Abstract

The reflection–transmission method for determining the optical constants of a film on a substrate has been examined and a computer program for the solution of the equation system has been written. A new method involving only transmittance data in a limited wavelength region has been worked out and has been found to give accurate results. The computer program for this method gives directly n and k curves from the transmission curve.

© 1968 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. Mayer, Physik dünner Schichten (Wissenschaftliche Verlagsgesellschaft mbH., Stuttgart1950), Teil I, pp. 207 and 218.
  2. O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths Scientific Publications, London, 1950), pp. 136 and 139.
  3. Ref. 1, p. 207.
  4. Ref. 2, p. 136.
  5. D. Beaglehole, Proc. Phys. Soc. 85, 1007 (1965).
    [CrossRef]
  6. S. Tolansky, Multiple Beam Interferometry (Clarendon Press, Oxford, 1948).
  7. H. E. Bennett, W. F. Koehler, J. Opt. Soc. Am. 50, 1 (1960).
    [CrossRef]
  8. O. S. Heavens, in Physics of Thin FilmsG. Hass, R. E. Thun, Eds. (Academic Press Inc., 1964), Vol. 2, p. 217.
  9. J. M. Bennett, M. J. Booty, Appl. Opt. 5, 41 (1966).
    [CrossRef] [PubMed]
  10. F. Abelès, M. L. Theye, Surface Sci. 5, 325 (1966).
    [CrossRef]
  11. H. E. Bennett, J. O. Porteus, J. Opt. Soc. Am. 51, 123 (1961).
    [CrossRef]
  12. H. E. Bennett, J. Opt. Soc. Am. 53, 1389 (1963).
    [CrossRef]
  13. J. S. Plaskett, P. N. Schatt, J. Chem. Phys. 38, 612 (1963).
    [CrossRef]
  14. S. Marda, G. Thyagarajan, P. N. Schatz, J. Chem. Phys. 39, 3473 (1963).
  15. D. M. Roessler, Brit. J. Appl. Phys. 16, 1119 (1965).
    [CrossRef]
  16. W. M. McKeemon, L. Tesler, Comm. ACM 6, 315 (1963).
    [CrossRef]

1966 (2)

1965 (2)

D. Beaglehole, Proc. Phys. Soc. 85, 1007 (1965).
[CrossRef]

D. M. Roessler, Brit. J. Appl. Phys. 16, 1119 (1965).
[CrossRef]

1963 (4)

W. M. McKeemon, L. Tesler, Comm. ACM 6, 315 (1963).
[CrossRef]

H. E. Bennett, J. Opt. Soc. Am. 53, 1389 (1963).
[CrossRef]

J. S. Plaskett, P. N. Schatt, J. Chem. Phys. 38, 612 (1963).
[CrossRef]

S. Marda, G. Thyagarajan, P. N. Schatz, J. Chem. Phys. 39, 3473 (1963).

1961 (1)

1960 (1)

Abelès, F.

F. Abelès, M. L. Theye, Surface Sci. 5, 325 (1966).
[CrossRef]

Beaglehole, D.

D. Beaglehole, Proc. Phys. Soc. 85, 1007 (1965).
[CrossRef]

Bennett, H. E.

Bennett, J. M.

Booty, M. J.

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths Scientific Publications, London, 1950), pp. 136 and 139.

O. S. Heavens, in Physics of Thin FilmsG. Hass, R. E. Thun, Eds. (Academic Press Inc., 1964), Vol. 2, p. 217.

Koehler, W. F.

Marda, S.

S. Marda, G. Thyagarajan, P. N. Schatz, J. Chem. Phys. 39, 3473 (1963).

Mayer, H.

H. Mayer, Physik dünner Schichten (Wissenschaftliche Verlagsgesellschaft mbH., Stuttgart1950), Teil I, pp. 207 and 218.

McKeemon, W. M.

W. M. McKeemon, L. Tesler, Comm. ACM 6, 315 (1963).
[CrossRef]

Plaskett, J. S.

J. S. Plaskett, P. N. Schatt, J. Chem. Phys. 38, 612 (1963).
[CrossRef]

Porteus, J. O.

Roessler, D. M.

D. M. Roessler, Brit. J. Appl. Phys. 16, 1119 (1965).
[CrossRef]

Schatt, P. N.

J. S. Plaskett, P. N. Schatt, J. Chem. Phys. 38, 612 (1963).
[CrossRef]

Schatz, P. N.

S. Marda, G. Thyagarajan, P. N. Schatz, J. Chem. Phys. 39, 3473 (1963).

Tesler, L.

W. M. McKeemon, L. Tesler, Comm. ACM 6, 315 (1963).
[CrossRef]

Theye, M. L.

F. Abelès, M. L. Theye, Surface Sci. 5, 325 (1966).
[CrossRef]

Thyagarajan, G.

S. Marda, G. Thyagarajan, P. N. Schatz, J. Chem. Phys. 39, 3473 (1963).

Tolansky, S.

S. Tolansky, Multiple Beam Interferometry (Clarendon Press, Oxford, 1948).

Appl. Opt. (1)

Brit. J. Appl. Phys. (1)

D. M. Roessler, Brit. J. Appl. Phys. 16, 1119 (1965).
[CrossRef]

Comm. ACM (1)

W. M. McKeemon, L. Tesler, Comm. ACM 6, 315 (1963).
[CrossRef]

J. Chem. Phys. (2)

J. S. Plaskett, P. N. Schatt, J. Chem. Phys. 38, 612 (1963).
[CrossRef]

S. Marda, G. Thyagarajan, P. N. Schatz, J. Chem. Phys. 39, 3473 (1963).

J. Opt. Soc. Am. (3)

Proc. Phys. Soc. (1)

D. Beaglehole, Proc. Phys. Soc. 85, 1007 (1965).
[CrossRef]

Surface Sci. (1)

F. Abelès, M. L. Theye, Surface Sci. 5, 325 (1966).
[CrossRef]

Other (6)

S. Tolansky, Multiple Beam Interferometry (Clarendon Press, Oxford, 1948).

O. S. Heavens, in Physics of Thin FilmsG. Hass, R. E. Thun, Eds. (Academic Press Inc., 1964), Vol. 2, p. 217.

H. Mayer, Physik dünner Schichten (Wissenschaftliche Verlagsgesellschaft mbH., Stuttgart1950), Teil I, pp. 207 and 218.

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths Scientific Publications, London, 1950), pp. 136 and 139.

Ref. 1, p. 207.

Ref. 2, p. 136.

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 (12)

Fig. 1
Fig. 1

The V-W attachment is placed in the sample beam of a double beam spectrophotometer. The first mirror is concave in order to maintain the focus on the sample and the detector. A and A′ are identical mirrors, B is the sample. RB = (IA/IA′)1/2.

Fig. 2
Fig. 2

The reflectance and transmittance of a sample as a function of the refractive index of the film. ns = 1.5 and d/λ = 0.1.

Fig. 3
Fig. 3

Schematic typical behavior of the refractive index of a metal and an insulator as a function of frequency.

Fig. 4
Fig. 4

The optical constants of a 100-Å thick film of CdI2 obtained from reflection and transmission measurements.

Fig. 5
Fig. 5

The imaginary part of the dielectric constant 2 = 2nk for a 410-Å thick Cu film. The curve is obtained from experimental R and T values.

Fig. 6
Fig. 6

The transmittance and the ratio of the reflectance from the film side and the substrate side of the sample as a function of the refractive index of the film. ns = 1.5 and d/λ = 0.1.

Fig. 7
Fig. 7

The transmittance for two samples of the same material as a function of the refractive index of the film. The solid line refers to d/λ = 0.1 and the broken line to d/λ = 0.2.

Fig. 8
Fig. 8

The K-K analysis is done on the amplitude (Tf)1/2 inside the substrate instead of the more directly measured amplitude (T)1/2. The correction for the substrate is made when approximative values of n and k has been obtained and thus Rb can be calculated. The interference in the substrate has to be neglected due to the thickness variations.

Fig. 9
Fig. 9

Integration contour to obtain the dispersion relation between the phase shift and the transmitted amplitude through a single film.

Fig. 10
Fig. 10

The transmittance and the phase shift on transmission for a film without substrate as a function of the refractive index. The refractive index behind the film is 1.5 and d/λ = 0.1.

Fig. 11
Fig. 11

The optical constants of a 100-Å thick CdI2 film obtained from K-K analysis of transmission data (solid line) and from combination of reflection and transmission data (broken line). The meaning of the arrows is discussed in the text.

Fig. 12
Fig. 12

The imaginary part of the dielectric constant 2 = 2nk for a 410-Å thick Cu film. The solid line is the result from K-K analysis, the broken line is obtained from R and T data. The vertical lines are discussed in the text.

Equations (19)

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

lim d / λ 0 T = 0.960 ,
{ F ( T , n , k ) = 0 G ( R , n , k ) = 0
Re { ln [ ( T f ) 1 / 2 exp ( i θ ) ] / ( ν - ν 0 ) }
θ ( ν 0 ) = 2 ν 0 π P 0 ln ( T f ) 1 / 2 ν 2 - ν 0 2 d ν - 2 π ν 0 d .
θ 0 ν 1 = 2 ν 0 π 0 ν 1 ln ( T f ) 1 / 2 ν 2 - ν 0 2 d ν = A ( ν 0 ) ln | ν 1 + ν 0 ν 1 - ν 0 |
θ ν 2 = 2 ν 0 π ν 2 ln ( T f ) 1 / 2 ν 2 - ν 0 2 d ν = B ( ν 0 ) ln | ν 2 + ν 0 ν 2 - ν 0 | ,
A ( ν 0 ) = 1 π { ln [ T f ( ν 1 ) ] 1 / 2 - 2 ν 0 ln | ν 1 - ν 0 ν 1 + ν 0 | 0 ν 1 ln [ T f ( ν ) ] 1 / 2 ν 2 - ν 0 2 d ν } ,
B ( ν 0 ) = - 1 π { ln [ T f ( ν 2 ) ] 1 / 2 + 2 ν 0 ln | ν 2 - ν 0 ν 2 + ν 0 | ν 2 ln [ T f ( ν ) ] 1 / 2 ν 2 - ν 0 2 d ν } .
T = T f n s × T s × 1 / [ 1 - R b ( 1 - T s ) ] ,
1 ν - ν 0 = lim s 0 + ν - ν 0 ( ν - ν 0 ) 2 + s 2
R = a 1 exp ( 4 π k / λ ) d + b 1 cos ( 4 π n / λ ) d + c 1 sin ( 4 π n / λ ) d + f 1 exp [ - ( 4 π k / λ ) ] d a 2 exp ( 4 π k / λ ) d + b 2 cos ( 4 π n / λ ) d + c 2 sin ( 4 π n / λ ) d + f 2 exp [ - ( 4 π k / λ ) ] d ,
T = a a 2 exp ( 4 π k / λ ) d + b 2 cos ( 4 π n / λ ) d + c 2 sin ( 4 π n / λ ) d + f 2 exp [ - ( 4 π k / λ ) ] d ,
a 1 = [ ( n - 1 ) 2 + k 2 ] { ( n s 2 + 1 ) ( n 2 + k 2 + n s 2 ) + 4 n n s 2 + ( n s 2 - 1 ) [ ( n 2 + k 2 - n s 2 ) cos y s - 2 k n s sin y s ] } , a 2 = [ ( n + 1 ) 2 + k 2 ] { ( n s 2 + 1 ) ( n 2 + k 2 + n s 2 ) + 4 n n s 2 + ( n s 2 - 1 ) [ ( n 2 + k 2 - n s 2 ) cos y s - 2 k n s sin y s ] } , b 1 = - 2 { ( n s 2 + 1 ) ( n 2 + k 2 - 1 ) ( n 2 + k 2 - n s 2 ) + 8 k 2 n s 2 + ( n s 2 - 1 ) [ ( n 2 + k 2 - 1 ) ( n 2 + k 2 + n s 2 ) cos y s + 4 n k n s sin y s ] } , b 2 = - 2 { ( n s 2 + 1 ) ( n 2 + k 2 - 1 ) ( n 2 + k 2 - n s 2 ) - 8 k 2 n s 2 + ( n s 2 - 1 ) [ ( n 2 + k 2 - 1 ) ( n 2 + k 2 + n s 2 ) cos y s - 4 n k n s sin y s ] } , c 1 = 4 { k [ - ( n s 2 + 1 ) ( n 2 + k 2 - n s 2 ) + 2 n s 2 ( n 2 + k 2 - 1 ) ] + ( n s 2 - 1 ) - [ k ( n 2 + k 2 + n s 2 ) cos y s + n n s ( n 2 + k 2 - 1 ) sin y s ] } , c 2 = 4 { k [ ( n s 2 + 1 ) ( n 2 + k 2 - n s 2 ) + 2 n s 2 ( n 2 + k 2 - 1 ) ] + ( n s 2 - 1 ) [ k ( n 2 + k 2 + n s 2 ) cos y s + n n s ( n 2 + k 2 - 1 ) sin y s ] } , f 1 = [ ( n + 1 ) 2 + k 2 ] { ( n s 2 + 1 ) ( n 2 + k 2 + n s 2 ) - 4 n n s 2 + ( n s 2 - 1 ) [ ( n 2 + k 2 - n s 2 ) cos y s + 2 k n s sin y s ] } , f 2 = [ ( n - 1 ) 2 + k 2 ] { ( n s 2 + 1 ) ( n 2 + k 2 + n s 2 ) - 4 n n s 2 + ( n s 2 - 1 ) [ ( n 2 + k 2 - n s 2 ) cos y s + 2 k n s sin y s ] } , a = 32 n s 2 ( n 2 + k 2 ) , and y s = 4 π n s ( d s / λ ) .
R b = ( g 2 2 + h 2 2 ) e 2 α 1 + ( g 1 2 + h 1 2 ) e - 2 α 1 + A cos 2 γ 1 - B sin 2 γ 1 e 2 α 1 + ( g 1 2 + h 1 2 ) ( g 2 2 + h 2 2 ) e - 2 α 1 + C cos 2 γ 1 + D sin 2 γ 1 ,
A = 2 ( g 1 g 2 + h 1 h 2 ) B = 2 ( g 1 h 2 - g 2 h 1 ) , C = 2 ( g 1 g 2 - h 1 h 2 ) D = 2 ( g 1 h 2 + g 2 h 1 ) ,
g 1 = 1 - n 2 - k 2 ( n + 1 ) 2 + k 2 h 1 = 2 k ( n + 1 ) 2 + k 2 , g 2 = n 2 + k 2 - n s 2 ( n + n s ) 2 + k 2 h 2 = - 2 k n s ( n + n s ) 2 + k 2 , α 1 = 2 π k d / λ γ 1 = 2 π n d / λ .
{ n s T f = n s 16 ( n 2 + k 2 ) / ( C 2 + D 2 ) θ f = arctan ( k C + n D ) / ( k D - n C ) ,
C = e K { [ ( 1 + n ) ( n + n s ) - k 2 ] cos N + k ( 1 + 2 n + n s ) sin N } + e - K { [ ( 1 - n ) ( n - n s ) + k 2 ] cos N - k ( 1 - 2 n + n s ) sin N } , D = e K { [ ( 1 + n ) ( n + n s ) - k 2 ] sin N - k ( 1 + 2 n + n s ) cos N } - e - K { [ ( 1 - n ) ( n - n s ) + k 2 ] sin N + k ( 1 - 2 n + n s ) cos N } ,
K = 2 π k d / λ             and             N = 2 π n d / λ .

Metrics