Theoretical and experimental investigations of speckle features based on free-space surface scattering

Di Cai; Qiyong Xu; Zichun Le; Haolei Mao; Yujia Cao; Janan Zhou; Yipeng Mo; Jun Zhou

doi:10.1364/OE.521773

1. Introduction

Lasers have wide applications in optical imaging systems owing to their advantages, including large color gamut, high brightness, narrow spectrum, long lifetime, and high light efficiency [1–3]. For example, laser projectors have gradually been developed and then replaced traditional projection systems [4]. However, the screen used in laser projection and imaging systems is usually rough relative to the laser wavelength; as a result, speckle is introduced.

Speckle is a phenomenon caused by interference of coherent or partially coherent light, and is usually defined and quantified by speckle contrast C [5]:

(1)$$C = \frac{{{\sigma _I}}}{{\bar{I}}},$$

where ${\sigma _I}$ and $\bar{I}$ represent the standard deviation of the light intensity and the average light intensity, respectively. Given that speckle is perceived by viewers as a random granular pattern, significantly degrading image quality [6], the suppression of speckle becomes crucial. To achieve this goal, it is essential to quantitatively analyze the speckle [7–12], including the classification of speckle [7] and the calculation of speckle contrast [9–11], which is very important either for the suppression of speckle or the utilization of speckle features for extracting information such as in bio-oximetry [12]. In recent years, many speckle reduction methods have been investigated to eliminate the influence of speckle on projected images [13–21], including the use of special screen [13,18], diffractive optical element [14,15,22], diffuser [16], multimode fiber [19] and liquid light guide [20]. Different methods of speckle suppression are usually divided into the introduction of wavelength [22–24], angular [15,22], and polarization diversities. In addition, speckle can also be reduced by temporal averaging, for instance, by moving the screen [21,25].

Among the above-mentioned method, the wavelength diversity is crucial and common-used, as it is determined by the nature of the laser source, which is essential in practical applications. Goodman has discussed the impact of laser source coherence on the speckle contrast in free-space optical path, and the speckle contrast is expressed as [5]:

(2)$$C = \sqrt {\frac{1}{{\sqrt {1 + 2{\pi ^2}{{\left( {\frac{{\delta \lambda }}{{\bar{\lambda }}}} \right)}^2}{{\left( {\frac{{{\sigma_h}}}{{\bar{\lambda }}}} \right)}^2}{{({\cos {\theta_o} + \cos {\theta_i}} )}^2}} }}} ,$$

where δλ and λ are the wavelength bandwidth and the center wavelength of the laser diode, respectively; σ_h is the standard deviation of the surface fluctuation, which is usually described by the effective surface roughness; and θ_i and θ_o are the incidence and observation angles, respectively. Afterwards, a study has extended the research to imaging optical path and revealed an inherent correlation between wavelength diversity and angular diversity [26,27]. However, it is important to note that for free-space optical path, Eq. (2) is only valid when the incidence angle is equal to the observation angle. Moreover, there are currently no experimental investigations to verify the theoretical prediction. A more universal theoretical model to describe speckle contrast in free-space optical path has yet to be developed, which is essential for describing non-imaging optical systems with lasers as light sources.

In the present study, an accurate and universal theoretical model for describing the speckle contrast in free-space optical path was established. The model presents unexpected findings, such as the high sensitivity of the speckle contrast to variations in the observation angle when the observation angle ${\theta _o} \approx {0^ \circ }.$ Subsequently, an experimental scheme was designed and used to verify the developed theoretical model. We believe that the developed theoretical model has enriched the speckle suppression theory, and combination with the theoretical method for the imaging optical path, a more universal speckle suppression theoretical system can be formed. Furthermore, since in certain scenarios, such as imaging through scattering media or biological tissue [28], speckle can be regarded as a preservation medium for valid information rather than noise, the findings on the speckle features and the factors affecting speckle evolution have potential to apply in the optical systems that involve speckle characteristics of surface scattering.

2. Theoretical model

Suppose that the spectrum of light incident on a scattering surface is constant over the entire scattering surface, which is expressed as the power spectral density $\hat{g}(v).$ The total intensity of a point on the scattering surface can be obtained by integrating over the entire frequency spectrum:

(3)$$I({x,y} )= \int_0^\infty {\hat{g}({ - v} )I({x,y;v} )dv} ,$$

where $I(x,y;v)$ is the speckle intensity at point $(x,y)$ when the light frequency is $v.$ Here, $I(x,y;v)$ follows the negative exponential statistics for a fully developed speckle. The speckle contrast can be calculated by [5]

(4)$$C = \sqrt {\int\limits_{ - \infty }^\infty {{K_{\hat{g}}}({\Delta v} ){{|{{\mu_A}({\overrightarrow {{q_1}} ,\overrightarrow {{q_2}} } )} |}^2}d\Delta v} } ,$$

where

(5)$${K_{\hat{g}}}({\Delta v} )= \int\limits_{ - \infty }^0 {\hat{g}(v )\hat{g}({v - \Delta v} )dv} ,$$

(6)$${\mu _A}({\overrightarrow {{q_1}} ,\overrightarrow {{q_2}} } )= {M_h}({\Delta \overrightarrow {{q_z}} } )\psi ({\Delta \overrightarrow {{q_t}} } ),$$

(7)$$|{\Delta \overrightarrow {{q_t}} } |= \left|{(\frac{{2\pi }}{{{\lambda_2}}} - \frac{{2\pi }}{{{\lambda_1}}})({\sin {\theta_o} - \sin {\theta_i}} )} \right|= \frac{{2\pi \Delta v}}{{\bar{v}\bar{\lambda }}}({\sin {\theta_o} - \sin {\theta_i}} ),$$

(8)$$|{\Delta \overrightarrow {{q_z}} } |= \left|{(\frac{{2\pi }}{{{\lambda_2}}} - \frac{{2\pi }}{{{\lambda_1}}})({\cos {\theta_o} + \cos {\theta_i}} )} \right|= \frac{{2\pi \Delta v}}{{\bar{v}\bar{\lambda }}}({\cos {\theta_o} + \cos {\theta_i}} ).$$

In Eqs. (5)–(8), ${K_{\hat{g}}}({\Delta v} )$ denotes the normalized power spectrum, ${\mu _A}({\overrightarrow {{q_1}} ,\overrightarrow {{q_2}} } )$ is the correlation function of the two speckle fields, ${M_h}({\Delta \overrightarrow {{q_z}} } )$ is the first-order characteristic function of the surface fluctuation h, and $\psi ({\Delta \overrightarrow {{q_t}} } )$ is the normalized Fourier transform of the intensity distribution on the scattered light spot. The $\overrightarrow {{q_1}}$ is the specific scattering vector of wavelength ${\lambda _1}$ (corresponding to ${v_1}$), and $\overrightarrow {{q_2}}$ is the specific scattering vector of wavelength ${\lambda _2}.$ Further, $\Delta \overrightarrow {{q_z}}$ and $\Delta \overrightarrow {{q_t}}$ are the normal and transverse components of the scattering vector of $\overrightarrow {{q_1}} - \overrightarrow {{q_2}} ,$ respectively. Finally, c represents the speed of light, and $\bar{v}$ is the center frequency of the light spectrum.

Assuming that both the fluctuation in the height of the screen and the light spectrum follow a Gaussian distribution, the power spectral density $\hat{g}(v)$ and the first-order characteristic function of the surface fluctuation ${M_h}({\Delta \overrightarrow {{q_z}} } )$ can be rewritten as

(9)$${|{{M_h}({\Delta \overrightarrow {{q_z}} } )} |^2} = \textrm{exp} ({ - \sigma_h^2\Delta q_z^2} ),$$

(10)$$\hat{g}(v )\approx \frac{2}{{\delta v\sqrt \pi }}\textrm{exp} \left[ { - {{\left( {\frac{{v + \bar{v}}}{{{{\delta v} / 2}}}} \right)}^2}} \right],$$

where $\delta v$ is the laser spectrum width in 1/e, $\Delta {q_z} = |{\Delta \overrightarrow {{q_z}} } |,$ and ${\sigma _h}$ is surface roughness of the scattering screen. In this study, the root-mean-square (RMS) roughness of the screen surface was treated as the effective surface roughness ${\sigma _h}.$ Therefore, the autocorrelation function of the normalized power spectrum ${K_{\hat{g}}}({\Delta v} )$ easily calculates as

(11)$${K_{\hat{g}}}({\Delta v} )= \sqrt {\frac{2}{{\pi \delta {v^2}}}} \textrm{exp} \left( { - \frac{{2\Delta {v^2}}}{{\delta {v^2}}}} \right).$$

From Eq. (6), it can be seen that the speckle contrast not only depends on the term of ${M_h}({\Delta \overrightarrow {{q_z}} } ),$ according to Goodman’s theoretical prediction [5] (Eq. (5–105) on page 155), but is also affected by $\psi ({\Delta \overrightarrow {{q_t}} } ).$ In this study, we take the term of $\psi ({\Delta \overrightarrow {{q_t}} } )$ into account; $\psi ({\Delta \overrightarrow {{q_t}} } )$ be calculated by

(12)$$\psi ({\Delta \overrightarrow {{q_t}} } )= \frac{{\int {\int_{ - \infty }^\infty {{{|{S({\alpha ,\beta } )} |}^2}\textrm{exp} ({ - j\Delta \overrightarrow {{q_t}} \cdot \overrightarrow {{\alpha_t}} } )d\alpha d\beta } } }}{{\int {\int_{ - \infty }^\infty {{{|{S({\alpha ,\beta } )} |}^2}d\alpha d\beta } } }},$$

where ${|{S({\alpha ,\beta } )} |^2}$ represents the intensity of the scattered light spot, as shown in Fig. 1; $\vec{\alpha } = (\alpha ,\beta ,z)$ is the coordinate system on the screen plane with the scattered light spot as the origin of the coordinate system; and $\overrightarrow {{\alpha _t}}$ represents the transverse component of $\overrightarrow \alpha .$ Assuming that the scattered light spot is circular, $\psi ({\Delta \overrightarrow {{q_t}} } )$ can be expressed as

(13)$$\psi ({\Delta \overrightarrow {{q_t}} } )= 2\frac{{{J_1}\left( {\frac{{D\Delta {q_t}}}{2}} \right)}}{{\frac{{D\Delta {q_t}}}{2}}},$$

where $\Delta {q_t} = |{\Delta \overrightarrow {{q_t}} } |$ and D represents the diameter of the scattered light-spot. Since the value of ${{D\Delta {q_t}} / 2}$ is not large, $\psi ({\Delta \overrightarrow {{q_t}} } )$ can be approximated as exponential form [29] for the sake of integration simplicity:

(14)$$\psi ({\Delta \overrightarrow {{q_t}} } )\approx \textrm{exp} \left[ { - \frac{7}{{25}}{{\left( {\frac{{D\Delta {q_t}}}{2}} \right)}^2}} \right],$$

where the constant coefficient 7/25 is the optimal fitting result, and the coefficient of determination ${R^2}$ exceeds 0.99. Substituting Eqs. (9), (11), and (14) into Eq. (4), the speckle contrast can be obtained as follows:

(15)$$C = \sqrt {\frac{1}{{\sqrt {1 + 2{\pi ^2}{{\left( {\frac{{\delta \lambda }}{{\bar{\lambda }}}} \right)}^2}{{\left( {\frac{{{\sigma_h}}}{{\bar{\lambda }}}} \right)}^2}{{({\cos {\theta_o} + \cos {\theta_i}} )}^2} + \frac{7}{{25}}{\pi ^2}{{\left( {\frac{{\delta \lambda }}{{\bar{\lambda }}}} \right)}^2}{{\left( {\frac{D}{{\bar{\lambda }}}} \right)}^2}{{({\sin {\theta_o} - \sin {\theta_i}} )}^2}} }}} .$$

The equation ${{\delta v} / {\bar{v}}} = {{\delta \lambda } / {\bar{\lambda }}}$ is used here. Equation (15) shows the quantitative relationship between the speckle contrast C and five parameters: wavelength $\lambda $ (and $\delta \lambda $), screen surface roughness ${\sigma _h},$ light-spot diameter D, incidence angle ${\theta _i}$ and observation angle ${\theta _o}.$ The key difference of the developed theoretical model with Goodman's theory [5] is that Eq. (15) not only includes the surface scattering term, but also shows the effect of light-spot size. This is conceivable in a free-space optical path, and since there is no imaging lens, the captured signal is the superposition of the scattered fields from all point sources on the scattering surface. Furthermore, the impact of angular diversity on factors ${\sigma _h}$ and D varies due to differences in the direction of the scattering vectors. When ${\theta _i} = {\theta _o},$ Eq. (15) can be reduced to Eq. (2).

Fig. 1. Schematic diagram of the reflection on a screen from a scattered light spot.

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In the case of significant speckle reduction, Eq. (15) can be approximated as:

(16)$$C \approx {\left( {\frac{{\sqrt {2{\pi^2}{\sigma_h}^2{{({\cos {\theta_o} + \cos {\theta_i}} )}^2} + \frac{7}{{25}}{\pi^2}{D^2}{{({\sin {\theta_o} - \sin {\theta_i}} )}^2}} }}{{\frac{{{{\bar{\lambda }}^2}}}{{\delta \lambda }}}}} \right)^{ - \frac{1}{2}}}.$$

Equation (16) describes the speckle reduction mechanism for surface scattering in free-space optical path. The “degrees of freedom” used for speckle reduction is derived from the roughness of the scattering surface and the light-spot size. The number of “degrees of freedom” is measured in terms of the laser coherence length.

At this stage, we have developed a model to characterize speckle in free-space optical path. In the experimental verifications, the incidence angle θ_i is set to 0 degree as the laser is typically incident vertically.

3. Experimental setup and simulations

An experimental setup is shown in Fig. 2. The center wavelength of laser (PL 520B1, OSRAM) $\overline \lambda = 520\textrm{nm,}$ and the wavelength bandwidth $\delta \lambda = 2\textrm{nm}\textrm{.}$ The laser beams were collimated and homogenized using a collimator-based on two lenses. The screens were sandpaper with different effective surface roughness values, and the thickness of sandpaper is 0.23 mm and the surface color is silver-white. A CMOS camera (DCC3260 M, Thorlabs) was used to capture the speckle image. The camera had 1936 × 1216 pixels, and the pixel size was 5.86 × 5.86 µm². In addition, as shown in Fig. 2, S₁ is the distance from the laser to the diaphragm; D₁ is the aperture of an adjustable diaphragm; S₂ is the distance from the diaphragm to lens 1; D₂ is the aperture of lens 1; F₁ is the focal length of lens 1; D₃ is the aperture of lens 2; F₂ is the focal length of lens 2; S₅ is the distance from lens 2 to the screen; S₆ is the distance from the screen to the detector array of camera, which is also called the observation distance. In our experiments, S₁ = 200 mm, S₂ = 100 mm, D₂ = 50.8 mm, F₁ = 100 mm, D₃ = 25.4 mm, F₂ = 45 mm, S₅ = 1100 mm, and S₆ = 500 mm. The diameter D of the light-spot on the screen is adjusted by the aperture of the diaphragm, D₁.

Fig. 2. Experimental setup of a laser speckle detection system.

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We obtained the values of σ_h for two screens used in the experiments by measuring their RMS surface roughness using a stylus surface profiler (Dektak-XT, Bruker). The RMS surface roughness of the used two screens was σ_h = 2.972 µm, and σ_h = 35.199 µm. Finally, the spot diameters were set to 5 mm and 15 mm, respectively, and numerical simulations were performed based on Eq. (15) with the four parameters sets listed in Table 1.

Table 1. Parameters setting for numerical simulation.

View Table

The numerical simulation results are shown in Fig. 3. It is evident that the speckle contrast decreases as the observation angle increases. Specifically, the initial maximum speckle contrast depends on surface roughness, the final minimum speckle contrast is determined by the light-spot diameter, and the rate of decrease of speckle contrast is positively correlated with ${D / {{\sigma _h}}}.$ Under our experimental parameter settings, i.e., the screen surface roughness is at the laser wavelength level and the light-spot diameter is at the millimeter level, an interesting phenomenon can be easily observed, that is, the speckle contrast decreases rapidly within the observation angle of 10 degrees or less. This is completely different from our perception of imaging optical path. In an imaging optical path, the angular diversity is mainly determined by the projection numerical aperture $N{A_{pro}}$ and the imaging numerical aperture $N{A_{cam}}$ [5]:

(17)$$C = \sqrt {\frac{{N{A_{cam}}}}{{N{A_{pro}}}}} .$$

The speckle contrast is determined by the resolution of the projection and imaging space, regardless of the spatial position of the observer relative to the object.

Fig. 3. Variation of speckle contrast with observation angle for four sets of different parameters.

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It is worth emphasizing that the most important parameters in the developed theoretical model remain the wavelength and linewidth of the laser. When a laser with very high coherence is used, the other parameters are useless and $C \approx 1.$

4. Results and discussion

In order to avoid multiple observation angles within the camera's detector area, the shooting distance was set to 500 mm. Figure 4 shows the experimental results when the light-spot diameters are 5 mm and 15 mm, respectively. It can be seen that the light-spot diameter increases by a factor of 3 when the observation angle is approximately equal to 0, however, the speckle contrast remains constant. This is consistent with the developed theory that the speckle contrast depends on the surface roughness in this case. The speckle contrast then decreases as the observation angle increases, and the rate of speckle contrast reduction is faster for greater D. Eventually the scattering contrast maintains at ∼0.21-0.23. These experimental phenomena are similar to the theoretical results. Nevertheless, the rate of decrease in speckle contrast and the minimum speckle contrast are less than theoretically calculated values. It means that the actual effect of light-spot diameter on speckle contrast is weaker than the theoretical results.

Fig. 4. Experimental results for light-spot diameters D with (a) D = 5 mm and (b) D = 15 mm, respectively.

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In order to quantify the difference between the theoretical and experimental results, it is assumed that other mechanisms for reducing speckle, such as polarization and intrinsic system errors, are independent of the parameters investigated in this study [14], and then the speckle suppression efficiency k is used to describe the change of the speckle contrast, which is calculated as follows:

(18)$$k = {{{C_0}} / C}$$

where C₀ and C are the initial and final speckle contrasts, respectively. As no other settings were changed during the experiments, other factors such as polarization can be eliminated in this way. Figure 5 shows the numerical simulation results and experimental results of the speckle suppression efficiency with the light-spot diameter equal to 5 mm. The curves in Fig. 5(a) and Fig. 5(b) have similar shapes but the efficiency of speckle suppression differs by 2 to 3 times. In the case we are interested in, Eq. (15) can be rewritten as:

(19)$$\left\{ \begin{array}{ll} {{C_0} \sim {K_{{\sigma_h}}} \cdot \sqrt {\frac{1}{{{\sigma_h}}}}} ,&{\theta_o} \approx {0^0}\\ {C \sim {K_{D,\textrm{ }{\theta_o}}} \cdot \sqrt {\frac{1}{D}} \cdot \sqrt {\frac{1}{{\sin {\theta_o}}}}} ,&{\theta_o} > {0^0} \end{array}\right. ,$$

where ${K_{{\sigma _h}}}$ is the correction parameter for screen roughness, and ${K_{D,\textrm{ }{\theta _o}}}$ is the correction parameter for light-spot diameter and observation angle. The difference between the theory and the experiment can be described in two parts. The first part of difference is caused by C₀. Due to the blocking of the light by the camera, it is not possible to set the observation angle exactly equal to 0 degree in the experiment, which results in the initial speckle contrast C₀ lower than its true value. Furthermore, when the speckle particle size is smaller than the detector pixel area, intensity-based integration occurs [5], resulting in multiple speckle particles being averaged, therefore further reducing the measured speckle contrast. Finally, the incident light used in the experiment was not strictly a plane wave, and according to a related study [30], this could lead to a further reduction in the measured speckle contrast. The second part of difference is caused by the scattering of surface. In the experiment, it can be found that the diffuse reflection of the laser light from the rough screen is still concentrated around the 0-degree angle space, instead of being evenly distributed throughout the space. This means that as the observation angle increases, the amount of light contributing to speckle reduction decreases rapidly. As a result, we observed a minimum speckle level of only about 22% in the experiments, which is significantly higher than the theoretical value. Four speckle images taken during the experiment and their normalized intensity of the light field in different angular spaces are shown in Fig. 6.

Fig. 5. (a) Simulation results and (b) experimental results of the speckle suppression efficiency with the spot diameter D = 5 mm.

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Fig. 6. Speckle images at observation angles of (a) 0, (b) 5, (c) 10 and (d) 15 degrees, and (e) the normalized light field intensity distributions in different angular spaces for surface roughness ${\sigma _h} = 2.971{\mathrm{\mu} \mathrm{m}}\textrm{.}$.

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Although the complexity of the speckle field increases with the observation angle, speckle features can still be observed within a limited range of observation angles, which meet the theoretical expectations. The theoretical results can also be used to predict or customize the scattering field for various scenarios, which has potential applications in the field of scattering medium imaging [31] or non-field-of-view imaging [32], e.g., localization of hidden objects by variations in the scattering field.

5. Conclusions

In this study, we establish a universal theoretical model of speckle reduction in free-space optical path. The theoretical results indicate that the speckle contrast is determined by five parameters: wavelength, screen surface roughness, light-spot diameter, incidence angle and observation angle. Among the above-mentioned parameters, the wavelength is a crucial one, and the coherence length of the laser determines the number of degrees of freedom that affects the other parameters to be regulated for speckle reduction. Finally, an experimental verification was conducted. Although the results were influenced by the experimental conditions, the phenomena consistent with the theory were still observed. This study enriches the theory of speckle reduction based on free-space surface scattering, which can be directly applied for eliminating the speckle noise in non-imaging optics. In the future research, the modeling of human eye pupil as an imaging system [33] and the consideration of complex incident wavefronts can be combined with the developed theoretical model for more realistic working scenarios. Moreover, capability of the developed theory for predicting or customizing the speckle features may have potential for much wider range of applications, for example, recent research hotspots of the scattering medium imaging and the non-field-of-view imaging.

Funding

National Natural Science Foundation of China (61975183); Natural Science Foundation of Zhejiang Province (LQ23F050007).

Disclosures

The authors declare no conflicts of interest.

Data availability

The data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Dependent variable	Independent variable	Other parameters
Dependent variable	Independent variable	$θ_{i}$	$δ λ / λ$	$σ_{h}$ (µm)	D (mm)
Speckle contrast (C)	Observation angle ( $θ_{o}$ )	0°	2/520	2.972	5
				2.972	15
				35.199	5
				35.199	15

Theoretical and experimental investigations of speckle features based on free-space surface scattering

Abstract

1. Introduction

2. Theoretical model

3. Experimental setup and simulations

4. Results and discussion

5. Conclusions

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Tables (1)

Equations (19)

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