Speckle phenomena in optics12/25/2023 To bypass the unmodulated light, we write a binary phase diffraction-grating on the SLM and work with the light diffracted to the first-order. However, a small portion of reflected light from the SLM is unmodulated. The pixels on the SLM can modulate the incident light’s phase between the values of 0 and 2 π in increments of 2 π/170. 1 (a), a linearly polarized monochromatic laser beam with a wavelength of λ = 642 nm uniformly illuminates a phase-only reflective SLM (Hamamatsu LCoS X10468). In our experimental setup, illustrated in Fig. While this method could modify the speckle contrast, via C 0, the intensity PDF itself could not be directly controlled using this method. In our recent work, 30 we demonstrated that it is possible to dramatically and controllably alter the intensity correlation function of a speckle pattern by introducing nonlocal correlations into the speckle pattern instead. 8,22–33 Because the local correlation function is effectively the diffraction-limited point spread function of a system, this approach can be quite limiting in terms of the range of possible correlation functions. Typically, the spatial intensity correlation function of a speckle pattern can be modified by altering the local correlation function via amplitude modulation of its Fourier components. 1,3,4,20 C NL(Δ r) represents the nonlocal (long-range) correlation function, 21 and it vanishes when the Siegert relation holds: C I(Δ r) ≡ C 0| C E(Δ r)| 2. Here, C L(Δ r) is known as the local (short-range) correlation function, and it is related to the field correlation function by C L(Δ r) = C 0| C E(Δ r)| 2, where C 0 = ⟨ I 2⟩/⟨ I⟩ 2 − 1 is related to the speckle contrast. Furthermore, they only possess short-ranged spatial intensity correlations which are determined by the average speckle grain shape, which in turn is dictated by the diffraction limit. Rayleigh speckles-the most common type of speckle patterns-obey a circular-Gaussian field PDF which results in a negative exponential intensity PDF. Additionally, in a fully developed speckle pattern, the amplitude and phase profiles are statistically independent. In a fully developed speckle pattern, therefore, the phase PDF is uniformly distributed between 0 and 2 π. For example, a speckle pattern is said to be “fully developed” if its joint PDF is circularly invariant. On the other hand, speckle patterns are categorized by the joint probability density function (PDF) of their complex-valued field. Ergodicity requires the statistical properties of two spatial positions-separated by more than one speckle grain size-to be independent and identical to those of the ensemble. In this context, stationarity requires the statistical properties of an ensemble of speckle patterns to be the same as those of an individual speckle pattern within the ensemble. On the one hand, the spatial-distribution of light in a speckle pattern is sufficiently complicated that speckles are described by a statistically stationary and ergodic random process. A speckle pattern is characterized by the twofold complexity of its optical field. Because of their speckled appearance, random light fields are commonly referred to as speckle patterns. Spatially random light fields have the hallmark appearance of intricate-yet highly irregular-mosaics of diffraction-limited speckle grains. This work provides a versatile methodology for creating complex light fields and controlling their statistical properties with varied applications in microscopy, imaging, and optical manipulation. In addition to our experimental demonstration, we explore both the theoretical and practical limitations on the extent to which the intensity PDF and the spatial intensity correlations can be manipulated concurrently in a speckle pattern. Irrespective of their distinct statistical properties, however, all of these speckles are created by appropriately encoding high-order correlations into the phase front of a monochromatic laser beam with a spatial light modulator. The various families of tailored speckle patterns-created by our method-can exhibit radically different topologies, statistics, and variable degrees of spatial order. In this work, we develop an experimental method for customizing the intensity probability density function (PDF) of speckle patterns while simultaneously introducing nonlocal spatial correlations among the speckle grains. Random light fields-commonly known as speckles-demonstrate Rayleigh intensity statistics and only possess local correlations which occur within the individual speckle grains.
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