1. Introduction

As early as the 1960s, researchers started noticing that fine-scale speckle patterns would appear after the laser was transmitted/reflected by objects with ’random’ roughness (In fact, most materials in the real world are rough on the scale of an optical wavelength). Inspired by the intensive research on the characteristics of speckles, scientific community has discovered a wealth of interesting physics and developed a diversity of significant technologies, ranging from fundamental phenomena such as memory effect [1,2] and Anderson localization [3,4], to applications in imaging [5–8], random lasing [9,10] and compact spectrometer [11,12], etc.

In order to describe the unpredictable scattering process and the randomly distributed speckles in a quantitative way, the statistics theory of light scattering is established by taking the incidence light as monochromatic, making the characteristics of speckles perceptible and understandable [13]. Despite the random distribution, the patterns show spatial invariance, with speckles at different positions on the observation plane holding the same shape, size and possessing the same degree of the first- or higher-order coherence [14]. The translation invariance of the morphology and statistics makes monochromatic speckles a promising transmitter or carrier of optical information, thus speckles are widely used as an illumination source or recordable signal in ghost imaging [15,16], imaging hidden objects based on memory effect (ME) [5,7], and other metrology methods [17–19]. On the other hand, studies on monochromatic speckles also provide a series of suppression methods in holographic imaging [20–24], optical coherence tomography (OCT) imaging [25,26] and optical communication [27], in which speckles are treated as noise and required to be suppressed or eliminated.

The past decades witnessed the rapid development of wide-spectrum light source, including femtosecond laser [28,29], light-emitting-diode (LED) [30,31] and so on. The extensive applications of these new sources demand figurative and statistical research on speckles produced by illumination of wide-spectrum. For example, LED is widely used as an illumination source to recover the image of objects hidden behind the scattering layer based on ME. The scattered speckles are roughly simplified as monochromatic, while the difference between speckles produced by different wavelengths [32–34], which will affect the visibility and field of view (FOV) of imaging, is ignored. Besides, in ghost imaging with true thermal light [35–37], a narrow band filter is employed to acquire good monochromaticity, with great energy loss. If the morphology and statistics of wide-spectrum speckles is clearly revealed, the possibility of utilizing a wider spectrum extent and energy will be open.

J. W. Goodman has built voluminous theory to elaborate the statistical properties of speckles [14,38]. He also noticed multi-wavelength speckles and pointed out that the correlation between speckles produced by two incidence lasers with different wavelengths reaches a peak on the optical axis, which means the trajectory of ballistic light [39,40]. Although the wide-spectrum speckle patterns require more figurative description, he gives us a shoulder to stand on for seeing further.

In this paper, we analyze morphology and statistics of speckles produced by a point-like wide-spectrum source, the intensity distribution of which is treated as the incoherent superposition of many discrete monochromatic patterns. To achieve that, the case involving scattering fields with two different wavelengths, which is the simplest and most pivotal model in wide spectrum issues, is considered. First- and second-order coherence are derived, respectively. Different from monochromatic speckles, both the degree of first-/second-order coherence of two different monochromatic fields decrease as the observation position getting far away from the optical axis. When extended to the wide-spectrum case, size and shape of speckles are transversely varying. The speckle patterns, which are resulted from the first-order coherence, appear in radial morphology with respect to the optical axis. As for second-order coherence, the peak value and visibility appear reducing radially, and the amount of reduction is related to the observation position, the center wavelength, and the features of the scattering layer. To present these figurative results, we propose a phase plate model to describe the scattering layer, with the scattering induced phase being wavelength-depending, which can also be used directly on other issues of wide-spectrum scattering process.

This paper is arranged as follows. In Sec. 2, the assumption and principle of phase plate are expressed. In Sec. 3, the first-order coherence of the speckle fields with different wavelengths is analyzed. Then the morphology of the wide-spectrum speckles is presented via simulation. In Sec. 4, we discuss the second-order coherence of speckle fields with different wavelengths, with factors affecting distribution and visibility analyzed.

2. Phase plate model

When light with a certain degree of coherence is reflected/transmitted by a medium with a rough surface or random thickness, the wavefront of the exiting light will be spatially modulated with random phases. Thus randomly distributed speckles formed via interference can be observed at a distance from the medium. In most literatures, phase screen model is used to describe the random modulation introduced by scattering. With that operation, morphology and statistics of the monochromatic speckles are studied intensively. However, for any given medium, the induced phase and the output wavefronts are actually wavelength related, as is not included by the phase screen model. That is, the model is not suit for wide spectrum cases.

Here we propose a phase plate, with a diagrammatic sketch shown in Fig. 1. Instead of solely imposing a random phase, we introduce random thickness (path length) on every point, which will cause wavelength-related phase modulation, denoted as $\varphi (\alpha,\beta,\lambda )$, with $(\alpha,\beta )$ being the position on the scattering plane and $\lambda$ being the wavelength of illumination light. To properly represent the effects of the scattering procedure, the following assumptions are considered in the model.

 

Fig. 1. Sketch diagram of optical scattering. A point source is set on the optical axis. After being scattered by the medium, the intensity distribution of the light field is detected with a camera. An aperture is set on the output plane of the scattering medium. The distances from the scattering medium to the light source and the camera are much larger than the aperture size. A phase plate model describes the scattering, with introduced path length on every point randomly distributed.

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The phase plate model describes those kinds of inhomogeneous medium such as ground glass, the fluctuation of whose surface or thickness is random enough compared to the optical wavelength, and the effect of dispersion and absorption can be neglected. It can describe the scattering process of monochromatic and wide-spectrum light.

3. Morphology of wide-spectrum speckles

3.1 First order coherence

Here, the first-order coherence between the scattering fields produced by light with two different wavelengths is derived. With no loss of generality, we consider a point-like source, and the illumination intensity of the two wavelengths is the same. Configuration shown in Fig. 1 is considered. The light field imprinted on the front surface of scattering medium at $(\alpha,\beta )$ can be expressed as

Take the illumination area as circular and the transmission ratio as spatially uniform, ${\left | {P(\alpha,\beta )} \right |}^2 =1$, within the involved area. We can get

There exists a circle with the radius $R$ in the observation plane. Outside the circle, $|g^{(1)} (x,y,{\lambda _1},{\lambda _2})|$ is 0. That is, first-order coherence between fields of $\lambda _1$ and $\lambda _2$ vanishes. When both $z_0$ and $D$ are given, the radius of the circle will decreases when the $\delta \lambda$ increases. Eq. (7) can be extended to the wide-spectrum case, and a conclusion can be obtained that when the spectrum band of the illumination source is wider, the visibility of the speckles on the whole observation plane will decrease, and that of the peripheral speckles will decrease faster, which will be presented via simulation results in the following.

3.2 Intensity distribution

To show the morphology of wide-spectrum speckle patterns, we performed a numerical simulation based on the phase plate model. Consider the setup shown in Fig. 1. A point-like wide-spectrum source illuminates a scattering layer, which is $0.5 m$ from the source. The average size of the grain on the layer is $40\mu m \times 40\mu m$ and the random thickness of the layer is subjected to a uniform distribution from $1$ to $11\mu m$. Thus when light passes through the layer at a certain position, a wavelength-related phase will be introduced. An aperture is placed on the rear surface of the scattering layer with the size of 4mm$\times$ 4mm. The speckles are recorded by a camera, which is $2 m$ away from the aperture. The pixel size of the camera is 20$\mu$m$\times$20$\mu$m.

The wide-spectrum of the illumination light is simulated by the incoherent superposition of a sequence of discrete monochromatic light, with a discretization step of 1nm. The intensity flux of the light with each wavelength is set as the same. Thus the wide-spectrum speckle patterns is the incoherent superposition of a sequence of discrete monochromatic speckle patterns with the same average intensity. The center wavelength is set as $\lambda =600nm$, and patterns for spectral bandwidth of 20nm and 35nm are generated, respectively, as shown in Fig. 2. The field of view is 3cm $\times$ 3cm.

 

Fig. 2. Speckle patterns produced by a wide spectrum point source. (a) shows results of light with the bandwidth of $20nm$, and (b) of $35nm$. The white circles roughly mark the areas within which the speckles show high contrast.

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It can be seen that those speckles close to the optical axis possess more circular shape and higher contrast. The patterns far away from the optical axis appear radial divergence and feature a reduced contrast, consistent with the theoretical results in part II.A. In order to quantitatively estimate the range of high contrast speckles via Eq. (7), we select the two wavelengths with the largest difference under a certain bandwidth. With $\lambda _1=591nm$ and $\lambda _2=610nm$, we get $R=1.16cm$, as marked with white dashed circle in Fig. 2(a), for bandwidth of 20nm. With $\lambda _1=583nm$ and $\lambda _2=617nm$, we get $R=0.64cm$, as marked in Fig. 2(b). We also calculated the contrast of different regions. For speckles from the light of 20nm bandwidth, the average contrast is 0.99 at the center part and 0.93 on the marked circle. For the light of 35nm bandwidth, those values are reduced to 0.96 and 0.94, respectively. This reduction can be explained by Eq. (7), in which a wider-spectrum band means a smaller $\Omega$. Thus the visibility of the speckles for the whole observation plane is decreased.

The morphology and the spatially varying visibility of wide-spectrum speckle patterns can also be explained by a picture of speckles expansion. The speckle patterns produced by light of approaching wavelengths will show similarity in structure. From Eq. (2), the induced phase of the exited light from the scattering layer gets smaller with increasing wavelength. In order to achieve constructive interference, a longer propagation distance is required. Therefore, considering a certain bright speckle from $\lambda _1$, the counterpart speckle for larger wavelength $\lambda _2$ will appear larger and farther apart from the center point, as is sketched in Fig. 3. Such expansion also shows more apparent when the considered position is farther away from the center.

 

Fig. 3. Diagram of speckles superposition generated by two approaching wavelengths, where the solid blue lines enclose the speckles generated by $\lambda _1$, and the red dot lines surround those by $\lambda _2$. $(x_1,y_1)$ and $(x_2,y_2)$ show two considered points in the light field. With the subscript being 1, intensity contributed by wavelength $\lambda _1$ at the point is considered and subscript 2 for that of $\lambda _2$. While $d_{12}$ denotes the distance between $(x_1,y_1)$ and $(x_2,y_2)$.

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4. Statistics of wide-spectrum speckles

4.1 Second-order coherence

Apart from the intensity distribution of wide-spectrum speckles, which can be seen as the first-order statistics, here we analyze higher-order statistics, among which the second-order coherence is fundamental [41]. For illumination of wide-spectrum, the intensity distribution can be expressed as $I_{w}(x,y)=\sum _{l = 1}^n I(x,y,\lambda _l)$, with $n$ being the number of different wavelengths involved. Then the second-order coherence of the wide-spectrum speckle fields can be written as

Based on our assumptions, the speckle fields produced by light with two different wavelengths is still subjected to a multivariate zero-mean Gaussian distribution [13]. According to complex Gaussian moment theorem, we can obtain

According to Eq. (3) and Eq. (5), Eq. (10) can be simplified to

Equation (11) is obtained based on point source. Now, let’s discuss the case of plane wave illumination. Since a plane wave can be taken as a point source at infinity, for light waves with fixed wavelength, whether plane wave or point-source is used, the speckle patterns show similar regulations. That is, with $z$ tends to infinity, the quadratic phase factor in Eq. (1) can be ignored. Then $\frac {z_0+z}{z}$ tends to 1, in Eq. (11). In the next part, we will analyze the results of point light source and plane wave illumination.

Equation (11) shows that the value of $g^{(2)}(x_1,y_1,\lambda _1;x_2,y_2,\lambda _2)$ depends not only on $d_{12}$, but also on the observation position, the spectrum bandwidth of the light, and the thickness of scattering layer, which are significantly different from the monochromatic case. In the following part, the factors affecting the statistical properties of wide-spectrum speckles will be analyzed based on numerical simulation results.

4.2 Statistical properties

As shown in Eq. (9), the particularity of the second-order coherence of the wide-spectrum speckles is dominated by the superposition of the cross-correlation between the speckle fields produced by light with different wavelengths, in which the case involving speckles with two different wavelengths is fundamental. Thus we performed numerical simulation on the second-order coherence of two different wavelengths to present and analyze the statistical properties of the wide-spectrum speckles.

4.2.1 Effects of the observation position

The simulation is performed in the same scenario as shown in Fig. 1. The point source is a monochromatic source with tunable wavelength. Initial wavelength is $\lambda _1 = 600nm$. Another wavelength is $\lambda _2= \lambda _1 + \delta \lambda$, where $\delta \lambda$ increases with a step size of $1nm$. Other parameters are consistent with Fig. 2. The $g^{(2)}(x_1,y_1;x_2,y_2)$ of monochromatic speckles reach 2 when $x_1=y_1~\&~x_2=y_2$, and will decrease to 1 in any other case. Thus on the issue of wide-spectrum, the second-order coherence $g^{(2)}(\lambda _1,\lambda _2)$ will also be analyzed in two cases, including $g^{(2)}$ at the same point and that between two different points in the observation plane.

In the case that $x_1 = x_2=x$, $y_1 = y_2=y$, $g^{(2)}(x,y;\lambda _1,\lambda _2)$ is calculated and plotted in Fig. 4(a). The solid lines represent theoretical results, which obtained via Eq. (11). The dotted lines are numerical simulation results. Three different observation positions are tested respectively and the calculated results are shown by different colors. It can be seen from each line that $g^{(2)}$ decreases with the increasing of $\delta \lambda$, which can be extended to a conclusion that speckle fields with a wider spectrum possess a smaller peak value of $g^{(2)}$. Besides, for the considered points farther away from the optical axis, the faster decorrelation appears with the increasing of $\delta \lambda$, which shows that $g^{(2)}$ of the central part is larger than that of the peripheral part, being consistent with the conclusion pointed out by Goodman [38].

 

Fig. 4. (a) $g^{(2)}$ of different speckle fields at the same position. Different colored lines represent results for different positions. (b) $g^{(2)}$ between point $(x_1,y_1)$ in speckle fields of $\lambda _{1}$ and point $(x_2,y_2)$ in speckle fields of $\lambda _{2}$. Different colored lines represent results for different $(x_1,y_1)$. Here $d_{12}=$0.42mm $(x_2-x_1=0.3mm,y_2-y_1=0.3mm)$ is kept unchanged. (c)(d) comparison between point source and plane wave illumination. The black curves in (a) and (c) are quite the same, while the green curves in (b) and (d) are the same.

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Except for $g^{(2)}$ at the same coordinate, that between fields at $(x_1,y_1)$ from $\lambda _1$ and at $(x_2,y_2)$ from $\lambda _1\rm {+}\delta \lambda$ are also calculated. The correlation of monochromatic speckles vanishes when the distance between the paired observation points is larger than the average size of the speckles. For an intuitive comparison, the paired observation points with the distance of $d_{12}=$0.42mm$(x_2-x_1=0.3mm,y_2-y_1=0.3mm)$, which is greater than the average size of speckles ($0.27mm$), is selected in our simulation. As shown in Fig. 4(b). When $\delta \lambda$ is near $0$, which means the illumination light can be treated as monochromatic, the correlation vanishes. With increased $\delta \lambda$, positive correlation appears.

This can be explained upon that the speckles produced by the monochromatic light with a longer wavelength can be seen as a slight expansion of the speckles with a slightly shorter wavelength, when $\delta \lambda$ is small enough compared with both $\lambda _1$ and $\lambda _2$. Every speckle is radially shifted, as illustrated in Fig. 2. With the increasing of wavelength, the shifting distance between corresponding speckles gradually increases. As a result, the originally uncorrelated observation points begin to be correlated. When the shifting distance increase further and is larger than $d_{12}$, correlation decreases. Besides, the expansion of the scattering field causes the peripheral speckles shifting faster. Thus the correlation appears earlier when the paired observation points are peripheral. At the same time, for the peripheral speckles, the translation distance of $d_{12}$ can be achieved by changing a smaller $\delta \lambda$, so it contains higher $g^{(2)}$. By contrast, $g^{(2)}$ increases slower and reaches a lower peak for the center area. As an extreme, $g^{(2)}$ turns out to be 1 all the time when the exact center is involved.

Figure 4(c) and (d) show the results of plane wave and point source illumination while other parameters are kept the same. As shown in Fig. 4(c) and (d), the trend of curves are the same between both cases. For sure, there are difference between two cases. For plane wave illumination, the light wave incident on the front surface of the scattering medium has the same phase. However, for point source illumination, quadratic phase will introduce additional phase difference (specifically, the farther away from the optical axis, the greater the phase change.). Due to such additional phase difference, the correlation between speckles will be reduced. Therefore, in the case of plane wave illumination, the curve drops more slowly as shown in Fig. 4(c), or higher peak in second-order coherence as shown in Fig. 4(d).

4.2.2 Influences of the center wavelength

When an optical wave passes through a layer with a certain thickness or propagates freely for the same distance, the cumulated phase depends on its wavelength. Thus wide-spectrum light of different center wavelengths will produce different speckle fields, even if the bandwidth is the same. Here, we analyze the decreasing of second-order coherence with the central wavelength of the illumination source of the same spectrum band. As the same operation mentioned before, the wide-spectrum light is simplified as a point-like source with two different wavelengths, referred to as $\lambda _1$ and $\lambda _1\rm {+}\delta \lambda$. The center wavelength is indicated by $\lambda _1$.

The simulation is also performed in the same scenario shown by Fig.1. Without losing generality, the observation position is located on the optical axis, the second-order correlation between speckle fields produced by the light of $\lambda _1$ and $\lambda _1\rm {+}\delta \lambda$ are plotted, as shown in Fig. 5. Three curves with different colors are calculated under $\lambda _1\rm {=}400nm, \lambda _1\rm {=}500nm, \lambda _1\rm {=}600nm$, respectively. It shows that when $\lambda _1$ is larger, the correlation of speckle fields decreases slower with the same $\delta \lambda$. That is because when light of longer wavelength passes through the same distance, the phase difference between two wavelengths of the same spectrum bandwidth $\delta \lambda$ is smaller. Therefore, the evolution of speckle patterns is slower. In multispectral imaging, a narrow band filter is used to improve the visibility of the recorded images, accompanied with attenuation of energy. Now, with Eq. (11) or Fig. 5, the correlation of speckle patterns generated by different wavelengths can be estimated. Thus optimization of filtering bandwidth to balance between flux and contrast can be conducted for better imaging quality.

 

Fig. 5. Decorrelation with the increasing of $\delta \lambda$ under different center wavelengths. $g^{(2)}$ between patterns from $\lambda _1$ and $\lambda _1\rm {+}\delta \lambda$ is plotted, calculated for the center point. The considered central wavelengths are $400nm, 500nm$ and $600nm$.

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4.2.3 Influences of scattering thickness

According to the theory of multi-channel interference, the phase of the wavefront exiting from the scattering medium determines the intensity distribution of the speckles. Thus the thickness distribution also affects the degree of $g^{(2)}$ of the wide-spectrum speckle fields. Towards that, we set $\lambda _1\rm {=}600nm$, $\lambda _2\rm {=}\lambda _1\rm {+}\delta \lambda$, and the range of thickness as [$2\mu m,10\mu m$], [$1\mu m,13\mu m$] and [$3\mu m,18\mu m$], respectively. $g^{(2)}$ at the center of the receiving plane is shown in Fig. 6(a). With the increasing range of thickness ($\Delta L$), the value of ${\left | {\Omega } \right |^2}$ will decrease, so will $g^{(2)}$. By contrast, if $\Delta L$ is a constant, $g^{(2)}$ remains unchanged (as shown in Eq. (5) and Fig. 6(b)), even if the total thickness is changed. In essence, it comes from the fluctuation range of phases. When a constant phase is added to the light field, it will not change the speckle patterns. In imaging experiments, for the scattering medium with a wide range of thickness, a source of better monochromaticity is required to ensure the speckles contrast and imaging quality, according to our results.

 

Fig. 6. Decorrelation under different scattering thicknesses. $g^{(2)}$ is calculated for the center point on receiving plane. (a) The ranges of thickness are set as [$2\mu m,10\mu m$], [$1\mu m,13\mu m$] and [$3\mu m,18\mu m$], respectively. (b) The ranges of thickness are set as [$1\mu m,9\mu m$], [$2\mu m,10\mu m$] and [$4\mu m,12\mu m$].

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5. Discussions and conclusion

Here, we mainly focused on phases induced by scattering, and that caused by propagation is not discussed, although included in simulations. Of course, the path of free propagation does also affect the value of $g^{(2)}$. However, for two fixed points in the receiving plane, if the considered points are farther away from the scattering medium, the relative difference between the two points will be smaller. Then first-/second- order coherence will appear less variation with wavelength. Since paraxial approximation is made, the phase induced by scattering is dominant.

Let us analyze the difference between phase screen models with and without thickness. For existed phase screen model without considering thickness, the induced phase by scattering, through a certain point, keeps the same for all the wavelengths. Therefore, $\Delta \varphi$=0, and $\Omega$ turns out to be unity all the time. In other words, there is no difference in the scattering medium for different wavelengths. According to Eq. (11), the phase difference will be only caused by propagation, with the rate of decorrelation slower.

The above discussion are all based on a single coherent point source. Increasing the spectral width of the light source will reduce the speckle contrast on the receiving plane. The more outward, the lower the contrast. If the illumination source contains two or more separated incoherent points, patterns from different source points will show spatial shifting. Therefore, the incoherent superposition will also cause decrease in correlation and the contrast of speckles. Further study on this can help to clarify wide-spectrum scattering phenomenon and provide tool for image reconstruction under such scenarios.

In summary, by introducing the phase plate model, we show the morphology of wide-spectrum speckles. The speckle patterns appear radial divergence, with the contrast of the central part higher than the peripheral area. From the first-order coherence of the two wavelengths, the visibility of speckles is related to the wavelength and transverse distance. For wide-spectrum speckles, $g^{(2)}$ decreases with the increasing of the distance from the observation point to the optical axis, which means $g^{(2)}$ no longer contains spatial invariance. At the same time, the structure of patterns is determined by the total phase differences among the detection plane, and we also analyze the effects of the central wavelength, the range of thickness on $g^{(2)}$ in numerical simulation. Our results on the morphology and statistical characteristics of wide-spectrum speckles can be of great significance in speckle metrology, wide-spectrum imaging against scattering, and so on.