Abstract

The flow-graph approach (FGA) is applied to optical analysis of isotropic stratified planar structures (ISPS’s) at inclined light incidence. Conditions for the presence of coherent and noncoherent light interaction within ISPS’s are determined. Examples of the use of FGA for calculation of the transmission and the reflection of two-layer ISPS’s for different types of light interaction are given. The advantages of the use of FGA for optical analysis of ISPS’s are discussed.

© 1994 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 1, pp. 166–170.
  2. D. K. Cheng, Field and Wave Electromagnetics (Addison-Wesley, Reading, Mass., 1989), Chap. 8, pp. 406–416.
  3. C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, New York, 1989), Chap. 8, pp. 414–424.
  4. L. Levi, Applied Optics: A Guide to Modern Optical Systems Design (Wiley, New York, 1968), Chap. 8, pp. 367–369.
  5. F. A. Jenkins, H. E. White, Fundamentals of Optics, 3rd ed. (McGraw-Hill, New York, 1957), Chap. 4, pp. 44–62.
  6. S. Ramo, J. R. Whinnery, T. van Duzer, Fields and Waves in Communication Electronics (Wiley, New York, 1965), Chap. 9, pp. 524–526.
  7. D. B. Kushev, “Signal flow graphs in the analysis of the transmission and reflection of plane-parallel layers,” Zh. Prikl. Spektrosk. 22, 746–754 (1975).
  8. M. H. Dunn, “Use of flow graphs in the analysis of optical cavities,” Appl. Opt. 10, 1393–1397 (1971).
    [CrossRef] [PubMed]
  9. R. Swanepoel, “Determination of the thickness and optical constants of amorphous silicon,” J. Phys. E. 16, 1214–1222 (1983).
    [CrossRef]

1983 (1)

R. Swanepoel, “Determination of the thickness and optical constants of amorphous silicon,” J. Phys. E. 16, 1214–1222 (1983).
[CrossRef]

1975 (1)

D. B. Kushev, “Signal flow graphs in the analysis of the transmission and reflection of plane-parallel layers,” Zh. Prikl. Spektrosk. 22, 746–754 (1975).

1971 (1)

Balanis, C. A.

C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, New York, 1989), Chap. 8, pp. 414–424.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 1, pp. 166–170.

Cheng, D. K.

D. K. Cheng, Field and Wave Electromagnetics (Addison-Wesley, Reading, Mass., 1989), Chap. 8, pp. 406–416.

Dunn, M. H.

Jenkins, F. A.

F. A. Jenkins, H. E. White, Fundamentals of Optics, 3rd ed. (McGraw-Hill, New York, 1957), Chap. 4, pp. 44–62.

Kushev, D. B.

D. B. Kushev, “Signal flow graphs in the analysis of the transmission and reflection of plane-parallel layers,” Zh. Prikl. Spektrosk. 22, 746–754 (1975).

Levi, L.

L. Levi, Applied Optics: A Guide to Modern Optical Systems Design (Wiley, New York, 1968), Chap. 8, pp. 367–369.

Ramo, S.

S. Ramo, J. R. Whinnery, T. van Duzer, Fields and Waves in Communication Electronics (Wiley, New York, 1965), Chap. 9, pp. 524–526.

Swanepoel, R.

R. Swanepoel, “Determination of the thickness and optical constants of amorphous silicon,” J. Phys. E. 16, 1214–1222 (1983).
[CrossRef]

van Duzer, T.

S. Ramo, J. R. Whinnery, T. van Duzer, Fields and Waves in Communication Electronics (Wiley, New York, 1965), Chap. 9, pp. 524–526.

Whinnery, J. R.

S. Ramo, J. R. Whinnery, T. van Duzer, Fields and Waves in Communication Electronics (Wiley, New York, 1965), Chap. 9, pp. 524–526.

White, H. E.

F. A. Jenkins, H. E. White, Fundamentals of Optics, 3rd ed. (McGraw-Hill, New York, 1957), Chap. 4, pp. 44–62.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 1, pp. 166–170.

Appl. Opt. (1)

J. Phys. E. (1)

R. Swanepoel, “Determination of the thickness and optical constants of amorphous silicon,” J. Phys. E. 16, 1214–1222 (1983).
[CrossRef]

Zh. Prikl. Spektrosk. (1)

D. B. Kushev, “Signal flow graphs in the analysis of the transmission and reflection of plane-parallel layers,” Zh. Prikl. Spektrosk. 22, 746–754 (1975).

Other (6)

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 1, pp. 166–170.

D. K. Cheng, Field and Wave Electromagnetics (Addison-Wesley, Reading, Mass., 1989), Chap. 8, pp. 406–416.

C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, New York, 1989), Chap. 8, pp. 414–424.

L. Levi, Applied Optics: A Guide to Modern Optical Systems Design (Wiley, New York, 1968), Chap. 8, pp. 367–369.

F. A. Jenkins, H. E. White, Fundamentals of Optics, 3rd ed. (McGraw-Hill, New York, 1957), Chap. 4, pp. 44–62.

S. Ramo, J. R. Whinnery, T. van Duzer, Fields and Waves in Communication Electronics (Wiley, New York, 1965), Chap. 9, pp. 524–526.

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Figures (7)

Fig. 1
Fig. 1

Example of a flow graph.

Fig. 2
Fig. 2

Schematic of the transmission of light through a two-layer ISPS (top) and the corresponding flow graph (bottom).

Fig. 3
Fig. 3

Schematic (top) and flow graph (bottom) of the reflection by a two-layer ISPS.

Fig. 4
Fig. 4

Transmission spectrum T(λ) of the model system for s- and p-polarized monochromatic light for different angles of light incidence Φ.

Fig. 5
Fig. 5

Reflection spectrum R(λ) of the model system for different polarizations and angles of incidence of monochromatic light.

Fig. 6
Fig. 6

Transmission spectrum T(λ) of the model system at Φ = 0° for slit width Δλ = 0 nm and Δλ = 4 nm.

Fig. 7
Fig. 7

Reflection spectrum R(λ) of the model system at Φ = 0° for Δλ = 0 nm and Δλ = 4 nm.

Equations (20)

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G = P n [ 1 - L n ( 1 ) + L n ( 2 ) - + ( - 1 ) i L n ( i ) + ] 1 - L ( 1 ) + L ( 2 ) - + ( - 1 ) i L ( i ) + ,
P 1 = a 12 a 23 a 37 a 78 a 89 ,             P 2 = a 12 a 25 a 54 a 46 a 67 a 78 a 89 ,
L 1 = a 25 a 54 a 42 ,             L 2 = a 54 a 46 a 65 ,             L 3 = a 67 a 78 a 86 , L 4 = a 23 a 37 a 78 a 86 a 65 a 54 a 42 .
G = P 1 ( 1 - L 2 ) + P 2 1 - L 1 - L 2 - L 3 - L 4 + L 1 L 3
t ˙ s j j + 1 = 2 N ˙ j cos Φ ˙ j N ˙ j cos Φ ˙ j + N ˙ j + 1 cos Φ ˙ j + 1 , t ˙ p j j + 1 = 2 N ˙ j cos Φ ˙ j N ˙ j cos Φ ˙ j + 1 + N ˙ j + 1 cos Φ ˙ j , r ˙ s j j + 1 = N ˙ j cos Φ ˙ j - N ˙ j + 1 cos Φ ˙ j + 1 N ˙ j cos Φ ˙ j + N ˙ j + 1 cos Φ ˙ j + 1 , r ˙ p j j + 1 = N ˙ j cos Φ ˙ j + 1 - N ˙ j + 1 cos Φ ˙ j N ˙ j cos Φ ˙ j + 1 + N ˙ j + 1 cos Φ ˙ j ,
T j j + 1 = Re ( N ˙ j + 1 cos Φ ˙ j + 1 ) Re ( N ˙ j cos Φ ˙ j ) t ˙ j j + 1 t * j j + 1 = P j + 1 P j τ j j + 1 2 , R j j + 1 = r ˙ j j + 1 r * j j + 1 = ρ j j + 1 2 ,
f ˙ j = exp [ - i ( 2 π λ ) ( P j - i Q j ) d j ] , F j = f ˙ j f * j = exp ( - 4 π Q j d j λ ) = exp ( - α j d j ) ,
P j = 2 2 { [ ( n j 2 - k j 2 - sin 2 Φ ) 2 + 4 n j 2 k j 2 ] 1 / 2 + ( n j 2 - k j 2 - sin 2 Φ ) } 1 / 2 , Q j = 2 2 { [ ( n j 2 - k j 2 - sin 2 Φ ) 2 + 4 n j 2 k j 2 ] 1 / 2 - ( n j 2 - k j 2 - sin 2 Φ ) } 1 / 2
T n = ( T s + T p ) / 2 ,             R n = ( R s + R p ) / 2 ,
d s < λ 2 cos Φ s / ( 4 π n s Δ λ ) .
d > λ 2 / ( 4 P Δ λ ) .
G 1 = abcde 1 - bjfi - dlhk - bcdlhgfi + bjfidlhk .
G 2 = mpn + mabjfon ( 1 - dlhk ) + mabcdlhgfon 1 - bjfi - dlhk - bcdlhgfi + bjfidlhk .
t ˙ 02 = t ˙ 01 t ˙ 12 f ˙ 1 1 - r ˙ 10 r ˙ 12 f ˙ 1 2             r 02 = r ˙ 01 + t ˙ 01 t ˙ 10 r ˙ 12 f ˙ 1 2 1 - r ˙ 10 r ˙ 12 f ˙ 1 2 , r 20 = r ˙ 21 + t ˙ 21 t ˙ 12 r ˙ 10 f ˙ 1 2 1 - r ˙ 10 r ˙ 12 f ˙ 1 2 , T 02 = P 2 cos Φ × τ 01 2 τ 12 2 F 1 - 1 + ρ 01 2 ρ 12 2 F 1 + 2 ρ 01 ρ 12 cos ( Δ 01 + Δ 12 + 2 δ 1 ) , R 02 = ρ 01 2 F 1 - 1 + ρ 12 2 F 1 + 2 ρ 01 ρ 12 cos ( - Δ 01 + Δ 12 + 2 δ 1 ) F 1 - 1 + ρ 01 2 ρ 12 2 F 1 + 2 ρ 01 ρ 12 cos ( Δ 01 + Δ 12 + 2 δ 1 ) , R 20 = ρ 01 2 F 1 + ρ 12 2 F 1 - 1 + 2 ρ 01 ρ 12 cos ( Δ 01 - Δ 12 + 2 δ 1 ) F 1 - 1 + ρ 01 2 ρ 12 2 F 1 + 2 ρ 01 ρ 12 cos ( Δ 01 + Δ 12 + 2 δ 1 ) , T = T 02 T 20 F 2 1 - R 20 R 20 F 2 2 = ( τ 01 τ 12 τ 20 ) 2 a 1 + b 1 cos x + c 1 sin x ,
a 1 = ( F 1 F 2 ) - 1 - ( ρ 01 ρ 20 ) 2 F 1 F 2 + ρ 12 2 ( ρ 01 2 F 1 F 2 - 1 - ρ 20 2 F 1 - 1 F 2 ) , b 1 = 2 ρ 01 ρ 12 ρ 20 [ ( ρ 20 F 2 ) - 1 cos Δ 1 - ρ 20 F 2 cos Δ 2 ] , c = - 2 ρ 01 ρ 12 ρ 20 [ ( ρ 20 F 2 ) - 1 sin Δ 1 - ρ 20 F 2 sin Δ 2 ] , x = 2 δ 1 = 4 π P 1 d 1 / λ ,             Δ 1 = Δ 01 + Δ 12 , Δ 2 = Δ 01 - Δ 12 , R = R 02 + ( 1 + Q 2 2 P 2 2 ) T 02 R 20 F 2 2 1 - R 20 R 20 F 2 2 = a 2 + b 2 cos x + c 2 sin x a 3 + b 3 cos x + c 3 sin x + ( 1 + Q 2 2 P 2 2 ) ( P 2 cos Φ ) 2 ( τ 01 2 τ 12 2 ρ 20 ) 2 F 2 a 1 + b 1 cos x + c 1 sin x ,
a 2 = ρ 01 2 F 1 - 1 + ρ 12 2 F 1 , b 2 = 2 ρ 01 ρ 12 cos Δ 2 , a 3 = F 1 - 1 + ρ 01 2 ρ 12 2 F 1 , b 3 = 2 ρ 01 ρ 12 cos Δ 1 , c 2 = 2 ρ 01 ρ 12 sin Δ 2 , c 3 = - 2 ρ 01 ρ 12 sin Δ 1 ,
n 1 = 3 × 10 5 λ 2 + 2.6 , k 1 = λ 4 π × 10 1.5 × ( 10 6 / λ 2 ) - 8 , n 2 = 1.51 , k 2 = 0 ,             ( λ in nanometers ) d 1 = 1000 nm ,
T = λ - Δ λ / 2 λ + Δ λ / 2 T ( λ ) d λ Δ λ = ( τ 01 τ 12 τ 20 ) 2 ( x 2 - x 1 ) x 1 x 2 d x a 1 + b 1 cos x + c 1 sin x = 2 ( τ 01 τ 12 τ 23 ) 2 ( x 2 - x 1 ) ( a 1 2 - b 1 2 - c 1 2 ) 1 / 2 × tan - 1 ( a 1 - b 1 ) tan x 2 + c 1 ( a 1 2 - b 1 2 - c 1 2 ) 1 / 2 | x 1 x 2 ,
x 1 = 4 π P 11 a 1 λ + Δ λ 2 ,             P 11 = P 1 λ + Δ λ / 2 , x 2 = 4 π P 12 a 1 λ - Δ λ 2 ,             P 12 = P 1 λ - Δ λ / 2 .
R = 1 ( x 2 - x 1 ) { b 2 c 3 - c 2 b 3 b 3 2 - c 3 2 × ln ( a 3 + b 3 cos x 2 + c 3 sin x 2 a 3 + b 3 cos x 1 + c 3 sin x 1 ) + b 2 b 3 + c 2 c 3 b 3 2 + c 3 2 ( x 2 - x 1 ) + ( 1 + Q 0 2 P 0 2 ) × ( P 0 τ 01 2 τ 12 2 ρ 20 cos ϕ ) 2 ( a 2 - b 2 b 3 + c 2 c 3 b 3 2 - c 3 2 a 3 ) × 2 F 2 ( a 3 2 - b 3 2 - c 3 2 ) 1 / 2 × tan - 1 ( a 3 - b 3 ) tan x 2 + c 3 ( a 3 2 - b 3 2 - c 3 2 ) 1 / 2 | x 1 x 2 } .

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