Abstract

When the reflected or transmitted field of a frustrated total internal reflection is returned along its path to the point of incidence, it produces a third auxiliary field. When any combination of two of the three fields is returned along their respective paths to the point of incidence, interference-enhanced frustrated total internal reflection (IEFTIR) occurs. Exact IEFTIR solutions are obtained for the case of two similar dielectric slabs separated by a dielectric gap. The general results include, for both s and p polarizations, the transmitted and reflected fields and the transmission and reflection coefficients for each of three posssible modes. Special features of the solution are discussed regarding fast optical switching and variable transmission/reflection applications.

© 1980 Optical Society of America

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References

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  1. I. Newton, Opticks (W. Innys, London, 1730), Qu. 29.
  2. R. H. Good and T. J. Nelson, Classical Theory of Electric and Magnetic Fields (Academic, New York, 1971), pp. 380–383.
  3. Z. Knittl, Optics of Thin Films (Wiley, New York, 1976), pp. 229–238.
  4. N. J. Harrick, Internal Reflection Spectroscopy (Interscience, New York, 1967).
  5. N. J. Harrick, “Use of Frustrated Total Internal Reflection to Measure Film Thickness and Surface Reliefs,” J. Appl. Phys. 33, 2774–2775 (1962).
    [Crossref]
  6. N. J. Harrick, “A Continuously Variable Optical Beam Splitter and Intensity Controller,” Appl. Opt. 2, 1203–1204 (1963).
    [Crossref]
  7. , Laser Focus 14110 (Feb.1978).
  8. E. L. Steele, W. C. Davis, and R. L. Treuthart, “A Laser Output Coupler Using Frustated Total Internal Reflection,” Appl. Opt. 5, 5–8(1966).
    [Crossref] [PubMed]

1978 (1)

, Laser Focus 14110 (Feb.1978).

1966 (1)

1963 (1)

1962 (1)

N. J. Harrick, “Use of Frustrated Total Internal Reflection to Measure Film Thickness and Surface Reliefs,” J. Appl. Phys. 33, 2774–2775 (1962).
[Crossref]

Davis, W. C.

Good, R. H.

R. H. Good and T. J. Nelson, Classical Theory of Electric and Magnetic Fields (Academic, New York, 1971), pp. 380–383.

Harrick, N. J.

N. J. Harrick, “A Continuously Variable Optical Beam Splitter and Intensity Controller,” Appl. Opt. 2, 1203–1204 (1963).
[Crossref]

N. J. Harrick, “Use of Frustrated Total Internal Reflection to Measure Film Thickness and Surface Reliefs,” J. Appl. Phys. 33, 2774–2775 (1962).
[Crossref]

N. J. Harrick, Internal Reflection Spectroscopy (Interscience, New York, 1967).

Knittl, Z.

Z. Knittl, Optics of Thin Films (Wiley, New York, 1976), pp. 229–238.

Nelson, T. J.

R. H. Good and T. J. Nelson, Classical Theory of Electric and Magnetic Fields (Academic, New York, 1971), pp. 380–383.

Newton, I.

I. Newton, Opticks (W. Innys, London, 1730), Qu. 29.

Steele, E. L.

Treuthart, R. L.

Appl. Opt. (2)

J. Appl. Phys. (1)

N. J. Harrick, “Use of Frustrated Total Internal Reflection to Measure Film Thickness and Surface Reliefs,” J. Appl. Phys. 33, 2774–2775 (1962).
[Crossref]

Laser Focus (1)

, Laser Focus 14110 (Feb.1978).

Other (4)

I. Newton, Opticks (W. Innys, London, 1730), Qu. 29.

R. H. Good and T. J. Nelson, Classical Theory of Electric and Magnetic Fields (Academic, New York, 1971), pp. 380–383.

Z. Knittl, Optics of Thin Films (Wiley, New York, 1976), pp. 229–238.

N. J. Harrick, Internal Reflection Spectroscopy (Interscience, New York, 1967).

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Figures (7)

FIG. 1
FIG. 1

Production of frustrated total internal reflection. n2 is the index of refraction of the gap.

FIG. 2
FIG. 2

Production of interference-enhanced frustrated total internal reflection. n2 is the index of refraction of the gap.

FIG. 3
FIG. 3

Schematic representation of frustrated total internal reflection.

FIG. 4
FIG. 4

Schematic representation of interference-enhanced frustrated total internal reflection, (a) represents the T mode, (b) represents the R mode, (c) represents the A mode.

FIG. 5
FIG. 5

IEFTIR transmission coefficient surface as a function of gap width d/λ and phase angle δ for θi = 45° and n = 1.4624. The left column contains the s-polarization solutions. The right column contains the p-polarization solutions, (a) contains the T-mode solutions, (b) contains the R-mode solutions, (c) contains the A-mode solutions. All absorption coefficients were zero.

FIG. 6
FIG. 6

FTIR transmission coefficient vs d/λ at θi = 45° and n = 1.4624 for p polarization (solid line) and s polarization (dashed line). Absorption coefficient is zero.

FIG. 7
FIG. 7

IEFTIR devices as they might be practically constructed.

Equations (113)

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sin θ r = n sin θ i ,
n = n 1 / n 2 .
E A x = A cos θ i exp [ i ( k A · x ω t ) ] , E A z = A sin θ i exp [ i ( k A · x ω t ) ] , B A y = n 1 A exp [ i ( k A · x ω t ) ] ,
A cos θ i + B cos θ i = C cos θ r + D cos θ r , A n B n = C D .
C α cos θ r + D ( 1 / α ) cos θ r = E β cos θ i , C α D ( 1 / α ) = E β n ,
β E A = τ p = 4 n cos θ i cos θ r ( α + 1 / α ) 2 n cos θ i cos θ r ( α 1 / α ) ( cos 2 θ i + n 2 cos 2 θ r ) ,
B β A = ρ p = ( α 1 / α ) ( cos 2 θ i n 2 cos 2 θ r ) ( α + 1 / α ) 2 n cos θ i cos θ r ( α 1 / α ) ( cos 2 θ i + n 2 cos 2 θ r ) .
β F 2 = τ p B 2 ,
A 2 = ρ p β B 2 .
F 1 = τ B 1 ,
τ = τ p e i ϕ ( 1 K ) ,
A 1 cos θ i + B 1 cos θ i = C 1 cos θ r + D 1 cos θ r , n A 1 + n B 1 = C 1 + D 1 .
C 1 α cos θ r + D 1 ( 1 / α ) cos θ r = E 1 β cos θ i + F 1 ( 1 / β ) cos θ i , C 1 α + D 1 ( 1 / α ) = E 1 n β + F 1 n ( 1 / β ) .
β E 1 A 1 = 4 n cos θ i cos θ r + ( τ / β ) [ ( α 1 / α ) ( cos 2 θ i + n 2 cos 2 θ r ) + ( α + 1 / α ) 2 n cos θ i cos θ r ] ( τ / β ) 4 n cos θ i cos θ r + [ ( α 1 / α ) ( cos 2 θ i + n 2 cos 2 θ r ) ( α + 1 / α ) 2 n cos θ i cos θ r ] ,
B 1 β A 1 = ( α 1 / α ) ( cos 2 θ i n 2 cos 2 θ r ) ( τ / β ) 4 n cos θ i cos θ r + [ ( α 1 / α ) ( cos 2 θ i + n 2 cos 2 θ r ) ( α + 1 / α ) 2 n cos θ i cos θ r ] .
B 2 = e i χ ( 1 K B ) B 1 ,
e i Ψ A 2 A 1 = ( 1 K B ) ( α 1 / α ) 2 ( cos 2 θ i n 2 cos 2 θ r ) 2 / [ ( α 1 / α ) ( cos 2 θ i + n 2 cos 2 θ r ) ( α + 1 / α ) 2 n cos θ i cos θ r ] ( τ / β ) 4 n cos θ i cos θ r + [ ( α 1 / α ) ( cos 2 θ i + n 2 cos 2 θ r ) ( α + 1 / α ) 2 n cos θ i cos θ r ] ,
k C z = i k A ( sin 2 θ i sin 2 θ c ) 1 / 2 ,
cos θ r = i n ( sin 2 θ i sin 2 θ c ) 1 / 2 ,
α = e γ = exp [ k A d ( sin 2 θ i sin 2 θ c ) 1 / 2 ] .
δ = ϕ k A z d ,
x = 2 ( cos 2 θ i n 2 | cos θ r | 2 ) ,
y = 4 n cos θ i | cos θ r | .
β E 1 A 1 = y ( y cosh γ + ix sinh γ ) + ( 1 K ) ( cos δ + i sin δ ) y ( y cosh γ i x sinh γ ) ( 1 K ) y 2 ( cos δ + i sin δ ) + ( x sinh γ i y cosh γ ) 2 ,
e i Ψ A 2 A 1 = ( 1 K B ) ( x 2 + y 2 ) sinh 2 γ ( 1 K ) y 2 ( cos δ + i sin δ ) + ( x sinh γ i y cosh γ ) 2 .
T = [ y 2 cosh γ ( 1 K ) ( y 2 cosh γ cos δ + x y sinh γ sin δ ) ] 2 + [ x y sinh γ + ( 1 K ) ( x y sinh γ cos δ y 2 cosh γ sin δ ) ] 2 [ x 2 sinh 2 γ y 2 cosh 2 γ + ( 1 K ) y 2 cos δ ] 2 + [ 2 x y sinh γ cosh γ ( 1 K ) y 2 sin δ ] 2 ,
R = [ ( 1 K B ) ( x 2 + y 2 ) sinh 2 γ ] 2 [ x 2 sinh 2 γ y 2 cosh 2 γ + ( 1 K ) y 2 cos δ ] 2 + [ 2 x y sinh γ cosh γ ( 1 K ) y 2 sin δ ] 2 .
tan γ = y ( cos δ 1 ) / x sin δ ,
tanh γ = y ( 1 cos δ ) / x sin δ ,
T = ( δ δ 0 ) 2 ( y 4 cosh 2 γ 0 + x 2 y 2 sinh 2 γ 0 ) ( x 2 + y 2 ) 2 sinh 4 γ 0 + ( δ δ 0 ) 2 ( y 4 cosh 2 γ 0 + x 2 y 2 sinh 2 γ 0 ) + .
T = ( δ δ 0 ) 2 ( δ δ 0 ) 2 + γ 0 4 ( x 2 + y 2 ) 2 / y 4 .
tan γ = y ( 1 + cos δ ) x sin δ ,
tanh γ = y ( 1 + cos δ ) x sin δ ,
T = ( δ δ 0 ) 2 ( x 2 + y 2 ) sinh 2 γ 0 ( x 2 sinh 2 γ 0 + y 2 cosh 2 γ 0 ) + y 4 + ( δ δ 0 ) 2 ( x 2 + y 2 ) sinh 2 γ 0 ( x 2 sinh 2 γ 0 + y 2 cosh 2 γ 0 ) +
y 4 ( x 2 + y 2 ) sinh 2 γ 0 ( x 2 sinh 2 γ 0 + y 2 cosh 2 γ 0 ) ,
y 2 / ( x 2 + y 2 ) γ 0 2
sin 2 γ = y 2 / ( x 2 + y 2 ) .
sinh 2 γ = y 2 / ( x 2 + y 2 ) .
T = ( δ δ 0 ) 2 + sin 4 γ 0 ( x 2 y 2 ) 2 / ( y 4 cos 2 γ 0 + x 2 y 2 sin 2 γ 0 ) + ( δ δ 0 ) 2 + .
Γ max = 4 ( x 2 y 2 ) / ( 3 y 2 x 2 ) ( x 2 + y 2 ) .
T = ( δ δ 0 ) 2 y 4 / [ ( x 2 y 2 ) sin 2 γ 0 ( x 2 sin 2 γ 0 + y 2 cos 2 γ 0 ) ] + ( δ δ 0 ) 2 + .
Γ min = y 4 / x 2 ( x 2 y 2 ) .
E A y = A exp [ i ( k A · x ω t ) ] , B A x = n A cos θ i exp [ i ( k A · x ω t ) ] , B A z = n A sin θ i exp [ i ( k A · x ω t ) ] ,
A + B = C + D , A n cos θ i + B n cos θ i = C cos θ r + D cos θ r .
C α + D ( 1 / α ) = E β , C α cos θ r + D ( 1 / α ) cos θ r = E n β cos θ i .
β E A = τ s = 4 n cos θ i cos θ r ( α + 1 / α ) 2 n cos θ i cos θ r ( α 1 / α ) ( n 2 cos 2 θ i + cos 2 θ r ) ,
B β A = ρ s = ( α 1 / α ) ( n 2 cos 2 θ i cos 2 θ r ) ( α + 1 / α ) 2 n cos θ i cos θ r ( α 1 / α ) ( n 2 cos 2 θ i + cos 2 θ r ) .
β F 2 = τ s B 2 ,
A 2 = ρ s β B 2 .
A 1 + B 1 = C 1 + D 1 , A 1 n cos θ i + B 1 n cos θ i = C 1 cos θ r + D 1 cos θ r .
C 1 α + D 1 ( 1 / α ) = E 1 β + F 1 ( 1 / β ) , C 1 α cos θ r + D 1 ( 1 / α ) cos θ r = E 1 n β cos θ i + F 1 n ( 1 / β ) cos θ i .
τ = τ s e i ϕ ( 1 K ) ,
β E 1 A 1 = 4 n cos θ i cos θ r + ( τ / β ) [ ( α 1 / α ) ( n 2 cos 2 θ i + cos 2 θ r ) + ( α + 1 / α ) 2 n cos θ i cos θ r ] ( τ / β ) 4 n cos θ i cos θ r + [ ( α 1 / α ) ( n 2 cos 2 θ i + cos 2 θ r ) ( α + 1 / α ) 2 n cos θ i cos θ r ] .
e i Ψ A 2 A 1 = ( 1 K B ) ( α 1 / α ) 2 ( n 2 cos 2 θ i cos 2 θ r ) 2 / [ ( α 1 / α ) ( n 2 cos 2 θ i + cos 2 θ r ) ( α + 1 / α ) 2 n cos θ i cos θ r ] ( τ / β ) 4 n cos θ i cos θ r + [ ( α 1 / α ) ( n 2 cos 2 θ i + cos 2 θ r ) ( α + 1 / α ) 2 n cos θ i cos θ r ] .
x = 2 ( n 2 cos 2 θ i | cos θ r | 2 ) ,
β A 2 = τ p E 2 ,
F 2 = β ρ p E 2 .
E 2 = ( 1 K E ) exp ( i ϕ E ) E 1 ,
F 1 = ( 1 K F ) exp ( i ϕ F ) F 2 ,
F 1 = β ρ E 1 ,
ρ = ( 1 K ) β e i δ ρ p ,
B 1 A 1 = ( ρ / β ) [ ( α 1 / α ) ( cos 2 θ i + n 2 cos 2 θ r ) + ( α + 1 / α ) 2 n cos θ i cos θ r ] ( α 1 / α ) ( cos 2 θ i n 2 cos 2 θ r ) ( ρ / β ) ( α 1 / α ) ( cos 2 θ i n 2 cos 2 θ r ) [ ( α 1 / α ) ( cos 2 θ i + n 2 cos 2 θ r ) ( α + 1 / α ) 2 n cos θ i cos θ r ] ,
β E 1 A 1 = 4 n cos θ i cos θ r ( ρ / β ) ( α 1 / α ) ( cos 2 θ i n 2 cos 2 θ r ) [ ( α 1 / α ) ( cos 2 θ i + n 2 cos 2 θ r ) ( α + 1 / α ) 2 n cos θ i cos θ r ] .
β A 2 = ( 1 K E ) exp ( i ϕ E ) τ p E 1 ,
e i Ψ A 2 A 1 = ( 1 K E ) τ p 4 n cos θ i cos θ r ( ρ / β ) ( α 1 / α ) ( cos 2 θ i n 2 cos 2 θ r ) [ ( α 1 / α ) ( cos 2 θ i + n 2 cos 2 θ r ) ( α + 1 / α ) 2 n cos θ i cos θ r ]
z = 2 ( cos 2 θ i + n 2 | cos θ r | 2 ) ,
B 1 A 1 = ( 1 K ) ( cos δ + i sin δ ) z sinh γ ( x sinh γ + iy cosh γ ) sinh γ ( x sinh γ i y cosh γ ) ( 1 K ) ( cos δ + i sin δ ) z 2 sinh 2 γ ( x sinh γ i y cosh γ ) 2 ,
e i Ψ A 2 A 1 = ( 1 K E ) y 2 ( 1 K ) ( cos δ + i sin δ ) z 2 sinh 2 γ ( x sinh γ i y cosh γ ) 2 ,
T = ( x 2 + y 2 ) sinh 2 γ [ { ( 1 K ) ( x sinh γ cos δ y cosh γ sin δ ) x sinh γ } 2 + { ( 1 K ) ( x sinh γ sin δ + y cosh γ cos δ ) + y cosh γ } 2 ] [ ( 1 K ) ( x 2 + y 2 ) sinh 2 γ cos δ x 2 sinh 2 γ + y 2 cosh 2 γ ] 2 + [ ( 1 K ) ( x 2 + y 2 ) sinh 2 γ sin δ + 2 x y sinh γ cosh γ ] 2 ,
R = ( 1 K E ) 2 y 4 [ ( 1 K ) ( x 2 + y 2 ) sinh 2 γ cos δ x 2 sinh 2 γ + y 2 cosh 2 γ ] 2 + [ ( 1 K ) ( x 2 + y 2 ) sinh 2 γ sin δ + 2 x y sinh γ cosh γ ] 2 .
B 1 A 1 = ( ρ / β ) [ ( α 1 / α ) ( n 2 cos 2 θ i + cos 2 θ r ) + ( α + 1 / α ) 2 n cos θ i cos θ r ] ( α 1 / α ) ( n 2 cos 2 θ i cos 2 θ r ) ( ρ / β ) ( α 1 / α ) ( n 2 cos 2 θ i cos 2 θ r ) [ ( α 1 / α ) ( n 2 cos 2 θ i cos 2 θ r ) ( α + 1 / α ) 2 n cos θ i cos θ r ] ,
e i Ψ A 2 A 1 = ( 1 K E ) τ s 4 n cos θ i cos θ r ( ρ / β ) ( α 1 / α ) ( n 2 cos 2 θ i cos 2 θ r ) [ ( α 1 / α ) ( n 2 cos 2 θ i + cos 2 θ r ) ( α + 1 / α ) 2 n cos θ cos θ r ] .
z = 2 ( n 2 cos 2 θ i + | cos θ r | 2 ) ,
β E 1 = τ p A 1 ,
B 1 = β ρ p A 1 .
E 2 = ( 1 K E ) exp ( i ϕ E ) E 1 ,
B 2 = ( 1 K B ) exp ( i ϕ B ) B 1 ,
B 2 = ρ A 1 ,
E 2 = τ A 1 ,
ρ = ( 1 K B ) β e i ϕ B ρ p ,
τ = ( 1 K E ) e i ϕ E τ p / β .
A 2 cos θ i + B 2 cos θ i = C 2 cos θ r + D 2 cos θ r ,
A 2 n + B 2 n = C 2 + D 2 .
C 2 ( 1 / α ) cos θ r + D 2 α cos θ r = E 2 ( 1 / β ) cos θ i + F 2 β cos θ i , C 2 ( 1 / α ) + D 2 α = E 2 n ( 1 / β ) + F 2 n β .
β F 2 A 1 = ( τ / β ) ( α 1 / α ) ( cos 2 θ i n 2 cos 2 θ r ) ρ 4 n cos θ i cos θ r ( α 1 / α ) ( cos 2 θ i + n 2 cos 2 θ r ) ( α + 1 / α ) 2 n cos θ i cos θ r ,
A 2 A 1 = ρ ( α 1 / α ) ( cos 2 θ i n 2 cos 2 θ r ) ( τ / β ) 4 n cos θ i cos θ r ( α 1 / α ) ( cos 2 θ i + n 2 cos 2 θ r ) ( α + 1 / α ) 2 n cos θ i cos θ r .
β F 2 A 1 = i y z sinh γ [ ( 1 K E ) e i Ψ E + ( 1 K B ) e i Ψ B ] ( x sinh γ i y cosh γ ) 2 ,
A 2 A 1 = ( 1 K E ) e i Ψ E y 2 ( 1 K B ) e i Ψ B z 2 sinh 2 γ ( x sinh γ i y cosh γ ) 2 .
T = y 2 ( x 2 + y 2 ) sinh 2 γ { [ ( 1 K E ) cos Ψ E + ( 1 K B ) cos Ψ B ] 2 + [ ( 1 K E ) sin Ψ E + ( 1 K B ) sin Ψ B ] 2 } ( x 2 sinh 2 γ + y 2 cosh 2 γ ) 2 ,
R = [ ( 1 K E ) y 2 cos Ψ E ( 1 K B ) ( x 2 y 2 ) sinh 2 γ cos Ψ B ] 2 + [ ( 1 K E ) y 2 sin Ψ E ( 1 K B ) ( x 2 y 2 ) sinh 2 γ sin Ψ B ] 2 ( x 2 sinh 2 γ + y 2 cosh 2 γ ) 2 .
A 2 + B 2 = C 2 + D 2 , A 2 n cos θ i + B 2 n cos θ i = C 2 cos θ r + D 2 cos θ r .
C 2 ( 1 / α ) + D 2 α = E 2 ( 1 / β ) + F 2 β , C 2 ( 1 / α ) cos θ r + D 2 α cos θ r = E 2 n ( 1 / β ) cos θ i + F 2 n β cos θ i .
β F 2 A 1 = ( τ / β ) ( α 1 / α ) ( n 2 cos 2 θ i cos 2 θ r ) ρ 4 n cos θ i cos θ r ( α 1 / α ) ( n 2 cos 2 θ i cos 2 θ r ) ( α + 1 / α ) 2 n cos θ i cos θ r ,
A 2 A 1 = ρ ( α 1 / α ) ( n 2 cos 2 θ i cos 2 θ r ) ( τ / β ) 4 n cos θ i cos θ r ( α 1 / α ) ( n 2 cos 2 θ i cos 2 θ r ) ( α + 1 / α ) 2 n cos θ i cos θ r .
cos θ r = n ( sin 2 θ c sin 2 θ i ) 1 / 2 ,
α = e i γ = exp [ i k A d ( sin 2 θ c sin 2 θ i ) 1 / 2 ] .
x = 2 ( cos 2 θ i + n 2 cos 2 θ r ) ,
y = 4 n cos θ i cos θ r .
β E 1 A 1 = y ( i x sin γ y cos γ ) + ( 1 K ) ( cos δ + i sin δ ) y ( i x sin γ + y cos γ ) ( 1 K ) ( cos δ + i sin δ ) y 2 ( i x sin γ y cos γ ) 2 ,
e i Ψ A 2 A 1 = ( 1 K B ) sin 2 γ ( x 2 y 2 ) ( 1 K ) ( cos δ + i sin δ ) y 2 ( i x sin γ y cos γ ) 2 .
T = { y 2 cos γ [ ( 1 K ) cos δ 1 ] x y ( 1 K ) sin γ sin δ } 2 + { y 2 ( 1 K ) cos γ sin δ + x y sin γ [ ( 1 K ) cos δ + 1 ) ] } 2 { y 2 [ ( 1 K ) cos δ cos 2 γ ] + x 2 sin 2 γ } 2 + [ y 2 ( 1 K ) sin δ + 2 x y sin γ cos γ ] 2 .
R = [ ( 1 K B ) sin 2 γ ( x 2 y 2 ) ] 2 { y 2 [ ( 1 K ) cos δ cos 2 γ ] + x 2 sin 2 γ } 2 + [ y 2 ( 1 K ) sin δ + 2 x y sin γ cos γ ] 2 .
x = 2 ( n 2 cos 2 θ i + cos 2 θ r ) .
z = 2 ( cos 2 θ i n 2 cos 2 θ r ) ,
B 1 A 1 = z sin γ [ ( 1 K ) ( cos δ + i sin δ ) ( x sin γ i y cos γ ) x sin γ i y cos γ ] ( 1 K ) ( cos δ + i sin δ ) z 2 sin 2 γ + ( y cos γ i x sin γ ) 2 ,
e i Ψ A 2 A 1 = ( 1 K E ) y 2 ( 1 K ) ( cos δ + i sin δ ) z 2 sin 2 γ + ( y cos γ i x sin γ ) 2 ,
T = ( x 2 y 2 ) sin 2 γ { [ ( 1 K ) ( y cos γ cos δ x sin γ sin δ ) + y cos γ ] 2 + [ ( 1 K ) ( x sin γ cos δ + y cos γ sin δ ) x sin γ ] 2 } [ ( 1 K ) ( y 2 x 2 ) sin 2 γ cos δ y 2 cos 2 γ + x 2 sin 2 γ ] 2 + [ ( 1 K ) ( y 2 x 2 ) sin 2 γ sin δ + 2 x y sin γ cos γ ] 2 .
R = ( 1 K E ) 2 y 4 [ ( 1 K ) ( y 2 x 2 ) sin 2 γ cos δ y 2 cos 2 γ + x 2 sin 2 γ ] 2 + [ ( 1 K ) ( y 2 x 2 ) sin 2 γ sin δ + 2 x y sin γ cos γ ] 2 .
z = 2 ( n 2 cos 2 θ i cos 2 θ r ) ,
β F 2 A 1 = i y z sin γ [ ( 1 K B ) e i Ψ B + ( 1 K E ) e i Ψ E ] ( y cos γ i x sin γ ) 2 ,
A 2 A 1 = ( 1 K E ) e i Ψ E y 2 ( 1 K E ) e i Ψ B z 2 sin 2 γ ( y cos γ i x sin γ ) 2 .
T = y 2 ( x 2 y 2 ) sin 2 γ { [ ( 1 K B ) cos Ψ B + ( 1 K E ) cos Ψ E ] 2 + [ ( 1 K E ) sin Ψ B + ( 1 K E ) sin Ψ E ] 2 } ( y 2 cos 2 γ x 2 sin 2 γ ) 2 + ( 2 x y sin γ cos γ ) 2 .
R = [ ( 1 K E ) y 2 cos Ψ E ( 1 K B ) ( x 2 y 2 ) sin 2 γ cos Ψ B ] 2 + [ ( 1 K E ) y 2 sin Ψ E ( 1 K B ) ( x 2 y 2 ) sin 2 γ sin Ψ B ] 2 ( y 2 cos 2 γ x 2 sin 2 γ ) 2 + ( 2 x y sin γ cos γ ) 2 .