Abstract

Stimulated Raman scattering (SRS) is an important limiting factor for achieving high peak power intensity in fiber amplifier systems. It was proposed to use partially coherent light to increase the SRS threshold significantly. In this paper, the SRS threshold of partially coherent light in silica fibers is investigated by both experiments and theoretical analysis, which show that the SRS threshold is independent on light coherency when the bandwidth of the light is much narrower than 30 nm.

© 2015 Optical Society of America

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References

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  2. V. Fomin, A. Ferin, M. Abramov, I. Samartsev, and V. Gapontsev, “Ultra-high power single mode fiber laser system with non-uniformly configure fiber-to-fiber rod multimode amplifier,” US Patent 20140314106 A1, (2014).
  3. A. Klenke, S. Hädrich, T. Eidam, J. Rothhardt, M. Kienel, S. Demmler, T. Gottschall, J. Limpert, and A. Tünnermann, “22 GW peak-power fiber chirped-pulse-amplification system,” Opt. Lett. 39(24), 6875–6878 (2014).
    [Crossref] [PubMed]
  4. H. Zhang, X. Shen, D. Chen, C. Zheng, P. Yan, and M. Gong, “High energy and high peak power nanosecond pulses generated by fiber amplifier,” IEEE Photonics Technol. Lett. 26(22), 2295–2298 (2014).
    [Crossref]
  5. P. S. Teh, R. J. Lewis, S. U. Alam, and D. J. Richardson, “200 W Diffraction limited, single-polarization, all-fiber picosecond MOPA,” Opt. Express 21(22), 25883–25889 (2013).
    [Crossref] [PubMed]
  6. K. Meng, H. Zhang, M. Liu, D. Li, P. Yan, and M. Gong, “670 kW nanosecond all-fiber super-irradiation pulsed amplifiers at high repetition rates,” J. Opt. 16(10), 105202 (2014).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  13. D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, “Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(3), 035601 (2001).
    [Crossref] [PubMed]
  14. M. Lisak, L. Helczynski, and D. Anderson, “Relation between different formalisms describing partially incoherent wave propagation in nonlinear optical media,” Opt. Commun. 220(4-6), 321–323 (2003).
    [Crossref]
  15. L. Helczynski, D. Anderson, R. Fedele, B. Hall, and M. Lisak, “Propagation of partially incoherent light in nonlinear media via the Wigner transform method,” IEEE J. Sel. Top. Quantum Electron. 8(3), 408–412 (2002).
    [Crossref]
  16. A. Picozzi, J. Garnier, T. Hansson, P. Suret, S. Randoux, G. Millot, and D. N. Christodoulides, “Optical wave turbulence: Towards a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics,” Phys. Rep. 542(1), 1–132 (2014).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]

2014 (4)

A. Klenke, S. Hädrich, T. Eidam, J. Rothhardt, M. Kienel, S. Demmler, T. Gottschall, J. Limpert, and A. Tünnermann, “22 GW peak-power fiber chirped-pulse-amplification system,” Opt. Lett. 39(24), 6875–6878 (2014).
[Crossref] [PubMed]

H. Zhang, X. Shen, D. Chen, C. Zheng, P. Yan, and M. Gong, “High energy and high peak power nanosecond pulses generated by fiber amplifier,” IEEE Photonics Technol. Lett. 26(22), 2295–2298 (2014).
[Crossref]

K. Meng, H. Zhang, M. Liu, D. Li, P. Yan, and M. Gong, “670 kW nanosecond all-fiber super-irradiation pulsed amplifiers at high repetition rates,” J. Opt. 16(10), 105202 (2014).
[Crossref]

A. Picozzi, J. Garnier, T. Hansson, P. Suret, S. Randoux, G. Millot, and D. N. Christodoulides, “Optical wave turbulence: Towards a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics,” Phys. Rep. 542(1), 1–132 (2014).
[Crossref] [PubMed]

2013 (1)

2011 (1)

A. Picozzi and J. Garnier, “Incoherent soliton turbulence in nonlocal nonlinear media,” Phys. Rev. Lett. 107(23), 233901 (2011).
[Crossref] [PubMed]

2006 (1)

2003 (1)

M. Lisak, L. Helczynski, and D. Anderson, “Relation between different formalisms describing partially incoherent wave propagation in nonlinear optical media,” Opt. Commun. 220(4-6), 321–323 (2003).
[Crossref]

2002 (2)

L. Helczynski, D. Anderson, R. Fedele, B. Hall, and M. Lisak, “Propagation of partially incoherent light in nonlinear media via the Wigner transform method,” IEEE J. Sel. Top. Quantum Electron. 8(3), 408–412 (2002).
[Crossref]

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, “Statistical theory for incoherent light propagation in nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(33 Pt 2A), 035602 (2002).
[Crossref] [PubMed]

2001 (1)

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, “Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(3), 035601 (2001).
[Crossref] [PubMed]

1991 (1)

1982 (1)

A. T. Georges, “Theory of stimulated raman scattering in a chaotic incoherent pump field,” Opt. Commun. 41(1), 61–66 (1982).
[Crossref]

Alam, S. U.

Anderson, D.

M. Lisak, L. Helczynski, and D. Anderson, “Relation between different formalisms describing partially incoherent wave propagation in nonlinear optical media,” Opt. Commun. 220(4-6), 321–323 (2003).
[Crossref]

L. Helczynski, D. Anderson, R. Fedele, B. Hall, and M. Lisak, “Propagation of partially incoherent light in nonlinear media via the Wigner transform method,” IEEE J. Sel. Top. Quantum Electron. 8(3), 408–412 (2002).
[Crossref]

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, “Statistical theory for incoherent light propagation in nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(33 Pt 2A), 035602 (2002).
[Crossref] [PubMed]

Bashkansky, M.

Chen, D.

H. Zhang, X. Shen, D. Chen, C. Zheng, P. Yan, and M. Gong, “High energy and high peak power nanosecond pulses generated by fiber amplifier,” IEEE Photonics Technol. Lett. 26(22), 2295–2298 (2014).
[Crossref]

Christodoulides, D. N.

A. Picozzi, J. Garnier, T. Hansson, P. Suret, S. Randoux, G. Millot, and D. N. Christodoulides, “Optical wave turbulence: Towards a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics,” Phys. Rep. 542(1), 1–132 (2014).
[Crossref] [PubMed]

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, “Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(3), 035601 (2001).
[Crossref] [PubMed]

Codemard, C.

Coskun, T. H.

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, “Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(3), 035601 (2001).
[Crossref] [PubMed]

Demmler, S.

Dupriez, P.

Eidam, T.

Eugenieva, E. D.

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, “Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(3), 035601 (2001).
[Crossref] [PubMed]

Fedele, R.

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, “Statistical theory for incoherent light propagation in nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(33 Pt 2A), 035602 (2002).
[Crossref] [PubMed]

L. Helczynski, D. Anderson, R. Fedele, B. Hall, and M. Lisak, “Propagation of partially incoherent light in nonlinear media via the Wigner transform method,” IEEE J. Sel. Top. Quantum Electron. 8(3), 408–412 (2002).
[Crossref]

Garnier, J.

A. Picozzi, J. Garnier, T. Hansson, P. Suret, S. Randoux, G. Millot, and D. N. Christodoulides, “Optical wave turbulence: Towards a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics,” Phys. Rep. 542(1), 1–132 (2014).
[Crossref] [PubMed]

A. Picozzi and J. Garnier, “Incoherent soliton turbulence in nonlocal nonlinear media,” Phys. Rev. Lett. 107(23), 233901 (2011).
[Crossref] [PubMed]

Georges, A. T.

A. T. Georges, “Theory of stimulated raman scattering in a chaotic incoherent pump field,” Opt. Commun. 41(1), 61–66 (1982).
[Crossref]

Gong, M.

H. Zhang, X. Shen, D. Chen, C. Zheng, P. Yan, and M. Gong, “High energy and high peak power nanosecond pulses generated by fiber amplifier,” IEEE Photonics Technol. Lett. 26(22), 2295–2298 (2014).
[Crossref]

K. Meng, H. Zhang, M. Liu, D. Li, P. Yan, and M. Gong, “670 kW nanosecond all-fiber super-irradiation pulsed amplifiers at high repetition rates,” J. Opt. 16(10), 105202 (2014).
[Crossref]

Gottschall, T.

Hädrich, S.

Hall, B.

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, “Statistical theory for incoherent light propagation in nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(33 Pt 2A), 035602 (2002).
[Crossref] [PubMed]

L. Helczynski, D. Anderson, R. Fedele, B. Hall, and M. Lisak, “Propagation of partially incoherent light in nonlinear media via the Wigner transform method,” IEEE J. Sel. Top. Quantum Electron. 8(3), 408–412 (2002).
[Crossref]

Hansson, T.

A. Picozzi, J. Garnier, T. Hansson, P. Suret, S. Randoux, G. Millot, and D. N. Christodoulides, “Optical wave turbulence: Towards a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics,” Phys. Rep. 542(1), 1–132 (2014).
[Crossref] [PubMed]

Helczynski, L.

M. Lisak, L. Helczynski, and D. Anderson, “Relation between different formalisms describing partially incoherent wave propagation in nonlinear optical media,” Opt. Commun. 220(4-6), 321–323 (2003).
[Crossref]

L. Helczynski, D. Anderson, R. Fedele, B. Hall, and M. Lisak, “Propagation of partially incoherent light in nonlinear media via the Wigner transform method,” IEEE J. Sel. Top. Quantum Electron. 8(3), 408–412 (2002).
[Crossref]

Kienel, M.

Kim, J.

Klenke, A.

Lewis, R. J.

Li, D.

K. Meng, H. Zhang, M. Liu, D. Li, P. Yan, and M. Gong, “670 kW nanosecond all-fiber super-irradiation pulsed amplifiers at high repetition rates,” J. Opt. 16(10), 105202 (2014).
[Crossref]

Limpert, J.

Lisak, M.

M. Lisak, L. Helczynski, and D. Anderson, “Relation between different formalisms describing partially incoherent wave propagation in nonlinear optical media,” Opt. Commun. 220(4-6), 321–323 (2003).
[Crossref]

L. Helczynski, D. Anderson, R. Fedele, B. Hall, and M. Lisak, “Propagation of partially incoherent light in nonlinear media via the Wigner transform method,” IEEE J. Sel. Top. Quantum Electron. 8(3), 408–412 (2002).
[Crossref]

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, “Statistical theory for incoherent light propagation in nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(33 Pt 2A), 035602 (2002).
[Crossref] [PubMed]

Liu, M.

K. Meng, H. Zhang, M. Liu, D. Li, P. Yan, and M. Gong, “670 kW nanosecond all-fiber super-irradiation pulsed amplifiers at high repetition rates,” J. Opt. 16(10), 105202 (2014).
[Crossref]

Meng, K.

K. Meng, H. Zhang, M. Liu, D. Li, P. Yan, and M. Gong, “670 kW nanosecond all-fiber super-irradiation pulsed amplifiers at high repetition rates,” J. Opt. 16(10), 105202 (2014).
[Crossref]

Millot, G.

A. Picozzi, J. Garnier, T. Hansson, P. Suret, S. Randoux, G. Millot, and D. N. Christodoulides, “Optical wave turbulence: Towards a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics,” Phys. Rep. 542(1), 1–132 (2014).
[Crossref] [PubMed]

Mitchell, M.

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, “Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(3), 035601 (2001).
[Crossref] [PubMed]

Nilsson, J.

Picozzi, A.

A. Picozzi, J. Garnier, T. Hansson, P. Suret, S. Randoux, G. Millot, and D. N. Christodoulides, “Optical wave turbulence: Towards a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics,” Phys. Rep. 542(1), 1–132 (2014).
[Crossref] [PubMed]

A. Picozzi and J. Garnier, “Incoherent soliton turbulence in nonlocal nonlinear media,” Phys. Rev. Lett. 107(23), 233901 (2011).
[Crossref] [PubMed]

Randoux, S.

A. Picozzi, J. Garnier, T. Hansson, P. Suret, S. Randoux, G. Millot, and D. N. Christodoulides, “Optical wave turbulence: Towards a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics,” Phys. Rep. 542(1), 1–132 (2014).
[Crossref] [PubMed]

Reintjes, J.

Richardson, D. J.

Rothhardt, J.

Sahu, J. K.

Segev, M.

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, “Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(3), 035601 (2001).
[Crossref] [PubMed]

Semenov, V. E.

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, “Statistical theory for incoherent light propagation in nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(33 Pt 2A), 035602 (2002).
[Crossref] [PubMed]

Shen, X.

H. Zhang, X. Shen, D. Chen, C. Zheng, P. Yan, and M. Gong, “High energy and high peak power nanosecond pulses generated by fiber amplifier,” IEEE Photonics Technol. Lett. 26(22), 2295–2298 (2014).
[Crossref]

Suret, P.

A. Picozzi, J. Garnier, T. Hansson, P. Suret, S. Randoux, G. Millot, and D. N. Christodoulides, “Optical wave turbulence: Towards a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics,” Phys. Rep. 542(1), 1–132 (2014).
[Crossref] [PubMed]

Teh, P. S.

Tünnermann, A.

Yan, P.

H. Zhang, X. Shen, D. Chen, C. Zheng, P. Yan, and M. Gong, “High energy and high peak power nanosecond pulses generated by fiber amplifier,” IEEE Photonics Technol. Lett. 26(22), 2295–2298 (2014).
[Crossref]

K. Meng, H. Zhang, M. Liu, D. Li, P. Yan, and M. Gong, “670 kW nanosecond all-fiber super-irradiation pulsed amplifiers at high repetition rates,” J. Opt. 16(10), 105202 (2014).
[Crossref]

Zhang, H.

K. Meng, H. Zhang, M. Liu, D. Li, P. Yan, and M. Gong, “670 kW nanosecond all-fiber super-irradiation pulsed amplifiers at high repetition rates,” J. Opt. 16(10), 105202 (2014).
[Crossref]

H. Zhang, X. Shen, D. Chen, C. Zheng, P. Yan, and M. Gong, “High energy and high peak power nanosecond pulses generated by fiber amplifier,” IEEE Photonics Technol. Lett. 26(22), 2295–2298 (2014).
[Crossref]

Zheng, C.

H. Zhang, X. Shen, D. Chen, C. Zheng, P. Yan, and M. Gong, “High energy and high peak power nanosecond pulses generated by fiber amplifier,” IEEE Photonics Technol. Lett. 26(22), 2295–2298 (2014).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

L. Helczynski, D. Anderson, R. Fedele, B. Hall, and M. Lisak, “Propagation of partially incoherent light in nonlinear media via the Wigner transform method,” IEEE J. Sel. Top. Quantum Electron. 8(3), 408–412 (2002).
[Crossref]

IEEE Photonics Technol. Lett. (1)

H. Zhang, X. Shen, D. Chen, C. Zheng, P. Yan, and M. Gong, “High energy and high peak power nanosecond pulses generated by fiber amplifier,” IEEE Photonics Technol. Lett. 26(22), 2295–2298 (2014).
[Crossref]

J. Opt. (1)

K. Meng, H. Zhang, M. Liu, D. Li, P. Yan, and M. Gong, “670 kW nanosecond all-fiber super-irradiation pulsed amplifiers at high repetition rates,” J. Opt. 16(10), 105202 (2014).
[Crossref]

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

Opt. Commun. (2)

M. Lisak, L. Helczynski, and D. Anderson, “Relation between different formalisms describing partially incoherent wave propagation in nonlinear optical media,” Opt. Commun. 220(4-6), 321–323 (2003).
[Crossref]

A. T. Georges, “Theory of stimulated raman scattering in a chaotic incoherent pump field,” Opt. Commun. 41(1), 61–66 (1982).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rep. (1)

A. Picozzi, J. Garnier, T. Hansson, P. Suret, S. Randoux, G. Millot, and D. N. Christodoulides, “Optical wave turbulence: Towards a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics,” Phys. Rep. 542(1), 1–132 (2014).
[Crossref] [PubMed]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (2)

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, “Statistical theory for incoherent light propagation in nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(33 Pt 2A), 035602 (2002).
[Crossref] [PubMed]

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, “Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(3), 035601 (2001).
[Crossref] [PubMed]

Phys. Rev. Lett. (1)

A. Picozzi and J. Garnier, “Incoherent soliton turbulence in nonlocal nonlinear media,” Phys. Rev. Lett. 107(23), 233901 (2011).
[Crossref] [PubMed]

Other (4)

I. P. G. Photonics, “IPG Photonics successfully tests world’s first 10 kilowatt single-mode production laser.” 2009, http://www.ipgphotonics.com /Collateral/Documents/English-US/PR_FinaI_10kW_SM laser, pdf (2009).

V. Fomin, A. Ferin, M. Abramov, I. Samartsev, and V. Gapontsev, “Ultra-high power single mode fiber laser system with non-uniformly configure fiber-to-fiber rod multimode amplifier,” US Patent 20140314106 A1, (2014).

G. P. Agrawal, Nonlinear Fiber Optics (AcademicPress, 2007), 4th ed.

A. Shirakawa, Y. Suzuki, S. Arisa, M. Chen, C. B. Olausson, J. K. Lyngso, and J. Broeng, “High-peak power pulse amplification by SRS-suppressed photonic bandgap fiber,” in Proceedings of the Pacific Rim Conf. Lasers Electro-Optics, CLEO – Tech. Digest 7–8 (2013).

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Figures (10)

Fig. 1
Fig. 1 Schematic diagram of the experimental setup for measuring SRS threshold of partially coherent light.
Fig. 2
Fig. 2 Output spectrum from third stage amplifier at full pump power of P3 = 320 mW.
Fig. 3
Fig. 3 Signal pulse shapes before 500 m fiber (position A) at pump power of P3 = 140 mW, 220 mW and 320 mW.
Fig. 4
Fig. 4 Signal and Raman pulse energies vary with pump power P3. Blue curve is for signal pulse energy before 500 m fiber (position A). Green curve is for signal pulse energy after 500 m fiber (position B). Orange curve is for SRS pulse energy after 500 m fiber (position B).
Fig. 5
Fig. 5 Input signal pulse shapes (purple) at position A, output signal pulse shapes (red) and output Raman pulse shapes (green) at position B at pump powers of (a) P3 = 140 mW, (b) P3 = 220 mW and (c) P3 = 320 mW.
Fig. 6
Fig. 6 Spectra after 500 m fiber (position B) at different pump powers.
Fig. 7
Fig. 7 SRS critical power Pcr vs. SRS suppression ratio x (dB). Solid circles are for theoretical predictions. Solid triangles are for experimental data.
Fig. 8
Fig. 8 Coherent light pulse width of 10 ns, peak power of 30 W evolves in SM fiber with length of 300 m. (a) Input pulse evolution (b) Raman pulse evolution (c) Output signal spectrum (d) Output Raman spectrum.
Fig. 9
Fig. 9 Partially coherent light with coherent time of 3 ps, pulse width of 1 ns, peak power of 30 W evolves in SM fiber with length of 300 m. (a) Input pulse evolution (b) Raman pulse evolution (c) WTF distribution for input pulse (d) WTF distribution for output signal pulse (e) WTF distribution for output Raman pulse.
Fig. 10
Fig. 10 Pulse shapes before and after leading-edge peak elimination by SRS. Blue is for input pulse with a leading-edge peak by gain saturation. Orange is for output pulse after peak elimination by SRS.

Equations (4)

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P c r = ( 16 x 10 ln ( 10 ) ) K A e f f g R L e f f Δ ν R + Δ ν p Δ ν R
A p z + i β 2 p 2 2 A p t 2 + α p 2 A p = i γ p [ | A p | 2 + ( 2 + δ R f R ) | A s | 2 ] A p g p 2 | A s | 2 A p A s z d A s t + i β 2 s 2 2 A s t 2 + α s 2 A s = i γ s [ | A s | 2 + ( 2 + δ R f R ) | A p | 2 ] A s + g s 2 | A p | 2 A s
ρ i ( t , ω , z ) = ( 1 2 π ) + e i ω τ A i * ( t + τ 2 , z ) A i ( t τ 2 , z ) d τ
ρ p z = β 2 p ω ρ p t α p ρ p 2 γ p [ | A p | 2 + ( 2 + δ R f R ) | A s | 2 ] sin ( 1 2 t ω ) ρ p g p | A s | 2 cos ( 1 2 t ω ) ρ p ρ s z = ( d + β 2 s ω ) ρ s t α s ρ s 2 γ s [ | A s | 2 + ( 2 + δ R f R ) | A p | 2 ] sin ( 1 2 t ω ) ρ s + g s | A p | 2 cos ( 1 2 t ω ) ρ s

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