Abstract

Coherent anti-Stokes Raman scattering (CARS) is a well-known Raman scattering process that occurs when Stokes, anti-Stokes and pump waves are properly phase-matched. Using a quantum-field approach with Langevin noise sources, we calculate the noise figure for wavelength conversion between the Stokes and anti-Stokes waves in CARS and show its dependence on phase mismatch. Under phase matched conditions, the minimum noise figure is approximately 3 dB, with a correction that depends on the pump frequency, Stokes shift, refractive indices, and nonlinear susceptibilities. We calculate the photon statistics of CARS and show that the photon number distribution is non-Gaussian. Our findings may be significant for currently pursued applications of CARS including wavelength conversion in data transmission and spectroscopic detection of dilute biochemical species.

© 2006 Optical Society of America

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References

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  1. P.D. Maker and R.W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. 137, A801 (1965).
    [Crossref]
  2. H. Vogt, “Coherent and hyper-Raman techniques” in Topics in Applied Physics (eds. M. Cardona and G. Guntherodt) vol. 50, p. 207, Springer-Verlag (1982).
  3. A.M. Zheltikov and P. Radi, “Non-linear Raman spectroscopy 75 years after the Nobel Prize for the discovery of Raman scattering and 40 years after the first CARS experiments,” J. of Raman Spectrosc. 36, 92–94 (2005).
    [Crossref]
  4. G. Beadie, Z.E. Sariyanni, Y. V. Rostovtsev, T. Opatrny, J. Reintjes, and M.O. Scully, “Towards a FAST CARS anthrax detector: coherence preparation using simultaneous femtosecond laser pulses,” Opt. Commun. 244, 423–430 (2005).
    [Crossref]
  5. K.P. Knutsen, J.C. Johnson, A.E. Miller, P.B. Petersen, and R.J. Saykally, “High-spectral-resolution multiplex CARS spectroscopy using chirped pulses,” SPIE Proc. 5323, 230–239 (2004).
    [Crossref]
  6. K. Ishii and H. Hamaguchi, “Picosecond time-resolved multiplex CARS spectroscopy using optical Kerr gating,” Chem. Phys. Lett. 367, 672–677 (2003).
    [Crossref]
  7. E.O. Potma, C.L. Evans, and X.S. Xie, “Heterodyne coherent anti-Stokes Raman scattering (CARS) imaging,” Opt. Lett. 31, 241–243 (2006).
    [Crossref] [PubMed]
  8. F. Vestin, M. Afzelius, C. Brackmann, and P-E. Bengtsson, “Dual-broadband rotational CARS thermometry in the product gas of hydrocarbon flames,” Proc. of the Combustion Inst. 30, 1673–1680 (2004).
    [Crossref]
  9. N. Djaker, P-F. Lenne, and H. Rigneault, “Vibrational imaging by coherent anti-Stokes Raman scattering (CARS) microscopy,” SPIE Proc. 5463, 133–139 (2004).
    [Crossref]
  10. R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, “Anti-Stokes Raman conversion in Silicon waveguides,” Opt. Express 11, 2862–2872 (2003).
    [Crossref] [PubMed]
  11. J. Perina, “Photon statistics in Raman scattering with frequency mismatch,” Optica Acta 28, 1529 (1981).
    [Crossref]
  12. J. Perina, “Photon statistics in Raman scattering of intense coherent light,” Optica Acta 28, 325 (1981).
    [Crossref]
  13. M. Karska and J. Perina, “Photon statistics in stimulated Raman scattering of squeeze light,” J. Mod. Optics 37, 195 (1990).
    [Crossref]
  14. P.L. Voss and P. Kumar, “Raman-noise-induced noise-figure limit for χ(3) parametric amplifiers,” Opt. Lett. 29, 445 (2004).
    [Crossref] [PubMed]
  15. R. Tang, P.L. Voss, J. Lasri, P. Devgan, and P. Kumar, “Noise-figure of fiber-optical parametric amplifiers and wavelength converters: experimental investigation,” Opt. Lett. 29, 2372 (2004).
    [Crossref] [PubMed]
  16. K.K. Das, G. S. Agarwal, Y.M. Golubev, and M.O. Scully, “Langevin analysis of fundamental noise limits in coherent anti-Stokes Raman spectroscopy,” Phys. Rev. A 71, 013802 (2005).
    [Crossref]
  17. R.W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, 1992).
  18. A. Yariv, Quantum Electronics, 3rd ed. (Wiley: New York1989), Chapter 5.
  19. H.A. Haus, Electromagnetic Noise and Quantum Optical Measurement (Springer2000), p. 217–223.
  20. W.H. Louisell, Quantum Statistical Properties of Radiation (John Wiley & Sons1973).

2006 (1)

2005 (3)

K.K. Das, G. S. Agarwal, Y.M. Golubev, and M.O. Scully, “Langevin analysis of fundamental noise limits in coherent anti-Stokes Raman spectroscopy,” Phys. Rev. A 71, 013802 (2005).
[Crossref]

A.M. Zheltikov and P. Radi, “Non-linear Raman spectroscopy 75 years after the Nobel Prize for the discovery of Raman scattering and 40 years after the first CARS experiments,” J. of Raman Spectrosc. 36, 92–94 (2005).
[Crossref]

G. Beadie, Z.E. Sariyanni, Y. V. Rostovtsev, T. Opatrny, J. Reintjes, and M.O. Scully, “Towards a FAST CARS anthrax detector: coherence preparation using simultaneous femtosecond laser pulses,” Opt. Commun. 244, 423–430 (2005).
[Crossref]

2004 (5)

K.P. Knutsen, J.C. Johnson, A.E. Miller, P.B. Petersen, and R.J. Saykally, “High-spectral-resolution multiplex CARS spectroscopy using chirped pulses,” SPIE Proc. 5323, 230–239 (2004).
[Crossref]

F. Vestin, M. Afzelius, C. Brackmann, and P-E. Bengtsson, “Dual-broadband rotational CARS thermometry in the product gas of hydrocarbon flames,” Proc. of the Combustion Inst. 30, 1673–1680 (2004).
[Crossref]

N. Djaker, P-F. Lenne, and H. Rigneault, “Vibrational imaging by coherent anti-Stokes Raman scattering (CARS) microscopy,” SPIE Proc. 5463, 133–139 (2004).
[Crossref]

P.L. Voss and P. Kumar, “Raman-noise-induced noise-figure limit for χ(3) parametric amplifiers,” Opt. Lett. 29, 445 (2004).
[Crossref] [PubMed]

R. Tang, P.L. Voss, J. Lasri, P. Devgan, and P. Kumar, “Noise-figure of fiber-optical parametric amplifiers and wavelength converters: experimental investigation,” Opt. Lett. 29, 2372 (2004).
[Crossref] [PubMed]

2003 (2)

R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, “Anti-Stokes Raman conversion in Silicon waveguides,” Opt. Express 11, 2862–2872 (2003).
[Crossref] [PubMed]

K. Ishii and H. Hamaguchi, “Picosecond time-resolved multiplex CARS spectroscopy using optical Kerr gating,” Chem. Phys. Lett. 367, 672–677 (2003).
[Crossref]

1990 (1)

M. Karska and J. Perina, “Photon statistics in stimulated Raman scattering of squeeze light,” J. Mod. Optics 37, 195 (1990).
[Crossref]

1981 (2)

J. Perina, “Photon statistics in Raman scattering with frequency mismatch,” Optica Acta 28, 1529 (1981).
[Crossref]

J. Perina, “Photon statistics in Raman scattering of intense coherent light,” Optica Acta 28, 325 (1981).
[Crossref]

1965 (1)

P.D. Maker and R.W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. 137, A801 (1965).
[Crossref]

Afzelius, M.

F. Vestin, M. Afzelius, C. Brackmann, and P-E. Bengtsson, “Dual-broadband rotational CARS thermometry in the product gas of hydrocarbon flames,” Proc. of the Combustion Inst. 30, 1673–1680 (2004).
[Crossref]

Agarwal, G. S.

K.K. Das, G. S. Agarwal, Y.M. Golubev, and M.O. Scully, “Langevin analysis of fundamental noise limits in coherent anti-Stokes Raman spectroscopy,” Phys. Rev. A 71, 013802 (2005).
[Crossref]

Beadie, G.

G. Beadie, Z.E. Sariyanni, Y. V. Rostovtsev, T. Opatrny, J. Reintjes, and M.O. Scully, “Towards a FAST CARS anthrax detector: coherence preparation using simultaneous femtosecond laser pulses,” Opt. Commun. 244, 423–430 (2005).
[Crossref]

Bengtsson, P-E.

F. Vestin, M. Afzelius, C. Brackmann, and P-E. Bengtsson, “Dual-broadband rotational CARS thermometry in the product gas of hydrocarbon flames,” Proc. of the Combustion Inst. 30, 1673–1680 (2004).
[Crossref]

Boyd, R.W.

R.W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, 1992).

Brackmann, C.

F. Vestin, M. Afzelius, C. Brackmann, and P-E. Bengtsson, “Dual-broadband rotational CARS thermometry in the product gas of hydrocarbon flames,” Proc. of the Combustion Inst. 30, 1673–1680 (2004).
[Crossref]

Claps, R.

Das, K.K.

K.K. Das, G. S. Agarwal, Y.M. Golubev, and M.O. Scully, “Langevin analysis of fundamental noise limits in coherent anti-Stokes Raman spectroscopy,” Phys. Rev. A 71, 013802 (2005).
[Crossref]

Devgan, P.

Dimitropoulos, D.

Djaker, N.

N. Djaker, P-F. Lenne, and H. Rigneault, “Vibrational imaging by coherent anti-Stokes Raman scattering (CARS) microscopy,” SPIE Proc. 5463, 133–139 (2004).
[Crossref]

Evans, C.L.

Golubev, Y.M.

K.K. Das, G. S. Agarwal, Y.M. Golubev, and M.O. Scully, “Langevin analysis of fundamental noise limits in coherent anti-Stokes Raman spectroscopy,” Phys. Rev. A 71, 013802 (2005).
[Crossref]

Hamaguchi, H.

K. Ishii and H. Hamaguchi, “Picosecond time-resolved multiplex CARS spectroscopy using optical Kerr gating,” Chem. Phys. Lett. 367, 672–677 (2003).
[Crossref]

Haus, H.A.

H.A. Haus, Electromagnetic Noise and Quantum Optical Measurement (Springer2000), p. 217–223.

Ishii, K.

K. Ishii and H. Hamaguchi, “Picosecond time-resolved multiplex CARS spectroscopy using optical Kerr gating,” Chem. Phys. Lett. 367, 672–677 (2003).
[Crossref]

Jalali, B.

Johnson, J.C.

K.P. Knutsen, J.C. Johnson, A.E. Miller, P.B. Petersen, and R.J. Saykally, “High-spectral-resolution multiplex CARS spectroscopy using chirped pulses,” SPIE Proc. 5323, 230–239 (2004).
[Crossref]

Karska, M.

M. Karska and J. Perina, “Photon statistics in stimulated Raman scattering of squeeze light,” J. Mod. Optics 37, 195 (1990).
[Crossref]

Knutsen, K.P.

K.P. Knutsen, J.C. Johnson, A.E. Miller, P.B. Petersen, and R.J. Saykally, “High-spectral-resolution multiplex CARS spectroscopy using chirped pulses,” SPIE Proc. 5323, 230–239 (2004).
[Crossref]

Kumar, P.

Lasri, J.

Lenne, P-F.

N. Djaker, P-F. Lenne, and H. Rigneault, “Vibrational imaging by coherent anti-Stokes Raman scattering (CARS) microscopy,” SPIE Proc. 5463, 133–139 (2004).
[Crossref]

Louisell, W.H.

W.H. Louisell, Quantum Statistical Properties of Radiation (John Wiley & Sons1973).

Maker, P.D.

P.D. Maker and R.W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. 137, A801 (1965).
[Crossref]

Miller, A.E.

K.P. Knutsen, J.C. Johnson, A.E. Miller, P.B. Petersen, and R.J. Saykally, “High-spectral-resolution multiplex CARS spectroscopy using chirped pulses,” SPIE Proc. 5323, 230–239 (2004).
[Crossref]

Opatrny, T.

G. Beadie, Z.E. Sariyanni, Y. V. Rostovtsev, T. Opatrny, J. Reintjes, and M.O. Scully, “Towards a FAST CARS anthrax detector: coherence preparation using simultaneous femtosecond laser pulses,” Opt. Commun. 244, 423–430 (2005).
[Crossref]

Perina, J.

M. Karska and J. Perina, “Photon statistics in stimulated Raman scattering of squeeze light,” J. Mod. Optics 37, 195 (1990).
[Crossref]

J. Perina, “Photon statistics in Raman scattering with frequency mismatch,” Optica Acta 28, 1529 (1981).
[Crossref]

J. Perina, “Photon statistics in Raman scattering of intense coherent light,” Optica Acta 28, 325 (1981).
[Crossref]

Petersen, P.B.

K.P. Knutsen, J.C. Johnson, A.E. Miller, P.B. Petersen, and R.J. Saykally, “High-spectral-resolution multiplex CARS spectroscopy using chirped pulses,” SPIE Proc. 5323, 230–239 (2004).
[Crossref]

Potma, E.O.

Radi, P.

A.M. Zheltikov and P. Radi, “Non-linear Raman spectroscopy 75 years after the Nobel Prize for the discovery of Raman scattering and 40 years after the first CARS experiments,” J. of Raman Spectrosc. 36, 92–94 (2005).
[Crossref]

Raghunathan, V.

Reintjes, J.

G. Beadie, Z.E. Sariyanni, Y. V. Rostovtsev, T. Opatrny, J. Reintjes, and M.O. Scully, “Towards a FAST CARS anthrax detector: coherence preparation using simultaneous femtosecond laser pulses,” Opt. Commun. 244, 423–430 (2005).
[Crossref]

Rigneault, H.

N. Djaker, P-F. Lenne, and H. Rigneault, “Vibrational imaging by coherent anti-Stokes Raman scattering (CARS) microscopy,” SPIE Proc. 5463, 133–139 (2004).
[Crossref]

Rostovtsev, Y. V.

G. Beadie, Z.E. Sariyanni, Y. V. Rostovtsev, T. Opatrny, J. Reintjes, and M.O. Scully, “Towards a FAST CARS anthrax detector: coherence preparation using simultaneous femtosecond laser pulses,” Opt. Commun. 244, 423–430 (2005).
[Crossref]

Sariyanni, Z.E.

G. Beadie, Z.E. Sariyanni, Y. V. Rostovtsev, T. Opatrny, J. Reintjes, and M.O. Scully, “Towards a FAST CARS anthrax detector: coherence preparation using simultaneous femtosecond laser pulses,” Opt. Commun. 244, 423–430 (2005).
[Crossref]

Saykally, R.J.

K.P. Knutsen, J.C. Johnson, A.E. Miller, P.B. Petersen, and R.J. Saykally, “High-spectral-resolution multiplex CARS spectroscopy using chirped pulses,” SPIE Proc. 5323, 230–239 (2004).
[Crossref]

Scully, M.O.

G. Beadie, Z.E. Sariyanni, Y. V. Rostovtsev, T. Opatrny, J. Reintjes, and M.O. Scully, “Towards a FAST CARS anthrax detector: coherence preparation using simultaneous femtosecond laser pulses,” Opt. Commun. 244, 423–430 (2005).
[Crossref]

K.K. Das, G. S. Agarwal, Y.M. Golubev, and M.O. Scully, “Langevin analysis of fundamental noise limits in coherent anti-Stokes Raman spectroscopy,” Phys. Rev. A 71, 013802 (2005).
[Crossref]

Tang, R.

Terhune, R.W.

P.D. Maker and R.W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. 137, A801 (1965).
[Crossref]

Vestin, F.

F. Vestin, M. Afzelius, C. Brackmann, and P-E. Bengtsson, “Dual-broadband rotational CARS thermometry in the product gas of hydrocarbon flames,” Proc. of the Combustion Inst. 30, 1673–1680 (2004).
[Crossref]

Vogt, H.

H. Vogt, “Coherent and hyper-Raman techniques” in Topics in Applied Physics (eds. M. Cardona and G. Guntherodt) vol. 50, p. 207, Springer-Verlag (1982).

Voss, P.L.

Xie, X.S.

Yariv, A.

A. Yariv, Quantum Electronics, 3rd ed. (Wiley: New York1989), Chapter 5.

Zheltikov, A.M.

A.M. Zheltikov and P. Radi, “Non-linear Raman spectroscopy 75 years after the Nobel Prize for the discovery of Raman scattering and 40 years after the first CARS experiments,” J. of Raman Spectrosc. 36, 92–94 (2005).
[Crossref]

Chem. Phys. Lett. (1)

K. Ishii and H. Hamaguchi, “Picosecond time-resolved multiplex CARS spectroscopy using optical Kerr gating,” Chem. Phys. Lett. 367, 672–677 (2003).
[Crossref]

J. Mod. Optics (1)

M. Karska and J. Perina, “Photon statistics in stimulated Raman scattering of squeeze light,” J. Mod. Optics 37, 195 (1990).
[Crossref]

J. of Raman Spectrosc. (1)

A.M. Zheltikov and P. Radi, “Non-linear Raman spectroscopy 75 years after the Nobel Prize for the discovery of Raman scattering and 40 years after the first CARS experiments,” J. of Raman Spectrosc. 36, 92–94 (2005).
[Crossref]

Opt. Commun. (1)

G. Beadie, Z.E. Sariyanni, Y. V. Rostovtsev, T. Opatrny, J. Reintjes, and M.O. Scully, “Towards a FAST CARS anthrax detector: coherence preparation using simultaneous femtosecond laser pulses,” Opt. Commun. 244, 423–430 (2005).
[Crossref]

Opt. Express (1)

Opt. Lett. (3)

Optica Acta (2)

J. Perina, “Photon statistics in Raman scattering with frequency mismatch,” Optica Acta 28, 1529 (1981).
[Crossref]

J. Perina, “Photon statistics in Raman scattering of intense coherent light,” Optica Acta 28, 325 (1981).
[Crossref]

Phys. Rev. (1)

P.D. Maker and R.W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. 137, A801 (1965).
[Crossref]

Phys. Rev. A (1)

K.K. Das, G. S. Agarwal, Y.M. Golubev, and M.O. Scully, “Langevin analysis of fundamental noise limits in coherent anti-Stokes Raman spectroscopy,” Phys. Rev. A 71, 013802 (2005).
[Crossref]

Proc. of the Combustion Inst. (1)

F. Vestin, M. Afzelius, C. Brackmann, and P-E. Bengtsson, “Dual-broadband rotational CARS thermometry in the product gas of hydrocarbon flames,” Proc. of the Combustion Inst. 30, 1673–1680 (2004).
[Crossref]

SPIE Proc. (2)

N. Djaker, P-F. Lenne, and H. Rigneault, “Vibrational imaging by coherent anti-Stokes Raman scattering (CARS) microscopy,” SPIE Proc. 5463, 133–139 (2004).
[Crossref]

K.P. Knutsen, J.C. Johnson, A.E. Miller, P.B. Petersen, and R.J. Saykally, “High-spectral-resolution multiplex CARS spectroscopy using chirped pulses,” SPIE Proc. 5323, 230–239 (2004).
[Crossref]

Other (5)

H. Vogt, “Coherent and hyper-Raman techniques” in Topics in Applied Physics (eds. M. Cardona and G. Guntherodt) vol. 50, p. 207, Springer-Verlag (1982).

R.W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, 1992).

A. Yariv, Quantum Electronics, 3rd ed. (Wiley: New York1989), Chapter 5.

H.A. Haus, Electromagnetic Noise and Quantum Optical Measurement (Springer2000), p. 217–223.

W.H. Louisell, Quantum Statistical Properties of Radiation (John Wiley & Sons1973).

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Figures (3)

Fig. 1.
Fig. 1.

The noise figure for signal conversion with perfect phase-matching is shown for interaction length L=2 cm, a gain coefficient 2gS=(30 cm/GW)×IP and r=1.2.

Fig. 2.
Fig. 2.

The noise figure variation with the phase-mismatch for both Stokes and anti-Stokes scattering with L=2 cm, r=1.2, a gain coefficient 30 cm/GW and a pump intensity of 17 MW/cm2 (gS =0.25 cm -1).

Fig. 3.
Fig. 3.

Photon number probability distribution for the Stokes output with an anti-Stokes input of 400 photons (mean number) and a gain parameter gS ×L=0.33, assuming perfect phasematching. The mean photon number at the Stokes output is 45; the best fit Gaussian for this mean value is plotted for comparison. The Stokes at the output with no anti-Stokes input is also shown (only spontaneous emission present). The noise reservoir is taken to be in the ground state.

Equations (70)

Equations on this page are rendered with MathJax. Learn more.

d a ̂ s d x = g s a ̂ s + r g s exp ( j ( Δ β x + 2 φ P ) ) a ̂ AS + j 2 g S N ̂ +
d a ̂ AS + dx = r 2 g S a ̂ AS + r g S exp ( j ( Δ β · x + 2 φ P ) ) a ̂ S
+ j exp ( j ( Δ β · x + 2 φ P ) ) 2 r 2 g S N ̂ + .
a ̂ S ( x ) = A ( x ) a ̂ S ( 0 ) + B ( x ) a ̂ AS + ( 0 ) + 0 x d x N S ( x x ) N ̂ + ( x )
a ̂ AS + ( x ) = C ( x ) a ̂ AS + ( 0 ) + D ( x ) a ̂ S ( 0 ) + 0 x d x N AS ( x x ) N ̂ + ( x ) .
a ̂ S + ( x ) a ̂ S ( x ) = B ( x ) 2 a 2 Converted signal + B ( x ) 2 + N S , TOT d . c . term ,
Δ ( a ̂ S + ( x ) a ̂ S ( x ) ) 2 = B ( x ) 2 ( N S , TOT + A ( x ) 2 + B ( x ) 2 ) a 2
+ A ( x ) 2 ( B ( x ) 2 + N S , TOT ) .
SNR ( x ) = ( a S + ( x ) a ̂ S ( x ) SIGNAL ) 2 Δ ( a ̂ S + ( x ) a ̂ S ( x ) ) 2
= B ( x ) 2 R 2 B ( x ) 2 ( N S , TOT + A ( x ) 2 + B ( x ) 2 ) R · Δ f + A ( x ) 2 ( B ( x ) 2 + N S , TOT ) ( Δ f ) 2 .
F AS S = SNR ( 0 ) SNR ( L ) = R Δ f SNR ( L )
= 1 + N S , TOT + A ( L ) 2 B ( L ) 2 + A ( L ) 2 ( B ( L ) 2 + N S , TOT ) B ( L ) 2 ( R Δ f ) ,
F AS S , MIN = 1 + N S , TOT + A ( L ) 2 B ( L ) 2 .
F S AS = 1 + C ( L ) 2 + N AS , TOT D ( L ) 2 ( 1 + 1 R Δ f ) ,
F S AS , MIN = 1 + C ( L ) 2 + N AS , TOT D ( L ) 2 .
F AS S , MIN = 1 + ( 1 r 2 ) 2 ,
F AS S , MIN = 2 + r 2 ( 1 r 2 ) 2 .
F S AS , MIN = { 2 + 1 ( r g S x ) 2 , g S ( r 2 1 ) x 1 2 + ( r 2 1 ) 2 r 2 , g S ( r 2 1 ) x 1 .
a ̂ S ( x ) = A a ̂ S ( 0 ) + B a ̂ AS + ( 0 ) + N S N ̂ +
a ̂ AS + ( x ) = D a ̂ S ( 0 ) + C a ̂ AS + ( 0 ) + N AS N ̂ + ,
ρ ̂ = ( 1 q ) n = 0 q n n R R n ,
p ( n o ) = e μ a 2 ( 1 λ ) λ n o L n o ( μ a 2 ( λ 1 ) λ ) ,
d A S d x = g S ( ω S n S ) A P 2 A S + κ ( ω S n S ) A P 2 e j Δ β x A AS *
d A AS * dx = α AS ( ω AS n AS ) A P 2 A AS * κ ( ω AS n AS ) ( A P * ) 2 e j Δ β x A S ,
d a ̂ S d x = g S a ̂ S + κ e 2 j φ P e j Δ β x a ̂ AS +
d a ̂ AS + d x = α AS a ̂ AS + κ e 2 j φ P e j Δ β x a ̂ S ,
d a ̂ S dx = g S a ̂ S + F ̂ G .
a ̂ S INH = x d x exp ( g S ( x x ) ) F ̂ G ( x ) .
[ a ̂ S INH ( x ) , F ̂ G + ( x ) ] = ( 1 2 ) C ,
[ F ̂ G ( x ) , F ̂ G + ( x ) ] = 2 g S δ ( x x ) .
κ e j ( Δ β x + 2 φ P ) [ a ̂ AS + , a ̂ AS ] κ e j ( Δ β x + 2 φ P ) [ a ̂ S , a ̂ S + ]
+ exp ( j θ S ) 2 g S [ N ̂ + , a ̂ AS ] + exp ( j θ AS ) 2 α AS [ a ̂ S , N ̂ ] = 0 .
a ̂ S ( x ) = exp ( j θ S ) 2 g S 0 x d x P S ( x x ) N ̂ + ( x )
a ̂ AS + ( x ) = exp ( j θ AS ) 2 α AS 0 x d x P AS ( x x ) N ̂ + ( x ) ,
θ S θ AS = Δ β x + 2 φ P + π ,
d a ̂ S dx = g S a ̂ S + r g S exp ( j ( Δ β x + 2 φ P ) ) a ̂ AS + j 2 g S N ̂ +
d a ̂ AS + d x = r 2 g S a ̂ AS + r g S exp ( j ( Δ β · x + 2 φ P ) ) a ̂ S
+ j exp ( j ( Δ β · x + 2 φ P ) ) 2 r 2 g S N ̂ + .
a ̂ S = b ̂ S exp ( j Δ β x 2 ) , a ̂ AS + = b ̂ AS + exp ( j Δβ x 2 ) .
d d x b ̂ s = ( g S j ( Δ β 2 ) ) b ̂ s + r g s θ b ̂ A S + j 2 g s e j Δ β x 2 N ̂ + ( x )
d d x b ̂ A S + = ( j ( Δ β 2 ) r 2 g s ) b ̂ AS + r g 2 θ * b ̂ s
U ̂ ( + ) = b ̂ S + C ( + ) b ̂ A S + , U ̂ ( ) = b ̂ S + C ( ) b ̂ A S + ,
C ( ± ) = ( ( r 2 + 1 ) g s i Δ β ) ± ( ( r 2 + 1 ) g s i Δ β ) 2 4 r 2 g s 2 2 r g s θ ,
d d x U ̂ ( ± ) = Λ ( ± ) U ̂ ( ± ) + j 2 g s ( r θ * C ( ± ) 1 ) e j Δ β · x 2 N ̂ + ( x ) ,
Λ ( ± ) = g s ( 1 r 2 ) ( g s ( 1 + r 2 ) i Δ β ) 2 4 r 2 g s 2 2 .
U ̂ ( ± ) ( x ) = exp ( Λ ( ± ) x ) U ̂ ( ± ) ( 0 )
+ j 2 g s ( r θ * C ( ± ) 1 ) 0 x d x exp ( Λ ( ± ) ( x x ) ) e j Δ β · x 2 N ̂ + ( x ) .
b ̂ s ( x ) = C ( ) exp ( Λ ( + ) x ) C ( + ) exp ( Λ ( ) x ) C ( ) C ( + ) b ̂ s ( 0 )
+ C ( ) C ( + ) exp ( Λ ( + ) x ) exp ( Λ ( ) x ) C ( ) C ( + ) b ̂ AS + ( 0 )
+ j 2 g s 0 x d x [ ( r θ * C ( + ) 1 ) C ( ) exp ( Λ ( + ) ( x x ) ) C ( ) C ( + )
( r θ * C ( ) 1 ) C ( + ) exp ( Λ ( ) ( x x ) ) C ( ) C ( + ) ] e j Δ β · x 2 N ̂ + ( x ) ,
b ̂ AS + ( x ) = exp ( Λ ( + ) x ) exp ( Λ ( ) x ) C ( + ) C ( ) b ̂ s ( 0 )
+ C ( + ) exp ( Λ ( + ) x ) C ( ) exp ( Λ ( ) x ) C ( + ) C ( ) b ̂ AS + ( 0 )
+ j 2 g s 0 x d x [ ( r θ * C ( + ) 1 ) exp ( Λ ( + ) ( x x ) ) C ( + ) C ( )
( r θ * C ( ) 1 ) exp ( Λ ( ) ( x x ) ) C ( + ) C ( ) ] e j Δ β · x 2 N ̂ + ( x ) .
P ˜ ( k ) = n = 0 p ( n ) e j k n = e j k n ,
p ( n o ) = 1 2 π 0 2 π d k e jk n o e jk n .
exp ( jk a ̂ S + ( x ) a ̂ S ( x ) ) = R 0 AS a S 0 exp ( jk a ̂ S + ( x ) a ̂ S ( x ) ) 0 S a AS 0 R .
exp ( ξ a ̂ + a ̂ ) = n = 0 ( e ξ 1 ) n n ! ( a ̂ + ) n a ̂ n ,
exp ( ξ a ̂ + a ̂ ) = e ξ n = 0 ( 1 e ξ ) n n ! a ̂ n ( a ̂ + ) n .
exp ( ξ a ̂ S + ( x ) a ̂ S ( x ) ) = e ξ n = 0 ( 1 e ξ ) n n ! a ̂ S ( x ) n ( a ̂ S + ( x ) ) n
= e ξ n = 0 ( 1 e ξ ) n n ! η n ( 1 + ξ ) n ( ξ ) n .
exp ( ξ a ̂ S + ( x ) a ̂ S ( x ) ) = e ξ + η 1 η n = 0 ( 1 e ξ ) n n ! ( 1 + ξ ) n ( ξ ) n
= 1 + ( η 1 ) e ξ η exp ( ξ ξ ( 1 + ξ ) ) .
exp ( ξ a ̂ S + ( x ) a ̂ S ( x ) ) = 1 + ( η 1 ) e ξ η n = 0 ( e ξ 1 ) n n ! ( ξ ) n ( 1 + ξ ) n
= 1 + ( η 1 ) e ξ η n = 0 ( e ξ 1 ) n n ! ( ξ ) n
= 1 + ( η 1 ) e ξ η exp ( ξ ( 1 e ξ ) ) .
exp ( j k a ̂ S + ( x ) a ̂ S ( x ) ) = e ξ e jk 1 + η ( e jk 1 ) exp ( ξ 1 + η ( e jk 1 ) ) .
p ( n o a ) = e ξ dz 2 π i z n o ( z z o ) exp ( ξ η z z o ) ,
p ( n o a ) = e ξ n = 0 ( ξ η ) n η n ! z n o d z 2 π i ( z z o ) n + 1 = e ξ n = 0 n o ( n o n ) ( ξ η ) n η n ! z o n o n .

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