Analysis and observation of finite beamBragg diffraction by a thick planar phase grating

Michael R. Wang

A numerical-impulse-response technique for studying the propagation and diffraction of finite-widthbeams in planar phase gratings is described. It can account for both symmetric and asymmetricdiffractions. The grating-length-to-beam-width ratio is shown to govern the extent of beam-profiledistortion and selectivity sidelobe suppression. Trade-offs between diffraction efficiency and beamprofile have also been demonstrated. Theoretical results have been verified by experimental observa-tions in a planar waveguide geometry of diffracted beams that change from a single diffraction peak tomultiple peaks as the grating-length-to-beam-width ratio increases.Key words: Grating, diffraction, finite beam. r 1996 Optical Society of America

1. Introduction

Bragg diffraction by a phase grating is recognized asan important scheme in the implementation of avariety of optical devices, including filters, wave-length division multiplexers and demultiplexers,couplers, opticalmemories, and optical interconnects.Important considerations in the design of thesedevices are not only wavelength dispersion, effi-ciency, cross talk, and fanout issues but also theeffect of finite beam diffraction when the laser beamsizes are comparable with, or smaller than, thegrating interaction length. The effects of finitebeam Bragg diffraction have been considered forthick planar gratings and symmetrical diffractiongeometries.1–5 A solution to asymmetric diffractionwith the capability of handling a variety of incidentbeam types is, however, necessary to account forvarious device geometries and application require-ments.Here I report on the results of a numerical study

and experimental observation of asymmetric finiteGaussian beam diffraction by a planar phase grating.An impulse-response technique is used in conjunc-tion with coupled-wave theory to study the depen-dence of diffraction efficiency, wavelength selectivity,and angular selectivity on the incident beam andgrating parameters. The grating-length-to-beam-

The author is with the Department of Electrical and ComputerEngineering, University of Miami, Coral Gables, Florida 33124.Received 7 February 1995; revised manuscript received 28

August 1995.0003-6935@96@040582-11$06.00@0r 1996 Optical Society of America

582 APPLIED OPTICS @ Vol. 35, No. 4 @ 1 February 1996

width ratio 1q 5 d@v02 is found to be an importantparameter in determining device performance. Inthe 1 , q , 5 regime, with d 5 300 µm, and a centerwavelength of 632.8 nm, several basic results arepredicted. They include a departure from the plane-wave behavior of 1sinc22 dependence for diffractionefficiency h with a corresponding modification in thesidelobe features, beam-profile distortion for a givend@v0 ratio and diffraction angles ad, a decrease in hwith increasing q and diffraction angle ad, andfinally an increase in the selectivity bandwidth atlonger center wavelengths. The analysis presentedhere can also be used to predict finite-beam contribu-tions for other center wavelengths, grating interac-tion lengths, and non-Gaussian incident beam pro-files.The approach used to analyze the finite-beam

Bragg diffraction by a planar phase grating for agiven d@v0 ratio is a twofold process. First, theplanar phase grating is partitioned into gratingsubsections, with each subsection length being muchsmaller than the incident-beam size. The incidentGaussian beam is likewise partitioned into a seriesof d-function inputs, each with different amplitudesand spatial locations. Second, one can apply theresults of plane-wave theory to each subsection bycalculating a weighted diffraction efficiency, basedon the spatial location of the propagating diffractedand undiffracted beams, for each impulse opticalinput. Finally, all the impulse responses aresummed to yield the resulting output beam profilesas a function of grating length, incident beam size,incident and diffracted angle orientation, and grat-ing modulation index. The numerical computation

technique is described in detail below. Calculationsdemonstrate that the beam profiles vary dramati-cally within the grating region. The trade-offs be-tween diffraction efficiency and beam profile are alsoanalyzed. Also considered are the effects of subsec-tion grating size to the computation error, the effectsof detuning from the Bragg condition, and the resul-tant angular and wavelength selectivities.

2. Numerical Solution with a Finite-ElementImpulse-Response Technique

The theory of two-dimensional plane-wave diffrac-tion by a transmission phase grating has beendescribed.6 Here I summarize several importantplane-wave diffraction results for general slantedtransmission gratings and then derive the requiredamplitude weighting factors for the finite-beamanalysis.

A. Review of Plane-Wave-Diffraction Theory

For the case of a monochromatic plane wave that isincident at an oblique angle on a lossless slantedphase grating bounded by two identical homoge-neous media with bulk indices equal to that of thegrating, as shown in Fig. 1, a grating can be de-scribed by a periodic index of refraction:

n 5 n0 1 Dn cos1K · r2 Dn 9 n0, 112

where n0 is the index of refraction of the unperturbedregion, Dn is the amplitude of index modulation, andK is the grating vector:

K 5 K sin fx 1 K cos fz, 122

where K 5 12p2@L 1L is the grating period2 and f isthe grating slant angle. For different surroundingbulk indices the Fresnel equation can be used totreat the interface refraction problem. For an inci-dent plane wave the propagation vector k can bewritten as

k 5 k0 sin ax 1 k0 cos az, 132

where k0 5 12pn02@l is the propagation constant inthe medium of index n0 and a is the angle of obliqueincidence. Here the total electric field in the grat-

Fig. 1. Model of a Bragg transmission phase grating withslanted grating fringes and an oblique incident angle.

ing region is just the superposition of incident anddiffracted plane waves6:

E 5 rR1z2exp12 jk · r2 1 sS1z2exp12 jkd · r2, 142

where R1z2 and S1z2 are position-dependent coeffi-cients that describe the coupling, or energy inter-change between the two propagating waves. Here rand s are the relevant beam unit polarization vec-tors, whereas kd is the propagation vector of thediffracted wave. The phase-matching condition forthe diffraction process is given by

kd 5 k 6 K, 152

whereas the diffracted beam angle ad is given by

tan ad 5kdxkdz

. 162

Because E is a solution to the vector wave equa-tion, R1z2 and S1z2 can be determined and are given by

R1z2 5 exp12 q

2Cdz2cos312 1

q2

Cd2

14k8

2

CiCd21@2

z4Ein

1jq

Cd1 q2

Cd2

14k8

2

CiCd21@2

exp12 q

2Cdz2

3 sin312 1q2

Cd2

14k8

2

CiCd21@2

z4Ein, 172

S1z2 52 j2k8

Cd1 q2

Cd2

14k8

2

CiCd21@2

exp12 q

2Cdz2

3 sin312 1q2

Cd2

14k8

2

CiCd21@2

4Ein, 182

where

k8 5 5k for s-polarized wave

k1k · kd2 for p-polarized wave, 192

and

k 5pDn

l110a2

Ci 5kzk0

5 cos a, 110b2

Cd 5kdzk0

5 cos ad. 110c2

Here an s-polarized wave defines an incident beamwhose polarization is perpendicular to the incidentplane, whereas the p-polarized wave has a polariza-tion parallel to the plane. Ci and Cd are the cosines

1 February 1996 @ Vol. 35, No. 4 @ APPLIED OPTICS 583

of incident and diffracted angles a and ad, respec-tively, whereas dephasing constant q is expressed as

q 5k02 2 kd2

2k05 2

K2

4pn0l 6 K cos1f 2 a2

f 2 a , 90° 1.90°2 1112

and is used to characterize the response of R1z2 andS1z2 to slight variations in incident angle or beamwavelengths near the Bragg condition 1kd 5 k02. Einis the incident field amplitude for either s or ppolarization. For the oblique angle-of-incidencecase, conservation of energy6 requires that

0Ci 0 0R1z2 02 1 0Cd 0 0S1z2 02 5 0Ci 0 0Ein 02. 1122

From Eq. 1122 it is convenient to define amplitudeweighting factors Q1u and Q1d, which are the undif-fracted and diffracted beam amplitudes, normalizedto the incident-beam amplitude. For a given grat-ing interaction length z, these factors can be ex-pressed as

Q1u1z2 5R1z2

Ein

, 1132

Q1d1z2 5 1Cd

Ci21@2 S1z2

Ein

. 1142

Therefore Eq. 1122 can be simplified to

0Q1u1z2 02 1 0Q1d1z2 02 5 1. 1152

The diffraction efficiency of the plane wave is thenwritten simply as

h 5 0Q1d1z2 02. 1162

The above expressions are valid only in the plane-wave diffraction case in which there is no energyvariation in the x direction. In the finite-beam-diffraction case in which there are beam-profilevariations along the x direction, x-dependent coeffi-cientsR1x, z2 and S1x, z2must be used in Eq. 142. Thistechnique has been used previously to obtain ananalytical solution of the finite-beam-diffraction prob-lem under symmetric diffraction 1ad 5 2a2 geometry.4Here we introduce a finite-element numerical tech-nique to analyze the finite-beam-diffraction problemin a general asymmetric geometry 1ad fi 2a2. Thesymmetric diffraction 1ad 5 2a2 is only a special caseof the numerical solution presented here, in whichthe numerical results have been shown to match theanalytical solutions4 well. The numerical solutionof the finite-beam Bragg diffraction is obtained bypartitioning the phase grating into many smallsubsections, so that the grating subsection interac-tion length is considered to be small compared withthe spatial variation in the incident beam profile.Therefore a plane-wave analysis can be used in partto treat each subsection grating diffraction, thereby

584 APPLIED OPTICS @ Vol. 35, No. 4 @ 1 February 1996

avoiding the use of x-dependent coefficients. Theseeffects are in fact accounted for in the spatiallyresolved beam propagation and convolution analysisdescribed below.Note that in the Bragg regime all higher diffrac-

tion orders other than the first order can be ignored,provided that the following equation is satisfied7:

j 52pld0n0L2

5 10 1172

with d . d0. Here j is a dimensionless parameterand d0 is the minimum grating interaction lengththat satisfies the Bragg diffraction condition. Thiscondition restricts the values of d, L, and n0 that canbe used in the subsection grating partition analysis.Even though higher-order diffractions might existfor some special cases in which d is close to d0 with jlarger than 10, we can still ignore higher diffractionorders, because the analysis of the finite beamdiffraction is for Bragg diffraction with a long grat-ing interaction length 1much longer than d02, and thepartition of gratings does nothing more than in-crease computation accuracy.

B. Finite-Element Algorithm for Partitioned Gratings

The idea of the finite-element numerical technique isto divide the planar grating 1with d . d02 into mul-tiple-grating subsections, with each subsection hav-ing a grating interaction length Dd such that theprojection of Dd onto the incident-beam-profile axis,z 5 Dd 0tan a 2 tan ad 0cos a, is much smaller thanthe laser beam spot size v0. For example, Dd 53 µm, a 5 0°, and ad 5 30° produce z 5 1.73 µm,which is only ,6.92% of an incident Gaussian beamwidth of v0 5 25 µm. The calculations in Section 4indicate that a ratio z@v0 of less than 10% can resultin less than a 0.2% computation error. Hence theuse of Dd 5 3 µm or smaller for larger ad angles andv0 5 25 µm or larger can result in excellent computa-tional accuracy. Note that small beam widths mayeventually result in a large beam divergence angle,owing to the Gaussian-beam propagation properties.The divergence angle is generally small near thebeam waist. The choice of the minimum beamwidth for computation must ensure that the beam-divergence effect can be ignored within the grating-interaction region.After the grating partition the plane-wave results

are applied to analyze the diffraction from eachgrating subsection. This is done by calculating theamplitudes and efficiencies of the diffracted andundiffracted rays in one subsection and using thoseas the inputs for the next subsection. In effect theinputs for each subsection are weighted by theprevious subsection outputs.The diffraction process from a given grating subsec-

tion is shown in Fig. 2. As indicated, the use ofamplitude weighting factors in the grating subsec-tion can simplify the process of determining theresulting diffracted and undiffracted beam ampli-

tudes. Equations 172, 182, and 1122–1162 are usedwhereincident angle a is shown as in Fig. 21a2. Forincident angles ad as shown in Fig. 21b2, where theincident and diffracted beams are interchanged, anew set of weighting factors can be defined. Herefor two grating subsections the diffracted beam fromthe first section becomes the incident beam for thesecond section. The same is true for the undif-fracted beam. Weighting factors Q2u1z2 and Q2d1z2,valid for waves with incident propagation 3shown inFig. 21b24, are then given as

Q2u1z2 5 exp12 q

2Cd2cos312 1

q2

Cd2

14k8

2

CiCd21@2

z42

jq

Cd1 q2

Cd2

14k8

2

CiCd21@2

exp12 q

2Cd2

q

2Cdz2

3 sin312 1q2

Cd2

14k8

2

CiCd21@2

z4 , 1182

Q2d1z2 52 j2k8

1CiCd21@21 q2

Cd2

14k8

2

CiCd21@2

exp12 q

2Cdz2

3 sin312 1q2

Cd2

14k8

2

CiCd21@2

z4 . 1192

Fig. 2. Definition for weighting factors 1a2Q1u andQ1d at incidentangle a and 1b2 Q2u and Q2d at incident angle ad, 1c2 origin of thetotal diffracted and undiffracted beam amplitudes.

It can be seen that term Q2u1z2 differs from Q1u1z2 byonly a sign in the second term, whereas Q2d1z2 is thesame asQ1d1z2. The difference comes from the physi-cal interchange of the incident and diffracted angles,whereas the definitions for a and ad, as shown in Fig.21a2, and the factors Ci, Cd, and q remain unchanged.For plane-wave diffraction in a grating of interac-

tion length d1 1 d2, we require that the diffractionfrom a grating of length d1 1 d2 equal that ofsuccessive diffractions from gratings of length d1 andthen d2, as shown in Fig. 21c2. The following ampli-tude relationships are expected:

Q1u1d1 1 d22 5 Q1u1d12Q1u1d22 1 Q1d1d12Q2d1d22,

120a2

Q1d1d1 1 d22 5 Q1d1d12Q2u1d22 1 Q1u1d12Q1d1d22.

120b2

These relationships can be confirmed by Eqs. 1132,1142, 1182, and 1192. The results from Fig. 21c2 suggestthat, in the quasi-plane-wave-diffraction case, it ispossible to divide the grating into many subsections,with the diffraction from each subsection calculatedwith the plane-wave theory and the total gratingdiffraction considered as the cumulative effect fromall these subsections. The subgrating length Ddshould be larger than value d0 set by Eq. 1172 tosatisfy the Bragg condition and therefore to suppresshigher diffractive orders, even in subsection gratingdiffractions.The use of plane-wave theory to treat the finite-

beam-diffraction case, as presented above, requiresthat 1a2 the variation of the transverse incident beamprofile be slow compared with spatial shift z causedby the propagation of undiffracted and diffractedbeams in the subgrating region and along the inci-dent-beam-distribution axis, and 1b2 the phase-matching condition of Eq. 152 is satisfied. Hence, thesmaller the subgrating interaction length or diffrac-tion angle, the better the accuracy in the plane-waveapproximation to the finite-beam case. Note, how-ever, that subgrating interaction length Dd mustalso satisfy the Bragg criterion as given in Eq. 1172.This limits the lower boundary of Dd to d0.The Bragg condition can be satisfied for each

grating subsection of length Dd as long as Dd islarger than d0. From Eq. 1172 we find that, for f 590°, n 5 1.5, and l 5 632.8 nm, a value of d and Ddlarger than d0 3with d0 , 6 µm 1when the Bragg angleis ,10°2 or d0 , 2.3 µm 1when the Bragg angle is.15°24 easily satisfies this requirement. Hence Iconsider only the large-angle-diffraction problemusing transmission phase gratings. The small-angle diffraction problem 1Bragg angle, #10°2 hasbeen investigated previously8 with other numericaltechniques.To describe fully the process of diffraction for each

grating subsection, we begin with a single ray inci-dent on the Bragg grating, as shown in Fig. 3. After

1 February 1996 @ Vol. 35, No. 4 @ APPLIED OPTICS 585

Fig. 3. Model of the impulse-response technique for the finite beam Bragg grating diffraction. The impulse-response amplitudefunction is obtained by successively weighting the diffracted and undiffracted beams at each grating subsection.

successive diffraction by multiple-grating subsec-tions, a set of diffracted rays in the kd propagationdirection and a set of undiffracted rays in the kpropagation direction are obtained. The change inpropagation direction and beam amplitude is consid-ered at each interval in the diffraction process. Atany diffraction point P the incident rays can becharacterized by field amplitude E11x, z2 propagatingin the k direction and E21x, z2 propagating in the kddirection. Originating from a single incident beam,E1 and E2 are coherently coupled inside the gratingand can therefore use the coupling equations derivedabove. It is assumed that s- and p-polarized wavesdo not couple in the grating region and can be treatedseparately, based on Eqs. 172, 182, 1182, and 1192.After diffraction at point P the new E1 beam can be

represented by the summation of beams generatedby the undiffracted portion of E11x, z2 and the dif-fracted portion of E21x, z2, which propagate in phasefrom the coherent Bragg coupling. Similarly, thenew E2 can be obtained by the summation of beamsgenerated by the diffracted portion of E11x, z2 and theundiffracted portion of E21x, z2. After diffraction thenew E1 propagates from position 1x, z2 to 1x 1 Dx1,z 1 Dz2, whereas the new E2 propagates from posi-tion 1x, z2 to 1x 1 Dx2, z 1 Dz2 for the next subsequentsubsection grating diffraction. Each beam diffrac-tion is then weighted by the coefficients in Eqs. 1132,1142, 1182, and 1192. Therefore we have

E11x1 Dx1, z1 Dz25Q1u1Dz2E11x, z21Q2d1Dz2E21x, z2,

121a2

E21x1 Dx2, z1 Dz25Q1d1Dz2E11x, z21Q2u1Dz2E21x, z2.

121b2

586 APPLIED OPTICS @ Vol. 35, No. 4 @ 1 February 1996

k8 is chosen in these equations for s- or p-polarizedincident beams.A set of spatially separated diffracted and undif-

fracted beams from all grating subsections resultswhen an impulse-response transfer function of thegrating is applied to an incident ray of unit ampli-tude, as shown in Fig. 3. Here we letG1x82 representthe spatially digitized impulse-response function forbeams with wave vector k, where x8 is the axis alongwhich the incident beam profile f 1x82 is defined.H1x92 represents a similar response function for abeamwithwave vectorkd, where x9 is the correspond-ing axis perpendicular to kd. Finally, h1x92 and g1x82are the digitized representations for diffracted andundiffracted beam profiles, respectively, which areobtained by convolving the impulse-response func-tions H1x92 and G1x82 with the corresponding spatiallydigitized representation F1x82 for the incident beamprofile f 1x82. This process is illustrated graphicallyin Fig. 4. The relationships between the coordi-nates 1x8, z82, 1x9, z92, and 1x, z2 are given in transforma-tion equations

1x8z82 5 1cos a 2sin a

sin a cos a 21xz2 , 1222

1x9z92 5 1cos ad 2sin ad

sin ad cos ad21xz2 . 1232

To digitize the incident beam profile for the convo-lution calculation, spacing Dx8, equal to z when Ddequals Dz, is chosen, as shown in Fig. 3, to satisfy

Dx8 5 0Dx1 2 Dx2 0cos a, 1242

where

Dx1 5 Dz tan a, 125a2

Dx2 5 Dz tan ad. 125b2

The undiffracted beam g1x82 in the x8–z8 coordinateuses the digitized spacing Dx8, whereas diffractedbeam h1x92, in the x9–z9 coordinate system, uses adigitized spacing of

Dx9 5 0Dx1 2 Dx2 0cos ad. 1262

The spacings are different because the same0Dx1 2 Dx1 0 is projected onto different x8 and x9 axes,respectively. The location of the vertical axis z9 ischosen to pass through the origin in the x–z coordi-nate for a direct comparison of the incident anddiffracted beam profiles.With these axis orientations and digitized spac-

ings, the digitized amplitude profiles for h1x92 andg1x82 can be expressed by the convolution notationsh3nDx94 and g3nDx84, respectively. The convolutioncalculation is then given by

h3nDx94 5 om52`

`

F 3mDx84H 31n2 m2Dx94, 1272

g3nDx84 5 om52`

`

F 3mDx84G31n2 m2Dx84, 1282

where m and n are integer values for each digitized

Fig. 4. Convolution process for determining diffracted and undif-fracted beam profiles: 1a2 Incident Gaussian beam, 1b2 convolu-tion process of the digitized incident beam F1x82 with the impulse-response function H1x92 of the grating, and 1c2 resultant digitizeddiffracted beam are illustrated schematically as an example.

input and F 3mDx84 is the digitized profile for theincident laser beam given by

F 3mDx84 5 f 1x82 0 x85mDx8. 1292

The above digitizations and convolution calculationsapply to any incident beam profiles as long as thefinite-beam-diffraction criteria set above are met.For Gaussian-beam incidence the beam amplitudeprofile is represented as

f 1x82 5 exp12 x82

v022 , 1302

where v0 is the spot size of the incident laser beam.The corresponding intensity distributions are propor-tional to the square of these profiles.

3. Behavior of Gaussian Beams inside theGrating Region

To examine the behavior of a Gaussian incidentbeam in the periodically index-modulated gratingmedium, a set of intensity profiles for undiffractedand diffracted beams was computed at differentgrating interaction lengths. The following param-eters were used for the calculation: bulk index,n0 5 1.5; incident-beam wavelength, l 5 0.6328 µm;Gaussian-incident-beam size, v0 5 25 µm; incidentangle, a 5 0°; grating slanted angle, f 5 105°;diffracted beam angle, ad 5 30°; and the index ofmodulation, Dn 5 0.002. The undiffracted and dif-fracted beam profiles are examined after diffractionand propagation in the grating for d 5 50 µm, and100–700 µm, in 100-µm increments. This is plottedin Figs. 51a2 and 51b2, respectively. Note that all thebeam-profile plots are represented after conversionto the proper viewing coordinate system, as specifiedin Fig. 3. Within the 700-µm region the beam-divergence angle is less than 0.05° and thus thebeam-divergence effect can be ignored.The primary results in Fig. 5 indicate distortion of

the original beam profile as the beam propagatesthrough the grating. In Fig. 5 we see that theundiffracted beam exhibits a double peak with moreenergy shifted toward the peak closer to the dif-fracted beams as d increases. The diffracted beamshows significantly greater changes in beam profile,being transformed successively from a single peak toa double peak and to triple peaks as d increases.In this case the profiles are shifted toward theundiffracted beam. As d increases further 1withinthe allowed range where the beam-divergence effectis ignored2, more intensity peaks become visible 1notshown2. Note that the diffracted beam profiles forall cases maintain their symmetry with respect totheir corresponding axes located at the center of theprofile. For the a 5 0° incident case the diffractedbeam profiles are bound within a region defined bythe coordinates

1z 2 d2tan ad 2 v0 # x # z tan ad 1 v0, 1312

1 February 1996 @ Vol. 35, No. 4 @ APPLIED OPTICS 587

whereas the undiffracted beam is bound by

2v0 # x # d tan ad 1 v0. 1322

Note that the diffracted single peak, which isnearly Gaussian in nature, occurs at smaller dvalues than other multiple-peak cases with a fixedindex of modulation. Thus it is possible to avoidmultiple peaks so that the near-Gaussian diffractedbeam can propagate without significant beam distor-tion and divergence. When the index of modulationDn is increased, similar beam profiles occur forsmaller d values with the product dDn and q factorremaining unchanged. These results are predictedfrom the k8d term in Eqs. 172, 182, 1182, and 1192. Thefull width at half-maximum diffracted and undif-fracted beams, however, change, given their addi-tional geometric dependence on parameter d. Thecontinuous energy interchange between diffractedand undiffracted beams through the grating subsec-tion diffraction process, as depicted in Fig. 3, resultsin an energy redistribution and is believed to be thecause of the Gaussian-beam-profile distortion.

4. Trade-off between Diffracted Beam Profile andDiffraction Efficiency

It is clear from the above discussion that gratinginteraction length d affects both the resulting diffrac-

1a2

1b2

Fig. 5. Distortion of 1a2 undiffracted and 1b2 diffracted beamprofiles as the beams propagate through the grating. For thecalculation the following parameters are used: n0 5 1.5, l 5

632.8 nm, v0 5 25 µm, a 5 0°, ad 5 30°, and Dn 5 0.001.

588 APPLIED OPTICS @ Vol. 35, No. 4 @ 1 February 1996

tion efficiency and the diffracted and undiffractedbeam profiles. A trade-off therefore exists betweenoptimized efficiency and preservation of the Gauss-ian beam profile after diffraction.Based on Eqs. 1132, 1142, 1182, and 1192, and finite

Gaussian-beam-diffraction calculations, diffractionefficiency h is a function of parameter q 5 1d@v02,incident angle a, diffracted angle ad, and normalizedparameter p 5 1Dnd2@l. Diffraction efficiency h canbe obtained by dividing the diffracted beam energyby the incident beam energy and expressed, inconvolution notation, as

h 5

on52`

`

0h3nDx94 02

om52`

`

F 3mDx842

. 1332

In the asymmetric diffraction case 1ad fi 2a2 thediffraction efficiency no longer depends only on thegrating strength and the geometric parameters.4The dependence on parameters such as a, ad, q, andp in our case makes the system design more compli-cated.Figure 6 shows plots of diffraction efficiency versus

parameter p for various diffraction angles ad. Theincident angle a is set at 0°. In Fig. 6, q is set to 1,

1a2

1b2

Fig. 6. Grating diffraction efficiency as a function of parameter pat various diffraction angles ad for a 5 0° and 1a2 q 5 1 and 1b2 q 5

5. The highest efficiency achievable for single-peak diffraction isidentified by dashed curves.

whereas in Fig. 61b2 it is set to 5. Here the diffrac-tion efficiency decreases with increasing q for fixed p.This trend shows that the finite-beam effect hasbecome significant at large q values. In Fig. 6significant finite-beam effects can also be identifiedat larger diffraction angles ad.The finite-beam effect at large q and ad not only

lowers the diffraction efficiency but also introducesmultiple diffraction peaks, which result in non-Gaussian diffracted beam propagation and beamspreading. In a practical application this couldresult in significant cross talk. Tomaintain a singlediffraction peak that is nearly Gaussian in profile,smaller p values must be used. The lower dashedcurves in Fig. 6 show the boundaries that differenti-ate the single and double diffraction peaks for largead angles. Single peaks are located to the left of thedemarcation lines. Clearly there is a trade-off indiffraction efficiency and diffraction line shape. Forsmaller ad angles the effect of multiple diffractionpeaks may never occur or occur at large p values.The upper dashed curves, which identify the effi-ciency peaks at small p and smaller angles ad, can beseen in both Figs. 61a2 and 61b2. Hence the highestdiffraction efficiencies with single-peak Gaussiandiffraction profiles can be achieved with reasonablysmall p values.In considering the aforementioned single-peak

limitation, I show in Fig. 7 the highest diffractionefficiencies achievable for single-peak diffraction atvarious diffraction angles. Curves are plotted for qvalues of 1, 3, and 5. A large q lowers the diffractionefficiency and introduces significant finite-beam ef-fects. In all three cases the bending and lowering ofthe efficiency curves at efficiencies of i60% aredominated primarily by the increase in diffractionangle ad. For h u 60% multiple-peak diffractioneffects dominate. The earlier cutoff at larger qvalues implies that to achieve high efficiency andsingle-peak behavior at large ad, the incident beamwidth v0 should not be much smaller than gratinginteraction length d. However, larger diffractionangles are expected to provide better angular andwavelength selectivities. Therefore, to achieve high

Fig. 7. Optimized single-peak diffraction efficiency at various qand ad values with a 5 0°. A smaller ad cutoff for larger qdiffraction is observed.

diffraction efficiencies at large ad, q should assume avalue of u1.Based on the computational method used to de-

rive these results, it is found that the smaller thegrating subsection Dd the smaller the value of z 5Dd 0tan a 2 tan ad 0cos a. This provides better accu-racy but longer computation time. Figure 8 showsthe error in calculated efficiency for a givenGaussian-beam-diffraction case using different z, q, and advalues. Given an incident beam size, v0 5 100 µm,incident angle a - 0°, grating interaction length d 5300 µm 1for q 5 32 or d 5 100 µm 1for q 5 12, errors of,0.2% are obtained when z is less than 10% of v0.The ideal curves are plotted, assuming z 5 0.025v0.Although the calculations are performed at specificv0 and d values, the results are applicable to other v0and d values.

5. Dephasing Effects and Angular and WavelengthSelectivity

Angular and wavelength selectivities are importantfigures of merit for phase grating-based devices.Finite beam diffraction with significant changes indiffracted and undiffracted beam energy distribu-tions is expected to affect the selectivity features.Here the angular and wavelength selectivities areexamined for various parametric conditions.For a fixed incident angle a, diffracted angle ad,

and center wavelength l0 the wavelength selectivi-ties can be calculated through the wavelength-dependent changes in the dephasing constant q.The equations developed in Section 2 indicate astrong dependence of dephasing and wavelengthselectivity on d, ad, l0, and q.Figure 9 shows a wavelength-selectivity plot for

d 5 300 µm, q 5 3, a 5 0°, ad 5 30°, and l0 5 632.8nm 1solid curve2. A similar calculation at ad 5 45° isshown as the dashed curve in Fig. 9. Several dis-tinct features appear that are not predicted by theinfinite-plane-wave Bragg diffraction theory. First,at smaller diffraction angles ad, the wavelength-selectivity response curve approaches the 1sinc22 func-tional dependence as in the plane-wave case. As adincreases the overlap between the diffracted andincident waves becomes smaller and the finite-beameffect becomes more pronounced. As a result thepeak efficiency decreases, as shown in Fig. 7, whereas

Fig. 8. Efficiency computation errors as a function of z@v0.

1 February 1996 @ Vol. 35, No. 4 @ APPLIED OPTICS 589

the sidelobe maxima decrease and the minima in-crease. As ad increases further the sidelobe fea-tures disappear and the selectivity curve appears asa smooth single-lobe response, as shown by thedashed curve in Fig. 9.A similar trend is seen in Fig. 10 for ad 5 30° but

different q values. As q approaches 0 the diffrac-tion features approach their plane-wave counter-parts. Thus as q increases the sidelobe featuresdecrease, similar to the results of Fig. 9 for varyingad. The disappearance of the sidelobe features hasbeen found to contribute to the slight broadening inthe selectivity bandwidth compared with the plane-wave case.The dependence of thewavelength-selectivity band-

width 1full width at half-maximum2 on the gratinginteraction length d and diffraction angle ad issimilar to that of the plane-wave case. As thegrating interaction length d increases or the diffrac-tion angle ad increases, the selectivity bandwidthdecreases. In Fig. 11 I summarize the dependenceof the wavelength-selectivity bandwidth on diffrac-tion angle ad, q, and d. For 1 , q , 5 the wave-length-selectivity bandwidth is not dramatically al-

Fig. 9. Wavelength-selectivity curves with peak, bandwidth, andsidelobe features varying with diffraction angle ad. For thecalculation we set d 5 300 µm, q 5 3, a 5 0°, and l0 5 632.8 nm;Dn was adjusted to obtain maximum achievable diffraction effi-ciency.

Fig. 10. Wavelength-selectivity dependence on the grating-length-to-beam-width ratio q. As q approaches 0 the responseapproaches a plane-wave 1sinc22 dependence. The calculationuses d 5 300 µm, a 5 0°, ad 5 30°, and l0 5 632.8 nm; Dn wasagain adjusted for maximum diffraction efficiency.

590 APPLIED OPTICS @ Vol. 35, No. 4 @ 1 February 1996

tered for large d. For ad * 30° the bandwidth doesnot change significantly, because, although the band-width should broaden slightly with the disappear-ance of distinct sidelobes, it simultaneously de-creases with increasing ad. From calculated resultsa wavelength-selectivity bandwidth of less than 1nm can be achieved for d larger than 1.8 mm and adlarger than 30°.The wavelength-selectivity bandwidth is also ex-

pected to vary, based on the choice of center wave-length l0. For a fixed incident angle a and dif-fracted angle ad, Ci and Cd from Eq. 1102 remainunchanged. Therefore the wavelength-dependentparameters are k 5 1pDn2@l0 or k8, from Eq. 192, and q.However, k8d is kept constant by adjusting Dn fordifferent center wavelengths l0 to keep the peakefficiencies identical. Therefore the remaining l-de-pendent parameter is q, the dephasing constant.From Eq. 1112, q 5 0 at l 5 l0 implies that

K 5 64pn0 cos1f 2 a2

l0

, 1342

Hence we have

q 5 2K2

4pn01l 2 l02 1352

or

q 5 24pn0 cos21f 2 a2

l02

Dl. 1362

Therefore, for q, which gives h 5 11@22hpeak, ratio1DlBW2@l0

2 is a constant, and we have

DlBW1

l012

5DlBW2

l022

5 · · · 5DlBWn

l0n2, 1372

A calculation at a 5 0°, ad 5 30°, d 5 300 µm, q 5 3and the center wavelengths l0 of 632.8, 850, 1300,and 1550 nm results in a plot of the center-wavelength-dependent bandwidth, as given in Fig.

Fig. 11. Wavelength-selectivity bandwidth as a function of q, d,and ad. A small-selectivity bandwidth occurs at small q, large d,and large ad values; a 5 0° and l0 5 632.8 nm were used for thiscalculation.

12. The results show that the longer the centerwavelength, the poorer the selectivity at a fixed d, q,and ad. To improve the wavelength selectivity atlonger wavelengths, such as l0 5 1.3 or 1.55 µm,grating interaction length d or diffraction angle admust be scaled accordingly.Figure 13 shows calculated angular-selectivity

curves at a 5 0°, d 5 300 µm, ad 5 30°, and l0 5632.8 nm at different q values. The bandwidth andsidelobe features are similar to the wavelength-selectivity plots of Fig. 10 because they are bothderived through dephasing factor q. Angular-selectivity bandwidth Da is less than 0.2° in thiscase.

6. Experimental Observations of Finite BeamDiffraction

Finite beam diffraction by a planar phase grating, asdescribed above, results in a number of features thatare different from the plane-wave-diffraction caseincluding a beam-profile distortion for both dif-fracted and undiffracted beams, a change in theselectivity bandwidth and sidelobe features, and adecrease in diffraction efficiency. Experimental ob-servations of the finite beam diffraction with doubleintensity peaks for both diffracted and undiffractedbeams have been reported previously by a symmetricdiffraction geometry 1ad 5 2a2 and at a 633-nm laserwavelength.9 Here the experimental observation isreported in planar waveguide geometry with visiblelaser wavelengths and under asymmetric diffractiongeometry 1ad fi 2a2.Optical beam diffraction in a planar waveguide

geometry, as shown in Fig. 14, has the advantage ofachieving long grating interaction lengths becausethe grating interaction length is independent of thegrating layer thickness. Instead it depends on thedimension of the predefined grating recording region.Because of the long grating interaction length, andthus excellent angular and wavelength selectivities,a number of waveguide devices including fanoutsand wavelength division multiplexers and demulti-plexers have been demonstrated.10 Also, because ofthe long grating interaction length, the features ofthe finite beam diffraction are expected to be easilyobserved in the waveguide geometry.

Fig. 12. Center wavelength-dependent selectivity bandwidth.

To observe the finite beam diffraction, we firstfabricated a polymer microstructure waveguide byspin coating a thin layer 1,3 µm2 of photolime gelatinon top of a glass substrate. After waveguide indexprofile tuning and film hardening,10 the waveguidewas found to support a single-mode propagation at543.0-, 594.1-, 611.9-, and 632.8-nm wavelengths.Followed by local sensitization, four holographicgratings were selectively defined within the sensi-tized waveguide region with each grating selectivelydeflecting, in the Bragg condition, one laser wave-length in the waveguide plane. The resulting four-channel waveguide-wavelength-division demulti-plexer is shown in Fig. 15. The grating interactionlength for this device is ,0.8 mm, whereas the laserbeam width at the grating region is ,110 µm. Fordiffraction the incident guided beams of polariza-tions in the waveguide plane were excited by prismcoupling into the polymermicrostructure waveguide.For guided beams of 543.0-nm 1green2 and 594.1-nm1yellow2wavelengths, incident angles a to the gratingwere 0° whereas diffraction angles ad were 20° and30°, respectively. There were no observable changesin the diffracted beam profiles. However, for guidedbeams of 611.9-nm 1orange2 and 632.8-nm 1red2 wave-

Fig. 13. Angular selectivity as a function of the grating-length-to-beam-width ratio q. For the calculation, parameters d5 300 µm,a 5 0°, ad 5 30°, and l0 5 632.8 nm were used.

Fig. 14. Schematic of waveguide diffraction geometry for finite-beam-diffraction observation and for the realization of a wave-guide-wavelength-division demultiplexer.

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lengths under the same incident angle, the corre-sponding diffracted beam angles were 40° and 50°,respectively. These diffracted beamswere widened,and two intensity peaks were clearly observable,especially for the red beam at the 50° diffractionangle. The observation of the multiple intensitypeaks for the diffracted beam is in agreement withthe finite-beam-diffraction phenomena describedabove. Note that the waveguide geometry intro-duces nothing more than a modified coupling con-stant10,11:

k 52p2

l02 e

2`

`

n01 y2Dn1 y2 0E1 y2 02dy, 1382

where n01 y2 is the waveguide index distribution,Dn1 y2 is the grating index distribution in the wave-guide depth direction, and E1 y2 is the fundamentalguided mode field distribution. Thus this experi-ment is a qualitative means of verifying the beam-distortion phenomena of the finite beam Bragg dif-fraction.

7. Conclusions

The effects of finite-beam size on the grating diffrac-tion process with relatively long grating interactionlengths have been investigated. The grating-length-to-beam-width ratio has been found to be an impor-tant parameter in the analysis of Gaussian-beampropagation, diffraction efficiency, and angular@wavelength-selectivity bandwidth. The techniqueis useful for treating both symmetric and asymmet-

Fig. 15. Waveguide-wavelength-division-demultiplexer deviceshowing diffracted-beam-profile broadening for the beams at 40°and 50° diffraction angles and two intensity peaks for the beam atthe 50° angle.

592 APPLIED OPTICS @ Vol. 35, No. 4 @ 1 February 1996

ric diffraction geometries. Results indicate that, forlarge q and ad values, finite-beam effects are signifi-cant and the diffracted beam may experience mul-tiple energy-distribution peaks as a result of strongcoupling inside the grating. A design trade-off alsoexists between diffraction efficiency and the need tomaintain a single diffracted beam profile for practi-cal applications, especially at large q and ad values.The spread and ultimate disappearance of sidelobefeatures should also be important in device design,especially in the design of waveguide-wavelength-division demultiplexers where minimal cross talk isrequired. The beam-profile distortion, being a fun-damental phenomenon of finite-beam diffraction,has been repeatedly observed in waveguide diffrac-tion geometry. The numerical technique describedhere should prove valuable for the design and optimi-zation of devices and systems for optical communica-tion, optical interconnection, and optical memoryapplications.

The experimental observation of finite-beam dif-fraction was performed when the author was withthe Physical Optics Corporation, Torrance, Calif.

References1. R. S. Chu and T. Tamir, ‘‘Bragg diffraction of Gaussian beams

by periodically modulated media,’’ J. Opt. Soc. Am. 66,220–226 119762.

2. R. S. Chu and T. Tamir, ‘‘Diffraction of Gaussian beams byperiodically modulated media for incidence close to a Braggangle,’’ J. Opt. Soc. Am. 66, 1438–1440 119762.

3. R. S. Chu, J. A. Kong, and T. Tamir, ‘‘Diffraction of Gaussianbeams by a periodically modulated layer,’’ J. Opt. Soc. Am. 67,1555–1561 119772.

4. M. G. Moharam, T. K. Gaylord, and R. Magnusson, ‘‘Braggdiffraction of finite beams by thick gratings,’’ J. Opt. Soc. Am.70, 300–304 119802.

5. D. H. McMahon, ‘‘Relative efficiency of optical Bragg diffrac-tion as a function of interaction geometry,’’ IEEE Trans.Sonics Ultrason. SU-16, 41–44 119692.

6. H. Kogelnik, ‘‘Coupled wave theory for thick hologram grat-ings,’’ Bell Syst. Tech. J. 48, 2909–2947 119692.

7. A. Yariv and P. Yeh, Optical Waves in Crystal 1Wiley, NewYork, 19842, p. 356.

8. D. Yevick and L. Thylen, ‘‘Analysis of gratings by the beam-propagation method,’’ J. Opt. Soc. Am. 72, 1084–1089 119822.

9. M. R. B. Forshaw, ‘‘Diffraction of a narrow laser beam by athick hologram: experimental results,’’ Opt. Commun. 12,279–281 119742.

10. R. T. Chen, M. R. Wang, G. J. Sonek, and T. Jannson, ‘‘Opticalinterconnection using polymer microstructure waveguides,’’Opt. Eng. 30, 622–628 119912.

11. R. P. Kenan, ‘‘Theory of diffraction of guided optical waves bythick holograms,’’ J. Appl. Phys. 46, 4545–4551 119752.