Phase-space measurement and coherence synthesis of optical beams

Summary form only given. Phase-space optics allows simultaneous visualization of both space (x) and spatial frequency (k) information. This is in distinct contrast with normal measurements, such as normal images and Fourier transforms, which record intensities in x-space or k-space only. For coherent beams, which are fully described by a 2D complex function (e.g. amplitude and phase), a phase-space description is useful but redundant. For partially coherent beams, on the other hand, each position x in the beam can have its own local power spectrum, so that a 4D description is often necessary. This is particularly true for propagation, as the beam coherence determines the evolution of its intensity. While a variety of theories has been developed to describe phase-space properties [1], there has been very little progress on the experimental front. Pinhole (Hartmann) or lenslet (Shack-Hartmann) arrays are most commonly used, but the arrays force a trade-off between spatial and angular sampling, usually resulting in poor resolution [2] (and often reduced dynamic range [3]) in both domains. Here, we demonstrate an alternative method for obtaining 4D phase-space distributions quickly, without sacrificing resolution in either dimension [4].The experimental setup is shown in Fig. 1. For measurement, we record a spatial spectrogram (windowed Wigner distribution function) by using a Spatial Light Modulator (SLM) to scan an aperture across the transverse field of the beam. The method is simple, fast, and free of mechanical scanning errors and artifacts [5]. For coherence control, we use a second SLM as a dynamically changing local diffuser with spatially varying statistics, allowing design and creation of arbitrary phase-space distributions. An example is shown Fig. 2, in which each small region in {x} has a variable Gaussian distribution in {k}. Traditional measurements along the marginals of the 4D distribution, i.e. the intensity and power spectrum projections 1(x) = f f(x- k)dk and S(k) = f f(x, k)dx, miss the coherence properties within the volume of phase space. Measurement and control of such higher-dimensional beams will have applications in coherence holography, encoding, illumination, and display.