Abstract

In the present paper, we have presented a Maxwellian boundary-type solution for total internal reflection with unbounded incident waves at an interface between two nonabsorbing media, in which the instantaneous, time varying, and time averaged radiant fluxes have been determined at all points in the two media. Solutions for the s and p polarizations were found for which the instantaneous tangential E and H components and normal components of the radiant flux were continuous in crossing the interface. From these radiant fluxes, it was possible to derive equations for the flow lines, to determine the instantaneous radiant fluxes along these flow lines, and to see how the methods of propagation differed in the two media and for the two polarizations. At the interface, the flow lines and their radiant fluxes experience unusual reflection and refraction processes, follow curved flow lines in the second medium, and return into the first medium with boundary conditions, which are mirror images of those at the points of incidence. These unfamiliar processes in the second medium are due to inhomogeneous waves, whose properties have not been understood. When these instantaneous solutions are extended to time varying and time averaged radiant fluxes, it is interesting to see how incident planes of constant radiant flux and phase experience such complex processes in the second medium and are still able to generate other reflected planes of constant radiant flux and phase in the first medium. These ideas prescribe specific detailed functions for the E and H fields and radiant fluxes in the second medium, which help to answer many long standing questions about the physical processes involved in total internal reflection.

© 1978 Optical Society of America

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References

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  1. H. K. V. Lotsch, J. Opt. Soc. Am. 58, 551 (1968);Optik 32, 116, 189 (1970);Optik 32, 299, 553 (1971).
    [CrossRef]
  2. H. Geiger, K. Scheel, Handbuch der Physik (Julius Springer, Berlin, 1928), Vol. 20, p. 228;C. Schaefer, Einfuhrung die theoretische Physik (W. de Gruyter, Berlin, 1949), Vol. 3, p. 406;M. Born, E. Wolf, Principles of Optics (Pergamon, New York,1959), p. 47.
  3. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 112.
  4. W. Culshaw, D. S. Jones, Proc. Phys. Soc. London, Sect. B 66, 859 (1954);H. Osterberg, L. W. Smith, J. Opt. Soc. Am. 1073 (1964).
    [CrossRef]
  5. C. Schaefer, G. Gross, Ann. Phys. 32, 648 (1910).
    [CrossRef]
  6. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), p. 34.
  7. M. Born, Optik (Julius Springer, Berlin, 1933), p. 41.
  8. H. Geiger, K. Scheel, Handbuch der Physik (Springer, Berlin, 1928), Vol. 20, p. 194.
  9. A. Eichenwald, J. Russ. Phys. Chem. 4, 137 (1919).
  10. M. Born, Optik (Julius Springer, Berlin, 1933), p. 31.
  11. G. G. Stokes, Cambridge Math. J. 8, 1 (1849).
  12. F. A. Jenkins, H. E. White, Fundamentals of Physical Optics (McGraw-Hill, New York, 1937), p. 390.
  13. M. Born, Optik (Springer, Berlin, 1933), p. 29.

1968 (1)

1954 (1)

W. Culshaw, D. S. Jones, Proc. Phys. Soc. London, Sect. B 66, 859 (1954);H. Osterberg, L. W. Smith, J. Opt. Soc. Am. 1073 (1964).
[CrossRef]

1919 (1)

A. Eichenwald, J. Russ. Phys. Chem. 4, 137 (1919).

1910 (1)

C. Schaefer, G. Gross, Ann. Phys. 32, 648 (1910).
[CrossRef]

1849 (1)

G. G. Stokes, Cambridge Math. J. 8, 1 (1849).

Born, M.

M. Born, Optik (Springer, Berlin, 1933), p. 29.

M. Born, Optik (Julius Springer, Berlin, 1933), p. 31.

M. Born, Optik (Julius Springer, Berlin, 1933), p. 41.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 112.

Culshaw, W.

W. Culshaw, D. S. Jones, Proc. Phys. Soc. London, Sect. B 66, 859 (1954);H. Osterberg, L. W. Smith, J. Opt. Soc. Am. 1073 (1964).
[CrossRef]

Eichenwald, A.

A. Eichenwald, J. Russ. Phys. Chem. 4, 137 (1919).

Geiger, H.

H. Geiger, K. Scheel, Handbuch der Physik (Springer, Berlin, 1928), Vol. 20, p. 194.

H. Geiger, K. Scheel, Handbuch der Physik (Julius Springer, Berlin, 1928), Vol. 20, p. 228;C. Schaefer, Einfuhrung die theoretische Physik (W. de Gruyter, Berlin, 1949), Vol. 3, p. 406;M. Born, E. Wolf, Principles of Optics (Pergamon, New York,1959), p. 47.

Gross, G.

C. Schaefer, G. Gross, Ann. Phys. 32, 648 (1910).
[CrossRef]

Jenkins, F. A.

F. A. Jenkins, H. E. White, Fundamentals of Physical Optics (McGraw-Hill, New York, 1937), p. 390.

Jones, D. S.

W. Culshaw, D. S. Jones, Proc. Phys. Soc. London, Sect. B 66, 859 (1954);H. Osterberg, L. W. Smith, J. Opt. Soc. Am. 1073 (1964).
[CrossRef]

Lotsch, H. K. V.

Schaefer, C.

C. Schaefer, G. Gross, Ann. Phys. 32, 648 (1910).
[CrossRef]

Scheel, K.

H. Geiger, K. Scheel, Handbuch der Physik (Springer, Berlin, 1928), Vol. 20, p. 194.

H. Geiger, K. Scheel, Handbuch der Physik (Julius Springer, Berlin, 1928), Vol. 20, p. 228;C. Schaefer, Einfuhrung die theoretische Physik (W. de Gruyter, Berlin, 1949), Vol. 3, p. 406;M. Born, E. Wolf, Principles of Optics (Pergamon, New York,1959), p. 47.

Stokes, G. G.

G. G. Stokes, Cambridge Math. J. 8, 1 (1849).

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), p. 34.

White, H. E.

F. A. Jenkins, H. E. White, Fundamentals of Physical Optics (McGraw-Hill, New York, 1937), p. 390.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 112.

Ann. Phys. (1)

C. Schaefer, G. Gross, Ann. Phys. 32, 648 (1910).
[CrossRef]

Cambridge Math. J. (1)

G. G. Stokes, Cambridge Math. J. 8, 1 (1849).

J. Opt. Soc. Am. (1)

J. Russ. Phys. Chem. (1)

A. Eichenwald, J. Russ. Phys. Chem. 4, 137 (1919).

Proc. Phys. Soc. London, Sect. B (1)

W. Culshaw, D. S. Jones, Proc. Phys. Soc. London, Sect. B 66, 859 (1954);H. Osterberg, L. W. Smith, J. Opt. Soc. Am. 1073 (1964).
[CrossRef]

Other (8)

F. A. Jenkins, H. E. White, Fundamentals of Physical Optics (McGraw-Hill, New York, 1937), p. 390.

M. Born, Optik (Springer, Berlin, 1933), p. 29.

M. Born, Optik (Julius Springer, Berlin, 1933), p. 31.

H. Geiger, K. Scheel, Handbuch der Physik (Julius Springer, Berlin, 1928), Vol. 20, p. 228;C. Schaefer, Einfuhrung die theoretische Physik (W. de Gruyter, Berlin, 1949), Vol. 3, p. 406;M. Born, E. Wolf, Principles of Optics (Pergamon, New York,1959), p. 47.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 112.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), p. 34.

M. Born, Optik (Julius Springer, Berlin, 1933), p. 41.

H. Geiger, K. Scheel, Handbuch der Physik (Springer, Berlin, 1928), Vol. 20, p. 194.

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Figures (10)

Fig. 1
Fig. 1

Reflection and refraction for s polarization at interface between two absorbing media with angles of incidence below critical angle.

Fig. 2
Fig. 2

Components of incident, reflected, and transmitted radiant fluxes for s polarization along interface (in units of λ1) for n1 = 1.5, n2 = 1.0, φ = 45°, and z = t = 0.

Fig. 3
Fig. 3

Radiant flux flow lines zi(s), zr(s), and zt(s) for s polarization with n1 = 1.5, n2 = 1.0, φ = 45°, and t = 0 when x1si = x1sr = x1st= −0.1 λ1, −0.2087λ1, −0.25λ1, and −0.28λ1.

Fig. 4
Fig. 4

Radiant flux and radiant flux flow lines for s polarization with x1s= −0.1λ1 and −0.2087λ1, when m1 = 1.5, n2 = 1.0, φ = 45°, and t = 0.

Fig. 5
Fig. 5

Radiant flux and radiant flux flow lines for s polarization with x1s = −0.2087λ1 and −0.28λ1 when n1 = 1.5, n2 = 1.0, φ = 45°, and t = 0.

Fig. 6
Fig. 6

Components of incident, reflected, and transmitted radiant fluxes for s polarization along interface (in units of λ1) for n1 = 1.5, n2 = 1.0, φ = 45°, and z = t = 0, but with an advance in phase Δ = 45°.

Fig. 7
Fig. 7

Radiant flux and radiant flux flow lines for two cycles of s polarization with x1s = − 0.1 λ1 and −0.2087λ1 when n1 = 1.5, n2 = 1.0, φ=45°,and t = 0.

Fig. 8
Fig. 8

Radiant flux and radiant flux flow line for p polarization with x1p = −0.0639λ1, when n1 = 1.5, n2 = 1.0, φ = 45°, and t = 0.

Fig. 9
Fig. 9

Time averages of s and p radiant fluxes in second medium at different distances from interface with Ep = Es,n1 = 1.5, n2 = 1.0, and φ = 45°.

Fig. 10
Fig. 10

Time averages of s and p reflected radiant fluxes in first medium, when a perfectly black absorbing surface is moved from z = 2.0λ1, to z = 0 with Ep = Es, m = 1.5, n2 = 1.0, and φ = 45°.

Equations (36)

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Cu r l E = 1 c H t , Cu r l H = c E t , div D = div H = 0 ,
E y z = 1 c H x t , E y x = 1 c H z t , H x z H z x = c E y t .
2 E y x 2 + 2 E y z 2 = c 2 2 E y t 2 ,
E y i = E s exp { i [ ω t ω c n 1 ( x sin φ + z cos φ ) ] } , H x i = n 1 cos φ E s exp { i [ ω t ω c n 1 ( x sin φ + z cos φ ) ] } , H z i = n 1 sin φ E s exp { i [ ω t ω c n 1 ( x sin φ + z cos φ ) ] } , E y r = R s exp { i [ ω t ω c n 1 ( x sin φ z cos φ ) ] } , H x r = n 1 cos φ R s exp { i [ ω t ω c n 1 ( x sin φ z cos φ ) ] } , H z r = n 1 sin φ R s exp { i [ ω t ω c n 1 ( x sin φ z cos φ ) ] } , E y t = D s exp { i [ ω t ω c n 2 ( x sin ψ + z cos ψ ) ] } , H x t = n 2 cos ψ D s exp { i [ ω t ω c n 2 ( x sin ψ + z cos ψ ) ] } , H z t = n 2 sin ψ D s exp { i [ ω t ω c n 2 ( x sin ψ + z cos ψ ) ] } .
n 1 sin φ = n 2 sin ψ ,
sin ψ = n 1 n 2 sin φ > 1 , cos ψ = i ( n 1 2 n 2 2 sin 2 ψ 1 ) 1 / 2
R s = E s exp ( i δ s r ) , D s = 2 n 1 cos φ ( n 1 2 n 2 2 ) 1 / 2 E s exp ( i δ s t ) ,
tan δ s r 2 = tan δ s t = ( n 1 2 sin 2 φ n 2 2 ) 1 / 2 n 1 cos φ .
E y i = E s cos [ ω t 2 π λ 1 ( x sin φ + z cos φ ) ] , H x i = n 1 cos φ E s cos [ ω t 2 π λ 1 ( x sin φ + z cos φ ) ] , H z i = n 1 sin φ E s cos [ ω t 2 π λ 1 ( x sin φ + z cos φ ) ] , E y r = E s cos [ ω t 2 π λ 1 ( x sin φ z cos φ ) + δ s r ] , H x r = n 1 cos φ E s cos [ ω t 2 π λ 1 ( x sin φ z cos φ ) + δ s r ] , H z r = n 1 sin φ E s cos [ ω t 2 π λ 1 ( x sin φ z cos φ ) + δ s r ] , E y t = 2 n 1 cos φ ( n 1 2 n 2 2 ) 1 / 2 E s exp [ 2 π λ 1 z ( sin 2 φ n 2 2 n 1 2 ) 1 / 2 ] × cos ( ω t 2 π λ 1 x sin φ + δ s t ) , H x t = 2 n 1 cos φ ( n 1 2 sin 2 φ n 2 2 ) 1 / 2 ( n 1 2 n 2 2 ) 1 / 2 E s × exp [ 2 π λ 1 z ( sin 2 φ n 2 2 n 1 2 ) 1 / 2 ] sin ( ω t 2 π λ 1 x sin φ + δ s t ) , H z t = 2 n 1 2 sin φ cos φ ( n 1 2 n 2 2 ) 1 / 2 E s exp [ 2 π λ 1 z ( sin 2 φ n 2 2 n 1 2 ) 1 / 2 ] cos ( ω t 2 π λ 1 x sin φ + δ s t ) .
P x i ( s ) = c 4 π n 1 sin φ E s 2 cos 2 [ ω t 2 π λ 1 ( x sin φ + z cos φ ) ] , P z i ( s ) = c 4 π n 1 cos φ E s 2 cos 2 [ ω t 2 π λ 1 ( x sin φ + z cos φ ) ] , P x r ( s ) = c 4 π n 1 sin φ E s 2 cos 2 [ ω t 2 π λ 1 ( x sin φ z cos φ ) + δ s r ] , P z r ( s ) = c 4 π n 1 cos φ E s 2 cos 2 [ ω t 2 π λ 1 ( x sin φ z cos φ ) + δ s r ] , P x t ( s ) = c 4 π 4 n 1 3 sin φ cos 2 φ n 1 2 n 2 2 E s 2 × exp [ 4 π λ 1 z ( sin 2 φ n 2 2 n 1 2 ) 1 / 2 ] cos 2 ( ω t 2 π x λ 1 sin φ + δ s t ) , P z t ( s ) = c 4 π 4 n 1 2 cos 2 φ ( n 1 2 sin 2 φ n 2 2 ) 1 / 2 n 1 2 n 2 2 E s 2 × exp [ 4 π λ 1 z ( sin 2 φ n 2 2 n 1 2 ) 1 / 2 ] × cos ( ω t 2 π x λ 1 sin φ + δ s t ) sin ( ω t 2 π x λ 1 sin φ + δ s t ) .
( dz dx ) i = P z i ( s ) P x i ( s ) = cot φ , ( dz dx ) r = P z r ( s ) P x r ( s ) = cot φ , ( dz dx ) t = P z t ( s ) P x t ( s ) = ( n 1 2 sin 2 φ n 2 2 ) 1 / 2 n 1 sin φ × tan ( ω t 2 π x λ 1 sin φ + δ s t ) .
z i ( s ) = ( x x 1 s i ) cot φ , z r ( s ) = ( x x 1 s r ) cot φ , z t ( s ) = λ 1 ( n 1 2 sin 2 φ n 2 2 ) 1 / 2 2 π n 1 sin 2 φ { log e [ cos ( ω t 2 π λ 1 x sin φ + δ s t ) ] log e [ cos ( ω t 2 π λ 1 x 1 s t sin φ + δ s t ) ] } ,
| P z i ( s ) | in = | P z r ( s ) | in + | P z t ( s ) | in ,
| P z r ( s ) | out = | P z i ( s ) | out + | P z t ( s ) | out .
2 π λ 1 x 1 s sin φ + δ s t = π 2
| P i ( s ) | = c 4 π n 1 E s 2 cos 2 [ ω t 2 π λ 1 ( x sin φ + z cos φ ) ] , | P r ( s ) | = c 4 π n 1 E s 2 cos 2 [ ω t 2 π λ 1 ( x sin φ z cos φ ) + δ s r ] , | P t ( s ) | = c 4 π 4 n 1 2 cos 2 φ n 1 2 n 2 2 E s 2 exp [ 4 π z λ 1 ( sin 2 φ n 2 2 n 1 2 ) 1 / 2 ] cos ( ω t 2 π x λ 1 sin φ + δ s r ) [ n 1 2 sin 2 φ n 2 2 sin 2 ( ω t 2 π x λ 1 sin φ + δ s t ) ] 1 / 2 .
2 π λ 1 x 1 s sin φ + δ s r = π 2 ,
2 π x λ 1 sin φ + δ s t = 0 ,
2 π x 2 s λ 1 sin φ = π 2 ,
| P i ( s ) | in cos φ = | P r ( s ) | in cos ϕ + | P t ( s ) | in cos φ ,
| P r ( s ) | out cos φ = | P i ( s ) | out cos ϕ + | P t ( s ) | out cos φ ,
Δ x = Δ 2 π sin φ λ 1 = 0.1768 λ 1
Δ x = ω t 2 π sin φ λ 1 = V 1 sin φ t = 0.1768 λ 1 .
| P i ( s ) | ¯ = c 4 π n 1 2 E s 2 , | P r ( s ) | ¯ = c 4 π n 1 2 E s 2 , P x t ( s ) ¯ = c 4 π 2 n 1 3 sin φ cos 2 φ n 1 2 n 2 2 E s 2 exp [ 4 π λ 1 z ( sin 2 φ n 2 2 n 1 2 ) 1 / 2 ] , P z t ( s ) ¯ = 0 .
| P i ( s ) ¯ | in = c 4 π n 1 ( ½ + 1 π sin 2 δ s t ) E s 2 , | P r ( s ) ¯ | in = c 4 π n 1 ( ½ + 1 π sin 2 δ s t ) E s 2 , P x t ( s ) in ¯ = c 4 π 2 n 1 3 sin φ cos 2 φ n 1 2 n 2 2 E s 2 , P z t ( s ) in ¯ = c 4 π 4 n 1 2 cos 2 φ ( n 1 2 sin 2 φ n 2 2 ) 1 / 2 π ( n 1 2 n 2 2 ) E s 2 .
H y z = c E x t , H y x = c E z t , E x z E z x = 1 c H y y .
H y i = n 1 E p cos [ ω t 2 π λ 1 ( x sin φ + z cos φ ) ] , E x i = cos φ E p cos [ ω t 2 π λ 1 ( x sin φ + z cos φ ) ] , E z i = sin φ E p cos [ ω t 2 π λ 1 ( x sin φ + z cos φ ) ] , H y r = n 1 E p cos [ ω t 2 π λ 1 ( x sin φ z cos φ ) + δ p r ] , E x r = cos φ E p cos [ ω t 2 π λ 1 ( x sin φ z cos φ ) + δ p r ] , E z r = sin φ E p cos [ ω t 2 π λ 1 ( x sin φ z cos φ ) + δ p r ] , H y t = 2 n 1 n 2 2 cos φ ( n 1 2 n 2 2 ) 1 / 2 [ ( n 1 2 + n 2 2 ) sin 2 φ n 2 2 ] 1 / 2 E p × exp [ 2 π λ 1 z ( sin 2 φ n 2 2 n 1 2 ) 1 / 2 ] cos ( ω t 2 π λ 1 x sin φ + δ p t ) , E x t = 2 n 1 cos φ ( n 1 2 sin 2 φ n 2 2 ) 1 / 2 ( n 1 2 n 2 2 ) 1 / 2 [ ( n 1 2 + n 2 2 ) sin 2 φ n 2 2 ] 1 / 2 E p × exp [ 2 π λ 1 z ( sin 2 φ n 2 2 n 1 2 ) 1 / 2 ] sin ( ω t 2 π λ 1 x sin φ + δ p t ) , E z t = 2 n 1 2 sin φ cos φ ( n 1 2 n 2 2 ) 1 / 2 [ ( n 1 2 + n 2 2 ) sin 2 φ n 2 2 ] 1 / 2 E p × exp [ 2 π λ 1 z ( sin 2 φ n 2 2 n 1 2 ) 1 / 2 ] cos ( ω t 2 π λ 1 x sin φ + δ p t ) ,
tan δ p r 2 = tan δ p t = n 1 ( n 1 2 sin 2 φ n 2 2 ) 1 / 2 n 2 2 cos φ .
P x i ( p ) = c 4 π n 1 sin φ E p 2 cos 2 [ ω t 2 π λ 1 ( x sin φ + z cos φ ) ] , P z i ( p ) = c 4 π n 1 cos φ E p 2 cos 2 [ ω t 2 π λ 1 ( x sin φ + z cos φ ) ] , P x r ( p ) = c 4 π n 1 sin φ E p 2 cos 2 [ ω t 2 π λ 1 ( x sin φ z cos φ ) + δ p r ] , P z r ( p ) = c 4 π n 1 cos φ E p 2 cos 2 [ ω t 2 π λ 1 ( x sin φ z cos φ ) + δ p r ] , P x t ( p ) = c 4 π 4 n 1 3 n 2 2 sin φ cos 2 φ ( n 1 2 n 2 2 ) [ ( n 1 2 + n 2 2 ) sin 2 φ n 2 2 ] E p 2 × exp [ 4 π λ 1 z ( sin 2 φ n 2 2 n 1 2 ) 1 / 2 ] cos 2 [ ω t 2 π λ 1 x sin φ + δ p t ) , P z t ( p ) = c 4 π 4 n 1 2 n 2 2 cos 2 φ ( n 1 2 sin 2 φ n 2 2 ) 1 / 2 ( n 1 2 n 2 2 ) [ ( n 1 2 + n 2 2 ) sin 2 φ n 2 2 ] E p 2 × exp [ 4 π λ 1 z ( sin 2 φ n 2 2 n 1 2 ) 1 / 2 ] × sin ( ω t 2 π λ 1 x sin φ + δ p t ) cos ( ω t 2 π λ 1 x sin φ + δ p t ) .
Δ x = λ 1 2 π sin φ ( δ p r δ s r )
Δ x = λ 1 2 π sin φ ( δ p t δ s t )
( dz dx ) t = P z t ( p ) P x t ( p ) = ( n 1 2 sin 2 φ n 2 2 ) 1 / 2 n 1 sin φ tan ( ω t 2 π λ 1 x sin φ + δ p t ) , z t ( p ) = λ 1 ( n 1 2 sin 2 φ n 2 2 ) 1 / 2 2 π n 1 sin 2 φ { log e [ cos ( ω t 2 π λ 1 x sin φ + δ p t ) ] log e [ cos ( ω t 2 π λ 1 x 1 p sin φ + δ p t ) ] } .
P x t ( s ) ¯ = c 4 π 2 n 1 3 sin φ cos 2 φ n 1 2 n 2 2 E s 2 exp [ 4 π λ 1 z ( sin 2 φ n 2 2 n 1 2 ) 1 / 2 ] , P z t ( s ) ¯ = c 4 π 2 n 1 2 cos φ ( n 1 2 sin 2 φ n 2 2 ) 1 / 2 π ( n 1 2 n 2 2 ) E s 2 exp [ 4 π λ 1 z ( sin 2 φ n 2 2 n 1 2 ) 1 / 2 ] .
| P i ( s ) ¯ | out = c 4 π n 1 ( ½ 1 π sin 2 δ s t ) E s 2 , | P r ( s ) ¯ | out = c 4 π n 1 ( ½ + 1 π sin 2 δ s t ) ( 1 f s ) E s 2 , P x t ( s ) out ¯ = c 4 π 2 n 1 3 sin φ cos 2 φ n 1 2 n 2 2 E s 2 × { 1 exp [ 4 π λ 1 z ( sin 2 φ n 2 2 n 1 2 ) 1 / 2 ] } , P z t ( s ) ¯ = c 4 π 4 n 1 2 cos 2 φ ( n 1 2 sin 2 φ n 2 2 ) 1 / 2 π ( n 1 2 n 2 2 ) × E s 2 { 1 exp [ 4 π λ 1 z ( sin 2 φ n 2 2 n 1 2 ) 1 / 2 ] } ,
f s = 4 n 1 cos φ ( n 1 2 sin 2 φ n 2 2 ) 1 / 2 π ( n 1 2 n 2 2 ) ( ½ + 1 π sin 2 δ s t ) × exp [ 4 π λ 1 z ( sin 2 φ n 2 2 n 1 2 ) 1 / 2 ] .
| P i ( s ) | = c 4 π n 1 2 E s 2 , | P r ( s ) | = c 4 π n 1 2 [ 1 ( ½ + 1 π sin 2 δ s t ) f s ] E s 2 , P x t ( s ) = c 4 π 2 n 1 3 sin φ cos 2 φ n 1 2 n 2 2 × { 1 ½ exp [ 4 π λ 1 z ( sin 2 φ n 2 2 n 1 2 ) 1 / 2 ] } E s 2 , P z t ( s ) = c 4 π 2 n 1 3 cos 2 φ ( n 1 2 sin 2 φ n 2 2 ) 1 / 2 π ( n 1 2 n 2 2 ) E s 2 × exp [ 4 π λ 1 z ( sin 2 φ n 2 2 n 1 2 ) 1 / 2 ] } .

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