Abstract

This paper extends the investigation of zone plates to the temporal case. By exploiting the space–time duality between the paraxial diffraction of beams in space and the linear dispersion of optical pulses, we present the time-domain analog of a multiple imaging system. The temporal system is created by using a digital chirp signal generator together with two dispersive delay lines. The key motivation is the potential application of the temporal zone plate to multiple compressions of pulses in dispersive lines.

© 2008 Optical Society of America

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References

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  1. A. Papoulis, “Pulse compression, fiber communication and diffraction: a unified approach,” J. Opt. Soc. Am. A 11, 3-13 (1994).
    [CrossRef]
  2. B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951-1963 (1994).
    [CrossRef]
  3. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
    [CrossRef]
  4. J. Jones, “On the propagation of a pulse through a dispersive medium,” Am. J. Phys. 42, 43-46 (1974).
    [CrossRef]
  5. C. V. Bennet and B. H. Kolner, “Upconversion time microscope demostrating 103× magnification of femtosecond waveforms,” Opt. Lett. 24, 783-785 (1999).
    [CrossRef]
  6. G. P. Agrawal, Applications of Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).
  7. T. Jannson, “Real-time Fourier transformation in dispersive optical fibers,” Opt. Lett. 8, 232-234 (1983).
    [CrossRef] [PubMed]
  8. M. A. Muriel, J. Azaña, and A. Carballar, “Real-time Fourier transformer based on fiber gratings,” Opt. Lett. 24, 1-3 (1999).
    [CrossRef]
  9. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).
  10. F. Le Chevalier, Principles of Radar and Sonar Signal Processing (Artech House, 2001).
  11. B. H. Kolner and M. Nazarathy, “Temporal imaging with a time lens,” Opt. Lett. 14, 630-632 (1989).
    [CrossRef] [PubMed]
  12. B. H. Kolner, “Active pulse compression using an integrated electro-optic phase modulator,” Appl. Phys. Lett. 52, 1122-1124 (1988).
    [CrossRef]
  13. A. A. Godil, B. A. Auld, and D. M. Bloom, “Time-lens producing 1.9 ps optical pulses,” Appl. Phys. Lett. 62, 1047-1049 (1993).
    [CrossRef]
  14. L. Kh. Mouradian, F. Louradour, V. Messager, A. Barthelemy, and C. Froehly, “Spectro-temporal imaging of femtosecond events,” IEEE J. Quantum Electron. 36, 795-801 (2000).
    [CrossRef]
  15. J. Ojeda-Castañeda and C. Gomez-Reino, Selected Papers on Zone Plates (SPIE Milestone Series, Vol. MS128) (SPIE Optical Engineering Press, 1996), and references therein.
  16. L. Chantada, C. R. Fernandez-Pousa, M. T. Flores-Arias, and C. Gomez-Reino, “Focusing of a train of pulses in phase opposition through a linear dispersive medium,” J. Opt. A, Pure Appl. Opt. 7, 767-773 (2005).
    [CrossRef]
  17. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2005).
  18. C. V. Bennet and B. H. Kolner, “Aberrations in temporal imaging,” IEEE J. Quantum Electron. 37, 20-32 (2001).
    [CrossRef]
  19. J. D. McKinney, D. Seo, D. E. Leaird, and A. M. Weiner, “Photonically assisted generation of arbitrary millimetre wave and microwave electromagnetic waveforms via direct space-to-time optical pulse shaping,” J. Lightwave Technol. 12, 3020-3028 (2003).
    [CrossRef]
  20. R. T. Watts, “Novel diagnostic technologies for optical communications systems,” Ph.D. dissertation (The University of Auckland, 2008).

2005 (1)

L. Chantada, C. R. Fernandez-Pousa, M. T. Flores-Arias, and C. Gomez-Reino, “Focusing of a train of pulses in phase opposition through a linear dispersive medium,” J. Opt. A, Pure Appl. Opt. 7, 767-773 (2005).
[CrossRef]

2003 (1)

J. D. McKinney, D. Seo, D. E. Leaird, and A. M. Weiner, “Photonically assisted generation of arbitrary millimetre wave and microwave electromagnetic waveforms via direct space-to-time optical pulse shaping,” J. Lightwave Technol. 12, 3020-3028 (2003).
[CrossRef]

2001 (1)

C. V. Bennet and B. H. Kolner, “Aberrations in temporal imaging,” IEEE J. Quantum Electron. 37, 20-32 (2001).
[CrossRef]

2000 (1)

L. Kh. Mouradian, F. Louradour, V. Messager, A. Barthelemy, and C. Froehly, “Spectro-temporal imaging of femtosecond events,” IEEE J. Quantum Electron. 36, 795-801 (2000).
[CrossRef]

1999 (2)

1994 (2)

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

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

1993 (1)

A. A. Godil, B. A. Auld, and D. M. Bloom, “Time-lens producing 1.9 ps optical pulses,” Appl. Phys. Lett. 62, 1047-1049 (1993).
[CrossRef]

1989 (1)

1988 (1)

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

1983 (1)

1974 (1)

J. Jones, “On the propagation of a pulse through a dispersive medium,” Am. J. Phys. 42, 43-46 (1974).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Applications of Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

Auld, B. A.

A. A. Godil, B. A. Auld, and D. M. Bloom, “Time-lens producing 1.9 ps optical pulses,” Appl. Phys. Lett. 62, 1047-1049 (1993).
[CrossRef]

Azaña, J.

Barthelemy, A.

L. Kh. Mouradian, F. Louradour, V. Messager, A. Barthelemy, and C. Froehly, “Spectro-temporal imaging of femtosecond events,” IEEE J. Quantum Electron. 36, 795-801 (2000).
[CrossRef]

Bennet, C. V.

Bloom, D. M.

A. A. Godil, B. A. Auld, and D. M. Bloom, “Time-lens producing 1.9 ps optical pulses,” Appl. Phys. Lett. 62, 1047-1049 (1993).
[CrossRef]

Carballar, A.

Chantada, L.

L. Chantada, C. R. Fernandez-Pousa, M. T. Flores-Arias, and C. Gomez-Reino, “Focusing of a train of pulses in phase opposition through a linear dispersive medium,” J. Opt. A, Pure Appl. Opt. 7, 767-773 (2005).
[CrossRef]

Fernandez-Pousa, C. R.

L. Chantada, C. R. Fernandez-Pousa, M. T. Flores-Arias, and C. Gomez-Reino, “Focusing of a train of pulses in phase opposition through a linear dispersive medium,” J. Opt. A, Pure Appl. Opt. 7, 767-773 (2005).
[CrossRef]

Flores-Arias, M. T.

L. Chantada, C. R. Fernandez-Pousa, M. T. Flores-Arias, and C. Gomez-Reino, “Focusing of a train of pulses in phase opposition through a linear dispersive medium,” J. Opt. A, Pure Appl. Opt. 7, 767-773 (2005).
[CrossRef]

Froehly, C.

L. Kh. Mouradian, F. Louradour, V. Messager, A. Barthelemy, and C. Froehly, “Spectro-temporal imaging of femtosecond events,” IEEE J. Quantum Electron. 36, 795-801 (2000).
[CrossRef]

Godil, A. A.

A. A. Godil, B. A. Auld, and D. M. Bloom, “Time-lens producing 1.9 ps optical pulses,” Appl. Phys. Lett. 62, 1047-1049 (1993).
[CrossRef]

Gomez-Reino, C.

L. Chantada, C. R. Fernandez-Pousa, M. T. Flores-Arias, and C. Gomez-Reino, “Focusing of a train of pulses in phase opposition through a linear dispersive medium,” J. Opt. A, Pure Appl. Opt. 7, 767-773 (2005).
[CrossRef]

J. Ojeda-Castañeda and C. Gomez-Reino, Selected Papers on Zone Plates (SPIE Milestone Series, Vol. MS128) (SPIE Optical Engineering Press, 1996), and references therein.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2005).

Jannson, T.

Jones, J.

J. Jones, “On the propagation of a pulse through a dispersive medium,” Am. J. Phys. 42, 43-46 (1974).
[CrossRef]

Kolner, B. H.

C. V. Bennet and B. H. Kolner, “Aberrations in temporal imaging,” IEEE J. Quantum Electron. 37, 20-32 (2001).
[CrossRef]

C. V. Bennet and B. H. Kolner, “Upconversion time microscope demostrating 103× magnification of femtosecond waveforms,” Opt. Lett. 24, 783-785 (1999).
[CrossRef]

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

B. H. Kolner and 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]

Le Chevalier, F.

F. Le Chevalier, Principles of Radar and Sonar Signal Processing (Artech House, 2001).

Leaird, D. E.

J. D. McKinney, D. Seo, D. E. Leaird, and A. M. Weiner, “Photonically assisted generation of arbitrary millimetre wave and microwave electromagnetic waveforms via direct space-to-time optical pulse shaping,” J. Lightwave Technol. 12, 3020-3028 (2003).
[CrossRef]

Louradour, F.

L. Kh. Mouradian, F. Louradour, V. Messager, A. Barthelemy, and C. Froehly, “Spectro-temporal imaging of femtosecond events,” IEEE J. Quantum Electron. 36, 795-801 (2000).
[CrossRef]

McKinney, J. D.

J. D. McKinney, D. Seo, D. E. Leaird, and A. M. Weiner, “Photonically assisted generation of arbitrary millimetre wave and microwave electromagnetic waveforms via direct space-to-time optical pulse shaping,” J. Lightwave Technol. 12, 3020-3028 (2003).
[CrossRef]

Messager, V.

L. Kh. Mouradian, F. Louradour, V. Messager, A. Barthelemy, and C. Froehly, “Spectro-temporal imaging of femtosecond events,” IEEE J. Quantum Electron. 36, 795-801 (2000).
[CrossRef]

Mouradian, L. Kh.

L. Kh. Mouradian, F. Louradour, V. Messager, A. Barthelemy, and C. Froehly, “Spectro-temporal imaging of femtosecond events,” IEEE J. Quantum Electron. 36, 795-801 (2000).
[CrossRef]

Muriel, M. A.

Nazarathy, M.

Ojeda-Castañeda, J.

J. Ojeda-Castañeda and C. Gomez-Reino, Selected Papers on Zone Plates (SPIE Milestone Series, Vol. MS128) (SPIE Optical Engineering Press, 1996), and references therein.

Papoulis, A.

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
[CrossRef]

Seo, D.

J. D. McKinney, D. Seo, D. E. Leaird, and A. M. Weiner, “Photonically assisted generation of arbitrary millimetre wave and microwave electromagnetic waveforms via direct space-to-time optical pulse shaping,” J. Lightwave Technol. 12, 3020-3028 (2003).
[CrossRef]

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
[CrossRef]

Watts, R. T.

R. T. Watts, “Novel diagnostic technologies for optical communications systems,” Ph.D. dissertation (The University of Auckland, 2008).

Weiner, A. M.

J. D. McKinney, D. Seo, D. E. Leaird, and A. M. Weiner, “Photonically assisted generation of arbitrary millimetre wave and microwave electromagnetic waveforms via direct space-to-time optical pulse shaping,” J. Lightwave Technol. 12, 3020-3028 (2003).
[CrossRef]

Am. J. Phys. (1)

J. Jones, “On the propagation of a pulse through a dispersive medium,” Am. J. Phys. 42, 43-46 (1974).
[CrossRef]

Appl. Phys. Lett. (2)

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

A. A. Godil, B. A. Auld, and D. M. Bloom, “Time-lens producing 1.9 ps optical pulses,” Appl. Phys. Lett. 62, 1047-1049 (1993).
[CrossRef]

IEEE J. Quantum Electron. (3)

L. Kh. Mouradian, F. Louradour, V. Messager, A. Barthelemy, and C. Froehly, “Spectro-temporal imaging of femtosecond events,” IEEE J. Quantum Electron. 36, 795-801 (2000).
[CrossRef]

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

C. V. Bennet and B. H. Kolner, “Aberrations in temporal imaging,” IEEE J. Quantum Electron. 37, 20-32 (2001).
[CrossRef]

J. Lightwave Technol. (1)

J. D. McKinney, D. Seo, D. E. Leaird, and A. M. Weiner, “Photonically assisted generation of arbitrary millimetre wave and microwave electromagnetic waveforms via direct space-to-time optical pulse shaping,” J. Lightwave Technol. 12, 3020-3028 (2003).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

L. Chantada, C. R. Fernandez-Pousa, M. T. Flores-Arias, and C. Gomez-Reino, “Focusing of a train of pulses in phase opposition through a linear dispersive medium,” J. Opt. A, Pure Appl. Opt. 7, 767-773 (2005).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Lett. (4)

Other (7)

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

F. Le Chevalier, Principles of Radar and Sonar Signal Processing (Artech House, 2001).

G. P. Agrawal, Applications of Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2005).

J. Ojeda-Castañeda and C. Gomez-Reino, Selected Papers on Zone Plates (SPIE Milestone Series, Vol. MS128) (SPIE Optical Engineering Press, 1996), and references therein.

R. T. Watts, “Novel diagnostic technologies for optical communications systems,” Ph.D. dissertation (The University of Auckland, 2008).

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Figures (3)

Fig. 1
Fig. 1

(a) Positive and (b) negative temporal generalized zone plate of phase.

Fig. 2
Fig. 2

Block diagram of temporal imaging system based on temporal zone plate (TZP) produced by a digital chirp signal generator.

Fig. 3
Fig. 3

Temporal Fresnel zone plate of phase.

Equations (34)

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O ( t ) z = i β 2 2 2 O ( t ) t 2 ,
β ( ω ) = β 0 + β 1 ( ω ω 0 ) + β 2 2 ( ω ω 0 ) 2 +
h ξ ( t ) = ( 2 π i ξ ) 1 2 exp ( i t 2 2 ξ ) ,
f ( t ) = α t
α = B T ,
ψ G + ( t ) = { 1 if t 2 2 j t 2 < t 2 2 j + ε 1 if t 2 2 j + ε t 2 < t 2 2 j + 2 } for ε < 2 α ,
ψ G ( t ) = { 1 if t 2 2 j + 1 t 2 < t 2 2 j + 1 + ε 1 if t 2 2 j + 1 + ε t 2 < t 2 2 j + 3 } for ε < 2 α .
ψ + ( t ) = α ε 1 + n = 1 2 n π [ sin ( n π α ε ) cos ( n π α t 2 ) + [ 1 cos ( n π α ε ) ] sin ( n π α t 2 ) ] ,
ψ ( t ) = α ε 1 + n = 1 2 ( 1 ) n n π [ sin ( n π α ε ) cos ( n π α t 2 ) + [ 1 cos ( n π α ε ) ] sin ( n π α t 2 ) ] .
ψ + ( t ) = α ε 1 + n = 1 [ a n exp ( i n π α t 2 ) + a * n exp ( i n π α t 2 ) ] ,
ψ ( t ) = α ε 1 + n = 1 [ b n exp ( i n π α t 2 ) + b * n exp ( i n π α t 2 ) ] ,
a n = i n π [ 1 exp { i n π α ε } ] , b n = ( 1 ) n a n .
t ( x , y ) = exp [ ± i k 2 f ( x 2 + y 2 ) ] ,
exp ( ± i n π α t 2 ) = exp ( ± i n ω c 2 T t 2 ) ,
ω c = 2 π B
f t n = T n .
E ( t ) = + O ( t ) h ζ , ζ ( t , t ) d t ,
h ζ ( t , t ) = ( i 2 π ζ ) 1 2 exp [ i ( t t ) 2 2 ζ ] ,
D ( t ) = h ζ ( t , t ) ψ + ( t ) .
h ζ , ζ ( t , t ) = + D ( t ) h ζ ( t , t ) d t ,
h ζ ( t , t ) = ( i 2 π ζ ) 1 2 exp [ i ( t t ) 2 2 ζ ] ,
h ζ ( t , t ) = exp ( i t 2 2 ζ ) exp ( i t 2 2 ζ ) i 2 π ( ζ ζ ) 1 2 { ( α ϵ 1 ) exp [ i 2 ( 1 ζ + 1 ζ ) t 2 ] exp [ + i ( t ζ + t ζ ) t ] d t + n = 1 a n exp [ i ( 1 ζ + 1 ζ + n ω c T ) t 2 ] exp [ i ( t ζ + t ζ ) t ] d t + n = 1 a * n exp [ i ( 1 ζ + 1 ζ n ω c T ) t 2 ] exp [ i ( t ζ + t ζ ) t ] d t } ,
1 ζ + 1 ζ = ± n ω c T
1 ζ + 1 ζ = ± ω c f t n ,
1 d + 1 d = 1 f n ,
f t 1 = B α .
f 1 = r 1 2 λ ,
+ exp { i ( t ζ + t ζ ) t } d t = δ ( t ζ + t ζ ) = ζ δ ( t + ζ ζ t ) ,
E ( t ) = 1 i 2 π ( ζ ζ ) 1 2 exp ( i t 2 2 ζ ) exp ( i t 2 2 ζ ) ( α ϵ 1 + n = 1 a n + n = 1 a * n ) O ( ζ ζ t ) .
ϵ = 1 α ,
ψ F + ( t ) = { 1 if t 2 2 j t 2 < t 2 2 j + 1 1 if t 2 2 j + 1 t 2 < t 2 2 j + 2 } ,
ψ F + ( t ) = q = 1 4 ( 2 q 1 ) π sin [ ( 2 q 1 ) π t 2 ϵ ] = q = 1 a 2 q 1 exp [ i ( 2 q 1 ) π t 2 ϵ ] + q = 1 a * 2 q 1 exp [ i ( 2 q 1 ) π t 2 ϵ ] ,
a 2 q 1 = 2 i ( 2 q 1 ) π ,
E F ( t ) = 1 i 2 π M t 1 2 exp ( i t 2 2 ζ ) exp ( i t 2 2 ζ ) ( q = 1 a 2 q 1 + q = 1 a * 2 q 1 ) O ( t M t ) .

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