Abstract

The physical optics of the laser-schlieren technique for the measurement of rate processes in shock waves is examined in detail. The method utilizes the Kirchhoff-Huygens integral with the usual thin lens, paraxial, and Fresnel approximations, all of which are appropriate for the typical laser schlieren experiment. The resolution and sensitivity of the technique are defined for all detector separations, and a reliable method for locating the time origin in the schlieren signal is provided. Diffraction is found to have a significant effect on the shock front generated signal, and geometrical optics treatments of this signal are shown to be inadequate.

© 1981 Optical Society of America

Full Article  |  PDF Article

Corrections

J. H. Kiefer, "Physical optics of the laser-schlieren shock tube technique: errata," Appl. Opt. 20, 2042-2042 (1981)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-20-12-2042

References

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  1. J. H. Kiefer, R. W. Lutz, Phys. Fluids 8, 1393 (1965).
    [CrossRef]
  2. J. H. Kiefer, R. W. Lutz, J. Chem. Phys. 44, 658 (1966).
    [CrossRef]
  3. R. G. Macdonald, G. Burns, R. K. Boyd, J. Chem. Phys. 66, 3598 (1977).
    [CrossRef]
  4. J. E. Dove, H. Teitelbaum, Chem. Phys. 6, 431 (1974).
    [CrossRef]
  5. J. H. Kiefer, J. C. Hajduk, “Shock Tubes and Waves,” in Proceedings, Twelfth International Symposium on Shock Tubes and Waves, Jerusalem (Magnes Press, Jerusalem, 1980), p. 97.
  6. T. Tanzawa, Y. Hidaka, W. C. Gardiner, (in Ref. 5), p. 555.
  7. J. H. Kiefer, “The Laser-Schlieren Technique in Shock Tube Kinetics,” in Shock Waves in Chemistry, A. Lifshitz, Ed. (Dekker, New York, to be published).
  8. P. C. T. deBoer, Phys. Fluids 6, 962 (1963).
    [CrossRef]
  9. J. E. Dove, H. Teitelbaum, “Shock Tube and Shock Wave Research,” in Proceedings, Eleventh International Symposium on Shock Tubes and Waves, Seattle (U. Washington Press, Seattle, 1978), p. 474.
  10. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1965).
  11. D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972).
  12. M. V. Klein, Optics (Wiley, New York, 1970).
  13. C. J. S. M. Simpson, T. R. D. Chandler, A. C. Strawson, J. Chem. Phys. 52, 2214 (1969).
    [CrossRef]
  14. M. J. Lighthill, Fourier Analysis and Generalized Functions (Cambridge, U.P., London, 1958).
    [CrossRef]
  15. A. L. Besse, Rev. Sci. Instrum. 42, 721 (1971).
    [CrossRef]
  16. J. Gruat, E. Rieutord, C.R. Acad. Sci. Paris 259, 1619 (1964).
  17. D. Simpson, J. Smy, J. Phys. D. 4, 1487 (1971).
    [CrossRef]
  18. B. Schmidt, Arch. Mech. 28, 809 (1976).
  19. C. deBoer, “Cadre: An Algorithm for Numerical Quadrature,” in Mathematical Software, J. R. Rice, Ed. (Academic, New York, 1971).
  20. T. Tanzawa, W. C. Gardiner, Seventeenth International Symposium on Combustion (Combustion Institute, Pittsburgh, 1979), p. 563.
  21. W. C. Gardiner, private communication.
  22. A. T. Mazzella, P. C. T. deBoer, at Fifth International Shock Tube Symposium, White Oak, (1965), p. 439.
  23. P. C. T. DeBoer, J. A. Miller, J. Chem. Phys. 64, 4233 (1976).
    [CrossRef]
  24. J. J. Bertin, E. S. Ehrhardt, W. C. Gardiner, T. Tanzawa, “Modern Developments in Shock Tube Research,” in Proceedings, Tenth International Shock Tube Symposium, Kyoto, Japan (1975), p. 595.

1977 (1)

R. G. Macdonald, G. Burns, R. K. Boyd, J. Chem. Phys. 66, 3598 (1977).
[CrossRef]

1976 (2)

B. Schmidt, Arch. Mech. 28, 809 (1976).

P. C. T. DeBoer, J. A. Miller, J. Chem. Phys. 64, 4233 (1976).
[CrossRef]

1974 (1)

J. E. Dove, H. Teitelbaum, Chem. Phys. 6, 431 (1974).
[CrossRef]

1971 (2)

A. L. Besse, Rev. Sci. Instrum. 42, 721 (1971).
[CrossRef]

D. Simpson, J. Smy, J. Phys. D. 4, 1487 (1971).
[CrossRef]

1969 (1)

C. J. S. M. Simpson, T. R. D. Chandler, A. C. Strawson, J. Chem. Phys. 52, 2214 (1969).
[CrossRef]

1966 (1)

J. H. Kiefer, R. W. Lutz, J. Chem. Phys. 44, 658 (1966).
[CrossRef]

1965 (1)

J. H. Kiefer, R. W. Lutz, Phys. Fluids 8, 1393 (1965).
[CrossRef]

1964 (1)

J. Gruat, E. Rieutord, C.R. Acad. Sci. Paris 259, 1619 (1964).

1963 (1)

P. C. T. deBoer, Phys. Fluids 6, 962 (1963).
[CrossRef]

Bertin, J. J.

J. J. Bertin, E. S. Ehrhardt, W. C. Gardiner, T. Tanzawa, “Modern Developments in Shock Tube Research,” in Proceedings, Tenth International Shock Tube Symposium, Kyoto, Japan (1975), p. 595.

Besse, A. L.

A. L. Besse, Rev. Sci. Instrum. 42, 721 (1971).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1965).

Boyd, R. K.

R. G. Macdonald, G. Burns, R. K. Boyd, J. Chem. Phys. 66, 3598 (1977).
[CrossRef]

Burns, G.

R. G. Macdonald, G. Burns, R. K. Boyd, J. Chem. Phys. 66, 3598 (1977).
[CrossRef]

Chandler, T. R. D.

C. J. S. M. Simpson, T. R. D. Chandler, A. C. Strawson, J. Chem. Phys. 52, 2214 (1969).
[CrossRef]

deBoer, C.

C. deBoer, “Cadre: An Algorithm for Numerical Quadrature,” in Mathematical Software, J. R. Rice, Ed. (Academic, New York, 1971).

DeBoer, P. C. T.

P. C. T. DeBoer, J. A. Miller, J. Chem. Phys. 64, 4233 (1976).
[CrossRef]

P. C. T. deBoer, Phys. Fluids 6, 962 (1963).
[CrossRef]

A. T. Mazzella, P. C. T. deBoer, at Fifth International Shock Tube Symposium, White Oak, (1965), p. 439.

Dove, J. E.

J. E. Dove, H. Teitelbaum, Chem. Phys. 6, 431 (1974).
[CrossRef]

J. E. Dove, H. Teitelbaum, “Shock Tube and Shock Wave Research,” in Proceedings, Eleventh International Symposium on Shock Tubes and Waves, Seattle (U. Washington Press, Seattle, 1978), p. 474.

Ehrhardt, E. S.

J. J. Bertin, E. S. Ehrhardt, W. C. Gardiner, T. Tanzawa, “Modern Developments in Shock Tube Research,” in Proceedings, Tenth International Shock Tube Symposium, Kyoto, Japan (1975), p. 595.

Gardiner, W. C.

J. J. Bertin, E. S. Ehrhardt, W. C. Gardiner, T. Tanzawa, “Modern Developments in Shock Tube Research,” in Proceedings, Tenth International Shock Tube Symposium, Kyoto, Japan (1975), p. 595.

W. C. Gardiner, private communication.

T. Tanzawa, W. C. Gardiner, Seventeenth International Symposium on Combustion (Combustion Institute, Pittsburgh, 1979), p. 563.

T. Tanzawa, Y. Hidaka, W. C. Gardiner, (in Ref. 5), p. 555.

Gruat, J.

J. Gruat, E. Rieutord, C.R. Acad. Sci. Paris 259, 1619 (1964).

Hajduk, J. C.

J. H. Kiefer, J. C. Hajduk, “Shock Tubes and Waves,” in Proceedings, Twelfth International Symposium on Shock Tubes and Waves, Jerusalem (Magnes Press, Jerusalem, 1980), p. 97.

Hidaka, Y.

T. Tanzawa, Y. Hidaka, W. C. Gardiner, (in Ref. 5), p. 555.

Kiefer, J. H.

J. H. Kiefer, R. W. Lutz, J. Chem. Phys. 44, 658 (1966).
[CrossRef]

J. H. Kiefer, R. W. Lutz, Phys. Fluids 8, 1393 (1965).
[CrossRef]

J. H. Kiefer, J. C. Hajduk, “Shock Tubes and Waves,” in Proceedings, Twelfth International Symposium on Shock Tubes and Waves, Jerusalem (Magnes Press, Jerusalem, 1980), p. 97.

J. H. Kiefer, “The Laser-Schlieren Technique in Shock Tube Kinetics,” in Shock Waves in Chemistry, A. Lifshitz, Ed. (Dekker, New York, to be published).

Klein, M. V.

M. V. Klein, Optics (Wiley, New York, 1970).

Lighthill, M. J.

M. J. Lighthill, Fourier Analysis and Generalized Functions (Cambridge, U.P., London, 1958).
[CrossRef]

Lutz, R. W.

J. H. Kiefer, R. W. Lutz, J. Chem. Phys. 44, 658 (1966).
[CrossRef]

J. H. Kiefer, R. W. Lutz, Phys. Fluids 8, 1393 (1965).
[CrossRef]

Macdonald, R. G.

R. G. Macdonald, G. Burns, R. K. Boyd, J. Chem. Phys. 66, 3598 (1977).
[CrossRef]

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972).

Mazzella, A. T.

A. T. Mazzella, P. C. T. deBoer, at Fifth International Shock Tube Symposium, White Oak, (1965), p. 439.

Miller, J. A.

P. C. T. DeBoer, J. A. Miller, J. Chem. Phys. 64, 4233 (1976).
[CrossRef]

Rieutord, E.

J. Gruat, E. Rieutord, C.R. Acad. Sci. Paris 259, 1619 (1964).

Schmidt, B.

B. Schmidt, Arch. Mech. 28, 809 (1976).

Simpson, C. J. S. M.

C. J. S. M. Simpson, T. R. D. Chandler, A. C. Strawson, J. Chem. Phys. 52, 2214 (1969).
[CrossRef]

Simpson, D.

D. Simpson, J. Smy, J. Phys. D. 4, 1487 (1971).
[CrossRef]

Smy, J.

D. Simpson, J. Smy, J. Phys. D. 4, 1487 (1971).
[CrossRef]

Strawson, A. C.

C. J. S. M. Simpson, T. R. D. Chandler, A. C. Strawson, J. Chem. Phys. 52, 2214 (1969).
[CrossRef]

Tanzawa, T.

J. J. Bertin, E. S. Ehrhardt, W. C. Gardiner, T. Tanzawa, “Modern Developments in Shock Tube Research,” in Proceedings, Tenth International Shock Tube Symposium, Kyoto, Japan (1975), p. 595.

T. Tanzawa, W. C. Gardiner, Seventeenth International Symposium on Combustion (Combustion Institute, Pittsburgh, 1979), p. 563.

T. Tanzawa, Y. Hidaka, W. C. Gardiner, (in Ref. 5), p. 555.

Teitelbaum, H.

J. E. Dove, H. Teitelbaum, Chem. Phys. 6, 431 (1974).
[CrossRef]

J. E. Dove, H. Teitelbaum, “Shock Tube and Shock Wave Research,” in Proceedings, Eleventh International Symposium on Shock Tubes and Waves, Seattle (U. Washington Press, Seattle, 1978), p. 474.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1965).

Arch. Mech. (1)

B. Schmidt, Arch. Mech. 28, 809 (1976).

C.R. Acad. Sci. Paris (1)

J. Gruat, E. Rieutord, C.R. Acad. Sci. Paris 259, 1619 (1964).

Chem. Phys. (1)

J. E. Dove, H. Teitelbaum, Chem. Phys. 6, 431 (1974).
[CrossRef]

J. Chem. Phys. (4)

J. H. Kiefer, R. W. Lutz, J. Chem. Phys. 44, 658 (1966).
[CrossRef]

R. G. Macdonald, G. Burns, R. K. Boyd, J. Chem. Phys. 66, 3598 (1977).
[CrossRef]

C. J. S. M. Simpson, T. R. D. Chandler, A. C. Strawson, J. Chem. Phys. 52, 2214 (1969).
[CrossRef]

P. C. T. DeBoer, J. A. Miller, J. Chem. Phys. 64, 4233 (1976).
[CrossRef]

J. Phys. D. (1)

D. Simpson, J. Smy, J. Phys. D. 4, 1487 (1971).
[CrossRef]

Phys. Fluids (2)

J. H. Kiefer, R. W. Lutz, Phys. Fluids 8, 1393 (1965).
[CrossRef]

P. C. T. deBoer, Phys. Fluids 6, 962 (1963).
[CrossRef]

Rev. Sci. Instrum. (1)

A. L. Besse, Rev. Sci. Instrum. 42, 721 (1971).
[CrossRef]

Other (13)

M. J. Lighthill, Fourier Analysis and Generalized Functions (Cambridge, U.P., London, 1958).
[CrossRef]

J. E. Dove, H. Teitelbaum, “Shock Tube and Shock Wave Research,” in Proceedings, Eleventh International Symposium on Shock Tubes and Waves, Seattle (U. Washington Press, Seattle, 1978), p. 474.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1965).

D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972).

M. V. Klein, Optics (Wiley, New York, 1970).

J. H. Kiefer, J. C. Hajduk, “Shock Tubes and Waves,” in Proceedings, Twelfth International Symposium on Shock Tubes and Waves, Jerusalem (Magnes Press, Jerusalem, 1980), p. 97.

T. Tanzawa, Y. Hidaka, W. C. Gardiner, (in Ref. 5), p. 555.

J. H. Kiefer, “The Laser-Schlieren Technique in Shock Tube Kinetics,” in Shock Waves in Chemistry, A. Lifshitz, Ed. (Dekker, New York, to be published).

J. J. Bertin, E. S. Ehrhardt, W. C. Gardiner, T. Tanzawa, “Modern Developments in Shock Tube Research,” in Proceedings, Tenth International Shock Tube Symposium, Kyoto, Japan (1975), p. 595.

C. deBoer, “Cadre: An Algorithm for Numerical Quadrature,” in Mathematical Software, J. R. Rice, Ed. (Academic, New York, 1971).

T. Tanzawa, W. C. Gardiner, Seventeenth International Symposium on Combustion (Combustion Institute, Pittsburgh, 1979), p. 563.

W. C. Gardiner, private communication.

A. T. Mazzella, P. C. T. deBoer, at Fifth International Shock Tube Symposium, White Oak, (1965), p. 439.

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Figures (11)

Fig. 1
Fig. 1

Schematic of typical LS apparatus. Rotating mirror is used to calibrate angular sensitivity.

Fig. 2
Fig. 2

Digitally recorded schlieren signal in krypton. Conditions were: P1 = 18.53 Torr, u = 0.8054 mm/μsec, ψ = 0.16, ϕ = 13.352 (sinϕ = 0.707). Modulation at maximum is 0.58, and the time scale is in 0.5 μsec units, which is about one unit in τ.

Fig. 3
Fig. 3

Schlieren signal for krypton. Conditions were: P1 = 13.56 Torr, u = 0.8696 mm/μsec,, ψ = 0.16, ϕ = 10.153 (sinϕ = −0.666). Modulation at maximum is 0.45, and the time scale is in units of 0.5 μsec, which is about one unit in τ.

Fig. 4
Fig. 4

Coordinate system (see text).

Fig. 5
Fig. 5

Signals generated by a plane shock front. Vertical scale is ΔS/S0P0 sinϕ. Here ψ = 0.06. Solid line is ρ = 1.5, and the dashed line ρ = ∞.

Fig. 6
Fig. 6

Curved shock profile of deBoer.

Fig. 7
Fig. 7

Calculated LS signals for the curved shock front. Here zc/a0 = 0.5. Right-hand column shows the sum of signals from the two detector halves.

Fig. 8
Fig. 8

Modulations at maximum and at minimum for curved shock signals. For these z c / a 0 = 2.1 P 1 - 1 / 2 and ψ = 0.6. P1 and zc/a0 values given on the graph refer to Ar with a density ratio of 3.0.

Fig. 9
Fig. 9

Curved shock signals at τ = 0, showing the behavior for various ψ. Here zc/a0 is as in Fig. 8.

Fig. 10
Fig. 10

Zero crossing times for curved shock signals. Here z c / a 0 = 2.1 P 1 - 1 / 2, ψ = 0.6. P1 and zc/a0 values given on the graph refer to Ar with a density ratio of 3.0.

Fig. 11
Fig. 11

Plot of zero crossing times against the modulation at minimum. Points shown are those used in the construction of Fig. 10. This figure may be used to locate the time origin by measuring the (absolute) experimental modulation at negative maximum (−ΔS/S0P0 min) and reading τc. Time origin is then located at a time t = −a0τc/u measured from zero crossing.

Equations (47)

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T ( x , y ) = exp - 2 π i λ - w / 2 w / 2 n ( x , y , z ) d z ,
P ( x , y ) = P 0 ( 2 π a 0 2 ) exp - [ 2 ( x 2 + y 2 ) a 0 2 ]
E ( x ¯ , y ) = i 2 λ A P 1 / 2 ( x , y ) ( cos α + cos γ ) T ( x , y ) × exp ( - 2 π i R / λ ) R d x d y .
E ( x ¯ , y ¯ ) = i exp - 2 γ i D / λ λ D - - P 1 / 2 ( x , y ) T ( x , y ) × exp - { π i λ D [ ( x ¯ - x ) 2 + ( y ¯ - y ) 2 ] d x d y } .
T ( x , y ) = T ( ω ) = exp - 2 π i λ - w / 2 w / 2 n ( z , ω ) d z ,
P ( x ¯ , y ¯ ) d x ¯ d y ¯ = 2 P 0 π a λ D a 0 exp - ( 2 y ¯ 2 a 2 ) | - T ( ω ) × exp - [ x 2 a 0 2 + π i λ D ( x 2 - 2 x x ¯ ] d x | 2 d x ¯ d y ¯ .
a = a 0 [ 1 + ( λ D π a 0 2 ) 2 ] 1 / 2
P ( x ¯ , y ¯ ) = P 0 ( 2 π a 2 ) exp - [ 2 ( x ¯ 2 + y ¯ 2 ) a 2 ] ,
α = x / a 0 , γ = π a 0 x ¯ / λ D , ψ = π a 0 2 / λ D .
P ( γ , y ¯ ) d ¯ γ d y ¯ = P 0 ( 2 π 2 a ) exp - ( 2 y ¯ 2 a 2 ) | - T ( ω ) × exp - [ α 2 + i ( ψ α 2 - 2 α γ ) ] d α | 2 d γ d y ¯ .
P ( γ ) d γ = P 0 ( 2 π 3 ) 1 / 2 | - T ( ω ) × exp - [ α 2 + i ( ψ α 2 - 2 α γ ) ] d α | 2 d γ .
Δ S = S 0 - r r P ( x ¯ ) sgn ( x ¯ ) d x ¯ ,
Δ S S 0 P 0 = ( 2 π 3 ) 1 / 2 - ρ ρ sgn ( γ ) | - T ( ω ) × exp [ α 2 + i ( ψ α 2 - 2 α γ ) ] d α | 2 d γ ,
Δ S = S 0 - r r sgn ( x ¯ ) - r 2 - x ¯ 2 r 2 - x ¯ 2 P ( x ¯ , y ¯ ) d x ¯ d y ¯ .
Δ S S 0 P 0 = ( 2 π 3 ) 1 / 2 - ρ ρ sgn ( γ ) Φ { [ 2 ( ρ 2 - γ 2 ) 1 + ψ 2 ] 1 / 2 } | - T ( ω ) × exp - [ α 2 + i ψ α 2 - 2 α γ ) ] d α | 2 d γ ,
Δ x = - w / 2 w / 2 d z - w / 2 z ( d n / d x ) d η = w ( d n / d x ) / 2.
θ = - w / 2 w / 2 ( d n / d x ) d η = w ( d n / d x ) ,
T ( ω ) = exp - 2 π i w n ( ω ) / λ .
- sgn ( γ ) exp [ 2 i α γ - 2 i β γ ] d γ = i / ( α - β )
Δ S S 0 P 0 = ( 2 π 3 ) 1 / 2 - d α - d β T ( α ) T * ( β ) ( i α - β ) × exp - [ i ψ ( α 2 - β 2 ) + α 2 + β 2 ] ,
T ( α ) T * ( β ) = exp - ( 2 π i w λ { n [ a 0 ( α + τ ) ] - n [ a 0 ( β + τ ) ] } ) .
Δ S S 0 P 0 = i 2 ( 2 π 3 ) 1 / 2 - d s - d t T T * s exp - [ i ψ s t + ( s 2 + t 2 ) / 2 ] .
n [ a 0 ( δ + τ ) ] n ( a 0 τ ) + a 0 n ( a 0 τ ) δ + a 0 2 n ( a 0 τ ) δ 2 / 2 + a 0 3 n ( a 0 τ ) δ 3 / 6 + .
w { n [ a 0 ( s + t + 2 τ ) / 2 ] - n [ a 0 ( t - s + 2 τ ) / 2 ] } = a 0 θ s + a 0 2 θ s t / 2 + a 0 3 θ s 3 / 24 + a 0 3 θ s t 2 / 8 + 0 ( a 0 4 θ ) .
T T * 1 - 2 π i λ ( a 0 θ s + a 0 2 θ s t / 2 + ) .
Δ S S 0 P 0 a 0 λ ( 2 π ) 1 / 2 - d s - d t [ θ + a 0 2 θ ( s 2 / 24 + t 2 / 8 ) ] × exp - [ i ψ s t + ( s 2 + t 2 ) / 2 ] .
Δ S S 0 P 0 = 2 2 π a 0 λ ( 1 + ψ 2 ) 1 / 2 [ θ + θ a 0 2 6 ( 1 + ψ 2 ) + 0 ( θ IV a 0 4 ) ] .
S ¯ = Δ S / S 0 P 0 θ = 2 2 π a 0 / λ ( 1 + ψ 2 ) 1 / 2 ,
Δ x > a 0 / 6 ( 1 + ψ 2 ) .
S ¯ / Δ x 2 12 π / λ
Δ S S 0 P 0 = 2 2 π a 0 λ [ θ + θ a 0 2 8 + ] .
P ( γ ) = P 0 ( 2 π 3 ) 1 / 2 - d α - T ( α ) T * ( β ) × exp - [ α 2 + β 2 + i ψ ( α 2 - β 2 ) - 2 i γ ( α - β ) ] d β .
w { n [ a 0 ( s + t + 2 τ ) / 2 ] - n [ a 0 ( t - s + 2 τ ) / 2 ] } = a 0 θ s + a 0 3 θ s 3 / 24 + ,
P ( γ ) d γ = 1 2 P 0 ( 2 π 3 ) 1 / 2 - d t - × exp - [ ( s 2 + t 2 ) 2 + ψ s t - 2 i γ s + 2 π i a 0 θ s γ ] d s .
P ( x ¯ ) d x ¯ = 2 P 0 λ D - exp - ( 2 x 2 / a 0 2 ) × exp - 2 [ ( π a 0 λ D ) 2 ( x ¯ - x - θ D ) 2 ] d x d x ¯ .
Δ S = S 0 - sgn ( x ¯ ) P ( x ¯ ) d x ¯ .
Δ S S 0 P 0 = ( 2 π a 0 2 ) - Φ [ 2 π a 0 ( x + θ D ) λ D ] exp - ( 2 x 2 a 0 2 ) d x .
Δ S S 0 P 0 = 4 λ - θ exp - ( 2 x 2 a 0 2 ) d x ,
Δ S S 0 P 0 = i ( 2 π ) 3 / 2 - exp ( - α 2 ) d α - exp ( - β 2 ) T ( α ) T * ( β ) × exp - [ i ψ ( α 2 - β 2 ) ] sin 2 [ ( α - β ) ρ ] α - β d β .
T ( α ) T * ( β ) = exp - i ϕ [ H ( α + τ ) - H ( β + τ ) ] ,
T ( α ) T * ( β ) exp - [ i ψ ( α 2 - β 2 ) ] = cos { ψ ( α 2 - β 2 ) + ϕ [ H ( α + τ ) - H ( β + τ ) ] } - i sin { ψ ( α 2 - β 2 ) + ϕ [ H ( α + τ ) - H ( β + τ ) ] } .
Δ S S 0 P 0 = ( 2 π ) 3 / 2 - exp ( - α 2 ) d α - exp ( - β 2 ) sin { ψ ( α 2 - β 2 ) + ϕ [ H ( α + τ ) - H ( β + τ ) ] } sin 2 [ ( α - β ) ρ ] d β α - β .
Δ S S 0 P 0 = 2 ( 2 π ) 3 / 2 - τ exp ( - α 2 ) d α - - τ × exp ( - β 2 ) sin [ ψ ( α 2 - β 2 ) + ϕ ] sin 2 [ ( α - β ) ρ ] d β α - β .
sin [ ψ ( α 2 - β 2 ) + ϕ ] = sin ϕ cos [ ψ ( α 2 - β 2 ) ] + cos ϕ sin [ ψ ( α 2 - β 2 ) ] ,
Δ S S 0 P 0 = 2 ( 2 π ) 3 / 2 sin ϕ - τ exp ( - α 2 ) d α - - τ × exp ( - β 2 ) cos [ ψ ( α 2 - β 2 ) ] sin 2 [ ( α - β ) ρ ] d β α - β .
2 z w = ( ω z c ) 1 / 2 [ A - B ω z c - C ( ω z c ) 2 ] ( 0 ω z c ) ,
T ( ω ) = exp - i ϕ { [ H ( ω ) - H ( ω - z c ) ] ( 2 z / w ) + H ( ω - z c ) } ,

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