Abstract

It is analytically shown that weak initial spectral phase modulations cause a pulse-contrast degradation at the output of nonlinear chirped-pulse amplification systems. The Kerr-nonlinearity causes an energy-transfer from the main pulse to side-pulses during nonlinear amplification. The relative intensities of these side-pulses can be described in terms of Bessel-functions. It is shown that the intensities of the pulses are dependent on the magnitude of the accumulated nonlinear phase-shift (i.e., the B-integral), the depth and period of the initial spectral phase-modulation and the slope of the linear stretching chirp. The results are applicable to any type of laser amplifier that is based on the technique of chirped-pulse amplification. The analytical results presented in this paper are of particular importance for high peak-power laser applications requiring high pulse-contrasts, e.g. high field physics.

©2008 Optical Society of America

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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  28. V. Bagnoud, J. D. Zuegel, N. Forget, and C. Le Blanc, “High-dynamic-range temporal measurements of short pulses amplified by OPCPA,” Opt. Express 15, (2007).
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2008 (3)

2007 (4)

2006 (3)

S. P. D. Mangles, A. G. R. Thomas, M. C. Kaluza, O. Lundh, F. Lindau, A. Persson, Z. Najmudin, C.-G. Wahlström, C. D. Murphy, C. Kamperidis, K. L. Lancaster, E. Divall, and K. Krushelnick, “Effect of laser contrast ratio on electron beam stability in laser wakefield acceleration experiments,” Plasma Phys. Control. Fusion 48, B83B90 (2006).
[Crossref]

J. Limpert, F. Röser, T. Schreiber, and A. Tünnermann, “High-Power Ultrafast Fiber Laser Systems,” IEEE J. Sel. Top. Quantum Electron. 12, 233–244 (2006).
[Crossref]

J. Limpert, O. Schmidt, J. Rothhardt, F. Röser, T. Schreiber, A. Tünnermann, S. Ermeneux, P. Yvernault, and F. Salin, “Extended single-mode photonic crystal fiber laser,” Opt. Express 14, 2715–2720 (2006).
[Crossref] [PubMed]

2005 (2)

F. Tavella, K. Schmid, N. Ishii, A. Marcinkevicius, L. Veisz, and F. Krausz, “High-dynamic range pulse-contrast measurements of a broadband optical parametric chirped-pulse amplifier,” Appl. Phys. B 81, 753756 (2005).
[Crossref]

L. Shah, Z. Liu, I. Hartl, G. Imeshev, G. C. Cho, and M. E. Fermann, “High energy femtosecond Yb cubicon fiber amplifier,” Opt. Express 13, 4717–4722 (2005).
[Crossref] [PubMed]

2004 (1)

M. Kaluza, J. Schreiber, M. I. K. Santala, G. D. Tsakiris, K. Eidmann, J. Meyer-ter-Vehn, and K. J. Witte, “Influence of the Laser Prepulse on Proton Acceleration in Thin-Foil Experiments,” Phys. Rev. Lett. 93, 0450031-4 (2004).
[Crossref]

2001 (3)

2000 (2)

M. I. K. Santala, M. Zepf, I. Watts, F. N. Beg, E. Clark, M. Tatarakis, K. Krushelnick, A. E. Dangor, T. McCanny, I. Spencer, R. P. Singhal, K.W. D. Ledingham, S. C. Wilks, A. C. Machacek, J. S. Wark, R. Allott, R. J. Clarke, and P. A. Norreys, “Effect of the Plasma Density Scale Length on the Direction of Fast Electrons in Relativistic Laser-Solid Interactions,” Phys. Rev. Lett. 84, 1459–1462 (2000).
[Crossref] [PubMed]

E. Zeek, R. Bartels, M. M. Murname, H. C. Kapteyn, S. Backus, and G. Vdovin, “Adaptive pulse compression for transform-limited 15-fs high-energy pulse generation,” Opt. Lett. 25, 587–589 (2000).
[Crossref]

1999 (2)

1998 (2)

M. Zepf, G. D. Tsakiris, G. Pretzler, I. Watts, D. M. Chambers, P. A. Norreys, U. Andiel, A. E. Dangor, K. Eidmann, C. Gahn, A. Machacek, J. S. Wark, and K. Witte, “Role of the plasma scale length in the harmonic generation from solid targets,” Phys. Rev. E 58, R5253– R5256 (1998).
[Crossref]

K. Ennser, M. Ibsen, M. Durkin, M. N. Zervas, and R. I. Laming, “Influence of Nonideal Chirped Fiber Grating Characteristics on Dispersion Cancellation,” IEEE Photon. Technol. Lett. 10, 1476–1478 (1998).
[Crossref]

1997 (1)

1995 (1)

A. Galvanauskas, M. Ferman, and D. Harter, “All-fiber femtosecond pulse amplification circuit using chirped bragg gratings,” Appl. Phys. Lett. 66, 1053 (1995).
[Crossref]

1994 (2)

1985 (1)

D. M. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56, 219–221 (1985).
[Crossref]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables,” Generating Function for the Bessel-function, formula 9.1.41 (Dover Publications, 1970).

Agrawal, G. P.

Akbulut, M.

T. Yilmaz, L. Vaissie, M. Akbulut, T. Booth, J. Jasapara, M. J. Andrejco, A. D. Yablon, C. Headley, and D. J. DiGovanni, “Large-mode-area Er-doped fiber chirped pulse amplification system for high-energy sub-picosecond pulses at 1.55 m,” Proc. SPIE 6873, 687354 (2008).

Akhmanov, S. A.

S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulse (American Institute of Physics, 1992), p. 93–98.

Albert, O.

Allott, R.

M. I. K. Santala, M. Zepf, I. Watts, F. N. Beg, E. Clark, M. Tatarakis, K. Krushelnick, A. E. Dangor, T. McCanny, I. Spencer, R. P. Singhal, K.W. D. Ledingham, S. C. Wilks, A. C. Machacek, J. S. Wark, R. Allott, R. J. Clarke, and P. A. Norreys, “Effect of the Plasma Density Scale Length on the Direction of Fast Electrons in Relativistic Laser-Solid Interactions,” Phys. Rev. Lett. 84, 1459–1462 (2000).
[Crossref] [PubMed]

Andiel, U.

M. Zepf, G. D. Tsakiris, G. Pretzler, I. Watts, D. M. Chambers, P. A. Norreys, U. Andiel, A. E. Dangor, K. Eidmann, C. Gahn, A. Machacek, J. S. Wark, and K. Witte, “Role of the plasma scale length in the harmonic generation from solid targets,” Phys. Rev. E 58, R5253– R5256 (1998).
[Crossref]

Andrejco, M. J.

T. Yilmaz, L. Vaissie, M. Akbulut, T. Booth, J. Jasapara, M. J. Andrejco, A. D. Yablon, C. Headley, and D. J. DiGovanni, “Large-mode-area Er-doped fiber chirped pulse amplification system for high-energy sub-picosecond pulses at 1.55 m,” Proc. SPIE 6873, 687354 (2008).

Backus, S.

Bagnoud, V.

V. Bagnoud, J. D. Zuegel, N. Forget, and C. Le Blanc, “High-dynamic-range temporal measurements of short pulses amplified by OPCPA,” Opt. Express 15, (2007).
[Crossref] [PubMed]

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]

Bartels, R.

Beg, F. N.

M. I. K. Santala, M. Zepf, I. Watts, F. N. Beg, E. Clark, M. Tatarakis, K. Krushelnick, A. E. Dangor, T. McCanny, I. Spencer, R. P. Singhal, K.W. D. Ledingham, S. C. Wilks, A. C. Machacek, J. S. Wark, R. Allott, R. J. Clarke, and P. A. Norreys, “Effect of the Plasma Density Scale Length on the Direction of Fast Electrons in Relativistic Laser-Solid Interactions,” Phys. Rev. Lett. 84, 1459–1462 (2000).
[Crossref] [PubMed]

Booth, T.

T. Yilmaz, L. Vaissie, M. Akbulut, T. Booth, J. Jasapara, M. J. Andrejco, A. D. Yablon, C. Headley, and D. J. DiGovanni, “Large-mode-area Er-doped fiber chirped pulse amplification system for high-energy sub-picosecond pulses at 1.55 m,” Proc. SPIE 6873, 687354 (2008).

Braun, A.

Bromage, J.

Cao, X. D.

Chambaret, J. P.

Chambers, D. M.

M. Zepf, G. D. Tsakiris, G. Pretzler, I. Watts, D. M. Chambers, P. A. Norreys, U. Andiel, A. E. Dangor, K. Eidmann, C. Gahn, A. Machacek, J. S. Wark, and K. Witte, “Role of the plasma scale length in the harmonic generation from solid targets,” Phys. Rev. E 58, R5253– R5256 (1998).
[Crossref]

Cheriaux, G.

Chirkin, A. S.

S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulse (American Institute of Physics, 1992), p. 93–98.

Cho, G. C.

Clark, E.

M. I. K. Santala, M. Zepf, I. Watts, F. N. Beg, E. Clark, M. Tatarakis, K. Krushelnick, A. E. Dangor, T. McCanny, I. Spencer, R. P. Singhal, K.W. D. Ledingham, S. C. Wilks, A. C. Machacek, J. S. Wark, R. Allott, R. J. Clarke, and P. A. Norreys, “Effect of the Plasma Density Scale Length on the Direction of Fast Electrons in Relativistic Laser-Solid Interactions,” Phys. Rev. Lett. 84, 1459–1462 (2000).
[Crossref] [PubMed]

Clarke, R. J.

M. I. K. Santala, M. Zepf, I. Watts, F. N. Beg, E. Clark, M. Tatarakis, K. Krushelnick, A. E. Dangor, T. McCanny, I. Spencer, R. P. Singhal, K.W. D. Ledingham, S. C. Wilks, A. C. Machacek, J. S. Wark, R. Allott, R. J. Clarke, and P. A. Norreys, “Effect of the Plasma Density Scale Length on the Direction of Fast Electrons in Relativistic Laser-Solid Interactions,” Phys. Rev. Lett. 84, 1459–1462 (2000).
[Crossref] [PubMed]

Dangor, A. E.

M. I. K. Santala, M. Zepf, I. Watts, F. N. Beg, E. Clark, M. Tatarakis, K. Krushelnick, A. E. Dangor, T. McCanny, I. Spencer, R. P. Singhal, K.W. D. Ledingham, S. C. Wilks, A. C. Machacek, J. S. Wark, R. Allott, R. J. Clarke, and P. A. Norreys, “Effect of the Plasma Density Scale Length on the Direction of Fast Electrons in Relativistic Laser-Solid Interactions,” Phys. Rev. Lett. 84, 1459–1462 (2000).
[Crossref] [PubMed]

M. Zepf, G. D. Tsakiris, G. Pretzler, I. Watts, D. M. Chambers, P. A. Norreys, U. Andiel, A. E. Dangor, K. Eidmann, C. Gahn, A. Machacek, J. S. Wark, and K. Witte, “Role of the plasma scale length in the harmonic generation from solid targets,” Phys. Rev. E 58, R5253– R5256 (1998).
[Crossref]

Didenko, N. V.

DiGovanni, D. J.

T. Yilmaz, L. Vaissie, M. Akbulut, T. Booth, J. Jasapara, M. J. Andrejco, A. D. Yablon, C. Headley, and D. J. DiGovanni, “Large-mode-area Er-doped fiber chirped pulse amplification system for high-energy sub-picosecond pulses at 1.55 m,” Proc. SPIE 6873, 687354 (2008).

Ditmire, T.

Divall, E.

S. P. D. Mangles, A. G. R. Thomas, M. C. Kaluza, O. Lundh, F. Lindau, A. Persson, Z. Najmudin, C.-G. Wahlström, C. D. Murphy, C. Kamperidis, K. L. Lancaster, E. Divall, and K. Krushelnick, “Effect of laser contrast ratio on electron beam stability in laser wakefield acceleration experiments,” Plasma Phys. Control. Fusion 48, B83B90 (2006).
[Crossref]

Dorrer, C.

Durkin, M.

K. Ennser, M. Ibsen, M. Durkin, M. N. Zervas, and R. I. Laming, “Influence of Nonideal Chirped Fiber Grating Characteristics on Dispersion Cancellation,” IEEE Photon. Technol. Lett. 10, 1476–1478 (1998).
[Crossref]

Eidam, T.

Eidmann, K.

M. Kaluza, J. Schreiber, M. I. K. Santala, G. D. Tsakiris, K. Eidmann, J. Meyer-ter-Vehn, and K. J. Witte, “Influence of the Laser Prepulse on Proton Acceleration in Thin-Foil Experiments,” Phys. Rev. Lett. 93, 0450031-4 (2004).
[Crossref]

M. Zepf, G. D. Tsakiris, G. Pretzler, I. Watts, D. M. Chambers, P. A. Norreys, U. Andiel, A. E. Dangor, K. Eidmann, C. Gahn, A. Machacek, J. S. Wark, and K. Witte, “Role of the plasma scale length in the harmonic generation from solid targets,” Phys. Rev. E 58, R5253– R5256 (1998).
[Crossref]

Ennser, K.

K. Ennser, M. Ibsen, M. Durkin, M. N. Zervas, and R. I. Laming, “Influence of Nonideal Chirped Fiber Grating Characteristics on Dispersion Cancellation,” IEEE Photon. Technol. Lett. 10, 1476–1478 (1998).
[Crossref]

Ermeneux, S.

Felix, C.

Ferman, M.

A. Galvanauskas, M. Ferman, and D. Harter, “All-fiber femtosecond pulse amplification circuit using chirped bragg gratings,” Appl. Phys. Lett. 66, 1053 (1995).
[Crossref]

Fermann, M. E.

Fittinghoff, D.

Forget, N.

V. Bagnoud, J. D. Zuegel, N. Forget, and C. Le Blanc, “High-dynamic-range temporal measurements of short pulses amplified by OPCPA,” Opt. Express 15, (2007).
[Crossref] [PubMed]

Gahn, C.

M. Zepf, G. D. Tsakiris, G. Pretzler, I. Watts, D. M. Chambers, P. A. Norreys, U. Andiel, A. E. Dangor, K. Eidmann, C. Gahn, A. Machacek, J. S. Wark, and K. Witte, “Role of the plasma scale length in the harmonic generation from solid targets,” Phys. Rev. E 58, R5253– R5256 (1998).
[Crossref]

Galvanauskas, A.

A. Galvanauskas, G. C. Cho, A. Hariharan, M. E. Fermann, and D. Harter, “Generation of high-energy femtosecond pulses in multi-core Yb-fiber chirped-pulse amplification systems,” Opt. Lett. 26, 935–937 (2001).
[Crossref]

A. Galvanauskas, “Mode-scalable Fiber-Based Chirped Pulse Amplification Systems,” IEEE J. Sel. Top. Quantum Electron. 7, 504–517 (2001).
[Crossref]

A. Galvanauskas, M. Ferman, and D. Harter, “All-fiber femtosecond pulse amplification circuit using chirped bragg gratings,” Appl. Phys. Lett. 66, 1053 (1995).
[Crossref]

Guo, T.

Hariharan, A.

Harter, D.

A. Galvanauskas, G. C. Cho, A. Hariharan, M. E. Fermann, and D. Harter, “Generation of high-energy femtosecond pulses in multi-core Yb-fiber chirped-pulse amplification systems,” Opt. Lett. 26, 935–937 (2001).
[Crossref]

A. Galvanauskas, M. Ferman, and D. Harter, “All-fiber femtosecond pulse amplification circuit using chirped bragg gratings,” Appl. Phys. Lett. 66, 1053 (1995).
[Crossref]

Hartl, I.

Headley, C.

T. Yilmaz, L. Vaissie, M. Akbulut, T. Booth, J. Jasapara, M. J. Andrejco, A. D. Yablon, C. Headley, and D. J. DiGovanni, “Large-mode-area Er-doped fiber chirped pulse amplification system for high-energy sub-picosecond pulses at 1.55 m,” Proc. SPIE 6873, 687354 (2008).

Ibsen, M.

K. Ennser, M. Ibsen, M. Durkin, M. N. Zervas, and R. I. Laming, “Influence of Nonideal Chirped Fiber Grating Characteristics on Dispersion Cancellation,” IEEE Photon. Technol. Lett. 10, 1476–1478 (1998).
[Crossref]

Imeshev, G.

Ishii, N.

F. Tavella, K. Schmid, N. Ishii, A. Marcinkevicius, L. Veisz, and F. Krausz, “High-dynamic range pulse-contrast measurements of a broadband optical parametric chirped-pulse amplifier,” Appl. Phys. B 81, 753756 (2005).
[Crossref]

Jasapara, J.

T. Yilmaz, L. Vaissie, M. Akbulut, T. Booth, J. Jasapara, M. J. Andrejco, A. D. Yablon, C. Headley, and D. J. DiGovanni, “Large-mode-area Er-doped fiber chirped pulse amplification system for high-energy sub-picosecond pulses at 1.55 m,” Proc. SPIE 6873, 687354 (2008).

Kaluza, M.

M. Kaluza, J. Schreiber, M. I. K. Santala, G. D. Tsakiris, K. Eidmann, J. Meyer-ter-Vehn, and K. J. Witte, “Influence of the Laser Prepulse on Proton Acceleration in Thin-Foil Experiments,” Phys. Rev. Lett. 93, 0450031-4 (2004).
[Crossref]

Kaluza, M. C.

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S. P. D. Mangles, A. G. R. Thomas, M. C. Kaluza, O. Lundh, F. Lindau, A. Persson, Z. Najmudin, C.-G. Wahlström, C. D. Murphy, C. Kamperidis, K. L. Lancaster, E. Divall, and K. Krushelnick, “Effect of laser contrast ratio on electron beam stability in laser wakefield acceleration experiments,” Plasma Phys. Control. Fusion 48, B83B90 (2006).
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M. I. K. Santala, M. Zepf, I. Watts, F. N. Beg, E. Clark, M. Tatarakis, K. Krushelnick, A. E. Dangor, T. McCanny, I. Spencer, R. P. Singhal, K.W. D. Ledingham, S. C. Wilks, A. C. Machacek, J. S. Wark, R. Allott, R. J. Clarke, and P. A. Norreys, “Effect of the Plasma Density Scale Length on the Direction of Fast Electrons in Relativistic Laser-Solid Interactions,” Phys. Rev. Lett. 84, 1459–1462 (2000).
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Opt. Express (6)

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M. Kaluza, J. Schreiber, M. I. K. Santala, G. D. Tsakiris, K. Eidmann, J. Meyer-ter-Vehn, and K. J. Witte, “Influence of the Laser Prepulse on Proton Acceleration in Thin-Foil Experiments,” Phys. Rev. Lett. 93, 0450031-4 (2004).
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M. I. K. Santala, M. Zepf, I. Watts, F. N. Beg, E. Clark, M. Tatarakis, K. Krushelnick, A. E. Dangor, T. McCanny, I. Spencer, R. P. Singhal, K.W. D. Ledingham, S. C. Wilks, A. C. Machacek, J. S. Wark, R. Allott, R. J. Clarke, and P. A. Norreys, “Effect of the Plasma Density Scale Length on the Direction of Fast Electrons in Relativistic Laser-Solid Interactions,” Phys. Rev. Lett. 84, 1459–1462 (2000).
[Crossref] [PubMed]

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S. P. D. Mangles, A. G. R. Thomas, M. C. Kaluza, O. Lundh, F. Lindau, A. Persson, Z. Najmudin, C.-G. Wahlström, C. D. Murphy, C. Kamperidis, K. L. Lancaster, E. Divall, and K. Krushelnick, “Effect of laser contrast ratio on electron beam stability in laser wakefield acceleration experiments,” Plasma Phys. Control. Fusion 48, B83B90 (2006).
[Crossref]

Proc. SPIE (1)

T. Yilmaz, L. Vaissie, M. Akbulut, T. Booth, J. Jasapara, M. J. Andrejco, A. D. Yablon, C. Headley, and D. J. DiGovanni, “Large-mode-area Er-doped fiber chirped pulse amplification system for high-energy sub-picosecond pulses at 1.55 m,” Proc. SPIE 6873, 687354 (2008).

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[PubMed]

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[Crossref]

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Figures (4)

Fig. 1.
Fig. 1. (a) Spectral amplitude-modulation due to a main pulse which is accompanied by a weak post-pulse (constrast of 40dB, delay of 10 ps), e.g. generated by a double internal reflection (b), and the modulated stretched pulse (c) with a full width at half maximum of 500 ps (ϕ (2)=33ps 2); (d) Spectral phase-modulation, i.e. a smooth intensity spectrum with a sinusoidal weak spectral phase-modulation (modulation depth is d=0.02, spectral modulation frequency is 10 ps) that corresponds to a transform limited multi-pulse in the time-domain (e), numerical and analytical result of the modulated stretched pulse (f) where the same stretching parameters as in (c) are used.

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Fig. 2.
Fig. 2. Spectrum after the amplifcation stage in a nonlinear CPA-system. The modulations have their origin in the weak spectral phase modulation at the input. The corresponding stretched pulse at the input of the nonlinear amplifier is shown in Fig. 1(f).

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Fig. 3.
Fig. 3. (a) Numerical and analytical result of the pulses at the output of the nonlinear CPA-system for the configuration shown in Fig.1(d–f); (b) intensity-distribution for the case of an inital sech 2 profile. The parameters of the phase-modulation and ϕ (2) are the same as in (a).

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Fig. 4.
Fig. 4. (a) Total intensity of the side-pulses relative to the intensity of the main pulse, 1-J 2 0(a), as a function of the product of B-integral and modulation-depth d times 1.4 and for different ratios (Δt)2/(2ϕ (2)) in the sine of the parameter a.

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Equations (18)

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A st ( T ) = 1 2 π ∫ d Ω exp ( − i Ω T ) exp ( i ϕ ( 2 ) 2 Ω 2 ) exp ( id cos ( Δ t Ω − b ) ) A ~ 0 ( Ω ) .
exp ( id cos ( x − b ) ) = ∑ m = − ∞ ∞ J m ( d ) i m exp ( − imx ) exp ( + imb ) .
Ω s = T + m Δ t ϕ ( 2 ) .
A st ( T ) = exp ( − i T 2 2 ϕ ( 2 ) ) − i 2 π ϕ ( 2 ) ∑ m = − ∞ ∞ J m ( d ) i m A ~ 0 ( T + m Δ t ϕ ( 2 ) ) exp ( i ( − m Δ t T ϕ ( 2 ) − m 2 ( Δ t ) 2 2 ϕ ( 2 ) + mb ) ) ,
∣ A st ( T ) ∣ 2 ≈ ∣ A ~ 0 ( T ϕ ( 2 ) ) ∣ 2 2 π ϕ ( 2 ) [ J 0 2 ( d ) + J 0 ( d ) J 1 ( d ) 4 sin ( ( Δ t ) 2 2 ϕ ( 2 ) ) cos ( Δ t T ϕ ( 2 ) − b ) ] .
A amp ( T ) = A st ( T ) exp ( gL 2 ) exp ( i γ L eff ∣ A st ( T ) ∣ 2 ) .
exp ( i γ L eff ∣ A st ( T ) ∣ 2 ) ≈ exp ( ipB 2 d sin ( ( Δ t ) 2 2 ϕ ( 2 ) ) cos ( Δ t T ϕ ( 2 ) − b ) ) .
A amp ( T ) = exp ( gL 2 ) − i 2 π ϕ ( 2 ) A ~ 0 ( T ϕ ( 2 ) ) exp ( − i T 2 2 ϕ ( 2 ) ) exp ( ia cos ( Δ t T ϕ ( 2 ) − b ) ) .
a = pB 2 d sin ( ( Δ t ) 2 2 ϕ ( 2 ) ) .
A amp ( T ) = exp ( gL 2 ) − i 2 π ϕ ( 2 ) ∑ m = − ∞ ∞ i m J m ( a ) A ~ 0 ( T ϕ ( 2 ) ) exp ( i φ m ( T ) ) ,
φ m ( T ) = − T 2 2 ϕ ( 2 ) − m Δ t T ϕ ( 2 ) + mb
T s = ϕ ( 2 ) Ω − m Δ t .
A amp ( Ω ) = ∫ d T exp ( i Ω T ) A amp ( T )
= exp ( gL 2 ) ∑ m = − ∞ ∞ i m J m ( a ) A ~ 0 ( Ω − m Δ t ϕ ( 2 ) ) exp [ i ( Ω T s + φ m ( T s ) ) ]
Ω T s + φ m ( T s ) = ϕ ( 2 ) 2 Ω 2 − m Δ t Ω + m 2 ( Δ t ) 2 2 ϕ ( 2 ) + mb .
1 2 π ∫ d Ω exp ( − i Ω T ) exp ( − im Δ t Ω ) A ~ 0 ( Ω − m Δ t ϕ ( 2 ) )
A out ( T ) = exp ( gL 2 ) ∑ m = − ∞ ∞ i m J m ( a ) A 0 ( T + m Δ t ) exp ( i φ m out ( T ) )
φ m out ( T ) = − m Δ t T ϕ ( 2 ) − m 2 ( Δ t ) 2 2 ϕ ( 2 ) + mb .

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