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, M. Cardona and G. Guntherodt, eds., (Springer-Verlag, 1982) Vol. 50, p. 207.
  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. Raman Spectrosc. 36, 92-4 (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," in Multiphoton Microscopy in the Biomedical Sciences IV, A. Periasamy and P. T. So, eds, Proc. SPIE 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, 672-7 (2003).
    [CrossRef]
  7. .E.O. Potma, C.L. Evans, X.S. Xie, "Heterodyne coherent anti-Stokes Raman scattering (CARS) imaging," Opt. Lett. 31, 241-3 (2006).
    [CrossRef] [PubMed]
  8. F. Vestin, M. Afzelius, C. Brackmann, P-E. Bengtsson, "Dual-broadband rotational CARS thermometry in the product gas of hydrocarbon flames," Proc. of the Combustion Inst. 30, 1673-80 (2004).
    [CrossRef]
  9. N. Djaker, P-F. Lenne, H. Rigneault, "Vibrational imaging by coherent anti-Stokes Raman scattering (CARS) microscopy," in Femtosecond Laser Applications in Biology, S. Avrillier, and J.-M. Tualle, eds., 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 York 1989), Chap. 5.
  19. H. A. Haus, Electromagnetic Noise and Quantum Optical Measurement (Springer 2000), p. 217 - 223.
  20. W. H. Louisell, Quantum Statistical Properties of Radiation (John Wiley & Sons 1973).

2006 (1)

2005 (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. Raman Spectrosc. 36, 92-4 (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]

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]

2004 (2)

2003 (2)

.K. Ishii and H. Hamaguchi, "Picosecond time-resolved multiplex CARS spectroscopy using optical Kerr gating," Chem. Phys. Lett. 367, 672-7 (2003).
[CrossRef]

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

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]

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]

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.

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, 672-7 (2003).
[CrossRef]

Ishii, K.

.K. Ishii and H. Hamaguchi, "Picosecond time-resolved multiplex CARS spectroscopy using optical Kerr gating," Chem. Phys. Lett. 367, 672-7 (2003).
[CrossRef]

Jalali, B.

Karska, M.

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

Kumar, P.

Lasri, J.

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]

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]

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. Raman Spectrosc. 36, 92-4 (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]

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]

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]

Voss, P. L.

Xie, X.S.

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. Raman Spectrosc. 36, 92-4 (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, 672-7 (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. 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. Raman Spectrosc. 36, 92-4 (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]

Other (8)

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

A. Yariv, Quantum Electronics, 3rd ed. (Wiley: New York 1989), Chap. 5.

H. A. Haus, Electromagnetic Noise and Quantum Optical Measurement (Springer 2000), p. 217 - 223.

W. H. Louisell, Quantum Statistical Properties of Radiation (John Wiley & Sons 1973).

F. Vestin, M. Afzelius, C. Brackmann, P-E. Bengtsson, "Dual-broadband rotational CARS thermometry in the product gas of hydrocarbon flames," Proc. of the Combustion Inst. 30, 1673-80 (2004).
[CrossRef]

N. Djaker, P-F. Lenne, H. Rigneault, "Vibrational imaging by coherent anti-Stokes Raman scattering (CARS) microscopy," in Femtosecond Laser Applications in Biology, S. Avrillier, and J.-M. Tualle, eds., SPIE Proc. 5463, 133-139 (2004).
[CrossRef]

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

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," in Multiphoton Microscopy in the Biomedical Sciences IV, A. Periasamy and P. T. So, eds, Proc. SPIE 5323, 230-239 (2004).
[CrossRef]

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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)

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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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