Abstract

A linearized perturbation quantum theory of second-order soliton propagation is developed. The theory shows that the quantum fluctuations of photon number, phase, momentum, and position at an arbitrary propagation distance are linear combinations of these fluctuations at zero distance. The evolutions of second-order soliton quantum fluctuations are evaluated and compared with the quantum-fluctuation evolutions of a fundamental soliton. Based on this theory, the squeezing effect of a second-order soliton is studied. It is shown that, like a fundamental soliton, a second-order soliton also exhibits squeezing along propagation when a proper combination of the number and phase operators is detected.

© 1999 Optical Society of America

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  1. A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
    [CrossRef]
  2. L. F. Mollenauer and R. H. Stolen, “The soliton laser,” Opt. Lett. 9, 13–15 (1984).
    [CrossRef] [PubMed]
  3. J. P. Gordon and H. A. Haus, “Random walk of coherently amplified solitons in optical fiber transmission,” Opt. Lett. 11, 665–667 (1986).
    [CrossRef] [PubMed]
  4. H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A 13, 2226–2243 (1976).
    [CrossRef]
  5. J. N. Hollenhorst, “Quantum limits on resonant-mass gravitational-radiation detectors,” Phys. Rev. D 19, 1669–1679 (1979).
    [CrossRef]
  6. T. Hirano and M. Matsuoka, “Broadband squeezing of light by pulse excitation,” Opt. Lett. 15, 1153–1155 (1990).
    [CrossRef] [PubMed]
  7. K. Bergman and H. A. Haus, “Squeezing in fibers with optical pulses,” Opt. Lett. 16, 663–665 (1991).
    [CrossRef] [PubMed]
  8. M. Rosenbluth and R. M. Shelby, “Squeezed optical solitons,” Phys. Rev. Lett. 66, 153–156 (1991).
    [CrossRef]
  9. 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]
  10. P. D. Drummond and S. J. Carter, “Quantum field theory of squeezing in solitons,” J. Opt. Soc. Am. B 4, 1565–1673 (1987).
    [CrossRef]
  11. Y. Lai and H. A. Haus, “Quantum theory of solitons in optical fibers. I. Time-dependent Hartree approximation,” Phys. Rev. A 40, 844–853 (1989).
    [CrossRef] [PubMed]
  12. Y. Lai and H. A. Haus, “Quantum theory of solitons in optical fibers. II. Exact solution,” Phys. Rev. A 40, 854–856 (1989).
    [CrossRef] [PubMed]
  13. H. A. Haus, K. Watanabe, and Y. Yamamoto, “Quantum-nondemolition measurement of optical solitons,” J. Opt. Soc. Am. B 6, 1138–1148 (1989).
    [CrossRef]
  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. Y. Lai, “Quantum theory of soliton propagation: a unified approach based on the linearization approximation,” J. Opt. Soc. Am. B 10, 475–484 (1993).
    [CrossRef]
  16. J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear wave in dispersive media,” Prog. Theor. Phys. Suppl. 55, 284–306 (1974).
    [CrossRef]
  17. G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1995), Chap. 5.
  18. H. A. Haus, F. I. Khatri, W. S. Wang, E. P. Ippen, and K. R. Tamura, “Interaction of soliton with sinusoidal wave packet,” IEEE J. Quantum Electron. 32, 917–924 (1996).
    [CrossRef]
  19. R. H. Stolen, L. F. Mollenauer, and W. J. Tomlinson, “Observation of pulse restoration at the soliton period in optical fibers,” Opt. Lett. 8, 186–188 (1983).
    [CrossRef] [PubMed]
  20. L. F. Mollenauer, R. H. Stolen, J. P. Gordon, and W. J. Tomlinson, “Extreme picosecond pulse narrowing by means of soliton effect in single-mode optical fibers,” Opt. Lett. 8, 289–291 (1983).
    [CrossRef] [PubMed]
  21. J. P. Gordon, “Dispersive perturbations of solitons of the nonlinear Schrodinger equation,” J. Opt. Soc. Am. B 9, 91–97 (1992).
    [CrossRef]
  22. D. J. Kaup, “Perturbation theory for solitons in optical fibers,” Phys. Rev. A 42, 5689–5694 (1990).
    [CrossRef] [PubMed]
  23. K. J. Blow and N. J. Doran, “The asymptotic dispersion of soliton pulses in lossy fibres,” Opt. Commun. 52, 367–370 (1985).
    [CrossRef]
  24. G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1995), Chap. 8.
  25. P. D. Drummond and A. D. Hardman, “Simulation of quantum effects in Raman-active waveguides,” Europhys. Lett. 21, 279–284 (1993).
    [CrossRef]
  26. Y. Lai and S.-S. Yu, “General quantum theory of nonlinear optical-pulse propagation,” Phys. Rev. A 51, 817–829 (1995).
    [CrossRef] [PubMed]

1996 (1)

H. A. Haus, F. I. Khatri, W. S. Wang, E. P. Ippen, and K. R. Tamura, “Interaction of soliton with sinusoidal wave packet,” IEEE J. Quantum Electron. 32, 917–924 (1996).
[CrossRef]

1995 (1)

Y. Lai and S.-S. Yu, “General quantum theory of nonlinear optical-pulse propagation,” Phys. Rev. A 51, 817–829 (1995).
[CrossRef] [PubMed]

1993 (2)

P. D. Drummond and A. D. Hardman, “Simulation of quantum effects in Raman-active waveguides,” Europhys. Lett. 21, 279–284 (1993).
[CrossRef]

Y. Lai, “Quantum theory of soliton propagation: a unified approach based on the linearization approximation,” J. Opt. Soc. Am. B 10, 475–484 (1993).
[CrossRef]

1992 (1)

1991 (2)

K. Bergman and H. A. Haus, “Squeezing in fibers with optical pulses,” Opt. Lett. 16, 663–665 (1991).
[CrossRef] [PubMed]

M. Rosenbluth and R. M. Shelby, “Squeezed optical solitons,” Phys. Rev. Lett. 66, 153–156 (1991).
[CrossRef]

1990 (3)

1989 (3)

H. A. Haus, K. Watanabe, and Y. Yamamoto, “Quantum-nondemolition measurement of optical solitons,” J. Opt. Soc. Am. B 6, 1138–1148 (1989).
[CrossRef]

Y. Lai and H. A. Haus, “Quantum theory of solitons in optical fibers. I. Time-dependent Hartree approximation,” Phys. Rev. A 40, 844–853 (1989).
[CrossRef] [PubMed]

Y. Lai and H. A. Haus, “Quantum theory of solitons in optical fibers. II. Exact solution,” Phys. Rev. A 40, 854–856 (1989).
[CrossRef] [PubMed]

1987 (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]

P. D. Drummond and S. J. Carter, “Quantum field theory of squeezing in solitons,” J. Opt. Soc. Am. B 4, 1565–1673 (1987).
[CrossRef]

1986 (1)

1985 (1)

K. J. Blow and N. J. Doran, “The asymptotic dispersion of soliton pulses in lossy fibres,” Opt. Commun. 52, 367–370 (1985).
[CrossRef]

1984 (1)

1983 (2)

1979 (1)

J. N. Hollenhorst, “Quantum limits on resonant-mass gravitational-radiation detectors,” Phys. Rev. D 19, 1669–1679 (1979).
[CrossRef]

1976 (1)

H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A 13, 2226–2243 (1976).
[CrossRef]

1974 (1)

J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear wave in dispersive media,” Prog. Theor. Phys. Suppl. 55, 284–306 (1974).
[CrossRef]

1973 (1)

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[CrossRef]

Bergman, K.

Blow, K. J.

K. J. Blow and N. J. Doran, “The asymptotic dispersion of soliton pulses in lossy fibres,” Opt. Commun. 52, 367–370 (1985).
[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]

P. D. Drummond and S. J. Carter, “Quantum field theory of squeezing in solitons,” J. Opt. Soc. Am. B 4, 1565–1673 (1987).
[CrossRef]

Doran, N. J.

K. J. Blow and N. J. Doran, “The asymptotic dispersion of soliton pulses in lossy fibres,” Opt. Commun. 52, 367–370 (1985).
[CrossRef]

Drummond, P. D.

P. D. Drummond and A. D. Hardman, “Simulation of quantum effects in Raman-active waveguides,” Europhys. Lett. 21, 279–284 (1993).
[CrossRef]

P. D. Drummond and S. J. Carter, “Quantum field theory of squeezing in solitons,” J. Opt. Soc. Am. B 4, 1565–1673 (1987).
[CrossRef]

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]

Gordon, J. P.

Hardman, A. D.

P. D. Drummond and A. D. Hardman, “Simulation of quantum effects in Raman-active waveguides,” Europhys. Lett. 21, 279–284 (1993).
[CrossRef]

Hasegawa, A.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[CrossRef]

Haus, H. A.

H. A. Haus, F. I. Khatri, W. S. Wang, E. P. Ippen, and K. R. Tamura, “Interaction of soliton with sinusoidal wave packet,” IEEE J. Quantum Electron. 32, 917–924 (1996).
[CrossRef]

K. Bergman and H. A. Haus, “Squeezing in fibers with optical pulses,” Opt. Lett. 16, 663–665 (1991).
[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]

Y. Lai and H. A. Haus, “Quantum theory of solitons in optical fibers. II. Exact solution,” Phys. Rev. A 40, 854–856 (1989).
[CrossRef] [PubMed]

Y. Lai and H. A. Haus, “Quantum theory of solitons in optical fibers. I. Time-dependent Hartree approximation,” Phys. Rev. A 40, 844–853 (1989).
[CrossRef] [PubMed]

H. A. Haus, K. Watanabe, and Y. Yamamoto, “Quantum-nondemolition measurement of optical solitons,” J. Opt. Soc. Am. B 6, 1138–1148 (1989).
[CrossRef]

J. P. Gordon and H. A. Haus, “Random walk of coherently amplified solitons in optical fiber transmission,” Opt. Lett. 11, 665–667 (1986).
[CrossRef] [PubMed]

Hirano, T.

Hollenhorst, J. N.

J. N. Hollenhorst, “Quantum limits on resonant-mass gravitational-radiation detectors,” Phys. Rev. D 19, 1669–1679 (1979).
[CrossRef]

Ippen, E. P.

H. A. Haus, F. I. Khatri, W. S. Wang, E. P. Ippen, and K. R. Tamura, “Interaction of soliton with sinusoidal wave packet,” IEEE J. Quantum Electron. 32, 917–924 (1996).
[CrossRef]

Kaup, D. J.

D. J. Kaup, “Perturbation theory for solitons in optical fibers,” Phys. Rev. A 42, 5689–5694 (1990).
[CrossRef] [PubMed]

Khatri, F. I.

H. A. Haus, F. I. Khatri, W. S. Wang, E. P. Ippen, and K. R. Tamura, “Interaction of soliton with sinusoidal wave packet,” IEEE J. Quantum Electron. 32, 917–924 (1996).
[CrossRef]

Lai, Y.

Y. Lai and S.-S. Yu, “General quantum theory of nonlinear optical-pulse propagation,” Phys. Rev. A 51, 817–829 (1995).
[CrossRef] [PubMed]

Y. Lai, “Quantum theory of soliton propagation: a unified approach based on the linearization approximation,” J. Opt. Soc. Am. B 10, 475–484 (1993).
[CrossRef]

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

Y. Lai and H. A. Haus, “Quantum theory of solitons in optical fibers. II. Exact solution,” Phys. Rev. A 40, 854–856 (1989).
[CrossRef] [PubMed]

Y. Lai and H. A. Haus, “Quantum theory of solitons in optical fibers. I. Time-dependent Hartree approximation,” Phys. Rev. A 40, 844–853 (1989).
[CrossRef] [PubMed]

Matsuoka, M.

Mollenauer, L. F.

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]

Rosenbluth, M.

M. Rosenbluth and R. M. Shelby, “Squeezed optical solitons,” Phys. Rev. Lett. 66, 153–156 (1991).
[CrossRef]

Satsuma, J.

J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear wave in dispersive media,” Prog. Theor. Phys. Suppl. 55, 284–306 (1974).
[CrossRef]

Shelby, R. M.

M. Rosenbluth and R. M. Shelby, “Squeezed optical solitons,” Phys. Rev. Lett. 66, 153–156 (1991).
[CrossRef]

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]

Stolen, R. H.

Tamura, K. R.

H. A. Haus, F. I. Khatri, W. S. Wang, E. P. Ippen, and K. R. Tamura, “Interaction of soliton with sinusoidal wave packet,” IEEE J. Quantum Electron. 32, 917–924 (1996).
[CrossRef]

Tappert, F.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[CrossRef]

Tomlinson, W. J.

Wang, W. S.

H. A. Haus, F. I. Khatri, W. S. Wang, E. P. Ippen, and K. R. Tamura, “Interaction of soliton with sinusoidal wave packet,” IEEE J. Quantum Electron. 32, 917–924 (1996).
[CrossRef]

Watanabe, K.

Yajima, N.

J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear wave in dispersive media,” Prog. Theor. Phys. Suppl. 55, 284–306 (1974).
[CrossRef]

Yamamoto, Y.

Yu, S.-S.

Y. Lai and S.-S. Yu, “General quantum theory of nonlinear optical-pulse propagation,” Phys. Rev. A 51, 817–829 (1995).
[CrossRef] [PubMed]

Yuen, H. P.

H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A 13, 2226–2243 (1976).
[CrossRef]

Appl. Phys. Lett. (1)

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[CrossRef]

Europhys. Lett. (1)

P. D. Drummond and A. D. Hardman, “Simulation of quantum effects in Raman-active waveguides,” Europhys. Lett. 21, 279–284 (1993).
[CrossRef]

IEEE J. Quantum Electron. (1)

H. A. Haus, F. I. Khatri, W. S. Wang, E. P. Ippen, and K. R. Tamura, “Interaction of soliton with sinusoidal wave packet,” IEEE J. Quantum Electron. 32, 917–924 (1996).
[CrossRef]

J. Opt. Soc. Am. B (5)

Opt. Commun. (1)

K. J. Blow and N. J. Doran, “The asymptotic dispersion of soliton pulses in lossy fibres,” Opt. Commun. 52, 367–370 (1985).
[CrossRef]

Opt. Lett. (6)

Phys. Rev. A (5)

H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A 13, 2226–2243 (1976).
[CrossRef]

Y. Lai and H. A. Haus, “Quantum theory of solitons in optical fibers. I. Time-dependent Hartree approximation,” Phys. Rev. A 40, 844–853 (1989).
[CrossRef] [PubMed]

Y. Lai and H. A. Haus, “Quantum theory of solitons in optical fibers. II. Exact solution,” Phys. Rev. A 40, 854–856 (1989).
[CrossRef] [PubMed]

D. J. Kaup, “Perturbation theory for solitons in optical fibers,” Phys. Rev. A 42, 5689–5694 (1990).
[CrossRef] [PubMed]

Y. Lai and S.-S. Yu, “General quantum theory of nonlinear optical-pulse propagation,” Phys. Rev. A 51, 817–829 (1995).
[CrossRef] [PubMed]

Phys. Rev. D (1)

J. N. Hollenhorst, “Quantum limits on resonant-mass gravitational-radiation detectors,” Phys. Rev. D 19, 1669–1679 (1979).
[CrossRef]

Phys. Rev. Lett. (2)

M. Rosenbluth and R. M. Shelby, “Squeezed optical solitons,” Phys. Rev. Lett. 66, 153–156 (1991).
[CrossRef]

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]

Prog. Theor. Phys. Suppl. (1)

J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear wave in dispersive media,” Prog. Theor. Phys. Suppl. 55, 284–306 (1974).
[CrossRef]

Other (2)

G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1995), Chap. 5.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1995), Chap. 8.

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Figures (6)

Fig. 1
Fig. 1

Evolution of the second-order soliton.

Fig. 2
Fig. 2

Covariances of the solitons’ quantum operators versus propagation distance: (a) photon number, (b) phase, (c) momentum, and (d) position. The dashed curves denote the fundamental soliton; the solid curves denote the second-order soliton.

Fig. 3
Fig. 3

Block diagram of the balanced homodyne detector.

Fig. 4
Fig. 4

Optimum detection angle versus propagation distance; the dashed curve denotes the fundamental soliton, and the solid curve denotes the second-order soliton.

Fig. 5
Fig. 5

Same as Fig. 4 but for the optimum squeezing ratio.

Fig. 6
Fig. 6

Ratio of quantum output intensity to classical output intensity from the optimum detector versus propagation distance; the detection angle is obtained from Fig. 4, and n0=1.

Equations (71)

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i zU(z, τ)=-2τ2U(z, τ)-2U*(z, τ)U(z, τ)U(z, τ),
U0(z, τ)=n02×expip0τ+θ0+n024z-p02z×sechn02(τ-T0-2p0 z).
i zUˆ(z, τ)=-2τ2Uˆ(z, τ)-2Uˆ(z, τ)Uˆ(z, τ)Uˆ(z, τ),
[Uˆ(z, τ), Uˆ(z, τ)]=δ(τ-τ),
[Uˆ(z, τ),Uˆ(z, τ)]=[Uˆ(z, τ),Uˆ(z, τ)]=0.
zuˆ(z, τ)=Puˆ(z, τ),
uˆ(z, τ)=uˆ1(z, τ)uˆ2(z, τ),
uˆ1(z, τ)=Re{uˆ(z, τ)},uˆ2(z, τ)=Im{uˆ(z, τ)},
P=0-P1P20,
P1=2τ2-n024+2|U˜0(0, τ)|2,
P2=2τ2-n024+6|U˜0(0, τ)|2.
fn(τ)=fn(τ)0,fθ(τ)=0fθ(τ),
fT(τ)=fT(τ)0,fp(τ)=0fp(τ),
uˆ(z, τ)=Δnˆ(z)fn(τ)+ΔTˆ(z)fT(τ)+Δθˆ(z)fθ(τ)+Δpˆ(z)fp(τ)+continuum.
U1(z, τ)=2n0 cosh32n0(τ-2p0z-T0)+3 exp(i2n02z)coshn02(τ-2p0z-T0)cosh[2n0(τ-2p0z-T0)]+4 cosh[n0(τ-2p0z-T0)]+3 cos(2n02z)exp[ip0(τ-p0z)]×expin024z+θ0,
Uˆ(z, τ)=U˜1(z, τ)+uˆ(z, τ),
zuˆ(z, τ)=i2τ2-n024+4|U˜1(z, τ)|2
×uˆ(z, τ)+i2U12(z, τ)uˆ(z, τ),
[uˆ(z, τ),uˆ(z, τ)]=δ(τ-τ),
[uˆ(z, τ),uˆ(z, τ)]=[uˆ(z, τ),uˆ(z, τ)]=0.
uˆ(z, τ)=Δnˆ0un(z, τ)+Δθˆ0uθ(z, τ)+Δpˆ0up(z, τ)+ΔTˆ0uT(z, τ)+Δuˆc(z, τ).
un(z, τ)=Un0θ0=p0=T0=0=[2F(n02z, n0τ/2)+τn0Fτ(n02z, n0τ/2)+4n02zFz(n02z, n0τ/2)+izn02F(n02z, n0τ/2)]exp(in02z/4),
uθ(z, τ)=Uθ0θ0=p0=T0=0=i2n0F(n02z, n0τ/2)exp(in02z/4),
up(z, τ)=Up0θ0=p0=T0=0=[i2n0τF(n02z, n0τ/2)-2n02zFτ(n02z, n0τ/2)]exp(in02z/4),
uT(z, τ)=UT0θ0=p0=T0=0=-n02Fτ(n02z, n0τ/2)exp(in02z/4),
F(z, τ)=cosh(3τ)+3 exp(i2z)cosh(τ)cosh(4τ)+4 cosh(2τ)+3 cos(2z),
Fτ=Fτ,
Fz=Fz.
u¯(z, τ)=iu(z, τ).
g¯(τ)|f(τ)=-dτ{Re[g¯(τ)]Re[f(τ)]+Im[g¯(τ)]Im[f(τ)]}.
u¯n(0, τ)|uθ(0, τ)=-u¯θ(0, τ)|un(0, τ)=4-dτF2(0, τ),
u¯p(0, τ)|uT(0, τ)=-u¯T(0, τ)|up(0, τ)=4n0-dτF2(0, τ),
u¯x(0, τ)|uy(0, τ)=0,
u¯A(0, τ)|Δuˆc(0, τ)=0;A=n,θ, p, T.
Δnˆ0=Δnˆ(z=0)=u¯θ(0, τ)|uˆ(0, τ)-4-dτF2(0, τ),
Δθˆ0=Δθˆ(z=0)=u¯n(0, τ)|uˆ(0, τ)4-dτF2(0, τ),
Δpˆ0=Δpˆ(z=0)=u¯T(0, τ)|uˆ(0, τ)-4n0-dτF2(0, τ),
ΔTˆ0=ΔTˆ(z=0)=u¯p(0, τ)|uˆ(0, τ)4n0-dτF2(0, τ).
Δnˆ(z)=u¯0(0, τ)|exp(-in02z/4)uˆ(z, τ)-4-dτF2(0, τ),
Δθˆ(z)=u¯n(0, τ)|exp(-in02z/4)uˆ(z, τ)4-dτF2(0, τ),
Δpˆ(z)=u¯T(0, τ)|exp(-in02z/4)uˆ(z, τ)-4n0-dτF2(0, τ),
ΔTˆ(z)=u¯p(0, τ)|exp(-in02z/4)uˆ(z, τ)4n0-dτF2(0, τ).
Δnˆ(z)=1-4-dτF2(0, τ)×[Δnˆ0u¯θ(0, τ)|exp(-in02z/4)un(z, τ)+Δθˆ0u¯θ(0, τ)|exp(-in02z/4)uθ(z, τ)+Δpˆ0u¯θ(0, τ)|exp(-in02z/4)up(z, τ)+ΔTˆ0u¯θ(0, τ)|exp(-in02z/4)uT(z, τ)+u¯θ(0, τ)|exp(-in02z/4)Δuˆc(z, τ)],
Δθˆ(z)=14-dτF2(0, τ)×[Δnˆ0u¯n(0, τ)|exp(-in02z/4)un(z, τ)+Δθˆ0u¯n(0, τ)|exp(-in02z/4)uθ(z, τ)+Δpˆ0u¯n(0, τ)|exp(-in02z/4)up(z, τ)+ΔTˆ0u¯n(0, τ)|exp(-in02z/4)uT(z, τ)+u¯n(0, τ)|exp(-in02z/4)Δuˆc(z, τ)],
Δpˆ(z)=1-4n0-dτF2(0, τ)×[Δnˆ0u¯T(0, τ)|exp(-in02z/4)un(z, τ)+Δθˆ0u¯T(0, τ)|exp(-in02z/4)uθ(z, τ)+Δpˆ0u¯T(0, τ)|exp(-in02z/4)up(z, τ)+ΔTˆ0u¯T(0, τ)|exp(-in02z/4)u¯T(z, τ)+u¯T(0, τ)|exp(-in02z/4)Δuˆc(z, τ)],
ΔTˆ(z)=14n0-dτF2(0, τ)×[Δnˆ0u¯p(0, τ)|exp(-in02z/4)un(z, τ)+Δθˆ0u¯p(0, τ)|exp(-in02z/4)uθ(z, τ)+Δpˆ0u¯p(0, τ)|exp(-in02z/4)up(z, τ)+ΔTˆ0u¯p(0, τ)|exp(-in02z/4)uT(z, τ)+u¯p(0, τ)|exp(-in02z/4)Δuˆc(z, τ)].
Δnˆ(z)Δθˆ(z)Δpˆ(z)ΔTˆ(z)=A11(z)A12(z)A13(z)A14(z)A21(z)A22(z)A23(z)A24(z)A31(z)A32(z)A33(z)A34(z)A41(z)A42(z)A43(z)A44(z)Δnˆ(0)Δθˆ(0)Δpˆ(0)ΔTˆ(0)+Δnˆc(z)Δθˆc(z)Δpˆc(z)ΔTˆc(z),
uˆ(0, τ)uˆ(0, τ)=uˆ(0, τ)uˆ(0, τ)=uˆ(0, τ)uˆ(0, τ)=0,
uˆ(0, τ)uˆ(0, τ)=δ(τ-τ).
Δnˆ02=-dτ|uθ(0, τ)|244-dτF2(0, τ)2,
Δθˆ02=-dτ|un(0, τ)|244-dτF2(0, τ)2,
Δpˆ02=-dτ|uT(0, τ)|244n0-dτF2(0, τ)2,
ΔTˆ02=-dτ|up(0, τ)|244n0-dτF2(0, τ)2.
Δnˆ2(z)Δθˆ2(z)Δpˆ2(z)ΔTˆ2(z)
=|A11(z)|2|A12(z)|2|A13(z)|2|A14(z)|2|A21(z)|2|A22(z)|2|A23(z)|2|A24(z)|2|A31(z)|2|A32(z)|2|A33(z)|2|A34(z)|2|A41(z)|2|A42(z)|2|A43(z)|2|A44(z)|2
×Δnˆ(0)2Δθˆ(0)2Δpˆ(0)2ΔTˆ(0)2.
Mˆ(z)=fL(z, τ)|uˆ(z, τ).
-dτ|fL(z, τ)|2=1.
aˆθL(z)=cos(θL) 12n0Δnˆ(z)+sin(θL)n0Δθˆ(z).
aˆθL(z)=f(z, τ)|uˆ(z, τ),
f(z, τ)=1-dτF2(0, τ)×-cos(θL)8n0u¯θ(0, τ)+sin(θL)n04u¯n(0, τ)×expi n02z4.
fθL(z, τ)=f(z, τ)-dτ|f(z, τ)|21/2,
MˆθL(z)=fθL(z, τ)|uˆ(z, τ)=aˆθL(z)-dτ|f(z, τ)|21/2.
MˆθL2(z)=14-dτF2(0, τ)2A(z)cos2(θL)+B(z)sin2(θL)+C(z)sin(θL)cos(θL)D cos2(θL)+E sin2(θL)+F sin(θL)cos(θL),
A(z)=14n0Δnˆ2(z),
B(z)=n0Δθˆ2(z),
C(z)=A11(z)A21(z)Δnˆ02+A12(z)A22(z)Δθˆ02+A13(z)A23(z)Δpˆ02+A14(z)A24(z)ΔTˆ02,
D=1n0u¯θ(0, τ)|u¯θ(0, τ),
E=4n0u¯n(0, τ)|u¯n(0, τ),
F=-4u¯n(0, τ)|u¯θ(0, τ).
R(z)=MˆθLopt2(z)MˆθLopt2(0)

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