Intraplane to interplane optical interconnectswith a high diffraction efficiency electro-optic grating

Degui Sun, Chunhe Zhao, and Ray T. Chen

We report on a new optical interconnect architecture for three-dimensional, multiple electro-optic grat-ings with LiNbO3 used in conjunction with substrate guided waves. First the operating mechanism ofthe system is studied in detail, and the momentum mismatch in the operating process of the system isalso demonstrated. We then derive a new method for calculating coupling efficiency by introducing acompensation for the mismatch. This theoretical research allows the new optical interconnect archi-tecture to provide a higher design accuracy and an optimized coupling efficiency, even though it is underthe case of momentum mismatch. We achieve this result by introducing a substrate guided wave with45° bouncing angle and 100-V applied voltage. The successful design and its theoretical analysis will behelpful for research on the grating coupler. © 1997 Optical Society of America

Key words: Optical interconnection, electro-optic grating, momentum mismatch, phase compensa-tion, coupling efficiency.

1. Introduction

The development of modern optical information pro-cessing technologies such as optical interconnection,memory, and computing requires multiplexed paral-lelism, high density, high switching speed and effi-ciency, and better controllability.1–3 One of themostpivotal elements for the realization of these applica-tions is electro-optic ~EO! modulated phase gratingthat has been used to perform switching, numericaloperations, and interconnections for computing andinformation processing.4–6The EO modulated phase grating has received at-

tention because of its important applications in opti-cal interconnection, communication, memory, andcomputing.7–11 However, the study of EO gratingwith high diffraction efficiency and short interactionlength has not been investigated carefully becausethe momentum matching for high diffraction effi-ciency based on the Bragg diffraction theory is noteasy to realize.12 In this paper, we investigate amicrostructural EO grating based on the LiNbO3crystal for a novel intraplane-to-interplane intercon-

The authors are with the Department of Electrical and Com-puter Engineering, University of Texas at Austin, Austin, Texas78712-1084.Received 2 October 1995; revised manuscript received 22 July

1996.0003-6935y97y030629-06$10.00y0© 1997 Optical Society of America

nect application. Substrate guided waves are em-ployed to provide the signal, which may be data,image, etc., to be routed. In this research, not onlydo we study the operating mechanism of the systemin detail and verify the momentum mismatch in theoperation, but we also propose a new method for cal-culating coupling efficiencies by introducing a com-pensation for the mismatch. At the same time, anoptimized architecture is obtained. Therefore theresult presented provides not only a higher designaccuracy, but also a high coupling efficiency. In Sec-tion 2, the bouncing angle u1 is selected in accordancewith the total interreflection condition that the mi-crostructural EO device requires without activatingthe EO grating. The diffraction angle that can breakthe total interreflection condition and the periodicityof the EO grating are discussed. Furthermore, westudy the momentum mismatch of the system anddiscuss its influence on the coupling efficiency basedon the Bragg diffraction theory ~Section 3!. To ob-tain the accurate design, the electric-field distribu-tion and index modulation are briefly studied inSection 4. On that basis, in Section 5 a compensa-tion method for the mismatch is proposed and ana-lyzed, and the coupling efficiency is optimized byintegration for index modulation. In Section 6, weprovide our conclusions.

2. Determination of the Bouncing Angle in LiNbO3

The purpose of this research is to study and design anEO grating with a short interaction length as shown

20 January 1997 y Vol. 36, No. 3 y APPLIED OPTICS 629

in Fig. 1. LiNbO3 is used to produce indexmodulationby the use of microelectrodes at various spots markedwith Gi, whereas polymer is used to fulfill such re-quired functions as wavelength redistribution anddata storage and reading. When the EO grating isactivated, the optical beams are coupled with highefficiency into the polymeric material when an appro-priate voltage is applied. Otherwise the beams aretotal-internally reflected at these spots. This oper-ating process is shown schematically in Fig. 2 inwhich the coupling process is performed between thefirst layer and the second layer. The second layerfunctions as an index modulation layer of grating,and its thickness Le is the effective modulation depthof LiNbO3 crystal. Of course the second layer doesnot exist when the EO grating is not activated andthe bouncing is within the first layer. Therefore thefirst requirement is achieved by a bouncing angle u1in LiNbO3 that is bigger than its critical angle uc, andthe second requirement is achieved by a diffractionangle u2 in the second layer of LiNbO3 that is smallerthan the critical angle uc. Therefore the bouncingbeam within the LiNbO3 substrate can be controlledby the given activated EO grating. In this research,we used ITO material as electrodes to obtain higherdiffraction efficiency. Along the cutting direction ofLiNbO3 as shown in Figs. 1 and 2, the refractiveindex of LiNbO3 is a function of the angle u betweenthe optical beam and the x axis and is defined asfollows12:

1ne

2~u!5sin2 u

no2 1

cos2 u

ne2 , (1)

and the critical angle for the total internal reflectioncan be calculated by

uc 5 arcsinF npne~u1!

G, (2)

where np 5 1.543 is the refractive index of the pho-topolymer material. If we set u1 5 45°, we can ob-tain ne~45°! 5 2.24 according to Eq. ~1! and also uc 543.49° according to Eq. ~2!. Apparently u1 is biggerthan uc, which meets the first requirement for theoperations of the system.

Fig. 1. Construction of LiNbO3-based EO diffraction grating.

630 APPLIED OPTICS y Vol. 36, No. 3 y 20 January 1997

3. The Electric-Field Distribution and its Influence onIndex Modulation

As illustrated in Fig. 1, the electrode grating vector isalong the z direction and the light wave travelswithin the z–x plane. When voltage V is appliedacross the electrodes and if the width of the spacingbetween the electrodes is 2a, we have

]2V]z2

1]2V]x2

5 0, (3a)

εz]2V]z2

1 εx]2V]x2

5 0, (3b)

where εz and εx are dielectric constants in directionsz and x, respectively. By our using the coordinatetransformation,

x9 5 Îεzyεxx. (4)

Eq. ~3b! can also be written as

]2V]z2

1]2V]x92

5 0. (5)

By our using the following coordinate transformationagain,

z 5 a cosh u cos v, (6a)

x9 5 a sinh u sin v, (6b)

the solutions for Eq. ~5! can be written as13

Ez 5 2SUapD cosh u sin vcosh2 u 2 cos2 v

, (7a)

Ex 5 2SUapD sinh u cos vcosh2 2 cos2 v

Îεzyεx. (7b)

Eqs. ~6! and ~7! determine the distribution of electric-field components in directions z and x. Further-more, together with Eq. ~1!, we can solve for the indexmodulation of LiNbO3 induced by the electric field asfollows:

Dne~u1! 5ne

3 sin2 u1Dno 1 no3 cos2 u1Dne

~no2 cos2 u1 1 ne

2 sin2 u1!3y2 , (8)

Fig. 2. Relationships among the unmodulated LiNbO3 sublayer1, the modulated layer 2, and the photopolymer memory layer 3.

where Dne and Dno are calculated by

Dne 5 2 12 ne

3g33Ez, (9a)

Dno 5 2 12 no

3g13Ez, (9b)

where g33 and g13 are the EO coefficients of theLiNbO3 gratings. Because we study the couplingcase in x direction, we discuss only the index modu-lation in x direction by taking z 5 0. By combiningEqs. ~8! and ~9! we obtain the index modulation curveDne~u1, x! with respect to x, as shown in Fig. 3. Notethat the index modulation Dne~u1, x! dramatically de-creases along x ~depth! direction. Therefore it isnecessary to use the integration to optimize the cou-pling efficiency.

4. Coupling Conditions and Momentum Mismatch ofthe Coupling System

In this system, the coupling of optical beams underthe action of the EO grating is a key operation. Be-cause some factors supporting the operation, such asthe grating direction and period, the directions andamplitudes of wave vectors, and the index modula-tion of the LiNbO3 crystal, are all limited by thesystem itself, coupling conditions and efficiency areworthy of a detailed study. As shown in Fig. 4, K1,K2, and Kg indicate the incident wave vector, the

Fig. 3. Distribution curve of index modulation as a function of x.

Fig. 4. Wave vectors and modulation scheme of an x-cut LiNbO3

crystal.

diffracted wave vector, and the grating vector, respec-tively. In z direction, we define

b1 52p

lne~u1!sin u1, (10a)

b2 52p

lne~u2!sin u2, (10b)

kg 52p

L. (10c)

In x direction, we define

a1 52p

lne~u1!cos u1, (11a)

a2 52p

lne~u2!cos u2, (11b)

where l and L are the wavelength and the gratingperiod, respectively. Based on the above equations,the momentummismatches in both z and x directionsare defined as follows:

Db 5 b1 2 b2 2 mkg, (12)

Da 5 a2 2 a1. (13)

In the Bragg diffraction theory, the vectors that areonly in the direction of the grating vector are requiredto be matched, i.e., Db 5 0.12 According to Eqs.~10a!, ~10b! and ~12!, when Db 5 0, we can obtain therelationship between the grating period and the dif-fraction angle for the Bragg condition:

L 5l

ne~u1!sin u1 2 ne~u2!sin u2. (14)

Based on Eqs. ~1! and ~14!, we obtain the relationshipbetween grating period L and diffraction angle u2 asshown in Fig. 5 in which u2 is always smaller than thecritical angle of 43.49° and that agrees with the sec-ond requirement of the system operation. It is cer-tain that the value of @ne~u2!cos u2 2 ne~u1!cos u1#functions as the grating period L or diffraction angleu2. Note that all the allowable values of the gratingperiod in Fig. 5 are able to be implemented practi-cally. In terms of Bragg diffraction theory, if bothDb and Da are zero, three vectors, K1, K2, and Kg arematched, which is the perfect Bragg diffraction, and

Fig. 5. Relationship between grating period and diffraction angle.

20 January 1997 y Vol. 36, No. 3 y APPLIED OPTICS 631

the diffraction efficiency of 100% can be realized the-oretically. Then with Eqs. ~11a!, ~11b! and ~13!, weobtain the distribution curves of Daly2p 5 @ne~u2!cosu2 2 ne~u2!cos u1# as a function of the diffraction angleu2, as shown in Fig. 6. Note that Daly2p 5@ne~u2!cos u2 2 ne~u2!cos u1# is always larger than 0,which means that the momentum matching condi-tion defined by Eqs. ~12! and ~13! among the threevectors K1, K2, and Kg does not exist at all. How-ever, Da is an important factor in influencing thecoupling efficiency in accordance with the Bragg dif-fraction theory. A bigger Da can induce the multi-mode coupling, so we have to study the multimodeextension of grating coupling from the compensationfor the momentum mismatch Da.

5. Compensation for Momentum Mismatch andOptimization of Coupling Efficiency

Assuming A1~x! and A2~x! are the components of thecomplex amplitudes of the normalized modes of inci-dent light wave and diffracted light wave in x direc-tion, we have12

dA1~x!dx

5 2ik12A2~x!exp~iDax!, (15a)

dA2~x!dx

5 2ik*12A1~x!exp~2Dax!, (15b)

k12 5v2m

2Îa1a2

P1 z DεP2, (15c)

where k12 is the coupling constant, v 5 2pcyl is theangular frequency of the light wave, P1 and P2 areunit wave vectors in incident wave and diffractedwave directions, m is the permeability of LiNbO3, andDε is the increment of dielectric constant ε. With Eq.~1! and optical principle, we can obtain the incrementof dielectric constant as

Dε 5 2εononeno

3 cos3 u1Dne 1 ne3 sin3 u1Dno

~ne2 sin2 u1 1 no

2 cos2 u1!2 . (16)

Note that here we are discussing the codirectionalcoupling. The coupled-mode Eqs. ~3a! and ~3b! areconsistent with the energy conservation in x direc-

Fig. 6. Relationship between Daly2p and diffraction angle.

632 APPLIED OPTICS y Vol. 36, No. 3 y 20 January 1997

tion, i.e.,

ddx

~uA1u2 1 uA2u2! 5 0. (17)

WhenDa is large, Eq. ~17! cannot be satisfied, and theset of coupling Eqs. ~15a!, ~15b!, and ~15c! is no longervalid, and the mode extension has to be used to an-alyze the coupling procedure of grating. In terms ofthe mode expansion principle, the TE modes of thediffracted optical waves in the region x $ 0 is repre-sented by the sum of the normal modes as14,15

A2~x! 5 (m52}

}

exp~2iDax! fm2~z!, (18)

where

fm2~z! 5 hm2~z!yÎDm2, (19a)

Dm2 5 *0

L

uhm2~z!u2dz, (19b)

hm2~z! 5 Dne~u1, x!exp~2ib2z!. (19c)

As shown in Fig. 7~a!, the diffracted wave repre-sented by Eq. ~18! produces an interaction with grat-ing in the region of interaction depth from 2Le to 0.The interaction depth has its inherent spatial fre-quency spectrum that covers all the diffraction modesand functions as a phase compensation region beforethe diffracted wave forms all the possible modes. Asa phase compensation, this frequency spectrum canbe calculated by

A2~Dr! 5 *2Ley2

Ley2

fm2~z!exp~2iDrx!dx, (20)

Fig. 7. Frequency spectrum of interaction depth of grating: ~a!diffraction theorem; ~b! spatial frequency spectrum.

where Dr is the spatial angular frequency in x direc-tion. Then for the energy of diffracted wave, we ob-tain the transmittance spectrum of the interactiondepth as

t~Dr! 5 Fsin~LeDr!LeDr

G2. (21)

The transmittance spectrum curve is shown in Fig.7~b!. The coupling efficiency of the EO modulationgrating can be calculated in accordance with Fig. 7~b!,which depends on the compensation of Dr to the mis-match Da. For example, when the grating period L5 8 mm, we obtain Da ' 0.6 3 106 from Eq. ~13!.Then from Fig. 7~b! this corresponds to a transmit-tance value of approximately 44%. By consideringthe phase compensation for themismatch Da, one canchange the set of coupling Eqs. ~15a! and ~15b! to

dA1~x!dx

5 2ik12A2~x!, (22a)

dA2~x!dx

5 2ik*12A1~x!. (22b)

Finally we obtain the coupling efficiency of grating as

h 5 Fsin~LeDa!

LeDa G2 sin2~k12Le!. (23)

As we know, Le depends on the applied voltage andindex modulation Dne~u1!, which again is induced bythe applied voltage. Because both the interactionlengthLe and the indexmodulation Dne~u1! depend onthe applied voltage and k12 functions as both theindex modulation Dne~u1! and the diffraction angle u2,we must optimize the coupling efficiency by takinginto account the total contribution from all the im-portant parameters as follows:

k12 54p2nonel2Îa1a2

no3 cos3u1Dne 1 ne

3 sin3u1Dno~ne

2 sin2u1 1 no2 cos2u1!

2

3 cos~u1 2 u2!, (24)

k12Le 5 *2Le

0

k12dx, (25)

*2Le

0

k12dx 54p2no

4ne4 cos~u1 2 u2!

l2~ne2 sin2u1 1 no

2 cos2u1!2Îa1a2

3 ~r33 cos3u1 1 r13 sin

3u1! *2Le

0

Ezdx. (26)

With Eqs. ~23!–~26! we obtain the coupling efficiencycurve as shown in Fig. 8. We can find from thisfigure that, when L equals approximately 8 mm ~aswe selected in our project!, the coupling efficiency h isapproximately 20%. In this simulation, we use theapplied voltage of 100 V. According to Fig. 4, thediffraction angle corresponding to the period value of

8 mm is approximately 41.35°, which is smaller thanthe critical angle for total internal reflection.

6. Summary and Conclusions

We report on the theory in the design of a new mi-crostructural high diffraction efficiency EO grating bythe use of substrate guided wave. In this research,the operating mechanism under momentum mis-match of the system is explained under the couplingtheory, and the coupling efficiency is calculated in themodified formula. In addition, the relation of theperiodicity and the diffraction angle is delineated,and the relationship of the grating period of the EOgrating and momentum mismatch in the vertical di-rection Da is discussed. The most important aspectin this research is a new conclusion that a gratingcoupler requires only the momentum match in thedirection of the grating vector, i.e., Db 5 0, whereas inthe perpendicular direction of the grating vector, themismatch Da can be compensated by the mode ex-pansion and Fourier transform of the diffracted wavefor the effective interaction depth. With the methodin which the interaction depth can be controlled, thisresearch numerically optimizes the coupling effi-ciency of the EO grating and the expected results areobtained.

The authors thank T. C. Lee and his fellows fortheir helpful discussions on this research. We alsothankMaggie Li andHuajun Tang for their help withthis research. This research is currently sponsoredby the Office of Naval Research, the Defense Ad-vanced Research Projects Agency, and the advancedtechnology program of the State of Texas.

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