Abstract

Extending the work of Epstein, the theoretical basis and the properties of equivalent layers in multilayer configurations are presented. The results are applied to the design of practical long-, short-, and bandpass filters.

© 1966 Optical Society of America

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References

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  1. A. Herpin, Compt. Rend. 225, 182 (1947).
  2. L. I. Epstein, J. Opt. Soc. Am. 42, 806 (1952).
    [Crossref]
  3. M. Born and E. Wolf, Principles of Optics, 2d ed. (Pergamon Press, New York, 1964), p. 51–70.
  4. L. Young, Appl. Opt. 5, 77 (1966).
    [Crossref] [PubMed]
  5. A. Thelen, J. Opt. Soc. Am. 53, 1267 (1963).
  6. S. D. Smith, J. Opt. Soc. Am. 48, 43 (1958).
    [Crossref]
  7. J. S. Seely, J. Opt. Soc. Am. 54, 342 (1964).
    [Crossref]
  8. S. D. Smith, J. Opt. Soc. Am. 54, 1459 A (1964).
    [Crossref]
  9. M. Auwarter, Process for the Manufacture of Thin Films, (U. S. Patent2 920 002, 5Jan.1960).

1966 (1)

1964 (2)

J. S. Seely, J. Opt. Soc. Am. 54, 342 (1964).
[Crossref]

S. D. Smith, J. Opt. Soc. Am. 54, 1459 A (1964).
[Crossref]

1963 (1)

A. Thelen, J. Opt. Soc. Am. 53, 1267 (1963).

1958 (1)

1952 (1)

1947 (1)

A. Herpin, Compt. Rend. 225, 182 (1947).

Auwarter, M.

M. Auwarter, Process for the Manufacture of Thin Films, (U. S. Patent2 920 002, 5Jan.1960).

Born, M.

M. Born and E. Wolf, Principles of Optics, 2d ed. (Pergamon Press, New York, 1964), p. 51–70.

Epstein, L. I.

Herpin, A.

A. Herpin, Compt. Rend. 225, 182 (1947).

Seely, J. S.

Smith, S. D.

S. D. Smith, J. Opt. Soc. Am. 54, 1459 A (1964).
[Crossref]

S. D. Smith, J. Opt. Soc. Am. 48, 43 (1958).
[Crossref]

Thelen, A.

A. Thelen, J. Opt. Soc. Am. 53, 1267 (1963).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 2d ed. (Pergamon Press, New York, 1964), p. 51–70.

Young, L.

Appl. Opt. (1)

Compt. Rend. (1)

A. Herpin, Compt. Rend. 225, 182 (1947).

J. Opt. Soc. Am. (5)

Other (2)

M. Auwarter, Process for the Manufacture of Thin Films, (U. S. Patent2 920 002, 5Jan.1960).

M. Born and E. Wolf, Principles of Optics, 2d ed. (Pergamon Press, New York, 1964), p. 51–70.

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Figures (10)

Fig. 1
Fig. 1

Equivalent indices for the system L/2HL/2. nL = 1 and nH/nL is parameter nH/nL = 1.25; 1.50; 1.75; 2.0; 2.5; 3.0. The curves with the wider stop band have the higher nH/nL-values.

Fig. 2
Fig. 2

Enlarged first part of Fig. 1.

Fig. 3
Fig. 3

Equivalent thickness of the system described in Fig. 1.

Fig. 4
Fig. 4

Equivalent index for the system LMHHML. nL = 1.0, n M = ( n L · n H ) 1 2, and nH/nL = 1.25; 1.5; 1.75; 2.0; 2.5; 3.0. Again, the curves with the wider stop band have the higher nH/nL-values.

Fig. 5
Fig. 5

Equivalent index for an inhomogeneous system where the index varies with the Jacobian elliptic function dn. As before, n(0) = 1.0 and nmax/nmin = 1.25; 1.50; 1.75; 2.0; 2.5; 3.0.

Fig. 6
Fig. 6

Comparison of the filters 1.52 | [(H/2)L(H/2)]15| 1.0 and 1.52 | 1.05[(h/2)L(H/2)]3 [(H/2)L(H/2)]12| 1.0 with nH = 2.3, nL = 1.56.

Fig. 7
Fig. 7

Transmittance of the narrow-band-pass filter 4.0 |(HLH)2(HLHLHLHLH)2(HLH)| 1.0 with nH = 4.0, nL = 1.8, and nL = 1.6.

Fig. 8
Fig. 8

Transmittance of the narrow-band-pass filter 1.34 |(HLH) (HLHLHLH)2HH (HLHLHLH)2 (HLH)| 1.0 with nH = 4.0, nL = 1.8, and nL = 2.2.

Fig. 10
Fig. 10

Experimental verification (dotted line) of the design given in Fig. 6 (solid line). Coating materials are TiO2 and Si2O3.9

Fig. 9
Fig. 9

Transmittance of the narrow-band-pass filter 1.7 |(HLH) [(6H/5) (3L/5) (6H/5) (3L/5) (6H/5) (3L/5) (6H/5) (3L/5) (6H/5)]3 (HLH)| 1.0 with nH = 4.0, nL = 1.8, and nL = 2.2

Equations (31)

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n [ 1 2 ( z a + z b ) + x ] = n [ 1 2 ( z a + z b ) x ] ; x 1 2 ( z b z a ) .
M ( λ 0 / λ ) = { M 11 ( λ 0 / λ ) j M 12 ( λ 0 / λ ) j M 21 ( λ 0 / λ ) M 22 ( λ 0 / λ ) } ,
M 11 = ( 2 π / λ ) n 2 M 12 M 12 = ( 2 π / λ ) M 11 M 21 = ( 2 π / λ ) n 2 M 22 M 22 = ( 2 π / λ ) M 21
M 11 ( z a ) = 1 , M 12 ( z a ) = 0 , M 21 ( z a ) = 0 , M 22 ( z a ) = 1.
u = π λ 0 · 0 z n ( x ) d x ,
M 11 = ( 2 λ 0 / λ ) n M 12 M 12 = ( 2 λ 0 / λ ) ( 1 / n ) M 11 M 21 = ( 2 λ 0 / λ ) n M 22 M 22 = ( 2 λ 0 / λ ) ( 1 / n ) M 21 ,
N E ( λ 0 / λ ) = + ( M 21 / M 12 ) 1 2
Γ E ( λ 0 / λ ) = arc cos M 11 = arc cos M 22 ,
N E [ n = A · f ( u ) ] = A · N E [ n = f ( u ) ] .
Γ E [ n = A · f ( u ) ] = Γ E [ n = f ( u ) ] .
N E ( Ge ZnS ) = 1.75 N E ( TiO 2 CaF 2 ) , Γ E ( Ge ZnS ) = Γ E ( TiO 2 CaF 2 ) ;
N E [ n = f ( u ) ] = 1 / N E [ n = 1 / f ( u ) ] .
Γ E [ n = f ( u ) ] = Γ E [ n = 1 / f ( u ) ] .
N E ( λ 0 / λ 0 ) = ( u a u b n d u / u a u b d u n ) 1 2 .
N E 1 = ( N E 2 n S ) 1 2 ,
Γ E 1 = ( 2 υ + 1 ) · 90 ° ; υ = 0 , 1 , 2 , .
M = M E υ = { cos υ Γ E j ( sin υ Γ E / N E ) j N E sin υ Γ E cos υ Γ E } .
cos υ Γ E 1 , sin υ Γ E 0 , N E 0 or .
M 11 · M 22 + M 12 · M 21 = 1 cos 2 Γ E + M 12 M 21 = 1 N E = ( M 21 / M 12 ) 1 2 = sin Γ E / M 12 = M 21 / sin Γ E .
M E υ = { 1 j υ M 12 0 1 } or { 1 0 j υ M 21 1 }
T = 4 n 0 n S / [ ( n 0 + n S ) 2 + ( υ n 0 n S M 12 ) 2 ] when M 21 = 0
T = 4 n 0 n S / [ ( n 0 + n S ) 2 + ( υ M 21 ) 2 ] when M 12 = 0.
substrate n S H L H H L H L H L H H L H air m M .
eqivalent layer I : H L H eqivalent layer II : H L H L H L H eqivalent layer III ( = I ) : H L H .
N E 1 = ( n S · N E 2 ) 1 2 , Γ E 1 = ( 2 υ + 1 ) · 90 ° , υ = 0 , 1 ,
N E 3 = ( n M · N E 2 ) 1 2 , Γ E 3 = ( 2 υ + 1 ) · 90 ° , υ = 0 , 1 , .
A B A B B A = ( A B ) 2 υ 1 A ,
Δ λ λ = 1 π arc sin 2 ( n A / n B 1 ) ( n A / n B ) υ if ( n A n B ) υ 0.
Δ λ λ = 1 π arc sin 2 ( 1 n B / n A ) ( n B / n A ) υ if ( n B n A ) 1 υ 0.
n a = ( n A n B ) 1 2 if ( n A / n B ) υ 0.
n a = n A ( 1 n A / n B + n A 2 / n B 2 ) 1 2 if ( n B n A ) 1 υ 0.