Abstract

A theoretical analysis of the measurement accuracy in sinusoidal phase modulating (SPM) interferometry is presented. The measurement accuracy is dependent on multiplicative and additive noise. The characteristics of SPM interferometry in the presence of this noise are made clear. Theoretical results show clearly that SPM interferometry has a high measurement accuracy of the order of 1 nm.

© 1986 Optical Society of America

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References

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  1. O. Sasaki, H. Okazaki, “Sinusoidal Phase Modulating Interferometry for Surface Profile Measurement,” Appl. Opt. 25, (1986), same issue.
    [CrossRef] [PubMed]
  2. R. Crane, “Interference Phase Measurement,” Appl. Opt. 8, 538 (1969).
  3. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital Wavefront Measuring Interferometer for Testing Optical Surfaces and Lenses,” Appl. Opt. 13, 2693 (1974).
    [CrossRef] [PubMed]

1986 (1)

O. Sasaki, H. Okazaki, “Sinusoidal Phase Modulating Interferometry for Surface Profile Measurement,” Appl. Opt. 25, (1986), same issue.
[CrossRef] [PubMed]

1974 (1)

1969 (1)

R. Crane, “Interference Phase Measurement,” Appl. Opt. 8, 538 (1969).

Brangaccio, D. J.

Bruning, J. H.

Crane, R.

R. Crane, “Interference Phase Measurement,” Appl. Opt. 8, 538 (1969).

Gallagher, J. E.

Herriott, D. R.

Okazaki, H.

O. Sasaki, H. Okazaki, “Sinusoidal Phase Modulating Interferometry for Surface Profile Measurement,” Appl. Opt. 25, (1986), same issue.
[CrossRef] [PubMed]

Rosenfeld, D. P.

Sasaki, O.

O. Sasaki, H. Okazaki, “Sinusoidal Phase Modulating Interferometry for Surface Profile Measurement,” Appl. Opt. 25, (1986), same issue.
[CrossRef] [PubMed]

White, A. D.

Appl. Opt. (3)

O. Sasaki, H. Okazaki, “Sinusoidal Phase Modulating Interferometry for Surface Profile Measurement,” Appl. Opt. 25, (1986), same issue.
[CrossRef] [PubMed]

R. Crane, “Interference Phase Measurement,” Appl. Opt. 8, 538 (1969).

J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital Wavefront Measuring Interferometer for Testing Optical Surfaces and Lenses,” Appl. Opt. 13, 2693 (1974).
[CrossRef] [PubMed]

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Figures (13)

Fig. 1
Fig. 1

Standard deviation of X divided by its mean.

Fig. 2
Fig. 2

Standard deviation of phase β.

Fig. 3
Fig. 3

Mean of amplitude |N|.

Fig. 4
Fig. 4

Mean of error ɛz in determination of z.

Fig. 5
Fig. 5

Standard deviation of ɛz in determination of z.

Fig. 6
Fig. 6

Mean of α calculated by numerical integration.

Fig. 7
Fig. 7

Standard deviation of α calculated by numerical integration.

Fig. 8
Fig. 8

Mean of α obtained from the approximate equations.

Fig. 9
Fig. 9

Standard deviation of α obtained from the approximate equations.

Fig. 10
Fig. 10

Standard deviation of α obtained from the approximate equations for various values of z.

Fig. 11
Fig. 11

Mean of α at ρA = 10 for various values of σz.

Fig. 12
Fig. 12

Mean of α at ρA = 100 for various values of σz

Fig. 13
Fig. 13

Standard deviation of α at ρA = 10 and 100 for various values of σz.

Equations (89)

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s ( t ) = s 0 cos [ z cos ( ω c t + θ ) + α + n M ( t ) ] + n A ( t ) .
ρ M = ( z 2 / 2 ) / σ M 2 ,             ρ A = ( s 0 2 / 2 ) / σ A 2 .
N M ( l ) = ( 1 / M ) m = 0 M - 1 exp [ j n M ( m Δ t ) ] exp ( - j 2 π m l / M ) , N A ( l ) = ( 1 / M ) m = 0 M - 1 n A ( m Δ t ) exp ( - j 2 π m l / M ) ,
Δ t = ( 2 π / ω c ) ( 1 / K ) ,
N M ( 0 ) = N exp ( j β ) ,             N A ( m M / K ) = A m exp ( j ϕ m ) ,
F ( 2 m ω c ) = ( - 1 ) m N s 0 cos ( α + β ) J 2 m ( z ) exp ( j 2 m θ ) + A 2 m exp ( j ϕ 2 m ) , F [ ( 2 m - 1 ) ω c ] = ( - 1 ) m N s 0 × sin ( α + β ) J 2 m - 1 ( z ) exp [ j ( 2 m - 1 ) θ ] + A 2 m - 1 exp ( j ϕ 2 m - 1 )             m = 0 , ± 1 , ± 2 , .
N M ( 0 ) = ( 1 / M ) m = 0 M - 1 ( cos n m + j sin n m ) = X + j Y ,
p ( n 0 , , n M - 1 ) = p ( - n 0 , , - n M - 1 ) .
β = tan - 1 ( X / Y ) = - - tan - 1 ( sin n 0 + + sin n M - 1 cos n 0 + + cos n M - 1 ) × p ( n 0 , p M - 1 ) d n 0 d M - 1 .
β = - - - tan - 1 ( sin n 0 + + sin n M - 1 cos n 0 + + cos n M - 1 ) × p ( n 0 , , n M - 1 ) d n 0 d n M - 1 = - β .
β = 0.
X = exp ( - σ M 2 / 2 ) ,
X 2 = ( 2 / M ) exp ( - σ M 2 ) k = 0 M - 1 [ 1 - ( k / M ) ] cosh [ C ( k Δ t ) ] - ( 1 / 2 M ) [ 1 + exp ( - σ M 2 ) ] ,
Y = 0 ,
Y 2 = ( 2 / M ) exp ( - σ M 2 ) k = 0 M - 1 [ 1 - ( k / M ) ] sinh [ C ( k Δ t ) ] + ( 1 / 2 M ) [ 1 - exp ( - σ M 2 ) ] ,
C ( τ ) = σ M 2 exp ( - 2 π f 0 τ ) ,
var { X } 1 / 2 / X 1 ,
X X .
β 2 Y 2 / X 2 .
β 2 Y 2 / X 2 .
N = ( X 2 + Y 2 ) 1 / 2 = X + ( 1 / 2 ) ( Y 2 / X ) - ( 1 / 8 ) ( Y 4 / X 3 ) + .
N X + ( 1 / 2 ) ( Y 2 / X ) .
z ˜ = z + ɛ z ,
F ( 3 ω c ) / F ( ω c ) = J 3 ( z ˜ ) / J 1 ( z ˜ ) = γ [ J 3 ( z ) / J 1 ( z ) ] .
γ = [ J 1 ( z ) / J 3 ( z ) ] s 0 J 3 ( z ) sin α exp ( j 3 θ ) + A 3 exp ( j ϕ 3 ) - s 0 J 1 ( z ) sin α exp ( j θ ) + A 1 exp ( j ϕ 1 ) .
J 3 ( z ˜ ) / J 1 ( z ˜ ) [ J 3 ( z ) / J 1 ( z ) ] ( 1 + P 31 ɛ z + Q 31 ɛ z 2 ) ,
P 31 = D 3 - D 1 , Q 31 = ( 1 / 2 ) ( E 3 - E 1 ) - D 3 D 1 + D 1 2 , D k = [ J k ( z ) / z ] / J z ( z ) , E k = [ 2 J k ( z ) / 2 z ] / J k ( z ) .
γ 1 + P 31 ɛ z + Q 31 ɛ z 2 .
γ 1 + P 31 ɛ z + Q 31 z 2 ,
γ 2 1 + 2 P 31 ɛ z + ( P 31 2 + 2 Q 31 ) ɛ z 2
ɛ z = ( 1 / P 31 3 ) [ ( P 31 2 + 2 Q 31 ) ( γ - 1 ) - Q 31 ( γ 2 - 1 ) ]
ɛ z 2 = ( 1 / P 31 2 ) [ - 2 ( γ - 1 ) + ( γ 2 - 1 ) ] .
P ( A m ) = ( 1 / Ψ 0 ) A m exp ( - A m 2 / 2 Ψ 0 ) ,
p ( ϕ m ) = 1 / 2 π             - π ϕ m π ,
α ˜ = α + ɛ α ,
tan α ˜ = - F ( ω c ) / J 1 ( z ) sgn { - Re [ F ( ω c ) ] cos θ } F ( 2 ω c ) / J 2 ( z ) sgn { - Re [ ( 2 ω c ) ] cos 2 θ } ,
F ( ω c ) = - s 0 J 1 ( z ) sin α exp ( j θ ) + A 1 exp ( j ϕ 1 ) , F ( 2 ω c ) = - s 0 J 2 ( z ) cos α exp ( j 2 θ ) + A 2 exp ( j ϕ 2 ) .
η t = tan ( α + ɛ α ) = a 0 + a 1 ɛ α + a 2 ɛ α 2 + ,
η t a 0 + a 1 ɛ α + a 2 ɛ α 2 ,
η t 2 a 0 2 + 2 a 0 a 1 ɛ α + ( a 1 2 + 2 a 0 a 1 ) ɛ α 2 .
ɛ α = ( 1 / a 1 3 ) [ ( a 1 2 + 2 a 0 a 2 ) ( η t - a 0 ) - a 2 ( η t 2 - a 0 2 ) ] ,
ɛ α 2 = ( 1 / a 1 2 ) [ - 2 a 0 ( η t - a 0 ) + ( η t 2 - a 0 2 ) ] .
η t = s 0 J 1 ( z ) sin α + A 1 exp ( j ϕ 1 ) / J 1 ( z ) · sgn { s 0 J 1 ( z ) sin α + A 1 cos ϕ 1 } - s 0 J 2 ( z ) cos α + A 2 exp ( j ϕ 2 ) / J 2 ( z ) · sgn { s 0 J 2 ( z ) cos α - A 2 cos ϕ 2 } .
C 1 = s 0 J 1 ( z ) sin α A 1 ,             C 2 = s 0 J 2 ( z ) cos α A 2 .
η t = tan α ( 1 + B 1 2 + 2 B 1 cos ϕ 1 ) 1 / 2 ( 1 + B 2 2 - 2 B 2 cos ϕ 2 ) 1 / 2 ,
η t = tan α [ 1 + ( 1 / 2 ) B 1 2 + B 1 cos ϕ 1 - ( 1 / 2 ) B 1 2 cos 2 ϕ 1 - ] × [ 1 - ( 1 / 2 ) B 2 2 + B 2 cos ϕ 2 + ( 3 / 2 ) B 2 2 cos 2 ϕ 2 + ] .
η t tan α [ 1 + ( 1 / 4 ) A 1 2 / C 1 2 ] [ 1 + ( 1 / 4 ) A 2 2 / C 2 2 ] .
η t 2 tan 2 α [ 1 + A 1 2 / C 1 2 ] [ 1 + A 2 2 / C 2 2 ] .
ɛ α = ( 1 / 8 M ρ A ) ( 1 - 2 sin 2 α ) [ ( cot α / J 1 2 ) + ( tan α / J 2 2 ) ] ,
ɛ α 2 = ( 1 / 4 M ρ A ) [ ( cos 2 α / J 1 2 ) + ( sin 2 α / J 2 2 ) ] .
η c = cot ( α + ɛ α ) = b 0 + b 1 ɛ α + b 2 ɛ α + ,
η c = 1 / η t .
ɛ α = ( 1 / b 1 3 ) [ ( b 1 2 + 2 b 0 b 2 ) ( η c - b 0 ) - b 2 ( η c 2 - b 0 2 ) ] ,
ɛ α 2 = ( 1 / b 1 2 ) [ - 2 b 0 ( η c - b 0 ) + ( η c 2 - b 0 2 ) ] .
α = - ( 1 / 8 M ρ A ) ( 1 - 2 cos 2 α ) [ cot α / J 1 2 ) + ( tan α / J 2 2 ) ] ,
α 2 = ( 1 / 4 M ρ A ) [ ( cos 2 α / J 1 2 ) + ( sin 2 α / J 2 2 ) ] .
z ^ = z + Δ z ,
tan ( α + α ) = F ( ω c ) / J 1 ( z ^ ) · sgn { Re [ F ( ω c ) ] } F ( 2 ω c ) / J 2 ( z ^ ) · sgn { Re [ - F ( 2 ω c ) ] } = η t ,
J 2 ( z ^ ) / J 1 ( z ^ ) [ J 2 ( z ) / J 1 ( z ) ] ( 1 + P 21 Δ z + Q 21 Δ z 2 ) .
η t ( 1 + P 21 Δ z + Q 21 Δ z 2 ) η t .
η t ( 1 + Q 21 σ z 2 ) η t ,
η t 2 [ 1 + ( P 21 2 + 2 Q 21 ) σ z 2 ] η t 2 .
α = α + σ z 2 { cos α sin α ( Q 21 - P 21 2 sin 2 α ) + ( 1 / 8 M ρ A ) × [ Q 21 - ( 6 Q 21 + 4 P 21 2 ) sin 2 α ] × [ ( cot α / J 1 2 ) + ( tan α / 2 2 ) ] } ,
α 2 = α 2 + σ z 2 { sin 2 α cos 2 α P 21 2 + ( 1 / 4 M ρ A ) ( 2 P 21 2 + 3 Q 21 ) × [ ( cos 2 α / J 1 2 ) + ( sin 2 α / J 2 2 ) ] } .
cot α = 1 / η t = η c ,
J 1 ( z ^ ) / J 2 ( z ^ ) [ J 1 ( z ) / J 2 ( z ) ] ( 1 + P 12 Δ z + Q 12 Δ z 2 ) .
η c ( 1 + Q 12 σ z 2 ) η c , η c 2 [ 1 + ( P 12 2 + 2 Q 12 ) σ z 2 ] η c 2 .
α = α - σ z 2 { sin α cos α ( Q 12 - P 12 2 sin 2 α ) + ( 1 / 8 M ρ A ) × [ Q 21 - ( 6 Q 12 + 4 P 12 2 ) sin 2 α ] × [ ( cot α / J 1 2 ) + ( tan α / J 2 2 ) ] } ,
α 2 = α 2 + σ z 2 { sin 2 α cos 2 α P 12 2 + ( 1 / 4 M ρ A ) ( 2 P 12 2 + 3 Q 12 ) × [ ( cos 2 α / J 1 2 ) + ( sin 2 α / J 2 2 ) ] } .
ϕ ( u , n p ) = exp ( j u n p ) = exp ( - u 2 σ M 2 / 2 ) .
N M ( 0 ) = exp ( - σ m 2 / 2 ) .
X = exp ( - σ M 2 / 2 ) ,
Y = 0.
ϕ ( u 1 , u 2 ; n p , n q ) = exp [ j ( u 1 , n p + u 2 , n q ) ] = exp [ - ( 1 / 2 ) ( R 11 u 1 2 + 2 R 12 u 1 u 2 + R 22 u 2 2 ) ] ,
X 2 + Y 2 = ( 1 / M 2 ) p = 1 M - 1 q = 1 M - 1 exp [ j ( n p - n q ) ] = ( 1 / M 2 ) p = 1 M - 1 q = 1 M - 1 exp ( - σ M 2 ) exp [ C ( p - q ) Δ t ] .
X 2 = ( 1 / M 2 ) p = 1 M - 1 q = 1 M - 1 ( 1 / 4 ) [ exp ( j n p ) + exp ( - j n p ) ] × [ exp ( j n q ) + exp ( - j n q ) ] .
X 2 = ( 1 / M 2 ) p = 1 M - 1 q = 1 M - 1 exp ( - σ M 2 ) cosh [ C ( p - q ) Δ t ] .
Y 2 = X 2 + Y 2 - X 2 .
s ( m Δ t ) = s 0 cos [ ω c m Δ t + α + n M ( m Δ t ) ] + n A ( m Δ t ) m = 0 , 1 , , M - 1.
F ( ω c ) = ( s 0 / 2 ) N exp [ j ( α + β ) ] + A 1 exp ( j ϕ 1 ) .
tan α ˜ = ( s 0 / 2 ) sin α + A 1 sin ϕ 1 ( s 0 / 2 ) cos α + A 1 cos ϕ 1 = η t .
η t = tan α 1 + ( 2 A 1 / s 0 ) ( sin ϕ 1 / sin α ) 1 + ( 2 A 1 / s 0 ) ( cos ϕ 1 / cos α ) .
η t = tan α [ 1 + ( 1 / M ρ A cos 2 α ) ] .
η t 2 = tan 2 α [ 1 + ( 3 / M ρ A cos 2 α ) + ( 1 / M ρ A sin 2 α ) ] .
α = 0 ,
α 2 = 1 / M ρ A .
cot α ˜ = ( s 0 / 2 ) cos α + A 1 cos ϕ 1 ( s 0 / 2 ) sin α + A 1 sin ϕ 1 = η c .
η c = cot α [ 1 + ( 1 / M ρ A sin 2 α ) ] ,
η c 2 = cot 2 α [ 1 + ( 3 / M ρ A sin 2 α ) + ( 1 / M ρ A cos 2 α ) ] .

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