Abstract

Multiwavelength interferometry (MWI) is a well established technique in the field of optical metrology. Previously, we have reported a theoretical analysis of the method of excess fractions that describes the mutual dependence of unambiguous measurement range, reliability, and the measurement wavelengths. In this paper wavelength, selection strategies are introduced that are built on the theoretical description and maximize the reliability in the calculated fringe order for a given measurement range, number of wavelengths, and level of phase noise. Practical implementation issues for an MWI interferometer are analyzed theoretically. It is shown that dispersion compensation is best implemented by use of reference measurements around absolute zero in the interferometer. Furthermore, the effects of wavelength uncertainty allow the ultimate performance of an MWI interferometer to be estimated.

© 2012 Optical Society of America

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References

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  1. A. A. Michelson and J. R. Benoit, “Détermination expérimentale de la valeur du mètre en longueurs d’ondes lumineuses,” Trav. Et Mem. Bur. Int. Poids es Mes. 11, 1–42 (1895).
  2. R. Benoît, “Application des phénomènes d’interférence à des déterminations métrologiques,” Phys. Radium 7, 57–68 (1898).
  3. M. Born and E. Wolf, Principles of Optics, 7th ed., (Cambridge University, 2006).
  4. J. C. Wyant, “Testing aspherics using two-wavelength holography,” Appl. Opt. 10, 2113–2118 (1971).
    [CrossRef]
  5. C. R. Tilford, “Analytical procedure for determining lengths from fractional fringes,” Appl. Opt. 16, 1857–1860 (1977).
    [CrossRef]
  6. K. Falaggis, D. P. Towers, and C. E. Towers, “Method of excess fractions with application to absolute distance metrology: Theoretical analysis,” Appl. Opt. 50, 5484–5498 (2011).
    [CrossRef]
  7. A. Hurwitz, “Über eine besondere Art der Kettenbruch-Entwicklung reeller Grössen,” Acta Math. Acad. Sci. Hung. 12, 367–405 (1889).
  8. C. E. Towers, D. P. Towers, and J. D. C. Jones, “Optimum frequency selection in multifrequency interferometry,” Opt. Lett. 28, 887–889 (2003).
    [CrossRef]
  9. K. Falaggis, D. P. Towers, and C. E. Towers, “Multiwavelength interferometry: Extended range metrology,” Opt. Lett. 34, 950–952 (2009).
    [CrossRef]
  10. K. Falaggis, D. P. Towers, and C. E. Towers, “Optimum wave-length selection for the method of excess fractions,” Proc. SPIE 7063, 70630V (2008).
    [CrossRef]
  11. K. Falaggis and C. E. Towers, “Absolute metrology by phase and frequency modulation for multiwavelength interferometry,” Opt. Lett. 36, 2928–2930 (2011).
    [CrossRef]

2011 (2)

2009 (1)

2008 (1)

K. Falaggis, D. P. Towers, and C. E. Towers, “Optimum wave-length selection for the method of excess fractions,” Proc. SPIE 7063, 70630V (2008).
[CrossRef]

2003 (1)

1977 (1)

1971 (1)

1898 (1)

R. Benoît, “Application des phénomènes d’interférence à des déterminations métrologiques,” Phys. Radium 7, 57–68 (1898).

1895 (1)

A. A. Michelson and J. R. Benoit, “Détermination expérimentale de la valeur du mètre en longueurs d’ondes lumineuses,” Trav. Et Mem. Bur. Int. Poids es Mes. 11, 1–42 (1895).

1889 (1)

A. Hurwitz, “Über eine besondere Art der Kettenbruch-Entwicklung reeller Grössen,” Acta Math. Acad. Sci. Hung. 12, 367–405 (1889).

Benoit, J. R.

A. A. Michelson and J. R. Benoit, “Détermination expérimentale de la valeur du mètre en longueurs d’ondes lumineuses,” Trav. Et Mem. Bur. Int. Poids es Mes. 11, 1–42 (1895).

Benoît, R.

R. Benoît, “Application des phénomènes d’interférence à des déterminations métrologiques,” Phys. Radium 7, 57–68 (1898).

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed., (Cambridge University, 2006).

Falaggis, K.

Hurwitz, A.

A. Hurwitz, “Über eine besondere Art der Kettenbruch-Entwicklung reeller Grössen,” Acta Math. Acad. Sci. Hung. 12, 367–405 (1889).

Jones, J. D. C.

Michelson, A. A.

A. A. Michelson and J. R. Benoit, “Détermination expérimentale de la valeur du mètre en longueurs d’ondes lumineuses,” Trav. Et Mem. Bur. Int. Poids es Mes. 11, 1–42 (1895).

Tilford, C. R.

Towers, C. E.

Towers, D. P.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed., (Cambridge University, 2006).

Wyant, J. C.

Acta Math. Acad. Sci. Hung. (1)

A. Hurwitz, “Über eine besondere Art der Kettenbruch-Entwicklung reeller Grössen,” Acta Math. Acad. Sci. Hung. 12, 367–405 (1889).

Appl. Opt. (3)

Opt. Lett. (3)

Phys. Radium (1)

R. Benoît, “Application des phénomènes d’interférence à des déterminations métrologiques,” Phys. Radium 7, 57–68 (1898).

Proc. SPIE (1)

K. Falaggis, D. P. Towers, and C. E. Towers, “Optimum wave-length selection for the method of excess fractions,” Proc. SPIE 7063, 70630V (2008).
[CrossRef]

Trav. Et Mem. Bur. Int. Poids es Mes. (1)

A. A. Michelson and J. R. Benoit, “Détermination expérimentale de la valeur du mètre en longueurs d’ondes lumineuses,” Trav. Et Mem. Bur. Int. Poids es Mes. 11, 1–42 (1895).

Other (1)

M. Born and E. Wolf, Principles of Optics, 7th ed., (Cambridge University, 2006).

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Figures (2)

Fig. 1.
Fig. 1.

Absolute value of the residual error of a four wavelength system with sf=4.25, α2=49/50, and β1=1/50 for the case of ε0=ε1=ε2=0. The residual error is shown over a subregion (top) and over the entire unambiguous measurement range (below).

Fig. 2.
Fig. 2.

Minimum value of the residual error in the interval m0=[1,M] for a two wavelength interferometer with sf=23/30π. The individual lower bounds of the residual error are calculated using ri=fract(qi/sf) as r1=0.1696 (q1=2), r2=0.0759 (q2=5), r3=0.0178 (q3=12), r4=0.0049 (q4=53), r5=0.0018 (q5=224). The critical points in between are given as r2r1=0.2456 (q2q1=3), r3r2=0.0937 (q3q2=7), r4r3=0.0227 (q4q3=41), and r5r4=0.0031 (q5q4=171).

Equations (64)

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OPD=u0λ0,OPD=u1λ1,,OPD=uN1λN1,
R(m0)=i=1N1|ri(m0)|2,
ri(m0)={fract[(m0+ε0)sf+E0i](fori=N1)fract[(1αi)(m0+ε0)sf+E0i](fori<N1),
Λ0(N1)=λN1λ0λN1λ0.
σR(N1)(σε0)2+1R2i=1N1(ri(ε0,εi))2(σεi)2.
σRNσε.
δ=1sf1+i=1N2(1αi)2,
ψsf=|1Tsf|,
x0=1sfskqk<1,
S=|sk+1|Q=|qk+1|if{!k||1/V|<6σR|1/T|},
T=pk+1/x0,V=pk+2/x0,W=|qk|forfract(x(k+1))0T=pk+1/x0,V=+,W=0forfract(x(k+1))=0,
sfα=fract[(1αN2)Ssf],sfβ(N3)=fract[β(N3)Sα],sfβ(N2)=fract[β(N2)Sβ(N3)],sfβ1=fract[β1Sβ2],
ψα=1Tα11+(1αN2)2,
ψβn=1Tβn11+(βn)2,
βn=1αn1αn+1,
min{δ,ψsf,ψα(N2),ψβ(N3),,ψβ1}6σR.
R(m0=c1|qk|+c2|qk+1|)=|fract(c1sgn[qk]T+c2sgn[qk+1]V)|,
sgn(x)={+1forx>00forx=01forx<0.
R(m0)=|1/Tsf|(form0<QW),R(m0)=|1/Tsf||1/Vsf|(form0<Q),R(m0)=|1/Vsf|(form0=Q),
UMR=(QsfWsf)λ0.
|1/Tsf||1/Vsf|6σR,
UMR=Qsfλ0.
R(m0=|qk+1||qk|)=|1|T|+1|V|||1T|.
Qsfλ0(QsfWsf)λ0112/(1+5)=2.618,
(1α(N2))=β(N3)==βn==β1=β,
ψα(N2)=ψβ(N3)==ψβ1=ψβ,
δ=1sf1+i=1N2β2n,
1λN1=1λ01sf1λ0,
1λn=1λ0β(N1n)sf1λ0.
1λn=1λ0(1sf)(Nn)1λ0,
R(m0)=|fract[m0sfε0sf+ε0ε1]|,
fract[μ0sf]=fract[ε0sf+ε0ε1].
λ0λ0#ε0ε0#λ1λ1#ε1ε1#,
sfsf#,
fract[μ0sf]fract[ε0#sf+ε0#ε1#].
Δr1(m0=μ0)=|R(m0)|{ε0,ε1}R(m0)|{ε0#,ε1#}|,
Δr1(m0=μ0)=|fract[μ0ε0#sf+ε0#ε1#]|.
Δr1(m0=μ0)=|fract[μ0ε0#sf+μ0#+ε0#sf#]|.
μ0<|sf#sfsf#sf|,
Δr1(m0=μ0#)=|fract[μ0#+ε0#S]|,
S=sf#sfsf#sf.
ΔR(m0=μ0#)=|fract[μ0#+ε0#S]|.
ΔR(m0=μ0#)=i=1N1(fract[μ0#+ε0#S0i])2,
R(m0)=i=1N1|fract[ri(m0)si(m0)]|2,
si(m0)=fract[m0+ε0#Σ],
Σ=sf*sfsf*sf,
1Σ=1S1Θ.
ΔR(m0=μ0#)=i=1N1(μ0#+ε0#Θ0i)2,
δ6σR,
δ6σR+ΔR,
m0Θδ6NσεN1.
εiεi+Δmi+Δεi,
OPD=(mi+εi+Δmi+Δεi)λiOPDref=(miref+εiref+Δmi+Δεi)λi.
OPDOPDref=[(mimiref)+(εiεiref)]λi.
λiλi#λiλi*εiεi#andεiεi*Δmi+ΔεiΔmi#+Δεi#Δmi+ΔεiΔmi*+Δεi*,
OPDOPDref=[(mi#miref#)+(εi#εiref#)]λi#.
OPDλi#OPDrefλi*=OPDλi#(OPDrefλi#OPDrefΩ)=(mi#miref#)+(εi#εiref#)+OPDrefΩ,
Ω=λi*λi#λi*λi#.
(εi#εiref#)=fract[(εi#+Δεi#)(εiref*+Δεiref*)OPDrefΩ].
1Γ=1Ω1Ξ,
(εi#εiref#)=fract[(εi#+Δεi#)(εiref*+Δεiref*)OPDrefΓOPDrefΞ].
OPDrefΩ0.
OPDrefΩ9.82·104,
(εi#εiref#)fract[(εi#+Δεi#)(εiref*+Δεiref*)].

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