Abstract

A general expression is derived for a Fabry-Perot spectrometer when one or more of the functions entering into its description lack the inherent radial symmetry of the fringe pattern about the etalon optical axis. Calculations of the effects of these asymmetric functions applicable to densitometry of photographic fringe records and vidicon scanning of fringe patterns are given. The theory is also applied to interference filters in order to quantitatively explain their behavior as they are tilted in order to scan a spectrum. Rather simple approximations to the theory are also derived for interference filters that show that both the inverse of the peak transmission and the square of the width (normalized to the unbroadened filter) are proportional to the square of the product of the tilt and field of view angles over the width, in excellent agreement with experimental measurements.

© 1974 Optical Society of America

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Errata

G. Hernandez, "Analytical description of a Fabry-Perot spectrometer. 3:Off-axis behavior and interference filters: erratum," Appl. Opt. 18, 3364-3365 (1979)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-18-20-3364

References

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  1. G. Hernandez, Appl. Opt. 5, 1745 (1966).
    [CrossRef] [PubMed]
  2. G. Hernandez, Appl. Opt. 9, 1591 (1970).
    [CrossRef] [PubMed]
  3. P. Jacquinot, Ch. Dufour, J. Rech. Centre Nat., Rech. Sci. 6, 91 (1948).
  4. G. G. Shepherd, C. W. Lake, J. R. Miller, L. L. Cogger, Appl. Opt. 4, 267 (1965).
    [CrossRef]
  5. J. G. Hirschberg, P. Platz, Appl. Opt. 4, 1375 (1965).
    [CrossRef]
  6. R. H. Eather, D. L. Reasoner, Appl. Opt. 8, 227 (1969).
    [CrossRef] [PubMed]
  7. G. T. Best, Appl. Opt. 6, 287 (1967).
    [CrossRef] [PubMed]
  8. M. Abramowitz, I. A. Stegun, Eds. Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964), AMS 55.
  9. G. Hernandez, J. P. Turtle, in Aurora and Airglow, B. M. McCormac, Ed (Reinhold, New York, 1967), p. 435.
  10. M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964), p. 70.
  11. P. H. Lissberger, W. L. Wilcock, J. Opt. Soc. Am. 49, 126 (1959).
    [CrossRef]
  12. J. O. Stoner, J. Opt. Soc. Am. 56, 370 (1966).
    [CrossRef]

1970 (1)

1969 (1)

1967 (1)

1966 (2)

1965 (2)

1959 (1)

1948 (1)

P. Jacquinot, Ch. Dufour, J. Rech. Centre Nat., Rech. Sci. 6, 91 (1948).

Best, G. T.

Born, M.

M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964), p. 70.

Cogger, L. L.

Dufour, Ch.

P. Jacquinot, Ch. Dufour, J. Rech. Centre Nat., Rech. Sci. 6, 91 (1948).

Eather, R. H.

Hernandez, G.

G. Hernandez, Appl. Opt. 9, 1591 (1970).
[CrossRef] [PubMed]

G. Hernandez, Appl. Opt. 5, 1745 (1966).
[CrossRef] [PubMed]

G. Hernandez, J. P. Turtle, in Aurora and Airglow, B. M. McCormac, Ed (Reinhold, New York, 1967), p. 435.

Hirschberg, J. G.

Jacquinot, P.

P. Jacquinot, Ch. Dufour, J. Rech. Centre Nat., Rech. Sci. 6, 91 (1948).

Lake, C. W.

Lissberger, P. H.

Miller, J. R.

Platz, P.

Reasoner, D. L.

Shepherd, G. G.

Stoner, J. O.

Turtle, J. P.

G. Hernandez, J. P. Turtle, in Aurora and Airglow, B. M. McCormac, Ed (Reinhold, New York, 1967), p. 435.

Wilcock, W. L.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964), p. 70.

Appl. Opt. (6)

J. Opt. Soc. Am. (2)

J. Rech. Centre Nat., Rech. Sci. (1)

P. Jacquinot, Ch. Dufour, J. Rech. Centre Nat., Rech. Sci. 6, 91 (1948).

Other (3)

M. Abramowitz, I. A. Stegun, Eds. Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964), AMS 55.

G. Hernandez, J. P. Turtle, in Aurora and Airglow, B. M. McCormac, Ed (Reinhold, New York, 1967), p. 435.

M. Born, E. Wolf, Principles of Optics (Macmillan, New York, 1964), p. 70.

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Figures (9)

Fig. 1
Fig. 1

Geometry for an aperture displaced in relation to the fringe axis. Details in text.

Fig. 2
Fig. 2

Transmission of an aperture 0.0349-rad radius as it is displaced away from the axis in 0.0524-rad steps. Tick marks show aperture center.

Fig. 3
Fig. 3

Effects of an offset aperture in a conventional Fabry-Perot spectrometer. A perfectly centered aperture profile is shown as comparison.

Fig. 4
Fig. 4

Finite aperture effects appropriate to photographic densitometry and/or vidicon scanning of Fabry-Perot fringes. The unscanned function is shown for comparison.

Fig. 5
Fig. 5

Effects of tilting an interference filter at arbitrary fields of view (marked in the upper right-hand side of the graphs). Tilting done in 3° steps. Perfect etalon case.

Fig. 6
Fig. 6

Effects of tilting an interference filter at arbitrary fields of view. Same conditions as Fig. 5 except for a microscopic surface defects limited etalon.

Fig. 7
Fig. 7

Laboratory measurement of an interference filter with the same properties as those calculated in Figs. 5 and 6.

Fig. 8
Fig. 8

Comparison between laboratory and calculated filters: (a) shift; (b) relative peak transmission; and (c) relative width (FWHH).

Fig. 9
Fig. 9

Transmission and broadening properties of a number of interference filters given by the approximate theory. The complete theory filters (□ and ○) are included as comparison.

Equations (42)

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A ( x ) = ( 1 R 2 ) [ 2 π ( 1 2 R cos x + R 2 ) ] 1
= ( 2 π ) 1 ( 1 + 2 n = 1 R n cos n x )
Ш = ( x / 2 π ) * L ( x ) ,
G ( x ) = ( G π 1 / 2 ) 1 exp [ ( x G 1 ) 2 ] ,
F ( x ) = Δ σ ( 4 π f ) 1 ( x Δ σ / 4 π f ) ,
T ( x ) = H ( x ) t 1 exp ( x t 1 ) ,
L ( x ) = π 1 ln ( 1 / R ) { [ ln ( 1 / R ) ] 2 + x 2 } 1 ,
Ш ( x ) = n = n = + δ ( x n ) ,
( x ) = | 1 | x | < 1 2 , ( 1 2 ) | x | = 1 2 , 0 | x | > 1 2 ,
H ( x ) = | 0 | x | < 0 , ( 1 2 ) | x | = 0 , 1 | x | > 0 ,
sinc ( x ) = ( π x ) 1 sin ( π x )
Y ( x ) = A ( x ) * G ( x ) * F ( x ) * T ( x ) ,
Y ( x ) = ( 2 π ) 1 [ 1 + 2 n = 1 b n cos ( n x d n ) ] ,
Y ( x ) * τ ( x ) = ( 2 π ) 1 { 1 + 2 n = 1 b n ( A n 2 + B n 2 ) 1 / 2 cos [ n x d n tan 1 ( B / A n ) ] } ,
A n = x l x u τ ( x ) cos n x d x ,
B n = x l x u τ ( x ) sin n x d x .
τ ( x ) = ( x u x l ) 1 { ( 2 x x u x l ) / [ 2 ( x u x l ) ] }
A n = [ n ( x u x 1 ) ] 1 ( sin n x u sin n x l ) ,
B n = [ n ( x u x 1 ) ] 1 ( cos n x l cos n x u ) .
Y ( x ) * τ ( x ) = ( 2 π ) 1 ( 1 + 2 n = 1 b n sinc [ n Δ x ( 2 π ) 1 ] × cos { n [ x ( x l + Δ x / 2 ) ] d n } ) .
Y ( x ) * τ ( x ) = ( 2 π ) 1 { 1 + 2 n = 1 b n sinc [ n Δ x ( 2 π ) 1 ] × cos [ n ( x x c ) d n ] } .
R 2 = ( x α ) 2 + y 2 = ρ 2 + α 2 2 α ρ cos θ .
τ ( ρ ) = ( π ) 1 cos 1 [ ( ρ 2 + α 2 R 2 ) / 2 α ρ ] .
τ ( z ) = 1 H ( z + 1 ) + π 1 [ ( z / 2 ) cos 1 ( z ) ] .
Y ( x ) = Ш ( x / 2 π ) * G ( x ) .
A ( x ) = L ( x ) * Ш ( x / 2 π ) L ( x ) = π 1 ln ( 1 / R ) { [ ln ( 1 / R ) ] 2 + x 2 } 1 .
ω 0 = 2 ln ( 1 / R ) ,
τ 0 = [ π ln ( 1 / R ) ] 1 = 2 ( π ω 0 ) 1 .
L ( x ) * F ( x ) = ( π y ) 1 { tan 1 [ ( x y ) / ln ( 1 / R ) ] tan 1 [ ( x y Δ y ) ] / ln ( 1 / R ) ] } ,
ω / ω 0 = [ ( Δ y / ω 0 ) 2 + 1 ] 1 / 2 ,
τ / τ 0 = T = ( ω 0 / Δ y ) tan 1 ( Δ y / ω 0 )
[ 1 + ( Δ y / ω 0 ) 2 ] 1 ,
T [ ( Δ y / ω 0 ) 2 ln 2 + 1 ] 1 ,
ω / ω 0 [ ( Δ y / ω 0 ) 2 ln 2 + 1 ] 1 / 2 .
y = 2 π [ sec ( θ + ϕ / 2 ) 1 ] ,
Δ y = 2 π [ sec ( θ + ϕ / 2 ) sec ( θ ϕ / 2 ) ] ,
y + Δ y / 2 = π ( θ 2 + ϕ 2 / 4 ) ,
Δ y = 2 π θ ϕ .
1 / T 1 = [ k θ ϕ σ 0 / ( m σ ) ] 2 ,
( ω / ω 0 ) 2 1 = [ k θ ϕ σ 0 / ( m σ ) ] 2 ,
( 1 / T ) 1 ( ω / ω 0 ) 2 1 | = [ k θ ϕ σ 0 / ( n 2 m σ ) ] 2 ,
n = sin ( tilt ) / sin { cos 1 [ σ 0 / ( σ 0 + σ ) ] } .

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