Abstract

The problem of how plane unbounded electromagnetic waves in an absorbing medium are reflected and transmitted at an interface between an absorbing medium and a nonabsorbing medium is a problem of much current interest. In this paper, we will present a Maxwellian boundary-type solution to this problem in which the forms of the E and H components are determined from Maxwell’s equations and the boundary conditions, and the radiant fluxes, represented by the Poynting vector, evaluated in both media. From these radiant fluxes, it is possible to determine the forms of the radiant flux flow lines and to see how the propagation characteristics differ in the two media. In the second medium, we again find, as in total internal reflection [ A. I. Mahan and C. V. Bitterli, Appl. Opt. 17, 509 ( 1978)], that inhomogeneous nontransverse waves appear, whose planes of constant phase are normal to the refracted rays and whose planes of constant amplitude are parallel to these refracted rays. Because of these inhomogeneous waves, the second medium, although nonabsorbing under more familiar conditions, now exhibits refractive indices and absorption coefficients, which are functions of the more conventional optical constants and the angle of incidence, and new laws for reflection and transmission appear along the interface for both polarizations. These equations have been applied specifically to an ocean–air interface for a frequency of 100 Hz, and extensive calculations were carried out in which the radiant fluxes and their associated radiant flux flow line forms were determined along the interface and at other points outside the interface for both polarizations under steady state, time changing, and time average conditions. The radiant flux flow lines in the air above the ocean take the forms of up and over radiant fluxes, some of which are trapped above the interface and the remaining flow lines come out of the interface and return to the interface. These ideas are extensions of our earlier work on total internal reflection and show how the radiant fluxes and their associated flow lines in the second medium can be changed markedly by simply making the incident medium absorbing.

© 1981 Optical Society of America

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References

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  1. A. I. Mahan, C. V. Bitterli, Appl. Opt. 17, 509 (1978).
    [CrossRef] [PubMed]
  2. K. Uller, Beitrage zur Theorie de electromagnetishe Strahlung, Dissertation Rostock (1903); J. Zenneck, Ann. Phys. (Leipzig), 23, 846 (1907).
  3. J. Picht, Ann. Phys. (Leipzig), 3, 433 (1929); M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 48; F. A. Jenkins, H. E. White, Physical Optics (McGraw-Hill, New York, 1937), pp. 395–398.
  4. C. Schaefer, Einfuhrung die theoretische Physik (W. de Gruyter, Berlin, 1949), Vol. 3, p. 414; M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 48.
  5. R. H. Williams, “Propagation between Conducting and Non-Conducting Media,” Engineering Experiment Station, U. of New Mexico, Albuquerque, Technical Report EE-50 (Sept.1961).
  6. H. Geiger, K. Scheel, in Handbuch der Physik, S. Flugge, Ed. (Julius Springer, Berlin, 1928), Vol. 20, p. 202.
  7. M. Born, Optik (Julius Springer, Berlin, 1933), pp. 30–36.
  8. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1951), p. 1.
  9. Ref. 6, p. 152.
  10. Ref. 6, p. 194.
  11. Ref. 7, p. 260.
  12. Ref. 6, pp. 202, 204, and 207.
  13. These equations will be found to differ from those of Williams.5
  14. A. Eichenwald, Russ. J. Phys. Chem. 4, 137 (1919).
  15. J. A. Saxton, J. A. Lane, Wireless Eng.October (1952), p. 269.
  16. Ref. 6, p. 209.
  17. Ref. 6, p. 200.
  18. Ref. 7, p. 122.

1978

1952

J. A. Saxton, J. A. Lane, Wireless Eng.October (1952), p. 269.

1929

J. Picht, Ann. Phys. (Leipzig), 3, 433 (1929); M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 48; F. A. Jenkins, H. E. White, Physical Optics (McGraw-Hill, New York, 1937), pp. 395–398.

1919

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

Bitterli, C. V.

Born, M.

M. Born, Optik (Julius Springer, Berlin, 1933), pp. 30–36.

Eichenwald, A.

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

Geiger, H.

H. Geiger, K. Scheel, in Handbuch der Physik, S. Flugge, Ed. (Julius Springer, Berlin, 1928), Vol. 20, p. 202.

Lane, J. A.

J. A. Saxton, J. A. Lane, Wireless Eng.October (1952), p. 269.

Mahan, A. I.

Picht, J.

J. Picht, Ann. Phys. (Leipzig), 3, 433 (1929); M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 48; F. A. Jenkins, H. E. White, Physical Optics (McGraw-Hill, New York, 1937), pp. 395–398.

Saxton, J. A.

J. A. Saxton, J. A. Lane, Wireless Eng.October (1952), p. 269.

Schaefer, C.

C. Schaefer, Einfuhrung die theoretische Physik (W. de Gruyter, Berlin, 1949), Vol. 3, p. 414; M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 48.

Scheel, K.

H. Geiger, K. Scheel, in Handbuch der Physik, S. Flugge, Ed. (Julius Springer, Berlin, 1928), Vol. 20, p. 202.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1951), p. 1.

Uller, K.

K. Uller, Beitrage zur Theorie de electromagnetishe Strahlung, Dissertation Rostock (1903); J. Zenneck, Ann. Phys. (Leipzig), 23, 846 (1907).

Williams, R. H.

R. H. Williams, “Propagation between Conducting and Non-Conducting Media,” Engineering Experiment Station, U. of New Mexico, Albuquerque, Technical Report EE-50 (Sept.1961).

Ann. Phys. (Leipzig)

J. Picht, Ann. Phys. (Leipzig), 3, 433 (1929); M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 48; F. A. Jenkins, H. E. White, Physical Optics (McGraw-Hill, New York, 1937), pp. 395–398.

Appl. Opt.

Russ. J. Phys. Chem.

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

Wireless Eng.

J. A. Saxton, J. A. Lane, Wireless Eng.October (1952), p. 269.

Other

Ref. 6, p. 209.

Ref. 6, p. 200.

Ref. 7, p. 122.

K. Uller, Beitrage zur Theorie de electromagnetishe Strahlung, Dissertation Rostock (1903); J. Zenneck, Ann. Phys. (Leipzig), 23, 846 (1907).

C. Schaefer, Einfuhrung die theoretische Physik (W. de Gruyter, Berlin, 1949), Vol. 3, p. 414; M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 48.

R. H. Williams, “Propagation between Conducting and Non-Conducting Media,” Engineering Experiment Station, U. of New Mexico, Albuquerque, Technical Report EE-50 (Sept.1961).

H. Geiger, K. Scheel, in Handbuch der Physik, S. Flugge, Ed. (Julius Springer, Berlin, 1928), Vol. 20, p. 202.

M. Born, Optik (Julius Springer, Berlin, 1933), pp. 30–36.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1951), p. 1.

Ref. 6, p. 152.

Ref. 6, p. 194.

Ref. 7, p. 260.

Ref. 6, pp. 202, 204, and 207.

These equations will be found to differ from those of Williams.5

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Figures (16)

Fig. 1
Fig. 1

Reflection and refraction at an interface between absorbing and nonabsorbing media.

Fig. 2
Fig. 2

Reflection and refraction at an interface between absorbing and nonabsorbing media indicating waveforms.

Fig. 3
Fig. 3

Refractive index n2 for an ocean–air interface for a frequency of 100 Hz as a function of the angle of incidence.

Fig. 4
Fig. 4

Angle of refraction at an ocean–air interface for a frequency of 100 Hz as a function of the angle of incidence.

Fig. 5
Fig. 5

Absorption coefficient k2 for an ocean–air interface for a frequency of 100 Hz as a function of the angle of incidence.

Fig. 6
Fig. 6

Rs1 and Ds1 at an ocean–air interface for a frequency of 100 Hz as a function of the angle of incidence.

Fig. 7
Fig. 7

Phase changes δ s r and δ s t after reflection and transmission at an ocean–air interface for a frequency of 100 Hz as a function of the angle of incidence.

Fig. 8
Fig. 8

Components of Poynting vectors for s polarization at z = t = 0 and ϕ = 45° along an ocean–air interface for a frequency of 100 Hz.

Fig. 9
Fig. 9

Radiant flux flow lines along an ocean–air interface for a frequency of 100 Hz at = 45°.

Fig. 10
Fig. 10

Components of Poynting vectors for s polarization at z = 0, 0.2λ2, t = 0, and ϕ = 45° along an ocean–air interface for a frequency of 100 Hz.

Fig. 11
Fig. 11

Components of Poynting vectors for s polarization at z = 0, ωt = 0°, 90°, and ϕ = 45° along an ocean–air interface for a frequency of 100 Hz.

Fig. 12
Fig. 12

Time averages of components of Poynting vectors for s polarization at z = 0 and ϕ = 45° along an ocean–air interface for a frequency of 100 Hz.

Fig. 13
Fig. 13

Rp1 and Dp1 at an ocean–air interface for a frequency of 100 Hz as a function of the angle of incidence.

Fig. 14
Fig. 14

Phase changes δ p r and δ p t after reflection and transmission at an ocean–air interface for a frequency of 100 Hz as a function of the angle of incidence.

Fig. 15
Fig. 15

Components of Poynting vectors for p polarization at z = t = 0 and ϕ = 45° along an ocean–air interface for a frequency of 100 Hz.

Fig. 16
Fig. 16

Time averages of components of Poynting vectors for p polarization at z = 0 and ϕ = 45° along an ocean–air interface for a frequency of 100 Hz.

Equations (38)

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C u r l E = - 1 c H t ,             C u r l H = 4 π σ c E + c E t , div D = div H = 0 ,
E y i z = 1 c H x i t ,             E y i x = - 1 c H z i t , H x i z - H z i x = 4 π σ 1 c E y i + 1 c E y i t .
2 E y i x 2 + 2 E y i z 2 = 4 π σ 1 c 2 E y i t + 1 c 2 2 E y i t 2 .
E y i = E s exp | - ω c k 1 d i | exp ( i | ω t - w c n 1 d i | ) , H x i = - ( n 1 - i k 1 ) cos ϕ E s exp | - ω c k 1 d i | exp ( i | ω t - ω c n 1 d i | ) , H z i = ( n 1 - i k 1 ) sin ϕ E s exp | - ω c k 1 d i | exp ( i | ω t - ω c n 1 d i | ) , E y r = R s exp | - ω c k 1 d r | exp ( i | ω t - ω c n 1 d r | ) , H x r = ( n 1 - i k 1 ) cos ϕ R s exp | - ω c k 1 d r | exp ( i | ω t - ω c n 1 d r | ) , H z r = ( n 1 - i k 1 ) sin ϕ R s exp | - ω c k 1 d r | exp ( i | ω t - ω c n 1 d r | ) .
d i = x sin ϕ + z cos ϕ ,             d r = x sin ϕ - z cos ϕ ,
n 1 = { 1 + [ 1 2 + ( 16 π 2 σ 1 2 / ω 2 ) ] 1 / 2 2 } 1 / 2 , k 1 = { - 1 + [ 1 2 + ( 16 π 2 σ 1 2 / ω 2 ) ] 1 / 2 2 } 1 / 2 .
E y i ( O ) = E s exp ( i ω t ) , E y i ( A ) = E s exp | - ω c k 1 x 1 sin ϕ | exp ( i | ω t - ω c n 1 x 1 sin ϕ | ) .
E y t ( G ) = D s exp [ i | ω t - ω c ( x 2 sin ψ + z 2 cos ψ | ] , E y t ( F ) = D s exp | - ω c k 2 d a t | exp ( i | ω t - ω c n 2 d p t | ) ,
d a t = x cos ψ - z sin ψ ,             d p t = x sin ψ + z cos ψ .
k 2 = ( k 1 sin ϕ ) / cos ψ .
n 2 = ( 2 + k 2 2 ) 1 / 2 ,
H x t = - ( n 2 cos ψ + i k 2 sin ψ ) D s exp | - ω c k 2 d a t | × exp ( i | ω t - ω c n 2 d p t | ) , H z t = ( n 2 sin ψ - i k 2 cos ψ ) D s exp | - ω c k 2 d a t | × exp ( i | ω t - ω c n 2 d p t | ) .
E y i + E y r = E y t ,             H x i + H x r = H x t ,
n 2 sin ψ = n 1 sin ϕ ,             k 2 = ( k 1 sin ϕ ) / cos ψ .
n 2 2 = ½ { ( n 1 2 + k 1 2 ) sin 2 ϕ + n 20 2 + [ ( n 1 2 + k 1 2 ) sin 2 ϕ + n 20 2 2 - 4 n 1 2 n 20 2 sin 2 ϕ ] 1 / 2 } . k 2 2 = n 2 2 k 1 2 sin 2 ϕ n 2 2 - n 1 2 sin 2 ϕ .
n 20 2 = 2
n 2 = n 20             and             k 2 = 0 ,
E s + R s = D s , - ( n 1 - i k 1 ) cos ϕ E s + ( n 1 - i k 1 ) cos ϕ R s = - ( n 2 cos ψ + i k 2 sin ψ ) D s .
R s = ( n 1 - i k 1 ) cos ϕ - ( n 2 cos ψ + i k 2 sin ψ ) ( n 1 - i k 1 ) cos ϕ + ( n 2 cos ψ + i k 2 sin ψ ) E s , D s = 2 ( n 1 - i k 1 ) cos ϕ ( n 1 - i k 1 ) cos ϕ + ( n 2 cos ψ + i k 2 sin ψ ) E s .
R s = [ ( n 1 cos ϕ - n 2 cos ψ ) 2 + ( k 1 cos ϕ + k 2 sin ψ ) 2 ( n 1 cos ϕ + n 2 cos ψ ) 2 + ( k 1 cos ϕ - k 2 sin ψ ) 2 ] 1 / 2 E s exp ( - i δ s r ) , D s = 2 ( n 1 2 + k 1 2 ) 1 / 2 cos ϕ ( n 1 cos ϕ + n 2 cos ϕ ) 2 + ( k 1 cos ϕ - k 2 sin ψ ) 2 1 / 2 E s exp ( - i δ s t ) ,
tan δ s r = 2 cos ϕ ( n 1 k 2 sin ψ + n 2 k 1 cos ψ ) ( n 1 2 + k 1 2 ) cos 2 ϕ - n 2 2 cos 2 ψ - k 2 2 sin 2 ψ , tan δ s t = n 1 k 2 sin ψ + n 2 k 1 cos ψ ( n 1 2 + k 1 2 ) cos ϕ + n 1 n 2 cos ψ - k 1 k 2 sin ψ .
E y i = E s exp | - ω c k 1 d i | cos θ i , H x i = - ( n 1 2 + k 1 2 ) 1 / 2 cos ϕ E s exp | - w c k 1 d i | cos ( θ i - Δ 1 ) , H z i = ( n 1 2 + k 1 2 ) 1 / 2 sin ϕ E s exp | - ω c k 1 d i | cos ( θ i - Δ 1 ) , E y r = R s 1 exp | - ω c k 1 d r | cos θ r s , H x r = ( n 1 2 + k 1 2 ) 1 / 2 cos ϕ R s 1 exp | - ω c k 1 d r | cos ( θ r s - Δ 1 ) , H z r = ( n 1 2 + k 1 2 ) 1 / 2 sin ϕ R s 1 exp | - ω c k 1 d r | cos ( θ r s - Δ 1 ) , E y t = D s 1 exp | - ω c k 2 d a t | cos θ t s , H x t = - ( n 2 2 cos 2 ψ + k 2 2 sin 2 ψ ) 1 / 2 D s 1 exp | - ω c k 2 d a t | cos ( θ t s + Δ 2 ) , H z t = ( n 2 2 sin 2 ψ + k 2 2 cos 2 ψ ) 1 / 2 D s 1 exp | - ω c k 2 d a t | cos ( θ t s - Δ 1 ) ,
θ i = ω t - ω c n 1 d i ,             θ r s = ω t - ω c n 1 d r - δ s r , θ t s = ω t - ω c n 2 d p t - δ s t ,
tan Δ 1 = k 1 / n 1 ,             tan Δ 2 = n 1 k 2 2 / n 2 2 k 1 .
P x i ( s ) = c 4 π ( n 1 2 + k 1 2 ) 1 / 2 sin ϕ E s 2 exp | - 2 ω c k 1 d i | × cos θ i cos ( θ i - Δ 1 ) , P z i ( s ) = c 4 π ( n 1 2 + k 1 2 ) 1 / 2 cos ϕ E s 2 exp | - 2 ω c k 1 d i | × cos θ i cos ( θ i - Δ 1 ) , P x r ( s ) = c 4 π ( n 1 2 + k 1 2 ) 1 / 2 sin ϕ R s 1 2 exp | - 2 ω c k 1 d r | × cos θ r s cos ( θ r s - Δ 1 ) , P z r ( s ) = - c 4 π ( n 1 2 + k 1 2 ) 1 / 2 cos ϕ R s 1 2 exp | - 2 ω c k 1 d r | × cos θ r s cos ( θ r s - Δ 1 ) , P x t ( s ) = c 4 π ( n 1 2 + k 1 2 ) 1 / 2 sin ϕ D s 1 2 exp | - 2 ω c k 2 d a t | × cos θ t s cos ( θ t s - Δ 1 ) , P z t ( s ) = c 4 π ( n 2 2 - n 20 2 n 1 2 n 2 2 sin 2 ϕ ) 1 / 2 D s 1 2 exp | - 2 ω c k 2 d a t | × cos θ t s cos ( θ t s + Δ 2 ) ,
| d z d x | i = | d z d z | t = P z i ( s ) P x i ( s ) = cot ϕ ,             | d z d x | r = P z r ( s ) P x r ( s ) = - cot ϕ , P z t ( s ) P x t ( s ) = ( n 2 4 - n 20 2 n 1 2 sin 2 ϕ ) 1 / 2 n 2 ( n 1 2 + k 1 2 ) 1 / 2 sin ϕ cos ( θ t s + Δ 2 ) cos ( θ t s - Δ 1 )
R s 1 = n 1 cos ϕ - n 20 cos ψ n 1 cos ϕ + n 20 cos ψ E s D s 1 = 2 n 1 cos ϕ n 1 cos ϕ + n 20 cos ψ δ s r = δ s t = 0 , E s ,
R s 1 = ( n 1 2 + k 1 2 + n 20 2 - 2 n 1 n 20 n 1 2 + k 1 2 + n 20 2 + 2 n 1 n 20 ) 1 / 2 E s , D s 1 = 2 ( n 1 2 + k 1 2 ) 1 / 2 ( n 1 2 + k 1 2 + n 20 2 - 2 n 1 n 20 ) 1 / 2 E s , tan δ s r = 2 n 20 k 1 n 1 2 + k 1 2 - n 20 2 , tan δ s t = n 20 k 1 n 1 2 + k 1 2 + n 1 n 20 .
Δ x = ω t 2 π sin ψ λ 2 = V 2 sin ψ t = 0.3421 λ 2 .
P x i ( s ) = c 4 π n 1 2 sin ϕ E s 2 exp | - 2 ω c k 1 d i | , P z i ( s ) = c 4 π n 1 2 cos ϕ E s 2 exp | - 2 ω c k 1 d i | , P x r ( s ) = c 4 π n 1 2 sin ϕ R s 1 2 exp | - 2 ω c k 1 d r | , P z r ( s ) = - c 4 π n 1 2 cos ϕ R s 1 2 exp | - 2 ω c k 1 d r | , P x t ( s ) = c 4 π n 1 2 sin ϕ D s 1 2 exp | - 2 ω c k 2 d a t | , P z t ( s ) = c 4 π n 2 2 k 1 ( n 2 2 - n 20 2 n 1 2 n 2 2 sin 2 ϕ ) 1 / 2 2 ( n 1 2 k 2 4 + n 2 4 k 1 2 ) 1 / 2 × D s 1 2 exp | - 2 ω c k 2 d a t | .
H y i z = - 4 π σ 1 c E x i - 1 c E x i t , H y 1 x = 4 π σ 1 c E z i + 1 c E z i t , E x i x - E z i x = - 1 c H y i t .
H y i = ( n 1 - i k 1 ) E p exp | - ω c k 1 d i | exp ( i | ω t - ω c n 1 d i | ) , E x i = cos ϕ E p exp | - ω c k 1 d i | exp ( i | ω t - ω c n 1 d i | ) , E z i = - sin ϕ E p exp | - ω c k 1 d i | exp ( i | ω t - ω c n 1 d i | ) , H y r = ( n 1 - i k 1 ) R p exp | - ω c k 1 d r | exp ( i | ω t - ω c n 1 d r | ) , E x r = - cos ϕ R p exp | - ω c k 1 d r | exp ( i | ω t - ω c n 1 d r | ) , E z r = - sin ϕ R p exp | - ω c k 1 d r | exp ( i | ω t - ω c n 1 d r | ) , H y t = n 20 2 ( n 2 2 - k 2 2 ) 1 / 2 D p exp | - ω c k 2 d a t | exp ( i | ω t - ω c n 2 d p t | ) , E x t = ( n 2 cos ψ + i k 2 sin ψ ) ( n 2 2 - k 2 2 ) 1 / 2 D p exp | - ω c k 2 d a t | × exp ( i | ω t - ω c n 2 d p t | ) , E z t = - ( n 2 sin ψ - i k 2 cos ψ ) ( n 2 2 - k 2 2 ) 1 / 2 D p exp | - ω c k 2 d a t | × exp ( i | ω t - ω c n 2 d p t | ) .
R p = n 20 2 cos ϕ - ( n 1 - i k 1 ) ( n 2 cos ψ + i k 2 sin ψ ) n 20 2 cos ϕ + ( n 1 - i k 1 ) ( n 2 cos ψ + i k 2 sin ψ ) E p , D p = 2 ( n 1 - i k 1 ) ( n 2 2 - k 2 2 ) 1 / 2 cos ϕ n 20 2 cos ϕ + ( n 1 - i k 1 ) ( n 2 cos ψ + i k 2 sin ψ ) E p .
R p = [ ( n 20 2 cos ϕ - n 1 n 2 cos ψ - k 1 k 2 sin ψ ) 2 + ( n 1 k 2 sin ψ - n 2 k 1 cos ψ ) 2 ( n 20 2 cos ϕ + n 1 n 2 cos ψ + k 1 k 2 sin ψ ) 2 + ( n 1 k 2 sin ψ - n 2 k 1 cos ψ ) 2 ] 1 / 2 E p exp ( - i δ p r ) , D p = 2 ( n 1 2 + k 1 2 ) 1 / 2 ( n 2 2 - k 2 2 ) cos ϕ [ ( n 20 2 cos ϕ + n 1 n 2 cos ψ + k 1 k 2 sin ψ ) 2 + ( n 1 k 2 sin ψ - n 2 k 1 cos ψ ) 2 ] 1 / 2 E p exp ( - i δ p t ) ,
tan δ p r = 2 n 20 2 cos ϕ ( n 1 k 2 sin ψ - n 2 k 1 cos ψ ) n 20 4 cos 2 ϕ - ( n 1 2 + k 1 2 ) ( n 2 2 cos 2 ψ + k 2 2 sin 2 ψ ) , tan δ p t = n 20 2 k 1 cos ϕ + k 2 ( n 1 2 + k 1 2 ) sin ψ n 1 n 20 2 cos ϕ + n 2 ( n 1 2 + k 1 2 ) cos ψ .
P x i ( p ) = c 4 π ( n 1 2 + k 1 2 ) 1 / 2 sin ϕ E p 2 exp | - 2 ω c k 1 d i | cos θ i cos ( θ i - Δ 1 ) , P z i ( p ) = c 4 π ( n 1 2 + k 1 2 ) 1 / 2 cos ϕ E p 2 exp | - 2 ω c k 1 d i | cos θ i cos ( θ i - Δ 1 ) , P x r ( p ) = c 4 π ( n 1 2 + k 1 2 ) 1 / 2 sin ϕ R p 1 2 exp | - 2 ω c k 1 d r | × cos θ r p cos ( θ r p - Δ 1 ) , P z r ( p ) = - c 4 π ( n 1 2 + k 1 2 ) 1 / 2 cos ϕ R p 1 2 exp | - 2 ω c k 1 d r | × cos θ r p cos ( θ r p - Δ 1 ) , P x t ( p ) = c 4 π n 20 2 ( n 1 2 + k 1 2 ) 1 / 2 n 2 2 - k 2 2 sin ϕ D p 1 2 exp | - 2 ω c k 2 d a t | × cos θ t p cos ( θ t p - Δ 1 ) , P z t ( p ) = c 4 π n 20 2 ( n 2 2 - n 20 2 n 1 2 n 2 2 sin 2 ϕ ) 1 / 2 n 2 2 - k 2 2 D p 1 2 exp | - 2 ω c k 2 d a t | × cos θ t p cos ( θ t p + Δ 2 ) .
Δ x = λ 2 2 π sin ψ ( δ p t - δ s t ) .
P x i ( p ) = c 4 π n 1 2 sin ϕ E p 2 exp | - 2 ω c k 1 d i | , P z i ( p ) = c 4 π n 1 2 cos ϕ E p 2 exp | - 2 ω c k 1 d i | , P x r ( p ) = c 4 π n 1 2 sin ϕ R p 1 2 exp | - 2 ω c k 1 d r | , P z r ( p ) = - c 4 π n 1 2 cos ϕ R p 1 2 exp | - 2 ω c k 1 d r | , P x t ( p ) = c 4 π n 20 2 n 1 2 ( n 2 2 - k 2 2 ) sin ϕ D p 1 2 exp | - 2 ω c k 2 d a t | , P z t ( p ) = c 4 π n 20 2 n 2 2 k 1 ( n 2 2 - n 20 2 n 2 1 n 2 2 sin 2 ϕ ) 1 / 2 2 ( n 2 2 - k 2 2 ) ( n 2 4 k 1 2 + n 1 2 k 2 2 ) 1 / 2 × D p 1 2 exp | - 2 ω c k 2 d a t | .

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