Coherence theory of a laser beam passing through a moving diffuser

Gaoming Li; Yishen Qiu; Hui Li

doi:10.1364/OE.21.013032

1. Introduction

The laser display technique has seen widespread applications due to its extensive color coverage and high luminance. However, one major drawback with this technique is that the image quality is often affected by speckles resulting from coherent light. Various methods of speckle reduction have been proposed [1–5], among which the employment of a moving diffuser is simple and effective and has been adopted since the early days of holographic and optical information processing. In 1970, Lowenthal et al. developed the theory of speckle reduction with the use of a moving diffuser [6]. Later they proposed using a two-diffuser system to eliminate speckles and realized it experimentally [7]. Goodman developed the coherence theory of monochromatic light passing through a moving diffuser [8]. According to the theory, both the temporal and spatial coherence of monochromatic light passing through a moving diffuser decrease because of the random phase distribution of the moving diffuser. Since monochromatic light has infinite coherence length, the theory only takes into consideration the mean time it takes the diffuser to move past a phase correlation area, making it unsuitable for explaining speckle-related coherence where the coherence time of laser becomes an equally important characteristic temporal scale.

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 [ $(α_{1}, β_{1})$ and $(α_{2}, β_{2})$ ] is given by [6]:

\tilde{γ} (Δ α, Δ β, τ) = \exp {- σ_{ϕ}^{2} [1 - \exp {- \frac{{(Δ α + v τ)}^{2} + Δ β^{2})}{r_{ϕ}^{2}}}]} (Δ α = α_{1} - α_{2}, Δ β = β_{1} - β_{2})

where

τ

,

σ_{ϕ}^{}

,

r_{ϕ}

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

τ_{c}

is defined as [6]:

\frac{τ_{c}}{τ_{0}} = \int_{- \infty}^{+ \infty} γ d t = \int_{- \infty}^{+ \infty} \frac{\tilde{γ} - e^{- σ_{ϕ}^{2}}}{1 - e^{- σ_{ϕ}^{2}}} d t = \frac{e^{- σ_{ϕ}^{2}}}{1 - e^{- σ_{ϕ}^{2}}} \int_{- \infty}^{+ \infty} (\exp [σ_{ϕ}^{2} \exp (- π t^{2})] - 1) d t

where

τ_{0} = \sqrt{π r_{ϕ}^{2}} / v

is the mean time it takes the diffuser to move past a phase correlation area. Plotting

τ_{c} / τ_{0}

versus

σ_{ϕ}

(Fig. 1) shows the coherence time decreases with the increase of phase covariance. In addition, for fixed

σ_{ϕ}

the coherence time is inversely proportional to the diffuser speed:

Fig. 1 Normalized coherence time as a function of the diffuser’s phase covariance

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τ_{c} \propto τ_{0} = \sqrt{π r_{ϕ}^{2}} / v .

The spatial coherence area $A_{c}$ is given by [8]

A_{c} = \iint \frac{\tilde{γ} - e^{- σ_{ϕ}^{2}}}{1 - e^{- σ_{ϕ}^{2}}} d Δ α d Δ β = \frac{e^{- σ_{ϕ}^{2}}}{1 - e^{- σ_{ϕ}^{2}}} \int \int_{- \infty}^{\infty} (\exp {σ_{ϕ}^{2} \exp [- {(\frac{\sqrt{Δ α^{2} + Δ β^{2}}}{r_{ϕ}})}^{2}]} - 1) d Δ α d Δ β

The $A_{c} / π r_{ϕ}^{2}$ vs $σ_{ϕ}$ plot (Fig. 2) shows that $A_{c}$ decreases in sigmoidal fashion with the increase of $σ_{ϕ}$ . In addition, $A_{c}$ has the same order of magnitude as $π r_{ϕ}^{2}$ only for sufficiently small values of $σ_{ϕ}$ [8]. 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 $- \infty$ to $+ \infty$ with respect to time [8]. 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.

Fig. 2 Normalized coherence area of light scattered from diffuser as a function of phase covariance.

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3. The new coherence theory

Consider a system shown in Fig. 3. 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) = a_{0} \exp [j ϕ_{d} (α - v t, β)] \exp [j θ (α, β; t)] \exp (- j ω t)

where

ω

,

ϕ_{d} (α - v t, 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.

Fig. 3 A laser beam scattered by a moving diffuser

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For finite observation time, the mutual coherence function is given by

\begin{array}{l} \tilde{Γ} (α_{1}, β_{1}, α_{2}, β_{2}; τ; T) = \frac{1}{T} \int_{0}^{T} {| a_{0} |}^{2} \exp [j ϕ_{d} (α_{1} - v t, β_{1})] \exp [- j ϕ_{d} \cdot (α_{2} - v (t + τ), β_{2})] \cdot \\ \begin{matrix}  \end{matrix} \begin{matrix}  \end{matrix} \exp [j θ (α_{1}, β_{1}; t)] \exp [- j θ (α_{2}, β_{2}; t + τ)] d t \end{array}

where

τ

is the time delay and T is the observation time (response time of the detector). We use

τ_{0}

and

τ_{l}

to denote the characteristic temporal scales of

ϕ_{d} (α - v t, 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

ϕ_{d} (α - v t, β)

and

θ (α, β, t)

, Eq. (6) is not integrable when

τ_{0}

has the same order of magnitude as

τ_{l}

.

On the other hand, when $τ_{l} > > τ_{0}$ the random variation of $θ (α, β, t)$ is slow compared to that of $ϕ_{d} (α - v t, β)$ and the corresponding exponential term can be moved out of the integral in Eq. (6).

\begin{array}{l} \tilde{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)] \cdot \\ \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \frac{1}{T} \int_{0}^{T} {| a_{0} |}^{2} \exp [j ϕ_{d} (α_{1} - v t, β_{1})] \exp [- j ϕ_{d} (α_{2} - v (t + τ), β_{2})] d t \end{array}

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

τ_{0} < < T ~ τ_{l}

Unlike Goodman’ explanation, Eq. (4) & (7) don’t require infinite observation time.

In most practical applications, the laser has a finite coherence time. Using $τ_{i}$ to denote the time it takes the diffuser to move past the i-th phase correlation area, Eq. (6) may be rewritten as:

\begin{array}{l} \tilde{Γ} (α_{1}, β_{1}, α_{2}, β_{2}; τ; T) = \frac{1}{T} \sum_{i = 1}^{N} \int_{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), \\ \begin{matrix}  \end{matrix} β_{2})] \exp [j θ (α_{1}, β_{1}; t_{i})] \exp [- j θ (α_{2}, β_{2}; t_{i} + τ)] d t_{i}^{} \end{array}

Given the typical values of

τ_{i} ~ τ_{0} = \sqrt{π r_{ϕ}^{2}} / v

(

10^{- 3} - 10^{- 5} s

) and

τ_{l}

(

10^{- 8} - 10^{- 11} s

), the factor

\exp [j ϕ_{d} (α_{1} - v t_{i}, β_{1})] \exp [- j ϕ_{d} (α_{2} - v (t_{i} + τ), β_{2})]

is slowly varying compared with the fluctuation time of

θ (α_{1}, β_{1}; t_{i})

. Moving it out of the integral of Eq. (9), we have

\begin{array}{l} \tilde{Γ} (α_{1}, β_{1}, α_{2}, β_{2}; τ; T) = \frac{1}{T} \sum_{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} \cdot \\ \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \frac{1}{τ_{i}} \int_{τ_{i}}^{} \exp [j θ (α_{1}, β_{1}; t_{i})] \exp [- j θ (α_{2}, β_{2}; t_{i} + τ)] d t_{i}^{} \end{array}

where

d t_{i}

and

τ_{i}

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

τ_{i}

. Since

τ_{i} > > τ_{l}

, the random phase change of the laser is statistically steady and may be described by an Gaussian process in a finite observation time

τ_{i}

. Assuming the laser to be a pure cross-spectral Gaussian source, we have

\frac{1}{τ_{1}} \int_{τ_{1}}^{} (\cdot \cdot \cdot) d t_{1}^{} = \frac{1}{τ_{2}} \int_{τ_{2}}^{} (\cdot \cdot \cdot) d t_{2}^{} = \cdot \cdot \cdot = \frac{1}{τ_{N}} \int_{τ_{N}}^{} (\cdot \cdot \cdot) d t_{N}^{} = μ (Δ α, Δ β) \exp (- \frac{τ^{2}}{τ_{l}^{2}})

where

μ (Δ α, Δ β)

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

\begin{array}{l} \tilde{Γ} (α_{1}, β_{1}, α_{2}, β_{2}; τ; T) = {\frac{1}{T} \sum_{i = 1}^{N} {| a_{0} |}^{2} \exp [j ϕ_{d} (α_{1} - v t_{i}, β_{1})] \cdot \exp [- j ϕ_{d} (α_{2} - v (t_{i} + τ), β_{2})] τ_{i}} \\ \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \cdot \exp (- \frac{τ^{2}}{τ_{l}^{2}}) \end{array}

When the observation time is much longer than the mean time it takes the diffuser to move past a phase correlation area, namely

T > > τ_{i} > > τ_{l}

The first factor of Eq. (12) may be expressed as an integral with respect to coarse-grained time

\begin{array}{l} \tilde{Γ} (α_{1}, β_{1}, α_{2}, β_{2}; τ; T) = {\frac{1}{T} \int_{0}^{T} {| a_{0} |}^{2} \exp [j ϕ_{d} (α_{1} - v t, β_{1})] \exp [- j ϕ_{d} (α_{2} - v (t + τ), β_{2}) \cdot d t} \exp (- \frac{τ^{2}}{τ_{l}^{2}}) \\ \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} = {| a_{0} |}^{2} \bar{\exp [j ϕ_{d} (α_{1} - v t, β_{1})] \exp [- j ϕ_{d} (α_{2} - v (t + τ), β_{2})]} \cdot \exp (\frac{τ^{2}}{τ_{l}^{2}}) \end{array}

Assuming the height related function of the diffuser surface to be Gaussian, the normalized mutual coherence of light passing through a moving diffuser can be written as [8]

\tilde{γ} (Δ α, Δ β, τ) = \exp {- σ_{ϕ}^{2} [1 - \exp {- \frac{{(Δ α + v τ)}^{2} + Δ β^{2})}{r_{ϕ}^{2}}}]} \exp (- \frac{τ^{2}}{τ_{l}^{2}})

Compared with Eq. (1), the Eq. (15) contains an extra factor describing the laser temporal coherence. By setting $Δ α = Δ β = 0$ 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 $τ_{l}$ is much shorter than $τ_{c}$ , 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], i.e.

\frac{1}{T} \int_{0}^{2 T} (1 - \frac{τ}{2 T}) | γ (τ) |^{2} d τ \to 0,

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

The previous discussions about the spatial coherence area are valid only when the diffuser is a pure scatterer. In the presence of a coherent direct component, the spatial coherence cannot be characterized by a localized coherence area. As a result, the speckle contrast on a screen illuminated by transmitted light through a moving diffuser is no longer given by $1 / \sqrt{N}$ , where N is the ratio of the resolution area of the human eye to the coherence area incident on the screen [6]. In such cases, a method based on the theory of eigenvalues can be developed to calculate the speckle contrast. First, we rewrite the relation between the normalized mutual coherence function and mutual covariance based on Eq. (2)

\tilde{γ} {=(1-e}^{- σ_{ϕ}^{2}}) γ {+e}^{- σ_{ϕ}^{2}}

The eigenvalue equation with kernel

γ

reads

\iint_{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})

where λ_i and

ψ_{i}

(i = 1,2,……N) are the i-th eigenvalue and eigenfunction of Eq. (18), respectively, and A represents the minimum area on the screen that can be resolved by the human eye. The total number of eigenfunctions is given by

N \approx A / A_{c}

with A_c being the coherence area of the transmitted light illuminating on the screen.

In the representation furnished by the eigenfunctions of $γ$ , $\tilde{γ}$ takes the form of a diagonal matrix

\tilde{γ} = ({1-e}^{- σ_{ϕ}^{2}}) (\begin{matrix} λ_{1} \\ ... \\ λ_{i} \\ ... \\ λ_{N} \end{matrix}) {+e}^{- σ_{ϕ}^{2}} (\begin{matrix} 1 \\ 0 \\ ... \\ 0 \\ 0 \end{matrix})

which satisfies

T r (\frac{γ}{A}) = \sum_{i = 1}^{N} λ_{i} = 1

The speckle contrast on the screen as observed by the human eye is given by

C = \frac{σ}{\bar{I}} = \sqrt{{(1 - e^{- σ_{ϕ}^{2}})}^{2} \sum_{i = 1}^{N} λ_{i}^{2} + 2 (1 - e^{- σ_{ϕ}^{2}}) e^{- σ_{ϕ}^{2}} λ_{1} + e^{-2 σ_{ϕ}^{2}}}

when

λ_{1} = λ_{2} = \cdot \cdot \cdot \cdot \cdot \cdot = λ_{N} =1/ N

, Eq. (21) reduces to

C = (1 - e^{- σ_{ϕ}^{2}}) \sqrt{\frac{1}{N} + \frac{2}{N (e^{σ_{ϕ}^{2}} - 1)} + \frac{1}{{(e^{σ_{ϕ}^{2}} - 1)}^{2}}}

for a pure scatterer where

σ_{ϕ} > > 1

, Eq. (22) reduces to

C = 1 / \sqrt{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 τ_c ~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 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 $σ_{ϕ} > > 1$ 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.

Fig. 4 (a) the study of coherence by Michelson interferometer (b) diffuser with $σ_{ϕ} > > 1$

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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), 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)).

Fig. 5 the coherence length of light scattered from dynamic diffuser vs. the speed of dynamic diffuser.

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In addition, reduction of speckle contrast in the interference patterns (Fig. 6) 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.

Fig. 6 Interference pattern of light scattered from a moving diffuser obtained by Michelson interferometer ( $v = 0.12 m / s$ )

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6. Conclusion

A general coherence theory of light transmitted through an moving diffuser is presented that takes into consideration the temporal coherence of the laser. The theory shows that the coherence time is in general determined only by the laser, while the coherence area is closely related to the phase correlation area of the moving diffuser. The conclusions have been confirmed by coherent experiments using a Michelson interferometer. The experiments also showed speckle contrast reduction as a result of the decrease in the spatial coherence of light passing through a moving diffuser. In addition, a method based on the theory of eigenvalues was developed to calculate the speckle contrast on a screen illuminated by light transmitted through a moving diffuser.

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

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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]

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Coherence theory of a laser beam passing through a moving diffuser

Abstract

1. Introduction

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

3. The new coherence theory

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

5. Experiment

6. Conclusion

Acknowledgments

References and links

Cited By

Figures (6)

Equations (22)

Optics Express