plate does not have to bedetermined.The disadvantage of a grating structure was seen in

Subsection 4.A. The pattern incident upon theDMSM results from interference between discretegrating orders, and high grating frequencies preventthese orders from overlapping. Thus there is a limitto the mode discrimination achievable with a periodicgrating. Apseudorandom phase plate has a continu-ous angular plane-wave spectrum and hence does notsuffer from this problem. The effect of these twophase plates on the resonator properties is examinedbelow.

1. Periodic-Grating Phase PlateThe ideal grating pattern provides very high modaldiscrimination without requiring excessively smallfeatures on the DMSM. We performed a compara-tive study between sine-wave and square-wave phasegratings of similar frequency, phase, and depth ofmodulation. Although the square-wave grating maybe simpler to fabricate, it appears to have an inherentdisadvantage compared with the sine-wave grating.Figure 13 shows the calculated modal threshold gainsof the fundamental and the second-order modes thatare due to finite feature size on the DMSM. Thecavity configuration is the same as in Subsection 4.A120th-order super-Gaussian, 2v0 5 1.2 mm, d1 5 1.3mm2, the frequency of the grating is 3.75 mm21, andthe peak-to-peak grating modulation is p rad. Themodal threshold gain is plotted for various minimumfeature sizes on the DMSM. As expected, the re-quired modal gain for the fundamental mode in-creases with increasing feature size. However, therequired gain is always higher for the square wavethan for the sine wave. Thus, for a particular funda-mental-mode loss, the feature-size requirements forthe square-wave phase pattern are more stringentthan for the sine-wave pattern. Because the discrimi-nation between the fundamental and the second-order modes is similar for the two grating types, thereis an advantage to using a sine-wave grating insteadof using a square-wave grating.The reason for the improved performance of the

sine-wave grating is apparent when the angular

Fig. 13. Fundamental and second-order TEM10 modal thresholdgains versus feature size for square-wave and sine-wave gratingphase plates.

plane-wave spectra of the two gratings are compared.The operation of the mode-selecting mirror can beviewed as a linear superposition of blazed gratings inwhich each grating retroreflects a particular incidentplane-wave component back along its original angle ofincidence. The larger the angular plane-wave spec-trum, the smaller the features required on the DMSM.For a lossless fundamental mode, these gratings mustefficiently phase conjugate all components of theincident angular plane-wave spectrum. The square-wave phase pattern contains many more high spatialfrequencies than does the sine-wave phase patternwith a similar modulation depth. This can be seenby comparing the asymptotic behavior of the diffrac-tion order intensity. The power in the qth diffractionorder of a square wave is proportional to 1@q2.However, the power in the qth diffraction order from asine-wave phase grating 1assuming unity area andunity illumination intensity2 is given by

Pq 5 0 Jq1m@22 02, 1182

where Jq is the qth-order Bessel function andm is themodulation index. The asymptotic behavior of thisfunction for large q and q . m is given by

0 Jq1m@22 02 ,1

2pq1em

4q 22q, q : 1, 1192

where e is the irrational number 2.71828 · · · . Thefalloff of the higher-order harmonics is much sharperfor the sine-wave grating, and there is very littlepower contained in these high-frequency terms.Hence the feature-size requirement on the DMSM isless demanding than for the square-wave grating.

2. Pseudorandom Phase PlateThe mode-discrimination ability of the periodic grat-ing is limited by the discrete nature of the diffractionorders. Grating frequencies that do not produceoverlap of the diffraction orders have poor modaldiscrimination. A pseudorandom phase plate, how-ever, does not have discrete orders and so arbitrarilylarge bandwidths can be used.It was postulated that the degree of modal discrimi-

nation was related to the angular plane-wave spec-trum incident upon the mode-selecting mirror. Totest this, we performed a series of experiments byusing random phase plates with different angularplane-wave spectra. A Gerchberg–Saxton algorithmwas used to apply the phase-only constraint and theGaussian angular spectrum constraint simulta-neously.32 The resultant phase-only data had anapproximately Gaussian angular plane-wave spec-trum with the power spectral bandwidth defined asthe 1@e2 point of the Gaussian. Figure 14 shows theincrease of threshold gain to the second-order modewith increasing phase-plate bandwidth. The errorbars show the statistical variation in the simulation161 standard deviation2. Very high modal discrimi-nation can be obtained if the DMSM is presented witha sufficiently complex light field. It appears from

this test that, in theory, arbitrarily large modaldiscrimination can be obtained with a pseudorandomphase plate of sufficient spatial frequency bandwidth.There are some practical considerations that limit theuse of arbitrarily high bandwidths, however. Higherbandwidth phase plates result in more complexDMSM’s with finer feature-size requirements. Fig-ure 15 shows the effect of mode-selecting-mirrorfeature-size quantization on the modal gains of thefundamental and the second-order modes. For fun-damental mode losses of approximately 1%, featuresas large as 3 µm can be used for the low-bandwidthphase mask 118 mm212, whereas 1.0-µm features arerequired for a high-bandwidth phase mask 153 mm212.At these fine feature sizes, scalar diffraction theory isalso not appropriate, and better estimates of diffrac-tion efficiency must account for vector diffractiontheory effects.33 Finally, the alignment of the phaseplate to the DMSM becomes more critical with ahigher phase-plate bandwidth.To illustrate the potential of this technique, we

have considered a case study of a 10-cm-long lasercavity containing a random phase plate with a band-width of 44 mm21. The phase mask was placed inthe center of the cavity, and a 20th-order super-Gaussianwas chosen as the fundamental-mode profile.A mode-selecting mirror was designed with a mini-mum feature size of 2 µm and 16 phase-quantizationlevels. The resulting theoretical fundamental-cavity-mode profile is shown in Fig. 161a2 for a 1.2-mm beamsize. The finite feature size and phase quantization

Fig. 14. Modal threshold gain of second-order mode as a functionof bandwidth of a pseudorandom phase plate.

Fig. 15. Modal threshold gain of fundamental and second-ordermodes as functions of minimum DMSM feature size when apseudorandom phase plate is used.

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Fig. 16. Theoretical performance of a 10-cm laser cavity containing a random phase plate: 1a2 two-dimensional fundamental-modeintensity profile, 1b2 laser gain required for overcoming the diffractive losses to the second-order mode for various fundamental beam sizes.The result from a conventional Fabry–Perot cavity is shown for comparison.

of themode-selectingmirror produce small nonunifor-mities in the beam profile and result in a fundamentalmode loss of approximately 1.3%. The gain requiredfor overcoming the losses to the second-order modewas 5.1 1corresponding to a loss of greater than 80%2.For comparison, a stable Fabry–Perot cavity with thesame cavity length, beam size, and fundamental-mode loss has a second-order modal gain of only 1.08,corresponding to a loss of just 7.2%.Figure 161b2 shows the second-order modal gain as a

function of output beam spot size for this cavity. If amodal gain of 1.33 is sufficient to discriminate againstthe second-order mode 1corresponding to a loss of25%2, beam diameters of up to 8.5 mm can be used inthis 10-cm-long cavity. It is therefore possible toextract power from a wide gain medium while stillmaintaining a very small cavity length and smallfundamental-mode loss.

5. Conclusion

Diffractive optical elements were used as the endmirrors of an optical resonator. Because these ele-ments can produce any phase profile, many desirableproperties that are impossible with conventional opti-cal elements can be incorporated into the opticalresonator. We have shown that any real fundamen-tal-mode profile can be produced by a single diffrac-tive mirror and that complex-mode profiles are pos-sible with two diffractive mirrors. In addition, thecavity can be designed to have a negligible theoreticalfundamental-mode loss while simultaneously provid-ing high modal discrimination against higher-ordermodes. Finite phase-quantization andminimum fea-ture-size constraints were shown to affect both theshape of the fundamental mode and the loss. For themode profiles studied, the minimum feature size wasfound to be quite large, making fabrication of thediffractive elements relatively easy.The modal discrimination was enhanced when an

additional transparent phase plate was included inthe cavity. It was shown that there is an optimumfrequency, phase, and modulation depth for a sinusoi-

2508 APPLIED OPTICS @ Vol. 34, No. 14 @ 10 May 1995

dal phase grating used to discriminate between cavitymodes. The modal discrimination of pseudorandomphase plates, however, appeared to increase as thespatial frequency bandwidth of the phase plate wasincreased. The limitation on modal discriminationin this case was dictated by practical considerations.The higher-bandwidth plates require smaller DMSMfeature sizes and thus have a higher fundamental-mode loss for a given DMSM feature size.

This work was supported by the National ScienceFoundation under grant ECS-9109029-01.

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