Abstract

The problems of spatial diffraction and optical dispersion in the time domain was extended by analogy to optical ultrafast pulse filtering. An analog setup in the 4-f spatial-filter configuration is proposed, and several applications such as the temporal signal convolver or correlator and the joint transform processor are suggested, followed by ways to generate the temporal filter. The implementation aspects and an example of the speed superiority of this approach are also discussed.

© 1992 Optical Society of America

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References

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  1. J. Shamir, “Fundamental speed limitations on parallel processing,” Appl. Opt. 26, 1567 (1987).
    [CrossRef] [PubMed]
  2. A. W. Lohmann, A. S. Marathay, “Globality and speed of optical parallel processors,” Appl. Opt. 28, 3838–3842 (1989).
    [CrossRef] [PubMed]
  3. S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, A. P. Sukhorukov, “Nonstationary nonlinear optical effects and ultrafast light pulse formation,” IEEE J. Quantum Electron. QE-4, 598–605 (1968).
    [CrossRef]
  4. D. Grischkowsky, A. C. Balant, “Optical pulse compression based on enhanced frequency chirping,” Appl. Phys. Lett. 41, 1–3 (1982).
    [CrossRef]
  5. D. Grischkowsky, “Optical pulse compression,” Appl. Phys. Lett. 25, 566–568 (1974).
    [CrossRef]
  6. B. H. Kolner, “Active pulse compression using an integrated electro-optic phase modulator,” Appl. Phys. Lett. 52, 1122–1124 (1988).
    [CrossRef]
  7. E. B. Treacy, “Optical pulse compression with diffraction grating,” IEEE J. Quantum Electron. QE-5, 454–458 (1969).
    [CrossRef]
  8. G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989), pp. 164–170.
  9. H. G. Winful, “Pulse compression in optical fiber filters,” Appl. Phys. Lett. 46, 527–529 (1985).
    [CrossRef]
  10. B. H. Kolner, M. Nazarathy, “Temporal imaging with a time lens,” Opt. Lett. 14, 630–632 (1989).
    [CrossRef] [PubMed]
  11. B. H. Kolner, M. Nazarathy, “Temporal imaging with a time lens: erratum,” Opt. Lett. 15, 655 (1990).
    [CrossRef] [PubMed]
  12. A. VanderLugt, “Signal detection by complex spatial filter,” IEEE Trans. Inf. Theory IT-10, 139–143 (1964).
    [CrossRef]
  13. M. Born, Optik—Eine Lehrbuch Der Elektromagnetischen Lichtheorie (Springer-Verlag, New York, 1965), pp. 81–82.
  14. J. L. Horner, P. Gianino, “Phase only matched filtering,” Appl. Opt. 23, 812–816 (1984).
    [CrossRef] [PubMed]
  15. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 7.
  16. A. Lohmann, D. P. Paris, “Binary Fraunhofer holograms generated by computer,” Appl. Opt. 6, 1739–1748 (1967).
    [CrossRef] [PubMed]
  17. C. S. Weaver, J. W. Goodman, “A technique for optically convolving two functions,” Appl. Opt. 5, 1248–1249 (1966).
    [CrossRef] [PubMed]
  18. A. W. Lohmann, D. P. Paris, “Computer generated spatial filters for coherent optical data processing,” Appl. Opt. 7, 651–655 (1968).
    [CrossRef] [PubMed]

1990 (1)

1989 (2)

1988 (1)

B. H. Kolner, “Active pulse compression using an integrated electro-optic phase modulator,” Appl. Phys. Lett. 52, 1122–1124 (1988).
[CrossRef]

1987 (1)

1985 (1)

H. G. Winful, “Pulse compression in optical fiber filters,” Appl. Phys. Lett. 46, 527–529 (1985).
[CrossRef]

1984 (1)

1982 (1)

D. Grischkowsky, A. C. Balant, “Optical pulse compression based on enhanced frequency chirping,” Appl. Phys. Lett. 41, 1–3 (1982).
[CrossRef]

1974 (1)

D. Grischkowsky, “Optical pulse compression,” Appl. Phys. Lett. 25, 566–568 (1974).
[CrossRef]

1969 (1)

E. B. Treacy, “Optical pulse compression with diffraction grating,” IEEE J. Quantum Electron. QE-5, 454–458 (1969).
[CrossRef]

1968 (2)

S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, A. P. Sukhorukov, “Nonstationary nonlinear optical effects and ultrafast light pulse formation,” IEEE J. Quantum Electron. QE-4, 598–605 (1968).
[CrossRef]

A. W. Lohmann, D. P. Paris, “Computer generated spatial filters for coherent optical data processing,” Appl. Opt. 7, 651–655 (1968).
[CrossRef] [PubMed]

1967 (1)

1966 (1)

1964 (1)

A. VanderLugt, “Signal detection by complex spatial filter,” IEEE Trans. Inf. Theory IT-10, 139–143 (1964).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989), pp. 164–170.

Akhmanov, S. A.

S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, A. P. Sukhorukov, “Nonstationary nonlinear optical effects and ultrafast light pulse formation,” IEEE J. Quantum Electron. QE-4, 598–605 (1968).
[CrossRef]

Balant, A. C.

D. Grischkowsky, A. C. Balant, “Optical pulse compression based on enhanced frequency chirping,” Appl. Phys. Lett. 41, 1–3 (1982).
[CrossRef]

Born, M.

M. Born, Optik—Eine Lehrbuch Der Elektromagnetischen Lichtheorie (Springer-Verlag, New York, 1965), pp. 81–82.

Chirkin, A. S.

S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, A. P. Sukhorukov, “Nonstationary nonlinear optical effects and ultrafast light pulse formation,” IEEE J. Quantum Electron. QE-4, 598–605 (1968).
[CrossRef]

Drabovich, K. N.

S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, A. P. Sukhorukov, “Nonstationary nonlinear optical effects and ultrafast light pulse formation,” IEEE J. Quantum Electron. QE-4, 598–605 (1968).
[CrossRef]

Gianino, P.

Goodman, J. W.

Grischkowsky, D.

D. Grischkowsky, A. C. Balant, “Optical pulse compression based on enhanced frequency chirping,” Appl. Phys. Lett. 41, 1–3 (1982).
[CrossRef]

D. Grischkowsky, “Optical pulse compression,” Appl. Phys. Lett. 25, 566–568 (1974).
[CrossRef]

Horner, J. L.

Khokhlov, R. V.

S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, A. P. Sukhorukov, “Nonstationary nonlinear optical effects and ultrafast light pulse formation,” IEEE J. Quantum Electron. QE-4, 598–605 (1968).
[CrossRef]

Kolner, B. H.

Kovrigin, A. I.

S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, A. P. Sukhorukov, “Nonstationary nonlinear optical effects and ultrafast light pulse formation,” IEEE J. Quantum Electron. QE-4, 598–605 (1968).
[CrossRef]

Lohmann, A.

Lohmann, A. W.

Marathay, A. S.

Nazarathy, M.

Paris, D. P.

Shamir, J.

Sukhorukov, A. P.

S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, A. P. Sukhorukov, “Nonstationary nonlinear optical effects and ultrafast light pulse formation,” IEEE J. Quantum Electron. QE-4, 598–605 (1968).
[CrossRef]

Treacy, E. B.

E. B. Treacy, “Optical pulse compression with diffraction grating,” IEEE J. Quantum Electron. QE-5, 454–458 (1969).
[CrossRef]

VanderLugt, A.

A. VanderLugt, “Signal detection by complex spatial filter,” IEEE Trans. Inf. Theory IT-10, 139–143 (1964).
[CrossRef]

Weaver, C. S.

Winful, H. G.

H. G. Winful, “Pulse compression in optical fiber filters,” Appl. Phys. Lett. 46, 527–529 (1985).
[CrossRef]

Appl. Opt. (6)

Appl. Phys. Lett. (4)

D. Grischkowsky, A. C. Balant, “Optical pulse compression based on enhanced frequency chirping,” Appl. Phys. Lett. 41, 1–3 (1982).
[CrossRef]

D. Grischkowsky, “Optical pulse compression,” Appl. Phys. Lett. 25, 566–568 (1974).
[CrossRef]

B. H. Kolner, “Active pulse compression using an integrated electro-optic phase modulator,” Appl. Phys. Lett. 52, 1122–1124 (1988).
[CrossRef]

H. G. Winful, “Pulse compression in optical fiber filters,” Appl. Phys. Lett. 46, 527–529 (1985).
[CrossRef]

IEEE J. Quantum Electron. (2)

E. B. Treacy, “Optical pulse compression with diffraction grating,” IEEE J. Quantum Electron. QE-5, 454–458 (1969).
[CrossRef]

S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, A. P. Sukhorukov, “Nonstationary nonlinear optical effects and ultrafast light pulse formation,” IEEE J. Quantum Electron. QE-4, 598–605 (1968).
[CrossRef]

IEEE Trans. Inf. Theory (1)

A. VanderLugt, “Signal detection by complex spatial filter,” IEEE Trans. Inf. Theory IT-10, 139–143 (1964).
[CrossRef]

Opt. Lett. (2)

Other (3)

M. Born, Optik—Eine Lehrbuch Der Elektromagnetischen Lichtheorie (Springer-Verlag, New York, 1965), pp. 81–82.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 7.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989), pp. 164–170.

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Figures (14)

Fig. 1
Fig. 1

Temporal optical pulse compressor.

Fig. 2
Fig. 2

2-f system: (a) spatial configuration, (b) analogous temporal configuration.

Fig. 3
Fig. 3

4-f system: (a) spatial configuration, (b) analogous temporal configuration.

Fig. 4
Fig. 4

Typical output signal of a 4-f system with a temporal filter. Note that the x axis is −T′. Convolution and correlation appear along the positive and negative diffraction orders, respectively.

Fig. 5
Fig. 5

System for recording the temporal filter information.

Fig. 6
Fig. 6

Joint transform temporal processor: an example for correlation is given. The final result is along the diffraction orders of the output.

Fig. 7
Fig. 7

Variation of the dispersive group-velocity for fused silica.

Fig. 8
Fig. 8

Signal propagation through a 2-f system: the solid curve is for normalized input, the dashed curve is for the normalized signal just before the lens, and the dotted curve is for the normalized signal just before the filter. The time scale is 14.5 ps per pixel.

Fig. 9
Fig. 9

Regular 1:1 imaging: the input signal is on the left and the normalized output signal when no filter is used is on the right.

Fig. 10
Fig. 10

Matched filter correlation: same plot as in Fig. 9 but when a matched filter is used.

Fig. 11
Fig. 11

Phase-only filter correlation: same plot as in Fig. 9 but when a phase-only matched filter is used.

Fig. 12
Fig. 12

Same as Fig. 8 but for the joint transform case.

Fig. 13
Fig. 13

Joint transform correlation: input signals are on tile left and the joint transform output signal is on the right. Note that the correlation is observed along the diffraction orders.

Fig. 14
Fig. 14

Differential operator: the input is the same as in Fig. 9 but with a differential filter.

Equations (38)

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u ( x , z = 0 ) = U ( ω ) exp ( i 2 π ω x ) d ω .
u ( x , z 0 ) exp ( i k z ) U ( ω ) × exp ( - i π λ z g ω 2 ) exp ( i 2 π g ω x ) d ω .
u ( t , z = 0 ) = U ( μ ) exp ( i 2 π μ t ) d μ = u 0 ( t ) .
exp { i 2 π [ μ t - β ( μ ) z ] } .
β ( μ ) = β 0 + β 1 μ + β 2 μ 2 2 .
exp ( - i 2 π β 0 z ) exp [ i 2 π μ ( t - β 1 z ) ] exp ( - i β 2 z μ 2 ) .
u ( t , z ) = - U ( μ ) exp ( - i π β 2 z μ 2 ) × exp [ i 2 π μ ( t - β 1 z ) ] d μ ,
t x , β 2 λ , μ ω ,
u ( t , z ) u ( x , z ) , U ( μ ) U ( ω ) .
L ( x ) = exp ( - i π x 2 λ f ) .
L ( t ) = exp [ - i π ( t - t 1 τ ) 2 ] ,
L ( t ) = - L f ( μ ) exp ( i 2 π μ t ) d μ ,
L f ( μ ) = τ exp ( - i π / 4 ) exp ( - i 2 π μ t 1 ) exp [ i π ( μ t ) 2 ] .
u ( t , f - 0 ) = - U ( μ ) exp [ i ϕ ( μ , f ) ] exp ( i 2 π μ t ) d μ , ϕ ( μ , t ) = - 2 π β 1 f μ - π μ 2 β 2 f .
u ( t , f + 0 ) = - - U ( μ ) exp [ i ϕ ( μ , f ) ] L f ( ρ ) × exp [ i 2 π ( μ + ρ ) t ] × exp ( - i 2 π ρ t 1 ) d ρ d μ .
u ( t , 2 f ) = - - U ( μ ) exp { i [ ϕ ( μ , f ) + ϕ ( μ + ρ , f ) + π ( ρ τ ) 2 ] } × exp { i 2 π ( ρ + μ ) t } × exp ( - i 2 π ρ t 1 ) d ρ d μ .
- 2 π ( 2 μ + ρ ) f β 1 + π ρ 2 ( τ 2 - β 2 f ) - 2 π β 2 f μ ( μ + ρ ) .
τ 2 = β 2 f .
u ( t , 2 f ) = U ( t - 2 f β 1 τ 2 ) ,
t 1 = β 1 f ,
U 2 f ( μ ) = exp ( - i 2 π μ 2 β 1 f ) u ( - τ 2 μ , 0 ) .
u ( t , 4 f ) = - - U 2 f ( μ ) exp { i [ ϕ ( μ , f ) + ϕ ( ρ + μ , f ) + π ( ρ τ ) 2 ] } × exp [ 2 π i ( ρ + μ ) t ] exp ( - 2 π i ρ t 1 ) d ρ d μ .
τ 2 = β 2 f .
u ( t , 4 f ) = u ( 4 f β 1 - t , 0 ) = u 0 ( 4 f β 1 - t ) ,
t 1 = 3 β 1 f .
τ 2 = β 2 f ,             t 1 = 2 β 1 f + β 1 f ,
M = - f / f = - ( τ / τ ) 2 .
P ( t - 2 f β 1 τ 2 ) = P ( T ) = A ( T ) exp [ i 2 π ϕ ( T ) ] ,
H ( T ) = A ( T ) { exp ( i 2 π α 0 T ) + exp [ i 2 π ϕ ( T ) ] } × { exp ( - i 2 π α 0 T ) + exp [ - i 2 π ϕ ( T ) ] } = A ( T ) ( 2 + 2 cos { 2 π [ α 0 T + ϕ ( T ) ] } ) ,
u ( t , 4 f ) = f ( T - α 0 ) * u ( T ) + f ( T + α 0 ) u ( T ) + O ( T ) ,
u ( t , 0 ) = u 1 ( t + t 0 , 0 ) + u 2 ( t - t 0 , 0 ) ,
u ( t , 2 f - 0 ) = U 1 ( T ) exp ( i 2 π t 0 T ) + U 2 ( T ) exp ( - i 2 π t 0 T ) ,
u ( t , 2 f + 0 ) = u ( t , 2 f - 0 ) 2 .
u ( t , 4 f ) = u 1 ( T , 0 ) u 2 ( T - t 0 , 0 ) + u 1 ( T - t 0 , 0 ) u 2 ( T , 0 ) + O ( T ) .
W f = τ 2 d t ,
d t = τ 2 0.2 β 1 f .
β 1 f d t = 120 points
F ( t ) = k t - 2 β 1 f τ 2

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