2620 OPTICS LETTERS / Vol. 30, No. 19 / October 1, 2005

Optical circuit design based on a wavefront-matching method

T. Hashimoto, T. Saida, I. Ogawa, M. Kohtoku, T. Shibata, and H. TakahashiNTT Photonics Laboratories, Nippon Telegraph and Telephone Corp., Atsugi 243-0198, Japan

Received April 7, 2005; revised manuscript received May 29, 2005; accepted May 31, 2005

We propose an optical circuit design method for coherent waves as a boundary value problem. The methodproduces a very compact circuit in which the refractive index pattern is automatically synthesized for giveninput and output fields with a numerical calculation. We employ the method to design a 1.3/1.55 �m wave-length demultiplexer and also describe the features of a circuit generated by use of the method. © 2005Optical Society of America

OCIS codes: 230.7390, 290.4210.

The rapidly increasing power of computer technologyenables us to calculate a refractive index distributionas a hologram or as a solution to a kind of inverseproblem, thus allowing us to obtain a desired outputlight field from a given input field. We use themethod with spatially spreading diffractive elements,where we expand the light field in a lateral directionor adopt a weak interaction along the propagationaxis to obtain a good description of the light field’s be-havior with a linear calculation (e.g., Refs. 1 and 2),and this makes the optical circuit large. In this paperwe describe a novel optical circuit design method thatprovides us with a compact refractive index distribu-tion that transfers a given input optical field to a de-sired output optical field. We also discuss the differ-ence between this circuit and conventional circuitswith respect to the circuit construction procedure.

To simplify the description, we assumed that lightpropagates in a direction z and ignored the backwardreflection. Pairs of cross-sectional fields at the inletand outlet boundary �z=zin,z=zout� of the object re-gion are given as the symbols �j�x� ,�j�x� (j=1, 2,…),where x is the lateral coordinate and j is the index ofthe pairs, and we consider a rectangular region [asshown in Fig. 1(a), for example]. We also define�j�x ,z� , �j�x ,z� (j=1, 2,…) as the field values at posi-tion �x ,z� propagating forward, from the inlet bound-ary, and propagating back (or time reversed), fromthe outlet boundary, respectively. The fields have aunit norm and propagate in accordance with theequation

i�z��x,z� = L�x,z���x,z� �1�

�1

2��− �x

2 + �2

− k02n�x,z�2���x,z�, �2�

where we consider a monochromatic scalar wave forsimplicity. Here k0 is the wavenumber in vacuum,and � is the propagation constant in the paraxial ap-proximation; n�x ,z� is the refractive index distribu-tion, described with the variation �n�x ,z� from thereference refractive index value nref=� /k0 as n�x ,z�

3

=nref+�n�x ,z�. L is an evolution operator, and when

0146-9592/05/192620-3/$15.00 ©

�n�x ,z� is very small, �L /��n�−k0. Now we considerthe energy

E�z� = �j

��j�x,z� − �j�x,z��2 �3�

that we intend to minimize by changing the refrac-tive index distribution to obtain the desired circuit.Using the above expression, we obtain the variationat position p in the energy,

�E

�n�p�= �

j

���j��j + 2 Re ���j��j + ���j��j

�n�p�, �4�

� q�p� 2k0�j

Im �j�p�*�j�p�, �5�

where the angle brackets denote the inner product inthe Hilbert space with � and � regarded as functionsof x. We omit z, since the terms for a fixed refractiveindex distribution are independent of the position z.We can omit the variations of the norms, since thenorms are invariant for a change of the refractive in-dex. Applying a first-order Born approximation to�� ��, we obtain Eq. (5). The coefficient on the right-hand side is derived from �L /��n�−k0. To minimizethe energy E, we change the refractive index distri-bution n�p� as

n�p� ⇐ n�p� − ���q�p� �6�

and calculate �j�x ,z� , �j�x ,z� with the replacedn�x ,z�. We repeat the procedures until we reach thelocal minimum condition q�0, where ��� is a positiveconstant and the condition can be interpreted asmatching the phases of the wavefronts propagatingfrom the inlet and outlet boundaries at each positionin the region. We can obtain the general proceduresimply by replacing the fields with the incoming andoutgoing fields at position p in Eq. (6).

Figure 1 shows a numerical demonstration of themethod applied to an optical circuit design by usingsilica planar light-wave circuit (PLC) technology,where the circuit has a fine planar binary patterncorresponding to the refractive index distributionand is fabricated with semiconductor fabrication

4,5

techniques.

2005 Optical Society of America

nm

October 1, 2005 / Vol. 30, No. 19 / OPTICS LETTERS 2621

To take the actual circuit into consideration and toperform the calculation, we introduced a refractiveindex distribution with a binary value and improvedthe computational complexity as follows. We considertwo equations for the evolution of each j th pair,

i�z�j = L�j, �7�

i�z�j = �L + �L��j, �8�

with boundary conditions at different positions de-scribed as �j=�j�x� �z=zout� ,�j=�j�x� �z=zin�. Wechose the variation value of the evolution operator�L�−k0 ��n to obtain a monotonous decrease in thedifference between these two fields along the z direc-tion, described as

�zE�z� = 2k0�j� ��n Im �*� dx 0. �9�

Although this provides a replacement proceduresimilar to Eqs. (5) and (6), we calculate only one fieldat a refractive index change and alternate the roles ofthe fields in the calculation in the backward and for-ward directions. In contrast, the previously used pro-cedure requires the calculation of both fields after achange in the refractive index, and therefore ourmethod halves the computational complexity. To ap-ply the method to a discrete refractive index value,we adopt a replacing algorithm that changes the signin n�x ,z�=nref±s according to the opposite sign of q inEq. (5), where s is half the refractive index differencebetween the binary index distribution and nref is thecenter value. Using the above algorithm, we calcu-lated an optical 1.31/1.55 �m wavelength demulti-plexer as shown in Fig. 1(a) after 20 reciprocations ofthe index distribution calculation, with the initial in-dex distribution taking random binary values. Fig-ures 1(b) and 1(c) show the intensities of the1.31/1.55 �m wavelength fields propagating in thecircuit. Here we used the finite differential beampropagation method to obtain the optical fields, andthe circuits are designed for a silica PLC formed of

Fig. 1. Calculated result designed for a 1.31/1.55 �m wavnref+s; black, nref−s. (b) Light-wave field intensity, �=1310

1 �m1 �m square pixels with a binary refractive

index distribution of nref=1.45 and s=nref0.0075/2.The inlet boundary fields for �=1.31 and �=1.55 �m wavelength signals are 7 �m diameterGaussian fields centered at x=0 �m, and the outletboundary fields are 7 �m diameter Gaussian fieldscentered at x=−20 and x=20 �m, respectively. Thecalculation mesh was 0.11 �m in both the propaga-tion and the lateral directions. In a usual waveguidetype PLC, the propagating light is confined aroundthe core of the waveguide, whereas the fields in Figs.1(b) and 1(c) are distributed in the circuit and aremultiple-scattered or multiple-diffracted at the re-fractive index pixels. However, each field convergesat the outlet boundary through media with almostrandom refractive index distributions. Figure 2shows the change in the circuit output performancewith the calculation reciprocations. The transmit-tances of the output and the cross talk are the over-lap integrals between the output field with the givenwavelength and the set outlet boundary field andthat of the wrong pair, respectively. In terms of opti-cal circuit design, we realized a much more compactoptical circuit than that of usual waveguide PLCs,and this circuit achieved very high conversion as amultilayered kinoform.

gth demultiplexer. (a) Refractive index distribution. White,. (c) Optical field intensity, �=1550 nm.

Fig. 2. Output change along calculation reciprocation

elen

count.

2622 OPTICS LETTERS / Vol. 30, No. 19 / October 1, 2005

These properties can be explained by investigatingthe construction procedure. The procedure describedwith Eqs. (5) and (6) can be summarized as follows.We considered a system where the refractive indexdistribution has a degree of freedom. We repeatedlyapplied the refractive index distribution change inEq. (5), with the renormalized optical field propagat-ing through the refractive index distribution, andmade the optical field take the higher-order interac-tion with the changed refractive index distribution inthe wave propagation calculation. Because of this thesystem can be regarded as being different from bothan optical waveguide and a conventional volume ho-logram. This is because a typical hologram stores im-ages as linearly overloaded refractive indexmodulations,2 whereas we consider all of the renor-malized optical fields to be consistently combinedwith the refractive index in the local minimum con-dition in our system. This makes the circuit compactand results in high conversion efficiency. As regardsdiffraction tomography, an approach that takes mul-tiple scattering into consideration has already beenproposed.6 Although this approach corresponds to ourapproach expressed as Eqs. (5) and (6), a system con-sidered in terms of tomography is usually determinedalmost entirely by given scattering data, and thissuppresses the internal interaction among the refrac-tive index values in the numerical calculation. On theother hand, our object has a very small number ofinput–output pairs; this yields an internal degree offreedom of the refractive index values at each point,and the system spontaneously converges with a re-fractive index distribution that suppresses the degreeof freedom of the local phase change of the light-wavefield. The suppression of the phase change of thelight-wave field is observed as a spectrum, as shownin Fig. 3. Here the phase of the local light-wave field

Fig. 3. Wavelength dependence of transmittance.

changes with changes in wavelength, and this de-grades the transmittance when the wavelength devi-ates from its designed value.

We can expect other characteristics, for example,the number of channels that can be stored (input–output pairs), to be explained in terms of a neuralnetwork system, since the system can be regarded asa neural network, and the study area has various ap-proaches for estimating such properties. The corre-spondence can easily be shown by discretization ofthe evolution equation (1) as follows. With everycross-sectional refractive index layer in a discretizedstep of the z direction in Eq. (1), the optical wavetravels with the superposition of the diffraction fromthe neighboring part caused by the Laplacian in Land taking its phase weight from the refractive indexvalue. Therefore our system can be regarded as amultilayered linear complex-valued perceptron. Itcan also be regarded as a Boltzmann machinelikenetwork in which the index values are interactingwith the optical fields. Equations (5) and (6) natu-rally correspond to the error backpropagationmethod, where the error propagates as the phase dif-ference between the inlet and the outlet opticalwaves as shown in Eq. (5). Figure 2 corresponds tothe learning curves. This formalism provides us witha statistical physics viewpoint and clarifies the differ-ence between the optical circuits that were discussedabove by using those phase diagrams such as those inRef. 7. Here we should consider that the boundarycondition (input–output pair field) affects the systemin the same way as an external field.

We hope that the optical circuit described abovewill be investigated from various viewpoints andused to design optical circuits such as those fabri-cated with PLC technology.

The authors thank T. Kitoh, M. Yanagisawa, Y.Abe, S. Asakawa, M. Kobayashi, R. Nagase, S. Su-zuki, T. Ohyama, T. Kitagawa, M. Okuno, Y. Hibino,and H. Toba.

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