## Abstract

In the framework of phase-shifting interferometry, the characteristic polynomial theory is extended to deal with nonlinear and nonuniform phase-shift miscalibration. A general procedure for designing algorithms that are insensitive to those errors is presented. It is also shown how analytical expressions for the residual phase errors can be obtained. Finally, it is demonstrated that the coefficients of any algorithm can be given a Hermitian symmetry.

© 1998 Optical Society of America

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### Figures (3)

Fig. 1

Characteristic diagram canceling the harmonic $m=-1$ in the presence of a quadratic miscalibration. The diagram is shown for the special case of $N=6.$

Fig. 2

Characteristic diagram of the six-sample algorithm described by Eq. (36).

Fig. 3

Characteristic diagram of the nine-sample algorithm described by Eq. (42).

### Equations (59)

$δk=kδ(1+∊1+k∊2+k2∊3+…).$
$PN(x)=1+ζ-1x+ζ-2x2+…+ζ-(N-1)xN-1=ζ∏l=0,l≠1N-1(x-ζl)$
$P(x)=ζ1/2PN(x)x-ζ-1ζ-ζ-12.$
$P(x)=ζ-3/2-ζ3/2x-ζ-3/2xN+ζ3/2xN+1(ζ-ζ-1)2+∑r=1Nζ-(r-1/2)xr.$
$S(ϕ)=S1(ϕ)+S-1(ϕ)$
$ϕ*=ϕ+arg[P(ζ)]+Δϕ,$
$RDP(ζ)P(ζ)=0.$
$δk-p=(k-p)δ(1+∊1+(k-p)∊2+(k-p)2∊3+…).$
$DQ(ζ)Q(ζ)=DP(ζ)P(ζ)-p,$
$p=RDP(ζ)P(ζ).$
$P(x)=α∏r=1M-1(x-ξr),$
$RDP(ζ)P(ζ)=M-12.$
$ϕ=arctanI0-I6-7(I2-I4)4(I1+I5)-8I3.$
$RDP(ζ)P(ζ)=0,$
$RD2P(ζ)P(ζ)=0.$
$Q(x)=[P(x)xp](x-ξ)=R(x)(x-ξ)$
$RDQ(ζ)Q(ζ)=RDR(ζ)R(ζ)+12=0.$
$tan(ψ/2)=I[DR(ζ)/R(ζ)]R[D2R(ζ)/R(ζ)].$
$A=RD2R(ζ)R(ζ)+iIDR(ζ)R(ζ),$
$DP(ζ)P(ζ)=2-i3,$
$D2P(ζ)P(ζ)=3-4i3,$
$DR(ζ)R(ζ)=DP(ζ)P(ζ)-p=-12-i3,$
$Q(x)x-5/2=(1-5i3)(x-1)(x-ζ-1)3×x+3738+5i338,$
$ϕ=arctan×3[-5(I0-I5)+6(I1-I4)+17(I2-I3)]I0+I5-26(I1+I4)+25(I2+I3),$
$-δ2∊1∊2ID3Q(ζ)Q(ζ)=-δ2∊1∊2I-265i396,$
$=-δ2∊2∊2-7i312∊1exp(-2iϕ),$
$DP(ζ)P(ζ)=72+i,$
$D2P(ζ)P(ζ)=14+7i,$
$DR(ζ)R(ζ)=DP(ζ)P(ζ)-p=-12+i,$
$Q(x)xp=-1+i2(x-1)(x+i)3(x+1)3×x+45-3i5,$
$ϕ=-arctan(1/2)(I0-I8)-(I1-I7)-7(I2-I6)-9(I3-I5)-(I0+I8)-4(I1+I7)-4(I2+I6)+4(I3+I5)+10I4.$
$P(x)=∑k=0M-1ckxk=cM-1∏r=1M-1(x-ξr)$
$ck*=cM-k-1,$
$(x*)M-1P1x*=∑k=0M-1ck(x*)M-k-1=∑k=0M-1cM-k-1(x*)k,$
$(x*)M-1P1x*=[P(x)]*.$
$exp(-2iψ)=(-1)M-1∏r=1M-1ξr.$
$cM-1=a1+(-1)M-1∏r=1M-1ξr,$