Abstract

Equations are developed for the flow of radiant power, transmittance, and absorptance of an absorbing multilayer, in terms of its characteristic matrix and the admittance of the surrounding media. This is applied to the design of bandpass filters and absorbing coatings. Some uv bandpass filters which contain several aluminum films are designed.

© 1969 Optical Society of America

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References

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  1. P. H. Berning, A. F. Turner, J. Opt. Soc. Amer. 47, 230 (1957).
    [CrossRef]
  2. W. Geffcken, “Interference Light Filter,” Deutches Reich Patentschrift716153 (1942).
  3. C. Dufour, J. Phys. Rad 11, 413 (1950).
    [CrossRef]
  4. A. Hermansen, Nature 174, 218 (1954).
    [CrossRef]
  5. A. Hermansen, Det Kongelige Danske Videnskabernes Selskab 29, #13 (1955).
  6. H. Wolter, in Handbuch der Physik, S. Flugge, Ed. (Springer, Berlin, 1956), Vol. 24, p. 500.
  7. D. J. Schroeder, J. Opt. Soc. Amer. 52, 1380 (1962).
    [CrossRef]
  8. A. Turner, in Radiative Transfer from Solid Materials, H. Blau, H. Fischer, Eds. (The Macmillan Company, New York, 1962), p. 24.
  9. A. Turner, “Principles of Enhancing Absorption in Thin Films,” in Symposium on Problems of Thermal Imaging at Ft. Belvoir, Va. (1956).
  10. R. Schmidt, K. Park, Appl. Opt. 4, 917 (1965).
    [CrossRef]
  11. H. Pohlack, in Optik und Spektroscopie aller Wellenlängen (Akademie-Verlag, Berlin, 1959), p. 369.
  12. H. Börsch, M. Lambeck, H. Wenzel, Z. Angew. Phys. 13, 548 (1961).
  13. Max Born, Emil Wolf, Principles of Optics (Pergamon Press, London, 1959), p. 33.
  14. P. Berning, in Physics of Thin Films, G. Hass, Ed. (Academic Press Inc., New York1963), p. 77.
  15. G. Ragan, Microwave Transmission Circuits (Dover Publications, New York, 1965), Sec. 2.11.
  16. P. Kard, Sov. Phys.–Dokl. 7, 256 (1956).
  17. P. Baumeister, V. Costich, S. Pieper, Appl. Opt. 4, 911 (1965).
    [CrossRef]
  18. J. Apfel, J. Opt. Soc. Amer. 56, 553A (1966).
  19. F. Abelès, in Advanced Optical Techniques, A.C.S. Van Heel, Ed. (North-Holland Publishing Company, Amsterdam, 1967), p. 178.
  20. L. Hadley, D. Dennison, J. Opt. Soc. Amer. 37, 451 (1948).
  21. R. Maier, Thin Solid Films 1, 31 (1967).
    [CrossRef]
  22. D. Gray, Ed. American Institute of Physics Handbook, 2nd edition (McGraw-Hill Book Company, Inc., New York, 1963), Sec. 6g.
  23. V. Costich, “Interference Filters for the Ultraviolet,” Ph.D. Thesis, Institute of Optics, University of Rochester, Rochester, New York, 1965.

1967 (1)

R. Maier, Thin Solid Films 1, 31 (1967).
[CrossRef]

1966 (1)

J. Apfel, J. Opt. Soc. Amer. 56, 553A (1966).

1965 (2)

1962 (1)

D. J. Schroeder, J. Opt. Soc. Amer. 52, 1380 (1962).
[CrossRef]

1961 (1)

H. Börsch, M. Lambeck, H. Wenzel, Z. Angew. Phys. 13, 548 (1961).

1957 (1)

P. H. Berning, A. F. Turner, J. Opt. Soc. Amer. 47, 230 (1957).
[CrossRef]

1956 (1)

P. Kard, Sov. Phys.–Dokl. 7, 256 (1956).

1955 (1)

A. Hermansen, Det Kongelige Danske Videnskabernes Selskab 29, #13 (1955).

1954 (1)

A. Hermansen, Nature 174, 218 (1954).
[CrossRef]

1950 (1)

C. Dufour, J. Phys. Rad 11, 413 (1950).
[CrossRef]

1948 (1)

L. Hadley, D. Dennison, J. Opt. Soc. Amer. 37, 451 (1948).

Abelès, F.

F. Abelès, in Advanced Optical Techniques, A.C.S. Van Heel, Ed. (North-Holland Publishing Company, Amsterdam, 1967), p. 178.

Apfel, J.

J. Apfel, J. Opt. Soc. Amer. 56, 553A (1966).

Baumeister, P.

Berning, P.

P. Berning, in Physics of Thin Films, G. Hass, Ed. (Academic Press Inc., New York1963), p. 77.

Berning, P. H.

P. H. Berning, A. F. Turner, J. Opt. Soc. Amer. 47, 230 (1957).
[CrossRef]

Born, Max

Max Born, Emil Wolf, Principles of Optics (Pergamon Press, London, 1959), p. 33.

Börsch, H.

H. Börsch, M. Lambeck, H. Wenzel, Z. Angew. Phys. 13, 548 (1961).

Costich, V.

P. Baumeister, V. Costich, S. Pieper, Appl. Opt. 4, 911 (1965).
[CrossRef]

V. Costich, “Interference Filters for the Ultraviolet,” Ph.D. Thesis, Institute of Optics, University of Rochester, Rochester, New York, 1965.

Dennison, D.

L. Hadley, D. Dennison, J. Opt. Soc. Amer. 37, 451 (1948).

Dufour, C.

C. Dufour, J. Phys. Rad 11, 413 (1950).
[CrossRef]

Geffcken, W.

W. Geffcken, “Interference Light Filter,” Deutches Reich Patentschrift716153 (1942).

Hadley, L.

L. Hadley, D. Dennison, J. Opt. Soc. Amer. 37, 451 (1948).

Hermansen, A.

A. Hermansen, Det Kongelige Danske Videnskabernes Selskab 29, #13 (1955).

A. Hermansen, Nature 174, 218 (1954).
[CrossRef]

Kard, P.

P. Kard, Sov. Phys.–Dokl. 7, 256 (1956).

Lambeck, M.

H. Börsch, M. Lambeck, H. Wenzel, Z. Angew. Phys. 13, 548 (1961).

Maier, R.

R. Maier, Thin Solid Films 1, 31 (1967).
[CrossRef]

Park, K.

Pieper, S.

Pohlack, H.

H. Pohlack, in Optik und Spektroscopie aller Wellenlängen (Akademie-Verlag, Berlin, 1959), p. 369.

Ragan, G.

G. Ragan, Microwave Transmission Circuits (Dover Publications, New York, 1965), Sec. 2.11.

Schmidt, R.

Schroeder, D. J.

D. J. Schroeder, J. Opt. Soc. Amer. 52, 1380 (1962).
[CrossRef]

Turner, A.

A. Turner, in Radiative Transfer from Solid Materials, H. Blau, H. Fischer, Eds. (The Macmillan Company, New York, 1962), p. 24.

A. Turner, “Principles of Enhancing Absorption in Thin Films,” in Symposium on Problems of Thermal Imaging at Ft. Belvoir, Va. (1956).

Turner, A. F.

P. H. Berning, A. F. Turner, J. Opt. Soc. Amer. 47, 230 (1957).
[CrossRef]

Wenzel, H.

H. Börsch, M. Lambeck, H. Wenzel, Z. Angew. Phys. 13, 548 (1961).

Wolf, Emil

Max Born, Emil Wolf, Principles of Optics (Pergamon Press, London, 1959), p. 33.

Wolter, H.

H. Wolter, in Handbuch der Physik, S. Flugge, Ed. (Springer, Berlin, 1956), Vol. 24, p. 500.

Appl. Opt. (2)

Det Kongelige Danske Videnskabernes Selskab (1)

A. Hermansen, Det Kongelige Danske Videnskabernes Selskab 29, #13 (1955).

J. Opt. Soc. Amer. (4)

D. J. Schroeder, J. Opt. Soc. Amer. 52, 1380 (1962).
[CrossRef]

P. H. Berning, A. F. Turner, J. Opt. Soc. Amer. 47, 230 (1957).
[CrossRef]

J. Apfel, J. Opt. Soc. Amer. 56, 553A (1966).

L. Hadley, D. Dennison, J. Opt. Soc. Amer. 37, 451 (1948).

J. Phys. Rad (1)

C. Dufour, J. Phys. Rad 11, 413 (1950).
[CrossRef]

Nature (1)

A. Hermansen, Nature 174, 218 (1954).
[CrossRef]

Sov. Phys.–Dokl. (1)

P. Kard, Sov. Phys.–Dokl. 7, 256 (1956).

Thin Solid Films (1)

R. Maier, Thin Solid Films 1, 31 (1967).
[CrossRef]

Z. Angew. Phys. (1)

H. Börsch, M. Lambeck, H. Wenzel, Z. Angew. Phys. 13, 548 (1961).

Other (11)

Max Born, Emil Wolf, Principles of Optics (Pergamon Press, London, 1959), p. 33.

P. Berning, in Physics of Thin Films, G. Hass, Ed. (Academic Press Inc., New York1963), p. 77.

G. Ragan, Microwave Transmission Circuits (Dover Publications, New York, 1965), Sec. 2.11.

F. Abelès, in Advanced Optical Techniques, A.C.S. Van Heel, Ed. (North-Holland Publishing Company, Amsterdam, 1967), p. 178.

H. Pohlack, in Optik und Spektroscopie aller Wellenlängen (Akademie-Verlag, Berlin, 1959), p. 369.

W. Geffcken, “Interference Light Filter,” Deutches Reich Patentschrift716153 (1942).

A. Turner, in Radiative Transfer from Solid Materials, H. Blau, H. Fischer, Eds. (The Macmillan Company, New York, 1962), p. 24.

A. Turner, “Principles of Enhancing Absorption in Thin Films,” in Symposium on Problems of Thermal Imaging at Ft. Belvoir, Va. (1956).

H. Wolter, in Handbuch der Physik, S. Flugge, Ed. (Springer, Berlin, 1956), Vol. 24, p. 500.

D. Gray, Ed. American Institute of Physics Handbook, 2nd edition (McGraw-Hill Book Company, Inc., New York, 1963), Sec. 6g.

V. Costich, “Interference Filters for the Ultraviolet,” Ph.D. Thesis, Institute of Optics, University of Rochester, Rochester, New York, 1965.

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Figures (27)

Fig. 1
Fig. 1

The arrangement of the layers in a stack that is bounded on the left by the incident medium and on the right by the emergent medium of admittance Ŷ.

Fig. 2
Fig. 2

The arrangement of the absorbing stack (shown as the cross-hatched layers), the matching stack (stack A), and the antireflection stack (stack ) in a multilayer bandpass filter.

Fig. 3
Fig. 3

The computed spectral transmittance (lower curve) and optical density (upper curve) of a bandpass filter. Its design is listed in Table II.

Fig. 4
Fig. 4

The radiant power flow ratio, Ψ, (curve c) reflectance from the incident side (curve a) and reflectance from the emergent side (curve b) of a bandpass filter. Its design is listed in Table II.

Fig. 5
Fig. 5

The contours of constant radiant power flow Ψ are plotted in the complex amplitude reflectance plane for a palladium film of physical thickness 200 Å, optical constant njk = 2.3 − j2.70 at λ0 = 5461 Å.

Fig. 6
Fig. 6

The absorbing stack, which is bisected to form the half filter, is cross-hatched. The admittance Ŷa and power flow ratio Ψa are measured at the left interface of this half filter.

Fig. 7
Fig. 7

Isotransmittance (in percent) contours on the complex admittance plane for a silver film 0.05 waves (300 Å) in thickness. The optical constants of the silver are: n = 0.06, k = 3.75, at λ0 = 6000 Å.

Fig. 8
Fig. 8

The same isotransmittance curves depicted in Fig. 7, but the scale of the abscissa is semilogarithmic.

Fig. 9
Fig. 9

The isotransmittance contours of the silver film cited in the caption to Fig. 7 as plotted in the amplitude reflectance plane. u is the real part of the complex reflectance Γ ^ and w is its imaginary part.

Fig. 10
Fig. 10

Isotransmittance (in percent) contours in the amplitude reflectance plane for an aluminum film which is 0.0986 waves in thickness. This corresponds to a physical thickness of 250 Å at λ0 = 2536 Å and its optical constants are n = 0.20 and k = 2.80.

Fig. 11
Fig. 11

A parametric plot of the admittance (in the reflectance plane) of the stack designated in Table VI as stack 1. The T = 30% contour (from Fig. 10) is shown as a dashed line in Figs. 1113. The points a, b, c, d, and e corresponds to λ0/λ = 0.90, 0.95, 1.0, 1.05, and 1.10, respectively.

Fig. 12
Fig. 12

A parametric plot of the admittance (in the reflectance plane) of the stack designated in Table VI as stack 2. The T = 30% contour (from Fig. 10) is shown as a dashed line in Figs. 1113. The points a, b, c, d, and e corresponds to λ0/λ = 0.90, 0.95, 1.0, 1.05, and 1.10, respectively.

Fig. 13
Fig. 13

A parametric plot of the admittance (in the reflectance plane) of the stack designated in Table VI as stack 3. The T = 30% contour (from Fig. 10) is shown as a dashed line in Figs. 11–13). The points a, b, c, d, and e corresponds to λ0/λ = 0.90, 0.95, 1.0, 1.05, and 1.10, respectively.

Fig. 14
Fig. 14

The optical density vs frequency of bandpass filters. Their designs are designated as 1 (solid curve, 1), 2 (dashed curve), and 3 (solid curve, 3) in Table VI.

Fig. 15
Fig. 15

The computed radiant reflectance R, transmittance T, and power flow ratio Ψ vs frequency of a bandpass filter. The design of this filter is cited in Table VI and is designated as 1 in that Table.

Fig. 16
Fig. 16

The computed radiant reflectance R, transmittance T, and power flow ratio Ψ vs frequency of a bandpass filter. The design of this filter is cited in Table VI and is designated as 2 in that Table.

Fig. 17
Fig. 17

The computed radiant reflectance R, transmittance T, and power flow ratio Ψ vs frequency of a bandpass filter. The design of this filter is cited in Table VI and is designated as 3 in that Table.

Fig. 18
Fig. 18

The computed Ψmax at 2536 Å for the stack: MDM vs the optical thickness of the dielectric spacer D. The M layers are aluminum, 250 Å in thickness, and n = 0.2 − j2.8. The cryolite has an index of 1.36.

Fig. 19
Fig. 19

The computed spectral transmittance of a two-M type of bandpass filter of the design: air (L H)10 1.828L M D M 1.828L (H L)8 quartz where L is QWOT at 2536 Å, index 1.36, D is QWOT at 3900 Å, index 1.36, H is a QWOT at 2536 Å, index 1.55, M is aluminum, 250 Å in thickness and n ^ = 0.2 − j 2.8 at 2536 Å for the nondispersive calculation (dashed curve). The index of quartz is 1.47. The dispersive calculation (solid curve) uses the optical constants for aluminum of W. Hunter, J. Phys. 25, 154 (1964).

Fig. 20
Fig. 20

The spectral transmittance curve of Fig. 19 plotted on an optical density scale.

Fig. 21
Fig. 21

The computed spectral transmittance (dashed curve) of a two-M type of bandpass filter, of the design: air (LH)10 1.828LM 0.5DHH 0.5DM 1.828L (HL)8 quartz where H,L,M, etc., have the same meaning as in to Fig. 19. The solid curve is the same as Fig. 20.

Fig. 22
Fig. 22

The computed spectral transmittance (optical density) of a bandpass filter of the design: air (HL)7H 1.845LM1D1M2D2M2D2D1M1 1.845L (HL)9 quartz where H is a QWOT at 2536 Å and index 1.55. L, D1, and D2 are cryolite films (index 1.36) of QWOT 2536 Å, 4100 Å, and 3900 Å, respectively. M1 and M2 represent aluminum films of thickness 150 Å and 250 Å, respectively. The dashed curve is computed with dispersive optical constants for the aluminum [W. Hunter, J. Phys 25, 154 (964)] and the solid curve with n ^ = 0.2 − j2.8.

Fig. 23
Fig. 23

The computed spectral transmittance (optical density) of a bandpass filter of the design: air (HL)7H 1.845LM1 0.5D1HH 0.5D1M2 0.5D2HH 0.5D2M2 0.5D1HH 0.5D1M1 1.845L (HL)9 quartz, where H, L, D1D2, etc., have the same meaning as in Fig. 22. The optical constants of the aluminum films are dispersive [W. Hunter, J. Phys. 25, 154 (1964)].

Fig. 24
Fig. 24

Computed maximum radiant power flow ratio Ψmax vs wavenumber for a gold film of physical thickness 20 nm (circles), 40 m (triangles) and 60 nm (squares). The dispersive optical constants are obtained from Ref. 22.

Fig. 25
Fig. 25

Computed maximum radiant power flow ratio Ψmax vs wavenumber for a copper film of physical thickness 20 nm (circles), 40 nm (triangles) and 60 nm (squares). The dispersive optical constants are obtained from Ref. 22.

Fig. 26
Fig. 26

Computed maximum radiant power flow ratio Ψmax vs wavenumber for a silver film of physical thickness 20 nm (circles), 40 nm (triangles) and 60 nm (squares). The dispersive optical constants are obtained from Ref. 22.

Fig. 27
Fig. 27

Computed maximum radiant power flow ratio Ψmax vs wavenumber for an aluminum film of physical thickness 20 nm (circles), 40 nm (triangles) and 60 nm (squares). The dispersive optical constants are obtained from Ref. 22.

Tables (6)

Tables Icon

Table I The Coefficients that Appear in Eq. (8)a

Tables Icon

Table II The Design of a Near Infrared Bandpass Filtera

Tables Icon

Table III The Design of an Absorbing Coatinga

Tables Icon

Table IV The Coefficients that Appear in Eq. (20)a

Tables Icon

Table V The Coefficients Which Appear in Eq. (24)a

Tables Icon

Table VI The Design of Matching Stacks and Bandpass Filtersa

Equations (25)

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Ψ a = S b S a - 1 .
S a = c ( 8 π ) - 1 Re ( E a H a * ) ,
[ F ^ a H ^ a ] = [ a ^ 1 a ^ 3 a ^ 2 a ^ 4 ] [ E ^ b H ^ b ] ,
Ψ a = x { Re [ ( a ^ 1 + Y ^ a ^ 3 ) ( a ^ 2 + Y ^ a ^ 4 ) * ] } - 1 ;
Ψ a = x ( A 1 B 1 + A 2 B 2 ) - 1 ,
A 1 = a 1 + a 3 x - b 3 z , A 2 = b 1 + b 3 x + a 3 z ,
B 1 = a 2 + a 4 x - b 4 z , B 2 = b 2 + b 4 x + a 4 z .
Ψ a = x [ D 10 + D 10 x + D 11 z + D 12 ( x 2 + z 2 ) ] - 1 .
x max = F D 12 - 1 ,
F = ( D 12 D 10 - 0.25 D 11 2 ) 1 2 ,
z max = - 0.5 D 11 D 12 - 1 .
Ψ max = ( D 10 + 2 D 12 x max ) - 1 = ( D 10 + 2 F ) - 1 .
A = ( 1 - R ) ( 1 - Ψ ) .
Γ ^ = u + j w = ( 1 - Y ^ ) ( 1 + Y ^ ) - 1 ,
Ψ a = ( 1 - ρ 2 ) ( e 2 ρ 2 + e 10 ρ sin α + e 01 ρ cos α + e 0 ) - 1 ,
T = Ψ a 2 ( 1 + z a 2 x a - 2 ) - 1 .
T = x 2 / G = x 2 [ ( A 1 2 + A 2 2 ) ( B 1 2 + B 2 2 ) ] - 1 ,
g 1 = a 1 a 4 + a 2 a 3 - b 1 b 4 - b 2 b 3 , h 1 = a 1 b 4 + a 2 b 3 + a 3 b 2 + a 4 b 1 , g 2 = 2 ( a 1 a 2 - b 1 b 2 ) , h 2 = 2 ( a 1 b 2 + a 2 b 1 ) , g 3 = 2 ( a 3 a 4 - b 3 b 4 ) , h 3 = 2 ( a 3 b 4 + a 4 b 3 ) ,
[ g 1 + j h 1 g 3 + j h 3 g 2 + j h 2 g 1 + j h 1 ] .
T = x 2 / H ,
H = C 0 + C 10 x + C 01 z + C 11 x z + C 20 x 2 + C 02 z 2 + C 12 x ( x 2 + z 2 ) + C 21 z ( x 2 + z 2 ) + C 22 ( x 2 + z 2 ) 2 .
T / z = 0 = C 01 + C 11 x + 2 C 02 z + 2 C 12 x z + C 21 x 2 + 3 C 21 z 2 + 4 C 22 z ( x 2 + z 2 ) ,
T / x = 0 = 2 ( C 0 + C 01 z + C 02 z 2 + C 21 z 3 + C 22 z 4 ) + x ( C 10 + C 11 z + C 12 z 2 ) - C 12 x 3 - 2 C 22 x 4 .
T = [ 1 - ( u 2 + w 2 ) ] 2 P - 1 ,
P = Q 0 + Q 10 u + Q 01 w + Q 11 u w + Q 20 u 2 + Q 02 w 2 + Q 12 u w 2 + Q 21 u 2 w + Q 22 u 2 w 2 + Q 30 u 3 + Q 03 w 3 + Q 40 u 4 + Q 04 w 4 .

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