Abstract

Abstract: We present the general coherence theory for laser beams passing through a moving diffuser. The temporal coherence of laser beams passing through a moving diffuser depends on two characteristic temporal scales: the laser coherence time and the mean time it takes the diffuser to move past a phase correlation area. In most applications, the former is much shorter than the latter. Our theoretical analysis shows the spatial coherence area of light scattered from a moving diffuser decreases while the coherence time remains unchanged. The conclusion has been confirmed by experiments using a Michelson interferometer and it is not in accordance with the original coherence theory in which both the temporal and spatial coherence of light scattered by a moving diffuser decrease. We also developed a method based on the theory of eigenvalues to calculate the speckle contrast on a screen illuminated by light transmitted through a moving diffuser.

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References

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  1. J. I. Trisnadi, “Hadamard speckle contrast reduction,” Opt. Lett.29(1), 11–13 (2004).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  3. S. V. Govorkov and L. A. Spinelli, “Speckle reduction in laser illuminated projection displays having a one dimensional spatial light modulator, ” U.S. Patent, 7,413,311 (Aug.19,2008).
  4. S. C. Shin, S. S. Yoo, S. Y. Lee, C.-Y. Park, S.-Y. Park, J. W. Kwon, and S.-G. Lee, “Removal of Hot Spot Speckle on Rear Projection Screen Using the Rotating Screen System,” J. Display Technol.2(1), 79–84 (2006).
    [CrossRef]
  5. Y.-H. Park, K.-H. Ha, J.-O. Kim, and Y.-K. Mun, “Speckle reduction laser and laser display apparatus having the same, ” U.S. patent, 7,489,714(Feb.10, 2009).
  6. S. Lowenthal and D. J. H. Arsenault, “Relation entre le deplacement fini d’un diffuseur mobile, eclaire par un laser, et le rapport signal sur bruit dans l’eclaisremenst observe distance finie oudans ou dans un plan image,” Opt. Commun.2(4), 184–188 (1970).
    [CrossRef]
  7. S. Lowenthal and E. Joyeu, “Speckle removal by a slowly moving diffuser associated with a motionless Diffuser,” J. Opt. Soc. Am.61(7), 847–851 (1971).
    [CrossRef]
  8. J. W. Goodman, “Speckle Phenonmena in optics: theory and applications,” (Roberts & Company, Englewood, 2007).
  9. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge: Cambridge, 1995).

2006 (1)

2004 (1)

1998 (1)

1971 (1)

1970 (1)

S. Lowenthal and D. J. H. Arsenault, “Relation entre le deplacement fini d’un diffuseur mobile, eclaire par un laser, et le rapport signal sur bruit dans l’eclaisremenst observe distance finie oudans ou dans un plan image,” Opt. Commun.2(4), 184–188 (1970).
[CrossRef]

Arsenault, D. J. H.

S. Lowenthal and D. J. H. Arsenault, “Relation entre le deplacement fini d’un diffuseur mobile, eclaire par un laser, et le rapport signal sur bruit dans l’eclaisremenst observe distance finie oudans ou dans un plan image,” Opt. Commun.2(4), 184–188 (1970).
[CrossRef]

Halldórsson, T.

Joyeu, E.

Kwon, J. W.

Lee, S. Y.

Lee, S.-G.

Lowenthal, S.

S. Lowenthal and E. Joyeu, “Speckle removal by a slowly moving diffuser associated with a motionless Diffuser,” J. Opt. Soc. Am.61(7), 847–851 (1971).
[CrossRef]

S. Lowenthal and D. J. H. Arsenault, “Relation entre le deplacement fini d’un diffuseur mobile, eclaire par un laser, et le rapport signal sur bruit dans l’eclaisremenst observe distance finie oudans ou dans un plan image,” Opt. Commun.2(4), 184–188 (1970).
[CrossRef]

Park, C.-Y.

Park, S.-Y.

Pétursson, P. R.

Shin, S. C.

Trisnadi, J. I.

Tschudi, T.

Wang, L. L.

Yoo, S. S.

Appl. Opt. (1)

J. Display Technol. (1)

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

S. Lowenthal and D. J. H. Arsenault, “Relation entre le deplacement fini d’un diffuseur mobile, eclaire par un laser, et le rapport signal sur bruit dans l’eclaisremenst observe distance finie oudans ou dans un plan image,” Opt. Commun.2(4), 184–188 (1970).
[CrossRef]

Opt. Lett. (1)

Other (4)

J. W. Goodman, “Speckle Phenonmena in optics: theory and applications,” (Roberts & Company, Englewood, 2007).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge: Cambridge, 1995).

Y.-H. Park, K.-H. Ha, J.-O. Kim, and Y.-K. Mun, “Speckle reduction laser and laser display apparatus having the same, ” U.S. patent, 7,489,714(Feb.10, 2009).

S. V. Govorkov and L. A. Spinelli, “Speckle reduction in laser illuminated projection displays having a one dimensional spatial light modulator, ” U.S. Patent, 7,413,311 (Aug.19,2008).

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Figures (6)

Fig. 1
Fig. 1

Normalized coherence time as a function of the diffuser’s phase covariance

Fig. 2
Fig. 2

Normalized coherence area of light scattered from diffuser as a function of phase covariance.

Fig. 3
Fig. 3

A laser beam scattered by a moving diffuser

Fig. 4
Fig. 4

(a) the study of coherence by Michelson interferometer (b) diffuser with σ ϕ >>1

Fig. 5
Fig. 5

the coherence length of light scattered from dynamic diffuser vs. the speed of dynamic diffuser.

Fig. 6
Fig. 6

Interference pattern of light scattered from a moving diffuser obtained by Michelson interferometer ( v=0.12m/s )

Equations (22)

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γ ˜ (Δα,Δβ,τ)=exp{ σ ϕ 2 [ 1exp{ (Δα+vτ) 2 +Δ β 2 ) r ϕ 2 } ] }(Δα= α 1 α 2 ,Δβ= β 1 β 2 )
τ c τ 0 = + γ dt= + γ ˜ - e σ ϕ 2 1 e σ ϕ 2 dt= e σ ϕ 2 1 e σ ϕ 2 + (exp[ σ ϕ 2 exp(π t 2 )]1) dt
τ c τ 0 = π r ϕ 2 /v .
A c = γ ˜ - e σ ϕ 2 1 e σ ϕ 2 dΔαdΔβ= e σ ϕ 2 1 e σ ϕ 2 ( exp{ σ ϕ 2 exp[ ( Δ α 2 +Δ β 2 r ϕ ) 2 ] }1 ) dΔαdΔβ
a(α,β;t)= a 0 exp[j ϕ d (αvt,β)]exp[jθ(α,β;t)]exp(jωt)
Γ ˜ ( α 1 , β 1 , α 2 , β 2 ;τ;T)= 1 T 0 T | a 0 | 2 exp[j ϕ d ( α 1 vt, β 1 )]exp[j ϕ d ( α 2 v(t+τ), β 2 )] exp[jθ( α 1 , β 1 ;t)]exp[jθ( α 2 , β 2 ;t+τ)]dt
Γ ˜ ( α 1 , β 1 , α 2 , β 2 ;τ;T)=exp[j[θ( α 2 , β 2 ;t)θ( α 1 , β 1 ;t+τ)] 1 T 0 T | a 0 | 2 exp[j ϕ d ( α 1 vt, β 1 )]exp[j ϕ d ( α 2 v(t+τ), β 2 )] dt
τ 0 <<T~ τ l
Γ ˜ ( α 1 , β 1 , α 2 , β 2 ;τ;T)= 1 T i=1 N τ i | a 0 | 2 exp[j ϕ d ( α 1 v t i , β 1 )]exp[j ϕ d ( α 2 v( t i +τ), β 2 )]exp[jθ( α 1 , β 1 ; t i )]exp[jθ( α 2 , β 2 ; t i +τ)]d t i
Γ ˜ ( α 1 , β 1 , α 2 , β 2 ;τ;T)= 1 T i=1 N | a 0 | 2 exp[j ϕ d ( α 1 v t i , β 1 )]exp[j ϕ d ( α 2 v( t i +τ), β 2 )] τ i 1 τ i τ i exp[jθ( α 1 , β 1 ; t i )]exp[jθ( α 2 , β 2 ; t i +τ)]d t i
1 τ 1 τ 1 ()d t 1 = 1 τ 2 τ 2 ()d t 2 = = 1 τ N τ N ()d t N =μ(Δα,Δβ)exp( τ 2 τ l 2 )
Γ ˜ ( α 1 , β 1 , α 2 , β 2 ;τ;T)={ 1 T i=1 N | a 0 | 2 exp[j ϕ d ( α 1 v t i , β 1 )] exp[j ϕ d ( α 2 v( t i +τ), β 2 )] τ i } exp( τ 2 τ l 2 )
T>> τ i >> τ l
Γ ˜ ( α 1 , β 1 , α 2 , β 2 ;τ;T)={ 1 T 0 T | a 0 | 2 exp[j ϕ d ( α 1 vt, β 1 )]exp[j ϕ d ( α 2 v(t+τ), β 2 )dt}exp( τ 2 τ l 2 ) = | a 0 | 2 exp[j ϕ d ( α 1 vt, β 1 )]exp[j ϕ d ( α 2 v(t+τ), β 2 )] ¯ exp( τ 2 τ l 2 )
γ ˜ (Δα,Δβ,τ)=exp{ σ ϕ 2 [ 1exp{ (Δα+vτ) 2 +Δ β 2 ) r ϕ 2 } ] }exp( τ 2 τ l 2 )
1 T 0 2T (1- τ 2T )|γ(τ) | 2 dτ 0,
γ ˜ =(1-e - σ ϕ 2 )γ +e - σ ϕ 2
Σ A -1 γ( x 1 - x 2 , y 1 - y 2 ) ψ i ( x 2 , y 2 )d x 2 d y 2 = λ i ψ i ( x 2 , y 2 )
γ ˜ =( 1-e - σ ϕ 2 )( λ 1 ... λ i ... λ N ) +e - σ ϕ 2 ( 1 0 ... 0 0 )
Tr( γ A )= i=1 N λ i =1
C= σ I ¯ = (1 e - σ ϕ 2 ) 2 i=1 N λ i 2 +2(1 e - σ ϕ 2 ) e - σ ϕ 2 λ 1 + e -2 σ ϕ 2
C=(1 e - σ ϕ 2 ) 1 N + 2 N( e σ ϕ 2 1) + 1 ( e σ ϕ 2 1) 2

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