Abstract

The refractive index measured by the Abeles method can have a serious error because of a film transition layer. The effect is different depending on which side of the film the layer exists. Calculation is chiefly made for a surface transition layer which is more realistic than an inner one in the case of deposited films. The following error characteristics are elucidated for a thin surface transition layer: (1) The error is a cotangent function of the film thickness. (2) It is proportional to the transition layer thickness. (3) It increases as the refractive index along the transition layer gets closer to the averaged index of the air and the film. (4) It is almost independent of the substrate index ns except when ns is close to the film index.

© 1984 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. F. Abeles, “La Determination de l’Indice et de l’Epaisseur des Couches Minces Transparentes,” J. Phys. Radium 11, 310 (1950).
    [CrossRef]
  2. M. Hacskaylo, “Refractive Index of Thin Dielectric Films,” J. Opt. Soc. Am. 54, 198 (1964).
    [CrossRef]
  3. R. J. King, S. P. Talim, “A Comparison of Thin Film Measurement by Guided Waves, Ellipsometry and Reflectometry,” Opt. Acta 28, 1107 (1981).
    [CrossRef]
  4. O. S. Heavens, S. D. Smith, “Dielectric Thin Films,” J. Opt. Soc. Am. 47, 469 (1957).
    [CrossRef]
  5. G. Koppelmann, K. Krebs, “Die Optischen Eigenschaften Dielectricher Schichten mit Kleinen Homogenitätsstörungen,” Z. Phys. 163, 539 (1961).
    [CrossRef]
  6. P. Rouard, P. Bousquet, “Optical Constants of Thin Films,” Prog. Opt. 4, 147 (1965).
  7. G. Rasigni, F. Varnier, J. P. Palmari, N. Mayani, M. Rasigni, A. Llebaria, “Study of Surface Roughness for Thin Films of CaF2 Deposited on Glass-Substrates,” Opt. Commun. 46, 294 (1983).
    [CrossRef]
  8. J. E. Goell, R. D. Standley, “Effect of Refractive Index Gradients on Index Measurement by the Abeles Method,” Appl. Opt. 11, 2502 (1972).
    [CrossRef] [PubMed]

1983 (1)

G. Rasigni, F. Varnier, J. P. Palmari, N. Mayani, M. Rasigni, A. Llebaria, “Study of Surface Roughness for Thin Films of CaF2 Deposited on Glass-Substrates,” Opt. Commun. 46, 294 (1983).
[CrossRef]

1981 (1)

R. J. King, S. P. Talim, “A Comparison of Thin Film Measurement by Guided Waves, Ellipsometry and Reflectometry,” Opt. Acta 28, 1107 (1981).
[CrossRef]

1972 (1)

1965 (1)

P. Rouard, P. Bousquet, “Optical Constants of Thin Films,” Prog. Opt. 4, 147 (1965).

1964 (1)

1961 (1)

G. Koppelmann, K. Krebs, “Die Optischen Eigenschaften Dielectricher Schichten mit Kleinen Homogenitätsstörungen,” Z. Phys. 163, 539 (1961).
[CrossRef]

1957 (1)

1950 (1)

F. Abeles, “La Determination de l’Indice et de l’Epaisseur des Couches Minces Transparentes,” J. Phys. Radium 11, 310 (1950).
[CrossRef]

Abeles, F.

F. Abeles, “La Determination de l’Indice et de l’Epaisseur des Couches Minces Transparentes,” J. Phys. Radium 11, 310 (1950).
[CrossRef]

Bousquet, P.

P. Rouard, P. Bousquet, “Optical Constants of Thin Films,” Prog. Opt. 4, 147 (1965).

Goell, J. E.

Hacskaylo, M.

Heavens, O. S.

King, R. J.

R. J. King, S. P. Talim, “A Comparison of Thin Film Measurement by Guided Waves, Ellipsometry and Reflectometry,” Opt. Acta 28, 1107 (1981).
[CrossRef]

Koppelmann, G.

G. Koppelmann, K. Krebs, “Die Optischen Eigenschaften Dielectricher Schichten mit Kleinen Homogenitätsstörungen,” Z. Phys. 163, 539 (1961).
[CrossRef]

Krebs, K.

G. Koppelmann, K. Krebs, “Die Optischen Eigenschaften Dielectricher Schichten mit Kleinen Homogenitätsstörungen,” Z. Phys. 163, 539 (1961).
[CrossRef]

Llebaria, A.

G. Rasigni, F. Varnier, J. P. Palmari, N. Mayani, M. Rasigni, A. Llebaria, “Study of Surface Roughness for Thin Films of CaF2 Deposited on Glass-Substrates,” Opt. Commun. 46, 294 (1983).
[CrossRef]

Mayani, N.

G. Rasigni, F. Varnier, J. P. Palmari, N. Mayani, M. Rasigni, A. Llebaria, “Study of Surface Roughness for Thin Films of CaF2 Deposited on Glass-Substrates,” Opt. Commun. 46, 294 (1983).
[CrossRef]

Palmari, J. P.

G. Rasigni, F. Varnier, J. P. Palmari, N. Mayani, M. Rasigni, A. Llebaria, “Study of Surface Roughness for Thin Films of CaF2 Deposited on Glass-Substrates,” Opt. Commun. 46, 294 (1983).
[CrossRef]

Rasigni, G.

G. Rasigni, F. Varnier, J. P. Palmari, N. Mayani, M. Rasigni, A. Llebaria, “Study of Surface Roughness for Thin Films of CaF2 Deposited on Glass-Substrates,” Opt. Commun. 46, 294 (1983).
[CrossRef]

Rasigni, M.

G. Rasigni, F. Varnier, J. P. Palmari, N. Mayani, M. Rasigni, A. Llebaria, “Study of Surface Roughness for Thin Films of CaF2 Deposited on Glass-Substrates,” Opt. Commun. 46, 294 (1983).
[CrossRef]

Rouard, P.

P. Rouard, P. Bousquet, “Optical Constants of Thin Films,” Prog. Opt. 4, 147 (1965).

Smith, S. D.

Standley, R. D.

Talim, S. P.

R. J. King, S. P. Talim, “A Comparison of Thin Film Measurement by Guided Waves, Ellipsometry and Reflectometry,” Opt. Acta 28, 1107 (1981).
[CrossRef]

Varnier, F.

G. Rasigni, F. Varnier, J. P. Palmari, N. Mayani, M. Rasigni, A. Llebaria, “Study of Surface Roughness for Thin Films of CaF2 Deposited on Glass-Substrates,” Opt. Commun. 46, 294 (1983).
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. (2)

J. Phys. Radium (1)

F. Abeles, “La Determination de l’Indice et de l’Epaisseur des Couches Minces Transparentes,” J. Phys. Radium 11, 310 (1950).
[CrossRef]

Opt. Acta (1)

R. J. King, S. P. Talim, “A Comparison of Thin Film Measurement by Guided Waves, Ellipsometry and Reflectometry,” Opt. Acta 28, 1107 (1981).
[CrossRef]

Opt. Commun. (1)

G. Rasigni, F. Varnier, J. P. Palmari, N. Mayani, M. Rasigni, A. Llebaria, “Study of Surface Roughness for Thin Films of CaF2 Deposited on Glass-Substrates,” Opt. Commun. 46, 294 (1983).
[CrossRef]

Prog. Opt. (1)

P. Rouard, P. Bousquet, “Optical Constants of Thin Films,” Prog. Opt. 4, 147 (1965).

Z. Phys. (1)

G. Koppelmann, K. Krebs, “Die Optischen Eigenschaften Dielectricher Schichten mit Kleinen Homogenitätsstörungen,” Z. Phys. 163, 539 (1961).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (14)

Fig. 1
Fig. 1

Relative error of refractive index measured by the Abeles method (Sputtered Nb2O5 film on Corning 7059 glass substrate: λ = 0.633 μm; n, refractive index of film; d, film thickness, ϕB, incident angle in film)

Fig. 2
Fig. 2

Four-layered structure for analyzing the effect of a uniform transition layer.

Fig. 3
Fig. 3

Multilayered structure for graded-index cases.

Fig. 4
Fig. 4

Refractive-index error vs film thickness. Transition layer is at the air–film interface. The parameter is the thickness of the transition layer. The marks 2θ = on the abscissa are for ease of comparison with Fig. 1.

Fig. 5
Fig. 5

Same as Fig. 4 except that the parameter is the transition layer index.

Fig. 6
Fig. 6

Refractive-index error vs film thickness. Transition layer is at the film–substrate interface. The parameter is the thickness of the transition layer.

Fig. 7
Fig. 7

Same as Fig. 6 except that the parameter is the transition layer index.

Fig. 8
Fig. 8

(a) Refractive-index error when the transition layer index varies in entire functions with thickness. (b) Refractive-index error when the transition layer index varies in irrational functions with thickness.

Fig. 9
Fig. 9

Refractive-index error when the transition layer index varies linearly with thickness.

Fig. 10
Fig. 10

Refractive-index error vs surface index, the parameter being the film thickness. nt varies linearly with thickness.

Fig. 11
Fig. 11

Integral of |ntnm| along the transition layer; (b) shows the minimum value.

Fig. 12
Fig. 12

Refractive-index error vs transition layer thickness, the parameter being film thickness. The index nt varies linearly with thickness.

Fig. 13
Fig. 13

Refractive-index error vs substrate index, the parameter being film thickness. nt varies linearly with thickness.

Fig. 14
Fig. 14

Refractive-index error vs film thickness when the substrate has a smaller index than the film.

Equations (32)

Equations on this page are rendered with MathJax. Learn more.

[ E i - 1 + E i - 1 - ] = 1 t i [ exp ( j θ i ) r i exp ( - j θ i ) r i exp ( j θ i ) exp ( - j θ i ) ] [ E i + E i - ] ,
r i = n i - 1 cos ϕ i - n i cos ϕ i - 1 n i - 1 cos ϕ i + n i cos ϕ i - 1 = sin 2 ϕ i - 1 - sin 2 ϕ i sin 2 ϕ i - 1 + sin 2 ϕ i .
t i = 2 n i - 1 cos ϕ i n i - 1 cos ϕ i + n i cos ϕ i - 1 ,
ϕ i = sin - 1 [ ( n i - 1 / n i ) sin ϕ i - 1 ] ,
θ i = [ ( 2 π n i / λ ) cos ϕ i ] d i .
[ E 0 + E 0 - ] = 1 t 01 t 12 [ exp ( j θ 1 ) r 01 exp ( - j θ 1 ) r 01 exp ( j θ 1 ) exp ( - j θ 1 ) ] [ exp ( j θ 2 ) r 12 exp ( - j θ 2 ) r 12 exp ( j θ 2 ) exp ( - j θ 2 ) ] [ E 2 + E 2 - ] ,
E 0 + E 0 - = r 01 + r 12 exp ( j 2 θ 1 ) + r 2 s exp [ j 2 ( θ 1 + θ 2 ) ] + r 01 r 12 r 2 s exp ( j 2 θ 2 ) 1 + r 01 r 12 exp ( j 2 θ 1 ) + r 12 r 2 s exp ( j 2 θ 2 ) + r 01 r 2 s exp [ j 2 ( θ 1 + θ 2 ) ] .
E 0 + E 0 - r 01 + r 12 exp ( j 2 θ 1 ) + r 2 s exp [ j 2 ( θ 1 + θ 2 ) ] .
r 01 + r 12 1 + r 01 r 12 = r 02 .
r 01 + r 12 r 02 .
r 01 + r 12 + r 2 s r 0 s .
R s = r 0 s 2 ,
R f R s = r 01 2 + r 12 2 + r 2 s 2 + 2 r 01 r 12 cos 2 θ 1 + 2 r 12 r 2 s cos 2 θ 2 + 2 r 01 r 2 s cos 2 ( θ 1 + θ 2 ) r 0 s 2 = 1 - 4 r 01 r 12 r 0 s 2 sin 2 θ 2 - 4 r 12 r 2 s r 0 s 2 sin 2 θ 2 - 4 r 01 r 2 s r 0 s 2 sin 2 ( θ 1 + θ 2 ) ,
r 12 = 0 ,             r 01 = r 02 ,             θ 1 = 0
R f R s = 1 - 4 r 02 r 2 s r 0 s 2 sin 2 θ 2 .
r 02 = 0 ,
ϕ 00 + ϕ 20 = π / 2 ,
r 02 = sin 2 ϕ 0 - sin 2 ϕ 2 sin 2 ϕ 0 + sin 2 ϕ 2 .
tan ϕ 00 = n 2 n 0 = n 2 ,
R f R s ( 1 - 4 r 02 r 2 s r 0 s 2 sin 2 θ 2 ) - 4 r 01 r 2 s r 0 s 2 sin 2 θ 2 · θ 1 ,
r 02 sin θ 2 + 2 θ 1 r 01 cos θ 2 = 0.
r 02 sin θ 2 f 0 ( ϕ 0 ) , 4 π n 1 cos ϕ 1 r 01 cos θ 2 f 1 ( ϕ 0 ) , d 1 / λ δ ;
f 0 ( ϕ 0 ) + f 1 ( ϕ 0 ) δ = 0.
f 0 ( ϕ 00 ) + f 0 ( ϕ 00 ) Δ ϕ 0 + f 1 ( ϕ 00 ) δ = 0 ,
f 0 ( ϕ 00 ) = 0 , f 0 ( ϕ 00 ) = r 02 sin θ 2 ϕ 00 = sec 2 ϕ 00 2 tan 2 ϕ 00 sin ( 2 π λ n 2 d 2 cos ϕ 20 ) , f 1 ( ϕ 00 ) = 4 π n 1 cos ϕ 10 sin 2 ϕ 00 - sin 2 ϕ 10 sin 2 ϕ 00 + sin 2 ϕ 10 cos ( 2 π λ n 2 d 2 cos ϕ 20 ) .
Δ ϕ 0 = - f 1 ( ϕ 00 ) f 0 ( ϕ 00 ) δ .
sec 2 ϕ 00 · Δ ϕ 0 = Δ n 2 .
Δ n 2 = ( 16 π d 1 λ ) ( n 1 2 - 1 ) ( n 1 2 - n 2 2 ) n 2 2 - 1 n 2 n 1 2 n 2 2 + n 1 2 - n 2 2 1 + n 2 2 ( n 1 2 + n 1 2 n 2 2 + n 1 2 - n 2 2 ) 2 × cot ( 2 π d 2 λ n 2 2 1 + n 2 2 ) .
R f R s ( 1 - 4 r 01 r 1 s r 0 s 2 sin 2 θ 1 ) - 4 r 01 r 2 s r 0 s 2 sin 2 θ 1 · θ 2 - 4 r 2 s r 0 s 2 ( r 01 cos 2 θ 1 + r 12 ) · θ 2 2 .
Δ ϕ 0 = - g 3 ( θ 00 ) g 0 ( ϕ 00 ) δ 2 ,
g 0 ( ϕ 00 ) = 1 2 tan 2 ϕ 00 cos 2 ϕ 00 sin 2 ϕ 10 - sin 2 ϕ s 0 sin 2 ϕ 10 + sin 2 ϕ s 0 sin 2 ( 2 π λ n 1 d 1 cos ϕ 10 ) g 3 ( ϕ 00 ) = ( 2 π n 2 ) 2 cos 2 ϕ 20 sin 2 ϕ 10 - sin 2 ϕ 20 sin 2 ϕ 10 + sin 2 ϕ 20 sin 2 ϕ 20 - sin 2 ϕ s 0 sin 2 ϕ 20 + sin 2 ϕ s 0 , δ = d 2 / λ .
Δ n 1 = ( 4 π d 2 λ ) 2 ( n s 2 - n 2 2 ) ( n 2 2 - n 1 2 ) ( n 2 2 - 1 ) ( n s 2 - n 1 2 ) ( n 1 2 - 1 ) ( n s 2 - 1 ) × n 1 ( n 1 2 n 2 2 + n 2 2 - n 1 2 ) ( n s 2 + n 1 2 n s 2 + n s 2 - n 1 2 ) 2 [ n 2 2 n s 2 ( n 1 2 + 1 ) - n 1 2 ( n 2 2 + n s 2 ) ] ( 1 + n 1 2 ) ( n 2 2 + n 1 2 n 2 2 + n 2 2 - n 1 2 ) 2 ( n 2 2 n 1 2 n s 2 + n s 2 - n 1 2 + n s 2 - n 1 2 n 2 2 + n 2 2 - n 1 2 ) 2 cosec 2 ( 2 π λ d 1 n 1 2 1 + n 1 2 ) .

Metrics