Abstract

The equivalent properties of a symmetric three-layer structure are derived by a nontraditional method that provides useful insights and a simplified application to some thin-film design problems. The equations derived from this method may be used to design a three-layer replacement for a single layer in an interference coating. The equations are especially helpful for cases that involve complex numbers, such as metal layers or above-critical angle propagation in dielectric layers. Several multilayer design problems are solved to demonstrate the application of this approach.

© 2002 Optical Society of America

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References

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  1. L. I. Epstein, “The design of optical filters,” JOSA 42, 806–810 (1952).
    [CrossRef]
  2. A. Herpin, “Calcu du pouvoir refecteur d’un systeme stratifie quelconque,” Comptes Rendus Academies des Science 225, 182–183 (1947).
  3. K. C. Park, “The extreme values of reflectivity and the conditions for zero reflection from thin film dielectirc films on metal,” Appl. Opt. 3, 877–881 (1964).
    [CrossRef]
  4. M. C. Ohmer, “Design of three-layer equivalent films,” JOSA 68, 137–139 (1978).
    [CrossRef]
  5. K. D. Mielenz, “Simple calculus for all-dielectric interference filters of the Fabry-Perot type,” JOSA 50, 1014–1016 (1960).
    [CrossRef]
  6. C. Dufour, A. Herpin, “Application des methodes matricielles au calcu d’ensembles complexes de couches minces alternees,” Opt. Acta 1, 1–8 (1954).
    [CrossRef]
  7. A. Thelen, “Multilayer filters with wide transmittance bands,” JOSA 53, 1266–1270 (1963).
    [CrossRef]
  8. H. G. Lotz, “Computer-aided multilayer design of optical filters with wide transmittance bands using SiO2 and TiO2,” Appl. Opt. 26, 4487–4490 (1987).
    [CrossRef] [PubMed]
  9. C. Monga, “Anti-reflection coatings for grazing incidence angles,” J. Mod. Opt. 36, 381–387 (1989).
    [CrossRef]
  10. P. H. Berning, A. F. Turner, “Induced transmission in absorbing films applied to band pass filter design,” JOSA 47, 230–239 (1957).
    [CrossRef]
  11. B. V. Landau, P. H. Lissberger, “Theory of induced transmittance filters in terms of the concept of equivalent layers,” JOSA 62, 1258–1264 (1972).
    [CrossRef]
  12. H. A. Macleod, “A new approach to the design of metal-dielectric thin-film optical coatings,” Opt. Acta 25, 93–106 (1978).
    [CrossRef]
  13. M. Banning, “Practical methods of making and using multilayer filters,” JOSA 37, 792–797 (1947).
    [CrossRef]
  14. L. Li, J. A. Dobrowolski, “High-performance thin-film polarizing beam splitter operating at angles greater than the critical angle,” Appl. Opt. 39, 2754–2771 (2000).
    [CrossRef]

2000 (1)

1989 (1)

C. Monga, “Anti-reflection coatings for grazing incidence angles,” J. Mod. Opt. 36, 381–387 (1989).
[CrossRef]

1987 (1)

1978 (2)

M. C. Ohmer, “Design of three-layer equivalent films,” JOSA 68, 137–139 (1978).
[CrossRef]

H. A. Macleod, “A new approach to the design of metal-dielectric thin-film optical coatings,” Opt. Acta 25, 93–106 (1978).
[CrossRef]

1972 (1)

B. V. Landau, P. H. Lissberger, “Theory of induced transmittance filters in terms of the concept of equivalent layers,” JOSA 62, 1258–1264 (1972).
[CrossRef]

1964 (1)

1963 (1)

A. Thelen, “Multilayer filters with wide transmittance bands,” JOSA 53, 1266–1270 (1963).
[CrossRef]

1960 (1)

K. D. Mielenz, “Simple calculus for all-dielectric interference filters of the Fabry-Perot type,” JOSA 50, 1014–1016 (1960).
[CrossRef]

1957 (1)

P. H. Berning, A. F. Turner, “Induced transmission in absorbing films applied to band pass filter design,” JOSA 47, 230–239 (1957).
[CrossRef]

1954 (1)

C. Dufour, A. Herpin, “Application des methodes matricielles au calcu d’ensembles complexes de couches minces alternees,” Opt. Acta 1, 1–8 (1954).
[CrossRef]

1952 (1)

L. I. Epstein, “The design of optical filters,” JOSA 42, 806–810 (1952).
[CrossRef]

1947 (2)

A. Herpin, “Calcu du pouvoir refecteur d’un systeme stratifie quelconque,” Comptes Rendus Academies des Science 225, 182–183 (1947).

M. Banning, “Practical methods of making and using multilayer filters,” JOSA 37, 792–797 (1947).
[CrossRef]

Banning, M.

M. Banning, “Practical methods of making and using multilayer filters,” JOSA 37, 792–797 (1947).
[CrossRef]

Berning, P. H.

P. H. Berning, A. F. Turner, “Induced transmission in absorbing films applied to band pass filter design,” JOSA 47, 230–239 (1957).
[CrossRef]

Dobrowolski, J. A.

Dufour, C.

C. Dufour, A. Herpin, “Application des methodes matricielles au calcu d’ensembles complexes de couches minces alternees,” Opt. Acta 1, 1–8 (1954).
[CrossRef]

Epstein, L. I.

L. I. Epstein, “The design of optical filters,” JOSA 42, 806–810 (1952).
[CrossRef]

Herpin, A.

C. Dufour, A. Herpin, “Application des methodes matricielles au calcu d’ensembles complexes de couches minces alternees,” Opt. Acta 1, 1–8 (1954).
[CrossRef]

A. Herpin, “Calcu du pouvoir refecteur d’un systeme stratifie quelconque,” Comptes Rendus Academies des Science 225, 182–183 (1947).

Landau, B. V.

B. V. Landau, P. H. Lissberger, “Theory of induced transmittance filters in terms of the concept of equivalent layers,” JOSA 62, 1258–1264 (1972).
[CrossRef]

Li, L.

Lissberger, P. H.

B. V. Landau, P. H. Lissberger, “Theory of induced transmittance filters in terms of the concept of equivalent layers,” JOSA 62, 1258–1264 (1972).
[CrossRef]

Lotz, H. G.

Macleod, H. A.

H. A. Macleod, “A new approach to the design of metal-dielectric thin-film optical coatings,” Opt. Acta 25, 93–106 (1978).
[CrossRef]

Mielenz, K. D.

K. D. Mielenz, “Simple calculus for all-dielectric interference filters of the Fabry-Perot type,” JOSA 50, 1014–1016 (1960).
[CrossRef]

Monga, C.

C. Monga, “Anti-reflection coatings for grazing incidence angles,” J. Mod. Opt. 36, 381–387 (1989).
[CrossRef]

Ohmer, M. C.

M. C. Ohmer, “Design of three-layer equivalent films,” JOSA 68, 137–139 (1978).
[CrossRef]

Park, K. C.

Thelen, A.

A. Thelen, “Multilayer filters with wide transmittance bands,” JOSA 53, 1266–1270 (1963).
[CrossRef]

Turner, A. F.

P. H. Berning, A. F. Turner, “Induced transmission in absorbing films applied to band pass filter design,” JOSA 47, 230–239 (1957).
[CrossRef]

Appl. Opt. (3)

Comptes Rendus Academies des Science (1)

A. Herpin, “Calcu du pouvoir refecteur d’un systeme stratifie quelconque,” Comptes Rendus Academies des Science 225, 182–183 (1947).

J. Mod. Opt. (1)

C. Monga, “Anti-reflection coatings for grazing incidence angles,” J. Mod. Opt. 36, 381–387 (1989).
[CrossRef]

JOSA (7)

P. H. Berning, A. F. Turner, “Induced transmission in absorbing films applied to band pass filter design,” JOSA 47, 230–239 (1957).
[CrossRef]

B. V. Landau, P. H. Lissberger, “Theory of induced transmittance filters in terms of the concept of equivalent layers,” JOSA 62, 1258–1264 (1972).
[CrossRef]

M. C. Ohmer, “Design of three-layer equivalent films,” JOSA 68, 137–139 (1978).
[CrossRef]

K. D. Mielenz, “Simple calculus for all-dielectric interference filters of the Fabry-Perot type,” JOSA 50, 1014–1016 (1960).
[CrossRef]

M. Banning, “Practical methods of making and using multilayer filters,” JOSA 37, 792–797 (1947).
[CrossRef]

L. I. Epstein, “The design of optical filters,” JOSA 42, 806–810 (1952).
[CrossRef]

A. Thelen, “Multilayer filters with wide transmittance bands,” JOSA 53, 1266–1270 (1963).
[CrossRef]

Opt. Acta (2)

C. Dufour, A. Herpin, “Application des methodes matricielles au calcu d’ensembles complexes de couches minces alternees,” Opt. Acta 1, 1–8 (1954).
[CrossRef]

H. A. Macleod, “A new approach to the design of metal-dielectric thin-film optical coatings,” Opt. Acta 25, 93–106 (1978).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Transmittance of two multilayer systems with second-, third-, and fourth-order reflection bands suppressed. Each configuration has ten periods centered at λ o = 500 nm. The solid curve is the ideal, five-layer, three-material configuration n S (L M 2H M L)10 n 0, and the dashed curve is the seven-layer, two-material equivalent stack n S (1.177L 0.317H 0.318L 2.317H 0.318L 0.317H 1.177L)10.

Fig. 2
Fig. 2

Continuum of relative layer thickness solutions of effective equivalent index N = 3.23. The graph is generated by means of p-plane effective refractive indices for the low- and the high-index layers with θ o = 79° light incidence. Only odd multiples of Γ = 90° equivalent phase thickness are suitable for simulating the single-layer antireflection coating.

Fig. 3
Fig. 3

Reflectance (p plane) for the trilayer solutions with effective equivalent index N = 3.23 satisfying the quarter wave (Solution 1) 0.8774H 1.2526L 0.8774H and the three quarter wave (Solution 2) 1.226H 0.7474L 1.226H phase thickness conditions. The refractive indices are given in the text.

Fig. 4
Fig. 4

Transmittance versus wavelength for the induced transmission filter n S HLHLHL [76.7L 75.12M 76.7L] LHLHLH n S . The notation and refractive indices are given in the text.

Fig. 5
Fig. 5

Transmittance (p plane) and reflectance (s plane) in decibels versus wavelength for a polarizer designed with an equivalent layer. The design is n 0 (11.81L 9.57H 11.81L)25 n 0, with n 0 = 1.8, n L = 1.45, n H = 2.45, and θ0 = 75°.

Equations (34)

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Mm11m12m12m22=j=13cos βji sin βj/njinj sin βjcos βj,
βj=2πnjdj/λ.
M=cos Γi sin Γ/NiN sin Γcos Γ,
M=xQ-P-1,
m11=m22=x cos βq-cos βp=cos Γ,
m21=xnq sin βq+np sin βp=N sin Γ,
m12=x sin βq/nq+sin βp/np=sin Γ/N.
sin βp=ρN2-nq2sin ΓnqN1-ρ2,
ρnq/np.
tan βq=ρnp2-N2sin βpN2-nq2cos Γ+cos βp.
M=xP-Q-1,
sin βq=ρnp2-N2sin ΓnpN1-ρ2,
tan βp=N2-nq2sin βqρnp2-N2cos Γ+cos βq.
nsL M 2H M LPn0,
nM=nLnH1/2=1.885.
M  0.3212H 0.3212L 0.3212H.
nSL 0.3212H 0.3212L 2.6424H 0.3212L 0.3212H LP n0.
nS1.177L 0.317H 0.318L 2.317H 0.318L 0.317H 1.177LP n0.
np=n2/n2-y21/2,
Np=nSpnop1/2=3.23.
βL=2πdL/λnL2-y21/2.
m11=m22=x cos βL-cosh αM=cos Γ,
m21=xnL sin βL-k sinh αM=N sin Γ,
m12=x sin βL/nL+sinh αM/k=sin Γ/N,
dM=λ2πksinh-1k1-N/nL2sin ΓN1+k/nL2,
dL=λ2πnLtan-1nLN2+k2sin ΓNnL2+k2cos Γ+cosh αM,
N=nLnH2PnS, for P=1, 2, 3,.
nSHLHLHL76.7L 75.12M 76.7L LHLHLHnS,
nH>n0>nL, with θo>sin-1nL/no.
m11=x cosh αL-cos βH=cos Γ, m21=nH2-y21/2 sin βH-xy2-nL21/2 sinh αL=N sin Γ, m12=sin βH/nH2-y21/2+x sinh αL/y2-nL21/2=sin Γ/N,
y=n0 sin θ0, βH=2πdH/λnH2-y21/2, αL=2πdL/λy2-nL21/2.
dH=λ2πnH2-y21/2×sin-1nH2-y21/2N2-nL2+y2sin ΓNnH2-nL2,
dL=λ2πy2-nL21/2tanh-1×nL2-y2N2-nH2+y2sin ΓNy2-nL21/2nH2-nL2cos βH+cos Γ.
n011.81L 9.57H 11.81L25 n0,

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