Abstract

The impact of high-frequency spectral phase modulation on the temporal intensity of optical pulses is derived analytically and simulated in two different regimes. The temporal contrast of an optical pulse close to the Fourier-transform limit is degraded by a pedestal related to the power spectral density of the spectral phase modulation. When the optical pulse is highly chirped, its intensity modulation is directly related to the spectral phase variations with a transfer function depending on the second-order dispersion of the chirped pulse. The metrology of the spectral phase of an optical pulse using temporal-intensity measurements performed after chirping the pulse is studied. The effect of spatial averaging is also discussed.

© 2008 Optical Society of America

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References

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  1. C. Dorrer and I. A. Walmsley, "Concepts for the temporal characterization of short optical pulses," Eurasip J. Appl. Signal Process. 2005, 1541?1553 (2005).
    [CrossRef]
  2. K.-H. Hong, B. Hou, J. A. Nees, E. Power, and G. A. Mourou, "Generation and measurement of >108 intensity contrast ratio in a relativistic khz chirped-pulse amplified laser," Appl. Phys. B 81, 447?457 (2005).
    [CrossRef]
  3. S. Backus, C. G. DurfeeIII, M. M. Murnane, and H. C. Kapteyn, "High power ultrafast lasers," Rev. Sci. Instrum. 69, 1207?1223 (1998).
    [CrossRef]
  4. A. Dubietis, R. Butkus, and A. P. Piskarskas, "Trends in chirped pulse optical parametric amplification," IEEE J. Sel. Top. Quantum Electron. 12, 163?172 (2006).
    [CrossRef]
  5. I. Walmsley, L. Waxer, and C. Dorrer, "The role of dispersion in ultrafast optics," Rev. Sci. Instrum. 72, 1?29 (2001).
    [CrossRef]
  6. J. C. Stover, Optical Scattering: Measurement and Analysis, 2nd ed., Vol. PM24 (SPIE, Bellingham, WA, 1995).
    [CrossRef]
  7. G. Chériaux, P. Rousseau, F. Salin, J. P. Chambaret, B. Walker, and L. F. Dimauro, "Aberration-free stretcher design for ultrashort-pulse amplification," Opt. Lett. 21, 414?416 (1996).
    [CrossRef] [PubMed]
  8. V. Bagnoud and F. Salin, "Influence of optical quality on chirped-pulse amplification: Characterization of a 150-nm-bandwidth stretcher," J. Opt. Soc. Am. B 16, 188?193 (1999).
    [CrossRef]
  9. C. Dorrer, "Chromatic dispersion characterization by direct instantaneous frequency measurement," Opt. Lett. 29, 204?206 (2004).
    [CrossRef] [PubMed]

2006

A. Dubietis, R. Butkus, and A. P. Piskarskas, "Trends in chirped pulse optical parametric amplification," IEEE J. Sel. Top. Quantum Electron. 12, 163?172 (2006).
[CrossRef]

2005

C. Dorrer and I. A. Walmsley, "Concepts for the temporal characterization of short optical pulses," Eurasip J. Appl. Signal Process. 2005, 1541?1553 (2005).
[CrossRef]

K.-H. Hong, B. Hou, J. A. Nees, E. Power, and G. A. Mourou, "Generation and measurement of >108 intensity contrast ratio in a relativistic khz chirped-pulse amplified laser," Appl. Phys. B 81, 447?457 (2005).
[CrossRef]

2004

2001

I. Walmsley, L. Waxer, and C. Dorrer, "The role of dispersion in ultrafast optics," Rev. Sci. Instrum. 72, 1?29 (2001).
[CrossRef]

1999

1998

S. Backus, C. G. DurfeeIII, M. M. Murnane, and H. C. Kapteyn, "High power ultrafast lasers," Rev. Sci. Instrum. 69, 1207?1223 (1998).
[CrossRef]

1996

Backus, S.

S. Backus, C. G. DurfeeIII, M. M. Murnane, and H. C. Kapteyn, "High power ultrafast lasers," Rev. Sci. Instrum. 69, 1207?1223 (1998).
[CrossRef]

Bagnoud, V.

Butkus, R.

A. Dubietis, R. Butkus, and A. P. Piskarskas, "Trends in chirped pulse optical parametric amplification," IEEE J. Sel. Top. Quantum Electron. 12, 163?172 (2006).
[CrossRef]

Chambaret, J. P.

Chériaux, G.

Dimauro, L. F.

Dorrer, C.

C. Dorrer and I. A. Walmsley, "Concepts for the temporal characterization of short optical pulses," Eurasip J. Appl. Signal Process. 2005, 1541?1553 (2005).
[CrossRef]

C. Dorrer, "Chromatic dispersion characterization by direct instantaneous frequency measurement," Opt. Lett. 29, 204?206 (2004).
[CrossRef] [PubMed]

I. Walmsley, L. Waxer, and C. Dorrer, "The role of dispersion in ultrafast optics," Rev. Sci. Instrum. 72, 1?29 (2001).
[CrossRef]

Dubietis, A.

A. Dubietis, R. Butkus, and A. P. Piskarskas, "Trends in chirped pulse optical parametric amplification," IEEE J. Sel. Top. Quantum Electron. 12, 163?172 (2006).
[CrossRef]

Durfee, C. G.

S. Backus, C. G. DurfeeIII, M. M. Murnane, and H. C. Kapteyn, "High power ultrafast lasers," Rev. Sci. Instrum. 69, 1207?1223 (1998).
[CrossRef]

Hong, K.-H.

K.-H. Hong, B. Hou, J. A. Nees, E. Power, and G. A. Mourou, "Generation and measurement of >108 intensity contrast ratio in a relativistic khz chirped-pulse amplified laser," Appl. Phys. B 81, 447?457 (2005).
[CrossRef]

Hou, B.

K.-H. Hong, B. Hou, J. A. Nees, E. Power, and G. A. Mourou, "Generation and measurement of >108 intensity contrast ratio in a relativistic khz chirped-pulse amplified laser," Appl. Phys. B 81, 447?457 (2005).
[CrossRef]

Kapteyn, H. C.

S. Backus, C. G. DurfeeIII, M. M. Murnane, and H. C. Kapteyn, "High power ultrafast lasers," Rev. Sci. Instrum. 69, 1207?1223 (1998).
[CrossRef]

Mourou, G. A.

K.-H. Hong, B. Hou, J. A. Nees, E. Power, and G. A. Mourou, "Generation and measurement of >108 intensity contrast ratio in a relativistic khz chirped-pulse amplified laser," Appl. Phys. B 81, 447?457 (2005).
[CrossRef]

Murnane, M. M.

S. Backus, C. G. DurfeeIII, M. M. Murnane, and H. C. Kapteyn, "High power ultrafast lasers," Rev. Sci. Instrum. 69, 1207?1223 (1998).
[CrossRef]

Nees, J. A.

K.-H. Hong, B. Hou, J. A. Nees, E. Power, and G. A. Mourou, "Generation and measurement of >108 intensity contrast ratio in a relativistic khz chirped-pulse amplified laser," Appl. Phys. B 81, 447?457 (2005).
[CrossRef]

Piskarskas, A. P.

A. Dubietis, R. Butkus, and A. P. Piskarskas, "Trends in chirped pulse optical parametric amplification," IEEE J. Sel. Top. Quantum Electron. 12, 163?172 (2006).
[CrossRef]

Power, E.

K.-H. Hong, B. Hou, J. A. Nees, E. Power, and G. A. Mourou, "Generation and measurement of >108 intensity contrast ratio in a relativistic khz chirped-pulse amplified laser," Appl. Phys. B 81, 447?457 (2005).
[CrossRef]

Rousseau, P.

Salin, F.

Walker, B.

Walmsley, I.

I. Walmsley, L. Waxer, and C. Dorrer, "The role of dispersion in ultrafast optics," Rev. Sci. Instrum. 72, 1?29 (2001).
[CrossRef]

Walmsley, I. A.

C. Dorrer and I. A. Walmsley, "Concepts for the temporal characterization of short optical pulses," Eurasip J. Appl. Signal Process. 2005, 1541?1553 (2005).
[CrossRef]

Waxer, L.

I. Walmsley, L. Waxer, and C. Dorrer, "The role of dispersion in ultrafast optics," Rev. Sci. Instrum. 72, 1?29 (2001).
[CrossRef]

Appl. Phys. B

K.-H. Hong, B. Hou, J. A. Nees, E. Power, and G. A. Mourou, "Generation and measurement of >108 intensity contrast ratio in a relativistic khz chirped-pulse amplified laser," Appl. Phys. B 81, 447?457 (2005).
[CrossRef]

Eurasip J. Appl. Signal Process.

C. Dorrer and I. A. Walmsley, "Concepts for the temporal characterization of short optical pulses," Eurasip J. Appl. Signal Process. 2005, 1541?1553 (2005).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

A. Dubietis, R. Butkus, and A. P. Piskarskas, "Trends in chirped pulse optical parametric amplification," IEEE J. Sel. Top. Quantum Electron. 12, 163?172 (2006).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Lett.

Rev. Sci. Instrum.

I. Walmsley, L. Waxer, and C. Dorrer, "The role of dispersion in ultrafast optics," Rev. Sci. Instrum. 72, 1?29 (2001).
[CrossRef]

S. Backus, C. G. DurfeeIII, M. M. Murnane, and H. C. Kapteyn, "High power ultrafast lasers," Rev. Sci. Instrum. 69, 1207?1223 (1998).
[CrossRef]

Other

J. C. Stover, Optical Scattering: Measurement and Analysis, 2nd ed., Vol. PM24 (SPIE, Bellingham, WA, 1995).
[CrossRef]

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Figures (5)

Fig. 1.
Fig. 1.

Temporal intensity of a pulse with sinusoidal spectral phase modulation of period 2π/τ with τ=30 ps on a logarithmic scale in the case of (a) an amplitude δφ 0=0.01 rad and (b) an amplitude δφ 0=0.1 rad.

Fig. 2.
Fig. 2.

(a) Normalized power spectral density for PSD1 (blue line) and PSD2 (red line). (b) Spectral intensity of the pulse (black continuous line) and realizations of the spectral phase with a standard deviation of 0.1 rad and normalized power spectral densities PSD1 (blue continuous line) and PSD2 (red continuous line). The realizations of the spectral phase have been vertically separated to ease comparison.

Fig. 3.
Fig. 3.

Temporal intensity of the pulse on a logarithmic scale for spectral phase modulation with normalized power spectral density PSD1 and standard deviation (a) 0.01 rad and (b) 0.1 rad, and with normalized power spectral density PSD2 and standard deviation (c) 0.01 rad and (d) 0.1 rad.

Fig. 4.
Fig. 4.

Temporal intensity of a pulse with sinusoidal spectral phase modulation having a period 2π/τ with τ=10 ps and amplitude 0.1 rad after dispersion (a)φ?=τ2/3π, (b) φ?=τ2/2π, and (c) φ?=τ2/π. (d) Displays a comparison between the simulated temporal intensity and the intensity calculated analytically for φ?=τ2/3π and φ?=τ2/π.

Fig. 5.
Fig. 5.

(a) Spectral intensity (black line) and phase (red line) of a pulse corresponding to a FWHM of 6 nm centered at 1053 nm and a continuous PSD with |δ φ ˜ (t)|2=exp(-t 2/T 2 2 with T 2=20 ps and a standard deviation of 0.1 rad. (b)–(d) represent close-ups of the temporal intensity of the pulse after second-order dispersion φ?=τ2/3π, φ?=τ2/2π, and φ?=τ2/π, where τ=10 ps, respectively.

Equations (21)

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E ~ ( ω ) = I ~ ( ω ) exp [ i φ ( ω ) ] ,
E ~ ( ω ) = I ~ ( ω ) exp [ i φ 0 ( ω ) ] exp [ i δ φ ( ω ) ] = E ~ 0 ( ω ) exp [ i δ φ ( ω ) ] .
E ~ ( ω ) = E ~ 0 ( ω ) [ 1 + i δ φ ( ω ) ] .
E ( t ) = E 0 ( t ) + i E 0 δ φ ~ ( t ) ,
I ( t ) = I 0 ( t ) + E 0 δ φ ˜ ( t ) 2 2 I m { E 0 * ( t ) [ E 0 δ φ ˜ ( t ) ] } .
I ( t ) = I 0 ( t ) + E δ φ ~ ( t ) 2 .
I ( t ) = I 0 ( t ) + δ φ 0 2 [ I 0 ( t τ ) + I 0 ( t + τ ) ] 4 .
I ( t ) = I 0 ( t ) + n δ φ 0 , n 2 [ I 0 ( t τ n ) + I 0 ( t + τ n ) ] 4 .
I ( t ) = I 0 ( t ) + I 0 δ φ ~ 2 ( t ) ,
I ( t ) = I 0 ( t ) + ε 0 δ φ ~ ( t ) 2 ,
E ~ ( ω ) = I ~ ( ω ) exp [ i φ 0 ( ω ) ] exp [ i δ φ ( ω ) ] .
E ~ ( ω ) = E 0 ~ ( ω ) [ 1 + i n α n cos ( τ n ω + β n ) ] .
E ( t ) = E 0 ( t ) + i 2 n α n [ E 0 ( t τ n ) exp ( i β n ) + E 0 ( t + τ n ) exp ( i β n ) ] .
I ( t ) = I 0 ( t ) i 2 E 0 ( t ) n α n [ E 0 * ( t τ n ) exp ( i β n )
+ E 0 * ( t + τ n ) exp ( i β n ) ] + i 2 E 0 * ( t ) ×
n α n [ E 0 ( t τ n ) exp ( i β n ) + E 0 ( t + τ n ) exp ( i β n ) ] .
E 0 ( t ) E 0 * ( t τ n ) = 1 φ I ˜ 0 ( t φ ) exp ( it τ n φ ) exp ( i τ n 2 2 φ ) ,
I ( t ) = 1 φ I ˜ 0 ( t φ ) [ 1 + 2 n α n cos ( t τ n φ + β n ) sin ( τ n 2 2 φ ) ] .
I ( x , y , t ) = I 0 ( x , y , t ) + E t δ φ ~ ( x , y , t ) 2 ,
I ( x , y , t ) = 1 φ I ˜ 0 ( x , y , t φ ) { 1 + 2 n α n ( x , y )
× cos [ t τ n ( x , y ) φ + β n ( x , y ) ] sin [ τ n 2 ( x , y ) 2 φ ] } .

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