Abstract

When the radiation source in a Michelson interferometer is placed in front of or behind the focal plane of the collimator, distortions arise in the spectral line shapes. The appearance and behavior of these distortions in a cube-corner interferometer are treated in this work. We shall also present a simple and fast method to calculate the true line position of a distorted line, and an easy rule of thumb to connect the off-focus shift and the amount of distortion. Calculation of the phase-error curves in the signal domain is treated as well. This calculation gives a possibility of correcting the distorted interferogram.

© 1992 Optical Society of America

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References

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  1. J. Kauppinen, P. Saarinen, “Line shape distortions in misaligned cube corner interferometers,” Appl. Opt. (to be published).
  2. G. Guelachvili, “Distortions in Fourier spectra and diagnosis,” in Spectrometric Techniques, G. Vanasse, ed. (Academic, New York, 1981), Vol. 2, pp. 1–62.
  3. J. Kauppinen, T. Kärkkäinen, E. Kyrö, “Correcting errors in the optical path difference in Fourier spectroscopy: a new accurate method,” Appl. Opt. 17, 1587–1594 (1978).
    [CrossRef] [PubMed]

1978 (1)

Guelachvili, G.

G. Guelachvili, “Distortions in Fourier spectra and diagnosis,” in Spectrometric Techniques, G. Vanasse, ed. (Academic, New York, 1981), Vol. 2, pp. 1–62.

Kärkkäinen, T.

Kauppinen, J.

Kyrö, E.

Saarinen, P.

J. Kauppinen, P. Saarinen, “Line shape distortions in misaligned cube corner interferometers,” Appl. Opt. (to be published).

Appl. Opt. (1)

Other (2)

J. Kauppinen, P. Saarinen, “Line shape distortions in misaligned cube corner interferometers,” Appl. Opt. (to be published).

G. Guelachvili, “Distortions in Fourier spectra and diagnosis,” in Spectrometric Techniques, G. Vanasse, ed. (Academic, New York, 1981), Vol. 2, pp. 1–62.

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Figures (9)

Fig. 1
Fig. 1

Point of the radiation source lying at the intersection of the focal plane and the effective axis, producing the correct spectrum. Any point in the focal plane aside from the effective axis gives a distorted spectrum, in which every wave number is multiplied by cos α.

Fig. 2
Fig. 2

Light penetrating the lens through a circle of radius r (centered in the effective axis), producing the same spectrum as a similar radiation source lying in the focal plane but with a shift s from the effective axis.

Fig. 3
Fig. 3

Correspondence between the radius r and the shift s.

Fig. 4
Fig. 4

Change in the off-axis line shape E(v, s) during the r integration.

Fig. 5
Fig. 5

Off-focus distorted line shape, convolved with the transform of the boxcar truncation (apodization) function and the derivative curve of the convolved line shape; the dashed line represents the undistorted boxcar line shape.

Fig. 6
Fig. 6

Derivative of the convolved off-focus line shape with a zero in the middle of the original undistorted line shape. This zero is insensitive to the amount of distortion provided that condition (8) holds (see text). This fact offers us a simple means of determining the true position of a distorted spectral line.

Fig. 7
Fig. 7

Fourier transforms of undistorted and distorted spectral lines situated in the origin and in vR. The latter ones are obtained from the former ones by multiplying by an exponential wave, exp(ivRx). Because the transform of the distorted line has a nonvanishing imaginary part even in the origin, a phase shift arises.

Fig. 8
Fig. 8

Off-focus distorted line shapes with various degrees of distortion (with the off-focus shift Δ varied) and corresponding epsi curves ɛ(x).

Fig. 9
Fig. 9

(A) Off-focus distortion is the difference between the distorted and the undistorted spectral lines. (B) Extra noise generated in a real measurement near a spectral line by the off-focus distortion is obtained by the off-focus distortion being convolved by the transform of the interferogram truncation function.

Equations (50)

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v ( ρ ) = v 0 cos α = v 0 [ 1 + ( ρ / f ) 2 ] 1 / 2 ,
v ( ρ ) v 0 ( 1 + ρ 2 2 f 2 ) .
E ( v , s ) = { 2 π v 0 s < R 0 , ρ ( v ) R 0 - s 2 π v 0 1 π arccos [ [ ρ ( v ) ] 2 + s 2 - R 0 2 2 ρ ( v ) s ] ρ ( v ) ( R 0 - s , R 0 + s ) 0 , elsewhere
ρ ( v ) f [ 2 ( 1 - v / v 0 ) ] 1 / 2 .
Δ v = v ( 0 ) - v ( R 0 ) v 0 R 0 2 2 f 2 .
2 π r d r π R L 2 = 2 r d r R L 2
E Δ ( v ) = 2 R L 2 0 R L E [ v , s ( r ) ] r d r .
s ( r ) = Δ tan β = Δ r f + Δ Δ f r .
E Δ ( v ) = 2 R L 2 0 R L E ( v , Δ f r ) r d r .
Δ < f ( R 0 / R L ) .
v 0 = z + 1 2 Δ v = z + v 0 R 0 2 4 f 2 v 0 = z 1 - ( R 0 / 2 f ) 2 .
f ( x ) = A ( x ) exp [ i ( x ) ] ,
A ( x ) = f ( x )
( x ) = arctan { Im [ f ( x ) ] Re [ f ( x ) ] } .
ϕ ( x ) = 2 π v R x + ( x ) .
ϕ ( x ) = ( 2 j - 1 ) π 2 = ( j - 1 2 ) π .
( j - 1 / 2 ) π = 2 π v R x j + ( x j ) .
( j - 1 / 2 ) π = 2 π v R x j R .
x j - x j R = - ( x j ) 2 π v R .
ɛ ( x j ) = x j - x j R = - 1 2 π v R arctan { Im [ f ( x ) ] Re [ f ( x ) ] } .
ϕ ( x j ) = ϕ 0 + 2 π v R x j + ( x j ) = ϕ 0 + 2 π v R × [ x j - ɛ ( x j ) ] = ϕ 0 + 2 π 0 x j v true ( y ) d y .
v true ( x j ) = v R [ 1 - d ɛ ( x j ) d x j ] .
N max 1.28 K 2 ( 1 - 1 / 2 K ) 2 A ,
K = Δ f R L R 0
K < 1
( S N ) = A N max = 1 1.28 K 2 ( 1 - ½ K ) 2 ,
K = 1 - { 1 - 2 [ 1.28 ( S / N ) ] 1 / 2 } 1 / 2 .
Δ max = R 0 R L f ( 1 - { 1 - 2 [ 1.28 ( S / N ) ] 1 / 2 } 1 / 2 ) .
v ( R 0 + s ) v 0 [ 1 - ( R 0 + s ) 2 2 f 2 ] = v ,
s ( r ) = f [ 2 ( 1 - v / v 0 ) ] 1 / 2 - R 0 = ρ ( v ) - R 0
r = f Δ [ ρ ( v ) - R 0 ] .
v ( s - R 0 ) = v 0 [ 1 - ( s - R 0 ) 2 2 f 2 ] = v
r = f Δ [ ρ ( v ) + R 0 ] .
R L f Δ [ ρ ( v ) - R 0 ] ,
E Δ ( v ) = 2 f 2 Δ 2 R L 2 ρ ( v ) - R 0 m ( v ) E ( v , s ) s d s ,
m ( v ) = min { Δ f R L , ρ ( v ) + R 0 } .
v { v 0 [ 1 - ( R 0 + s ) 2 2 f 2 ] , v 0 [ 1 - ( R 0 - s ) 2 2 f 2 ] } ,
E ( v , s ) = 2 π v 0 1 π arccos { [ ρ ( v ) ] 2 + s 2 - R 0 2 2 ρ ( v ) s } ,
E Δ ( v ) = 1 v 0 ( 2 f R L Δ ) 2 ρ ( v ) - R 0 m ( v ) arccos { [ ρ ( v ) ] 2 + s 2 - R 0 2 2 ρ ( v ) s } s d s .
v [ R 0 - s ( r ) ] = v 0 { 1 - [ ( R 0 - s ( r ) ] 2 2 f 2 } = v s = R 0 - f [ 2 ( 1 - v / v 0 ) ] 1 / 2 ,
r = f Δ [ R 0 - ρ ( v ) ] .
E [ v , s ( r ) ] = 2 v 0 arccos { [ ρ ( v ) 2 + [ s ( r ) ] 2 - R 0 2 2 ρ ( v ) s ( r ) }
v ( s - R 0 ) = v 0 [ 1 - ( s - R 0 ) 2 2 f 2 ] = v .
r = f Δ [ ρ ( v ) + R 0 ] .
r = f Δ [ R 0 - ρ ( v ) ]
E Δ ( v ) = 2 R L 2 0 R L 2 π v 0 r d r = 2 π v 0 ,
r = f Δ [ R 0 - ρ ( v ) ]
r = f Δ [ ρ ( v ) + R 0 ]
E Δ ( v ) = 2 R L 2 0 ( f / Δ ) [ R 0 - ρ ( v ) ] 2 π v 0 r d r = 2 R L 2 ( f / Δ ) [ R 0 - ρ ( v ) ] ( f / Δ ) m ( v ) E ( v , Δ f r ) r d r = 2 R L 2 2 π v 0 1 2 f 2 Δ 2 [ R 0 - ρ ( v ) ] 2 + 2 f 2 Δ 2 R L 2 R 0 - ρ ( v ) m ( v ) E ( v , s ) s d s .
E Δ ( v ) = 2 π v 0 ( f R L Δ ) 2 [ R 0 - ρ ( v ) ] 2 + 1 v 0 ( 2 f R L Δ ) 2 R 0 - ρ ( v ) m ( v ) arccos { [ ρ ( v ) ] 2 + s 2 - R 0 2 2 ρ ( v ) s } s d s .

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