Abstract

An expression for the frequency dependence of the phase of reflections from quarter-wave multilayers is deduced. The result is included in a general formula for the resolving power of single half-wave filters and extended to the synthesis of multi-half-wave, or “coupled,” filters.

© 1964 Optical Society of America

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References

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  1. P. W. Baumeister and F. A. Jenkins, J. Opt. Soc. Am. 47, 57 (1957).
    [Crossref]
  2. P. Giacomo, Rev. Opt. 35, 317 (1956).
  3. J. M. Stone, J. Opt. Soc. Am. 43, 927 (1953).
    [Crossref]
  4. S. D. Smith, J. Opt. Soc. Am. 48, 43 (1958).
    [Crossref]
  5. O. S. Heavens, Rept. Progr. Phys. 23, 12 (1960).
    [Crossref]
  6. J. S. Seeley, Proc. Phys. Soc. 78, 998 (1961).
    [Crossref]
  7. J. S. Seeley and J. C. Williams, Proc. IEE 109B, Suppl. 23, 827 (1962).
  8. C. H. Dufour, Rev. Opt. 31, 1 (1952).
  9. P. H. Lissberger and J. Ring, Opt. Acta 2, 45 (1955).
    [Crossref]
  10. L. Young, Trans. IRE MTT-10, 339 (1962).
    [Crossref]
  11. H. J. Riblet, Trans. IRE MTT-6, 359 (1958).
    [Crossref]
  12. J. C. Williams (unpublished).

1962 (2)

J. S. Seeley and J. C. Williams, Proc. IEE 109B, Suppl. 23, 827 (1962).

L. Young, Trans. IRE MTT-10, 339 (1962).
[Crossref]

1961 (1)

J. S. Seeley, Proc. Phys. Soc. 78, 998 (1961).
[Crossref]

1960 (1)

O. S. Heavens, Rept. Progr. Phys. 23, 12 (1960).
[Crossref]

1958 (2)

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

H. J. Riblet, Trans. IRE MTT-6, 359 (1958).
[Crossref]

1957 (1)

1956 (1)

P. Giacomo, Rev. Opt. 35, 317 (1956).

1955 (1)

P. H. Lissberger and J. Ring, Opt. Acta 2, 45 (1955).
[Crossref]

1953 (1)

1952 (1)

C. H. Dufour, Rev. Opt. 31, 1 (1952).

Baumeister, P. W.

Dufour, C. H.

C. H. Dufour, Rev. Opt. 31, 1 (1952).

Giacomo, P.

P. Giacomo, Rev. Opt. 35, 317 (1956).

Heavens, O. S.

O. S. Heavens, Rept. Progr. Phys. 23, 12 (1960).
[Crossref]

Jenkins, F. A.

Lissberger, P. H.

P. H. Lissberger and J. Ring, Opt. Acta 2, 45 (1955).
[Crossref]

Riblet, H. J.

H. J. Riblet, Trans. IRE MTT-6, 359 (1958).
[Crossref]

Ring, J.

P. H. Lissberger and J. Ring, Opt. Acta 2, 45 (1955).
[Crossref]

Seeley, J. S.

J. S. Seeley and J. C. Williams, Proc. IEE 109B, Suppl. 23, 827 (1962).

J. S. Seeley, Proc. Phys. Soc. 78, 998 (1961).
[Crossref]

Smith, S. D.

Stone, J. M.

Williams, J. C.

J. S. Seeley and J. C. Williams, Proc. IEE 109B, Suppl. 23, 827 (1962).

J. C. Williams (unpublished).

Young, L.

L. Young, Trans. IRE MTT-10, 339 (1962).
[Crossref]

J. Opt. Soc. Am. (3)

Opt. Acta (1)

P. H. Lissberger and J. Ring, Opt. Acta 2, 45 (1955).
[Crossref]

Proc. IEE (1)

J. S. Seeley and J. C. Williams, Proc. IEE 109B, Suppl. 23, 827 (1962).

Proc. Phys. Soc. (1)

J. S. Seeley, Proc. Phys. Soc. 78, 998 (1961).
[Crossref]

Rept. Progr. Phys. (1)

O. S. Heavens, Rept. Progr. Phys. 23, 12 (1960).
[Crossref]

Rev. Opt. (2)

P. Giacomo, Rev. Opt. 35, 317 (1956).

C. H. Dufour, Rev. Opt. 31, 1 (1952).

Trans. IRE (2)

L. Young, Trans. IRE MTT-10, 339 (1962).
[Crossref]

H. J. Riblet, Trans. IRE MTT-6, 359 (1958).
[Crossref]

Other (1)

J. C. Williams (unpublished).

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Figures (7)

Fig. 1
Fig. 1

The amplitude and phase of the reflection coefficient, ρe, for a quarter-wave multilayer as a function of frequency, ν. ν0 is the frequency at which the phase thickness of the layers is π/2. Frequencies ν1 and ν2 mark the stop-band and are defined by ν2ν0=ν1ν0=(2ν0/π) ν 2 - ν 0 = ν 1 - ν 0 = ( 2 ν 0 / π ) sin - 1 { 4 / [ 2 + n + ( 1 / n ) ] } 1 2, n being the ratio of the refractive indices.

Fig. 2
Fig. 2

Boundary conditions for the electric vector.

Fig. 3
Fig. 3

Multilayers: (a) symmetrical: x layers of n1, x−1 layers of n2; (b) x layers of n1 and n2, incidence to high index layer; (c) as (b) but incidence to low index layer. It is assumed that n=(n1/n2)>1.5, n0n2, nLn1.

Fig. 4
Fig. 4

The dispersive phase change as function of n and x. Thick line (a), limiting value. Shaded areas indicate range of −(n2/n0) (/) for the boundary conditions specified in Eqs. (6) and (7) when nx is less than the limiting value. Circles indicate constructed values8 for n=1.67, nL=1.52.

Fig. 5
Fig. 5

Single half-wave filters. (a) Symmetrical multilayers, half-wave layer of low index. (b) Half-wave layer of high index.

Fig. 6
Fig. 6

Resolving power of cryolite/zinc sulfide filters [Fig. 5(a)] as function of x, using Eq. (9). 0—experimental date of Lissberger and Ring.9 X—experimental data of Stone.3

Fig. 7
Fig. 7

Synthesis of multi-half-wave filters containing m half-wave layers. (a) Ripple function, T=1/{1+h2[Tm( ν ¯)]2}, (b) flat function, T=1/[1+( ν ¯)2m], where ν ¯=(νν0)/δν and Tm( ν ¯) is the Tchebyshev function:

Tables (1)

Tables Icon

Table I Limiting dispersive phase change.

Equations (26)

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[ E A A H A A ] = [ M ] [ E B B H B B ]
[ m 11 j m 12 j m 21 m 22 ]
ϱ = E r E i = m 11 + j n L m 12 - j ( m 21 / n 0 ) - ( n L / n 0 ) m 22 m 11 + j n L m 12 + j ( m 21 / n 0 ) + ( n L / n 0 ) m 22 ,
τ = E e E i = 2 m 11 + j n L m 12 + j ( m 21 / n 0 ) + ( n L / n 0 ) m 22 .
[ cos ψ ( j / n ) sin ψ j n sin ψ cos ψ ]
[ 1 2 sin θ j / n j n 1 2 sin θ ]
θ = π ν / ν 0 .
[ m 11 j m 12 j m 21 m 22 ] = ( 1 2 sin θ 1 - x x - 1 n x - 1 + n x - 2 + j n 2 n x j n x n 2 1 2 sin θ 1 - x x - 1 n x - 1 + n x - 2 + )             for Fig . 3 ( a ) ( 1 n x j sin θ 2 n 2 - x 1 - x n x - 1 + n x - 2 + j n 1 sin θ 2 x 1 - x n x - 1 + n x - 2 + n x )             for Fig . 3 ( b ) ( n x j sin θ 2 n 2 - x 1 - x n x - 1 + n x - 2 + j n 1 sin θ 2 x 1 - x n x - 1 + n x - 2 + 1 n x )             for Fig . 3 ( c )
tan ϕ = m 11 [ 1 + ( n L / n 0 ) ] n L m 12 + ( m 21 / n 0 ) - m 11 [ 1 - ( n L / n 0 ) ] n L m 12 - ( m 21 / n 0 ) .
tan ϕ lim = ( n 0 n 2 1 - 2 x - 1 n - 1 + n - 2 + ) sin θ .
1 - 2 x - 1 n - 1 + n - 2 + ( 1 - 1 n ) - 1 - 1 = 1 n - 1             for             n x > 15 ,
tan ϕ lim = [ n 0 / n 2 ( n - 1 ) ] sin θ ,
tan ϕ max 2 m 11 n 0 m 12 + ( m 21 / n 0 ) = n 0 1 - x x - 1 n x - 1 + n x - 2 + n 2 [ n x + ( n 0 / n 2 ) 2 ( 1 / n x ) ] sin θ ,
tan ϕ min = 1 - x x - 1 n x - 1 + n x - 2 + n x + ( 1 / n x ) sin θ .
tan ϕ lim 2 n 0 m 12 / m 22 = [ n 0 / n 2 ( n - 1 ) ] sin θ ,
tan ϕ lim 2 m 21 / n 0 m 11 = [ n 1 / n 0 ( n - 1 ) ] sin θ .
d ϕ / d θ = - K ,
[ - 1 j / n sin θ j n sin θ - 1 ]
sin θ = ( 1 + n L n 0 ) / [ n 2 n 2 x n 0 + n L n 2 n 2 x + ( n L n 2 n x + n 2 n x n 0 ) 1 - x x - 1 n x - 1 + n x - 2 + ]
sin θ = ( 1 + n L n 0 ) / [ n 2 n 2 x + 1 n 0 + n L n 2 n 2 x + 1 + ( n L n 2 n x + 1 + n 2 n x + 1 n 0 ) - x x - 1 n x - 1 + n x - 2 + ]
sin θ = ( n 0 / n 2 ) [ 1 + ( n L / n 0 ) ] n 2 x [ 1 - ( n 2 / n 0 ) ( d ϕ / d θ ) ]
sin θ = ( n 0 / n 2 ) [ 1 + ( n L / n 0 ) ] n 2 x + 1 [ 1 - ( n 2 / n 0 ) ( d ϕ / d θ ) ]
r 1 2 = 2 n γ t · sin π 2 m r 2 2 = 4 ( n ) 2 sin 3 π 2 m · sin π 2 m t 2 ( γ 2 + sin 2 π m ) r i 2 = 4 ( n ) 2 sin ( 2 i - 1 ) π 2 m · sin ( 2 i - 3 ) π 2 m t 2 ( γ 2 + sin 2 ( i - 1 ) π m ) }             where γ = sinh [ ( 1 / m ) sinh ( - 11 / h ) ] for the ripple function , Fig . 7 ( a ) . r 1 2 = 2 n h 1 / m t · sin π 2 m r 2 2 = 4 ( n ) 2 h 1 / 2 m t 2 · sin 3 π 2 m · sin π 2 m r i 2 = 4 ( n ) 2 h 1 / 2 m t 2 · sin ( 2 i - 1 ) π 2 m · sin ( 2 i - 3 ) π 2 m }             for the flat function , Fig . 7 ( b ) .
[ 0 j / r j r 0 ] .
[ ( n x / p ) sin θ j / n x j n x ( n x / p ) sin θ ] ,
T m ( ν ¯ ) = { cos ( m cos - 1 ν ¯ ) for ν ¯ 1 ; cosh ( m cosh - 1 ν ¯ ) for ν ¯ 1.