Abstract

The use of different multiple-aperture pupils for recording each image in speckle photography is proposed. The introduction of suitable spatial frequency carriers, by internally modulating imaged speckles, allows one to selectively isolate or combine the spectral content of different images into spatially separated regions in the Fourier plane. Theoretical and experimental results extend the speckle photography technique to the depiction of several specklegrams of multiple uniform in-plane displacements. In this case, because different pupils are considered for recording, the cross-correlation functions for the amplitudes and intensities in the image plane are calculated on the basis of the statistical properties of the object. Also, the ensemble-average intensity in the Fourier plane is analytically derived, and fringe visibility is investigated.

© 2000 Optical Society of America

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References

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  1. M. Françon, “Information processing using speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), pp. 171–202.
  2. I. Yamaguchi, “Speckle displacement and decorrelation in the diffraction and image fields for small object deformation,” Opt. Acta 28, 1359–1376 (1981).
    [CrossRef]
  3. I. Yamaguchi, “Fringe formations in deformation and vibration measurements using laser light,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1985), Vol. XXII, pp. 271–339.
  4. D. W. Li, J. B. Chen, F. P. Chiang, “Statistical analysis of one-beam subjective laser-speckle interferometry,” J. Opt. Soc. Am. A 2, 657–666 (1985).
    [CrossRef]
  5. M. Sjödahl, “Calculation of speckle displacement, decorrelation, and object-point location in imaging systems,” Appl. Opt. 34, 7998–8010 (1995).
    [CrossRef] [PubMed]
  6. G. H. Kaufmann, “Numerical processing of speckle photography data by Fourier transform,” Appl. Opt. 20, 4277–4280 (1981).
    [CrossRef] [PubMed]
  7. R. Meynart, “Diffraction halo in speckle photography,” Appl. Opt. 23, 2235–2236 (1984).
    [CrossRef] [PubMed]
  8. F. D. Chiang, R. P. Khetan, “Strain analysis by one-beam laser speckle interferometry. 2. Multiaperture method,” Appl. Opt. 18, 2175–2186 (1979).
    [CrossRef] [PubMed]
  9. Y. Y. Hung, R. E. Rowlands, I. M. Daniel, “Speckle-shearing interferometric technique: a full-field strain gauge,” Appl. Opt. 14, 618–622 (1975).
    [CrossRef] [PubMed]
  10. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), pp. 11–76.
  11. L. Angel, M. Tebaldi, M. Trivi, N. Bolognini, “Optical operations based on speckle modulation by using a photorefractive crystal,” Opt. Commun. 168, 55–64 (1999).
    [CrossRef]
  12. K. Nakagawa, T. Takatsuji, T. Minemoto, “Measurement of the displacement distribution by speckle photography using a BSO crystal,” Opt. Commun. 76, 206–212 (1990).
    [CrossRef]
  13. K. Nakagawa, T. Minemoto, “Readout properties of the specklegram recorded in photorefractive Bi12SiO20 crystal,” Appl. Opt. 30, 2386–2392 (1991).
    [CrossRef] [PubMed]
  14. S. Bosco, M. Trivi, “Correlation-like algorithm for superresolution analysis,” Opt. Commun. 115, 444–448 (1995).
    [CrossRef]

1999

L. Angel, M. Tebaldi, M. Trivi, N. Bolognini, “Optical operations based on speckle modulation by using a photorefractive crystal,” Opt. Commun. 168, 55–64 (1999).
[CrossRef]

1995

1991

1990

K. Nakagawa, T. Takatsuji, T. Minemoto, “Measurement of the displacement distribution by speckle photography using a BSO crystal,” Opt. Commun. 76, 206–212 (1990).
[CrossRef]

1985

1984

1981

G. H. Kaufmann, “Numerical processing of speckle photography data by Fourier transform,” Appl. Opt. 20, 4277–4280 (1981).
[CrossRef] [PubMed]

I. Yamaguchi, “Speckle displacement and decorrelation in the diffraction and image fields for small object deformation,” Opt. Acta 28, 1359–1376 (1981).
[CrossRef]

1979

1975

Angel, L.

L. Angel, M. Tebaldi, M. Trivi, N. Bolognini, “Optical operations based on speckle modulation by using a photorefractive crystal,” Opt. Commun. 168, 55–64 (1999).
[CrossRef]

Bolognini, N.

L. Angel, M. Tebaldi, M. Trivi, N. Bolognini, “Optical operations based on speckle modulation by using a photorefractive crystal,” Opt. Commun. 168, 55–64 (1999).
[CrossRef]

Bosco, S.

S. Bosco, M. Trivi, “Correlation-like algorithm for superresolution analysis,” Opt. Commun. 115, 444–448 (1995).
[CrossRef]

Chen, J. B.

Chiang, F. D.

Chiang, F. P.

Daniel, I. M.

Françon, M.

M. Françon, “Information processing using speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), pp. 171–202.

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), pp. 11–76.

Hung, Y. Y.

Kaufmann, G. H.

Khetan, R. P.

Li, D. W.

Meynart, R.

Minemoto, T.

K. Nakagawa, T. Minemoto, “Readout properties of the specklegram recorded in photorefractive Bi12SiO20 crystal,” Appl. Opt. 30, 2386–2392 (1991).
[CrossRef] [PubMed]

K. Nakagawa, T. Takatsuji, T. Minemoto, “Measurement of the displacement distribution by speckle photography using a BSO crystal,” Opt. Commun. 76, 206–212 (1990).
[CrossRef]

Nakagawa, K.

K. Nakagawa, T. Minemoto, “Readout properties of the specklegram recorded in photorefractive Bi12SiO20 crystal,” Appl. Opt. 30, 2386–2392 (1991).
[CrossRef] [PubMed]

K. Nakagawa, T. Takatsuji, T. Minemoto, “Measurement of the displacement distribution by speckle photography using a BSO crystal,” Opt. Commun. 76, 206–212 (1990).
[CrossRef]

Rowlands, R. E.

Sjödahl, M.

Takatsuji, T.

K. Nakagawa, T. Takatsuji, T. Minemoto, “Measurement of the displacement distribution by speckle photography using a BSO crystal,” Opt. Commun. 76, 206–212 (1990).
[CrossRef]

Tebaldi, M.

L. Angel, M. Tebaldi, M. Trivi, N. Bolognini, “Optical operations based on speckle modulation by using a photorefractive crystal,” Opt. Commun. 168, 55–64 (1999).
[CrossRef]

Trivi, M.

L. Angel, M. Tebaldi, M. Trivi, N. Bolognini, “Optical operations based on speckle modulation by using a photorefractive crystal,” Opt. Commun. 168, 55–64 (1999).
[CrossRef]

S. Bosco, M. Trivi, “Correlation-like algorithm for superresolution analysis,” Opt. Commun. 115, 444–448 (1995).
[CrossRef]

Yamaguchi, I.

I. Yamaguchi, “Speckle displacement and decorrelation in the diffraction and image fields for small object deformation,” Opt. Acta 28, 1359–1376 (1981).
[CrossRef]

I. Yamaguchi, “Fringe formations in deformation and vibration measurements using laser light,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1985), Vol. XXII, pp. 271–339.

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Acta

I. Yamaguchi, “Speckle displacement and decorrelation in the diffraction and image fields for small object deformation,” Opt. Acta 28, 1359–1376 (1981).
[CrossRef]

Opt. Commun.

L. Angel, M. Tebaldi, M. Trivi, N. Bolognini, “Optical operations based on speckle modulation by using a photorefractive crystal,” Opt. Commun. 168, 55–64 (1999).
[CrossRef]

K. Nakagawa, T. Takatsuji, T. Minemoto, “Measurement of the displacement distribution by speckle photography using a BSO crystal,” Opt. Commun. 76, 206–212 (1990).
[CrossRef]

S. Bosco, M. Trivi, “Correlation-like algorithm for superresolution analysis,” Opt. Commun. 115, 444–448 (1995).
[CrossRef]

Other

I. Yamaguchi, “Fringe formations in deformation and vibration measurements using laser light,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1985), Vol. XXII, pp. 271–339.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), pp. 11–76.

M. Françon, “Information processing using speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), pp. 171–202.

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Figures (8)

Fig. 1
Fig. 1

Experimental setups for (a) recording and (b) analysis of the specklegram. R, diffuser; L1, imaging lens; P, pupil mask; S’s, specklegrams; L2, Fourier lens; F, Fourier plane.

Fig. 2
Fig. 2

Diffraction pattern schemes corresponding to the four pupils employed.

Fig. 3
Fig. 3

Average intensity profile for a spot centered at (α, β) in the UV plane.

Fig. 4
Fig. 4

Double-exposed specklegrams through pupils without common apertures.

Fig. 5
Fig. 5

Double-exposed specklegrams through identical pupils.

Fig. 6
Fig. 6

Double-exposed specklegrams through pupils with common and noncommon apertures.

Fig. 7
Fig. 7

(a) Triple-exposed and (b) quadruple-exposed specklegrams for different uniform in-plane displacements between images recorded through the pupils illustrated schematically in Fig. 2.

Fig. 8
Fig. 8

(a) Triple-exposed and (b) quadruple-exposed real experiments for different uniform in-plane displacements between images.

Equations (55)

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A0k(x, y)=IWζ(x-xk, y-yk)×exp[iϕ(x-xk, y-yk)],
Ak(X, Y)=A0k(x, y)Kk(X, Y; x, y)dxdy,
Kk(X, Y; x, y)=Pk(u, v)exp-i2πλWxZ0+XZCu+yZ0+YZCvdudv
Ak(Xa, Ya)[Al(Xb, Yb)]*
IW|ζ|2Kk(Xa, Ya; x+xk, y+yk)×[Kl(Xb, Yb; x+xl, y+yl)]*dxdy,
Ak(Xa, Ya)[Al(Xb, Yb)]*
Pk(u, v)exp-i2πλWxkZ0+XaZCu+ykZ0+YaZCvexp-i2πλWZ0(xu+yv)dudv×Pl(u, v)expi2πλWxlZ0+XbZCu+ylZ0+YbZCvexpi2πλWZ0(xu+yv)dudvdxdyexp-i2πλWZ0[x(u-u)+y(v-v)]dxdy×Pk(u, v)exp-i2πλWxkZ0+XaZCu+ykZ0+YaZCv×Pl(u, v)expi2πλWxlZ0+XbZCu+ylZ0+YbZCvdudvdudv.
exp{-i(2π/λWZ0)[x(u-u)+y(v-v)]}dxdy=δ(u-u, v-v),
Ak(Xa, Ya)[Al(Xb, Yb)]*Pk(u, v)Pl(u, v)exp-i2πλWxkZ0+XaZCu-xlZ0+XbZCu×exp-i2πλWykZ0+YaZCv-ylZ0+YbZCvδ(u-u, v-v)dudvdudvF{Pkl(u, v)}Xa-XbλWZC-ΔxklλWZ0,Ya-YbλWZC-ΔyklλWZ0,
Pkl(u, v)Pk(u, v)Pl(u, v),
(Δxkl, Δykl)(xl-xk, yl-yk),
Ik=Ik(X, Y)=Ak(X, Y)[Ak(X, Y)]*F{Pk(u, v)}(0, 0),
Ik(Xa, Ya)Il(Xb, Yb)=IkIl+|Ak(Xa, Ya)[Al(Xb, Yb)]*|2,
Ik(Xa, Ya)Il(Xb, Yb)
F{Pk(u, v)}(0, 0)F{Pl(u, v)}(0, 0)+F{Pkl(u, v)}Xa-XbλWZC-ΔxklλWZ0,Ya-YbλWZC-ΔyklλWZ02.
Ik(Xa, Ya)Il(Xb, Yb)IkIl1+F{|Pkl(u, v)|2}Xa-XbλWZC-ΔxklλWZ0,Ya-YbλWZC-ΔyklλWZ02|Pk(ξ, χ)|2dξdχ|Pl(ξ, χ)|2dξdχ,
I(X, Y)=k=1NIk(X, Y).
Gf(U, V)=IRk=1NGk(U, V),
Gk(U, V)=Ik(X, Y)exp-i2πλRf(XU+YV)dXdY.
Gf(U1, V1)[Gf(U2, V2)]*=IRk,l=1NIk(Xa, Ya)Il(Xb, Yb)×exp-i2πλRf(XaU1-XbU2+YaV1-YbV2)dXadYadXbdYb.
If(U, V)=Gf(U, V)[Gf(U, V)]*=IRk,l=1NIk(Xa, Ya)Il(Xb, Yb)×exp-i2πλRf[U(Xa-Xb)+V(Ya-Yb)]dXadYadXbdYb.
If(U, V)=δ(U, V)+14d(Xa+Xb)d(Ya+Yb)×k,l=1NF{Pkl(u, v)}×X+ΔXklλWZC,Y+ΔYklλWZC2×exp-i2πλRf(UX+VY)dXdY,
(ΔXkl, ΔYkl)=-ZCZ0(Δxkl, Δykl)
If(U, V)=k,l=1NFF{Pkl(u, v)}×X+ΔXklλWZC,Y+ΔYklλWZC2UλRf,VλRf=(λWZC)2k,l=1Nexp-i2πλRf(UΔXkl+VΔYkl)×F{|F{Pkl(u, v)}(X, Y)|2}(ϑU, ϑV),
If(U, V)=k=1NF{|F{Pk(u, v)}(X, Y)|2}(ϑU, ϑV)+2k,l=1(k<l)Ncos2πλRf(UΔXkl+VΔYkl)×F{|F{Pkl(u, v)}(X, Y)|2}(ϑU, ϑV),
F{|F{Pk(u, v)}(X, Y)|2}(ϑU, ϑV)
|Pk(ξ, η)|2|Pk(ξ-ϑU, η-ϑV)|2dξdη
If(U, V)=F{|F{P1(u, v)}(X, Y)|2}(ϑU, ϑV)+F{|F{P2(u, v)}(X, Y)|2}(ϑU, ϑV)+2 cos2πλRf(UΔX12+VΔY12)×F{|F{P12(u, v)}(X, Y)|2}(ϑU, ϑV).
If(U, V)=4 cos2πλRf(UΔX12+VΔY12)×F{|F{P1(u, v)}(X, Y)|2}(ϑU, ϑV),
Ak(Xa, Ya)[Al(Xb, Yb)]*
=j=1qklexp[-i2π(ujklηabkl+vjklζabkl)]×F{a(u, v)}(ηabkl, ζabkl),
(ηabkl, ζabkl)(1/λWZC)(Xa-Xb-ΔXkl, Ya-Yb-ΔYkl).
Ik=qkF{ak(u, v)}(0, 0)
|Ak(Xa, Ya)[Al(Xb, Yb)]*|2=qkl+2i,j=1(i<j)qklcos{2π[(uikl-ujkl)ηabkl+(vikl-vjkl)ζabkl]}|F{a(u, v)}(ηabkl, ζabkl)|2.
Ik(Xa, Ya)Il(Xb, Yb)=qkql[F{a(u, v)}(0, 0)]2+qkl+2i,j=1(i<j)qklcos{2π[(uikl-ujkl)ηabkl+(vikl-vjkl)ζabkl]}|F{a(u, v)}(ηabkl, ζabkl)|2.
F{|F{Pk(u, v)}(X, Y)|2}(ϑU, ϑV)=FFj=1qka(u-ujk, v-vjk)(X, Y)2(ϑU, ϑV)=FF{a(u, v)}(X, Y)j=1qkexp[-i2π(Xujk+Yvjk)]2×(ϑU, ϑV)=F|F{a(u, v)}(X, Y)|2×qk+2r,s=1(r<s)qkcos{2π[X(urk-usk)+Y(vrk-vsk)]}(ϑU, ϑV),
F{|F{Pkl(u, v)}(X, Y)|2}(ϑU, ϑV)=F|F{a(u, v)}(X, Y)|2qkl+2r,s=1(r<s)qkcos{2π[X(urkl-uskl)+Y(vrkl-vskl)]}(ϑU, ϑV).
If(U, V)=If0(U, V)+IfL(U, V),
If0(U, V)=k=1Nqk+2k,l=1(k<l)Nqkl cos2πλRf(UΔXkl+VΔYkl)×F{|F{a(u, v)}(X, Y)|2}(ϑU, ϑV),
IfL(U, V)=k=1Nr,s=1(rs)qkF{|F{a(u, v)}(X, Y)|2}×(ϑU+urk-usk, ϑV+vrk-vsk)+2k,l=1(k<l)Ncos2πλRf(UΔXkl+VΔYkl)×r,s=1(rs)qklF{|F{a(u, v)}(X, Y)|2}×(ϑU+urkl-uskl, ϑV+vrkl-vskl).
(U, V)F{|F{a(u, v)}(X, Y)|2}(ϑU, ϑV)
If0(U, V)=k=1Nqk+2k,l=1(k<l)Nqkl cos2πλRf(UΔXkl+VΔYkl)(U, V),
IfL(U, V)=k=1Nr,s=1(rs)qk(U-Ursk, V-Vrsk)+2k,l=1(k<l)Ncos2πλRf(UΔXkl+VΔYkl)×r,s=1(rs)qkl(U-Urskl, V-Vrskl),
(Ursk, Vrsk)(λRf/λWZC)(usk-urk, vsk-vrk),
(Urskl, Vrskl)(λRf/λWZC)(uskl-urkl, vskl-vrkl).
(U, V)=D22cos-12ρD-2ρD1-2ρD21/2×cylρD,
ρ=U2+V2,D=(2 f/ZC)(λR/λW)D,
cyl(ρ/D)=1if0ρD/20otherwise.
Iαβ(U, V)=k=1Nqαβk+2k,l=1(k<l)Nqαβklcos2πλRf(UΔXkl+VΔYkl)×(U-α, V-β).
Iαβ(U, V)=qαβ1+qαβ2+2qαβ12cos2πλRf(UΔX12+VΔY12)×(U-α, V-β).
Vαβ=Iαβ(U, V)max-Iαβ(U, V)minIαβ(U, V)max+Iαβ(U, V)min,
Iαβ(U, V)max={qαβ1+qαβ2+2qαβ12}(U-α, V-β),
Iαβ(U, V)min={qαβ1+qαβ2-2qαβ12}(U-α, V-β).
Vαβ=2qαβ12qαβ1+qαβ2.
I00(U, V)={8+4 cos(2πUΔ/λRf)+2 cos(2πVΔ/λRf)+4 cos[2π(U+V)Δ/λRf]}(U, V).

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