Abstract

A Gaussian-wave theory is developed for the classical and quantum analysis of the z-scan method that is often used to measure third-order nonlinearities. The theory allows us to compute the transmittance in the z scan and the associated regimes of amplitude squeezing. The classical limits of our theory are in perfect agreement with the previous theoretical results. We show that amplitude squeezing of 1.2dB can be obtained using the z scan with a careful selection of the signal power and the aperture size.

© 2006 Optical Society of America

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  1. D. Levandovsky, M. Vasilyev, and P. Kumar, "Perturbation theory of quantum solitons: continuum evolution and optimum squeezing by spectral filtering," Opt. Lett. 24, 43-45 (1999).
    [CrossRef]
  2. F. Konig, S. Spalter, I. L. Shumay, A. Sizmann, T. Fauster, and G. Leuchs, "Fibre-optic photon-number squeezing in the normal group-velocity dispersion regime," J. Mod. Opt. 45, 2425-2431 (1998).
    [CrossRef]
  3. S. Spalter, M. Burk, U. Strossner, A. Sizmann, and G. Leuchs, "Propagation of quantum properties of sub-picosecond solitons in a fiber," Opt. Express 2, 77-83 (1998).
    [CrossRef]
  4. S. Spalter, M. Burk, U. Strossner, M. Bohm, A. Sizmann, and G. Leuchs, "Photon number squeezing of spectrally filtered sub-picosecond optical solitons," Europhys. Lett. 38, 335-340 (1997).
    [CrossRef]
  5. A. Mecozzi and P. Kumar, "Sub-Poissonian light by spatial soliton filtering," Quantum Semiclassic. Opt. 10, L21-L26 (1998).
    [CrossRef]
  6. M. Sheik-Bahae, A. A. Said, and E. W. Stryland, "High-sensitivity, single-beam n2 measurements," Opt. Lett. 14, 955-957 (1989).
    [CrossRef] [PubMed]
  7. M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. van Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
    [CrossRef]
  8. R. W. Boyd, Nonlinear Optics (Academic, 1992).
  9. K. G. Köprülü and O. Aytür, "Analysis of Gaussian-beam degenerate optical parametric amplifiers for the generation of quadrature-squeezed states," Phys. Rev. A 60, 4122-4134 (1999).
    [CrossRef]
  10. K. G. Köprülü and O. Aytür, "Analysis of amplitude-squeezed light generation with Gaussian-beam degenerate optical parametric amplifiers," J. Opt. Soc. Am. B 18, 846-854 (2001).
    [CrossRef]
  11. D. Weaire, B. S. Wherrett, D. A. B. Miller, and S. D. Smith, "Effect of low-power nonlinear refraction on laser beam propagation in InSb," Opt. Lett. 4, 331-333 (1979).
    [CrossRef] [PubMed]
  12. One must keep in mind that the assumption of a purely dispersive nonlinearity is usually valid when working far away from resonances in any medium. Of course, the Kramers-Kronig relation must be satisfied. In such a case, the linearization approximation is usually valid, and quantum fluctuations can be treated in a similar way as classical fluctuations, provided the contribution of the vacuum noise from all relevant modes is accounted for.
  13. S. J. Carter, P. D. Drummond, M. D. Reid, and R. M. Shelby, "Squeezing of quantum solitons," Phys. Rev. Lett. 58, 1841-1844 (1987).
    [CrossRef] [PubMed]
  14. H. A. Haus and Y. Lai, "Quantum theory of soliton squeezing: a linearized approach," J. Opt. Soc. Am. B 7, 386-392 (1990).
    [CrossRef]
  15. L. Boivin, F. X. Kärtner, and H. A. Haus, "Analytical solution to the quantum field theory of self-phase modulation with a finite response time," Phys. Rev. Lett. 73, 240-243 (1994).
    [CrossRef] [PubMed]
  16. J. H. Shapiro and L. Boivin, "Raman-noise limit on squeezing in continuous-wave four-wave mixing," Opt. Lett. 20, 925-927 (1995).
    [CrossRef] [PubMed]

2001 (1)

1999 (2)

D. Levandovsky, M. Vasilyev, and P. Kumar, "Perturbation theory of quantum solitons: continuum evolution and optimum squeezing by spectral filtering," Opt. Lett. 24, 43-45 (1999).
[CrossRef]

K. G. Köprülü and O. Aytür, "Analysis of Gaussian-beam degenerate optical parametric amplifiers for the generation of quadrature-squeezed states," Phys. Rev. A 60, 4122-4134 (1999).
[CrossRef]

1998 (3)

F. Konig, S. Spalter, I. L. Shumay, A. Sizmann, T. Fauster, and G. Leuchs, "Fibre-optic photon-number squeezing in the normal group-velocity dispersion regime," J. Mod. Opt. 45, 2425-2431 (1998).
[CrossRef]

A. Mecozzi and P. Kumar, "Sub-Poissonian light by spatial soliton filtering," Quantum Semiclassic. Opt. 10, L21-L26 (1998).
[CrossRef]

S. Spalter, M. Burk, U. Strossner, A. Sizmann, and G. Leuchs, "Propagation of quantum properties of sub-picosecond solitons in a fiber," Opt. Express 2, 77-83 (1998).
[CrossRef]

1997 (1)

S. Spalter, M. Burk, U. Strossner, M. Bohm, A. Sizmann, and G. Leuchs, "Photon number squeezing of spectrally filtered sub-picosecond optical solitons," Europhys. Lett. 38, 335-340 (1997).
[CrossRef]

1995 (1)

1994 (1)

L. Boivin, F. X. Kärtner, and H. A. Haus, "Analytical solution to the quantum field theory of self-phase modulation with a finite response time," Phys. Rev. Lett. 73, 240-243 (1994).
[CrossRef] [PubMed]

1990 (2)

H. A. Haus and Y. Lai, "Quantum theory of soliton squeezing: a linearized approach," J. Opt. Soc. Am. B 7, 386-392 (1990).
[CrossRef]

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. van Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

1989 (1)

1987 (1)

S. J. Carter, P. D. Drummond, M. D. Reid, and R. M. Shelby, "Squeezing of quantum solitons," Phys. Rev. Lett. 58, 1841-1844 (1987).
[CrossRef] [PubMed]

1979 (1)

Aytür, O.

K. G. Köprülü and O. Aytür, "Analysis of amplitude-squeezed light generation with Gaussian-beam degenerate optical parametric amplifiers," J. Opt. Soc. Am. B 18, 846-854 (2001).
[CrossRef]

K. G. Köprülü and O. Aytür, "Analysis of Gaussian-beam degenerate optical parametric amplifiers for the generation of quadrature-squeezed states," Phys. Rev. A 60, 4122-4134 (1999).
[CrossRef]

Bohm, M.

S. Spalter, M. Burk, U. Strossner, M. Bohm, A. Sizmann, and G. Leuchs, "Photon number squeezing of spectrally filtered sub-picosecond optical solitons," Europhys. Lett. 38, 335-340 (1997).
[CrossRef]

Boivin, L.

J. H. Shapiro and L. Boivin, "Raman-noise limit on squeezing in continuous-wave four-wave mixing," Opt. Lett. 20, 925-927 (1995).
[CrossRef] [PubMed]

L. Boivin, F. X. Kärtner, and H. A. Haus, "Analytical solution to the quantum field theory of self-phase modulation with a finite response time," Phys. Rev. Lett. 73, 240-243 (1994).
[CrossRef] [PubMed]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, 1992).

Burk, M.

S. Spalter, M. Burk, U. Strossner, A. Sizmann, and G. Leuchs, "Propagation of quantum properties of sub-picosecond solitons in a fiber," Opt. Express 2, 77-83 (1998).
[CrossRef]

S. Spalter, M. Burk, U. Strossner, M. Bohm, A. Sizmann, and G. Leuchs, "Photon number squeezing of spectrally filtered sub-picosecond optical solitons," Europhys. Lett. 38, 335-340 (1997).
[CrossRef]

Carter, S. J.

S. J. Carter, P. D. Drummond, M. D. Reid, and R. M. Shelby, "Squeezing of quantum solitons," Phys. Rev. Lett. 58, 1841-1844 (1987).
[CrossRef] [PubMed]

Drummond, P. D.

S. J. Carter, P. D. Drummond, M. D. Reid, and R. M. Shelby, "Squeezing of quantum solitons," Phys. Rev. Lett. 58, 1841-1844 (1987).
[CrossRef] [PubMed]

Fauster, T.

F. Konig, S. Spalter, I. L. Shumay, A. Sizmann, T. Fauster, and G. Leuchs, "Fibre-optic photon-number squeezing in the normal group-velocity dispersion regime," J. Mod. Opt. 45, 2425-2431 (1998).
[CrossRef]

Hagan, D. J.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. van Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

Haus, H. A.

L. Boivin, F. X. Kärtner, and H. A. Haus, "Analytical solution to the quantum field theory of self-phase modulation with a finite response time," Phys. Rev. Lett. 73, 240-243 (1994).
[CrossRef] [PubMed]

H. A. Haus and Y. Lai, "Quantum theory of soliton squeezing: a linearized approach," J. Opt. Soc. Am. B 7, 386-392 (1990).
[CrossRef]

Kärtner, F. X.

L. Boivin, F. X. Kärtner, and H. A. Haus, "Analytical solution to the quantum field theory of self-phase modulation with a finite response time," Phys. Rev. Lett. 73, 240-243 (1994).
[CrossRef] [PubMed]

Konig, F.

F. Konig, S. Spalter, I. L. Shumay, A. Sizmann, T. Fauster, and G. Leuchs, "Fibre-optic photon-number squeezing in the normal group-velocity dispersion regime," J. Mod. Opt. 45, 2425-2431 (1998).
[CrossRef]

Köprülü, K. G.

K. G. Köprülü and O. Aytür, "Analysis of amplitude-squeezed light generation with Gaussian-beam degenerate optical parametric amplifiers," J. Opt. Soc. Am. B 18, 846-854 (2001).
[CrossRef]

K. G. Köprülü and O. Aytür, "Analysis of Gaussian-beam degenerate optical parametric amplifiers for the generation of quadrature-squeezed states," Phys. Rev. A 60, 4122-4134 (1999).
[CrossRef]

Kumar, P.

Lai, Y.

Leuchs, G.

S. Spalter, M. Burk, U. Strossner, A. Sizmann, and G. Leuchs, "Propagation of quantum properties of sub-picosecond solitons in a fiber," Opt. Express 2, 77-83 (1998).
[CrossRef]

F. Konig, S. Spalter, I. L. Shumay, A. Sizmann, T. Fauster, and G. Leuchs, "Fibre-optic photon-number squeezing in the normal group-velocity dispersion regime," J. Mod. Opt. 45, 2425-2431 (1998).
[CrossRef]

S. Spalter, M. Burk, U. Strossner, M. Bohm, A. Sizmann, and G. Leuchs, "Photon number squeezing of spectrally filtered sub-picosecond optical solitons," Europhys. Lett. 38, 335-340 (1997).
[CrossRef]

Levandovsky, D.

Mecozzi, A.

A. Mecozzi and P. Kumar, "Sub-Poissonian light by spatial soliton filtering," Quantum Semiclassic. Opt. 10, L21-L26 (1998).
[CrossRef]

Miller, D. A. B.

Reid, M. D.

S. J. Carter, P. D. Drummond, M. D. Reid, and R. M. Shelby, "Squeezing of quantum solitons," Phys. Rev. Lett. 58, 1841-1844 (1987).
[CrossRef] [PubMed]

Said, A. A.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. van Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

M. Sheik-Bahae, A. A. Said, and E. W. Stryland, "High-sensitivity, single-beam n2 measurements," Opt. Lett. 14, 955-957 (1989).
[CrossRef] [PubMed]

Shapiro, J. H.

Sheik-Bahae, M.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. van Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

M. Sheik-Bahae, A. A. Said, and E. W. Stryland, "High-sensitivity, single-beam n2 measurements," Opt. Lett. 14, 955-957 (1989).
[CrossRef] [PubMed]

Shelby, R. M.

S. J. Carter, P. D. Drummond, M. D. Reid, and R. M. Shelby, "Squeezing of quantum solitons," Phys. Rev. Lett. 58, 1841-1844 (1987).
[CrossRef] [PubMed]

Shumay, I. L.

F. Konig, S. Spalter, I. L. Shumay, A. Sizmann, T. Fauster, and G. Leuchs, "Fibre-optic photon-number squeezing in the normal group-velocity dispersion regime," J. Mod. Opt. 45, 2425-2431 (1998).
[CrossRef]

Sizmann, A.

F. Konig, S. Spalter, I. L. Shumay, A. Sizmann, T. Fauster, and G. Leuchs, "Fibre-optic photon-number squeezing in the normal group-velocity dispersion regime," J. Mod. Opt. 45, 2425-2431 (1998).
[CrossRef]

S. Spalter, M. Burk, U. Strossner, A. Sizmann, and G. Leuchs, "Propagation of quantum properties of sub-picosecond solitons in a fiber," Opt. Express 2, 77-83 (1998).
[CrossRef]

S. Spalter, M. Burk, U. Strossner, M. Bohm, A. Sizmann, and G. Leuchs, "Photon number squeezing of spectrally filtered sub-picosecond optical solitons," Europhys. Lett. 38, 335-340 (1997).
[CrossRef]

Smith, S. D.

Spalter, S.

S. Spalter, M. Burk, U. Strossner, A. Sizmann, and G. Leuchs, "Propagation of quantum properties of sub-picosecond solitons in a fiber," Opt. Express 2, 77-83 (1998).
[CrossRef]

F. Konig, S. Spalter, I. L. Shumay, A. Sizmann, T. Fauster, and G. Leuchs, "Fibre-optic photon-number squeezing in the normal group-velocity dispersion regime," J. Mod. Opt. 45, 2425-2431 (1998).
[CrossRef]

S. Spalter, M. Burk, U. Strossner, M. Bohm, A. Sizmann, and G. Leuchs, "Photon number squeezing of spectrally filtered sub-picosecond optical solitons," Europhys. Lett. 38, 335-340 (1997).
[CrossRef]

Strossner, U.

S. Spalter, M. Burk, U. Strossner, A. Sizmann, and G. Leuchs, "Propagation of quantum properties of sub-picosecond solitons in a fiber," Opt. Express 2, 77-83 (1998).
[CrossRef]

S. Spalter, M. Burk, U. Strossner, M. Bohm, A. Sizmann, and G. Leuchs, "Photon number squeezing of spectrally filtered sub-picosecond optical solitons," Europhys. Lett. 38, 335-340 (1997).
[CrossRef]

Stryland, E. W.

van Stryland, E. W.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. van Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

Vasilyev, M.

Weaire, D.

Wei, T. H.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. van Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

Wherrett, B. S.

Europhys. Lett. (1)

S. Spalter, M. Burk, U. Strossner, M. Bohm, A. Sizmann, and G. Leuchs, "Photon number squeezing of spectrally filtered sub-picosecond optical solitons," Europhys. Lett. 38, 335-340 (1997).
[CrossRef]

IEEE J. Quantum Electron. (1)

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. van Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

J. Mod. Opt. (1)

F. Konig, S. Spalter, I. L. Shumay, A. Sizmann, T. Fauster, and G. Leuchs, "Fibre-optic photon-number squeezing in the normal group-velocity dispersion regime," J. Mod. Opt. 45, 2425-2431 (1998).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Express (1)

Opt. Lett. (4)

Phys. Rev. A (1)

K. G. Köprülü and O. Aytür, "Analysis of Gaussian-beam degenerate optical parametric amplifiers for the generation of quadrature-squeezed states," Phys. Rev. A 60, 4122-4134 (1999).
[CrossRef]

Phys. Rev. Lett. (2)

S. J. Carter, P. D. Drummond, M. D. Reid, and R. M. Shelby, "Squeezing of quantum solitons," Phys. Rev. Lett. 58, 1841-1844 (1987).
[CrossRef] [PubMed]

L. Boivin, F. X. Kärtner, and H. A. Haus, "Analytical solution to the quantum field theory of self-phase modulation with a finite response time," Phys. Rev. Lett. 73, 240-243 (1994).
[CrossRef] [PubMed]

Quantum Semiclassic. Opt. (1)

A. Mecozzi and P. Kumar, "Sub-Poissonian light by spatial soliton filtering," Quantum Semiclassic. Opt. 10, L21-L26 (1998).
[CrossRef]

Other (2)

R. W. Boyd, Nonlinear Optics (Academic, 1992).

One must keep in mind that the assumption of a purely dispersive nonlinearity is usually valid when working far away from resonances in any medium. Of course, the Kramers-Kronig relation must be satisfied. In such a case, the linearization approximation is usually valid, and quantum fluctuations can be treated in a similar way as classical fluctuations, provided the contribution of the vacuum noise from all relevant modes is accounted for.

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Figures (6)

Fig. 1
Fig. 1

Typical z-scan setup for measuring the third-order nonlinearity.

Fig. 2
Fig. 2

Transmittance as a function of the position of the nonlinear medium for different values of the nonlinear phase shift.

Fig. 3
Fig. 3

Output power versus input power for different values of the normalized aperture radius ( ξ c = 0.5 ) . Both axes are normalized with respect to P 0 , which is the amount of power that gives a nonlinear phase shift of 1 rad ( Φ nl = 1 ) .

Fig. 4
Fig. 4

Fano factor in decibels as a function of the aperture radius for Φ nl = 1 and ξ c = 0.25 .

Fig. 5
Fig. 5

Minimum Fano factor and the required aperture radius as a function of the position of the nonlinear medium ( Φ nl = 4 ) .

Fig. 6
Fig. 6

Minimum Fano factor, required aperture radius, and required position of the nonlinear sample as a function of the nonlinear phase shift Φ nl .

Equations (52)

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E s ( r , t ) = 1 2 A s ( ρ , z ) exp [ i ( k s z ω t ) ] + c.c. ,
A s ( ρ , z ) z + 1 2 i k s 2 A s ( ρ , z ) = i n 2 ω c A s ( ρ , z ) 2 A s ( ρ , z ) ,
A s ( ρ , z ) = A s 0 1 + i 2 z z 0 exp ( ρ 2 w 0 2 1 + i 2 z z 0 ) ,
A s ( r , ξ ) ξ i 2 A s ( r , ξ ) = i n 2 ω z 0 c A s ( r , ξ ) 2 A s ( r , ξ ) ,
A s ( r , ξ ) = A s 0 1 + i 2 ξ exp ( r 2 2 1 + i 2 ξ ) .
ξ in = ( z c l 2 ) z 0 ,
ξ out = ( z c + l 2 ) z 0 .
A s ( r , ξ ) 2 = A s 0 2 1 + 4 ξ c 2 exp ( r 2 1 + 4 ξ c 2 ) ,
A s ( r , ξ ) ξ i 2 A s ( r , ξ ) = i γ 1 + 4 ξ c 2 exp ( r 2 1 + 4 ξ c 2 ) A s ( r , ξ ) ,
γ = n 2 ω z 0 c A s 0 2 .
γ = n 2 I P 4 π λ 2 .
A s ( r , ξ ) = n = 0 A n ( ξ ) G n ( r , ξ ) ,
G n ( r , ξ ) = L n ( r 2 1 + 4 ξ 2 ) 1 1 + i 2 ξ exp ( r 2 2 1 + i 2 ξ ) × exp ( i 2 n tan 1 2 ξ )
2 0 G n ( r , ξ ) G m * ( r , ξ ) r d r = δ m n .
d d ξ A = i γ 1 + 4 ξ c 2 T A ,
A T ( ξ ) = [ A 0 ( ξ ) A 1 ( ξ ) A 2 ( ξ ) ] ,
T m n = [ 2 m n 1 ( m + n ) ! m ! n ! ] exp ( i 2 ( n m ) tan 1 2 ξ c ) .
A ( ξ out ) = exp ( i Φ nl 1 + 4 ξ c 2 T ) A ( ξ in ) ,
Φ nl = γ l z 0
A ( ξ in ) = [ 1 0 0 ] T .
t = 0 r a A s ( r , ξ d ) 2 r d r 0 r a G 0 ( r , ξ d ) 2 r d r ,
S = r a 2 ( 1 + 4 ξ d 2 ) ,
tan 1 ( 2 ξ d ) = π 2 .
t = n = 0 ( 1 ) n A n ( ξ out ) 2 .
P out = [ 1 exp ( 2 S 2 ) ] t ( Φ nl , S ) P in ,
d P out d P in = P out P in .
d P out d P in < P out P in ,
A ̂ s ( r , ξ ) ξ i 2 A ̂ s ( r , ξ ) = i n 2 ω z 0 c A ̂ s ( r , ξ ) A ̂ s ( r , ξ ) A ̂ s ( r , ξ ) ,
A ̂ s = A s + Δ A ̂ s ,
Δ A ̂ s ( r , ξ ) ξ i 2 Δ A ̂ s ( r , ξ ) = i 2 n 2 ω z 0 c A s 2 Δ A ̂ s ( r , ξ ) + i n 2 ω z 0 c ( A s ) 2 Δ A ̂ s ( r , ξ ) .
( A s ( r , ξ ) ) 2 = A s ( r , ξ ) 2 exp ( i 2 n 2 ω z 0 ( ξ ξ in ) c A s ( r , ξ ) 2 ) .
Δ A ̂ s ( r , ξ ) = n = 0 Δ A ̂ n ( ξ ) G n ( r , ξ ) ,
d d ξ Δ A ̂ = i 2 γ 1 + 4 ξ c 2 T Δ A ̂ + i γ 1 + 4 ξ c 2 U exp ( i 2 γ ( ξ ξ in ) 1 + 4 ξ c 2 T ) Δ A ̂ ,
U m n = [ 2 m n 1 ( m + n ) ! m ! n ! ] exp ( i 2 ( m + n ) tan 1 2 ξ c ) .
Δ A ̂ = exp ( i γ ( ξ ξ in ) 1 + 4 ξ c 2 T ) Δ A ̂ .
d d ξ Δ A ̂ = i γ 1 + 4 ξ c 2 T Δ A ̂ + i γ 1 + 4 ξ c 2 U Δ A ̂ .
Δ A ̂ ( ξ ) = M ( ξ ) Δ A ̂ ( ξ in ) + N ( ξ ) Δ A ̂ ( ξ in ) ,
d 2 d ξ 2 M = d 2 d ξ 2 N = 0 .
M ( ξ in ) = I ,
N ( ξ in ) = 0 ,
M ( ξ out ) = I + i Φ n l 1 + 4 ξ c 2 T ,
N ( ξ out ) = i Φ n l 1 + 4 ξ c 2 U .
Δ A ̂ ( ξ out ) = M Δ A ̂ ( ξ in ) + N Δ A ̂ ( ξ in ) ,
M = exp ( i Φ n l 1 + 4 ξ c 2 T ) ( I + i Φ n l 1 + 4 ξ c 2 T ) ,
N = exp ( i Φ n l 1 + 4 ξ c 2 T ) ( i Φ n l 1 + 4 ξ c 2 U ) .
Δ A ̂ n ( ξ in ) = c ̂ n ,
n ̂ = 0 A ̂ s ( r , ξ ) A ̂ s ( r , ξ ) r d r .
n ̂ n = 0 A n 2 ,
Δ n ̂ n = 0 A n Δ A ̂ n + A n * Δ A ̂ n .
Δ n ̂ = n = 0 [ ( m = 0 q m * M m n + q m N m n * ) c ̂ n + ( m = 0 q m * N m n + q m M m n * ) c ̂ n ] ,
F = ( Δ n ̂ ) 2 n ̂ = n = 0 m = 0 q m * M m n + q m N m n * 2 n = 0 q n 2 .
q n = 0 r a A s ( r , ξ d ) G n * ( r , ξ d ) r d r .

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