Abstract

A nonorthogonal theory is used to study the non-Hermitian character of the excess noise in gain-guided transient Raman amplifiers. We calculated the output of the gain-guided transient Raman scattering based on a nonorthogonal mode expansion. We found that the excess noise in a transient gain-guided amplifier increases nonlinearly from 0 (and the Petermann factor nonlinearly from 1) and that the evolution depends mainly on the integrated energy of the pump laser pulse.

© 1999 Optical Society of America

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References

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  1. K. Petermann, “Calculated spontaneous emission factor for double-heterostructure injection lasers with gain-induced waveguiding,” IEEE J. Quantum Electron. QE-15, 566–570 (1979).
    [Crossref]
  2. H. A. Haus and S. Kawakami, “On the excess spontaneous emission factor in gain guided lasers amplifiers,” IEEE J. Quantum Electron. QE-21, 63–69 (1985).
    [Crossref]
  3. A. E. Siegman, “Excess spontaneous emission in a non-Hermitian optical system. I. Laser amplifiers,” Phys. Rev. A 39, 1253–1263 (1989).
    [Crossref] [PubMed]
  4. R. A. London, M. Strauss, and M. D. Rosen, “Modal analysis of x-ray laser coherence,” Phys. Rev. Lett. 65, 563–566 (1990).
    [Crossref] [PubMed]
  5. W. A. Hamel and J. P. Woerdman, “Observation of enhanced fundamental linewidth of a laser due to nonorthogonality of its longitudinal eigenmodes,” Phys. Rev. Lett. 13, 1506–1509 (1990).
    [Crossref]
  6. S. J. Kuo, D. T. Smithey, and M. G. Raymer, “Beam-pointing fluctuation in gain guided amplifiers,” Phys. Rev. Lett. 66, 2605–2608 (1991).
    [Crossref] [PubMed]
  7. J. G. Wessel, K. S. Repasky, and J. L. Carlsten, “Competition between spontaneous scattering and stimulated scattering in an injection-seeded Raman amplifier,” Phys. Rev. A 53, 1854–1861 (1996).
    [Crossref] [PubMed]
  8. K. S. Repasky, J. K. Brasscai, and J. G. Wessel, “Influence of gain and index guiding on the mode structure and performance of a Raman amplifier,” Phys. Rev. A 56, 859–869 (1997).
    [Crossref]
  9. P. R. Battle, J. G. Wessel, and J. L. Carlsten, “Mode structure and the noise performance of a gain-guided amplifier,” Phys. Rev. A 50, 2587–2593 (1994).
    [Crossref] [PubMed]
  10. P. R. Battle, J. G. Wessel, and J. L. Carlsten, “Gain-guiding effects in an amplifier with focused gain,” Phys. Rev. A 48, 707–716 (1993).
    [Crossref] [PubMed]
  11. P. R. Battle, J. G. Wessel, and J. L. Carlsten, “Excess noise in a focused gain amplifier,” Phys. Rev. Lett. 70, 1607–1610 (1993).
    [Crossref] [PubMed]
  12. S. Lögl, M. Scherm, and M. Maier, “Stimulated Raman scattering with a Gaussian pump beam in H2 gas,” Phys. Rev. A 52, 657–670 (1995).
    [Crossref]
  13. I. V. Deutsch, J. C. Garrison, and E. M. Wright, “Excess noise in gain-guided amplifiers,” J. Opt. Soc. Am. B 8, 1244–1251 (1991).
    [Crossref]
  14. B. N. Perry, P. Rabinowitz, and P. S. Bomse, “Stimulated Raman scattering with a tightly focused pump beam,” Opt. Commun. 10, 146–148 (1985).
  15. M. G. Raymer and J. Mostowski, “Stimulated Raman scattering: unified treatment of spontaneous initiation and spatial propagation,” Phys. Rev. A 24, 1980–1993 (1981).
    [Crossref]
  16. B. N. Perry, P. Rabinowitz, and M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27, 1989–2002 (1983).
    [Crossref]
  17. M. D. Duncan, R. Mahon, and L. L. Tankersley, “Transient stimulated Raman amplification in hydrogen,” J. Opt. Soc. Am. B 5, 37–52 (1988).
    [Crossref]
  18. C. S. Wang, “Theory of stimulated Raman scattering,” Phys. Rev. 182, 482–494 (1969).
    [Crossref]
  19. R. J. Heeman and H. P. Godfried, “Gain reduction measurement in transient stimulated Raman scattering,” IEEE J. Quantum Electron. 31, 358–364 (1995).
    [Crossref]

1997 (1)

K. S. Repasky, J. K. Brasscai, and J. G. Wessel, “Influence of gain and index guiding on the mode structure and performance of a Raman amplifier,” Phys. Rev. A 56, 859–869 (1997).
[Crossref]

1996 (1)

J. G. Wessel, K. S. Repasky, and J. L. Carlsten, “Competition between spontaneous scattering and stimulated scattering in an injection-seeded Raman amplifier,” Phys. Rev. A 53, 1854–1861 (1996).
[Crossref] [PubMed]

1995 (2)

S. Lögl, M. Scherm, and M. Maier, “Stimulated Raman scattering with a Gaussian pump beam in H2 gas,” Phys. Rev. A 52, 657–670 (1995).
[Crossref]

R. J. Heeman and H. P. Godfried, “Gain reduction measurement in transient stimulated Raman scattering,” IEEE J. Quantum Electron. 31, 358–364 (1995).
[Crossref]

1994 (1)

P. R. Battle, J. G. Wessel, and J. L. Carlsten, “Mode structure and the noise performance of a gain-guided amplifier,” Phys. Rev. A 50, 2587–2593 (1994).
[Crossref] [PubMed]

1993 (2)

P. R. Battle, J. G. Wessel, and J. L. Carlsten, “Gain-guiding effects in an amplifier with focused gain,” Phys. Rev. A 48, 707–716 (1993).
[Crossref] [PubMed]

P. R. Battle, J. G. Wessel, and J. L. Carlsten, “Excess noise in a focused gain amplifier,” Phys. Rev. Lett. 70, 1607–1610 (1993).
[Crossref] [PubMed]

1991 (2)

I. V. Deutsch, J. C. Garrison, and E. M. Wright, “Excess noise in gain-guided amplifiers,” J. Opt. Soc. Am. B 8, 1244–1251 (1991).
[Crossref]

S. J. Kuo, D. T. Smithey, and M. G. Raymer, “Beam-pointing fluctuation in gain guided amplifiers,” Phys. Rev. Lett. 66, 2605–2608 (1991).
[Crossref] [PubMed]

1990 (2)

R. A. London, M. Strauss, and M. D. Rosen, “Modal analysis of x-ray laser coherence,” Phys. Rev. Lett. 65, 563–566 (1990).
[Crossref] [PubMed]

W. A. Hamel and J. P. Woerdman, “Observation of enhanced fundamental linewidth of a laser due to nonorthogonality of its longitudinal eigenmodes,” Phys. Rev. Lett. 13, 1506–1509 (1990).
[Crossref]

1989 (1)

A. E. Siegman, “Excess spontaneous emission in a non-Hermitian optical system. I. Laser amplifiers,” Phys. Rev. A 39, 1253–1263 (1989).
[Crossref] [PubMed]

1988 (1)

1985 (2)

B. N. Perry, P. Rabinowitz, and P. S. Bomse, “Stimulated Raman scattering with a tightly focused pump beam,” Opt. Commun. 10, 146–148 (1985).

H. A. Haus and S. Kawakami, “On the excess spontaneous emission factor in gain guided lasers amplifiers,” IEEE J. Quantum Electron. QE-21, 63–69 (1985).
[Crossref]

1983 (1)

B. N. Perry, P. Rabinowitz, and M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27, 1989–2002 (1983).
[Crossref]

1981 (1)

M. G. Raymer and J. Mostowski, “Stimulated Raman scattering: unified treatment of spontaneous initiation and spatial propagation,” Phys. Rev. A 24, 1980–1993 (1981).
[Crossref]

1979 (1)

K. Petermann, “Calculated spontaneous emission factor for double-heterostructure injection lasers with gain-induced waveguiding,” IEEE J. Quantum Electron. QE-15, 566–570 (1979).
[Crossref]

1969 (1)

C. S. Wang, “Theory of stimulated Raman scattering,” Phys. Rev. 182, 482–494 (1969).
[Crossref]

Battle, P. R.

P. R. Battle, J. G. Wessel, and J. L. Carlsten, “Mode structure and the noise performance of a gain-guided amplifier,” Phys. Rev. A 50, 2587–2593 (1994).
[Crossref] [PubMed]

P. R. Battle, J. G. Wessel, and J. L. Carlsten, “Gain-guiding effects in an amplifier with focused gain,” Phys. Rev. A 48, 707–716 (1993).
[Crossref] [PubMed]

P. R. Battle, J. G. Wessel, and J. L. Carlsten, “Excess noise in a focused gain amplifier,” Phys. Rev. Lett. 70, 1607–1610 (1993).
[Crossref] [PubMed]

Bomse, P. S.

B. N. Perry, P. Rabinowitz, and P. S. Bomse, “Stimulated Raman scattering with a tightly focused pump beam,” Opt. Commun. 10, 146–148 (1985).

Brasscai, J. K.

K. S. Repasky, J. K. Brasscai, and J. G. Wessel, “Influence of gain and index guiding on the mode structure and performance of a Raman amplifier,” Phys. Rev. A 56, 859–869 (1997).
[Crossref]

Carlsten, J. L.

J. G. Wessel, K. S. Repasky, and J. L. Carlsten, “Competition between spontaneous scattering and stimulated scattering in an injection-seeded Raman amplifier,” Phys. Rev. A 53, 1854–1861 (1996).
[Crossref] [PubMed]

P. R. Battle, J. G. Wessel, and J. L. Carlsten, “Mode structure and the noise performance of a gain-guided amplifier,” Phys. Rev. A 50, 2587–2593 (1994).
[Crossref] [PubMed]

P. R. Battle, J. G. Wessel, and J. L. Carlsten, “Excess noise in a focused gain amplifier,” Phys. Rev. Lett. 70, 1607–1610 (1993).
[Crossref] [PubMed]

P. R. Battle, J. G. Wessel, and J. L. Carlsten, “Gain-guiding effects in an amplifier with focused gain,” Phys. Rev. A 48, 707–716 (1993).
[Crossref] [PubMed]

Deutsch, I. V.

Duncan, M. D.

Garrison, J. C.

Godfried, H. P.

R. J. Heeman and H. P. Godfried, “Gain reduction measurement in transient stimulated Raman scattering,” IEEE J. Quantum Electron. 31, 358–364 (1995).
[Crossref]

Hamel, W. A.

W. A. Hamel and J. P. Woerdman, “Observation of enhanced fundamental linewidth of a laser due to nonorthogonality of its longitudinal eigenmodes,” Phys. Rev. Lett. 13, 1506–1509 (1990).
[Crossref]

Haus, H. A.

H. A. Haus and S. Kawakami, “On the excess spontaneous emission factor in gain guided lasers amplifiers,” IEEE J. Quantum Electron. QE-21, 63–69 (1985).
[Crossref]

Heeman, R. J.

R. J. Heeman and H. P. Godfried, “Gain reduction measurement in transient stimulated Raman scattering,” IEEE J. Quantum Electron. 31, 358–364 (1995).
[Crossref]

Kawakami, S.

H. A. Haus and S. Kawakami, “On the excess spontaneous emission factor in gain guided lasers amplifiers,” IEEE J. Quantum Electron. QE-21, 63–69 (1985).
[Crossref]

Kuo, S. J.

S. J. Kuo, D. T. Smithey, and M. G. Raymer, “Beam-pointing fluctuation in gain guided amplifiers,” Phys. Rev. Lett. 66, 2605–2608 (1991).
[Crossref] [PubMed]

Lögl, S.

S. Lögl, M. Scherm, and M. Maier, “Stimulated Raman scattering with a Gaussian pump beam in H2 gas,” Phys. Rev. A 52, 657–670 (1995).
[Crossref]

London, R. A.

R. A. London, M. Strauss, and M. D. Rosen, “Modal analysis of x-ray laser coherence,” Phys. Rev. Lett. 65, 563–566 (1990).
[Crossref] [PubMed]

Mahon, R.

Maier, M.

S. Lögl, M. Scherm, and M. Maier, “Stimulated Raman scattering with a Gaussian pump beam in H2 gas,” Phys. Rev. A 52, 657–670 (1995).
[Crossref]

Mostowski, J.

M. G. Raymer and J. Mostowski, “Stimulated Raman scattering: unified treatment of spontaneous initiation and spatial propagation,” Phys. Rev. A 24, 1980–1993 (1981).
[Crossref]

Newstein, M.

B. N. Perry, P. Rabinowitz, and M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27, 1989–2002 (1983).
[Crossref]

Perry, B. N.

B. N. Perry, P. Rabinowitz, and P. S. Bomse, “Stimulated Raman scattering with a tightly focused pump beam,” Opt. Commun. 10, 146–148 (1985).

B. N. Perry, P. Rabinowitz, and M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27, 1989–2002 (1983).
[Crossref]

Petermann, K.

K. Petermann, “Calculated spontaneous emission factor for double-heterostructure injection lasers with gain-induced waveguiding,” IEEE J. Quantum Electron. QE-15, 566–570 (1979).
[Crossref]

Rabinowitz, P.

B. N. Perry, P. Rabinowitz, and P. S. Bomse, “Stimulated Raman scattering with a tightly focused pump beam,” Opt. Commun. 10, 146–148 (1985).

B. N. Perry, P. Rabinowitz, and M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27, 1989–2002 (1983).
[Crossref]

Raymer, M. G.

S. J. Kuo, D. T. Smithey, and M. G. Raymer, “Beam-pointing fluctuation in gain guided amplifiers,” Phys. Rev. Lett. 66, 2605–2608 (1991).
[Crossref] [PubMed]

M. G. Raymer and J. Mostowski, “Stimulated Raman scattering: unified treatment of spontaneous initiation and spatial propagation,” Phys. Rev. A 24, 1980–1993 (1981).
[Crossref]

Repasky, K. S.

K. S. Repasky, J. K. Brasscai, and J. G. Wessel, “Influence of gain and index guiding on the mode structure and performance of a Raman amplifier,” Phys. Rev. A 56, 859–869 (1997).
[Crossref]

J. G. Wessel, K. S. Repasky, and J. L. Carlsten, “Competition between spontaneous scattering and stimulated scattering in an injection-seeded Raman amplifier,” Phys. Rev. A 53, 1854–1861 (1996).
[Crossref] [PubMed]

Rosen, M. D.

R. A. London, M. Strauss, and M. D. Rosen, “Modal analysis of x-ray laser coherence,” Phys. Rev. Lett. 65, 563–566 (1990).
[Crossref] [PubMed]

Scherm, M.

S. Lögl, M. Scherm, and M. Maier, “Stimulated Raman scattering with a Gaussian pump beam in H2 gas,” Phys. Rev. A 52, 657–670 (1995).
[Crossref]

Siegman, A. E.

A. E. Siegman, “Excess spontaneous emission in a non-Hermitian optical system. I. Laser amplifiers,” Phys. Rev. A 39, 1253–1263 (1989).
[Crossref] [PubMed]

Smithey, D. T.

S. J. Kuo, D. T. Smithey, and M. G. Raymer, “Beam-pointing fluctuation in gain guided amplifiers,” Phys. Rev. Lett. 66, 2605–2608 (1991).
[Crossref] [PubMed]

Strauss, M.

R. A. London, M. Strauss, and M. D. Rosen, “Modal analysis of x-ray laser coherence,” Phys. Rev. Lett. 65, 563–566 (1990).
[Crossref] [PubMed]

Tankersley, L. L.

Wang, C. S.

C. S. Wang, “Theory of stimulated Raman scattering,” Phys. Rev. 182, 482–494 (1969).
[Crossref]

Wessel, J. G.

K. S. Repasky, J. K. Brasscai, and J. G. Wessel, “Influence of gain and index guiding on the mode structure and performance of a Raman amplifier,” Phys. Rev. A 56, 859–869 (1997).
[Crossref]

J. G. Wessel, K. S. Repasky, and J. L. Carlsten, “Competition between spontaneous scattering and stimulated scattering in an injection-seeded Raman amplifier,” Phys. Rev. A 53, 1854–1861 (1996).
[Crossref] [PubMed]

P. R. Battle, J. G. Wessel, and J. L. Carlsten, “Mode structure and the noise performance of a gain-guided amplifier,” Phys. Rev. A 50, 2587–2593 (1994).
[Crossref] [PubMed]

P. R. Battle, J. G. Wessel, and J. L. Carlsten, “Excess noise in a focused gain amplifier,” Phys. Rev. Lett. 70, 1607–1610 (1993).
[Crossref] [PubMed]

P. R. Battle, J. G. Wessel, and J. L. Carlsten, “Gain-guiding effects in an amplifier with focused gain,” Phys. Rev. A 48, 707–716 (1993).
[Crossref] [PubMed]

Woerdman, J. P.

W. A. Hamel and J. P. Woerdman, “Observation of enhanced fundamental linewidth of a laser due to nonorthogonality of its longitudinal eigenmodes,” Phys. Rev. Lett. 13, 1506–1509 (1990).
[Crossref]

Wright, E. M.

IEEE J. Quantum Electron. (3)

K. Petermann, “Calculated spontaneous emission factor for double-heterostructure injection lasers with gain-induced waveguiding,” IEEE J. Quantum Electron. QE-15, 566–570 (1979).
[Crossref]

H. A. Haus and S. Kawakami, “On the excess spontaneous emission factor in gain guided lasers amplifiers,” IEEE J. Quantum Electron. QE-21, 63–69 (1985).
[Crossref]

R. J. Heeman and H. P. Godfried, “Gain reduction measurement in transient stimulated Raman scattering,” IEEE J. Quantum Electron. 31, 358–364 (1995).
[Crossref]

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

Opt. Commun. (1)

B. N. Perry, P. Rabinowitz, and P. S. Bomse, “Stimulated Raman scattering with a tightly focused pump beam,” Opt. Commun. 10, 146–148 (1985).

Phys. Rev. (1)

C. S. Wang, “Theory of stimulated Raman scattering,” Phys. Rev. 182, 482–494 (1969).
[Crossref]

Phys. Rev. A (8)

S. Lögl, M. Scherm, and M. Maier, “Stimulated Raman scattering with a Gaussian pump beam in H2 gas,” Phys. Rev. A 52, 657–670 (1995).
[Crossref]

M. G. Raymer and J. Mostowski, “Stimulated Raman scattering: unified treatment of spontaneous initiation and spatial propagation,” Phys. Rev. A 24, 1980–1993 (1981).
[Crossref]

B. N. Perry, P. Rabinowitz, and M. Newstein, “Wave propagation in media with focused gain,” Phys. Rev. A 27, 1989–2002 (1983).
[Crossref]

A. E. Siegman, “Excess spontaneous emission in a non-Hermitian optical system. I. Laser amplifiers,” Phys. Rev. A 39, 1253–1263 (1989).
[Crossref] [PubMed]

J. G. Wessel, K. S. Repasky, and J. L. Carlsten, “Competition between spontaneous scattering and stimulated scattering in an injection-seeded Raman amplifier,” Phys. Rev. A 53, 1854–1861 (1996).
[Crossref] [PubMed]

K. S. Repasky, J. K. Brasscai, and J. G. Wessel, “Influence of gain and index guiding on the mode structure and performance of a Raman amplifier,” Phys. Rev. A 56, 859–869 (1997).
[Crossref]

P. R. Battle, J. G. Wessel, and J. L. Carlsten, “Mode structure and the noise performance of a gain-guided amplifier,” Phys. Rev. A 50, 2587–2593 (1994).
[Crossref] [PubMed]

P. R. Battle, J. G. Wessel, and J. L. Carlsten, “Gain-guiding effects in an amplifier with focused gain,” Phys. Rev. A 48, 707–716 (1993).
[Crossref] [PubMed]

Phys. Rev. Lett. (4)

P. R. Battle, J. G. Wessel, and J. L. Carlsten, “Excess noise in a focused gain amplifier,” Phys. Rev. Lett. 70, 1607–1610 (1993).
[Crossref] [PubMed]

R. A. London, M. Strauss, and M. D. Rosen, “Modal analysis of x-ray laser coherence,” Phys. Rev. Lett. 65, 563–566 (1990).
[Crossref] [PubMed]

W. A. Hamel and J. P. Woerdman, “Observation of enhanced fundamental linewidth of a laser due to nonorthogonality of its longitudinal eigenmodes,” Phys. Rev. Lett. 13, 1506–1509 (1990).
[Crossref]

S. J. Kuo, D. T. Smithey, and M. G. Raymer, “Beam-pointing fluctuation in gain guided amplifiers,” Phys. Rev. Lett. 66, 2605–2608 (1991).
[Crossref] [PubMed]

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Figures (3)

Fig. 1
Fig. 1

Real part of the eigenvalues for the nonorthogonal mode n=0 and l=0, 1, 2. The laser power is 40 MW/cm2, and the Raman dephasing time is T2=l/Γ=1.0 ns. It can be seen that the evolutions of the eigenvalues are nonlinear.

Fig. 2
Fig. 2

Petermann factor for the lowest-order nonorthogonal modes as a function of local time for pulses with rectangular (solid curve), Gaussian (dashed–dotted curve), and triangular (dotted curve) profiles. The three pulses have the same energy (10-4 J). The width (FWHM) of the Gaussian pulse is 200 ps, and the width (FWHM) of the triangular pulse is 250 ps. The Raman dephasing time is T2=1/Γ=1.0 ns.

Fig. 3
Fig. 3

Logarithm of the Stokes energy versus pump energy. The pump pulse with a Gaussian profile has a width of 0.4 ns (FWHM). The solid and the dashed curves were calculated according to the nonorthogonal theory and the plane-wave theory, respectively.

Equations (31)

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E(r, t)=p(r, t)+s(r, t),
p(r, t)=Ep(r, t)exp[i(ωpt-kp·r)]+Ep*(r, t)exp[-i(ωpt-kp·r)],
s(r, t)=Es(-)(r, t)exp{i(ωst-ks·r)}+Es(+)(r, t)exp{-i(ωst-ks·r)}
ddtσij=iωijσij+ik(djkσk-dkiσkj)E,
Enonorthogonal=Eorthogonal
T2-2ikzEs(-)(z, rT, t)=-2kκ2Q+Ep(z, rT, t),
dQ+(z, rT, t)dt=-ΓQ+(z, rT, t)+iκ1Ep*(z, rT, t)×Es(-)(z, rT, t)+F+(z, rT, t),
F+(r, t)F(r, t)=(2Γ/ρ)δ(t-t)δ(r-r),
F(r, t)F(r, t)=0,
Q+(r, 0)Q(r, 0)=1/ρδ(r-r),
Ep=A(t)Ep(z, rT),
=A(t)1ωg(z)1π×exp-rT2ωg2(z)+ikpz+rT22R-tan-1zz0,
Es(-)=(z, rT, t)=A(t)n,lan+(z, t)ϕnl(z, rT, t),
Q+(z, rT, t)=Ep*(z, rT)n,lan+(z, t)Ψnl(z, rT, t).
(T2-2ikz)ϕnl(z, rT, t)=-2kκ2|Ep(z, rT, t)|2Ψnl(z, rT, t)+λnli4kkpωg2(z)ϕnl,
dΨnl(z, rT, t)dt=iκ1|A(t)|2ϕnl,
dΨnl(z, rT, t)dα=iκ1ϕnl.
Ψnl(z, rT, t)=iκ1ϕnlα(t).
(T2-2ikz)ϕnl(z, rT, t)=-2ikκ1κ2|Ep(z, rT, t)|2α(t)ϕnl+λnli4kkpωg2(z)ϕnl.
ϕnl(z, rT, t)=p exp[-2ip tan-1(z/z0)]Upl(z, rT)χn,pl(t),
[2ip-λnl(t)]χnl(t)+μG(t)pχnl(t)Kp,pl(μ)=0,
μ=k/(k+kp)
G(t)=κ1κ2α(t)z0πωg2(0).
d2rTϕml*ϕnj=δljpχm,pl*χn,pj=Bn,ml(t),
d2rTφml*ϕnl=δm,n,
n,lzanl+ϕnl=λnl2kpωg2(z)anl+ϕnl,
n,lψnlanl+t=-Γn,lanl+ψnl+F+.
anl+(θ, t)aml(θ, t)=anl+(θ, 0)aml(θ, 0)exp(-2Γt)×exp[(λnl+λml*)(θ-θ0)]+2Γρ-texp[-2Γ(t-t)]×1α2(t)dt×φnl*1|Ep(z, rT)|2φnld2rT.
anl+(θ, t)aml(θ, t)=1ρa(t)2κ122Bnml*λnl+λml*×exp[(λnl+λml*)(θ-θ0)],
P(θ, t)=c2πd2rTE(-)(θ, t)E(+)(θ, t)=cA2(t)πρα2(t)κ12n,m,l 1λnl+λml*|Bmnl(t)|2×exp[(λnl+λml*)(θ-θ0)].
PcA2(t)2πρα2(t)κ12 Re(λ00)|B000|2 exp[2 Re(λ00)(θ-θ0)].

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