Off-Bragg analysis of the diffractionefficiency of transmission photorefractive holograms

Koutarou Nonaka

Formulas for calculating the diffraction efficiency of gratings recorded in a photorefractive medium aregiven. The analysis uses coupled-wave theory for photorefractive hologram gratings and takes intoaccount the photorefractive phase shift and fringe-bending effect. General solutions for diffracted ~sig-nal! and undiffracted ~reference! waves are derived in a closed-form expression. By use of the derivedformulas the diffraction efficiency for angle mismatch from the Bragg condition can easily be evaluatedas compared with numerical-analysis methods. The diffraction efficiency is also quantified in terms ofmedium parameters and recording and reconstruction conditions. © 1997 Optical Society of America

Key words: Holography, diffraction efficiency, photorefractive crystal, special functions.

1. Introduction

The photorefractive medium is widely used for opticalcomputing and optical data storage as an ideal holo-graphic medium.1 This is because it can record vol-ume holograms with high light sensitivity andwithout the need for a development process.

The diffraction efficiency of a photorefractive holo-gram grating has already been calculated by manyresearchers.2–5 Vahey2 derived the diffraction-efficiency formula for the on-Bragg condition for loss-less and lossy transmission-type holograms. Theresult successfully explained the experimental dataon temporal variation of the diffraction efficiency dur-ing grating formation obtained by Amodei et al.6Hong and Saxena3 obtained analytic expressions forthe diffraction efficiency of a weak reading beam thathad a polarization different from that of the recordingbeams. Maniloff and Johnson4 and Zhou et al.5 eval-uated the storage capacity and derived the diffractionefficiency for the on-Bragg condition for multiplexedphotorefractive holograms. However, all these stud-ies focused on the on-Bragg readout conditions. Thediffraction properties for off-Bragg conditions are im-portant for the evaluation of cross-talk noise, wave-

The author is with the Integrated Information and Energy Sys-tem Laboratories, Nippon Telephone and Telegraph ~NTT!, 3-9-11Midori-cho, Musashino-shi, Tokyo, 180 Japan.

Received 4 November 1996; revised manuscript received 13 Feb-ruary 1997.

0003-6935y97y204792-09$10.00y0© 1997 Optical Society of America

4792 APPLIED OPTICS y Vol. 36, No. 20 y 10 July 1997

length, and angular sensitivities for multiplexedholographic recording.

Off-Bragg analysis has already been reported byHofmeister et al.,7 Tao et al.,8 and Vre et al.9Hofmeister et al.7 calculated the diffraction efficiencyof a reflection photorefractive hologram; they did notconsider the transmission photorefractive hologram.Tao et al.8 calculated the diffraction efficiency oftransmission and reflection photorefractive holo-grams and measured angle selectivity; they did notpresent an analytical formula except for the specialcase of the zero photorefractive phase shift. Vre etal.9 reported the experimental and theoretical resultsof the diffraction efficiency for the off-Bragg condition.The fringe-bending effect caused by nonlinear two-wave mixing is also reported in their paper. How-ever, they similarly did not present an analyticalformula for calculating the diffraction efficiency.The analytical formula for calculating the diffractionefficiency for the off-Bragg condition is essential foreasily evaluating the cross-talk noise, wavelength,and angular sensitivities for multiplexed holographicrecording. Calculation of the diffraction efficiency byuse of an analytical formula is faster and easier thanis numerically solving the coupled-wave equations.Furthermore, the analytical formula directly ex-presses diffraction properties.

The purpose of this paper is to derive analyticalformulas for calculating the diffraction efficiency of atransmission photorefractive hologram under off-Bragg readout conditions and to obtain a closed-formexpression for the diffraction efficiency. I show gen-eral solutions for diffracted ~signal! and undiffracted

~reference! waves and discuss the diffraction proper-ties for off-Bragg conditions.

2. Analytical Model

A. Grating Recording Model

Let us use the transmission-type recording geometryshown in Fig. 1~a!. The z axis is set perpendicular tothe recording-medium surface. The medium has athickness T and absorption coefficient a for the lightintensity. The reference ~pump! beam Rw and theobject ~probe! beam Sw irradiate the medium at anincident angle u0, which is measured in the medium.In this recording geometry, the intensities of the two

Fig. 1. Schematic illustration of transmission-type recordinggeometry: ~a! recording geometry and ~b! intensity distribu-tions of the reference ~pump! and object ~probe! beams. In di-agram ~b! Rw shows the reference beam and Sw the object beam;Sw0 shows the intensity of the object beam at the medium sur-face, and Rw0 shows that of the reference beam. The absorptioncoefficient of the medium is a, and the coupling constant betweenthe two beams is g.

beams, Rw and Sw, within the volume of the mediumare given by10

Rw 5 Rw0

1 1 1yr1 1 exp~gz!yr

exp~2azycos u0!, (1a)

Sw 5 Sw0

1 1 r1 1 r exp~2gz!

exp~2azycos u0!, (1b)

where Rw0 is the intensity of the reference beam atthe medium surface and Sw0 is that of the objectbeam. These expressions, Rw and Sw, are illus-trated schematically in Fig. 1~b!. The modulation ofthe interference pattern m~z! is defined by

m~z! 52~RwSw!1y2

Rw 1 Sw. (2)

Using Eqs. ~1! and ~2! we obtain m~z! as follows:

m~z! 52

Îr exp~2gzy2! 1 exp~gzy2!yÎr, (3)

where r and g are given by

r 5Rw0

Sw0, (4)

g 52pDnS

l cos u0sin wg , (5)

where l is the wavelength of the reference and objectbeams, DnS is the modulation depth ~amplitude! ofthe refractive index during recording, and wg is thephase difference between the optical intensity gratingand the induced index grating. The term r is therecording-beam intensity ratio ~RS ratio!, and g is theintensity coupling constant between two beams dur-ing recording.

The phases of the object wS and the reference wavewR are given by10

wS~z! 5b

gln

1 1 r1 1 r exp~2gz!

, (6a)

wR~z! 5 2b

gln

1 1 1yr1 1 exp~gz!yr

, (6b)

where

b 5pDnS

l cos u0cos wg . (7)

The phase difference between wS~z! and wR~z! is

w~z! 5 wS~z! 2 wR~z!

5 cot wg lnÎr 1 1yÎr

Îr exp~2gzy2! 1 exp~gzy2!yÎr.

(8)

Equation ~8! represents the fringe-bending effect.We assume that the index grating is fixed within thephotorefractive medium.

10 July 1997 y Vol. 36, No. 20 y APPLIED OPTICS 4793

B. Reconstruction Model

The analytical model we use for reconstruction isshown in Fig. 2. The z axis is set perpendicular tothe medium surface; the x axis is set in the incidentplane parallel to the medium surface. The bent ho-lographic grating is aligned periodically within themedium with a period L. The medium is the sameas that for Fig. 1.

The refractive index n~x, z! in the medium is writ-ten by

n~x, z! 5 n0 1 1y2(DnLm~z! exp~iwg!

3 exp$i@K z r 1 w~z!#% 1 c.c.), (9)

where n0 is the average refractive index of the me-dium, DnL is the modulation depth ~amplitude! of therefractive index during reconstruction, m~z! is theindex-modulation depth given by Eq. ~3!, w~z! is thephase difference between the reference and objectbeams, c.c. is the complex conjugate of the precedingterm, and K is the grating vector ~magnitude of 2pyL!.The grating vector K is determined by the equationK 5 kRW 2 kSW, where kRW and kSW @see Fig. 1~a!# arethe wave vectors of the reference and signal waves,respectively, in the medium during recording. In thedetermination of K the fringe-bending effect is ignoredbecause this effect is represented by w~z!. The wave-length of the reference beam is l, and the Bragg angleis u0, which is also the incident angle of the referencebeam during recording.

We assume that the index grating is a sinusoidalshape, as stated in Eq. ~9!. However, a large modula-tion depth results in a nonsinusoidal shape of the re-fractive-index grating, so a high-order harmonic isobserved.11–13 In this case our analysis becomes invalid.

3. Derivation of the Formula

A. Derivation of the Diffraction-Efficiency Formula

The analysis method used is similar to that previ-ously reported by Nonaka.14 Using Eq. ~9! and Ko-

Fig. 2. Model of a hologram grating during reconstruction.

4794 APPLIED OPTICS y Vol. 36, No. 20 y 10 July 1997

gelnik’s theory,15 we obtain modified coupled-waveequations as follows:

cr

dRdz

1 a9R 5 FRS, (10a)

cs

dSdz

1 a9S 5 FSR, (10b)

where the coupling functions FR and FS are

FR 512

i exp~iwg!Gm~z!exp@iw~z!#exp~2iDkzz!, (10c)

FS 512

i exp~2iwg!Gm~z!exp@2iw~z!#exp~iDkzz!, (10d)

where cr and cs are the obliquity factors, a9 is theabsorption coefficient of the medium for the ampli-tude of the light beam, G is the amplitude couplingconstant, and Dkz is the z component of the wave-vector mismatch Dk. These are given by

G 5pDnL

l, (11a)

cr 5 cos u, (11b)

cs 5 cos u 2 2 cos~f 2 u!cos f, (11c)

a9 5 ay2, (11d)

Dkz 5 K@cos~f 1 Du! 2 cosf#. (11e)

The angle f is the grating-slant angle between grat-ing vector K ~magnitude of 2pyL! and the z axis.15

The grating vector K is determined by the equationK 5 kRW 2 kSW @see Fig. 1~a!#, and the tilt of thegrating8 ~a special case of bending of the grating!caused by the photorefractive effect is represented byw~z!. The wave-vector mismatch Dk is given by

Dk 5 kR 2 kS 2 K, (11f )

where kR and kS are the wave vectors of the referenceand the signal waves, respectively, in the mediumduring reconstruction. Equation ~11e! gives themismatch from the Bragg condition, where Du standsfor the angle mismatch from the Bragg angle u0.Therefore, as stated in Eqs. ~10a!–~10d!, the wavecoupling between the reference wave and the signalwave in the medium becomes weak as a result of thewave-vector mismatch Dk.

Differentiating Eq. ~10b! with respect to z andcombining it with Eq. ~10a!, we obtain the second-order differential equation with respect to the sig-

nal wave S:

d2Sdt2 1

1t~t 2 1! H~1 1 i cot wg!t

1 Fhr 1 hs 212

2 iS12

cot wg 1 DDGJ dSdt

11

t2~t 2 1!2 H2b2t2 1 ~b2 2 hs 1 ihs cot wg!t

1 Fhrhs 112

hs 2 ihsS12

cot wg 1 DDGJ S 5 0.

(12a)

Applying the same procedure to Eq. ~10a!, we obtainthe differential equation with respect to the referencewave R:

d2Rdt2 1

1t~t 2 1!

H~1 2 i cot wg!t

1 Fhr 1 hs 212

1 iS12

cot wg 1 DDG J dRdt

11

t2~t 2 1!2 H2b2t2 1 ~b2 2 hr 2 ihr cot wg!t

1 Fhrhs 112

hr 1 ihrS12

cot wg 1 DDG JR 5 0,

(12b)

where

t 51

1 1 @exp~gz!yr#, (13)

hr 5a9

crg, (14a)

hs 5a9

csg, (14b)

D 5Dkz

g, (15a)

b2 5G2

crcsg2 . (15b)

The general solutions of Eqs. ~12a! and ~12b! are

S~z! 5 S t1 2 tD

hs

@C1 2F1~nS , nS9, wS , t!

1C2t12wS

2F1~uS1ÎD, uS2ÎD, 22wS , t!#,(16a)

R~z! 5 S t1 2 tD

hr

@C3 2F1~nR , nR9, wR , t!

1C4t12wR

2F1~uR1ÎD, uR2ÎD, 22wR , t!#,(16b)

respectively, where 2F1~. . .! is the hypergeometricfunction, C1, C2, C3, and C4 are constants, and

uS 512

2 hs 1 hr 2 iD, (17a)

uR 512

2 hr 1 hs 1 iD, (17b)

wS 512

1 hs 2 hr 1 iS12

cot wg 1 DD, (17c)

wR 512

1 hr 2 hs 2 iS12

cot wg 1 DD, (17d)

nS 5 i12

cot wg 1 ÎD, (17e)

nS9 5 i12

cot wg 2 ÎD, (17f )

nR 5 2i12

cot wg 1 ÎD, (17g)

nR9 5 2i12

cot wg 2 ÎD, (17h)

D 5 b2 214

cot2wg . (18)

The constants C1, C2, C3, and C4 are determined bythe boundary condition

R~0! 5 1, S~0! 5 0, (19)

and Eqs. ~10a!–~10d!. The results are

C1 5 2iScr

csD1y2

b exp~2iwg!

3Îr

1 1 rt0

2hs~12t0!2~1y2!1hr1i@~1y2!cotwg2D#

~1y2!2hs1hr2i@~1y2!cot wg1D#

3 2F1~uS 1 ÎD, uS 2 ÎD, 2 2 wS , t0!, (20a)

C2 5 iScr

csD1y2

b exp~2iwg!Îr

1 1 r

3t0

2~1y2!2hr1i@~1y2!cotwg1D#~1 2 t0!2~1y2!1hr1i@~1y2!cotwg2D#

~1y2! 2 hs 1 hr 2 i@~1y2!cot wg 1 D#

3 2 F1~nS , nS9, wS , t0!, (20b)

10 July 1997 y Vol. 36, No. 20 y APPLIED OPTICS 4795

C3 5 2t0

12hr~1 2 t0!~1y2!1hs2i@~1y2!cotwg2D#

~1y2! 2 hr 1 hs 1 i@~1y2!cot wg 1 D#

3F12wR

t02F1~uR 1 ÎD, uR 2 ÎD, 2 2 wR , t0!

1uR

2 2 D2 2 wR

2 F1~uR 1 ÎD 1 1, uR

2 ÎD1 1, 3 2 wR , t0!G, (19a)

C4 5 2b2t0

~1y2!2hs2i@~1y2!cotwg1D#~1 2 t0!~1y2!1hs2i@~1y2!cotwg2D#

$~1y2! 2 hr 1 hs 1 i@~1y2!cot wg 1 D#%wR

3 2F1~nR 1 1, nR9 1 1, wR 1 1, t0!, (19b)

where

t0 51

1 1 1yr. (21)

The diffraction efficiency h of transmission hologramsis given by

h 5ucsucr

S~T!S*~T!, (22)

where the asterisk stands for the complex conjugate.

B. Special Cases of the Derived Formula

When wg 5 0, the phase difference between the ref-erence and object beams w~z! becomes

w~z! 5 w0z, (23a)

where

w0 5r 2 1r 1 1

pDns

l cos u0. (23b)

Equation ~23a! is the limiting value of Eq. ~8! when wgapproaches 0. The coupling functions FS and FRbecome

FS 5 iG1

Îr 1 1yÎrexp~2iw0z!exp~iDkzz!, (24a)

FR 5 iG1

Îr 1 1yÎrexp~iw0z!exp~2iDkzz!, (24b)

4796 APPLIED OPTICS y Vol. 36, No. 20 y 10 July 1997

respectively. The coupled-wave equations become

d2Sdz2 1 ~ar 1 as 1 id!

dSdz

1 ~aras 1 iasd 1 B2!S 5 0,

(25a)

d2Rdz2 1 ~ar 1 as 2 id!

dRdz

1 ~ar as 2 iard 1 B2!R 5 0,

(25b)

where

ar 5a9

cr, (26a)

as 5a9

cs, (26b)

d 5 w0 2 Dkz , (26c)

B2 5G2

crcs3

r~1 1 r!2 . (26d)

The general solutions S~z! and R~z! are

S~z! 5 iGÎr

cs~1 1 r!~p1 2 p2!@exp~p1z! 2 exp~p2 z!#,

(27a)

R~z! 5~q1 1 ar!exp~q2 z! 2 ~q2 1 ar!exp~q1z!

~q1 2 q2!, (27b)

respectively, where

p1 512

$2~ar 1 as 1 id! 1 @~ar 2 as 1 id!2 2 4B2#1y2%,

(28a)

p2 512

$2~ar 1 as 1 id! 2 @~ar 2 as 1 id!2 2 4B2#1y2%,

(28b)

q1 512

$2~ar 1 as 2 id! 1 @~as 2 ar 2 id!2 2 4B2#1y2%,

(28c)

q2 512

$2~ar 1 as 2 id! 2 @~as 2 ar 2 id!2 2 4B2#1y2%.

(28d)

The diffraction efficiency h is obtained from Eq. ~22!by replacement of S~T! with Eq. ~27a!. When r 5 1,our analysis coincides with Kogelnik’s theory. Fur-thermore, when a 5 0, Eqs. ~27a! and ~27b! become

S~z! 5GÎr

cs~1 1 r!

2$sin@~1y2!~d2 1 4B2!1y2z#% @2sin~1y2!dz 1 i cos~1y2!dz#

~d2 1 4B2!1y2 , (29a)

R~z! 5exp@i~1y2!dz# $~d2 1 4B2!1y2cos@~1y2!~d2 1 4B2!1y2z# 2 id sin@~1y2!~d2 1 4B2!1y2z#%

~d2 1 4B2!1y2 , (29b)

respectively. The diffraction efficiency h is then

h 54B2

d2 1 4B2 sin2F12

~d2 1 4B2!1y2TG. (30)

Equation ~30! is similar to the formula derived byHeaton et al.16 The diffraction efficiency has no di-mensional value. Although their formula has a di-mension of inverse meters, Eq. ~30! here has nodimension. Therefore, their formula seems to be inerror.

The next special case is when wg 5 90°. In thiscase, the coupling functions FS and FR are

FS 5 2G1

Îr exp~2gzy2! 1 exp~gzy2!yÎrexp~iDkzz!

(31a)

FR 5 1G1

Îr exp~2gzy2! 1 exp~gzy2!yÎrexp~2iDkzz!,

(31b)

respectively. The modulation function m~z! withinEqs. ~31!, that is, Eq. ~3!, is in the same form as Eq.~3! in Nonaka,14 when =r, gy2, and G are replacedwith r0, a, and 2G, respectively. Using Kogelniks’dephasing measure15 q instead of Dkz, we can find thediffraction-efficiency formula of this case, as reportedin Appendix A in Nonaka.14

Furthermore, when Dkz 5 0 and cr 5 cs 5 c, thesignal-wave function S~z! and reference-wave func-tion R~z! are

S~z! 5 exp~2a9zyc!sin (2G

cg$tan21@exp~gzy2!yÎr#

2 tan21~1yÎr!%), (32a)

R~z! 5 exp~2a9zyc!cos (2G

cg$tan21@exp~gzy2!yÎr#

2 tan21~1yÎr!%), (32b)

respectively. The diffraction efficiency h is S~T!S~T!,which has already been obtained by Hong and Sax-ena.3

4. Evaluation of the Derived Formula

A. Evaluation of Bent Holographic Gratings

The photorefractive phase w~z! of Eq. ~8! as a functionof Z is plotted in Fig. 3 for r~RS ratio! 5 100. Thephotorefractive phase shift wg is taken as a parame-ter. A new variable Z is defined by

Z 5gz

2 sin wg5

pDnS

l cos u0z. (33)

Therefore, w~z! becomes a function of only wg, Z, andr. When wg 5 0, w~z! is a linear function of Z, asstated in Eqs. ~23!. This means that the fringe sur-

faces tilt toward the signal beam.8 On the otherhand, when wg 5 90°, w~z! is a constant of zero: Thefringe surfaces do not tilt and do not bend. Withother values of wg, w~z! first increases and then de-creases with Z. The fringe-bending effect appears inthe curves.

B. Evaluation of the Diffraction Efficiency

The dependence of the diffraction efficiency h on theindex amplitude Dn ~5DnS 5 DnLy2! is shown in Fig.4 for r 5 100, with wg as a parameter. For wg 5 90°,when Dn , 80 3 1026, the coupling constant G isproportional to the amplitude of the index Dn, asstated in Eqs. ~5! and ~11a!. Wave coupling betweenthe signal and reference beams increases as G in-creases, so the diffracted signal-beam intensity in-creases with Dn. Therefore, h increases as Dnincreases. However, h starts to decrease as Dn in-creases further. This occurs because the light en-ergy of the signal beam is transferred to the referencebeam in this Dn region. As wg decreases, the peaksof the curves shift toward the large Dn direction andreach Dn 5 100 3 1026 when wg 5 0°.

Next, we show signal- and reference-wave propa-gation within the photorefractive medium. In Fig.5, the signal-beam intensity IS and the reference-beam intensity IR within the medium are plottedagainst the propagation distance into the medium zfor T 5 5 mm. IS versus z corresponds to h versus T.Therefore, Fig. 5 also shows the diffraction efficiencyh as a function of the medium thickness T. Thephotorefractive phase shift wg is taken as a parame-ter. The diffraction efficiency first increases andthen decreases with increasing medium thickness T~or distance z!. For wg 5 90°, when T is small ~T ,2 mm!, the wave coupling between the signal and

Fig. 3. Photorefractive phase w~z! within the medium as a func-tion of Z for various values of the photorefractive phase shift wg,which is the phase difference between the optical intensity gratingand the induced index grating. The input intensity ratio of therecording beams r is 100.

10 July 1997 y Vol. 36, No. 20 y APPLIED OPTICS 4797

reference beams becomes tight as T increases, so thediffracted signal-beam intensity increases with T.Therefore, h increases as T increases. On the otherhand, for T . 2 mm, the energy transfer from thesignal beam to the reference beam becomes large as Tincreases and wg decreases, so h ~or IS! decreases as Tincreases and wg decreases. For wg 5 0°, our resultcoincides with Kogelnik’s result that h shows adamped oscillation with T. When T becomes large,the light absorption during reconstruction also in-

Fig. 4. Dependence of the diffraction efficiency h on the indexamplitude Dn ~5DnS 5 DnLy2! for various values of wg. Thecalculation conditions are as follows: The medium parametersare a medium thickness of T 5 5 mm, a refractive index of n0 5 2.4,an absorption coefficient of a 5 200 m21, a grating period of L 5 5mm, and a slant angle of the grating of f 5 90°; the recordingconditions are a wavelength of the recording beam of l 5 0.532 mm,an incident angle of the recording beam of u0 5 3.05°, a recording-beam intensity ratio of r 5 100; the reconstruction conditions area wavelength of the reference beam of l 5 0.532 mm, an incidentangle of the reference beam of u 5 3.05°, and an angle mismatchfrom the Bragg condition of Du 5 0.

Fig. 5. Signal-beam intensity IS and reference-beam intensity IR

within the medium for various values of wg. IS versus z corre-sponds to the diffraction efficiency h as a function of the mediumthickness T. The parameters used are the same as those for Fig.4, except for an index-modulation depth of Dn 5 80 3 1026 ~5DnS

5 DnLy2!.

4798 APPLIED OPTICS y Vol. 36, No. 20 y 10 July 1997

creases. Therefore, a decrease in h caused by anincreasing T includes the effect of light absorption.From Fig. 5, in this case we obtain an optimum me-dium thickness of 2 to 2.5 mm for a high diffractionefficiency.

The angle mismatch from the Bragg condition isknown to strongly affect the diffraction efficiency.Here, we show the signal- and reference-beam inten-sity distributions within the medium for the off-Braggcondition. Figure 6 shows the distributions for Du 50.05°. The calculation conditions are the same asthose for Fig. 5, except for Du. The signal-beam in-tensity IS is smaller and the reference-beam intensityIR is larger compared with those of Fig. 5, which arefor the on-Bragg condition of Du 5 0°. This occursbecause the energy transfer between the signal beamand the reference beam becomes weak as a result ofthe Bragg mismatch. In Subsection 4.C we show thediffraction efficiency under off-Bragg conditions.

C. Evaluation of Mismatch from the Bragg Condition

The diffraction efficiency h is plotted against the an-gle mismatch Du in Fig. 7 for various values of wg. Achange in the angle from the Bragg angle causes asharp change in the diffraction efficiency. When wg

5 0°, the mismatch curve becomes symmetric withrespect to Du 5 20.02°. This means that the fringesurfaces tilt toward the signal beam8 at an angle of0.02° with respect to the z axis. The peaks of thecurves shift in the 1Du direction as wg increases.When wg 5 90°, the mismatch curve becomes sym-metric with respect to Du 5 0. The values of h at Du5 0.05° are the same as those of IS at T 5 5 mm inFig. 6.

The mismatch curves for large Dn are shown in Fig.8. The parameters used in the analysis are thesame as those for Fig. 7, except for a value of Dn 5200 3 1026. The large Dn gives rise to strong wavecoupling, so the fringe-bending effect strongly affectsthe diffraction efficiency.9 The behavior of the mis-

Fig. 6. Signal- and reference-beam intensities for the off-Braggcondition of Du 5 0.05°. The parameters used are the same asthose for Fig. 5, except for Du.

match curves is more complex than that shown inFig. 7: The increasing Dn broadens the peaks of themismatch curves. Furthermore, we find asymmet-ric curves and a curve with many peaks. Broad-ening of the peaks and asymmetric curves wereobtained experimentally by Tao et al.8 and Vre et al.9However, when wg 5 0°, the mismatch curve becomessymmetric with respect to Du 5 20.05°. On theother hand, when wg 5 90°, the mismatch curve be-comes symmetric with respect to Du 5 0°. In thiscase, the fringe surfaces do not tilt or bend. Asshown in Figs. 7 and 8, Dn greatly affects the mis-match curves.

The angle mismatch Du has an effect similar to thatof wavelength mismatch Dl on the behavior of theholograms; there is a close relation between the angle

Fig. 7. Diffraction efficiency h as a function of the angle mismatchDu from the Bragg angle u0 for various values of wg. The param-eters used are the same as those for Fig. 4, except for a value ofDn 5 80 3 1026 ~5DnS 5 DnLy2! and Du.

Fig. 8. Angle mismatch curves for large values of Dn ~5200 31026!. The parameters used are the same as those for Fig. 7,except for Dn.

mismatch and the wavelength mismatch of holo-grams.15 The angular and wavelength mismatchesplay important roles in evaluating cross talk betweenmultiplexed-hologram recorded pages.

In this paper, we considered the photorefractivephase shift and the fringe-bending effect in the anal-ysis. The optical activity and birefringence17,18 ap-pearing in the sillenite crystals ~BSO, BTO, etc.! arenot considered. An off-Bragg analysis that takesthese effects into account is the next challenge. Un-fortunately, deriving an analytical formula to calcu-late the diffraction efficiency is very difficult.

5. Conclusion

Analytical formulas were derived to calculate the dif-fraction efficiency of transmission-type hologramgratings recorded in a photorefractive medium. Thephotorefractive phase shift and the fringe-bendingeffect are taken into account in this analysis. By useof this formula, the diffraction efficiency can be eval-uated easily in terms of medium parameters and re-cording and reconstruction conditions compared withnumerical-analysis methods. The angle mismatchfrom the Bragg condition was also quantified with thederived formula. These results will play an impor-tant role in evaluating the cross talk of multiplexed-hologram recording.

I would like to thank Tatsuya Kume and ManabuYamamoto for their helpful advice and encourage-ment.

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