Abstract

Standard one-dimensional nonlinear-wave equations are modified to accommodate the growth and coupling of nonlinear waves in droplets. The propagation direction of the nonlinear waves along the length of an optical cell is changed so that it is along the droplet rim. The model includes radiation losses because of nonzero absorption, leakage from the droplet, and depletion in generating other nonlinear waves. For multimode-laser input, the growth and decay of the first- through fourth-order Stokes stimulated Raman scattering (SRS) are calculated as a function of the phase matching of the four-wave mixing process and the model-dependent Raman gain coefficient. The Raman gain coefficient determines the delay time of the first-order SRS, while the phase matching determines the correlated temporal profiles of the multiorder SRS. Both the Raman gain and the phase matching are found to be enhanced in the droplet. The spatial distribution of the internal input-laser intensity is calculated by using the Lorenz–Mie formalism. The temporal profile of the input-laser intensity used in the calculations is identical to the experimentally observed laser time profile. The delay time and the correlated growth and decay of nonlinear waves resulting from the numerical simulation compare favorably with those of the experimental observations. Similar calculations are made for single-mode-laser input for which the stimulated Brillouin scattering (SBS) achieves its threshold before the SRS does and subsequently pumps the SRS.

© 1992 Optical Society of America

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References

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  1. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 93–104.
  2. E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681–681 (1946).
  3. S. C. Ching, H. M. Lai, and K. Young, “Dielectric microspheres as optical cavities: thermal spectrum and density of states,” J. Opt. Soc. Am. B 4, 1995–2003 (1987);“Dielectric microspheres as optical cavities: Einstein A and B coefficients and level shift,” J. Opt. Soc. Am. B 4, 2004–2009 (1987).
    [Crossref]
  4. J.-Z. Zhang, G. Chen, and R. K. Chang, “Pumping of stimulated Raman scattering by stimulated Brillouin scattering within a single droplet: input laser linewidth effects,” J. Opt. Soc. Am. B 7, 108–115 (1990).
    [Crossref]
  5. S.-X. Qian, S. Yuan, Y. Li, H. Wang, X. Huang, and Z. Yu, “Comparison between the temporal characteristics of picosecond SRS from the cell and SRO from the droplet,” Opt. Commun. 74, 414–418 (1990).
    [Crossref]
  6. R. G. Pinnick, A. Biswas, P. Chylek, R. L. Armstrong, H. Latifi, E. Creegan, V. Srivastava, M. Jarzembski, and G. Fernandez, “Stimulated Raman scattering in micrometer-sized droplets: time-resolved measurements,” Opt. Lett. 13, 494–496 (1988).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  10. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984), pp. 201–202.
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  13. D. S. Benincasa, P. W. Barber, J.-Z. Zhang, W.-F. Hsieh, and R. K. Chang, “Spatial distribution of the internal and near-field intensities of large cylindrical and spherical scatterers,” Appl. Opt. 26, 1348–1356 (1987).
    [Crossref] [PubMed]
  14. A. Penzkofer, A. Laubereau, and W. Kaiser, “High intensity Raman interactions,” J. Prog. Quantum Electron. 6, 55–140 (1979).
    [Crossref]
  15. D. von der Linde, M. Maier, and W. Kaiser, “Quantitative investigations of the stimulated Raman effect using sub-nanosecond pulses,” Phys. Rev. 178, 11–17 (1969).
    [Crossref]
  16. Calculated from the measured values of the spontaneous Raman cross section and the linewidth of 5 M NH4NO3dissolved in water.
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  20. T. R. Lettieri and R. E. Preston, “Observation of sharp resonances in the spontaneous Raman spectrum of a single optically levitated microdroplet,” Opt. Commun. 54, 349–352, (1985).
    [Crossref]
  21. J. B. Snow, S.-X. Qian, and R. K. Chang, “Stimulated Raman scattering from individual water and ethanol droplets at morphology-dependent resonances,” Opt. Lett. 10, 37–39 (1985).
    [Crossref] [PubMed]
  22. H. Chew, “Radiation and lifetimes of atoms inside dielectric particles,” Phys. Rev. A 38, 3410–3416 (1990).
    [Crossref]
  23. P. A. Fleury and R. Y. Chiao, “Dispersion of hypersonic waves in liquids,” J. Acoust. Soc. Am. 39, 751–752 (1966).
    [Crossref]
  24. C. Hu and J. R. Whinnery, “New thermooptical measurement method and a comparison with other methods,” Appl. Opt. 12, 72–79 (1973).
    [Crossref] [PubMed]
  25. J. Stone, “Measurement of absorption of light in low-loss liquids,” J. Opt. Soc. Am. 62, 327–333 (1972).
    [Crossref]
  26. Assumed to be of the same order of magnitude as the water value.

1991 (1)

A. J. Campillo, J. D. Eversole, and H-B. Lin, “Cavity quantum elecrodynamic enhancement of stimulated emission in microdroplets,” Phys. Rev. Lett. 67, 437–440 (1991).
[Crossref] [PubMed]

1990 (3)

J.-Z. Zhang, G. Chen, and R. K. Chang, “Pumping of stimulated Raman scattering by stimulated Brillouin scattering within a single droplet: input laser linewidth effects,” J. Opt. Soc. Am. B 7, 108–115 (1990).
[Crossref]

S.-X. Qian, S. Yuan, Y. Li, H. Wang, X. Huang, and Z. Yu, “Comparison between the temporal characteristics of picosecond SRS from the cell and SRO from the droplet,” Opt. Commun. 74, 414–418 (1990).
[Crossref]

H. Chew, “Radiation and lifetimes of atoms inside dielectric particles,” Phys. Rev. A 38, 3410–3416 (1990).
[Crossref]

1988 (3)

1987 (2)

1985 (3)

1979 (1)

A. Penzkofer, A. Laubereau, and W. Kaiser, “High intensity Raman interactions,” J. Prog. Quantum Electron. 6, 55–140 (1979).
[Crossref]

1973 (2)

1972 (1)

1969 (1)

D. von der Linde, M. Maier, and W. Kaiser, “Quantitative investigations of the stimulated Raman effect using sub-nanosecond pulses,” Phys. Rev. 178, 11–17 (1969).
[Crossref]

1966 (1)

P. A. Fleury and R. Y. Chiao, “Dispersion of hypersonic waves in liquids,” J. Acoust. Soc. Am. 39, 751–752 (1966).
[Crossref]

1946 (1)

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681–681 (1946).

Armstrong, R. L.

Barber, P. W.

Benincasa, D. S.

Benner, R. E.

S. C. Hill and R. E. Benner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Particles, P. W. Barber and R. K. Chang, eds. (World Scientific, Singapore, 1988), pp. 3–61.

Biswas, A.

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 93–104.

Campillo, A. J.

A. J. Campillo, J. D. Eversole, and H-B. Lin, “Cavity quantum elecrodynamic enhancement of stimulated emission in microdroplets,” Phys. Rev. Lett. 67, 437–440 (1991).
[Crossref] [PubMed]

Chang, R. K.

Chen, G.

Chew, H.

H. Chew, “Radiation and lifetimes of atoms inside dielectric particles,” Phys. Rev. A 38, 3410–3416 (1990).
[Crossref]

Chiao, R. Y.

P. A. Fleury and R. Y. Chiao, “Dispersion of hypersonic waves in liquids,” J. Acoust. Soc. Am. 39, 751–752 (1966).
[Crossref]

Ching, S. C.

Chylek, P.

Creegan, E.

Eversole, J. D.

A. J. Campillo, J. D. Eversole, and H-B. Lin, “Cavity quantum elecrodynamic enhancement of stimulated emission in microdroplets,” Phys. Rev. Lett. 67, 437–440 (1991).
[Crossref] [PubMed]

Fernandez, G.

Fleury, P. A.

P. A. Fleury and R. Y. Chiao, “Dispersion of hypersonic waves in liquids,” J. Acoust. Soc. Am. 39, 751–752 (1966).
[Crossref]

Hale, G. M.

Haus, H. A.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984), pp. 201–202.

Hill, S. C.

S. C. Hill and R. E. Benner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Particles, P. W. Barber and R. K. Chang, eds. (World Scientific, Singapore, 1988), pp. 3–61.

P. W. Barber and S. C. Hill, Light Scattering by Particles (World Scientific, Singapore, 1990), pp. 228–229.

Hsieh, W. F.

Hsieh, W.-F.

Hu, C.

Huang, X.

S.-X. Qian, S. Yuan, Y. Li, H. Wang, X. Huang, and Z. Yu, “Comparison between the temporal characteristics of picosecond SRS from the cell and SRO from the droplet,” Opt. Commun. 74, 414–418 (1990).
[Crossref]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 93–104.

Jarzembski, M.

Kaiser, W.

A. Penzkofer, A. Laubereau, and W. Kaiser, “High intensity Raman interactions,” J. Prog. Quantum Electron. 6, 55–140 (1979).
[Crossref]

D. von der Linde, M. Maier, and W. Kaiser, “Quantitative investigations of the stimulated Raman effect using sub-nanosecond pulses,” Phys. Rev. 178, 11–17 (1969).
[Crossref]

Kiefer, W.

Lai, H. M.

Latifi, H.

Laubereau, A.

A. Penzkofer, A. Laubereau, and W. Kaiser, “High intensity Raman interactions,” J. Prog. Quantum Electron. 6, 55–140 (1979).
[Crossref]

Leach, D. H.

Lettieri, T. R.

T. R. Lettieri and R. E. Preston, “Observation of sharp resonances in the spontaneous Raman spectrum of a single optically levitated microdroplet,” Opt. Commun. 54, 349–352, (1985).
[Crossref]

Li, Y.

S.-X. Qian, S. Yuan, Y. Li, H. Wang, X. Huang, and Z. Yu, “Comparison between the temporal characteristics of picosecond SRS from the cell and SRO from the droplet,” Opt. Commun. 74, 414–418 (1990).
[Crossref]

Lin, H-B.

A. J. Campillo, J. D. Eversole, and H-B. Lin, “Cavity quantum elecrodynamic enhancement of stimulated emission in microdroplets,” Phys. Rev. Lett. 67, 437–440 (1991).
[Crossref] [PubMed]

Maier, M.

D. von der Linde, M. Maier, and W. Kaiser, “Quantitative investigations of the stimulated Raman effect using sub-nanosecond pulses,” Phys. Rev. 178, 11–17 (1969).
[Crossref]

Penzkofer, A.

A. Penzkofer, A. Laubereau, and W. Kaiser, “High intensity Raman interactions,” J. Prog. Quantum Electron. 6, 55–140 (1979).
[Crossref]

Pinnick, R. G.

Preston, R. E.

T. R. Lettieri and R. E. Preston, “Observation of sharp resonances in the spontaneous Raman spectrum of a single optically levitated microdroplet,” Opt. Commun. 54, 349–352, (1985).
[Crossref]

Purcell, E. M.

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681–681 (1946).

Qian, S.-X.

S.-X. Qian, S. Yuan, Y. Li, H. Wang, X. Huang, and Z. Yu, “Comparison between the temporal characteristics of picosecond SRS from the cell and SRO from the droplet,” Opt. Commun. 74, 414–418 (1990).
[Crossref]

J. B. Snow, S.-X. Qian, and R. K. Chang, “Stimulated Raman scattering from individual water and ethanol droplets at morphology-dependent resonances,” Opt. Lett. 10, 37–39 (1985).
[Crossref] [PubMed]

Querry, M. R.

Shen, Y. R.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), pp. 141–201.

Snow, J. B.

Srivastava, V.

Stone, J.

Thurn, R.

von der Linde, D.

D. von der Linde, M. Maier, and W. Kaiser, “Quantitative investigations of the stimulated Raman effect using sub-nanosecond pulses,” Phys. Rev. 178, 11–17 (1969).
[Crossref]

Wang, H.

S.-X. Qian, S. Yuan, Y. Li, H. Wang, X. Huang, and Z. Yu, “Comparison between the temporal characteristics of picosecond SRS from the cell and SRO from the droplet,” Opt. Commun. 74, 414–418 (1990).
[Crossref]

Whinnery, J. R.

Young, K.

Yu, Z.

S.-X. Qian, S. Yuan, Y. Li, H. Wang, X. Huang, and Z. Yu, “Comparison between the temporal characteristics of picosecond SRS from the cell and SRO from the droplet,” Opt. Commun. 74, 414–418 (1990).
[Crossref]

Yuan, S.

S.-X. Qian, S. Yuan, Y. Li, H. Wang, X. Huang, and Z. Yu, “Comparison between the temporal characteristics of picosecond SRS from the cell and SRO from the droplet,” Opt. Commun. 74, 414–418 (1990).
[Crossref]

Zhang, J.-Z.

Zheng, J.-B.

Appl. Opt. (4)

J. Acoust. Soc. Am. (1)

P. A. Fleury and R. Y. Chiao, “Dispersion of hypersonic waves in liquids,” J. Acoust. Soc. Am. 39, 751–752 (1966).
[Crossref]

J. Opt. Soc. Am. (1)

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

J. Prog. Quantum Electron. (1)

A. Penzkofer, A. Laubereau, and W. Kaiser, “High intensity Raman interactions,” J. Prog. Quantum Electron. 6, 55–140 (1979).
[Crossref]

Opt. Commun. (2)

S.-X. Qian, S. Yuan, Y. Li, H. Wang, X. Huang, and Z. Yu, “Comparison between the temporal characteristics of picosecond SRS from the cell and SRO from the droplet,” Opt. Commun. 74, 414–418 (1990).
[Crossref]

T. R. Lettieri and R. E. Preston, “Observation of sharp resonances in the spontaneous Raman spectrum of a single optically levitated microdroplet,” Opt. Commun. 54, 349–352, (1985).
[Crossref]

Opt. Lett. (4)

Phys. Rev. (2)

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681–681 (1946).

D. von der Linde, M. Maier, and W. Kaiser, “Quantitative investigations of the stimulated Raman effect using sub-nanosecond pulses,” Phys. Rev. 178, 11–17 (1969).
[Crossref]

Phys. Rev. A (1)

H. Chew, “Radiation and lifetimes of atoms inside dielectric particles,” Phys. Rev. A 38, 3410–3416 (1990).
[Crossref]

Phys. Rev. Lett. (1)

A. J. Campillo, J. D. Eversole, and H-B. Lin, “Cavity quantum elecrodynamic enhancement of stimulated emission in microdroplets,” Phys. Rev. Lett. 67, 437–440 (1991).
[Crossref] [PubMed]

Other (7)

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984), pp. 201–202.

S. C. Hill and R. E. Benner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Particles, P. W. Barber and R. K. Chang, eds. (World Scientific, Singapore, 1988), pp. 3–61.

P. W. Barber and S. C. Hill, Light Scattering by Particles (World Scientific, Singapore, 1990), pp. 228–229.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), pp. 141–201.

Calculated from the measured values of the spontaneous Raman cross section and the linewidth of 5 M NH4NO3dissolved in water.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 93–104.

Assumed to be of the same order of magnitude as the water value.

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Figures (12)

Fig. 1
Fig. 1

Schematic of the spatial distribution of the intensity (shaded areas) and the rays of the input-laser first- and second-order Stokes SRS waves in the droplet equatorial plane. The intensity maximum of the input-laser beam is concentrated on the shadow side of the droplet at ϕ = 0°. The generated first- and second-order Stokes SRS waves are on MDR’s, which are represented as two counterpropagating traveling waves or standing waves around the droplet rim. The droplet has radius a, and ϕ is the azimuthal angle.

Fig. 2
Fig. 2

(a) Radially averaged internal-intensity distribution and (b) internal-intensity distribution at r = 0.82a (continuous curve) and at r = 0.76a (dashed curve) of the input wave Iinput(ϕ) as a function of the azimuthal angle ϕ in the droplet equatorial plane. The droplet shadow face is located at ϕ = 0° + n360°, and the droplet illuminated face is located at ϕ = 180° + n360°, where n is an integer. The range of ϕ shown is from ϕ = 0° to ϕ = 800°. The first round trip around the droplet rim is from ϕ = 0° to ϕ = 360°, and the second round trip is from ϕ = 360° to ϕ = 720°.

Fig. 3
Fig. 3

Temporal profiles of the following experimentally observed pulses for aqueous NH4NO3 solution: (a) input laser Iinput(t), (b) first-order Stokes SRS I1S(t), (c) second-order Stokes SRS I2S(t), (d) third-order Stokes SRS I3S(t). The multiorder Stokes SRS is associated with the ν1 symmetric vibrational mode of the nitrate ions; the time resolution is ≈0.4 ns. (After Ref. 7.)

Fig. 4
Fig. 4

(a) Idealized square-shaped laser time profile of 7-ns duration; using Eq. (5), calculated time profiles of (b) I1S(t), (c) I2S(t), and (d) I3S(t). The parameters used in the calculations are listed in Table 1. The Raman gain coefficient of the droplet cavity is gSc = 0.31 cm/GW. The wave-vector mismatch for all FWM processes is Δk = 103π m−1, which corresponds to a coherence time of τcoh = 4 ps. The time-averaged input-laser intensity is 〈Iinputt = 0.8 GW/cm2.

Fig. 5
Fig. 5

Same as in Fig. 4, except that the Raman gain coefficient is increased to gSc = 0.45 cm/GW

Fig. 6
Fig. 6

Calculated time profiles of I1S(t) as a function of gSc with the square-shaped input-laser pulse with 7-ns duration and 〈Iinputt = 0.8 GW/cm2 kept fixed. The Raman gain coefficient is varied from gSc = 0.15 cm/GW to gSc = 0.65 cm/GW in increments of ΔgSc = 0.02 cm/GW. The wave-vector mismatch is fixed at Δk = 103πm−1, which corresponds to a coherence time of τcoh = 4 ps.

Fig. 7
Fig. 7

Same as in Fig. 5, except that the wave-vector mismatch is decreased to Δk ≈ 10π m−1, which corresponds to a coherence time of τcoh = 0.4 ns.

Fig. 8
Fig. 8

Calculated time profiles of I1S(t) as a function of Δk or τcoh with the square-shaped input-laser pulse with 7-ns duration and 〈Iinputt = 0.8 GW/cm2 kept fixed. The wave-vector mismatch is varied from Δk = 103π m−1 to Δk = 0.1π m−1. The corresponding variation of τcoh is from τcoh = 4 ps to τcoh = 40 ns. The Raman gain coefficient is fixed at gSc = 0.45 cm/GW.

Fig. 9
Fig. 9

(a) Experimentally observed laser pulse Iinput(t) (normalized) (after Ref. 7); using Eq. (5), calculated time profiles of (b) I1S(t), (c) I2S(t), and (d) I3S(t). To achieve reasonable agreement with the experimental time profiles (shown in Fig. 3), the Raman gain coefficient is adjusted to be gSc = 0.45 cm/GW and the wave-vector mismatch is adjusted to be Δk = 2.18 m−1 (corresponding to τcoh = 6.4 ns). All other parameters used in the calculations of Eq. (5) for nitrate ions in water droplets are listed in Table 1.

Fig. 10
Fig. 10

Schematic of the spatial distribution of the intensity (shaded areas) and the rays of the input-laser, SBS, and first-order Stokes SRS waves in the droplet equatorial plane. The intensity maximum of the input-laser beam is concentrated on the shadow side of the droplet. There is a spread of the input-laser wave vector kinput. The generated SBS and SRS waves are on MDR’s, which are represented as two counterpropagating traveling waves around the droplet rim. The wave vectors of the SBS (kSBS) and SRS (k1S) are always tangential to the droplet rim since both waves are on MDR’s. The Brillouin gain coefficient and the Brillouin frequency are functions of the angle between kSBS and kinput.

Fig. 11
Fig. 11

Temporal profiles of the experimentally observed pulses of (a) Iinput(t), (b) IB(t), and (c) I1S(t) for ethanol droplets, with a time resolution of ≈0.1 ns. (After Ref. 4.)

Fig. 12
Fig. 12

(a) Experimentally observed time profile (after Ref. 4) of the input-laser pulse Iinput(t) (normalized); calculated time profiles of (b) IB(t) and (c) I1S(t). All the other parameters used in the calculations of Eq. (11) for ethanol droplets are listed in Table 2. To achieve reasonable agreement with the experimental time profiles (shown in Fig. 11), the Raman gain coefficient is adjusted to be gSc = 2.0 cm/GW, and the Brillouin gain coefficient is adjusted to be gBc = 3.0 cm/GW. The wave-vector mismatch is Δk = 10π m−1 (corresponding to τcoh = 4 ps).

Tables (2)

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Table 1 Parameters for the Multimode Laser Pumping SRS

Tables Icon

Table 2 Parameters for the Single-Mode Laser Pumping SBS and SRS

Equations (15)

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I input ( ϕ ) r = m ( ν ) a [ m ( ν ) 1 ] a / m ( ν ) a I input ( r , ϕ ) d r .
I input t = 1 τ input 0 I input ( t ) d t ,
T input ( t ) = I input ( t ) I input t .
I input ( ϕ , t ) = I input ( ϕ ) r T input ( t ) .
d E 1 S a d ϕ = [ g S 2 ( I input I 2 S ) α 1 S 2 L 1 S 2 ] E 1 S g S 2 E input E 1 S * E 2 S exp ( i Δ k 2 S a ϕ ) j = 3 j final g S 2 E input E ( j 1 ) S * E j S exp ( i Δ k j S a ϕ ) ,
d E 2 S a d ϕ = [ g S 2 ν 2 S ν 1 S ( I 1 S I 3 S ) α 2 S 2 L 2 S 2 ] E 2 S + g S 2 ν 2 S ν 1 S E input * E 1 S E 1 S exp ( i Δ k 2 S a ϕ ) j = 3 j final g S 2 ν 2 S ν 1 S E input E ( j 2 ) S * E j S exp ( i Δ k j S a ϕ ) .
L j S = 2 π ν j S m ( ν j S ) Q .
g S c ( ω ) / g S ( ω ) = ρ c ( ω ) / V ρ vac ( ω ) ,
g S c ( x ) / g S ( x ) = ( 3 π / 4 ) ( 2 n + 1 ) Q / x 3 .
g S c ( x ) / g S c ( x ) = [ ( 2 n + 1 ) / ( 2 n + 1 ) ] ( Q / Q ) ( x / x ) 3 .
τ coh = l coh m ( ω ) c = ( π Δ k ) m ( ω ) c .
d E B a d ϕ = [ g B 2 I input g S 2 ν B ν 1 S I 1 S α B 2 L B 2 ] E B g S 2 ν B ν 1 S E 2 S * E 1 S E 1 S exp ( i Δ k 2 S B a ϕ ) j = 3 j final g S 2 ν B ν 1 S E j S * E 1 S E ( j 1 ) S exp ( i Δ k jSB a ϕ ) ,
d E 1 S a d ϕ = [ g s 2 ( I B + I input I 2 S ) α 1 S 2 L 1 S 2 ] E 1 S g s 2 E input E 1 S * E 2 S exp ( i Δ k 2 S a ϕ ) g s 2 E B E 1 S * E 2 S exp ( i Δ k 2 S B a ϕ ) j = 3 j final g S 2 E input E ( j 1 ) S * E j S exp ( i Δ k j S a ϕ ) j = 3 j final g S 2 E B E ( j 1 ) S * E j S exp ( i Δ k jSB a ϕ ) .
ν B ( ϕ B = π ) = ν input ν ac ( ϕ B = π ) ,
ν ac ( ϕ B = π ) = k ac ( ϕ B = π ) / 2 π = k input / π = 2 ν input m ( ν input ) .

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