Abstract

Graphical displays of complex reflectance for nonabsorbing multilayer optical coatings based upon circle diagrams are presented. Similar displays are then developed for absorbing films. The two types of display are used to analyze several common multilayers, and a repetitive structure with period characteristics is introduced.

© 1972 Optical Society of America

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References

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  1. P. H. Berning, Physics of Thin Films, G. Hass, Ed. (Academic Press, New York, 1963), Vol. 1.
  2. J. T. Cox, G. Hass, A. Thelen, J. Opt. Soc. Am. 52, 965 (1962).
    [CrossRef]
  3. A. Thelen, J. Opt. Soc. Am. 50, 509 (1960).
  4. A. Thelen, U.S. Patent3,185,020.
  5. F. Rock, U.S. Patent3,432,225.
  6. P. Giacomo, J. Phys. (Paris) 25, 855 (1964).
    [CrossRef]
  7. P. H. Berning, A. F. Turner, J. Opt. Soc. Am. 41, 230 (1957).
    [CrossRef]
  8. P. W. Baumeister, Appl. Opt. 8, 423 (1969).
    [CrossRef] [PubMed]
  9. P. W. Baumeister, V. R. Costich, S. C. Pieper, Appl. Opt. 4, 911 (1965).
    [CrossRef]
  10. R. Gelber, P. Baumeister, Appl. Opt. 9, 863 (1970).
    [CrossRef] [PubMed]

1970 (1)

1969 (1)

1965 (1)

1964 (1)

P. Giacomo, J. Phys. (Paris) 25, 855 (1964).
[CrossRef]

1962 (1)

1960 (1)

A. Thelen, J. Opt. Soc. Am. 50, 509 (1960).

1957 (1)

P. H. Berning, A. F. Turner, J. Opt. Soc. Am. 41, 230 (1957).
[CrossRef]

Baumeister, P.

Baumeister, P. W.

Berning, P. H.

P. H. Berning, A. F. Turner, J. Opt. Soc. Am. 41, 230 (1957).
[CrossRef]

P. H. Berning, Physics of Thin Films, G. Hass, Ed. (Academic Press, New York, 1963), Vol. 1.

Costich, V. R.

Cox, J. T.

Gelber, R.

Giacomo, P.

P. Giacomo, J. Phys. (Paris) 25, 855 (1964).
[CrossRef]

Hass, G.

Pieper, S. C.

Rock, F.

F. Rock, U.S. Patent3,432,225.

Thelen, A.

J. T. Cox, G. Hass, A. Thelen, J. Opt. Soc. Am. 52, 965 (1962).
[CrossRef]

A. Thelen, J. Opt. Soc. Am. 50, 509 (1960).

A. Thelen, U.S. Patent3,185,020.

Turner, A. F.

P. H. Berning, A. F. Turner, J. Opt. Soc. Am. 41, 230 (1957).
[CrossRef]

Appl. Opt. (3)

J. Opt. Soc. Am. (3)

J. T. Cox, G. Hass, A. Thelen, J. Opt. Soc. Am. 52, 965 (1962).
[CrossRef]

A. Thelen, J. Opt. Soc. Am. 50, 509 (1960).

P. H. Berning, A. F. Turner, J. Opt. Soc. Am. 41, 230 (1957).
[CrossRef]

J. Phys. (Paris) (1)

P. Giacomo, J. Phys. (Paris) 25, 855 (1964).
[CrossRef]

Other (3)

P. H. Berning, Physics of Thin Films, G. Hass, Ed. (Academic Press, New York, 1963), Vol. 1.

A. Thelen, U.S. Patent3,185,020.

F. Rock, U.S. Patent3,432,225.

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Figures (18)

Fig. 1
Fig. 1

Complex reflectance r of a high index film on an air/glass interface.

Fig. 2
Fig. 2

Complex reflectance of an unsupported high index film.

Fig. 3
Fig. 3

Circle diagram for 2.37 index film material in air or vacuum medium.

Fig. 4
Fig. 4

Circle diagram for 1.72 index film material in air or vacuum medium. Also serves for a 2.37 index film material in a medium of 1.38 index.

Fig. 5
Fig. 5

Wavelength dependence of reflectance for high index film on glass. The film has optical thickness of one quarter-wave at wavelength of 550 nm.

Fig. 6
Fig. 6

Graphical description of two-layer AR, or V coat for glass substrate. High index film causes the reflectance to progress from the substrate 1 to position 2 or 2′, whereupon the low index film advances reflectance to the origin 3.

Fig. 7
Fig. 7

Complex reflectance for a three-layer antireflection coating. The high index half-wave layer achromatizes the two quarter-wave layers.

Fig. 8
Fig. 8

Circle diagram for an immersed layer of 1.38 index as described in Fig. 9. Reflectance of the substructure and the thickness of the 1.38 layer determine the over-all reflectance within the webbed region.

Fig. 9
Fig. 9

Multilayer system used for the circle diagram of Fig. 8.

Fig. 10
Fig. 10

Diagram illustrating the definition of quantities used in analysis of three-part multilayer. Groups I and II may be single surfaces, single layers, or combinations of layers.

Fig. 11
Fig. 11

Graphical display of complex reflectance from a nickel film. Such a film placed upon any complex reflectance causes the reflectance to progress along the contours toward the opaque point.

Fig. 12
Fig. 12

Graphical display of complex reflectance from a silver film. Such a film placed upon any complex reflectance causes the reflectance to progress along the contours toward the opaque point.

Fig. 13
Fig. 13

Transmittance and reflectance of a variable attenuator which has uniform and low reflectance.

Fig. 14
Fig. 14

Graphical display of a metal–dielectric period composed of 160-Å dielectric (n = 1.5) layers.

Fig. 15
Fig. 15

Transmittance of metal–dielectric periods shown in Fig. 14 which are initiated at the origin. Each period is composed of a metal–dielectric–metal trilayer with 80-Å metal thickness.

Fig. 16
Fig. 16

Locus of reflectance values for the period composed of a 115-Å nickel layer and a dielectric layer of index 1.38.

Fig. 17
Fig. 17

Similar to Fig. 16 except reflectance is displayed for the substructure on which the nickel film is to be placed.

Fig. 18
Fig. 18

Contours of equipotential transmittance of 160-Å silver film at wavelength 550 nm. The cross indicates the reflectance required for peak potential transmittance.

Equations (16)

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r = ( n 1 n 2 ) / ( n 1 + n 2 ) ,
r = 0.2 e i 180 ° ,
r r * = 0.04 ,
r = ( r 1 + r 2 e i ϕ ) / ( 1 + r 1 r 2 e i ϕ ) ,
ϕ = 4 π n d / λ ,
n 2 = [ ( 1 s 1 ) / ( 1 + s 1 ) ] × [ ( 1 s 2 ) / ( 1 + s 2 ) ] .
r = ( 1 n ) / ( 1 + n ) .
ϕ = 4 π n d / λ
centers = r + ( n / n 0 ) [ r 3 r 3 * s * t 2 / ( 1 r 3 r 3 * s s * ) ] , radii = ( n / n 0 ) | [ r 3 t t * / ( 1 r 3 r 3 * s s * ) ] | ,
centers = r ( n / n 0 ) [ B * t 2 / ( B * s + B s * ) ] , radii = ( n / n 0 ) | [ t 2 / ( B * s + B s * ) ] | , B = sin ϕ + i cos ϕ .
r a = r 3 , f b a = r = s , ρ a = ϕ .
M = A × [ cos ϕ ( i / n ) sin ϕ i n sin ϕ cos ϕ ] × B ,
Γ = ( n 0 M 11 + n 0 n s M 12 M 21 n s M 22 ) / ( n 0 M 11 + n 0 n s M 11 + M 21 + n s M 22 ) ,
Γ = ( a tan ϕ + b ) / ( c tan ϕ + d ) .
r j 1 = ( a r j + b ) / ( c r j + d ) .
r = ( f + r ) / ( 1 + f r ) , f = ( 1 n ) / ( 1 + n ) ,

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