Abstract

The cross-correlation function between two light fields is recorded with the help of a new device. The proposed correlator exhibits ultrashort time resolution. The optical path difference between the two interfering beams does not have to be known with interferometric precision. The experimental dynamic range proved to be as large as 105. The device features imaging capabilities that could be applied to the analysis of two-dimensional images with ultrashort time resolution.

© 1994 Optical Society of America

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References

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  1. E. P. Ippen, C. V. Shank, “Techniques for measurement,” in Ultrashort Light Pulses: Picosecond Techniques and Applications, S. L. Shapiro, ed. (Springer-Verlag, Berlin, 1977), pp. 83–122.
  2. M. A. Duguay, J. W. Hansen, “Optical sampling of sub-nanosecond light pulses,” Appl. Phys. Lett. 13, 178–180 (1968).
    [CrossRef]
  3. J. E. Rothenberg, D. Grischkowsky, A. C. Balant, “Observation of the formation of the 0π pulse,” Phys. Rev. Lett. 53, 552–555 (1984).
    [CrossRef]
  4. A. Rebane, J. Aaviskoo, J. Kuhl, “Storage and time reversal of femtosecond light signals via persistent spectral hole burning holography,” Appl. Phys. Lett. 54, 93–95 (1989).
    [CrossRef]
  5. C. Joubert, M.-L. Roblin, R. Grousson, “Temporal reversal of picosecond optical pulses by holographic phase conjugation,” Appl. Opt. 28, 4604–4612 (1989).
    [CrossRef] [PubMed]
  6. H. J. Eichler, U. Klein, D. Langhans, “Coherence time measurement of picosecond pulses by light-induced grating method,” Appl. Phys. Lett. 21, 215–219 (1980).
  7. R. Trebino, E. K. Gustafson, A. E. Siegman, “Fourth-order partial-coherence effects in the formation of integrated intensity gratings with pulsed light sources,” J. Opt. Soc. Am. B 3, 1295–1304 (1986).
    [CrossRef]
  8. W. L. Nighan, T. Gong, L. Liou, P. M. Fauchet, “Self-diffraction: a new method for characterization of ultrashort laser pulses,” Opt. Commun. 69, 339–344 (1989).
    [CrossRef]
  9. A. S. L. Gomes, L. H. Acioli, C. de Araujo, J. Rios Leite, “Dispersion of coherence spikes of incoherent broadband dye lasers,” Opt. Commun. 73, 475–478 (1989).
    [CrossRef]
  10. V. Dominic, X. S. Yao, R. M. Pierre, J. Feinberg, “Measuring the coherence length of mode-locked laser pulses in real time,” Appl. Phys. 56, 521–523 (1990).
  11. A. Débarre, J. C. Keiler, J. L. Le Gouët, A. Richard, P. Tchénio, “An amplitude correlator for broadband laser source characterization,” Opt. Commun. 73, 309–313 (1989).
    [CrossRef]
  12. A. Débarre, J. C. Keiler, J. L. Le Gouët, P. Tchénio, “Field cross-correlation retrieval of optically stored data,” J. Opt. Soc. Am. B 8, 153–159 (1991).
    [CrossRef]
  13. M. A. Bouchene, A. Débarre, J. C. Keiler, J. L. Le Gouët, P. Tchénio, “Observation of 0π-pulse formation with incoherent light,” J. Opt. Soc. Am. B 9, 281–289 (1992).
    [CrossRef]
  14. J.-P. Maillard, D. Simons, “First results of an imaging FTS with a NICMOS camera,” in Proceedings of the ESA Workshop on Solar Physics and Astrophysics at Interferometric Resolution, L. Damé, T. D. Guyenne, eds. (European Space Agency, Noordwijk, The Netherlands, 1992), ESA SP-344, pp. 205–210.
  15. R. Hanbury Brown, R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature (London) 177, 27–29 (1956).
    [CrossRef]
  16. Z. Y. Ou, E. C. Gage, B. E. Magill, L. Mandel, “Fourth-order interference technique for determining the coherence time of a light beam,” J. Opt. Soc. Am. B 6, 100–103 (1989).
    [CrossRef]
  17. Y. Miyamoto, T. Kuga, M. Baba, M. Matsuoka, “Measurement of ultrafast optical pulses with two-photon interference,” Opt. Lett. 18, 900–902 (1993).
    [CrossRef] [PubMed]
  18. P. Juncar, J. Pinard, “Instrument to measure wave numbers of cw and pulsed laser lines: the sigmameter,” Rev. Sci. Instrum. 53, 939–948 (1982).
    [CrossRef]
  19. H. A. Haus, “Theory of mode locking with a slow saturable absorber,” IEEE J. Quantum Electron. QE-11, 736–746 (1975).
    [CrossRef]
  20. H. A. Haus, C. V. Shank, E. P. Ippen, “Shape of passively mode-locked laser pulses,” Opt. Commun. 15, 29–31 (1975).
    [CrossRef]

1993 (1)

1992 (1)

1991 (1)

1990 (1)

V. Dominic, X. S. Yao, R. M. Pierre, J. Feinberg, “Measuring the coherence length of mode-locked laser pulses in real time,” Appl. Phys. 56, 521–523 (1990).

1989 (6)

A. Débarre, J. C. Keiler, J. L. Le Gouët, A. Richard, P. Tchénio, “An amplitude correlator for broadband laser source characterization,” Opt. Commun. 73, 309–313 (1989).
[CrossRef]

W. L. Nighan, T. Gong, L. Liou, P. M. Fauchet, “Self-diffraction: a new method for characterization of ultrashort laser pulses,” Opt. Commun. 69, 339–344 (1989).
[CrossRef]

A. S. L. Gomes, L. H. Acioli, C. de Araujo, J. Rios Leite, “Dispersion of coherence spikes of incoherent broadband dye lasers,” Opt. Commun. 73, 475–478 (1989).
[CrossRef]

A. Rebane, J. Aaviskoo, J. Kuhl, “Storage and time reversal of femtosecond light signals via persistent spectral hole burning holography,” Appl. Phys. Lett. 54, 93–95 (1989).
[CrossRef]

C. Joubert, M.-L. Roblin, R. Grousson, “Temporal reversal of picosecond optical pulses by holographic phase conjugation,” Appl. Opt. 28, 4604–4612 (1989).
[CrossRef] [PubMed]

Z. Y. Ou, E. C. Gage, B. E. Magill, L. Mandel, “Fourth-order interference technique for determining the coherence time of a light beam,” J. Opt. Soc. Am. B 6, 100–103 (1989).
[CrossRef]

1986 (1)

1984 (1)

J. E. Rothenberg, D. Grischkowsky, A. C. Balant, “Observation of the formation of the 0π pulse,” Phys. Rev. Lett. 53, 552–555 (1984).
[CrossRef]

1982 (1)

P. Juncar, J. Pinard, “Instrument to measure wave numbers of cw and pulsed laser lines: the sigmameter,” Rev. Sci. Instrum. 53, 939–948 (1982).
[CrossRef]

1980 (1)

H. J. Eichler, U. Klein, D. Langhans, “Coherence time measurement of picosecond pulses by light-induced grating method,” Appl. Phys. Lett. 21, 215–219 (1980).

1975 (2)

H. A. Haus, “Theory of mode locking with a slow saturable absorber,” IEEE J. Quantum Electron. QE-11, 736–746 (1975).
[CrossRef]

H. A. Haus, C. V. Shank, E. P. Ippen, “Shape of passively mode-locked laser pulses,” Opt. Commun. 15, 29–31 (1975).
[CrossRef]

1968 (1)

M. A. Duguay, J. W. Hansen, “Optical sampling of sub-nanosecond light pulses,” Appl. Phys. Lett. 13, 178–180 (1968).
[CrossRef]

1956 (1)

R. Hanbury Brown, R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature (London) 177, 27–29 (1956).
[CrossRef]

Aaviskoo, J.

A. Rebane, J. Aaviskoo, J. Kuhl, “Storage and time reversal of femtosecond light signals via persistent spectral hole burning holography,” Appl. Phys. Lett. 54, 93–95 (1989).
[CrossRef]

Acioli, L. H.

A. S. L. Gomes, L. H. Acioli, C. de Araujo, J. Rios Leite, “Dispersion of coherence spikes of incoherent broadband dye lasers,” Opt. Commun. 73, 475–478 (1989).
[CrossRef]

Baba, M.

Balant, A. C.

J. E. Rothenberg, D. Grischkowsky, A. C. Balant, “Observation of the formation of the 0π pulse,” Phys. Rev. Lett. 53, 552–555 (1984).
[CrossRef]

Bouchene, M. A.

de Araujo, C.

A. S. L. Gomes, L. H. Acioli, C. de Araujo, J. Rios Leite, “Dispersion of coherence spikes of incoherent broadband dye lasers,” Opt. Commun. 73, 475–478 (1989).
[CrossRef]

Débarre, A.

Dominic, V.

V. Dominic, X. S. Yao, R. M. Pierre, J. Feinberg, “Measuring the coherence length of mode-locked laser pulses in real time,” Appl. Phys. 56, 521–523 (1990).

Duguay, M. A.

M. A. Duguay, J. W. Hansen, “Optical sampling of sub-nanosecond light pulses,” Appl. Phys. Lett. 13, 178–180 (1968).
[CrossRef]

Eichler, H. J.

H. J. Eichler, U. Klein, D. Langhans, “Coherence time measurement of picosecond pulses by light-induced grating method,” Appl. Phys. Lett. 21, 215–219 (1980).

Fauchet, P. M.

W. L. Nighan, T. Gong, L. Liou, P. M. Fauchet, “Self-diffraction: a new method for characterization of ultrashort laser pulses,” Opt. Commun. 69, 339–344 (1989).
[CrossRef]

Feinberg, J.

V. Dominic, X. S. Yao, R. M. Pierre, J. Feinberg, “Measuring the coherence length of mode-locked laser pulses in real time,” Appl. Phys. 56, 521–523 (1990).

Gage, E. C.

Gomes, A. S. L.

A. S. L. Gomes, L. H. Acioli, C. de Araujo, J. Rios Leite, “Dispersion of coherence spikes of incoherent broadband dye lasers,” Opt. Commun. 73, 475–478 (1989).
[CrossRef]

Gong, T.

W. L. Nighan, T. Gong, L. Liou, P. M. Fauchet, “Self-diffraction: a new method for characterization of ultrashort laser pulses,” Opt. Commun. 69, 339–344 (1989).
[CrossRef]

Grischkowsky, D.

J. E. Rothenberg, D. Grischkowsky, A. C. Balant, “Observation of the formation of the 0π pulse,” Phys. Rev. Lett. 53, 552–555 (1984).
[CrossRef]

Grousson, R.

Gustafson, E. K.

Hanbury Brown, R.

R. Hanbury Brown, R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature (London) 177, 27–29 (1956).
[CrossRef]

Hansen, J. W.

M. A. Duguay, J. W. Hansen, “Optical sampling of sub-nanosecond light pulses,” Appl. Phys. Lett. 13, 178–180 (1968).
[CrossRef]

Haus, H. A.

H. A. Haus, “Theory of mode locking with a slow saturable absorber,” IEEE J. Quantum Electron. QE-11, 736–746 (1975).
[CrossRef]

H. A. Haus, C. V. Shank, E. P. Ippen, “Shape of passively mode-locked laser pulses,” Opt. Commun. 15, 29–31 (1975).
[CrossRef]

Ippen, E. P.

H. A. Haus, C. V. Shank, E. P. Ippen, “Shape of passively mode-locked laser pulses,” Opt. Commun. 15, 29–31 (1975).
[CrossRef]

E. P. Ippen, C. V. Shank, “Techniques for measurement,” in Ultrashort Light Pulses: Picosecond Techniques and Applications, S. L. Shapiro, ed. (Springer-Verlag, Berlin, 1977), pp. 83–122.

Joubert, C.

Juncar, P.

P. Juncar, J. Pinard, “Instrument to measure wave numbers of cw and pulsed laser lines: the sigmameter,” Rev. Sci. Instrum. 53, 939–948 (1982).
[CrossRef]

Keiler, J. C.

Klein, U.

H. J. Eichler, U. Klein, D. Langhans, “Coherence time measurement of picosecond pulses by light-induced grating method,” Appl. Phys. Lett. 21, 215–219 (1980).

Kuga, T.

Kuhl, J.

A. Rebane, J. Aaviskoo, J. Kuhl, “Storage and time reversal of femtosecond light signals via persistent spectral hole burning holography,” Appl. Phys. Lett. 54, 93–95 (1989).
[CrossRef]

Langhans, D.

H. J. Eichler, U. Klein, D. Langhans, “Coherence time measurement of picosecond pulses by light-induced grating method,” Appl. Phys. Lett. 21, 215–219 (1980).

Le Gouët, J. L.

Liou, L.

W. L. Nighan, T. Gong, L. Liou, P. M. Fauchet, “Self-diffraction: a new method for characterization of ultrashort laser pulses,” Opt. Commun. 69, 339–344 (1989).
[CrossRef]

Magill, B. E.

Maillard, J.-P.

J.-P. Maillard, D. Simons, “First results of an imaging FTS with a NICMOS camera,” in Proceedings of the ESA Workshop on Solar Physics and Astrophysics at Interferometric Resolution, L. Damé, T. D. Guyenne, eds. (European Space Agency, Noordwijk, The Netherlands, 1992), ESA SP-344, pp. 205–210.

Mandel, L.

Matsuoka, M.

Miyamoto, Y.

Nighan, W. L.

W. L. Nighan, T. Gong, L. Liou, P. M. Fauchet, “Self-diffraction: a new method for characterization of ultrashort laser pulses,” Opt. Commun. 69, 339–344 (1989).
[CrossRef]

Ou, Z. Y.

Pierre, R. M.

V. Dominic, X. S. Yao, R. M. Pierre, J. Feinberg, “Measuring the coherence length of mode-locked laser pulses in real time,” Appl. Phys. 56, 521–523 (1990).

Pinard, J.

P. Juncar, J. Pinard, “Instrument to measure wave numbers of cw and pulsed laser lines: the sigmameter,” Rev. Sci. Instrum. 53, 939–948 (1982).
[CrossRef]

Rebane, A.

A. Rebane, J. Aaviskoo, J. Kuhl, “Storage and time reversal of femtosecond light signals via persistent spectral hole burning holography,” Appl. Phys. Lett. 54, 93–95 (1989).
[CrossRef]

Richard, A.

A. Débarre, J. C. Keiler, J. L. Le Gouët, A. Richard, P. Tchénio, “An amplitude correlator for broadband laser source characterization,” Opt. Commun. 73, 309–313 (1989).
[CrossRef]

Rios Leite, J.

A. S. L. Gomes, L. H. Acioli, C. de Araujo, J. Rios Leite, “Dispersion of coherence spikes of incoherent broadband dye lasers,” Opt. Commun. 73, 475–478 (1989).
[CrossRef]

Roblin, M.-L.

Rothenberg, J. E.

J. E. Rothenberg, D. Grischkowsky, A. C. Balant, “Observation of the formation of the 0π pulse,” Phys. Rev. Lett. 53, 552–555 (1984).
[CrossRef]

Shank, C. V.

H. A. Haus, C. V. Shank, E. P. Ippen, “Shape of passively mode-locked laser pulses,” Opt. Commun. 15, 29–31 (1975).
[CrossRef]

E. P. Ippen, C. V. Shank, “Techniques for measurement,” in Ultrashort Light Pulses: Picosecond Techniques and Applications, S. L. Shapiro, ed. (Springer-Verlag, Berlin, 1977), pp. 83–122.

Siegman, A. E.

Simons, D.

J.-P. Maillard, D. Simons, “First results of an imaging FTS with a NICMOS camera,” in Proceedings of the ESA Workshop on Solar Physics and Astrophysics at Interferometric Resolution, L. Damé, T. D. Guyenne, eds. (European Space Agency, Noordwijk, The Netherlands, 1992), ESA SP-344, pp. 205–210.

Tchénio, P.

Trebino, R.

Twiss, R. Q.

R. Hanbury Brown, R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature (London) 177, 27–29 (1956).
[CrossRef]

Yao, X. S.

V. Dominic, X. S. Yao, R. M. Pierre, J. Feinberg, “Measuring the coherence length of mode-locked laser pulses in real time,” Appl. Phys. 56, 521–523 (1990).

Appl. Opt. (1)

Appl. Phys. (1)

V. Dominic, X. S. Yao, R. M. Pierre, J. Feinberg, “Measuring the coherence length of mode-locked laser pulses in real time,” Appl. Phys. 56, 521–523 (1990).

Appl. Phys. Lett. (3)

H. J. Eichler, U. Klein, D. Langhans, “Coherence time measurement of picosecond pulses by light-induced grating method,” Appl. Phys. Lett. 21, 215–219 (1980).

M. A. Duguay, J. W. Hansen, “Optical sampling of sub-nanosecond light pulses,” Appl. Phys. Lett. 13, 178–180 (1968).
[CrossRef]

A. Rebane, J. Aaviskoo, J. Kuhl, “Storage and time reversal of femtosecond light signals via persistent spectral hole burning holography,” Appl. Phys. Lett. 54, 93–95 (1989).
[CrossRef]

IEEE J. Quantum Electron. (1)

H. A. Haus, “Theory of mode locking with a slow saturable absorber,” IEEE J. Quantum Electron. QE-11, 736–746 (1975).
[CrossRef]

J. Opt. Soc. Am. B (4)

Nature (London) (1)

R. Hanbury Brown, R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature (London) 177, 27–29 (1956).
[CrossRef]

Opt. Commun. (4)

W. L. Nighan, T. Gong, L. Liou, P. M. Fauchet, “Self-diffraction: a new method for characterization of ultrashort laser pulses,” Opt. Commun. 69, 339–344 (1989).
[CrossRef]

A. S. L. Gomes, L. H. Acioli, C. de Araujo, J. Rios Leite, “Dispersion of coherence spikes of incoherent broadband dye lasers,” Opt. Commun. 73, 475–478 (1989).
[CrossRef]

A. Débarre, J. C. Keiler, J. L. Le Gouët, A. Richard, P. Tchénio, “An amplitude correlator for broadband laser source characterization,” Opt. Commun. 73, 309–313 (1989).
[CrossRef]

H. A. Haus, C. V. Shank, E. P. Ippen, “Shape of passively mode-locked laser pulses,” Opt. Commun. 15, 29–31 (1975).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. Lett. (1)

J. E. Rothenberg, D. Grischkowsky, A. C. Balant, “Observation of the formation of the 0π pulse,” Phys. Rev. Lett. 53, 552–555 (1984).
[CrossRef]

Rev. Sci. Instrum. (1)

P. Juncar, J. Pinard, “Instrument to measure wave numbers of cw and pulsed laser lines: the sigmameter,” Rev. Sci. Instrum. 53, 939–948 (1982).
[CrossRef]

Other (2)

J.-P. Maillard, D. Simons, “First results of an imaging FTS with a NICMOS camera,” in Proceedings of the ESA Workshop on Solar Physics and Astrophysics at Interferometric Resolution, L. Damé, T. D. Guyenne, eds. (European Space Agency, Noordwijk, The Netherlands, 1992), ESA SP-344, pp. 205–210.

E. P. Ippen, C. V. Shank, “Techniques for measurement,” in Ultrashort Light Pulses: Picosecond Techniques and Applications, S. L. Shapiro, ed. (Springer-Verlag, Berlin, 1977), pp. 83–122.

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Figures (7)

Fig. 1
Fig. 1

Schematic diagram of the dual-channel-field cross correlator: S, R, signal and reference light fields with orthogonal polarization directions; BS, beam sputter; P, analyzer; λ/4, quarter-wave plate; A, B, UDT 455 photoelectric detectors.

Fig. 2
Fig. 2

Scheme of the test setup. The optical delay line on path R is adjusted with the help of a motorized translation stage.

Fig. 3
Fig. 3

Continuous scan of the optical delay by the piezoactuator. A path-difference variation of one wavelength leads to a 2π change in phase shift between the signal and the reference pulses. It can also be expressed as a delay variation of λ/c = 2.07 fs. (a) Oscillatory behavior of the correlation signal when a single detector is used. (b) Elimination of the fluctuations by dual-channel detection.

Fig. 4
Fig. 4

Autocorrelation profiles obtained in a single scan of the translation stage without any average: (a) single-channel detection; (b) dual-channel detection.

Fig. 5
Fig. 5

Averaged correlation profiles for different values of the optical density inserted on the signal path: (a) experimental (solid) and theoretical (dashed) curves for D = 0 plotted together. The average was taken over six scans, (b) Signals of the field cross correlation averaged over approximately 20 times for (1) D = 2, (2) D = 4,and(3)D = 5.

Fig. 6
Fig. 6

Dependence of the square deviation σS2 on the average intensity 〈S〉 is linear and the slope was obtained as k1 = 0.29.

Fig. 7
Fig. 7

Background of the average intensity 〈S〉 versus the relative time delay measured as 〈S0〉 = 0.15 for D = 5.

Equations (32)

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E ( r , t ) = E ( t ) exp i ( ω t k r ) + c . c . ,
G ( T ) = | d t E ( t ) E * ( t T ) | / d t | E ( t ) | 2 .
τ C = d t G ( t ) .
d t E ( t ) E * ( t T ) exp [ i φ ( T ) ] ,
E S ( t ) = 0 d τ E ( t τ ) L ( τ ) ,
g ( T ) = | d t E S ( t ) E R * ( t ) | / [ d t | E S ( t ) | 2 d t | E R ( t ) | 2 ] 1 / 2 ,
g ( T ) = | 0 d τ G ( T τ ) L ( τ ) | .
A = d t | t 1 R ( t ) + r 1 S ( t ) | 2 .
B = d t | [ r 2 R ( t ) t 2 S ( t ) ] sin θ + i [ r 2 R ( t ) + t 2 S ( t ) ] cos θ | 2 .
exp ( i φ 1 ) = t 1 r 1 * / | t 1 r 1 | ,
exp ( i φ 2 ) = t 2 r 2 * / | t 2 r 2 | .
exp ( i ϕ ) = d t R ( t ) S * ( t ) / | d t R ( t ) S * ( t ) | .
A = A 0 + A 1 + 2 ( A 0 A 1 ) 1 / 2 g ( T ) cos ( φ 1 + ϕ ) , B = B 0 + B 1 + 2 ( B 0 B 1 ) 1 / 2 g ( T ) sin ( φ 1 + ϕ ) ,
A 0 = d t | t 1 | 2 | R ( t ) | 2 , A 1 = d t | r 1 | 2 | S ( t ) | 2 B 0 = d t | r 2 | 2 | R ( t ) | 2 , B 1 = d t | t 2 | 2 | S ( t ) | 2 .
φ 2 2 θ = φ 1 + π / 2 .
g ( T ) = [ ( A A 0 A 1 ) 2 / 4 A 0 A 1 + ( B B 0 B 1 ) 2 / 4 B 0 B 1 ] 1 / 2 ,
g ( T ) = [ ( a a 0 a 1 ) 2 / 4 a 0 a 1 + ( b b 0 b 1 ) 2 / 4 b 0 b 1 ] 1 / 2 .
[ ( a a 0 a 1 ) 2 / 4 a 0 a 1 + ( b b 0 b 1 ) 2 / 4 b 0 b 1 ] 1 / 2 ,
X = ( a a ) / a 0 , Y = ( b b ) / b 0 ,
I ( T ) = [ ( a a 0 a 1 ) 2 / 4 a 0 a 1 + ( b b 0 b 1 ) 2 / 4 b 0 b 1 ] 1 / 2 ,
I ( T ) = { g ( T ) 2 + g ( T ) [ X a 0 / a 1 cos ( Ψ ) + Y b 0 / b 1 sin ( Ψ ) ] + ( X 2 a 0 / a 1 + Y 2 b 0 / b 1 ) / 4 } 1 / 2 ,
σ S 2 = S 2 S 2 .
X 2 = σ X 2 , Y 2 = σ Y 2 .
S = g 2 + ( 1 / 4 ) ( σ X 2 a 0 / a 1 + σ Y 2 b 0 / b 1 ) ,
S 2 = g 4 + g 2 { σ X 2 [ cos 2 ( Ψ ) + 1 / 2 ] a 0 / a 1 + σ Y 2 [ sin 2 ( Ψ ) + 1 / 2 ] b 0 / b 1 } + ( 3 / 16 ) [ σ X 4 ( a 0 / a 1 ) 2 + σ Y 4 ( b 0 / b 1 ) 2 ] + ( 1 / 8 ) σ X 2 σ Y 2 a 0 b 0 / ( a 1 b 1 ) .
σ S 2 = k 1 S k 0 ,
k 1 = σ X 2 cos 2 ( Ψ ) a 0 / a 1 + σ Y 2 sin 2 ( Ψ ) b 0 / b 1 ,
k 0 = ( 1 / 4 ) [ σ X 2 cos 2 ( Ψ ) a 0 / a 1 + σ Y 2 sin 2 ( Ψ ) b 0 / b 1 ] × ( σ X 2 a 0 / a 1 + σ Y 2 b 0 / b 1 ) ( 1 / 8 ) [ σ X 4 ( a 0 / a 1 ) 2 + σ Y 4 ( b 0 / b 1 ) 2 ] .
k 1 = ( 1 / 2 ) ( σ X 2 a 0 / a 1 + σ Y 2 b 0 / b 1 ) ,
k 0 = ( 1 / 4 ) ( σ X 2 σ Y 2 a 0 b 0 / a 1 b 1 ) .
S 0 = ( 1 / 4 ) ( σ X 2 a 0 / a 1 + σ Y 2 b 0 / b 1 ) .
k 1 / S 0 = 2 .

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