Abstract

Signal processing in general, and optical signal processing in particular, make extensive use of linear transformations. The temporal nature of many optical signals (e.g., in optical communication systems) makes the realization of temporal transformations a desired extension. We present a system making possible the realization of arbitrary temporal linear transformation. The system supports real-time changes of the realized transformation. The mathematical analysis is derived, and computer simulations are presented.

© 2004 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. C. Froehly, B. Colombeau, M. Vampouille, “Shaping and analysis of picosecond light pulses,” in Progress in Optics, Vol. XX, E. Wolf, ed. (Elsevier North-Holland, Amsterdam, 1983), pp. 63–153.
  2. M. C. Nuss, M. Li, T. H. Chiu, A. M. Weiner, A. Partovi, “Time-to-space mapping of femtosecond pulses,” Opt. Lett. 19, 664–666 (1994).
    [CrossRef] [PubMed]
  3. A. W. Lohmann, D. Mendlovic, “Temporal perfect-shuffle optical processor,” Opt. Lett. 17, 882–884 (1992).
    [CrossRef]
  4. T. Mazurenko, “Time-domain Fourier transform holography and possible applications in signal processing,” Opt. Eng. 31, 739–749 (1992).
    [CrossRef]
  5. J. Azana, “Time-to-frequency conversion using a single time lens,” Opt. Commun. 217, 205–209 (2003).
    [CrossRef]
  6. S. A. Akhmanov, A. S. Chirkin, N. K. 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]
  7. B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951–1963 (1994).
    [CrossRef]
  8. H. Hakimi, F. Hakimi, K. L. Hall, K. A. Rauschenbach, “A new wide-band pulse-restoration technique for digital fiber-optic communication systems using temporal gratings,” IEEE Photon. Technol. Lett. 11, 1048–1950 (1999).
    [CrossRef]
  9. D. Mendlovic, Z. Zalevsky, P. Andreas, “A novel device for achieving negative or positive dispersion and its applications,” Optik (Stuttgart) 110, 45–50 (1999).
  10. P. Andreas, J. Lancis, W. D. Furlan, “White light Fourier transformer with low chromatic aberration,” Appl. Opt. 31, 4682–4689 (1992).
    [CrossRef]
  11. J. Lancis, P. Andreas, W. D. Furlan, A. Pons, “All-diffractive, achromatic Fourier transform setup,” Opt. Lett. 19, 402–404 (1994).
    [PubMed]

2003 (1)

J. Azana, “Time-to-frequency conversion using a single time lens,” Opt. Commun. 217, 205–209 (2003).
[CrossRef]

1999 (2)

H. Hakimi, F. Hakimi, K. L. Hall, K. A. Rauschenbach, “A new wide-band pulse-restoration technique for digital fiber-optic communication systems using temporal gratings,” IEEE Photon. Technol. Lett. 11, 1048–1950 (1999).
[CrossRef]

D. Mendlovic, Z. Zalevsky, P. Andreas, “A novel device for achieving negative or positive dispersion and its applications,” Optik (Stuttgart) 110, 45–50 (1999).

1994 (3)

1992 (3)

P. Andreas, J. Lancis, W. D. Furlan, “White light Fourier transformer with low chromatic aberration,” Appl. Opt. 31, 4682–4689 (1992).
[CrossRef]

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]

1968 (1)

S. A. Akhmanov, A. S. Chirkin, N. K. 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]

Akhmanov, S. A.

S. A. Akhmanov, A. S. Chirkin, N. K. 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]

Andreas, P.

Azana, J.

J. Azana, “Time-to-frequency conversion using a single time lens,” Opt. Commun. 217, 205–209 (2003).
[CrossRef]

Chirkin, A. S.

S. A. Akhmanov, A. S. Chirkin, N. K. 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]

Chiu, T. H.

Colombeau, B.

C. Froehly, B. Colombeau, M. Vampouille, “Shaping and analysis of picosecond light pulses,” in Progress in Optics, Vol. XX, E. Wolf, ed. (Elsevier North-Holland, Amsterdam, 1983), pp. 63–153.

Drabovich, N. K.

S. A. Akhmanov, A. S. Chirkin, N. K. 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]

Froehly, C.

C. Froehly, B. Colombeau, M. Vampouille, “Shaping and analysis of picosecond light pulses,” in Progress in Optics, Vol. XX, E. Wolf, ed. (Elsevier North-Holland, Amsterdam, 1983), pp. 63–153.

Furlan, W. D.

Hakimi, F.

H. Hakimi, F. Hakimi, K. L. Hall, K. A. Rauschenbach, “A new wide-band pulse-restoration technique for digital fiber-optic communication systems using temporal gratings,” IEEE Photon. Technol. Lett. 11, 1048–1950 (1999).
[CrossRef]

Hakimi, H.

H. Hakimi, F. Hakimi, K. L. Hall, K. A. Rauschenbach, “A new wide-band pulse-restoration technique for digital fiber-optic communication systems using temporal gratings,” IEEE Photon. Technol. Lett. 11, 1048–1950 (1999).
[CrossRef]

Hall, K. L.

H. Hakimi, F. Hakimi, K. L. Hall, K. A. Rauschenbach, “A new wide-band pulse-restoration technique for digital fiber-optic communication systems using temporal gratings,” IEEE Photon. Technol. Lett. 11, 1048–1950 (1999).
[CrossRef]

Khokhlov, R. V.

S. A. Akhmanov, A. S. Chirkin, N. K. 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]

Kovrigin, A. I.

S. A. Akhmanov, A. S. Chirkin, N. K. 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]

Lancis, J.

Li, M.

Lohmann, A. W.

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

Mazurenko, T.

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

Mendlovic, D.

D. Mendlovic, Z. Zalevsky, P. Andreas, “A novel device for achieving negative or positive dispersion and its applications,” Optik (Stuttgart) 110, 45–50 (1999).

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

Nuss, M. C.

Partovi, A.

Pons, A.

Rauschenbach, K. A.

H. Hakimi, F. Hakimi, K. L. Hall, K. A. Rauschenbach, “A new wide-band pulse-restoration technique for digital fiber-optic communication systems using temporal gratings,” IEEE Photon. Technol. Lett. 11, 1048–1950 (1999).
[CrossRef]

Sukhorukov, A. P.

S. A. Akhmanov, A. S. Chirkin, N. K. 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]

Vampouille, M.

C. Froehly, B. Colombeau, M. Vampouille, “Shaping and analysis of picosecond light pulses,” in Progress in Optics, Vol. XX, E. Wolf, ed. (Elsevier North-Holland, Amsterdam, 1983), pp. 63–153.

Weiner, A. M.

Zalevsky, Z.

D. Mendlovic, Z. Zalevsky, P. Andreas, “A novel device for achieving negative or positive dispersion and its applications,” Optik (Stuttgart) 110, 45–50 (1999).

Appl. Opt. (1)

IEEE J. Quantum Electron. (2)

S. A. Akhmanov, A. S. Chirkin, N. K. 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]

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

IEEE Photon. Technol. Lett. (1)

H. Hakimi, F. Hakimi, K. L. Hall, K. A. Rauschenbach, “A new wide-band pulse-restoration technique for digital fiber-optic communication systems using temporal gratings,” IEEE Photon. Technol. Lett. 11, 1048–1950 (1999).
[CrossRef]

Opt. Commun. (1)

J. Azana, “Time-to-frequency conversion using a single time lens,” Opt. Commun. 217, 205–209 (2003).
[CrossRef]

Opt. Eng. (1)

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

Opt. Lett. (3)

Optik (Stuttgart) (1)

D. Mendlovic, Z. Zalevsky, P. Andreas, “A novel device for achieving negative or positive dispersion and its applications,” Optik (Stuttgart) 110, 45–50 (1999).

Other (1)

C. Froehly, B. Colombeau, M. Vampouille, “Shaping and analysis of picosecond light pulses,” in Progress in Optics, Vol. XX, E. Wolf, ed. (Elsevier North-Holland, Amsterdam, 1983), pp. 63–153.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Device for controllable positive dispersion.

Fig. 2
Fig. 2

Suggested setup.

Fig. 3
Fig. 3

Mask of the tunable filter for frequency 1: (a) real value, (b) absolute value.

Fig. 4
Fig. 4

Simulation results of both filters: frequency 1 (solid curve), frequency 2 (dashed curve).

Equations (49)

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

ϕ = β ( ν ) z = β 0 z + β 1 z ( 2 π ν ) + β 2 z 2   ( 2 π ν ) 2 +   ,
u ( t ,   z ) = - U ( ν ) exp - i   β 2 ( h ) 2   z ( 2 π ν ) 2 × exp { i 2 π ν [ t - β 1 ( h ) z ] } exp ( i 2 π ν 0 t ) d ν .
d = Z 1 Z 2 .
V ( λ ) = - λ O Z 1 d λ O Z 1 - λ d ,
D ( λ ) = λ λ O Z 2 d 2 λ O 2 Z 1 Z 2 - λ λ O Z 2 d + λ 2 d 2 ,
l ( λ ) = h 1 V ( λ )   [ V ( λ ) + d ] ,
P ( λ ,   h 1 ) = - ( h 1 2 + d 2 ) 1 / 2 + { [ l ( λ ) - h 1 ] 2 + d 2 } 1 / 2 + [ l 2 ( λ ) + D 2 ( λ ) ] 1 / 2 .
ϕ = 2 π λ   P ( ω ,   h ) = 2 π ν c   P ( ω ,   h ) = ω cz   P ( ω ,   h ) z ,
β 1 ( h ) = P ( ω 0 ,   h ) cz + ω 0 P ω ω = ω 0 cz ,
β 2 ( h ) = ω 0 2 P ω 2 ω = ω 0 2 cz + 2   P ω ω = ω 0 cz .
u ( t ,   z ,   h ) = - U ( ν ) exp - i   β 2 ( h ) 2   z ( 2 π ν ) 2 × exp { i 2 π ν [ t - β 1 ( h ) z ] } exp ( i 2 π ν 0 t ) d ν .
u ( t ,   z ) = exp ( i 2 π ν 0 t ) × - - A ( h ) U ( ν ) exp - i   β 2 ( h ) 2   z ( 2 π ν ) 2 × exp [ - i 2 π ν β 1 ( h ) z ] exp ( i 2 π ν t ) d ν d h ;
B z ( h ,   ν ) = β 1 ( h ) z + π ν β 2 ( h ) z
H ( ν ) = - A ( h ) exp [ - i 2 π ν B z ( h ,   ν ) ] d h
u ( t ,   z ) = exp ( i 2 π ν 0 t ) - H ( ν ) U ( ν ) exp ( i 2 π ν t ) d ν = exp ( i 2 π ν 0 t ) F - 1 { H ( ν ) U ( ν ) } ,
= - | H ( ν ) - H d ( ν ) | 2 d ν = - { | H ( ν ) | 2 + | H d ( ν ) | 2 - 2   Re [ H ( ν ) H d * ( ν ) ] } d ν .
A δ ( h ) = A ( h ) + δ A ( h ) H δ ( ν ) = H ( ν ) + δ H ( ν ) ,
Δ = δ - = 0 .
Re - [ H * ( ν ) δ H ( ν ) - δ H ( ν ) H d * ( ν ) ] d ν = 0 .
α z ( h 1 ,   h 2 ,   ν ) = B z ( h 1 ,   ν ) - B z ( h 2 ,   ν ) ,
M ( h ,   ν ) = - A * ( h 1 ) exp [ i 2 π ν α z ( h 1 ,   h ,   ν ) ] d h 1 ,
P ( h ) = - { M ( h ,   ν ) - H d * ( ν ) exp [ - i 2 π ν B z ( h ,   ν ) ] } d ν .
Re - δ A ( h ) P ( h ) d h = 0 .
P ( h ) = 0 .
- A * ( h 1 ) I ( h 1 ,   h ) d h 1
= - H d * ( ν ) exp [ - i 2 π ν B z ( h ,   ν ) ] d ν ,
I ( h 1 ,   h ) = -   exp [ i 2 π ν α z ( h 1 ,   h ,   ν ) ] d ν .
I ( h 1 ,   h ) = - i   exp ( i π / 4 ) ( 2 π ) 1 / 2   1 [ β 2 ( h ) z - β 2 ( h 1 ) z ] 1 / 2   × exp i 2 [ β 1 ( h ) z - β 1 ( h 1 ) z ] 2 β 2 ( h ) z - β 2 ( h 1 ) z .
β 1 ( h - h 1 ) β 1 ( h ) - β 1 ( h 1 ) + B 1 ,
β 2 ( h - h 1 ) β 2 ( h ) - β 2 ( h 1 ) + B 2 .
I ( h 1 ,   h ) - i   exp ( i π / 4 ) 2 π   1 [ β 2 ( h - h 1 ) z - B 2 ] 1 / 2   × exp i 2 [ β 1 ( h - h 1 ) z - B 1 ] 2 β 2 ( h - h 1 ) z - B 2 .
K z ( h ) = 1 β 2 ( h ) z - B 2   exp - i 2 [ β 1 ( h ) z - B 1 ] 2 β 2 ( h ) z - B 2 .
exp ( i π / 4 ) 2 π   [ A ( h )   *   K z ( h ) ]
= - H d ( ν ) exp [ i 2 π ν B z ( h ,   ν ) ] d ν ,
G z ( h ) = 2 π exp ( i π / 4 )   - H d ( ν ) exp [ i 2 π ν B z ( h ,   ν ) ] d ν ;
A ( h )   *   K z ( h ) = G z ( h ) .
A ˜ ( μ ) K ˜ z ( μ ) = G ˜ z ( μ ) = - G z ( h ) exp ( - i 2 π h μ ) d h ,
A ( h ) = -   G ˜ z ( μ ) K ˜ z ( μ )   exp ( i 2 π h μ ) d μ = - -   1 K ˜ z ( μ )   G z ( h ) exp [ i 2 π μ ( h - h ) ] d h d μ .
- ( N - 1 ) 2   Δ h , - ( N - 3 ) 2   Δ h ,   ,   ( N - 3 ) 2   Δ h ,   ( N - 1 ) 2   Δ h .
A ( h ) = n = 0 N - 1 a n rect h Δ h + ( N - 1 ) 2 - n ,
rect ( x ) = 1 , | x | 1 / 2 0 , | x | > 1 / 2 .
- A * ( h 1 ) I ( h 1 ,   h ) d h 1
= - n = 0 N - 1 a n * rect h 1 Δ h 1 + ( N - 1 ) 2 - n I ( h 1 ,   h ) d h 1 = n = 0 N - 1 a n * - rect h 1 Δ h 1 + ( N - 1 ) 2 - n I ( h 1 ,   h ) d h 1 .
K ( n ,   h ) = - rect h 1 Δ h 1 + ( N - 1 ) 2 - n I ( h 1 ,   h ) d h 1 = ( n - N / 2 ) Δ h ( n - N / 2 + 1 ) Δ h I ( h 1 ,   h ) d h 1 ,
G ( h ) = - H d * ( ν ) exp [ - i 2 π ν B z ( h ,   ν ) ] d ν ;
n = 0 N - 1 a n * K ( n ,   h ) = G ( h ) .
n = 0 N - 1 a n * K ( n ,   m ) = G ( m ) .
A ¯ K = G ¯ ,
A ¯ = G ¯ K - 1 .

Metrics