Abstract

The formulas for a fast computation method are derived for the derivatives of transmittance and reflectance with respect to the thickness d, the refractive index n, and the extinction coefficient k of any layer in an absorbing multilayer stack.

© 1985 Optical Society of America

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References

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  1. P. W. Baumeister, “Methods of Altering the Characteristics of a Multi-Layer Stack,” J. Opt. Soc. Am. 52, 1149 (1962).
    [CrossRef]
  2. C. J. van der Laan, H. J. Frankena, “Fast Computation Method for Derivatives of Multilayer Stack Reflectance,” Appl. Opt. 17, 538 (1978).
    [CrossRef]
  3. Z. Knittl, Optics of Thin Films (Wiley, New York, 1976).
  4. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975).
  5. K. O. Peng, “Theory of Absorbing Thin Films,” Internal Report, N.V. Optische Industrie De Oude Delft, The Netherlands (1984).
  6. G. W. DeBell, “Computational Methods for Optical Thin Films,” Proc. Soc. Photo-Opt. Instrum. Eng. 140, 2 (1978).
  7. D. Kossel, K. Deutscher, K. Hirschberg, “Interference Photocathodes,” Phys. Thin Films 5, 1 (1969).
  8. O. Arnon, “Loss Mechanisms in Dielectric Optical Interference Devices,” Appl. Opt. 16, 2147 (1977).
    [CrossRef] [PubMed]

1978 (2)

C. J. van der Laan, H. J. Frankena, “Fast Computation Method for Derivatives of Multilayer Stack Reflectance,” Appl. Opt. 17, 538 (1978).
[CrossRef]

G. W. DeBell, “Computational Methods for Optical Thin Films,” Proc. Soc. Photo-Opt. Instrum. Eng. 140, 2 (1978).

1977 (1)

1969 (1)

D. Kossel, K. Deutscher, K. Hirschberg, “Interference Photocathodes,” Phys. Thin Films 5, 1 (1969).

1962 (1)

Arnon, O.

Baumeister, P. W.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975).

DeBell, G. W.

G. W. DeBell, “Computational Methods for Optical Thin Films,” Proc. Soc. Photo-Opt. Instrum. Eng. 140, 2 (1978).

Deutscher, K.

D. Kossel, K. Deutscher, K. Hirschberg, “Interference Photocathodes,” Phys. Thin Films 5, 1 (1969).

Frankena, H. J.

Hirschberg, K.

D. Kossel, K. Deutscher, K. Hirschberg, “Interference Photocathodes,” Phys. Thin Films 5, 1 (1969).

Knittl, Z.

Z. Knittl, Optics of Thin Films (Wiley, New York, 1976).

Kossel, D.

D. Kossel, K. Deutscher, K. Hirschberg, “Interference Photocathodes,” Phys. Thin Films 5, 1 (1969).

Peng, K. O.

K. O. Peng, “Theory of Absorbing Thin Films,” Internal Report, N.V. Optische Industrie De Oude Delft, The Netherlands (1984).

van der Laan, C. J.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975).

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

Phys. Thin Films (1)

D. Kossel, K. Deutscher, K. Hirschberg, “Interference Photocathodes,” Phys. Thin Films 5, 1 (1969).

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

G. W. DeBell, “Computational Methods for Optical Thin Films,” Proc. Soc. Photo-Opt. Instrum. Eng. 140, 2 (1978).

Other (3)

Z. Knittl, Optics of Thin Films (Wiley, New York, 1976).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975).

K. O. Peng, “Theory of Absorbing Thin Films,” Internal Report, N.V. Optische Industrie De Oude Delft, The Netherlands (1984).

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Figures (2)

Fig. 1
Fig. 1

Absorbing multilayer stack.

Fig. 2
Fig. 2

Reflectance and the derivatives with respect to the parameters of the photoemissive layer (j = 2) as a function of wavelength λ for a stack, n0 = 1.52, n1=2.85, k1 = 0, d1 = 93.618 nm, n2 = 2, k2 = 1.65, d2 = 26.25 nm, nsub = 1.0, ksub = 0; λ0 = 420 nm, θ0 = 0.

Equations (20)

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n 0 sin ( θ 0 ) = n ̂ q sin ( θ ̂ q ) ( q = 1 , 2 , , f + 1 ) .
n ̂ q = n q + i k q ,
M = q = 1 f M q .
M q = ( cos ( δ ̂ q ) i sin ( δ ̂ q ) / N ̂ q i N ̂ q sin ( δ ̂ q ) cos ( δ ̂ q ) ) ,
δ ̂ q = 2 π λ n ̂ q cos ( ϕ ̂ q ) d q ,
N ̂ q s = n ̂ q cos ( ϕ ̂ q )
N ̂ q p = n ̂ q cos ( ϕ ̂ q )
M = ( m ̂ 11 m ̂ 12 m ̂ 21 m ̂ 22 ) ,
E 0 = m ̂ 11 + N ̂ m ̂ 12 , H 0 = m ̂ 21 N ̂ m ̂ 22 ,
R = ( N 0 E 0 H 0 ) ( N 0 E 0 H 0 ) * ( N 0 E 0 + H 0 ) ( N 0 E 0 + H 0 ) * , T = 4 N 0 real ( N ̂ ) ( N 0 E 0 + H 0 ) ( N 0 E 0 + H 0 ) * ,
R x j = 4 N 0 { real ( E 0 H 0 * ) [ N 0 2 ( E 0 E 0 * ) x j + ( H 0 H 0 * ) x j ] ( N 0 2 E 0 E 0 * + H 0 H 0 * ) [ real ( E 0 H 0 * ) ] x j } / [ N 0 2 E 0 E 0 * + H 0 H 0 * + 2 N 0 real ( E 0 H 0 * ) ] ; T x j = 4 N 0 real ( N ̂ ) { N 0 2 ( E 0 E 0 * ) x j + ( H 0 H 0 * ) x j + 2 N 0 [ real ( E 0 H 0 * ) ] x j } / [ N 0 2 E 0 E 0 * + H 0 H 0 * + 2 N 0 real ( E 0 H 0 * ) ] + T real ( N ̂ ) [ real ( N ̂ ) ] x j δ ( j ) ,
δ ( j ) = { 1 if j = f + 1 , 0 elsewhere .
E 0 x j = m ̂ 11 x j + N ̂ m ̂ 12 x j H 0 x j = m ̂ 21 x j + N ̂ m ̂ 22 x j } if j f + 1 , E 0 x j = m ̂ 12 N ̂ x j H 0 x j = m ̂ 22 N ̂ x j } if j = f + 1 .
M = A j M j C j ,
C j = ( ĉ 11 ĉ 12 ĉ 21 ĉ 22 ) = q = j + 1 f M q ,
A j = ( â 11 â 12 â 21 â 22 ) = M ( M j C j ) 1 .
A j = M ( C j 1 ) 1 .
m ̂ 11 x j = â 11 ĉ 11 m ̂ 11 , j x j + â 12 ĉ 11 m ̂ 21 , j x j + â 11 ĉ 21 m ̂ 12 , j x j + â 12 ĉ 21 m ̂ 22 , j x j , m ̂ 12 x j = â 11 ĉ 12 m ̂ 11 , j x j + â 12 ĉ 12 m ̂ 21 , j x j + â 11 ĉ 22 m ̂ 12 , j x j + â 12 ĉ 22 m ̂ 22 , j x j , m ̂ 21 x j = â 21 ĉ 11 m ̂ 11 , j x j + â 22 ĉ 11 m ̂ 21 , j x j + â 21 ĉ 21 m ̂ 12 , j x j + â 22 ĉ 21 m ̂ 22 , j x j , m ̂ 22 x j = â 21 ĉ 12 m ̂ 11 , j x j + â 22 ĉ 12 m ̂ 21 , j x j + â 21 ĉ 22 m ̂ 12 , j x j + â 22 ĉ 22 m ̂ 22 , j x j . }
m ̂ 11 , j x j = m ̂ 22 , j x j = sin ( δ ̂ j ) δ ̂ j x j , m ̂ 12 , j x j = i [ N ̂ j 1 cos ( δ ̂ j ) δ ̂ j x j N ̂ j 2 sin ( δ ̂ j ) N ̂ j x j ] , m ̂ 21 , j x j = i [ N ̂ j cos ( δ ̂ j ) δ ̂ j x j + sin ( δ ̂ j ) N ̂ j x j ] . }
δ ̂ j d j = 2 π λ n ̂ j cos ( δ ̂ j ) , δ ̂ j n j = 2 π λ d j cos ( δ ̂ j ) , δ ̂ j k j = i δ ̂ j n j , N ̂ j d j = 0 , N ̂ j n j = { cos 1 ( ϕ j ) ( for s polarization ) , cos 2 ( ϕ ̂ j ) sin 2 ( ϕ ̂ j ) cos 3 ( ϕ ̂ j ) ( for p polarization ) , N ̂ j k j = i N ̂ j n j . }

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