<title>Phase-space formalism and ray-tracing modeling of photometric quantities</title>

Abstract

In this paper, Radiometric Ray Tracing (R2T), based on phase-space formalism, is formulated. INTRODUCTIONIn this presentation, the phase-space formation is applied to ray-tracing modeling of photometric quantities, such as radiance (luminance), emissivity (intensity) radiant (luminant) intensity, and power (flux). Based on spatial coherence and Fourier formalism, Radiometric Ray Tracing (R2T) has been proposed by the author [l3] The R2T facilitates the provision of input ray-tracing based on quasi-homogeneous source optical intensity and spatial coherence. In this paper, we expand this model to a four-dimensional (4-D) phase-space defined by Cartesian coordinates (x,y) and directional unit vector components (kx, ky). While the first ones define surface area, the second one (1) defines the so-called Fourier area.The fundamental connection between photometric quantities and 4-D phase space is based on the fact that brightness (or radiance, luminance) is a 4-D phase-space density. Thus, an arbitrary bundle of rays passing through a given plane (x,y) can be presented as a multiplicity of points in phase space (i.e., a single ray is represented isomorphically by a single point in phase space). In particular, a Lambertian source is represented by uniform 4-D distribution in phase space.The fundamental relation between luminance and phase-space density, or 4-D ray density, can be applied to evaluate photometric quantities by numerically calculating a number of rays passing through selected phase-space domains. For example, luminance can be calculated as a number of rays located in the phase-space elementary cell defined by Heisenberg's uncertainty relation. Emissivity, on the other hand, can be calculated by integrating elementary cells through directional vector space (kx, kr). As a result, the presented formalism allows precise connection of Monte-Carlo ray-tracing for a multiplicity of rays, with basic photometric (radiometric) quantities. LUMINANCE AS PHASE-SPACE DENSITYFor the purposes of Radiometric-Ray-Tracing (R2T), we define three basic radiometric quantities (power (P)), radiant intensity (J), and radiance (B)) with respect to 4-D phase-space (x,y; k, ky), where (x,y)-Cartesian coordinates of a given plane (physical or not), and the (kxky).CarteSian coordinate of directional cosine space, with a directional unit vector, k, and directional cosinuses (kg, k, kz): = (kx,ky,kz) (,kz); = (kx,ky); (Ict = k = 1 (2-1) as shown in Figure 2-1. 36 SPIE Vol. 3140. 0277-786X1971$10.00 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 06/25/2016 Terms of Use: http://spiedigitallibrary.org/ss/TermsOfUse.aspx

Performance metrics for integrated lighting systems

“…Therefore, ray-tracing, in phase-space formulation, is called Radiometric Ray Tracing (R 2 T), as in Refs. [4][5] . According to Liouville theorem [6] , the phasespace density is invariant to ray transformation from input plane (x o ,y o ), to output plane (x,y), in the form:…”

“…1, we can also describe its location in directional cosine space: (p x , p y , p z ), where r p is a directional unit vector (in Ref. [4] , it was denoted by r k ), in the form:…”

<title>Superanamorphic backlighting remote imaging display (SABRID)</title>

“…In the case of Radiometric Ray-Tracing (RET), discussed in detail in Ref; [2], not only ray tracing but also ray counting according to radiometric principles is provided. POC applies a combination of OptiCAD and MatLab.…”

Ray tracing analysis of display illumination systems

“…The list of quasihomogeneous sources includes LEDs, lasers, CCFLs, and bulbs which are typically used in illumination systems. Based on physical photometry, spatial coherence, and Fourier optics, radiometric ray tracing (R 2 T) expands the standard raytracing algorithms by adding information about the optical intensity and spatial coherence distribution of quasihomogeneous sources [5][6][7] . The R 2 T formalism is not based on complex wave amplitude characteristics of typical diffraction formulae but rather on standard photometric quantities such as brightness, (or luminance) radiant intensity, emissivity, and optical flux.…”

“…For example, luminance (in nits) can be calculated as a number of rays located in the phase-space elementary cell defined by Heisenberg's uncertainty relation. Emissivity on the other hand, can be calculated by performing double integration over the entire Fourier area [7] relevant to the problem. The presented formalism allows precise connection of Monte-Carlo ray tracing for a multiplicity of rays, with basic photometric (radiometric) quantities.…”