Abstract

On the basis of the recently introduced definition of energy spectrum of a partially temporally coherent pulse train [Opt. Lett. 29, 394 (2004) ], we show that the well-known time–frequency uncertainty relationship applicable to a fully temporally coherent pulse train is still correct for a partially temporally coherent pulse train. The condition under which the uncertainty relationship reaches equality is derived. An example is presented to illustrate the validity of the uncertainty relationship.

© 2006 Optical Society of America

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References

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  1. C. Rulliere, Femtosecond Laser Pulses: Principles and Experiments (Springer, 1998), p. 30.
  2. D. J. Kane and R. Trebino, "Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating," IEEE J. Quantum Electron. 29, 571-579 (1993).
    [CrossRef]
  3. C. Iaconis and I. A. Walmsley, "Self-referencing spectral interferometry for measuring ultrashort optical pulses," IEEE J. Quantum Electron. 35, 501-509 (1999).
    [CrossRef]
  4. 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]
  5. 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]
  6. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  7. X. Ka, Advanced Quantum Mechanics (Advanced Education Press, 1999).
  8. T. S. Blyth and E. F. Robertson, Further Linear Algebra (Springer, 2002), Chap. 1.
    [CrossRef]

2005 (1)

2004 (1)

1999 (1)

C. Iaconis and I. A. Walmsley, "Self-referencing spectral interferometry for measuring ultrashort optical pulses," IEEE J. Quantum Electron. 35, 501-509 (1999).
[CrossRef]

1993 (1)

D. J. Kane and R. Trebino, "Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating," IEEE J. Quantum Electron. 29, 571-579 (1993).
[CrossRef]

Agrawal, G. P.

Blyth, T. S.

T. S. Blyth and E. F. Robertson, Further Linear Algebra (Springer, 2002), Chap. 1.
[CrossRef]

Iaconis, C.

C. Iaconis and I. A. Walmsley, "Self-referencing spectral interferometry for measuring ultrashort optical pulses," IEEE J. Quantum Electron. 35, 501-509 (1999).
[CrossRef]

Ka, X.

X. Ka, Advanced Quantum Mechanics (Advanced Education Press, 1999).

Kane, D. J.

D. J. Kane and R. Trebino, "Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating," IEEE J. Quantum Electron. 29, 571-579 (1993).
[CrossRef]

Lajunen, H.

Mandel, L.

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

Ponomarenko, S. A.

Robertson, E. F.

T. S. Blyth and E. F. Robertson, Further Linear Algebra (Springer, 2002), Chap. 1.
[CrossRef]

Rulliere, C.

C. Rulliere, Femtosecond Laser Pulses: Principles and Experiments (Springer, 1998), p. 30.

Tervo, J.

Trebino, R.

D. J. Kane and R. Trebino, "Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating," IEEE J. Quantum Electron. 29, 571-579 (1993).
[CrossRef]

Vahimaa, P.

Walmsley, I. A.

C. Iaconis and I. A. Walmsley, "Self-referencing spectral interferometry for measuring ultrashort optical pulses," IEEE J. Quantum Electron. 35, 501-509 (1999).
[CrossRef]

Wolf, E.

IEEE J. Quantum Electron. (2)

D. J. Kane and R. Trebino, "Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating," IEEE J. Quantum Electron. 29, 571-579 (1993).
[CrossRef]

C. Iaconis and I. A. Walmsley, "Self-referencing spectral interferometry for measuring ultrashort optical pulses," IEEE J. Quantum Electron. 35, 501-509 (1999).
[CrossRef]

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

Opt. Lett. (1)

Other (4)

C. Rulliere, Femtosecond Laser Pulses: Principles and Experiments (Springer, 1998), p. 30.

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

X. Ka, Advanced Quantum Mechanics (Advanced Education Press, 1999).

T. S. Blyth and E. F. Robertson, Further Linear Algebra (Springer, 2002), Chap. 1.
[CrossRef]

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Equations (36)

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( Δ t ) 2 ( Δ ν ) 2 ( 1 4 π ) 2 ,
I ( t ) = e ( t ) 2 , S ( ν ) = E ( ν ) 2 ,
( Δ t ) 2 = I ( t ) ( t t ¯ ) 2 d t I ( t ) d t , ( Δ ν ) 2 = S ( ν ) ( ν ν ¯ ) 2 d ν S ( ν ) d ν ,
t ¯ = I ( t ) t d t I ( t ) d t
ν ¯ = S ( ν ) ν d ν S ( ν ) d ν
W ( ν 1 , ν 2 ) = Γ ( t 1 , t 2 ) exp [ i 2 π ( ν 1 t 1 ν 2 t 2 ) ] d t 1 d t 2 .
Γ ( t 1 , t 2 ) = n λ n u n ( t 1 ) u n * ( t 2 ) ,
W ( ν 1 , ν 2 ) = n λ n U n ( ν 1 ) U n * ( ν 2 ) ,
( Δ t ) 2 = n λ n ( Δ t n ) 2 n λ n , ( Δ ν ) 2 = n λ n ( Δ ν n ) 2 n λ n ,
( Δ t n ) 2 = u n ( t ) 2 ( t t ¯ ) 2 d t
( Δ ν n ) 2 = U n ( ν ) 2 ( ν ν ¯ ) 2 d ν .
( Δ t n ) 2 ( Δ ν n ) 2 ( 1 4 π ) 2 ,
u n ( t ) = u n ( 0 ) exp [ t ¯ 2 ( t t ¯ ) 2 4 ( Δ t n ) 2 ] exp ( i 2 π ν ¯ t ) , u n ( 0 ) = exp [ t ¯ 2 4 ( Δ t n ) 2 ] exp ( i φ n ) [ 2 π ( Δ t n ) 2 ] 1 4 ,
( Δ t ) 2 ( Δ ν ) 2 = n λ n ( Δ t n ) 2 n λ n n λ n ( Δ ν n ) 2 n λ n [ n λ n ( Δ t n ) 2 λ n ( Δ ν n ) 2 ] 2 ( n λ n ) 2 ( 1 4 π ) 2 ,
λ n ( Δ t n ) 2 λ n ( Δ ν n ) 2 = C ,
u n ( t ) = u n ( 0 ) ψ ( t ) , u n ( 0 ) = exp [ t ¯ 2 4 ( Δ t ) 2 ] exp ( i φ n ) [ 2 π ( Δ t ) 2 ] 1 4 ,
ψ ( t ) = exp [ t ¯ 2 ( t t ¯ ) 2 4 ( Δ t ) 2 ] exp ( i 2 π ν ¯ t ) .
Γ ( t 1 , t 2 ) = A ψ ( t 1 ) ψ * ( t 2 ) , A = exp [ t ¯ 2 2 ( Δ t ) 2 ] n λ n [ 2 π ( Δ t ) 2 ] 1 2 ,
γ ( t 1 , t 2 ) = Γ ( t 1 , t 2 ) [ Γ ( t 1 , t 1 ) Γ ( t 2 , t 2 ) ] 1 2 = ψ ( t 1 ) ψ * ( t 2 ) ψ ( t 1 ) ψ ( t 2 ) .
e ( t ) = c ( t , 0 ) e ( 0 ) ,
Γ ( t 1 , t 2 ) = e ( 0 ) 2 c ( t 1 , 0 ) c * ( t 2 , 0 ) .
c ( t , 0 ) = c 0 ψ ( t ) ,
e ( t ) = c 0 ψ ( t ) e ( 0 ) = ψ ( t ) e ( 0 ) ,
Γ ( t 1 , t 2 ) = Γ 0 exp [ t 1 2 + t 2 2 2 T 2 ( t 1 t 2 ) 2 2 T c 2 i 2 π ν 0 ( t 1 t 2 ) ] ,
W ( ν 1 , ν 2 ) = W 0 exp [ ( ν 1 ν 0 ) 2 + ( ν 2 ν 0 ) 2 2 M 2 ( ν 1 ν 2 ) 2 2 M c 2 ] ,
M = 1 π ( 1 4 T 2 + 1 2 T c 2 ) 1 2 , W 0 = Γ 0 T M , M c = T c M T ,
I ( t ) = Γ 0 exp ( t 2 T 2 ) , S ( ν ) = W 0 exp [ ( ν ν 0 ) 2 M 2 ] .
t ¯ = 0 , ν ¯ = ν 0 , ( Δ t ) 2 = T 2 2 , ( Δ ν ) 2 = M 2 2 ,
( Δ ν ) 2 = ( 1 4 π ) 2 1 ( Δ t ) 2 + 1 4 π 2 T c 2 ,
γ ( t 1 , t 2 ) = Γ ( t 1 , t 2 ) I ( t 1 ) I ( t 2 ) = exp [ ( t 1 t 2 ) 2 2 T c 2 i 2 π ν 0 ( t 1 t 2 ) ] .
τ c 2 = γ ( t 1 , t 2 ) 2 ( t 1 t 2 ) 2 d t 1 d t 2 γ ( t 1 , t 2 ) 2 d t 1 d t 2 = T c 2 2 .
β ( ν 1 , ν 2 ) = W ( ν 1 , ν 2 ) S ( ν 1 ) S ( ν 2 ) ,
ν c 2 = β ( ν 1 , ν 2 ) 2 ( ν 1 ν 2 ) 2 d ν 1 d ν 2 β ( ν 1 , ν 2 ) 2 d ν 1 d ν 2 .
β ( ν 1 , ν 2 ) = exp [ ( ν 1 ν 2 ) 2 2 M c 2 ] , ν c 2 = M c 2 2 .
( Δ ν ) 2 = ( 1 4 π ) 2 1 ( Δ t ) 2 + 1 8 π 2 τ c 2 .
( Δ ν ) 2 ( 1 4 π ) 2 1 ( Δ t ) 2 ,

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