Abstract

Speckle suppression in a two-diffuser system is examined. An analytical expression for the speckle space–time correlation function is derived, so that the speckle suppression mechanism can be investigated statistically. The grain size of the speckle field illuminating the second diffuser has a major impact on the speckle contrast after temporal averaging. It is shown that, when both the diffusers are rotating, the one with the lower rotating speed determines the period of the speckle correlation function. The coherent length of the averaged speckle intensity is shown to equal the mean speckle size of the individual speckle pattern before averaging. Numerical and experimental results are presented to verify our analysis in the context of speckle reduction.

© 2013 Optical Society of America

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References

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2013

2011

2010

2008

2006

2004

K. Kasazumi, Y. Kitaoka, and K. M. A. K. Yamamoto, “A practical laser projector with new illumination optics for reduction of speckle noise,” Jpn. J. Appl. Phys. 43, 5904–5906 (2004).
[CrossRef]

1999

1993

1992

1990

1982

1980

J. D. Farina and L. M. Narducci, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
[CrossRef]

1979

P. de Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

1976

1971

1962

I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. Theory 8, 194–195 (1962).
[CrossRef]

1961

Akram, M. N.

Allen, G.

Cao, H.

Chen, X.

Churnside, J. H.

de Santis, P.

P. de Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

Dufresne, E. R.

Farina, J. D.

J. D. Farina and L. M. Narducci, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
[CrossRef]

Fujiwara, K.

Goodman, J. W.

E. G. Rawson, A. B. Nafarrate, R. E. Norton, and J. W. Goodman, “Speckle-free rear-projection screen using two close screens in slow relative motion,” J. Opt. Soc. Am. 66, 1290–1294 (1976).
[CrossRef]

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications, 1st ed. (Roberts & Company, 2006), pp. 125–149.

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications, 1st ed. (Roberts & Company, 2006), pp. 47–53.

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications, 1st ed. (Roberts & Company, 2006), pp. 127–135.

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications, 1st ed. (Roberts & Company, 2006), pp. 188–194.

J. W. Goodman, Statistical Optics (Wiley, 2000), pp. 170–187.

Gori, F.

P. de Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

Guattari, G.

P. de Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

Hansen, R. S.

Hanson, S. G.

Joyeux, D.

Kartashov, V.

Kasazumi, K.

K. Kasazumi, Y. Kitaoka, and K. M. A. K. Yamamoto, “A practical laser projector with new illumination optics for reduction of speckle noise,” Jpn. J. Appl. Phys. 43, 5904–5906 (2004).
[CrossRef]

Katagiri, B.

Kawakami, T.

Kelly, D. P.

Kirchner, M.

Kitaoka, Y.

K. Kasazumi, Y. Kitaoka, and K. M. A. K. Yamamoto, “A practical laser projector with new illumination optics for reduction of speckle noise,” Jpn. J. Appl. Phys. 43, 5904–5906 (2004).
[CrossRef]

Kuratomi, Y.

Lapchuk, A.

Leifer, I.

Leushacke, L.

Li, D.

Lowenthal, S.

Nafarrate, A. B.

Narducci, L. M.

J. D. Farina and L. M. Narducci, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
[CrossRef]

Norton, R. E.

O’Donnell, K. A.

Oldham, W. G.

Ouyang, G.

Palma, C.

P. de Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

Partlo, W. N.

Rawson, E. G.

Redding, B.

Reed, I. S.

I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. Theory 8, 194–195 (1962).
[CrossRef]

Richmond, C. N.

Rose, B.

Satoh, H.

Sekiya, K.

Sheridan, J. T.

Song, J.

Spencer, C. J. D.

Suzuki, Y.

Tomiyama, T.

Tong, Z.

Uchida, T.

Voelz, D.

Wang, K.

Welford, W. T.

Xiao, X.

Yamamoto, K. M. A. K.

K. Kasazumi, Y. Kitaoka, and K. M. A. K. Yamamoto, “A practical laser projector with new illumination optics for reduction of speckle noise,” Jpn. J. Appl. Phys. 43, 5904–5906 (2004).
[CrossRef]

Yan, X.

Yang, H.

Yoshimura, T.

Yun, S.

Yura, H. T.

Yurlov, V.

Appl. Opt.

IRE Trans. Inf. Theory

I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. Theory 8, 194–195 (1962).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Jpn. J. Appl. Phys.

K. Kasazumi, Y. Kitaoka, and K. M. A. K. Yamamoto, “A practical laser projector with new illumination optics for reduction of speckle noise,” Jpn. J. Appl. Phys. 43, 5904–5906 (2004).
[CrossRef]

Opt. Commun.

P. de Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

J. D. Farina and L. M. Narducci, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
[CrossRef]

Opt. Express

Opt. Lett.

Other

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications, 1st ed. (Roberts & Company, 2006), pp. 125–149.

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications, 1st ed. (Roberts & Company, 2006), pp. 127–135.

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications, 1st ed. (Roberts & Company, 2006), pp. 47–53.

J. W. Goodman, Statistical Optics (Wiley, 2000), pp. 170–187.

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications, 1st ed. (Roberts & Company, 2006), pp. 188–194.

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Figures (9)

Fig. 1.
Fig. 1.

Optical configuration in which dynamic speckled speckles are formed. D1: the first diffuser; D2: the second diffuser. Both can be rotated around the optical axis with angular speed ω1 and ω2, respectively. L: thin lens with diameter 2a0 and focal length f.

Fig. 2.
Fig. 2.

Speckle contrast (C) for doubly scattered speckle as a function of M for various values of N.

Fig. 3.
Fig. 3.

Comparison of the temporal correlation functions. (a) Plots of μ1(p,p,ω2τ) and μ2(p,p,ω1τ,ω2τ,v), as a function of ω2τ. Compared with μ1, μ2 has a faster decorrelation rate and this decorrelation rate is additionally controllable by the value of v. (b) Plot of μ2(p,p,ω1τ,ω2τ,v) as a function of ω2τ, for different states of ω1τ. As demonstrated, the period of μ2 is controllable by different choices of ω1τ. μ2 has the highest decorrelation rate when the two diffusers are rotating in opposite directions.

Fig. 4.
Fig. 4.

Simulation results of speckle suppression by doubly scattering system. (a) Speckle contrast as a function of M for various values of N. (b) Demonstrating the effect of correlation function period on the performance of speckle suppression.

Fig. 5.
Fig. 5.

Resulting intensity pattern from averaging 1000 speckle patterns when ω1τ=3° with ω2τ=0 (a), and when ω1τ=3° with ω2τ=0.3° (c). Red lines in (b) and (d): intensity profile along “Line 128,” marked in red in (a) and (c), respectively. Blue line in (b) and (d): reference Gaussian intensity profiles without the diffusers presented.

Fig. 6.
Fig. 6.

Optical configuration used to observe the interfering fringes of partially coherent intensity: OM: opaque mask with two pinholes of diameter dp each. The two pinholes are equidistant from the optical axis, with separation s=(sx,sy); L1: thin lens of focal length f1. The other symbols are the same as those in Fig. 1.

Fig. 7.
Fig. 7.

Fringe profiles for different partially coherent beams propagated through two pinholes shown against pixel number. Diameter of L is chosen at (a) 2a0=15mm, (b) 2a0=8mm, and (c) 2a0=4mm. These cases correspond to the following values of γ: (a) 0.09, (b) 0.5, and (c) 0.84, respectively. Simulation (red dots) and analytical results (blue continuous line) are overlaid for comparison.

Fig. 8.
Fig. 8.

Measured speckle contrast (C) as a function of camera exposure time (T). Fits to the experimental data (dots) using the model C=aTb are included.

Fig. 9.
Fig. 9.

Experimentally demonstrating speckle reduction when projecting picture in the singly and doubly scattering systems. The contrasts of the intensity in the marked area are (a) 0.465, (b) 0.192, (c) 0.155, and (d) 0.075.

Equations (18)

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I(r1,t)I(r2,t+τ)=A(r1,t)A*(r1,t)A(r2,t+τ)A*(r2,t+τ),
μ2(p1,p2,ω1τ,ω2τ,v)=I(p1,t)I(p2,t+τ)I(p1,t)I(p2,t+τ)I(p1,t)I(p2,t+τ)=Γ12(p1,p2,ω1,ω2,τ)Γ0=4v2[1+v(1v)cos(ω1τω2τ)]2exp{2v(p2p1cosω2τ+Rp1sinω2τ)2ρ22[1+v(1v)cos(ω1τω2τ)]}×exp{2(1v)[1cos(ω1τω2τ)](p12+p22)ρ22[1+v(1v)cos(ω1τω2τ)]},
v=ζ2(ρ12+ζ2)=1N+1,
ρ1=λdiπa0,
ρ2=λdoπa0.
μ1(p1,p2,ω2τ)=I(p1,t)I(p2,t+τ)I(p1,t)I(p2,t+τ)I(p1,t)I(p2,t+τ)=exp[(p2p1cosω2τ+Rp1sinω2τ)2ρ22].
C=N+M+1NM,
σc=ρ2=λdoπa0.
If(x,y,z)=18λ2z2[πdp2exp(sx2+sy24r02)]2[J1(kdp2zx2+y2)(kdp2zx2+y2)]2{1+γcos[kz(sxx+syy)]},
γ=exp[(sx2+sy2)2σc2].
U(r,t)=exp(r2w02)exp(ikr22R),
A*(p2,t2)A(p1,t1)=U*(r2,t2)U(r1,t1)exp[iϕ(r2)iϕ(r1)]T*(r2,p2)T(r1,p1)d2r1d2r2,
T(r,p)=exp[iπλ(r2di+p2do)]exp[(rρ1+pρ2)2],
r1=r1cosω2τRr1sinω2τ,
R=(0110),
(ξ1η1)=(ξ1cosω1τη1sinω1τξ1sinω1τ+η1cosω1τ).
exp[iϕ(r2)iϕ(r1)]=δ(r2r1).
U*(r2,t2)U(r1,t1)=exp(r22r02)exp(r12r02).

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