Introducing non-local correlations into speckles

Laser speckles have become a fundamental component of the modern optics-research toolbox. Not only are speckle patterns the basis of numerous imaging techniques, but also, they are employed to generate optical potentials for cold atoms and colloidal particles. The ability to manipulate a speckle pattern's spatial intensity correlations, particularly long-range (non-local) ones, is essential in numerous applications. A typical fully-developed speckle pattern, however, only possesses short-ranged (local) intensity correlations which are determined by the spatial field correlations. Here we experimentally demonstrate and theoretically develop a general method for creating fully-developed speckles with strong non-local intensity correlations. The functional form of the spatial intensity correlations can be arbitrarily tailored without altering the field correlations. Our approach provides a versatile and utilitarian framework for enhancing and controlling non-local correlations in speckle patterns.


I. INTRODUCTION
A bedrock principle of optical coherence theory is the Siegert equation, which relates the intensity and the field correlation functions. It was used in the Hanbury Brown-Twiss stellar interferometry experiment to infer the spatial field correlation length of a signal from the measurement of its spatial intensity correlation length: in order to extract the angular diameter of radio stars [1]. In quantum optics, photon bunching or anti-bunching violates the Siegert equation, and this violation has been widely explored in studies of non-classical light such as squeezed light [2,3].
However, there is a dearth of investigation focused on tailoring a random medium in order to enhance or control the long-range intensity correlations of its scattered waves. The simplest "scattering" structure -which can be controlled with facility-is a spatial light modulator (SLM). Although the incident light is only scattered once by the SLM, arbitrary correlations can be encoded among the SLM pixels. Such correlations can be significantly stronger and more versatile than the correlations built among partial waves during the process of multiple scattering in a random medium.
Beyond the scope of mesoscopic transport, the ability to manipulate the spatial intensity correlations of speckle patterns has a plethora of potential applications. For example in speckle-based fluorescence microscopy, C I (∆r) corresponds to the point spread function [28][29][30][31][32][33][34], and thus customizing C I (∆r) enables one to engineer the point spread function. Furthermore, laser speckle patterns with designed intensity correlations can be used as bespoke disordered optical-potentials in transport studies of cold atoms [35], colloidal particles [36], and active media [37]. While there have been previous studies dedicated to altering speckle patterns and their spatial intensity correlations C I (∆r) of speckle patterns [38][39][40][41][42][43][44][45][46][47][48][49], generally this has been accomplished by using the Siegert relation and modulating the spatial field correlations. It is much more challenging to violate the Siegert relation and control the intensity correlations without changing the field correlations. Such a modification requires the field and intensity to fluctuate spatially on different length scales. Even in our recent demonstrations of speckle patterns with arbitrary intensity probability density functions, the field and the intensity still have the same correlation length [50,51]. Although the speckled speckles produced by double scattering have C I (∆r) = |C E (∆r)| 2 , the difference C I (∆r) − |C E (∆r)| 2 representing the non-local intensity correlations C N L (∆r) is rather small [52][53][54].
In this article, we not only enhance but also manipulate the long-range intensity correlations of speckle patterns. We show that the intensity correlation length can arXiv:1807.00671v1 [physics.optics] 2 Jul 2018 be augmented such that it significantly exceeds the field correlation length: with non-local intensity correlations comparable in strength to the local intensity correlations. Furthermore, we demonstrate that it is possible to arbitrarily tailor the long-range intensity correlation function -for example making it anisotropic and oscillatingwhile keeping the field correlation function isotropic and untouched. Finally, a theoretical analysis reveals that the non-local intensity correlations in the far-field speckle patterns originate from high-order phase correlations encoded into the near-field by the SLM.

II. EXPERIMENTAL SETUP
Experimentally, a phase-only reflective SLM (Hamamatsu LCoS X10468) and a CCD camera (Allied Vision Prosilica GC660) are juxtaposed at the front and back focal planes of a lens with focal length f = 500 mm. The SLM is uniformly illuminated by a linearly-polarized monochromatic laser beam at wavelength λ = 642 nm (Coherent OBIS). We only use the central part of the phase modulating region of the SLM, and partition it into a square array of 32 × 32 macro-pixels, each consisting of 16 × 16 pixels. The remaining illuminated pixels outside the central square diffract the laser beam away from the CCD camera via a phase grating. While to a good approximation the field on the camera is a Fourier transform of the SLM field, we use an experimentally measured field-transmission matrix (T-matrix) to relate the light field on the SLM and camera planes in order to be more precise and general.

III. ISOTROPIC CORRELATIONS
First, we will demonstrate how to increase the intensity correlation length of the speckles in the far-field of the SLM without altering the speckle field correlation length. We begin by measuring a generic Rayleigh speckle pattern -such as the one shown in Fig. 1(a,c)when a random phase pattern is displayed upon the SLM. In this case, the speckle field obeys a circular Gaussian probability density function for the complex amplitudes, and possesses only short-range intensity correlations, C I (∆r) = |C E (∆r)| 2 , as confirmed in Fig. 1(e). To proceed, we magnify the speckle intensity pattern numerically by a factor α, in order to increase the intensity correlation length by the same factor α. A nonlinearoptimization algorithm [55,56] is used to determine a phase pattern -which upon application to the SLM-generates the enlarged speckle intensity pattern on the camera plane. To facilitate the convergence to a numerical solution, we reduce the area we attempt to control -on the camera plane-to the central quarter of the region representing the Fourier transform of the phase modulating region of the SLM [51]. Numerically we minimize the difference between the target intensity pattern and the intensity pattern obtained after applying the T-matrix to the SLM phase array. Since the SLM does not change the field amplitude, the spatial field correlation function in the Fourier plane remains identical to that of the unmagnified speckle pattern and therefore, so do the local intensity correlations C L (∆r) = |C E (∆r)| 2 [57].
After numerically finding the appropriate twodimensional (2D) SLM phase-patterns, we shift to the experimental setup and display them on the SLM: recording the speckle patterns incident upon the CCD camera. Figure 1(b,d) presents one demonstration of an "enlarged Rayleigh" speckle pattern. The intensity fluctuates on a length scale α = 2.5 times longer than that of the Rayleigh pattern in Fig. 1(a). While the width of C I (∆r) is increased 2.5 times, |C E (∆r)| 2 remains the same as that of the original Rayleigh speckles, as shown in Fig.  1(f). This means that the speckle field, more precisely, the phase of the field plotted in Fig. 1(d), fluctuates faster in space than the intensity. Still, the phase pattern is significantly modified relative to that of a Rayleigh speckle pattern such as in Fig. 1(c). It exhibits distinct topological features such as elongated equiphase lines, which can be see in Fig. 1(d). Nevertheless, these features are masked by the spatial averaging inherent to the spatial field correlation function. The dramatic difference between C I (∆r) and |C E (∆r)| 2 demonstrates the profound non-local intensity correlations present in the speckle pattern. It is important to note, however, that the non-local correlations are largely independent from the other statistical properties of the speckles: such as the intensity probability density function.
Since the Rayleigh speckles are magnified by the same factor α = 2.5 in both x and y directions, the intensity correlation functions, both C L and C N L , are isotropic and depend only on ∆r = |∆r| = (∆x) 2 + (∆y) 2 . Figure 1(f) compares C N L (∆r) to C L (∆r) and C I (∆r). Unlike C L , C N L does not decay monotonically with ∆r, instead it rises to its maximum when C L almost dies out, and subsequently C N L dominates the functional form of C I (∆r). The maximum value of C N L is comparable to that of C L at ∆r = 0. In this example, the speckle intensity correlations become long-ranged but remain isotropic, namely, the correlation lengths are identical in both the x and y directions. We can easily make the correlations anisotropic, by setting the amplification factor in x different from that in y. This allows us to tune the intensity correlation lengths in x and y separately without modifying the field correlation length.

IV. ANISOTROPIC CORRELATIONS
Next, we will demonstrate how to synthesize speckles with significantly more complex spatial intensity correlations. Figure 2(a) shows C I (∆r) with an oscillating nonlocal correlation function C N L (∆r) = (1/10) cos[(∆x + ∆y)/10], where x and y are spatial coordinates. To generate speckles possessing such correlations, we first find speckle intensity patterns I(r) which adhere to the desired C I (∆r). Since the Fourier transform of I(r) is related to that of C I (∆r) by F[C I (∆r) + 1] = |F[I(r)]| 2 , |F[I(r)]| is known. As plotted in Fig. 2(b), it is a sparse function. We then solve for I(r) with a Gerchberg-Saxton algorithm [58,59]. Starting with a Rayleigh speckle intensity pattern, J(r), we modify the amplitude of its Fourier components, such that |F[J(r)]| is equal to | F[C I (∆r) + 1]|, without altering the phase values. The inverse Fourier transform of the modified Fourier spectrum gives a complex valued function for J(r). Since intensity values must be positive real numbers, we ignore the phase values and setJ(r) = |J(r)|. Cyclical repetition of this process will eventually result in a spatial intensity pattern which adheres to the desired correlation function. Starting with different initial Rayleigh speckle patterns will produce uncorrelated intensity patterns that satisfy the same C I (∆r). Using the T-matrix based non-linear optimization algorithm discussed previously, we obtain the SLM phase patterns to create the desired intensity patterns on the camera. Fig-Figure 2. Creating speckle patterns with spatially oscillating, anisotropic long-range intensity correlations. The spatial intensity correlation function CI (∆r) (a), determines the Fourier amplitude profile of I(r) (b). An experimentally generated speckle intensity-pattern I(r) (c) possessing the correlations given in (a), and the corresponding phase profile θ(r) (d). The local intensity correlation function CL(∆r) (e) has a maximum value of 1, while the non-local intensity correlation function CL(∆r) (f) has a maximum/minimum value of ±0.1. The correlation functions in (a,e,f) are obtained by averaging over 100 speckle patterns. The origins in (a,b,e,f) are located at the center of the plots. ure 2(c) presents one such intensity pattern recorded experimentally. Its phase profile is predicted by the measured T-matrix and shown in (d). The local intensity correlation function C L (∆r) = |C E (∆r)| 2 , shown in Fig.  2(e), remains isotropic and identical to that of the original Rayleigh speckles. However, the non-local correlation function C N L (∆r), plotted in (f), oscillates along the diagonal direction.
A useful feature of our method is its ability to vary the contrast of the speckle intensity without altering the functional form of the long-range intensity correlation function. For the example given in Fig. 2, we can adjust the magnitude of the zeroth-order spatial frequency component in (b), in order to change the constant background of the speckle intensity pattern in real space and thus modify the speckle contrast. Speckle patterns with identically shaped, i.e. congruent, C N L but different intensity contrasts are presented in Fig. 3 (a,c). Given that the speckle contrast is directly related to the second moment of the intensity probability density function, this property illustrates the relative independence of the nonlocal correlations with respect to the intensity probability density function. Although the above method excels at generating speckle patterns when the desired non-local correlation function has sparse Fourier components, it fails to converge to a speckle pattern when the desired non-local correlation function is sparse in real space, such as the one shown in Fig. 4(a). While the correlations are positive at (0, 100 µm) and (0, -100 µm), they become negative at (100 µm, 0) and (-100 µm, 0). Rather than producing a random intensity pattern, the Gerchberg-Saxton algorithm converges to a more or less ordered pattern, g(r) in (d), which adheres to the desired C I (∆r) in (a). To produce a speckle intensity pattern, we simply convolve g(r) with a speckle pattern without non-local correlations, such as J(r) in (e), and obtain I(r) = g(r) J(r). This results in a speckle pattern with F[I(r)] = F[J(r)]F[g(r)], and F[C I (∆r)] ∼ = F[C J (r)]F[C g (r)]. Since the local correlation length of the convolving speckle pattern is set by the diffraction limit, its correlation function can be approximated by a δ function [60]. Consequently, F[C I (∆r)] ≈ F[C g (r)], and I(r) possesses the same intensity correlations as g(r). Once the target intensitypattern I(r) is obtained, a corresponding speckle-pattern can be created experimentally using the machinery of our T-matrix based nonlinear optimization algorithm. Fig-ure 3 (f) is an experimentally generated speckle intensity pattern I(r). The corresponding local and non-local intensity correlation functions are shown in (b) and (c). Just as before, one has the freedom to increase or decrease the speckle contrast of the target pattern. This is accomplished by convolving g(r) with either a super-Rayleigh or a sub-Rayleigh speckle intensity pattern; which has either a higher or lower contrast yet the same correlation length as a Rayleigh speckle pattern [50].

V. ORIGINS OF NON-LOCAL CORRELATIONS
In this section, we conduct a theoretical analysis to illustrate that the non-local intensity correlations introduced into the speckle patterns, C N L (∆r) = C I (∆r) − |C E (∆r)| 2 , originate from high-order correlations encoded in the phase patterns of the SLM. For simplicity, we consider a 1D speckle field E(r) and its spatial Fourier components ε(ρ), where ρ corresponds to the spatial position on the SLM plane.
The spatial field correlation function is therefore given by: Taking the absolute-value squared of this expression gives the local intensity correlation function C L (∆r): With an expression for the local correlations in hand, we turn to the spatial intensity correlations: Grouping the summation into four terms according to the number of different ρ's summed over and spatial averaging gives: C I (∆r) = C 1 (∆r) + C 2 (∆r) + C 3 (∆r) + C 4 (∆r) − 1 (4) where: Since C 1 and C 3 are on the order of 1/L, they are negligible for large L, and C I is dominated by C 2 and C 4 : Comparing the expression of C L (∆r) to that of C 2 (∆r)− 1, we notice their difference scales as 1/L. When L is large, C L (∆r) C 2 (∆r) − 1, and C I (∆r) = C L (∆r) + C 4 (∆r) Therefore, the non-local correlation function C N L (∆r) C 4 (∆r), and the non-local correlations originate from the fourth-order correlations between different Fourier components of the speckle fields.

VI. DISCUSSION AND CONCLUSION
Based on the results of the previous section we expect the speckles with tailored intensity correlations to gradually lose the non-local correlations as they axially propagate away from the Fourier plane of the SLM. This can be understood using the Fresnel approximation, where the axial propagation of a field pattern adds a quadratic phase to its spatial Fourier spectrum [9]. Because the non-local intensity correlations result from high-order correlations encoded into the phases of the Fourier components, the phase parabola accompanying axial-propagation erodes away such correlations as the tailored speckles propagate: eventually only the local intensity correlations remain. The erosion of non-local correlations upon axial-propagation occurs on a length-scale corresponding to the Rayleigh range of the system.
In conclusion, we have presented a general approach for introducing strong non-local intensity-correlations into speckle patterns, making the intensity fluctuate on a much longer length scale than the field. Specifically, the generated speckle patterns can have a tailored C I (∆r), with |C E (∆r)| 2 remaining fixed. In essence, we show that a speckle pattern with a desired C I (∆r) can be synthesized via encoding the requisite fourth-order correlations into the light field reflected from the SLM. Encoding such correlations using the T-matrix of the optical system is a versatile approach, since it can readily be incorporated into a broad range of optical systems with copious potential applications.