Abstract

The simulation of a homogeneous quarterwave layer by a thin-film assembly requires that the complex amplitude reflections of both layer systems have to coincide at the design wavelength. Since the complex amplitude reflection consists of a real and an imaginary part (magnitude and phase), this requirement leads to two conditions for the simulating film assembly. The addition of a third condition, namely, the coincidence of the slopes of the magnitude curves at the design wavelength, yields a good approximation of the complex amplitude reflection of the homogeneous quarterwave layer over a broad wavelength range and is, therefore, called a wideband simulation. The three conditions for the wideband simulation can be fulfilled by three independent parameters. These can be the layer thicknesses of a three-layer system consisting of two materials.

© 1985 Optical Society of America

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References

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  1. R. Herrmann, “Simulating Quarterwave Layers,” in Technical Digest, Topical Meeting on Optical Interference Coatings (Optical Society of America, Washington, D.C., 1984), paper TuA-A2.
  2. W. Geffcken, “Schicht zur Änderung des Reflexionsvermögens aus einer Mehrzahl abwechselnd übereinanderliegender Teilschichten aus zwei Stoffen von verschiedener Brechzahl,” German patentDRP 742,463 (1942).
  3. L. I. Epstein, “Design of Optical Filters,” J. Opt. Soc. Am. 42, 806 (1952).
    [CrossRef]
  4. P. H. Berning, “Use of Equivalent Films in the Design of Infrared Multilayer Antireflection Coatings,” J. Opt. Soc. Am. 52, 431 (1962).
    [CrossRef]
  5. A. Thelen, “Equivalent Layers in Multilayer Filters,” J. Opt. Soc. Am. 56, 1533 (1966).
    [CrossRef]
  6. C. Ufford, P. Baumeister, “Graphical Aids in the Use of Equivalent Index in Multilayer-Filter Design,” J. Opt. Soc. Am. 64, 329 (1974).
    [CrossRef]
  7. M. C. Ohmer, “Design of Three-Layer Equivalent Films,” J. Opt. Soc. Am. 68, 137 (1978).
    [CrossRef]
  8. F. Rock, “Antireflection Coating and Assembly having Synthesized Layer of Index of Refraction,” U.S. Patent3,432,225.
  9. P. H. Berning, “Theory and Calculations of Optical Thin Films,” in Physics of Thin Films, G. Hass, Ed. (Academic, New York, 1963), Vol. 1.
  10. J. H. Apfel, “Graphics in Optical Coating Design,” Appl. Opt. 11, 1303 (1972).
    [CrossRef] [PubMed]
  11. A. J. Vermeulen, “Some Phenomena Connected with the Optical Monitoring of Thin-Film Deposition, and Their Application to Optical Coatings,” Opt. Acta 18, 531 (1971).
    [CrossRef]
  12. K. Rabinovitch, A. Pagis, “Multilayer Antireflection Coatings: Theoretical Model and Design Parameters,” Appl. Opt. 14, 1326 (1975).
    [CrossRef] [PubMed]
  13. K. Rabinovitch, M. Drucker, “Parameters for Optimization of Multilayer Antireflection Coatings,” Appl. Opt. 18, 553 (1979).
    [PubMed]
  14. W. H. Southwell, “Coating Design Using Very Thin High and Low Index Layers,” in Technical Digest, Topical Meeting on Optical Interference Coatings (Optical Society of America, Washington, D.C., 1984), paper TuA-A3.
  15. W. T. Boord, H. Y. B. Mar, M. C. Ohmer, “Low Absorption TIJ/KCl/TIJ Antireflection Coatings for KCl surfaces,” Opt. Eng. 18, 586 (1979).
    [CrossRef]
  16. H. A. Macleod, Thin Film Optical Filters (Hilger, London, 1969).
  17. A. D. Baer, “Design of Three-Layer Antireflectance Coatings,” Natl. Bur. Stand. (U.S.) Spec. Publ. 462, 221 (1976).

1979 (2)

W. T. Boord, H. Y. B. Mar, M. C. Ohmer, “Low Absorption TIJ/KCl/TIJ Antireflection Coatings for KCl surfaces,” Opt. Eng. 18, 586 (1979).
[CrossRef]

K. Rabinovitch, M. Drucker, “Parameters for Optimization of Multilayer Antireflection Coatings,” Appl. Opt. 18, 553 (1979).
[PubMed]

1978 (1)

1976 (1)

A. D. Baer, “Design of Three-Layer Antireflectance Coatings,” Natl. Bur. Stand. (U.S.) Spec. Publ. 462, 221 (1976).

1975 (1)

1974 (1)

1972 (1)

1971 (1)

A. J. Vermeulen, “Some Phenomena Connected with the Optical Monitoring of Thin-Film Deposition, and Their Application to Optical Coatings,” Opt. Acta 18, 531 (1971).
[CrossRef]

1966 (1)

1962 (1)

1952 (1)

Apfel, J. H.

Baer, A. D.

A. D. Baer, “Design of Three-Layer Antireflectance Coatings,” Natl. Bur. Stand. (U.S.) Spec. Publ. 462, 221 (1976).

Baumeister, P.

Berning, P. H.

P. H. Berning, “Use of Equivalent Films in the Design of Infrared Multilayer Antireflection Coatings,” J. Opt. Soc. Am. 52, 431 (1962).
[CrossRef]

P. H. Berning, “Theory and Calculations of Optical Thin Films,” in Physics of Thin Films, G. Hass, Ed. (Academic, New York, 1963), Vol. 1.

Boord, W. T.

W. T. Boord, H. Y. B. Mar, M. C. Ohmer, “Low Absorption TIJ/KCl/TIJ Antireflection Coatings for KCl surfaces,” Opt. Eng. 18, 586 (1979).
[CrossRef]

Drucker, M.

Epstein, L. I.

Geffcken, W.

W. Geffcken, “Schicht zur Änderung des Reflexionsvermögens aus einer Mehrzahl abwechselnd übereinanderliegender Teilschichten aus zwei Stoffen von verschiedener Brechzahl,” German patentDRP 742,463 (1942).

Herrmann, R.

R. Herrmann, “Simulating Quarterwave Layers,” in Technical Digest, Topical Meeting on Optical Interference Coatings (Optical Society of America, Washington, D.C., 1984), paper TuA-A2.

Macleod, H. A.

H. A. Macleod, Thin Film Optical Filters (Hilger, London, 1969).

Mar, H. Y. B.

W. T. Boord, H. Y. B. Mar, M. C. Ohmer, “Low Absorption TIJ/KCl/TIJ Antireflection Coatings for KCl surfaces,” Opt. Eng. 18, 586 (1979).
[CrossRef]

Ohmer, M. C.

W. T. Boord, H. Y. B. Mar, M. C. Ohmer, “Low Absorption TIJ/KCl/TIJ Antireflection Coatings for KCl surfaces,” Opt. Eng. 18, 586 (1979).
[CrossRef]

M. C. Ohmer, “Design of Three-Layer Equivalent Films,” J. Opt. Soc. Am. 68, 137 (1978).
[CrossRef]

Pagis, A.

Rabinovitch, K.

Rock, F.

F. Rock, “Antireflection Coating and Assembly having Synthesized Layer of Index of Refraction,” U.S. Patent3,432,225.

Southwell, W. H.

W. H. Southwell, “Coating Design Using Very Thin High and Low Index Layers,” in Technical Digest, Topical Meeting on Optical Interference Coatings (Optical Society of America, Washington, D.C., 1984), paper TuA-A3.

Thelen, A.

Ufford, C.

Vermeulen, A. J.

A. J. Vermeulen, “Some Phenomena Connected with the Optical Monitoring of Thin-Film Deposition, and Their Application to Optical Coatings,” Opt. Acta 18, 531 (1971).
[CrossRef]

Appl. Opt. (3)

J. Opt. Soc. Am. (5)

Natl. Bur. Stand. (U.S.) Spec. Publ. (1)

A. D. Baer, “Design of Three-Layer Antireflectance Coatings,” Natl. Bur. Stand. (U.S.) Spec. Publ. 462, 221 (1976).

Opt. Acta (1)

A. J. Vermeulen, “Some Phenomena Connected with the Optical Monitoring of Thin-Film Deposition, and Their Application to Optical Coatings,” Opt. Acta 18, 531 (1971).
[CrossRef]

Opt. Eng. (1)

W. T. Boord, H. Y. B. Mar, M. C. Ohmer, “Low Absorption TIJ/KCl/TIJ Antireflection Coatings for KCl surfaces,” Opt. Eng. 18, 586 (1979).
[CrossRef]

Other (6)

H. A. Macleod, Thin Film Optical Filters (Hilger, London, 1969).

R. Herrmann, “Simulating Quarterwave Layers,” in Technical Digest, Topical Meeting on Optical Interference Coatings (Optical Society of America, Washington, D.C., 1984), paper TuA-A2.

W. Geffcken, “Schicht zur Änderung des Reflexionsvermögens aus einer Mehrzahl abwechselnd übereinanderliegender Teilschichten aus zwei Stoffen von verschiedener Brechzahl,” German patentDRP 742,463 (1942).

F. Rock, “Antireflection Coating and Assembly having Synthesized Layer of Index of Refraction,” U.S. Patent3,432,225.

P. H. Berning, “Theory and Calculations of Optical Thin Films,” in Physics of Thin Films, G. Hass, Ed. (Academic, New York, 1963), Vol. 1.

W. H. Southwell, “Coating Design Using Very Thin High and Low Index Layers,” in Technical Digest, Topical Meeting on Optical Interference Coatings (Optical Society of America, Washington, D.C., 1984), paper TuA-A3.

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Figures (8)

Fig. 1
Fig. 1

Schematic diagram of a three-layer AR coating. The quarterwave layer adjacent to the substrate has to be simulated.

Fig. 2
Fig. 2

Amplitude reflection curves of a homogeneous quarterwave layer, —; amplitude reflection curve of the simulating layer system according to Rock, - - -. The homogeneous layer has a refractive index of 1.64, and the substrate has a refractive index of 1.52. The simulating system consists of two layers of MgF2 (n = 1.38) and ZrO2 (n = 2.08). The ZrO2 layer adjacent to the substrate has a reduced optical thickness (nd0) of 0.0507 and the MgF2 layer a thickness of 0.0681.

Fig. 3
Fig. 3

Amplitude reflection curves of the homogeneous quarterwave layer of Fig. 2, —; amplitude reflection curves of the corresponding equivalent layer system, - - -. The equivalent layer system consists of three layers ZrO2/MgF2/ZrO2 with the reduced optical thicknesses 0.0738/0.0963/0.0738.

Fig. 4
Fig. 4

Schematic illustration of the simulation problem with three independent layer thicknesses.

Fig. 5
Fig. 5

Amplitude reflection curves of the homogeneous quarterwave layer of Figs. 2 and 3, —; amplitude reflection curves of the best approximated three-layer system according to the wideband simulation, - - -. The materials of this layer system are ZrO2/MgF2/ZrO2, and the reduced optical thicknesses are 0.0313/0.0865/0.0557/substrate.

Fig. 6
Fig. 6

Curve A, spectral reflection curve of the homogeneous AR coating of Fig. 1. Curve B, spectral reflection curve of an AR coating with the quarterwave layer adjacent to the substrate being simulated according to Rock by two layers. The homogeneous system consists of MgF2, ZrO2, and a fictitious material with refractive index of 1.64. The design of curve B is listed in Table I.

Fig. 7
Fig. 7

Curve A, AR coating with homogeneous layers. Curve B, AR coating with the quarterwave layer adjacent to the substrate being replaced by an equivalent system. The design of curve C is listed in Table I.

Fig. 8
Fig. 8

Curve A, AR coating with homogeneous layers. Curve B, AR coating with the quarterwave layer adjacent to the substrate being simulated according to the wideband simulation described in Sec. III. The design of curve D is listed in Table I.

Tables (1)

Tables Icon

Table I Coating Designs of Figs. 68

Equations (19)

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r = r 0 exp ( i ϕ ) ,
δ = 2 π n d / λ .
Y Q = Y ,
Y = C / B ,
| B C | = | M 11 i M 12 i M 21 M 22 | | 1 n s | .
Y Q = N 2 / n s .
( N 2 / n s ) · M 11 + i N 2 M 12 = n s M 22 + i M 21 .
M 22 M 11 = N 2 n s 2 ,
M 21 M 12 = N 2 .
M 11 = c ( 1 ρ q r p r 1 ρ p q ) , M 12 = 1 n r · c ( r + ρ q p 1 ρ p q r ) , M 21 = n r · c ( r + 1 ρ q + p ρ p q r ) , M 22 = c ( 1 1 ρ q r p r ρ p q ) ,
p = tan δ p , q = tan δ q , r = tan δ r ,
p r + A 1 q r + A 2 q p = 1 , p + r + B 1 q + B 2 pqr = 0.
A 1 = ( n s n q ) 2 ( N n r ) 2 n r n q · ( n s 2 N 2 ) A 2 = ( n s n r ) 2 ( N n q ) 2 n r n q · ( n s 2 N 2 ) , B 1 = ( n r n q ) 2 ( N n r ) 2 n r n q · ( n r 2 N 2 ) B 2 = n r 4 ( N n q ) 2 n r n q · ( n r 2 N 2 ) .
p = ± A 1 r 2 + B 1 B 2 r 2 A 2 ;
q = p + r B 1 + B 2 p r .
A 1 q r = 1 r + B 1 q = 0 .
q = ± n q ( n r 2 N 2 ) ( N 2 n s 2 ) ( n q 2 N 2 ) ( n s 2 n q 2 N 2 n r 2 ) , r = ± n r ( n q 2 N 2 ) ( N 2 n s 2 ) ( n r 2 N 2 ) ( n s 2 n q 2 N 2 n r 2 ) .
N = n o n s .
r = ± n r ( N 2 n q 2 ) n r 4 ( N n q ) 2 , q = ± n q n r 2 N 2 ( N 2 n q 2 ) [ n r 4 ( N n q ) 2 ] .

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