Abstract

We study the properties of quasi-stationary, partially coherent, plane-wave optical pulses in the space–time and space–frequency domains. A generalized van Cittert–Zernike theorem in time is derived to describe the propagation of the coherence function of quasi-stationary pulses. The theory is applied to rectangular pulses chopped from a stationary light source, and the evolution characteristics of such pulse trains with different states of coherence are discussed and illustrated with numerical examples.

© 2006 Optical Society of America

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References

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  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  2. J. W. Goodman, Statistical Optics (Wiley, 1985).
  3. B. Cairns and E. Wolf, "The instantaneous cross-spectral density of non-stationary wavefields," Opt. Commun. 62, 215-218 (1986).
    [CrossRef]
  4. M. Bertolotti, A. Ferrari, and L. Sereda, "Coherence properties of nonstationary polychromatic light sources," J. Opt. Soc. Am. B 12, 341-347 (1995).
    [CrossRef]
  5. L. Sereda, M. Bertolotti, and A. Ferrari, "Coherence properties of nonstationary light wave fields," J. Opt. Soc. Am. A 15, 695-705 (1998).
    [CrossRef]
  6. P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, "Partially coherent Gaussian pulses," Opt. Commun. 204, 53-58 (2002).
    [CrossRef]
  7. S. A. Ponomarenko, G. P. Agrawal, and E. Wolf, "Energy spectrum of a nonstationary ensemble of pulses," Opt. Lett. 29, 394-396 (2004).
    [CrossRef] [PubMed]
  8. H. Lajunen, J. Tervo, and P. Vahimaa, "Overall coherence and coherent-mode expansion of spectrally partially coherent pulses," J. Opt. Soc. Am. A 21, 2117-2123 (2004).
    [CrossRef]
  9. H. Lajunen, P. Vahimaa, and J. Tervo, "Theory of spatially and spectrally partially coherent pulses," J. Opt. Soc. Am. A 22, 1536-1545 (2005).
    [CrossRef]
  10. R. A. Silverman, "Locally stationary random processes," Proc. IRE Trans. Inf. Theory 3, 182-187 (1957).
    [CrossRef]
  11. W. H. Carter and E. Wolf, "Coherence and radiometry with quasihomogeneous planar sources," J. Opt. Soc. Am. 67, 785-796 (1977).
    [CrossRef]
  12. E. Wolf and W. H. Carter, "A radiometric generalization of the van Cittert-Zernike theorem for fields generated by sources of arbitrary state of coherence," Opt. Commun. 16, 297-302 (1976).
    [CrossRef]
  13. J. Lancis, V. Torres-Company, E. Silvestre, and P. Andrés, "Space-time analogy for partially coherent plane-wave-type pulses," Opt. Lett. 30, 2973-2975 (2005).
    [CrossRef] [PubMed]
  14. C. Dorrer, "Temporal van Cittert-Zernike theorem and its application to the measurement of chromatic dispersion," J. Opt. Soc. Am. B 21, 1417-1422 (2004).
    [CrossRef]
  15. H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, "Spectrally partially coherent pulse trains in dispersive media," Opt. Commun. 255, 12-22 (2005).
    [CrossRef]
  16. Q. Lin, L. Wang, and S. Zhu, "Partially coherent light pulse and its propagation," Opt. Commun. 219, 65-70 (2003).
    [CrossRef]
  17. Use can be made of "Siegman's lemma" int^inf_inf exp(−ax2+bx)dx=π/aexp(b2/4a), valid for any complex numbers a and b with R{a}>0. See A. E. Siegman, Lasers (University Science Books, 1986), p. 783.
  18. The hat superscript implies that the function depends in time on t and tau, as opposed to t1 and t2, or in frequency on w and varpi, as opposed to w1 and w2. However, for functions of only a single variable t or tau, or w or varpi, the symbol is not used.
  19. Since Γ(t1,t2) has to remain finite, the Dirac delta function δ(t2−t1) is to be interpreted as a limiting case of a narrow function with a very high but normalizable value.
  20. H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, "Spectral coherence properties of temporally modulated stationary light sources," Opt. Express 11, 1894-1899 (2003).
    [CrossRef] [PubMed]
  21. Q. Lin and S. Wang, "Propagation characteristics of chopped light pulses through dispersive media," J. Opt. 26, 247-249 (1995).
    [CrossRef]

2005

2004

2003

2002

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, "Partially coherent Gaussian pulses," Opt. Commun. 204, 53-58 (2002).
[CrossRef]

1998

1995

M. Bertolotti, A. Ferrari, and L. Sereda, "Coherence properties of nonstationary polychromatic light sources," J. Opt. Soc. Am. B 12, 341-347 (1995).
[CrossRef]

Q. Lin and S. Wang, "Propagation characteristics of chopped light pulses through dispersive media," J. Opt. 26, 247-249 (1995).
[CrossRef]

1986

B. Cairns and E. Wolf, "The instantaneous cross-spectral density of non-stationary wavefields," Opt. Commun. 62, 215-218 (1986).
[CrossRef]

1977

1976

E. Wolf and W. H. Carter, "A radiometric generalization of the van Cittert-Zernike theorem for fields generated by sources of arbitrary state of coherence," Opt. Commun. 16, 297-302 (1976).
[CrossRef]

1957

R. A. Silverman, "Locally stationary random processes," Proc. IRE Trans. Inf. Theory 3, 182-187 (1957).
[CrossRef]

Agrawal, G. P.

Andrés, P.

Bertolotti, M.

Cairns, B.

B. Cairns and E. Wolf, "The instantaneous cross-spectral density of non-stationary wavefields," Opt. Commun. 62, 215-218 (1986).
[CrossRef]

Carter, W. H.

W. H. Carter and E. Wolf, "Coherence and radiometry with quasihomogeneous planar sources," J. Opt. Soc. Am. 67, 785-796 (1977).
[CrossRef]

E. Wolf and W. H. Carter, "A radiometric generalization of the van Cittert-Zernike theorem for fields generated by sources of arbitrary state of coherence," Opt. Commun. 16, 297-302 (1976).
[CrossRef]

Dorrer, C.

Ferrari, A.

Friberg, A. T.

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, "Partially coherent Gaussian pulses," Opt. Commun. 204, 53-58 (2002).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 1985).

Lajunen, H.

Lancis, J.

Lin, Q.

Q. Lin, L. Wang, and S. Zhu, "Partially coherent light pulse and its propagation," Opt. Commun. 219, 65-70 (2003).
[CrossRef]

Q. Lin and S. Wang, "Propagation characteristics of chopped light pulses through dispersive media," J. Opt. 26, 247-249 (1995).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Pääkkönen, P.

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, "Partially coherent Gaussian pulses," Opt. Commun. 204, 53-58 (2002).
[CrossRef]

Ponomarenko, S. A.

Sereda, L.

Siegman, A. E.

Use can be made of "Siegman's lemma" int^inf_inf exp(−ax2+bx)dx=π/aexp(b2/4a), valid for any complex numbers a and b with R{a}>0. See A. E. Siegman, Lasers (University Science Books, 1986), p. 783.

Silverman, R. A.

R. A. Silverman, "Locally stationary random processes," Proc. IRE Trans. Inf. Theory 3, 182-187 (1957).
[CrossRef]

Silvestre, E.

Tervo, J.

Torres-Company, V.

Turunen, J.

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, "Spectrally partially coherent pulse trains in dispersive media," Opt. Commun. 255, 12-22 (2005).
[CrossRef]

H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, "Spectral coherence properties of temporally modulated stationary light sources," Opt. Express 11, 1894-1899 (2003).
[CrossRef] [PubMed]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, "Partially coherent Gaussian pulses," Opt. Commun. 204, 53-58 (2002).
[CrossRef]

Vahimaa, P.

Wang, L.

Q. Lin, L. Wang, and S. Zhu, "Partially coherent light pulse and its propagation," Opt. Commun. 219, 65-70 (2003).
[CrossRef]

Wang, S.

Q. Lin and S. Wang, "Propagation characteristics of chopped light pulses through dispersive media," J. Opt. 26, 247-249 (1995).
[CrossRef]

Wolf, E.

S. A. Ponomarenko, G. P. Agrawal, and E. Wolf, "Energy spectrum of a nonstationary ensemble of pulses," Opt. Lett. 29, 394-396 (2004).
[CrossRef] [PubMed]

B. Cairns and E. Wolf, "The instantaneous cross-spectral density of non-stationary wavefields," Opt. Commun. 62, 215-218 (1986).
[CrossRef]

W. H. Carter and E. Wolf, "Coherence and radiometry with quasihomogeneous planar sources," J. Opt. Soc. Am. 67, 785-796 (1977).
[CrossRef]

E. Wolf and W. H. Carter, "A radiometric generalization of the van Cittert-Zernike theorem for fields generated by sources of arbitrary state of coherence," Opt. Commun. 16, 297-302 (1976).
[CrossRef]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Wyrowski, F.

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, "Spectrally partially coherent pulse trains in dispersive media," Opt. Commun. 255, 12-22 (2005).
[CrossRef]

H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, "Spectral coherence properties of temporally modulated stationary light sources," Opt. Express 11, 1894-1899 (2003).
[CrossRef] [PubMed]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, "Partially coherent Gaussian pulses," Opt. Commun. 204, 53-58 (2002).
[CrossRef]

Zhu, S.

Q. Lin, L. Wang, and S. Zhu, "Partially coherent light pulse and its propagation," Opt. Commun. 219, 65-70 (2003).
[CrossRef]

J. Opt.

Q. Lin and S. Wang, "Propagation characteristics of chopped light pulses through dispersive media," J. Opt. 26, 247-249 (1995).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Commun.

B. Cairns and E. Wolf, "The instantaneous cross-spectral density of non-stationary wavefields," Opt. Commun. 62, 215-218 (1986).
[CrossRef]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, "Partially coherent Gaussian pulses," Opt. Commun. 204, 53-58 (2002).
[CrossRef]

E. Wolf and W. H. Carter, "A radiometric generalization of the van Cittert-Zernike theorem for fields generated by sources of arbitrary state of coherence," Opt. Commun. 16, 297-302 (1976).
[CrossRef]

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, "Spectrally partially coherent pulse trains in dispersive media," Opt. Commun. 255, 12-22 (2005).
[CrossRef]

Q. Lin, L. Wang, and S. Zhu, "Partially coherent light pulse and its propagation," Opt. Commun. 219, 65-70 (2003).
[CrossRef]

Opt. Express

Opt. Lett.

Proc. IRE Trans. Inf. Theory

R. A. Silverman, "Locally stationary random processes," Proc. IRE Trans. Inf. Theory 3, 182-187 (1957).
[CrossRef]

Other

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

J. W. Goodman, Statistical Optics (Wiley, 1985).

Use can be made of "Siegman's lemma" int^inf_inf exp(−ax2+bx)dx=π/aexp(b2/4a), valid for any complex numbers a and b with R{a}>0. See A. E. Siegman, Lasers (University Science Books, 1986), p. 783.

The hat superscript implies that the function depends in time on t and tau, as opposed to t1 and t2, or in frequency on w and varpi, as opposed to w1 and w2. However, for functions of only a single variable t or tau, or w or varpi, the symbol is not used.

Since Γ(t1,t2) has to remain finite, the Dirac delta function δ(t2−t1) is to be interpreted as a limiting case of a narrow function with a very high but normalizable value.

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Figures (2)

Fig. 1
Fig. 1

Normalized average intensity distributions of pulses chopped from different sources after propagation of distance z = 20 km in a dispersive medium with a = 25 km ps 2 . The initial duration of all the pulses is T = 10 ps , and the temporal coherence times T c = 2 Ω 0 are assumed to be 1 ps (dotted curve), 4 ps (dashed–dotted curve), and 40 ps (dashed curve). The solid curve corresponds to the quasi-stationary approximation in the case of T c = 1 ps .

Fig. 2
Fig. 2

Normalized average intensity distributions of pulses chopped from the same stationary source with different initial durations after propagation of distance z = 20 km in a dispersive medium with a = 25 km ps 2 . The temporal coherence time of all pulses is T c = 10 ps , and the initial durations T are assumed to be 40 ps (dotted curve), 10 ps (dashed–dotted curve), and 4 ps (dashed curve). The solid curve corresponds to the quasi-stationary approximation in the case of T = 40 ps .

Equations (55)

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Γ ( z 1 , z 2 , t 1 , t 2 ) = U * ( z 1 , t 1 ) U ( z 2 , t 2 ) ,
γ ( z 1 , z 2 , t 1 , t 2 ) = Γ ( z 1 , z 2 , t 1 , t 2 ) I ( z 1 , t 1 ) I ( z 2 , t 2 ) .
U ̃ ( z , ω ) = 1 2 π U ( z , t ) exp ( i ω t ) d t ,
W ( z 1 , z 2 , ω 1 , ω 2 ) = U ̃ * ( z 1 , ω 1 ) U ̃ ( z 2 , ω 2 ) ,
μ ( z 1 , z 2 , ω 1 , ω 2 ) = W ( z 1 , z 2 , ω 1 , ω 2 ) S ( z 1 , ω 1 ) S ( z 2 , ω 2 ) ,
W ( z 1 , z 2 , ω 1 , ω 2 ) = 1 ( 2 π ) 2 Γ ( z 1 , z 2 , t 1 , t 2 ) exp [ i ( ω 1 t 1 ω 2 t 2 ) ] d t 1 d t 2 ,
Γ ( z 1 , z 2 , t 1 , t 2 ) = W ( z 1 , z 2 , ω 1 , ω 2 ) exp [ i ( ω 1 t 1 ω 2 t 2 ) ] d ω 1 d ω 2 .
2 z 2 U ̃ ( z , ω ) + β 2 ( ω ) U ̃ ( z , ω ) = 0 ,
U ̃ ( z , ω ) = U ̃ ( 0 , ω ) A ( z , ω ) ,
W ( z 1 , z 2 , ω 1 , ω 2 ) = W ( 0 , 0 , ω 1 , ω 2 ) A * ( z 1 , ω 1 ) A ( z 2 , ω 2 ) ,
U ( z , t ) = 1 2 π U ( 0 , t ) A ( z , t t ) d t ,
A ( z , t ) = A ( z , ω ) exp ( i ω t ) d ω .
Γ ( z 1 , z 2 , t 1 , t 2 ) = 1 ( 2 π ) 2 Γ ( 0 , 0 , t 1 , t 2 ) A * ( z 1 , t 1 t 1 ) A ( z 2 , t 2 t 2 ) d t 1 d t 2 ,
A ( z , ω ) = exp { i [ β ( ω 0 ) + ( ω ω 0 ) v g + a ( ω ω 0 ) 2 ] z } ,
A ( z , t ) = i π a z exp { i [ β ( ω 0 ) z ω 0 t ] } exp [ i ( t z v g ) 2 4 a z ] ,
Γ ( z 1 , z 2 , t 1 , t 2 ) = exp { i [ β ( ω 0 ) ( z 1 z 2 ) ω 0 ( t 1 t 2 ) ] } 4 π a ( z 1 z 2 ) 1 2 Γ ( 0 , 0 , t 1 , t 2 ) exp [ i ω 0 ( t 1 t 2 ) ] × exp { i [ t 1 ( t 1 z 1 v g ) ] 2 4 a z 1 i [ t 2 ( t 2 z 2 v g ) ] 2 4 a z 2 } d t 1 d t 2 .
Γ ( z 1 , z 2 , t 1 , t 2 ) = exp { i [ β ( ω 0 ) ω 0 v g ] ( z 1 z 2 ) } Γ ( 0 , 0 , t 1 z 1 v g , t 2 z 2 v g ) .
τ = t 2 t 1 ,
Γ ( t 1 , t 2 ) = I ( t 1 ) I ( t 2 ) γ ( t 1 , t 2 ) I ( t ) γ ( τ ) = Γ ̂ ( t , τ ) ,
t = ( t 1 + t 2 ) 2
ω = ( ω 1 + ω 2 ) 2 ,
ϖ = ω 2 ω 1 ,
W ( ω 1 , ω 2 ) = W ̂ ( ω , ϖ ) = 1 ( 2 π ) 2 Γ ̂ ( t , τ ) exp [ i ( ϖ t + ω τ ) ] d t d τ ,
W ̂ ( ω , ϖ ) = W 1 ( ω ) W 2 ( ϖ ) ,
W 1 ( ω ) = 1 2 π γ ( τ ) exp ( i ω τ ) d τ ,
W 2 ( ϖ ) = 1 2 π I ( t ) exp ( i ϖ t ) d τ .
S ( ω ) = W ̂ ( ω , 0 ) = W 1 ( ω ) W 2 ( 0 ) ,
W 2 ( 0 ) = 1 2 π I ( t ) d t ,
μ ( ω 1 , ω 2 ) = W 1 ( ω ) W 1 ( ω 1 ) W 1 ( ω 2 ) W 2 ( ϖ ) W 2 ( 0 ) = S ( ω ) S ( ω 1 ) S ( ω 2 ) W 2 ( ϖ ) W 2 ( 0 ) ,
μ ( ω 1 , ω 2 ) μ ( ϖ ) = W 2 ( ϖ ) W 2 ( 0 ) ,
W ̂ ( ω , ϖ ) S ( ω ) μ ( ϖ ) ,
Γ ̂ ( z , z , t , τ ) = 1 4 π a z Γ ̂ ( 0 , 0 , t , τ ) exp [ i ( ω 0 1 2 a v g ) ( τ τ ) ] exp [ i 2 a z ( t τ + t τ t τ t τ ) ] d t d τ ,
Γ ( 0 , 0 , t 1 , t 2 ) = Γ ̂ ( 0 , 0 , t , τ ) = I ( 0 , t ) γ ( 0 , 0 , τ ) ,
Γ ̂ ( z , z , t , τ ) = 1 4 π a z exp [ i ( ω 0 + t z v g 2 a z ) τ ] I ( 0 , t ) exp ( i τ t 2 a z ) d t × γ ( 0 , 0 , τ ) exp [ i ( ω 0 + t z v g 2 a z ) τ ] d τ
I ( z , t ) = I 0 4 π a z γ ( 0 , 0 , τ ) exp [ i ( ω 0 + t z v g 2 a z ) τ ] d τ ,
I 0 = I ( 0 , t ) d t
Γ ̂ ( z , z , t , τ ) = I ( z , t ) I 0 exp [ i ( ω 0 + t z v g 2 a z ) τ ] I ( 0 , t ) exp ( i τ t 2 a z ) d t .
γ ̂ ( z , z , t , τ ) = exp [ i ( ω 0 + t z v g 2 a z ) τ ] I ( 0 , t ) I 0 exp ( i τ t 2 a z ) d t ,
γ ( z , z , t 1 , t 2 ) I 0 ( 0 , t ) exp ( i t 2 t 1 2 a z t ) d t ,
Γ ( 0 , 0 , t 1 , t 2 ) = M * ( t 1 ) U 0 * ( t 1 ) M ( t 2 ) U 0 ( t 2 ) = M * ( t 1 ) M ( t 2 ) Γ s ( τ ) ,
Γ s ( τ ) = S s ( ω ) exp ( i ω τ ) d ω ,
S s ( ω ) = S 0 exp [ ( ω ω 0 ) 2 Ω 0 2 ] ,
Γ s ( τ ) = Γ 0 exp ( τ 2 T c 2 ) exp ( i ω 0 τ ) ,
M ( t ) = { 1 when 0 < t < T , 0 otherwise .
Γ ( 0 , 0 , t 1 , t 2 ) = Γ 0 exp [ ( t 2 t 1 ) 2 T c 2 ] exp [ i ω 0 ( t 2 t 1 ) ] ,
I ( t ) = { Γ 0 when 0 < t < T , 0 otherwise ,
γ ( τ ) = exp ( τ 2 T c 2 ) exp ( i ω 0 τ ) ,
W 1 ( ω ) = π T c 2 π exp [ ( ω ω 0 ) 2 4 T c 2 ] ,
W 2 ( ϖ ) = Γ 0 T 2 π sinc ( ϖ T 2 π ) exp ( i ϖ T 2 ) .
S ( ω ) = S 0 T 2 π exp [ ( ω ω 0 ) 2 Ω 0 2 ] ,
μ ( ϖ ) = sinc ( ϖ T 2 π ) exp ( i ϖ T 2 ) ,
S ( ω ) = S 0 ( T 4 π ) 2 sinc 2 [ T ( ω ω 0 ) 2 π ] ,
Γ ( z , z , t , τ ) = I ( z , t ) sinc ( T τ 4 π a z ) exp [ i τ ( ω 0 + t z v g 2 a z T 4 a z ) ] ,
I ( z , t ) = Γ 0 T T c π 4 π a z exp [ ( t z v g 4 a z T c ) 2 ] ,
I ( z , t ) = Γ 0 T 2 4 π a z sinc 2 [ T ( t z v g ) 4 π a z ]

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