Abstract

A conventional approach to short-exposure imaging through the turbulent atmosphere is generalized to imaging of nonisoplanatic objects. A new approximate formula for a short-exposure modulation transfer function is obtained, based on the path-integral representation of the field in a random medium. This formula is free of some of the drawbacks of previous approximations and describes the influence of diffraction effects and medium fluctuation distribution along the propagation path. Results of computation of short-exposure structure functions and short-exposure modulation transfer functions are presented.

© 1993 Optical Society of America

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  1. D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966).
    [CrossRef]
  2. J. R. Dunphy, J. R. Kerr, “Turbulence effects on target illumination by laser sources, phenomenological analysis and experimental results,” Appl. Opt. 16, 1345–1358 (1977).
    [CrossRef] [PubMed]
  3. A. I. Kon, “Light focusing in a turbulent medium,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 15, 533–538 (1970).
  4. H. T. Yura, “Short-term average optical beam spread in a turbulent medium,” J. Opt. Soc. Am. 63, 567–572 (1973).
    [CrossRef]
  5. M. Tavis, H. Yura, “Short-term average irradiance profile of an optical beam in a turbulent medium,” Appl. Opt. 15, 2922–2931 (1976).
    [CrossRef] [PubMed]
  6. S. E. Lutomirskii, W. C. Woodie, R. G. Buser, “urbulence degraded beam quality: improvement obtained with a tilt-corrected aperture,” Appl. Opt. 16, 665–673 (1977).
    [CrossRef]
  7. G. C. Valley, “Long- and short-term Strehl ratios for turbulence with finite inner and outer scales,” Appl. Opt. 18, 984–987 (1979).
    [CrossRef] [PubMed]
  8. Yu. A. Kravtsov, Z. I. Feizullin, “Some consequences of the Huygens–Kirchhoff principle for inhomogeneous medium,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 12, 886–893 (1969).
  9. I. Last, M. Tur, “he plane-wave short-term structure with finite turbulence scales: an empirical approach,” Waves Random Media 1, 35–42 (1991).
    [CrossRef]
  10. R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 68, 1424–1444 (1980).
    [CrossRef]
  11. V. P. Lukin, Atmospheric Adaptive Optics (Nauka, Novosibirsk, 1986) (in Russian).
  12. G. C. Valley, “Isoplanatic degradation of tilt correction and short-term imaging systems,” Appl. Opt. 19, 574–577 (1980).
    [CrossRef] [PubMed]
  13. D. L. Fried, “Anisoplanatism in adaptive optics,” J. Opt. Soc. Am. 72, 52–61 (1982).
    [CrossRef]
  14. F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. 19, pp. 281–376.
    [CrossRef]
  15. S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics 4. Wave Propagation through Random Media (Springer-Verlag, Berlin1989).
  16. V. I. Klyatskin, Stochastic Equations and Waves in Random Media (Nauka, Moscow, 1980) (in Russian).
  17. V. I. Klyatskin, V. I. Tatarskii, “heory of light beam propagation in random media,” Sov. Phys. JETP 31, 335–339 (1970).
  18. V. U. Zavorotny, V. I. Klyatskin, V. I. Tatarskii, “Strong intensity fluctuations of electro-magnetic waves in random media,” Sov. Phys. JETP 46, 252–260 (1977).
  19. R. Dashen, “Path integrals for waves in random media,” J. Math. Phys. 20, 894–920 (1979).
    [CrossRef]
  20. M. I. Charnotskii, “Strong intensity fluctuations of finite light beams in turbulent atmosphere,” in Proceedings of the Fifth Symposium on Laser Propagation in the Atmosphere (Institute of Atmospheric Optics, Tomsk, 1979), pp. 74–78 (in Russian).
  21. I. G. Yakushkin, “Intensity fluctuation during small-angle scattering of wave fields,” Radiophys. Quantum Electron. 28, 365–389, (1985).
    [CrossRef]

1991

I. Last, M. Tur, “he plane-wave short-term structure with finite turbulence scales: an empirical approach,” Waves Random Media 1, 35–42 (1991).
[CrossRef]

1985

I. G. Yakushkin, “Intensity fluctuation during small-angle scattering of wave fields,” Radiophys. Quantum Electron. 28, 365–389, (1985).
[CrossRef]

1982

1980

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 68, 1424–1444 (1980).
[CrossRef]

G. C. Valley, “Isoplanatic degradation of tilt correction and short-term imaging systems,” Appl. Opt. 19, 574–577 (1980).
[CrossRef] [PubMed]

1979

1977

1976

1973

1970

A. I. Kon, “Light focusing in a turbulent medium,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 15, 533–538 (1970).

V. I. Klyatskin, V. I. Tatarskii, “heory of light beam propagation in random media,” Sov. Phys. JETP 31, 335–339 (1970).

1969

Yu. A. Kravtsov, Z. I. Feizullin, “Some consequences of the Huygens–Kirchhoff principle for inhomogeneous medium,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 12, 886–893 (1969).

1966

Buser, R. G.

Charnotskii, M. I.

M. I. Charnotskii, “Strong intensity fluctuations of finite light beams in turbulent atmosphere,” in Proceedings of the Fifth Symposium on Laser Propagation in the Atmosphere (Institute of Atmospheric Optics, Tomsk, 1979), pp. 74–78 (in Russian).

Dashen, R.

R. Dashen, “Path integrals for waves in random media,” J. Math. Phys. 20, 894–920 (1979).
[CrossRef]

Dunphy, J. R.

Fante, R. L.

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 68, 1424–1444 (1980).
[CrossRef]

Feizullin, Z. I.

Yu. A. Kravtsov, Z. I. Feizullin, “Some consequences of the Huygens–Kirchhoff principle for inhomogeneous medium,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 12, 886–893 (1969).

Fried, D. L.

Kerr, J. R.

Klyatskin, V. I.

V. U. Zavorotny, V. I. Klyatskin, V. I. Tatarskii, “Strong intensity fluctuations of electro-magnetic waves in random media,” Sov. Phys. JETP 46, 252–260 (1977).

V. I. Klyatskin, V. I. Tatarskii, “heory of light beam propagation in random media,” Sov. Phys. JETP 31, 335–339 (1970).

V. I. Klyatskin, Stochastic Equations and Waves in Random Media (Nauka, Moscow, 1980) (in Russian).

Kon, A. I.

A. I. Kon, “Light focusing in a turbulent medium,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 15, 533–538 (1970).

Kravtsov, Yu. A.

Yu. A. Kravtsov, Z. I. Feizullin, “Some consequences of the Huygens–Kirchhoff principle for inhomogeneous medium,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 12, 886–893 (1969).

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics 4. Wave Propagation through Random Media (Springer-Verlag, Berlin1989).

Last, I.

I. Last, M. Tur, “he plane-wave short-term structure with finite turbulence scales: an empirical approach,” Waves Random Media 1, 35–42 (1991).
[CrossRef]

Lukin, V. P.

V. P. Lukin, Atmospheric Adaptive Optics (Nauka, Novosibirsk, 1986) (in Russian).

Lutomirskii, S. E.

Roddier, F.

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. 19, pp. 281–376.
[CrossRef]

Rytov, S. M.

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics 4. Wave Propagation through Random Media (Springer-Verlag, Berlin1989).

Tatarskii, V. I.

V. U. Zavorotny, V. I. Klyatskin, V. I. Tatarskii, “Strong intensity fluctuations of electro-magnetic waves in random media,” Sov. Phys. JETP 46, 252–260 (1977).

V. I. Klyatskin, V. I. Tatarskii, “heory of light beam propagation in random media,” Sov. Phys. JETP 31, 335–339 (1970).

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics 4. Wave Propagation through Random Media (Springer-Verlag, Berlin1989).

Tavis, M.

Tur, M.

I. Last, M. Tur, “he plane-wave short-term structure with finite turbulence scales: an empirical approach,” Waves Random Media 1, 35–42 (1991).
[CrossRef]

Valley, G. C.

Woodie, W. C.

Yakushkin, I. G.

I. G. Yakushkin, “Intensity fluctuation during small-angle scattering of wave fields,” Radiophys. Quantum Electron. 28, 365–389, (1985).
[CrossRef]

Yura, H.

Yura, H. T.

Zavorotny, V. U.

V. U. Zavorotny, V. I. Klyatskin, V. I. Tatarskii, “Strong intensity fluctuations of electro-magnetic waves in random media,” Sov. Phys. JETP 46, 252–260 (1977).

Appl. Opt.

Izv. Vyssh. Uchebn. Zaved. Radiofiz.

A. I. Kon, “Light focusing in a turbulent medium,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 15, 533–538 (1970).

Yu. A. Kravtsov, Z. I. Feizullin, “Some consequences of the Huygens–Kirchhoff principle for inhomogeneous medium,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 12, 886–893 (1969).

J. Math. Phys.

R. Dashen, “Path integrals for waves in random media,” J. Math. Phys. 20, 894–920 (1979).
[CrossRef]

J. Opt. Soc. Am.

Proc. IEEE

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 68, 1424–1444 (1980).
[CrossRef]

Radiophys. Quantum Electron.

I. G. Yakushkin, “Intensity fluctuation during small-angle scattering of wave fields,” Radiophys. Quantum Electron. 28, 365–389, (1985).
[CrossRef]

Sov. Phys. JETP

V. I. Klyatskin, V. I. Tatarskii, “heory of light beam propagation in random media,” Sov. Phys. JETP 31, 335–339 (1970).

V. U. Zavorotny, V. I. Klyatskin, V. I. Tatarskii, “Strong intensity fluctuations of electro-magnetic waves in random media,” Sov. Phys. JETP 46, 252–260 (1977).

Waves Random Media

I. Last, M. Tur, “he plane-wave short-term structure with finite turbulence scales: an empirical approach,” Waves Random Media 1, 35–42 (1991).
[CrossRef]

Other

V. P. Lukin, Atmospheric Adaptive Optics (Nauka, Novosibirsk, 1986) (in Russian).

M. I. Charnotskii, “Strong intensity fluctuations of finite light beams in turbulent atmosphere,” in Proceedings of the Fifth Symposium on Laser Propagation in the Atmosphere (Institute of Atmospheric Optics, Tomsk, 1979), pp. 74–78 (in Russian).

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. 19, pp. 281–376.
[CrossRef]

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics 4. Wave Propagation through Random Media (Springer-Verlag, Berlin1989).

V. I. Klyatskin, Stochastic Equations and Waves in Random Media (Nauka, Moscow, 1980) (in Russian).

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Figures (18)

Fig. 1
Fig. 1

Normalized short-exposure structure function D SE B dependence on normalized separation R for different values of aperture Fresnel number N = ka2L, for a horizontal path. Dashed curve, long-exposure structure function.

Fig. 2
Fig. 2

Comparison of short-exposure structure functions DSE in two approximations for different values of aperture Fresnel number N = ka2/L, for a horizontal path. Dashed curves, Fried’s approximation (43); solid curves, our approximation (40); dotted curve, long-exposure structure function.

Fig. 3
Fig. 3

Normalized short-exposure structure function D SE B dependence on normalized separation R/2a for different values of aperture radius a, for a vertical path and zenith observations. Wavelength λ = 500 nm; dashed curve, long-exposure structure function.

Fig. 4
Fig. 4

Same as Fig. 3 but for nadir observations.

Fig. 5
Fig. 5

Short-exposure structure function D SE B dependence on separation R for different values of aperture Fresnel number N = ka2/L, calculated with the use of uniform normalization, for a horizontal path. The maximum separation value for each curve is the aperture diameter. Dashed curve, long-exposure structure function.

Fig. 6
Fig. 6

Short-exposure structure function D SE B dependence on separation R for different values of aperture radius a, calculated with the use of uniform normalization, for a vertical path and zenith observations. The maximum separation value for each curve is the pupil diameter. Wavelength λ = 500 nm. Dashed curve, long-exposure structure function.

Fig. 7
Fig. 7

Same as Fig. 6 but for nadir observations.

Fig. 8
Fig. 8

MTF dependence on normalized angular frequency f/(2ka) for different values of long-exposure mean-square phase difference in aperture diameter d = D(L, 2a) and for aperture Fresnel number N = 10, for a horizontal path. Dashed curves, long-exposure MTF’s; solid curves, short-exposure MTF’s.

Fig. 9
Fig. 9

Same as Fig. 8 but for aperture Fresnel number N = 1.

Fig. 10
Fig. 10

Same as Fig. 8 but for aperture Fresnel number N = 0.1.

Fig. 11
Fig. 11

MTF dependence on normalized angular frequency f/(2ka) for different values of long-exposure mean-square phase difference in aperture diameter d = D(L, 2a), for vertical-path and zenith observations. Wavelength λ = 500 nm. Aperture radius a = 0.2 m corresponds to a large effective Fresnel number. Dashed curves, long-exposure MTF’s; solid curves, short-exposure MTF’s.

Fig. 12
Fig. 12

Same as Fig. 11, but aperture radius a = 0.01 m corresponds to a small effective Fresnel number.

Fig. 13
Fig. 13

Same as Fig. 11 but for nadir observations. Aperture radius a = 1.0 m corresponds to a large effective Fresnel number.

Fig. 14
Fig. 14

Same as Fig. 13, but aperture radius a = 0.2 m corresponds to a small effective Fresnel number.

Fig. 15
Fig. 15

MTF dependence on normalized aperture radius for different values of spatial angular frequency f for a horizontal path. Coherence radius Fresnel number Q = 10. Dotted curves, free-space propagation; dashed curves, long-exposure MTF’s; solid curves, short-exposure MTF’s.

Fig. 16
Fig. 16

Same as Fig. 15 but for coherence radius Fresnel number Q = 1.

Fig. 17
Fig. 17

Same as Fig. 15 but for coherence radius Fresnel number Q = 0.1.

Fig. 18
Fig. 18

MTF dependence on aperture radius for different values of the angular size of object details (in arcseconds) for vertical-path and zenith observations. Coherence radius r0 = 0.1 m. Dotted curves, free space; dashed curves, long-exposure MTF’s; solid curves, short-exposure MTF’s.

Equations (45)

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I ( R 2 ) = d R 1 O ( R 1 ) P ( R 1 , R 2 + R 1 L 1 L ) ,
P ( R 1 , R 2 ) = k 4 ( 4 π 2 ) 2 L 2 L 1 2 d R d R A ( R + R 2 ) A ( R R 2 ) × g ( R 1 , R + R 2 ) g ( R 1 , R R 2 ) exp ( i k L 1 R 2 R ) ,
G 0 ( x 1 , R 1 ; x 2 , R 2 ) = k 2 π i | x 2 x 1 | exp [ i k ( R 2 R 1 ) 2 2 | x 2 x 1 | ]
I LE ( R 2 ) = I ( R 2 ) = d R 1 O ( R 1 ) P LE ( R 2 + R 1 L 1 L ) ,
P LE ( R 2 ) = k 4 ( 4 π 2 ) 2 L 2 L 1 2 d R C A ( R ) exp ( D ( L , R ) / 2 ) × exp ( i k L 1 R 2 R ) ,
C A ( R ) = d R A ( R + R 2 ) A ( R R 2 ) ,
D ( L , R ) = π k 2 0 L d x d p Φ ( x , p ) { 1 cos [ pr ( 1 x / L ) ] }
g ( R 1 , R + R 2 ) g ( R 1 , R R 2 ) = exp ( D ( L , R ) / 2 ) .
Î LE ( q ) = Ô ( q L 1 L ) P ̂ LE ( q ) ,
Î LE ( q ) = d R I LE ( R ) exp ( i qr ) ,
Ô ( q ) = d R 1 O ( R 1 ) exp ( i qr ) ,
P ̂ LE ( q ) = d R P LE ( R ) exp ( i qr ) = k 2 4 π 2 L 2 C A ( q l k ) × exp [ 1 2 D ( L , q l k ) ]
R C ( R 1 ) = d R 2 R 2 P ( R 1 , R 2 ) d R 2 P ( R 1 , R 2 ) ,
P SE ( R 1 , R 2 ) = P [ R 1 , R 2 + R C ( R 1 ) ]
I SE ( R 2 ) = d R 1 O ( R 1 ) P SE ( R 1 , R 2 + R 1 L 1 L ) .
I SE ( R 2 ) = k 4 ( 4 π 2 ) 2 L 2 L 1 2 d R 1 O ( R 1 ) d R d R × exp [ i k L 1 ( R 2 + R 1 L 1 L ) R ] × A ( R + R 2 ) A ( R R 2 ) × exp [ i k L 1 R R C ( R 1 ) ] g ( R 1 , R + R 2 ) g ( R 1 , R R 2 ) .
g ( R 1 , R ) = 2 π i L k D s ( ξ ) exp [ i k 2 0 L ( d s d ξ ) 2 d ξ ] × exp { i k 2 0 L [ ξ , R 1 ξ L + R ( 1 ξ L ) + s ( x ) ] d ξ } × δ [ s ( 0 ) ] δ [ s ( L ) ] ,
D s ( ξ ) exp [ i k 2 0 L ( d s d ξ ) 2 d ξ ] δ [ s ( 0 ) ] = 1 ,
R C ( R 1 ) = k L 1 4 π 0 L d x x d R Ã 2 ( R ) × d R ( x , R + R 1 x L + R L x L ) × sin ( k R 2 L 2 x ( L x ) ) .
à ( R ) = A ( R ) [ A 2 ( R ) d R ] 1 / 2 .
P SE ( R 1 , R 2 ) = k 2 4 π 2 L 1 2 D t ( ξ ) D s ( ξ ) × d R d R A ( R + R 2 ) A ( R R 2 ) × exp ( i k 0 L d s d ξ d t d ξ d ξ ) × exp { 1 2 D SE [ L , R , R , s ( ξ ) , t ( ξ ) ] } × exp ( i k L 1 R 2 R ) δ [ t ( 0 ) ] δ [ t ( L ) ] δ [ s ( 0 ) ] δ [ s ( L ) ] ,
D SE [ L , R , R , s ( ξ ) , t ( ξ ) ] = 2 π k 2 0 L d x d p Φ ( x , p ) ( M 2 [ x , p , R , t , ( x ) ] + N 2 ( x , p , R ) 2 M [ x , p , R , t , ( x ) ] N ( x , p , R ) × cos { Ψ [ x , p , R , s , ( x ) ] } ) ,
M ( x , p , R , t ) = sin { p [ R ( 1 x / L ) + t ] / 2 } ,
N ( x , p , R ) = ( 1 x / L ) 2 ( p R ) J a [ p ( 1 x / L ) ] × cos [ p 2 x ( 1 x / L ) 2 k ] ,
Ψ ( x , p , R , s ) = p [ R ( 1 x / L ) + s ] ,
J a ( p ) = ( 2 π ) 2 d R Ã 2 ( R ) exp ( i p R ) ,
Î SE ( q ) = Ô ( q l L ) P ̂ SE ( q ) ,
P ̂ SE ( q ) = d R d R A ( R + q l 2 k ) A ( R q l 2 k ) D t ( ξ ) D s ( ξ ) × exp ( i k 0 L d s d ξ d t d ξ d ξ ) × exp { D SE [ L , R , q L 1 k , s ( ξ ) , t ( ξ ) ] / 2 } × δ [ t ( 0 ) ] δ [ t ( L ) ] δ [ s ( 0 ) ] δ [ s ( L ) ] .
P ̂ SE A ( q ) = k 2 4 π 2 L 2 d R d R A ( R + q L 1 k ) A ( R q L 1 k ) × exp [ 1 2 D SE A ( L , R , q L 1 k ) ] ,
D SE A ( L , R , R ) = 2 π k 2 0 L d x d p Φ ( x , p ) { M 2 [ x , p , R , 0 ) + N 2 ( x , p , R ) 2 M ( x , p , R , 0 ) N ( x , p , R ) cos [ Ψ ( x , p , R , 0 ) ] } .
P ̂ SE B ( q ) = k 2 4 π 2 L 2 C A ( q L 1 k ) exp [ 1 2 D SE B ( L , q L 1 k ) ] ,
D SE B ( L , R ) = 2 π k 2 0 L d x d p Φ ( x , p ) × [ M ( x , p , R , 0 ) N ( x , p , R ) ] 2 .
D SE F ( L , R ) = 2 π k 2 0 L d x d p Φ ( x , p ) × [ M 2 ( x , p , R , 0 ) N 2 ( x , p , R ) ] ,
Φ ( x , p ) = 0.033 C 2 ( x ) p 11 / 3 , H ( x , R ) = 0.435 C ( x ) r 5 / 3 ,
D ( L , R ) = 0.73 r 5 / 3 k 2 0 L C 2 ( x ) ( 1 x / L ) 5 / 3 d x ,
D ( L , R ) = 6.88 ( r / r 0 ) 5 / 3 .
à ( R ) = { ( π a 2 ) 1 / 2 if | R | a 0 if | R | > a ,
C A ( R ) = 2 a 2 { cos 1 ( r / 2 a ) ( r / 2 a ) [ 1 ( r / 2 a ) 2 ] 1 / 2 } ,
J a ( p ) = 2 J 1 ( p a ) / ( p a ) ,
D SE B ( R ) = 0.751 D 0 ( L , 2 a ) 0 1 d t S ( t ) ( 1 t ) 5 / 3 × d p p 8 / 3 { 1 J 0 ( 2 ζ p ) 8 ζ J 1 ( ζ p ) J 1 ( p ) × cos [ p 2 t 2 N ( 1 t ) ] + 8 ζ 2 J 1 2 ( p ) cos 2 [ p 2 t 2 N ( 1 t ) ] } .
D SE ( L , R ) = 0.751 D 0 ( L , 2 a ) 0 1 d t S ( t ) ( 1 t ) 5 / 3 d p p 8 / 3 × [ 1 J 0 ( 2 ζ p ) 8 ζ J 1 ( ζ p ) J 1 ( p ) + 8 ζ 2 J 1 2 ( p ) ] .
D ( L , R ) = 0.751 D 0 ( L , 2 a ) 0 1 d t S ( t ) ( 1 t ) 5 / 3 × d p p 8 / 3 [ 1 J 0 ( 2 ζ p ) ] ,
D SE F ( L , R ) = 0.751 D 0 ( L , 2 a ) 0 1 d t S ( t ) ( 1 t ) 5 / 3 d p p 8 / 3 × { 1 J 0 ( 2 ζ p ) 8 ζ 2 J 1 2 ( p ) cos 2 [ p 2 t 2 N ( 1 t ) ] } ,
D SE F ( R ) = 0.751 D 0 ( L , 2 a ) 0 1 d t S ( t ) d p p 8 / 3 × [ 1 J 0 ( 2 ζ p ) 8 ζ 2 J 1 2 ( p ) ] .
C 2 ( h ) = C 2 ( h 0 ) ( h / h 0 ) 2 / 3 exp ( h / h 1 ) ,

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