Abstract

In this paper a general method is presented for calculating the theoretical speckle contrast of a sum of correlated speckle patterns, motivated by the need to suppress the presence of speckle in laser projection displays. The method is applied to a specific example, where correlated speckle patterns are created by sequentially passing light through partially overlapping areas on a diffuser, before being projected onto a screen. This design makes it possible to find a simple expression for the correlation between speckle patterns. When the set of correlations involves symmetry, it is shown that the expression for the speckle contrast becomes simpler. The difference in performance between discretely and continuously varying speckle patterns is also investigated. In an example with speckle reduction by a rotating sinusoidal grating, it is found that continuous variation gives a speckle contrast that is 0.61 times the contrast obtained by discretely summing the maximum number of independent patterns.

© 2012 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2006).
  2. S. V. Egge, M. N. Akram, V. Kartashov, K. Welde, Z. Tong, U. Österberg, and A. Aksnes, “Sinusoidal rotating grating for speckle reduction in laser projectors: feasibility study,” Opt. Eng. 50, 083202 (2011).
    [CrossRef]
  3. J. W. Goodman, Introduction to Fourier Optics3rd ed. (McGraw-Hill, 2005).
  4. V. Kartashov and M. N. Akram, “Speckle suppression in projection displays by using a motionless changing diffuser,” J. Opt. Soc. Am. A 27, 2593–2601 (2010).
    [CrossRef]
  5. J. W. Goodman, “Probability density function of the sum of N partially correlated speckle patterns,” Opt. Commun. 13, 244–247 (1975).
    [CrossRef]
  6. R. Barakat, “The brightness distribution of the sum of two correlated speckle patterns,” Opt. Commun. 8, 14–16 (1973).
    [CrossRef]
  7. J. W. Goodman, Statistical Optics (Wiley, 1985).
  8. J. Ohtsubo and T. Asakura, “Statistical properties of the sum of partially-developed, correlated speckle patterns,” Appl. Phys. A 17, 159–164 (1978).
    [CrossRef]
  9. J. Ohtsubo and T. Asakura, “Statistical properties of the sum of two partially correlated speckle patterns,” Appl. Phys. 14, 183–187 (1977).
    [CrossRef]
  10. A. Valberg, Light Vision Color (Wiley, 2005).
  11. R. J. Larsen and M. L. Marx, An Introduction to Mathematical Statistics and Its Applications, 3rd ed. (Prentice-Hall, 2001).
  12. J. W. Goodman, “Some fundamental properties of speckle,” J. Opt. Soc. Am. 66, 1145–1150 (1976).
    [CrossRef]
  13. J. F. Power, “Fresnel diffraction model for the point spread of a laser light profile microscope (LPM),” Appl. Phys. B 78, 693–703 (2004).
    [CrossRef]
  14. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, 1968).
  15. C. Meneses-Fabian and G. Rodriguez-Zurita, “Carrier fringes in the two-aperture common-path interferometer,” Opt. Lett. 36, 642–644 (2011).
    [CrossRef]

2011 (2)

S. V. Egge, M. N. Akram, V. Kartashov, K. Welde, Z. Tong, U. Österberg, and A. Aksnes, “Sinusoidal rotating grating for speckle reduction in laser projectors: feasibility study,” Opt. Eng. 50, 083202 (2011).
[CrossRef]

C. Meneses-Fabian and G. Rodriguez-Zurita, “Carrier fringes in the two-aperture common-path interferometer,” Opt. Lett. 36, 642–644 (2011).
[CrossRef]

2010 (1)

2004 (1)

J. F. Power, “Fresnel diffraction model for the point spread of a laser light profile microscope (LPM),” Appl. Phys. B 78, 693–703 (2004).
[CrossRef]

1978 (1)

J. Ohtsubo and T. Asakura, “Statistical properties of the sum of partially-developed, correlated speckle patterns,” Appl. Phys. A 17, 159–164 (1978).
[CrossRef]

1977 (1)

J. Ohtsubo and T. Asakura, “Statistical properties of the sum of two partially correlated speckle patterns,” Appl. Phys. 14, 183–187 (1977).
[CrossRef]

1976 (1)

1975 (1)

J. W. Goodman, “Probability density function of the sum of N partially correlated speckle patterns,” Opt. Commun. 13, 244–247 (1975).
[CrossRef]

1973 (1)

R. Barakat, “The brightness distribution of the sum of two correlated speckle patterns,” Opt. Commun. 8, 14–16 (1973).
[CrossRef]

Akram, M. N.

S. V. Egge, M. N. Akram, V. Kartashov, K. Welde, Z. Tong, U. Österberg, and A. Aksnes, “Sinusoidal rotating grating for speckle reduction in laser projectors: feasibility study,” Opt. Eng. 50, 083202 (2011).
[CrossRef]

V. Kartashov and M. N. Akram, “Speckle suppression in projection displays by using a motionless changing diffuser,” J. Opt. Soc. Am. A 27, 2593–2601 (2010).
[CrossRef]

Aksnes, A.

S. V. Egge, M. N. Akram, V. Kartashov, K. Welde, Z. Tong, U. Österberg, and A. Aksnes, “Sinusoidal rotating grating for speckle reduction in laser projectors: feasibility study,” Opt. Eng. 50, 083202 (2011).
[CrossRef]

Asakura, T.

J. Ohtsubo and T. Asakura, “Statistical properties of the sum of partially-developed, correlated speckle patterns,” Appl. Phys. A 17, 159–164 (1978).
[CrossRef]

J. Ohtsubo and T. Asakura, “Statistical properties of the sum of two partially correlated speckle patterns,” Appl. Phys. 14, 183–187 (1977).
[CrossRef]

Barakat, R.

R. Barakat, “The brightness distribution of the sum of two correlated speckle patterns,” Opt. Commun. 8, 14–16 (1973).
[CrossRef]

Egge, S. V.

S. V. Egge, M. N. Akram, V. Kartashov, K. Welde, Z. Tong, U. Österberg, and A. Aksnes, “Sinusoidal rotating grating for speckle reduction in laser projectors: feasibility study,” Opt. Eng. 50, 083202 (2011).
[CrossRef]

Goodman, J. W.

J. W. Goodman, “Some fundamental properties of speckle,” J. Opt. Soc. Am. 66, 1145–1150 (1976).
[CrossRef]

J. W. Goodman, “Probability density function of the sum of N partially correlated speckle patterns,” Opt. Commun. 13, 244–247 (1975).
[CrossRef]

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2006).

J. W. Goodman, Introduction to Fourier Optics3rd ed. (McGraw-Hill, 2005).

J. W. Goodman, Statistical Optics (Wiley, 1985).

Kartashov, V.

S. V. Egge, M. N. Akram, V. Kartashov, K. Welde, Z. Tong, U. Österberg, and A. Aksnes, “Sinusoidal rotating grating for speckle reduction in laser projectors: feasibility study,” Opt. Eng. 50, 083202 (2011).
[CrossRef]

V. Kartashov and M. N. Akram, “Speckle suppression in projection displays by using a motionless changing diffuser,” J. Opt. Soc. Am. A 27, 2593–2601 (2010).
[CrossRef]

Larsen, R. J.

R. J. Larsen and M. L. Marx, An Introduction to Mathematical Statistics and Its Applications, 3rd ed. (Prentice-Hall, 2001).

Marx, M. L.

R. J. Larsen and M. L. Marx, An Introduction to Mathematical Statistics and Its Applications, 3rd ed. (Prentice-Hall, 2001).

Meneses-Fabian, C.

Ohtsubo, J.

J. Ohtsubo and T. Asakura, “Statistical properties of the sum of partially-developed, correlated speckle patterns,” Appl. Phys. A 17, 159–164 (1978).
[CrossRef]

J. Ohtsubo and T. Asakura, “Statistical properties of the sum of two partially correlated speckle patterns,” Appl. Phys. 14, 183–187 (1977).
[CrossRef]

Österberg, U.

S. V. Egge, M. N. Akram, V. Kartashov, K. Welde, Z. Tong, U. Österberg, and A. Aksnes, “Sinusoidal rotating grating for speckle reduction in laser projectors: feasibility study,” Opt. Eng. 50, 083202 (2011).
[CrossRef]

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, 1968).

Power, J. F.

J. F. Power, “Fresnel diffraction model for the point spread of a laser light profile microscope (LPM),” Appl. Phys. B 78, 693–703 (2004).
[CrossRef]

Rodriguez-Zurita, G.

Tong, Z.

S. V. Egge, M. N. Akram, V. Kartashov, K. Welde, Z. Tong, U. Österberg, and A. Aksnes, “Sinusoidal rotating grating for speckle reduction in laser projectors: feasibility study,” Opt. Eng. 50, 083202 (2011).
[CrossRef]

Valberg, A.

A. Valberg, Light Vision Color (Wiley, 2005).

Welde, K.

S. V. Egge, M. N. Akram, V. Kartashov, K. Welde, Z. Tong, U. Österberg, and A. Aksnes, “Sinusoidal rotating grating for speckle reduction in laser projectors: feasibility study,” Opt. Eng. 50, 083202 (2011).
[CrossRef]

Appl. Phys. (1)

J. Ohtsubo and T. Asakura, “Statistical properties of the sum of two partially correlated speckle patterns,” Appl. Phys. 14, 183–187 (1977).
[CrossRef]

Appl. Phys. A (1)

J. Ohtsubo and T. Asakura, “Statistical properties of the sum of partially-developed, correlated speckle patterns,” Appl. Phys. A 17, 159–164 (1978).
[CrossRef]

Appl. Phys. B (1)

J. F. Power, “Fresnel diffraction model for the point spread of a laser light profile microscope (LPM),” Appl. Phys. B 78, 693–703 (2004).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Commun. (2)

J. W. Goodman, “Probability density function of the sum of N partially correlated speckle patterns,” Opt. Commun. 13, 244–247 (1975).
[CrossRef]

R. Barakat, “The brightness distribution of the sum of two correlated speckle patterns,” Opt. Commun. 8, 14–16 (1973).
[CrossRef]

Opt. Eng. (1)

S. V. Egge, M. N. Akram, V. Kartashov, K. Welde, Z. Tong, U. Österberg, and A. Aksnes, “Sinusoidal rotating grating for speckle reduction in laser projectors: feasibility study,” Opt. Eng. 50, 083202 (2011).
[CrossRef]

Opt. Lett. (1)

Other (6)

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, 1968).

J. W. Goodman, Introduction to Fourier Optics3rd ed. (McGraw-Hill, 2005).

J. W. Goodman, Statistical Optics (Wiley, 1985).

A. Valberg, Light Vision Color (Wiley, 2005).

R. J. Larsen and M. L. Marx, An Introduction to Mathematical Statistics and Its Applications, 3rd ed. (Prentice-Hall, 2001).

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2006).

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 (9)

Fig. 1.
Fig. 1.

Diffraction spot being sequentially displayed at two partially overlapping positions on a diffuser. The light on the screen can assume different spatial distributions depending on the imaging system.

Fig. 2.
Fig. 2.

Pair of diffraction spots displayed at two partially overlapping positions on the diffuser.

Fig. 3.
Fig. 3.

Pair of diffraction spots displayed at three partially overlapping positions on the diffuser.

Fig. 4.
Fig. 4.

Pair of diffraction spots displayed at N partially overlapping positions on the diffuser.

Fig. 5.
Fig. 5.

Example of a rotating diffraction pattern where only perpendicular positions overlap on both sides.

Fig. 6.
Fig. 6.

Three pairs of diffraction spots displayed at N partially overlapping positions on the diffuser.

Fig. 7.
Fig. 7.

Rotation of the qth diffraction order on the diffuser. After p incremental rotations the diffraction order will partially overlap with its original position when sq(p)<1. The area of the diffraction spot is denoted by A, and the area of intersection is denoted by Aoq(p).

Fig. 8.
Fig. 8.

The solid curve shows the formula for speckle contrast derived in this paper, the dotted curve shows the contrast obtained by independent speckle patterns, and the dashed curve shows the correlation between neighboring positions of the diffraction pattern, all plotted against the number of summed speckle patterns. The parameter d/R1=0.15 has been used, and a correlation of 1 signifies total overlap.

Fig. 9.
Fig. 9.

In order to obtain a more uniform amplitude distribution over each diffraction spot, the diffuser was positioned a sufficient distance in front of the Fourier plane. The focal length of the lens system is denoted by f, the distance between the grating and the lens system is denoted by z1, and the distance between the lens system and the diffuser is denoted by z2.

Tables (2)

Tables Icon

Table 1. Contribution to the Different Correlations When the Diffraction Pattern Is Displayed at N Odd Positions on the Diffuser and All Positions Are Allowed to Overlapa

Tables Icon

Table 2. Contribution to the Different Correlations When the Diffraction Pattern Is Displayed at N Even Positions on the Diffuser and All Positions Are Allowed to Overlapa

Equations (63)

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

Is=n=1NIn,
C=σsIs¯,
Is¯=n=1NIn¯=n=1NIn¯,
σs2=n=1Nm=1Nσnm.
σs2=n=1Nm=1Nσnσmρnm.
σs2=n=1Nm=1Nσnσm|μnm|2.
C(N)=σsIs¯=n=1Nm=1NIn¯Im¯|μnm|2n=1NIn¯.
C(N)=N+2n<m|μnm|2N,
A(n)=1Nkak(n)eiϕk(n).
μnm=E[A(n)A(m)*](E[|A(n)|2]E[|A(m)|2])1/2,
E[A(1)A(2)*]=E[1Nk,l=1Nak(1)al(2)ei(ϕk(1)ϕl(2))]=1Nk,l=1NE[ak(1)al(2)ei(ϕk(1)ϕl(2))]=1Nk,l=1NE[ak(1)al(2)]E[ei(ϕk(1)ϕl(2))],
ϕk(1)=ϕk(2):E[ei(ϕk(1)ϕk(2))]=E[ei0]=1,
ϕk(1)ϕl(2):E[ei(ϕk(1)ϕl(2))]=E[eiϕk(1)]E[eiϕl(2)]=0,
E[A(1)A(2)*]=1NkoverlapE[ak(1)ak(2)]=1NkoverlapE[(ak(1))2]=NoNE[a2]=AoAE[a2],
E[|A(1)|2]=E[A(1)A(1)*]=1Nk,l=1NE[ak(1)al(1)ei(ϕk(1)ϕl(1))]=1Nk,l=1NE[ak(1)al(1)]E[ei(ϕk(1)ϕl(1))]=1Nk=1NE[(ak(1))2]=E[a2],
μ12=AoAE[a2](E[a2]E[a2])1/2=AoA.
n<m|μnm|2=μ122.
C=1+μ1222=1+Ar22,
μ12=2AoA=2Ar.
C(2)=1+μ1222=1+4Ar22.
μ=AoA=Ar.
n<m|μnm|2=3μ2.
C(3)=1+2μ23=1+2Ar23.
n<mμnm2=N(μ(1))2,
C(N)=1+2(μ(1))2N=1+2(Ar(1))2N.
n<mμnm2=Np=1(N1)/2(μ(p))2.
C(N)=1+2p=1(N1)/2(μ(p))2N,
n<mμnm2=p=1(N/2)1[N(μ(p))2]+N2(μ(N/2))2=N{p=1(N/2)1[(μ(p))2]+12(μ(N/2))2}.
C(N)=1+2p=1(N/2)1[(μ(p))2]+(μ(N/2))2N,
C(N)=1+2p=0(N1)/2[(μ(p))2]+P(N)(μ(N/2))2N,
=1+2p=0(N1)/2[(Ar(p))2]+4P(N)(Ar(N/2))2N,
A(n)=1qmax2Nk=1qmax2Nak(n)eiϕk(n)=1qmax2Nq=1qmaxJqk=1+(q1)2Nq2Nbk(n)eiϕk(n),
E[A(n)A(m)*]=1qmax2NkoverlapE[(ak(n))2]=1qmax2Nq=1qmax{Jq2koverlapqth pairE[(bk(n))2]}.
E[A(n)A(m)*]=1qmax2Nq=1qmax{Jq2×2Noq(p)E[b2]}=E[b2]qmaxq=1qmaxJq2Aoq(p)A,
E[A(n)A(m)*]=1qmax2Nq=1qmax{Jq2×4Noq(N/2)E[b2]}=E[b2]qmaxq=1qmaxJq22Aoq(N/2)A.
E[|A(n)|2]=1qmax2Nk=1qmax2NE[(ak(n))2]=1qmax2Nq=1qmax{Jq2k=1+(q1)2Nq2NE[(bk(n))2]}=1qmax2Nq=1qmax{Jq2×2NE[b2]}=E[b2]qmaxq=1qmaxJq2.
μ(p)={q=1qmaxJq2Arq(p)/q=1qmaxJq2;1p<N2,2q=1qmaxJq2Arq(p)/q=1qmaxJq2;p=N2.
C(N)=1N{1+2p=0(N1)/2[(q=1qmaxJq2Arq(p)q=1qmaxJq2)2]+4P(N)(q=1qmaxJq2Arq(N/2)q=1qmaxJq2)2}1/2.
Arq(p)={2π[arccos(qs1(p))qs1(p)1(qs1(p))2];qs1(p)<1,0;qs1(p)1.
C(N)1+8.13p=1(N1)/2(q=13Jq2Arq(p))2N.
U(r,θ)=kD2i4feeik(z1+z2)eiξ/2ξexp[ik(fz1)2ffer2]×q=Jq(m2)[L1(ξ,η)+iL2(ξ,η)],
ξ=kD2(z2f)4ffe,
η=kD2fe[r22qλfeΛrcos(θpΔα)+(qλfeΛ)2]1/2,
limNC(N)0.346dR1.
μq(p)={0;q1,1;q=0,
C(N)=1+(N1)I02N,
C(N,3)0.98×C(N,2),
limNC(N,3)0.97limNC(N,2).
C(N)=1+2p=1(N1)/2[μ(s1(p))]2N,
SN=2Np=1(N1)/2{μ[2R1dsin(pπ2N)]}2,
SN=2Np=1(N1)/2[μ(2R1dsinxp)]2.
Δx=xpxp1=pπ2N(p1)π2N=π2N.
limNSN=4π0π/4[μ(2R1dsinx)]2dx,
S=2dπR102R1/d[μ(s1)]21sin2x(s1)ds1.
S=2dπR101[μ(s1)]21(s1d2R1)2ds1,
S8.13dπR101[q=13Jq2Arq(s1)]21(s1d2R1)2ds1=8.13dπR101/3[q=13Jq2Arq(s1)]21(s1d2R1)2ds1+8.13dπR11/31/2[q=12Jq2Arq(s1)]21(s1d2R1)2ds1+8.13dπR11/21[J12Ar1(s1)]21(s1d2R1)2ds1,
1[s1d/(2R1)]21.
S0.0463×8.13dπR1,
limNC(N)=S0.346dR1.
S0.04631(d2R1)2×8.13dπR1.
CU[1(d2R1)2]1/4CL.
ΔC=CCLC=1CLC<1CLCU=1[1(d2R1)2]1/4.
dR121(1tΔC)4.

Metrics