Abstract

The design, theory, operation, and characteristics of an optically phase-locked electronic speckle pattern interferometer (OPL-ESPI) are described. The OPL-ESPI system couples an optical phase-locked loop with an ESPI system to generate real-time equal Doppler speckle contours of moving objects from unstable sensor platforms. In addition, the optical phase-locked loop provides the basis for a new ESPI video signal processing technique which incorporates local oscillator phase shifting coupled with video sequential frame subtraction.

© 1987 Optical Society of America

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References

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  1. A. E. Ennos, “Speckle Interferometry,” in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer-Verlag, New York, 1975), pp. 203–253.
    [CrossRef]
  2. T. J. Cookson, J. N. Butters, H. C. Pollard, “Pulsed Lasers in Electronic Speckle Pattern Interferometry,” Opt. Laser Technol. 119 (June1978).
  3. O. J. Lokberg, “Chopped Laser Light in Electronic Speckle Pattern Interferometry,” Appl. Opt. 18, 2377 (1979).
    [CrossRef] [PubMed]
  4. D. B. Neumann, H. W. Rose, “Improvement of Recorded Holographic Fringes by Feedback Control,” Appl. Opt. 6, 1097 (1967).
    [CrossRef] [PubMed]
  5. H. W. Rose, H. D. Pruett, “Stabilization of Holographic Fringes by FM Feedback,” Appl. Opt. 7, 87 (1968).
    [CrossRef] [PubMed]
  6. D. R. MacQuigg, “Hologram Fringe Stabilization Method,” Appl. Opt. 16, 291 (1977).
    [CrossRef] [PubMed]
  7. C. C. Aleksoff, “Temporally Modulated Holography,” Appl. Opt. 10, 1329 (1971).
    [CrossRef] [PubMed]
  8. O. J. Lokberg, K. Hogmoen, “Vibration Phase Mapping using Electronic Speckle Pattern Interferometry,” Appl. Opt. 15, 2701 (1976).
    [CrossRef] [PubMed]
  9. O. J. Lokberg, O. K. Ledang, “Vibration of Flutes Studied by Electronic Speckle Pattern Interferometry,” Appl. Opt. 23, 3052 (1984).
    [CrossRef] [PubMed]
  10. K. Hogmoen, O. J. Lokberg, “Detection and Measurement of Small Vibrations using Electronic Speckle Pattern Interferometry,” Appl. Opt. 16, 1869 (1977).
    [CrossRef] [PubMed]
  11. K. Hogmoen, H. M. Pedersen, “Measurement of Small Vibrations using Electronic Speckle Pattern Interferometry: Theory,” J. Opt. Soc. Am. 67, 1578 (1977).
    [CrossRef]
  12. H. K. Philipp, A. L. Scholtz, E. Bonek, W. R. Leeb, “Costas Loop Experiments for a 10.6 Micron Communications Receiver,” IEEE Trans. Commun. COM-31, 1000 (1983).
    [CrossRef]
  13. H. K. Philipp, A. L. Scholtz, E. Bonek, W. R. Leeb, “Infrared Homodyne Receiver with Acousto-optically Controlled Local Oscillator,” Electron. Lett. 19, 234 (1983).
    [CrossRef]
  14. H. K. Philipp, A. L. Scholtz, E. Bonek, W. R. Leeb, “Frequency Synchronization and Phase Locking of CO2 Lasers,” Appl. Phys. Lett. 41, 592 (1982).
    [CrossRef]
  15. R. C. Cumming, “The Serrodyne Frequency Translator,” Proc. IRE 175 (Feb.1957).
  16. C. J. Tranter, Bessel Functions With Some Physical Applications (English Universities Press, 1968), p. 36.
  17. J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer-Verlag, New York, 1975), pp. 9–75.
    [CrossRef]
  18. L. Ek, N. E. Molin, “Detection of the Nodal Lines and the Amplitude of Vibration by Speckle Interferometry,” Opt. Commun. 2, 419 (1971).
    [CrossRef]
  19. K. Creath, G. A. Slettemoen, “Vibration-Observation Techniques for Digital Speckle-Pattern Interferometry,” J. Opt. Soc. Am. A 2, 1629 (1985).
    [CrossRef]
  20. S. Nakadate, T. Yatagai, H. Saito, “Electronic Speckle Pattern Interferometry using Digital Image Processing Techniques,” Appl. Opt. 19, 1879 (1980).
    [CrossRef] [PubMed]

1985 (1)

1984 (1)

1983 (2)

H. K. Philipp, A. L. Scholtz, E. Bonek, W. R. Leeb, “Costas Loop Experiments for a 10.6 Micron Communications Receiver,” IEEE Trans. Commun. COM-31, 1000 (1983).
[CrossRef]

H. K. Philipp, A. L. Scholtz, E. Bonek, W. R. Leeb, “Infrared Homodyne Receiver with Acousto-optically Controlled Local Oscillator,” Electron. Lett. 19, 234 (1983).
[CrossRef]

1982 (1)

H. K. Philipp, A. L. Scholtz, E. Bonek, W. R. Leeb, “Frequency Synchronization and Phase Locking of CO2 Lasers,” Appl. Phys. Lett. 41, 592 (1982).
[CrossRef]

1980 (1)

1979 (1)

1978 (1)

T. J. Cookson, J. N. Butters, H. C. Pollard, “Pulsed Lasers in Electronic Speckle Pattern Interferometry,” Opt. Laser Technol. 119 (June1978).

1977 (3)

1976 (1)

1971 (2)

L. Ek, N. E. Molin, “Detection of the Nodal Lines and the Amplitude of Vibration by Speckle Interferometry,” Opt. Commun. 2, 419 (1971).
[CrossRef]

C. C. Aleksoff, “Temporally Modulated Holography,” Appl. Opt. 10, 1329 (1971).
[CrossRef] [PubMed]

1968 (1)

1967 (1)

1957 (1)

R. C. Cumming, “The Serrodyne Frequency Translator,” Proc. IRE 175 (Feb.1957).

Aleksoff, C. C.

Bonek, E.

H. K. Philipp, A. L. Scholtz, E. Bonek, W. R. Leeb, “Costas Loop Experiments for a 10.6 Micron Communications Receiver,” IEEE Trans. Commun. COM-31, 1000 (1983).
[CrossRef]

H. K. Philipp, A. L. Scholtz, E. Bonek, W. R. Leeb, “Infrared Homodyne Receiver with Acousto-optically Controlled Local Oscillator,” Electron. Lett. 19, 234 (1983).
[CrossRef]

H. K. Philipp, A. L. Scholtz, E. Bonek, W. R. Leeb, “Frequency Synchronization and Phase Locking of CO2 Lasers,” Appl. Phys. Lett. 41, 592 (1982).
[CrossRef]

Butters, J. N.

T. J. Cookson, J. N. Butters, H. C. Pollard, “Pulsed Lasers in Electronic Speckle Pattern Interferometry,” Opt. Laser Technol. 119 (June1978).

Cookson, T. J.

T. J. Cookson, J. N. Butters, H. C. Pollard, “Pulsed Lasers in Electronic Speckle Pattern Interferometry,” Opt. Laser Technol. 119 (June1978).

Creath, K.

Cumming, R. C.

R. C. Cumming, “The Serrodyne Frequency Translator,” Proc. IRE 175 (Feb.1957).

Ek, L.

L. Ek, N. E. Molin, “Detection of the Nodal Lines and the Amplitude of Vibration by Speckle Interferometry,” Opt. Commun. 2, 419 (1971).
[CrossRef]

Ennos, A. E.

A. E. Ennos, “Speckle Interferometry,” in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer-Verlag, New York, 1975), pp. 203–253.
[CrossRef]

Goodman, J. W.

J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer-Verlag, New York, 1975), pp. 9–75.
[CrossRef]

Hogmoen, K.

Ledang, O. K.

Leeb, W. R.

H. K. Philipp, A. L. Scholtz, E. Bonek, W. R. Leeb, “Costas Loop Experiments for a 10.6 Micron Communications Receiver,” IEEE Trans. Commun. COM-31, 1000 (1983).
[CrossRef]

H. K. Philipp, A. L. Scholtz, E. Bonek, W. R. Leeb, “Infrared Homodyne Receiver with Acousto-optically Controlled Local Oscillator,” Electron. Lett. 19, 234 (1983).
[CrossRef]

H. K. Philipp, A. L. Scholtz, E. Bonek, W. R. Leeb, “Frequency Synchronization and Phase Locking of CO2 Lasers,” Appl. Phys. Lett. 41, 592 (1982).
[CrossRef]

Lokberg, O. J.

MacQuigg, D. R.

Molin, N. E.

L. Ek, N. E. Molin, “Detection of the Nodal Lines and the Amplitude of Vibration by Speckle Interferometry,” Opt. Commun. 2, 419 (1971).
[CrossRef]

Nakadate, S.

Neumann, D. B.

Pedersen, H. M.

Philipp, H. K.

H. K. Philipp, A. L. Scholtz, E. Bonek, W. R. Leeb, “Costas Loop Experiments for a 10.6 Micron Communications Receiver,” IEEE Trans. Commun. COM-31, 1000 (1983).
[CrossRef]

H. K. Philipp, A. L. Scholtz, E. Bonek, W. R. Leeb, “Infrared Homodyne Receiver with Acousto-optically Controlled Local Oscillator,” Electron. Lett. 19, 234 (1983).
[CrossRef]

H. K. Philipp, A. L. Scholtz, E. Bonek, W. R. Leeb, “Frequency Synchronization and Phase Locking of CO2 Lasers,” Appl. Phys. Lett. 41, 592 (1982).
[CrossRef]

Pollard, H. C.

T. J. Cookson, J. N. Butters, H. C. Pollard, “Pulsed Lasers in Electronic Speckle Pattern Interferometry,” Opt. Laser Technol. 119 (June1978).

Pruett, H. D.

Rose, H. W.

Saito, H.

Scholtz, A. L.

H. K. Philipp, A. L. Scholtz, E. Bonek, W. R. Leeb, “Infrared Homodyne Receiver with Acousto-optically Controlled Local Oscillator,” Electron. Lett. 19, 234 (1983).
[CrossRef]

H. K. Philipp, A. L. Scholtz, E. Bonek, W. R. Leeb, “Costas Loop Experiments for a 10.6 Micron Communications Receiver,” IEEE Trans. Commun. COM-31, 1000 (1983).
[CrossRef]

H. K. Philipp, A. L. Scholtz, E. Bonek, W. R. Leeb, “Frequency Synchronization and Phase Locking of CO2 Lasers,” Appl. Phys. Lett. 41, 592 (1982).
[CrossRef]

Slettemoen, G. A.

Tranter, C. J.

C. J. Tranter, Bessel Functions With Some Physical Applications (English Universities Press, 1968), p. 36.

Yatagai, T.

Appl. Opt. (9)

Appl. Phys. Lett. (1)

H. K. Philipp, A. L. Scholtz, E. Bonek, W. R. Leeb, “Frequency Synchronization and Phase Locking of CO2 Lasers,” Appl. Phys. Lett. 41, 592 (1982).
[CrossRef]

Electron. Lett. (1)

H. K. Philipp, A. L. Scholtz, E. Bonek, W. R. Leeb, “Infrared Homodyne Receiver with Acousto-optically Controlled Local Oscillator,” Electron. Lett. 19, 234 (1983).
[CrossRef]

IEEE Trans. Commun. (1)

H. K. Philipp, A. L. Scholtz, E. Bonek, W. R. Leeb, “Costas Loop Experiments for a 10.6 Micron Communications Receiver,” IEEE Trans. Commun. COM-31, 1000 (1983).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (1)

L. Ek, N. E. Molin, “Detection of the Nodal Lines and the Amplitude of Vibration by Speckle Interferometry,” Opt. Commun. 2, 419 (1971).
[CrossRef]

Opt. Laser Technol. (1)

T. J. Cookson, J. N. Butters, H. C. Pollard, “Pulsed Lasers in Electronic Speckle Pattern Interferometry,” Opt. Laser Technol. 119 (June1978).

Proc. IRE (1)

R. C. Cumming, “The Serrodyne Frequency Translator,” Proc. IRE 175 (Feb.1957).

Other (3)

C. J. Tranter, Bessel Functions With Some Physical Applications (English Universities Press, 1968), p. 36.

J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer-Verlag, New York, 1975), pp. 9–75.
[CrossRef]

A. E. Ennos, “Speckle Interferometry,” in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer-Verlag, New York, 1975), pp. 203–253.
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Schematic representation of OPL-ESPI system.

Fig. 2
Fig. 2

Laboratory OPL-ESPI system.

Fig. 3
Fig. 3

Bessel functions of the first kind.

Fig. 4
Fig. 4

Image processing block diagram.

Fig. 5
Fig. 5

Sawtooth phase modulation waveforms.

Fig. 6
Fig. 6

Illustration of image processing procedure.

Fig. 7
Fig. 7

Speaker cone equal Doppler contours.

Fig. 8
Fig. 8

Effect of changing lock point location on speaker cone equal Doppler contours.

Fig. 9
Fig. 9

Effect of increasing vibrational amplitude on speaker cone equal Doppler contours.

Fig. 10
Fig. 10

Piezoelectric translator imagery.

Fig. 11
Fig. 11

Minimum amplitude sensitivity of OPL-ESPI system.

Tables (1)

Tables Icon

Table I Calculation of Amplitude Variations for Fig. 9

Equations (68)

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f = f 0 + ( 2 π ) - 1 d θ ( t ) / d t .
H ( r , t , τ ) = U L ( r , t ) + U I ( r , t + τ ) 2 β ,
U L ( r , t ) = [ H L ( r ) ] 1 / 2 exp { i [ ω 0 t + θ L ( r ) + α L ( r , t ) + β ( t ) ] } ,
U I ( r , t ) = [ H I ( r ) ] 1 / 2 exp { i [ ω 0 t + θ L ( r ) + α I ( r , t ) + β ( t ) ] } ,
H ( r , t , τ ) = H b ( r ) + 2 [ H L ( r ) H I ( r ) ] 1 / 2 × Γ ( τ ) Re exp ( i { [ α I ( r , t ) - α L ( r , t ) ] + θ I ( r ) - θ L ( r ) } ) ,
α L ( r , t ) = α I ( r 0 , t ) , θ L ( r ) = θ I ( r 0 ) .
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 ) .
d ( r , t ) = A ( r ) sin [ Ω t + ϕ ( r ) ] ,
α I ( r , t ) = 4 π d ( r , t ) / λ = 4 π [ A ( r ) / λ ] sin [ Ω t + ϕ ( r ) ] .
α I ( r 0 , t ) - α I ( r , t ) = A ( r 0 ) sin [ Ω t + ϕ ( r 0 ) ] - A ( r ) sin [ Ω t + ϕ ( r ) ] = B ( r , r 0 ) sin [ Ω t + γ ( r , r 0 ) ] ,
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 ) ] } ) ,
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 ) } ) .
exp [ i x sin ( y ) ] = n = - J n ( x ) exp ( i n y ) ,
H ( r , r 0 , t ) = H b ( r ) + 2 [ H L ( r ) H I ( r ) ] 1 / 2 × Re n = - J n [ B ( r , r 0 ) ] × exp ( i { n [ Ω t + γ ( 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 [ H L ( r ) H I ( r ) ] 1 / 2 Re n = - J n [ B ( r , r 0 ) ] × t 0 - T / 2 t 0 + T / 2 exp ( i { n [ Ω t + γ ( r , r 0 ) ] + Θ ( r , r 0 ) } ) d t ] = T { H b ( r ) + 2 [ H L ( r ) H I ( r ) ] 1 / 2 n = - J n [ B ( r , r 0 ) ] × cos { n [ Ω t 0 + γ ( r , r 0 ) ] + Θ ( r , r 0 ) } sinc ( n F T ) } ,
sinc ( n T F ) = sin ( π n T F ) / π n T F ,
sinc ( n F T ) = [ 1 n = 0 , 0 n 0.
E ( r , r 0 ) = T { H b ( r ) + 2 [ H L ( r ) ] 1 / 2 J 0 [ B ( r , r 0 ) ] cos [ Θ ( r , r 0 ) ] }
E ( r , r 0 ) = T { H b ( r ) + 2 [ H L ( r ) H I ( r ) ] 1 / 2 cos [ Θ ( r , r 0 ) ] } ,
B ( r , r 0 ) = [ M 1 2 ( r , r 0 ) + M 2 2 ( r , r 0 ) ] 1 / 2 ,
M 1 ( r , r 0 ) = 8 π [ A ( r 0 ) A ( r ) ] 1 / 2 sin [ Φ ( r , r 0 ) ] / λ , M 2 ( r , r 0 ) = 4 π [ A ( r ) - A ( r 0 ) ] / λ .
E ( r , r 0 ) = T ( H b ( r ) + 2 [ H L ( r ) H I ( r ) ] 1 / 2 J 0 { [ M 1 2 ( r , r 0 ) + M 2 2 ( r , r 0 ) ] 1 / 2 } × cos [ Θ ( r , r 0 ) ] ) .
J 0 { [ M 1 2 + M 2 2 - 2 M 1 M 2 cos ( θ ) ] 1 / 2 } = n = - J n ( M 1 ) J n ( M 2 ) cos ( n θ )
J 0 [ ( M 1 2 + M 2 2 ) 1 / 2 ] = n = - ( - 1 ) n J 2 n ( M 1 ) J 2 n ( M 2 ) .
E ( r , r 0 ) = T { H b ( r ) + 2 [ H L ( r ) H I ( r ) ] 1 / 2 × n = - ( - 1 ) n J 2 n [ M 1 ( r , r 0 ) ] × J 2 n [ M 2 ( r , r 0 ) ] cos [ Θ ( r , r 0 ) ] } .
α I ( r , t ) α I ( r , t = t 0 ) + ( t - t 0 ) d α i ( r , t = t 0 ) / d t 4 π [ d ( r , t 0 ) + ( t - t 0 ) v ( r , t 0 ) ] / λ ,
v ( r , t 0 ) = A ( r ) Ω cos [ Ω t 0 + ϕ ( r ) ]
E ( r , r 0 , t ) = T { H b ( r ) + 2 [ H L ( r ) H L ( r ) H I ( r ) ] 1 / 2 × cos [ 4 π D ( r , r 0 , t 0 ) t / λ + Θ ( r , r 0 ) ] × sinc [ 2 V ( r , r 0 , t 0 ) T / λ ] } ,
D ( r , r 0 , t 0 ) = d ( r 0 , t 0 ) - d ( r , t 0 ) , V ( r , r 0 , t 0 ) = v ( r , t 0 ) - v ( r , t 0 ) .
C ( r , r 0 ) = σ E / E E ,
σ E 2 = E 2 ( r , r 0 ) E - E ( r , r 0 ) E 2
P ( H , θ ) = [ ( 4 π σ 2 ) - 1 exp ( - H / 2 σ 2 ) H > 0 , 0 otherwise .
P ( H , θ ) = P ( H ) P ( θ ) ,
P ( H ) = [ ( 2 σ 2 ) - 1 exp ( - H / 2 σ 2 ) H > 0 , 0 otherwise .
P ( θ ) = [ ( 2 π ) - 1 - π < θ < π , 0 otherwise .
P [ Θ ( r , r 0 ) ] = [ ( 2 π ) - 1 - π < Θ ( r , r 0 ) < π , 0 otherwise .
H n H = n ! H H n .
σ H 2 = H 2 H - H H 2 = H H 2 .
cos [ Θ ( r , r 0 ) ] Θ = 0 , cos 2 [ Θ ( r , r 0 ) ] Θ = ½ .
E E = { H L + H I H + 2 H 1 1 / 2 H H L 1 / 2 × cos [ Θ ( r , r 0 ) ] Θ J 0 [ B ( r , r 0 ) ] } T ,
E E = T { H I H + H L } .
E 2 E = T 2 { H L + H I ) 2 H + 4 H L H I H cos 2 [ Θ ( r , r 0 ) ] Θ J 0 2 [ B ( r , r 0 ) ] + 4 ( H L + H I ) ( H L H I ) 1 / 2 H cos [ Θ ( r , r 0 ) ] Θ × J 0 2 [ B ( r , r 0 ) ] } = T 2 { H L 2 + H I 2 H + 2 H L H I H + 2 H L H I H J 0 2 [ B ( r , r 0 ) ] } .
σ E 2 = T 2 { H I H + 2 H I H H L J 0 2 [ B ( r , r 0 ) ] } .
C ( r , r 0 ) = [ 1 + Q ] - 1 [ 1 + 2 Q J 0 2 [ B ( r , r 0 ) ] ] 1 / 2 = [ 1 + Q ] - 1 [ 1 + 2 Q J 0 2 { [ M 1 2 ( r , r 0 ) + M 2 2 ( r , r 0 ) ] 1 / 2 } ] 1 / 2 ,
Q = H L / H I H
C ( r , r 0 ) = [ 1 + Q ] - 1 × [ 1 + 2 Q { n = - ( - 1 ) n J 2 n ( M 1 ( r , r 0 ) ] × J 2 n [ M 2 ( r , r 0 ) ] } 2 ] 1 / 2 .
C ( r , r 0 , t 0 , T ) = [ 1 + Q ] - 1 { 1 + 2 Q sinc 2 [ 2 V ( r , r 0 , t 0 ) T / λ ] } 1 / 2 .
[ 1 + Q ] - 1 < C ( r , r 0 ) < [ 1 + 2 Q ] 1 / 2 [ 1 + Q ] - 1 .
G c = [ 1 + 2 Q ] 1 / 2 .
J n [ M 2 ( r , r 0 ) ] = [ 1 n = 0 , 0 n 0 ,
B ( r , r 0 ) = M 1 ( r , r 0 )
C ( r , r 0 ) = [ 1 + Q ] - 1 { 1 + 2 Q J 0 2 { 8 π A ( r 0 ) sin [ Φ ( r , r 0 ) ] / λ } ) 1 / 2 .
J n [ M 1 ( r , r 0 ) ] = [ 1 n = 0 , 0 n 0 ,
B ( r , r 0 ) = M 2 ( r , r 0 )
C ( r , r 0 ) = [ 1 + Q ] - 1 [ 1 + 2 Q J 0 2 { 4 π [ A ( r 0 ) - A ( r ) ] / λ } ] 1 / 2 .
C ( r , r 0 ) = [ 1 + Q ] - 1 [ 1 + 2 Q ( J 0 { 4 π [ A ( r 0 ) - A ( r ) ] / λ } × J 0 { 8 π [ A ( r ) A ( r 0 ) ] 1 / 2 sin [ Φ ( r , r 0 ) ] / λ } ) 2 ] 1 / 2 .
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 ) + 2 π f 0 t + saw - ( 2 π f 0 t ) } ) ,
2 π f 0 t = saw + ( 2 π f 0 t ) .
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 ) + saw + ( 2 π f 0 t ) + saw - ( 2 π f 0 t ) } )
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 ) + saw + ( 2 π f 0 t + π ) + saw - ( 2 π f 0 t ) } ) .
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 ) } )
E ( r , r 0 ) = T [ H b ( r ) ] ± 2 [ H I ( r ) ] 1 / 2 cos [ Θ ( r , r 0 ) ] J 0 [ B ( r , r 0 ) ]
E ( r , r 0 ) = 4 T [ H L ( r ) H I ( r ) ] 1 / 2 cos [ Θ ( r , r 0 ) ] J 0 [ B ( r , r 0 ) ] .
ν d = 2 V max / λ .
V max = A max Ω ,
A max = ν d λ / 2 Ω = 14 μ m .

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