Does the structure of light influence the speckle size?

It is well known that when a laser is reflected from a rough surface or transmitted through a diffusive medium, a speckle pattern will be formed at a given observation plane. An important parameter of speckle is its size, which for the case of homogeneous illumination, well-known relations for its computation have been derived. This is not the case for structured light beams of non-homogeneous intensity and phase distribution. Here, we propose and demonstrate, using Hermite- and Laguerre-Gaussian light modes, that the mean size of the speckle generated by these structured light beams can be measured assuming a homogeneous illumination. We further provide with mathematical expressions that relate the speckle size to the generalised definition of "spot size". To reinforce our assessment, we compare the mean speckle size generated by structured light modes with that generated by wave fronts of constant phase and amplitude and show that in both cases the mean speckle size is almost identical. Our findings reveal a fundamental property of speckle, which will be of great relevance in many speckle-based applications and will pave the way towards the development of novel applications.


Speckle size produced by circular and rectangular aperture of homogeneous-intensity
The mean speckle size can be measured through the autocorrelation function, which in terms of the intensity of the beam illuminating the rough surface is given by 1 where, |P(x, y)| 2 is the intensity function describing the area illuminated by the light beam. For the specific case of a circular aperture of radius R and uniform intensity given by, where ρ = x 2 + y 2 and circ(ρ/R)=1 for |ρ| ≤ R and zero otherwise, we obtain, after substituting into Eq. 1, the well-known expression 2 , In the above equation, z f is the distance between the rough surface and the observation plane and J 1 (ξ ) is the Bessel function of the first kind and order 1. The mean radius of the speckle can then be definedn as the value √ ∆u 2 + ∆v 2 ≡ ∆s circ for which J 1 (ξ ) first becomes zero, which happens when the argument is equal to 1.22π. Hence, the mean diameter of the generated speckle is, The above equation can be written in terms of the total illuminated area as, Figure S1. Intensity distribution of an LG 2 1 mode, whose spot size is illustrated by the transparent circle in (a), which is also indicated by the dashed vertical lines shown in the 1-Dimensional intensity profile in (b). Intensity distribution of an HG 23 mode, whose spot size is illustrated by the transparent rectangle (c), also illustrated in the 1-Dimensional intensity profile shown in (d).
which is Eq.2 of our main manuscript.
For the case when the illuminating wave has a rectangular shape of dimensions L 1 × L 2 , the intensity function is mathematically given by, where, rect(x)=1 for |x| ≤ 1 and zero otherwise. In this case, the autocorrelation function, obtained by inserting Eq. 6 into Eq. 1 will take the form 1, 2 , where sinc(ξ ) is defined as sin(πξ )/(πξ ). Again, the mean speckle size can then be taken as the value ∆u (or ∆v) for which sinc 2 (ξ ) first become zero, which yields, In terms of the total illuminated area A = L x L y , the mean speckle size is then defined as which is Eq. 4 in our main text.

The spot size in Laguerre-and Hermite-Gaussian modes
The generalised definition of "spot size", taken as the maximum area to where the beam's intensity still has a significant value, provides with the perfect means to measure the total area illuminated by the LG p and HG nm modes (see Fig. S1). For the case of LG p modes, the radius σ of the spot size (see Fig. S1(a) and S1(b)) can be derived using the standard deviation as 3 , where, I LG (x, y, z) is the intensity of a given LG p mode. Both integrals in Eq. 10 are straightforward and the final result provides with the spot size radius in terms of both modal indices (see 3 for a detailed derivation), namely, Hence, the spot size for any LG p mode, as function of the modal indices takes the form,  Figure S2. Mean speckle size generated by the subset of LG p modes analysed in our main text. Here modes with the same spot size have been coded with the same colour. Notice that the mean speckle size labelled with the same colour is almost identical.
which is equation 7 of our main file. For the case of HG nm modes (S1(c) and S1(d)), the spot size can be found by first computing the standard deviation along the horizontal and vertical directions 4 , that is, where, I 0m (x, y, z) and I n0 (x, y, z) are the intensities of the HG 0m (x, y, z) and HG n0 (x, y, z) modes, respectively. The integrals in Eq. 13 can be computed in a straight forwad way, yielding the final result 3 , from which, the spot size can be computed as, which is Eq. 9 in our main file.

Speckle size for Laguerre-Gaussian modes of similar areas
A closer inspection of Eq. 12 reveals that the spot size of certain LG p modes will be the same for particular combinations of p and . For example, the mode LG 2 0 has the same area as the mode LG 0 1 , namely, A p = 3πω(z) 2 . In the same way the modes LG 0 4 , LG 2 3 and LG 4 2 will also have the same area, which in this case will take the value A p = 9πω(z) 2 . Incidentally, modes with the same spot size will propagate in an identical manner, experiencing the same diffraction properties. It is expected then that such modes should produce a speckle pattern with the same mean size. This was corroborated experimentally and shown in Fig. S2 where a detailed list of the mean speckle size for the subset of LG p analysed in the main text is presented. For the sake of clarity, modes with the same spot size are represented with the same colour. For example, the modes LG 0 3 , LG 2 2 and LG 4 1 , with the same spot size, have almost identical mean speckle size, namely, 35.25 µm, 35.22µm and 35.23µm, respectively.