Abstract

We experimentally investigate the gain response of a frequency-degenerate but polarization-nondegenerate traveling-wave optical parametric amplifier that consists of a type II phase-matched potassium titanyl phosphate crystal pumped by a frequency-doubled Q-switched mode-locked Nd:YAG laser. Both the optical phase-sensitive and phase-insensitive configurations of the parametric amplifier are studied. Experimental results are in excellent agreement with the theory of an optical parametric amplifier when the Gaussian-beam nature of the various fields is taken into account. In the phase-sensitive configuration a gain of >100 (20 dB) could be easily obtained in the amplified quadrature, which is limited only by the available pump power. Because of gain-induced diffraction and phase fluctuations, however, maximum deamplification in the orthogonal quadrature is limited to <0.5 (-3 dB).

© 1997 Optical Society of America

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  1. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interaction between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
    [CrossRef]
  2. G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
    [CrossRef]
  3. G. A. Massey and J. C. Johnson, “Gain limitations in optical parametric amplifiers,” IEEE J. Quantum Electron. QE-15, 201–203 (1979).
    [CrossRef]
  4. R. A. Baumgartner and R. L. Byer, “Optical parametric amplification,” IEEE J. Quantum Electron. QE-15, 432–444 (1979).
    [CrossRef]
  5. S. X. Dou, D. Josse, and J. Zyss, “Comparison of collinear and one-beam noncritical noncollinear phase matching in optical parametric amplification,” J. Opt. Soc. Am. B 9, 1312–1319 (1992).
    [CrossRef]
  6. R. E. Slusher, P. Grangier, A. LaPorta, B. Yurke, and M. J. Potasek, “Pulsed squeezed light,” Phys. Rev. Lett. 59, 2566–2569 (1987); P. Kumar, O. Aytür, and J. Huang, “Squeezed light generation with an incoherent pump,” Phys. Rev. Lett. 64, 1015–1018 (1990); T. Hirano and M. Matsuoka, “Generation of broadband squeezed states pumped by cw mode-locked pulses,” Appl. Phys. B APPCDL 55, 233–241 (1992); P. D. Townsend and R. Loudon, “Quantum-noise reduction at frequencies up to 0.5 GHz using pulsed parametric amplification,” Phys. Rev. A PLRAAN 45, 458–467 (1992); D. T. Smithey, M. Beck, and M. G. Raymer, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. PRLTAO 70, 1244–1247 (1993).
    [CrossRef] [PubMed]
  7. C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D 26, 1817–1839 (1982).
    [CrossRef]
  8. Z. Y. Ou, S. F. Pereira, and H. J. Kimble, “Quantum noise reduction in optical amplification,” Phys. Rev. Lett. 70, 3239–3242 (1993); J. A. Levenson, I. Abram, T. Rivera, and P. Grangier, “Reduction of quantum-noise in optical parametric amplification,” J. Opt. Soc. Am. B 10, 2233–2238 (1993).
    [CrossRef] [PubMed]
  9. K. Bencheikh, O. Lopez, I. Abram, and J. A. Levenson, “Improvement of photodetection quantum efficiency by noiseless optical preamplification,” Appl. Phys. Lett. 66, 399–401 (1995).
    [CrossRef]
  10. A. LaPorta and R. E. Slusher, “Squeezing limits at high parametric gains,” Phys. Rev. A 44, 2013–2022 (1991).
    [CrossRef]
  11. O. Aytür and P. Kumar, “Squeezed-light generation with a mode-locked and Q-switched laser and detection using a matched local oscillator,” Opt. Lett. 17, 529–531 (1992).
    [CrossRef]
  12. C. Kim and P. Kumar, “Quadrature-squeezed light detection using a self-generated matched local oscillator,” Phys. Rev. Lett. 73, 1605–1608 (1994).
    [CrossRef] [PubMed]
  13. C. Kim, R.-D Li, and P. Kumar, “Deamplification response of a traveling-wave phase-sensitive optical parametric-amplifier,” Opt. Lett. 19, 132–134 (1994).
    [CrossRef]
  14. J. Q. Yao and T. S. Fahlen, “Calculations of optimum phase match parameters for the biaxial crystal KTiOPO4,” J. Appl. Phys. 55, 65–68 (1984).
    [CrossRef]
  15. R. W. Boyd, Nonlinear Optics (Academic, New York, 1992), Chap. 2.
  16. C. Kim and P. Kumar, “Equivalence of twin beams and squeezed states in a nondegenerate optical parametric amplifier,” in Vol. 23 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 193; J. A. Levenson, I. Abram, T. Rivera, P. Fayolle, J. C. Garreau, and P. Grangier, “Quantum optical cloning amplifier,” Phys. Rev. Lett. 70, 267–270 (1993).
    [CrossRef] [PubMed]
  17. R.-D. Li, S.-K. Choi, and P. Kumar, “Gaussian-wave theory of sub-Poissonian light generation by means of travelling-wave parametric deamplification,” Quantum Semiclass. Opt. 7, 705–713 (1995);, “Corrigendum,” Quantum Semiclass. Opt. 8, 381 (1996).
    [CrossRef]
  18. The group-velocity mismatch between the signal and the pump pulses in KTP is small enough that, for the ~120-ps signal and the ~85-ps pump pulse widths, there is negligible change in their overlap as they propagate through the KTP crystal.
  19. A. Yariv, Optical Electronics (Saunders, Philadelphia, Pa., 1991), Chap. 8.

1995 (1)

K. Bencheikh, O. Lopez, I. Abram, and J. A. Levenson, “Improvement of photodetection quantum efficiency by noiseless optical preamplification,” Appl. Phys. Lett. 66, 399–401 (1995).
[CrossRef]

1994 (2)

C. Kim and P. Kumar, “Quadrature-squeezed light detection using a self-generated matched local oscillator,” Phys. Rev. Lett. 73, 1605–1608 (1994).
[CrossRef] [PubMed]

C. Kim, R.-D Li, and P. Kumar, “Deamplification response of a traveling-wave phase-sensitive optical parametric-amplifier,” Opt. Lett. 19, 132–134 (1994).
[CrossRef]

1992 (2)

1991 (1)

A. LaPorta and R. E. Slusher, “Squeezing limits at high parametric gains,” Phys. Rev. A 44, 2013–2022 (1991).
[CrossRef]

1984 (1)

J. Q. Yao and T. S. Fahlen, “Calculations of optimum phase match parameters for the biaxial crystal KTiOPO4,” J. Appl. Phys. 55, 65–68 (1984).
[CrossRef]

1982 (1)

C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D 26, 1817–1839 (1982).
[CrossRef]

1979 (2)

G. A. Massey and J. C. Johnson, “Gain limitations in optical parametric amplifiers,” IEEE J. Quantum Electron. QE-15, 201–203 (1979).
[CrossRef]

R. A. Baumgartner and R. L. Byer, “Optical parametric amplification,” IEEE J. Quantum Electron. QE-15, 432–444 (1979).
[CrossRef]

1968 (1)

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
[CrossRef]

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interaction between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Abram, I.

K. Bencheikh, O. Lopez, I. Abram, and J. A. Levenson, “Improvement of photodetection quantum efficiency by noiseless optical preamplification,” Appl. Phys. Lett. 66, 399–401 (1995).
[CrossRef]

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interaction between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Aytür, O.

Baumgartner, R. A.

R. A. Baumgartner and R. L. Byer, “Optical parametric amplification,” IEEE J. Quantum Electron. QE-15, 432–444 (1979).
[CrossRef]

Bencheikh, K.

K. Bencheikh, O. Lopez, I. Abram, and J. A. Levenson, “Improvement of photodetection quantum efficiency by noiseless optical preamplification,” Appl. Phys. Lett. 66, 399–401 (1995).
[CrossRef]

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interaction between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Boyd, G. D.

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
[CrossRef]

Byer, R. L.

R. A. Baumgartner and R. L. Byer, “Optical parametric amplification,” IEEE J. Quantum Electron. QE-15, 432–444 (1979).
[CrossRef]

Caves, C. M.

C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D 26, 1817–1839 (1982).
[CrossRef]

Dou, S. X.

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interaction between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Fahlen, T. S.

J. Q. Yao and T. S. Fahlen, “Calculations of optimum phase match parameters for the biaxial crystal KTiOPO4,” J. Appl. Phys. 55, 65–68 (1984).
[CrossRef]

Johnson, J. C.

G. A. Massey and J. C. Johnson, “Gain limitations in optical parametric amplifiers,” IEEE J. Quantum Electron. QE-15, 201–203 (1979).
[CrossRef]

Josse, D.

Kim, C.

C. Kim and P. Kumar, “Quadrature-squeezed light detection using a self-generated matched local oscillator,” Phys. Rev. Lett. 73, 1605–1608 (1994).
[CrossRef] [PubMed]

C. Kim, R.-D Li, and P. Kumar, “Deamplification response of a traveling-wave phase-sensitive optical parametric-amplifier,” Opt. Lett. 19, 132–134 (1994).
[CrossRef]

Kleinman, D. A.

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
[CrossRef]

Kumar, P.

LaPorta, A.

A. LaPorta and R. E. Slusher, “Squeezing limits at high parametric gains,” Phys. Rev. A 44, 2013–2022 (1991).
[CrossRef]

Levenson, J. A.

K. Bencheikh, O. Lopez, I. Abram, and J. A. Levenson, “Improvement of photodetection quantum efficiency by noiseless optical preamplification,” Appl. Phys. Lett. 66, 399–401 (1995).
[CrossRef]

Li, R.-D

Lopez, O.

K. Bencheikh, O. Lopez, I. Abram, and J. A. Levenson, “Improvement of photodetection quantum efficiency by noiseless optical preamplification,” Appl. Phys. Lett. 66, 399–401 (1995).
[CrossRef]

Massey, G. A.

G. A. Massey and J. C. Johnson, “Gain limitations in optical parametric amplifiers,” IEEE J. Quantum Electron. QE-15, 201–203 (1979).
[CrossRef]

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interaction between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Slusher, R. E.

A. LaPorta and R. E. Slusher, “Squeezing limits at high parametric gains,” Phys. Rev. A 44, 2013–2022 (1991).
[CrossRef]

Yao, J. Q.

J. Q. Yao and T. S. Fahlen, “Calculations of optimum phase match parameters for the biaxial crystal KTiOPO4,” J. Appl. Phys. 55, 65–68 (1984).
[CrossRef]

Zyss, J.

Appl. Phys. Lett. (1)

K. Bencheikh, O. Lopez, I. Abram, and J. A. Levenson, “Improvement of photodetection quantum efficiency by noiseless optical preamplification,” Appl. Phys. Lett. 66, 399–401 (1995).
[CrossRef]

IEEE J. Quantum Electron. (2)

G. A. Massey and J. C. Johnson, “Gain limitations in optical parametric amplifiers,” IEEE J. Quantum Electron. QE-15, 201–203 (1979).
[CrossRef]

R. A. Baumgartner and R. L. Byer, “Optical parametric amplification,” IEEE J. Quantum Electron. QE-15, 432–444 (1979).
[CrossRef]

J. Appl. Phys. (2)

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
[CrossRef]

J. Q. Yao and T. S. Fahlen, “Calculations of optimum phase match parameters for the biaxial crystal KTiOPO4,” J. Appl. Phys. 55, 65–68 (1984).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Lett. (2)

Phys. Rev. (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interaction between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Phys. Rev. A (1)

A. LaPorta and R. E. Slusher, “Squeezing limits at high parametric gains,” Phys. Rev. A 44, 2013–2022 (1991).
[CrossRef]

Phys. Rev. D (1)

C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D 26, 1817–1839 (1982).
[CrossRef]

Phys. Rev. Lett. (1)

C. Kim and P. Kumar, “Quadrature-squeezed light detection using a self-generated matched local oscillator,” Phys. Rev. Lett. 73, 1605–1608 (1994).
[CrossRef] [PubMed]

Other (7)

R. W. Boyd, Nonlinear Optics (Academic, New York, 1992), Chap. 2.

C. Kim and P. Kumar, “Equivalence of twin beams and squeezed states in a nondegenerate optical parametric amplifier,” in Vol. 23 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 193; J. A. Levenson, I. Abram, T. Rivera, P. Fayolle, J. C. Garreau, and P. Grangier, “Quantum optical cloning amplifier,” Phys. Rev. Lett. 70, 267–270 (1993).
[CrossRef] [PubMed]

R.-D. Li, S.-K. Choi, and P. Kumar, “Gaussian-wave theory of sub-Poissonian light generation by means of travelling-wave parametric deamplification,” Quantum Semiclass. Opt. 7, 705–713 (1995);, “Corrigendum,” Quantum Semiclass. Opt. 8, 381 (1996).
[CrossRef]

The group-velocity mismatch between the signal and the pump pulses in KTP is small enough that, for the ~120-ps signal and the ~85-ps pump pulse widths, there is negligible change in their overlap as they propagate through the KTP crystal.

A. Yariv, Optical Electronics (Saunders, Philadelphia, Pa., 1991), Chap. 8.

Z. Y. Ou, S. F. Pereira, and H. J. Kimble, “Quantum noise reduction in optical amplification,” Phys. Rev. Lett. 70, 3239–3242 (1993); J. A. Levenson, I. Abram, T. Rivera, and P. Grangier, “Reduction of quantum-noise in optical parametric amplification,” J. Opt. Soc. Am. B 10, 2233–2238 (1993).
[CrossRef] [PubMed]

R. E. Slusher, P. Grangier, A. LaPorta, B. Yurke, and M. J. Potasek, “Pulsed squeezed light,” Phys. Rev. Lett. 59, 2566–2569 (1987); P. Kumar, O. Aytür, and J. Huang, “Squeezed light generation with an incoherent pump,” Phys. Rev. Lett. 64, 1015–1018 (1990); T. Hirano and M. Matsuoka, “Generation of broadband squeezed states pumped by cw mode-locked pulses,” Appl. Phys. B APPCDL 55, 233–241 (1992); P. D. Townsend and R. Loudon, “Quantum-noise reduction at frequencies up to 0.5 GHz using pulsed parametric amplification,” Phys. Rev. A PLRAAN 45, 458–467 (1992); D. T. Smithey, M. Beck, and M. G. Raymer, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. PRLTAO 70, 1244–1247 (1993).
[CrossRef] [PubMed]

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Figures (10)

Fig. 1
Fig. 1

Schematic of the experimental setup to measure the gain response of the traveling-wave OPA. In the PSA configuration the pump phase is controlled by applying a voltage to the PZT.

Fig. 2
Fig. 2

Schematic of the pump-beam (solid curves) and the signal-beam (dashed curves) phase fronts plotted as a function of z inside the nonlinear medium. The bounding hyperbolas mark the contours of the field radii of the two beams at the ±e-1 level.

Fig. 3
Fig. 3

(a) Q-switched envelopes at the input and the output of the OPA. The pump-beam phase is adjusted to either amplify or deamplify the input signal beam. The pump power was set to obtain a peak PSA gain of 6. (b) Peak gain of the OPA plotted as a function of the pump-beam phase for GPSA4 in both the PSA and the PIA configurations.

Fig. 4
Fig. 4

Dependence of the peak OPA gain on pump power for KTP crystals of length 1.57 mm (triangles), 3.25 mm (diamonds), and 5.21 mm (squares). The OPA is configured as a PSA in (a) and as a PIA in (b) with z0=41.1 mm. In both (a) and (b) a relative pump-power value of 100 corresponds to the mode-locked pump-pulse peak power of 46 kW corresponding to an intensity of 1.5 GW/cm2 at the beam waist in the crystal. The solid and the dashed curves are theoretical fits to Eqs. (24) and (27), respectively, in (a) and Eqs. (26) and (29), respectively, in (b).

Fig. 5
Fig. 5

PSA gain of the OPA plotted as a function of the PIA gain. The solid line is a theoretical fit to the parametric plot of Eqs. (24) and (26), the dashed line to that of Eqs. (27) and (29), and the dotted line to that of Eqs. (11) and (16). The dotted–dashed line is a plot of Eq. (30) without any adjustable parameter. The various theoretical plots are virtually indistinguishable from one another.

Fig. 6
Fig. 6

Dependence of the peak deamplification factor, GPSD, on pump power for KTP crystals of length (a) 1.57 mm, (b) 3.25 mm, and (c) 5.21 mm. The OPA is configured as a PSA with z0 =41.1 mm. The pump-power calibration is the same as that in Fig. 4. In each case the solid curves are theoretical fits to Eq. (25) for the measured values of l/z0 and δθ21/2, whereas the dashed curves are fits to Eq. (28) for the measured values of δθ21/2. See text for description of the dotted–dashed and dotted curves.

Fig. 7
Fig. 7

(a) Representative one-dimensional spatial profiles for the signal beam. The dashed curve is the input profile, and the solid curves are profiles of the amplified and the deamplified quadratures, as marked, for a PSA gain of 8. The fullwidth at half-maximum, shown by the horizontal dotted line, for the amplified quadrature is much smaller (by almost a factor of 2) than that for the input signal. (b) A qualitative sketch of the gain-induced diffraction phenomenon that is responsible for the decrease in the peak deamplification factor.

Fig. 8
Fig. 8

Dependence of the peak OPA gain on pump power for KTP crystals of length 1.57 mm, 3.25 mm, and 5.21 mm. The OPA is configured as a PSA in (a) and as a PIA in (b) with z0 9.19 mm. The solid and the dashed curves are theoretical fits to Eqs. (24) and Eq. (27), respectively, in (a) and Eqs. (26) and Eq. (29), respectively, in (b).

Fig. 9
Fig. 9

Dependence of the peak deamplification factor on pump power for KTP crystals of length (a) 1.57 mm, (b) 3.25 mm, and (c) 5.21 mm. The OPA is configured as a PSA with z0 9.19 mm. The solid curves are theoretical fits taking into account the spatio-temporal profile of the pulses in the Gaussian-wave theory.

Fig. 10
Fig. 10

(a) PSA gain of the OPA plotted as a function of the PIA gain for z09.19 mm. (b) An expanded plot of the low-gain data in (a). In both cases, the solid curve is a plot of Eq. (30) without any adjustable parameter.

Equations (45)

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Ez+12ik2E=KEpE* exp(iϕp),
Ez+12ik2E=KEpE* exp(iϕp).
E(E+E)/2,
E(E-E)/2,
Ez+12ik2E=K exp(iϕp)EpE*,
Ez+12ik2E=K exp[i(ϕp+π)]EpE*,
Ep(ρ, z)=Ep01+2iz/z0exp-ρ2/2a021+2iz/z0,
E()(ρ, z)=E0(0)(ρ, z)1+2iz/z0exp-ρ2/4a021+2iz/z0,
E0(ρ, l/2)=12exp(ζ)+exp(-ζ)+[exp(ζ)-exp(-ζ)]exp(iθ)-i lz0{f1(ζ, Φ)+f1(-ζ,-Φ)+[f1(-ζ,-Φ)-f1(ζ, Φ)]exp(iθ)}+lz02{f2(ζ, Φ)×[1+exp(iθ)]+f2(-ζ,-Φ)×[1-exp(iθ)]}E0(-l/2),
ζ=KEp0l exp(-ρ2/2a02),
Φ=KEp0l,
f1(ζ, Φ)=exp(ζ)1ζln Φζ-12-12ζ+exp(-ζ)12ζ+32+2ζ-2-1ζln Φζ,
f2(ζ, Φ)=exp(-ζ)34-12ζ+34ζ2+ln Φζζ-2+4ζ+6ζ2+ln Φζ2+2ζ+1ζ2+exp(ζ)-316ζ-134+2ζ-34ζ2+ln Φζ8ζ2+23ζ-2+8ζ-6ζ2-ln Φζ85ζ3+10ζ2+283ζ+1ζ2,
E0(ρ,-l/2)=E0(ρ,-l/2)=Ein2.
|E0(ρ, l/2)|2exp(2ζ)+lz02[f1(-ζ,-Φ)2+2 exp(ζ)f2(ζ, Φ)]|E0(-l/2)|2,
|E0(ρ, l/2)|2exp(2ζ)+lz02[f1(-ζ,-Φ)2+2 exp(ζ)f2(ζ, Φ)]+δθ24[exp(-2ζ)-exp(2ζ)]×|E0(-l/2)|2.
GPSA=|E(ρ, l/2)|2dρ|E(ρ,-l/2)|2dρexp(2Φ)-12Φ.
|E0(ρ, l/2)|2exp(-2ζ)+lz02[f1(ζ, Φ)2+2 exp(-ζ)f2(-ζ,-Φ)]+δθ24[exp(2ζ)-exp(-2ζ)]×|E0(-l/2)|2.
GPSD=|E(ρ, l/2)|2dρ|E(ρ,-l/2)|2dρ1-exp(-2Φ)2Φ+1Φlz020Φ[f1(ζ, Φ)2+2 exp(-ζ)f2(-ζ,-Φ)]dζ+δθ22Φsinh2 Φ.
E0(ρ, l/2)Eincosh(ζ)-i2lz0[f1(ζ, Φ)+f1(-ζ,-Φ)]+12lz02×[f2(ζ, Φ)+f2(-ζ,-Φ)],
E0(ρ, l/2)Ein exp(iθ)sinh(ζ)-i2lz0×[f1(-ζ,-Φ)-f1(ζ, Φ)]+12lz02×[f2(ζ, Φ)-f2(-ζ,-Φ)].
|E0(ρ, l/2)|2|Ein|2cosh2(ζ)+14lz02{[f1(ζ, Φ)+f1(-ζ,-Φ)]2+4 cosh(ζ)×[f2(ζ, Φ)+f2(-ζ,-Φ)]},
|E0(ρ, l/2)|2|Ein|2sinh2(ζ)+14lz02{[f1(ζ, Φ)-f1(-ζ,-Φ)]2+4 sinh(ζ)×[f2(ζ, Φ)-f2(-ζ,-Φ)]},
GPIA=|E(ρ, l/2)|2dρ|E(ρ,-l/2)|2dρ12+14exp(2Φ)-exp(-2Φ)2Φ.
E(l/2)=E(-l/2)cosh(Φ)+E*(-l/2)exp(iϕp)sinh(Φ),
E(l/2)=E*(-l/2)exp(iϕp)sinh(Φ)+E(-l/2)cosh(Φ).
GPSA=exp(2Φ),
GPSD=exp(-2Φ),
GPIA=12+14[exp(2Φ)+exp(-2Φ)].
GPSD=exp(-2Φ)+δθ24[exp(2Φ)-exp(-2Φ)].
Ep(ρ, z)=Ep01+2iz/z0exp-ρ2/2a021+2iz/z0-t2,
E()(ρ, z)=E0(0)(ρ, z)1+2iz/z0exp-ρ2/4a021+2iz/z0-t22,
GPSA1π1/2aI1/20{exp[2aI1/2 exp(-t2)]-1}dt,
GPSD1π1/2aI1/20{1-exp[-2aI1/2 exp(-t2)]}dt+2lz0200Φ[f1(ζ, Φ; t)2+2 exp[-ζ exp(-t2)]f2(-ζ,-Φ; t)]×exp(-t2)dζdt+δθ240{exp[2aI1/2 exp(-t2)]+exp[-2aI1/2 exp(-t2)]-2}dt,
GPIA12+14π1/2aI1/20{exp[2aI1/2 exp(-t2)]-exp[-2aI1/2 exp(-t2)]}dt.
GPSA=2π1/20 exp[-t2+2aI1/2 exp(-t2)]dt,
GPSD=2π1/20 exp[-t2-2aI1/2 exp(-t2)]dt+δθ240 exp(-t2){exp[2aI1/2 exp(-t2)]-exp[-2aI1/2 exp(-t2)]}dt,
GPIA=12+12π1/20{exp[-t2+2aI1/2 exp(-t2)]+exp[-t2-2aI1/2 exp(-t2)]}dt,
GPSA=2GPIA-1+[(2GPIA-1)2-1]1/2.
GPSD=1π1/2Φ0{1-exp[-2Φ exp(-t2)]}dt+δθ240{exp[2Φ exp(-t2)]+exp[-2Φ exp(-t2)]-2}dt,
δIout=GPSDδIin+Iinδθ24A+δEp0Ep0(B-C),
A=0{exp[2Φ exp(-t2)]+exp[-2Φ exp(-t2)]-2}dt,
B=2π1/20 exp[-t2-2Φ exp(-t2)]dt,
C=1π1/2Φ0{1-exp[-2Φ exp(-t2)]}dt.
δθ21/2δθ41/4=2GPSDA1/2δIout2Iout2-δIin2Iin2-14BGPSD-12 δIp2Ip221/8,

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