Aperiodic Distributed-Parameter Waveguides forIntegrated Optics

K. 0. Hill

Aperiodic distributed-parameter waveguides are considered for use as reflectors and filters in integratedoptics. Coupled-mode theory is used to predict the effects on modal propagation of aperiodic perturba-tions induced in a slab waveguide. In particular, it is shown that aperiodic perturbations can couple for-ward-traveling bound modes to backward-traveling bound modes; and that the frequency dependence ofsuch a coupling process can be controlled through the aperiodicity. It is suggested that aperiodic cou-pling may be used to realize optical-waveguide reflectors and filters having predetermined spectral-re-sponse characteristics. As examples, the spectral-response characteristics of a 'narrow band-stop filter, abroadband-stop filter, and a narrow band-pass filter are calculated and presented.

1. Introduction

Recently there has been much effort directedtoward the development of various guided-wave op-tical components for use in integrated optics. Inte-grated optics components are pictured as performinga large class of optical operations, many of whichbear a close similarity, in principle, to operations ac-complished by microwave elements and circuits.Among the elements that play a role in integratedoptics are reflectors and filters.

The use of aperiodic perturbations of certainwaveguide parameters as reflectors and filters in in-tegrated optics is the subject of this paper. It isshown that reflectors and filters can be designed thathave a tailored frequency-response characteristic,and examples of such designs are given. Aperiodicperturbations share the mode-coupling property ofperiodic perturbations, but additionally allow controlof the frequency-response characteristic of the ele-ment through the aperiodicity. The periodic pertur-bation of optical-waveguide parameters along the di-rection of propagation of electromagnetic energy in awaveguide has been shown to result in reflection ofthe propagating energy when certain resonance con-ditions obtain.' Laser action using this type of dis-tributed feedback has been previously demonstratedin thin-film lasers.2 4 Also, the reflecting propertyof a periodic perturbation of a waveguide parameterhas been used to fabricate thin-film waveguide filterswith a high-frequency cutoff characteristics

The author is with the Communications Research Centre, De-partment of Communications, Ottawa, KiN 8T5.

Received 2 January 1974.

In general, strong reflections occur only when acharacteristic period of the perturbation is such thatit couples two contradirectional modes of the wave-guiding structure. In distributed-feedback lasers,these two modes consist of forward- and backward-traveling bound modes, whereas in high frequencycutoff filters operating in the cutoff regime, the for-ward mode is a bound mode of the waveguide, andthe backward mode is a radiation mode. The cou-pling from bound mode to bound mode is stronglyfrequency dependent and narrow band in character,whereas that from bound mode to radiation modeexhibits a sharp transition region that begins at thatfrequency at which coupling first occurs.

Many of the principles governing the behavior ofaperiodically perturbed waveguide devices can beadequately illustrated by studying bound mode tocontradirectional bound mode coupling in these de-vices. In the following, we treat exclusively thistype of coupling and give the conditions that mustbe satisfied for strong mode-to-mode coupling via anaperiodic perturbation, As examples, we give de-signs based on coupled-mode theory for (1) a narrowband-stop filter; (2) a broadband-stop filter; and (3)a narrow band-pass filter.

II. Theory and Results

Coupled-mode theory for treating mode conversioncaused by stationary imperfections of a dielectric-slab waveguide has been given by Marcuse.6 Yarivhas treated the general time-dependent case.7 Be-cause we are concerned here with stationary aperiod-ic perturbations, we use results given by Marcuse asour starting point and employ his notation for ease ofreference. Time-dependent coupled-mode theorymust be used, however, to treat certain classes of de-vices, for example, those based on acoustooptic in-

August 1974 / Vol. 13, No. 8 / APPLIED OPTICS 1853

teractions. These devices may find application inthe implementation of tunable optical filters due tothe ease whereby the period of the perturbing acous-tic field can be controlled. Aperiodicity can be in-troduced by frequency modulating the acousticsource.

Let us consider an unperturbed symmetric-slabwaveguide in free space. We assume that the funda-mental solutions to Maxwell's equations E and E, re-spectively, the bound and continuous modes of theunperturbed waveguide, are known. Under the in-fluence of a time-independent perturbation, the fieldin the perturbed waveguide Ey can always be ex-pressed in terms of a spatially varying linear combi-nation of this complete orthogonal set:

E = E C(Z)E + fg(pz)E(p)dp, (1)n Jo

where the bound and continuous unperturbed-modecontents of Ey are given, respectively, by the discretecoefficients, Cn and the continuous coefficient g as afunction of distance z measured along the directionof propagation in the slab waveguide. The firstsummation in Eq. (1) is over the bound mode set,and the second summation is over the even and oddmodes of the continuum.

The refractive-index distribution describing theperturbed waveguide is written

n2(n,z) = n 2(x) + An2(x,z), (2)

where n2 (x,z) is the perturbation component,no2(x) is the refractive-index distribution of the un-perturbed slab waveguide, and x is a spatial coordi-nate measured along a direction normal to the slab.The solutions to the wave equation with n2 setequal to zero are thus En and e.

The modal coefficients C and g in Eq. (1) havebeen given by Marcuse in the form of a set of inte-gral equations.6 The set may be simplified by as-suming that the perturbation n2 only couplesbound modes. Then the second summation in Eq.(1) can be neglected, and Cm may be shown to begiven by

Cm Am + Bm exp(2iomz)

+ 2i f {exp[2i0m(z - l)] - 1}F,,()dt, (3)

where

Fm(Z) =2wji [ZCn(Z) f Em*An2Endx], (4)

fim is the z-direction propagation constant of themth bound mode of the unperturbed slab waveguidewith the integer m running over the enumerablebound mode set, k2 = W2Eo/gzo, and A and B are con-stants to be determined from initial conditions. Thepower-normalization factor

P = 12 lx

is constrained to be independent of mode-order num-ber.

The analysis to this point is sufficiently general toallow the treatment of both intramode and inter-mode coupling. For simplicity, we restrict the re-maining comments to intramode coupling. The per-turbation n2 is thus assumed not to change themode-order number during the coupling process.Consider that for a restricted range of frequenciesonly the qth mode of the structure interacts stronglywith the perturbation. (We find below the criterionthat must hold for a strong interaction of this type tooccur.) Thus, we can write

FBz) = -k2Cq(Z)f Eq*n2EqdX JEq |2dx. (5)

The coefficient Cq contains information about boththe forward- and backward-traveling components ofthe qth mode. An intramode perturbation in thecontext defined above serves to couple these twocomponents together to the exclusion of all others.To derive the coupled-mode integral equations ap-plicable in the case of intramode coupling, let uswrite the mode coefficient for the qth mode as thesum of two components parts,

Cq = Cq + Cq, (6)

where Cq+ is associated with the forward-travelingcomponent of the qth mode and will be a slowlyvarying complex function of z whenever the interac-tion length associated with the perturbation is longcompared to the wavelength of the electromagneticfield in the guide. In contrast, Cq - is a complexfunction characterized by a slowly varying amplitudeand a rapidly varying phase, being associated withthe backward-traveling component of the qth mode.By definition we write

Cq- = Cq;s xp(2iqz), (7)

where Cq;s- represents the slowly varying amplitudeof the backward-traveling component of the qthmode. With the aid of this definition of Cq;s- andEqs. (3), (5), and (6), we find

(8)Cq+ A - 2i/3q [F,(C)]sd

and

C-'s B + fig zw[exp(-2i0,1 )F(t)],dt, (9)

where []s means that the slowly varying componentof the quantity in the brackets is to be taken as theintegrand.

In Eqs. (8) and (9), A and B are determined byinitial conditions. Let us assume that the perturba-tion An2 extends in the guide over the interval z fromO to L, and that the qth mode is incident on this re-gion as a forward-traveling mode. In the half-space

1854 APPLIED OPTICS / Vol. 13, No. 8 / August 1974

z < 0 the mode coefficient Cq generally containsboth forward and backward components. For z > Lonly the forward component will exist. Therefore,we can write the boundary conditions

Cq+(O) = A, (10)

as well as

Cq;sf(L) = 0

and so determine A and B with the aid of Eqs. (8)and (9). The coupled-mode integrals' given by Eqs.(8) and (9) with boundary conditions (10) and (11)can be solved numerically, at least in certain in-stances, by using an iterative technique. An ade-quate set of regularly spaced values for C+ andCq;.- is assumed in the interval 0 to L, and this setis used to determine [F(D)]s and [exp(-2iq¢)F(v)]swith the aid of Eq. (5). Integrating these two func-tions numerically as required by Eqs. (8) and (9)yields refined values for Cq- and Cq,.s' that are thenused as the new starting point in the process. Theprocess is continued until sufficient consistency isobtained and additional refinement ceases to offersignificant improvement in the solution. This nu-merical process can be conveniently started by as-suming initial values Cq+ = A and Cq;s- = 0 for all 0<z <L.

A broadband reflector or broadband-stop filtermay be implemented by a refractive-index perturba-tion in the guide interval 0 z L that is of theform

An2 = Ano2{exp[if (a + z)z] + c.c.}, (12)

where the spatial frequency f(a + yz) of the period-icity varies linearly with z. (A refractive index per-turbation is used for illustrative purposes only; inpractice, filters are more likely to be implemented ascorrugated waveguides.) For this refractive indexperturbation, Eqs. (8) and (9) become

Cq+(z) A + (k2 o) fJ Cq;,f(C) exp{[2i3q

- if(a + )]C}dC (13)and

Cq;s (z) B - (kS2

) ,C~Q+(C) exp{[if(Y

+ 7y) - 2iq]k}dC, (14)

respectively. The integrals above are maximizedwhen the range of integration contains a region of in-tegrand stationary phase. The position of the centerof the region of stationary phase with respect to theperturbed region 0 z L of the guide is thereforean important indicator of the strength of the interac-tions that can occur through the aperiodic perturba-tion between the forward and backward componentsof the qth mode. The region of stationary phase iscentered at zr where

Zr = (2/3q fiy)/2fy. (15)

The width of this region AZr is also of significanceand is given by

AZ r = (2 Trfy)' . (16)

It can be readily shown that for those cases whereAZr << L, strong interactions occur only when 0 <Zr < L, but for AZr > > L they will occur for

4zL < 1.r_ Z _

(17)

These are the strong-interaction criteria for linearlyaperiodic refractive index perturbations and intra-mode coupling; the concepts used in their derivationmay be readily extended to more general perturba-tions. The sense in which we have just used theterm strong interactions presupposes not only thatthe foregoing criteria are met but also that both theperturbation and the effective length of the per-turbed region are of sufficient magnitude to effectsuch interactions. The parameter S defined by

S = [(k2 no 2)/2fq] *Z for Zr << L (18)

and

S = [(k2 Ano2 )/2/3q1 L for Az, >> L (19)

is a measure of the strength of the perturbation atresonance. We define the reflectivity of the per-turbed portion of the guide R by

(20)R = BA|2and the transmittance T by

T = 1 -R. (21)

Note that, from Eqs. (13) and (14), R as a functionof 6 is under certain circumstances closely related tothe Fourier transform of the perturbation; in thosecases the form of R is very similar to that of thespectrum of the perturbation, an attribute that canbe usefully employed in design work.

Figure 1 is a plot of R vs a for two reflector de-signs, one narrow band and the other broadband;(k2 Afno2 /20q) 19.1 cm-' in both cases for in therange illustrated in the figure. For the narrow banddesign, f = 12.55 X 104 cm-', a = 1, y = 10-6 cm-l(i.e., - 0), and L = 7.12 X 10-2 cm. For thebroadband design, f = 12.39 X 104 cm-', a = 1.01, y= 0.01 cm-, and L = 0.3 cm. We note that S isthe same for both designs, consequently at the centerof their respective bands, the two designs have ap-proximately the same reflectivity.

Figure 2 is a plot of R vs for a bandpass filterdesign. The parameters are the same as for thebroadband design described above, but the phase ofthe spatial aperiodicity was changed suddenly by180° in the center of the structure giving rise to thepassband. This type of filter could find use as amode-controlling etalon in thin-film lasers (for ex-ample, semiconductor lasers).

It is worth noting that for sufficiently large valuesof S, the numerical procedure described above forsolving the coupled mode integral equations may not

August 1974 / Vol. 13, No. 8 / APPLIED OPTICS 1855

(I )

6.27 6.28 6.29a (UNITS OF 10

4cms'l)

Fig. 1. Variation of the reflectivity R as a function of the z-di-rection propagation constant a for (a) narrow band-stop filter and(b) broadband-stop filter. The design constants are found in the

text.

1.0-

1-l-

R

16t-

103

624 I I I 6.25 6.26 6.27 6.28 629,O (UNITS OF 104 CM-I)

I I6.30 6.31

Fig. 2. Variation of the reflectivity R as a function of the z-di-rection propagation constant for a narrow band-pass filter.

The design constants are found in the text.

converge. A method for obtaining convergence maybe to turn on the perturbation gradually during thenumerical process and obtain stepwise convergenceuntil the desired valve of S is reached.

111. Discussion

There are many other ways that can be envisagedof utilizing an aperiodic distributed-parameter wave-

guide to perform the functions of filtering and re-flecting guided electromagnetic energy. For exam-ple, structures where the aperiodic perturbation cou-ples a bound mode to a radiation mode may proveuseful in practice to control the sharpness of cutoff inhigh-frequency cutoff filters. Also, broadband modeconverters that change the mode-order number of aforward-traveling bound mode can be designed usingaperiodically perturbed waveguides. This lattertype of mode control used in conjunction with a sud-den transition in waveguide parameters can serve asa filter. Consider a device where the input-side seg-ment of the guide supports two modes and the out-put-side segment supports one mode that is similarin transverse distribution to the lowest-order modesupported by the input guide. Such a filter gradual-ly converts the first-order mode propagating in theupstream guide, by means of an appropriate pertur-bation, to the second-order mode of the guide.When this second-order mode is incident on the sin-gle-mode output guide, it will not excite the boundmode efficiently, losing energy to the continuum.Mode conversion will, of course, occur in the inputguide only provided that the frequency of the elec-tromagnetic field obeys the resonance condition de-manded by the perturbation. The spectral responseof the filter is bandstop, but it can readily be oper-ated as a bandpass filter by initially exciting the sec-ond-order mode of the input guide. The aperiodicityof the perturbation provides the design flexibility forshaping the frequency-response characteristic of thefilter. The filter has the advantage in its implemen-tation of a low-frequency aperiodicity and the disad-vantage of a two-guide design.

In summary, periodically perturbed waveguidingstructures offer the possibility of obtaining reflectorand filter elements with tailored frequency-responsecharacteristics for use in integrated optics. The useof coupled-mode theory has provided a means of de-signing such elements and lends insight into modecoupling phenomena. Refinement of the methodmay allow the design of matched filters for disper-sion compensation.8

Since original submission of this paper the manu-facture and evaluation of high-quality periodic cor-rugated waveguide filters have been reported. 9 Bycarefully controlling the phase fronts of one or bothof the interfering light beams, as used in Ref. 9 towrite periodic corrugations, aperiodic feedbackstructures can be implemented.

References1. H. Stoll and A. Yariv, Opt. Commun. 8, 5 (1973).2. H. Kogelnik and C. V. Shank, Appl. Phys. Lett. 18, 152 (1971).3. P. Zory, Appl. Phys. Lett. 22, 125 (1973).4. K. 0. Hill and A. Watanabe, Appl. Opt. 12, 430 (1973).5. F. W. Dabby, M. A. Saifi, and A. Kestenbaum, Appl. Phys.

Lett. 22, 190 (1973).6. D. Marcuse, Bell Syst. Tech. J. 48, 3187 (1969).7. A. Yariv, IEEE J. Quantum Electron. QE-9, 919 (1973).8. A. Watanabe, Communications Research Centre, Department of

Communications, Ottawa; private communication.9. D. C. Flanders, R. V. Schmidt, and C. V. Shank, in OSA Inte-

grated Optics Conference, New Orleans, Digest of TechnicalPapers (Optical Society of America, Washington, D.C., 1974).

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