Abstract

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

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References

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2005 (1)

O. Moriwaki, T. Kitoh, T. Sakamoto, and A. Okada, "Novel PLC-based optical correlator for multiple phase-modulated labels," IEEE Photon. Technol. Lett. 17, 489-491 (2005).
[CrossRef]

2003 (2)

2001 (1)

1999 (1)

J. M. Heaton and R. M. Jenkins, "General matrix theory of self-imaging in multimode interference (MMI) couplers," IEEE Photon. Technol. Lett. 11, 212-214 (1999).
[CrossRef]

1997 (1)

1995 (1)

L. B. Soldano and E. C. M. Pennings, "Optical multimode interference devices based on self-imaging: principles and applications," J. Lightwave Technol. 13, 615-627 (1995).
[CrossRef]

1994 (2)

M. Reck and A. Zeilinger, "Experimental realization of any discrete unitary operator," Phys. Rev. Lett. 73, 58-61 (1994).
[CrossRef] [PubMed]

P. A. Besse, M. Bachmann, H. Nelchior, L. B. Soldano, and M. K. Smit, "Optical bandwidth and fabrication tolerances of multimode interference couplers," J. Lightwave Technol. 12, 1004-1009 (1994).
[CrossRef]

1992 (1)

G. R. Hadley, "Wide-angle beam propagation using Pade approximant operators," Opt. Lett 17, 1426-1428 (1992).
[CrossRef] [PubMed]

1991 (1)

1987 (1)

1981 (1)

1973 (1)

A. Yariv, "Coupled-mode theory for guided-wave optics," IEEE J. Quantum Electron. 9, 919-933 (1973).
[CrossRef]

Bachmann, M.

P. A. Besse, M. Bachmann, H. Nelchior, L. B. Soldano, and M. K. Smit, "Optical bandwidth and fabrication tolerances of multimode interference couplers," J. Lightwave Technol. 12, 1004-1009 (1994).
[CrossRef]

Besse, P. A.

P. A. Besse, M. Bachmann, H. Nelchior, L. B. Soldano, and M. K. Smit, "Optical bandwidth and fabrication tolerances of multimode interference couplers," J. Lightwave Technol. 12, 1004-1009 (1994).
[CrossRef]

Boone, B. G.

B. G. Boone, Signal Processing Using Optics (Oxford, 1998).

Brady, D. J.

Erdogan, T.

Goldhar, J.

S.-Y. Tseng, Y. Kim, C. Richardson, and J. Goldhar, "Optical processor using waveguide holograms in multimode interference (MMI) couplers," in Conference on Lasers and Electrooptics on CD-ROM (Optical Society of America, 2005), paper JTuC70.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Gupta, A. R.

Hadley, G. R.

G. R. Hadley, "Wide-angle beam propagation using Pade approximant operators," Opt. Lett 17, 1426-1428 (1992).
[CrossRef] [PubMed]

Heaton, J. M.

J. M. Heaton and R. M. Jenkins, "General matrix theory of self-imaging in multimode interference (MMI) couplers," IEEE Photon. Technol. Lett. 11, 212-214 (1999).
[CrossRef]

Jannson, T.

Jenkins, R. M.

J. M. Heaton and R. M. Jenkins, "General matrix theory of self-imaging in multimode interference (MMI) couplers," IEEE Photon. Technol. Lett. 11, 212-214 (1999).
[CrossRef]

Kim, Y.

S.-Y. Tseng, Y. Kim, C. Richardson, and J. Goldhar, "Optical processor using waveguide holograms in multimode interference (MMI) couplers," in Conference on Lasers and Electrooptics on CD-ROM (Optical Society of America, 2005), paper JTuC70.

Kitoh, T.

O. Moriwaki, T. Kitoh, T. Sakamoto, and A. Okada, "Novel PLC-based optical correlator for multiple phase-modulated labels," IEEE Photon. Technol. Lett. 17, 489-491 (2005).
[CrossRef]

Lee, B.-T.

Marhic, M. E.

Moriwaki, O.

O. Moriwaki, T. Kitoh, T. Sakamoto, and A. Okada, "Novel PLC-based optical correlator for multiple phase-modulated labels," IEEE Photon. Technol. Lett. 17, 489-491 (2005).
[CrossRef]

Nakayama, J.

Nelchior, H.

P. A. Besse, M. Bachmann, H. Nelchior, L. B. Soldano, and M. K. Smit, "Optical bandwidth and fabrication tolerances of multimode interference couplers," J. Lightwave Technol. 12, 1004-1009 (1994).
[CrossRef]

Okada, A.

O. Moriwaki, T. Kitoh, T. Sakamoto, and A. Okada, "Novel PLC-based optical correlator for multiple phase-modulated labels," IEEE Photon. Technol. Lett. 17, 489-491 (2005).
[CrossRef]

Pennings, E. C. M.

L. B. Soldano and E. C. M. Pennings, "Optical multimode interference devices based on self-imaging: principles and applications," J. Lightwave Technol. 13, 615-627 (1995).
[CrossRef]

Psaltis, D.

Reck, M.

M. Reck and A. Zeilinger, "Experimental realization of any discrete unitary operator," Phys. Rev. Lett. 73, 58-61 (1994).
[CrossRef] [PubMed]

Richardson, C.

S.-Y. Tseng, Y. Kim, C. Richardson, and J. Goldhar, "Optical processor using waveguide holograms in multimode interference (MMI) couplers," in Conference on Lasers and Electrooptics on CD-ROM (Optical Society of America, 2005), paper JTuC70.

Sakamoto, T.

O. Moriwaki, T. Kitoh, T. Sakamoto, and A. Okada, "Novel PLC-based optical correlator for multiple phase-modulated labels," IEEE Photon. Technol. Lett. 17, 489-491 (2005).
[CrossRef]

Shin, S.-Y.

Siegman, A. E.

Smit, M. K.

P. A. Besse, M. Bachmann, H. Nelchior, L. B. Soldano, and M. K. Smit, "Optical bandwidth and fabrication tolerances of multimode interference couplers," J. Lightwave Technol. 12, 1004-1009 (1994).
[CrossRef]

Soldano, L. B.

L. B. Soldano and E. C. M. Pennings, "Optical multimode interference devices based on self-imaging: principles and applications," J. Lightwave Technol. 13, 615-627 (1995).
[CrossRef]

P. A. Besse, M. Bachmann, H. Nelchior, L. B. Soldano, and M. K. Smit, "Optical bandwidth and fabrication tolerances of multimode interference couplers," J. Lightwave Technol. 12, 1004-1009 (1994).
[CrossRef]

Tseng, S.-Y.

S.-Y. Tseng, Y. Kim, C. Richardson, and J. Goldhar, "Optical processor using waveguide holograms in multimode interference (MMI) couplers," in Conference on Lasers and Electrooptics on CD-ROM (Optical Society of America, 2005), paper JTuC70.

Tsutsumi, K.

Yariv, A.

A. Yariv, "Coupled-mode theory for guided-wave optics," IEEE J. Quantum Electron. 9, 919-933 (1973).
[CrossRef]

A. Yariv, Optical Electronics in Modern Communications (Oxford, 1997).

Zeilinger, A.

M. Reck and A. Zeilinger, "Experimental realization of any discrete unitary operator," Phys. Rev. Lett. 73, 58-61 (1994).
[CrossRef] [PubMed]

Appl. Opt. (2)

IEEE J. Quantum Electron. (1)

A. Yariv, "Coupled-mode theory for guided-wave optics," IEEE J. Quantum Electron. 9, 919-933 (1973).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

J. M. Heaton and R. M. Jenkins, "General matrix theory of self-imaging in multimode interference (MMI) couplers," IEEE Photon. Technol. Lett. 11, 212-214 (1999).
[CrossRef]

O. Moriwaki, T. Kitoh, T. Sakamoto, and A. Okada, "Novel PLC-based optical correlator for multiple phase-modulated labels," IEEE Photon. Technol. Lett. 17, 489-491 (2005).
[CrossRef]

J. Lightwave Technol. (2)

P. A. Besse, M. Bachmann, H. Nelchior, L. B. Soldano, and M. K. Smit, "Optical bandwidth and fabrication tolerances of multimode interference couplers," J. Lightwave Technol. 12, 1004-1009 (1994).
[CrossRef]

L. B. Soldano and E. C. M. Pennings, "Optical multimode interference devices based on self-imaging: principles and applications," J. Lightwave Technol. 13, 615-627 (1995).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Lett (1)

G. R. Hadley, "Wide-angle beam propagation using Pade approximant operators," Opt. Lett 17, 1426-1428 (1992).
[CrossRef] [PubMed]

Opt. Lett. (3)

Phys. Rev. Lett. (1)

M. Reck and A. Zeilinger, "Experimental realization of any discrete unitary operator," Phys. Rev. Lett. 73, 58-61 (1994).
[CrossRef] [PubMed]

Other (4)

A. Yariv, Optical Electronics in Modern Communications (Oxford, 1997).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

B. G. Boone, Signal Processing Using Optics (Oxford, 1998).

S.-Y. Tseng, Y. Kim, C. Richardson, and J. Goldhar, "Optical processor using waveguide holograms in multimode interference (MMI) couplers," in Conference on Lasers and Electrooptics on CD-ROM (Optical Society of America, 2005), paper JTuC70.

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Figures (13)

Fig. 1
Fig. 1

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

Fig. 2
Fig. 2

(Color online) Schematic of a matrix–vector multiplier using MWHs. Pi (0) and Pi ( L ) are the amplitudes of the i th single-mode access waveguide at the input and output, respectively. U is the unitary transform matrix of the device.

Fig. 3
Fig. 3

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

Fig. 4
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).

Fig. 5
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.

Fig. 6
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.

Fig. 7
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).

Fig. 8
Fig. 8

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

Fig. 9
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. 10
Fig. 10

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

Fig. 11
Fig. 11

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

Fig. 12
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.

Fig. 13
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.

Equations (21)

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E ( x , z , t ) = 1 M A i ( z ) ϕ i ( x ) exp ( i β i z + iωt ) ,
+ ϕ i ϕ j d x = 2 ωμ | β i | δ i j ,
Δ n ( x , z ) = f 0 ( x ) + i j f i j ( x ) exp [ i ( β i β j ) z ] .
d A i d z = κ i j A i j ,
κ i i = i ω ϵ 0 n 0 2 f 0 ( x ) ϕ i 2 d x ,
κ i j = i ω ϵ 0 n 0 2 f i j ( x ) ϕ i ϕ j d x ( i j ) .
d A d z = K A ,
A ( z ) = exp ( K z ) A ( 0 ) = H ( z ) A ( 0 ) .
P ( L ) = ( V B V T ) P ( 0 ) ,
P ( L ) = ( V B H V T ) P ( 0 ) = U P ( 0 ) .
H = B T V T U V .
ϕ i ϕ j exp [ i ( β i β j ) z ] + c .c .
P ( L img ) = exp ( K L img ) P ( 0 ) .
[ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] .
P out = P ( 0 ) + [ ( R + S ) P ( 0 ) ] [ exp ( i C l ) 1 ] ( R + S ) ,
[ S P ( 0 ) ] 2 [ 1 cos ( C l ) ] ,
( κ L ) 2 sin 2 ( δ L ) ( δ L ) 2 , δ κ ,
δ = π λ 4 n 0 W 2 ( k 2 j 2 ) ,
L min 2 π δ = 8 n 0 W 2 λ ( k 2 j 2 ) 8 n 0 W 2 3 λ .
1 max | c | 2 | a | 2 ,
C ( ± Δ x ) = C 0 Φ 2 ( x ) Φ 2 ( x ± Δ x ) d x Φ 4 ( x ) d x ,

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