Abstract

The use of the quarter-wave optical thickness concept in the design of certain types of multilayer optical filters, or in the thickness determination of optical thickness of monolayers from transmission or reflection data, can present appreciable errors if dispersion is present in the layer optical constants. Moderate amounts of dispersion can cause significant shifts in the location of the global reflection or transmission extrema from what would be predicted using simple optical thickness considerations. Examples, including the use of measured index of refraction values for CeO2, are presented.

© 1975 Optical Society of America

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References

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  1. P. H. Berning, in Physics of Thin Films, G. Hass, Ed. (Academic, New York, 1963), Vol. 3, p. 90.
  2. F. Abelès, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1968), Vol. 2, p. 256.
  3. G. Hass, J. B. Ramsey, R. Thun, J. Opt. Soc. Am. 48, 325 (1958).
  4. A. Smakula, Opt. Acta 9, 205 (1962). Appropriate material is reproduced in American Institute of Physics Handbook, D. E. Gray, Ed. (McGraw-Hill, New York, 1972), p. 6–14.
    [CrossRef]
  5. H. Mumann, Z. Phys. 80, 161 (1933).
    [CrossRef]
  6. L. N. Hadley, D. M. Dennison, J. Opt. Soc. Am. 37, 451 (1947); J. Opt. Soc. Am. 38, 546 (1948).
    [CrossRef] [PubMed]
  7. See, for example, M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964), p. 63.

1962

A. Smakula, Opt. Acta 9, 205 (1962). Appropriate material is reproduced in American Institute of Physics Handbook, D. E. Gray, Ed. (McGraw-Hill, New York, 1972), p. 6–14.
[CrossRef]

1958

G. Hass, J. B. Ramsey, R. Thun, J. Opt. Soc. Am. 48, 325 (1958).

1947

1933

H. Mumann, Z. Phys. 80, 161 (1933).
[CrossRef]

Abelès, F.

F. Abelès, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1968), Vol. 2, p. 256.

Berning, P. H.

P. H. Berning, in Physics of Thin Films, G. Hass, Ed. (Academic, New York, 1963), Vol. 3, p. 90.

Born, M.

See, for example, M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964), p. 63.

Dennison, D. M.

Hadley, L. N.

Hass, G.

G. Hass, J. B. Ramsey, R. Thun, J. Opt. Soc. Am. 48, 325 (1958).

Mumann, H.

H. Mumann, Z. Phys. 80, 161 (1933).
[CrossRef]

Ramsey, J. B.

G. Hass, J. B. Ramsey, R. Thun, J. Opt. Soc. Am. 48, 325 (1958).

Smakula, A.

A. Smakula, Opt. Acta 9, 205 (1962). Appropriate material is reproduced in American Institute of Physics Handbook, D. E. Gray, Ed. (McGraw-Hill, New York, 1972), p. 6–14.
[CrossRef]

Thun, R.

G. Hass, J. B. Ramsey, R. Thun, J. Opt. Soc. Am. 48, 325 (1958).

Wolf, E.

See, for example, M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964), p. 63.

J. Opt. Soc. Am.

Opt. Acta

A. Smakula, Opt. Acta 9, 205 (1962). Appropriate material is reproduced in American Institute of Physics Handbook, D. E. Gray, Ed. (McGraw-Hill, New York, 1972), p. 6–14.
[CrossRef]

Z. Phys.

H. Mumann, Z. Phys. 80, 161 (1933).
[CrossRef]

Other

See, for example, M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964), p. 63.

P. H. Berning, in Physics of Thin Films, G. Hass, Ed. (Academic, New York, 1963), Vol. 3, p. 90.

F. Abelès, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1968), Vol. 2, p. 256.

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Figures (3)

Fig. 1
Fig. 1

Calculated radiant transmittance of a CeO2 monolayer of quarter-wave optical thickness 550 nm using an experimentally measured dispersion index of refraction (Table I). Dotted lines are transmission curves of monolayers with constant indices of refraction ranging from 2.1 to 2.5 and the same physical thickness as the CeO2 layer. A glass-air second interface is also included in the calculation.

Fig. 2
Fig. 2

Same as Fig. 1, except that the physical thickness of all layers is doubled to achieve a half-wave thickness for CeO2 at 550 nm.

Fig. 3
Fig. 3

Shifts toward shorter wavelengths in the local radiant transmittance minimum of a dispersive monolayer having a quarter-wave optical thickness of 550 nm for various values of n1 and b in Eq. (4) used for the index of refraction of the layer. n0 = 1.0, ns = 1.521.

Tables (1)

Tables Icon

Table I Experimentally Determined Index of Refraction of Cerium Dioxide

Equations (7)

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λ 0 = 4 n t .
1 R = T = 4 n s / { n 0 [ ( 1 + n s n 0 ) 2 cos 2 β + ( n s n + n n 0 ) 2 sin 2 β ] } ,
β = ( 2 π n t ) / λ .
d R d σ = 0 = 8 n 0 n s tan β ( 1 + tan 2 β ) { ( n 0 + n s ) 2 + [ ( n 0 n s / n ) ] 2 tan 2 β } 2 × [ ( n n 0 2 n s 2 n 3 ) ( d n d σ ) tan β + ( n 0 2 n s 2 n 2 + n 2 n 0 2 n s 2 ) d β d σ ] ,
d β d σ = β [ 1 n ( d n d σ ) + 1 σ ] ,
σ = 1 / λ .
n = n 1 [ 1 b ( λ λ 0 ) ] ,

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