The effect of detector size to the broadening of the penumbra--a computer simulated study

The effect of detector size to the broadening of the measured beam penumbra has been a subject of numerous studies. Based on measured data, linear and quadratic curves have been proposed to describe the relationship between the measured penumbra width, between 10%-90% and 20%-80% intensities, and the detector size. Extrapolations of these curves to zero detector size also suggest that the inherent penumbras can be deduced. However, due to experimental noise, especially when a small ionization chamber is used, and the inherent penumbra is not known, it is difficult to discern the superiority of either model. In this study, one dimensional convolution using a thin circular disk shape detector was employed to analyze the effect of detector size to the broadening of the penumbra. A set of beams with different inherent penumbra widths, ranging from 0 to 350 in arbitrary unit, was first generated. Each beam was then convoluted with the response function of the detectors with different sizes from 10 to 280, in arbitrary unit. The result is an output signal with penumbra that is wider than the inherent penumbra. The plots of penumbra widths to detector radii are a family of concave parabolic curves with different inherent penumbra widths. The concave portions of the curve represent results from scanning with detectors equal to or smaller than the penumbra width, and the linear portions represent results from scanning with detectors much larger than the penumbra. The curves are nothing but different scales of one curve. The extrapolations of the curves to the penumbra axis when the detector radius approaches zero give the deduced inherent penumbra widths. The deduced inherent penumbra widths approximate the inherent penumbra widths satisfactorily. From the graphs provided, the inherent penumbra can be deduced using the detector radius and the measured penumbra width.