Abstract

The impact of high-frequency spectral phase modulation on the temporal contrast of ultrafast pulses is evaluated. Expressions are derived for the low-intensity pedestal produced by optical component surface roughness within pulse stretchers and compressors. Phase noise, added across the near-field of a spectrally dispersed beam, produces space–time coupling in the far-field or focal plane. The pedestal is swept across an area in the focal plane many times the size of the diffraction-limited spot. Simulations are performed for generic stretchers and compressors that show fundamentally different forms of temporal contrast degradation at focus.

© 2012 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2011 (1)

2009 (1)

E. Sidick, “Power spectral density specification and analysis of large optical surfaces,” SPIE 7390, 73900L (2009).

2008 (1)

2007 (1)

2005 (1)

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).

2001 (1)

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

1999 (1)

1997 (1)

A. Antonetti, F. Blasco, J. P. Chambaret, G. Chériaux, G. Darpentingy, C. Le Blanc, P. Rousseau, S. Ranc, G. Rey, and F. Salin, “A laser system producing 5×1019  W/cm2 at 10 Hz,” Appl. Phys. B 65, 197–204 (1997).

1996 (2)

M. M. Wefers and K. A. Nelson, “Space-time profiles of shaped ultrafast optical waveforms,” IEEE J. Quantum Electron. 32, 161–172 (1996).
[CrossRef]

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]

1993 (1)

S. Luan, M. H. R. Hutchinson, R. A. Smith, and F. Zhou, “High dynamic range third-order correlation measurement of picosecond laser pulse shapes,” Meas. Sci. Technol. 4, 1426–1429 (1993).
[CrossRef]

1992 (1)

1990 (1)

E. L. Church, P. Z. Takacs, and T. A. Leonard, “Prediction of BRDFs from surface profile measurements,” SPIE 1165, 136–150 (1990).

1988 (1)

P. Maine, D. Strickland, P. Bado, M. Pessot, and G. Mourou, “Generation of ultrahigh peak power pulses by chirped pulse amplification,” IEEE J. Quantum Electron. 24, 398–403 (1988).
[CrossRef]

1972 (1)

1969 (1)

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

1836 (1)

H. F. Talbot, “Facts relating to optical science. No. IV,” Philos. Mag. Ser. 3 9, 401–407 (1836).
[CrossRef]

Antonetti, A.

A. Antonetti, F. Blasco, J. P. Chambaret, G. Chériaux, G. Darpentingy, C. Le Blanc, P. Rousseau, S. Ranc, G. Rey, and F. Salin, “A laser system producing 5×1019  W/cm2 at 10 Hz,” Appl. Phys. B 65, 197–204 (1997).

Bado, P.

P. Maine, D. Strickland, P. Bado, M. Pessot, and G. Mourou, “Generation of ultrahigh peak power pulses by chirped pulse amplification,” IEEE J. Quantum Electron. 24, 398–403 (1988).
[CrossRef]

Bagnoud, V.

Blasco, F.

A. Antonetti, F. Blasco, J. P. Chambaret, G. Chériaux, G. Darpentingy, C. Le Blanc, P. Rousseau, S. Ranc, G. Rey, and F. Salin, “A laser system producing 5×1019  W/cm2 at 10 Hz,” Appl. Phys. B 65, 197–204 (1997).

Bromage, J.

Chambaret, J. P.

A. Antonetti, F. Blasco, J. P. Chambaret, G. Chériaux, G. Darpentingy, C. Le Blanc, P. Rousseau, S. Ranc, G. Rey, and F. Salin, “A laser system producing 5×1019  W/cm2 at 10 Hz,” Appl. Phys. B 65, 197–204 (1997).

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]

Chekhlov, O.

Chériaux, G.

A. Antonetti, F. Blasco, J. P. Chambaret, G. Chériaux, G. Darpentingy, C. Le Blanc, P. Rousseau, S. Ranc, G. Rey, and F. Salin, “A laser system producing 5×1019  W/cm2 at 10 Hz,” Appl. Phys. B 65, 197–204 (1997).

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]

Church, E. L.

E. L. Church, P. Z. Takacs, and T. A. Leonard, “Prediction of BRDFs from surface profile measurements,” SPIE 1165, 136–150 (1990).

Collier, J.

Crouse, R. F.

Dainty, J. C.

Darpentingy, G.

A. Antonetti, F. Blasco, J. P. Chambaret, G. Chériaux, G. Darpentingy, C. Le Blanc, P. Rousseau, S. Ranc, G. Rey, and F. Salin, “A laser system producing 5×1019  W/cm2 at 10 Hz,” Appl. Phys. B 65, 197–204 (1997).

Dimauro, L. F.

Divall, E.

Dorrer, C.

C. Dorrer and J. Bromage, “Impact of high-frequency spectral phase modulation on the temporal profile of short optical pulses,” Opt. Express 163058–3068 (2008).
[CrossRef]

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

Ertel, K.

Goodman, J. W.

J. W. Goodman, Statistical Optics, Wiley Series in Pure and Applied Optics (Wiley, 1985).

Hawkes, S.

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).

Hooker, C.

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).

Huang, H.

Hutchinson, M. H. R.

S. Luan, M. H. R. Hutchinson, R. A. Smith, and F. Zhou, “High dynamic range third-order correlation measurement of picosecond laser pulse shapes,” Meas. Sci. Technol. 4, 1426–1429 (1993).
[CrossRef]

Kessler, T. J.

Latimer, P.

Le Blanc, C.

A. Antonetti, F. Blasco, J. P. Chambaret, G. Chériaux, G. Darpentingy, C. Le Blanc, P. Rousseau, S. Ranc, G. Rey, and F. Salin, “A laser system producing 5×1019  W/cm2 at 10 Hz,” Appl. Phys. B 65, 197–204 (1997).

Leonard, T. A.

E. L. Church, P. Z. Takacs, and T. A. Leonard, “Prediction of BRDFs from surface profile measurements,” SPIE 1165, 136–150 (1990).

Luan, S.

S. Luan, M. H. R. Hutchinson, R. A. Smith, and F. Zhou, “High dynamic range third-order correlation measurement of picosecond laser pulse shapes,” Meas. Sci. Technol. 4, 1426–1429 (1993).
[CrossRef]

Maine, P.

P. Maine, D. Strickland, P. Bado, M. Pessot, and G. Mourou, “Generation of ultrahigh peak power pulses by chirped pulse amplification,” IEEE J. Quantum Electron. 24, 398–403 (1988).
[CrossRef]

Mourou, G.

P. Maine, D. Strickland, P. Bado, M. Pessot, and G. Mourou, “Generation of ultrahigh peak power pulses by chirped pulse amplification,” IEEE J. Quantum Electron. 24, 398–403 (1988).
[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).

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).

Nelson, K. A.

M. M. Wefers and K. A. Nelson, “Space-time profiles of shaped ultrafast optical waveforms,” IEEE J. Quantum Electron. 32, 161–172 (1996).
[CrossRef]

Parry, B.

Pessot, M.

P. Maine, D. Strickland, P. Bado, M. Pessot, and G. Mourou, “Generation of ultrahigh peak power pulses by chirped pulse amplification,” IEEE J. Quantum Electron. 24, 398–403 (1988).
[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).

Rajeev, P. P.

Ranc, S.

A. Antonetti, F. Blasco, J. P. Chambaret, G. Chériaux, G. Darpentingy, C. Le Blanc, P. Rousseau, S. Ranc, G. Rey, and F. Salin, “A laser system producing 5×1019  W/cm2 at 10 Hz,” Appl. Phys. B 65, 197–204 (1997).

Rey, G.

A. Antonetti, F. Blasco, J. P. Chambaret, G. Chériaux, G. Darpentingy, C. Le Blanc, P. Rousseau, S. Ranc, G. Rey, and F. Salin, “A laser system producing 5×1019  W/cm2 at 10 Hz,” Appl. Phys. B 65, 197–204 (1997).

Rousseau, P.

A. Antonetti, F. Blasco, J. P. Chambaret, G. Chériaux, G. Darpentingy, C. Le Blanc, P. Rousseau, S. Ranc, G. Rey, and F. Salin, “A laser system producing 5×1019  W/cm2 at 10 Hz,” Appl. Phys. B 65, 197–204 (1997).

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]

Salin, F.

Sidick, E.

E. Sidick, “Power spectral density specification and analysis of large optical surfaces,” SPIE 7390, 73900L (2009).

Smith, R. A.

S. Luan, M. H. R. Hutchinson, R. A. Smith, and F. Zhou, “High dynamic range third-order correlation measurement of picosecond laser pulse shapes,” Meas. Sci. Technol. 4, 1426–1429 (1993).
[CrossRef]

Strickland, D.

P. Maine, D. Strickland, P. Bado, M. Pessot, and G. Mourou, “Generation of ultrahigh peak power pulses by chirped pulse amplification,” IEEE J. Quantum Electron. 24, 398–403 (1988).
[CrossRef]

Takacs, P. Z.

E. L. Church, P. Z. Takacs, and T. A. Leonard, “Prediction of BRDFs from surface profile measurements,” SPIE 1165, 136–150 (1990).

Talbot, H. F.

H. F. Talbot, “Facts relating to optical science. No. IV,” Philos. Mag. Ser. 3 9, 401–407 (1836).
[CrossRef]

Tang, Y.

Treacy, E. B.

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

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]

Waxer, L.

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

Wefers, M. M.

M. M. Wefers and K. A. Nelson, “Space-time profiles of shaped ultrafast optical waveforms,” IEEE J. Quantum Electron. 32, 161–172 (1996).
[CrossRef]

Zhou, F.

S. Luan, M. H. R. Hutchinson, R. A. Smith, and F. Zhou, “High dynamic range third-order correlation measurement of picosecond laser pulse shapes,” Meas. Sci. Technol. 4, 1426–1429 (1993).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B (2)

A. Antonetti, F. Blasco, J. P. Chambaret, G. Chériaux, G. Darpentingy, C. Le Blanc, P. Rousseau, S. Ranc, G. Rey, and F. Salin, “A laser system producing 5×1019  W/cm2 at 10 Hz,” Appl. Phys. B 65, 197–204 (1997).

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).

IEEE J. Quantum Electron. (3)

M. M. Wefers and K. A. Nelson, “Space-time profiles of shaped ultrafast optical waveforms,” IEEE J. Quantum Electron. 32, 161–172 (1996).
[CrossRef]

P. Maine, D. Strickland, P. Bado, M. Pessot, and G. Mourou, “Generation of ultrahigh peak power pulses by chirped pulse amplification,” IEEE J. Quantum Electron. 24, 398–403 (1988).
[CrossRef]

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

J. Opt. Soc. Am. (1)

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

Meas. Sci. Technol. (1)

S. Luan, M. H. R. Hutchinson, R. A. Smith, and F. Zhou, “High dynamic range third-order correlation measurement of picosecond laser pulse shapes,” Meas. Sci. Technol. 4, 1426–1429 (1993).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Philos. Mag. Ser. 3 (1)

H. F. Talbot, “Facts relating to optical science. No. IV,” Philos. Mag. Ser. 3 9, 401–407 (1836).
[CrossRef]

Rev. Sci. Instrum. (1)

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

SPIE (2)

E. L. Church, P. Z. Takacs, and T. A. Leonard, “Prediction of BRDFs from surface profile measurements,” SPIE 1165, 136–150 (1990).

E. Sidick, “Power spectral density specification and analysis of large optical surfaces,” SPIE 7390, 73900L (2009).

Other (1)

J. W. Goodman, Statistical Optics, Wiley Series in Pure and Applied Optics (Wiley, 1985).

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Figures (13)

Fig. 1.
Fig. 1.

(a) Generic schematic of a system that imprints spectral phase noise on an optical pulse propagating with a finite beam size. The near-field spatial coordinates are ( x , y ) and the far-field spatial coordinates are ( u , v ) . Three important scale lengths are (1)  γ Δ ω , the spatial width of the dispersed spectrum; (2)  W , the beam size; and (3)  l C , the correlation length of the phase noise. (b) Gaussian spectrum plotted against the near-field coordinate x ; γ = 0.44 mm / THz and Δ ω = 100 THz , corresponding to a 17 fs pulse width. (c) Simulated phase screen ϕ ( x , y ) and beam locations for optical frequencies marked in (b). Scale lengths are W = 10 mm , l C = 0.5 mm , and γ Δ ω = 44 mm .

Fig. 2.
Fig. 2.

(a) Near-field average intensity in the x t plane calculated using Eq. (19). The main pulse, I 0 ( x , y , t ) is the red line at t = 0 ps . [(b),(c),(d)] Analytic (thin blue) and numeric (jagged red) values of intensity calculated at different x values.

Fig. 3.
Fig. 3.

(a) Far-field average intensity in the u t plane calculated using Eq. (18). Space–time coupling in the noise-dependent term follows the diagonal black line, u = t / γ . [(b),(c),(d)] Analytic (thin blue) and numeric (jagged red) values of intensity calculated at different u values.

Fig. 4.
Fig. 4.

(a) Far-field average intensity in the v t plane calculated using Eq. (17). [(b),(c),(d)] Analytic (thin blue) and numeric (jagged red) values of intensity calculated at different v values.

Fig. 5.
Fig. 5.

On-axis temporal intensities for Gaussian functional forms and a range of near-field beam sizes ( W ). The on-axis far-field contrast increases with the beam size.

Fig. 6.
Fig. 6.

On-axis temporal intensities for Gaussian functional forms. Near-field results for (a) one phase screen and (b) an ensemble average of pulses for 25 phase screens. Far-field results for (c) one phase screen and (d) an ensemble average of pulses for 25 phase screens.

Fig. 7.
Fig. 7.

Average on-axis intensity calculated either by averaging in time (rectangular window, T ) or over an ensemble of phase maps ( N ). (a) Near-field intensity: T = 200 fs , N = 200 . (b) Far-field intensity: T = 40 fs , N = 200 .

Fig. 8.
Fig. 8.

Intensity probability distributions at t = 250 fs for (a) the near-field and (b) the far-field.

Fig. 9.
Fig. 9.

On-axis temporal intensity calculated in (a) near-field and (b) far-field for a Lorentzian PSD with σ = 0.04 radians, l C = 100 μm , and S = 1.55 .

Fig. 10.
Fig. 10.

Near-field quantities after spectrally dispersed propagation through a sinusoidal phase screen with a modulation period and amplitude of 450 μm and 0.5 nm, respectively. (a) The spectrum, (b) spectral phase, and (c) temporal intensity immediately after the screen. [(d),(e),(f)] The same quantities after propagating a distance of 1 m. There is complete conversion of phase-to-amplitude modulation at the peak wavelength of 910 nm.

Fig. 11.
Fig. 11.

Far-field intensity calculated in the u t plane for two phase screens. (a) Simulation results where phase screens were coincident and (b) simulation results where phase screens were separated by Δ z = 1 m . Each spectral field after the first phase screens was propagated to the second using a scalar Fresnel propagation code. The details of the intensity structure between (a) and (b) are different, but any differences between the average profiles are insignificant.

Fig. 12.
Fig. 12.

(a) Schematic of a four-grating compressor, showing the input and output beams (black) and three spectral components. Modulation on the surfaces of G 2 and G 3 produces spectral phase noise. A roof mirror can be used along the line A A to retroreflect the beam, halving the required number of gratings. (b) Far-field average intensity calculated using Lorentzian PSDs in the u t plane at t = 0 . G 2 and G 3 have the same spatiospectral coefficient, γ = 0.44 mm / THz . The black line corresponds to u = t / γ and tracks the peak intensity from the PSD term.

Fig. 13.
Fig. 13.

(a) Schematic of an Öffner stretcher, showing the input beam (black) and three spectral components. (b) Far-field average intensity calculated using the same Lorentzian PSD for each optic in the u t plane at t = 0 . The black lines correspond to u = t / γ for each optic.

Tables (1)

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Table 1. Default Simulation Parameters Used in This Paper, Unless Explicitly Specified

Equations (37)

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x 0 ( ω ) = γ ω ,
C ( x , x , y , y ) = ϕ ( x , y ) ϕ * ( x , y ) .
C ( Δ x , Δ y ) = σ 2 exp ( Δ x 2 + Δ y 2 2 l C 2 ) ,
E ˜ ( x , y , ω ) = E 0 ˜ ( x , y , ω ) e i Φ ( x , y , ω ) ,
Φ ( x , y , ω ) = ϕ ( x x 0 , y ) = ϕ ( x γ ω , y ) .
E ˜ ( x , y , ω ) = E ˜ 0 ( x , y , w ) [ 1 + i ϕ ( x γ ω , y ) ] .
E ( x , y , t ) = d ω E ˜ ( x , y , ω ) e i ω t = E 0 ( x , y , t ) + i d ω E ˜ 0 ( x , y , ω ) ϕ ( x γ ω , y ) e i ω t .
Ĕ ( u , v , t ) = d x d y E ( x , y , t ) e i u x e i v y , = Ĕ 0 ( u , v , t ) + i d ω d x d y E ˜ 0 ( x , y , ω ) ϕ ( x γ ω , y ) e i ( ω t + u x + v y ) .
I ( u , v , t ) = I 0 ( u , v , t ) + d ω d ω d x d x d y d y E ˜ 0 ( x , y , ω ) E ˜ 0 * ( x , y , ω ) × ϕ ( x γ ω , y ) ϕ * ( x γ ω , y ) e i ( ω ω ) t e i ( x x ) u e i ( y y ) v .
I ( u , v , t ) = I 0 ( u , v , t ) + d u d v I 0 ( u , v , t + γ u γ u ) PSD ( u u , v v ) .
I ( x , y , t ) = I 0 ( x , y , t ) + d ω d ω E ˜ 0 ( x , y , ω ) E ˜ 0 * ( x , y , ω ) × ϕ ( x γ ω , y ) ϕ * ( x γ ω , y ) e i ( ω ω ) t .
C x y ( Δ ω ) = ϕ ( x γ ω , y ) ϕ * ( x γ ω , y ) .
I ( x , y , t ) = I 0 ( x , y , t ) + d t I 0 ( x , y , t ) PSD x y ( t t ) .
PSD x y ( t ) = 1 γ d v PSD ( t / γ , v ) .
PSD ( u , v ) = σ 2 l C 2 2 π exp [ l C 2 ( u 2 + v 2 ) / 2 ] .
I 0 ( u , v , t ) = I P exp ( Δ ω 2 t 2 W 2 u 2 W 2 v 2 ) ,
I ( u , v , t ) = I 0 ( u , v , t ) + I P σ 2 l C 2 2 W ( l C 2 / 2 + γ 2 Δ ω 2 + W 2 ) 1 / 2 × exp [ ( γ Δ ω 2 t W 2 u ) 2 l C 2 / 2 + γ 2 Δ ω 2 + W 2 ] exp ( W 2 u 2 l C 2 v 2 / 2 Δ ω 2 t 2 ) .
I ( u , v , t ) = I 0 ( u , v , t ) + I P σ 2 l C 2 γ Δ ω ( l C 2 W ) exp [ W ( u + t γ ) 2 l C 2 v 2 2 l C 2 t 2 2 γ 2 ] .
I ( x , y , t ) = I 0 ( x , y , t ) + I P ( x , y , t ) σ 2 l C 2 γ Δ ω exp ( l C 2 t 2 2 γ 2 ) .
p ( I ) = 1 I exp ( r I I ) I 0 ( 2 r I / I ) ,
p ( I ) 1 I exp ( I I ) .
PSD ( u , v ) = ( S 1 ) σ 2 l C 2 2 π ( 1 + l C 2 u 2 + l C 2 v 2 ) ( S + 1 ) / 2 .
PSD x y ( t ) = σ 2 l C Γ ( S / 2 ) π γ Γ ( S 1 2 ) ( 1 + l C 2 t 2 / γ 2 ) S / 2 ,
Φ total ( x , y , ω ) = n ϕ n ( x γ n ω , y ) .
Φ total ( x , y , ω ) Φ total * ( x , y , ω ) = n C n ( Δ x γ n Δ ω , Δ y ) .
I ( u , v , t ) = I 0 ( u , v , t ) + n d u d v I 0 ( u , v , t + γ n u γ n u ) PSD n ( u u , v v ) .
I ( u , v , t ) = I 0 ( u , v , t ) + 1 2 3 d Ω d Δ ω d X d Δ x d Y d Δ y × E 0 ˜ ( X + Δ x 2 , Y + Δ y 2 , Ω + Δ ω 2 ) E ˜ 0 * ( X Δ x 2 , Y Δ y 2 , Ω Δ ω 2 ) × C ( Δ x γ Δ ω , Δ y ) e i Δ ω t e i Δ x u e i Δ y v ,
C ( Δ x , Δ y ) = ϕ ( x , y ) ϕ * ( x , y ) .
d Ω d X d Y d t d t E 0 ( X + Δ x 2 , Y + Δ y 2 , t ) E 0 * ( X Δ x 2 , Y Δ y 2 , t ) e i ( Ω + Δ ω 2 ) t e i ( Ω Δ ω 2 ) t .
2 d X d Y d t E 0 ( X + Δ x 2 , Y + Δ y 2 , t ) E 0 * ( X Δ x 2 , Y Δ y 2 , t ) e i ω t .
2 3 d t d u d v I 0 ( u , v , t ) e i Δ ω t e i Δ x u e i Δ y v .
I ( u , v , t ) = I 0 ( u , v , t ) + d u d v d t d Δ ω d Δ x d Δ y I 0 ( u , v , t ) × C ( Δ x γ Δ ω , Δ y ) e i Δ ω ( t t ) e i Δ x ( u u ) e i Δ y ( v v ) .
PSD ( u , v ) = d Δ x d Δ y C ( Δ x , Δ y ) e i Δ x u e i Δ y v .
I ( u , v , t ) = I 0 ( u , v , t ) + d u d v I 0 ( u , v , t + γ u γ u ) PSD ( u u , v v ) .
PSD ( u , v ) d v = d Δ x d Δ y C ( Δ x , Δ y ) e i Δ x u ( d v e i Δ y v ) .
PSD x y ( t ) = d Δ ω C x y ( Δ ω ) e i Δ ω t ,
d v PSD ( t / γ , v ) = γ P S D x y ( t ) .

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