Abstract

The performance of the optically phase-locked electronic speckle pattern interferometer (OPL-ESPI) in the presence of random noise displacements is analyzed. It is shown that a sufficient condition for high visibility of signal speckle contours is that the signal pattern lie within a region of completely spatially coherent noise, i.e., where the random components of the surface vibrations have equal amplitudes and are in phase, and that the region contain the OPL-ESPI system lock point. Under these conditions the effects of the noise vibrations within the region are canceled by the action of the OPL-ESPI optical phase-locked loop, and the complete signal speckle contour pattern is visible. The single-frame speckle contrast and the root-mean-square exposure for sequentially subtracted frames are evaluated for stationary, coherence-separable noise displacements that obey Gauss–Markov temporal statistics. It is shown that these functions decrease as the noise temporal bandwidth and the mean-square displacement difference between the observation point and the lock point increase. Experimental results that illustrate the theoretical development are presented.

© 1989 Optical Society of America

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References

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  1. O. J. Lokberg, G. A. Slettemoen, “Basic electronic speckle pattern interferometry,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1987). Vol. 10, Chap. 8.
  2. R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, London, 1983).
  3. S. E. Moran, R. L. Law, P. N. Craig, W. M. Goldberg, “Optically phase-locked electronic speckle pattern interferometer,” Appl. Opt. 26, 475–491 (1987).
    [CrossRef] [PubMed]
  4. O. J. Lokberg, J. T. Malmo, “Long-distance electronic speckle pattern interferometry,” Opt. Eng. 27, 150–156 (1988).
    [CrossRef]
  5. G. A. Slettemoen, “First-order statistics of displayed speckle patterns in electronic speckle pattern interferometry,”J. Opt. Soc. Am. 71, 474–482 (1981).
    [CrossRef]
  6. G. A. Slettemoen, “General analysis of fringe contrast in electronic speckle pattern interferometry,” Opt. Acta 26, 313–327 (1979).
    [CrossRef]
  7. G. A. Slettemoen, “Optimal signal processing in electronic speckle pattern interferometry,” Opt. Commun. 23, 213–216 (1977).
    [CrossRef]
  8. K. Hogmoen, H. M. Pedersen, “Measurement of small vibrations using electronic speckle pattern interferometry: theory,”J. Opt. Soc. Am. 67, 1578–1583 (1977).
    [CrossRef]
  9. K. A. Stetson, “A rigorous treatment of the fringes of hologram interferometry,” Optik 29, 386–400 (1969).
  10. I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1965).
  11. K. A. Stetson, “Effects of beam modulation on fringe loci and localization in time-average hologram interferometry,”J. Opt. Soc. Am. 60, 1378–1388 (1970).
    [CrossRef]
  12. N.-E. Molin, K. A. Stetson, “Measuring combination mode vibration patterns by hologram interferometry,”J. Phys. E 2, 609–612 (1969).
    [CrossRef]
  13. K. A. Stetson, “Holographic vibration analysis,” in Holographic Nondestructive Testing, R. K. Erf, ed. (Academic, New York, 1974), Chap. 7.
    [CrossRef]
  14. A. D. Wilson, “Characteristic functions for time-average holography,”J. Opt. Soc. Am. 60, 1068–1071 (1972).
    [CrossRef]
  15. A. D. Wilson, D. H. Strope, “Time-average holographic interferometry of a circular plate vibrating simultaneously in two rationally related modes,”J. Opt. Soc. Am. 60, 1162–1165 (1970).
    [CrossRef]
  16. A. D. Wilson, “Computed time-average holographic interferometric fringes of a circular plate vibrating simultaneously in two rationally or irrationally related modes,”J. Opt. Soc. Am. 61, 924–929 (1971).
    [CrossRef]

1988

O. J. Lokberg, J. T. Malmo, “Long-distance electronic speckle pattern interferometry,” Opt. Eng. 27, 150–156 (1988).
[CrossRef]

1987

1981

1979

G. A. Slettemoen, “General analysis of fringe contrast in electronic speckle pattern interferometry,” Opt. Acta 26, 313–327 (1979).
[CrossRef]

1977

G. A. Slettemoen, “Optimal signal processing in electronic speckle pattern interferometry,” Opt. Commun. 23, 213–216 (1977).
[CrossRef]

K. Hogmoen, H. M. Pedersen, “Measurement of small vibrations using electronic speckle pattern interferometry: theory,”J. Opt. Soc. Am. 67, 1578–1583 (1977).
[CrossRef]

1972

1971

1970

1969

N.-E. Molin, K. A. Stetson, “Measuring combination mode vibration patterns by hologram interferometry,”J. Phys. E 2, 609–612 (1969).
[CrossRef]

K. A. Stetson, “A rigorous treatment of the fringes of hologram interferometry,” Optik 29, 386–400 (1969).

Craig, P. N.

Goldberg, W. M.

Gradshteyn, I. S.

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

Hogmoen, K.

Jones, R.

R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, London, 1983).

Law, R. L.

Lokberg, O. J.

O. J. Lokberg, J. T. Malmo, “Long-distance electronic speckle pattern interferometry,” Opt. Eng. 27, 150–156 (1988).
[CrossRef]

O. J. Lokberg, G. A. Slettemoen, “Basic electronic speckle pattern interferometry,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1987). Vol. 10, Chap. 8.

Malmo, J. T.

O. J. Lokberg, J. T. Malmo, “Long-distance electronic speckle pattern interferometry,” Opt. Eng. 27, 150–156 (1988).
[CrossRef]

Molin, N.-E.

N.-E. Molin, K. A. Stetson, “Measuring combination mode vibration patterns by hologram interferometry,”J. Phys. E 2, 609–612 (1969).
[CrossRef]

Moran, S. E.

Pedersen, H. M.

Ryzhik, I. M.

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

Slettemoen, G. A.

G. A. Slettemoen, “First-order statistics of displayed speckle patterns in electronic speckle pattern interferometry,”J. Opt. Soc. Am. 71, 474–482 (1981).
[CrossRef]

G. A. Slettemoen, “General analysis of fringe contrast in electronic speckle pattern interferometry,” Opt. Acta 26, 313–327 (1979).
[CrossRef]

G. A. Slettemoen, “Optimal signal processing in electronic speckle pattern interferometry,” Opt. Commun. 23, 213–216 (1977).
[CrossRef]

O. J. Lokberg, G. A. Slettemoen, “Basic electronic speckle pattern interferometry,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1987). Vol. 10, Chap. 8.

Stetson, K. A.

K. A. Stetson, “Effects of beam modulation on fringe loci and localization in time-average hologram interferometry,”J. Opt. Soc. Am. 60, 1378–1388 (1970).
[CrossRef]

N.-E. Molin, K. A. Stetson, “Measuring combination mode vibration patterns by hologram interferometry,”J. Phys. E 2, 609–612 (1969).
[CrossRef]

K. A. Stetson, “A rigorous treatment of the fringes of hologram interferometry,” Optik 29, 386–400 (1969).

K. A. Stetson, “Holographic vibration analysis,” in Holographic Nondestructive Testing, R. K. Erf, ed. (Academic, New York, 1974), Chap. 7.
[CrossRef]

Strope, D. H.

Wilson, A. D.

Wykes, C.

R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, London, 1983).

Appl. Opt.

J. Opt. Soc. Am.

J. Phys. E

N.-E. Molin, K. A. Stetson, “Measuring combination mode vibration patterns by hologram interferometry,”J. Phys. E 2, 609–612 (1969).
[CrossRef]

Opt. Acta

G. A. Slettemoen, “General analysis of fringe contrast in electronic speckle pattern interferometry,” Opt. Acta 26, 313–327 (1979).
[CrossRef]

Opt. Commun.

G. A. Slettemoen, “Optimal signal processing in electronic speckle pattern interferometry,” Opt. Commun. 23, 213–216 (1977).
[CrossRef]

Opt. Eng.

O. J. Lokberg, J. T. Malmo, “Long-distance electronic speckle pattern interferometry,” Opt. Eng. 27, 150–156 (1988).
[CrossRef]

Optik

K. A. Stetson, “A rigorous treatment of the fringes of hologram interferometry,” Optik 29, 386–400 (1969).

Other

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

K. A. Stetson, “Holographic vibration analysis,” in Holographic Nondestructive Testing, R. K. Erf, ed. (Academic, New York, 1974), Chap. 7.
[CrossRef]

O. J. Lokberg, G. A. Slettemoen, “Basic electronic speckle pattern interferometry,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1987). Vol. 10, Chap. 8.

R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, London, 1983).

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Figures (8)

Fig. 1
Fig. 1

OPL-ESPI system block diagram. Abbreviations: EOM, electro-optic modulator; VCSO, voltage-controlled sawtooth oscillator; PMT, photomultiplier tube; MO, microscope objective; PH, pinhole; PBS, polarization beam splitter; BS, beam splitter; M, mirror; MX, electronic mixer; BPF, bandpass filter; RFT, radio-frequency transformer; SW, field-effect-transistor switch; LPF, low-pass filter; PF, polarizing filter; SM, steering mirror; OVCO, optical voltage-controlled oscillator.

Fig. 2
Fig. 2

Single-frame speckle contrast as a function of noise fringe function for Fs(r, r0) = 1 and various values of the parameter Q.

Fig. 3
Fig. 3

Single-frame and composite fringe functions for primary speckle contours, B(r, r0) = 0, as a function of the spatial and temporal coherence parameters L(r, r0) and c.

Fig. 4
Fig. 4

Single-frame noise fringe functions and associated errors for signal frequencies of (A) 3 kHz and (B) 300 Hz.

Fig. 5
Fig. 5

Composite noise fringe functions and associated errors for signal frequencies of (A) 3 kHz and (B) 300 Hz.

Fig. 6
Fig. 6

Experimental geometry for illustrating effects of background noise on equal-Doppler speckle contours.

Fig. 7
Fig. 7

Speaker cone equal Doppler contours due to (A) 5-kHz sinusoid, (B) 48-Hz PZT-driven tilting motion, and (C) the sum of 5-kHz sinusoid and 48-Hz PZT-driven tilt.

Fig. 8
Fig. 8

Equal Doppler contours generated by (A1–D1) Gauss–Markov noise (G–M) of varying amplitude, (A2–D2) the sum of Gauss–Markov noise and 48-Hz PZT-driven tilting motion, and (A3–D3) the sum of Gauss–Markov noise and 5-kHz sinusoid.

Equations (130)

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C ( r , r 0 ) = σ E ( r , r 0 ) / E ( r , r 0 ) E ,
H ± ( r , r 0 , t ) = H B ( r ) ± 2 [ H L ( r ) H I ( r ) ] 1 / 2 × Re ( exp { i [ α I ( r 0 , t ) - α I ( r , t ) + Θ ( r , r 0 ) ] } ) ,
Θ ( r , r 0 ) = θ I ( r ) - θ I ( r 0 )
H B ( r ) = H I ( r ) + H L ( r ) .
d ( r , t ) = A ( r ) sin [ Ω t + ϕ ( r ) ] + d n ( r , t ) ,
α I ( r , t ) = 4 π d ( r , t ) / λ = 4 π { A ( r ) sin [ Ω t + ϕ ( r ) ] + d n ( r , t ) } / λ .
H ± ( r , r 0 , t ) = H B ( r ) ± 2 [ H L ( r ) H I ( r ) ] 1 / 2 × Re [ exp ( i { B ( r , r 0 ) sin [ Ω t + γ ( r , r 0 ) ] + Θ ( r , r 0 ) + 4 π [ d n ( r 0 , t ) - d n ( r , t ) ] / λ } ) ] ,
B ( r , r 0 ) = 4 π { A ( r 0 ) 2 + A ( r ) 2 - 2 A ( r 0 ) A ( r ) × cos [ 2 Φ ( r , r 0 ) ] } 1 / 2 / λ , Φ ( r , r 0 ) = [ ϕ ( r 0 ) - ϕ ( r ) ] / 2 , γ ( r , r 0 ) = ϕ ( r 0 ) + arctan ( - A ( r ) sin [ Φ ( r , r 0 ) ] / × { A ( r 0 ) - A ( r ) cos [ Φ ( r , r 0 ) ] } ) .
E ± ( r , r 0 , t 0 , T ) = t 0 - T / 2 t 0 + T / 2 H ± ( r , r 0 , t ) d t = T H B ( r ) ± 2 T [ H L ( r ) H I ( r ) ] 1 / 2 × f s ( r , r 0 , t 0 , T ) ,
f s ( r , r 0 , t 0 ) = T - 1 t 0 - T / 2 t 0 + T / 2 cos { B ( r , r 0 ) sin [ Ω t + γ ( r , r 0 ) ] + Θ ( r , r 0 ) + 2 k D ( r , r 0 , t ) } d t ,
D ( r , r 0 , t ) = d n ( r 0 , t ) - d n ( r , t ) ,
E ± ( r , r 0 ) d , H , Θ = T H B ( r ) H .
E ± 2 ( r , r 0 , t 0 ) d , H , Θ = T 2 H B 2 ( r ) H + 2 T 2 H L ( r ) H I ( r ) H F s ( r , r 0 , t 0 , T ) ,
F s ( r , r 0 , t 0 , T ) = 2 f s 2 ( r , r 0 , t 0 ) d , Θ = 2 T - 2 t 0 - T / 2 t 0 + T / 2 cos { B ( r , r 0 ) sin [ Ω t 1 + γ ( r , r 0 ) ] + Θ ( r , r 0 ) + 2 k D ( r , r 0 , t 1 ) } × cos { B ( r , r 0 ) sin [ Ω t 2 + γ ( r , r 0 ) ] + Θ ( r , r 0 ) + 2 k D ( r 0 , r , t 2 ) } d , Θ d t 1 d t 2 .
F s ( r , r 0 , t 0 , T ) = T - 2 t 0 - T / 2 t 0 + T / 2 cos ( B ( r , r 0 ) { sin [ Ω t 1 + γ ( r , r 0 ) ] - sin [ Ω t 2 + γ ( r , r 0 ) ] } + 2 k [ D ( r , r 0 , t 1 ) - D ( r , r 0 , t 2 ) ] ) + cos { B ( r , r 0 ) { sin [ Ω t 1 + γ ( r , r 0 ) ] + sin [ Ω t 2 + γ ( r , r 0 ) ] } + 2 Θ ( r , r 0 ) + 2 k [ D ( r , r 0 , t 1 ) + D ( r , r 0 , t 2 ) } d , Θ d t 1 d t 2 .
F s ( r , r 0 , t 0 , T ) = T - 2 t 0 - T / 2 t 0 + T / 2 cos ( B ( r , r 0 ) { sin [ Ω t 1 + γ ( r , r 0 ) ] - sin [ Ω t 2 + γ ( r , r 0 ) ] } + 2 k [ D ( r , r 0 , t 1 ) - D ( r , r 0 , t 2 ) ] ) d d t 1 d t 2 .
cos ( X ) = cos ( X ) exp [ - ½ Var ( X ) ] .
E ± 2 ( r , r 0 , t 0 ) d , H , Θ = T 2 H B 2 ( r ) H ± 2 T 2 H L ( r ) H I ( r ) H F s ( r , r 0 , t 0 ) ,
F s ( r , r 0 , t 0 , T ) = T - 2 Re [ t 0 - T / 2 t 0 + T / 2 exp ( i B ( r , r 0 ) × { sin [ Ω t 1 + γ ( r , r 0 ) ] - sin [ Ω t 2 + γ ( r , r 0 ) ] } ) exp { - 2 k 2 × [ C D ( r , r 0 , t 1 , t 1 ) + C D ( r , r 0 , t 2 , t 2 ) - 2 C D ( r , r 0 , t 1 , t 2 ) ] } d t 1 d t 2 ] ,
C D ( r 0 , r , t 1 , t 2 ) = C d ( r 0 , r 0 , t 1 , t 2 ) + C d ( r , r , t 1 , t 2 ) - 2 C d ( r , r 0 , t 1 , t 2 )
C D ( r , r 0 , t 1 , t 1 ) + C D ( r , r 0 , t 2 , t 2 ) - 2 C D ( r , r 0 , t 1 , t 2 ) = [ C d ( r 0 , r 0 , t 1 , t 1 ) + C d ( r , r , t 1 , t 1 ) - 2 C d ( r , r 0 , t 1 , t 1 ) ] + [ C d ( r 0 , r 0 , t 2 , t 2 ) + C d ( r , r , t 2 , t 2 ) - 2 C d ( r , r 0 , t 2 , t 2 ) ] - 2 [ C d ( r 0 , r 0 , t 1 , t 2 ) + C d ( r , r , t 1 , t 2 ) - C d ( r , r 0 , t 1 , t 2 ) - C d ( r , r 0 , t 2 , t 1 ) ] .
E c ( r , r 0 , t 0 , T ) = E + ( r , r 0 , t 0 , T ) - E - ( r , r 0 , t 0 + T , T ) = 2 [ H L ( r ) H I ( r ) ] 1 / 2 [ Re ( t 0 - T / 2 t 0 + T / 2 exp { i [ B ( r , r 0 ) × sin [ Ω t + γ ( r , r 0 ) ] + Θ ( r , r 0 ) + 2 k D ( r , r 0 , t ) ] } d t ) + Re ( t 0 + T / 2 t 0 + 3 T / 2 exp { i [ B ( r , r 0 ) sin [ Ω t + γ ( r , r 0 ) ] + Θ ( r , r 0 ) + 2 k D ( r , r 0 , t ) ] } d t ) ] .
E c ( r , r 0 , t 0 , T ) d , H , Θ = 0.
E c 2 ( r , r 0 , t 0 , T ) d , H , Θ = 2 T 2 H L ( r ) H I ( r ) H [ F s ( r , r 0 , t 0 , T ) + F s ( r , r 0 , t 0 + T , T ) + 2 F c ( r , r 0 , t 0 , t 0 + T , T ) ] ,
F c ( r , r 0 , t 0 , t 0 + T , T ) = T - 2 Re [ t 0 - T / 2 t 0 + T / 2 t 0 + T / 2 t 0 + 3 T / 2 exp ( i B ( r , r 0 ) × { sin [ Ω t 1 + γ ( r , r 0 ) ] - sin [ Ω t 2 + γ ( r , r 0 ) ] } ) × exp { - 2 k 2 [ C D ( r , r 0 , t 1 , t 1 ) + C D ( r , r 0 , t 2 , t 2 ) - 2 C D ( r , r 0 , t 1 , t 2 ) ] } d t 1 d t 2 ] .
C ( r , r 0 , t 0 , T ) = [ E 2 ( r , r 0 , t 0 ) E - E ( r , r 0 , t 0 ) E 2 ] 1 / 2 / E E = [ T 2 H B 2 ( r ) H - T 2 H B ( r ) H 2 ] + 2 T 2 H L ( r ) × H I ( r ) H F s ( r , r 0 , t 0 , T ) ] 1 / 2 / T H B ( r ) H .
C ( r , r 0 , t 0 , T ) = ( 1 + Q ) - 1 [ 1 + 2 Q F s ( r , r 0 , t 0 , T ) ] 1 / 2 ,
[ E c 2 ( r , r 0 , t 0 , T ) d , H , Θ ] 1 / 2 = 2 T [ H L ( r ) H I ( r ) H ] 1 / 2 [ F s ( r , r 0 , t 0 , T ) + F s ( r , r 0 , t 0 + T , T ) + 2 F c ( r , r 0 , t 0 , t 0 + T , T ) ] 1 / 2 .
F s ( r , r 0 , t 0 , T ) = F 0 ( r , r 0 ) = J 0 2 [ B ( r , r 0 ) ] .
F c ( r , r 0 , t 0 , t 0 + T , T ) = T - 2 Re [ t 0 - T / 2 t 0 + T / 2 t 0 + T / 2 t 0 + 3 T / 2 exp ( i B ( r , r 0 ) × { sin [ Ω t 1 + γ ( r , r 0 ) ] - sin [ Ω t 2 + γ ( r , r 0 ) ] } ) × exp { - 4 k 2 [ C D ( r , r 0 , 0 ) - C D ( r , r 0 , t 1 - t 2 ) ] } d t 1 d t 2 ] .
exp [ i x sin ( y ) ] = n = - J n ( x ) exp ( i n y )
v 1 = t 1 - t 0 , v 2 = t 2 - t 0 - T ,
F c ( r , r 0 , μ 1 , T ) = Re { n , m = - J m [ B ( r , r 0 ) ] J n [ B ( r , r 0 ) ] × exp [ i ( m - n ) μ ] exp ( - i n Ω T ) × T - 2 - T / 2 T / 2 exp { i ( m Ω v 1 - n Ω v 2 ) } exp { - 4 k 2 [ C D ( r , r 0 , 0 ) - C D ( r , r 0 , v 1 - v 2 - T ) ] } d v 1 d v 2 } ,
μ = Ω t 0 + γ ( r , r 0 ) .
F c ( r , r 0 , T ) = ( 2 π ) - 1 - π π F c ( r , r 0 , μ , T ) d μ .
( 2 π ) - 1 - π π exp [ i ( m - n ) μ ] d μ = { 1 , n = m 0 , n m ,
F c ( r , r 0 , T ) = Re { m = - J m 2 [ B ( r , r 0 ) ] exp ( - i m Ω T ) × T - 2 - T / 2 T / 2 exp [ i m Ω ( v 1 - v 2 ) ] × exp { - 4 k 2 [ C D ( r , r 0 , 0 ) - C D ( r , r 0 , v 1 - v 2 - T ) ] } d v 1 d v 2 } .
F c ( r , r 0 , T ) = Re ( m = - J m 2 [ B ( r , r 0 ) ] T - 2 × - T T ( T - τ ) exp [ - i m Ω ( τ + T ) ] × exp { 4 k 2 [ C D ( r , r 0 , 0 ) - C D ( r , r 0 , τ + T ) ] } d τ ) .
m = - J n 2 [ B ( r , r 0 ) ] exp ( - i m Ω τ ) = J 0 [ 2 B ( r , r 0 ) sin ( Ω τ / 2 ) ] ,
F c ( r , r 0 , T ) = T - 2 - T T ( T - τ ) J 0 { 2 B ( r , r 0 ) } sin [ Ω ( τ + T ) / 2 ] } × exp { - 4 k 2 [ C D ( r , r 0 , 0 ) - C D ( r , r 0 , τ + T ) ] } d τ .
F s ( r , r 0 , T ) = T - 2 - T T ( T - τ ) J 0 [ 2 B ( r , r 0 ) sin ( Ω τ / 2 ) ] × exp { - 4 k 2 [ C D ( r , r 0 , 0 ) - C D ( r , r 0 , τ ) ] } d τ .
F c , s ( r , r 0 ) = F N , c , s ( r , r 0 ) F 0 ( r , r 0 ) + E c , s ( r , r 0 ) ,
F N , c ( r , r 0 ) = N - 1 ( q = 0 N - 1 [ ( q + 1 / 2 ) / N ] exp { - 4 k 2 [ C D ( r , r 0 , 0 ) - C D ( r , r 0 , q T s ) ] } + q = N 2 N - 1 [ 2 - ( q + 1 / 2 ) / N ] × exp { - 4 k 2 [ C D ( r , r 0 , 0 ) - C D ( r , r 0 , q T s ) ] } )
F N , s ( r , r 0 ) = 2 N - 1 q = 0 N - 1 [ 1 - ( q + 1 / 2 ) / N ] × exp { - 4 k 2 [ C D ( r , r 0 , 0 ) - C D ( r , r 0 , q T s ) ] } ,
T s = 2 π / Ω
N T / T s < N + 1.
exp [ - 4 k 2 C D ( r , r 0 , 0 ) ] + ( 1 / 2 N 2 ) { 1 - exp [ - 4 k 2 C D ( r , r 0 , 0 ) ] } F N , c ( r , r 0 ) 1
exp [ - 4 k 2 C D ( r , r 0 , 0 ) ] + [ 2 ( 1 - 1 / 2 N ) / N ] × { 1 - exp [ - 4 k 2 C D ( r , r 0 , 0 ) ] } F N , s ( r , r 0 ) 1.
C ( r , r 0 ) = ( 1 + Q ) - 1 [ 1 + 2 Q F N , s ( r , r 0 ) F 0 ( r , r 0 ) ] 1 / 2 ,
[ E c 2 ( r , r 0 , T ) d , H , Θ ] 1 / 2 = 2 2 T [ H L ( r ) H I ( r ) H ] 1 / 2 J 0 [ B ( r , r 0 ) ] × [ F N , s ( r , r 0 ) + F N , c ( r , r 0 ) ] 1 / 2 .
C d ( r 0 , r , τ ) = C s p ( r , r 0 ) C t ( τ ) ,
C sp ( r , r 0 ) = C d ( r , r 0 , t , t ) = d n ( r , t ) d n ( r 0 , t ) d
C D ( r , r 0 , 0 ) - C D ( r , r 0 , τ ) = [ C sp ( r 0 , r 0 ) + C sp ( r , r ) - 2 C sp ( r , r 0 ) ] [ C t ( 0 ) - C t ( τ ) ] .
F c ( r , r 0 ) = T - 2 - T T ( T - τ ) J 0 { 2 B ( r , r 0 ) sin [ Ω ( τ + T ) / 2 ] } × exp { - 16 π 2 L ( r , r 0 ) [ C t ( 0 ) - C t ( τ + T ) ] } d τ
F s ( r , r 0 ) = T - 2 - T T ( T - τ ) J 0 [ 2 B ( r , r 0 ) sin ( Ω τ / 2 ) ] × exp { - 16 π 2 L ( r , r 0 ) [ C t ( 0 ) - C t ( τ ) ] } d τ ,
F N , c ( r , r 0 ) = N - 1 ( q = 0 N - 1 [ ( q + 1 / 2 ) / N ] exp { - 16 π 2 L ( r , r 0 ) } × [ 1 - C t ( q T s ) ] } + q = N 2 N - 1 [ 2 - ( 1 + 1 / 2 ) / N ] × exp { - 16 π 2 L ( r , r 0 ) [ 1 - C t ( q T s ) ] } ) ,
F N , s ( r , r 0 ) = 2 N - 1 q = 0 N - 1 [ 1 - ( q + 1 / 2 ) / N ] × exp { - 16 π 2 L ( r , r 0 ) [ 1 - C t ( q T s ) ] } ,
L ( r , r 0 ) = { C sp ( r 0 , r 0 ) + C sp ( r , r ) - 2 C sp ( r , r 0 ) } / λ 2 .
L ( r , r 0 ) = C D ( r , r 0 , 0 ) / λ 2 , C sp ( r 0 , r 0 ) = d n 2 ( r 0 , t ) d , C sp ( r , r ) = d n 2 ( r , t ) d .
L ( r , r 0 ) = d n ( r , t ) - d n ( r 0 , t ) 2 d / λ 2 = D 2 ( r , r 0 , t ) d / λ 2 .
C t ( τ ) = exp ( - c τ ) ,
S ( ω ) = c / π ( c 2 + ω 2 ) ,
F c ( r , r 0 ) = T - 1 - T T ( T - τ ) J 0 { 2 B ( r , r 0 ) sin [ Ω ( τ + T ) / 2 ] } × exp { - 16 π 2 L ( r , r 0 ) [ 1 - exp ( - c τ + T ) ] } d τ ,
F s ( r , r 0 ) = T - 2 - T T ( T - τ ) J 0 [ 2 B ( r , r 0 ) sin ( Ω τ / 2 ] × exp { - 16 π 2 L ( r , r 0 ) [ 1 - exp ( - c τ ) ] } d τ .
F N , c ( r , r 0 ) = N - 1 ( q = 0 N - 1 [ ( q + 1 / 2 ) / N ] exp { - 16 π 2 L ( r , r 0 ) × [ 1 - exp ( - c q T s ) ] } + q = N 2 N - 1 [ 2 - ( q + 1 / 2 ) / N ] × exp { - 16 π 2 L ( r , r 0 ) [ 1 - exp ( - c q T s ) ] } ) ,
F N , s ( r , r 0 ) = 2 N - 1 q = 0 N - 1 [ 1 - ( q + 1 / 2 ) / N ] × exp { - 16 π 2 L ( r , r 0 ) [ 1 - exp ( - c q T s ) ] } .
lim L F c , s ( r , r 0 ) = 0.
lim L F N , c ( r , r 0 ) = 1 / 2 N 2 ,
lim L F N , s ( r , r 0 ) = 2 ( 1 - 1 / 2 N ) / N .
lim L N F N , c , s ( r , r 0 ) = lim L F c , s ( r , r 0 ) .
lim c F c , s ( r , r 0 ) = exp [ - 16 π 2 L ( r , r 0 ) ] J 0 2 [ B ( r , r 0 ) ] .
lim c F c , s ( r , r 0 ) = exp [ - 16 π 2 L ( r , r 0 ) .
lim c F N , c ( r , r 0 ) = exp [ - 16 π 2 L ( r , r 0 ) ] ( 1 - 1 / 2 N 2 ) + 1 / 2 N 2
lim c F N , s ( r , r 0 ) = exp [ - 16 π 2 L ( r , r 0 ) ] ( 1 - 1 / N 2 ) + 2 ( 1 - 1 / 2 N ) / N .
lim c N F N , s ( r , r 0 ) = lim c F s ( r , r 0 ) .
lim L E c / E s = 1 / 2 ( 2 N - 1 ) .
X = B ( r , r 0 ) { sin [ Ω t 1 + γ ( r , r 0 ) ] - sin [ Ω t 2 + γ ( r , r 0 ) ] } × 2 k [ D ( r , r 0 , t 1 ) - D ( r , r 0 , t 2 ) ] .
X d = B ( r , r 0 ) { sin [ Ω t 1 + γ ( r , r 0 ) ] - sin [ Ω t 2 + γ ( r , r 0 ) ] } .
X 2 d = X d 2 + 2 k X d [ D ( r , r 0 , t 1 ) - D ( r , r 0 , t 2 ) ] d + 4 k 2 [ D ( r , r 0 , t 1 ) - D ( r , r 0 , t 2 ) ] 2 d .
σ X 2 = X 2 d = X d 2 = 4 k 2 [ D ( r , r 0 , t 1 ) - D ( r , r 0 , t 2 ) ] 2 d = 4 k 2 [ C D ( r , r 0 , t 1 , t 1 ) + C D ( r , r 0 , t 2 , t 2 ) - 2 C D ( r , r 0 , t 1 , t 2 ) ] .
C D ( r , r 0 , t 1 , t 1 ) = C D ( r , r 0 , t 2 , t 2 ) = C D ( r , r 0 , 0 ) ,
σ X 2 = 8 k 2 [ C D ( r , r 0 , 0 ) - C D ( r , r 0 , t 1 - t 2 ) ] .
C D ( r , r 0 , t 1 , t 1 ) + C D ( r , r 0 , t 2 , t 2 ) - 2 C D ( r , r 0 , t 1 , t 2 ) .
C D ( r , r 0 , t 1 , t 2 ) = [ d n ( r 0 , t 1 ) - d n ( r , t 1 ) ] [ d n ( r 0 , t 2 ) - d n ( r , t 2 ) ] d = d n ( r 0 , t 1 ) d n ( r 0 , t 2 ) d + d n ( r , t 1 ) d n ( r , t 2 d - d n ( r 0 , t 1 ) d n ( r , t 2 ) d - d n ( r , t 1 ) d n ( r 0 , t 2 ) d = C d ( r 0 , r 0 , t 1 , t 2 ) + C d ( r , r , t 1 , t 2 ) - C d ( r , r 0 , t 1 , t 2 ) - C d ( r , r 0 , t 2 , t 1 ) .
C D ( r , r 0 , t 1 , t 1 ) + C D ( r , r 0 , t 2 , t 2 ) - 2 C D ( r , r 0 , t 1 , t 2 ) = [ C d ( r 0 , r 0 , t 1 , t 1 ) + C d ( r , r , t 1 , t 1 ) - 2 C d ( r , r 0 , t 1 , t 1 ) ] + [ C d ( r 0 , r 0 , t 2 , t 2 ) + C d ( r , r , t 2 , t 2 ) - 2 C d ( r , r 0 , t 2 , t 2 ) ] - 2 [ C d ( r 0 , r 0 , t 1 , t 2 ) + C d ( r , r , t 1 , t 2 ) - C d ( r , r 0 , t 1 , t 2 ) - C d ( r , r 0 , t 2 , t 1 ) ] .
C D ( r , r 0 , τ ) = C d ( r 0 , r 0 , τ ) + C d ( r , r , τ ) - C d ( r , r 0 , τ ) - C d ( r , r 0 , - τ ) ,
C D ( r , r 0 , τ ) = C d ( r 0 , r 0 , τ ) + C d ( r , r , τ ) - 2 C d ( r , r 0 , τ ) .
C D ( r , r 0 , t 1 , t 1 ) + C D ( r , r 0 , t 2 , t 2 ) - 2 C D ( r , r 0 , t 1 , t 2 ) = 2 [ C D ( r , r 0 , t 1 , t 1 ) - C D ( r , r 0 , t 1 , t 2 ) ] = 2 [ C d ( r 0 , r 0 , 0 ) + C d ( r , r , 0 ) - 2 C d ( r , r 0 , 0 ) - C d ( r 0 , r 0 , τ ) - C d ( r , r , τ ) + 2 C d ( r , r 0 , τ ) ] .
C D ( r , r 0 , 0 ) - C D ( r , r 0 , τ ) = [ C sp ( r 0 , r 0 ) + C sp ( r , r ) - 2 C sp ( r , r 0 ) ] [ C t ( 0 ) - C t ( τ ) ] .
C D ( r , r 0 , q T s ) - C D [ r , r 0 , ( q + y ) T s ] .
C D ( r , r 0 , q T s ) - C D [ r , r 0 , ( q + y ) T s ] = [ C sp ( r , r 0 ) + C sp ( r , r ) - 2 C sp ( r , r 0 ) ] { C t ( q T s ) - C t [ ( q + y ) T s ] } .
F c , s ( r , r 0 ) = F N , c , s ( r , r 0 ) F 0 ( r , r 0 ) + E c , s ( r , r 0 ) ,
F c ( r , r 0 ) = T - 2 - T T ( T - τ ) J 0 { 2 B ( r , r 0 ) sin [ Ω ( τ + T ) / 2 ] } × exp { - 4 k 2 [ C D ( r , r 0 , 0 ) - C D ( r , r 0 , τ + T ) ] } d τ .
F c ( r , r 0 ) = T s T - 1 ( 0 T / T s ( x T s / T ) J 0 [ 2 B ( r , r 0 ) sin ( π x ) ] × exp { - 4 k 2 [ C D ( r , r 0 , 0 ) - C D ( r , r 0 , x T s ) ] } d x + T / T s 2 T / T s ( 2 - x T s / T ) J 0 [ 2 B ( r , r 0 ) sin ( π x ) ] × exp { - 4 k 2 [ C D ( r , r 0 , 0 ) - C D ( r , r 0 , x T s ) ] } d x ) ,
T s = 2 π / Ω
N T / T s < N + 1.
F c ( r , r 0 ) = T s T - 1 ( 0 N ( x T s / T ) J 0 [ 2 B ( r , r 0 ) sin ( π x ) ] × exp { - 4 k 2 [ C D ( r , r 0 , 0 ) - C D ( r , r 0 , x T s ) ] } d x + N 2 N ( 2 - x T s / T ) J 0 [ 2 B ( r , r 0 ) sin ( π x ) ] × exp { - 4 k 2 [ C D ( r , r 0 , 0 ) - C D ( r , r 0 , x T s ) ] } d x ) + E c 1 ( r , r 0 ) ,
F c 1 ( r , r 0 ) = T s T - 1 ( N T / T s ( x T s / T ) J 0 [ 2 B ( r , r 0 ) sin ( π x ) ] × exp { - 4 k 2 [ C D ( r , r 0 , 0 ) - C D ( r , r 0 , x T s ) ] } d x - N T / T s ( 2 - x T s / T ) J 0 [ 2 B ( r , r 0 ) sin ( π x ) ] × exp { - 4 k 2 [ C D ( r , r 0 , 0 ) - C D ( r , r 0 , x T s ) ] } d x - 2 N 2 T / T s ( 2 - x T s / T ) J 0 [ 2 B ( r , r 0 ) sin ( π x ) ] × exp { - 4 k 2 [ C D ( r , r 0 , 0 ) - C D ( r , r 0 , x T s ) ] } d x ) .
F c ( r , r 0 ) = T s T - 1 ( q = 0 N - 1 q q + 1 ( x T s / T ) J 0 [ 2 B ( r , r 0 ) sin ( π x ) ] × exp { - 4 k 2 [ C D ( r , r 0 , 0 ) - C D ( r , r 0 , x T s ) ] } d x + q = N 2 N - 1 q q + 1 ( 2 - x T s / T ) J 0 [ 2 B ( r , r 0 ) sin ( π x ) ] × exp { - 4 k 2 [ C D ( r , r 0 , 0 ) - C D ( r , r 0 , x T s ) ] } d x ) + E c 1 ( r , r 0 ) .
F c ( r , r 0 ) = T s T - 1 [ q = 0 N - 1 exp { - 4 k 2 [ C D ( r , r 0 , 0 ) - C D ( r , r 0 , q T s ) ] } × 0 1 [ ( y + q ) T s / T ] J 0 [ 2 B ( r , r 0 ) sin π y ) ] exp ( - 4 k 2 { C D ( r , r 0 , q T s ) - C D [ r , r 0 , ( q + y ) T s ] } ) d y + q = N 2 N - 1 exp { - 4 k 2 [ C D ( r , r 0 , 0 ) - C D ( r , r 0 , q T s ) ] } × 0 1 [ 2 - ( y + q ) T s / T ] J 0 [ 2 B ( r , r 0 ) sin ( π y ) ] exp ( - 4 k 2 { C D ( r , r 0 , q T s ) - C D [ r , r 0 , ( q + y ) T s ] } ) d y ] + E c 1 ( r , r 0 ) .
F c ( r , r 0 ) = T s T - 1 ( q = 0 N - 1 exp { - 4 k 2 [ C D ( r , r 0 , 0 ) - C D ( r , r 0 , q T s ) ] } × 0 1 [ ( y + q ) T s / T ] J 0 [ 2 B ( r , r 0 ) sin ( π y ) ] d y + q = N 2 N - 1 exp { - 4 k 2 [ C D ( r , r 0 , 0 ) - C D ( r , r 0 , q T s ) ] } × 0 1 [ 2 - ( y + q ) T s / T ] J 0 [ 2 B ( r , r 0 ) sin ( π y ) ] d y ) + E c 1 ( r , r 0 ) + E c 2 ( r , r 0 ) ,
E c 2 ( r , r 0 ) = T s T - 1 { q = 0 N - 1 exp { - 4 k 2 [ C D ( r , r 0 , 0 ) - C D ( r , r 0 , q T s ) ] } × 0 1 [ ( y + q ) T s / T ] J 0 × [ 2 B ( r , r 0 ) sin ( π y ) ] [ exp { - 4 k 2 [ C D ( r , r 0 , q T s ) - C D [ r , r 0 , ( q + y ) T s ] } ) - 1 ] d y + q = N 2 N - 1 exp { - 4 k 2 [ C D ( r , r 0 , 0 ) - C D ( r , r 0 , q T s ) ] } × 0 1 [ 2 - ( y + q ) T s / T ] J 0 [ 2 B ( r , r 0 ) sin ( π y ) ] × [ exp ( - 4 k 2 { C D ( r , r 0 , q T s ) - C D [ r , r 0 , ( q + y ) T s ] } ) - 1 ] d y } .
0 1 J 0 [ 2 B ( r , r 0 ) sin ( π y ) ] d y = J 0 2 [ B ( r , r 0 ) ]
0 1 y J 0 [ 2 B ( r , r 0 ) sin ( π y ) ] d y = J 0 2 [ B ( r , r 0 ) / 2 ]
F c ( r , r 0 ) = T s T - 1 ( q = 0 N - 1 [ ( q + 1 / 2 ) T s / T ] exp { - 4 k 2 [ C D ( r , r 0 , 0 ) - C D ( r , r 0 , q T s ) ] } + q = N 2 N - 1 [ 2 - ( q + 1 / 2 ) T s / T ] × exp { - 4 k 2 [ C D ( r , r 0 , 0 ) - C D ( r , r 0 , q T s ) ] } ) × J 0 2 [ B ( r , r 0 ) ] + E c 1 ( r , r 0 ) + E c 2 ( r , r 0 ) .
F c ( r , r 0 ) = F N , c ( r , r 0 ) J 0 2 [ B ( r , r 0 ) ] + E c 1 ( r , r 0 ) + E c 2 ( r , r 0 ) + E c 3 ( r , r 0 ) ,
F N , c ( r , r 0 ) = N - 1 ( q = 0 N - 1 [ ( q + 1 / 2 ) / N ] exp { - 4 k 2 [ C D ( r , r 0 , 0 ) - C D ( r , r 0 , q T s ) ] } + q = N 2 N - 1 [ 2 - ( q + 1 / 2 ) / N ] × exp { - 4 k 2 [ C D ( r , r 0 , 0 ) - C D ( r , r 0 , q T s ) ] } ) ,
E c 3 ( r , r 0 ) = ( q = 0 N - 1 { ( T s / T ) [ ( q + 1 / 2 ) T s / T ] - N - 1 [ ( q + 1 / 2 ) / N ] } exp { - 4 k 2 [ C D ( r , r 0 , 0 ) - C D ( r , r 0 , q T s ) ] } + q = N 2 N - 1 { ( T s / T ) [ 2 - ( q + 1 / 2 ) T s / T ] - N - 1 [ 2 - ( q + 1 / 2 ) / N ] } × exp { - 4 k 2 [ C D ( r , r 0 , 0 ) - C D ( r , r 0 , q T s ) ] } ) × J 0 2 [ B ( r , r 0 ) ] .
F 0 ( r , r 0 ) = T - 2 - T T ( T - τ ) J 0 [ 2 B ( r , r 0 ) sin ( Ω τ / 2 ) ] × exp { - 4 k 2 [ C D ( r , r 0 , 0 ) - C D ( r , r 0 , τ ) ] } d τ
F s ( r , r 0 ) = F N , s ( r , r 0 ) J 0 2 [ B ( r , r 0 ) ] + E s 1 ( r , r 0 ) + E s 2 ( r , r 0 ) + E s 3 ( r , r 0 ) ,
F N , s ( r , r 0 ) = 2 N - 1 q = 0 N - 1 [ 1 - ( q + 1 / 2 ) / N ] × exp { - 4 k 2 [ C D ( r , r 0 , 0 ) - C D ( r , r 0 , q T s ) ] }
E s 1 ( r , r 0 ) = 2 T s T - 1 N T / T s ( 1 - x T s / T ) J 0 [ 2 B ( r , r 0 ) sin ( π x ) ] × exp { - 4 k 2 [ C D ( r , r 0 , 0 ) - C D ( r , r 0 , x T s ) ] } d x ,
E s 2 ( r , r 0 ) = 2 T s T - 1 q = 0 N - 1 exp { - 4 k 2 [ C D ( r , r 0 , 0 ) - C D ( r , r 0 , q T s ) } 0 1 [ 1 - ( y + q ) T s / T ] × J 0 [ 2 B ( r , r 0 ) sin ( π y ) ] [ exp ( - 4 k 2 { C D ( r , r 0 , q T s ) - C D [ r , r 0 , ( q + y ) T s ] } ) - 1 ] d y ,
E s 3 ( r , r 0 ) = ( q = 0 N - 1 { ( 2 T s / T ) } [ 1 - ( q + 1 / 2 ) T s / T ] × ( 2 / N ) [ 1 - ( q + 1 / 2 ) / N ] } exp { - 4 k 2 [ C D ( r , r 0 , 0 ) - C D ( r , r 0 , q T s ) ] } ) J 0 2 [ B ( r , r 0 ) ] .
E c 1 ( r , r 0 ) = T s T - 1 ( N T / T s ( x T s / T ) J 0 [ 2 B ( r , r 0 ) sin ( π x ) ] × exp { - 16 π 2 L ( r , r 0 ) [ 1 - exp ( - c x T s ) ] } d x - N T / T s ( 2 - x T s / T ) J 0 [ 2 B ( r , r 0 ) sin ( π x ) ] × exp { - 16 π 2 L ( r , r 0 ) [ 1 - exp ( - c x T s ) ] } d x + 2 N 2 T / T s ( 2 - x T s / T ) J 0 [ 2 B ( r , r 0 ) sin ( π x ) ] × exp { - 16 π 2 L ( r , r 0 ) [ 1 - exp ( - c x T s ) ] } d x ) ,
E c 2 ( r , r 0 ) = T s T - 1 [ q = 0 N - 1 exp { - 16 π 2 L ( r , r 0 ) [ 1 - exp ( - c q T s ) ] } × 0 1 [ ( x + q ) T s / T ] J 0 [ 2 B ( r , r 0 ) sin ( π x ) ] × ( exp { - 16 π 2 L ( r , r 0 ) [ 1 - exp ( - c x T s ) ] × exp ( - c q T s ) } - 1 ) d x + q = 0 N - 1 exp { - 16 π 2 L ( r , r 0 ) × [ 1 - exp ( - c q T s ) ] } 0 1 [ 2 - ( x + q ) T s / T ] × J 0 [ 2 B ( r , r 0 ) sin ( π x ) ] ( exp { - 16 π 2 L ( r , r 0 ) × [ 1 - exp ( - c x T s ) ] exp ( - c q T s ) } - 1 ) d x ] ,
E c 3 ( r , r 0 ) = ( q = 0 N - 1 { ( T s / T ) [ ( q + 1 / 2 ) T s / T ] - N - 1 [ ( q + 1 / 2 ) / N ] × exp { - 16 π 2 L ( r , r 0 ) [ q - exp ( - c q T s ) ] } + q = N 2 N - 1 { T s / T [ 2 - ( q + 1 / 2 ) T s / T ] - N - 1 [ 2 - ( q + 1 / 2 ) / N ] } exp { - 16 π 2 L ( r , r 0 ) × [ 1 - exp ( - c q T s ) ] } ) J 0 2 [ B ( r , r 0 ) ] ,
E s 1 ( r , r 0 ) = 2 T s T - 1 N T / T s ( 1 - x T s / T ) J 0 [ B ( r , r 0 ) sin ( π x ) ] × exp { - 32 π 2 L ( r , r 0 ) [ 1 - exp ( - c x T s ) ] } d x ,
E s 2 ( r , r 0 ) = 2 T s T - 1 q = 0 N - 1 exp { - 16 π 2 L ( r , r 0 ) [ 1 - exp ( - c q T s ) ] } × 0 1 [ 1 - ( x + q ) ( T s / T ] J 0 [ 2 B ( r , r 0 ) sin ( π x ) ] × ( exp { - 16 π 2 L ( r , r 0 ) [ 1 - exp ( - c x T s ) ] × exp ( - c q T s ) } - 1 ) d x ,
E s 3 ( r , r 0 ) = ( q = 0 N - 1 { ( 2 T s / T ) [ 1 - ( q + 1 / 2 ) T s / T ] - ( 2 / N ) [ 1 - ( q + 1 / 2 ) / N ] } exp { - 16 π 2 L ( r , r 0 ) × [ 1 - exp ( - c q T s ) ] } ) J 0 [ B ( r , r 0 ) ] .
f ( r , r 0 , t 0 , T ) = i = 1 N f i ( r , r 0 , t 0 , T ) ,
α I ( r , t ) = ( 4 π / λ ) i = 1 N A i ( r ) sin [ Ω i t + ϕ ( r ) ] .
H ± ( r , r 0 , t ) = H B ( r ) ± 2 [ H L ( r ) H I ( r ) ] 1 / 2 × Re [ exp ( i i = 1 N { B i ( r , r 0 ) sin [ Ω i t + γ i ( r , r 0 ) ] } + Θ ( r , r 0 ) ) ] .
E ± ( r , r 0 , t 0 , T ) = t 0 - T / 2 t 0 + T / 2 H ± ( r , r 0 , t ) d t = T H B ( r ) ± 2 T [ H L ( r ) H I ( r ) ] 1 / 2 f ( r , r 0 , t 0 , T ) ,
f ( r , r 0 , t 0 , T ) = Re { t 0 - T / 2 t 0 + T / 2 exp [ i ( I = 1 N { B i ( r , r 0 ) × sin [ Ω i t + γ i ( r , r 0 ) ] } + Θ ( r , r 0 ) ) ] d t } .
f ( r , r 0 , t 0 , T ) = Re [ t 0 - T / 2 t 0 + T / 2 i = 1 N n = - J n [ B i ( r , r 0 ) ] × exp ( i { n [ Ω i t + γ i ( r , r 0 ) ] + Θ ( r , r 0 ) } ) d t ] = l m n J l [ B 1 ( r , r 0 ) ] J m [ B 2 ( r , r 0 ) ] J n [ B N ( r , r 0 ) ] × t 0 - T / 2 t 0 + T / 2 exp { i [ l Ω 1 + m Ω 2 + + n Ω N ) t + ( l γ 2 ( r , r 0 ) + m γ 2 ( r , r 0 ) + + n γ N ( r , r 0 ) + Θ ( r , r 0 ) ] } d t = l m n J l [ B 1 ( r , r 0 ) ] J m [ B 2 ( r , r 0 ) ] × J n [ B N ( r , r 0 ) ] cos { ( l Ω 1 + m Ω 2 + + n Ω N ) t 0 + ( l γ 1 ( r , r 0 ) + m γ 2 ( r , r 0 ) + + n γ N ( r , r 0 ) + Θ ( r , r 0 ) ] } sinc [ ( l F 1 + m F 2 + + n F N ) T ] .
f i ( r , r 0 , t 0 , T ) = n = - J n [ B i ( r , r 0 ) ] cos { n [ Ω i t + γ i ( r , r 0 ) + Θ ( r , r 0 ) } sinc ( n F i T ) .
sinc [ ( l F 1 + m F 2 + + n F n ) T ] = { 0 l , m , n 0 1 l , m , n = 0 .
f ( r , r 0 , t 0 , T ) = cos [ Θ ( r , r 0 ) ] j = 1 N J 0 [ B j ( r , r 0 ) ] , f i ( r , r 0 , t 0 , T ) = cos [ Θ ( r , r 0 ) ] J 0 [ B i ( r , r 0 ) ] ,
E ± ( r , r 0 , t 0 , T ) = T H B ( r ) ± 2 T [ H L ( r ) H I ( r ) ] 1 / 2 × cos [ Θ ( r , r 0 ) ] f ( r , r 0 , t 0 , T ) .

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