Abstract

Optical signal processing can be done with time-lens devices. A temporal processor based on chirp-z transformers is suggested. This configuration is more compact than a conventional 4-f temporal processor. On the basis of implementation aspects of such a temporal processor, we did a performance analysis. This analysis leads to the conclusion that an ultrafast optical temporal processor can be implemented.

© 1995 Optical Society of America

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References

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  1. A. W. Lohmann, D. Mendlovic, “Temporal perfect-shuffle optical processor,” Opt. Lett. 17, 882–884 (1992).
    [CrossRef]
  2. A. W. Lohmann, D. Mendlovic, “Temporal filtering with time lenses,” Appl. Opt. 31, 6212–6219 (1992).
    [CrossRef] [PubMed]
  3. T. Mazurenko, “Time-domain Fourier-transform holography and possible applications in signal processing,” Opt. Eng. 31, 739–749 (1992).
    [CrossRef]
  4. 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]
  5. A. Papoulis, “Pulse compression, fiber communication, and diffraction: a unified approach,” J. Opt. Soc. Am. A 11, 3–13 (1994).
    [CrossRef]
  6. B. H. Kolner, “Space–time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951–1963 (1994).
    [CrossRef]
  7. B. H. Kolner, M. Nazarathy, “Temporal imaging with a time lens,” Opt. Lett. 14, 630–632 (1989).
    [CrossRef] [PubMed]
  8. B. H. Kolner, “Active pulse compression using an integrated electro-optic phase modulator,” Appl. Phys. Lett. 52, 1122–1124 (1988).
    [CrossRef]
  9. A. VanderLugt, Optical Signal Processing (Wiley, New York, 1992).
  10. G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989), Chap. 3.
  11. H. G. Winful, “Pulse compression in optical fiber filters,” Appl. Phys. Lett. 46, 527–529 (1985).
    [CrossRef]

1994

B. H. Kolner, “Space–time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951–1963 (1994).
[CrossRef]

A. Papoulis, “Pulse compression, fiber communication, and diffraction: a unified approach,” J. Opt. Soc. Am. A 11, 3–13 (1994).
[CrossRef]

1992

A. W. Lohmann, D. Mendlovic, “Temporal filtering with time lenses,” Appl. Opt. 31, 6212–6219 (1992).
[CrossRef] [PubMed]

A. W. Lohmann, D. Mendlovic, “Temporal perfect-shuffle optical processor,” Opt. Lett. 17, 882–884 (1992).
[CrossRef]

T. Mazurenko, “Time-domain Fourier-transform holography and possible applications in signal processing,” Opt. Eng. 31, 739–749 (1992).
[CrossRef]

1989

1988

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

1985

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

1968

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]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989), Chap. 3.

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]

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]

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.

B. H. Kolner, “Space–time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951–1963 (1994).
[CrossRef]

B. H. Kolner, M. Nazarathy, “Temporal imaging with a time lens,” Opt. Lett. 14, 630–632 (1989).
[CrossRef] [PubMed]

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

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. W.

A. W. Lohmann, D. Mendlovic, “Temporal perfect-shuffle optical processor,” Opt. Lett. 17, 882–884 (1992).
[CrossRef]

A. W. Lohmann, D. Mendlovic, “Temporal filtering with time lenses,” Appl. Opt. 31, 6212–6219 (1992).
[CrossRef] [PubMed]

Mazurenko, T.

T. Mazurenko, “Time-domain Fourier-transform holography and possible applications in signal processing,” Opt. Eng. 31, 739–749 (1992).
[CrossRef]

Mendlovic, D.

A. W. Lohmann, D. Mendlovic, “Temporal filtering with time lenses,” Appl. Opt. 31, 6212–6219 (1992).
[CrossRef] [PubMed]

A. W. Lohmann, D. Mendlovic, “Temporal perfect-shuffle optical processor,” Opt. Lett. 17, 882–884 (1992).
[CrossRef]

Nazarathy, M.

Papoulis, A.

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]

VanderLugt, A.

A. VanderLugt, Optical Signal Processing (Wiley, New York, 1992).

Winful, H. G.

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

Appl. Opt.

Appl. Phys. Lett.

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.

B. H. Kolner, “Space–time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951–1963 (1994).
[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]

J. Opt. Soc. Am. A

Opt. Eng.

T. Mazurenko, “Time-domain Fourier-transform holography and possible applications in signal processing,” Opt. Eng. 31, 739–749 (1992).
[CrossRef]

Opt. Lett.

A. W. Lohmann, D. Mendlovic, “Temporal perfect-shuffle optical processor,” Opt. Lett. 17, 882–884 (1992).
[CrossRef]

B. H. Kolner, M. Nazarathy, “Temporal imaging with a time lens,” Opt. Lett. 14, 630–632 (1989).
[CrossRef] [PubMed]

Other

A. VanderLugt, Optical Signal Processing (Wiley, New York, 1992).

G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989), Chap. 3.

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Figures (8)

Fig. 1
Fig. 1

Spatial processors: (a) 2-f system; the output of this system is a scaled Fourier transform of the input image. (b) 4-f system; this is a basic Fourier optics image processor, which contains two 2-f systems in cascade. (b) is a 1:1 imaging setup capable of Fourier-plane filtering.

Fig. 2
Fig. 2

Optical temporal 2-f system.

Fig. 3
Fig. 3

Optical temporal 4-f processor.

Fig. 4
Fig. 4

Chirp-z transformer: (a) spatial configuration, (b) temporal analog configuration.

Fig. 5
Fig. 5

2-Chirp-z processor: (a) spatial configuration, (b) temporal analog configuration.

Fig. 6
Fig. 6

Schematic illustration of the input temporal signal with a total width of Δt and a finest resolution of 1/Δω.

Fig. 7
Fig. 7

Input signal is composed of NPs resolution points, each with a Gaussian shape and half-width T0.

Fig. 8
Fig. 8

Time distance between two input signals.

Equations (52)

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u ( x , z 0 ) C exp ( i k z ) U ( μ ) exp ( i πλ z μ 2 ) × exp ( i 2 πμ x ) d μ .
exp { i 2 π [ ω t β ( ω ) z ] } ,
β ( ω ) β 0 + β 1 ω + β 2 ( ω 2 / 2 ) .
u ( t , z ) = U ( ω ) exp ( i πβ 2 z ω 2 ) exp [ i 2 πω ( t β 1 z ) ] d ω .
L ( t ) = exp [ i π ( t t 1 T ) 2 ] ,
τ 2 = β 2 f ,
t 1 = β 1 f
u ( t , 2 f ) = U 0 ( t 2 f β 1 τ 2 ) ,
τ 2 = β 2 f ,
t 1 = 3 β 1 f .
u ( t , 4 f ) = u 0 ( 4 f β 1 t ) .
u ( t , 0 + ) = u ( t ) L 1 ( t ) .
L 1 ( t ) = exp ( i π t 2 β 2 f ) ,
L 1 f ( ω ) = ( β 2 f ) 1 / 2 exp ( i π / 4 ) exp ( i πβ 2 f ω 2 ) .
u ( t , 0 + ) = U ( ω ) L 1 f ( μ ) exp ( i 2 πω t ) exp ( i 2 πμ t ) d ω d μ .
u ( t , f ) = U ( ω ) L 1 f ( μ ) exp { i ϕ [ ( μ + ω ) , f ] } × exp ( i 2 πω t ) exp ( i 2 πμ t ) d ω d μ , ϕ ( ω , f ) 2 πβ 1 f ω πω 2 β 2 f .
u ( t , f ) = C U ( t β 1 f β 2 f ) exp [ i π ( t β 1 f ) 2 β 2 f ] ,
L 2 ( t ) = exp [ i π ( t β 1 f ) 2 β 2 f ] ,
u ( t , f ) = C U ( t β 1 f β 2 f ) ,
u ( t , f + ) = U ( t β 1 f β 2 f ) L 3 ( t ) .
L 3 ( t ) = exp [ i π ( t β 1 f ) 2 β 2 f ] ,
u ( t ¯ , f + ) = U ( t ¯ β 2 f ) exp [ i π t ¯ β 2 f ] t ¯ = def t β 1 f .
u ( t , 2 f ) = u ( 2 β 1 f t ) exp [ i π ( t 2 β 1 f ) 2 β 2 f ] .
L 4 ( t ) = exp [ i π ( t 2 β 1 f ) 2 β 2 f ] ,
u ( t , 2 f ) = u ( 2 β 1 f t ) .
s ( t , z = 0 ) = exp ( t 2 / 2 T 0 2 ) .
s ( t , z ) = ( T 0 2 T 0 2 i β 2 z ) 1 / 2 exp [ ( t β 1 z ) 2 2 ( T 0 2 i β 2 z ) ] .
T 1 = T 0 [ 1 + ( z β 2 T 0 2 ) 2 ] 1 / 2 .
T 1 = z β 2 / T 0 .
Δ t 1 = Δ t + β 2 f Δ ω .
Δ t 1 < β 1 f ,
δ T > Δ t 1 .
Δ t 2 = β 2 f Δ ω .
FT [ u ( t , f ) ] = FT [ U ( t f β 1 τ 2 ) ] = C u ( β 2 f ω ) .
Δ ω 2 = Δ t / β 2 f .
Δ t F = Δ t 2 ,
Δ ω F > Δ t / β 2 f .
Δ t < f ( β 1 β 2 Δ ω ) .
Δ ω < β 1 / β 2 .
Δ t < Δ ω M β 2 f .
NP s = Δ t Δ ω M ,
Δ t = ( 1 / c 1 ) f ( β 1 β 2 Δ ω ) , c 1 > 1 ,
Δ t = ( 1 / d 1 ) Δ ω M β 2 f , d 1 > 1 ,
NP s = MIN [ ( 1 / c 1 ) f ( β 1 β 2 Δ ω M ) Δ ω M , Δ ω M 2 β 2 f ( 1 / d 1 ) ] , c 1 , d 1 > 1 .
N = Δ t Δ ω M / δ T .
δ T > Δ t + β 2 f Δ ω M .
δ T = c 2 [ ( 1 / c 1 ) f ( β 1 β 2 Δ ω M ) + f β 2 Δ ω M ] , c 2 > 1 ,
δ T = d 2 [ ( 1 / d 1 ) f β 2 Δ ω M + f β 2 Δ ω M ] , d 2 > 1 .
N = MIN { ( β 1 β 2 Δ ω M ) Δ ω M c 2 [ β 1 + β 2 Δ ω M ( c 1 1 ) ] , Δ ω M d 2 ( d 1 + 1 ) } , c 1 , c 2 , d 1 , d 2 > 1 .
u ( t , 2 f ) = U ( t 2 f β 1 β 2 f ) ,
Δ t 2 = β 2 f Δ ω,
Δ ω 2 = Δ t / β 2 f .

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