Optical speckle arises from the interference between random distributions of plane wave components, such as generated by light scattering from rough surfaces or propagating through turbid diffusers41. For example, when a laser is incident on an object such as ground glass or scattering screen, the transmitted or reflected light would be observed with fine-scale granular pattern. According to the Huygens-Fresnel principle, the optical speckle resulting from scattering coherent light can be considered as the interference caused by different scattering points that act as individual new nearly-spherical wave sources. Since the solid angle subtended by the detecting system is sufficiently small, each spherical wave in the volume of space around the viewing aperture is approximated by a plane wave. Hence the plane-wave approximation is widely used to simulate the optical speckle mathematically42, 43. In this work we model the optical speckle as a superposition of a large number of plane waves with random phases and directions, as shown in Fig.1a. The intensity pattern of the speckle has a grainy appearance, where the bright spots and dark specks arise from the constructive and destructive interference respectively. In particular, the centre of each dark speck is a phase singularity, and in 3 dimensions these dark filaments thread themselves through the speckle field creating highly complicated networks of vortex lines and loops44,45,46. Intuitively, the angular spectrum of light field can be mapped to direction space of wave vectors (k-space), i.e., with amplitude correspondence to the k-spectrum, where each point represents a plane wave, to which is assigned random transverse projected components (kx and ky), as shown in Fig.1b. The corresponding nonzero radial component \(k_{r} = \sqrt {k_{x}^{2} + k_{y}^{2} }\) produces a modification of average axial component \(\left\langle {k_{z} } \right\rangle = \sqrt {k_{0}^{2} – \left\langle {k_{r}^{2} } \right\rangle }\), where \(\left\langle {…} \right\rangle\) denotes the statistic expectation over the k-spectrum.

Figure 1
figure 1

Optical speckle in free space and k-space. (a) Superposition between a sufficiently large set of randomly phased and directed plane waves is an approximation to the optical speckle created by scattering a laser beam from a diffuser. (b) k-spectrum of optical speckle and the projection of one of the points in direction space of wave vectors.

To characterize the propagation speed of optical speckle, we introduce the phase and group velocities which are averaged across all the wave components velocity which we have previously shown corresponds to the time the light or photons take to travel from plane to plane. Different from the conventional definition of group velocity47, the spatially average group velocity refers to the traveling energy envelope of a group of plane waves with a small spread in directions (i.e., spatial components in k-spectrum) rather than in frequencies or wavenumbers (i.e., temporal components in frequency spectrum). The spatially averaged phase velocity is then given as \(v_{\phi } = c \cdot {{k_{0} } \mathord{\left/ {\vphantom {{k_{0} } {\left\langle {k_{z} } \right\rangle }}} \right. \kern-0pt} {\left\langle {k_{z} } \right\rangle }}\) by the average value of kz. For a structured beam in free-space it seems sensible that the average group velocity and phase velocity have the same relation as in the theory of hollow waveguides, i.e., \(v_{\phi } v_{g,z} = c^{2}\). The condition is best satisfied when we assume that the radial projection of wave vector kr in optical speckle analyzed here is independent on its angular frequency . The resultant spatially average group velocity along z is thus given as

$$v_{g,z} = c \cdot \sqrt {1 – {{\left\langle {k_{r}^{2} } \right\rangle } \mathord{\left/ {\vphantom {{\left\langle {k_{r}^{2} } \right\rangle } {k_{0}^{2} }}} \right. \kern-0pt} {k_{0}^{2} }}} ,$$

(1)

meaning that structured beams with a nonzero expectation value of \(k_{r}^{2}\), of which optical speckle is one example, will experience a reduced propagation speed, i.e., vg,z<c.

We emphasize that the optical field considered here is quasi-chromatic, i.e., the frequencies of the wave group are clustered in a very narrow region around the main frequency. The field endowed with fixed \({k}_{r}\) still experiences group velocity dispersion (GVD) when the input beam is pulsed. This is another distinction between the effect of structured slow light and the group velocity control with spacetime wave packets, which results in dispersion-free propagation48, 49. However, in the structured slow light the amount of GVD is insignificant compared with the differentiable group delay \(\tau_{DGD} = L\left| {\frac{1}{c} – \frac{1}{{v_{g} }}} \right|\) acquired by this pulse, where L is the axial propagation distance. It can be shown that for the pulse of spectral bandwidth of and spatial wavevector kr the ratio of pulse broadening to the differentiable group delay \(\tau_{DGD}\) is proportional to \(\frac{\Delta \omega }{{\omega_{0} }}\), which is at the quasi-monochromatic regime is negligible, i.e. \(\frac{\Delta \tau }{{\tau_{DGD} }} \sim \frac{\Delta \omega }{{\omega_{0} }} \ll 1\)49, 50.

As introduced earlier, to experimentally generate an optical speckle with kr components that are independent of k0 requires diffractive elements, e.g., superposed grating patterns uploaded on SLM. For a single randomized plane wave produced by a hologram of grating pattern with fringe separation d, the resultant transverse components kx (ky) is 2/dx (2/dy), and kr is independent of wavelength. Each plane-wave hologram is assigned with three individual variables: polar angle, azimuthal angle and phase offset, where the polar angles are distributed with a Gaussian profile, and both azimuthal angles and phase offsets are uniform noise. The resulting phase hologram uploaded on SLM comprises the wavevectors of optical speckle by combining the grating patterns.

To model such optical speckle numerically, we define a finite two-dimensional grid in transverse k-space, where each point describes a plane wave tilted by x and y with respect to the propagation axis. According to the central limit theorem51, the superpositions of infinitely many waves tend to Gaussian random functions52. The ensembles of plane harmonics are asymptotically Gaussian, which means that the probability density distribution of each tilted direction (x and y) follows a 2D Gaussian distribution, as shown in Fig.2a. Our simulation for optical speckle is based on a superposition of 2000 plane waves randomly distributed in direction and phase, and each of which has a Gaussian amplitude in profile. Their distribution in k-space is subject to a Gaussian density distribution, characterized by a divergence of sin in free space, where is the standard deviation of the tilted angles of the wave vectors. A typical example for =5 is calculated in Fig.2b. The resultant intensity profile of the optical speckle in the far field is shown in Fig.2c. By performing 2D Fourier transform for the complex amplitude of the speckle field, its k-spectrum is obtained as shown in Fig.2d, where the coordinates are divided by the initial wavenumber k0. It can be seen that the k-spectrum of optical speckle has a 2D Gaussian density envelope, which depends on the distribution of tilted directions of wave vectors in Fig.2b. More importantly, in the paraxial regime, the effect of light propagating over z in free space is simply a phase change in the components of its angular spectrum, and then since the k-spectrum is equivalent to the modulus of angular spectrum mathematically, the k-spectrum of optical speckle is propagation-invariant, which means that its slowing persists over arbitrarily long ranges.

Figure 2
figure 2

An example of numerically generating optical speckle. (a) Gaussian probability density distribution of tilted directions in k-space. (b) Direction points with Gaussian density of standard deviation of 5. (c) Intensity profile of optical speckle created by the interference between Gaussian random waves with directions as (b). (d) Calculated k-spectrum of speckle field by 2D Fourier transform of its complex amplitude.

Optical speckle is usually characterized by its lateral size, which refers to the lowest length scale at which beam is correlated53. Particularly, for a fully developed speckle field created by a scattering surface, the size of speckle increases with the distance from the surface to observation plane54, 55. From the perspective of plane wave interference, the larger the tilted angle, a greater the transverse phase varying gradient and then the denser the interference fringes. In a Fourier sense, statistical properties with high complexity in real space correspond to an expanded angular spectrum. This means that the k-spectrum range of speckles is negatively correlated with speckle size.

To evaluate the degree of slowing for this numerically creating optical speckle, we divide the k-spectrum in Fig.2d radially according to the evenly equidistant 1000 scales on the established \({{k_{r}^{2} } \mathord{\left/ {\vphantom {{k_{r}^{2} } {k_{0}^{2} }}} \right. \kern-0pt} {k_{0}^{2} }}\) axis. By summing and normalizing all the amplitudes with the individual ring regions divided from k-spectrum, each ring is calculated as a value point with global normalized probability along the \({{k_{r}^{2} } \mathord{\left/ {\vphantom {{k_{r}^{2} } {k_{0}^{2} }}} \right. \kern-0pt} {k_{0}^{2} }}\) axis, as shown in Fig.3. Physically, each discrete point represents the probability of a plane wave that appears within a \({{\Delta k_{r}^{2} } \mathord{\left/ {\vphantom {{\Delta k_{r}^{2} } {k_{0}^{2} }}} \right. \kern-0pt} {k_{0}^{2} }}\) ring region of k-space, where \(\Delta k_{r}^{2}\) is the division value on axis. In this case (=5), the value \({{\left\langle {k_{r}^{2} } \right\rangle } \mathord{\left/ {\vphantom {{\left\langle {k_{r}^{2} } \right\rangle } {k_{0}^{2} }}} \right. \kern-0pt} {k_{0}^{2} }}\) is calculated as 0.022465, and then the spatially average group velocity of such optical speckle is calculated by Eq.(1) as \(v_{g,z} \approx 0.9887c\). This means that the propagation speed of an optical speckle with the Gaussian divergence with standard deviation of 5 corresponds to a slowing of 1.13% in free space.

Figure 3
figure 3

Statistical distribution of tilted components in optical speckle. The discrete points represent the probability distribution of the k-spectrum components calculated along the scales of radial proportion square. The solid curve is the theoretical probability density distribution of radial proportion square from an ideal continuous Gaussian angular spectrum.

In addition to the discrete sampling of the k-spectrum of optical speckle as an example, a continuous probability density distribution of radial proportion square \({{k_{r}^{2} } \mathord{\left/ {\vphantom {{k_{r}^{2} } {k_{0}^{2} }}} \right. \kern-0pt} {k_{0}^{2} }}\) can be deduced from the Gaussian angular spectrum mathematically as

$$p\left( {{{k_{r}^{2} } \mathord{\left/ {\vphantom {{k_{r}^{2} } {k_{0}^{2} }}} \right. \kern-0pt} {k_{0}^{2} }}} \right) = {{\sqrt {2\pi \cdot {{k_{r}^{2} } \mathord{\left/ {\vphantom {{k_{r}^{2} } {k_{0}^{2} }}} \right. \kern-0pt} {k_{0}^{2} }}} } \mathord{\left/ {\vphantom {{\sqrt {2\pi \cdot {{k_{r}^{2} } \mathord{\left/ {\vphantom {{k_{r}^{2} } {k_{0}^{2} }}} \right. \kern-0pt} {k_{0}^{2} }}} } {\sin \sigma_{\theta } }}} \right. \kern-0pt} {\sin \sigma_{\theta } }} \cdot \exp \left( { – \frac{{{{k_{r}^{2} } \mathord{\left/ {\vphantom {{k_{r}^{2} } {k_{0}^{2} }}} \right. \kern-0pt} {k_{0}^{2} }}}}{{2\sin^{2} \sigma_{\theta } }}} \right),$$

(2)

where sin again refers to the divergence of optical speckle. Figure3 shows a good fit between the theoretical curve of Eq.(2) and the sampling points from a typical k-spectrum in Fig.2d.

We perform a numerical analysis for the relation between slowing effect as a function of the divergence of the optical speckle. In particular, the divergence refers to the spreading angle, which describes the standard deviation of the tilted angles of the wave vectors, as shown in the inset of Fig.4. By gradually adjusting from 0.5 to 5 at 0.5 intervals, we calculate the value \({{\left\langle {k_{r}^{2} } \right\rangle } \mathord{\left/ {\vphantom {{\left\langle {k_{r}^{2} } \right\rangle } {k_{0}^{2} }}} \right. \kern-0pt} {k_{0}^{2} }}\) and corresponding slowing, as plotted in Fig.4. For each case, the differences in the generation of Gaussian distributed random numbers within a diverging range would result in variation of the slowing predicted, and hence the error bar is derived from performing the calculation 8 times using the method of Fig.3. As anticipated, the predicted slowing effect becomes greater as the divergence increases.

Figure 4
figure 4

Numerically quantifying slowing effect of optical speckle. (a) Expectation values of radial proportion square and (b) degree of slowing under different divergence of optical speckle. The inset is a schematic of divergence of optical speckle where is the half spreading angle that describes the tilted plane-wave components.

Beyond the numerical simulations described above, the theoretical expression of slowing effect also deduced. According to the probability density distribution of radial proportion square \({{k_{r}^{2} } \mathord{\left/ {\vphantom {{k_{r}^{2} } {k_{0}^{2} }}} \right. \kern-0pt} {k_{0}^{2} }}\) in Eq.(2), its expected value is calculated as

$${{\left\langle {k_{r}^{2} } \right\rangle } \mathord{\left/ {\vphantom {{\left\langle {k_{r}^{2} } \right\rangle } {k_{0}^{2} }}} \right. \kern-0pt} {k_{0}^{2} }} = \frac{{\int_{0}^{\infty } {p\left( {{{k_{r}^{2} } \mathord{\left/ {\vphantom {{k_{r}^{2} } {k_{0}^{2} }}} \right. \kern-0pt} {k_{0}^{2} }}} \right)} \cdot {{k_{r}^{2} } \mathord{\left/ {\vphantom {{k_{r}^{2} } {k_{0}^{2} }}} \right. \kern-0pt} {k_{0}^{2} }} \cdot d\left( {{{k_{r}^{2} } \mathord{\left/ {\vphantom {{k_{r}^{2} } {k_{0}^{2} }}} \right. \kern-0pt} {k_{0}^{2} }}} \right)}}{{\int_{0}^{\infty } {p\left( {{{k_{r}^{2} } \mathord{\left/ {\vphantom {{k_{r}^{2} } {k_{0}^{2} }}} \right. \kern-0pt} {k_{0}^{2} }}} \right)} \cdot d\left( {{{k_{r}^{2} } \mathord{\left/ {\vphantom {{k_{r}^{2} } {k_{0}^{2} }}} \right. \kern-0pt} {k_{0}^{2} }}} \right)}} = 3\sin^{2} \sigma_{\theta } ,$$

(3)

where the infinite of upper limit in the integral is only mathematically meaningful for its normalization among the whole space, while more strictly in physics, the upper limit should be 1 since kr<k0. Clearly, \({{\left\langle {k_{r}^{2} } \right\rangle } \mathord{\left/ {\vphantom {{\left\langle {k_{r}^{2} } \right\rangle } {k_{0}^{2} }}} \right. \kern-0pt} {k_{0}^{2} }}\) is proportional to square of the divergence of optical speckle, see the solid curve in Fig.4a. Using Eq.(1), for small angles , the degree of slowing of optical speckle is theoretically calculated as

$${{v_{g,z} } \mathord{\left/ {\vphantom {{v_{g,z} } c}} \right. \kern-0pt} c} – 1 = \sqrt {1 – 3\sin^{2} \sigma_{\theta } } – 1.$$

(4)

Figure4b indicates the agreement between the theoretical curve and the mean values of each result calculated by discrete statistical method. Note that Eq.(4) is only applicable for the low-NA case to ensure paraxial approximation. Significantly, the slowing of the optical speckle can reach of order 1% even with a small beam divergence. Over the range of several meters, the temporal delay of optical speckle is thus predicted to be enhanced by three orders of magnitude for the same traveling distance compared to the previously measured Bessel or focused beams27.

To anticipate the observable slowing in a practical detecting system, we consider the role that the aperture of the detector plays. The NA is a restriction on k-space when the optical speckle is observed by a detector or our eyes, as shown in the inset of Fig.5. When considering the restriction on complete spatial harmonics collecting of optical speckle by the detecting system, the upper limit of the integral in Eq.(3) is replaced by NA2 from infinite. In the initialization settings of calculation here, the beam waist of Gaussian-distributed intensity profile of the optical speckle is set to 2mm, and its half spreading angle is set to 5. Figure5 shows the calculated degree of slowing under different NA, where the dashed line predicted by Eq.(4) refers to the ideal case without restriction of NA, and the solid curve is predicted by modified Eq.(3), and the data points are obtained with 8 calculations by filtering the complex amplitude of speckle field in k-spectrum. Since the NA is a restriction of maximum range of angular spectrum, the angular spectrum outside this range is filtered out, which is analogue to a low-pass filtering while the higher components in k-spectrum give a greater slowing. This means that reducing the NA of the detection system will obviously reduce the corresponding slowing effect, which is seen in Fig.5. In contrast, a beam aperture, i.e., a transverse restriction to the propagation of light in real space, would not impact the slowing effect drastically since whole spatial harmonics can pass the aperture, but the restriction of beam aperture would reduce the resolution of the k-spectrum of field due to the correspondence of the maximum of beam size to the minimum of k-space. Note that the structured slowing effect analyzed in this work is a global property. However, when one observes the local grain of optical speckle, the structured slowing effect is preserved even within a small region of interest, as all transverse kr could contribute to the light behavior in this region.

Figure 5
figure 5

Restrictions of practical system on slowing effect of optical speckle where the itself speckle has a divergence of 5 and the detector has a limiting NA. The degree of slowing calculated under different numerical apertures (NA) of detecting system. The inset is a schematic of a detecting system for observing the plane-to-plane propagation of optical speckle.

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