Abstract

It is a well-known fact that one grating can act as an imaging element for another grating when the first is illuminated with an extended monochromatic light source. The conditions for image formation in such a system are studied when the finite size and position of the broad light source are considered. From the presented analysis, expressions for the location and the depth of focus of such images can be derived.

© 2000 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  8. A. Olszak, L. Wronkowski, “Analysis of the Fresnel field of a double diffraction system in the case of two amplitude diffraction gratings under partially coherent illumination,” Opt. Eng. 36, 2149–2157 (1997).
    [CrossRef]
  9. E. Keren, O. Kafri, “Diffraction effects in moiré deflectometry,” J. Opt. Soc. Am. 2, 111–120 (1985).
    [CrossRef]
  10. G. J. Swanson, E. N. Leith, “Lau effect and grating imaging,” J. Opt. Soc. Am. 72, 552–555 (1982).
    [CrossRef]

1997

A. Olszak, L. Wronkowski, “Analysis of the Fresnel field of a double diffraction system in the case of two amplitude diffraction gratings under partially coherent illumination,” Opt. Eng. 36, 2149–2157 (1997).
[CrossRef]

1989

1985

1982

1979

F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1979).
[CrossRef]

1948

E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. Phys. (Leipzig) 6, 417–423 (1948).
[CrossRef]

1836

F. Talbot, “Facts relating to optical science. No. IV,” Philos. Mag. 9, 401–407 (1836).

Bar-Ziv, E.

Gori, F.

F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1979).
[CrossRef]

Hershey, R.

Kafri, O.

Keren, E.

Lau, E.

E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. Phys. (Leipzig) 6, 417–423 (1948).
[CrossRef]

Leith, E. N.

Liu, L.

Olszak, A.

A. Olszak, L. Wronkowski, “Analysis of the Fresnel field of a double diffraction system in the case of two amplitude diffraction gratings under partially coherent illumination,” Opt. Eng. 36, 2149–2157 (1997).
[CrossRef]

Swanson, G. J.

Talbot, F.

F. Talbot, “Facts relating to optical science. No. IV,” Philos. Mag. 9, 401–407 (1836).

Wronkowski, L.

A. Olszak, L. Wronkowski, “Analysis of the Fresnel field of a double diffraction system in the case of two amplitude diffraction gratings under partially coherent illumination,” Opt. Eng. 36, 2149–2157 (1997).
[CrossRef]

Ann. Phys. (Leipzig)

E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. Phys. (Leipzig) 6, 417–423 (1948).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1979).
[CrossRef]

Opt. Eng.

A. Olszak, L. Wronkowski, “Analysis of the Fresnel field of a double diffraction system in the case of two amplitude diffraction gratings under partially coherent illumination,” Opt. Eng. 36, 2149–2157 (1997).
[CrossRef]

Philos. Mag.

F. Talbot, “Facts relating to optical science. No. IV,” Philos. Mag. 9, 401–407 (1836).

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Figures (3)

Fig. 1
Fig. 1

Optical system used to obtain pseudoimages of the first grating G1 on the observation plane P.

Fig. 2
Fig. 2

Maps of the modulation of the pseudoimages as a function of Z1 (horizontal axis) and Z2 (vertical axis): (a) images of order (1, -1), (b) images of order (1, -2), (c) images of order (1, -3).

Fig. 3
Fig. 3

Angle subtended on any point of the observation plane by the light source, a single period of the first grating, and a single period of the second grating. The first of these angles must be much greater than the other two in order to obtain clearly visible self-images.

Equations (80)

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Ψ(x, xi)=ϕ(x-xi),
t1(x)=nan exp(inq1x),
f1(x, xi)=t1(x)ϕ(x-xi)=nan exp(inq1x)ϕ(x-xi).
F1(kx, xi)=FT[f1(x, xi)]
=nan exp[-i(kx-nq1)xi]×Φ(kx-nq1),
f2(x, xi)=nan- expikxx-kx2 z12k×exp[-i(kx-nq1)xi]Φ(kx-nq1)dkx.
t2(x)=mbm exp(imq2x).
f3(x, xi)=f2(x, xi)t2(x)=nmanbm exp(imq2x)×- expikxx-kx2 z12k×exp[-i(kx-nq1)xi]Φ(kx-nq1)dkx.
F3(kx, xi)=FT[f3(x, xi)]=nmanbm exp[-i(kx-nq1-mq2)xi]×Φ(kx-nq1-mq2)×exp-i(kx-mq2)2 z12k.
f4(x, xi)=nmanbm exp[i(nq1+mq2)x]×exp-i(nq1+mq2)2 z22kexp-in2q12 z12k×- expikxx-(nq1+mq2)×z2k-nq1 z1kexp-ikx2 z1+z22k×exp(-ikxxi)Φ(kx)dkx.
Ii(x, xi)=|f4(x, xi)|2=nnmmanan*,bmbm* exp{i[q1(n-n)+q2(m-m)]x}exp-i(q1n+q2m)2 z22k×exp-iq12n2 z12kexpi(q1n+q2m)2×z22kexpiq12n2 z12k×- expikxx-(nq1+mq2)×z2k-nq1 z1kexp-ikx2 z1+z22k×exp(-ikxxi)Φ(kx)dkx×- exp-ikxx-(nq1+mq2)×z2k-nq1 z1kexpikx2 z1+z22k×exp(ikxxi)Φ*(kx)dkx.
ϕ(x)=expix2 k2z0,
Φ(kx)=exp-ikx2 z02k.
Ii(x, xi)=nnmmanan*bmbm*×exp{i[q1(n-n)+q2(m-m)]x}×expi z0zT (n2-n2)q12 z1+z22k×expi z0+z1zT (m2-m2)q22 z22k×expi z0zT (nm-nm)q1q2 z22k×exp-i x-xizT [(n-n)q1(z1+z2)+(m-m)q2z2],
IS(x)=-S/2S/2Ii(x, xi)dxi.
IS(x)=nnmmanan*bmbm*×expixq1 z0zT (n-n)+q2 z0+z1zT (m-m)×expi z0zT (n2-n2)q12 z1+z22k×expi z0+z1zT (m2-m2)q22 z22k×expi2 z0zT (nm-nm)q1q2 z22k×sincSzT [(n-n)q1(z1+z2)+(m-m)q2z2],
R=q2q1,Z0=q12z0πk,Z1=q12z1πk,
Z2=q12z2πk,ZT=q12zTπk,
IS(x)=nnmmanan*bmbm*×expix q1Zt [Z0(n-n)+R(Z0+Z1)×(m-m)]expi π2 Z0ZT (n2-n2)(Z1+Z2)×expi π2 Z0+Z1ZT (m2-m2)R2Z2×expiπ Z0ZT (nm-nm)RZ2×sincSZT q1[(n-n)(Z1+Z2)+(m-m)RZ2].
I(x)=nnmmanan*bmbm* exp{iq1x[(n-n)+R(m-m)]}expi π2 (Z1+Z2)(n2-n2)×expi π2 Z2R2(m2-m2)×exp[iπRZ2(nm-nm)]×δ([(Z1+Z2)(n-n)+RZ2(m-m)]).
Z1+Z2RZ2=Q,
(Z1+Z2)nI+RZ2mI0,
|nI|Sq1πZT RZ2,|mI|Sq1πZT (Z1+Z2).
|nI||mI|, |mI||nI|ZTSq1 1(Z1+Z2)nI+RZ2mI2.
ISI0+j0dj expix Z1Z2 q1, jnI×sincj Sq1ZT [(Z1+Z2)nI+RZ2mI].
MRe(d1)sincSq1ZT [(Z1+Z2)nI+RZ2mI].
Sq1ZT [(Z1+Z2)nI+RZ2mI]=π2.
(Z10+Z20)nI+RZ20mI=0,
ΔZ1=π2 ZTSq1 1nI,
ΔZ2=π2 ZTSq1 1nI+RmI,
d1=expi π2 RZ1nImI×nan+nIan*mbm+mIbm*×exp(-iπRZ1nIm);
pd=2RnI.
a0=b0=12,a2n+1=b2n+1=1π (-1)n2n+1,
a2n=b2n=0(n0).
M1π2nImI cosπ2 RZ1nImI×sincSq1ZT [(Z1+Z2)nI+RZ2mI].
M1πnI sincSq1ZT [(Z1+Z2)nI+RZ2mI]×m 1π2(2m+1+mI)(2m+1)×cos[πRZ1nI(2m+1-mI/2)].
Z1=2lRnImI(lN).
1πnI m 1π2(2m+1+mI)(2m+1)×cos[πRZ1nI(2m+1-mI/2)].
IS(x)=n,m exp[iq1(αn+βm)x]sinc(An+Bm)cnm,
α=Z0/ZT,
β=R(Z0+Z1)/ZT,
A=Sq1(Z1+Z2)/ZT,
B=Sq1RZ2/ZT,
cnm=expi π2 Z0ZT (Z1+Z2)n2×expi π2 Z0+Z1ZT R2Z2m2expiπ Z0ZT RZ2nm×n,man+nan*bm+mbm*×exp-iπ Z0ZT (Z1+Z2)nn×exp-iπ Z0+Z1ZT R2Z2mm×exp-iπ Z0ZT RZ2(nm-nm),
cnm=expi π2 AZ0Sq1 n2expi π2 RZ2(βm2+2αnm)×n,man+nan*,bm+mbm*×exp-iπ AZ0Sq1 nn×exp{-iπRZ2[βmm+α(nm-nm)].}
SZTp1Z1+Z2,SZTp2Z2
|A|>π1,|B|>π2,1,21;
|sinc(nA)|<1nπ0,|sinc(mB)| <2mπ0,
IS(x)I0+n,m0 exp(iαxn)×exp(iβxm)sinc(An+Bm)cnm,
|cnm| K/|nm|,
qI=q1(αnI+βmI).
I(x)=I0+jcjni,jmI sinc[j(AnI+BmI)]×exp(ijqIx)+I2(x),
sinc[j(AnI+BmI)]1,
|AnI+BmI| <π2nL,nL1;
r2=mnI-nmInI,
|An+Bm| = |r1(AnI+BmI)+r2B|>r2B|-|r1(AnI+BmI).
|r2B|>BnI
|An+Bm|>BnI-|r1(AnI+BmI)|.
|r1(AnI+BmI)| <π2,
|An+Bm|>BnI -π2.
|An+Bm|>π2nI-π2.
|nI|12,
|nm|n2=r12nI2nL2nI2.
nL2nI2|nImI|,
|mI||nI|nL2
|mI|1/1,
|nI|/|mI|nL2.
|AnI+BmI| <π2nL,nL1,
|nI|1/2,|mI|1/1,
|mI|/|nI|,|nI|/|mI|nL2.
|AnI+BmI|=0.
A=A0+ΔA,B=B0+ΔB.
|AnI+BmI| = |ΔAnI+ΔBmI|=π2nL.
|ΔAnI+ΔBmI|π/2,
|ΔAnI+ΔBmI|2π2|nI||mI|,
(Z1+Z2)nI+RZ2mI0,
|nI|Sq1πZT RZ2,|mI|Sq1πZT (Z1+Z2),
|nI||mI|, |mI||nI|ZTSq1 1(Z1+Z2)nI+RZ2mI2.
ISI0+j0dj expix Z1Z2 q1jnI×sincj Sq1ZT [(Z1+Z2)nI+RZ2mI],
dj=expi π2 RZ1nImI j2nan+jnIan*×mbm+jmIbm* exp(-iπRZ1nImj).

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