Abstract

The quantum noise inherent in the initial stages of stimulated Raman scattering leads to a fundamental limit to the shot-to-shot directional stability of the Stokes beam from an unsaturated Raman generator. The probability that the centroid of the Stokes beam will propagate in a given direction is calculated theoretically by means of a coherent-mode expansion, and it is shown that the angular stability of the centroid is ~10% of the beam divergence angle for a uniformly pumped, cylindrical Raman generator with Fresnel-number unity.

© 1991 Optical Society of America

Full Article  |  PDF Article

Corrections

I. A. Walmsley, "Quantum noise limit to the beam-pointing stability in stimulated Raman generation: errata," J. Opt. Soc. Am. B 8, 2392-2393 (1991)
https://www.osapublishing.org/josab/abstract.cfm?uri=josab-8-11-2392

References

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  1. I. A. Walmsley and M. G. Raymer, “Observation of macroscopic quantum fluctuations in stimulated Raman scattering,” Phys. Rev. Lett. 50, 963 (1983).
    [CrossRef]
  2. M. G. Raymer, Z. W. Li, and I. A. Walmsley, “Temporal shape fluctuations in stimulated Raman scattering: coherent modes description,” Phys. Rev. Lett. 63, 1586 (1989).
    [CrossRef] [PubMed]
  3. D. C. MacPherson, R. C. Swanson, and J. L. Carlsten, “Quantum fluctuations and correlations in the stimulated Raman scattering spectrum,” Phys. Rev. A 39, 3487 (1989).
    [CrossRef] [PubMed]
  4. S.-J. Kuo, C. Radcewicz, and M. G. Raymer, “Quantum statistics of stimulated Raman scattering using a Gaussian-profile pump,” J. Opt. Soc. Am. A 4 (13), P16 (1987).
  5. J. Mostowski and M. G. Raymer, “The buildup of stimulated Raman scattering from spontaneous Raman scattering,” Opt. Commun. 36, 237 (1981).
    [CrossRef]
  6. R. J. Glauber and F. Haake, “The initiation of superfluorescence,” Phys. Lett. 68A, 29 (1978).
  7. M. D. Levenson, W. H. Richardson, and S. H. Perlmutter, “Stochastic noise in TEM00laser beam position,” Opt. Lett. 14, 779 (1989).
    [CrossRef] [PubMed]
  8. M. G. Raymer and I. A. Walmsley, in Proceedings of the Vth Rochester Conference on Coherence and Quantum Optics, L. Mandel and E. Wolf, eds. (Plenum, New York, 1984).
  9. See, for example, B. E. A. Saleh, Photoelectron Statistics (Springer-Verlag, Berlin, 1978).
    [CrossRef]
  10. M. G. Raymer, I. A. Walmsley, J. Mostowski, and B. Sobolewska, “Quantum theory of spatial and temporal coherence properties of stimulated Raman scattering,” Phys. Rev. A 32, 332 (1985).
    [CrossRef] [PubMed]
  11. W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973), p. 173.
  12. B. Ya. Zel’dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer, New York, 1985).
    [CrossRef]
  13. The application of this procedure to the Raman problem is described by M. G. Raymer and I. A. Walmsley, “The quantum coherence properties of stimulated Raman scattering,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1990), Vol. 28, pp. 216 et seq.
    [CrossRef]
  14. F. A. Hopf and E. A. Overman, “Fluctuations in nonlinear swept-gain amplifiers,” Phys. Rev. A 19, 1180 (1979); F. A. Hopf, “Phase waves in superfluorescence,” Phys. Rev. A 20, 2064 (1979).
    [CrossRef]
  15. J. C. Englund and C. M. Bowden, “Spontaneous generation of Raman solitons,” Phys. Rev. Lett. 57, 2661 (1986); C. M. Bowden and J. C. Englund, “Macroscopic manifestations of quantum noise,” Opt. Commun. 67, 71 (1988).
    [CrossRef] [PubMed]
  16. Although ΔKx from Eq. (23) and ΔK from the coherent-mode theory have a similar numerical value for F= 3.0, it should be noted that, for a rotationally symmetric distribution,ΔK=2ΔKx.

1989 (3)

M. G. Raymer, Z. W. Li, and I. A. Walmsley, “Temporal shape fluctuations in stimulated Raman scattering: coherent modes description,” Phys. Rev. Lett. 63, 1586 (1989).
[CrossRef] [PubMed]

D. C. MacPherson, R. C. Swanson, and J. L. Carlsten, “Quantum fluctuations and correlations in the stimulated Raman scattering spectrum,” Phys. Rev. A 39, 3487 (1989).
[CrossRef] [PubMed]

M. D. Levenson, W. H. Richardson, and S. H. Perlmutter, “Stochastic noise in TEM00laser beam position,” Opt. Lett. 14, 779 (1989).
[CrossRef] [PubMed]

1987 (1)

S.-J. Kuo, C. Radcewicz, and M. G. Raymer, “Quantum statistics of stimulated Raman scattering using a Gaussian-profile pump,” J. Opt. Soc. Am. A 4 (13), P16 (1987).

1986 (1)

J. C. Englund and C. M. Bowden, “Spontaneous generation of Raman solitons,” Phys. Rev. Lett. 57, 2661 (1986); C. M. Bowden and J. C. Englund, “Macroscopic manifestations of quantum noise,” Opt. Commun. 67, 71 (1988).
[CrossRef] [PubMed]

1985 (1)

M. G. Raymer, I. A. Walmsley, J. Mostowski, and B. Sobolewska, “Quantum theory of spatial and temporal coherence properties of stimulated Raman scattering,” Phys. Rev. A 32, 332 (1985).
[CrossRef] [PubMed]

1983 (1)

I. A. Walmsley and M. G. Raymer, “Observation of macroscopic quantum fluctuations in stimulated Raman scattering,” Phys. Rev. Lett. 50, 963 (1983).
[CrossRef]

1981 (1)

J. Mostowski and M. G. Raymer, “The buildup of stimulated Raman scattering from spontaneous Raman scattering,” Opt. Commun. 36, 237 (1981).
[CrossRef]

1979 (1)

F. A. Hopf and E. A. Overman, “Fluctuations in nonlinear swept-gain amplifiers,” Phys. Rev. A 19, 1180 (1979); F. A. Hopf, “Phase waves in superfluorescence,” Phys. Rev. A 20, 2064 (1979).
[CrossRef]

1978 (1)

R. J. Glauber and F. Haake, “The initiation of superfluorescence,” Phys. Lett. 68A, 29 (1978).

Bowden, C. M.

J. C. Englund and C. M. Bowden, “Spontaneous generation of Raman solitons,” Phys. Rev. Lett. 57, 2661 (1986); C. M. Bowden and J. C. Englund, “Macroscopic manifestations of quantum noise,” Opt. Commun. 67, 71 (1988).
[CrossRef] [PubMed]

Carlsten, J. L.

D. C. MacPherson, R. C. Swanson, and J. L. Carlsten, “Quantum fluctuations and correlations in the stimulated Raman scattering spectrum,” Phys. Rev. A 39, 3487 (1989).
[CrossRef] [PubMed]

Englund, J. C.

J. C. Englund and C. M. Bowden, “Spontaneous generation of Raman solitons,” Phys. Rev. Lett. 57, 2661 (1986); C. M. Bowden and J. C. Englund, “Macroscopic manifestations of quantum noise,” Opt. Commun. 67, 71 (1988).
[CrossRef] [PubMed]

Glauber, R. J.

R. J. Glauber and F. Haake, “The initiation of superfluorescence,” Phys. Lett. 68A, 29 (1978).

Haake, F.

R. J. Glauber and F. Haake, “The initiation of superfluorescence,” Phys. Lett. 68A, 29 (1978).

Hopf, F. A.

F. A. Hopf and E. A. Overman, “Fluctuations in nonlinear swept-gain amplifiers,” Phys. Rev. A 19, 1180 (1979); F. A. Hopf, “Phase waves in superfluorescence,” Phys. Rev. A 20, 2064 (1979).
[CrossRef]

Kuo, S.-J.

S.-J. Kuo, C. Radcewicz, and M. G. Raymer, “Quantum statistics of stimulated Raman scattering using a Gaussian-profile pump,” J. Opt. Soc. Am. A 4 (13), P16 (1987).

Levenson, M. D.

Li, Z. W.

M. G. Raymer, Z. W. Li, and I. A. Walmsley, “Temporal shape fluctuations in stimulated Raman scattering: coherent modes description,” Phys. Rev. Lett. 63, 1586 (1989).
[CrossRef] [PubMed]

Louisell, W. H.

W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973), p. 173.

MacPherson, D. C.

D. C. MacPherson, R. C. Swanson, and J. L. Carlsten, “Quantum fluctuations and correlations in the stimulated Raman scattering spectrum,” Phys. Rev. A 39, 3487 (1989).
[CrossRef] [PubMed]

Mostowski, J.

M. G. Raymer, I. A. Walmsley, J. Mostowski, and B. Sobolewska, “Quantum theory of spatial and temporal coherence properties of stimulated Raman scattering,” Phys. Rev. A 32, 332 (1985).
[CrossRef] [PubMed]

J. Mostowski and M. G. Raymer, “The buildup of stimulated Raman scattering from spontaneous Raman scattering,” Opt. Commun. 36, 237 (1981).
[CrossRef]

Overman, E. A.

F. A. Hopf and E. A. Overman, “Fluctuations in nonlinear swept-gain amplifiers,” Phys. Rev. A 19, 1180 (1979); F. A. Hopf, “Phase waves in superfluorescence,” Phys. Rev. A 20, 2064 (1979).
[CrossRef]

Perlmutter, S. H.

Pilipetsky, N. F.

B. Ya. Zel’dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer, New York, 1985).
[CrossRef]

Radcewicz, C.

S.-J. Kuo, C. Radcewicz, and M. G. Raymer, “Quantum statistics of stimulated Raman scattering using a Gaussian-profile pump,” J. Opt. Soc. Am. A 4 (13), P16 (1987).

Raymer, M. G.

M. G. Raymer, Z. W. Li, and I. A. Walmsley, “Temporal shape fluctuations in stimulated Raman scattering: coherent modes description,” Phys. Rev. Lett. 63, 1586 (1989).
[CrossRef] [PubMed]

S.-J. Kuo, C. Radcewicz, and M. G. Raymer, “Quantum statistics of stimulated Raman scattering using a Gaussian-profile pump,” J. Opt. Soc. Am. A 4 (13), P16 (1987).

M. G. Raymer, I. A. Walmsley, J. Mostowski, and B. Sobolewska, “Quantum theory of spatial and temporal coherence properties of stimulated Raman scattering,” Phys. Rev. A 32, 332 (1985).
[CrossRef] [PubMed]

I. A. Walmsley and M. G. Raymer, “Observation of macroscopic quantum fluctuations in stimulated Raman scattering,” Phys. Rev. Lett. 50, 963 (1983).
[CrossRef]

J. Mostowski and M. G. Raymer, “The buildup of stimulated Raman scattering from spontaneous Raman scattering,” Opt. Commun. 36, 237 (1981).
[CrossRef]

M. G. Raymer and I. A. Walmsley, in Proceedings of the Vth Rochester Conference on Coherence and Quantum Optics, L. Mandel and E. Wolf, eds. (Plenum, New York, 1984).

The application of this procedure to the Raman problem is described by M. G. Raymer and I. A. Walmsley, “The quantum coherence properties of stimulated Raman scattering,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1990), Vol. 28, pp. 216 et seq.
[CrossRef]

Richardson, W. H.

Saleh, B. E. A.

See, for example, B. E. A. Saleh, Photoelectron Statistics (Springer-Verlag, Berlin, 1978).
[CrossRef]

Shkunov, V. V.

B. Ya. Zel’dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer, New York, 1985).
[CrossRef]

Sobolewska, B.

M. G. Raymer, I. A. Walmsley, J. Mostowski, and B. Sobolewska, “Quantum theory of spatial and temporal coherence properties of stimulated Raman scattering,” Phys. Rev. A 32, 332 (1985).
[CrossRef] [PubMed]

Swanson, R. C.

D. C. MacPherson, R. C. Swanson, and J. L. Carlsten, “Quantum fluctuations and correlations in the stimulated Raman scattering spectrum,” Phys. Rev. A 39, 3487 (1989).
[CrossRef] [PubMed]

Walmsley, I. A.

M. G. Raymer, Z. W. Li, and I. A. Walmsley, “Temporal shape fluctuations in stimulated Raman scattering: coherent modes description,” Phys. Rev. Lett. 63, 1586 (1989).
[CrossRef] [PubMed]

M. G. Raymer, I. A. Walmsley, J. Mostowski, and B. Sobolewska, “Quantum theory of spatial and temporal coherence properties of stimulated Raman scattering,” Phys. Rev. A 32, 332 (1985).
[CrossRef] [PubMed]

I. A. Walmsley and M. G. Raymer, “Observation of macroscopic quantum fluctuations in stimulated Raman scattering,” Phys. Rev. Lett. 50, 963 (1983).
[CrossRef]

M. G. Raymer and I. A. Walmsley, in Proceedings of the Vth Rochester Conference on Coherence and Quantum Optics, L. Mandel and E. Wolf, eds. (Plenum, New York, 1984).

The application of this procedure to the Raman problem is described by M. G. Raymer and I. A. Walmsley, “The quantum coherence properties of stimulated Raman scattering,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1990), Vol. 28, pp. 216 et seq.
[CrossRef]

Zel’dovich, B. Ya.

B. Ya. Zel’dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer, New York, 1985).
[CrossRef]

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

S.-J. Kuo, C. Radcewicz, and M. G. Raymer, “Quantum statistics of stimulated Raman scattering using a Gaussian-profile pump,” J. Opt. Soc. Am. A 4 (13), P16 (1987).

Opt. Commun. (1)

J. Mostowski and M. G. Raymer, “The buildup of stimulated Raman scattering from spontaneous Raman scattering,” Opt. Commun. 36, 237 (1981).
[CrossRef]

Opt. Lett. (1)

Phys. Lett. (1)

R. J. Glauber and F. Haake, “The initiation of superfluorescence,” Phys. Lett. 68A, 29 (1978).

Phys. Rev. A (3)

M. G. Raymer, I. A. Walmsley, J. Mostowski, and B. Sobolewska, “Quantum theory of spatial and temporal coherence properties of stimulated Raman scattering,” Phys. Rev. A 32, 332 (1985).
[CrossRef] [PubMed]

F. A. Hopf and E. A. Overman, “Fluctuations in nonlinear swept-gain amplifiers,” Phys. Rev. A 19, 1180 (1979); F. A. Hopf, “Phase waves in superfluorescence,” Phys. Rev. A 20, 2064 (1979).
[CrossRef]

D. C. MacPherson, R. C. Swanson, and J. L. Carlsten, “Quantum fluctuations and correlations in the stimulated Raman scattering spectrum,” Phys. Rev. A 39, 3487 (1989).
[CrossRef] [PubMed]

Phys. Rev. Lett. (3)

J. C. Englund and C. M. Bowden, “Spontaneous generation of Raman solitons,” Phys. Rev. Lett. 57, 2661 (1986); C. M. Bowden and J. C. Englund, “Macroscopic manifestations of quantum noise,” Opt. Commun. 67, 71 (1988).
[CrossRef] [PubMed]

I. A. Walmsley and M. G. Raymer, “Observation of macroscopic quantum fluctuations in stimulated Raman scattering,” Phys. Rev. Lett. 50, 963 (1983).
[CrossRef]

M. G. Raymer, Z. W. Li, and I. A. Walmsley, “Temporal shape fluctuations in stimulated Raman scattering: coherent modes description,” Phys. Rev. Lett. 63, 1586 (1989).
[CrossRef] [PubMed]

Other (6)

W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973), p. 173.

B. Ya. Zel’dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer, New York, 1985).
[CrossRef]

The application of this procedure to the Raman problem is described by M. G. Raymer and I. A. Walmsley, “The quantum coherence properties of stimulated Raman scattering,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1990), Vol. 28, pp. 216 et seq.
[CrossRef]

Although ΔKx from Eq. (23) and ΔK from the coherent-mode theory have a similar numerical value for F= 3.0, it should be noted that, for a rotationally symmetric distribution,ΔK=2ΔKx.

M. G. Raymer and I. A. Walmsley, in Proceedings of the Vth Rochester Conference on Coherence and Quantum Optics, L. Mandel and E. Wolf, eds. (Plenum, New York, 1984).

See, for example, B. E. A. Saleh, Photoelectron Statistics (Springer-Verlag, Berlin, 1978).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Geometry of a Raman generator. The interaction volume is a uniformly pumped cylinder of radius a and length L. A laser pulse of mean frequency ωL is incident upon the left-hand face of the region. Radiation at the Stokes frequency ωS is generated by spontaneous scattering and builds up by means of stimulated scattering on propagation. The coordinate system in the left-hand (output) face of the region is given in the inset.

Fig. 2
Fig. 2

Examples of the intensity and phase distribution at the output face of the interaction region for F = 0.7. The intensity profile is quite smooth, with less than one node, indicating a single-peaked spatial frequency distribution. The phase is also quite smooth, but the gradients across the face produce a shift of the spatial frequency distribution, leading to an uncertainty in the direction of the beam. Note the screw dislocation in the phase front of the second realization (see Ref. 12).

Fig. 3
Fig. 3

Examples of the intensity and phase distribution at the output face of the interaction region for F = 3.0. Both the intensity and the phase profiles consist of several peaks, leading to a speckled appearance in the far-field pattern, which changes randomly from pulse to pulse.

Fig. 4
Fig. 4

Full probability density P(K) of the transverse wave vector K is shown in the upper plot. The radial probability distribution P(K) of the (radial) transverse wave vector K, for F = 1.0, is shown in the lower plot.

Fig. 5
Fig. 5

Radial density P(K) for several Fresnel numbers F, plotted on linear (a) and semilogarithmic (b) scales. The sharply peaked distributions for small Fresnel numbers indicate that beam pointing is quite stable from pulse to pulse.

Fig. 6
Fig. 6

Standard deviation of the transverse wave vector ΔK as function of Fresnel number F. The solid curve connects points calculated by means of the numerical theory, and the crosses are points calculated with Eq. (23) multiplied by a scale factor of 1.586. The beam wanders more from pulse to pulse as more coherent modes are excited, leading to larger values of ΔK. For large Fresnel numbers the directional fluctuations stabilize, indicating that the far-field intensity distribution is a speckle pattern. The asymptotic dependences of the coherent-mode theory and cell model on the Fresnel number are quite similar (2-D, two-dimensional).

Equations (37)

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F = π a 2 λ S L ,
E ^ ( - ) ( k , t ) = 1 2 π R d 2 ρ E ^ ( - ) ( ρ , t ) exp ( i k · ρ ) .
K ^ = - d t d 2 k k E ^ ( - ) ( k , t ) E ^ ( + ) ( k , t ) - d t d 2 k E ^ ( - ) ( k , t ) E ^ ( + ) ( k , t ) ,
K ^ = - i - d t R d 2 ρ E ^ ( - ) ( ρ , t ) E ^ ( + ) ( ρ , t ) - d t R d 2 ρ E ^ ( - ) ( ρ , t ) E ^ ( + ) ( ρ , t ) .
P ( K ) = : δ 2 ( K - K ^ ) : ,
C ( ξ ) = : exp ( i ξ · K ^ ) : .
P ( { ɛ ( k ) } ) = M exp [ - - d t K d 2 k × K d 2 k E ( k , t ) E * ( k , t ) / C ( k , k ) ] ,
C ( k , k ) = d t : E ^ ( - ) ( k , t ) E ^ ( + ) ( k , t ) : .
E ^ ( - ) ( ρ , t ) = ( 2 h ω a 2 c ) 1 / 2 m , p b ^ m , p ψ p ( t ) ϕ m ( ρ ) ,
b ^ m , p b ^ m , p = β m λ p δ m m δ p p ,
C ( ρ , ρ ) = a 2 c 2 π - d t E ^ ( - ) ( ρ , t ) E ^ ( + ) ( ρ , t ) ,
C ( ρ , ρ ) = m β m ϕ m ( ρ ) ϕ m * ( ρ ) .
A d 2 ρ C ( ρ , ρ ) ϕ m ( ρ ) = β m ϕ m ( ρ ) .
P ( b 1 , b 2 , b 3 , b 4 , ) = P ( b ) = n = 1 N 1 π Λ n exp { - b n 2 Λ n } = 1 π N 1 det ( Λ ) exp ( - b Λ - 1 b ) ,
K ^ = - i n m b ^ n b ^ m U n m n b ^ n b ^ n ,
U n m = A d 2 ρ ϕ n ( ρ ) ϕ m * ( ρ ) .
P ( K ) = 1 2 π Ξ d 2 ξ 1 n = 1 N ( γ n ( ξ ) - i ξ · K ) i = 1 N 1 γ n ( ξ ) - i ξ · K ,
P ( W i j ) = N 2 π W exp ( - W i j N 2 W ) ,
P ( f i j ) = ( N 2 - 1 ) ( 1 - f i j ) N 2 - 2 ,
P ( f i j , f k l ) = ( N 2 - 1 ) ( N 2 - 2 ) [ 1 - ( f i j + f k l ) ] N 2 - 3 ,
x ¯ = j = 1 N i = 1 N i 2 a N f i j .
Δ θ = Δ x ¯ 2 1 / 2 L = Δ K x k z ,
Δ K x a = 2 3 1 / 2 ( F 2 - 1 F 2 + 1 ) 1 / 2 .
P ( K ) = : δ 2 ( K - K ^ ) : ,
P ( K ) = d 2 { b } δ 2 ( K - b U b b b ) P ( b ) ,
d 2 { b } = i = 1 N d ( Re { b i } ) d ( Im ( b i } ) .
P ( K ) = d 2 { b } b b δ 2 [ b ( K 1 - U ) b ] P ( b ) ,
P ( K ) = d 2 { b } b b d 2 ξ 2 π exp [ b ( i ξ · K 1 - i ξ · U ) b ] P ( b ) .
P ( K ) = d 2 ξ 2 π d 2 { b } 1 π N ( i ξ · K ) × { exp [ b ( i ξ · K 1 - i ξ · U ) b ] exp ( - b Λ - 1 b ) det ( Λ ) } .
P ( K ) = 1 det ( Λ ) d 2 ξ 2 π ( i ξ · K ) × [ 1 det ( Λ - 1 + i ξ · U - i ξ · K 1 ) ] .
det ( Λ - 1 + i E · U - i ξ · K 1 ) = n = 1 N ( γ n ( ξ ) - i ξ · K ) ,
( i ξ · K ) [ 1 det ( Λ - 1 + i ξ · U - i ξ · K 1 ) ] = 1 n = 1 N [ γ n ( ξ ) - i ξ · K ] i = 1 N 1 γ i ( ξ ) - i ξ · K .
ϕ n ( ρ ) = exp ( i k θ ) g l ( k ) ( ρ ) exp [ - i F ( ρ 2 / a 2 ) ] ,
2 π 0 a d ρ ρ [ g l ( k ) ( ρ ) ] 2 = 1 ,
ξ · U n m = ξ · [ A d 2 ρ ϕ n * ( ρ ) ϕ m ( ρ ) ] = δ k , k + 1 [ ( ξ x - i ξ y ) ( A k k l l + i B k k l l ) + ( ξ x + i ξ y ) C k k l l ] - δ k , k - 1 { ( ξ x + i ξ y ) ( A k k l l - i B k k l l ) - ( ξ x - i ξ y ) C k k l l } ,
A k k l l = 0 a d ρ ρ g l ( k ) ( ρ ) g l ( k ) ( ρ ) ρ , B k k l l = 2 F a 2 0 a d ρ ρ 2 g l ( k ) ( ρ ) g l ( k ) ( ρ ) , C k k l l = k 0 a d ρ g l ( k ) ( ρ ) g l ( k ) ( ρ ) .
P ( K ) = 1 2 π P ( K ) = 1 2 π 0 d ξ ξ 0 2 π d θ 1 n = 1 N [ γ n ( ξ ) - i ξ K cos θ ] × i = 1 N 1 [ γ i ( ξ ) - i ξ K cos θ ] .

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