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

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. K. Hibino, B. Oreb, D. Farrant, K. Larkin, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. A 14, 918–930 (1997).
    [CrossRef]
  2. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35, 51–60 (1996).
    [CrossRef] [PubMed]
  3. P. de Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of data-sampling window,” Appl. Opt. 34, 4723–4730 (1995).
    [CrossRef]
  4. Y. Surrel, “Additive noise effect in digital phase detection,” Appl. Opt. 36, 271–276 (1997).
    [CrossRef] [PubMed]

1997 (2)

1996 (1)

1995 (1)

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
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
Fig. 2

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

Fig. 3
Fig. 3

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

Equations (59)

Equations on this page are rendered with MathJax. Learn more.

δk=kδ(1+1+k2+k23+).
Sm(ϕ)=αmk=0M-1ck exp[im(ϕ+δk)]=αm exp(imϕ)k=0M-1ck exp(imδk).
k=0M-1ck exp(imδk)
=k=0M-1ck exp[imkδ(1+1+k2+k23+)]=k=0M-1ck[exp(iδ)m]k1+imkδ1+imk2δ2+imk3δ3++(imkδ1)22!+(imk2δ2)22!+(imk3δ3)22!++2(imkδ1)(imk2δ2)2!+2(imkδ1)(imk3δ3)2!+2(imk2δ2)(imk3δ3)2!+.
Sm(ϕ)=αm exp(imϕ)P(ζm)+imδ1DP(ζm)+imδ2D2P(ζm)+imδ3D3P(ζm)++(imδ1)22!D2P(ζm)+(imδ2)22!D4P(ζm)+(imδ3)22!D6P(ζm)++2(imδ1)(imδ2)2!D3P(ζm)+2(imδ1)(imδ3)2!D4P(ζm)+2(imδ2)(imδ3)2!D5P(ζm)+.
PN(x)=1+ζ-1x+ζ-2x2++ζ-(N-1)xN-1=ζl=0,l1N-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.
ϕ=arctan14(I0+I1-IN-IN+1) sin(3π/N)sin2(2π/N)-r=1NIr sin[2π(r-12)/N]14(-I0+I1+IN-IN+1) cos(3π/N)sin2(2π/N)+r=1NIr cos[2π(r-12)/N].
S(ϕ)=S1(ϕ)+S-1(ϕ)
=exp(iϕ)P(ζ)1+iδ1 DP(ζ)P(ζ)+iδ2 D2P(ζ)P(ζ)+exp(-2iϕ)-iδ1 DP(ζ-1)P(ζ)-iδ2 D2P(ζ-1)P(ζ),
ϕ*=ϕ+arg[P(ζ)]+Δϕ,
Δϕ=δ1RDP(ζ)P(ζ)+2RD2P(ζ)P(ζ)+1 DP(ζ-1)P(ζ)+2 D2P(ζ-1)P(ζ)sin(2ϕ+ψ),
D(P1P2Pn)P1P2Pn=k DPkPk,
D2(P1P2Pn)P1P2Pn=k D2PkPk+2k<l DPkPkDPlPl.
P(x)=xp  DnP(x)P(x)=pn,
P(x)=x-ξ  n,DnP(x)P(x)=xx-ξ.
ζζ-ξ=11-exp(iψ)=12[1+i cotan(ψ/2)].
RDP(ζ)P(ζ)=0.
δk-p=(k-p)δ(1+1+(k-p)2+(k-p)23+).
DQ(ζ)Q(ζ)=DP(ζ)P(ζ)-p,
p=RDP(ζ)P(ζ).
P(x)=αr=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)
ξ=ζ exp(iψ).
RDQ(ζ)Q(ζ)=RDR(ζ)R(ζ)+12=0.
D2Q(ζ)Q(ζ)=D2R(ζ)R(ζ)+2 DR(ζ)R(ζ)12[1+i cotan(ψ/2)]+12[1+i cotan(ψ/2)],
RD2Q(ζ)Q(ζ)=RD2R(ζ)R(ζ)+2-12 12-2IDR(ζ)R(ζ) cotan(ψ/2)2+12=0,
tan(ψ/2)=I[DR(ζ)/R(ζ)]R[D2R(ζ)/R(ζ)].
A=RD2R(ζ)R(ζ)+iIDR(ζ)R(ζ),
ξ=ζ A2|A|2.
DP(ζ)P(ζ)=2-i3,
D2P(ζ)P(ζ)=3-4i3,
DR(ζ)R(ζ)=DP(ζ)P(ζ)-p=-12-i3,
D2R(ζ)R(ζ)=D2P(ζ)P(ζ)-2p DP(ζ)P(ζ)+p2=-34+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),
-δ212ID3Q(ζ)Q(ζ)=-δ212I-265i396,
-δ2222D4Q(ζ-1)Q(ζ)-δ212 D3Q(ζ-1)Q(ζ)exp(-2iϕ)
=-δ222-7i3121exp(-2iϕ),
δϕ=π2292653961+2 sin(2ϕ)+73121 cos(2ϕ),
DP(ζ)P(ζ)=72+i,
D2P(ζ)P(ζ)=14+7i,
DR(ζ)R(ζ)=DP(ζ)P(ζ)-p=-12+i,
D2R(ζ)R(ζ)=D2P(ζ)P(ζ)-2p DP(ζ)P(ζ)+p2=2-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-1r=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)]*.
(x*)M-1P1x*=cM-1r=1M-1(1-x*ξr)=(-1)M-1 cM-1cM-1*r=1M-1ξr×cM-1*r=1M-1(x*-ξr*)=(-1)M-1 cM-1cM-1*r=1M-1ξr[P(x)]*.
(-1)M-1 cM-1cM-1*r=1M-1ξr=1.
exp(-2iψ)=(-1)M-1r=1M-1ξr.
cM-1=a1+(-1)M-1r=1M-1ξr,

Metrics