Abstract

Presented is a comprehensive analysis of spectrum errors due to drive nonlinearities in a Michelson interferometer. The fringe-reference sampling case is treated for both repeatable and random nonlinearities. Partial results are given for the equal-time sampling case. The derived equations for the actual (or mean-square) spectrum error can be evaluated numerically for any choice of drive speed variation (or its power spectrum), input radiation spectrum, apodizing function, and filter response characteristic. Closed form solutions are obtained for special cases to show qualitative features of the spectrum error and to establish bounds on its magnitude. It is shown that when fringe-reference sampling is used with a filter having minimal distortion, the relative rms random error in the measured intensity of spectrum lines is typically of the same order as the rms, relative error in drive speed if the resolving power is about five times greater than needed to just resolve the lines and that this error varies as the square root of the resolved spectral interval.

© 1977 Optical Society of America

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References

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  1. J. Connes, Rev. Opt. 40, 45, 116, 171, 231 (1961).
  2. H. Sakai, Aspen International Conference on Fourier Spectroscopy (1970), AFCRL-71-0019, Special Report 114, Jan.1971.
  3. T. Nishiyama, T. Yamauchi, M. Ohno, M. Morii, N. Ura, K. Masutani, Jpn. J. Appl. Phys. 14, Suppl. 14-1, 67 (1975).
  4. E. Bell, R. Sanderson, Appl. Opt. 11, 688 (1972).
    [CrossRef] [PubMed]
  5. L. Mertz, Transformations in Optics (Wiley, New York, 1965), p. 25.

1975

T. Nishiyama, T. Yamauchi, M. Ohno, M. Morii, N. Ura, K. Masutani, Jpn. J. Appl. Phys. 14, Suppl. 14-1, 67 (1975).

1972

1961

J. Connes, Rev. Opt. 40, 45, 116, 171, 231 (1961).

Bell, E.

Connes, J.

J. Connes, Rev. Opt. 40, 45, 116, 171, 231 (1961).

Masutani, K.

T. Nishiyama, T. Yamauchi, M. Ohno, M. Morii, N. Ura, K. Masutani, Jpn. J. Appl. Phys. 14, Suppl. 14-1, 67 (1975).

Mertz, L.

L. Mertz, Transformations in Optics (Wiley, New York, 1965), p. 25.

Morii, M.

T. Nishiyama, T. Yamauchi, M. Ohno, M. Morii, N. Ura, K. Masutani, Jpn. J. Appl. Phys. 14, Suppl. 14-1, 67 (1975).

Nishiyama, T.

T. Nishiyama, T. Yamauchi, M. Ohno, M. Morii, N. Ura, K. Masutani, Jpn. J. Appl. Phys. 14, Suppl. 14-1, 67 (1975).

Ohno, M.

T. Nishiyama, T. Yamauchi, M. Ohno, M. Morii, N. Ura, K. Masutani, Jpn. J. Appl. Phys. 14, Suppl. 14-1, 67 (1975).

Sakai, H.

H. Sakai, Aspen International Conference on Fourier Spectroscopy (1970), AFCRL-71-0019, Special Report 114, Jan.1971.

Sanderson, R.

Ura, N.

T. Nishiyama, T. Yamauchi, M. Ohno, M. Morii, N. Ura, K. Masutani, Jpn. J. Appl. Phys. 14, Suppl. 14-1, 67 (1975).

Yamauchi, T.

T. Nishiyama, T. Yamauchi, M. Ohno, M. Morii, N. Ura, K. Masutani, Jpn. J. Appl. Phys. 14, Suppl. 14-1, 67 (1975).

Appl. Opt.

Jpn. J. Appl. Phys.

T. Nishiyama, T. Yamauchi, M. Ohno, M. Morii, N. Ura, K. Masutani, Jpn. J. Appl. Phys. 14, Suppl. 14-1, 67 (1975).

Rev. Opt.

J. Connes, Rev. Opt. 40, 45, 116, 171, 231 (1961).

Other

H. Sakai, Aspen International Conference on Fourier Spectroscopy (1970), AFCRL-71-0019, Special Report 114, Jan.1971.

L. Mertz, Transformations in Optics (Wiley, New York, 1965), p. 25.

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Figures (6)

Fig. 1
Fig. 1

The spectral response of a four-pole Butterworth filter. This filter produces very little distortion if the bandwidth W f is one and one-half times the highest spectrum frequency.

Fig. 2
Fig. 2

Spectrum ghost contributions of the first five terms in the expansion of the interferogram error corresponding to a sinusoidal velocity error. The ghosts occur at frequencies corresponding to integer values of m = (νν s )/ν ξ , where ν s is the frequency of the monochromatic line and ν ξ is the frequency of the velocity error. The values shown next to the ghost contributions are intensities normalized by (πr d ν s ξ0) n /n!.

Fig. 3
Fig. 3

The delta functions which are convolved with B(ν) to obtain the transfer function Y(ν,ν1) for a monochromatic source and a doublet-line source [(a) and (b), respectively] and the corresponding mean-square spectrum errors for a white power spectrum (c). The result for the doublet lines illustrates the formation of rms ghosts.

Fig. 4
Fig. 4

The function q(s,s a ), which defines the spectral variation of 〈E2(ν)〉 for the line-plus-continuum model.

Fig. 5
Fig. 5

Estimates of the normalized maximum rms spectrum error for the line-plus-continuum model. For the solid curves normalization is with respect to the mean spectral intensity at the line centers [(Eq. (57)]; for the dashed curves normalization is with respect to the apparent spectral intensity of the continuum. The results shown are for emission lines but are approximately correct for absorption lines provided |S l /d l |/S c ≲ 0.1.

Fig. 6
Fig. 6

A spectrum synthesized from delta function lines and rectangular bands (a) and the corresponding rms spectrum error for a random velocity error with a Gaussian power spectrum (b). The normalization constant K is defined by Eq. (58). The error is given by the uppermost solid curve. The uppermost dashed curve gives the error when aliasing is not accounted for. The lower pair of curves is defined similarly, except that they include only the error contributed by the lines. The resolved spectral interval is 0.005 times the nominal maximum frequency ν m .

Tables (1)

Tables Icon

Table I Mean-Square Spectrum Error for Doublet Line Source

Equations (69)

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r ( t ) = 2 v t + 2 0 t δ v ( t 1 d t ) 1 .
r ( x ) = x + 0 x ξ ( x 1 ) d x 1 ,
g ˜ ( x ) = f ( x ) * g [ x + 0 x ξ ( x 1 ) d x 1 ] ,
x n + 0 x n ξ ( x ) d x = ( n + c ) / ν L r n , n = 0 , 1 , 2 , ,
g ˜ ( x n ) = - f ( X ) g [ x n - X + 0 x n - X ξ ( x ) d x ] d X = - f ( X ) g [ x n - X + 0 x n ξ ( x ) d x - x n - X x n ξ ( x ) d x ] d X ,
g ˜ ( x n ) = - f ( X ) g [ r n - X - x n - X x n ξ ( x ) d x ] d X .
g ˜ ( r ) - f ( X ) g { r - X [ 1 - ξ ( r ) ] } d X .
( r ) = g ˜ ( r ) - f ( r ) * g ( r ) = - f ( X ) ( g { r - X [ 1 + ξ ( r ) ] } - g ( r - X ) ) d X .
( r ) = - f ( X ) { - X ξ ( r ) g ( r - X ) + [ X ξ ( r ) ] 2 g ( r - X ) / 2 ! - } d X = - ξ ( r ) { [ r f ( r ) ] * g ( r ) } + [ ξ 2 ( r ) / 2 ! ] { [ r 2 f ( r ) ] * g ( r ) } - .
( W g / W f ) ξ ( r ) 1 ,
( r ) = - ξ ( r + r d ) { [ ( r + r d ) f ( r + r d ) ] * g ( r ) } .
δ r ( x ) = 0 x ξ ( x 1 ) d x 1
( x ) = g ˜ ( x ) - f ( x ) * g ( x ) = f ( x ) * { g [ x + δ r ( x ) ] - g ( x ) } .
( x ) = f ( x + x d ) * { δ r ( x ) g ( x ) } .
δ r W g 1.
E ( ν ) = Re - b ( r ) ( r ) exp ( - i 2 π ν r ) d r ,
E ( ν ) = Re { M ( ν ) * [ F ( ν ) exp ( i 2 π r d ν ) ν S ( ν ) ] } = COS [ b ( r ) ξ ( r + r d ) ] * { [ F ( ν ) cos β ( ν ) - F ( ν ) ϕ ( ν ) sin β ( ν ) ] ν S ( ν ) } - SIN [ b ( r ) ξ ( r + r d ) ] * { [ F ( ν ) sin β ( ν ) + F ( ν ) ϕ ( ν ) cos β ( ν ) ] ν S ( ν ) } ,
S ( ν ) = COS [ b ( r ) ] * [ F ( ν ) cos β ( ν ) S ( ν ) ] - SIN [ b ( r ) ] * [ F ( ν ) sin β ( ν ) S ( ν ) ] .
E ( ν ) = - 2 π r d [ ν S ( ν ) ] * SIN [ b ( r ) ξ ( r + r d ) ] .
E ( ν ) = - 2 π r d [ ν S ( ν ) ] * { [ COS b ( r ) ] * SIN ξ ( r + r d ) + [ SIN b ( r ) ] * COS ξ ( r + r d ) } .
FOR ( g 1 ) = exp [ - i ( ψ 0 + 2 π r p ν ) ] S ( ν ) ,
FOR [ g 1 * f ( r + r d ) ] T } = ( F ( ν ) exp { - i [ ψ 0 + 2 π r p ν + β ( ν ) ] } S ( ν ) ) * FOR ( T ) = exp [ - i ( ψ 0 + 2 π r p ν ) ] ( { F ( ν ) exp [ - i β ( ν ) ] S ( ν ) } * FOR [ T 1 ( r ) ] ) ,
exp { - i [ ψ 0 + 2 π r p ν + β ( ν ) ] } { [ F ( ν ) S ( ν ) ] * COS ( T 1 ) - i [ F ( ν ) S ( ν ) ] * SIN ( T 1 ) } .
FOR ( T ) = [ F ( ν ) exp ( 2 π i r d ν ) ν FOR ( g ) ] * FOR [ ξ ( r + r d ) T ] .
FOR ( T ) = exp [ - i ( ψ 0 + 2 π r p ν ) ] { [ F ( ν ) exp ( 2 π i r d ν ) ν S ( ν ) ] * FOR [ ξ 1 ( r + r d ) T 1 ] } ,
F ( ν ) exp ( 2 π i r d ν ) = - i 2 π r d ,
- 2 π r d [ ν S ( ν ) ] * SIN [ T ( r + r p ) ξ ( r + r d + r p ) ] .
E ( ν ) = - 2 π r d [ ν S ( ν ) ] * SIN [ b ( r ) ξ ( r + r d ) ] ,
ξ ( r ) = ξ 0 sin 2 π ν ξ ( r - r 0 )
S ( ν ) = δ ( ν - ν s ) + δ ( ν + ν s ) .
E ( ν ) = π r d ν s ξ 0 { [ cos 2 π ( r d - r 0 ) ν ξ ] [ COS b ( r ) ] * [ - δ ( ν - ν s - ν ξ ) + δ ( ν - ν s + ν ξ ) + δ ( ν + ν s - ν ξ ) - δ ( ν + ν s + ν ξ ) ] + [ sin 2 π ( r d - r 0 ) ν ξ ] [ SIN b ( r ) ] * [ - δ ( ν - ν s - ν ξ ) - δ ( ν - ν s + ν ξ ) + δ ( ν + ν s - ν ξ ) + δ ( ν + ν s + ν ξ ) ] } .
( - 1 ) n [ ξ n ( r + r d ) / n ! ] { [ ( r + r d ) n f ( r + r d ) ] * g ( n ) ( r ) } .
E n ( ν ) = [ ( - π r d ν s ξ 0 ) n / n ! ] [ δ ( ν - ν ξ ) - δ ( ν + ν ξ ) ] * n - 1 [ δ ( ν - ν ξ ) - δ ( ν + ν ξ ) ] ,
E ( ν s + m ν ξ ) S ( ν s ) = ( - 1 ) m k = 0 ( - 1 ) k ( π r d ν s ξ 0 ) m + 2 k k ! ( m + k ) ! = ( - 1 ) m J m ( 2 π r d ν s ξ 0 ) . for m = 1 , 2 ,
E ( ν s ) S ( ν s ) = k = 1 ( - 1 ) k ( π r d ν s ξ 0 ) 2 l ( k ! ) 2 = J 0 ( 2 π r d ν s ξ 0 ) - 1.
g ˜ ( r ) = cos [ 2 π ν s ( r - r d ξ 0 sin 2 π ν ξ r ) ]
g ˜ ( r ) = m = - ( - 1 ) m J m ( 2 π ν s r d ξ 0 ) cos 2 π r ( ν s + m ν ξ ) .
E ( ν ) = 2 π [ ν S ( ν ) ] * SIN [ b ( x ) δ r ( x ) ] .
δ r ( x ) = ( ξ 0 / 2 π ν ξ ) sin 2 π ν ξ x .
g ˜ ( x ) = cos { 2 π ν s [ x + ( ξ 0 / 2 π ν ξ ) sin 2 π ν ξ x ] } .
g ˜ ( x ) = m = - J m ( 2 π ν s R ) cos 2 π x ( ν s + m ν ξ ) ,
E ( ν s + m ν ξ ) / S ( ν s ) = J m ( 2 π ν s R ) ; m = 1 , 2 , E ( ν s ) / S ( ν s ) = J 0 ( 2 π ν s R ) - 1.
{ h [ F ( ν ) , ϕ ( ν ) ] ν S ( ν ) } * [ COS or SIN of b ( r ) ] * [ COS or SIN of ξ ( r + r d ) or δ ( r ) ] ,
E ( ν , y ) = - - b ( r ) ξ ( r + r d + y ) e ( r + r d ) cos 2 π ν r d r ,
e ( r ) [ r f ( r ) ] * g ( r ) .
E 2 ( ν ) = - [ lim 1 Y - Y / 2 Y / 2 ξ ( r + r d + y ) ξ ( r 1 + r d + y ) d y ] b ( r ) b ( r 1 ) e ( r + r d ) e ( r 1 + r d ) cos 2 π ν r cos 2 π ν r 1 d r d r 1 .
C ξ ( r - r 1 ) = - P ξ ( ν 1 ) exp [ - i 2 π ν 1 ( r - r 1 ) ] d ν 1 .
E 2 ( ν ) = 2 0 P ξ ( ν 1 ) | - b ( r ) { [ ( r + r d ) f ( r + r d ) ] * g ( r ) } × exp ( i 2 π ν 1 r ) cos 2 π ν r d r | 2 d ν 1 .
P E ( ν , ν 1 ) = P ξ ( ν 1 ) Y ( ν , ν 1 ) 2 ,
E 2 ( ν ) = 2 0 P ξ ( ν 1 ) the right - hand side of Eq . with exp ( i 2 π ν 1 r ) in place of ξ ( r + r d ) 2 d ν 1
E 2 ( ν ) = 2 ( 2 π r d ) 2 0 P ξ ( ν 1 ) SIN [ b ( r ) exp ( i 2 π ν 1 r ) ] * [ ν S ( ν ) ] 2 d ν 1 ,
E 2 ( ν ) = 2 ( π r d ) 2 0 P ξ ( ν 1 ) B ( ν 1 ) * [ ( - ν + ν 1 ) S ( - ν + ν 1 ) + ( ν + ν 1 ) S ( ν + ν 1 ) ] 2 d ν 1
= 2 ( π r d ) 2 0 P ξ ( ν 1 ) Z ( - ν + ν 1 ) + Z ( ν + ν 1 ) 2 d ν 1 ; Z ( ν ) B ( ν ) * [ ν S ( ν ) ] ,
E 2 ( ν ) = 2 ( π r d ν s ) 2 0 P ξ ( ν 1 ) B ( ν 1 ) * [ - δ ( - ν + ν s + ν 1 ) + δ ( - ν - ν s + ν 1 ) - δ ( ν + ν s + ν 1 ) + δ ( ν - ν s + ν 1 ) ] 2 d ν 1 .
E 2 ( ν )             ~ P ξ ( ν s + ν ) + P ξ ( ν s - ν ) for ν 0 , ν s , ~ 4 P ξ ( ν s ) for ν = 0 , ~ P ξ ( 2 ν s ) for ν = ν s ,
S ( ν ) = i S l i [ δ ( ν - ν i ) + δ ( ν + ν i ) ] + S c for ν a ν ν b , 0 otherwise ,
E 2 ( s ) 2 ( π r d ) 2 ν b 3 P ξ = 1 3 ( 1 Δ ν S l 2 + σ S l 2 d l + 2 S l S c d l + S c 2 ) ( 1 - s a 3 ) + 2 ( S l S c d l + S c 2 ) q ( s , s a ) ,
q ( s , s a ) = 4 3 s 3 - ( 1 + s a 2 ) s + 1 3 ( 1 - s a 2 ) s for 0 s s a , 2 3 s 3 - s + 1 3 ( 1 - 2 s a 3 ) for s a s ( 1 - s a ) / 2 , - 2 3 s 3 + s a 2 s - 1 3 s a 3 for ( 1 - s a ) / 2 s ( 1 + s a ) / 2 , 2 3 s 3 - s + 1 3 for ( 1 + s a ) / 2 s 1.
q ( s , s a ) = 4 3 s 3 - ( 1 + s a 2 ) s + 1 3 ( 1 - s a 3 ) for 0 s ( 1 - s a ) / 2 , 0 for ( 1 - s a ) / 2 s s a , - 2 3 s 3 + s a 2 s - 1 3 s a 3 for s a s ( 1 + s a ) / 2 , 2 3 s 3 - s + 1 3 for ( 1 + s a ) / 2 s 1.
max E rms S ( ν i ) [ 3 2 ( 2 π r d ) 2 ( ν b 3 - ν a 3 ) P ξ ] 1 / 2 = ( Δ ν d l ) 1 / 2 [ 1 + Δ ν d l S c S l / d l ( 2 + S c S l / d l ) ] 1 / 2 ,
K = π σ ξ r d ν m .
r d = n / ( 8 W f , ) ,
2 π r d ν s ξ 0 = 2 π ν s σ ξ / W f ,
J 0 ( 2 π σ ξ ) - 1 - π 2 σ ξ 2 / 2 - 4.9 σ ξ 2 , - J 1 ( 2 π σ ξ ) - π σ ξ / 2 - 2.2 σ ξ , J 2 ( 2 π σ ξ ) π 2 σ ξ 2 / 4 2.5 σ ξ 2 .
σ ξ 2 [ P ˜ ξ ( 0 ) + 2 P ˜ ξ ( 2 ν s ) + 2 P ˜ ξ ( 4 ν s ) + ] ,
E rms ( ν s ) / S ( ν s ) π ( Δ ν / 2 ν s ) 1 / 2 σ ξ = π σ ξ / ( 2 R ) 1 / 2 ,
E rms / S ( ν i ) π σ ξ ( N / R ) 1 / 2 .
E rms S ( ν i ) π σ ξ ( Δ ν d l ν b - ν a ν b ) 1 / 2 ,
E rms / S ( ν i ) ( 2 / 3 ) 1 / 2 π σ ξ ( Δ ν / d l ) 1 / 2 .

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