Abstract

The determination of the absorption of very weakly absorbing thin films and film systems is a difficult problem. In the present work a previously suggested method is generalized and improved in order to apply it for the determination of the absorption of films and multilayers. In the case of single layers, a single measurement is adequate to determine also the thickness and refractive index of the film. The main advantages of the method described here are its simplicity, its self-calibrating nature, and its high sensitivity to absorption.

© 1978 Optical Society of America

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References

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  1. K. Kinosita, M. Yamamoto, Thin Solid Films 34, 283 (1976).
    [CrossRef]
  2. R. A. Hoffman, Appl. Opt. 13, 1405 (1974).
    [CrossRef] [PubMed]
  3. H. Ahrens, H. Welling, H. E. Scheel, Appl. Phys. 1, 69 (1973).
    [CrossRef]
  4. V. Sanders, Appl. Opt. 16, 19 (1977).
    [CrossRef] [PubMed]
  5. J. Shamir, P. Gräff, Appl. Opt. 14, 3053 (1975).
    [CrossRef] [PubMed]
  6. J. Shamir, Appl. Opt. 15, 120 (1976).
    [CrossRef] [PubMed]
  7. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975), pp. 51–70, 611–634.
  8. J. Shamir, R. Fox, Am. J. Phys. 35, 161 (1967).
    [CrossRef]
  9. J. Shamir, Am. J. Phys. 45, 1118 (1977).
    [CrossRef]

1977

1976

J. Shamir, Appl. Opt. 15, 120 (1976).
[CrossRef] [PubMed]

K. Kinosita, M. Yamamoto, Thin Solid Films 34, 283 (1976).
[CrossRef]

1975

1974

1973

H. Ahrens, H. Welling, H. E. Scheel, Appl. Phys. 1, 69 (1973).
[CrossRef]

1967

J. Shamir, R. Fox, Am. J. Phys. 35, 161 (1967).
[CrossRef]

Ahrens, H.

H. Ahrens, H. Welling, H. E. Scheel, Appl. Phys. 1, 69 (1973).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975), pp. 51–70, 611–634.

Fox, R.

J. Shamir, R. Fox, Am. J. Phys. 35, 161 (1967).
[CrossRef]

Gräff, P.

Hoffman, R. A.

Kinosita, K.

K. Kinosita, M. Yamamoto, Thin Solid Films 34, 283 (1976).
[CrossRef]

Sanders, V.

Scheel, H. E.

H. Ahrens, H. Welling, H. E. Scheel, Appl. Phys. 1, 69 (1973).
[CrossRef]

Shamir, J.

Welling, H.

H. Ahrens, H. Welling, H. E. Scheel, Appl. Phys. 1, 69 (1973).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975), pp. 51–70, 611–634.

Yamamoto, M.

K. Kinosita, M. Yamamoto, Thin Solid Films 34, 283 (1976).
[CrossRef]

Am. J. Phys.

J. Shamir, R. Fox, Am. J. Phys. 35, 161 (1967).
[CrossRef]

J. Shamir, Am. J. Phys. 45, 1118 (1977).
[CrossRef]

Appl. Opt.

Appl. Phys.

H. Ahrens, H. Welling, H. E. Scheel, Appl. Phys. 1, 69 (1973).
[CrossRef]

Thin Solid Films

K. Kinosita, M. Yamamoto, Thin Solid Films 34, 283 (1976).
[CrossRef]

Other

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975), pp. 51–70, 611–634.

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Figures (11)

Fig. 1
Fig. 1

The sample and waves propagating through it.

Fig. 2
Fig. 2

The ellipse and its measurable parameters.

Fig. 3
Fig. 3

Measurable parameters as functions of κ.

Fig. 4
Fig. 4

Δ1p map for a quarterwave film.

Fig. 5
Fig. 5

Dependence of Δ1 on n1 for small values of κ.

Fig. 6
Fig. 6

The same as Fig. 5 for large values of κ.

Fig. 7
Fig. 7

The same as Fig. 5 for various incidence angles.

Fig. 8
Fig. 8

The same as Fig. 5 for different values of n3.

Fig. 9
Fig. 9

Δ1 and absorption as functions of the wavelength λ for a theoretical laser mirror.

Fig. 10
Fig. 10

A schematic drawing of the measuring system (for details see text).

Fig. 11
Fig. 11

Actual results for a measured sample illustrating the measured parameters.

Equations (70)

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E 11 r 1 E 1 = | r 1 | exp ( j ρ 1 ) E 1 ,
E 13 t 1 E 1 = | t 1 | exp ( j ϕ 1 ) E 1
[ U ( o ) V ( o ) ] = [ M 11 M 12 M 12 M 22 ] [ U ( z s ) V ( z s ) ] ,
r 1 = ( M 11 + M 12 p 3 ) p 1 ( M 21 + M 22 p 3 ) ( M 11 + M 12 p 3 ) p 1 + ( M 21 + M 22 p 3 ) ,
t 1 = 2 p 1 ( M 11 + M 12 p 3 ) p 1 + ( M 21 + M 22 p 3 ) ,
p i n i cos θ i
a M 11 p 1 M 22 p 3 , b j ( M 12 p 3 p 1 M 21 ) , c M 11 p 1 + M 22 p 3 , d j ( M 12 p 3 p 1 + M 21 ) ,
r 1 | r 1 | exp ( j ρ 1 ) = ( a + j b ) / ( c + j d ) ,
t 1 | t 1 | exp ( j ϕ 1 ) = ( 2 p 1 ) / ( c + j d ) .
I n cos θ 2 c μ 0 | E | 2 ,
R 1 | r 1 | 2 = | a + j b | 2 | c + j d | 2 ,
T 1 p 3 p 1 | t 1 | 2 = 4 p 1 p 3 | c + j d | 2 .
| r 1 | = [ ( a 2 + b 2 ) / ( c 2 + d 2 ) ] 1 / 2 ,
ρ 1 = t g 1 ( b / a ) t g 1 ( d / c ) ,
| t 1 | = ( 2 p 1 ) / [ ( c 2 + d 2 ) 1 / 2 ] ,
ϕ 1 = t g 1 ( d / c ) ,
R 1 = ( a 2 + b 2 ) / ( c 2 + d 2 ) ,
T 1 = ( 4 p 1 p 3 ) / ( c 2 + d 2 ) .
E 33 r 3 E 3 = | r 3 | exp ( j ρ 3 ) E 3 ,
E 31 t 3 E 3 = | t 3 | exp ( j ϕ 3 ) E 3 .
p 1 p 3 , M 11 M 22 , M 12 M 21 , M 21 M 12 , M 22 M 11 ,
r 3 | r 3 | exp ( j ρ 3 ) = ( a + j b ) / ( c + j d ) ,
t 3 | t 3 | exp ( j ϕ 3 ) = ( 2 p 3 ) / ( c + j d ) .
R 3 | r 3 | 2 = ( | a + j b | 2 ) / ( | c + j d | 2 ) ,
T 3 p 1 / p 3 | t 3 | 2 = ( 4 p 1 p 3 ) / ( | c + j d | 2 ) ,
T 1 = T 3 T ,
ϕ 1 = ϕ 3 ϕ .
R 3 = ( a 2 + b 2 ) / ( c 2 + d 2 ) = R 1 ,
ρ 3 = t g 1 ( b / a ) t g 1 ( d / c ) = π t g 1 ( b / a ) t g 1 ( d / c ) .
ρ 3 = π ρ 1 + 2 ϕ .
E 1 = E 11 + E 31 = r 1 E 1 + t 3 E 3 ,
E 3 = E 13 + E 33 = t 1 E 1 + r 3 E 3 .
E 3 = η exp ( j Φ ) E 1 ,
E 1 = [ r 1 + η t 3 exp ( j Φ ) ] E 1 ,
E 3 = [ t 1 + η r 3 exp ( j Φ ) ] E 1 .
I 1 = n 1 cos θ 1 2 c μ 0 [ | r 1 | 2 + η 2 | t 3 | 2 + r 1 * η t 3 exp ( j Φ ) + r 1 η t 3 * exp ( j Φ ) ] | E 1 | 2 ,
I 3 = n 3 cos θ 3 2 c μ 0 [ | t 1 | 2 + η 2 | r 3 | 2 + t 1 * η r 3 exp ( j Φ ) + t 1 η r 3 * exp ( j Φ ) ] | E 1 | 2 .
I 1 = n 1 cos θ 1 2 c μ 0 [ | r 1 | 2 + η 2 | t 3 | 2 + 2 | r 1 | | t 3 | cos ( ϕ ρ 1 + Φ ) ] | E 1 | 2 ,
I 3 = n 3 cos θ 3 2 c μ 0 [ | t 1 | 2 + η 2 | r 3 | 2 + 2 | t 1 | | r 3 | cos ( ϕ + ρ 3 + Φ ) ] | E 1 | 2 .
I 4 = 4 n 4 cos θ 4 n 3 cos θ 3 ( n 3 cos θ 3 + n 4 cos θ 4 ) 2 I 3 D 0 I 3 .
X = F x I 1 ,
Y = F y I 4 .
F x = F y = 1.
I 4 = X 0 + A x cos ( Φ x + Φ ) = X ,
I 4 = Y 0 + A y cos ( Φ y + Φ ) = Y ,
X 0 = 1 2 c μ 0 n 1 cos θ 1 | E 1 | 2 ( | r 1 | 2 + η 2 | t 3 | 2 ) ,
Y 0 = 1 2 c μ 0 D 0 n 3 cos θ 3 | E 1 | 2 ( | t 1 | 2 + η 2 | r 3 | 2 ) ,
A x = n 1 cos θ 1 2 c μ 0 | E 1 | 2 η | r 1 | | t 3 | ,
A y = D 0 n 3 cos θ 3 2 c μ 0 | E 1 | 2 η | t 1 | | r 3 | ,
Φ x = ϕ ρ 1 ,
Φ y = ϕ + ρ 3 .
Δ = Φ y Φ x = ρ 1 + ρ 3 2 ϕ .
Δ = sin 1 ( y / A y ) ,
A y / A x = D 0 n 3 cos θ 3 n 1 cos θ 1 | t 1 | | r 3 | | t 3 | | r 1 | = D 0 ( R 3 R 1 ) 1 / 2 ,
Y 0 / X 0 = D 0 n 3 cos θ 3 n 1 cos θ 1 | t 1 | 2 + η 2 | r 3 | 2 | r 1 | 2 + η 2 | t 3 | 2 = D 0 n 1 cos θ 1 n 3 cos θ 3 1 + η 2 p η 2 + q ,
p | r 3 | 2 | t 1 | 2 = n 3 cos θ 3 n 1 cos θ 1 · R 3 T ,
q | r 1 | 2 | t 3 | 2 = n 1 cos θ 1 n 3 cos θ 3 · R 1 T .
Δ 1 Δ π ,
Δ 1 = 0.
p = n 3 cos θ 3 D 0 n 1 cos θ 1 Y 0 / X 0 ( η ) ,
1 / q = n 3 cos θ 3 D 0 n 1 cos θ 1 Y 0 / X 0 ( η = 0 ) ,
A y / A x = n 1 cos θ 1 n 3 cos θ 3 D 0 ( p / q ) 1 / 2 .
n 2 = n ( 1 j κ ) .
n 1 = n 3 .
F = F x / F y
I 4 / I 1 = F · Y / X .
F = n 3 cos θ 3 D 0 η 2 n 1 cos θ 1 · 1 Y 0 / X 0 Y 0 / X 0 ( 0 ) Y 0 / X 0 Y 0 / X 0 ( ) .
η 2 = | E 3 | 2 | E 1 | 2 = n 1 cos θ 1 n 3 cos θ 3 I 3 I 1 = n 3 cos θ 4 D 0 n 1 cos θ 1 I 4 I 1 ,
F = I 1 I 4 · 1 Y 0 / X 0 Y 0 / X 0 ( 0 ) Y 0 / X 0 Y 0 / X 0 ( ) .
A y / A x = [ Y 0 / X 0 ( 0 ) Y 0 / X 0 ( ) ] 1 / 2 ,

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