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(^1044) PROCEEDINGS OF THE IEEE, VOL. 6 2 , NO. 8, AUGUST 1974
Guided Wave Optics
HENRY F.TAYLOR AND AMNON YARIV, FELLOW, IEEE
Invited Paper
AAvmcr-Phenomena associated with the propagation and manipula- tion of light in thin-fh dielectric waveguides are presently the object of q d e n b l e reseuch effort, directed towrat possiile appliations in communicationsand information pmcesbg. The theory of dielec- lric waveguide modes is reviewed, and the topics of directionalcoupling, input-output coupling, modulation, md distniuted feedback laser sources are treated on the basis of coupled-mode themy. A summary of experimental results for each of the guided-wave optical phenomena covered by the theory is also presented.
I. INTRODUCTION
T
HE TRANSMISSION and manipulation of optical power has long been the basis of a considerable industry as well as of substantial academic endeavor. The advent of the laser in 1960 stimulated a great deal of interest in the study of the properties of confined Gaussian beams which are emitted by laser oscillators. Thespatial coherence of the laser field made it possible, forthe first time, to obtainsubstantial powers in optical beams whose diffraction spread very nearly approaches thetheoretical limit. Themanipulation of these beams, by and large, was still accomplished by the “classical” prelaser means. Opticalsystems incorporating laser sources today are comprised essentially of the same components to be found in conventional optical systems. The laser radiation in the form of a propagating Gaussian beam passes in succession through discrete opticalcomponents whichareindividually secured to a common bed. An alternative approach to the manipulationof radiation has longbeen used inthe microwave portion of thespectrum. Here the guiding and processing of electromagnetic propagat- ing beams is accomplishedwithin low-loss (metallic) wave- guides with cross-sectional dimensions comparable to the wavelength. These waveguides, which confine the radiation by repeated reflections from the walls, can propagate power at a given frequency in a number of spatially distinct modes with different phase (and group) velocities. The possibility of applying the microwave approach to laser beams was suggested by the demonstration of discrete propa- gating modesin optical fibers by Snitzer and Osterberg [ 11 (1961) and by the demonstration by Yariv and Leite [ 21 and by Bond et al.^ [^ 31 (1 963)^ of planar dielectric^ waveguides in GaAs p n junctions. A key experiment pointing theway to the control of such radiation was the modulation, by the electro- optic effect, of light guided by a p-n junction by Nelson and Reinhart [4] (1964). The rapiddevelopmentsin the field of
T& invitedpaper is one of a series planned on topics of general interesr-The Editor. Manuscript received November 26, 1913; revised February 12, 1974. The work^ reported here^ was^ supported by^ Materials Sciences, Advanced
Naval Research.
Research Projects Agency, and byAeronautics Programs, Office of
H. F. Taylor is withthe Naval Electronics Laboratory Center, Son Diego, Calif. 92152. A. Yariv is with the Department of Electrical Engineering, California Institute of Technology, Pasadena, Calif. 91 109.
epitaxialthin films by Alferov et al. 5, Hayashi et al. [6] (1969), and Kressel and Nelson [ 71 (1969) led to the application of dielectric waveguiding principles in order to lower, dramatically,the threshold of GaAs injection lasers. Continuous room temperature operation was then possible, as demonstrated by Hayashi et al. [ 81 (1970).It also showed the way to a new family of dielectric waveguide devices based on epitaxy, as illustrated by Hall et ~ l. [ 91 ( 1970). The main theme of the developments in thisfield since 1969 was thedemonstration of various guiding mechanisms and guided-wave phenomena. Here we may notethe work on prism couplers by Tien et al. [ 101 (1969), and the work on grating couplers by Dakss et al. [ 111 (1970), which effectively provided solutions to the problem of coupling Gaussian laser beams into and out of dielectric waveguides. The concept of distributedfeedback lasers, introduced by Kogelnik and Shank [ 121 (1 97 l), seemed especially attractive as applied to lasers in guided-wave configurations, asillustrated by the corrugated epitaxial GaAs-GaAlAs lasers of Nakamura et d. [ 131 (1973). The feasibility of optical directional coupling was illustrated by Somekh e t QZ. [ 141 (1 973) in power transfer experiments involving channel waveguides produced by ion implantation. The use of the term “guided wave optics” in the title rather than “integrated optics” stems from our desire to distinguish between the investigation of optical phenomena inwaveguides, the main topic here, and the integrationof a number of optical components in a single structure in order to perform complex functions [ 151 , [ 161, which we regard as the domain of inte- grated optics. Thepurpose of the present paperisto review the main theoretical and experimental progress which has takenplace in this field as well as to considersome of the more promising scientific and practicalpossibilitiesopened up by recent developments.
11. WAVEGUIDES AND WAVEGUIDE MODES
A. Materials and Fabrication Techniques A variety of materials and fabrication techniques have been used to make thin-film dielectric waveguides. Ion exchange in glass was one of the earliest methods reported [ 171. Amor- phous or polycrystalline film waveguides have been produced by vacuum deposition of ZnS [ 101 and Taz05 [ 181, and by sputtering of Ge [ 191. Organic materials for waveguides have included polyester and polyurethane epoxy resins [ 201, vinyl trimethyl silane and hexamethyl disiloxane [ 2 1 ], cyclohexyl methacralate [22], polyphenyl siloxane [23], andpoly- styrene [ 241. Films of nitrobenzene liquid 1251 and of nematic liquid crystals [ 261, [ 271 have also exhibited wave- guiding. Channel waveguides, which confine a beam in two dimensions, have beenfabricatedin sputtered glass films by sputter etching [ 281, [ 291, and in both fused silica [30] and
V: TAYLOR AND WAVE OPTICS 1045 " 1
I "2 - PROPAGATION " I = - Fig. 1. A planar dielectric waveguide. GaAs [ 141 by ionimplantation. Solid-statediffusion has produced planar waveguides inLiNb03 [ 3 11 ' and channel waveguides in CdS [ 321 , and ZnSe [ 331. Waveguiding has been observed in planar epitaxial layers of GaAs [ 91, [ 341, GaAs-AlGaAs [35], Si [36], ZnSe [37], ZnO [38], and gar- nets [ 3 9 ]. Two-dimensional confinement hasbeen demon- strated in mesa structures of epitaxial GaAs-AlGaAs [ 401, [4 1 ]. B. TheWaveguideModes [ 4 2 / , [43] Aprerequisite to an understanding of guided-wave interac- tions is a knowledge of the properties of the guided modes. A mode of a dielectric waveguide at a (radian) frequency w is a solution of Maxwell's propagation equation V Z E ( r ) + k 2 n Z ( r ) ~ ( r ) = o (1) subject to the continuity of the tangential components of E and H at the dielectric interfaces. In (1) the form of the field is taken as E(r, t ) =E(r)eiw' (2)
where o E kc, and the index of refraction n(r) is related to the
dielectric constant E @ ) by n2 ( r ) E E ( ~ ) / E o. Limiting ourselves to waves with phase fronts normal to the waveguide axis z , we have The basic features of the behavior of dielectric waveguides can be extracted from aplanar model in which no variation exists in one, say y , dimension. Channel waveguides, in which the waveguide dimensions are finite in both the x and the y directions, approach thebehavior of the planar guide when one dimension is considerably larger thantheother[44],[45]. Even whenthis is not the case, most of the phenomena dis- cussed below are only modified in a simple quantitative way when going from a planar to a channel waveguide. Because of thisand of the immense mathematical simplification which results, we will limit most of the treatment to planar wave- guides such as the one shown in Fig. 1. Putting a / a y = 0 in (3) and writing it separately for regions 1, 2, and 3 yields: region 1 region 2 az --(x, y ) + (k2n: - p ) E ( x , y ) = 0 axz region 3 a z
- E ( x , y ) + (kzn: - P ) E ( x , y ) = 0
a x 2 (4c) d b Fig. 2. Propagation constants, electricfielddistributions,and wave vector diagrams for the different types of waveguide modes: u is not physicallyrealizable; b and c are guided modes; d is a substrate radiation mode; and e is a radiation mode of the waveguide. where E ( x , y ) is some Cartesian component of E ( x , y ). Be- foreembarkingon a formalsolution of (4) we may learn a great deal about the physical nature of the solutionsby simple arguments. Let us consider thenature of thesolutions as a function of the propagation constant 0 at some fixed fre- quency w. Let us assume that n 2 > n 3 > n l. For 6 > knz, i.e., region a in Fig. 2, it follows directly from (4) that everywhere and E ( x ) is exponential in all three layers 1 , 2 , and 3 of the waveguides. Because of the need to match both E ( x ) and its derivatives (see Section 114) at the two interfaces, the resulting field distribution is as shownin Fig. 2(a). The field increases without bound away from the waveguide so that the solution is not physicallyrealizable and thus does not corre- spond to a real wave. For kn3 < < k n z , as in points b and c, it follows from (4)
that the solution is sinusoidalin region 2, since (1/E)@ E /
a x z ) < 0, but is exponential in regions 1 and 3. This makes it possible to have a solution E ( x ) which satisfies the boundary conditions while decaying exponentiallyin regions 1 and 3. These solutions areshown in Fig. 2(b)and (c). The energy carried by these modes is confined to the vicinity of the guid- ing layer 2 and we shall, consequently, refer to them as con- fined or guided modes. From the foregoing discussion it fol-
lows that a necessary condition for theirexistence is that k n l ,
kn3 < fi < knz so that confined modes are possible only when n2 > nl , n 3 , i.e., when the inner layer possesses the highest index of refraction. Solutions of (4) for knl < 0 < kn3 (region d ) correspond to exponential behavior in region 1 and to sinusoidal behavior in regions2and3asillustrated in Fig. 2(d). We shall refer to these modes as substrate radiation modes. For 0 < p < knl , as in region e, the solution for E ( x ) becomes sinusoidal in all
TAYLOR AND YARIV: GUIDED WAVE OPTICS 1047
Fig. 3. Dispersion characteristic for TE, and TE, modesof planar waveguide, illustrating the cutoff frequencies (wC), and ( w ~ ) ~.
equation
where
The normalization constant C is chosen so that the field repre- sented by (14) and (1 5 ) carries one watt per unit width in the y direction.
H,E: d x = - dx = I
or using n: E E ~ / E O
d x = - -. Pm
This condition determines the value of Cm as [ 461
111. COUPLINGBETWEEN WAVEGUIDE MODES A. TheCoupled-ModeEquations Many of the experimental situations of guided wave optics and especially those which involve exchange of power between modes can be treated by means of the coupled-mode approach. This formalism, introduced originally by Pierce [47], describes thetotal propagating disturbance in astructure as a sum of (usually two) unperturbed modes of the system whose ampli- tudes vary with distance z due to somecouplingbetween them. This point ofview is fruitful when the z variation is slow, andhas beenapplied tothe description of different guided-wave phenomena [ 461.Inthe following section we will reproduce some of the main features of this formalism. Consider two electromagnetic modes with, in general, differ- ent frequencies whose complex amplitudes are A and B. These are taken as the eigenmodes of theunperturbed medium so that they representpropagating disturbances _a ( z , I ) =Aei(Wa'-&z)b(z, t ) =Bei(wbr@bz)_* (19)
where A and B are the complex normalized amplitudes which in the unperturbed structure are independent of z. In the presence of aperturbation, power is exchanged be- tween.modes a and 6. Thecomplexamplitudes A and B in this case are no longer constant but willbe found to depend on z. They will be shown to obey relationsof the type
d A d z
_ - - a bB ~ - ~ A z
The phase mismatch constant A meritssome discussion. It is clear from the structure of (20) that a cumulative sustained exchange of power between modes a and b requires that A = 0. Otherwise the values of d A / d z , for example, fromdifferent parts of the propagation path interfere destructively. In what follows we will find that in most of the problems of interest to us we can visualize the process of power exchange as follows. Traveling mode b interactswiththeperturbationto yield a traveling polarization wave. This wave in turn drives mode a. Simultaneously,mode a interacts with theperturbationto drive mode b. The constant A is equal to the difference in the propagation constants of the driven waves andthe driving polarizations. The coupling coefficients Kab and K b a are determined by the physical situation under consideration and will be considered below. Before proceedingwith specific experimental situations we may draw some general conclusions which apply to the large number of phenomena whichare described by equations of the general form of (20).
B.CodirectionalCoupling We take up, first, the case where modes a and b carry electro- magnetic power in the same direction.It is extremelycon- venient to define, as was done in Section 11, A and B in such a way that I A ( z ) 1 ' and 1 B ( z ) I' correspond to the powercarried by mode Q and mode b , respectively. The conservation of total power is thus expressed as
d dz
which, using (20), is satisfied when
Kab = - K z a. (22)
If boundary conditions are such that a single mode, say b , is incident at z = 0 on the perturbed region z > 0, we have
b(0) Bo ~ ( 0 )= 0. (23)
Subject to these conditions, the solutionsof (20) become
1048 AUGUST IEEE,PROCEEDlNGS OF THE 1974
Under phase-matched condition (A = 0), a complete spatially periodicpower transferbetween modes a and b takesplace with a period n l ~ :
a ( z , t ) = Bo - eK a b i(wat-Paz) sin ( K Z )
b ( z , t ) = Boei(wb'-Pbz) cos K Z. (25)
A plot of the mode intensities ( a 1' and I b 1 ' is shown in
Fig. 4. This figure demonstrates the fact that for phase mis- match 1 A 1 >> I Kab I the power exchange between the modes is negligible. Specific physical situations which are describable in terms of this picture will be discussed further below.
C.ContradirectionalCoupling
K
In this case the propagation in the unperturbed medium is described by
Fig. 4. Distribution of power between modes for codirectional coupling under phase-matched and unmatched conditions.
a = Aei(wat+Baz)
b = Bei(wb'-@bz) (26) nlol
where A and B are constant. Mode u corresponds to a left ( - z ) traveling wave while b travels to ,the right. A time-space periodic perturbation can lead to power exchange between the
Conservationmodes. ofis total power now expressed iis 3 2
d
- ( / A I ' - IBI')=O dz
(27) Z = O 2.L
which is satisfied by (20) if we take
so that
In this case we take the mode b with an amplitude Bo to be incident at z = 0 on the perturbation region which occupies the space between z = 0 and z = L. Since mode a is generated by the perturbation we have u ( L ) = 0. With these boundary conditions, the solutionof (29) is given by
2jKabe-i(AZ/') A ( z ) = B o^
SL
- A s i n h - + i S ~ ~ s h -
SL
e i ( A z / ' ) B ( z ) = B o {A sinh [! ( z - L ) ] SL -A sinh - + is cosh -
SL
2 2
Fig. 5. Contradirectional transfer of powerfrom an incident forward wave with amplitude B ( z ) t o a reflected wave A (2).
cosh and sinh functions in (31), the incident mode power de- cays exponentially along the perturbation region. This decay, however, is due not to absorption but to reflection of power intothe backward traveling mode u. This case will be con- sidered inmore detail inSection VII,^ where^ we^ discuss coupling by waveguide corrugationsanddistributed feed- back lasers.
D. TheCouplingCoefficient The general behavior of the two coupled modes is described by (24) and (30) forthe case of codirectionalandcontra- directional coupling,respectively. Theform of theseequa- tions is independent of the numerical magnitude of the couplingcoefficient K. Thelatter, however, determinesthe interaction strength or, in practice, the distance over which a given fractional powerexchangebetween twomodestakes place. Consider coupling between, say,a TE mode
+iS cosh - ( Z - L ) [s 11
where SZd-', and K = I K = * I. _ _
Under phase-rnatching conditions (A = 0 ) we have
a ( * ) ( x ,Z , t ) = A ( * ) ( z ) e i ( w a r s f l a z )& ( a ) ( ~ )Y (32)
and a forward-traveling TM mode
b+(x, z , t ) = B+(z)ei(Wb'-Pbz)J f p ) ( x ) (33)
where A and B are the power-normalized mode amplitudes. The (+) and (-) superscriptsrefer to forward and backward waves, respectively, and &,,(x) and Jf,,(x) are the normalized mode profiles as given by (8) and ( 1 5). The wave equation for the perturbedcase is cosh [ K ( Z - L ) ] cosh ( K L ) '
B ( z ) = Bo (31) a'E, a'
V Z E y ( r , ~ ) = I L E3+1r,t, [Ppert(l, t ) I y (34)
case is shown in Fig. 5. For sufficiently large arguments of the where Ppert represents the deviation in the medium polarization
A plot of the mode powers I B ( z ) 1 ' and I A ( z ) 1 ' for t h i s
1050 PROCEEDINGS OF THE IEEE, AUGUST 1974
same sign, and contradirectional coupling, if these quantities are of opposite sign. Solutions to the coupled-mode equations for these two cases are given by (24) and (30), respectively.
IV. DIRECTIONALCOUPLING
.Exchange of power between guided modes of parallel wave- guides is known as directional coupling. It is anticipated that waveguide directional couplers will perform a number of use- fulfunctions inthin-film devices, includingpower division, modulation, switching, frequency selection,andpolarization selection. If two coupled waveguides have equal propagation constants (synchronous case), completetransfer of power fromone waveguide to the other is possible; if the propagation constants are unequal (asynchronous case), only a fraction of the power initiallypropagatingin one waveguide can be transferred to the other. Theoretical treatments of both synchronous [431, [44],[48] and asynchronous [49],(501 cases have been given. Experimentally, coupling has been observed between glass fibers [ 5 11, between planar films of glass [ 521, between
channel waveguides in glass [ 291 , and in GaAs [ 141 , [ 531.
Waveguide coupling can be treated theoretically by coupled- mode theory. Consider the case of the two planar waveguides illustratedin Fig. 6. Refractive indexdistributionsforthe two guides in the absence of coupling are given by na(x)^ and n b ( x ). The transverse electric field distribution for a particu- lar guided mode of waveguide u and a particular mode of waveguide b will be denoted by &y)and 8jb), and the propa- gation constants by Pa and o b. The field for the coupled-guide structurefor propagationin the positive z direction is ap- proximated by E , = A ( z ) 8 y ) ( x )ei(wr-8az) + B(z) &(b)(x)Y ei(wr-8bz). (49)
The perturbation polarizationresponsible for the coupling is calculatedby substituting (49) into(34), neglecting the variation of A and B. The result is
PWrt = -eiwreo [&F)A (2) (nf - n , ~ )e-ipaz
+ 8 j b ) B ( z ) (nz - n g ) e-isbz] (50)
where n,(x) is the refractive index for the two-guide structure. Substituting (50) into (35) andintegrating over x yields
d A d z
_ - - (^) a bB e - i A z + MaA
where
Theterms Ma and Mb represent small correctionsto pa and o b , so (5 1) reduces to the familiar form ( 2 0 ) , with A = pa - Ob - i(Ma - M b ). Thesolution is given by (24). If a power Po is initiallycoupled into guide b at z = 0, the interguide
"3 " ", (^) _ x_* Fig. 6. Spatialvariation of refractive index for uncoupledwaveguides na(x) and n b ( x ) , and for a parallel waveguide structure n,(x).
1
Fig. 7. Experimentalresultsdemonstratingdirectionalcoupling in GaAs waveguides.
distribution for z > 0 is given by
4K 4K2 + A2 [(4K2 + f 2 ) 1 ' 2 Z
Pa =Po - sinz
Complete power transfer occurs in a distance L = n/2K in the synchronous case (A = 0). In the asynchronous case (A f 0), both couplinglengthandmaximumpower transfer are less than in the synchronouscase. Couplingbetween waveguides withdifferentpropagation
constants # p ( b ) ) canbeimprovedbyaperiodic perturba-
tion in refractive index. Most efficient coupling by a sinusoidal perturbation requires that A = 0, where
and A is the period of the perturbation, measured in the z di- rection. Both codirectional and contradirectional power trans- fer are possible [ 541. The preceding discussion assumes that only two waveguides are involved in the coupling, but coupled-mode theory can also beapplied to problems involving morethantwo wave- guides. In the case of an array of equally spaced, synchronous
TAYLOR AND YARIV: GUIDED WAVE OPTICS 105 1
waveguides, the coupled-mode relationsare --+-&-=
where K is given by (52), and A , represents the mode ampli- tudeforthenth waveguide. If all of theincidentpower is initially in the zeroth guide for z = 0, the solution to (55) is
A , ( z ) = A o ( 0 ) (-i),J,(2i~z)(56)
for z > 0 [ 141, where J , is the Bessel function of order n. Ex- perimental results illustrating this behavior are given in Fig. 7.
V. INPUT-OUTPUTCOUPLING
Efficient coupling of light into and out of a thin film is an
important considerationin guided wave opticsexperiments and devices. Three commonly used coupling techniques are illustrated in Fig. 8. Efficient coupling by the "end-fire" tech- nique [Fig. 8(a)l requires that the exposed waveguide surface be smooth, in order to avoid excessive scattering of thein- cidentbeam.Good optical surfaces atthe edge of the film can usually be obtainedonly by cleaving acrystal. With the prism coupler [lo],[201, [ 5 5 1 [Fig. 8(b)l, light enters through the top surface of thefilm, which can generally be made very smooth.Inthistype of coupler,a narrow (e.g., 1000 A) gap of air orotherlowrefractive-index material separates the prism fromthe film. Opticalpower entersor
leaves the film by tunneling throughthe gap. Formost ef-
ficient power transfer, the phase velocity of the light wave in the prism must matchthat of aguided mode of the wave- guide. Mathematically, this condition can be written as follows
where x is the free-space wavelength, np is the refractive index of the prism, 0 is the angle between the incident beam in the prism and the normal to the surface of the fim, and 0 , is the propagation constant of aparticularmodein the film. By
varying 6, different modes of the film can be excited; in out-
put coupling, different modesareidentifiedby the angles at which they emerge from the film. Input coupling efficiencies of 88 percent have been reported [ 561. A phase grating deposited on the surface of the film [ F i g. 8(c)] can also provide efficientcoupling [ 111,[57]-[60]. In this case, the phase-match condition is
where n l is the refractive index of the air or other medium in which the incident beam propagates, A is the grating period, and 2 is the diffraction order of the grating (an integer). Light is coupledeither throughthetop surface of thefilm, as il- lustrated,orthroughthesubstrate. Bothcodirectionaland contradirectional coupling have been demonstrated. Once again, angular differencesin input and output beams corre- spond to different waveguide modes. Input efficiencies as high as 7 1 percent have been observed [ 6 1 ] , [ 621. A fourth type of coupler, not illustrated, makes use of the tapered edges of the film itself for deflecting an incident beam into, or a transmitted beam out of, the film [63]. Efficiencies of 40 percent have been reported for inputcoupling.
ibl
0 -' \ '< ' I m - \ L
\ '< ' I m - \ L
I 151 I Fig. 8. Techniques for coupling light into 8 thin-filmwaveguide. (a) Endfue. (b) Prism coupler. (c) Grating coupler.
A feature which is common to both prism and grating cou- plers is that, in both, power is exchangedbetweena guided mode and a continuum of^ modes.^ In^ the grating coupler, the continuum consists of the radiation modes of the waveguide itself. Inthe prism coupler, thecontinuum modes comprise the angular spectrum of plane waves which can propagate in the uniform material of the prism. Equations which describe coupling from one modeto a continuum are
where K~~~ is the coupling constant, A p and B are the respec- tive mode amplitudes for continuum and guided modes, f i and Po are the propagation constants, and up is the mode density (number of modes per unit f i ) forthecontinuum. The plus and minus signs in the second equation refer, respectively, to contradirectional andcodirectional coupling. Theexpression ( 5 9 ) is simplyageneralization of ( 2 0 ) , which describes cou- pling between two modes. Since theinteraction in the case of the gratingcoupler in- volves modes of the same waveguide, the calculation of the coefficients K B ~ follows the treatment of mode coupling by surfacea corrugation, given in Section 111-E. The prism coupler, on the other hand, can be thought of as two parallel waveguides, in^ which^ one of^ the^ "waveguides"^ (the prism itself) is of infinite extent and will thus support a continuum of guided modes. In either case, if the incident wave travels in amedium of refractive index n o , the modefields in that medium will have the form
TAYLOR AND YARIV:OPTICS GUIDED WAVE 1053
component of polarization of the medium is simply
i
This relation wlil be used in calculating the behavior of the various types of modulator.
B. WaveguideModulation
Thin-film modulators can be classified according to the man- ner in which a change in the dielectric tensor A ~ i j affects the propagation of power in waveguide modes. Forexample, a phase gratingset upinthe mediumby anelectroopticor acoustooptic interaction can diffract aguidedlightbeam. In the following discussion, the devices which have been demon- strated or proposedare divided into thesecategories: 1) rela- tive phase control, 2 ) polarization rotation, 3) grating diffrac- tion, 4) waveguide cutoff, 5) interguidecoupling, and 6) attenuation. In the relative phase modulator, the phase of guided modes of one polarization is changed with respect to that of modes of orthogonal polarization. The phase change results from a perturbation which affectstheexpectation values, in some coordinatesystem, of diagonal elements of the dielectric tensor. No coupling between modes is involved. For example, the differential equation of theamplitude of themth TE mode of a planar waveguide is, from (35) and (72),
where
The solution to (73) is
The relative phase difference &$ between TE and TM modes which are in phase at z = 0 is thus
A@ = - PTE + -((AE,)mm - < A € y ) m m ) z (76)
[^4^1
0
where PTE and PTM are propagation constants in the absence
of theperturbation.For spatiallyuniform changes inthe dielectric tensor,
so that
Then (76) reduces to the more familiar result
Bulk electrooptic modulators [ 671 are relative phase devices, and the sameprinciple has been used in thethh-film con- figuration. Fig. 9(a) is a schematic illustration of such ade-
I PDLARIZED INPUT BEAM
WAVEGUIDE (^) ANALYZER I
WAVEGUIDE ELECTRDDE ,
I ELECTRIC FIELD LINES tb) Fig. 9. Twoconfigurations for relative-phase electrooptic modulator. (a) Planar waveguide. (b) Channel waveguide.
vice. Alinearly polarized lightbeam is coupled into a wave- guide inwhich the relative phase of orthogonal polarization components is controlled by an applied electric field.The phase-modulated outputfromthe waveguide becomes in- tensity-modulated upon passing throughan analyzer, which transmits only aselectedpolarization component. If the polarization of theoutput beam is oriented at 45' tothe principal axes, and the analyzer is set to pass light with the same polarization as theinput beam, theintensity Z of the modulated output is
where Zm is the maximum intensity. An induced relative-phase
change .of f n radians is required for 90 ' rotation in polariza-
tion, giving rise to completeintensitymodulation.Forin- termediate values of the phase change, the light emerging from the waveguide will be ellipticallypolarized.The phase dif- ference is proportional to the interaction length and, in the Linear electrooptic crystals generally used for these devices, to the applied electric field. Waveguide electroopticmodulation was first observed a decade ago in diffusiondoped GaP diodes [ 4 ] , and subse- quentefforts have improved the performance andunder- standing of these devices [ 681, [69]. Waveguiding confines the light to the high-field region in the vicinity of the reverse- biased p n junction. Relative-phase modulation has also been observed inepitaxiallayers of GaAs in a Schottky barrier configuration [91. Best results to date for aplanar structure were obtained in a reverse-biased GaAs-Alx Gal -, As double- heterostructurediode grown by liquid epitaxy [ 7 0 ]. Only
10 V were required for an-radian relative-phase shift in a
1-mm-long device. Recently, modulation in planar waveguides produced by outdiffusion of Li from LiNbOB substrates has been reported [ 7 1 1. The modulating voltage was applied to parallel stripe electrodes deposited on the waveguide surface. In another device, illustrated in Fig. 9(b), channel waveguides which provided beam confinementintwo dimensions were fabricatedby masked diffusion of Se into CdS and Cd into ZnSe [ 7 2 ]. In the polarization rotation modulator, power is exchanged between modes of orthogonal polarization, which are coupled via theoffdiagonalelements of the dielectric tensor. We
1054 PROCEEDINGS OF THE IEEE, AUGUST 1914
consider coupling between the mth TE mode and the nth TM mode of a waveguide, with respective amplitudes A , and B,. The coupled-mode equations, obtained from (35) and (72), are
These relations have the form (20), and the solution for co- directional coupling is given by (24), with
and
CouplingbetweenTE and TM modes is efficient only if
&E and PTM arenearly equal.In particular, if A = 0 and
A,(O) = 0, we have from (25)
A,(z) = B,(O) - sin I K l z
K
IK I
B,(z) = B,(O) cos I K I z. (79) If K is real, as will be the case in magnetoqpticmodulation imaginary), thepolarization of alinearly polarized incident beam will rotate as a function of propagation dis- tance. If K is imaginaryreal), themodulation is of the relative-phase type with reference to axes oriented at 45' to the x and y axes. Thuselectrooptic coupling, even by off- diagonal elements of 7, is of the relative-phase type. In bulk magnetooptic devices, a modulating field applied in the direction of optical propagation causes a rotation in the polarization vector of the beam. The same principle has been used to obtain modulation in an epitaxial iron-garnet film on a gadolinium-garnet substrate [ 7 3 ]. However, in the thin-film device it was necessary to employ the electrode structure il- lustrated in Fig. 10 t o achieve phase-matching between TE and TM modes of the waveguide. Current in the electrodes sets up
a magnetic field with spatial period A. The phase-match con-
dition becomes
with 1 an integer corresponding to the diffraction order. Over 50 percent of the powercoupled into a TM mode was con- verted to TE modes in the experimental device. The use of a highly birefrigent prism for output coupling provided for wide angular separation of the two polarization components., The polarization of an optical guided wave can be altered by a shear acoustic wave which is polarized normal to the direc- tion of optical propagation. Conversion between TE and TM modes has been produced by bulk acoustic waves in glass and in polyurethane waveguides [ 741 , [ 7 5 1. In the spatial diffraction modulator, a light beam is deflected by a periodic perturbation in refractive index (phase grating). Theperturbation can be produced by anelectrooptic, mag- netooptic, or acoustooptic interaction.An important property of this grating is its momentum vector kg, which is directed normal to andcoplanar withthe rulings and is equal in
W L A R I Z E DBEAM INPUT \ PRISM C W P L E R
WAVEGUIDE
1. 1 SERPENTINE PRlSU C O U P L E R STRUCTURE
I 1 I I
lit (bl Fig. 10. Magnetoopticthin-fdmmodulator. The serpentineelectrode structureproduces a spatiallyperiodicmodulating field,which ro- tates the polarization of the guided wave.
OPTICAL IWUT I TRAVELING INTERMGITAL
T
F F
Fig. 1 1. Collinear acoustoopticmodulator, in which an opticalbeam is diffracted out of the waveguide and into the substrate.
magnitude to 27r/A, where A is the gratingperiod. Forthe planar geometry of previous examples, we have two phase- match conditions:
(&>y * l ( k g ) y = ( 8 d ) y (80) and
where &- and & are propagation vectors for incident and dif-
fracted beams, and 1 is the diffraction order (an integer).
In the collinear diffraction modulator, fli, &, and kg are all
in the z direction, so that (81) is the condition for most ef-
ficientpower transfer.Experimentally, exchange of power between two guided modes of a glass waveguide on a quartz substrate hasbeen induced by acollinear traveling acoustic wave [ 7 6 ]. The theoretical treatment of mode coupling by a periodic perturbation in refractive index, which was given in Section 111-E, is appropriate to this experimental situation. In a similar experimentalarrangement, illustratedin Fig. 11, power was deflected out of a waveguide and into its substrate
by a traveling acoustic wave [ 77 1. The theoretical treatment
1056 PROCEEDINGS OF THE IEEE, AUGUST 1914
slab waveguide, illustratedin Fig. 1, inwhich n 2 - n l >> n 2 - n 3 , the cutoff condition for both TEand TM modes is approximately given by
d m - - = -t 1 x 4
(91 1
where t is the width of the waveguide. In a planar device in epitaxiallayers of GaAs, an applied elec- tric field reduced the waveguide refractive index difference to a value belowcutoff [88]. In the presence of the field, the input beam coupled into guided modes and was lost by spread- ing; with no field, part of the input power was confinedby the waveguide. Inanotherexperiment, a voltage applied be- tween parallel electrodesonthe surface of abulk LiNb crystal caused a localized increase in refractiveindex sufficient for waveguiding [89]. Modulation based on interguide coupling has not been demonstrated experimentally, but Fig. 14 illustrates how such a device might operate [go]. Inthe absence of an applied electric field, thewaveguides have equal propagation constants, and the length of the coupling region is chosen such that light entering one guide emerges fromtheother.The applied field causes a reduction in both coupling length and coupling efficiency by destroying the phase matching initially present. A field large enough to reduce the coupling length by a factor of two switches the output to the initially excited guide. Ac- cording to (53), theappropriate values of K and A for this type of device are K = n/2L and A = &TIL, where K is the coupling constant, A is the induced difference in propagation constants between the two waveguides, and L is the length of the coupling region. If the refractive index for one waveguide is increased by An and that of the other is decreased by the same amount, it follows that A 4nAnlX, so that
AnL (^6) x 4
Attenuation of aguided wave by induced absorptionor scattering will modulateitsintensity. Mathematically, the attenuation is a result of an imaginary component in diagonal elements of the dielectric tensor. Modulation by scattering has recently been demonstratedin thin-film waveguides of nematic liquid crystals [261,[271. An applied electric field causes turbulence in the liquid, which becomes less transparent to the guided light wave.
C. Modulator Power Requirements As mentioned earlier, one of the main reasons that thin-film modulators are of practical interest is that they offer the pos- sibility of low-power operationat wide bandwidths. The external power P , requiredforoperating a modulator ata bandwidth B is given approximately by
P, = B W (93)
where W is the power supplied from an external source to
switch the device on or off. In the electrooptic case,
where Ea is the applied electric field and the volume integral is over all space. The change in refractive index An is related to the applied field by
An = i n 3 r E a (95)
ELECTRODE WAVEGUIDE
INCIDENT BEAM Fig. 14.Roposedelectroopticmodulator or switch. (a)Theapplied field increases the refractive index of one waveguide and decreases that of the other. (b) The applied field causes the coupling length t o be shortened, so that the output is transferred from one guide t o the other.
where r is the linear electrooptic coefficient. For relative phase, spatial diffraction, and interguide coupling modulators, we have
AnL//h = 4. (96)
From these results, we arrive at the following estimate for the modulation power:
where w and t are the transverse dimensions of the waveguide, and L is themodulationpathlength. This formula assumes coincidence between the optical guided wave and the applied field. Intheacoustooptic case, theinducedindex change An is related totheacousticintensity Z by anacoustic figure of
merit M = n 6 p z / p V : , where p is the appropriate photoelastic
tensor element, V, is the acoustic velocity, and p is the mass density [ 9 1]. The relation is
By integrating the expression
W = j l d v
and recalling (96), we derive the result
P, 2 (E) M (?) B.
The geometrical factor ( w t / L ) thus appears in the expressions for both electrooptic and acoustoopticdevices, indicating that a reduction in required power can be obtained by minimizing the effective crosssectional area wt of the device. Electricalpower requirements of only about 0.2 mW/MHz have been reported for two of the planar electrooptic devices: doubleheterostructure GaAs-Al,Gal -, As diodes [ 701 and LiNbO3 diffused waveguides [71 I. This is about two orders of magnitude less than in bulk LiNb03 modulators [92], and
TAYLOR AND YARIV: GUIDED WAVE OPTICS 1057
Fig. 15. Schematic diagram of GaAs-Gal-,AlxAs injection laser.
a factor of six better than bulk LiTa03 [93] (which requires close temperaturecontrol).Furtherreduction of powercan be anticipated as ways of providinggoodbeamconfinement in twodimensions are developed.Therecentfabrication of channel waveguides in GaAs-Al,Gal-,As [401 is^ encouraging from this standpoint.
VII. LASER SOURCES
At the present time, the most suitable sources for integrated and guided wave optics are the semiconductor injection lasers. This is due to anumber of factors:1) Thebasicplanarep- itaxialfabricationtechniquesused to makethese lasers are very similar to those used to make a large varietyof other thin- filmwaveguidecomponents. 2)Theelectricalcurrentexcita- tion is compatible for interfacing with conventional electronic circuits. 3)Theycan be modulated,at least up to theGHz level, by current modulation [94]. 4) The material used most commonly in these lasers, GaAsGaAlAs, is also an excellent material forfabricatingawholearrayofothercomponents [95]. It hasindeedbeenused to makethin-filmwaveguides, modulators,directionalcouplers,anddetectors. This opens the way t o the possibility of an integrated optical technology based on a single material,GaAs,anditsrelated alloys, in analogy withtherole playedby silicon inintegrated elec- tronics. The typical structure [ 51 -[ 71 used in fabricating injection lasers is illustrated in Fig. 15. The laser mode is confined in a dielectric waveguide formed by the inner GaAs layer and the Gal - , A l x A s layers bounding it which possess a lower index of refraction. Holesinjected fromthep Gal-,AlxAslayer intotheinner GaAs layerrecombineradiatively with elec- trons injected from the n Gal-,Al,As layer thus providing the coherent laser radiation.Thethreshold 'current intensityde- creases with smaller thicknessesof both the opticalConfinement distance and also of the distance t o which the electrons and holesareconfined. More recentadvances [96],[97] use additionallayersfor improved electronandoptical confine- ment.Continuousoperationwithlifetimes of afewthou- sand hours hasbeen demonstrated [98]. These lifetimes will have to increase by at least an order of magnitude before these
1 W s EPITAXIALLAYER G%,filo ,k EPITAXIAL LAYER
Fig. 16. Corrugatedwaveguide confauration for distributedfeedback laser in GaAs-GaXAl1,As.
A recentdevelopmentwhichaddresses itself to apossible solution to these problems is that of the corrugated waveguide laser [ 131. Sucha waveguide is illustratedbyFig.16.The feedback necessary for laser oscillation is provided, not by the conventionalendreflectors,butbya series of corrugations separated by half a wavelength which, in a manner equivalent to a multilayer dielectric mirror, reflect a propagating mode, thus coupling the forward and backward modes. The principle involvedhere, one of distributedfeedback, was formulated originallybyKogelnikandShank [ 121andapplied by them to dye lasers. Tounderstandtheprincipleinvolvedinthis classof lasers we may go back to the problem of contradirectional coupling considered in Section 111. If the coupling between the forward and the backwardmodes is due to surfacecorrugations,the coupling coefficient is given by (44). The coupled-mode equations become
dB dz
where A = 20 - 27r/A, 0 is thepropagationconstant of the unperturbedmode,and A is the periodof thecorrugation. Since here we deal with amplifying media rather than with a
passive guide, a is the distributed gain constant.
Assuming a solution of the form
A ( Z ) = a ( z ) e - i A @
B ( Z ) = b ( z ) e i A f l z (102)
where 2 4 e A, leads t o
da dz
- = - d b ( i ~ a)- b + x a. (^) (103) dz
Let us assume for a moment that a wave with amplitude b(0)
is incident at z = 0 on the cormgated section of length L , as shown in Fig. 5 , and we want to findtheamplituderatio a(O)/b(O) of the reflected t o incident waves at the input z = 0. This expression is obtainable directly from (30), except that, according to (103), we need to replaceeverywhere iA/2 by iA0 - a. The result is thus
usedbediodes can in mostpractical opticalsystems. 4 0 ) - K sinh (7L)
A number of problems s t i l l remain to be solved in injection (1 04) lasers inaddition to that of thelifetime. Theseincludethe
control of thetendancyto oscillate simultaneouslynum-a in + + ( - a + i M I 2. (105)
ber of modes and the difficulty of mirror fabrication in laser sourcesdesigned t o couplemonolithicallyintoothercom-Unlike (301, theratio a(O)/b(O) can now,for certain values of ponentsaon single opticalchip. 4 and a, infiity.goto This corresponds to afinitereflected
-- (^) - b(0) (-a + iu) sinh (715) + 7 cosh ( 7 L )
TAYLOR AND YARIV: GUIDED WAVE OPTICS 1059
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