Implementation of discrete unitary transformations by multimode

Implementation of discrete unitary transformations by multimode waveguide holograms Shuo-Yen Tseng, Younggu Kim, Christopher J. K. Richardson, and Julius Goldhar

Integration of holograms into multimode waveguides allows the implementation of arbitrary unitary mode transformations and unitary matrix–vector multiplication. Theoretical analysis is used to justify a design approach to implement specific functions in these devices. Based on this approach, a compact mode-order converter, a Hadamard transformer, and a spatial pattern generator– correlator are proposed and analyzed. Beam propagation simulations are used to verify the theoretical calculations and to address bandwidth, scalability, and fabrication criteria. Optical pattern generators were successfully fabricated using standard photolithographic techniques to demonstrate the feasibility of the devices. © 2006 Optical Society of America OCIS codes: 090.7330, 070.4550, 090.1760, 130.3120, 200.4860, 070.6020.

1. Introduction

There is interest in discrete optical signal processing using integrated optics. A particularly important class of operation that includes correlation, discrete Fourier transforms, and Hadamard transforms involves multiplication of vectors by unitary matrices. Many of these operations have been demonstrated with coherent spatial signals using conventional freespace holography, and this technology is described in various texts.1,2 Planar holographic techniques using slab waveguides combine both the advantages of traditional holography and integrated optics.3 Here, we focus on developing the theory necessary for designing holograms within guided-mode structures to create compact devices that perform unitary operations. Discrete Fourier and the Hadamard transforms have been proposed using single-mode star networks4,5 or multimode interference (MMI) waveguides.6 Also, it has been shown that arbitrary unitary operation can be realized with a combination of beam splitters, phase shifters, and mirrors.7 The holographic approach is another versatile technique for implementation of different transformations that can be readily imple-

The authors are with the Laboratory for Physical Sciences and Department of Electrical and Computer Engineering, University of Maryland, College Park, Maryland 20742. S.-Y. Tseng’s e-mail address is [email protected]. Received 12 July 2005; accepted 6 January 2006; posted 8 February 2006 (Doc. ID 63322). 0003-6935/06/204864-09$15.00/0 © 2006 Optical Society of America 4864

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mented using integrated optics. Linear transformations using a holographic matrix–vector multiplier incorporating a slab waveguide with integrated 2D lenses has been demonstrated in photorefractive waveguides under the assumption of undepleted inputs.8 By implementing holograms in a multimode channel waveguide, we take advantage of the discrete nature of the modes for efficient realization of discrete mathematical operations. In this paper, discrete unitary transformations using computer-generated volume holograms in multimode channel waveguides are analyzed. The general case of mode transformations in Fig. 1 is considered first. The multimode waveguide with multimode waveguide holograms (MWHs) transforms the input mode amplitude vector A(0) into the output mode amplitude vector A(L) through the unitary matrix H. Coupled-mode theory9 is used to show that arbitrary unitary mode transformations can be performed on coherent spatial signals using the MWHs. Next, an architecture for optical matrix–vector multiplication as shown in Fig. 2 is discussed, where the amplitudes of light in the access waveguides represent the input and output vectors. In this case, the MWH transforms the input vector P(0) into the output vector P(L) through the unitary matrix U. It is shown that arbitrary unitary matrix–vector multipliers can be implemented using MWHs in MMI devices10 without the need for waveguide lenses. The general design approach of MWHs is discussed, and an intuitive design method for the MWHs is outlined. For matrix–vector multipliers at imaging lengths10 of the MMIs, it is shown that the imaging

Fig. 1. (Color online) Schematic of a unitary mode transformer using MWHs. Ai(0) and Ai(L) are the amplitudes of the ith mode at the input and output, respectively. H is the unitary matrix representing the MWH.

property of MMIs can be used to simplify the calculation of MWHs to perform the desired transformations. A mode-order converter, a Hadamard transformer, and a spatial pattern generator– correlator11 based on MWHs are proposed and simulated. Spatial pattern generators were fabricated using polymer waveguides with integrated MWHs to demonstrate the principle. Beam propagation simulations are compared with experimental data to verify the accuracy of the model. Bandwidth, fabrication tolerances, and scalability are also discussed using a specific design example.

Fig. 2. (Color online) Schematic of a matrix–vector multiplier using MWHs. Pi(0) and Pi(L) are the amplitudes of the ith singlemode access waveguide at the input and output, respectively. U is the unitary transform matrix of the device.

⌬n共x, z兲 ⫽ f0共x兲 ⫹ 兺 fij共x兲exp关⫺i共␤i ⫺ ␤j兲z兴.

The first term adds different phase shifts to the modes depending on their overlap integrals, and the second term represents phase gratings that mix the modes. For the index perturbation to be a real function and physically realizable, we also require f0 to be a real function and fij ⫽ 共fji兲*. The coupled-mode theory involves substituting Eqs. (1)–(3) into the wave equation and collecting the synchronous terms to obtain the following set of M coupled equations: dAi ⫽ ␬ A, dz 兺 ij ij

2. Theory A. Unitary Mode Transformations Using Multimode Waveguide Holograms

(4)

where the coupling coefficients are given by

The propagation of an electric field in an M-mode multimode channel waveguide with a MWH in Fig. 1 is considered. The hologram is implemented as a weak perturbation to the index of refraction n0 of the core of the multimode waveguide. The analysis is based on a 2D representation of the multimode waveguide as described in Ref. 10. The electric field in the perturbed multimode waveguide can be written using the superposition of the unperturbed waveguide modes as M

E共x, z, t兲 ⫽ 兺 Ai共z兲␾i共x兲exp共⫺i␤iz ⫹ i␻t兲,

(3)

i⫽j

(1)

␬ii ⫽ ⫺i

␻⑀0n0 2

冕

␬ij ⫽ ⫺i

␻⑀0n0 2

冕

f0共x兲␾i2dx,

(5)

fij共x兲␾i␾jdx 共i ⫽ j兲 .

(6)

The coupled-mode equations in Eq. (4) can be expressed in matrix form as

1

where Ai represents the amplitude, ␾i共x兲 represents the lateral profile, ␤i is the propagation constant in the z direction corresponding to the ith guided mode, and ␻ is the optical frequency. The guided modes are normalized with the following condition9:

冕

⫹⬁

⫺⬁

␾i␾jdx ⫽

2␻␮ ␦, ⱍ␤iⱍ ij

(2)

such that the power carried by the ith mode is represented by |Ai|2. The hologram is chosen to be of the following form:

dA ⫽ KA, dz

(7)

where the ith row and jth column of the M ⫻ M matrix K is ␬ij, and A ⫽ 关A1共z兲A2共z兲, . . . , AM共z兲兴T. The solution to Eq. (7) is A共z兲 ⫽ exp共Kz兲A共0兲 ⫽ H共z兲A共0兲.

(8)

The requirements imposed on f0 and fij make Kz an anti-Hermitian matrix. As a result, its exponential map H is an M ⫻ M unitary matrix. The operation in Fig. 1 can thus be described as a unitary matrix H transforming the input mode amplitude vector A(0) 10 July 2006 兾 Vol. 45, No. 20 兾 APPLIED OPTICS

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into the output mode amplitude vector A(L). Since each component of K corresponds to an independent grating, changing the shape and strength of the hologram through f0 and fij allows us to generate arbitrary unitary matrices. One specific approach for calculating f0 and fij is described in Section 3. Although the theory describes unique transformations, in reality, the hologram for a particular unitary matrix is not unique. The grating shapes and strengths can be adjusted to suit specific design considerations as long as the overlap integrals in Eqs. (5) and (6) yield the desired coupling coefficients. Although a set of MWHs can be physically different but mathematically equivalent, one should not expect each member of this set to perform equally. Any figure of merit based on physical performance such as tolerance and bandwidth will be based on physical realizations of the structure such as waveguide dimension or index contrast. It is beyond the scope of this work to categorize or evaluate the various design approaches that are possible; rather, we focus on a specific design approach to demonstrate the concepts of this class of devices. B. Unitary Matrix–Vector Multiplication Using Multimode Waveguide Holograms

Consider the matrix–vector multiplier using MWHs in Fig. 2, N symmetric single-mode input and output waveguides are attached to the M-mode multimode waveguide of length L. This is similar to an N ⫻ N MMI coupler except that the device incorporates a hologram, and the self-imaging property is not required for the device to function. The input and output vectors are represented by the amplitudes of the input and the output waveguides. Consider the multimode waveguide behavior of the device without the hologram: The fields at the input and the output waveguides can be decomposed into the guided modes of the multimode section as in Eq. (1). Since Eq. (1) can be interpreted as a discrete spatial Fourier transform10 of the input vector, the amplitudes of the input waveguides P共0兲 ⫽ 关P1共0兲P2共0兲, . . . , PN共0兲兴T and output waveguides P共L兲 ⫽ 关P1共L兲P2共L兲, . . . , PN共L兲兴T can be related in matrix form as P共L兲 ⫽ 共VBVT 兲P共0兲,

(9)

where B is an M ⫻ M diagonal unitary matrix that describes the phase change resulting from the propagation of a distance of L of each guided mode, and V is an M ⫻ M unitary matrix that relates the vectors P at the input and output waveguides to mode amplitudes vectors. When the number of modes M in the multimode section is greater than the number of access waveguides N, the expressions for P(0) and P(L) can be zero padded to account for the virtual waveguides. When a hologram is incorporated into the multimode waveguide, Eq. (9) is combined with Eq. (8), which describes the unitary mode transformation in Fig. 1 to yield 4866

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P共L兲 ⫽ 共VBHVT 兲P共0兲 ⫽ UP共0兲.

(10)

This is a vector–matrix multiplication mapping an input vector P(0) to an output vector P(L) through a unitary matrix U determined by the MWH. To generate an arbitrary U, H is found by H ⫽ BTVTUV.

(11)

3. Design of Multimode Waveguide Holograms

From the previous derivations, the calculation of MWHs to perform the desired transformations involves determining the grating shapes, strengths, and periods in Eq. (3) such that the resulting MWH gives the desired matrix H in Eq. (8) for unitary mode transformation and in Eq. (11) for matrix–vector multiplication. Here, an intuitive design method in which the gratings are generated by the interference of propagating beams is discussed. This method involves using the interference patterns of the guided modes to calculate individual grating elements in Eq. (3). A special case of this method to design a matrix–vector multiplier using the imaging property of MMIs is also discussed. A. Multimode Waveguide Holograms Calculated Using the Guided Modes as the Basis Sets

Similar to the formation of a hologram by the interference of two waves, we consider the interference of guided modes i and j. The interference pattern that couples modes i and j can be described by ␾i␾j exp关⫺i共␤i ⫺ ␤j兲z兴 ⫹ c.c.

(12)

When the gratings in the hologram Eq. (3) are in the form of Eq. (12), that is, fij ⫽ gij␾i␾j, the coupling coefficients ␬ij are proportional to the grating strengths gij. By calculating the interference patterns between guided modes, gratings corresponding to each component of K in Eq. (7) are obtained. The hologram can then be determined by simply varying the grating strength according to the required coupling coefficients. B. Matrix–Vector Multipliers Calculated Using the Eigenvectors of Multimode Interference (A Special Case When the Multimode Waveguide Is at Imaging Length)

The general method outlined in Subsection 3.A can be used to design matrix–vector multipliers as long as Eq. (11) can be satisfied. Self-imaging of the multimode waveguide is not essential. Here, a simplified design approach is discussed when the length of the multimode waveguide is equal to its imaging length. The self-imaging property describes the reproduction of a single or multiple image of the input field profile at periodic intervals along the propagation direction of a multimode waveguide.10 First of all, by choosing the length of the multimode waveguide to be equal to the self-imaging length Limg, the problem can be simplified since the matrix B in Eq. (9) becomes the identity matrix. For an N ⫻ N MMI coupler, there

are N orthogonal eigenvectors that represent the patterns at input, output, and any intermediate imaging planes.12 These eigenvectors are obtained by sampling the guided-mode profiles corresponding to each access waveguide. Instead of using the M guided modes as a basis set to design the hologram as discussed in Subsection 3.A, the gratings are calculated directly from the interference patterns between these N eigenvectors. The strengths of these gratings are proportional to the coupling coefficients in the N ⫻ N matrix, K, which now directly relates the amplitudes of the input waveguides P(0) and output waveguides P共Limg兲 by P共Limg兲 ⫽ exp共KLimg兲P共0兲.

(13)

By writing the eigenvectors of MMI in the form of the mode amplitude vectors and going through the calculations from Eqs. (9)–(11), it can be shown that the hologram calculated using Eq. (13) is mathematically equivalent to the hologram calculated using the general method in Subsection 3.A. The advantage of this method is its simplicity when the beam propagation method is used to design the hologram as discussed later in the examples. 4. Design Examples of Multimode Waveguide Hologram Devices

Based on the design principle, three devices are designed and simulated using the 2D wide-angle beam propagation method (WA-BPM)13 using the Padé (3, 3) approximant operator. The effective indices of the waveguide core (1.516) and of the cladding region (1.0) are estimated using the effective-index method in accordance to the fabrication technique used later for experimental demonstration. In the simulations presented here, the hologram is modeled as smooth perturbations to the effective index of the waveguide core. A.

Mode-Order Converter

A mode-order converter based on MWHs using the configuration in Fig. 1 is proposed and simulated. The design method described in Subsection 3.A is used to calculate the hologram for an antidiagonal matrix in Eq. (8) that converts the mode-order in the multimode waveguide. The width of the waveguide is 16 ␮m and the length is 1.5 mm. The maximum index modulation depth is approximately 0.003. The slices of the BPM simulation are illustrated in Fig. 3 for four different mode conversion pairs. Figure 3(a) shows TE00 to TE30, Fig. 3(b) shows TE10 to TE20, Fig. 3(c) shows TE20 to TE10, and Fig. 3(d) shows TE30 to TE00. The advantage of the proposed approach is that the MWH mode converter is only 1.5 mm long. This device is more than ten times more compact than the mode converter made from multichannel branching waveguides.14 B.

Hadamard Transformer

The Hadamard transformer is capable of recognizing multiple phase-modulated labels as demonstrated

Fig. 3. Normalized intensity slices of the BPM simulation showing the evolution of modes along the proposed MWH mode-order converter.

recently using a star network.15 Using the matrix– vector multiplier configuration, a Hadamard transformer is proposed and simulated. The transfer matrix of a 4 ⫻ 4 Hadamard transformer is described by

冤

冥

1 1 1 1 1 ⫺1 1 ⫺1 . 1 1 ⫺1 ⫺1 1 ⫺1 ⫺1 1

(14)

The MWH is designed using the eigenvectors of the MMI coupler to calculate the grating as described in Subsection 3.B. The width of the multimode section of the MMI coupler is 16 ␮m, and the length is chosen to be at the first imaging length of 2052 ␮m. The maximum index modulation depth is approximately 0.003. It can be seen from the BPM simulation in Fig. 4 that four orthogonal phase-modulated spatial patterns are mapped into four different ports as described by the matrix in Eq. (14). This approach offers the advantage of compactness and does not require optical crossings as encountered in the star network approach.4,5 C.

Spatial Optical Pattern Generator–Correlator

The third device is a spatial optical pattern generator– correlator. The concept of a MWH optical processor is illustrated in Fig. 5. The N ⫻ N MMI coupler with MWHs in the figure is at the selfimaging length. Analogous to traditional holograms, one input port is designated as the reference bit, and the other ports are used to represent the image, which in this case is an N ⫺ 1 bit spatial pattern. When light is coupled into the reference bit port, as shown in Fig. 5(a), the stored N ⫺ 1 bit spatial pattern is reconstructed at the output ports. On the other hand, when the N ⫺ 1 bit spatial pattern is coupled 10 July 2006 兾 Vol. 45, No. 20 兾 APPLIED OPTICS

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Fig. 4. Normalized intensity slices of the BPM simulation of four phase-modulated patterns in a MWH Hadamard transformer. The Hadamard transformation is described by Eq. (14).

into the device as shown in Fig. 5(b), the reference bit is reconstructed with an amplitude that is proportional to the correlation of the incoming pattern and the stored pattern. The hologram is designed by calculating the interference pattern between the reference bit and the stored pattern using the N eigenvectors as the basis set as outlined in Subsection 3.B. With Eq. (13), the following solution is obtained for the output pattern Pout of the optical processor: Pout ⫽ P共0兲 ⫹ 关共R ⫹ S兲 · P共0兲兴关exp共iCl兲 ⫺ 1兴共R ⫹ S兲, (15) where P(0) is the input pattern, R is the reference bit, S is the stored pattern, l is the length of the hologram, and the coupling efficiency C is proportional to the index modulation depth. For pattern generation, P(0) ⫽ R, and it can be seen from Eq. (15)

Fig. 6. (Color online) Simulated correlation results of a MWH correlator with the stored pattern (1 1 1 1). Squares, MWH correlator; circles, ideal correlation.

that the intensity of the generated pattern S is proportional to 关1 ⫺ cos共Cl兲兴. For pattern correlation, since R and P(0) do not overlap at the input waveguides, the intensity of the generated reference bit R is proportional to

关S · P共0兲兴2关1 ⫺ cos共Cl兲兴,

(16)

which is proportional to the correlation of the stored pattern and the input pattern. A MWH processor with a 4-bit stored pattern (1 1 1 1) is designed and simulated. The correlation outputs (intensity of the generated reference bit) of different 4-bit intensity modulated input patterns are shown in Fig. 6. Close to ideal correlation results are also obtained with different stored patterns as well as with spatial patterns that have phase and amplitude modulation. The effect of the hologram length in the MMI coupler is also investigated. Holograms are written on different fractions of the total length of the MMI coupler. The correlation output of the matched pattern for different hologram lengths is recorded. The simulated data are fit with Eq. (16) and plotted in Fig. 7(a). The parameter C in Eq. (16) is proportional to the depth of index modulation. The index modulation depth is varied in the simulation with the fixed hologram length. The simulation result and the fit with Eq. (16) are shown in Fig. 7(b). 5. Experiment

Fig. 5. (Color online) Schematic of a MWH optical pattern generator– correlator using MMI. (a) The stored pattern is regenerated when light is coupled into the reference bit port. (b) The reference bit is reconstructed with an amplitude that is proportional to the correlation of the incoming pattern and the stored pattern. 4868

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The smooth index modulation used in the analytical theory is difficult to obtain in lithographically fabricated devices. The feasibility of lithographically fabricated devices is investigated by simulations of the MWH pattern generator– correlator with binary index modulation. Negligible differences are found in the generation– correlation properties.

Fig. 9. (Color online) Schematic of the experimental setup to measure the spatial pattern stored in the MWH pattern generator. The output spatial pattern is imaged with an IR camera.

Fig. 7. (Color online) Simulated effects of (a) hologram length and (b) index modulation depth on the reference bit output. Squares, WA-BPM simulation; curves, fit using Eq. (16).

The MWH pattern generators were fabricated using passive waveguides on a Si wafer. Three micrometers of SiO2 were used for the bottom cladding layer. The core consists of a 2.4 ␮m layer of Cyclotene™ (BCB), and the upper cladding is air. This asymmetric waveguide design was selected so that index variations for each hologram could be created lithographically by etching shallow indentations on the top of the Cyclotene™ surface (Fig. 8). The waveguide and hologram pattern were transferred using conventional photolithography using an i-line projection aligner and reactive ion etching (RIE). The device width is 16 ␮m, and the length of the device is

chosen at the first mirror-imaging length, 1026 ␮m. Devices were probed by coupling a cw laser at 1560 nm into the reference bit port as shown in Fig. 9. The output was imaged with an Electrophysics Micron Viewer 7290 IR camera. It can be seen from Fig. 10 that the patterns stored in the MWH pattern generator were reconstructed as predicted by coupled-mode theory and WA-BPM simulation. By varying the wavelength of the cw laser, the wavelength-dependent diffraction efficiency of the device was measured. The light was coupled into the reference bit port of a pattern generator with the 001 pattern. The intensity of the 1 bit in the generated pattern was recorded as a function of wavelength and plotted in Fig. 11. The results from the WA-BPM simulation are also plotted, and good agreement between the BPM model and the measured data can be seen in the figure. 6. Discussions

In the theoretical analysis, it is assumed that the Bragg conditions are satisfied for every grating element, and only the synchronous terms are collected to obtain Eq. (4). The validity of the condition and the effect of the cross talk between gratings can be considered qualitatively as follows. Considering two grating vectors kji ⫽ ␤j ⫺ ␤i and kki ⫽ ␤k ⫺ ␤i, when the jth mode is present, grating kji scatters it into the ith mode; a small fraction of its power is also scattered by the grating kki into the ith mode. The fraction of the ith mode from grating kki is the cross talk and can be estimated by16

共␬L兲2

sin2共␦L兲 , 共␦L兲2

␦ ⬎⬎ ␬,

(17)

where ␬ is the coupling coefficient of the grating, L is the length of the grating, and ␦ ⫽ ␤k ⫺ ␤j. For a multimode channel waveguide of width W and core index n0, ␦ can be estimated as10 ␦⫽

Fig. 8. Plan-view scanning electron microscope micrograph of a section of a MWH pattern generator. The shallow etched computergenerated pattern is visible on the surface of the device.

␲␭ 4n0W2

共k2 ⫺ j2兲,

(18)

where ␭ is the wavelength. Equations (17) and (18) give an estimate of the range of the coupling coefficients for the Bragg condition approximations to be valid. Also, it can be seen from Eq. (17) that when ␦L is sufficiently large, the cross talk is negligible. By 10 July 2006 兾 Vol. 45, No. 20 兾 APPLIED OPTICS

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Fig. 10. Experimental results of 3-bit spatial pattern generation by a MWH pattern generator.

use of the location of the second zero of Eq. (17) as a measurement of selectivity as in Ref. 16, the requirement on the minimum length of the hologram Lmin is Lmin

2␲ 8n0W2 8n0W2 ⱖ ⱖ ⫽ . ␦ 3␭ ␭共k2 ⫺ j2兲

(19)

The simulated and fabricated device in this paper are designed to be at the first mirror image length, 4n0W2兾␭, and the first direct image length, 8n0W2兾␭, of the multimode waveguides, with the MWHs covering the whole lengths of the devices. Although the self-imaging property is not essential for this class of devices, it offers a simplified design method for MWH as described in Subsection 3.B and satisfies the minimum length requirement of Eq. (19). The specific polymer waveguide system chosen for the experiment is used to investigate the fabrication

Fig. 11. (Color online) Wavelength dependence of the diffraction efficiency of a MWH pattern generator patterned with a 001 pattern. 4870

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Fig. 12. (Color online) BPM simulation and theoretical results on the effect of hologram lateral shift on the diffraction efficiency and the discrimination capability of the MWH pattern generator– correlator.

tolerances and scalability of the MWH devices. The spatial pattern generator– correlator described in Subsection 4.C and demonstrated in experiments is used in the simulations. First, the fabrication tolerance on the lateral shift between the MMI coupler and the hologram is investigated since they are fabricated in two different lithography steps. By use of the BPM model, the diffraction efficiency and the discrimination capability are calculated and plotted in Fig. 12. The discrimination capability is defined as 1⫺

maxⱍcⱍ2 , ⱍaⱍ2

(20)

where c is the cross correlation and a is the autocorrelation. From Eqs. (5) and (6), the lateral shift can be estimated as errors to the coupling coefficients due to the change in the overlap integrals. Since the MWH is designed using the eigenvectors of the MMI in this example, the error to the coupling coefficient is estimated using the change in the overlap integral of these eigenvectors due to the lateral shift. The input waveguides are well separated in this case, and the eigenvectors can be approximated using the guided modes of the single-mode access ports. This way, each coupling coefficient is affected by the same amount, and the effect of the lateral shift can be estimated as a change to the overall coupling efficiency C in Eq. (16). With these approximations, C can be written as a function of the lateral shift ⌬x as

C共⫾⌬x兲 ⫽ C0

冕

⌽2共x兲⌽2共x ⫾ ⌬x兲dx

冕

, ⌽4共x兲dx

(21)

ances of the polymer MMI coupler at the first mirror image length (N ⫽ M in Ref. 18) are analyzed. The calculated results are shown in Fig. 13. The optical bandwidth and fabrication tolerances are inversely proportional to the number of ports. It can be seen from the figure that devices with more than eight ports will be difficult to fabricate using current fabrication technology owing to the rapid rise in excess loss. The examples given here are only applicable to the specific material system chosen for this demonstration. It is worth noting that these limitations can be relaxed if a different material system with a smaller core or cladding index contrast is chosen. However, the multimode waveguide has to support at least the same number of guided modes to perform the same transformations. The trade-off will thus be increased device size. 7. Summary

Unitary transformations and matrix–vector multiplication by multimode waveguide holograms are described and analyzed. This class of devices is compact and versatile in performing different functionalities. A design method based on the interference of propagating beams is discussed. Theoretical calculations are verified by beam propagation simulations and experimental results. It was demonstrated that these devices can be fabricated by photolithographic techniques. The beam propagation model shows good accuracy and has been used to explore device performance. Fig. 13. (a) Bandwidth and (b) fabrication tolerance of the polymer MMI coupler used in our experiment. The practical scaling of these devices is limited by the proportionality between the number of ports and the sensitivity to wavelength detuning and fabrication tolerance.

where C0 is the coupling efficiency when the lateral shift is zero, and ⌽ is the mode of the single-mode access ports. Since the lateral shift only affects the overall coupling coefficient, it is not surprising that the discrimination capability in Fig. 12 is almost constant with the lateral shift in the range of ⫺0.5 to 0.5 ␮m offset. A theoretical prediction of the diffraction efficiency is obtained using Eq. (21) in Eq. (16) and plotted in Fig. 12. For applications in modern high-speed optical communications systems, the scalability and bandwidth of the MWHs are investigated. The MWH consists basically of multiplexed long-period gratings on a MMI coupler. The wide transmission bandwidth of long-period grating structure is well documented in the literature.17 From our simulations, the optical bandwidth and fabrication tolerances of the MMI coupler are the dominating factors in the performance of MWH devices. The scalability of MWH devices to perform transformations on large-sized input vectors is thus determined by the scalability of the MMI couplers. By use of the formalism introduced by Besse et al.,18 the bandwidth and fabrication toler-

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