Abstract

One can exploit the duality between the parabolic equations of diffraction and dispersion to create the time-domain equivalent of a spatial imaging system. A key element in this temporal imaging system is the time lens (quadratic phase modulator) with attributes analogous to the focal length and the f-number of conventional space lenses. These temporal analogs are analyzed and compared with their spatial counterparts in a generalized framework, from which we conclude that chirp rate and fractional bandwidth are the defining quantities in both systems.

© 1994 Optical Society of America

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References

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  1. S. A. Akhmanov, V. A. Vysloukh, A. S. Chirkin, “Self-action of wave packets in a nonlinear medium and femtosecond laser pulse generation,” Sov. Phys. Usp. 29, 642–677 (1987).
    [CrossRef]
  2. B. H. Kolner, M. Nazarathy, “Temporal imaging with a time lens,” Opt. Lett. 14, 630–632 (1989).
    [CrossRef] [PubMed]
  3. B. H. Kolner, “Space–time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951–1953 (1994).
    [CrossRef]
  4. B. H. Kolner, “Active pulse compression,” in Ultrafast Phenomena IV, T. Yajima, K. Yoshihara, C. B. Harris, S. Shionoya, eds. (Springer-Verlag, Berlin, 1988), pp. 47–49.
  5. E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. QE-5, 454–458 (1969).
    [CrossRef]
  6. D. Grischkowsky, A. C. Balant, “Optical pulse compression based on enhanced frequency chirping,” Appl. Phys. Lett. 41, 1–3 (1982).
    [CrossRef]
  7. J. E. Bjorkholm, E. H. Turner, D. B. Pearson, “Conversion of cw light into a train of subnanosecond pulses using frequency modulation and the dispersion of a near-resonant atomic vapor,” Appl. Phys. Lett 26, 564–566 (1975).
    [CrossRef]
  8. A. A. Godil, B. A. Auld, D. M. Bloom, “Time-lens producing 1.9 ps optical pulses,” Appl. Phys. Lett. 62, 1047–1049 (1993).
    [CrossRef]
  9. B. H. Kolner, “Active pulse compression using an integrated electro-optic phase modulator,” Appl. Phys. Lett. 52, 1122–1124 (1988).
    [CrossRef]
  10. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  11. C. V. Bennett, R. P. Scott, B. H. Kolner, “Temporal magnification and reversal of 100 Gb/s optical data with an up-conversion time microscope,” Appl. Phys. Lett. (to be published).

1994 (1)

B. H. Kolner, “Space–time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951–1953 (1994).
[CrossRef]

1993 (1)

A. A. Godil, B. A. Auld, 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]

1987 (1)

S. A. Akhmanov, V. A. Vysloukh, A. S. Chirkin, “Self-action of wave packets in a nonlinear medium and femtosecond laser pulse generation,” Sov. Phys. Usp. 29, 642–677 (1987).
[CrossRef]

1982 (1)

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

1975 (1)

J. E. Bjorkholm, E. H. Turner, D. B. Pearson, “Conversion of cw light into a train of subnanosecond pulses using frequency modulation and the dispersion of a near-resonant atomic vapor,” Appl. Phys. Lett 26, 564–566 (1975).
[CrossRef]

1969 (1)

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

Akhmanov, S. A.

S. A. Akhmanov, V. A. Vysloukh, A. S. Chirkin, “Self-action of wave packets in a nonlinear medium and femtosecond laser pulse generation,” Sov. Phys. Usp. 29, 642–677 (1987).
[CrossRef]

Auld, B. A.

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

Bennett, C. V.

C. V. Bennett, R. P. Scott, B. H. Kolner, “Temporal magnification and reversal of 100 Gb/s optical data with an up-conversion time microscope,” Appl. Phys. Lett. (to be published).

Bjorkholm, J. E.

J. E. Bjorkholm, E. H. Turner, D. B. Pearson, “Conversion of cw light into a train of subnanosecond pulses using frequency modulation and the dispersion of a near-resonant atomic vapor,” Appl. Phys. Lett 26, 564–566 (1975).
[CrossRef]

Bloom, D. M.

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

Chirkin, A. S.

S. A. Akhmanov, V. A. Vysloukh, A. S. Chirkin, “Self-action of wave packets in a nonlinear medium and femtosecond laser pulse generation,” Sov. Phys. Usp. 29, 642–677 (1987).
[CrossRef]

Godil, A. A.

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

Goodman, J. W.

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

Grischkowsky, D.

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

Kolner, B. H.

B. H. Kolner, “Space–time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951–1953 (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]

B. H. Kolner, “Active pulse compression,” in Ultrafast Phenomena IV, T. Yajima, K. Yoshihara, C. B. Harris, S. Shionoya, eds. (Springer-Verlag, Berlin, 1988), pp. 47–49.

C. V. Bennett, R. P. Scott, B. H. Kolner, “Temporal magnification and reversal of 100 Gb/s optical data with an up-conversion time microscope,” Appl. Phys. Lett. (to be published).

Nazarathy, M.

Pearson, D. B.

J. E. Bjorkholm, E. H. Turner, D. B. Pearson, “Conversion of cw light into a train of subnanosecond pulses using frequency modulation and the dispersion of a near-resonant atomic vapor,” Appl. Phys. Lett 26, 564–566 (1975).
[CrossRef]

Scott, R. P.

C. V. Bennett, R. P. Scott, B. H. Kolner, “Temporal magnification and reversal of 100 Gb/s optical data with an up-conversion time microscope,” Appl. Phys. Lett. (to be published).

Treacy, E. B.

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

Turner, E. H.

J. E. Bjorkholm, E. H. Turner, D. B. Pearson, “Conversion of cw light into a train of subnanosecond pulses using frequency modulation and the dispersion of a near-resonant atomic vapor,” Appl. Phys. Lett 26, 564–566 (1975).
[CrossRef]

Vysloukh, V. A.

S. A. Akhmanov, V. A. Vysloukh, A. S. Chirkin, “Self-action of wave packets in a nonlinear medium and femtosecond laser pulse generation,” Sov. Phys. Usp. 29, 642–677 (1987).
[CrossRef]

Appl. Phys. Lett (1)

J. E. Bjorkholm, E. H. Turner, D. B. Pearson, “Conversion of cw light into a train of subnanosecond pulses using frequency modulation and the dispersion of a near-resonant atomic vapor,” Appl. Phys. Lett 26, 564–566 (1975).
[CrossRef]

Appl. Phys. Lett. (3)

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

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

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

IEEE J. Quantum Electron. (2)

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

B. H. Kolner, “Space–time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951–1953 (1994).
[CrossRef]

Opt. Lett. (1)

Sov. Phys. Usp. (1)

S. A. Akhmanov, V. A. Vysloukh, A. S. Chirkin, “Self-action of wave packets in a nonlinear medium and femtosecond laser pulse generation,” Sov. Phys. Usp. 29, 642–677 (1987).
[CrossRef]

Other (3)

B. H. Kolner, “Active pulse compression,” in Ultrafast Phenomena IV, T. Yajima, K. Yoshihara, C. B. Harris, S. Shionoya, eds. (Springer-Verlag, Berlin, 1988), pp. 47–49.

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

C. V. Bennett, R. P. Scott, B. H. Kolner, “Temporal magnification and reversal of 100 Gb/s optical data with an up-conversion time microscope,” Appl. Phys. Lett. (to be published).

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Figures (5)

Fig. 1
Fig. 1

Prototypical time lens realized with a traveling-wave electro-optic (EO) modulator. An optical pulse E(0, t) entering the modulator at x = 0 and confined to a cusp of the driving field obtains a quadratic phase modulation after passage through the modulator.

Fig. 2
Fig. 2

Complex Gaussian electric field in one dimension (a) before and (b) after passage through a thin lens that produces the quadratic phase transformation.

Fig. 3
Fig. 3

Instantaneous transverse wave number ki produced by a thin lens in the paraxial approximation.

Fig. 4
Fig. 4

Instantaneous temporal frequency ωi produced by a time lens in the quadratic approximation.

Fig. 5
Fig. 5

Temporal imaging system as the time-domain analog of conventional spatial imaging. (a) Temporal imaging configuration. Input and output dispersions (shown here as diffraction grating pairs) play the role of free-space diffraction, while a quadratic phase modulator acts as a lens in the time domain. (b) Mathematical effects on waveforms in temporal imaging.

Equations (35)

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t l ( x , y ) = exp ( - i k 0 n Δ 0 ) exp [ i k 0 2 f ( x 2 + y 2 ) ]
f = 1 ( n - 1 ) ( 1 R 1 - 1 R 2 ) .
t l ( x , y ) = exp ( - i k 0 n Δ 0 ) exp [ i ϕ x ( x ) ] exp [ i ϕ y ( y ) ] ,
ϕ x ( x ) = k 0 x 2 2 f
ϕ ( x ) = ϕ 0 ( 0 ) + x d ϕ d x | x = 0 + x 2 2 ! d 2 ϕ d x 2 | x = 0 + ,
f = k 0 d 2 ϕ x / d x 2 .
k i d ϕ x d x = k 0 x f
d k i d x = d 2 ϕ x d x 2 = k 0 f ,
f = k 0 d k i / d x .
ϕ ( z , t ) = ω 0 t - k 0 z - ω 0 c 0 z Δ n ( z , t ) d z = ω 0 t - k 0 z - Γ ( z , t ) ,
Γ ( z , t ) Γ 0 [ 1 - ( ω m t ) 2 2 ] ,
H l ( t ) = exp ( - i Γ 0 ) exp [ i Γ 0 ( ω m t ) 2 2 ] .
H l ( t ) = exp ( - i Γ 0 ) exp ( i ω 0 t 2 2 f T ) ,
f T = ω 0 Γ 0 ω m 2 .
ϕ ( t ) = ϕ 0 ( t 0 ) + ( t - t 0 ) d ϕ d t | t = t 0 + ( t - t 0 ) 2 2 ! d 2 ϕ d t 2 | t = t 0 + .
f T = ω 0 d 2 ϕ / d t 2 .
ω i ω 0 + d ϕ d t ,
f T = ω 0 d ω i / d t .
1 ξ 1 d 2 β 1 d ω 2 + 1 ξ 2 d 2 β 2 d ω 2 = - ω 0 f T ,
ξ 2 d 2 β 2 d ω 2 = - f T ω 0 .
d β d ω = 1 v g ,
d 2 β d ω 2 = d d ω ( 1 v g )
ξ d 2 β d ω 2 = d d ω ( ξ v g ) = d τ g d ω
d τ g d ω = - f T ω 0 ,
f T = - ( ω 0 ω ) τ g .
f # f Δ x = k 0 Δ x ( d k i / d x ) .
Δ k Δ x d k i d x .
f # = k 0 Δ k .
f T # f T Δ t = ω 0 Δ t ( d ω i / d t ) .
Δ ω = Δ t d ω i d t ,
f T # = ω 0 Δ ω .
δ τ in = T 0 f T # = T 0 ω 0 Δ ω = 1 Δ f ,
f T # = f T Δ t = ω 0 Γ 0 ω m ,
δ τ in = T 0 ω 0 Γ 0 ω m = 1 Γ 0 f m .
SPACE TIME f = k 0 d k i / d x , f T = ω 0 d ω i / d t , f # = k 0 Δ k , f T # = ω 0 Δ ω .

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