Optical radiation flux illuminating a circular disk from an off-axis point source separated by a homogeneous plane-parallel plate

Abstract

In a number of techniques that measure weak fluxes of optical radiation, it is frequently necessary to keep a detector in a medium different from that of the radiating source by separating it from the source with a planar transparent window. However, sources such as systems of light-emitting diodes, large-fiber illuminators, and microscopic living objects that emit biological luminescence may sometimes be regarded as multiple-point sources. To estimate the fluxes of optical radiation illuminating a surface from a nonuniformly distributed multiple-point source, a method for calculating fluxes from a single off-axis point source is needed. A formula is derived to estimate a flux of temporally incoherent optical radiation incident on a circular disk from a single off-axis point source separated by a plane-parallel plate (PPP). This formula is expressed by a series of single integrals of some superposed elementary functions. These functions depend on the variables that characterize the point-source-plane-parallel-plate-circular-disk geometry and on the optical properties of the media that separate the source from the PPP and the PPP from the disk. The solution was obtained for isotropic media. For illustrative purposes some examples of the use of the formula are presented. The selected results are illustrated by three-dimensional surface plots and compared with the values of the fluxes calculated for radiation incident on the disk from a point source not separated by a PPP.

Optical systems with circular apertures protected by plane-parallel entrance windows in wide-angle radiation fields

“…This independence on the angle φ 1 allows us to calculate all of the inner integrals in Eq. (27) analytically. Thus, it simplifies the numerical procedures.…”

“…4(b) and 4(c) were obtained using Eq. (27) at T ν;F ;k θ 1;K A ν;k θ 1;K 1, and A ν;k θ 1;K 1, respectively. All calculations were performed for the optical window of n ν;2 1.47047 and H 2 0.2, positioned in the middle source-aperture distance and for the geometrical variables given in the relative units with respect to R 1.…”

“…Equation (3) was integrated in [27] for the optical radiation passing through the homogeneous isotropic plane-parallel plate and incident on a circular disk. However, the resulting formulas were expressed by the angles θ 3 and φ 3 , so that they are somewhat cumbersome for practical implementation, especially when the point-source radiation patterns are expressed by the angles θ 1 and φ 1 as shown in Fig.…”

“…Unless the input radiant fluxes from various source types can be calculated in a relatively simple manner using one of many methods for single-media propagation [17][18][19][20][21][22][23], the calculation of the output radiant fluxes for multiple media propagation will be significantly complicated [24][25][26][27]. Consequently, nowadays they are usually computed by Monte Carlo sampling procedures [17,19,24,25,28].…”

“…The methodological approach used in this investigation is similar to the method proposed in [27]; however, this paper generalizes, improves, and simplifies that method for practical convenience. This method utilizes a mathematical formula expressed generally by several double-integral equations that can be simplified to several single-integral equations for rotationally symmetrical radiation around the optical axes of point sources, which are perpendicular to the circular apertures of the optical systems.…”