Abstract

The effect of spatial coherence of a source on the spectrum of the emitted light is studied for a class of beamlike fields. The source is assumed to be planar, secondary, and quasi-homogeneous. We consider in detail the situation in which the source spectrum is a line with a Gaussian profile and the degree of spectral coherence is characterized by a Gaussian distribution. The changes that the spectrum undergoes as the emitted light propagates from the source plane to the far zone are illustrated by computed curves. Two factors are shown to contribute to changes in the spectrum: effects due to the finite size of the source, which tend to shift the emitted spectral line toward the shorter wavelengths (a blue shift), and effects due to source correlations, which tend to shift the line toward longer wavelengths (a red shift). The magnitude of the resulting shift in the far zone depends on the direction of observation. These results are in qualitative agreement with the recent experimental observations of Faklis and Morris [ Opt. Lett. 13, 4 ( 1988)].

© 1988 Optical Society of America

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References

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  1. E. Wolf, “Invariance of spectrum on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
    [CrossRef] [PubMed]
  2. E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature 326, 363–365 (1987).
    [CrossRef]
  3. E. Wolf, “Redshifts and blueshifts of spectral lines caused by source correlations,” Opt. Commun. 62, 12–16 (1987).
    [CrossRef]
  4. L. Mandel, “Concept of cross-spectral purity in coherence theory,”J. Opt. Soc. Am. 51, 1342–1350 (1961).
    [CrossRef]
  5. L. Mandel, E. Wolf, “Spectral coherence and the concept of cross-spectral purity,”J. Opt. Soc. Am. 66, 529–535 (1976).
    [CrossRef]
  6. F. Gori, R. Grella, “Shape-invariant propagation of polychromatic fields,” Opt. Commun. 49, 173–177 (1984).
    [CrossRef]
  7. E. Wolf, “Red shifts and blue shifts of spectral lines emitted by two correlated sources,” Phys. Rev. Lett. 58, 2646–2648 (1987).
    [CrossRef] [PubMed]
  8. G. M. Morris, D. Faklis, “Effects of source correlation on the spectrum of light,” Opt. Commun. 62, 5–11 (1987).
    [CrossRef]
  9. 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]
  10. W. H. Carter, E. Wolf, “Coherence and radiometry with quasihomogeneous planar sources,”J. Opt. Soc. Am. 67, 785–796 (1977).
    [CrossRef]
  11. The definitions of the cross-spectral density used in the present paper and in Refs. 10 and 12 differ trivially from each other by the interchange of the roles of their two spatial arguments.
  12. W. H. Carter, E. Wolf, “Fields generated by homogeneous and by quasi-homogeneous planar secondary sources,” Opt. Commun. 50, 131–136 (1984).
    [CrossRef]
  13. That the spectral transfer function (2.21) for propagation to the far zone is now also independent of ω follows at once from the fact, established in Ref. 1, that for a planar, secondary, quasi-homogeneous source that obeys the scaling law, μ˜(0)has necessarily the functional formμ˜(0)(ks⊥,ω)=1k2h˜(s⊥).
  14. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).
  15. Alternatively, the formula (5.4) may be derived from the general expression (5.3) if we approximate the function 2F1(−n, −n− 1, 1; x) by the last term in its series expansion (5.3a), with x= ρ2/a2. The series on the right-hand side can then be summed, with the result given by the expression (5.6).
  16. Stated somewhat differently, this effect is caused by the well-known factor 1/λ = ω/2πc in front of the diffraction integral in the mathematical representation of the Huygens–Fresnel principle. In the picturesque language of the elementary diffraction theory this fact is expressed by saying that the amplitudes of the secondary waves of Huygens’s principle are in the ratio 1/λ to the amplitude of the primary wave.
  17. The fluorescence of a dye laser would, of course, not exhibit frequency shifts unless appropriate source correlations were introduced.
  18. This prediction is in qualitative agreement with results of recent experiments carried out by D. Faklis, G. M. Morris, “Spectral shifts produced by source correlations,” Opt. Lett. 13, 4–6 (1988).
    [CrossRef] [PubMed]
  19. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. 8.5.2.
  20. See, for example, G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, 1960), p. 20, Eq. (5), with an obvious substitution.

1988 (1)

1987 (5)

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

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

E. Wolf, “Red shifts and blue shifts of spectral lines emitted by two correlated sources,” Phys. Rev. Lett. 58, 2646–2648 (1987).
[CrossRef] [PubMed]

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

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]

1986 (1)

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

1984 (2)

F. Gori, R. Grella, “Shape-invariant propagation of polychromatic fields,” Opt. Commun. 49, 173–177 (1984).
[CrossRef]

W. H. Carter, E. Wolf, “Fields generated by homogeneous and by quasi-homogeneous planar secondary sources,” Opt. Commun. 50, 131–136 (1984).
[CrossRef]

1977 (1)

1976 (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), Sec. 8.5.2.

Carter, W. H.

W. H. Carter, E. Wolf, “Fields generated by homogeneous and by quasi-homogeneous planar secondary sources,” Opt. Commun. 50, 131–136 (1984).
[CrossRef]

W. H. Carter, E. Wolf, “Coherence and radiometry with quasihomogeneous planar sources,”J. Opt. Soc. Am. 67, 785–796 (1977).
[CrossRef]

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.

Gori, F.

F. Gori, R. Grella, “Shape-invariant propagation of polychromatic fields,” Opt. Commun. 49, 173–177 (1984).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).

Grella, R.

F. Gori, R. Grella, “Shape-invariant propagation of polychromatic fields,” Opt. Commun. 49, 173–177 (1984).
[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]

Mandel, L.

Morris, G. M.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).

Watson, G. N.

See, for example, G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, 1960), p. 20, Eq. (5), with an obvious substitution.

Wolf, E.

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

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

E. Wolf, “Red shifts and blue shifts of spectral lines emitted by two correlated sources,” Phys. Rev. Lett. 58, 2646–2648 (1987).
[CrossRef] [PubMed]

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

W. H. Carter, E. Wolf, “Fields generated by homogeneous and by quasi-homogeneous planar secondary sources,” Opt. Commun. 50, 131–136 (1984).
[CrossRef]

W. H. Carter, E. Wolf, “Coherence and radiometry with quasihomogeneous planar sources,”J. Opt. Soc. Am. 67, 785–796 (1977).
[CrossRef]

L. Mandel, E. Wolf, “Spectral coherence and the concept of cross-spectral purity,”J. Opt. Soc. Am. 66, 529–535 (1976).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. 8.5.2.

J. Opt. Soc. Am. (3)

Nature (1)

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

Opt. Commun. (4)

E. Wolf, “Redshifts and blueshifts 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, R. Grella, “Shape-invariant propagation of polychromatic fields,” Opt. Commun. 49, 173–177 (1984).
[CrossRef]

W. H. Carter, E. Wolf, “Fields generated by homogeneous and by quasi-homogeneous planar secondary sources,” Opt. Commun. 50, 131–136 (1984).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. Lett. (3)

E. Wolf, “Red shifts and blue shifts of spectral lines emitted by two correlated sources,” Phys. Rev. Lett. 58, 2646–2648 (1987).
[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]

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

Other (8)

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. 8.5.2.

See, for example, G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, 1960), p. 20, Eq. (5), with an obvious substitution.

The definitions of the cross-spectral density used in the present paper and in Refs. 10 and 12 differ trivially from each other by the interchange of the roles of their two spatial arguments.

That the spectral transfer function (2.21) for propagation to the far zone is now also independent of ω follows at once from the fact, established in Ref. 1, that for a planar, secondary, quasi-homogeneous source that obeys the scaling law, μ˜(0)has necessarily the functional formμ˜(0)(ks⊥,ω)=1k2h˜(s⊥).

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).

Alternatively, the formula (5.4) may be derived from the general expression (5.3) if we approximate the function 2F1(−n, −n− 1, 1; x) by the last term in its series expansion (5.3a), with x= ρ2/a2. The series on the right-hand side can then be summed, with the result given by the expression (5.6).

Stated somewhat differently, this effect is caused by the well-known factor 1/λ = ω/2πc in front of the diffraction integral in the mathematical representation of the Huygens–Fresnel principle. In the picturesque language of the elementary diffraction theory this fact is expressed by saying that the amplitudes of the secondary waves of Huygens’s principle are in the ratio 1/λ to the amplitude of the primary wave.

The fluorescence of a dye laser would, of course, not exhibit frequency shifts unless appropriate source correlations were introduced.

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Figures (5)

Fig. 1
Fig. 1

Illustrating the notation.

Fig. 2
Fig. 2

Illustrating the effect of spectral coherence of the source on the normalized far-zone spectrum of the emitted light. The normalized source spectrum s(0)(ω) is a line of Gaussian profile with δ0/ω0 = 1/20. The degree of spectral coherence of the source has Gaussian form with σ = 10 λ0. The normalized far-zone spectrum s ( ) ( θ , ω ) s ( ) ( r s ˆ , ω ) is plotted (a) on axis, (b) at θ = 2°, and (c) at θ = 3°.

Fig. 3
Fig. 3

The far-zone spectral intensity S ( ) ( θ , ω ) S ( ) ( r s ˆ , ω ) as functions of θ for (a) ω = ω0, (b) ω = ω0 + δ0, (c) ω = ω0δ0, (d) ω = ω0 + 2δ0, and (e) ω = ω0 + 2δ0. The source parameters have the same values as in Fig. 2. The values of S(∞) are given in units of M = 2π2B(a/z)2 (σ0)2.

Fig. 4
Fig. 4

Evolution of the normalized spectrum s(ρ, z, ω) along the line θ = ρ/z = 3°. The normalized source spectrum s(0)(ω) is a line of Gaussian profile with δ0/ω0 = 1/20. The degree of spectral coherence of the source has Gaussian form, with σ = 10λ0. The normalized spectrum in the plane (a) z = 50a, (b) z = 100a, and (c) in the far zone. It is assumed that aσ is the source radius.

Fig. 5
Fig. 5

The normalized source spectrum s(0)(ω) and the normalized field spectrum s(ρ, z, ω) plotted (a) on axis, (b) off-axis distance ρ = 0.0524z, and (c) off-axis distance ρ = 0.034z, in a plane at a distance z = 100a from the source. The source parameters have the same values as in Fig. 4.

Equations (90)

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W ( 0 ) ( ρ 1 , ρ 2 , ω ) = [ S ( 0 ) ( ρ 1 , ω ) ] 1 / 2 [ S ( 0 ) ( ρ 2 , ω ) ] 1 / 2 μ ( 0 ) ( ρ 1 , ρ 2 , ω ) .
W ( 0 ) ( ρ 1 , ρ 2 , ω ) = S ( 0 ) [ ( ρ 1 + ρ 2 ) / 2 , ω ] μ ( 0 ) ( ρ 2 ρ 1 , ω ) .
S ( ) ( r s ˆ , ω ) = ( 2 π k r ) 2 S ˜ ( 0 ) ( 0 , ω ) μ ˜ ( 0 ) ( k s , ω ) cos 2 θ ,
k = ω / c ,
S ˜ ( 0 ) ( f , ω ) = 1 ( 2 π ) 2 S ( 0 ) ( ρ , ω ) exp ( i f · ρ ) d 2 ρ ,
μ ˜ ( 0 ) ( f , ω ) = 1 ( 2 π ) 2 μ ( 0 ) ( ρ , ω ) exp ( i f · ρ ) d 2 ρ ,
W ( r 1 , r 2 , ω ) = S ˜ ( 0 ) ( f 2 f 1 , ω ) μ ˜ ( 0 ) [ ( f 1 + f 2 ) / 2 , ω ] × exp [ i ( f 2 · r 2 f 1 * · r 1 ) ] d 2 f 1 d 2 f 2 ,
f j ( f j x , f j y , f j z ) , f j ( f j x , f j y , 0 ) ,
f j z = ( k 2 f j 2 ) 1 / 2 when f j 2 k 2
= i ( f j 2 k 2 ) 1 / 2 when f j 2 k 2
S ( r , ω ) = S ˜ ( 0 ) ( f 2 f 1 , ω ) μ ˜ ( 0 ) [ ( f 1 + f 2 ) / 2 , ω ] × exp [ i ( f 2 f 1 * ) · r ] d 2 f 1 d 2 f 2 .
S ( ρ , z , ω ) = S ˜ ( 0 ) ( f 2 f 1 , ω ) μ ˜ ( 0 ) [ ( f 1 + f 2 ) / 2 , ω ] × exp [ i ( f 2 f 1 ) · ρ ] exp [ i ( f 2 z f 1 z * ) z ] × d 2 f 1 d 2 f 2 .
| f j z | k f j 2 2 k ( j = 1 , 2 ) .
S ( ρ , z , ω ) = S ˜ ( 0 ) ( f 2 f 1 , ω ) μ ˜ ( 0 ) [ ( f 1 + f 2 ) / 2 , ω ] × exp [ i ( f 2 f 1 ) · ρ ] exp [ i z 2 k ( f 2 2 f 1 2 ) ] × d 2 f 1 d 2 f 2 .
S ˜ ( 0 ) ( f , ω ) = S ( 0 ) ( ω ) N ˜ ( f ) ,
N ˜ ( f ) = 1 ( 2 π ) 2 D exp ( i f · ρ ) d 2 ρ .
S ( ρ , z , ω ) = S ( 0 ) ( ω ) T ( ρ , z , ω ) ,
T ( ρ , z , ω ) = N ˜ ( f 2 f 1 ) μ ˜ ( 0 ) [ ( f 1 + f 2 ) / 2 , ω ] × exp [ i ( f 2 f 1 ) · ρ ] exp [ i z 2 k ( f 2 2 f 1 2 ) ] × d 2 f 1 d 2 f 2 .
f = f 2 f 1 , f = ( f 2 + f 1 ) / 2 .
T ( ρ , z , ω ) = N ˜ ( f ) μ ˜ ( 0 ) ( f , ω ) exp ( i f · ρ ) × exp [ i z k f · f ] d 2 f d 2 f .
T ( ρ , z , ω ) = μ ( 0 ) ( f z / k , ω ) N ˜ ( f ) exp ( i f · ρ ) d 2 f .
S ˜ ( 0 ) ( 0 , ω ) = A ( 2 π ) 2 S ( 0 ) ( ω ) ,
S ( ) ( r s ˆ , ω ) = S ( 0 ) ( ω ) T ( ) ( r s ˆ , ω ) ,
T ( ) ( r s ˆ , ω ) = k 2 A r 2 μ ˜ ( 0 ) ( k s ˆ , ω ) cos 2 θ
s ( ρ , z , ω ) = S ( ρ , z , ω ) 0 S ( ρ , z , ω ) d ω .
0 s ( ρ , z , ω ) d ω = 1.
s ( ρ , z , ω ) = S ( 0 ) ( ω ) T ( ρ , z , ω ) 0 S ( 0 ) ( ω ) T ( ρ , z , ω ) d ω ,
S ( ) ( r s ˆ , ω ) = k 2 S ( 0 ) ( ω ) μ ˜ ( 0 ) ( k s , ω ) 0 k 2 S ( 0 ) ( ω ) μ ˜ ( 0 ) ( k s , ω ) d ω .
μ ( 0 ) ( ρ , ω ) = h ( k ρ ) ( k = ω / c ) ,
T ( ρ , z , ω ) = h ( f z ) N ( f ) exp ( i f · ρ ) d 2 f .
s ( ρ , z , ω ) = s ( 0 ) ( ω )
μ ( 0 ) ( ρ 2 ρ 1 , ω ) = exp [ ( ρ 2 ρ 1 ) 2 / 2 σ 2 ] .
T ( ρ , z , ω ) = a 0 J 0 ( ρ u ) J 1 ( a u ) exp [ 1 2 ( u z k σ ) 2 ] d u .
T ( ρ , z , ω ) = 1 2 n = 0 1 ( n + 1 ) ! ( 1 2 ) n ( k σ a z ) 2 ( n + 1 ) × 2 F 1 ( n , n 1 , 1 ; ρ 2 / a 2 ) ,
F 2 ( n , n 1 , 1 ; x ) = k = 0 n ( n k ) ( n + 1 k ) x k .
μ ˜ ( 0 ) ( f , ω ) = σ 2 2 π exp ( σ 2 f 2 / 2 ) .
T ( ) ( r s ˆ , ω ) = 1 2 ( k a σ r ) 2 cos 2 θ exp { [ ( k σ ) 2 sin 2 θ ] / 2 } ,
T ( ) ( r s ˆ , ω ) 1 2 ( k a σ z ) 2 exp { [ ( k σ ) 2 sin 2 θ ] / 2 } ,
S ( 0 ) ( ω ) = B exp [ ( ω ω 0 ) 2 / 2 δ 0 2 ] ( δ 0 / ω 0 1 )
S ( ) ( r s ˆ , ω ) = B ω 2 2 ( a σ c z ) 2 exp [ ( ω ω 0 ) 2 / 2 δ 0 2 ] × exp { [ ( ω σ / c ) 2 sin 2 θ ] / 2 } ,
( ω ω 0 ) 2 2 δ 0 2 + ( ω σ / c ) 2 sin 2 θ 2 = α 2 ( ω ω 0 ) 2 + β 2 ,
ω 0 = ω 0 2 α 2 δ 0 2 ,
α 2 = 1 2 ( 1 δ 0 2 + σ 2 c 2 sin 2 θ ) ,
β 2 = ( ω 0 σ 2 α δ 0 c ) 2 sin 2 θ .
ω 0 = ω 0 1 + ( σ ƛ 0 ) 2 ( δ 0 ω 0 ) 2 sin 2 θ ,
α 2 = 1 2 δ 0 2 [ 1 + ( σ ƛ 0 ) 2 ( δ 0 ω 0 ) 2 sin 2 θ ] ,
β 2 = 1 2 ( σ ƛ 0 ) 2 sin 2 θ [ 1 + ( σ ƛ 0 ) 2 ( δ 0 ω 0 ) 2 sin 2 θ ] .
S ( ) ( r s ˆ , ω ) = 1 2 B ( ω ω 0 ) 2 ( σ ƛ 0 ) 2 ( a z ) 2 exp ( β 2 ) × exp [ α 2 ( ω ω 0 ) 2 ] .
s ( ) ( r s ˆ , ω ) = ω 2 M exp [ α 2 ( ω ω 0 ) 2 ] ,
M = 0 ω 2 exp [ α 2 ( ω ω 0 ) 2 ] d ω .
1 / α ω 0 ,
M ω 2 exp [ α 2 ( ω ω 0 ) 2 ] d ω .
δ 0 / ω 0 1 ,
M π α ( ω 0 2 + 1 2 α 2 ) .
s ˆ ( ) ( r s , ω ) = α π ω 2 ω 0 2 + 1 / 2 α 2 exp [ α 2 ( ω ω 0 ) 2 ] .
ω 0 ω 0
ω ¯ ( θ ) ω 0 [ 2 ( δ 0 ω 0 ) 2 + 1 1 + ( σ ƛ 0 ) 2 ( δ 0 ω 0 ) 2 sin 2 θ ] .
Δ ω ( θ ) = ω ¯ ( θ ) ω 0 .
Δ ω ( θ ) = ω 0 [ 2 ( δ 0 ω 0 ) 2 ( σ ƛ 0 ) 2 ( δ 0 ω 0 ) 2 sin 2 θ 1 + ( σ ƛ 0 ) 2 ( δ 0 ω 0 ) 2 sin 2 θ ] .
sin θ c = 2 ( ƛ 0 σ )
( ω 0 σ c ) 2 sin 2 θ 0 = 2.
sin θ 0 = 2 ƛ 0 σ .
θ c = θ 0 .
N ( ρ ) = { 1 when ρ < a 0 when ρ > a .
N ˜ ( f ) = a 2 4 π [ 2 J 1 ( a f ) a f ] ( f = | f | ) ,
T ( ρ , z , ω ) = a 2 2 π [ J 1 ( a f ) a f ] exp [ 1 2 ( f z k σ ) 2 ] × exp ( i f · ρ ) d 2 f .
f u cos θ , u sin θ , ρ ρ cos Φ , ρ sin Φ .
T ( ρ , z , ω ) = a 2 2 π 0 0 2 π [ J 1 ( a u ) a u ] exp [ 1 2 ( u z k σ ) 2 ] × exp [ i u ρ cos ( θ Φ ) ] u d u d θ .
0 2 π exp ( i x cos Ψ ) d Ψ = 2 π J 0 ( x ) ,
T ( ρ , z , ω ) = a 0 J 0 ( ρ u ) J 1 ( a u ) exp [ 1 2 ( u z k σ ) 2 ] d u ,
1 / α ω 0 .
1 + ( σ ƛ 0 ) 2 ( δ 0 ω 0 ) 2 sin 2 θ 1 2 ( ω 0 δ 0 ) 2 .
( σ ƛ 0 ) 2 sin 2 θ 1 2 ( ω 0 δ 0 ) 4 .
θ max ~ 2 λ max σ ,
λ max λ 0 ( ω 0 δ 0 ) 2 ,
ω 0 ω min ( ω 0 δ 0 ) 2 ,
ω 0 ω 0 δ ( ω 0 δ 0 ) 2
1 ( δ 0 ω 0 ) ( δ 0 ω 0 ) 2 .
δ 0 / ω 0 1 ,
M = ω 2 exp [ α 2 ( ω ω 0 ) 2 ] d ω .
M = ω 0 2 exp ( α 2 Ω 2 ) d Ω + 2 ω 0 Ω exp ( α 2 Ω 2 ) d Ω + Ω 2 exp ( α 2 Ω 2 ) d Ω .
M = π α ( ω 0 2 + 1 2 α 2 ) .
d s ( ) d ω = 2 ω C ( 1 + α 2 ω 0 ω α 2 ω 2 ) exp [ α 2 ( ω ω 0 ) 2 ] ,
C = α π 1 ω 0 2 + 1 / 2 α 2 .
α 2 ω 2 α 2 ω ω 0 1 = 0.
ω = ω 0 2 { 1 ± [ 1 + ( 2 / ω 0 α ) 2 ] 1 / 2 } .
ω ¯ = ω 0 2 { 1 + [ 1 + ( 2 / ω 0 α ) 2 ] 1 / 2 } .
ω ¯ ω 0 [ 1 + 1 ( ω 0 α ) 2 ] .
ω ¯ ( θ ) ω 0 [ 2 ( δ 0 ω 0 ) 2 + 1 1 + ( σ ƛ 0 ) 2 ( δ 0 ω 0 ) 2 sin 2 θ ] .
μ˜(0)(ks,ω)=1k2h˜(s).

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