Abstract

Any nonabsorbing multilayer system may be completely described by two effective interfaces. An analysis based on this representation is applied to deduce the properties of systems consisting of one or two half-wave layers surrounded by reflecting stacks. The salient features of the transmission characteristics are easily obtained. The conditions to be fulfilled in the design of band-pass filters of various band widths and of low-(frequency) pass filters are deduced. It is shown that double half-wave systems have theoretical advantages in band shape and peak transmission over single half-wave systems. This is confirmed experimentally.

© 1958 Optical Society of America

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References

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  1. F. Abelès, Ann. phys. 3, 504 (1948).
  2. H. D. Polster, J. Opt. Soc. Am. 42, 21 (1952).
    [Crossref]
  3. A. F. Turner and et al., “Infrared transmission filters,” Bausch & Lomb Optical Company Technical Reports, New York, 1952, Nos. 1–6.
  4. L. I. Epstein, J. Opt. Soc. Am. 42, 806 (1952).
    [Crossref]

1952 (2)

1948 (1)

F. Abelès, Ann. phys. 3, 504 (1948).

Abelès, F.

F. Abelès, Ann. phys. 3, 504 (1948).

Epstein, L. I.

Polster, H. D.

Turner, A. F.

A. F. Turner and et al., “Infrared transmission filters,” Bausch & Lomb Optical Company Technical Reports, New York, 1952, Nos. 1–6.

Ann. phys. (1)

F. Abelès, Ann. phys. 3, 504 (1948).

J. Opt. Soc. Am. (2)

Other (1)

A. F. Turner and et al., “Infrared transmission filters,” Bausch & Lomb Optical Company Technical Reports, New York, 1952, Nos. 1–6.

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Figures (14)

F. 1
F. 1

Representation by two effective interfaces.

F. 2
F. 2

Phase shifts on reflection from LH and H.

F. 3
F. 3

Computed transmittance of HLHHLH.

F. 4
F. 4

Computed transmittance of HHLHH and explanatory reflectance curves.

F. 5
F. 5

Analysis of double half-wave system HHLHH.

F. 6
F. 6

Computed phase curves for HHLHH.

F. 7
F. 7

T0(ω) and F for HHLHH (calculated).

F. 8
F. 8

Computed transmittance of HLLHLHLLH.

F. 9
F. 9

Computed transmittance of MLHHLHLHHLM.

F. 10
F. 10

Computed transmittance of HLHHLHLHLHHLH.

F. 11
F. 11

Computed transmittance of HLHHLHLHLHHLH.

F. 12
F. 12

Computed transmittance of HLHHLHHLH and reflectance of components.

F. 13
F. 13

Measured transmittance of HLLHLHLLH (full line) and HLHHLH (dotted).

F. 14
F. 14

Measured transmittance of three low pass filters HLHHLHHLH made with “edges” at 1.5, 1.9, and 3.3 μ.

Tables (1)

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Table I

Equations (17)

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T = T 1 ( ω ) T 2 ( ω ) ( 1 R ( ω ) ) 2 × 1 1 + 4 R ( ω ) ( 1 R ( ω ) 2 sin 2 ( ϕ 2 ( ω ) + ϕ 1 ( ω ) β ) 2 ω = λ 0 / λ
β = 2 π n d / λ R = ( R 1 R 2 ) 1 3 .
T ( ω ) = T 0 ( ω ) 1 1 + F ( ω ) sin 2 θ ( ω ) ,
T 0 ( ω ) = T 1 T 2 ( 1 R ) 2 , F ( ω ) = 4 R ( 1 R ) 2 ,
θ ( ω ) = ( ϕ 1 + ϕ 2 β ) 2
T 0 ω T 0 1 + F sin 2 θ ( F ω sin 2 θ + F sin 2 θ θ ω ) = 0 .
T 0 ω = 0
F ω sin 2 θ + F sin 2 θ θ ω = 0
( λ / 4 stack ) , ( N λ / 2 spacer ) , ( λ / 4 stack ) ,
T = T 0 1 1 + F sin 2 θ ( ω ) ,
HLHL H H LH HLH L L H HLHH HL .
T max = 1 ( 1 + A / T ) 2 ,
( λ / 4 stack ) , ( λ / 2 spacer ) , ( λ / 4 stack ) , ( λ / 2 spacer ) , ( λ / 4 stack )
T ( ω ) = T 1 T 2 ( ω ) ( 1 R ( ω ) ) 2 1 1 + F ( ω ) sin 2 ( ϕ 2 ( ω ) 2 β ( ω ) )
T 0 = T 1 T 2 ( ω ) ( 1 R ( ω ) ) 2 and ( ϕ 2 ( ω ) 2 β ( ω ) ) = θ .
T 0 ( ω ) = ( 1 R 1 ) ( 1 R 2 ) ( 1 R ) 2 = 1 ,
HLHHLHHLH HLLLH HLH .