Abstract

Scattering of polychromatic light by a medium whose dielectric susceptibility is a random function of position is considered within the accuracy of the first Born approximation. It is shown, in particular, that if the two-point spatial correlation function of the dielectric susceptibility has Gaussian form and the spectrum of the incident light has a Gaussian profile, the spectrum of the scattered light may be shifted toward the shorter or the longer wavelengths, depending on the angle of scattering. The results are analogous to those derived recently in connection with radiation from partially coherent sources [ Nature (London) 326, 363 ( 1987)].

© 1989 Optical Society of America

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Corrections

E. Wolf, J. T. Foley, and F. Gori, "Frequency shifts of spectral lines produced by scattering from spatially random media:errata," J. Opt. Soc. Am. A 7, 173-173 (1990)
https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-7-1-173

References

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  1. E. Wolf, “Invariance of spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
    [CrossRef] [PubMed]
  2. See also L. Mandel, “Concept of cross-spectral purity in coherence theory,”J. Opt. Soc. Am. 51, 1342–1350 (1961); L. Mandel, E. Wolf, “Spectral coherence and the concept of cross-spectral purity,”J. Opt. Soc. Am. 66, 529–535 (1976); F. Gori, R. Grella, “Shape invariant propagation of polychromatic fields,” Opt. Commun. 49, 173–177 (1984).
    [CrossRef]
  3. G. M. Morris, D. Faklis, “Effects of source correlation on the spectrum of light,” Opt. Commun. 62, 5–11 (1987).
    [CrossRef]
  4. E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature (London) 326, 363–365 (1987).
    [CrossRef]
  5. E. Wolf, “Red shifts and blue shifts of spectral lines caused by source correlations,” Opt. Commun. 62, 12–16 (1987).
    [CrossRef]
  6. Z. Dacic, E. Wolf, “Changes in the spectrum of partially coherent light beam propagating in free space,” J. Opt. Soc. Am. A 5, 1118–1126 (1988).
    [CrossRef]
  7. D. Faklis, G. M. Morris, “Spectral shifts produced by source correlations,” Opt. Lett. 13, 4–6 (1988).
    [CrossRef] [PubMed]
  8. F. Gori, G. Guattari, C. Palma, G. Padovani, “Observation of optical redshifts and blueshifts produced by source correlations,” Opt. Commun. 67, 1–4 (1988).
    [CrossRef]
  9. W. H. Knox, R. S. Knox, “Direct observation of the optical Wolf shift using white-light interferometry,” J. Opt. Soc. Am. A 4(13), P131 (1987).
  10. M. F. Bocko, D. H. Douglass, R. S. Knox, “Observation of frequency shifts of spectral lines due to source correlations,” Phys. Rev. Lett. 58, 2649–2651 (1987).
    [CrossRef] [PubMed]
  11. A. Gamliel, E. Wolf, “Spectral modulation by control of source correlations,” Opt. Commun. 65, 91–96 (1988); see also E. Wolf, “Red shifts and blue shifts of spectral lines emitted by two correlated sources,” Phys. Rev. Lett. 58, 2646–2648 (1987).
    [CrossRef] [PubMed]
  12. E. Wolf, “New Theory of partial coherence in the space-frequency domain. Part I: spectra and cross-spectra of steady state sources,”J. Opt. Soc. Am. 72, 343–351 (1982); “Part II: steady-state fields and higher-order correlations,” J. Opt. Soc. Am. A 3, 76–85 (1986).
    [CrossRef]
  13. See, for example, P. Roman, Advanced Quantum Theory (Addison-Wesley, Reading, Mass., 1965), Sec. 3.2.
  14. L. G. Shirley, N. George, “Diffuser radiation patterns over a large dynamic range. 1: Strong diffusers,” Appl. Opt. 27, 1850–1861 (1988), Sec. II.
    [CrossRef] [PubMed]
  15. See also J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty ed. (Springer, New York, 1984), Sec. 2.6.1.
  16. T. S. McKechnie, “Speckle reduction,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty ed. (Springer, New York, 1984), Sec. 4.3.
  17. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Secs. 2.2.1 and 2.2.3.
  18. It follows from Eqs. (4.8e) and (4.8c) that for a given value of θthe magnitude of this red shift depends on the product of the two ratios σ/ƛ0(the correlation length of the medium in units of ƛ0) and Γ0/ω0(the linewidth of the incident light in units of ω0). Since the linewidth of the incident light is Γ0, its coherence length, L0, may be defined by the expression L0= c/Γ0. Therefore (σ/ƛ0)(Γ0/ω0) = σ/L0, and the expression (4.8e) may be rewritten as α(θ)={1+[2(σL0)sin(θ2)]2}1/2. Hence, for fixed θ, α(θ) is a monotonically increasing function of the ratio σ/L0; consequently Eq. (4.8c) implies that the center frequency ω˜ of the Gaussian factor in Eq. (4.7) is a monotonically decreasing function of this ratio.
  19. For the sake of simplicity we do not show explicitly the dependence of Γ1(nor of the quantities ω˜,Γ˜, and αdefined below) on θthroughout the main part of this appendix.

1988 (5)

Z. Dacic, E. Wolf, “Changes in the spectrum of partially coherent light beam propagating in free space,” J. Opt. Soc. Am. A 5, 1118–1126 (1988).
[CrossRef]

D. Faklis, G. M. Morris, “Spectral shifts produced by source correlations,” Opt. Lett. 13, 4–6 (1988).
[CrossRef] [PubMed]

F. Gori, G. Guattari, C. Palma, G. Padovani, “Observation of optical redshifts and blueshifts produced by source correlations,” Opt. Commun. 67, 1–4 (1988).
[CrossRef]

A. Gamliel, E. Wolf, “Spectral modulation by control of source correlations,” Opt. Commun. 65, 91–96 (1988); see also E. Wolf, “Red shifts and blue shifts of spectral lines emitted by two correlated sources,” Phys. Rev. Lett. 58, 2646–2648 (1987).
[CrossRef] [PubMed]

L. G. Shirley, N. George, “Diffuser radiation patterns over a large dynamic range. 1: Strong diffusers,” Appl. Opt. 27, 1850–1861 (1988), Sec. II.
[CrossRef] [PubMed]

1987 (5)

W. H. Knox, R. S. Knox, “Direct observation of the optical Wolf shift using white-light interferometry,” J. Opt. Soc. Am. A 4(13), P131 (1987).

M. F. Bocko, D. H. Douglass, R. S. Knox, “Observation of frequency shifts of spectral lines due to source correlations,” Phys. Rev. Lett. 58, 2649–2651 (1987).
[CrossRef] [PubMed]

G. M. Morris, D. Faklis, “Effects of source correlation on the spectrum of light,” Opt. Commun. 62, 5–11 (1987).
[CrossRef]

E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature (London) 326, 363–365 (1987).
[CrossRef]

E. Wolf, “Red shifts and blue shifts of spectral lines caused by source correlations,” Opt. Commun. 62, 12–16 (1987).
[CrossRef]

1986 (1)

E. Wolf, “Invariance of spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
[CrossRef] [PubMed]

1982 (1)

1961 (1)

Bocko, M. F.

M. F. Bocko, D. H. Douglass, R. S. Knox, “Observation of frequency shifts of spectral lines due to source correlations,” Phys. Rev. Lett. 58, 2649–2651 (1987).
[CrossRef] [PubMed]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Secs. 2.2.1 and 2.2.3.

Dacic, Z.

Douglass, D. H.

M. F. Bocko, D. H. Douglass, R. S. Knox, “Observation of frequency shifts of spectral lines due to source correlations,” Phys. Rev. Lett. 58, 2649–2651 (1987).
[CrossRef] [PubMed]

Faklis, D.

D. Faklis, G. M. Morris, “Spectral shifts produced by source correlations,” Opt. Lett. 13, 4–6 (1988).
[CrossRef] [PubMed]

G. M. Morris, D. Faklis, “Effects of source correlation on the spectrum of light,” Opt. Commun. 62, 5–11 (1987).
[CrossRef]

Gamliel, A.

A. Gamliel, E. Wolf, “Spectral modulation by control of source correlations,” Opt. Commun. 65, 91–96 (1988); see also E. Wolf, “Red shifts and blue shifts of spectral lines emitted by two correlated sources,” Phys. Rev. Lett. 58, 2646–2648 (1987).
[CrossRef] [PubMed]

George, N.

Goodman, J. W.

See also J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty ed. (Springer, New York, 1984), Sec. 2.6.1.

Gori, F.

F. Gori, G. Guattari, C. Palma, G. Padovani, “Observation of optical redshifts and blueshifts produced by source correlations,” Opt. Commun. 67, 1–4 (1988).
[CrossRef]

Guattari, G.

F. Gori, G. Guattari, C. Palma, G. Padovani, “Observation of optical redshifts and blueshifts produced by source correlations,” Opt. Commun. 67, 1–4 (1988).
[CrossRef]

Knox, R. S.

M. F. Bocko, D. H. Douglass, R. S. Knox, “Observation of frequency shifts of spectral lines due to source correlations,” Phys. Rev. Lett. 58, 2649–2651 (1987).
[CrossRef] [PubMed]

W. H. Knox, R. S. Knox, “Direct observation of the optical Wolf shift using white-light interferometry,” J. Opt. Soc. Am. A 4(13), P131 (1987).

Knox, W. H.

W. H. Knox, R. S. Knox, “Direct observation of the optical Wolf shift using white-light interferometry,” J. Opt. Soc. Am. A 4(13), P131 (1987).

Mandel, L.

McKechnie, T. S.

T. S. McKechnie, “Speckle reduction,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty ed. (Springer, New York, 1984), Sec. 4.3.

Morris, G. M.

D. Faklis, G. M. Morris, “Spectral shifts produced by source correlations,” Opt. Lett. 13, 4–6 (1988).
[CrossRef] [PubMed]

G. M. Morris, D. Faklis, “Effects of source correlation on the spectrum of light,” Opt. Commun. 62, 5–11 (1987).
[CrossRef]

Padovani, G.

F. Gori, G. Guattari, C. Palma, G. Padovani, “Observation of optical redshifts and blueshifts produced by source correlations,” Opt. Commun. 67, 1–4 (1988).
[CrossRef]

Palma, C.

F. Gori, G. Guattari, C. Palma, G. Padovani, “Observation of optical redshifts and blueshifts produced by source correlations,” Opt. Commun. 67, 1–4 (1988).
[CrossRef]

Roman, P.

See, for example, P. Roman, Advanced Quantum Theory (Addison-Wesley, Reading, Mass., 1965), Sec. 3.2.

Shirley, L. G.

Wolf, E.

A. Gamliel, E. Wolf, “Spectral modulation by control of source correlations,” Opt. Commun. 65, 91–96 (1988); see also E. Wolf, “Red shifts and blue shifts of spectral lines emitted by two correlated sources,” Phys. Rev. Lett. 58, 2646–2648 (1987).
[CrossRef] [PubMed]

Z. Dacic, E. Wolf, “Changes in the spectrum of partially coherent light beam propagating in free space,” J. Opt. Soc. Am. A 5, 1118–1126 (1988).
[CrossRef]

E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature (London) 326, 363–365 (1987).
[CrossRef]

E. Wolf, “Red shifts and blue shifts of spectral lines caused by source correlations,” Opt. Commun. 62, 12–16 (1987).
[CrossRef]

E. Wolf, “Invariance of spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
[CrossRef] [PubMed]

E. Wolf, “New Theory of partial coherence in the space-frequency domain. Part I: spectra and cross-spectra of steady state sources,”J. Opt. Soc. Am. 72, 343–351 (1982); “Part II: steady-state fields and higher-order correlations,” J. Opt. Soc. Am. A 3, 76–85 (1986).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Secs. 2.2.1 and 2.2.3.

Appl. Opt. (1)

J. Opt. Soc. Am. (2)

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

Z. Dacic, E. Wolf, “Changes in the spectrum of partially coherent light beam propagating in free space,” J. Opt. Soc. Am. A 5, 1118–1126 (1988).
[CrossRef]

W. H. Knox, R. S. Knox, “Direct observation of the optical Wolf shift using white-light interferometry,” J. Opt. Soc. Am. A 4(13), P131 (1987).

Nature (London) (1)

E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature (London) 326, 363–365 (1987).
[CrossRef]

Opt. Commun. (4)

E. Wolf, “Red shifts and blue shifts of spectral lines caused by source correlations,” Opt. Commun. 62, 12–16 (1987).
[CrossRef]

G. M. Morris, D. Faklis, “Effects of source correlation on the spectrum of light,” Opt. Commun. 62, 5–11 (1987).
[CrossRef]

F. Gori, G. Guattari, C. Palma, G. Padovani, “Observation of optical redshifts and blueshifts produced by source correlations,” Opt. Commun. 67, 1–4 (1988).
[CrossRef]

A. Gamliel, E. Wolf, “Spectral modulation by control of source correlations,” Opt. Commun. 65, 91–96 (1988); see also E. Wolf, “Red shifts and blue shifts of spectral lines emitted by two correlated sources,” Phys. Rev. Lett. 58, 2646–2648 (1987).
[CrossRef] [PubMed]

Opt. Lett. (1)

Phys. Rev. Lett. (2)

E. Wolf, “Invariance of spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
[CrossRef] [PubMed]

M. F. Bocko, D. H. Douglass, R. S. Knox, “Observation of frequency shifts of spectral lines due to source correlations,” Phys. Rev. Lett. 58, 2649–2651 (1987).
[CrossRef] [PubMed]

Other (6)

See, for example, P. Roman, Advanced Quantum Theory (Addison-Wesley, Reading, Mass., 1965), Sec. 3.2.

See also J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty ed. (Springer, New York, 1984), Sec. 2.6.1.

T. S. McKechnie, “Speckle reduction,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty ed. (Springer, New York, 1984), Sec. 4.3.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Secs. 2.2.1 and 2.2.3.

It follows from Eqs. (4.8e) and (4.8c) that for a given value of θthe magnitude of this red shift depends on the product of the two ratios σ/ƛ0(the correlation length of the medium in units of ƛ0) and Γ0/ω0(the linewidth of the incident light in units of ω0). Since the linewidth of the incident light is Γ0, its coherence length, L0, may be defined by the expression L0= c/Γ0. Therefore (σ/ƛ0)(Γ0/ω0) = σ/L0, and the expression (4.8e) may be rewritten as α(θ)={1+[2(σL0)sin(θ2)]2}1/2. Hence, for fixed θ, α(θ) is a monotonically increasing function of the ratio σ/L0; consequently Eq. (4.8c) implies that the center frequency ω˜ of the Gaussian factor in Eq. (4.7) is a monotonically decreasing function of this ratio.

For the sake of simplicity we do not show explicitly the dependence of Γ1(nor of the quantities ω˜,Γ˜, and αdefined below) on θthroughout the main part of this appendix.

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Figures (4)

Fig. 1
Fig. 1

Illustrating the notation relating to the asymptotic approximation (2.9).

Fig. 2
Fig. 2

Spectrum [in units of N(r)] of light scattered at various angles θ, from a Gaussian correlated medium with rms width σ = 3ƛλ0, λ0= 5500Å (ω0 = 3.427 × 1015 sec−1). The spectrum of the incident light is a line with Gaussian profile of rms width Γ0 = 10−2ω0). The curve labeled θ = 0 also represents the spectrum [in units of N(r)] for the case when the scatterer is completely uncorrelated (σ = 0) [see Eq. (4.4)].

Fig. 3
Fig. 3

Height factor [H(θ)] [Eq. (4.8b)], with Γ0/ω0 = 10−2. The values of the correlation parameter σ0 are indicated on the curves.

Fig. 4
Fig. 4

Relative frequency shift z(θ), (a), and the difference ζ(θ) = z(θ) − zuncorr, (b), of the spectral line of light scattered at various angles θ from a Gaussian-correlated medium with rms width σ. The spectrum of the incident light is a line with a Gaussian profile of rms width Γ0 = 10−2ω0.

Equations (102)

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W ( i ) ( r 1 , r 2 , ω ) = U ( i ) * ( r 1 , ω ) U ( i ) ( r 2 , ω ) ,
U ( i ) ( r , ω ) = a ( ω ) exp ( i k s ^ 0 · r ) ,
k = ω c ,
S ( i ) ( r , ω ) W ( i ) ( r , r , ω ) = U ( i ) * ( r , ω ) U ( i ) ( r , ω )
S ( i ) ( ω ) S ( i ) ( r , ω ) = a * ( ω ) a ( ω )
U ( s ) ( r , ω ) = a ( ω ) V F ( r , ω ) G ( r , r , ω ) exp ( i k s ^ 0 · r ) d 3 r ,
F ( r , ω ) = ( ω / c ) 2 η ( r , ω ) ,
G ( r , r , ω ) = exp [ i k ( r - r ) ] r - r
G ( r , r , ω ) ~ e i k r r exp ( - i k s ^ · r )             ( k r ) .
U ( ) ( r s ^ , ω ) = a ( ω ) e i k r r V F ( r , ω ) exp [ - i k ( s ^ - s ^ 0 ) · r ] d 3 r ,
F ˜ ( K , ω ) = V F ( r , ω ) exp ( - i K · r ) d 3 r ,
U ( ) ( r s ^ , ω ) = a ( ω ) F ˜ [ k ( s ^ - s ^ 0 ) , ω ] e i k r r ,
U ( ) ( r s ^ , ω ) = a ( ω ) ( ω c ) 2 η ˜ [ k ( s ^ - s ^ 0 ) , ω ] e i k r r ,
S ( ) ( r s ^ , ω ) = U ( ) * ( r s ^ , ω ) U ( ) ( r s ^ , ω ) .
S ( ) ( r s ^ , ω ) = 1 r 2 ( ω c ) 4 η ˜ * [ k ( s ^ - s ^ 0 ) , ω ] η ˜ [ k ( s ^ - s ^ 0 ) , ω ] S ( i ) ( ω ) .
S ( ) ( r s ^ , ω ) = 1 r 2 ( ω c ) 4 η ˜ * [ k ( s ^ - s ^ 0 ) , ω ] η ˜ [ k ( s ^ - s ^ 0 ) , ω ] η S ( i ) ( ω ) .
η ˜ * ( K 1 , ω ) η ˜ ( K 2 , ω ) η = V V C η ( r 1 , r 2 , ω ) × exp [ - i ( K 2 · r 2 - K 1 · r 1 ) ] d 3 r 1 d 3 r 2 ,
C η ( r 1 , r 2 ω ) = η * ( r 1 , ω ) η ( r 2 , ω ) η
C ˜ η ( K 1 , K 2 , ω ) = V V C η ( r 1 , r 2 , ω ) × exp [ - i ( K 1 · r 1 + K 2 · r 2 ) ] d 3 r 1 d 3 r 2 ,
η ˜ * ( K 1 , ω ) η ˜ ( K 2 , ω ) η = C ˜ η ( - K 1 , K 2 , ω ) .
S ( ) ( r s ^ , ω ) = 1 r 2 ( ω / c ) 4 C ˜ η [ - k ( s ^ - s ^ 0 ) , k ( s ^ - s ^ 0 ) , ω ] S ( i ) ( ω ) .
C η ( r 1 , r 2 , ω ) = I ( ω ) δ ( 3 ) ( r 2 - r 1 ) when r 1 V , r 2 V = 0 otherwise ,
C ˜ η [ - k ( s ^ - s ^ 0 ) , k ( s ^ - s ^ 0 ) , ω ] = I ( ω ) V ,
[ S ( ) ( r s ^ , ω ) ] uncorr = V r 2 ( ω c ) 4 I ( ω ) S ( i ) ( ω ) .
C η ( r 1 , r 2 , ω ) = C η ( r 2 - r 1 , ω ) when r 1 V , r 2 V = 0 otherwise .
C ˜ η ( K 1 , K 2 , ω ) = V V C η ( r 1 - r 2 , ω ) exp [ - i ( K 1 · r 1 + K 2 · r 2 ) ] d 3 r 1 d 3 r 2 ,
C ˜ η [ - k ( s ^ - s ^ 0 ) , k ( s ^ - s ^ 0 ) , ω ] = V V C η ( r 2 - r 1 , ω ) exp [ - i k ( s ^ - s ^ 0 ) · ( r 2 - r 1 ) ] d 3 r 1 d 3 r 2 .
r = ( r 1 + r 2 ) / 2 ,             r = ( r 2 - r 1 )
C ˜ η [ - k ( s ^ - s ^ 0 ) , k ( s ^ - s ^ 0 ) , ω ] V C ˜ η [ k ( s ^ - s ^ 0 ) , ω ] ,
C ˜ η ( K , ω ) = V C η ( r , ω ) exp ( - i K · r ) d 3 r .
S ( ) ( r s ^ , ω ) V r 2 ( ω c ) 2 C ˜ η [ k ( s ^ - s ^ 0 ) , ω ] S ( i ) ( ω ) .
C η ( r , ω ) = A ( 2 π σ 2 ) 3 / 2 exp ( - r 2 / 2 σ 2 ) ,
C ˜ η ( K , ω ) = A exp [ - ( K σ ) 2 / 2 ] .
K 2 = 4 ( ω / c ) 2 sin 2 ( θ / 2 ) ,
C ˜ η [ k ( s ^ - s ^ 0 ) , ω ] = A exp [ - 2 ( ω c ) 2 σ 2 sin 2 ( θ 2 ) ] ,
S ( ) ( r s ^ , ω ) = A V r 2 ( ω c ) 4 exp [ - 2 ( ω c ) 2 σ 2 sin 2 ( θ 2 ) ] S ( i ) ( ω ) .
S ( i ) ( ω ) = B exp [ - ( ω - ω 0 ) 2 2 Γ 0 2 ] ,
S ( ) ( r , θ ; ω ) = N ( r ) H ( θ ) ω 4 exp { - 1 2 [ ω - ω ˜ ( θ ) Γ ˜ ( θ ) ] 2 } ,
N ( r ) = V A B c 4 r 2 ,
H ( θ ) = exp [ - 2 α 2 ( θ ) ( σ ƛ 0 ) 2 sin 2 ( θ 2 ) ] ,
ω ˜ ( θ ) = ω 0 α 2 ( θ ) ,
Γ ˜ ( θ ) = Γ 0 α ( θ ) ,
α ( θ ) = { 1 + [ 2 ( σ ƛ 0 ) ( Γ 0 ω 0 ) sin ( θ 2 ) ] 2 } 1 / 2
ƛ 0 = λ 0 2 π .
ω ˜ ( θ ) < ω 0 ,             Γ ˜ ( θ ) < Γ 0 ,
ω 0 ( θ ) = ω 0 2 α 2 ( θ ) { 1 + [ 1 + α 2 ( θ ) ( 4 Γ 0 ω 0 ) 2 ] 1 / 2 } .
z = λ 0 - λ 0 λ 0 = ω 0 - ω 0 ω 0 ,
z ( θ ) = 2 α 2 ( θ ) - { 1 + [ 1 + α 2 ( θ ) ( 4 Γ 0 ω 0 ) 2 ] 1 / 2 } 1 + [ 1 + α 2 ( θ ) ( 4 Γ 0 ω 0 ) 2 ] 1 / 2 .
z uncorr = 1 - [ 1 + ( 4 Γ 0 ω 0 ) 2 ] 1 / 2 1 + [ 1 + ( 4 Γ 0 ω 0 ) 2 ] 1 / 2 .
( 4 Γ 0 / ω 0 ) 2 1 ,             σ / ƛ 0 = O ( 1 )
z ( θ ) 4 ( Γ 0 ω 0 ) 2 [ ( σ ƛ 0 ) 2 sin 2 ( θ / 2 ) - 1 ] = 4 [ ( σ L 0 ) 2 sin 2 ( θ 2 ) - ( Γ 0 ω 0 ) 2 ] ,
z G ( θ ) ω 0 - ω ˜ ( θ ) ω ˜ ( θ ) ,
z G ( θ ) 4 ( σ L 0 ) 2 sin 2 ( θ 2 ) .
G ( ω - ω j ; Γ j ) = exp [ - ( ω - ω j ) 2 / 2 Γ j 2 ]             ( j = 1 , 2 ) ,
G ( ω - ω 1 ; Γ 1 ) G ( ω - ω 2 ; Γ 2 ) = G [ ω 1 - ω 2 ; ( Γ 1 2 + Γ 2 2 ) 1 / 2 ] G ( ω - ω ˜ ; Γ ˜ ) ,
ω ˜ = ω 1 Γ 2 2 + ω 2 Γ 1 2 Γ 1 2 + Γ 2 2 ,
1 Γ ˜ 2 = 1 Γ 1 2 + 1 Γ 2 2 .
G ( ω - ω 1 ; Γ 1 ) G ( ω - ω 2 ; Γ 2 ) = exp [ - g ( ω ) ] ,
g ( ω ) = 1 2 Γ 1 2 Γ 2 2 [ Γ 2 2 ( ω - ω 1 ) 2 + Γ 1 2 ( ω - ω 2 ) 2 ] = 1 2 Γ 1 2 Γ 2 2 ( a 2 ω 2 - 2 b ω + c )
a 2 = Γ 1 2 + Γ 2 2 ,
b = ω 1 Γ 2 2 + ω 2 Γ 1 2 ,
c = ω 1 2 Γ 2 2 + ω 2 2 Γ 1 2 .
g ( ω ) = a 2 2 Γ 1 2 Γ 2 2 ( ω - b a 2 ) 2 + 1 2 Γ 1 2 Γ 2 2 ( c - b 2 a 2 ) = 1 2 Γ ˜ 2 ( ω - ω ˜ ) 2 + 1 2 Γ 1 2 Γ 2 2 ( c - b 2 a 2 ) ,
1 Γ ˜ 2 = a 2 Γ 1 2 Γ 2 2 = 1 Γ 1 2 + 1 Γ 2 2 ,
ω ˜ = b a 2 = ω 1 Γ 2 2 + ω 2 Γ 1 2 Γ 1 2 + Γ 2 2 .
c - b 2 a 2 = Γ 1 2 Γ 2 2 Γ 1 2 + Γ 2 2 ( ω 1 - ω 2 ) 2 .
G ( ω - ω 1 ; Γ 1 ) G ( ω - ω 2 ; Γ 2 ) = exp [ - ( ω 1 - ω 2 ) 2 2 ( Γ 1 2 + Γ 2 2 ) ] exp [ - ( ω - ω ˜ ) 2 2 Γ ˜ 2 ] = G [ ω 1 - ω 2 ; ( Γ 1 2 + Γ 2 2 ) 1 / 2 ] G ( ω - ω ˜ ; Γ ˜ ) ,
ω ˜ = f 1 ω 1 + f 2 ω 2 ,
f 1 = Γ 2 2 Γ 1 2 + Γ 2 2 ,             f 2 = Γ 1 2 Γ 1 2 + Γ 2 2 .
ω 1 < ω ˜ < ω 2 .
Γ ˜ < Γ 1 ,             Γ ˜ < Γ 2 .
S ( ) ( r s ^ , ω ) = A V r 2 ( ω c ) 4 exp ( - ω 2 2 Γ 1 2 ) B exp [ - ( ω - ω 0 ) 2 2 Γ 0 2 ] ,
1 Γ 1 2 = 4 c 2 σ 2 sin 2 ( θ 2 ) .
G ( ω - ω j ; Γ j ) = exp [ - ( ω - ω j ) 2 2 Γ j 2 ] ,
N = V A B c 4 r 2 ,
S ( ) ( r s ^ , ω ) = N ω 4 G ( ω ; Γ 1 ) G ( ω - ω 0 ; Γ 0 ) .
G ( ω ; Γ 1 ) G ( ω - ω 0 ; Γ 0 ) = G [ ω 0 ; ( Γ 1 2 + Γ 2 2 ) 1 / 2 ] G ( ω - ω ˜ ; Γ ) ,
ω ˜ = ω 0 Γ 1 2 Γ 0 2 + Γ 1 2
1 Γ ˜ 2 = 1 Γ 0 2 + 1 Γ 1 2 .
ω ˜ ω 0 = 1 1 + ( Γ 0 / Γ 1 ) 2 ,
Γ ˜ Γ 0 = 1 [ 1 + ( Γ 0 / Γ 1 ) 2 ] 1 / 2 .
Γ 0 Γ 1 = 2 c σ Γ 0 sin ( θ 2 ) ,
Γ 0 Γ 1 = 2 ( σ ƛ 0 ) ( Γ 0 ω 0 ) sin ( θ 2 ) .
ω ˜ ω 0 = 1 α 2 ,
Γ ˜ Γ 0 = 1 α ,
α = { 1 + [ 2 ( σ ƛ 0 ) ( Γ 0 ω 0 ) sin ( θ 2 ) ] 2 } 1 / 2 .
Γ 0 2 + Γ 1 2 = Γ 1 2 [ 1 + ( Γ 0 Γ 1 ) 2 ] = Γ 1 2 ( Γ 0 Γ ˜ ) 2
Γ 0 2 + Γ 1 2 = α 2 Γ 1 2 .
G [ ω 0 ; ( Γ 0 2 + Γ 1 2 ) 1 / 2 ] = exp [ - ω 0 2 2 α 2 Γ 1 2 ]
G [ ω 0 ; ( Γ 0 2 + Γ 1 2 ) 1 / 2 ] = exp [ - 2 α 2 ( σ ƛ 0 ) 2 sin 2 ( θ 2 ) ] .
G ( ω ; Γ 1 ) G ( ω - ω 0 ; Γ 0 ) = exp [ - 2 α 2 ( σ ƛ 0 ) 2 sin 2 ( θ 2 ) ] G ( ω - ω ˜ ; Γ ˜ ) .
S ( ) ( r s ^ , ω ) = N ( r ) H ( θ ) ω 4 exp { - 1 2 [ ω - ω ˜ ( θ ) Γ ˜ ( θ ) ] 2 } ,
H ( θ ) = exp [ - 2 α 2 ( θ ) ( σ ƛ 0 ) 2 sin 2 ( θ 2 ) ] .
S ( ) ( r , θ ; ω ) = N ( r ) H ( θ ) f ( ω , θ ) ,
f ( ω , θ ) = ω 4 exp { - 1 2 [ ω - ω ˜ ( θ ) Γ ˜ ( θ ) ] 2 } ,
f ( ω , θ ) ω = [ 4 ω 3 - ω - ω ˜ Γ ˜ 2 ω 4 ] exp [ - 1 2 ( ω - ω ˜ Γ ˜ ) 2 ] .
f ( ω , θ ) ω = 0
4 - ω ± - ω ˜ Γ ˜ 2 ω ± = 0 ,
ω ± = 1 2 ω ˜ { 1 ± [ 1 + ( 4 Γ ˜ ω ˜ ) 2 ] 1 / 2 } .
ω ± = ω 0 2 α 2 { 1 ± [ 1 + α 2 ( 4 Γ 0 ω 0 ) 2 ] 1 / 2 } .
ω 0 ( θ ) = ω 0 2 α 2 { 1 + [ 1 + α 2 ( 4 Γ 0 ω 0 ) 2 ] 1 / 2 } .
α(θ)={1+[2(σL0)sin(θ2)]2}1/2.

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