Optical circuit design based on a wavefront ... - OSA Publishing

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

Optical circuit design based on a wavefrontmatching method T. Hashimoto, T. Saida, I. Ogawa, M. Kohtoku, T. Shibata, and H. Takahashi NTT 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 method produces a very compact circuit in which the refractive index pattern is automatically synthesized for given input and output fields with a numerical calculation. We employ the method to design a 1.3/ 1.55 ␮m wavelength demultiplexer and also describe the features of a circuit generated by use of the method. © 2005 Optical Society of America OCIS codes: 230.7390, 290.4210.

The rapidly increasing power of computer technology enables us to calculate a refractive index distribution as a hologram or as a solution to a kind of inverse problem, thus allowing us to obtain a desired output light field from a given input field. We use the method with spatially spreading diffractive elements, where we expand the light field in a lateral direction or adopt a weak interaction along the propagation axis to obtain a good description of the light field’s behavior with a linear calculation (e.g., Refs. 1 and 2), and this makes the optical circuit large. In this paper we describe a novel optical circuit design method that provides us with a compact refractive index distribution that transfers a given input optical field to a desired output optical field. We also discuss the difference between this circuit and conventional circuits with respect to the circuit construction procedure. To simplify the description, we assumed that light propagates in a direction z and ignored the backward reflection. Pairs of cross-sectional fields at the inlet and outlet boundary 共z = zin , z = zout兲 of the object region are given as the symbols ␺j共x兲 , ␸j共x兲 (j = 1, 2,…), where x is the lateral coordinate and j is the index of the pairs, and we consider a rectangular region [as shown in Fig. 1(a), for example]. We also define ␺j共x , z兲 , ␸j共x , z兲 (j = 1, 2,…) as the field values at position 共x , z兲 propagating forward, from the inlet boundary, and propagating back (or time reversed), from the outlet boundary, respectively. The fields have a unit norm and propagate in accordance with the equation i⳵z␺共x,z兲 = L共x,z兲␺共x,z兲 ⬇

1 2␤

共1兲

关− ⳵2x + ␤2

− k02n共x,z兲2兴␺共x,z兲,

共2兲

where we consider a monochromatic scalar wave for simplicity. Here k0 is the wavenumber in vacuum, and ␤ is the propagation constant in the paraxial approximation; n共x , z兲 is the refractive index distribution, described with the variation ⌬n共x , z兲 from the reference refractive index value nref = ␤ / k0 as n共x , z兲 = nref + ⌬n共x , z兲. L is an evolution operator,3 and when 0146-9592/05/192620-3/$15.00

⌬n共x , z兲 is very small, ⳵L / ⳵⌬n ⬇ −k0. Now we consider the energy E共z兲 = 兺 兩␸j共x,z兲 − ␺j共x,z兲兩2

共3兲

j

that we intend to minimize by changing the refractive index distribution to obtain the desired circuit. Using the above expression, we obtain the variation at position p in the energy,

␦E ␦n共p兲

=兺

␦具␸j兩␸j典 + 2 Re ␦具␸j兩␺j典 + ␦具␺j兩␺j典 ␦n共p兲

j

⬀ q共p兲 ⬅ 2k0 兺 Im ␸j共p兲*␺j共p兲,

,

共4兲 共5兲

j

where the angle brackets denote the inner product in the Hilbert space with ␸ and ␺ regarded as functions of x. We omit z, since the terms for a fixed refractive index distribution are independent of the position z. We can omit the variations of the norms, since the norms are invariant for a change of the refractive index. Applying a first-order Born approximation to 具␸ 兩 ␺典, we obtain Eq. (5). The coefficient on the righthand side is derived from ⳵L / ⳵⌬n ⬇ −k0. To minimize the energy E, we change the refractive index distribution n共p兲 as n共p兲 ⇐ n共p兲 − 兩␣兩q共p兲

共6兲

and calculate ␺j共x , z兲 , ␸j共x , z兲 with the replaced n共x , z兲. We repeat the procedures until we reach the local minimum condition q ⬃ 0, where 兩␣兩 is a positive constant and the condition can be interpreted as matching the phases of the wavefronts propagating from the inlet and outlet boundaries at each position in the region. We can obtain the general procedure simply by replacing the fields with the incoming and outgoing fields at position p in Eq. (6). Figure 1 shows a numerical demonstration of the method applied to an optical circuit design by using silica planar light-wave circuit (PLC) technology, where the circuit has a fine planar binary pattern corresponding to the refractive index distribution and is fabricated with semiconductor fabrication techniques.4,5 © 2005 Optical Society of America

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

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Fig. 1. Calculated result designed for a 1.31/ 1.55 ␮m wavelength demultiplexer. (a) Refractive index distribution. White, nref + s; black, nref − s. (b) Light-wave field intensity, ␭ = 1310 nm. (c) Optical field intensity, ␭ = 1550 nm.

To take the actual circuit into consideration and to perform the calculation, we introduced a refractive index distribution with a binary value and improved the computational complexity as follows. We consider two equations for the evolution of each j th pair, i ⳵ z␸ j = L ␸ j ,

共7兲

i⳵z␺j = 共L + ␦L兲␺j ,

共8兲

with boundary conditions at different positions described as ␸j = ␸j共x兲 共z = zout兲 , ␺j = ␺j共x兲 共z = zin兲. We chose the variation value of the evolution operator ␦L ⬃ −k0 ␦⌬n to obtain a monotonous decrease in the difference between these two fields along the z direction, described as

⳵zE共z兲 = 2k0 兺 j

冕

␦⌬n Im ␸*␺ dx 艋 0.

共9兲

Although this provides a replacement procedure similar to Eqs. (5) and (6), we calculate only one field at a refractive index change and alternate the roles of the fields in the calculation in the backward and forward directions. In contrast, the previously used procedure requires the calculation of both fields after a change in the refractive index, and therefore our method halves the computational complexity. To apply the method to a discrete refractive index value, we adopt a replacing algorithm that changes the sign in n共x , z兲 = nref ± s according to the opposite sign of q in Eq. (5), where s is half the refractive index difference between the binary index distribution and nref is the center value. Using the above algorithm, we calculated an optical 1.31/ 1.55 ␮m wavelength demultiplexer as shown in Fig. 1(a) after 20 reciprocations of the index distribution calculation, with the initial index distribution taking random binary values. Figures 1(b) and 1(c) show the intensities of the 1.31/ 1.55 ␮m wavelength fields propagating in the circuit. Here we used the finite differential beam propagation method to obtain the optical fields, and the circuits are designed for a silica PLC formed of 1 ␮m ⫻ 1 ␮m square pixels with a binary refractive

index distribution of nref = 1.45 and s = nref ⫻ 0.0075/ 2. The inlet boundary fields for ␭ = 1.31 and ␭ = 1.55 ␮m wavelength signals are 7 ␮m diameter Gaussian fields centered at x = 0 ␮m, and the outlet boundary fields are 7 ␮m diameter Gaussian fields centered at x = −20 and x = 20 ␮m, respectively. The calculation mesh was 0.11 ␮m in both the propagation and the lateral directions. In a usual waveguide type PLC, the propagating light is confined around the core of the waveguide, whereas the fields in Figs. 1(b) and 1(c) are distributed in the circuit and are multiple-scattered or multiple-diffracted at the refractive index pixels. However, each field converges at the outlet boundary through media with almost random refractive index distributions. Figure 2 shows the change in the circuit output performance with the calculation reciprocations. The transmittances of the output and the cross talk are the overlap integrals between the output field with the given wavelength and the set outlet boundary field and that of the wrong pair, respectively. In terms of optical circuit design, we realized a much more compact optical circuit than that of usual waveguide PLCs, and this circuit achieved very high conversion as a multilayered kinoform.

Fig. 2. count.

Output change along calculation reciprocation

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

These properties can be explained by investigating the construction procedure. The procedure described with Eqs. (5) and (6) can be summarized as follows. We considered a system where the refractive index distribution has a degree of freedom. We repeatedly applied the refractive index distribution change in Eq. (5), with the renormalized optical field propagating through the refractive index distribution, and made the optical field take the higher-order interaction with the changed refractive index distribution in the wave propagation calculation. Because of this the system can be regarded as being different from both an optical waveguide and a conventional volume hologram. This is because a typical hologram stores images as linearly overloaded refractive index modulations,2 whereas we consider all of the renormalized optical fields to be consistently combined with the refractive index in the local minimum condition in our system. This makes the circuit compact and results in high conversion efficiency. As regards diffraction tomography, an approach that takes multiple scattering into consideration has already been proposed.6 Although this approach corresponds to our approach expressed as Eqs. (5) and (6), a system considered in terms of tomography is usually determined almost entirely by given scattering data, and this suppresses the internal interaction among the refractive index values in the numerical calculation. On the other hand, our object has a very small number of input–output pairs; this yields an internal degree of freedom of the refractive index values at each point, and the system spontaneously converges with a refractive index distribution that suppresses the degree of freedom of the local phase change of the light-wave field. The suppression of the phase change of the light-wave field is observed as a spectrum, as shown in Fig. 3. Here the phase of the local light-wave field

changes with changes in wavelength, and this degrades the transmittance when the wavelength deviates 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 neural network system, since the system can be regarded as a neural network, and the study area has various approaches for estimating such properties. The correspondence can easily be shown by discretization of the evolution equation (1) as follows. With every cross-sectional refractive index layer in a discretized step of the z direction in Eq. (1), the optical wave travels with the superposition of the diffraction from the neighboring part caused by the Laplacian in L and taking its phase weight from the refractive index value. Therefore our system can be regarded as a multilayered linear complex-valued perceptron. It can also be regarded as a Boltzmann machinelike network in which the index values are interacting with the optical fields. Equations (5) and (6) naturally correspond to the error backpropagation method, where the error propagates as the phase difference between the inlet and the outlet optical waves as shown in Eq. (5). Figure 2 corresponds to the learning curves. This formalism provides us with a statistical physics viewpoint and clarifies the difference between the optical circuits that were discussed above by using those phase diagrams such as those in Ref. 7. Here we should consider that the boundary condition (input–output pair field) affects the system in the same way as an external field. We hope that the optical circuit described above will be investigated from various viewpoints and used to design optical circuits such as those fabricated with PLC technology. The authors thank T. Kitoh, M. Yanagisawa, Y. Abe, S. Asakawa, M. Kobayashi, R. Nagase, S. Suzuki, T. Ohyama, T. Kitagawa, M. Okuno, Y. Hibino, and H. Toba. References

Fig. 3. Wavelength dependence of transmittance.

1. T. W. Mossberg, Opt. Lett. 26, 414 (2001). 2. G. Barbastathis and D. J. Brady, Proc. IEEE, 87, 2098 (1999). 3. H. Rao, M. J. Steel, R. Scarmozzino, and R. M. Osgood, Jr., J. Lightwave Technol. 18, 1155 (2000). 4. M. Kawachi, Opt. Quantum Electron. 22, 391 (1990). 5. Y. Hibino, T. Maruno, and K. Okamoto, NTT Rev. 13, 4 (2001). 6. G. A. Tsihrintzis and A. J. Devaney, IEEE Trans. Inf. Theory 46, 1748 (2000). 7. T. Matsui, in Fluctuating Paths and Fields, W. Janke, A. Pelster, and M. Bachmann, eds. (World Scientific, 2001), and references therein.