Abstract

The problems of maximum allowable cone of incidence and peak shift with angle of incidence have been examined both theoretically and experimentally for multilayer Fabry–Perot interference filters. It is shown that both effects can be characterized by a monolayer with an effective refractive index. Explicit expressions for the effective index (which is intermediate between that of the high and low components), the peak transmission, and the half-width are given. The method is extended to the case of double-half-wave filters, and the effective index deduced.

© 1964 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. D. Smith and C. R. Pidgeon, Mém. Soc. Roy. Sci. Liège, cinquième série 9, 336 (1963).
  2. S. D. Smith, J. Opt. Soc. Am. 48, 43 (1958).
    [Crossref]
  3. P. W. Baumeister and F. A. Jenkins, J. Opt. Soc. Am. 47, 57 (1957).
    [Crossref]
  4. J. M. Stone, J. Opt. Soc. Am. 43, 927 (1953).
    [Crossref]
  5. J. S. Seeley, J. Opt. Soc. Am. 54, 342 (1964).
    [Crossref]
  6. J. H. Jaffe, J. Opt. Soc. Am. 43, 1170 (1953).
    [Crossref]
  7. J. S. Seeley, Proc. Phys. Soc. (London) 78, 998 (1961).
    [Crossref]

1964 (1)

1963 (1)

S. D. Smith and C. R. Pidgeon, Mém. Soc. Roy. Sci. Liège, cinquième série 9, 336 (1963).

1961 (1)

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

1958 (1)

1957 (1)

1953 (2)

Baumeister, P. W.

Jaffe, J. H.

Jenkins, F. A.

Pidgeon, C. R.

S. D. Smith and C. R. Pidgeon, Mém. Soc. Roy. Sci. Liège, cinquième série 9, 336 (1963).

Seeley, J. S.

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

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

Smith, S. D.

S. D. Smith and C. R. Pidgeon, Mém. Soc. Roy. Sci. Liège, cinquième série 9, 336 (1963).

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

Stone, J. M.

J. Opt. Soc. Am. (5)

Mém. Soc. Roy. Sci. Liège (1)

S. D. Smith and C. R. Pidgeon, Mém. Soc. Roy. Sci. Liège, cinquième série 9, 336 (1963).

Proc. Phys. Soc. (London) (1)

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

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

F. 1
F. 1

Computed dependence of reflection phase change upon frequency for two λ0/4 stacks, at normal incidence.

F. 2
F. 2

Computed dependence of reflection phase change upon external angle of incidence (parallel light) for two λ0/4 stacks, at ω = ω0.

F. 3
F. 3

Percent shift of peak frequency with scanning angle (using parallel light) for two Fabry–Perot filters. In both cases the monolayer curves fit the computed curves to ±2% in n*. Key. ⊙ Experiment, filter (a) ⊡ experiment, filter (b); (——) computation, filter (a) (≡ monolayer, n* = 2.5); (– – – –) computation, filter (b) (≡ monolayer, n* = 3.35).

F. 4
F. 4

Computed effect of external cone angle on transmittance for filter (b)−L/Ge/L/HLHLHHLHLH. (a) Φe = 0°, Δ ν 1 2 = 3.5 cm 1; (b) Φe = 12° Δ ν 1 2 = 4.1 cm 1. (c) Φe = 15°, Δ ν 1 2 = 4.4 cm 1; (d) Φe = 20°, Δ ν 1 2 = 5.1 cm 1.

F. 5
F. 5

Transmittance traces for two filters of type (b)— L/Ge/L/HLHLHHLHLH (H = PbTe, L = ZnS).

F. 6
F. 6

Effect of external cone angle on measured transmittance of filter (b). (a) Φe ≏ 3°, Δ ν 1 2 = 3.6 ± 0.4 cm 1; (b) Φe ≏ 8.5°, Δ ν 1 2 = 3.8 ± 0.4 cm 1. (c) Φe ≏ 17.5°, Δ ν 1 2 = 4.8 ± 0.4 cm 1.

F. 7
F. 7

Computed percent shift of peak frequency with scanning angle (parallel light) for DHW filter of type: HLHL | H H | LHLHLHLH L HL H H LHLH ( H = PbTe , L = ZnS ) . Calculated n* = 3.3.

F. 8
F. 8

Transmittance (parallel light) of DHW filter shown in Fig. 7. Dashed lines show phase changes on reflection from systems ① (—●—●—) and ② (– – – – – –).

Equations (53)

Equations on this page are rendered with MathJax. Learn more.

T = [ T 1 ( ω ) T 2 ( ω ) ] / [ 1 R ( ω ) ] 2 × 1 / 1 + { [ 4 R ( ω ) ] / [ 1 R ( ω ) ] 2 } sin 2 1 2 θ ,
θ = 1 ( ω , ϕ ) 2 ( ω , ϕ ) + 2 ω nd cos ϕ ,
Δ ω / ω 0 = [ ( 1 / cos ϕ * ) 1 ] 1 2 ( sin ϕ e / n * ) 2 ,
ω p = ( 1 p + 2 p + 2 N π ) / 2 nd cos ϕ ,
ω 0 = ( 1 0 + 2 0 + 2 N π ) / 2 nd ,
Δ ω / ω 0 = ω p ω 0 ω 0 = [ ( p 1 + p 2 + 2 N π 1 0 + 2 0 + 2 N π ) 1 cos ϕ 1 ] .
( E A A H A A ) = ( M ) ( E B B H B B ) .
M k = [ cos β k i n k sin β k i n k sin β k cos β k ] ,
( M ) = ( m 11 i m 12 i m 21 m 22 ) .
ρ e i = m 11 + i n p m 12 i ( m 21 / n 0 ) ( n p / n 0 ) m 22 m 11 + i n p m 12 + i ( m 21 / n 0 ) + ( n p / n 0 ) m 22 ,
tan lim = k sin ( π ω / ω 0 ) ,
k A = n 0 / n L ( n r 1 ) ,
k B = n 0 / n L ( n r 1 ) ,
k C = n H / n 0 ( n r 1 ) ,
ω 0 / ω = k π at ω = ω 0 .
M k = ( cos β k i / n k sin β k i n k sin β k cos β k ) ,
β k = 2 π ( λ 0 / 4 ) cos ϕ k λ 1 2 π λ 0 λ [ 1 1 2 ( sin ϕ e n k ) 2 ] .
M k ( B k i / n k i n k B k ) .
( | m 11 | i | m 12 | i | m 21 | | m 22 | ) = ( B H s = 0 x 1 n r x ( 2 s + 1 ) + B L s = 1 x n r x 2 s i n L n r x i n L n r x B H s = 0 x 1 n r x ( 2 s + 1 ) + B L s = 1 x n r x 2 s ) for A ,
( 1 / n r x i B H n H s = 0 x 1 n r x 2 s + i B L n L s = 0 x 1 ( 1 n r ) x 2 s i n L B L s = 0 x 1 n r x 2 s + i n H B H s = 0 x 1 ( 1 n r ) x 2 s n r x ) for B ,
( n r x as in ( B ) above as in ( B ) above 1 / n r x ) for C .
tan lim = 2 n 0 m 11 / m 21 ,
tan lim = 2 n 0 m 12 / m 22 ,
tan lim = 2 m 21 / n 0 m 11 .
tan lim = 2 n 0 n H [ B H s = 0 x 1 n r 2 s + B L s = 1 x 1 n r ( 2 s 1 ) ] ,
tan lim = 2 n 0 [ B H n H s = 0 x 1 n r 2 s + B L n L s = 0 x 1 ( 1 n r ) 2 x 2 s ] ,
tan lim = 2 n 0 [ n L B L s = 0 x 1 n r 2 s + n H B H s = 0 x 1 ( 1 n r ) 2 x 2 s ] .
s = 0 x 1 n r 2 s ( n r 2 n r 2 1 ) and s = 1 x 1 n r ( 2 s 1 ) ( n r n r 2 1 ) ,
s = 0 x 1 ( 1 n r ) 2 x 2 s ( 1 n r ) 2 for n r > 1 .
tan lim = π n 0 2 n H [ 1 n H 2 ( n r 2 n r 2 1 ) + 1 n L 2 ( n r n r 2 1 ) ] sin 2 ϕ e ,
tan lim = π n 0 2 [ 1 n H 3 ( n r 2 n r 2 1 ) + 1 n L 3 ( 1 n r ) 2 ] sin 2 ϕ e ,
tan lim = π 2 n 0 [ 1 n L ( n r 2 n r 2 1 ) + 1 n H ( 1 n r ) 2 ] sin 2 ϕ e .
/ ( sin 2 ϕ e ) = J .
Δ = ( 1 p 1 0 ) = ( 2 p 2 0 ) = ω Δ ω + ( sin 2 ϕ e ) Δ ( sin 2 ϕ e ) = k π ω 0 Δ ω + J sin 2 ϕ e ,
Δ ω ω 0 = 1 2 A ( 1 cos ϕ ) + J sin 2 ϕ e 1 2 A cos ϕ + k π ,
( n * ) 2 ( 1 2 A + k π ) / [ ( 1 2 A / n 2 ) + 2 J ] .
( a ) L / Ge / L / HLH LL HLH ,
( b ) L / Ge / L / HLHL HH LHLH ,
T ( ν ) = 1 Δ ν ν Δ ν ν T ( ν ) d ν ,
Q k ( 1 i π n k [ ( ν ν 0 ) / ν 0 ] i n k π [ ( ν ν 0 ) / ν 0 ] 1 )
T ( ν ) = [ 1 + 1 4 ( m 12 m 21 ) 2 ] 1 ,
T L L ( ν ) = { 1 + 1 4 [ n L n r 2 x ( 1 + 1 n r 1 ) π ( ν ν 0 ν 0 ) ] 2 } 1 ,
T H H ( ν ) = { 1 + 1 4 [ n H n r 2 x ( 1 + 1 n r 1 ) π ( ν ν 0 ν 0 ) ] 2 } 1 .
F k = 1 2 n k n r 2 x + 1 / ( n r 1 ) .
T ( ν ) = ν 0 π F k Δ ν { tan 1 [ F k π ( ν ν 0 ν 0 ) ] tan 1 × [ F k π ( ν ν 0 Δ ν ν 0 ) ] } .
ν m = ν 0 + 1 2 Δ ν .
T ̂ = ( π 2 F k Δ ν ν 0 ) 1 tan 1 ( π 2 F k Δ ν ν 0 ) .
Δ ν 1 2 = [ ( Δ ν ) 2 + 4 ν 0 2 / π 2 F k 2 ] 1 2 ,
HLHL 1 | M H H | N LHLH HL HL 2 H H LHLH ,
Δ 1 = k 1 π ( Δ ω / ω 0 ) + J 1 sin 2 ϕ c ,
Δ 2 = k 2 π ( Δ ω / ω 0 ) + J 2 sin 2 ϕ c ,
( n * ) 2 [ A + ( k 1 + k 2 ) π ] / [ ( A / n 2 ) + 2 ( J 1 + J 2 ) ] ,
HLHL H H LHLHLHLHLHL H H LHLH ( x = 2 , y = 6 ) ,