Speckle contrast in near field scattering limited by time coherence

Gaoming Li; Yishen Qiu; Hui Li; Yan Huang; Shou Liu; Zhiyun Huang

doi:10.1364/OE.19.003694

1. Introduction

Near field scattering (NFS) techniques recently was introduced by M. Giglio group [1–8]. In contrast to traditional small angle scattering techniques, the scattered light is collected in the near field of the sample by CCD camera. The NFS technique can be realized by three different conceptual schemes, they are homodyne NFS [1,2], heterodyne NFS [3] and Schlieren NFS [4]. For the NFS technique, the scattered intensity distribution can be derived by Fourier analysis of the speckle images in the near field. It has been widely used in nonequilibrium systems [5,6], complex fluids [7] and dynamics measurements [8].

In this letter, we will describe an interesting phenomenon about speckle contrast in NFS. When the illumination source with proper time coherence length is used, the curve of speckle contrast vs. longitudinal distance that is perpendicular to scattering surface appears minimum turning point. A proper analysis is presented based on optical coherence and the characteristic of NFS. Using this speckle contrast phenomenon, the correlation area of scattered light is easy gained which can be used in measuring the roughness of surface. The Asakura group introduced a method to acquire the roughness parameters of surface height variance and surface height correlation area [9,10], but their method requires to know either of roughness parameters in advance. Now, both parameters can be acquired by doing an extra measurement of the correlation area of scattered light in the same experiment. Furthermore, this phenomenon of speckle contrast is related with the time coherence of scattered light in our analysis, which is possible to be applied in biomedical system because of the decoherence in bio-tissue with multi-scattering.

2. Experiments

In order to study the longitudinal distance change rule of the speckle contrast in the near scattering field, the measurement setup is described in Fig. 1 . The 488nm CW TEM₀₀ Ar + laser (U.S., Spectra-Physics Inc., Model 177-G12), the 532nm CW TEM₀₀ green diode laser (China, Photon Technologies, Inc., GDL8100) and the 520nm green LED (China, HongKe Optoelectronic Co.,LTD, H10- A21GHC5-G10-3P) were used respectively, their coherence lengths are about 50mm, 2mm and 0.01mm. A variable optical expander was used before the scattering screen. The incident angle was set as 60° in order to avoid the influence of mirror reflecting into CCD camera. The length of major axis of the ellipse illumination spot was 2cm and the minor axis was 1cm, and white painted board was used as scattering screen in our experiment. The microscope (Japan, Nikon Inc., ST100) is reconstructed and laid perpendicularly to the scattering screen. Its micro objective magnification is 10X and N.A. is 0.25. The CCD camera (Japan, Nikon Inc., DS-U2) take the place of microscopic ocular and its active area is adjusted to image plane. The time of CCD exposure is 1/60s and the digital gain is 1.0X.

Fig. 1 Optical diagram of the speckle contrast measurement setup.

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Increasing the distance between the microscope and the scattering screen, the CCD camera acquired the speckle pattern with proportional spacing. Then the patterns are processed by computer to work out speckle contrast. When illumination sources with different time coherence length are adopted, the speckle contrast vs. longitudinal distance curves show the result as in Fig. 2 . The curves show an interesting phenomenon, which a minimum turning point appears while using the 532nm green diode laser as illumination source. But other illumination source of LED and Ar⁺ laser doesn’t represent this character. When the 488nm Ar + laser illuminates the scattering screen, the speckle contrast curve keeps steadily at a close distance and increases slowly with increasing distance. While taking the LED as illumination source, the speckle contrast keeps nearly constant.

Fig. 2 Speckle contrast of scattered light with different time coherence lengths varies with increasing distance. In order to show the phenomenon, the solid lines are the numerical fits.

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In order to further study this character of the 532nm green diode laser, different scattering screens (white painted board and white cardboard) are used in our experiments. Figure 3 shows their surface micrographs, the white painted board is rougher than the white cardboard. Figure 4 is the experimental results of different boards, the speckle contrast curve of the rough surface lies on top of the smooth surface’s speckle contrast curve, but the minimum point position of the rough surface is less than the smooth one.

Fig. 3 Scattering surfaces. (a) White painted board. (b) White cardboard.

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Fig. 4 Speckle contrast of scattered light from different surfaces varies with increasing distance. In order to show the phenomenon, the solid lines are the numerical fits.

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3. Theoretical analysis

Above experimental results can be explained based on optical coherence and the characteristic of speckle in the near field. The laser with TEM₀₀ mode has good space coherence, so its coherence area is restricted by the time coherence length. To simplify the analysis without injuring reasonability of experimental results, we only take the center area of illumination region into consideration. As shown in Fig. 5(a) , the relation between the coherent length δ of illumination source and the linear dimension $L_{t}$ of the equivalent time coherence area (ETCA) can be expressed as:

δ = \sqrt{z^{2} + L_{t}^{2}} - z \pm L_{t} sin α,

where α is the light incidence angle. z is the distance between the plane of the scattering spot and the observation plane. According to the geometry, the ETCA can be written as (see in Fig. 5(b)):

A_{t i m e} = \frac{1}{2} π (L_{x +} + L_{x -}) L_{y},

where

L_{x \pm}, L_{y}

is the x-axis and y-axis of the ETCA, they can be deduced from Eq. (1):

Fig. 5 (a) Schematic diagram of equivalent time coherence area. (b) Shape of ETCA.

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\begin{array}{l} L_{x \pm} = \frac{\pm (δ + z) \sin α + \sqrt{δ^{2} + 2 δ z + z^{2} \sin^{2} α}}{\cos^{2} α}, \\ L_{y} = \sqrt{δ^{2} + 2 δ z} . \end{array}

The shape of ETCA will become a circular for normal incidence or transmission.

On the other hand, according to the characteristic of speckle in the near field [11], when the distance between the plane of the scattering spot and the observation plane meets following equation:

z < 2 \frac{\sqrt{A_{s} A_{a}}}{λ},

where

A_{a}

is the correlation area of scattered light.

A_{s}

is the area of illumination region. There exists relationship that the effective scattering area is smaller than the actual illumination region, as shown in Fig. 6 . The effective scattering area can be expressed as [1]:

Fig. 6 Effective scattering area is smaller than the area of illumination region.

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A_{e f f} \propto \frac{λ^{2} z^{2}}{A_{a}} .

Therefore, the speckle contrast of scattered light can be expressed as (see Fig. 6) [11]:

C = \frac{\sqrt{\sum_{n = 1}^{N} {\bar{I}}_{n}^{2}}}{\sum_{n = 1}^{N} {\bar{I}}_{n}},

where

{\bar{I}}_{n}

is the intensity generated by the scattering waves from the nth ETCA.N is the number of the ETCA in the effective scattering area. Namely

N = \frac{A_{e f f}}{A_{t i m e}} .

The curves of $A_{t i m e}$ and $A_{e f f}$ vs. z are described in Fig. 7 based on Eq. (2) and 5. When the time coherence of light is poor (LED, for example), the curve of ETCA remains nearly a small value constant line. So the number of the ETCA in the effective scattering area is large and the speckle contrast keeps in low level. When the time coherence length is so large(Ar + laser, for example) that its ETCA is larger than the area of illumination region, the effective scattering area is less than the ETCA in different distance and the speckle contrast keeps steadily. When the time coherence length is in proper range (532nm diode laser, for example), as the slope of the $A_{t i m e} - z$ curve is less than that of the $A_{e f f} - z$ curve, the value of Nincreases according to Eq. (7) while the speckle contrast keeps decreasing. The effective scattering area remains unchanged after it equals to the area of illumination region, but the ETCA expands continuously with increasing distance. Then the value of N decreases and the speckle contrast starts to rise up. When the ETCA equals to the area of illumination region, the speckle contrast keeps steadily and rises slowly. They are concordant with experimental results (see Fig. 2).

Fig. 7 Effective scattering area and the equivalent time coherence area of illumination source with different coherence length vary with increasing distance. z_min is the position of minimum point of speckle contrast.

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Considering the significance of minimum point, we make further researches into the special phenomenon. When the coherence length is in proper range (approximately less than linear dimension of illumination region, see Fig. 8 , $δ^{2} < A_{s}$ ), it will appear a minimum turning point. According to Eq. (5), the position of minimum point can be given by:

z_{\min} \propto \frac{\sqrt{A_{s} A_{a}}}{λ} .

Then we have

A_{a} \propto {(z_{\min} λ)}^{2} / A_{s} .

Obviously, if the area of illumination region

A_{s}

and the wavelength λ are known, the correlation area of scattered light

A_{a}

can be easy gained by measuring the position

z_{\min}

.

Fig. 8 Effective scattering area and the equivalent time coherence area of different roughness surfaces vary with increasing distance.

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On the other hand, the minimum value of speckle contrast may be written formally based on Eq. (6). To simplify the qualitative analysis, we consider that the intensity of light is uniform and normally incidence. The speckle contrast can be expressed as:

C \approx \frac{\sqrt{\sum_{m = 0}^{M - 1} \frac{4 π^{2} δ^{2}}{z_{\min}^{2} {(1 + r_{m}^{2} / z_{\min}^{2})}^{3}} {(1 + \frac{δ z_{\min} \sqrt{1 + r_{m}^{2} / z_{\min}^{2}}}{2 r_{m}^{2}})}^{2}}}{\sum_{m = 0}^{M - 1} \frac{2 π δ}{z_{\min} {(1 + r_{m}^{2} / z_{\min}^{2})}^{3 / 2}} (1 + \frac{δ z_{\min} \sqrt{1 + r_{m}^{2} / z_{\min}^{2}}}{2 r_{m}^{2}})},

where

r_{m}

is the radius of the mth circle of ETCA (see Fig. 9 ). For details, please refer to appendix.

Fig. 9 the minimum value of speckle contrast is approximately estimated.

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Obviously, Eq. (8) and 10 show that the minimum of speckle contrast is decided by the coherence of light and scattering matter. If the scattering screen is rougher, the $A_{a}$ becomes smaller and the $z_{\min}$ gets shorter according to Eq. (8) (see Fig. 8). According to Eq. (10), the minimum value of speckle contrast of the rough scattering surface is larger than the smooth one. Which in accord well with the experimental results seeing in Fig. 4.

4. Application

The extremum of speckle contrast mentioned above provides a method to measure the roughness of surface, which is an extendable method introduced by the group of Asakura [9] [10]. As the correlation area of scattered light $A_{a}$ can easily derived by the measurement of $z_{\min}$ , it provides a method to get both parameters of $σ_{h}$ and $π r_{c}^{2}$ , which takes place of requiring the knowledge of two parameters in advance mentioned in reference 9 and 10. Suppose light incidence normally, the relationship between the correlation area of scattered light and roughness of surface can be expressed as [11]:

A_{a} = \frac{π r_{c}^{2} e^{- {(4 π / λ)}^{2} σ_{h}^{2}}}{1 - e^{- {(4 π / λ)}^{2} σ_{h}^{2}}} {E i [{(\frac{4 π}{λ})}^{2} σ_{h}^{2}] - ε - \ln [{(\frac{4 π}{λ})}^{2} σ_{h}^{2}]},

where

E i (x)

represents the exponential integral and ε is Euler’s constant. The speckle contrast can be deduced to [11]:

C = \sqrt{\frac{8 (N - 1) {N - 1 + \cosh [{(4 π / λ)}^{2} σ_{h}^{2}]} \sinh^{2} [{(4 π / λ)}^{2} σ_{h}^{2} / 2]}{N {(N - 1 + \exp [{(4 π / λ)}^{2} σ_{h}^{2}])}^{2}}},

where

N = A_{k} / A_{a}

,

A_{k}

is the area of point spread function of micro objective at scattering surface. There exists a speckle contrast maximum, which can be found out by changing

A_{k}

(actually changing the diaphragm of micro objective). By using Eq. (12) and expression

\partial C (N, σ_{h}) / \partial N = 0

, it supplies a measurement of the

σ_{h}

. Then

π r_{c}^{2}

can be acquired by uniting the other Eq. (11).

In addition, the value of minimum turning point of speckle contrast may estimate the time coherence of scattered light, which is likely to offer part information of volume scattering in biomedical system in terms of decoherence generated by multi-scattering.

5. Conclusion

In conclusion, the relationship between speckle contrast and longitudinal distance in the near scattering field is presented. The time coherence of light affects the changing rule of speckle contrast in the near field. When the laser with proper coherence length is used, the speckle contrast curve appears a minimum turning point. The position of minimum turning point is decided by the area of illumination region, wavelength and the correlation area of scattered light. So it is easy to gain the correlation area of scattered light in near field by measuring the position of extremum point and illumination region. By the relationship between the correlation area of scattered light and roughness of surface, it can be used in surface roughness measurement and has the potential applicative value in biomedical systems.

Appendix

The minimum value of speckle contrast may be written formally based on Equation (6). To refine the qualitative analysis of speckle contrast of scattered light from different surfaces, we consider that the intensity of light is uniform and normally incidence. The intensity of scattered light can be written as (see Fig.9):

I_{m} = \frac{S_{m + 1} \cos^{2} θ_{m}}{r_{m}^{2} + z_{\min}^{2}},

where

S_{m + 1}

is the size of the

(m + 1)

th circle of time coherence area,

θ_{m}

is the angle of the mth circle of time coherence area relativing to observation point, and

r_{m}

is the radius of the mth circle.

If $δ < < \sqrt{A_{s}}$ , $δ < < z$ , and $Δ θ_{m}$ is a small quantity, we can approximately deduce the speckle contrast based on Eq. (6) and (13) as below:

C \approx \frac{\sqrt{\sum_{m = 0}^{M - 1} \frac{4 π^{2} δ^{2}}{z_{\min}^{2} {(1 + r_{m}^{2} / z_{\min}^{2})}^{3}} {(1 + \frac{δ z_{\min} \sqrt{1 + r_{m}^{2} / z_{\min}^{2}}}{2 r_{m}^{2}})}^{2}}}{\sum_{m = 0}^{M - 1} \frac{2 π δ}{z_{\min} {(1 + r_{m}^{2} / z_{\min}^{2})}^{3 / 2}} (1 + \frac{δ z_{\min} \sqrt{1 + r_{m}^{2} / z_{\min}^{2}}}{2 r_{m}^{2}})},

where Mis the number of time coherence area. According to the geometry in Fig. 9, we have recursive relation on

r_{m}

:

\begin{array}{l} r_{0} = \sqrt{2 δ z_{\min} + δ^{2}}, \\ r_{1} \approx r_{0} + \frac{δ \sqrt{r_{0}^{2} + z_{\min}^{2}}}{r_{0}}, \\ ...... \\ r_{m + 1} \approx r_{m} + \frac{δ \sqrt{r_{m}^{2} + z_{\min}^{2}}}{r_{m}} . \end{array}

Acknowledgments

This work was supported by Region Major Program of Fujian Province Natural Science Foundation of China (No: 2009H4003), National Natural Science Foundation of China (Grant No. 61008062), Natural Science Foundation of Fujian Province of China (No:2009J01277 and 2010J01325) and Fujian Province Department of Education Fund of China (No: A10077).

References and links

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8. F. Croccolo, D. Brogioli, A. Vailati, M. Giglio, and D. S. Cannell, “Use of dynamic schlieren interferometry to study fluctuations during free diffusion,” Appl. Opt. 45(10), 2166–2173 (2006). [CrossRef] [PubMed]

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Speckle contrast in near field scattering limited by time coherence

Abstract

1. Introduction

2. Experiments

3. Theoretical analysis

4. Application

5. Conclusion

Appendix

Acknowledgments

References and links

Cited By

Figures (9)

Equations (15)

Optics Express