Recovery of Schell-model partially coherent beams

Abstract

Partially coherent light is often preferable to its completely coherent counterpart in applications such as imaging, sensing, and free-space optical communications. To fully exploit its advantages, techniques able to retrieve information carried by the beam are required. Here, we develop and experimentally demonstrate a phase-space optics technique for complete spatial analysis of widely used Schell-model beams. It allows for fast information recovery and can be applied for quantitative phase imaging of objects under partially coherent illumination. The further development of imaging techniques requires a more realistic model of illumination as partially coherent light instead of its limiting cases: the coherent or incoherent light used so far. Moreover, partially coherent beams have important advantages with respect to completely coherent ones in other relevant applications such as lithography, plasma confinement, and free-space communication, to name a few. The description of partially coherent light is inherently complex, and its characterization is difficult even in the quasi-monochromatic scalar paraxial approximation. Indeed, a two-dimensional (2D) partially coherent beam is described by a complexvalued 4D function, that is, mutual intensity (MI) defined as Γr 1 ; r 2 hf r 1 f r 2 i, where: r 1;2 is the position vector in a plane transverse to the beam propagation direction and h·i stands for ensemble averaging. The coherent case corresponds to Γ c r 1 ; r 2 f r 1 f r 2 . Schell-model partially coherent beams (SMBs) [1], described by Γr 1 ; r 2 f r 1 f r 2 γr 1 − r 2 Γ c r 1 ; r 2 γr 1 − r 2 , where γr is an equal-time complex degree of spatial coherence (DoC), are often used in practical applications. This kind of beam is generated, for example, when a partially coherent plane wave characterized by γr propagates through an object described by a complex modulation function, f r, as sketched in Fig. 1(a). We recall that, according with the van Cittert Zernike theorem, a partially coherent plane wave can be created by collimating light emitted from an incoherent source with intensity distribution I inc r. Specifically, γr ∝ R I inc r 0 exp−i2πr 0 r∕λf cl dr 0 , where λ is the wavelength and f cl is the focal length of the collimating lens [2]. Moreover, the generalization of structurally stable and spiral coherent beams to the partially coherent case is also described by the Schell model [3,4]. In the last decades, special attention has been paid to coherent vortex beams that carry orbital angular momentum (OAM) useful for different applications such as optical tweezers and free-space communications [5]. On the other hand, the SMB vortex has been found more robust than its coherent counterpart to distortions caused by turbulent atmosphere [6,7]. We underline that the valuable information carried by the SMB is often encoded into f r, while γr in most, but not all, cases is known.In this Letter we propose and experimentally demonstrate an iterative MI retrieval technique for arbitrary SMBs with a priori known ...

Rapid quantitative phase imaging for partially coherent light microscopy

“…3(b) and 3(c), respectively. The laser beam with low spatial coherence was generated by using a moving diffuser (Optotune, LSR-3005-12D-VIS), [23,24]. Note that the LED beam is both temporal and spatial partially coherent.…”

“…1(a)] is given by the expression [24] for a similar optical system. Here, we adapt the iterative algorithm developed in [24] for MI recovery to this particular case.…”

“…The proposed algorithm can also estimate the DSC, which results crucial to reconstruct the object wavefield. This requires an additional routine in the algorithm for the DSC determination, as for example the Richardson-Lucy deconvolution (RLD) reported in [26] for the Fourier transform and generalized in [24] for more complex transformations and systems.…”

“…As initial estimate γ (1) m (r o ) an arbitrary Gaussian function was used, which can be applied for any SMB because its DSC is such that γ m (r o ) is a real-positive function. Applying this algorithm the object wavefield and the DSC can be successfully recovered from the experimental data [24]. We recall that the DSC,…”

“…The DSC can be tuned in several ways, for example, by spatial filtering of the incoherent source or by using a moving diffuser in the case of a coherent source (e.g. laser) [22][23][24]. We underline that a pinhole of few microns (e.g.…”