Abstract

We propose a theoretical method to analyze and characterize the alignment of optical systems by using a singular-value decomposition (SVD) technique. One application of this method is to identify critical parameters for alignment systematically during alignment of two-dimensional device arrays. Another example application is to characterize the sensitivity or the accuracy of the misalignment-detection capability by evaluation of the singular values of optical systems. The information obtained through SVD-based techniques yields critical insights into the analysis of alignment.

© 2000 Optical Society of America

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  1. J. W. Goodman, F. J. Leonberger, S.-Y. Kung, R. A. Athale, “Optical interconnections for VLSI systems,” Proc. IEEE 72, 850–865 (1994).
    [CrossRef]
  2. M. Ishikawa, N. McArdle, “Optically interconnected parallel computing systems,” IEEE Comput. 31, 61–68 (1998).
    [CrossRef]
  3. V. N. Morozov, Y.-C. Lee, J. A. Neff, D. O’Brien, T. S. McLaren, H. Zhou, “Tolerance analysis for three-dimensional optoelectronic systems packaging,” Opt. Eng. 35, 2034–2044 (1996).
    [CrossRef]
  4. B. Robertson, D. Kabal, G. C. Boisset, Y. Liu, W. M. Robertson, M. R. Taghizadeh, D. V. Plant, “Design and operation of an in situ microchannel alignment-detection system,” Appl. Opt. 37, 5638–5376 (1998).
    [CrossRef]
  5. A. C. Walker, T.-Y. Yang, J. Gourlay, J. A. B. Dines, M. G. Forbes, S. M. Prince, D. A. Baillie, D. T. Neilson, R. Williams, L. C. Wilkinson, G. R. Smith, M. P. Y. Desmulliez, G. S. Buller, M. R. Taghizadeh, A. Waddie, I. Underwood, C. R. Stanley, F. Pottier, B. Vögele, W. Sibbett, “Optoelectronic systems based on InGaAs-complementary-metal-oxide-semiconductor smart-pixel arrays and free-space optical interconnects,” Appl. Opt. 37, 2822–2830 (1998).
    [CrossRef]
  6. D. Miyazaki, S. Masuda, K. Matsushita, “Self-alignment with optical microconnectors for free-space optical interconnections,” Appl. Opt. 37, 228–232 (1998).
    [CrossRef]
  7. K. Hirabayashi, T. Yamamoto, S. Hino, Y. Kohama, K. Tateno, “Optical beam direction compensating system for board-to-board free space optical interconnection in high-capacity ATM switch,” IEEE J. Lightwave Technol. 15, 874–882 (1997).
    [CrossRef]
  8. G. C. Boisset, B. Robertson, H. S. Hinton, “An approach to active alignment of free-space optical interconnects,” in Optical Computing, B. S. Wherrett, P. Chavel, eds., Vol. 139 of the Institute of Physics Conference Series (Institute of Physics, London, 1994), 227–230.
  9. D. J. Goodwill, D. Kabal, P. Palacharla, “Free space optical interconnect at 1.25Gb/s/channel using adaptive alignment,” in Digest of Topical Meeting on Optical Fiber Communication (Optical Society of America, Washington, D.C., 1999), paper WM22.
  10. K. Hashimoto, T. Noritsugu, “Performance and sensitivity in visual servoing,” in Proceedings of the 1999 IEEE International Conference on Robotics and Automation (Institute of Electrical and Electronics Engineers, New York, 1998), pp. 2321–2326.
  11. G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins University, Baltimore, Md., 1989).
  12. R. P. Paul, Robot Manipulators (MIT Press, Cambridge, Mass., 1981).
  13. F. B. McCormick, A. L. Lentine, R. L. Morrison, S. L. Walker, L. M. F. Chirovsky, L. A. D’Asaro, “Parallel operation of a 32 × 16 symmetric self-electrooptic effect device array,” IEEE Photonics Technol. Lett. 3, 232–234 (1991).
    [CrossRef]
  14. D. C. O’Shea, Elements of Modern Optical Design (Wiley, New York, 1985).
  15. T.-Y. Yang, J. Gourlay, A. C. Walker, “Adaptive alignment with 6-degrees of freedom in free-space optoelectronic interconnects,” in Digest of Topical Meeting on Optics in Computing (Optical Society of America, Washington, D.C., 1999), pp. 8–10.

1998 (4)

1997 (1)

K. Hirabayashi, T. Yamamoto, S. Hino, Y. Kohama, K. Tateno, “Optical beam direction compensating system for board-to-board free space optical interconnection in high-capacity ATM switch,” IEEE J. Lightwave Technol. 15, 874–882 (1997).
[CrossRef]

1996 (1)

V. N. Morozov, Y.-C. Lee, J. A. Neff, D. O’Brien, T. S. McLaren, H. Zhou, “Tolerance analysis for three-dimensional optoelectronic systems packaging,” Opt. Eng. 35, 2034–2044 (1996).
[CrossRef]

1994 (1)

J. W. Goodman, F. J. Leonberger, S.-Y. Kung, R. A. Athale, “Optical interconnections for VLSI systems,” Proc. IEEE 72, 850–865 (1994).
[CrossRef]

1991 (1)

F. B. McCormick, A. L. Lentine, R. L. Morrison, S. L. Walker, L. M. F. Chirovsky, L. A. D’Asaro, “Parallel operation of a 32 × 16 symmetric self-electrooptic effect device array,” IEEE Photonics Technol. Lett. 3, 232–234 (1991).
[CrossRef]

Athale, R. A.

J. W. Goodman, F. J. Leonberger, S.-Y. Kung, R. A. Athale, “Optical interconnections for VLSI systems,” Proc. IEEE 72, 850–865 (1994).
[CrossRef]

Baillie, D. A.

Boisset, G. C.

B. Robertson, D. Kabal, G. C. Boisset, Y. Liu, W. M. Robertson, M. R. Taghizadeh, D. V. Plant, “Design and operation of an in situ microchannel alignment-detection system,” Appl. Opt. 37, 5638–5376 (1998).
[CrossRef]

G. C. Boisset, B. Robertson, H. S. Hinton, “An approach to active alignment of free-space optical interconnects,” in Optical Computing, B. S. Wherrett, P. Chavel, eds., Vol. 139 of the Institute of Physics Conference Series (Institute of Physics, London, 1994), 227–230.

Buller, G. S.

Chirovsky, L. M. F.

F. B. McCormick, A. L. Lentine, R. L. Morrison, S. L. Walker, L. M. F. Chirovsky, L. A. D’Asaro, “Parallel operation of a 32 × 16 symmetric self-electrooptic effect device array,” IEEE Photonics Technol. Lett. 3, 232–234 (1991).
[CrossRef]

D’Asaro, L. A.

F. B. McCormick, A. L. Lentine, R. L. Morrison, S. L. Walker, L. M. F. Chirovsky, L. A. D’Asaro, “Parallel operation of a 32 × 16 symmetric self-electrooptic effect device array,” IEEE Photonics Technol. Lett. 3, 232–234 (1991).
[CrossRef]

Desmulliez, M. P. Y.

Dines, J. A. B.

Forbes, M. G.

Golub, G. H.

G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins University, Baltimore, Md., 1989).

Goodman, J. W.

J. W. Goodman, F. J. Leonberger, S.-Y. Kung, R. A. Athale, “Optical interconnections for VLSI systems,” Proc. IEEE 72, 850–865 (1994).
[CrossRef]

Goodwill, D. J.

D. J. Goodwill, D. Kabal, P. Palacharla, “Free space optical interconnect at 1.25Gb/s/channel using adaptive alignment,” in Digest of Topical Meeting on Optical Fiber Communication (Optical Society of America, Washington, D.C., 1999), paper WM22.

Gourlay, J.

Hashimoto, K.

K. Hashimoto, T. Noritsugu, “Performance and sensitivity in visual servoing,” in Proceedings of the 1999 IEEE International Conference on Robotics and Automation (Institute of Electrical and Electronics Engineers, New York, 1998), pp. 2321–2326.

Hino, S.

K. Hirabayashi, T. Yamamoto, S. Hino, Y. Kohama, K. Tateno, “Optical beam direction compensating system for board-to-board free space optical interconnection in high-capacity ATM switch,” IEEE J. Lightwave Technol. 15, 874–882 (1997).
[CrossRef]

Hinton, H. S.

G. C. Boisset, B. Robertson, H. S. Hinton, “An approach to active alignment of free-space optical interconnects,” in Optical Computing, B. S. Wherrett, P. Chavel, eds., Vol. 139 of the Institute of Physics Conference Series (Institute of Physics, London, 1994), 227–230.

Hirabayashi, K.

K. Hirabayashi, T. Yamamoto, S. Hino, Y. Kohama, K. Tateno, “Optical beam direction compensating system for board-to-board free space optical interconnection in high-capacity ATM switch,” IEEE J. Lightwave Technol. 15, 874–882 (1997).
[CrossRef]

Ishikawa, M.

M. Ishikawa, N. McArdle, “Optically interconnected parallel computing systems,” IEEE Comput. 31, 61–68 (1998).
[CrossRef]

Kabal, D.

B. Robertson, D. Kabal, G. C. Boisset, Y. Liu, W. M. Robertson, M. R. Taghizadeh, D. V. Plant, “Design and operation of an in situ microchannel alignment-detection system,” Appl. Opt. 37, 5638–5376 (1998).
[CrossRef]

D. J. Goodwill, D. Kabal, P. Palacharla, “Free space optical interconnect at 1.25Gb/s/channel using adaptive alignment,” in Digest of Topical Meeting on Optical Fiber Communication (Optical Society of America, Washington, D.C., 1999), paper WM22.

Kohama, Y.

K. Hirabayashi, T. Yamamoto, S. Hino, Y. Kohama, K. Tateno, “Optical beam direction compensating system for board-to-board free space optical interconnection in high-capacity ATM switch,” IEEE J. Lightwave Technol. 15, 874–882 (1997).
[CrossRef]

Kung, S.-Y.

J. W. Goodman, F. J. Leonberger, S.-Y. Kung, R. A. Athale, “Optical interconnections for VLSI systems,” Proc. IEEE 72, 850–865 (1994).
[CrossRef]

Lee, Y.-C.

V. N. Morozov, Y.-C. Lee, J. A. Neff, D. O’Brien, T. S. McLaren, H. Zhou, “Tolerance analysis for three-dimensional optoelectronic systems packaging,” Opt. Eng. 35, 2034–2044 (1996).
[CrossRef]

Lentine, A. L.

F. B. McCormick, A. L. Lentine, R. L. Morrison, S. L. Walker, L. M. F. Chirovsky, L. A. D’Asaro, “Parallel operation of a 32 × 16 symmetric self-electrooptic effect device array,” IEEE Photonics Technol. Lett. 3, 232–234 (1991).
[CrossRef]

Leonberger, F. J.

J. W. Goodman, F. J. Leonberger, S.-Y. Kung, R. A. Athale, “Optical interconnections for VLSI systems,” Proc. IEEE 72, 850–865 (1994).
[CrossRef]

Liu, Y.

B. Robertson, D. Kabal, G. C. Boisset, Y. Liu, W. M. Robertson, M. R. Taghizadeh, D. V. Plant, “Design and operation of an in situ microchannel alignment-detection system,” Appl. Opt. 37, 5638–5376 (1998).
[CrossRef]

Masuda, S.

Matsushita, K.

McArdle, N.

M. Ishikawa, N. McArdle, “Optically interconnected parallel computing systems,” IEEE Comput. 31, 61–68 (1998).
[CrossRef]

McCormick, F. B.

F. B. McCormick, A. L. Lentine, R. L. Morrison, S. L. Walker, L. M. F. Chirovsky, L. A. D’Asaro, “Parallel operation of a 32 × 16 symmetric self-electrooptic effect device array,” IEEE Photonics Technol. Lett. 3, 232–234 (1991).
[CrossRef]

McLaren, T. S.

V. N. Morozov, Y.-C. Lee, J. A. Neff, D. O’Brien, T. S. McLaren, H. Zhou, “Tolerance analysis for three-dimensional optoelectronic systems packaging,” Opt. Eng. 35, 2034–2044 (1996).
[CrossRef]

Miyazaki, D.

Morozov, V. N.

V. N. Morozov, Y.-C. Lee, J. A. Neff, D. O’Brien, T. S. McLaren, H. Zhou, “Tolerance analysis for three-dimensional optoelectronic systems packaging,” Opt. Eng. 35, 2034–2044 (1996).
[CrossRef]

Morrison, R. L.

F. B. McCormick, A. L. Lentine, R. L. Morrison, S. L. Walker, L. M. F. Chirovsky, L. A. D’Asaro, “Parallel operation of a 32 × 16 symmetric self-electrooptic effect device array,” IEEE Photonics Technol. Lett. 3, 232–234 (1991).
[CrossRef]

Neff, J. A.

V. N. Morozov, Y.-C. Lee, J. A. Neff, D. O’Brien, T. S. McLaren, H. Zhou, “Tolerance analysis for three-dimensional optoelectronic systems packaging,” Opt. Eng. 35, 2034–2044 (1996).
[CrossRef]

Neilson, D. T.

Noritsugu, T.

K. Hashimoto, T. Noritsugu, “Performance and sensitivity in visual servoing,” in Proceedings of the 1999 IEEE International Conference on Robotics and Automation (Institute of Electrical and Electronics Engineers, New York, 1998), pp. 2321–2326.

O’Brien, D.

V. N. Morozov, Y.-C. Lee, J. A. Neff, D. O’Brien, T. S. McLaren, H. Zhou, “Tolerance analysis for three-dimensional optoelectronic systems packaging,” Opt. Eng. 35, 2034–2044 (1996).
[CrossRef]

O’Shea, D. C.

D. C. O’Shea, Elements of Modern Optical Design (Wiley, New York, 1985).

Palacharla, P.

D. J. Goodwill, D. Kabal, P. Palacharla, “Free space optical interconnect at 1.25Gb/s/channel using adaptive alignment,” in Digest of Topical Meeting on Optical Fiber Communication (Optical Society of America, Washington, D.C., 1999), paper WM22.

Paul, R. P.

R. P. Paul, Robot Manipulators (MIT Press, Cambridge, Mass., 1981).

Plant, D. V.

B. Robertson, D. Kabal, G. C. Boisset, Y. Liu, W. M. Robertson, M. R. Taghizadeh, D. V. Plant, “Design and operation of an in situ microchannel alignment-detection system,” Appl. Opt. 37, 5638–5376 (1998).
[CrossRef]

Pottier, F.

Prince, S. M.

Robertson, B.

B. Robertson, D. Kabal, G. C. Boisset, Y. Liu, W. M. Robertson, M. R. Taghizadeh, D. V. Plant, “Design and operation of an in situ microchannel alignment-detection system,” Appl. Opt. 37, 5638–5376 (1998).
[CrossRef]

G. C. Boisset, B. Robertson, H. S. Hinton, “An approach to active alignment of free-space optical interconnects,” in Optical Computing, B. S. Wherrett, P. Chavel, eds., Vol. 139 of the Institute of Physics Conference Series (Institute of Physics, London, 1994), 227–230.

Robertson, W. M.

B. Robertson, D. Kabal, G. C. Boisset, Y. Liu, W. M. Robertson, M. R. Taghizadeh, D. V. Plant, “Design and operation of an in situ microchannel alignment-detection system,” Appl. Opt. 37, 5638–5376 (1998).
[CrossRef]

Sibbett, W.

Smith, G. R.

Stanley, C. R.

Taghizadeh, M. R.

Tateno, K.

K. Hirabayashi, T. Yamamoto, S. Hino, Y. Kohama, K. Tateno, “Optical beam direction compensating system for board-to-board free space optical interconnection in high-capacity ATM switch,” IEEE J. Lightwave Technol. 15, 874–882 (1997).
[CrossRef]

Underwood, I.

Van Loan, C. F.

G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins University, Baltimore, Md., 1989).

Vögele, B.

Waddie, A.

Walker, A. C.

Walker, S. L.

F. B. McCormick, A. L. Lentine, R. L. Morrison, S. L. Walker, L. M. F. Chirovsky, L. A. D’Asaro, “Parallel operation of a 32 × 16 symmetric self-electrooptic effect device array,” IEEE Photonics Technol. Lett. 3, 232–234 (1991).
[CrossRef]

Wilkinson, L. C.

Williams, R.

Yamamoto, T.

K. Hirabayashi, T. Yamamoto, S. Hino, Y. Kohama, K. Tateno, “Optical beam direction compensating system for board-to-board free space optical interconnection in high-capacity ATM switch,” IEEE J. Lightwave Technol. 15, 874–882 (1997).
[CrossRef]

Yang, T.-Y.

Zhou, H.

V. N. Morozov, Y.-C. Lee, J. A. Neff, D. O’Brien, T. S. McLaren, H. Zhou, “Tolerance analysis for three-dimensional optoelectronic systems packaging,” Opt. Eng. 35, 2034–2044 (1996).
[CrossRef]

Appl. Opt. (3)

IEEE Comput. (1)

M. Ishikawa, N. McArdle, “Optically interconnected parallel computing systems,” IEEE Comput. 31, 61–68 (1998).
[CrossRef]

IEEE J. Lightwave Technol. (1)

K. Hirabayashi, T. Yamamoto, S. Hino, Y. Kohama, K. Tateno, “Optical beam direction compensating system for board-to-board free space optical interconnection in high-capacity ATM switch,” IEEE J. Lightwave Technol. 15, 874–882 (1997).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

F. B. McCormick, A. L. Lentine, R. L. Morrison, S. L. Walker, L. M. F. Chirovsky, L. A. D’Asaro, “Parallel operation of a 32 × 16 symmetric self-electrooptic effect device array,” IEEE Photonics Technol. Lett. 3, 232–234 (1991).
[CrossRef]

Opt. Eng. (1)

V. N. Morozov, Y.-C. Lee, J. A. Neff, D. O’Brien, T. S. McLaren, H. Zhou, “Tolerance analysis for three-dimensional optoelectronic systems packaging,” Opt. Eng. 35, 2034–2044 (1996).
[CrossRef]

Proc. IEEE (1)

J. W. Goodman, F. J. Leonberger, S.-Y. Kung, R. A. Athale, “Optical interconnections for VLSI systems,” Proc. IEEE 72, 850–865 (1994).
[CrossRef]

Other (7)

D. C. O’Shea, Elements of Modern Optical Design (Wiley, New York, 1985).

T.-Y. Yang, J. Gourlay, A. C. Walker, “Adaptive alignment with 6-degrees of freedom in free-space optoelectronic interconnects,” in Digest of Topical Meeting on Optics in Computing (Optical Society of America, Washington, D.C., 1999), pp. 8–10.

G. C. Boisset, B. Robertson, H. S. Hinton, “An approach to active alignment of free-space optical interconnects,” in Optical Computing, B. S. Wherrett, P. Chavel, eds., Vol. 139 of the Institute of Physics Conference Series (Institute of Physics, London, 1994), 227–230.

D. J. Goodwill, D. Kabal, P. Palacharla, “Free space optical interconnect at 1.25Gb/s/channel using adaptive alignment,” in Digest of Topical Meeting on Optical Fiber Communication (Optical Society of America, Washington, D.C., 1999), paper WM22.

K. Hashimoto, T. Noritsugu, “Performance and sensitivity in visual servoing,” in Proceedings of the 1999 IEEE International Conference on Robotics and Automation (Institute of Electrical and Electronics Engineers, New York, 1998), pp. 2321–2326.

G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins University, Baltimore, Md., 1989).

R. P. Paul, Robot Manipulators (MIT Press, Cambridge, Mass., 1981).

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Figures (8)

Fig. 1
Fig. 1

Alignment problem for two-dimensional device arrays.

Fig. 2
Fig. 2

Modeling of photodetector array and optical-beam array.

Fig. 3
Fig. 3

Alignment index.

Fig. 4
Fig. 4

Schematic of the optical system model used in the simulation.

Fig. 5
Fig. 5

Alignment evaluation vector derived by theory and simulation.

Fig. 6
Fig. 6

Characterization of misalignment-detection capability with design criteria being the ratio of the maximum singular value to the minimum one.

Fig. 7
Fig. 7

Quadrant detectors on the detector array.

Fig. 8
Fig. 8

Condition number as a function of the distance between a pair of quadrant detectors.

Tables (1)

Tables Icon

Table 1 Condition Number as a Function of the Distance between a Pair of Quadrant Detectors with the Estimation Error of Matrix M

Equations (34)

Equations on this page are rendered with MathJax. Learn more.

A=UDVT,
σ1σ2σr0.
Rij=xc+ip cos ϕc cos θc+jpcos ϕc sin θc sin ψc-sin ϕc cos ψcyc+ip cos θc sin ϕc+jpsin ϕc sin θc sin ψc+cos ϕc cos ψczc-ip sin θc+jp cos θc sin ψc,
Iij=x, y, z|x-ip2+y-jp2z tan Δ2.
k=k1k2=ϕcjp+θczc-xc+-θcip+ψcjp+θcxc-ϕcyc+zctan Δ-ϕcip+ψczc-yc+-θcip+ψcjp+θcxc-ϕcyc+zctan Δ.
xc, yc, zc, ψc, θc, ϕc=xnx, yny, znz, ψnψ, θnθ, ϕnϕ
dk=Jdr,
J=-nx+nxnθθ tan Δ,nxnθθ tan Δ-nynϕϕ tan Δ,-ny-nynϕϕ tan Δnznθθ+nz tan Δ,nznψψ+nz tan Δjpnψ tan Δ,nψnzz+jpnψ tan Δnznθz+nxnθx-ipnθtan Δ,nxnθx-ipnθtan Δnϕjp-nϕnyy tan Δ,-nϕip-nϕnyy tan Δ.
J=UDVT,
U=u1, u2,  V=v1,, v6,D=DO,D=diagσ1, σ2.
q=σ1|v1|e+σ2|v2|e,
q¯=1.41×100, 1.41×100, 1.50×10-1,2.68×10-2, 2.68×10-2, 2.48×10-1,
2.00×10-4, 2.00×10-4, 1.80×10-3,1.20×10-3, 1.20×10-3, 8.00×10-4.
E=i ki-Jri2.
J=-1.0423-0.0423-0.0423-0.0423-0.0423-0.1316-0.0423-1.0423-0.0423-0.0423-0.0423-0.1316,
q=1.4740, 1.4740, 0.0598, 0.0598, 0.0598, 0.1861.
s=Mr,
condM=max σimin σi,
xc, yc, ϕc=nxx, nyy, nϕϕ,
M=nx0nx00ny0ny-p1xnϕp1ynϕ-pnxnϕpnynϕT
E=isi-Mri2.
Ax=b.
Ax+Δx=b+Δb.
A=UDVT,
U=u1,, ur, ur+1,, un,V=v1,, vr, vr+1,, vm,D=DOOO,  D=diagσ1,, σr,σ1σ2σr>0.
x˜=VTx,  b˜=UTb.
σix˜i=b˜ii=1,, r,
0=b˜ii=r+1,, m.
x˜i=b˜i/σii=1,, r,
x=j=1m ξjvj.
b=j=1r σjξjuj.
minxb=1σ1,  maxxb=1σr,
minΔxΔb=1σ1,  maxΔxΔb=1σr.
σrσ1ΔxxΔbbσ1σr,

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