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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 11 — Jun. 3, 2013
  • pp: 13032–13039
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Coherence theory of a laser beam passing through a moving diffuser

Gaoming Li, Yishen Qiu, and Hui Li  »View Author Affiliations


Optics Express, Vol. 21, Issue 11, pp. 13032-13039 (2013)
http://dx.doi.org/10.1364/OE.21.013032


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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.

© 2013 OSA

1. Introduction

In this paper, we present a modified coherence theory that takes into account the finite coherence time of the laser. According to our theory, when the laser coherence time is much shorter than the mean time it takes the diffuser to move past a phase correlation area, the spatial coherence area of light scattered from a moving diffuser will decrease while the temporal coherence remains unchanged. This conclusion was confirmed by coherent experiments using a Michelson interferometer. It can also be shown that our theory agrees with the original theory under the condition that the laser coherence time is much longer than the mean time it takes the diffuser to move past a phase correlation area. In addition, a method is developed based on the theory of eigenvalues to calculate the observed speckle contrast on a screen illuminated by light passing through a moving diffuser.

2. Original coherence theory of light transmitted through a moving diffuser

In the original theory, the diffuser is assumed to have uniform transparency and a statistically steady phase distribution described by an Gaussian random process. When a monochromatic light beam passes through a moving diffuser, the normalized mutual coherence function between two arbitrary points on the diffuser [(α1,β1) and (α2,β2)] is given by [6

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]

]:
γ˜(Δα,Δβ,τ)=exp{σϕ2[1exp{(Δα+vτ)2+Δβ2)rϕ2}]}(Δα=α1α2,Δβ=β1β2)
(1)
where τ, σϕ, rϕ and v are the time delay, phase covariance of the diffuser, radius of the phase correlation area and diffuser speed, respectively. The presence of a direct component of light passing through a moving diffuser would cause the integral of Eq. (1) (with respect to τ) to diverge. Therefore it is necessary to subtract off the direct component while calculating the coherence time and re-normalize the integrand so that it is equal to unity at τ=0 (Eq. (2)). Thus, the coherence time τc is defined as [6

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]

]:
τcτ0=+γdt=+γ˜-eσϕ21eσϕ2dt=eσϕ21eσϕ2+(exp[σϕ2exp(πt2)]1)dt
(2)
where τ0=πrϕ2/v is the mean time it takes the diffuser to move past a phase correlation area. Plotting τc/τ0 versus σϕ (Fig. 1
Fig. 1 Normalized coherence time as a function of the diffuser’s phase covariance
) shows the coherence time decreases with the increase of phase covariance. In addition, for fixed σϕ the coherence time is inversely proportional to the diffuser speed:

τcτ0=πrϕ2/v.
(3)

The spatial coherence area Ac is given by [8

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

]

Ac=γ˜-eσϕ21eσϕ2dΔαdΔβ=eσϕ21eσϕ2(exp{σϕ2exp[(Δα2+Δβ2rϕ)2]}1)dΔαdΔβ
(4)

The Ac/πrϕ2 vs σϕ plot (Fig. 2
Fig. 2 Normalized coherence area of light scattered from diffuser as a function of phase covariance.
) shows that Ac decreases in sigmoidal fashion with the increase of σϕ. In addition, Ac has the same order of magnitude as πrϕ2 only for sufficiently small values of σϕ [8

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

]. When the diffuser starts to move, one can expect the spatial coherence area to be given by Eq. (4), regardless of the speed. On the other hand, the spatial coherence area of light scattered from a motionless diffuser is throughout the entire light beam. This apparent discontinuity arises from the fact that the mutual coherence function (Eq. (1)) is obtained by integrating from to + with respect to time [8

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

]. However, the infinite coherence time of monochromatic light is logically inconsistent with the integration limits being infinities. In order to resolve this issue, a new coherence theory needs to be developed in which both the coherence time and integrated time are finite.

3. The new coherence theory

Consider a system shown in Fig. 3
Fig. 3 A laser beam scattered by a moving diffuser
. We assume that like before the phase distribution of diffuser is a statistically steady Guassian random process. The complex amplitude of light passing through a moving diffuser at any given point (α,β) and time t may be expressed as
a(α,β;t)=a0exp[jϕd(αvt,β)]exp[jθ(α,β;t)]exp(jωt)
(5)
where ω, ϕd(αvt,β), v and θ(α,β,t) are the laser center frequency, phase of the moving diffuser, diffuser speed and random phase of the incident polarized light, respectively.

For finite observation time, the mutual coherence function is given by
Γ˜(α1,β1,α2,β2;τ;T)=1T0T|a0|2exp[jϕd(α1vt,β1)]exp[jϕd(α2v(t+τ),β2)]exp[jθ(α1,β1;t)]exp[jθ(α2,β2;t+τ)]dt
(6)
where τis the time delay and T is the observation time (response time of the detector). We use τ0 and τl to denote the characteristic temporal scales of ϕd(αvt,β)and θ(α,β,t), respectively. The latter is the same as the coherence time of the laser. Due to the statistical correlation between ϕd(αvt,β) and θ(α,β,t), Eq. (6) is not integrable when τ0has the same order of magnitude as τl.

On the other hand, when τl>>τ0 the random variation of θ(α,β,t) is slow compared to that of ϕd(αvt,β) and the corresponding exponential term can be moved out of the integral in Eq. (6).

Γ˜(α1,β1,α2,β2;τ;T)=exp[j[θ(α2,β2;t)θ(α1,β1;t+τ)]1T0T|a0|2exp[jϕd(α1vt,β1)]exp[jϕd(α2v(t+τ),β2)]dt
(7)

Equation (7) shows the coherence of transmitted light depends on the phase correlation of the moving diffuser. The temporal and spatial coherence are described by Eq. (2) and Eq. (4), respectively, when
τ0<<T~τl
(8)
Unlike Goodman’ explanation, Eq. (4) & (7) don’t require infinite observation time.

In most practical applications, the laser has a finite coherence time. Using τi to denote the time it takes the diffuser to move past the i-th phase correlation area, Eq. (6) may be rewritten as:
Γ˜(α1,β1,α2,β2;τ;T)=1Ti=1Nτi|a0|2exp[jϕd(α1vti,β1)]exp[jϕd(α2v(ti+τ),β2)]exp[jθ(α1,β1;ti)]exp[jθ(α2,β2;ti+τ)]dti
(9)
Given the typical values of τi~τ0=πrϕ2/v (103105s) and τl (1081011s), the factor exp[jϕd(α1vti,β1)]exp[jϕd(α2v(ti+τ),β2)] is slowly varying compared with the fluctuation time of θ(α1,β1;ti). Moving it out of the integral of Eq. (9), we have
Γ˜(α1,β1,α2,β2;τ;T)=1Ti=1N|a0|2exp[jϕd(α1vti,β1)]exp[jϕd(α2v(ti+τ),β2)]τi1τiτiexp[jθ(α1,β1;ti)]exp[jθ(α2,β2;ti+τ)]dti
(10)
wheredtiand τi can be interpreted in the context of statistical physics as the fine-grained and coarse-grained time scales, respectively. The integral in Eq. (10) gives the normalized mutual coherence function of the laser for a finite observation time τi. Since τi>>τl, the random phase change of the laser is statistically steady and may be described by an Gaussian process in a finite observation time τi. Assuming the laser to be a pure cross-spectral Gaussian source, we have
1τ1τ1()dt1=1τ2τ2()dt2==1τNτN()dtN=μ(Δα,Δβ)exp(τ2τl2)
(11)
whereμ(Δα,Δβ) is the complex coherence factor of the laser and is usually equal to unity for the TEM00 mode commonly used in optical information processing and laser display. Substituting Eq. (11) into Eq. (10) gives

Γ˜(α1,β1,α2,β2;τ;T)={1Ti=1N|a0|2exp[jϕd(α1vti,β1)]exp[jϕd(α2v(ti+τ),β2)]τi}exp(τ2τl2)
(12)

γ˜(Δα,Δβ,τ)=exp{σϕ2[1exp{(Δα+vτ)2+Δβ2)rϕ2}]}exp(τ2τl2)
(15)

Compared with Eq. (1), the Eq. (15) contains an extra factor describing the laser temporal coherence. By setting Δα=Δβ=0 in Eq. (15), we see that the temporal coherence of light transmitted through a moving diffuser is determined only by the laser provided that the laser coherence time τl is much shorter than τc, which is a requirement satisfied in most applications involving a moving diffuser. On the other hand, when the incident beam is monochromatic with an infinite coherence time, the temporal coherence of the transmitted light is determined by the moving diffuser, in accordance with the original theory.

By setting τ=0 in Eq. (15), we see that the spatial coherence area becomes the same as that given by Eq. (4). It is important to note that in the new theory, Eq. (13) is a necessary and sufficient condition for Eq. (14) and (15) . Recall that the original theory is valid only when the observation time is much longer than the laser coherence time [9

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

], i.e.

1T02T(1-τ2T)|γ(τ)|2dτ0,
(16)

When the diffuser motion is sufficiently slow within a finite observation time period, the mutual coherence function should be given by Eq. (12) rather than Eq. (15). This results in an uncertainty in the coherence area which increases with decreasing diffuser speed. When the diffuser is stationary, The spatial coherence area is no longer given by making v=0 in Eq. (4), since the transmitted light is now coherent throughout the entire light beam.

4. The contrast of speckle on a screen illuminated by light transmitted through a moving diffuser

In the representation furnished by the eigenfunctions of γ, γ˜ takes the form of a diagonal matrix
γ˜=(1-e-σϕ2)(λ1...λi...λN)+e-σϕ2(10...00)
(19)
which satisfies

Tr(γA)=i=1Nλi=1
(20)

The speckle contrast on the screen as observed by the human eye is given by
C=σI¯=(1e-σϕ2)2i=1Nλi2+2(1e-σϕ2)e-σϕ2λ1+e-2σϕ2
(21)
whenλ1=λ2==λN=1/N, Eq. (21) reduces to
C=(1e-σϕ2)1N+2N(eσϕ21)+1(eσϕ21)2
(22)
for a pure scatterer where σϕ>>1, Eq. (22) reduces to C=1/N.

One explanation for the speckle contrast reduction is the decrease in the spatial coherence of light passing through the moving diffuser. Equivalently, it can also be attributed to the average effect of N randomly moving images over the observation time T. Note that τc ~T /N is the correlation time of this spatio-temporal correlation speckle pattern, as the diffuser moves faster, the correlation time decreases. It has the same meaning as τ0 in the article and should be differentiated from the laser coherence time.

5. Experiment

To confirm our theory, a Michelson-type optical arrangement shown in Fig. 4
Fig. 4 (a) the study of coherence by Michelson interferometer (b) diffuser with σϕ>>1
was used. The laser was a 488 nm CW TEM00 Ar + (U.S., Spectra-Physics Inc., Model 177-G12) with a coherence length of 4.6 cm. The diffuser (CA, Luminit Inc., LSD) had a phase covariance of σϕ>>1 and may be regarded as a pure scatterer. The diffuser was undergoing reciprocating linear motion throughout the experiment and the incident beam was normal to the diffuser. The diameter of the laser spot was parallel expanded to 5 mm by an expander lens (China, Daheng Group, Inc., GCO-2501) positioned in front of the diffuser. A convergent lens was used to collect light into a Michelson interferometer, and a CCD camera (Japan, Nikon Inc., DS-U2) was used to record interference patterns.

As the diffuser speed increased, the coherence length of light transmitted through the diffuser was obtained by adjusting the position of the reflector in the Michelson interferometer. The coherence length was measured by the following method: move the one of the reflectors in the Michelson setup forward to the point where the fringe disappears (Point 1). Then move it backward to the point where the fringe disappears again (Point 2). The coherence length will be equal to half the distance between points 1 and 2. Despite the 5% relative error in our results (Fig. 5
Fig. 5 the coherence length of light scattered from dynamic diffuser vs. the speed of dynamic diffuser.
), we are able to demonstrate the temporal coherence of the laser beam passing through the diffuser does not change with the diffuser speed, however, according to the original theory, the coherence time is inversely proportional to the diffuser speed (Eq. (2)).

In addition, reduction of speckle contrast in the interference patterns (Fig. 6
Fig. 6 Interference pattern of light scattered from a moving diffuser obtained by Michelson interferometer (v=0.12m/s)
) was observed as the diffuser speed increased, which is indicative of a decrease in the spatial coherence of light passing through the diffuser. These results agree with our theoretical predictions.

6. Conclusion

Acknowledgments

This work was supported by Program for Changjiang Scholars and Innovative Research Team in University (No: IRT1115), Key Industry Project of Fujian Province of China (No: 2012H6007)

References and links

1.

J. I. Trisnadi, “Hadamard speckle contrast reduction,” Opt. Lett. 29(1), 11–13 (2004). [CrossRef] [PubMed]

2.

L. L. Wang, T. Tschudi, T. Halldórsson, and P. R. Pétursson, “Speckle reduction in laser projection systems by diffractive optical elements,” Appl. Opt. 37(10), 1770–1775 (1998). [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).

OCIS Codes
(030.1640) Coherence and statistical optics : Coherence
(030.6140) Coherence and statistical optics : Speckle
(140.3295) Lasers and laser optics : Laser beam characterization

ToC Category:
Coherence and Statistical Optics

History
Original Manuscript: March 27, 2013
Revised Manuscript: May 7, 2013
Manuscript Accepted: May 14, 2013
Published: May 20, 2013

Citation
Gaoming Li, Yishen Qiu, and Hui Li, "Coherence theory of a laser beam passing through a moving diffuser," Opt. Express 21, 13032-13039 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-11-13032


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References

  1. J. I. Trisnadi, “Hadamard speckle contrast reduction,” Opt. Lett.29(1), 11–13 (2004). [CrossRef] [PubMed]
  2. L. L. Wang, T. Tschudi, T. Halldórsson, and P. R. Pétursson, “Speckle reduction in laser projection systems by diffractive optical elements,” Appl. Opt.37(10), 1770–1775 (1998). [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).

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