Abstract

The effects of extrinsic and inherent errors are analyzed for the integrated optical Givens rotation device. The extrinsic errors, caused by inaccurate voltage applied to the grating and inaccurate detection, are found to be important. The inherent errors caused by the propagation of these inaccuracies are detailed in algorithms for analog norm computation and in the QR algorithm for numerical linear algebra. A calibration procedure is developed to eliminate most of the errors.

© 1992 Optical Society of America

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References

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  1. T. K. Gaylord, E. I. Verriest, “Matrix triangularization using arrays of integrated optical Givens rotation devices,” Computer 20(12), 59–66 (1987).
    [CrossRef]
  2. M. M. Mirsalehi, T. K. Gaylord, E. I. Verriest, “Integrated-optical Givens rotation device,” Appl. Opt. 25, 1608–1614 (1986).
    [CrossRef] [PubMed]
  3. G. H. Golub, C. F. Van Loan, Matrix Computation (Johns Hopkins U. Press, Baltimore, Md., 1983).
  4. E. I. Verriest, E. N. Glytsis, T. K. Gaylord, “Performance analysis of Givens rotation integrated optical interdigitated-electrode cross-channel Bragg diffraction devices: intrinsic accuracy,” Appl. Opt. 29, 2556–2563 (1990).
    [CrossRef] [PubMed]
  5. E. I. Verriest, “Quaternions, quasors, and Q-planes: a representation for coherent optical fourports based on Bragg diffraction,” Appl. Opt. (to be published).
  6. D. Marcuse, Principles of Quantum Electronics (Academic, New York, 1980).
  7. R. A. Becker, L. M. Johnson, “Low-loss multiple-branching circuit in Ti-indiffused LiNbO3 channel waveguides,” Opt. Lett. 9, 246–248 (1984).
    [CrossRef] [PubMed]
  8. V. Pan, J. H. Reif, “Efficient parallel solution of linear systems,” in Proceedings of the 17th Annual ACM Symposium on Theory of Computing (Association for Computing Machinery, New York, 1985), pp. 143–152.
  9. T. Kailath, “Signal processing in the VLSI era,” in VLSI and Modern Signal Processing, S. Y. Kung, H. J. Whitehouse, T. Kailath, eds. (Prentice-Hall, Englewood Cliffs, N.J., 1985).
  10. B. Moslehi, J. W. Goodman, M. Tur, H. J. Shaw, “Fiber-optic lattice signal processing,” Proc. IEEE 72, 909–930 (1984).
    [CrossRef]
  11. A. K. Ghosh, “Matrix pre-conditioning on optical linear algebra processors,” in Proceedings of the 24th Allerton Conference on Communication, Control and ComputingU. Illinois Press, Urbana, Ill., 1986), pp. 184–193.
  12. F. M. Davidson, L. Boutsikaris, “Homodyne detection using photorefractive materials as beamsplitters,” Opt. Eng. 29, 369–377 (1990).
    [CrossRef]

1990

1987

T. K. Gaylord, E. I. Verriest, “Matrix triangularization using arrays of integrated optical Givens rotation devices,” Computer 20(12), 59–66 (1987).
[CrossRef]

1986

1984

B. Moslehi, J. W. Goodman, M. Tur, H. J. Shaw, “Fiber-optic lattice signal processing,” Proc. IEEE 72, 909–930 (1984).
[CrossRef]

R. A. Becker, L. M. Johnson, “Low-loss multiple-branching circuit in Ti-indiffused LiNbO3 channel waveguides,” Opt. Lett. 9, 246–248 (1984).
[CrossRef] [PubMed]

Becker, R. A.

Boutsikaris, L.

F. M. Davidson, L. Boutsikaris, “Homodyne detection using photorefractive materials as beamsplitters,” Opt. Eng. 29, 369–377 (1990).
[CrossRef]

Davidson, F. M.

F. M. Davidson, L. Boutsikaris, “Homodyne detection using photorefractive materials as beamsplitters,” Opt. Eng. 29, 369–377 (1990).
[CrossRef]

Gaylord, T. K.

Ghosh, A. K.

A. K. Ghosh, “Matrix pre-conditioning on optical linear algebra processors,” in Proceedings of the 24th Allerton Conference on Communication, Control and ComputingU. Illinois Press, Urbana, Ill., 1986), pp. 184–193.

Glytsis, E. N.

Golub, G. H.

G. H. Golub, C. F. Van Loan, Matrix Computation (Johns Hopkins U. Press, Baltimore, Md., 1983).

Goodman, J. W.

B. Moslehi, J. W. Goodman, M. Tur, H. J. Shaw, “Fiber-optic lattice signal processing,” Proc. IEEE 72, 909–930 (1984).
[CrossRef]

Johnson, L. M.

Kailath, T.

T. Kailath, “Signal processing in the VLSI era,” in VLSI and Modern Signal Processing, S. Y. Kung, H. J. Whitehouse, T. Kailath, eds. (Prentice-Hall, Englewood Cliffs, N.J., 1985).

Marcuse, D.

D. Marcuse, Principles of Quantum Electronics (Academic, New York, 1980).

Mirsalehi, M. M.

Moslehi, B.

B. Moslehi, J. W. Goodman, M. Tur, H. J. Shaw, “Fiber-optic lattice signal processing,” Proc. IEEE 72, 909–930 (1984).
[CrossRef]

Pan, V.

V. Pan, J. H. Reif, “Efficient parallel solution of linear systems,” in Proceedings of the 17th Annual ACM Symposium on Theory of Computing (Association for Computing Machinery, New York, 1985), pp. 143–152.

Reif, J. H.

V. Pan, J. H. Reif, “Efficient parallel solution of linear systems,” in Proceedings of the 17th Annual ACM Symposium on Theory of Computing (Association for Computing Machinery, New York, 1985), pp. 143–152.

Shaw, H. J.

B. Moslehi, J. W. Goodman, M. Tur, H. J. Shaw, “Fiber-optic lattice signal processing,” Proc. IEEE 72, 909–930 (1984).
[CrossRef]

Tur, M.

B. Moslehi, J. W. Goodman, M. Tur, H. J. Shaw, “Fiber-optic lattice signal processing,” Proc. IEEE 72, 909–930 (1984).
[CrossRef]

Van Loan, C. F.

G. H. Golub, C. F. Van Loan, Matrix Computation (Johns Hopkins U. Press, Baltimore, Md., 1983).

Verriest, E. I.

E. I. Verriest, E. N. Glytsis, T. K. Gaylord, “Performance analysis of Givens rotation integrated optical interdigitated-electrode cross-channel Bragg diffraction devices: intrinsic accuracy,” Appl. Opt. 29, 2556–2563 (1990).
[CrossRef] [PubMed]

T. K. Gaylord, E. I. Verriest, “Matrix triangularization using arrays of integrated optical Givens rotation devices,” Computer 20(12), 59–66 (1987).
[CrossRef]

M. M. Mirsalehi, T. K. Gaylord, E. I. Verriest, “Integrated-optical Givens rotation device,” Appl. Opt. 25, 1608–1614 (1986).
[CrossRef] [PubMed]

E. I. Verriest, “Quaternions, quasors, and Q-planes: a representation for coherent optical fourports based on Bragg diffraction,” Appl. Opt. (to be published).

Appl. Opt.

Computer

T. K. Gaylord, E. I. Verriest, “Matrix triangularization using arrays of integrated optical Givens rotation devices,” Computer 20(12), 59–66 (1987).
[CrossRef]

Opt. Eng.

F. M. Davidson, L. Boutsikaris, “Homodyne detection using photorefractive materials as beamsplitters,” Opt. Eng. 29, 369–377 (1990).
[CrossRef]

Opt. Lett.

Proc. IEEE

B. Moslehi, J. W. Goodman, M. Tur, H. J. Shaw, “Fiber-optic lattice signal processing,” Proc. IEEE 72, 909–930 (1984).
[CrossRef]

Other

A. K. Ghosh, “Matrix pre-conditioning on optical linear algebra processors,” in Proceedings of the 24th Allerton Conference on Communication, Control and ComputingU. Illinois Press, Urbana, Ill., 1986), pp. 184–193.

E. I. Verriest, “Quaternions, quasors, and Q-planes: a representation for coherent optical fourports based on Bragg diffraction,” Appl. Opt. (to be published).

D. Marcuse, Principles of Quantum Electronics (Academic, New York, 1980).

G. H. Golub, C. F. Van Loan, Matrix Computation (Johns Hopkins U. Press, Baltimore, Md., 1983).

V. Pan, J. H. Reif, “Efficient parallel solution of linear systems,” in Proceedings of the 17th Annual ACM Symposium on Theory of Computing (Association for Computing Machinery, New York, 1985), pp. 143–152.

T. Kailath, “Signal processing in the VLSI era,” in VLSI and Modern Signal Processing, S. Y. Kung, H. J. Whitehouse, T. Kailath, eds. (Prentice-Hall, Englewood Cliffs, N.J., 1985).

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Figures (6)

Fig. 1
Fig. 1

Schematic of the optical Givens rotation device. The x components are the input beams; the y components are the output beams.

Fig. 2
Fig. 2

Implementation of the integrated optical Givens rotation device.

Fig. 3
Fig. 3

Exact detection: Q-plane representation. The input quasor X is rotated to either X′ or X″.

Fig. 4
Fig. 4

Inexact detection: Q-plane representation. X1 falls within the band [− ∊, ∊]. The detector output is the quantized version of P(X1). X2 is in the first or second quadrant, outside the ∊ band. The detector output is the quantized P(X2′). X3 in the third or fourth quadrant similarly yields P(X3′).

Fig. 5
Fig. 5

Pipeline of rotation devices for computing the norm of a vector. The input vector is [a1, …, a N ] T . The computed norm is b ^ N . The b ^ N , i are the partial norms, and the e i are the detector errors.

Fig. 6
Fig. 6

Pipeline of rotation devices or the QR factorization. The input is the matrix A; the output is the rotated version R N A in triangular form, thus factoring A into [R N ] T [R N A]. The rotation R N is encoded in the rotation angles of the devices and can be read out by inputting the identity matrix into the pipeline.

Equations (35)

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[ y 1 y 2 ] = [ cos ψ - sin ψ sin ψ cos ψ ] [ x 1 x 2 ] .
Δ i = ( 2 η q λ / h c ) A R A S cos Γ 0 ,
i + = ½ ( η q λ / h c ) ( A R + A S cos Γ 0 ) 2 ( 1 + γ + ) ,
i - = ½ ( η q λ / h c ) ( A R - A S cos Γ 0 ) 2 ( 1 + γ - ) ,
Δ i = η λ q / ( 2 h c ) ( A R 2 + A S 2 cos 2 Γ 0 ) ( γ + - γ - ) + η λ q / ( h c ) A R A S cos Γ 0 ( 2 + γ + + γ - ) = 2 η λ q / ( h c ) A R A S cos Γ 0 + η λ q / ( 2 h c ) × [ ( γ + - γ - ) ( A R 2 + A S 2 cos 2 Γ 0 ) + 2 ( γ + + γ - ) A R A S cos Γ 0 ] .
Δ i ˜ 2 = γ 2 η 2 q 2 λ 2 / ( 2 h 2 c 2 ) × [ ( A R 2 + A S 2 cos 2 Γ 0 ) 2 + 4 A R 2 A S 2 cos 2 Γ 0 ] ,
λ 2 q 2 η 2 / ( 2 h 2 c 2 ) ( A R ± A S cos Γ 0 ) 4 ,
Quan [ ( X 2 - 2 ) 1 / 2 ] .
x ( b ) 2 = a R - b 2 + a I 2 a I 2 .
x sin ϕ ˜ A sin ϕ ˜ <
sin [ arg ( x ) ] / x ,
sin δ / A .
exp ( j Γ 0 ) [ σ 1 exp ( j Γ 2 ) 0 0 σ 2 ] × [ cos ψ exp ( j ζ a ) sin ψ exp ( j ζ b ) sin ψ cos ψ ] [ σ 3 exp ( j Γ 1 ) 0 0 σ 4 ] ,
y ^ 1 = σ 1 σ 4 cos ( Γ 0 + Γ 2 + ζ a ) sin ψ , y ^ 2 = σ 2 σ 4 cos Γ 0 cos ψ .
y ^ 1 = 0 , y ^ 2 = σ 2 σ 3 cos ( Γ 1 + ζ b ) .
[ σ 1 0 0 σ 2 ]             [ cos ψ - sin ψ sin ψ cos ψ ]             [ σ 3 0 0 σ 4 ] .
[ 0 y ^ 2 ] = [ cos ψ - sin ψ sin ψ cos ψ ]             [ x 1 x 2 ] + [ e 1 e 2 ] .
ψ = ψ - sin - 1 [ e 1 / ( x 1 2 + x 2 2 ) 1 / 2 ] ,
y ^ 2 = e 2 + ( x 1 2 + x 2 2 - e 1 2 ) 1 / 2 σ 1 + r 1 , 2 ( 1 - 1 2 / 2 ρ 2 ) .
y ^ 2 = r 1 , 2 ( 1 - 1 / 2 ρ 2 ) .
Var [ y ^ 2 ] = σ 2 ( 1 + 1 / 2 ρ 2 ) ,
b ^ N = b N [ 1 - i = 1 N - 1 ( i ρ ) 2 ] 1 / 2 + σ N b N [ 1 - 1 2 i = 1 N - 1 ( i ρ ) 2 ] + σ N .
bias / b N = ( N - 1 ) / 2 ρ 2 ,
Var / b N 2 = ( 1 + ( N - 1 ) / 2 ρ 2 ) / ρ 2 .
[ 0 · · · 0 b ^ N ] = [ R N ] [ a 1 a 2 · · · a N ] + [ e 1 e 2 · · · e N ] ,
R N [ c 1 , , c N ] = [ 0 · c ˜ 1 c ˜ N - 1 · · 0 b N 1 b N , N - 1 b N N ] + [ 0 0 e 1 N · · · · · · · · · 0 0 e N - 1 , N 0 0 0 ] .
{ R 2 R N - 1 R N } A = [ L \ E U ] = [ L + E U ] ,
R 2 R N - 1 R N = Q ,
A = Q ( L + E U ) .
Q Q + E Q , L + E U L + E L .
A ^ = ( Q + E Q ) ( L + E L ) = Q ( L + E U ) + Q ( E L - E U ) + E Q ( L + E L ) = A + Q ( E L - E U ) + E Q ( L + E L ) .
A ˜ i j , A ˜ k l = 2 σ 2 δ j l ( QQ ) i k + σ 2 δ i k ( L L ) j l + σ 4 δ i k δ j l = σ 2 ( A A ) j l δ i k + 2 σ 2 ( 1 + σ 2 ) δ i k δ j l ,
L L = A A + σ 2 I .
A ˜ , A ˜ = σ 2 A A + 2 σ 2 ( 1 + σ 2 ) I ,
A ˜ = [ A 2 + 2 ( 1 + σ 2 ) ] 1 / 2 .

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