Abstract

The results of a numerical study of both the passive and the active mode-structure properties and far-field behavior of a linear, positive-branch, confocal unstable resonator with a large equivalent Fresnel number in the ideal unaberrated, intracavity phase-aberrated, and intracavity phase-corrected states are presented. A simple, saturable, gain-medium model of a homogeneously broadened CO2–electron-beam discharge laser system is employed in the active cavity study. The active cavity results presented here show that, at least for the level of gain saturation considered, the passive cavity theory of phase-aberration sensitivity remains applicable in this more-physical situation. Furthermore, the ideal correction to phase-tilt and curvature aberrations, derived from passive cavity considerations, remains exact in the presence of saturable gain. Finally, the influence of a multidithered, zonal deformable mirror on the cavity-mode structure and far-field behavior is analyzed in order to account for recent experimental tests of the intracavity adaptive optics concept.

© 1983 Optical Society of America

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References

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  1. K E. Oughstun, “Intracavity adaptive optic compensation of phase aberrations. I: Analysis,” J. Opt. Soc. Am. 71, 862–872 (1981).
    [Crossref]
  2. K.E. Oughstun, “Intracavity adaptive optic compensation of phase aberrations.“ II: Passive cavity study for a small Neqresonator,” J. Opt. Soc. Am. 71, 1180–1192 (1981).
    [Crossref]
  3. J. M. Spinhirne, D. Anafi, R. Freeman, and H. R. Garcia, “Intracavity adaptive optics. 1: Astigmatism correction performance,” Appl. Opt. 20, 976–984 (1981).
    [Crossref] [PubMed]
  4. D. Anafi, J. M. Spinhirne, R. H. Freeman, and K. E. Oughstun, “Intracavity adaptive optics. 2: Tilt correction performance,” Appl. Opt. 20, 1926–1932 (1981).
    [Crossref] [PubMed]
  5. E. A. Sziklas and A. E. Siegman, “Mode calculations in unstable resonators with flowing saturable gain. 2: Fast Fourier transform method,” Appl. Opt. 14, 1874–1889 (1975).
    [Crossref] [PubMed]
  6. A. E. Siegman and H.Y. Miller, “Unstable optical resonator loss calculations using the Prony method,” Appl. Opt. 9, 2729–2736 (1970).
    [Crossref] [PubMed]
  7. Yu. A. Anan’ev, “Unstable resonators and their applications (review),” Sov. J. Quantum Electron. 1, 565–586 (1972).
    [Crossref]
  8. V. E. Sherstobitov and G. N. Vinokurov, “Properties of unstable resonators with large equivalent Fresnel numbers,” Sov. J. Quantum Electron. 2, 224–229 (1972).
    [Crossref]
  9. Yu. A. Anan’ev, “Establishment of oscillations in unstable resonators,” Sov. J. Quantum Electron. 5, 615–617 (1975).
    [Crossref]
  10. K. E. Oughstun, K. A. Bush, and P. A. Slaymaker, “Transverse mode structure properties in off-axis ring resonators,” J. Opt. Soc. Am. 71, 1598 (A) (1981).
  11. A. Maitland and M. H. Dunn, Laser Physics (North-Holland, Amsterdam, 1969), Chap. 8.
  12. G. P. Agrawal and M. Lax, “Effects of interference on gain saturation in laser resonators,” J. Opt. Soc. Am. 69, 1717–1719 (1979).
    [Crossref]
  13. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, London, 1975).
  14. K. E. Oughstun, “Theory of intracavity adaptive optic mode control,” Proc. Soc. Photo-Opt. Instrum. Eng.365 (to be published).
  15. The interior volume of an unstable cavity is indicated by the region a in Fig. 2 of A. E. Siegman, “Orthogonality properties of optical resonator eigenmodes,” Opt. Commun. 31, 369–373 (1979).
    [Crossref]
  16. Yu. A. Anan’ev and V. E. Sherstobitov, “Influences of the edge effects on the properties of unstable resonators,” Sov. J. Quantum Electron. 1, 263–267 (1971).
    [Crossref]
  17. K. E. Oughstun, “Intracavity compensation of quadratic phase aberrations,” J. Opt. Soc. Am. 72, 1529–1537 (1982).
    [Crossref]
  18. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, S9.6, Vol. 55 of National Bureau of Standards Applied Mathematics Series (U.S. Government Printing Office, Washington, D.C., 1972).

1982 (1)

1981 (5)

1979 (2)

G. P. Agrawal and M. Lax, “Effects of interference on gain saturation in laser resonators,” J. Opt. Soc. Am. 69, 1717–1719 (1979).
[Crossref]

The interior volume of an unstable cavity is indicated by the region a in Fig. 2 of A. E. Siegman, “Orthogonality properties of optical resonator eigenmodes,” Opt. Commun. 31, 369–373 (1979).
[Crossref]

1975 (2)

1972 (2)

Yu. A. Anan’ev, “Unstable resonators and their applications (review),” Sov. J. Quantum Electron. 1, 565–586 (1972).
[Crossref]

V. E. Sherstobitov and G. N. Vinokurov, “Properties of unstable resonators with large equivalent Fresnel numbers,” Sov. J. Quantum Electron. 2, 224–229 (1972).
[Crossref]

1971 (1)

Yu. A. Anan’ev and V. E. Sherstobitov, “Influences of the edge effects on the properties of unstable resonators,” Sov. J. Quantum Electron. 1, 263–267 (1971).
[Crossref]

1970 (1)

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, S9.6, Vol. 55 of National Bureau of Standards Applied Mathematics Series (U.S. Government Printing Office, Washington, D.C., 1972).

Agrawal, G. P.

Anafi, D.

Anan’ev, Yu. A.

Yu. A. Anan’ev, “Establishment of oscillations in unstable resonators,” Sov. J. Quantum Electron. 5, 615–617 (1975).
[Crossref]

Yu. A. Anan’ev, “Unstable resonators and their applications (review),” Sov. J. Quantum Electron. 1, 565–586 (1972).
[Crossref]

Yu. A. Anan’ev and V. E. Sherstobitov, “Influences of the edge effects on the properties of unstable resonators,” Sov. J. Quantum Electron. 1, 263–267 (1971).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, London, 1975).

Bush, K. A.

K. E. Oughstun, K. A. Bush, and P. A. Slaymaker, “Transverse mode structure properties in off-axis ring resonators,” J. Opt. Soc. Am. 71, 1598 (A) (1981).

Dunn, M. H.

A. Maitland and M. H. Dunn, Laser Physics (North-Holland, Amsterdam, 1969), Chap. 8.

Freeman, R.

Freeman, R. H.

Garcia, H. R.

Lax, M.

Maitland, A.

A. Maitland and M. H. Dunn, Laser Physics (North-Holland, Amsterdam, 1969), Chap. 8.

Miller, H.Y.

Oughstun, K E.

Oughstun, K. E.

K. E. Oughstun, “Intracavity compensation of quadratic phase aberrations,” J. Opt. Soc. Am. 72, 1529–1537 (1982).
[Crossref]

D. Anafi, J. M. Spinhirne, R. H. Freeman, and K. E. Oughstun, “Intracavity adaptive optics. 2: Tilt correction performance,” Appl. Opt. 20, 1926–1932 (1981).
[Crossref] [PubMed]

K. E. Oughstun, K. A. Bush, and P. A. Slaymaker, “Transverse mode structure properties in off-axis ring resonators,” J. Opt. Soc. Am. 71, 1598 (A) (1981).

K. E. Oughstun, “Theory of intracavity adaptive optic mode control,” Proc. Soc. Photo-Opt. Instrum. Eng.365 (to be published).

Oughstun, K.E.

Sherstobitov, V. E.

V. E. Sherstobitov and G. N. Vinokurov, “Properties of unstable resonators with large equivalent Fresnel numbers,” Sov. J. Quantum Electron. 2, 224–229 (1972).
[Crossref]

Yu. A. Anan’ev and V. E. Sherstobitov, “Influences of the edge effects on the properties of unstable resonators,” Sov. J. Quantum Electron. 1, 263–267 (1971).
[Crossref]

Siegman, A. E.

Slaymaker, P. A.

K. E. Oughstun, K. A. Bush, and P. A. Slaymaker, “Transverse mode structure properties in off-axis ring resonators,” J. Opt. Soc. Am. 71, 1598 (A) (1981).

Spinhirne, J. M.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, S9.6, Vol. 55 of National Bureau of Standards Applied Mathematics Series (U.S. Government Printing Office, Washington, D.C., 1972).

Sziklas, E. A.

Vinokurov, G. N.

V. E. Sherstobitov and G. N. Vinokurov, “Properties of unstable resonators with large equivalent Fresnel numbers,” Sov. J. Quantum Electron. 2, 224–229 (1972).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, London, 1975).

Appl. Opt. (4)

J. Opt. Soc. Am. (5)

Opt. Commun. (1)

The interior volume of an unstable cavity is indicated by the region a in Fig. 2 of A. E. Siegman, “Orthogonality properties of optical resonator eigenmodes,” Opt. Commun. 31, 369–373 (1979).
[Crossref]

Sov. J. Quantum Electron. (4)

Yu. A. Anan’ev and V. E. Sherstobitov, “Influences of the edge effects on the properties of unstable resonators,” Sov. J. Quantum Electron. 1, 263–267 (1971).
[Crossref]

Yu. A. Anan’ev, “Unstable resonators and their applications (review),” Sov. J. Quantum Electron. 1, 565–586 (1972).
[Crossref]

V. E. Sherstobitov and G. N. Vinokurov, “Properties of unstable resonators with large equivalent Fresnel numbers,” Sov. J. Quantum Electron. 2, 224–229 (1972).
[Crossref]

Yu. A. Anan’ev, “Establishment of oscillations in unstable resonators,” Sov. J. Quantum Electron. 5, 615–617 (1975).
[Crossref]

Other (4)

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, S9.6, Vol. 55 of National Bureau of Standards Applied Mathematics Series (U.S. Government Printing Office, Washington, D.C., 1972).

A. Maitland and M. H. Dunn, Laser Physics (North-Holland, Amsterdam, 1969), Chap. 8.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, London, 1975).

K. E. Oughstun, “Theory of intracavity adaptive optic mode control,” Proc. Soc. Photo-Opt. Instrum. Eng.365 (to be published).

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Figures (25)

Fig. 1
Fig. 1

Linear positive-branch confocal unstable cavity configuration with an intracavity adaptive deformable mirror.

Fig. 2
Fig. 2

(a) Intracavity mode and (b) far-field irradiance distributions for the unaberrated (ideal) passive cavity.

Fig. 3
Fig. 3

Integrated relative far-field intensity about the optic axis for (a) the outcoupled geometric mode, (b) the outcoupled diffractive mode of the unaberrated passive cavity, and (c) the outcoupled diffractive mode of the unaberrated ideal active cavity.

Fig. 4
Fig. 4

Intracavity phase-tilt aberrated irradiance distribution incident upon the convex feedback mirror (left) and the resulting far-field irradiance distribution (right) for several increasing values of the applied aberration strength.

Fig. 5
Fig. 5

Intracavity-mode phase tilt relative to (a) the critical-angle phase tilt and (b) the phase-tilt-aberration weighting factor as a function of the applied phase-tilt strength. The approximate geometric behavior is indicated by the dotted lines, and the exact numerical behavior is indicated by the solid curves.

Fig. 6
Fig. 6

Intracavity irradiance distribution incident upon the convex feedback mirror for the λ phase-tilt-perturbed (left) and intracavity-compensated (right) cavities.

Fig. 7
Fig. 7

First four eigenvalue magnitudes as functions of the applied phase-tilt-aberration strength for the aberrated and intracavity-correlated (indicated by the dotted curves) cavity configurations.

Fig. 8
Fig. 8

Geometric phase front of the mλ tilt aberrated cavity mode incident upon the convex feedback mirror.

Fig. 9
Fig. 9

Far-field intensity centroid coordinate along the x direction about the aligned optic axis as a function of the applied phase-tilt strength. The centroid coordinate has been normalized with respect to the radius rminG of the Airy disk that is due to the outcoupled geometric mode of the unaberrated cavity.

Fig. 10
Fig. 10

Integrated relative far-field intensity of the outcoupled mode structure for (a) the λ/8 and (b) the λ/2 phase-tilt-aberrated cavities taken about the optic axis (solid curves) and far-field beam centroid coordinates (short-dashed curves). The long-dashed curves represent the intracavity-corrected performance in each case.

Fig. 11
Fig. 11

(a) Far-field Strehl ratios and (b) relative beam quality as functions of the applied phase-tilt-aberration strength.

Fig. 12
Fig. 12

Intracavity-astigmatism-aberrated irradiance distribution incident upon the convex feedback mirror (left) and the resulting far-field irradiance distribution (right) for several increasing values of the applied aberration strength.

Fig. 13
Fig. 13

Phase-curvature-weighting factor as a function of the applied intracavity curvature-aberration strength as given by the approximate geometric result (dashed line) and as determined numerically from the aberrated phase of the cavity mode structure (solid curves).

Fig. 14
Fig. 14

Intracavity irradiance distribution incident upon the convex feedback mirror for the λ/4 astigmatism-perturbed (left) and intracavity-compensated (right) cavities.

Fig. 15
Fig. 15

Behavior of (a) the cavity magnification and (b) equivalent Fresnel number as a function of the applied phase-curvature-aberration strength.

Fig. 16
Fig. 16

First four eigenvalue magnitudes as functions of the applied astigmatism-aberration strength for the aberrated and intracavity-corrected (indicated by the dotted curves) cavity configurations.

Fig. 17
Fig. 17

(a) Far-field Strehl ratios and (b) relative beam quality as functions of the applied astigmatism-aberration strength.

Fig. 18
Fig. 18

Radial cross section of (a) the intracavity irradiance and phase distributions for both the ideal unaberrated active and passive cavity modes and (b) the degree of saturation I/Isat of the unaberrated active cavity.

Fig. 19
Fig. 19

Intracavity power, outcoupled power, and power in the bucket as a function of the applied phase-tilt-aberration strength for the active cavity.

Fig. 20
Fig. 20

Far-field irradiance patterns that are due to the (a) aberrated, (b) extracavity-compensated, and (c) intracavity-corrected modes of the active cavity for the λ/2 phase-tilt-perturbation case.

Fig. 21
Fig. 21

Intracavity power, outcoupled power, and power in the bucket as a function of the applied astigmatic-aberration strength for the active cavity.

Fig. 22
Fig. 22

Far-field irradiance patterns that are due to the (a) aberrated, (b) extracavity-compensated, and (c) intracavity-corrected modes of the active cavity for the λ/4 astigmatism case. Notice the scale change in the intracavity-corrected case.

Fig. 23
Fig. 23

Integrated relative far-field intensity of the outcoupled mode structure for the intentionally unaberrated baseline resonator and the intracavity adaptive optic resonator with best flat deformable mirror profiles DM1 and DM3.

Fig. 24
Fig. 24

Intracavity mode and far-field irradiance structures of the intracavity adaptive optic resonator with best flat deformable mirror profiles DM1 (left) and DM3 (right).

Fig. 25
Fig. 25

Time-averaged Strehl-intensity degradation as a function of the relative dither amplitude m = A/λ for several values of the total number N of zonal elements.

Tables (1)

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Table 1 Important Parameters for the Confocal Positive-Branch Unstable Resonator Cavity Treated in This Numerical Study

Equations (78)

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D calc = 2 G a 2 = 10.574 cm ,
γ 1 = | γ 1 | = 0.796 , arg ( γ ¯ 1 ) = 0.025 rad , ( l = 0 ) , γ 2 = | γ 2 | = 0.703 , arg ( γ 2 ) = 2.59 rad , ( l = 1 ) , γ 3 = | γ 3 | = 0.679 , arg ( γ 3 ) = 2.21 rad , ( l = 0 ) , γ 4 = | γ 4 | = 0.534 , arg ( γ 4 ) = 0.604 rad ,
G = 1 M = 0.695 ,
r min G = 0.913 λ f D ,
α ¯ 1 ( z 1 ) = 1 M 1 , ( m + 1 + ( M 1 ) z 1 z τ ) = 5.70 ,
m c = m c α ¯ 1 ( z 1 , δ 1 ) ,
m c m c α ¯ 1 ( z 1 ) = M α ¯ 1 2 ( z 1 ) N eq .
1 α ¯ 1 ( z 1 , δ 1 ) α ¯ 1 ( z ) , | δ 1 | < m c λ ,
α ¯ 1 ( z 1 , δ 1 ) α ¯ 1 ( z 1 ) , | δ 1 | > m c λ .
β 1 = δ 1 M ( 2 z 1 + z 2 + z 3 ) + z 2 + z 3 M ( 2 z 1 + 2 z 2 + z 3 ) + z 3 = 0.88402 δ 1 ,
m p ( 1 ) 0.228 α ¯ 1 ( z 1 ) ,
m p ( 1 ) < m c
δ N eq = ( Δ + Δ ) / λ = 2 α ¯ 1 ( z 1 ) m M .
Destructive interference : m I ( 1 ) = M 2 α ¯ 1 ( z 1 ) ( n + ½ ) ,
Constructive interference : m S ( 1 ) = M 2 α ¯ 1 ( z 1 ) n .
m I ( 1 ) M m p ( 1 )
x ¯ p G r min G = 2 m α ¯ 1 ( z 1 ) 0.9125 .
m 1 2 α ¯ 1 ( z 1 ) .
α ¯ 1 ( z 1 ) = 1 M 2 1 [ M 2 + ( 1 + ( M 1 ) z 1 z τ ) 2 ] = 2.997 ,
α ¯ 2 ( z 1 , δ 2 ) = α ¯ 2 ( z 1 , δ 2 ) ,
1 α ¯ 2 ( z 1 , δ 2 ) α ¯ 2 ( z 1 ) .
β 2 ( + δ 2 ) = Γ 1 ( δ 2 ) + [ Γ 1 2 ( δ 2 ) + 4 Γ 0 ( δ 2 ) Γ 2 ( δ 2 ) ] 1 / 2 2 Γ 2 ( δ 2 )
β 2 ( δ 2 ) = Γ 1 ( δ 2 ) + [ Γ 1 2 ( δ 2 ) 4 Γ 0 ( δ 2 ) Γ 2 ( δ 2 ) ] 1 / 2 2 Γ 2 ( δ 2 )
Δ M ( + δ 2 ) = Δ M ( δ 2 ) .
N eq c = N eq .
N eq p = N eq 2 m α ¯ 2 ( z 1 , δ 2 ) M 2 ,
N eq p ( θ ) = N eq 2 m α ¯ 2 ( z 1 , δ 2 ) M 2 cos ( 2 θ ) ,
m I ( 2 ) = M 2 2 α ¯ 2 ( z 1 , δ 2 ) | N eq N eq p |
M 2 2 α ¯ 2 ( z 1 ) | N eq N eq p | .
m p ( 2 ) 0.228 α ¯ 2 ( z 1 ) ,
m I ( 2 ) M 2 m p ( 2 )
g = g 0 1 + I / I sat ,
g 0 = 0.01 cm 1 , I sat = 3.404 W / cm 2 .
γ F = 0.725 ,
β 2 = α ¯ 2 ( z 1 , δ 2 ) β 2
g ( r ) = g 0 1 + I / I sat f ( r ) ,
f ( r ) = 1 a r 2 + b r 4 c r 6 ; r D 2
a = 1.5606 × 10 2 / cm 2 , b = 8.664 × 10 4 / cm 4 , c = 2.249 × 10 5 / cm 6 ,
Φ TOT ( r , θ , t ) = j = 1 N Φ j ( r , θ ; r j , θ j ) sin ( ω j t + ψ j ) ,
Φ j ( r , θ ; r j , θ j ) = A j k = 1 l = 0 k δ j k l ( r j , σ j , z c ) r k cos l ( θ θ j ) ,
i ( P ) 1 ( 2 π λ ) 2 ( Δ Φ ) 2 ,
i ( P , t ) 1 ( 2 π λ ) 2 2 1 2 j = 1 N m = 1 N A j A m × sin ( ω j t + ψ j ) sin ( ω m t + ψ m ) k = 1 n = 1 × [ ( 1 k + n + 2 k + n + 2 2 ( 1 k + 2 ) ( 1 n + 2 ) ( 1 2 ) ( k + 2 ) ( n + 2 ) ) δ j k 0 δ m n 0 + 1 k + n + 2 2 ( k + n + 2 ) l = 1 Min ( n , k ) δ j k l δ m n l cos l ( θ j θ m ) ] .
i ( P ) 1 ( 2 π λ ) 2 1 1 2 j = 1 N A j 2 k = 1 n = 1 × [ ( 1 k + n + 2 k + n + 2 2 ( 1 k + 2 ) ( 1 n + 2 ) ( 1 2 ) ( k + 2 ) ( n + 2 ) ) δ j k 0 δ j n 0 + 1 k + n + 2 2 ( k + n + 2 ) l = 1 Min ( n , k ) δ j k l δ j n l ] ,
1 k + n + 2 k + n + 2 2 ( 1 k + 2 ) ( 1 n + 2 ) ( 1 2 ) ( k + 2 ) ( n + 2 ) ,
i ( P , t ) = 1 ; sin ( ω j t + ψ j ) = 0 , j = 1 , 2 , , N ,
A j = m λ , j = 1 , 2 , , N ,
i ( P ) 1 2 π 2 m 2 { ( 1 + 2 ) a 2 2 α ¯ 1 2 ( z c ) j = 1 N r j 2 σ j 4 × exp [ 2 ( r j / σ j ) 2 ] + ( 1 6 3 ( 1 2 ) ( 1 2 ) 2 4 ) a 2 4 α ¯ 2 2 ( z c ) × j = 1 N 1 σ j 4 ( r j 2 σ j 2 1 ) 2 exp [ 2 ( r j / σ j ) 2 ] + } .
σ ( π N ) 1 / 2 a 2 .
i ( P ) 1 1 2 α ¯ 1 2 ( z c ) ( 1 + 2 ) m 2 N 2 exp ( N / 2 π ) × [ 1 + ( 1 6 3 ( 1 4 ) 1 + 2 4 ) α ¯ 2 2 ( z c ) α ¯ 1 2 ( z c ) ( N 4 π 1 ) 2 + ] 1 1 2 α ¯ 1 2 ( z c ) ( 1 + 2 ) m 2 N 2 exp ( N / 2 π ) ,
N 4 π .
α ¯ k = α ¯ k ( M , z , δ k , N eq ) .
α ¯ k α ¯ k ( M , z ) as δ k 0
α ¯ k ( M , z ) α ¯ k ( M , z , δ k , N eq ) 1
m c = m c α ¯ 1 M α ¯ 1 2 ( M , z ) N eq .
m p ( k ) = 0.228 α ¯ k , k = 1 , 2 .
m I ( k ) M k m p ( k )
m p ( 2 ) m p ( 1 ) ,
m I ( 2 ) m I ( 1 ) ,
α ¯ k = α ¯ k ( M , z , δ k , N eq , g 0 , I sat ) ,
δ k , in = δ k , out α ¯ k
ϕ ( r , θ ; r j , θ j ) = A j exp [ 1 σ j 2 ( r 2 + r j 2 ) ] × exp [ 2 σ j 2 r r j cos ( θ θ j ) ] ,
exp ( β cos θ ) = l = I l ( β ) exp ( i l θ ) = I 0 ( β ) + 2 l = 1 I l ( β ) cos ( l θ ) ,
I l ( β ) = ( 1 2 β ) ) l k = 0 1 k ! Γ ( k + l + 1 ) ( 1 4 β 2 ) k .
Φ ( r , θ ; r j , θ j ) = A j { k = 0 m = 0 n n = 0 ( 1 ) n k ! n ! Γ ( k + 1 ) σ j 2 ( n + 2 k ) × ( m n ) r j 2 ( k + n m ) r 2 ( k + m ) + 2 l = 1 k = 0 m = 0 n n = 0 ( 1 ) n k ! n ! Γ ( k + l + 1 ) σ j 2 ( n + l + 2 k ) × ( m n ) r j 2 ( k + n m ) + l r 2 ( k + m ) + l cos l ( θ θ j ) } ,
Φ j ( r , θ ; r j , θ j ) = A j { k = 0 m = 0 n n = 0 ( 1 ) n r j 2 ( k + n m ) k ! n ! Γ ( k + 1 ) σ j 2 ( n + 2 k ) × ( m n ) α ¯ 2 ( k + m ) ( z c ) r 2 ( k + m ) + 2 l = 1 k = 0 m = 0 n n = 0 ( 1 ) n r j 2 ( k + n m ) + l k ! n ! Γ ( k + l + 1 ) σ j 2 ( n + l + 2 k ) × ( m n ) α ¯ 2 ( k + m ) + l ( z c ) r 2 ( k + m ) + l cos l ( θ θ j ) } ,
α ¯ p ( z c ) = α ¯ p G ( z c ) δ p o ,
Φ j ( r , θ ; r j , θ j ) = A j k = 1 l = 0 k δ j k l ( r j , σ j , z c ) r k cos l ( θ θ j ) .
δ j 11 = 2 r j σ j 2 exp [ ( r j / σ j ) 2 ] α ¯ 1 ( z c ) ,
δ j 20 = 1 σ j 2 ( r j 2 σ j 2 1 ) exp [ ( r j / σ j ) 2 ] α ¯ 2 ( z c ) ,
δ j 22 = r j 2 σ j 4 exp [ ( r j / σ j ) 2 ] α ¯ 2 ( z c ) ,
δ j 31 = r j σ j 4 ( r j 2 σ j 2 2 ) exp [ ( r j / σ j ) 2 ] α ¯ 3 ( z c ) ,
δ j 33 = r j 3 3 σ j 6 exp [ ( r j / σ j ) 2 ] α ¯ 3 ( z c ) ,
δ j 40 = 1 4 σ j 4 ( r j 2 σ j 2 2 r j 2 σ j 2 + 2 ) exp [ ( r j / σ j ) 2 ] α ¯ 4 ( z c ) ,
Φ ¯ = 1 π ( 1 2 ) 1 0 2 π Φ TOT ( ρ , θ , t ) ρ d ρ d θ = 2 1 2 j = 1 N A j sin ( ω j t + ψ j ) k = 1 δ j k 0 1 k + 2 k + 2 .
Φ ¯ 2 = 1 π ( 1 2 ) 1 0 2 π Φ TOT 2 ( ρ , θ , t ) ρ d ρ d θ = 1 π ( 1 2 ) j = 1 N m = 1 N A j A m sin ( ω j t + ψ j ) sin ( ω m t + ψ m ) × k = 1 n = 1 l = 0 k p = 0 n δ j k l δ m n p 1 ρ k + n + 1 d ρ × 0 2 π cos l ( θ θ j ) cos p ( θ θ m ) d θ .
0 2 π cos l ( θ θ j ) cos p ( θ θ m ) d θ = π [ δ ( p + l ) cos ( l θ j + p θ m ) + δ ( p l ) cos ( l θ j p θ m ) ] ,
Φ ¯ 2 = 1 1 2 j = 1 N m = 1 N A j A m sin ( ω j t + ψ j ) sin ( ω m t + ψ m ) × k = 1 n = 1 1 k + n + 2 k + n + 2 [ δ j k 0 δ m n 0 + l = 0 Min ( n , k ) δ j k l δ m n l × cos l ( θ j θ m ) ] ,
i ( p , t ) 1 ( 2 π λ ) 2 2 1 2 j = 1 N m = 1 N × A j A m sin ( ω j t + ψ j ) sin ( ω m t + ψ m ) × k = 1 n = 1 [ ( 1 k + n + 2 k + n + 2 2 ( 1 k + 2 ) ( 1 n + 2 ) ( 1 2 ) ( k + 2 ) ( n + 2 ) ) × δ j k 0 δ m n 0 + 1 k + n + 2 2 ( k + n + 2 ) l = 1 Min ( n , k ) × δ j k l δ m n l cos l ( θ j θ m ) ] .