Abstract

In optical coating calculations, a 2 × 2 matrix formalism is generally used to describe the effect of each layer on the electromagnetic field. In current applications, the layers are considered homogeneous, and, therefore, the matrix used is restricted to this particular case. In this paper, a 2 × 2 matrix describing an inhomogeneous dielectric thin film is derived from Maxwell’s equations. The inhomogeneity is assumed to vary along the direction perpendicular to the film plane. No restriction is made on the amplitude of its variation. The matrix is illustrated in the case of a rugate filter designed according to Sossi’s Fourier transform technique. An improved approximation for the Fourier transform technique is then introduced.

© 1988 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964), p. 55.
  2. H. A. Macleod, Thin-Film Optical Filters (Macmillan, New York, 1986).
    [CrossRef]
  3. R. Jacobsson, J. O. Martensson, “Evaporated Inhomogeneous Thin Films,” Appl. Opt. 5, 29 (1966).
    [CrossRef] [PubMed]
  4. R. Jacobsson, “Inhomogeneous and Coevaporated Homogeneous Films for Optical Applications,” Phys. Thin Films 8, 51 (1975).
  5. R. Jacobsson, “Optical Properties of a Class of Inhomogeneous Thin Films,” Opt. Acta 10, 309 (1963).
    [CrossRef]
  6. R. Jacobsson, “Light Reflection from Films of Continuously Varying Refractive Index,” Prog. Opt. 5, 247 (1966).
    [CrossRef]
  7. E. Delano, “Fourier Synthesis of Multilayer Filters,” J. Opt. Soc. Am. 57, 1529 (1967).
    [CrossRef]
  8. L. Sossi, “On the Theory of the Synthesis of Multilayer Dielectric Light Filters,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 25, 171 (1976).
  9. L. Sossi, “A Method for the Synthesis of Multilayer Dielectric Interference Coatings,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 23, 229 (1974).
  10. L. Sossi, P. Kard, “On the Theory of the Reflection and Transmission of Light by a Thin Inhomogeneous Dielectric Film,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 17, 41 (1968).
  11. J. A. Dobrowolski, D. G. Lowe, “Optical Thin Film Synthesis Program Based on the Use of the Fourier Transforms,” Appl. Opt. 17, 3039 (1978).
    [CrossRef] [PubMed]
  12. D. W. Berreman, “Optics in Stratified and Anisotropic Media: 4 × 4-Matrix Formulation,” J. Opt. Soc. Am. 62, 502 (1972).
    [CrossRef]
  13. H. Bremmer, “The Propagation of Electromagnetic Waves through a Stratified Medium and its W.K.B. Approximation for Oblique Incidence,” Physica 15, 593 (1949).
    [CrossRef]
  14. L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1960), particularly p. 193 and following.
  15. W. H. Southwell, “Use of Gradient Index for Spectral Filters,” Proc. Soc. Photo-Opt. Instrum. Eng. 464, 110 (1984).

1984 (1)

W. H. Southwell, “Use of Gradient Index for Spectral Filters,” Proc. Soc. Photo-Opt. Instrum. Eng. 464, 110 (1984).

1978 (1)

1976 (1)

L. Sossi, “On the Theory of the Synthesis of Multilayer Dielectric Light Filters,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 25, 171 (1976).

1975 (1)

R. Jacobsson, “Inhomogeneous and Coevaporated Homogeneous Films for Optical Applications,” Phys. Thin Films 8, 51 (1975).

1974 (1)

L. Sossi, “A Method for the Synthesis of Multilayer Dielectric Interference Coatings,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 23, 229 (1974).

1972 (1)

1968 (1)

L. Sossi, P. Kard, “On the Theory of the Reflection and Transmission of Light by a Thin Inhomogeneous Dielectric Film,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 17, 41 (1968).

1967 (1)

1966 (2)

R. Jacobsson, J. O. Martensson, “Evaporated Inhomogeneous Thin Films,” Appl. Opt. 5, 29 (1966).
[CrossRef] [PubMed]

R. Jacobsson, “Light Reflection from Films of Continuously Varying Refractive Index,” Prog. Opt. 5, 247 (1966).
[CrossRef]

1963 (1)

R. Jacobsson, “Optical Properties of a Class of Inhomogeneous Thin Films,” Opt. Acta 10, 309 (1963).
[CrossRef]

1949 (1)

H. Bremmer, “The Propagation of Electromagnetic Waves through a Stratified Medium and its W.K.B. Approximation for Oblique Incidence,” Physica 15, 593 (1949).
[CrossRef]

Berreman, D. W.

Born, M.

M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964), p. 55.

Brekhovskikh, L. M.

L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1960), particularly p. 193 and following.

Bremmer, H.

H. Bremmer, “The Propagation of Electromagnetic Waves through a Stratified Medium and its W.K.B. Approximation for Oblique Incidence,” Physica 15, 593 (1949).
[CrossRef]

Delano, E.

Dobrowolski, J. A.

Jacobsson, R.

R. Jacobsson, “Inhomogeneous and Coevaporated Homogeneous Films for Optical Applications,” Phys. Thin Films 8, 51 (1975).

R. Jacobsson, “Light Reflection from Films of Continuously Varying Refractive Index,” Prog. Opt. 5, 247 (1966).
[CrossRef]

R. Jacobsson, J. O. Martensson, “Evaporated Inhomogeneous Thin Films,” Appl. Opt. 5, 29 (1966).
[CrossRef] [PubMed]

R. Jacobsson, “Optical Properties of a Class of Inhomogeneous Thin Films,” Opt. Acta 10, 309 (1963).
[CrossRef]

Kard, P.

L. Sossi, P. Kard, “On the Theory of the Reflection and Transmission of Light by a Thin Inhomogeneous Dielectric Film,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 17, 41 (1968).

Lowe, D. G.

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters (Macmillan, New York, 1986).
[CrossRef]

Martensson, J. O.

Sossi, L.

L. Sossi, “On the Theory of the Synthesis of Multilayer Dielectric Light Filters,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 25, 171 (1976).

L. Sossi, “A Method for the Synthesis of Multilayer Dielectric Interference Coatings,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 23, 229 (1974).

L. Sossi, P. Kard, “On the Theory of the Reflection and Transmission of Light by a Thin Inhomogeneous Dielectric Film,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 17, 41 (1968).

Southwell, W. H.

W. H. Southwell, “Use of Gradient Index for Spectral Filters,” Proc. Soc. Photo-Opt. Instrum. Eng. 464, 110 (1984).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964), p. 55.

Appl. Opt. (2)

Eesti NSV Tead. Akad. Toim. Fuus. Mat. (3)

L. Sossi, “On the Theory of the Synthesis of Multilayer Dielectric Light Filters,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 25, 171 (1976).

L. Sossi, “A Method for the Synthesis of Multilayer Dielectric Interference Coatings,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 23, 229 (1974).

L. Sossi, P. Kard, “On the Theory of the Reflection and Transmission of Light by a Thin Inhomogeneous Dielectric Film,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 17, 41 (1968).

J. Opt. Soc. Am. (2)

Opt. Acta (1)

R. Jacobsson, “Optical Properties of a Class of Inhomogeneous Thin Films,” Opt. Acta 10, 309 (1963).
[CrossRef]

Phys. Thin Films (1)

R. Jacobsson, “Inhomogeneous and Coevaporated Homogeneous Films for Optical Applications,” Phys. Thin Films 8, 51 (1975).

Physica (1)

H. Bremmer, “The Propagation of Electromagnetic Waves through a Stratified Medium and its W.K.B. Approximation for Oblique Incidence,” Physica 15, 593 (1949).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

W. H. Southwell, “Use of Gradient Index for Spectral Filters,” Proc. Soc. Photo-Opt. Instrum. Eng. 464, 110 (1984).

Prog. Opt. (1)

R. Jacobsson, “Light Reflection from Films of Continuously Varying Refractive Index,” Prog. Opt. 5, 247 (1966).
[CrossRef]

Other (3)

M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964), p. 55.

H. A. Macleod, Thin-Film Optical Filters (Macmillan, New York, 1986).
[CrossRef]

L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1960), particularly p. 193 and following.

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Figures (10)

Fig. 1
Fig. 1

Refractive index of an inhomogeneous single layer embedded in media of indices η = η0 and η = ηs.

Fig. 2
Fig. 2

Right-handed coordinate system showing the electromagnetic field in the case of a TE wave.

Fig. 3
Fig. 3

Refractive-index profile of the 50% reflector designed according to Sossi’s technique.

Fig. 4
Fig. 4

Refractive-index profile of the 90% reflector designed according to Sossi’s technique.

Fig. 5
Fig. 5

Reflectance of the filter presenting the refractive-index profile plotted in Fig. 3. The film is deposited onto glass and covered by glass. The circles indicate the result obtained by matrix multiplication, the squares correspond to the first order, the triangles to the second order, and the plus signs to the third-order approximations of the matrix presented in this paper. The solid line represents the desired reflectance.

Fig. 6
Fig. 6

Reflectance of the filter presenting the refractive-index profile plotted in Fig. 4. The film is deposited onto glass and covered by glass. The circles indicate the result obtained by matrix multiplication, the squares correspond to the seventh order, the triangles to the eighth order, and the plus signs to the ninth-order approximations of the matrix presented in this paper. The solid line represents the desired reflectance band.

Fig. 7
Fig. 7

Reflectance curves obtained for a 99% reflector with a 20-nm halfwidth designed according to Sossi’s technique (circles). The squares indicate the result obtained when using the - log T approximation. Both curves were computed using the matrix multiplication technique. The solid line corresponds to the desired reflectance.

Fig. 8
Fig. 8

Reflectance curves obtained for a 99.9% reflector with a 20-nm halfwidth designed according to Sossi’s technique (circles). The squares indicate the result obtained when using the - log T approximation. Both curves were computed using the matrix multiplication technique. The solid line corresponds to the desired reflectance.

Fig. 9
Fig. 9

Reflectance curves obtained for a 90% reflector with a 20-nm halfwidth designed according to Sossi’s technique (squares). The circles indicate the result obtained when using the - log T approximation. Both curves were computed using the matrix multiplication technique. The solid line corresponds to the desired reflectance.

Fig. 10
Fig. 10

Case of a 90% reflector designed using the Q ( σ ) = - log T The circles correspond to the reflectance obtained by matrix multiplication, the triangles correspond to the second order, the plus signs to the third order, the squares to the fourth order, and the crosses to the fifth-order approximation of the matrix presented in this paper. The solid line is the desired reflectance curve.

Tables (3)

Tables Icon

Table I Influence of the Order of the Computation on the Reflectance Value for Three Wavelengths In the Case of a 50% Reflector Designed According to Sossi’s Fourier Transform Technique

Tables Icon

Table II Influence of the Order of the Computation on the Reflectance Value for Three Wavelengths In the Case of a 90% Reflector Designed According to Sossi’s Fourier Transform Technique

Tables Icon

Table III Influence of the Order of the Computation on the Reflectance Value for Three Wavelengths in the Case of a 90 % Reflector Designed Using Q ( σ ) = - log T ( σ )

Equations (68)

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curl H = i ω E ,
curl E = - i ω μ H ,
div H = 0 ,
div E = 0 ,
E H ,             μ ,
H - E ,             μ .
E = [ E x ( z ) exp [ i ( ω t - k S y ) ] 0 0 ] ,
H = [ 0 H y ( z ) exp [ i ( ω t - k S y ) ] H z ( z ) exp [ i ( ω t - k S y ) ] ] ,
Z 0 H y ( z ) = - i k ( r - S 2 μ r ) E x ( z ) , E x ( z ) = - i k μ r Z 0 H y ( z ) , H z ( z ) = - ( S Z 0 μ r ) E x ( z ) ,
u ( s ) = cos ϕ ( z ) Z c ( z ) with Z c ( z ) = Z 0 μ r r , γ ( z ) = r μ r cos ϕ ( z )
E x ( z ) = - i k γ ( z ) u ( z ) H y ( z ) ;
H y ( z ) = - i k γ ( z ) u ( z ) E x ( z ) .
β = 2 π λ 0 z γ ( z 1 ) d z 1 ,
E ¯ x ( β ) = E x ( β ) u ( β ) , H ¯ y ( β ) =             H y ( β ) u ( β ) .
E ¯ x ( β ) - u ( β ) 2 u ( β ) E ¯ x ( β ) = - i H ¯ y ( β ) ,
H ¯ y ( β ) + u ( β ) 2 u ( β ) H ¯ y ( β ) = - i E ¯ x ( β ) ,
E ¯ y ( β ) + u ( β ) 2 u ( β ) E ¯ y ( β ) = i H ¯ x ( β ) ,
H ¯ x ( β ) - u ( β ) 2 u ( β ) H ¯ x ( β ) = i E ¯ y ( β ) ,
E ¯ y ( β ) = E y ( β ) u ( β ) , H ¯ x ( β ) = H x ( β ) u ( β ) .
E ¯ x ( β ) - η ( β ) 2 η ( β ) E ¯ x ( β ) = - i H ¯ y ( β ) ;
H ¯ y ( β ) + η ( β ) 2 η ( β ) H ¯ y ( β ) = - i E ¯ x ( β ) ,
E ¯ x ( β ) = A 1 F ( β ) + A 2 G ( β ) ; H ¯ y ( β ) = B 1 K ( β ) + B 2 L ( β ) ;
E ¯ x ( 0 ) = E ¯ 0 , H ¯ y ( 0 ) = H ¯ 0 ,
E ¯ x ( β ) = E ¯ 0 F ( β ) - i H ¯ 0 G ( β ) ,
H ¯ y ( β ) = H ¯ 0 L ( β ) - i E ¯ 0 K ( β ) .
E 0 = η ( β ) η ( 0 ) L ( β ) E x + i η ( β ) η ( 0 ) G ( β ) H y ,
H 0 = η ( 0 ) η ( β ) F ( β ) H y + i η ( β ) η ( 0 ) K ( β ) E x ,
P = [ L ( β ) η ( β ) η ( 0 ) i G ( β ) η ( β ) η ( 0 ) i K ( β ) η ( β ) η ( 0 ) F ( β ) η ( 0 ) η ( β ) ] ,
P [ E x H y ] = [ E 0 H 0 ] .
F ( β ) = r ( β ) F ( β ) - K ( β ) ,
K ( β ) = F ( β ) - r ( β ) K ( β ) , F ( 0 ) = 1 ,             K ( 0 ) = 0 ,
G ( β ) = r ( β ) G ( β ) + L ( β ) ,
L ( β ) = - G ( β ) - r ( β ) L ( β ) , G ( 0 ) = 0 ,             L ( 0 ) = 1 ,
r ( β ) = η ( β ) 2 η ( β ) .
F ( β ) = f ( β ) cos β + k ( β ) sin β , K ( β ) = f ( β ) sin β - k ( β ) cos β , G ( β ) = l ( β ) sin β + g ( β ) cos β , L ( β ) = l ( β ) cos β - g ( β ) sin β
S ( β ) = U ( β ) S ( β ) ,
U ( β ) = [ r ( β ) cos 2 β r ( β ) sin 2 β r ( β ) sin 2 β - r ( β ) cos 2 β ] ,
[ f ( β ) k ( β ) ] ,
[ f ( 0 ) k ( 0 ) ] = [ 1 0 ] ,
[ g ( β ) l ( β ) ] ,
[ g ( 0 ) l ( 0 ) ] = [ 0 1 ] .
A ( β ) = I + Q ( U ) + Q [ U × Q ( U ) ] + Q { U × Q ( U × Q ( U ) ] } + ,
Q ( U ) is Q ( U ) = 0 β U ( β ) d β .
C 1 ( β ) = 0 β r ( β 1 ) cos 2 β 1 d β 1 , S 1 ( β ) = 0 β r ( β 1 ) sin 2 β 1 d β 1 , C 2 ( β ) = 0 β 0 β 1 r ( β 1 ) r ( β 2 ) cos 2 ( β 2 - β 1 ) d β 2 d β 1 , S 2 ( β ) = 0 β 0 β 1 r ( β 1 ) r ( β 2 ) sin 2 ( β 2 - β 1 ) d β 2 d β 1 ,
C m ( β ) = 0 β 0 β 1 0 β m - 1 r ( β 1 ) r ( β 2 ) r ( β m ) cos 2 ( β m - β m - 1 + β 1 ) d β m d β 2 d β 1 , S m ( β ) = 0 β 0 β 1 0 β m - 1 r ( β 1 ) r ( β 2 ) r ( β m ) sin 2 ( β m - β m - 1 + β 1 ) d β m d β 2 d β 1 .
A ( β ) = [ 1 + C 1 ( β ) + C 2 ( β ) + S 1 ( β ) - S 2 ( β ) + S 3 ( β ) - S 1 ( β ) + S 2 ( β ) + S 3 ( β ) + 1 - C 1 ( β ) + C 2 ( β ) - C 3 ( β ) + ] .
f ( β ) = 1 + C 1 ( β ) + C 2 ( β ) + C 3 ( β ) + ,
g ( β ) = S 1 ( β ) + S 2 ( β ) + S 3 ( β ) + ,
l ( β ) = 1 - C 1 ( β ) + C 2 ( β ) - C 3 ( β ) + ,
k ( β ) = S 1 ( β ) - S 2 ( β ) + S 3 ( β ) - ,
[ η ( β ) η ( 0 ) cos β i sin β η ( β ) η ( 0 ) i η ( β ) η ( 0 ) sin β η ( 0 ) η ( β ) cos β ] .
ρ ( β ) = - 0 β r ( β 1 ) exp ( - 2 i β 1 ) d β 1 .
x π λ = β - β 0 2 ,
B 2 n ( σ ) = [ C 2 n ( σ ) - i S 2 n ( σ ) ] exp ( - i π σ 0 x ) , B 2 n + 1 ( σ ) = C 2 n + 1 ( σ ) - i S 2 n + 1 ( σ ) .
R = ( B 1 + B 3 + B 5 + ... ) · ( B 1 + B 3 + B 5 + ... ) * ( 1 + B 2 + B 4 + ... ) · ( 1 + B 2 + B 4 + ... ) * ,
T = 1 ( 1 + B 2 + B 4 + ... ) · ( 1 + B 2 + B 4 + ... ) * ,
1 / τ = 1 + B 2 + B 4 + B 6 + ,
1 T = 1 τ · 1 τ * = 1 + B 2 + B 2 * + B 2 B 2 * + .
τ = 1 - B 2 + B 2 2 + ,
T = 1 - B 2 - B 2 * + B 2 B 2 * + .
1 2 ( 1 T - T ) = B 2 + B 2 * .
Q ( σ ) exp i Φ ( σ ) = - + n ' ( x ) 2 n ( x ) exp ( - 2 i π σ x ) d x ,
Q ( σ ) = 0.5 [ 1 T ( σ ) - T ( σ ) ] .
n ' ( x ) n ( x ) = 2 [ - + Q ( σ ) exp ( 2 i π σ x ) d σ ] .
n ( x ) = n ( 0 ) exp 4 Q ( 1 / λ 0 ) × [ 0 x sin ( 2 π Δ σ u ) cos ( 2 π u / λ 0 ) π u d u ] ,
1 2 p ( T - p - T p ) = B 1 B 1 *
- log T - p - T p 2 p
- log T ( σ ) exp i Φ ( σ ) = - + n ' ( x ) 2 n ( x ) exp ( - 2 i π σ x ) d x .

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