Abstract

Explicit formulas are derived for the mean-square electric fields induced by plane electromagnetic radiation in a two-phase, three-phase, and N-phase stratified medium. The first (incident) and last phases are semi-infinite in extent. Boundaries separating phases are plane and parallel. Phases are isotropic with arbitrary optical constants. Simple relationships follow for special cases such as at the critical angle for a two-phase system. Equations for reflectance, transmittance, and phase changes on reflectance and transmittance are given. Details are given concerning the energy absorption process, especially in the two- and three-layer cases. Equations for the N-layer case are in terms of characteristic matrices which can be readily programmed for a computer.

© 1968 Optical Society of America

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References

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  1. N. J. Harrick, J. Opt. Soc. Am. 55, 851 (1965).
    [Crossref]
  2. W. N. Hansen, T. Kuwana, and R. A. Osteryoung, Anal. Chem. 38, 1810 (1966).
    [Crossref]
  3. aM. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), p. 61; bp. 609.
  4. J. A. Stratton, Electromagnetic Theory (McGraw–Hill Book Company, New York, 1941), pp. 492ff.
  5. A. Vašíček, Optics of Thin Films (North-Holland Publishing Company, Amsterdam, 1960), p. 330.
  6. A. Vašíček, Opt. Spectry. (USSR) 11, 128 (1961).
  7. J. A. Stratton, Ref. 4, p. 137.
  8. See Ref. 7, p. 136.
  9. See Ref. 7, p. 511.
  10. M. Born and E. Wolf, Ref. 3, p. 23.
  11. N. J. Harrick and F. K. du Pré, Appl. Opt. 5, 1739 (1966).
    [Crossref] [PubMed]
  12. F. Abelès, Ann. Phys. (Paris) 5, 596 (1950).

1966 (2)

W. N. Hansen, T. Kuwana, and R. A. Osteryoung, Anal. Chem. 38, 1810 (1966).
[Crossref]

N. J. Harrick and F. K. du Pré, Appl. Opt. 5, 1739 (1966).
[Crossref] [PubMed]

1965 (1)

1961 (1)

A. Vašíček, Opt. Spectry. (USSR) 11, 128 (1961).

1950 (1)

F. Abelès, Ann. Phys. (Paris) 5, 596 (1950).

Abelès, F.

F. Abelès, Ann. Phys. (Paris) 5, 596 (1950).

Born, M.

M. Born and E. Wolf, Ref. 3, p. 23.

aM. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), p. 61; bp. 609.

du Pré, F. K.

Hansen, W. N.

W. N. Hansen, T. Kuwana, and R. A. Osteryoung, Anal. Chem. 38, 1810 (1966).
[Crossref]

Harrick, N. J.

Kuwana, T.

W. N. Hansen, T. Kuwana, and R. A. Osteryoung, Anal. Chem. 38, 1810 (1966).
[Crossref]

Osteryoung, R. A.

W. N. Hansen, T. Kuwana, and R. A. Osteryoung, Anal. Chem. 38, 1810 (1966).
[Crossref]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw–Hill Book Company, New York, 1941), pp. 492ff.

J. A. Stratton, Ref. 4, p. 137.

Vašícek, A.

A. Vašíček, Opt. Spectry. (USSR) 11, 128 (1961).

A. Vašíček, Optics of Thin Films (North-Holland Publishing Company, Amsterdam, 1960), p. 330.

Wolf, E.

aM. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), p. 61; bp. 609.

M. Born and E. Wolf, Ref. 3, p. 23.

Anal. Chem. (1)

W. N. Hansen, T. Kuwana, and R. A. Osteryoung, Anal. Chem. 38, 1810 (1966).
[Crossref]

Ann. Phys. (Paris) (1)

F. Abelès, Ann. Phys. (Paris) 5, 596 (1950).

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

Opt. Spectry. (USSR) (1)

A. Vašíček, Opt. Spectry. (USSR) 11, 128 (1961).

Other (7)

J. A. Stratton, Ref. 4, p. 137.

See Ref. 7, p. 136.

See Ref. 7, p. 511.

M. Born and E. Wolf, Ref. 3, p. 23.

aM. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), p. 61; bp. 609.

J. A. Stratton, Electromagnetic Theory (McGraw–Hill Book Company, New York, 1941), pp. 492ff.

A. Vašíček, Optics of Thin Films (North-Holland Publishing Company, Amsterdam, 1960), p. 330.

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Figures (5)

Fig. 1
Fig. 1

Interaction of plane wave in a three-phase system. Sign conventions for parallel (|| or TM) and perpendicular (⊥ or TE) polarized radiation are illustrated—the indicated geometry is for zero phase change, in all cases. The symbol s refers to unit propagation vector.

Fig. 2
Fig. 2

Mean-square electric field in air at the surface of glass as a function of internal angle of incidence for ⊥ polarization, 〈Ey2〉, and for || polarization, 〈Ex2〉+〈Ez2〉. The indices are: n1=1.51, n2=1.0, and k2=0.

Fig. 3
Fig. 3

Mean-square electric field in air at the surface of silicon as a function of internal angle of incidence. The indices are: n1=3.4, n2=1.0, k2=0.

Fig. 4
Fig. 4

Dependence of mean-square fields in the last medium on the extinction coefficient. The optical parameters are: n1=4, n2=1.4, k2 varies, and θ1=20.5°.

Fig. 5
Fig. 5

Standing-wave fields in phase 1 for metallic reflection. The optical parameters are: n1=1.5, n2=0.3, k2=3, and θ1=45°.

Equations (119)

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r j k = μ k ξ j - μ j ξ k μ k ξ j + μ j ξ k ,             t j k = 2 μ k ξ j μ k ξ j + μ j ξ k ,
r = r 12 + r 23 e 2 i β 1 + r 12 r 23 e 2 i β ,             t E = t 12 t 23 e i β 1 + r 12 r 23 e 2 i β .
R = r 2 ,             T = μ 1 Re ξ 3 μ 3 ξ 1 t E 2 .
δ r = arg r ,             and             δ r = arg t E .
R = R 12 + R 23 e - 4 Im β + R 12 1 2 R 23 1 2 e - 2 Im β 2 cos ( δ 23 r - δ 12 r + 2 Re β ) 1 + R 12 R 23 e - 4 Im β + R 12 1 2 R 23 1 2 e - 2 Im β 2 cos ( δ 23 r + δ 12 r + 2 Re β ) .
T = μ 1 Re ξ 3 μ 3 ξ 1 t 12 2 t 23 2 e - 2 Im β 1 + R 12 R 23 e - 4 Im β + R 12 1 2 R 23 1 2 e - 2 Im β 2 cos ( δ 23 r + δ 12 r + 2 Re β ) ,
r j k = ˆ k ξ j - ˆ j ξ k ˆ k ξ j + ˆ j ξ k ,             t j k = H k 0 H j 0 = 2 ˆ k ξ j ˆ k ξ j + ˆ j ξ k ,
r = r 12 + r 23 e 2 i β 1 + r 12 r 23 e 2 i β ,             t H = t 12 t 23 e i β 1 + r 12 r 23 e 2 i β ,
R = r 2 ,             T = μ 3 Re ( ξ 3 / n ˆ 3 2 ) μ 1 cos θ 1 / n 1 t H 2 ,
t E = ( μ 3 1 μ 1 ˆ 3 ) 1 2 t H = μ 3 μ 1 n 1 n ˆ 3 t H ,
δ r = arg r ,             δ t = arg t E ,
R = R 12 + R 23 e - 4 Im β + R 12 1 2 R 23 1 2 e - 2 Im β 2 cos ( δ 23 r - δ 12 r + 2 Re β ) 1 + R 12 R 23 e - 4 Im β + R 12 1 2 R 23 1 2 e - 2 Im β 2 cos ( δ 23 r + δ 12 r + 2 Re β ) ,
T = μ 3 Re ( ξ 3 / n ˆ 3 2 ) μ 1 cos θ 1 / n 1 t 12 2 t 23 2 e - 2 Im β 1 + R 12 R 23 e - 4 Im β + R 12 1 2 R 23 1 2 e - 2 Im β 2 cos ( δ 23 r + δ 12 r + 2 Re β ) .
Re ( · S ) = - 1 2 σ E · E * = - σ E 2 ,
μ σ = n k ν ,
E 1 = E 1 0 t exp ( i k 1 t · r - i ω t ) + E 1 0 r exp ( i k 1 r · r - i ω t ) .
E 1 = E 1 0 t + E 1 0 r ( r , t = 0 ) .
E 1 0 r = r E 1 0 t = R 1 2 exp ( i δ r ) E 1 0 t ,
E 1 2 = 1 2 ( E 1 · E 1 * )
E 1 2 / E 1 0 t 2 = { exp [ i ( k 1 t · r - ω t ) ] + r exp [ i ( k 1 r · r - ω t ) ] } × { exp [ - i ( k 1 t · r - ω t ) ] + r * exp [ - i ( k 1 r · r - ω t ) ] }
E 1 2 = 1 2 ( 1 + R ) + R 1 2 cos [ δ r + ( k 1 r · r - k 1 t · r ) ] ( E 1 0 t ) 2 .
( k 1 r · r - k 1 t · r ) = 2 π λ n 1 [ ( x sin θ 1 - z cos θ 1 ) - ( x sin θ 1 + z cos θ 1 ) ] = - 4 π λ n 1 cos θ 1 z = - ( 4 π / λ ) ξ 1 z
E 1 2 = 1 2 ( 1 + R ) + R 1 2 cos [ δ r - ( 4 π / λ ) ξ 1 z ] ( E 1 0 t = 1 ) .
E 3 = E 3 0 t exp ( i k 3 t · r - i ω t ) ,
E 3 2 = 1 2 ( E 3 · E 3 * ) = 1 2 { E 3 0 t E 3 0 t * exp [ - 2 Im ( k 3 t · r ) ] = 1 2 t E 2 exp [ - ( 4 π / λ ) Im ξ 3 ( z - h ) ] ( E 1 0 t ) 2 .
E 2 = E 2 0 t exp ( i k 2 t · r - i ω t ) + E 2 0 r exp ( i k 2 r · r - i ω t ) .
E 2 t + E 2 r = E 1 t + E 1 r .
H TE 2 = ( ˆ 2 μ 2 ) 1 2 s 2 t × E 2 t + ( ˆ 2 μ 2 ) 1 2 s 2 r × E 2 r
H TE 2 x = ( ˆ 2 μ 2 ) 1 2 ( - cos θ 2 E 2 t + cos θ 2 E 2 r ) = ξ 2 μ 2 ( - E 2 t + E 2 r ) .
ξ 1 μ 2 ξ 2 μ 1 ( E 1 t - E 1 r ) = ( E 2 t - E 2 r ) ( r · z ˆ = 0 ) .
2 E 2 t = ( 1 + ξ 1 μ 2 ξ 2 μ 1 ) E 1 t + ( 1 - ξ 1 μ 2 ξ 2 μ 1 ) r E 1 t 2 E 2 r = ( 1 - ξ 1 μ 2 ξ 2 μ 1 ) E 1 t + ( 1 + ξ 1 μ 2 ξ 2 μ 1 ) r E 1 t } ( r · z ˆ = 0 )
E 2 = E 2 0 t exp ( i k 2 t · r - i ω t ) + E 2 0 r exp ( i k 2 r · r - i ω t )
E 2 E 1 0 t = exp ( i 2 π λ n 2 sin θ 2 x - i ω t ) [ ( 1 + r ) cos ( 2 π ξ 2 λ z ) + i ξ 1 μ 2 ξ 2 μ 1 ( 1 - r ) sin ( 2 π ξ 2 λ z ) ] y ˆ .
E 2 = exp [ i ( 2 π λ n 1 sin θ 1 x - ω t ) ] [ ( 1 + r ) cos ( 2 π ξ 2 λ z ) + i ξ 1 μ 2 ξ 2 μ 1 ( 1 - r ) sin ( 2 π ξ 2 λ z ) ]
E 2 2 = 1 2 ( E 2 · E 2 * ) = 1 2 ( E 2 E 2 * ) .
E 1 = E 1 0 t exp ( i k 1 t · r - i ω t ) + E 1 0 r exp ( i k 1 r · r - i ω t ) ,
H TM 1 = H TM 1 t + H TM 1 r = ( 1 μ 1 ) 1 2 ( s 1 t × E 1 t + s 1 r × E 1 r ) ,
H TM 1 r = r H TM 1 t = R 1 2 exp ( i δ r ) H TM 1 t ( r · z ˆ = 0 ) ,
E 1 r = - ( μ 1 1 ) 1 2 s 1 r × H TM 1 r
E 1 x r = - cos θ 1 r E 1 t
E 1 z r = sin θ , r E 1 t ( r · z ˆ = 0 )
E 1 r = r E 1 t ( r · z ˆ = 0 ) .
E 1 2 = 1 2 ( E 1 · E 1 * ) = E 1 r 2 + E 1 z 2 ,
E 1 x 2 = cos 2 θ 1 [ 1 2 ( 1 + R ) - R 1 2 cos ( δ r - 4 π ( z / λ ) ξ 1 ) ] ( E 1 0 t ) 2
E 1 z 2 = sin 2 θ 1 [ 1 2 ( 1 + R ) + R 1 2 × cos ( δ r - 4 π z λ ξ 1 ) ] ( E 1 0 t ) 2 .
E 2 = E 2 0 t exp η t + E 2 0 r exp η r ,
E 2 = - ( μ 2 ˆ 2 ) 1 2 ( s 2 t × H TM 2 0 t exp η t + s 2 r × H TM 2 0 r exp η r ) .
H TM 2 t = 1 2 [ ( 1 + ξ 1 ˆ 2 ξ 2 1 ) + ( 1 - ξ 1 ˆ 2 ξ 2 1 ) r ] H TM 1 t and H TM 2 r = 1 2 [ ( 1 - ξ 1 ˆ 2 ξ 2 1 ) + ( 1 + ξ 1 ˆ 2 ξ 2 1 ) r ] H TM 1 t } ( r · z ˆ = 0 ) .
H TM 1 t = ( 1 μ 1 ) 1 2 s 1 t × E 1 t
E 2 = - ( μ 2 ˆ 2 ) 1 2 { s 2 t × [ 1 2 ( 1 + ξ 1 ˆ 2 ξ 2 1 ) + ( 1 - ξ 1 ˆ 2 ξ 2 1 ) r ] ( 1 μ 1 ) 1 2 s 1 t × E 1 0 t exp η t + s 2 r × [ 1 2 ( 1 - ξ 1 ˆ 2 ξ 2 1 ) + ( 1 + ξ 1 ˆ 2 ξ 2 1 ) r ] ( 1 μ 1 ) 1 2 s 1 t × E 1 0 t exp η r } = ( cos θ 2 , 0 , - sin θ 2 ) E 1 0 t ( μ 2 1 μ 1 ˆ 2 ) 1 2 1 2 { [ ( 1 + ξ 1 ˆ 2 ξ 2 1 ) + ( 1 - ξ 1 ˆ 2 ξ 2 1 ) r ] exp η t } + ( - cos θ 2 , 0 , - sin θ 2 ) E 1 0 t ( μ 2 1 μ 1 ˆ 2 ) 1 2 1 2 { [ ( 1 - ξ 1 ˆ 2 ξ 2 1 ) + ( 1 + ξ 1 ˆ 2 ξ 2 1 ) r ] exp η r } }
E 2 x = exp ( i 2 π λ n ˆ 2 sin θ 2 x - i ω t ) [ cos θ 1 ( 1 - r ) cos ( 2 π λ ξ 2 z ) + i μ 2 ξ 2 μ 1 n ˆ 2 2 n 1 ( 1 + r ) sin ( 2 π λ ξ 2 z ) ] E 1 0 t and E 2 z = - exp ( i 2 π λ n ˆ 2 sin θ 2 x - i ω t ) [ μ 2 μ 1 n ˆ 2 sin θ 2 n ˆ 2 2 n 1 ( 1 + r ) cos ( 2 π λ ξ 2 z ) + n ˆ 2 sin θ 2 ξ 2 cos θ 1 ( 1 - r ) sin ( 2 π λ ξ 2 z ) ] E 1 0 t } .
E 2 2 = 1 2 ( E 2 · E 2 * ) = 1 2 ( E 2 x 2 + E 2 z 2 ) = E 2 x 2 + E 2 z 2 ,
H TM 3 t = H TM 1 t t H = ( 1 μ 1 ) 1 2 s 1 t × E 1 t t H ( r · z ˆ = h ) ,
E 3 t = - ( μ 3 ˆ 3 ) 1 2 s 3 t × H TM 3 t = ( cos θ 3 , 0 , - sin θ 3 ) × ( μ 3 1 ˆ 3 μ 1 ) 1 2 t H E 1 t ( r · z ˆ = h )
E 3 t = ( μ 3 1 ˆ 3 μ 1 ) 1 2 t H E 1 t = t E E 1 t ( r · z ˆ = h ) .
E 3 = E 3 0 t exp ( i k 3 t · r - i ω t ) ,
E 3 x = cos θ 3 t E E 1 0 t exp ( i 2 π λ × n ˆ 3 sin θ 3 - i ω t ) × exp [ i 2 π λ ξ 3 ( z - h ) ] E 3 z = - sin θ 3 t E E 1 0 t exp ( i 2 π λ × n ˆ 3 sin θ 3 - i ω t ) × exp [ i 2 π λ ξ 3 ( z - h ) ]
E 3 2 = 1 2 t E 2 exp [ - 4 π Im ξ 3 ( z - h ) / λ ] E 1 0 t 2
E 3 x 2 = 1 2 | ξ 3 n ˆ 3 t E | 2 exp [ - 4 π Im ξ 3 ( z - h ) / λ ] E 1 0 t 2
E 3 z 2 = 1 2 | n 1 sin θ 1 n ˆ 3 t E | 2 exp [ - 4 π Im ξ 3 ( z - h ) / λ ] E 1 0 t 2 .
R = r 2 ,             and             T = μ 1 μ 2 Re ξ 2 ξ 1 t E 2 ,
r = r 12 = μ 2 ξ 1 - μ 1 ξ 2 μ 2 ξ 1 + μ 1 ξ 2 ,             and             t E = t 12 = 2 μ 2 ξ 1 μ 2 ξ 1 + μ 1 ξ 2 .
R = r 2 ,             and             T = μ 2 μ 1 n 1 2 ξ 1 Re ξ 2 n ˆ 2 2 t H 2 ,
r = r 12 = ˆ 2 ξ 1 - 1 ξ 2 ˆ 2 ξ 1 + 1 ξ 2 t H = t 12 = 2 ˆ 2 ξ 1 ˆ 2 ξ 1 + 1 ξ 2 ,
t E = μ 2 μ 1 n 1 n ˆ 2 t H .
E 1 2 = [ 1 2 ( 1 + R ) + R 1 2 cos ( δ r - 4 π ( z / λ ) ξ 1 ) ] E 1 0 t 2 ,
E 2 2 = 1 2 t E 2 exp ( - 4 π ( z / λ ) Im ξ 2 ) E 1 0 t 2
E 2 2 = 1 2 t E 2 exp ( - 4 π Im ξ 2 ( z / λ ) ) E 1 0 t 2
E 1 x 2 = cos 2 θ 1 [ 1 2 ( 1 + R ) - R 1 2 cos ( δ r - 4 π ( z / λ ) ξ 1 ) ] E 1 0 t 2
E 1 z 2 = sin 2 θ 1 [ 1 2 ( 1 + R ) + R 1 2 cos ( δ r - 4 π ( z / λ ) ξ 1 ) ] E 1 0 t 2 ,
E 2 x 2 = 1 2 | ξ 2 n ˆ 2 t E | 2 exp ( - 4 π z λ Im ξ 2 ) E 1 0 t 2
E 2 z 2 = 1 2 | n 1 sin θ 1 n ˆ 2 t E | 2 exp ( - 4 π z λ Im ξ 2 ) E 1 0 t 2 .
E 1 2 = E 2 2 = 1 2 | 2 μ 2 ξ 1 μ 2 ξ 1 + μ 1 ξ 2 | 2 ( E 1 0 t ) 2 ( r · z ˆ = 0 )
E 2 2 = 1 2 | ( 1 μ 2 2 μ 1 ) 1 2 2 2 ξ 1 2 ξ 1 + 1 ξ 2 | 2 ( E 1 0 t ) 2 ( r · z ˆ = 0 ) .
E 2 = 2 ξ 1 2 ( ξ 1 + ξ 2 ) 2 for             θ 1 θ c = 2 ξ 1 2 n 1 2 - n 2 2 for             θ 1 θ c }             r · z ˆ = 0 μ = 1 E 1 0 t = 1
E 2 2 = 2 ( ξ 1 n 2 cos θ 1 + n 1 cos θ 2 ) 2 for             θ 1 θ c = 2 ξ 1 2 ( n 2 cos θ 1 ) 2 - ( n 1 cos θ 2 ) 2 for             θ 1 θ c } .             r · z ˆ = 0 μ = 1 E 1 0 t = 1
E 2 = 2 E x 2 = 0 E z 2 = 2 ( n 1 / n 2 ) 2 } .             E 1 0 t = 1 r · z ˆ = 0 θ = θ c
E 2 = E x 2 = 2 ( n 1 n 1 + n 2 ) 2 E z 2 = 0 } .             E 1 0 t = 1 r · z ˆ = 0 θ 1 = 0
[ U 1 V 1 ] = M 2 M 3 M N - 1 [ U N - 1 V N - 1 ] = M [ U N - 1 V N - 1 ] ,
M j = [ cos β j - i p j sin β j - i p j sin β j cos β j ] ,             ( TE polarization )
M j = [ cos β j - i q j sin β j - i q j sin β j cos β j ] ,             ( TM polarization )
r = E y 1 0 r E y 1 0 t = ( m 11 + m 12 p N ) p 1 - ( m 21 + m 22 p N ) ( m 11 + m 12 p N ) p 1 + ( m 21 + m 22 p N )
t E = E y N 0 t E y 1 0 t = 2 p 1 ( m 11 + m 12 p N ) p 1 + ( m 21 + m 22 p N )
r = H y 1 0 r H y 1 0 t = ( m 11 + m 12 q N ) q 1 - ( m 21 + m 22 q N ) ( m 11 + m 12 q N ) q 1 + ( m 21 + m 22 q N )
t H = H y N 0 t H y 1 0 t = 2 q 1 ( m 11 + m 12 q N ) q 1 + ( m 21 + m 22 q N )
t E = ( μ N / μ 1 ) ( n 1 / n ˆ N ) t H .
R = r 2 ,             δ r = arg r
T = μ 1 Re ( n ˆ N cos θ N ) μ N n 1 cos θ 1 t E 2 ,             δ t = arg t E
R = r 2 ,             δ r = arg r
T = μ N Re ( n ˆ N cos θ N / n ˆ N 2 ) μ 1 n 1 cos θ 1 / n 1 2 t H 2 ,             δ t = arg t E .
Q 2 ( z ) = N 2 ( z ) Q 1 ,
Q k ( z ) = N k ( z ) j = k - 1 2 N j Q 1 ,
Q k ( z ) = N k ( z ) j = k N - 1 M j Q N - 1 .
Q 1 = [ U 1 V 1 ] = [ E y 1 0 H x 1 0 ] = [ E y 1 0 t + E y 1 0 r H x 1 0 t + H x 1 0 r ] = [ 1 + r - p 1 ( 1 - r ) ] E y 1 0 t
Q N - 1 = [ U N - 1 V N - 1 ] = [ E y ( N - 1 ) 0 H x ( N - 1 ) 0 ] = [ E y ( N - 1 ) 0 t H x ( N - 1 ) 0 t ] = [ t E p N t E ] E y 1 0 t
N k ( z ) = [ cos ( 2 π λ ξ k ( z - z k - 1 ) ) i p k sin ( 2 π λ ξ k ( z - z k - 1 ) ) i p k sin ( 2 π λ ξ k ( z - z k - 1 ) ) cos ( 2 π λ ξ k ( z - z k - 1 ) ) ] .
Q k ( z ) [ U k ( z ) V k ( z ) ]
E k = U k ( z ) exp ( i 2 π λ n 1 sin θ 1 x - i ω t ) .
E k 2 = 1 2 E k E k * = 1 2 U k ( z ) 2 .
H TE k x = V k ( z ) exp ( i 2 π λ n 1 sin θ 1 x - i ω t )
H TE k z = W k ( z ) exp ( i 2 π λ n 1 sin θ 1 x - i ω t )
H TE k = ( V k ( z ) , 0 , W k ( z ) ) exp ( i 2 π λ n 1 sin θ 1 x - i ω t ) .
W k ( z ) = n 1 sin θ 1 U k ( z ) / μ k .
Q 1 = [ U 1 V 1 ] = [ H y 1 0 E x 1 0 ] = [ H y 1 0 t + H y 1 0 r E x 1 0 t + E x 1 0 r ] = [ 1 + r - q 1 ( 1 - r ) ] H y 1 0 t
Q N - 1 = [ U N - 1 V N - 1 ] = [ H y ( N - 1 ) 0 E x ( N - 1 ) 0 ] = [ H y ( N - 1 ) 0 t E x ( N - 1 ) 0 t ] = [ t H - q N t H ] H y 1 0 t
N k ( z ) = [ cos ( 2 π λ ξ k ( z - z k - 1 ) )             i q k sin ( 2 π λ ξ k ( z - z k - 1 ) ) i q k sin ( 2 π λ ξ k ( z - z k - 1 ) )             cos ( 2 π λ ξ k ( z - z k - 1 ) ) ]
H TM k = U k ( z ) exp ( i 2 π λ n 1 sin θ 1 x - i ω t )
E k x = V k ( z ) exp ( i 2 π λ n 1 sin θ 1 x - i ω t )
E k z = W k ( z ) exp ( i 2 π λ n 1 sin θ 1 x - i ω t ) ,
E k = ( V k ( z ) , 0 , W k ( z ) ) exp ( i 2 π λ n 1 sin θ 1 x - i ω t ) .
W k ( z ) = n 1 sin θ 1 U k ( z ) / ˆ k = μ k n 1 sin θ 1 U k ( z ) / n ˆ k 2 .
E k x 2 = 1 2 V k ( z ) 2
E k z 2 = 1 2 W k ( z ) 2
E k 2 = 1 2 ( V k ( z ) 2 + W k ( z ) 2 ) .
E j = E j 0 exp { i [ ( 2 π / λ ) ξ j z - ω t ] } ,
1 2 ( E j · E j * ) = 1 2 E j 0 2 exp [ i 2 π λ z ( ξ j - ξ j * ) ] = 1 2 E j 0 2 exp ( - 4 π λ Im ξ j z ) .
Im ξ j 0 ( time factor e - i ω t ) .
ξ j n ˆ j cos θ j = ( n ˆ j 2 - n 1 2 sin 2 θ 1 ) 1 2 = ( x + i y ) 1 2 ,
Re ξ j 0 , Im ξ j 0 ( time factor e - i ω t , Im n ˆ j 0 ) .