Abstract

The identification method is based on approximations of the reflection coefficient r by terms of the Bremmer series. One of these supplies an explicit representation of the refractive-index profile n, utilizing the Fourier transform Fr of r. Various procedures for the identification of thin-film systems using fast Fourier transform are derived from this relationship between Fr and n. One main issue of this paper is the determination of jumps within the refractive-index profile. For this purpose a numerical representation of Fr is subjected to waveform analysis. Theoretical and algorithmic aspects of the method are investigated, including a critical discussion in connection with numerical experiments.

© 1983 Optical Society of America

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References

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  1. H. Kaiser, H.-C. Kaiser, Appl. Opt. 20, 1043 (1981).
    [CrossRef] [PubMed]
  2. M. Reed, B. Simon, Methods of Modern Mathematical Physics II (Academic, New York, 1975).
  3. J. Hirsch, Opt. Acta 26, 1273 (1979).
    [CrossRef]
  4. E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974).
  5. D. K. Kahaner, J. Appl. Math. Phys. 29, 387 (1978).
    [CrossRef]
  6. H. Kaiser, H.-C. Kaiser, Linguistische Modelle und Algorithmen zur Strukturerkennung (Potsdamer Forschungen, Reihe B, Heft 27, Potsdam, 1981).

1981

1979

J. Hirsch, Opt. Acta 26, 1273 (1979).
[CrossRef]

1978

D. K. Kahaner, J. Appl. Math. Phys. 29, 387 (1978).
[CrossRef]

Brigham, E. O.

E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974).

Hirsch, J.

J. Hirsch, Opt. Acta 26, 1273 (1979).
[CrossRef]

Kahaner, D. K.

D. K. Kahaner, J. Appl. Math. Phys. 29, 387 (1978).
[CrossRef]

Kaiser, H.

H. Kaiser, H.-C. Kaiser, Appl. Opt. 20, 1043 (1981).
[CrossRef] [PubMed]

H. Kaiser, H.-C. Kaiser, Linguistische Modelle und Algorithmen zur Strukturerkennung (Potsdamer Forschungen, Reihe B, Heft 27, Potsdam, 1981).

Kaiser, H.-C.

H. Kaiser, H.-C. Kaiser, Appl. Opt. 20, 1043 (1981).
[CrossRef] [PubMed]

H. Kaiser, H.-C. Kaiser, Linguistische Modelle und Algorithmen zur Strukturerkennung (Potsdamer Forschungen, Reihe B, Heft 27, Potsdam, 1981).

Reed, M.

M. Reed, B. Simon, Methods of Modern Mathematical Physics II (Academic, New York, 1975).

Simon, B.

M. Reed, B. Simon, Methods of Modern Mathematical Physics II (Academic, New York, 1975).

Appl. Opt.

J. Appl. Math. Phys.

D. K. Kahaner, J. Appl. Math. Phys. 29, 387 (1978).
[CrossRef]

Opt. Acta

J. Hirsch, Opt. Acta 26, 1273 (1979).
[CrossRef]

Other

E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974).

H. Kaiser, H.-C. Kaiser, Linguistische Modelle und Algorithmen zur Strukturerkennung (Potsdamer Forschungen, Reihe B, Heft 27, Potsdam, 1981).

M. Reed, B. Simon, Methods of Modern Mathematical Physics II (Academic, New York, 1975).

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Figures (11)

Fig. 1
Fig. 1

Refractive-index profile of the considered layered medium.

Fig. 2
Fig. 2

Refractive-index profile [Eq. (46)] (Example 1).

Fig. 3
Fig. 3

Wave number dependence of the reflectivity for the refractive-index profile [Eq. (46)].

Fig. 4
Fig. 4

Refractive-index profile [Eq. (47)] (Example 2).

Fig. 5
Fig. 5

Wave number dependence of the reflectivity for the refractive-index profile [Eq. (47)].

Fig. 6
Fig. 6

Spectrum [Eq. (51)] (left) of the reflection coefficient of a monolayer 1//2//1.5 with the optical thickness d in comparison to its approximation [Eq. (66)] (right).

Fig. 7
Fig. 7

Discrete Fourier transform of the reflection coefficient for the monolayer 1//2//1.5 with the optical thickness d = 1 (N = 211, xD = 0.1).

Fig. 8
Fig. 8

Wave number dependence of the reflectivity for the refractive-index profile [Eq. (74)].

Fig. 9
Fig. 9

Discrete Fourier transform of the reflection coefficient for the refractive-index profile [Eq. (74)] (N = 211, xD = 0.1).

Fig. 10
Fig. 10

Reconstructed refractive-index profile in comparison with the original one (dahsed line) (Example 6).

Fig. 11
Fig. 11

Discrete Fourier transform of the reflection coefficient for the monolayer 1/12//1.5 with the optical thickness d = 1 (N = 211, xD = 0.06).

Tables (4)

Tables Icon

Table I Discrete Fourier Transform F x D of the Reflection Coefficient for the Refractive-Index Profile of Eq. (46), Performed with N = 211 and Various Step Size xD

Tables Icon

Table II Parameter Reconstruction for the Refractive-Index Profile of Eq. (46) from F x D ( N = 2 11 )

Tables Icon

Table III Parameter Reconstruction for the Refractive-Index Profile of Eq. (47) from F x D ( N = 2 11 )

Tables Icon

Table IV Parameter Reconstruction for the Discontinuous Refractive Index Profile of Eq. (74) from F x D ( N = 2 11 )

Equations (92)

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r = r ( k ) = j = 0 U 2 j + 1 ( 0 , k )
U 0 ( x , k ) = n ( 0 ) n ( x ) exp ( i k x ) , U 2 j ( x , k ) = 1 n ( x ) 0 x U 2 j 1 ( s , k ) × exp [ i k ( x s ) ] d n ( s ) d s d s , U 2 j + 1 ( x , k ) = 1 n ( x ) x U 2 j ( s , k ) exp [ i k ( x s ) ] d n ( s ) d s d s .
U 2 j + 1 ( x , k ) = ( 1 ) j + 1 n ( 0 ) n ( x ) exp ( i k x ) x a ( t 2 j + 1 ) exp ( 2 i k t 2 j + 1 ) × 0 t 2 j + 1 a ( t 2 j ) exp ( 2 i k t 2 j ) t 2 j a ( t 2 j 1 ) exp ( 2 i k t 2 j 1 ) × 0 t 3 a ( t 2 ) exp ( 2 i k t 2 ) t 2 a ( t 1 ) exp ( 2 i k t 1 ) d t 1 d t 2 d t 2 j + 1 ,
a ( x ) = d ln n ( x ) d x
a ( x ) = 0 for x < 0.
( F φ ) ( y ) = 1 2 π exp ( i x y ) φ ( x ) d x , φ S ,
F T , φ = T , F φ for all φ S , T S * .
n ( x ) a > 0 for all x 0 , 0 | d n d x ( x ) | d x = b , b < 2 a .
F r , φ = j = 0 F U 2 j + 1 , φ for all φ S .
( F 1 δ b ) ( y ) = exp ( i b y ) .
F U 1 , φ = U 1 , F φ = 0 a ( s ) exp ( 2 i k s ) d s , F φ = 0 a ( s ) exp ( 2 i k s ) , F φ d s = 0 a ( s ) F 1 δ 2 s , F φ d s = 0 a ( s ) δ 2 s , φ d s = 0 a ( s ) φ ( 2 s ) d s = a ( s ) φ ( 2 s ) d s = 1 2 a ( x 2 ) φ ( x ) d x .
( F U 1 ) ( x ) = 1 2 a ( x 2 ) .
( F U 3 ) ( x ) = 1 2 0 x / 2 0 x / 2 s a ( x 2 s ) a ( t ) a ( s + t ) d t d s ,
( F U 2 j + 1 ) ( x ) = 1 2 ( 1 ) j + 1 0 x / 2 0 x / 2 s j 0 s j 0 t j + s j s j 1 0 s 3 0 t 3 + s 3 s 2 0 s 2 0 t 2 + s 2 s 1 a ( x 2 s j ) a ( t 1 ) a ( t 1 + s 1 ) × l = 2 j a ( t l ) a ( t l + s l s l 1 ) × d t 1 d s 1 d t 2 d s 2 d t j d s j .
( F U 2 j + 1 ) ( x ) = 0 for x > 0 , j = 0,1,2 .
( F U 2 j + 1 ) ( x ) = 0 for x 2 ( j + 1 ) d , j = 0,1,2 .
n ( x ) = { n 0 = exp ( β ) for x < 0 exp ( α x + β ) for 0 x < d n g = exp ( α d + β ) for d x ,
j = 0 ( F U 2 j + 1 ) ( x )
( F U 1 ) ( x ) = α 4 ,
( F U 3 ) ( x ) = 1 4 ( α 2 ) 3 ( x 2 ) 2 ,
( F U 5 ) ( x ) = 1 24 ( α 2 ) 5 ( x 2 ) 4 ,
F r F U 1 .
| f ( x ) | 2 d x = 2 π | ( F f ) ( y ) | 2 d y .
2 0 R ( k ) d k = | r ( k ) | 2 d k = 2 π ( F r ) 2 ( x ) d x 2 π ( F U 1 ) 2 ( x ) x = π 4 0 [ d ln n ( x ) d x ] 2 d x .
F r j = 0 l F U 2 j + 1 .
F ( x ) : = ( F r ) ( x ) = 1 π Re 0 r ( k ) exp ( i k x ) d k
F ( x l ) F x ( x l ) : = Re { k D π j = 0 N 1 r j ω l j } , l = 0,1 , , N 1 ,
r j = r ( j k D ) , j = 0,1 , , N 1 ,
x l = l x D , l = 0,1 , , N 1 ,
ω = exp ( 2 π i N ) ,
N x D k D = 2 π .
F ( x ) = { 0 for x > 0 1 2 d d s [ ln n ( s ) ] | s = x 2 for x 0 ,
n ( x ) = n ( 0 ) exp ( 2 2 x 0 F ( s ) d s ) .
n = n ( x ) = exp ( c 0 + j = 1 m c j ( x y j ) + ) ,
j = 1 m c j = 0.
0 = y 1 < y 2 < < y m = d ,
ξ + p : = { ξ p for ξ 0 0 for ξ < 0 ,
y m + 1 : = 1 2 N x D .
y j = ( j 1 ) Δ , j = 1 , , m ,
Δ = x D 2 P
j = 1 l c j = 4 F ( 2 x ) for y l < x < y l + 1 , l = 1,2 , , m ,
j = 1 l c j = 4 F ( x D ( P l Q ) ) = 4 F x D ( x D ( P l Q ) ) ,
Q = [ P 2 ]
F x D ( x j ) = f ( c 1 , c 2 , , c m , x j ) , j = 0,1 , , N 1.
a ( x ) = α over ξ 1 x ξ 2 .
d r d x ( x ) = a ( x ) [ 1 r 2 ( x ) ] + 2 i k r ( x ) , ξ 1 x ξ 2 , r ( ξ 2 ) = r 0 ,
r ( k ) = r 0 k 1 cosh ( k 1 h ) ( i k r 0 + α ) sinh ( k 1 h ) k 1 cosh ( k 1 h ) + ( i k r 0 α ) sinh ( k 1 h ) for k < | α | , r ( k ) = r 0 ( 1 i k h ) α h ( 1 + i k h ) α h r 0 for k = | α | , r ( k ) = r 0 k 2 cos ( k 2 h ) ( i k r 0 + α ) sin ( k 2 h ) k 2 cos ( k 2 h ) + ( i k r 0 α ) sin ( k 2 h ) for k > | α | ,
r ( d ) = n ( d ) n g n ( d ) + n g = 0.
n ( x ) = 1.5 exp ( 1 2 x + 1 2 ( x 1 ) + )
I k = ( 0 , K ) K = N k D = 2 π x D
n ( x ) = 1.5 exp ( x + 2 ( x 1 2 ) + + ( x 1 ) + )
r ( k ) = r 0 = n 0 n g n 0 + n g ,
( F r ) ( x ) = n 0 n g n 0 + n g δ ( x ) .
r ( k ) = r 0 + r 1 exp ( 2 i k d ) 1 + r 0 r 1 exp ( 2 i k d ) ,
( F r ) ( x ) = r 0 δ ( x ) + ( r 0 1 r 0 ) j = 1 ( r 0 r 1 ) j δ ( x + 2 j d ) .
0 R ( k ) d k
0 z 1 < z 2 < < z M .
n ( x ) n 0 for x < 0 n ( x ) const for x d z M
ln n ( x ) = ln n 0 + 0 x A ( ξ ) d ξ + L = 1 M a L ( x z L ) + 0 ,
A ( x ) 0 for x < 0 and x d .
a ( x ) = A ( x ) + L = 1 M a L δ ( x z L ) .
n = n ( x ) , > 0 ,
n ( x ) n 0 for x < 0 ,
n n as 0 ( least squares convergence ) ,
n ( x ) = n ( x ) for all x outside an neighborhood of the jump points z L
F ( x ) = ( F r ) ( x ) ( F U 1 ) ( x ) = 1 2 a ( x 2 )
r ( k ) r ( k ) as 0 for each k .
r , φ r , φ as 0 for all φ S ,
F r , φ F r , φ as 0
a , φ a , φ as 0 for all φ S .
F ( x ) = ( F r ) ( x ) = 1 2 a ( x 2 ) .
( F r ) ( x ) ln n g n 0 δ ( x )
( F r ) ( x ) ln n 1 n 0 δ ( x ) ln n g n 1 δ ( x + 2 d )
δ L x D ( x ) = δ ( x + L x D ) = ( F g ) ( x ) , g = g ( k ) = exp ( i k L x D ) ,
G x D ( x l ) = Re { k D π j = 0 N 1 g ( j k D ) exp ( l j 2 π i N ) } = Re { 2 N x D j = 0 N 1 exp ( j 2 π i N ( l L ) ) } = { 2 x D for l = L 0 for l = 0,1 , , N 1 ; l L
2 b / x D at x = L x D ;
F ( x ) = 1 2 A ( x 2 ) L = 1 M a L 2 δ ( x 2 z L ) = 1 2 A ( x 2 ) L = 1 M a L δ ( x + 2 z L ) .
F x D ( l x D ) = 1 2 A ( l x D 2 ) 2 a L x D for l = 2 z L x D , L = 1 , , M ;
F x D ( l x D ) = 1 2 A ( l x D 2 ) for l = 0,1 , , N 1 , l 2 z L x D for all L = 1 , , M .
n ( x ) = n 0 exp ( 2 0 x A ( ξ ) d ξ ) L = 1 z L < x M exp ( 2 a L ) .
n 0 = 1 ; n 1 = 2 ; n g = 1.5 ; d = 1.
n ( x ) = 1.5 exp ( x 2 ) ,
M = 2 , z 1 = 0 , z 2 = 1 , a 1 = 0.20273 , a 2 = 0.25000 , A ( x ) = { 0.25 for 0 x < 1 0.00 for x 1.
z 1 = 0 , z 2 = 1.
A ( 0 ) = 0.1148 , A ( 1 ) = 0.1227.
a 1 = 0.2002 , a 2 = 0.2351.
A ( x ) = i = 1 m c j ( x y j ) + 0
0 = y 1 < y 2 < < y m < y m + 1 = N x D 2 ,
C j : = 2 ( c 1 + c 2 + + c j )
C j = 4 F x D ( ( y j + y j + 1 ) ) , j = 1,2 , , m .
y 1 = 0 , y 2 = 1 , y 3 = 2 ;
N = 2 11 , x D = 0.06.

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