Abstract

The problem of electromagnetic wave interaction with a stratified dielectric medium is solved in the framework of molecular optics, leading to a new derivation for the refraction and reflection laws at stratified media interfaces. In addition, the solution confirms the Lorentz–Lorenz refractive index formula characterizing the transverse propagation modes and the characteristic frequencies associated with the longitudinal mode. The analytic results presented in this study support the existence of another longitudinal mode propagating with the vacuum wave number within the medium, and they provide a new concept for the Ewald–Oseen extinction theorem. Other new results of this study are that (1) the relation between Fresnel horizontal and vertical reflection coefficients and the relation between the corresponding transmission coefficients are revealed and (2) a new concept is presented for the Brewster angle, and (3) the concept is introduced of multiple reflections in formulating the reflection coefficient at the interface between two different dielectric materials.

© 1996 Optical Society of America

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References

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  1. J. R. Wait, Electromagnetic Waves in Stratified Media, 2nd ed. (Pergamon, Oxford, 1970), Chap.1.
  2. L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985), Secs. 2.1, 2.2, 3.1.
  3. M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1975), Secs. 1.5, 2.4.
  4. L. Rosenfeld, Theory of Electrons (North-Holland, Amsterdam, 1951), Chap. VI.
  5. J. Van Kranendonk, J. E. Sipe, “Foundation of the macroscopic electromagnetic theory of dielectric media,”in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1977), Vol. XV.
    [CrossRef]
  6. B. Crosignani, P. Di Porto, M. Bertolotti, Statistical Properties of Scattered Light (Academic, New York, 1975), Sec. I.5.
  7. J. De Goede, P. Mazur, “On the extinction theorem in electrodynamics,” Physica (Utrecht) 58, 569–584 (1972).
    [CrossRef]
  8. P. P. Ewald, “Zur Begrundung der Kristalloptik,” Ann. Phys. (Leipzig) 49, 1–39 (1915).
  9. C. W. Oseen, “Uber die Wechselwirkung zwischen elektrischen Dipolen und uber Drehung der Polarizationsebene in Kristallen und Flussigheiten,” Ann. Phys. (Leipzig) 48, 1–56 (1915).
  10. E. Lalor, E. Wolf, “Exact solution of the equations of molecular optics for refraction and reflection of an electromagnetic wave on a semi-infinite dielectric,” J. Opt. Soc. Am. 62, 1165–1174 (1972).
    [CrossRef]
  11. G. C. Reali, “Reflection from dielectric materials,” Am. J. Phys. 50, 1133–1136 (1982).
    [CrossRef]
  12. G. C. Reali, “Exact solution of the equations of molecular optics for refraction and reflection of an electromagnetic wave on a semi-infinite dielectric,” J. Opt. Soc. Am. 72, 1121–1124 (1982).
    [CrossRef]
  13. G. C. Reali, “Reflection, refraction, and transmission of plane electromagnetic waves from a lossless dielectric slab,” Am. J. Phys. 60, 532–536 (1992).
    [CrossRef]
  14. F. Hynne, “An extinction theorem for optical scattering,” Mol. Phys. 41, 583–603 (1980).
    [CrossRef]
  15. R. K. Bullough, “Many-body optics. I. Dielectric constants and optical dispersion relations,” J. Phys. A 1, 409–430 (1968).
    [CrossRef]
  16. R. K. Bullough, “Many-body optics. III. The optical extinction theorem and ∊l(κω),” J. Phys. A 3, 708–725 (1970).
    [CrossRef]
  17. R. K. Bullough, “Many-body optics. IV. The total transverse response and ∊t(k, ω),” J. Phys. A 3, 726–750 (1970).
    [CrossRef]

1992 (1)

G. C. Reali, “Reflection, refraction, and transmission of plane electromagnetic waves from a lossless dielectric slab,” Am. J. Phys. 60, 532–536 (1992).
[CrossRef]

1982 (2)

1980 (1)

F. Hynne, “An extinction theorem for optical scattering,” Mol. Phys. 41, 583–603 (1980).
[CrossRef]

1972 (2)

1970 (2)

R. K. Bullough, “Many-body optics. III. The optical extinction theorem and ∊l(κω),” J. Phys. A 3, 708–725 (1970).
[CrossRef]

R. K. Bullough, “Many-body optics. IV. The total transverse response and ∊t(k, ω),” J. Phys. A 3, 726–750 (1970).
[CrossRef]

1968 (1)

R. K. Bullough, “Many-body optics. I. Dielectric constants and optical dispersion relations,” J. Phys. A 1, 409–430 (1968).
[CrossRef]

1915 (2)

P. P. Ewald, “Zur Begrundung der Kristalloptik,” Ann. Phys. (Leipzig) 49, 1–39 (1915).

C. W. Oseen, “Uber die Wechselwirkung zwischen elektrischen Dipolen und uber Drehung der Polarizationsebene in Kristallen und Flussigheiten,” Ann. Phys. (Leipzig) 48, 1–56 (1915).

Bertolotti, M.

B. Crosignani, P. Di Porto, M. Bertolotti, Statistical Properties of Scattered Light (Academic, New York, 1975), Sec. I.5.

Born, M.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1975), Secs. 1.5, 2.4.

Bullough, R. K.

R. K. Bullough, “Many-body optics. III. The optical extinction theorem and ∊l(κω),” J. Phys. A 3, 708–725 (1970).
[CrossRef]

R. K. Bullough, “Many-body optics. IV. The total transverse response and ∊t(k, ω),” J. Phys. A 3, 726–750 (1970).
[CrossRef]

R. K. Bullough, “Many-body optics. I. Dielectric constants and optical dispersion relations,” J. Phys. A 1, 409–430 (1968).
[CrossRef]

Crosignani, B.

B. Crosignani, P. Di Porto, M. Bertolotti, Statistical Properties of Scattered Light (Academic, New York, 1975), Sec. I.5.

De Goede, J.

J. De Goede, P. Mazur, “On the extinction theorem in electrodynamics,” Physica (Utrecht) 58, 569–584 (1972).
[CrossRef]

Di Porto, P.

B. Crosignani, P. Di Porto, M. Bertolotti, Statistical Properties of Scattered Light (Academic, New York, 1975), Sec. I.5.

Ewald, P. P.

P. P. Ewald, “Zur Begrundung der Kristalloptik,” Ann. Phys. (Leipzig) 49, 1–39 (1915).

Hynne, F.

F. Hynne, “An extinction theorem for optical scattering,” Mol. Phys. 41, 583–603 (1980).
[CrossRef]

Kong, J. A.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985), Secs. 2.1, 2.2, 3.1.

Lalor, E.

Mazur, P.

J. De Goede, P. Mazur, “On the extinction theorem in electrodynamics,” Physica (Utrecht) 58, 569–584 (1972).
[CrossRef]

Oseen, C. W.

C. W. Oseen, “Uber die Wechselwirkung zwischen elektrischen Dipolen und uber Drehung der Polarizationsebene in Kristallen und Flussigheiten,” Ann. Phys. (Leipzig) 48, 1–56 (1915).

Reali, G. C.

G. C. Reali, “Reflection, refraction, and transmission of plane electromagnetic waves from a lossless dielectric slab,” Am. J. Phys. 60, 532–536 (1992).
[CrossRef]

G. C. Reali, “Reflection from dielectric materials,” Am. J. Phys. 50, 1133–1136 (1982).
[CrossRef]

G. C. Reali, “Exact solution of the equations of molecular optics for refraction and reflection of an electromagnetic wave on a semi-infinite dielectric,” J. Opt. Soc. Am. 72, 1121–1124 (1982).
[CrossRef]

Rosenfeld, L.

L. Rosenfeld, Theory of Electrons (North-Holland, Amsterdam, 1951), Chap. VI.

Shin, R. T.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985), Secs. 2.1, 2.2, 3.1.

Sipe, J. E.

J. Van Kranendonk, J. E. Sipe, “Foundation of the macroscopic electromagnetic theory of dielectric media,”in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1977), Vol. XV.
[CrossRef]

Tsang, L.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985), Secs. 2.1, 2.2, 3.1.

Van Kranendonk, J.

J. Van Kranendonk, J. E. Sipe, “Foundation of the macroscopic electromagnetic theory of dielectric media,”in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1977), Vol. XV.
[CrossRef]

Wait, J. R.

J. R. Wait, Electromagnetic Waves in Stratified Media, 2nd ed. (Pergamon, Oxford, 1970), Chap.1.

Wolf, E.

Am. J. Phys. (2)

G. C. Reali, “Reflection from dielectric materials,” Am. J. Phys. 50, 1133–1136 (1982).
[CrossRef]

G. C. Reali, “Reflection, refraction, and transmission of plane electromagnetic waves from a lossless dielectric slab,” Am. J. Phys. 60, 532–536 (1992).
[CrossRef]

Ann. Phys. (Leipzig) (2)

P. P. Ewald, “Zur Begrundung der Kristalloptik,” Ann. Phys. (Leipzig) 49, 1–39 (1915).

C. W. Oseen, “Uber die Wechselwirkung zwischen elektrischen Dipolen und uber Drehung der Polarizationsebene in Kristallen und Flussigheiten,” Ann. Phys. (Leipzig) 48, 1–56 (1915).

J. Opt. Soc. Am. (2)

J. Phys. A (3)

R. K. Bullough, “Many-body optics. I. Dielectric constants and optical dispersion relations,” J. Phys. A 1, 409–430 (1968).
[CrossRef]

R. K. Bullough, “Many-body optics. III. The optical extinction theorem and ∊l(κω),” J. Phys. A 3, 708–725 (1970).
[CrossRef]

R. K. Bullough, “Many-body optics. IV. The total transverse response and ∊t(k, ω),” J. Phys. A 3, 726–750 (1970).
[CrossRef]

Mol. Phys. (1)

F. Hynne, “An extinction theorem for optical scattering,” Mol. Phys. 41, 583–603 (1980).
[CrossRef]

Physica (Utrecht) (1)

J. De Goede, P. Mazur, “On the extinction theorem in electrodynamics,” Physica (Utrecht) 58, 569–584 (1972).
[CrossRef]

Other (6)

J. R. Wait, Electromagnetic Waves in Stratified Media, 2nd ed. (Pergamon, Oxford, 1970), Chap.1.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985), Secs. 2.1, 2.2, 3.1.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1975), Secs. 1.5, 2.4.

L. Rosenfeld, Theory of Electrons (North-Holland, Amsterdam, 1951), Chap. VI.

J. Van Kranendonk, J. E. Sipe, “Foundation of the macroscopic electromagnetic theory of dielectric media,”in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1977), Vol. XV.
[CrossRef]

B. Crosignani, P. Di Porto, M. Bertolotti, Statistical Properties of Scattered Light (Academic, New York, 1975), Sec. I.5.

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Figures (2)

Fig. 1
Fig. 1

Problem configuration.

Fig. 2
Fig. 2

Geometry of a two-layered medium.

Equations (84)

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P ( r ) = ρ α [ E ex ( r ) + V σ ( r ) G ¯ ( r , r ) · P ( r ) d r ] ,
G ¯ ( r , r ) = k 2 ( Ī + k 2 ) g ( r , r ) = k 2 ( Ī + k 2 ) exp ( j k | r r | ) | r r | ,
P ( r ) = ρ α 1 4 π ρ α / 3 E ( r ) = η E ( r ) .
E i ( r ) = E i exp ( j k i · r ) = ( Ī k i k i k 2 ) · E i exp ( j k i · r ) = ( E υ i υ ̂ i + E h i ĥ i ) exp ( j k i · r ) ,
k i = k x i x ̂ + k y i y ̂ k z i z ̂ = k { sin θ i [ ( cos ϕ i ) x ̂ + ( sin ϕ i ) y ̂ ] ( cos θ i ) z ̂ } , ĥ i = z ̂ × k i | z ̂ × k i | = k y i x ̂ + k x i y ̂ k ρ = ( sin ϕ i ) x ̂ + ( cos ϕ i ) y ̂ , υ ̂ i = ĥ i × k i | ĥ i × k i | = k z i [ ( cos ϕ i ) x ̂ + ( sin ϕ i ) y ̂ ] k ρ z ̂ k , k ρ = k sin θ i .
E ex ( r ) = E i ( r ) + m n V m G ¯ ( r , r ) · P m ( r ) d r ,
P n ( r ) = ρ n α n [ E i ( r ) + V n σ ( r ) G ¯ ( r , r ) · P n ( r ) d r + m n V m G ¯ ( r , r ) · P m ( r ) d r ] .
P n ( r ) = ρ n α n [ E i ( r ) + 4 π 3 P n ( r ) + m = 1 N + 1 V m G ¯ ( r , r ) · P m ( r ) d r ] .
P n ( r ) = [ P n f exp ( j γ n z ) + P n b exp ( j β n z ) ] × exp [ j ( k x x + k y y ) ] = P n f exp ( j k n f · r ) + P n b exp ( j k n b · r ) , k n f = k x x ̂ + k y y ̂ γ n z ̂ ; k n b = k x x ̂ + k y y ̂ β n z ̂ ;
G ¯ ( r , r ) = 4 π δ ( r r ) z ̂ z ̂ + k 2 j π d q x d q y × ( Ī + k 2 ) × exp { j [ q x ( x x ) + q y ( y y ) ] } 2 λ × exp ( j λ | z z | ) ,
d r = d x d y d z , 1 2 π d x exp [ j ( k x q x ) x ] = δ ( k x q x ) , 1 2 π d y exp [ j ( k y q y ) y ] = δ ( k y q y ) ,
Ā n · [ P n f exp ( j γ n z ) + P n b exp ( j β n z ) ] × exp [ j ( k x x + k y y ) ] = α n ρ n E i exp [ j ( k x i x + k y i y k z i z ) ] + 2 π ρ n α n k 2 j k z m = 1 N d m d m 1 d z ( Ī + k 2 ) × exp [ j ( k x x + k y y + k z | z z | ) ] × [ P n f exp ( j γ n z ) + P n b exp ( j β n z ) ] ,
Ā n = Ī 4 π 3 ρ n α n Ī + 4 π ρ n α n z ̂ z ̂ ,
Ā n · [ P n f exp ( j γ n z ) + P n b exp ( j β n z ) ] = α n ρ n E i exp ( j k z i z ) + 2 π ρ n α n k 2 j k z i exp ( j k z i z ) × ( I + k r k r k 2 ) { d n z d z exp ( j k z i z ) × [ P n f exp ( j γ n z ) + P n b exp ( j β n z ) ] + m = n + 1 N d m d m 1 d z exp ( j k z i z ) [ P n f exp ( j γ n z ) + P n f exp ( j β n z ) ] } + 2 π ρ n α n k 2 j k z i ( I k i k i k 2 ) × exp ( j k z i z ) { z d n 1 d z exp ( j k z i z ) × [ P n f exp ( j γ n z ) + P n b exp ( j β n z ) ] + m = 1 n 1 d m d m 1 d z exp ( j k z i z ) [ P n f exp ( j γ n z ) + P n b exp ( j β n z ) ] } ,
k r = k x i x ̂ + k y i y ̂ + k z i z ̂ .
Ā n · [ P n f exp ( j γ n z ) + P n b exp ( j β n z ) ] = C i exp ( j k z i z ) + C r exp ( j k z i z ) + C ¯ n f · P n f exp ( j γ n z ) + C n b · P n b exp ( j β n z ) ,
C i = ρ n α n E i + 2 π k 2 ρ n α n k z i ( Ī k i k i k 2 ) · ( P n f k z i γ n exp [ j ( k z i γ n ) d n 1 ] + P n b k z i + β n exp [ j ( k z i β n ) d n 1 ] + m = 1 n 1 P m f k z i γ m { exp [ j ( k z i γ m ) d m 1 ] exp [ j ( k z i γ m ) d m ] } P m b k z i + β m { exp [ j ( k z i + β m ) d m ] exp [ j ( k z i + β m ) d m 1 ] } ) ,
C r = 2 π k 2 ρ n α n k z i ( Ī k r k r k 2 ) · ( P n f k z i + γ n exp [ j ( k z i + γ n ) d n ] + P n b k z i β n exp [ j ( k z i β n ) d n ] + m = n + 1 N P m f k z i + γ m { exp [ j ( k z i + γ m ) d m ] exp [ j ( k z i + γ m ) d m 1 ] } + P m b k z i β m × { exp [ j ( k z i β m ) d m exp [ j ( k z i β m ) d m 1 ] } )
C ¯ n f = 2 π k 2 ρ n α n k z i ( Ī k r k r / k 2 k z i + γ n + Ī k i k i / k 2 γ n k z i ) ,
C ¯ n b = 2 π k 2 ρ n α n k z i ( Ī k r k r / k 2 k z i β n + Ī k i k i / k 2 k z i + γ n ) .
[ Ā n + 2 π k 2 ρ n α n k z i ( Ī k r k r / k 2 k z i + γ n + I k i k i / k 2 k z i γ n ) ] · P n f = 0 ,
[ Ā n + 2 π k 2 ρ n α n k z i ( Ī k r k r / k 2 k z i β n + Ī k i k i / k 2 k z i + β n ) ] · P n b = 0 ,
E i + 2 π k 2 k z i ( Ī k i k i k 2 ) · { m = 1 n exp ( j k z i d m 1 ) × [ P m f k z i γ m exp ( j γ m d m 1 ) + P m b k z i + β m exp ( j β m d m 1 ) ] m = 1 n 1 exp ( j k z i d m ) × [ P m f k z i + γ m exp ( j γ m d m ) + P m b k z i + β m exp ( j β m d m ) ] } = 0 ,
( Ī k r k r k 2 ) · { m = n N exp ( j k z i d m ) [ P m f k z i + γ m × exp ( j γ m d m ) + P m b k z i β m exp ( j β m d m ) ] m = n + 1 N + 1 exp ( j k z i d m 1 ) [ P m f k z i + γ m × exp ( j γ m d m 1 ) + P m b k z i β m × exp ( j β m d m 1 ) ] } = 0 .
{ ( γ n 2 k z i 2 ) Ā n + 2 π ρ n α n [ γ n k z i ( k i k i k r k r ) + ( k i k i + k r k r ) k 2 Ī ] } · P n f = 0 .
k i = k n f + ( γ n k z i ) z ̂ , k r = k n f + ( γ n + k z i ) z ̂ .
k i k i = k n f k n f + ( γ n k z i ) 2 z ̂ z ̂ + ( γ n k z i ) × ( z ̂ k n f + k n f z ̂ ) , k r k r = k n f k n f + ( γ n + k z i ) 2 z ̂ z ̂ + ( γ n + k z i ) × ( z ̂ k n f + k n f z ̂ ) .
{ ( γ n 2 k z i 2 ) Ā n + 4 π ρ n α n [ k n f k n f ( γ n 2 k z i 2 ) z ̂ z ̂ k 2 Ī ] } · P n f = 0 .
[ ( k n f 2 k 2 ) ( I 4 π ρ n α n 3 ) Ī + 4 π ρ n α n ( k n f k n f k 2 Ī ) ] · P n f = 0 .
P n f = P nfh ĥ n f + P n f υ υ ̂ n f + P nfk k ̂ n f ,
[ ( k n f 2 k 2 ) ( 1 4 π ρ n α n 3 ) 4 π ρ n α n k 2 ] P nfh = 0 ,
[ ( k n f 2 k 2 ) ( 1 4 π ρ n α n 3 ) 4 π ρ n α n k 2 ] P n f υ = 0 ,
[ ( k n f 2 k 2 ) ( 1 + 8 π ρ n α n 3 ) ] P nfk = 0 .
( k n f 2 k 2 ) ( 1 4 π ρ n α n 3 ) 4 π ρ n α n k 2 = 0 .
1 + 8 π ρ n α n 3 = 0 ,
k n f = k .
( υ ̂ i υ ̂ i + ĥ i ĥ i ) · ( k z i 2 π k 2 E i + P 1 f k z i γ 1 + P 1 b k z i + γ 1 ) = 0 ,
( υ ̂ r υ ̂ r + ĥ r ĥ r ) · [ P 1 f exp ( j γ 1 d 1 ) k z i + γ 1 + P 1 b exp ( j γ 1 d 1 ) k z i γ 1 P 2 f exp ( j γ 2 d 1 ) k z i + γ 2 ] = 0 ,
( υ ̂ i υ ̂ i + ĥ i ĥ i ) · [ P 1 f exp ( j γ 1 d 1 ) k z i γ 1 + P 1 b exp ( j γ 1 d 1 ) k z i + γ 1 P 2 f exp ( j γ 2 d 1 ) k z i γ 2 ] = 0 .
P n b = P nbh ĥ n b + P n b υ υ ̂ n b + P nbk k ̂ n b ,
k z i 2 π k 2 E h i + P 1 f h k z i γ 1 + P 1 b h k z i + γ 1 = 0 ,
P 1 f h exp ( j γ 1 d 1 ) k z i + γ 1 + P 1 b h exp ( j γ 1 d 1 ) k z i γ 1 P 2 f h exp ( j γ 2 d 1 ) k z i + γ 2 = 0 ,
P 1 f h exp ( j γ 1 d 1 ) k z i γ 1 + P 1 b h exp ( j γ 1 d 1 ) k z i + γ 1 P 2 f h exp ( j γ 2 d 1 ) k z i γ 2 = 0 .
P 1 b h = R 12 exp ( 2 j γ 1 d 1 ) P 1 f h = R 01 R 02 R 01 R 02 1 exp ( 2 j γ 1 d 1 ) P 1 f h ,
R 0 n = k z i γ n k z i + γ n , n = 1 , 2 .
R 12 = γ 1 γ 2 γ 1 + γ 2 ,
P 1 f h = η 1 X 01 1 + R 01 R 12 exp ( 2 j γ 1 d 1 ) E h i ,
X 01 = 2 k z i k z i + γ 1 .
P 1 b h = η 1 X 01 R 12 exp ( 2 j γ 1 d 1 ) 1 + R 01 R 12 exp ( 2 j γ 1 d 1 ) E h i ,
P 2 f h = η 2 X 01 X 12 exp ( 2 j γ 1 d 1 ) 1 + R 01 R 12 exp ( 2 j γ 1 d 1 ) E h i ,
k z i 2 π k 2 E υ i + υ ̂ i · υ ̂ 1 f k z i γ 1 P 1 f υ + υ ̂ i · υ ̂ 1 b k z i + γ 1 P 1 b υ = 0 ,
( υ ̂ r · υ ̂ 1 f ) P 1 f υ exp ( j γ 1 d 1 ) k z i + γ 1 + ( υ ̂ r · υ ̂ 1 b ) P 1 b υ exp ( j γ 1 d 1 ) k z i γ 1 ( υ ̂ r · υ ̂ 2 f ) P 2 f υ exp ( j γ 2 d 1 ) k z i + γ 2 = 0 ,
( υ ̂ i · υ ̂ 1 f ) P 1 f υ exp ( j γ 1 d 1 ) k z i γ 1 + ( υ ̂ i · υ ̂ 1 b ) P 1 b υ exp ( j γ 1 d 1 ) k z i + γ 1 ( υ ̂ i · υ ̂ 2 f ) P 2 f υ exp ( j γ 2 d 1 ) k z i γ 2 = 0.
P 1 b υ = S 12 exp ( 2 j γ 1 d 1 ) P 1 f υ = S 01 S 02 S 01 S 02 1 exp ( 2 j γ 1 d 1 ) P 1 f υ ,
S 0 n = ( k z i γ n k z i + γ n ) ( υ ̂ r · υ ̂ n f υ ̂ i · υ ̂ n f ) , n = 1 , 2 .
υ ̂ i · υ ̂ n f = k z i γ n + k ρ 2 k k 1 = υ ̂ r · υ ̂ n b , υ ̂ r · υ ̂ n f = k z i γ n + k ρ 2 k k 1 = υ ̂ i · υ ̂ n b ,
S 0 n = k z i k n f 2 γ n k 2 k z i k n f 2 + γ n k 2 ,
S 0 n = R 0 n ( υ ̂ r · υ ̂ n f υ ̂ i · υ ̂ n f ) ,
P 1 f υ = η 1 Y 01 1 + S 01 S 12 exp ( 2 j γ 1 d 1 ) E υ i ,
Y 01 = X 01 υ ̂ i · υ ̂ 1 f .
Y 01 = 2 k z i k 1 f k k 1 f 2 k z i + k 2 γ 1 ,
P 1 b υ = η 1 Y 01 S 12 exp ( 2 j γ 1 d 1 ) 1 + S 01 S 12 exp ( 2 j γ 1 d 1 ) E υ i ,
P 2 f υ = η 2 Y 01 Y 12 exp ( 2 j γ 1 d 1 ) 1 + S 01 S 12 exp ( 2 j γ 1 d 1 ) E υ i ,
k z i 2 π k 2 E υ i + ( υ ̂ i · k ̂ 1 f ) k z i γ 1 P 1 f k + ( υ ̂ i · k ̂ 1 b ) k z i + γ 1 P 1 b k = 0 ,
( υ ̂ r · k ̂ 1 f ) P 1 f k exp ( j γ 1 d 1 ) k z i + γ 1 + ( υ ̂ r · k ̂ 1 b ) P 1 b k exp ( j γ 1 d 1 ) k z i γ 1 ( υ ̂ r · k ̂ 2 f ) P 2 f k exp ( j γ 2 d 1 ) k z i + γ 2 = 0 ,
( υ ̂ i · k ̂ 1 f ) P 1 f k exp ( j γ 1 d 1 ) k z i γ 1 + ( υ ̂ i · k ̂ 1 b ) P 1 b k exp ( j γ 1 d 1 ) k z i + γ 1 ( υ ̂ i · k ̂ 2 f ) P 2 f k exp ( j γ 2 d 1 ) k z i γ 2 = 0 .
υ ̂ i · k ̂ n f = k ρ k k n f ( k z i γ n ) = υ ̂ r · k ̂ n b , υ ̂ i · k ̂ n b = k ρ k k n f ( k z i + γ n ) = υ ̂ r · k ̂ n f ,
E υ i 2 π k ρ k z i ( P 1 f k + P 1 b k ) = 0 .
P 1 f k exp ( j γ 1 d 1 ) + P 1 b k exp ( j γ 1 d 1 ) k 1 f k 2 f P 2 f k exp ( j γ 2 d 1 ) = 0 .
k z i E υ i = 2 π k ρ P 1 f k .
P 1 b k = P 1 f k exp ( 2 j γ 1 d 1 ) .
k z i E υ i = 2 π k ρ P 1 f k [ 1 exp ( 2 γ 1 d 1 ) ] .
E r ( r ) = n = 1 2 V n G ¯ ( r , r ) · P m ( r ) d r .
E r ( r ) = E r exp [ j ( k x i x + k y i y + k z i z ) ] = E r exp ( j k r · r ) ,
E r = 2 π k 2 j k z i ( υ ̂ r υ ̂ r + ĥ r ĥ r ) · m = 1 2 d m d m 1 d z × [ P m f exp ( j γ m z ) + P m b exp ( j γ m z ) ] .
E r = 2 π k 2 k z i ( υ ̂ r υ ̂ r + ĥ r ĥ r ) · ( P 1 f k z i + γ i + P 1 b k z i γ i ) .
E r · ĥ r = 2 π k 2 k z i ( P 1 f h k z i + γ i + P 1 b h k z i γ i ) ,
E r · ĥ r = [ R 01 + R 12 exp ( 2 j γ 1 d 1 ) 1 + R 01 R 12 exp ( 2 j γ 1 d 1 ) ] E h i .
R 12 = R 10 + R 02 1 + R 10 R 02 ,
E r · υ ̂ r = 2 π k 2 k z i [ ( υ ̂ r · υ ̂ 1 f ) P 1 f υ k z i + γ 1 + ( υ ̂ r · υ ̂ 1 b ) P 1 b υ k z i γ 1 ] ,
E r · υ ̂ r = [ S 01 + S 12 exp ( 2 j γ 1 d 1 ) 1 + S 01 + S 12 exp ( 2 j γ 1 d 1 ) ] E υ i .
S 12 = S 10 + S 02 1 + S 10 S 12 ,
E r = 2 π k 2 k z i υ ̂ r ( υ ̂ r · k ̂ 1 f k z i + γ 1 P 1 f k ) ,
E r = E υ i υ ̂ r .

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