Abstract

Dispersive optical bistability in a dielectric sphere is modeled. The analysis is applicable to cases in which the incident frequency is near a morphology-dependent resonance of the sphere. The refractive index of the sphere is assumed to vary as m(r) = m0 + m2I(r), where I(r) is the internal intensity at position r. In general I(r) contains all the modes of the sphere. However, we first obtain a simplified analytical expression for bistability in which we assume that I(r) consists of a single near-resonant mode. We also analyze the bistability problem; in the analysis we include all the modes of the sphere in computing I(r). The agreement between the all-mode and the single-mode results is good when the incident frequency is within a few linewidths of ahigh-Q (>104) mode. We compare bistability in a dielectric sphere with that in a Fabry–Perot cavity. We use a quasi-steady-state approximation to calculate the time-dependent backscattering from a CS2 sphere near a resonance. The computed backscattered intensity has bistable characteristics.

© 1995 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
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1993

L. Collot, V. Lefevre-Seguin, M. Brune, J. M. Raimond, and S. Haroche, "Very high-Q whispering-gallery mode resonances observed on fused silica microspheres," Europhys. Lett. 23, 327–334 (1993).
[CrossRef]

K. Sasaki, H. Misawa, N. Kitamura, R. Fujisawa, and H. Masuhara, "Optical micromanipulation of a lasing polymer particle in water," Jpn. J. Appl. Phys. 32, L1144–L1147 (1993).
[CrossRef]

J. C. Swindal, D. H. Leach, and R. K. Chang, "Precession of morphology-dependent resonances in nonspherial droplets," Opt. Lett. 18, 191–193 (1993).
[CrossRef]

1992

D. Q. Chowdhury, S. C. Hill, and P. W. Barber, "Time dependence of internal intensity of a dielectric sphere on and near resonance," J. Opt. Soc. Am. A 9, 1364–1373 (1992).
[CrossRef]

M. M. Mazumder, S. C. Hill, and P. W. Barber, "Morphologydependent resonances in inhomogeneous spheres: comparison of the layered T-matrix method and the time-independent perturbation method," J. Opt. Soc. Am. A 9, 1844–1853 (1992).
[CrossRef]

E. E. M. Khaled, S. C. Hill, P. W. Barber, and D. Q. Chowdhury, "Near-resonance excitation of dielectric spheres with plane waves and off-axis Gaussian beams," Appl. Opt. 31, 1166–1169 (1992).
[CrossRef] [PubMed]

H.-B. Lin, J. D. Eversole, C. D. Merritt, and A. J. Campillo, "Cavity-modified spontaneous-emission rates in liquid microdroplets," Phys. Rev. A 45, 6756–6760 (1992).
[CrossRef] [PubMed]

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, "Whispering-gallery mode microdisk lasers," Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

V. S. Il'chenko and M. L. Gorodetskii, "Thermal nonlinear effects in optical whispering gallery microresonators," Laser Phys. 2, 1004–1009 (1992).

1991

1990

S. Arnold, T. R. O'Keeffe, K. M. Leung, L. M. Folan, T. Scalese, and A. Pluchino, "Optical bistability of an aerosol particle detected through light scattering: theory and experiment," Appl. Opt. 29, 3473–3478 (1990).
[CrossRef] [PubMed]

H.-B. Lin, J. D. Eversole, and A. J. Campillo, "Frequency pulling of stimulated Raman scattering in microdroplets," Opt. Lett. 15, 387–389 (1990).
[CrossRef] [PubMed]

H.-B. Lin, J. D. Eversole, and A. J. Campillo, "Vibrating orifice droplet generator for precision optical studies," Rev. Sci. Instrum. 61, 1018–1023 (1990).
[CrossRef]

H. M. Lai, P. T. Leung, and K. Young, "Limitations on the photon storage lifetime in electromagnetic resonances of highly transparent microdroplets," Phys. Rev. A 41, 5199–5204 (1990).
[CrossRef] [PubMed]

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, "Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets," Phys. Rev. A 41, 5187–5198 (1990).
[CrossRef] [PubMed]

1989

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Internal field of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance," J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

D. W. Hall, M. A. Newhouse, N. F. Borrelli, W. H. Dumbaugh, and D. L. Weidman, "Nonlinear optical susceptibilities of high-index glasses," Appl. Phys. Lett. 54, 1293–1295 (1989).
[CrossRef]

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, "Quality-factor and nonlinear properties of optical whispering-gallery modes," Phys. Lett. A 137, 393–397 (1989).
[CrossRef]

1987

1986

1985

1984

1981

1979

T. Bischofberger and Y. R. Shen, "Theoretical and experimental study of the dynamic behavior of a nonlinear Fabry–Perot interferrometer," Phys. Rev. A 19, 1169–1176 (1979).
[CrossRef]

Alexander, D. R.

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Internal field of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance," J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

Arnold, S.

Ashkin, A.

Baer, T.

Barber, P. W.

Barton, J. P.

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Internal field of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance," J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

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).
[CrossRef]

Bischofberger, T.

T. Bischofberger and Y. R. Shen, "Theoretical and experimental study of the dynamic behavior of a nonlinear Fabry–Perot interferrometer," Phys. Rev. A 19, 1169–1176 (1979).
[CrossRef]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 4.

Borrelli, N. F.

D. W. Hall, M. A. Newhouse, N. F. Borrelli, W. H. Dumbaugh, and D. L. Weidman, "Nonlinear optical susceptibilities of high-index glasses," Appl. Phys. Lett. 54, 1293–1295 (1989).
[CrossRef]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, Boston, Mass., 1992), pp. 267–269 and 337.

Braginsky, V. B.

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, "Quality-factor and nonlinear properties of optical whispering-gallery modes," Phys. Lett. A 137, 393–397 (1989).
[CrossRef]

Brune, M.

L. Collot, V. Lefevre-Seguin, M. Brune, J. M. Raimond, and S. Haroche, "Very high-Q whispering-gallery mode resonances observed on fused silica microspheres," Europhys. Lett. 23, 327–334 (1993).
[CrossRef]

Butcher, P. N.

P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, New York, 1990), pp. 306–308 and 314–315.
[CrossRef]

Campillo, A. J.

H.-B. Lin, J. D. Eversole, C. D. Merritt, and A. J. Campillo, "Cavity-modified spontaneous-emission rates in liquid microdroplets," Phys. Rev. A 45, 6756–6760 (1992).
[CrossRef] [PubMed]

H.-B. Lin, J. D. Eversole, and A. J. Campillo, "Vibrating orifice droplet generator for precision optical studies," Rev. Sci. Instrum. 61, 1018–1023 (1990).
[CrossRef]

H.-B. Lin, J. D. Eversole, and A. J. Campillo, "Frequency pulling of stimulated Raman scattering in microdroplets," Opt. Lett. 15, 387–389 (1990).
[CrossRef] [PubMed]

H.-B. Lin, A. L. Huston, B. L. Justus, and A. J. Campillo, "Some characteristics of a droplet whispering-gallery-mode laser," Opt. Lett. 11, 614–616 (1986).
[CrossRef] [PubMed]

Chang, R. K.

Chemla, D. S.

Chen, G.

Ching, S. C.

Chowdhury, D. Q.

Collot, L.

L. Collot, V. Lefevre-Seguin, M. Brune, J. M. Raimond, and S. Haroche, "Very high-Q whispering-gallery mode resonances observed on fused silica microspheres," Europhys. Lett. 23, 327–334 (1993).
[CrossRef]

Cotter, D.

P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, New York, 1990), pp. 306–308 and 314–315.
[CrossRef]

Dumbaugh, W. H.

D. W. Hall, M. A. Newhouse, N. F. Borrelli, W. H. Dumbaugh, and D. L. Weidman, "Nonlinear optical susceptibilities of high-index glasses," Appl. Phys. Lett. 54, 1293–1295 (1989).
[CrossRef]

Dziedzic, J. M.

Eberly, J. H.

P. W. Milonni and J. H. Eberly, Lasers (Wiley, New York, 1988).

Eversole, J. D.

H.-B. Lin, J. D. Eversole, C. D. Merritt, and A. J. Campillo, "Cavity-modified spontaneous-emission rates in liquid microdroplets," Phys. Rev. A 45, 6756–6760 (1992).
[CrossRef] [PubMed]

H.-B. Lin, J. D. Eversole, and A. J. Campillo, "Vibrating orifice droplet generator for precision optical studies," Rev. Sci. Instrum. 61, 1018–1023 (1990).
[CrossRef]

H.-B. Lin, J. D. Eversole, and A. J. Campillo, "Frequency pulling of stimulated Raman scattering in microdroplets," Opt. Lett. 15, 387–389 (1990).
[CrossRef] [PubMed]

Folan, L. M.

Fujisawa, R.

K. Sasaki, H. Misawa, N. Kitamura, R. Fujisawa, and H. Masuhara, "Optical micromanipulation of a lasing polymer particle in water," Jpn. J. Appl. Phys. 32, L1144–L1147 (1993).
[CrossRef]

Gibbs, H. M.

H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, New York, 1985).

Gorodetskii, M. L.

V. S. Il'chenko and M. L. Gorodetskii, "Thermal nonlinear effects in optical whispering gallery microresonators," Laser Phys. 2, 1004–1009 (1992).

Gorodetsky, M. L.

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, "Quality-factor and nonlinear properties of optical whispering-gallery modes," Phys. Lett. A 137, 393–397 (1989).
[CrossRef]

Hall, D. W.

D. W. Hall, M. A. Newhouse, N. F. Borrelli, W. H. Dumbaugh, and D. L. Weidman, "Nonlinear optical susceptibilities of high-index glasses," Appl. Phys. Lett. 54, 1293–1295 (1989).
[CrossRef]

Haroche, S.

L. Collot, V. Lefevre-Seguin, M. Brune, J. M. Raimond, and S. Haroche, "Very high-Q whispering-gallery mode resonances observed on fused silica microspheres," Europhys. Lett. 23, 327–334 (1993).
[CrossRef]

Haus, H. A.

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

Hill, S. C.

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 4.

Huston, A. L.

Ilchenko, V. S.

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, "Quality-factor and nonlinear properties of optical whispering-gallery modes," Phys. Lett. A 137, 393–397 (1989).
[CrossRef]

Il'chenko, V. S.

V. S. Il'chenko and M. L. Gorodetskii, "Thermal nonlinear effects in optical whispering gallery microresonators," Laser Phys. 2, 1004–1009 (1992).

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).

Justus, B. L.

Khaled, E. E. M.

Kitamura, N.

K. Sasaki, H. Misawa, N. Kitamura, R. Fujisawa, and H. Masuhara, "Optical micromanipulation of a lasing polymer particle in water," Jpn. J. Appl. Phys. 32, L1144–L1147 (1993).
[CrossRef]

Lai, H. M.

H. M. Lai, C. C. Lam, P. T. Leung, and K. Young, "The effect of perturbations on the widths of narrow morphology-dependent resonances in Mie scattering," J. Opt. Soc. Am. B 8, 1962–1973 (1991).
[CrossRef]

H. M. Lai, P. T. Leung, and K. Young, "Limitations on the photon storage lifetime in electromagnetic resonances of highly transparent microdroplets," Phys. Rev. A 41, 5199–5204 (1990).
[CrossRef] [PubMed]

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, "Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets," Phys. Rev. A 41, 5187–5198 (1990).
[CrossRef] [PubMed]

S. C. Ching, H. M. Lai, and K. Young, "Dielectric microspheres as optical cavities: Einstein A and B coefficients and level shift," J. Opt. Soc. Am. B 4, 2004–2009 (1987).
[CrossRef]

Lam, C. C.

Leach, D. H.

Lefevre-Seguin, V.

L. Collot, V. Lefevre-Seguin, M. Brune, J. M. Raimond, and S. Haroche, "Very high-Q whispering-gallery mode resonances observed on fused silica microspheres," Europhys. Lett. 23, 327–334 (1993).
[CrossRef]

Leung, K. M.

Leung, P. T.

H. M. Lai, C. C. Lam, P. T. Leung, and K. Young, "The effect of perturbations on the widths of narrow morphology-dependent resonances in Mie scattering," J. Opt. Soc. Am. B 8, 1962–1973 (1991).
[CrossRef]

H. M. Lai, P. T. Leung, and K. Young, "Limitations on the photon storage lifetime in electromagnetic resonances of highly transparent microdroplets," Phys. Rev. A 41, 5199–5204 (1990).
[CrossRef] [PubMed]

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, "Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets," Phys. Rev. A 41, 5187–5198 (1990).
[CrossRef] [PubMed]

Levi, A. F. J.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, "Whispering-gallery mode microdisk lasers," Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

Lin, H.-B.

H.-B. Lin, J. D. Eversole, C. D. Merritt, and A. J. Campillo, "Cavity-modified spontaneous-emission rates in liquid microdroplets," Phys. Rev. A 45, 6756–6760 (1992).
[CrossRef] [PubMed]

H.-B. Lin, J. D. Eversole, and A. J. Campillo, "Frequency pulling of stimulated Raman scattering in microdroplets," Opt. Lett. 15, 387–389 (1990).
[CrossRef] [PubMed]

H.-B. Lin, J. D. Eversole, and A. J. Campillo, "Vibrating orifice droplet generator for precision optical studies," Rev. Sci. Instrum. 61, 1018–1023 (1990).
[CrossRef]

H.-B. Lin, A. L. Huston, B. L. Justus, and A. J. Campillo, "Some characteristics of a droplet whispering-gallery-mode laser," Opt. Lett. 11, 614–616 (1986).
[CrossRef] [PubMed]

Logan, R. A.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, "Whispering-gallery mode microdisk lasers," Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

Long, M. B.

Masuhara, H.

K. Sasaki, H. Misawa, N. Kitamura, R. Fujisawa, and H. Masuhara, "Optical micromanipulation of a lasing polymer particle in water," Jpn. J. Appl. Phys. 32, L1144–L1147 (1993).
[CrossRef]

Mazumder, M. M.

McCall, S. L.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, "Whispering-gallery mode microdisk lasers," Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

Merritt, C. D.

H.-B. Lin, J. D. Eversole, C. D. Merritt, and A. J. Campillo, "Cavity-modified spontaneous-emission rates in liquid microdroplets," Phys. Rev. A 45, 6756–6760 (1992).
[CrossRef] [PubMed]

Miller, D. A. B.

Milonni, P. W.

P. W. Milonni and J. H. Eberly, Lasers (Wiley, New York, 1988).

Misawa, H.

K. Sasaki, H. Misawa, N. Kitamura, R. Fujisawa, and H. Masuhara, "Optical micromanipulation of a lasing polymer particle in water," Jpn. J. Appl. Phys. 32, L1144–L1147 (1993).
[CrossRef]

Newhouse, M. A.

D. W. Hall, M. A. Newhouse, N. F. Borrelli, W. H. Dumbaugh, and D. L. Weidman, "Nonlinear optical susceptibilities of high-index glasses," Appl. Phys. Lett. 54, 1293–1295 (1989).
[CrossRef]

O'Keeffe, T. R.

Pearton, S. J.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, "Whispering-gallery mode microdisk lasers," Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

Pluchino, A.

Qian, S.-X.

Raimond, J. M.

L. Collot, V. Lefevre-Seguin, M. Brune, J. M. Raimond, and S. Haroche, "Very high-Q whispering-gallery mode resonances observed on fused silica microspheres," Europhys. Lett. 23, 327–334 (1993).
[CrossRef]

Sasaki, K.

K. Sasaki, H. Misawa, N. Kitamura, R. Fujisawa, and H. Masuhara, "Optical micromanipulation of a lasing polymer particle in water," Jpn. J. Appl. Phys. 32, L1144–L1147 (1993).
[CrossRef]

Scalese, T.

Schaub, S. A.

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Internal field of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance," J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

Shen, Y. R.

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[CrossRef]

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[CrossRef]

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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Figures (6)

Fig. 1
Fig. 1

Graphical scheme used to solve Eqs. (11) and (13) for |cν|2. (a) The points at which the straight line intersects the Lorentzian curve are the solutions for |cν|2. As Iinc is gradually increased, |cν|2 follows the path 1–2–A–3–3′ –4. As Iinc is decreased, |cν|2 follows the path 4–3′ –C–2′ –2–1. (b) |cν|2E02 versus Iinc constructed from the solutions of |cν|2 obtained from (a). The hysteresis loop is obtained from the paths described in (a).

Fig. 2
Fig. 2

Ratio of the energy in the resonant mode to the total internal energy (Uν0/U0) inside a homogeneous CS2 sphere [refractive index m0 = 1.62 and m2I(r) = 0] as a function of the detuning of the circularly polarized incident plane wave. The detunings are expressed as the number of linewidths away from the resonance frequency of a mode. The parameters for the three different resonances are shown in Table 1. Points a and c indicate the detunings at which 90% of the total energy is in the TE30,1 and the TE68,5 modes, respectively. Points b and d show the detunings at which the energy fraction is 50% in those modes.

Fig. 3
Fig. 3

Total intensity and the intensity of the TE30,1 mode along the axis of a CS2 droplet. The incident frequency is detuned 5 linewidths from the resonance frequency of the TE30,1 mode. The incident field is a circularly polarized plane wave. The unperturbed resonance size parameter xν00 is 21.7364; Q0 is 5.5 × 105; m = m0 = 1.62.

Fig. 4
Fig. 4

Total internal energy density inside a CS2 droplet as a function of the incident intensity for two different modes, TE30,1 and TE68,5. The incident wave is a circularly polarized plane wave. The solid curves indicate the total energy density computed with all the modes of the sphere, and the dotted–dashed curves indicate the internal energy density computed with a single near-resonant mode. The refractive index of CS2 is 1.62, and the nonlinear index coefficient m2 is 10−20 m2/V2. Detunings in (a), (b), (c), and (d) correspond to the detunings at points a, b, c, and d in Fig. 2. In (a) the incident frequency is detuned so that xx30,100 = −12.7Δx30,1 where x30,100 and Δx30,1 are the unperturbed resonance frequency and the linewidth of the TE30,1 mode, respectively. (b) xx30,100 = −38Δx30,1; (c) xx68,500 = −2Δ x68,5; (d) xx68,500 = −6.7Δ x68,5.

Fig. 5
Fig. 5

(a) Schematic diagram showing the spectrally nondegenerate azimuthal modes corresponding to the TE654 mode of a distorted CS2 droplet with e = −10−3. The unperturbed Q = 106, ω0 = 18803.4 cm−1 for a = 4.5929 μm, and the laser frequency ωL is 18 797 cm−1. (b) Maximum values of the intensity-dependent refractive-index changes caused by Brillouin gain and the standard Kerr-type nonlinearity for a maximum internal intensity of 300 MW/cm2. The values used in the calculation are appropriate for CS2.

Fig. 6
Fig. 6

(a) Incident intensity (Iinc) and the backscattered intensity (Iscat) at r = 500a as a function of time. The incident field is a circularly polarized plane wave with a slowly varying envelope Iinc(t) = Ip exp{−[(t − 3T)/T]2} with T = 15 ns. The incident intensities are in units of kilowatts per square centimeter, and the scattered intensities are in units of milliwatts per square centimeter. The dashed curve represents the time profile of the incident pulse. The solid curves indicate the scattered intensities for different detunings δ = −2, −3, −5, −7 from the TE30,1 mode. The refractive index mo = 1.62, and the nonlinear refractive index coefficient m2 = 10−20 (m2/V2). The diameter of the sphere is ≃7 wavelengths, and Qo is 5.5 × 105. (b) Iscat as a function of Iinc for detuning δ = −5 for the same conditions as described in (a). (c) Iinc and σb as a function of time for the same conditions described in (a).

Tables (1)

Tables Icon

Table 1 Resonant Size Parameters, Linewidths, Q’s, and Radii for the Three Resonances Used in Fig. 1a

Equations (74)

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m ( r ) = m 0 + m 2 I ( r ) ,
E int ( r ) = E 0 n T n [ c n j n ( m k r ) X n , 1 ( θ , ϕ ) + d n 1 m k × j n ( m k r ) X n , 1 ( θ , ϕ ) ] , E scat ( r ) = E 0 n T n [ b n h n ( k r ) X n , 1 ( θ , ϕ ) + a n 1 k × [ h n ( k r ) X n , 1 ( θ , ϕ ) ] ] ,
X n , m ( θ , ϕ ) = i [ n ( n + 1 ) ] 1 / 2 r × Y n , m ( θ , ϕ ) ,
x ν 0 = x ν 00 [ 1 1 ( m 0 2 1 ) a 3 j ν ( m 0 x ν 00 ) 2 × V s 2 m 0 m 2 I ( r ) | j ν ( m 0 k ν r ) X ν , 1 | 2 d r 3 ] ,
E int ( r ) = E ν int ( r ) + E N int ( r ) .
E ν int ( r ) = E 0 c ν [ T ν j ν ( m 0 k r ) X ν , 1 ] = E 0 c ν F ν ( r ) ,
E N int ( r ) = E 0 { n ν T n c n j n ( m 0 k r ) X n , 1 + n T n d n 1 m 0 k × [ j n ( m 0 k ) X n , 1 ] } .
c ν = c ν 0 i Δ x / 2 ( x x ν 0 ) + i Δ x / 2 ,
I ( r ) = | E int | 2 = E 0 2 ( | c ν | 2 | F ν ( r ) | 2 + | E N ( r ) / E 0 | 2 + 2 c ν r Re [ F ν ( r ) E N * ( r ) / E 0 ] 2 c ν i Im [ F ν ( r ) E N * ( r ) / E 0 ] ) .
I ( r ) = | E ν ( r ) | 2 = E 0 2 | c ν | 2 4 π ( 2 ν + 1 ) | j ν ( m 0 k r ) X ν , 1 | 2 .
| c ν | 2 = | c ν 0 | 2 1 + 4 ( x x ν 0 Δ x ) 2 .
x x ν 0 Δ x = δ + K I inc | c ν | 2 ,
| c ν | 2 = 1 K I inc ( x x ν 0 Δ x ) δ K I inc .
T = T 0 1 + F sin 2 ( ϕ / 2 ) ,
ϕ = ϕ 0 + K I inc T ,
u = υ 1 + 4 ( δ + u ) 2 ,
υ = Q 0 | A 0 | 2 c ν 02 | E 0 | 2 ,
u = Q 0 | A 0 | 2 | c ν | 2 | E 0 | 2 ,
1 | A 0 | 2 = 16 π 2 m 0 m 2 ( 2 ν + 1 ) ( m 0 2 1 ) ( x ν 00 ) 3 j ν 2 ( m 0 x ν 00 ) × y = 0 x ν 00 | j ν ( m 0 y ) | 4 y 2 d y θ = 0 π | X ν , 1 | 4 sin θ d θ .
u 1 , 2 = 2 3 δ 1 3 ( δ 2 3 4 ) 1 / 2 .
u 1 | δ | 3 , υ 1 16 27 | δ | 3 .
I inc th = E 0 2 2 η 0 8 | A 0 | 2 | δ | 3 27 η 0 c ν 02 Q 0 .
c ν 02 = 2 Q 0 ( m 0 2 1 ) ( x ν 00 ) 3 j ν 2 ( m 0 x ν 00 ) ν 2 ( ν + 1 ) 2 ( 2 ν + 1 ) 2 .
I inc th = K ν | δ | 3 m 2 Q 0 2 ,
1 K ν = 108 π 2 m 0 η 0 ν 2 ( ν + 1 ) 2 ( m 0 2 1 ) 2 ( x ν 00 ) 6 j ν 4 ( m 0 x ν 00 ) ( 2 ν + 1 ) × y = 0 x ν 00 | j ν ( m 0 y ) | 4 y 2 d y θ = 0 π | X ν , 1 | 4 sin θ d θ ,
W bist = m 0 ( m 0 2 1 ) ω 0 u 1 V eff 8 π m 2 Q 0 2 ξ c ,
V eff = a 3 j ν 2 ( m 0 x ν 00 ) y = 0 x ν 00 | j ν ( m 0 y ) | 2 y 2 d y θ = 0 π | X ν , 1 | 2 sin θ d θ y = 0 x ν 00 | j ν ( m 0 y ) | 4 y 2 d y θ = 0 π | X ν , 1 | 4 sin θ d θ ,
c ν = c ν r + i c ν i = c ν 0 1 1 i 2 ( x x ν 0 Δ x ) .
x x ν 0 Δ x = δ + δ ,
δ = Q 0 ( m 0 2 1 ) a 3 j ν ( m 0 x ν 00 ) 2 V s 2 m 0 m 2 I ( r ) × | j ν ( m 0 k ν r ) X ν , 1 | 2 d r 3 .
2 ( δ + δ ) c ν r c ν i = 0 ,
c ν r + 2 ( δ + δ ) c ν i = c ν 0 .
2 δ = K 0 I inc ( I 0 + c ν r I 1 c ν i I 2 ) ,
K 0 = 16 π m 0 m 2 η 0 Q 0 ( m 0 2 1 ) x 3 j ν 2 ( m 0 x ν 00 ) ,
I 0 = y = 0 x θ = 0 π | E N / E 0 | 2 × j ν 2 ( m 0 x ν 00 y / x ) | X ν , 1 | 2 y 2 sin θ d θ d y ,
I 1 = y = 0 x θ = 0 π [ 2 Re ( F ν E N * / E 0 ) + c ν 0 | F ν | 2 ] × j ν 2 ( m 0 x ν 00 y / x ) | X ν , 1 | 2 y 2 sin θ d θ d y ,
I 2 = y = 0 x θ = 0 π 2 Im ( F ν E N * / E 0 ) × j ν 2 ( m 0 x ν 00 y / x ) | X ν , 1 | 2 y 2 sin θ d θ d y ,
( c ν r ) 3 + a 2 ( c ν r ) 2 + a 1 ( c ν r ) + a o = 0 ,
c ν i = c ν r [ 2 δ + K 0 I inc ( I 0 + I 1 c ν r ) ] 1 + K 0 I inc I 2 c ν r ,
a 0 = c ν 0 ( K 0 I inc ) 2 ( I 1 2 + I 2 2 ) ,
a 1 = 1 + ( 2 δ + K 0 I inc I 0 ) 2 2 K 0 I inc I 2 c ν 0 ( K 0 I inc ) 2 ( I 1 2 + I 2 2 ) ,
a 2 = 2 K 0 I inc I 2 + 2 K 0 I inc I 1 ( 2 δ + K 0 I inc I 0 ) c ν 0 ( K 0 I inc I 0 ) 2 ( K 0 I inc ) 2 ( I 1 2 + I 2 2 ) .
U = υ s ½ 0 m 0 2 | E ( r ) | 2 d V V s ,
U ν = 3 0 m 0 2 η 0 I inc 4 π x 3 2 π | c ν m 2 I ( r ) | 2 × y = 0 x θ = 0 π | T ν j ν ( m 0 y / x ) X ν , 1 | 2 y 2 d y sin θ d θ ,
U N = 3 0 m 0 2 η 0 I inc 4 π x 3 2 π y = 0 x θ = 0 π | E N ( r ) / E 0 | 2 y 2 d y × sin θ d θ ,
Δ ω = e 6 [ 1 3 m 2 n ( n + 1 ) ] ω 0 ,
Δ m ( ω ) = 1 2 π 2 P 0 d ω Δ g ( ω ) ω 2 ω 2 ,
Δ g ( ω ) = g 0 I 1 + 4 ( ω SBS ω Γ ) 2 ,
I scat = | E scat ( r , θ = π , ϕ ) | 2 2 η 0 .
σ b = | k F ( θ = π ) / E 0 | 2 π ( k a ) 2 ,
E scat ( r ) = F exp ( ikr ) r .
E ν scat ( r ) = E 0 T ν b ν h ν ( 1 ) ( k r ) X ν , 1 ,
I scat I inc | b ν | 2 4 π ( 2 ν + 1 ) | X π ) | 2 ( k r ) 2 , σ b | b ν | 2 4 π ( 2 ν + 1 ) | X ν , 1 ( θ = π ) | 2 ( k a ) 2 .
| b ν | 2 = | b ν 0 | 2 u υ ,
( r ) = m 2 ( r ) = m 0 2 + m 2 2 I 2 ( r ) + 2 m 0 m 2 I ( r ) .
1 ( r ) 2 m 0 m 2 I ( r ) .
x ν 0 = x ν 00 ( 1 V 2 G ) ,
V = [ h ν ( 1 ) ( x ν 00 ) j ν ( m 0 x ν 00 ) ] 2 V s 1 ( r ) | j ν ( m 0 k ν r ) X ν , 1 | 2 d r 3 ,
G = ( m 0 2 1 ) a 3 2 [ h ν ( 1 ) ( x ν 00 ) ] 2 ,
Δ x = x ν 0 Q p x ν 00 Q p = x ν 00 ( 1 Q 0 + C 1 + C 2 ) ,
C 1 = 2 ( m 0 2 1 ) x ν 00 Im [ D ν ( 1 ) ( z 0 ) ] ,
C 2 = 2 ( m 0 2 1 ) ( x ν 00 ) 3 n ν | D ν n ( 1 ) | 2 × | j n ( m 0 x ν 00 ) j n ( m 0 x ν 00 ) h n ( 1 ) ( x ν 00 ) m 0 j n ( m 0 x ν 00 ) h n ( 1 ) ( x ν 00 ) | ,
D ν n ( 1 ) = { 1 ( x ν 00 ) 2 j ν ( m 0 x ν 00 ) j n ( m 0 x ν 00 ) } × ϕ = 0 2 π ϕ = 0 π k ν r = 0 k ν r = x ν 00 2 m 0 m 2 I ( r ) j ν ( m 0 k ν r ) × j n ( m 0 k ν r ) ( k ν r ) 2 d ( k ν r ) × Y ν , 1 * ( θ , ϕ ) Y n , 1 ( θ , ϕ ) sin θ d θ d ϕ .
z 0 = x ν 00 i x ν 00 2 Q 0 .
W = stored energy photon lifetime = ω Q 0 V s 1 2 0 m 0 2 E int E int * d V ,
W bist = m 0 ( m 0 2 1 ) ω 0 u 1 V eff 8 π m 2 Q 0 2 ξ c ,
V eff = a 3 j ν 2 ( m 0 x ν 00 ) y = 0 x ν 00 | j ν ( m 0 y ) | 2 y 2 d y θ = 0 π | X ν , 1 | 2 sin θ d θ y = 0 x ν 00 | j ν ( m 0 y ) | 4 y 2 d y θ = 0 π | X ν , 1 | 4 sin θ d θ ,
δ x x ν 00 = V s ½ 0 1 ( r ) | E int ( r ) | 2 d V V s ½ 0 m 0 2 | E int ( r ) | 2 d V = 2 m 2 V s | E int ( r ) | 4 d V m 0 V s | E int ( r ) | 2 d V ,
E int ( r ) E 0 c ν T ν j ν ( m k r ) X ν , 1 = E 0 c ν E ν 0 ( r ) .
x ν 0 = x ν 00 ( 1 | c ν E 0 | 2 | A 0 c | 2 ) ,
1 | A 0 c | 2 = 2 m 2 V s | E ν 0 ( r ) | 4 d V m 0 V s | E ν 0 | 2 d V .
W bist c = 0 m 0 3 ω V eff u 1 4 m 2 Q 0 2 ξ c ,
V eff = ( V s | E ν 0 | 2 d V ) 2 V s | E ν 0 | 4 d V .
W bist c = 0 m 0 4 ω V eff 3 χ ( 3 ) Q 0 2 .

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