Abstract

An analytical procedure for the calculation of the derivatives of the reflectance R of a dielectric multilayer stack is presented. Considered are the derivatives (∂R/∂nk)d, (∂R/∂nk)D, (∂R/∂dk)n, and (∂R/∂Dk)n, nk, dk, and Dk being the refractive index, the thickness, and the effective optical thickness of the kth layer, respectively. These calculations lead to a computational algorithm which, as compared with the method using finite difference approximation, reduces the computer time by a factor of f (the total number of layers).

© 1978 Optical Society of America

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References

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  1. P. Baumeister, J. Opt. Soc. Am. 48, 955 (1958).
    [CrossRef]
  2. J. A. Dobrowolski, Appl. Opt. 4, 937 (1965).
    [CrossRef]
  3. P. Baumeister, J. Opt. Soc. Am. 52, 1149 (1962).
    [CrossRef]
  4. A. Vašíček, Optics of Thin Films (North-Holland, Amsterdam, 1960).
  5. H. A. Macleod, Thin-Film Optical Filters (A. Hilger, London, 1969).

1965 (1)

1962 (1)

1958 (1)

Baumeister, P.

Dobrowolski, J. A.

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters (A. Hilger, London, 1969).

Vašícek, A.

A. Vašíček, Optics of Thin Films (North-Holland, Amsterdam, 1960).

Appl. Opt. (1)

J. Opt. Soc. Am. (2)

Other (2)

A. Vašíček, Optics of Thin Films (North-Holland, Amsterdam, 1960).

H. A. Macleod, Thin-Film Optical Filters (A. Hilger, London, 1969).

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Figures (3)

Fig. 1
Fig. 1

Light path through a cross section of a stack consisting of f dielectric layers.

Fig. 2
Fig. 2

Block diagram of the computational procedure for the reflectance R (blocks 1–5) and the derivatives (∂R/∂zk)c (blocks 6–10). A dotted line connects a block in which a certain calculation is performed with another block from which necessary information can be obtained. f = the number of layers; k = 1,2, . . .f;j = 1,2,3,4; p,q = 1,2.

Fig. 3
Fig. 3

Reflectance R and the derivatives of R as a function of the wavenumber 1/λ (n = 1.52, n0 = 1.00, ϕ0 = 0) for a single layer, (a) Solid curve: R of a layer with D1 = λ0/4 = 0.25 μm, n1 = 2.30, and d1 = 0.25/2.30 μm. All derivatives are calculated from this curve. Dotted curve: R of a layer with the same thickness d1 = 0.25/2.30 μm but n1 = 2.35. (b) Solid curve: (∂R/∂d1)n. Dotted curve: (∂R/∂D1)n. (c) Solid curve: (∂R/∂n1)d. Dotted curve: (∂R/∂n1)d.

Tables (1)

Tables Icon

Table I Summary of the Formulas for (∂δk/∂zk)c, (∂Nkp/∂zk)c, and (∂Nks/∂zk)c Necessary for the Calculation of the Derivative (∂R/∂zk)c

Equations (30)

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n o sin ( ϕ o ) = n k sin ( ϕ k ) ( k = 1 , 2 , . . . , f + 1 ) .
M k = ( cos ( δ k ) i sin ( δ k ) / N k i N k sin ( δ k ) cos ( δ k ) ) ,
δ k = 2 π D k / λ .
D k = n k d k cos ( ϕ k ) .
N k p = n k / cos ( ϕ k )
N k s = n k cos ( ϕ k )
M = ( m 11 i m 12 i m 21 m 22 ) = k = 1 f M k ,
R = ( N o m 11 N m 22 ) 2 + ( N N o m 12 m 21 ) 2 ( N o m 11 + N m 22 ) 2 + ( N N o m 12 + m 21 ) 2 ,
H 1 = N o m 11 N m 22 , H 2 = N N o m 12 m 21 , H 3 = N o m 11 + N m 22 , H 4 = N N o m 12 + m 21 ,
R = ( H 1 2 + H 2 2 ) / ( H 3 2 + H 4 2 ) .
R z k = 2 [ ( H 3 2 + H 4 2 ) ( H 1 H 1 / z k + H 2 H 2 / z k ) ( H 1 2 + H 2 2 ) ( H 3 H 3 / z k + H 4 H 4 / z k ) ] ( H 3 2 + H 4 2 ) 2 .
H 1 / z k = N o m 11 / z k N m 22 / z k , H 2 / z k = N N o m 12 / z k m 21 / z k , H 3 / z k = N o m 11 / z k + N m 22 / z k , H 4 / z k = N N o m 12 / z k + m 21 / z k ,
M = A k M k C k ,
A k = ( a 11 i a 12 i a 21 a 22 ) = j = 1 k 1 M j ( k = 2 , 3 , . . . , f ) , A 1 = ( 1 0 0 1 ) ,
C k = ( c 11 i c 12 i c 21 c 22 ) = j = k + 1 f M j ( k = 1 , 2 , . . . , f 1 ) , C f = ( 1 0 0 1 ) .
m 11 / z k = [ ( a 11 c 11 a 12 c 21 ) sin ( δ k ) + ( a 11 c 21 / N k + a 12 c 11 N k ) cos ( δ k ) ] δ k / z k + ( a 11 c 21 / N k 2 a 12 c 11 ) sin ( δ k ) N k / z k , m 12 / z k = [ ( a 11 c 12 + a 12 c 22 ) sin ( δ k ) + ( a 11 c 22 / N k + a 12 c 12 N k ) cos ( δ k ) ] δ k / z k + ( a 11 c 22 / N k 2 a 12 c 12 ) sin ( δ k ) N k / z k , m 21 / z k = [ ( a 21 c 11 + a 22 c 21 ) sin ( δ k ) + ( a 21 c 21 / N k a 22 c 11 N k ) cos ( δ k ) ] δ k / z k + ( a 21 c 21 / N k 2 + a 22 c 11 ) sin ( δ k ) N k / z k , m 22 / z k = [ ( a 21 c 12 + a 22 c 22 ) sin ( δ k ) + ( a 21 c 22 / N k + a 22 c 12 N k ) cos ( δ k ) ] δ k / z k + ( a 21 c 22 / N k 2 a 22 c 12 ) sin ( δ k ) N k / z k .
P k = j = 1 k M j = ( p 11 i p 12 i p 21 p 22 ) .
P k C k = M .
c 11 = p 22 m 11 + p 12 m 21 , c 12 = p 22 m 12 p 12 m 22 , c 21 = p 11 m 21 p 21 m 11 , c 22 = p 11 m 22 + p 21 m 12 .
ϕ h / n k = tan ( ϕ k ) / n k .
( δ k / d k ) n = 2 π n k cos ( ϕ k ) / λ
( N k p / d k ) n = ( N k s / d k ) n = 0 ,
( δ k / D k ) n = 2 π / λ
( N k p / D k ) n = ( N k s / D k ) n = 0 ,
( δ k / n k ) d = 2 π d k [ cos ( ϕ k ) n 0 sin ( ϕ 0 ) ϕ k / n k ] / λ
( N k p / n k ) d = cos ( ϕ k ) + n 0 sin ( ϕ 0 ) ϕ k / n k cos 2 ( ϕ k )
( N k s / n k ) d = cos ( ϕ k ) n 0 sin ( ϕ 0 ) ϕ k / n k ,
( δ k / n k ) D = 0
( N k p / n k ) D = same as right - hand side of Eq . ( 25 )
( N k s / n k ) D = same as right - hand side of Eq . ( 26 )

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