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Noise in laser speckle correlation and imaging techniques

S. E. Skipetrov, J. Peuser, R. Cerbino, P. Zakharov, B. Weber, and F. Scheffold

Open Access Open Access

Abstract

We study the noise of the intensity variance and of the intensity correlation and structure functions measured in light scattering from a random medium in the case when these quantities are obtained by averaging over a finite number N of pixels of a digital camera. We show that the noise scales as 1/N in all cases and that it is sensitive to correlations of signals corresponding to adjacent pixels as well as to the effective time averaging (due to the finite integration time) and spatial averaging (due to the finite pixel size). Our results provide a guide to estimation of noise levels in such applications as multi-speckle dynamic light scattering, time-resolved correlation spectroscopy, speckle visibility spectroscopy, laser speckle imaging etc.

©2010 Optical Society of America

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Figures (7)
Fig. 1.
Fig. 1. Experimental setup. A flat sample is illuminated with a polarized expanded laser beam. The diffuse reflected light is detected in the cross polarization channel by imaging the surface of the sample with a digital camera. The size of individual speckle spots in the image plane can be adjusted by changing the aperture of the camera objective.

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Fig. 2.
Fig. 2. Schematic representation of the matrix of pixels of a digital camera. The full matrix is divided in a set of meta-pixels Nx × Ny = N pixels each. Spatially-varying statistical properties of the speckle pattern imaged by the camera (i.e., the variance of intensities c) are estimated by averaging over all pixels within the same meta-pixel. In our calculation (Sec. 2.3), we are taking into account correlations between intensities at neighboring pixels in all directions. A pixel which is not at the boundary of the square matrix (pixel 1 in the figure) has 8 neighbors: 4 neighbors of type 2 and 4 neighbors of type 3.

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Fig. 3.
Fig. 3. Left: False-color image of light intensities in a speckle pattern (60 × 60 pixels) obtained by using the smallest available aperture setting (f/# = 32). Intensity scale from 0 to 104 in arbitrary units. Right: Intensity correlation function obtained by the inverse Fourier transform of the speckle power spectrum (full frame 640×480 pixels). The data is quantitatively described by Eq. (13) with b = 1.24a and μ = 1.09.

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Fig. 4.
Fig. 4. Speckle parameter μ as a function of the speckle size b divided by the size of the camera pixel a. Symbols: experimental results for the case of scattering from a solid sample (Teflon). Solid line: Eq. (11). Dashed line: the approximate result in the small-speckle limit b ≪ a.

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Fig. 5.
Fig. 5. Average variance of the intensity fluctuations 〈c〉 and its noise σ 2 c as functions of the number of pixels N. Speckle pattern is recorded in the image plane for light reflected from a solid piece of Teflon. Exposure time is 1 ms, transport mean free path in Teflon l* ≃ 0.25 mm, camera pixel size a = 9.9 µm, magnification one. Left panel: analysis using a subset of pixels (all neighboring pixels omitted, full symbols) leads to a constant value of 〈c〉 [Eq. (16), dotted lines]. Analysis using all pixels (open symbols) is compared to the prediction of Eq. (28) (solid lines). Right panel: thick black and thin blue lines show predictions of Eqs. (18) and (29), respectively. The inset shows a comparison of the experimental values (symbols) and theoretical predictions for H(μ) = Nσ 2 c /〈c〉2: Eq. (18) (blue line) and Eq. (29) (red line).

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Fig. 6.
Fig. 6. Noise of the speckle correlation coefficient as a function of the time lag τ (symbols). Lines show predictions by Eqs. (39) (left) and (36) (right). Inset of the left panel shows the intensity correlation function g 2(τ).

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Fig. 7.
Fig. 7. Noise of the speckle structure coefficient as a function of time lag τ (symbols). As predicted by the theory, the noise is independent of τ. Lines show predictions of Eqs. (46) (left) and (44) (right). The inset of the right panel shows the intensity structure function d(τ).

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Equations (50)

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I α = 1 a 2 ∫ pixel α d 2 r I ( r ) ,
i = 1 N Σ α = 1 N I α ,
c = 1 N − 1 Σ α = 1 N ( I α − i ) 2 .
〈 i 〉 = 〈 I 〉 ,
〈 c 〉 = 〈 ( I − 〈 I 〉 ) 2 〉 = 〈 I 〉 2 .
K = 〈 c 〉 〈 I 〉 .
σ i 2 〈 i 〉 2 = 〈 i 2 〉 − 〈 i 〉 2 〈 i 〉 2 = 1 N ,
σ c 2 〈 c 〉 2 = 〈 c 2 〉 − 〈 c 〉 2 〈 c 〉 2 = 8 N × N − 3 4 N − 1 .
σ c 2 〈 c 〉 2 ∣ N → ∞ = 8 N .
P ( I α ) = 1 Γ ( μ ) ( μ 〈 I 〉 ) μ I α μ − 1 exp ( − μ I α 〈 I 〉 ) .
1 μ = 1 a 4 ∫ pixel d 2 r ∫ pixel d 2 r ' [ g 2 ( r − r ' ) − 1 ] ,
g 2 ( Δ r ) = 〈 I ( r ) I ( r + Δ r ) 〉 〈 I 〉 2 = 1 + exp ( − Δ r 2 b 2 ) ,
g 2 ( Δ r ) = 1 + [ 2 J 1 ( Δ r b ) ( Δ r b ) ] 2 .
〈 I α n 〉 = Γ ( μ + n ) Γ ( μ ) ( 〈 I 〉 μ ) n .
〈 i 〉 = 〈 I 〉 , σ i 2 〈 i 〉 2 = 1 μ N ,
〈 c 〉 = 〈 I 〉 2 μ ,
σ c 2 〈 c 〉 2 = 2 N − 1 [ 1 + 3 μ ( 1 − 1 N ) ] .
σ c 2 〈 c 〉 2 ∣ N → ∞ = 2 N ( 1 + 3 μ ) .
1 μ 2 = 1 a 4 ∫ pixel 1 d 2 r ∫ pixel 2 d 2 r ′ [ g 2 ( r − r ′ ) − 1 ] ,
1 μ 3 = 1 a 4 ∫ pixel 1 d 2 r ∫ pixel 3 d 2 r ′ [ g 2 ( r − r ′ ) − 1 ] ,
〈 i 2 〉 = 1 N 2 Σ α = 1 N Σ α ′ = 1 N 〈 I α I α ′ 〉 .
〈 I 2 〉 = 1 a 4 ∫ pixel d 2 r ∫ pixel d 2 r ' 〈 I ( r ) I ( r ' ) 〉
= 〈 I 〉 2 ( 1 + 1 μ ) ,
〈 I 1 I 2 〉 = 1 a 4 ∫ pixel 1 d 2 r ∫ pixel 2 d 2 r ' 〈 I ( r ) I ( r ' ) 〉
= 〈 I 〉 2 ( 1 + 1 μ 2 ) ,
〈 I 1 I 3 〉 = 1 a 4 ∫ pixel 1 d 2 r ∫ pixel 3 d 2 r ' 〈 I ( r ) I ( r ' ) 〉
= 〈 I 〉 2 ( 1 + 1 μ 3 ) ,
〈 i 2 〉 = 1 N 2 { N 〈 I 2 〉 + 4 ( N − N ) 〈 I 1 I 2 〉
+ 4 ( N − 1 ) 2 〈 I 1 I 3 〉 + [ N 2 − ( 3 N − 2 ) 2 ] 〈 I 〉 2 } .
σ i 2 〈 i 〉 2 = 1 N [ 1 μ + 4 μ 2 ( 1 − 1 N ) + 4 μ 3 ( 1 − 2 N + 1 N ) ] .
c = 1 N − 1 Σ α = 1 N I α 2 − i 2 1 − 1 N .
〈 c 〉 = 〈 I 〉 2 [ 1 μ − 4 μ 2 N ( N + 1 ) − 4 ( N − 1 ) μ 3 N ( N + 1 ) ] .
σ c 2 〈 c 〉 2 ≃ 2 N [ 1 + 3 μ + 12 μ 2 ] = H ( μ ) N ,
g 2 ( τ ) = 〈 I ( t ) I ( t + τ ) 〉 〈 I 〉 2 ,
1 v = 1 T 2 ∫ 0 T d t ∫ 0 T d t ' [ g 2 ( t − t ' ) − 1 ] .
1 v = 2 T ∫ 0 T [ g 2 ( τ ) − t ] ( 1 − τ / T )   d τ .
c ( τ ) = 1 N − 1 Σ α = 1 N [ I α ( t ) − i ( t ) ] [ I α ( t + τ ) − i ( t + τ ) ] ,
i ( t ) = 1 N Σ α = 1 N I α ( t )
〈 c ( τ ) 〉 = 〈 I 〉 2 [ g 2 ( τ ) − 1 ] ,
σ c ( τ ) 2 〈 c ( τ ) 〉 2 = g 2 ( τ ) 2 ( 3 − 2 N ) − 2 [ g 2 ( τ ) − 1 N ] ( N − 1 ) [ g 2 ( τ ) − 1 ] 2 .
σ c ( τ ) 2 〈 c ( τ ) 〉 2 ∣ N → ∞ = 1 N × g 2 ( τ ) [ 3 g 2 ( τ ) − 2 ] [ g 2 ( τ ) − 1 ] 2 .
〈 c ( τ ) 〉 = 〈 I 〉 2 [ 1 μ − 4 μ 2 N ( N + 1 ) − 4 ( N − 1 ) μ 3 N ( N + 1 ) ] [ g 2 ( τ ) − 1 ]
σ c ( τ ) 2 〈 c ( τ ) 〉 2 ≃ H ( μ ) 8 N × g 2 ( τ ) [ 3 g 2 ( τ ) − 2 ] [ g 2 ( τ ) − 1 ] 2 .
D ( τ ) = 〈 [ I ( t ) − I ( t + τ ) ] 2 〉 = 〈 I 〉 2 d ( τ ) ,
d ( τ ) = 〈 [ I ( t ) − I ( t + τ ) ] 2 〉 〈 I 〉 2 = 2 [ g 2 ( 0 ) − g 2 ( τ ) ] .
s ( τ ) = 1 N Σ α = 1 N [ I α ( t ) − I α ( t + τ ) ] 2 .
〈 s ( τ ) 〉 = D ( τ ) ,
σ s ( τ ) 2 〈 s ( τ ) 〉 2 = 5 N .
〈 s ( τ ) 〉 = D ( τ ) μ .
σ s ( τ ) 2 〈 s ( τ ) 〉 2 ≃ 5 8 N H ( μ ) .
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