Abstract

An approximate formula is derived for the spectrum ghosts caused by periodic drive speed variations in a Michelson interferometer. The solution represents the case of fringe-controlled sampling and is applicable when the reference fringes are delayed to compensate for the delay introduced by the electrical filter in the signal channel. Numerical results are worked out for several common low-pass filters. It is shown that the maximum relative ghost amplitude over the range of frequencies corresponding to the lower half of the filter band is typically 20 times smaller than the relative zero-to-peak velocity error, when delayed sampling is used. In the lowest quarter of the filter band it is more than 100 times smaller than the relative velocity error. These values are ten and forty times smaller, respectively, than they would be without delay compensation if the filter is a 6-pole Butterworth.

© 1979 Optical Society of America

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References

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  1. T. Nishiyama, T. Yamauchi, M. Ohno, M. Morii, N. Ura, K. Masutani, Jpn. J. Appl. Phys. 14, Suppl. 14-1, 67 (1975).
  2. A. S. Zachor, Appl. Opt. 16, 1412 (1977).
    [CrossRef] [PubMed]
  3. L. Mertz, Transformations in Optics (Wiley, New York, 1965), p. 22.
  4. Ref. 3, pp. 22–26.
  5. L. Weinberg, Network Analysis and Synthesis (McGraw-Hill, New York, 1962).
  6. W. G. Mankin, Opt. Eng. 17, 39 (1978).
    [CrossRef]

1978 (1)

W. G. Mankin, Opt. Eng. 17, 39 (1978).
[CrossRef]

1977 (1)

1975 (1)

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

Mankin, W. G.

W. G. Mankin, Opt. Eng. 17, 39 (1978).
[CrossRef]

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. 22.

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).

Ura, N.

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

Weinberg, L.

L. Weinberg, Network Analysis and Synthesis (McGraw-Hill, New York, 1962).

Yamauchi, T.

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

Zachor, A. S.

Appl. Opt. (1)

Jpn. J. Appl. Phys. (1)

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

Opt. Eng. (1)

W. G. Mankin, Opt. Eng. 17, 39 (1978).
[CrossRef]

Other (3)

L. Mertz, Transformations in Optics (Wiley, New York, 1965), p. 22.

Ref. 3, pp. 22–26.

L. Weinberg, Network Analysis and Synthesis (McGraw-Hill, New York, 1962).

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Figures (9)

Fig. 1
Fig. 1

Schematic representation of wavenumber and temporal frequency spectra of the interferogram and the effects of filter amplitude distortion. The normalization factor for temporal frequency is the average retardation velocity 〈dr/dt〉.

Fig. 2
Fig. 2

Locations of the ghosts represented in the first five terms in the Taylor expansion of the spectrum error [from Eq. (6)]. The ghosts occur at frequencies corresponding to integer values of m = (νν s )/ν ξ , where ν s is the frequency of the monochromatic source, and ν ξ is the frequency of the periodic velocity error.

Fig. 3
Fig. 3

Relative ghost amplitude corresponding to a sinusoidal velocity error of relative amplitude ξ0 = 0.01 for the case of an RC filter. Curve U is the error when conventional fringe-reference sampling is used. Curve T is the error for delayed fringe-reference sampling, equal to the quadrature sum of components A and P due to amplitude and phase distortion.

Fig. 4
Fig. 4

Same as Fig. 3, except the filter is a 2-pole Butterworth. P is zero at ν s /ν c = 1, where β′ changes sign.

Fig. 5
Fig. 5

Same as Fig. 3, except the filter is a 3-pole Butterworth.

Fig. 6
Fig. 6

Same as Fig. 3, except the filter is a 6-pole Butterworth.

Fig. 7
Fig. 7

Ghost amplitude for a 6-pole Butterworth filter. Curve a is the same as curve T of Fig. 6. Curves b and c are the corresponding results when the sampling delay equals the filter group delay at ν = 0.5ν c and ν = 0.6ν c , respectively.

Fig. 8
Fig. 8

Same as Fig. 3, except the filter is a 3-pole Bessel.

Fig. 9
Fig. 9

Comparison of the relative ghost amplitudes for different filters for the delayed sampling case. Except for the 3-pole Chebyshev, these are the same as curves T in the previous figures. The gain ripple for the Chebyshev is 0.5 dB.

Tables (1)

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Table I Sampling Delays Used in Calculating Filter Phase Distortion

Equations (25)

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i J 1 ( ξ 0 ν s / ν ξ ) { 1 - exp [ i ν ξ β ( ν s ) ] } ± ξ 0 ν s β ( ν s ) / 2 ,
i [ ξ 0 ν s F ( ν s ) / 2 ] exp [ - i β ( ν s ) ] ,
± [ ξ 0 ν s β ( ν s ) / 2 ] F ( ν s ) exp [ - i β ( ν s ) ]
ν s ξ 0 2 { [ β ( ν s ) ] 2 + [ F ( ν s ) F ( ν s ) ] 2 } 1 / 2 .
( r ) = n = 1 ( - 1 ) n [ ξ n ( r ) / n ! ] [ r n f ( r ) ] * [ g ( n ) ( r ) ] ,
ξ ( r ) = ξ 0 sin ( 2 π ν ξ r - ψ 0 ) ,
g ( r ) = 2 cos ( 2 π ν s r ) δ ( ν - ν s ) + δ ( ν + ν s ) ,
( r ) E ( ν ) = n = 1 { [ δ ( ν - ν s ) + δ ( ν + ν s ) ] [ ( i ν ξ 0 / 2 ) n / n ! ] F ( n ) ( ν ) } * ( exp [ - i ( ν / ν ξ ) ψ 0 ] { [ δ ( ν - ν ξ ) - δ ( ν + ν ξ ) ] * n - 1 [ δ ( ν - ν ξ ) - δ ( ν + ν ξ ) ] } ) ,
E ( ν ) = n = 1 [ i ν s ξ 0 / 2 ) n / n ! ] F ( n ) ( ν s ) D n ( ν ) ,
D n ( ν ) = ( - 1 ) ( n - m ) / 2 exp ( - i m ψ 0 ) ( n n + m 2 ) δ ( ν - ν s - m ν ξ ) ;
m = n , n - 2 , n - 4 , , 0 ( even n ) 1 ( odd n ) .
E m ( ν ) = exp ( - i m ψ o ) j = k ( - 1 ) j ( i ν s ξ 0 / 2 ) m + 2 j j ! ( m + j ) ! × F ( ν s ) ( m + 2 j ) δ ( ν - ν s - m ν ζ ) ;             k = 1 if m = 0 ; k = 0 m 0.
F ( ν ) = exp ( i 2 π r d ν ) F ( ν ) exp [ - i ϕ ( ν ) ] = F ( ν ) exp [ - i β ( ν ) ] ,
F ( n ) [ ( - i β ) n F + n ( - i β ) n - 1 F ] exp ( - i β )
E m ( ν ) = exp ( - i m ψ 0 ) F ( ν s ) exp [ - i β ( ν s ) ] δ ( ν - ν s - m ν ξ ) × { J m ( x ) + i x F ( ν s ) 2 F ( ν s ) β ( ν s ) [ J m - 1 ( x ) - J m + 1 ( x ) ] } ,             m = 1 , 2 , ,
E 0 ( ν ) = F ( ν s ) exp [ - i β ( ν s ) ] δ ( ν - ν s ) [ J 0 ( x ) - 1 - i x F ( ν s ) F ( ν s ) β ( ν s ) J 1 ( x ) ] ,
E ˜ 1 ( ν s ) = exp ( - i ψ 0 ) ν s ξ 0 2 [ β ( ν s ) + i F ( ν s ) F ( ν s ) ( 1 - 3 8 x 2 ) ] .
E ˜ 1 ( ν s ) = ν s ξ 0 2 { [ β ( ν s ) ] 2 + [ F ( ν s ) F ( ν s ) ] 2 ( 1 - 3 8 x 2 ) 2 } 1 / 2 .
Re [ E ˜ 1 ( ν s ) ] = E ˜ 1 ( ν s ) cos ( ψ 0 - ϕ D ) ; ϕ D = tan - 1 { F ( ν s ) / [ F ( ν s ) β ( ν s ) ] } .
E ˜ 1 ( ν s ) = ( J 1 2 ( x ) + { ν s ξ 0 F ( ν s ) 2 F ( ν s ) [ J 0 ( x ) - J 2 ( x ) ] } 2 ) 1 / 2
F Bu ( ν ) = k = 0 n - 1 [ i ν / ν c - exp [ i ( 2 k + 1 + n ) π / ( 2 n ) ] ] - 1
F Be ( ν ) = [ k = 0 n ( 2 n - k ) ! ( i α n ν / ν c ) k 2 n - k k ! ( n - k ) ! ] - 1
β ( ν ) = ϕ ( ν ) - ν ϕ ( 0 ) .
r d = ϕ ( 0 ) / 2 π
t d = ϕ ( 0 ) / 2 π d r / d t ) ,

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