Abstract

Thermal excitations on a germanium surface under simultaneous irradiation by two monochromatic optical beams, one strong and one weak, are predicted as functions of the angular separation and frequency difference between the beams, their relative polarization, their intensities, and pulse durations. Nonlinear optical reflection for Q-switched ruby laser pulses is then described. Weak reflected and diffracted beam intensities show tendencies in which the former is preferentially enhanced for a downshifted weak beam frequency, while the latter depends only on the shift magnitude. Both are suppressed for large shifts or large angular separations between input beams.

© 1978 Optical Society of America

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  1. C. H. Carmichael, G. N. Simpson, Nature 202, 787 (1964);W. R. Sooy, M. Geller, D. P. Bortfeld, Appl. Phys. Lett. 5, 54 (1964);M. Birnbaum, J. Appl. Phys. 36, 657 (1965);M. Birnbaum, T. L. Stocker, Br. J. Appl. Phys. 17, 461 (1966).
    [CrossRef]
  2. A. M. Bonch-Bruevich, V. P. Kovalev, G. S. Romanov, Ya. A. Imas, M. N. Libenson, Sov. Phys. Tech. Phys. 13, 507 (1968);I. Nagata, R. J. Galagali, S. Horiguchi, T. Sakai, T. Nakaya, J. Phys. D 3, 1305 (1970).
    [CrossRef]
  3. C. J. Kennedy, J. C. Matter, A. L. Smirl, H. Weichel, F. A. Hopf, S. V. Pappu, M. O. Scully, Phys. Rev. Lett. 32, 419 (1974);A. L. Smirl, J. C. Matter, A. Elci, M. O. Scully, Opt. Commun. 16, 118 (1976).
    [CrossRef]
  4. D. H. Auston, C. V. Shank, Phys. Rev. Lett. 32, 1120 (1974).
    [CrossRef]
  5. C. V. Shank, D. H. Auston, Phys. Rev. Lett. 34, 479 (1975);D. H. Auston, C. V. Shank, P. LeFur, Phys. Rev. Lett. 35, 1022 (1975).
    [CrossRef]
  6. T. A. Wiggins, A. Salik, Appl. Phys. Lett. 25, 438 (1974).
    [CrossRef]
  7. T. A. Wiggins, J. A. Bellay, A. H. Carrieri, Appl. Opt. 17, 526 (1978).
    [CrossRef] [PubMed]
  8. A. J. Alcock, P. B. Corkum, D. J. James, Appl. Phys. Lett. 27, 680 (1975).
    [CrossRef]
  9. S. A. Jamison, A. V. Nurmikko, H. J. Gerritsen, Appl. Phys. Lett. 39, 640 (1976).
    [CrossRef]
  10. R. M. Herman, M. A. Gray, Phys. Rev. Lett. 19, 824 (1967);D. H. Rank, C. W. Cho, N. D. Foltz, T. A. Wiggins, Phys. Rev. Lett. 19, 828 (1967);T. A. Wiggins, C. W. Cho, D. R. Dietz, N. D. Foltz, Phys. Rev. Lett. 20, 831 (1968);M. A. Gray, R. M. Herman, Phys. Rev. 181, 374 (1968);A. O. Creaser, R. M. Herman, Phys. Rev. Lett. 29, 147 (1972);SeeI P. Batra, R. H. Enns, D. Pohl, Phys. Status Solidi B 48, 10 (1971) for a general review of progress in this field.
    [CrossRef]
  11. Auston and Shank4 have found the solution for a similar transport equation at z = 0, in which their source term is spatially distributed and instantaneous in time, in contrast to the source term in the present treatment, which is localized in space and distributed in time. The solutions are compatible in the limits of localization both in space and time.

1978

1976

S. A. Jamison, A. V. Nurmikko, H. J. Gerritsen, Appl. Phys. Lett. 39, 640 (1976).
[CrossRef]

1975

C. V. Shank, D. H. Auston, Phys. Rev. Lett. 34, 479 (1975);D. H. Auston, C. V. Shank, P. LeFur, Phys. Rev. Lett. 35, 1022 (1975).
[CrossRef]

A. J. Alcock, P. B. Corkum, D. J. James, Appl. Phys. Lett. 27, 680 (1975).
[CrossRef]

1974

T. A. Wiggins, A. Salik, Appl. Phys. Lett. 25, 438 (1974).
[CrossRef]

C. J. Kennedy, J. C. Matter, A. L. Smirl, H. Weichel, F. A. Hopf, S. V. Pappu, M. O. Scully, Phys. Rev. Lett. 32, 419 (1974);A. L. Smirl, J. C. Matter, A. Elci, M. O. Scully, Opt. Commun. 16, 118 (1976).
[CrossRef]

D. H. Auston, C. V. Shank, Phys. Rev. Lett. 32, 1120 (1974).
[CrossRef]

1968

A. M. Bonch-Bruevich, V. P. Kovalev, G. S. Romanov, Ya. A. Imas, M. N. Libenson, Sov. Phys. Tech. Phys. 13, 507 (1968);I. Nagata, R. J. Galagali, S. Horiguchi, T. Sakai, T. Nakaya, J. Phys. D 3, 1305 (1970).
[CrossRef]

1967

R. M. Herman, M. A. Gray, Phys. Rev. Lett. 19, 824 (1967);D. H. Rank, C. W. Cho, N. D. Foltz, T. A. Wiggins, Phys. Rev. Lett. 19, 828 (1967);T. A. Wiggins, C. W. Cho, D. R. Dietz, N. D. Foltz, Phys. Rev. Lett. 20, 831 (1968);M. A. Gray, R. M. Herman, Phys. Rev. 181, 374 (1968);A. O. Creaser, R. M. Herman, Phys. Rev. Lett. 29, 147 (1972);SeeI P. Batra, R. H. Enns, D. Pohl, Phys. Status Solidi B 48, 10 (1971) for a general review of progress in this field.
[CrossRef]

1964

C. H. Carmichael, G. N. Simpson, Nature 202, 787 (1964);W. R. Sooy, M. Geller, D. P. Bortfeld, Appl. Phys. Lett. 5, 54 (1964);M. Birnbaum, J. Appl. Phys. 36, 657 (1965);M. Birnbaum, T. L. Stocker, Br. J. Appl. Phys. 17, 461 (1966).
[CrossRef]

Alcock, A. J.

A. J. Alcock, P. B. Corkum, D. J. James, Appl. Phys. Lett. 27, 680 (1975).
[CrossRef]

Auston, D. H.

C. V. Shank, D. H. Auston, Phys. Rev. Lett. 34, 479 (1975);D. H. Auston, C. V. Shank, P. LeFur, Phys. Rev. Lett. 35, 1022 (1975).
[CrossRef]

D. H. Auston, C. V. Shank, Phys. Rev. Lett. 32, 1120 (1974).
[CrossRef]

Bellay, J. A.

Bonch-Bruevich, A. M.

A. M. Bonch-Bruevich, V. P. Kovalev, G. S. Romanov, Ya. A. Imas, M. N. Libenson, Sov. Phys. Tech. Phys. 13, 507 (1968);I. Nagata, R. J. Galagali, S. Horiguchi, T. Sakai, T. Nakaya, J. Phys. D 3, 1305 (1970).
[CrossRef]

Carmichael, C. H.

C. H. Carmichael, G. N. Simpson, Nature 202, 787 (1964);W. R. Sooy, M. Geller, D. P. Bortfeld, Appl. Phys. Lett. 5, 54 (1964);M. Birnbaum, J. Appl. Phys. 36, 657 (1965);M. Birnbaum, T. L. Stocker, Br. J. Appl. Phys. 17, 461 (1966).
[CrossRef]

Carrieri, A. H.

Corkum, P. B.

A. J. Alcock, P. B. Corkum, D. J. James, Appl. Phys. Lett. 27, 680 (1975).
[CrossRef]

Gerritsen, H. J.

S. A. Jamison, A. V. Nurmikko, H. J. Gerritsen, Appl. Phys. Lett. 39, 640 (1976).
[CrossRef]

Gray, M. A.

R. M. Herman, M. A. Gray, Phys. Rev. Lett. 19, 824 (1967);D. H. Rank, C. W. Cho, N. D. Foltz, T. A. Wiggins, Phys. Rev. Lett. 19, 828 (1967);T. A. Wiggins, C. W. Cho, D. R. Dietz, N. D. Foltz, Phys. Rev. Lett. 20, 831 (1968);M. A. Gray, R. M. Herman, Phys. Rev. 181, 374 (1968);A. O. Creaser, R. M. Herman, Phys. Rev. Lett. 29, 147 (1972);SeeI P. Batra, R. H. Enns, D. Pohl, Phys. Status Solidi B 48, 10 (1971) for a general review of progress in this field.
[CrossRef]

Herman, R. M.

R. M. Herman, M. A. Gray, Phys. Rev. Lett. 19, 824 (1967);D. H. Rank, C. W. Cho, N. D. Foltz, T. A. Wiggins, Phys. Rev. Lett. 19, 828 (1967);T. A. Wiggins, C. W. Cho, D. R. Dietz, N. D. Foltz, Phys. Rev. Lett. 20, 831 (1968);M. A. Gray, R. M. Herman, Phys. Rev. 181, 374 (1968);A. O. Creaser, R. M. Herman, Phys. Rev. Lett. 29, 147 (1972);SeeI P. Batra, R. H. Enns, D. Pohl, Phys. Status Solidi B 48, 10 (1971) for a general review of progress in this field.
[CrossRef]

Hopf, F. A.

C. J. Kennedy, J. C. Matter, A. L. Smirl, H. Weichel, F. A. Hopf, S. V. Pappu, M. O. Scully, Phys. Rev. Lett. 32, 419 (1974);A. L. Smirl, J. C. Matter, A. Elci, M. O. Scully, Opt. Commun. 16, 118 (1976).
[CrossRef]

Imas, Ya. A.

A. M. Bonch-Bruevich, V. P. Kovalev, G. S. Romanov, Ya. A. Imas, M. N. Libenson, Sov. Phys. Tech. Phys. 13, 507 (1968);I. Nagata, R. J. Galagali, S. Horiguchi, T. Sakai, T. Nakaya, J. Phys. D 3, 1305 (1970).
[CrossRef]

James, D. J.

A. J. Alcock, P. B. Corkum, D. J. James, Appl. Phys. Lett. 27, 680 (1975).
[CrossRef]

Jamison, S. A.

S. A. Jamison, A. V. Nurmikko, H. J. Gerritsen, Appl. Phys. Lett. 39, 640 (1976).
[CrossRef]

Kennedy, C. J.

C. J. Kennedy, J. C. Matter, A. L. Smirl, H. Weichel, F. A. Hopf, S. V. Pappu, M. O. Scully, Phys. Rev. Lett. 32, 419 (1974);A. L. Smirl, J. C. Matter, A. Elci, M. O. Scully, Opt. Commun. 16, 118 (1976).
[CrossRef]

Kovalev, V. P.

A. M. Bonch-Bruevich, V. P. Kovalev, G. S. Romanov, Ya. A. Imas, M. N. Libenson, Sov. Phys. Tech. Phys. 13, 507 (1968);I. Nagata, R. J. Galagali, S. Horiguchi, T. Sakai, T. Nakaya, J. Phys. D 3, 1305 (1970).
[CrossRef]

Libenson, M. N.

A. M. Bonch-Bruevich, V. P. Kovalev, G. S. Romanov, Ya. A. Imas, M. N. Libenson, Sov. Phys. Tech. Phys. 13, 507 (1968);I. Nagata, R. J. Galagali, S. Horiguchi, T. Sakai, T. Nakaya, J. Phys. D 3, 1305 (1970).
[CrossRef]

Matter, J. C.

C. J. Kennedy, J. C. Matter, A. L. Smirl, H. Weichel, F. A. Hopf, S. V. Pappu, M. O. Scully, Phys. Rev. Lett. 32, 419 (1974);A. L. Smirl, J. C. Matter, A. Elci, M. O. Scully, Opt. Commun. 16, 118 (1976).
[CrossRef]

Nurmikko, A. V.

S. A. Jamison, A. V. Nurmikko, H. J. Gerritsen, Appl. Phys. Lett. 39, 640 (1976).
[CrossRef]

Pappu, S. V.

C. J. Kennedy, J. C. Matter, A. L. Smirl, H. Weichel, F. A. Hopf, S. V. Pappu, M. O. Scully, Phys. Rev. Lett. 32, 419 (1974);A. L. Smirl, J. C. Matter, A. Elci, M. O. Scully, Opt. Commun. 16, 118 (1976).
[CrossRef]

Romanov, G. S.

A. M. Bonch-Bruevich, V. P. Kovalev, G. S. Romanov, Ya. A. Imas, M. N. Libenson, Sov. Phys. Tech. Phys. 13, 507 (1968);I. Nagata, R. J. Galagali, S. Horiguchi, T. Sakai, T. Nakaya, J. Phys. D 3, 1305 (1970).
[CrossRef]

Salik, A.

T. A. Wiggins, A. Salik, Appl. Phys. Lett. 25, 438 (1974).
[CrossRef]

Scully, M. O.

C. J. Kennedy, J. C. Matter, A. L. Smirl, H. Weichel, F. A. Hopf, S. V. Pappu, M. O. Scully, Phys. Rev. Lett. 32, 419 (1974);A. L. Smirl, J. C. Matter, A. Elci, M. O. Scully, Opt. Commun. 16, 118 (1976).
[CrossRef]

Shank, C. V.

C. V. Shank, D. H. Auston, Phys. Rev. Lett. 34, 479 (1975);D. H. Auston, C. V. Shank, P. LeFur, Phys. Rev. Lett. 35, 1022 (1975).
[CrossRef]

D. H. Auston, C. V. Shank, Phys. Rev. Lett. 32, 1120 (1974).
[CrossRef]

Simpson, G. N.

C. H. Carmichael, G. N. Simpson, Nature 202, 787 (1964);W. R. Sooy, M. Geller, D. P. Bortfeld, Appl. Phys. Lett. 5, 54 (1964);M. Birnbaum, J. Appl. Phys. 36, 657 (1965);M. Birnbaum, T. L. Stocker, Br. J. Appl. Phys. 17, 461 (1966).
[CrossRef]

Smirl, A. L.

C. J. Kennedy, J. C. Matter, A. L. Smirl, H. Weichel, F. A. Hopf, S. V. Pappu, M. O. Scully, Phys. Rev. Lett. 32, 419 (1974);A. L. Smirl, J. C. Matter, A. Elci, M. O. Scully, Opt. Commun. 16, 118 (1976).
[CrossRef]

Weichel, H.

C. J. Kennedy, J. C. Matter, A. L. Smirl, H. Weichel, F. A. Hopf, S. V. Pappu, M. O. Scully, Phys. Rev. Lett. 32, 419 (1974);A. L. Smirl, J. C. Matter, A. Elci, M. O. Scully, Opt. Commun. 16, 118 (1976).
[CrossRef]

Wiggins, T. A.

Appl. Opt.

Appl. Phys. Lett.

T. A. Wiggins, A. Salik, Appl. Phys. Lett. 25, 438 (1974).
[CrossRef]

A. J. Alcock, P. B. Corkum, D. J. James, Appl. Phys. Lett. 27, 680 (1975).
[CrossRef]

S. A. Jamison, A. V. Nurmikko, H. J. Gerritsen, Appl. Phys. Lett. 39, 640 (1976).
[CrossRef]

Nature

C. H. Carmichael, G. N. Simpson, Nature 202, 787 (1964);W. R. Sooy, M. Geller, D. P. Bortfeld, Appl. Phys. Lett. 5, 54 (1964);M. Birnbaum, J. Appl. Phys. 36, 657 (1965);M. Birnbaum, T. L. Stocker, Br. J. Appl. Phys. 17, 461 (1966).
[CrossRef]

Phys. Rev. Lett.

C. J. Kennedy, J. C. Matter, A. L. Smirl, H. Weichel, F. A. Hopf, S. V. Pappu, M. O. Scully, Phys. Rev. Lett. 32, 419 (1974);A. L. Smirl, J. C. Matter, A. Elci, M. O. Scully, Opt. Commun. 16, 118 (1976).
[CrossRef]

D. H. Auston, C. V. Shank, Phys. Rev. Lett. 32, 1120 (1974).
[CrossRef]

C. V. Shank, D. H. Auston, Phys. Rev. Lett. 34, 479 (1975);D. H. Auston, C. V. Shank, P. LeFur, Phys. Rev. Lett. 35, 1022 (1975).
[CrossRef]

R. M. Herman, M. A. Gray, Phys. Rev. Lett. 19, 824 (1967);D. H. Rank, C. W. Cho, N. D. Foltz, T. A. Wiggins, Phys. Rev. Lett. 19, 828 (1967);T. A. Wiggins, C. W. Cho, D. R. Dietz, N. D. Foltz, Phys. Rev. Lett. 20, 831 (1968);M. A. Gray, R. M. Herman, Phys. Rev. 181, 374 (1968);A. O. Creaser, R. M. Herman, Phys. Rev. Lett. 29, 147 (1972);SeeI P. Batra, R. H. Enns, D. Pohl, Phys. Status Solidi B 48, 10 (1971) for a general review of progress in this field.
[CrossRef]

Sov. Phys. Tech. Phys.

A. M. Bonch-Bruevich, V. P. Kovalev, G. S. Romanov, Ya. A. Imas, M. N. Libenson, Sov. Phys. Tech. Phys. 13, 507 (1968);I. Nagata, R. J. Galagali, S. Horiguchi, T. Sakai, T. Nakaya, J. Phys. D 3, 1305 (1970).
[CrossRef]

Other

Auston and Shank4 have found the solution for a similar transport equation at z = 0, in which their source term is spatially distributed and instantaneous in time, in contrast to the source term in the present treatment, which is localized in space and distributed in time. The solutions are compatible in the limits of localization both in space and time.

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Figures (1)

Fig. 1
Fig. 1

Incident (S and W), reflected (SR and WR), and diffracted (D) light beams relative to the germanium surface.

Equations (42)

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E inc ( z , x , t ) = 1 2 E S exp [ i ( k S z ω S t ) ] + c . c . + 1 2 E W exp [ i ( k W z + Kx ω W t ) ] + c . c . ,
I ( 0 , x , t ) ~ 1 2 ( | E S | 2 + { E S * E W exp [ i ( Kx ω t ) ] + c . c . } )
I ( 0 , x , t ) = I S ( 1 + { E W / E S exp [ i ( Kx ω t ) ] + c . c . } )
( 1 R ) fI ( 0 , x , t ) δ ( z ) ,
ρ 0 c p T ( z , x , t ) t λ 2 T ( z , x , t ) = 2 ( 1 R ) fI ( 0 , x , t ) δ ( z ) ,
2 ( R T ) T ( 0 , x , t ) f I S ( 0 , t ) .
T ( z , x , t ) = 1 2 [ exp ( iKx ) dkT ( k , t ) exp ( ikz ) + c . c . ] .
ρ 0 c p T ( k , t ) t + λ ( K 2 + k 2 ) T ( k , t ) = 2 ( 1 R ) f I S ( t ) [ ( E W ) / ( E S ) ] π exp ( i ω t ) ,
δ ( z ) = 1 2 π exp ( ikz ) dk ,
T ( k , t ) = 2 ( 1 R ) f E W / E S π ρ 0 c p t d t I S ( t ) × exp [ λ ( k 2 + K 2 ) ρ 0 c p ( t t ) + i ω t ] ,
T ( z , x , t ) = [ ( 1 R ) f E W / E S π ρ 0 c p ] × exp [ i ( Kx ω t ) ] dk t d t I S ( t ) exp [ ikz λ ( K 2 + k 2 ) ρ 0 c p ( t t ) i ω ( t t ) ] + c . c .
T ( 0 , x , t ) = exp [ i ( Kx ω t ) ] [ ( 1 R ) f E W / E S ( π ρ 0 c p λ ) 1 / 2 ] t d t I S ( t ) × ( t t ) 1 / 2 exp [ ( λ K 2 ρ 0 c p + i ω ) ( t t ) ] + c . c .
D ( K , ω ) 0 dt I S ( t ) ( t ) 1 / 2 exp [ ( λ K 2 ρ 0 c p + i ω ) t ] 0 dt I S ( t ) ( t ) 1 / 2 .
0 dt I S ( t ) t 1 / 2 exp [ ( λ K 2 ρ 0 c p + i ω ) t ] 0 dt I S ( t ) t 1 / 2 .
D ( K , ω ) [ 1 + ( λ K 2 ρ 0 c p + i ω ) τ p / 2 ] 1 / 2
D ( K , ω ) = { 1 for | ( λ K 2 ρ 0 c p + i ω ) τ p / 2 | 1 const . [ ( λ K 2 ρ 0 c p + i ω ) τ p / 2 ] 1 / 2 for | ( λ K 2 ρ 0 c p + i ω ) τ p / 2 | 1
E Reflected ( 0 , x , t ) = ( n ik 1 n ik + 1 ) x , t , z = 0 [ E S ( 0 , t ) + E W ( 0 , x , t ) ] .
E WR ( 0 , x , t ) ( n 0 i k 0 1 n 0 i k 0 + 1 ) ( { 1 + 2 ( δ n i δ k ) ( n 0 i k 0 ) 2 1 2 ( δ n i δ k ) 2 [ ( n 0 i k 0 ) 2 1 ] ( n 0 i k 0 + 1 ) } E W ( 0 , x , t ) + { 2 ( δ n i δ k ) ( n 0 i k 0 ) 2 1 2 ( δ n i δ k ) 2 [ ( n 0 i k 0 ) 2 1 ] ( n 0 i k 0 + 1 ) } K E S ( 0 , t ) ) ,
( δ n 2 ) K = 2 δ n ( δ n ) K ,
δ n = n 2 I S + n 4 I S 2 + . . . ,
δ n K = I S ( d ( δ n ) d I S ) E W / E S D ( K , ω = 0 ) exp ( iKx )
δ n ( E W / E S ) s n D ( K , ω = 0 ) exp ( iKx ) ,
s n = ( n 2 + 2 I S n 4 + . . . n 2 + I S n 4 + . . . ) .
I WR = R 0 I W [ 1 + 4 Re { δ n [ 1 + s n D ( K , 0 ) ] i δ k [ 1 + s k D ( K , 0 ) ] ( n 0 i k 0 ) 2 1 } 4 Re { ( δ n i δ k ) { δ n [ 1 + 2 s n D ( K , 0 ) ] } i δ k [ 1 + 2 s k D ( K , 0 ) ] [ ( n 0 i k 0 ) 2 1 ] ( n 0 i k 0 + 1 ) } + 4 | δ n [ 1 + s n D ( K , 0 ) ] i δ k [ 1 + s k D ( K , 0 ) ] ( n 0 i k 0 ) 2 1 | 2 ]
| n 0 i k 0 1 n 0 i k 0 + 1 | 2
Δ R R 0 = 4 ( Re { δ n ( 1 + s n D ) i δ k ( 1 + s k D ) ( n 0 i k 0 ) 2 1 } Re { ( δ n i δ k ) [ δ n ( 1 + 2 s n D ) i δ k ( 1 + 2 s k D ) ] [ ( n 0 i k 0 ) 2 1 ] ( n 0 i k 0 + 1 ) } + | δ n ( 1 + s n D ) i δ k ( 1 + s k D ) ( n 0 i k 0 ) 2 1 | 2 )
E D ( 0 , x , t ) ( n 0 i k 0 1 n 0 i k 0 + 1 ) × [ 2 ( δ n i δ k ) ( n 0 i k 0 ) 2 1 ] K E S ( 0 , t )
( δ n ) K = δ n ( E W * / E S * ) s n D ( K , 0 ) exp ( iKx ) ,
I D = 4 R 0 I W | [ δ n s n i δ k s k ( n 0 i k 0 ) 2 1 ] D ( K , 0 ) | 2 ,
D 2 ( K , 0 ) ( 1 + λ K 2 τ p 2 ρ 0 c p ) 1 .
ω D = ω S ω = 2 ω S ω W .
I D ~ | D ( K , ω ) | 2 [ ( 1 + λ K 2 τ p 2 ρ 0 c p ) 2 + ( ω τ p 2 ) 2 ] 1 / 2
T ( k , t ) t + λ ( K 2 + k 2 ) ρ 0 c p T ( k , t ) = 2 ( 1 R ) f I S ( t ) E W / E S π ρ 0 c p exp ( i ω t ) .
s = [ λ ( K 2 + k 2 ) ] / ( ρ 0 c p )
g ( t ) = 2 ( 1 R ) f I S ( t ) E W / E S π ρ 0 c p exp ( i ω t ) ,
[ T ( t ) ] / ( t ) + sT ( t ) = g ( t ) .
G ( t , t ) / t + sG ( t , t ) = δ ( t t )
G ( t < t ) = 0 .
G ( t > t ) = exp [ s ( t t ) ]
T ( t ) = g ( t ) G ( t , t ) d t
T ( t ) = t g ( t ) exp [ s ( t , t ) ] d t .
T ( t ) = 2 ( 1 R ) f E W / W S π ρ 0 c p t d t I S ( t ) × exp [ λ ( K 2 + k 2 ) ρ 0 c p ( t t ) + i ω t ]

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