Journal of Double Star Observations V5, No4

This paper reviews the modeling of stellar images using diffraction theory applied to different types of telescope masks. The masks are projected by secondary mirror holder vanes (such as the spider type) or holes on the primary mirror which result in different configurations of single stellar images. Using Fast Fourier Transform, the image of binary stars with different magnitudes is calculated. Given the numerical results obtained, a discussion is presented on the best secondary vane configurations and on the effect of obstruction types for the separation of binary pairs with different magnitudes. Vol. 11 No. 4 November 1, 2015 Page 403 Journal of Double Star Observations Numerical Generation of Double Star Images for Different Types of Telescopes tor primary mirror) acting as an aperture stop where light is diffracted. Diffraction is the main cause for a star image to never show any detail of its surface. To minimize diffraction effects (and therefore obtain detailed images), large primaries are constructed with the additional benefit of increasing the amount of collected light. Wave fronts at the aperture can be represented as oscillations in a local incident electric field Einc(rA), where rA is the aperture coordinate in a predefined coordinate system. According to the Fresnel-Kirchhoff formulation of Huygens’ Principle (Klein, 1970), the electric field at a point r beyond the aperture will be given by where  is the light wavelength (assumed monochromatic), k = 2 /, dA is a differential element of area (on the aperture), and H(rA) is the aperture stop function. The value of  can be taken as the wavelength of yellow light (570 590 nm) to better approximate the eye’s maximum sensitivity (Barten, 1999). Eq. (1) is the mathematical formulation of all contributions coming from the aperture at the point of interest (r) which can be taken as the telescope focus. Another important comment about the application of Eq. (1) to reality is that it represents the contribution of a single frequency component of non-polarized light. The output field of Eq. (1) is polarized in accordance with the incident amplitude E(rA). Therefore, Eq. (1) works as a scalar model for light diffraction. The next step is to take a linear approximation for the path differences of the several contributions of diffracted fields. For point-like objects like stars – in which case the direction cosines of a target object at the primal focus are very small – the approximation is sufficient to render a finished simulated image. Eq. (1) can be transformed to Here EA is a phase factor proportional to the light intensity at the aperture and a and b are the position angles of the source. Therefore, the image intensity at the telescope focus can be modelled as the space Fourier transform of the aperture (xA, yA) having the pair (, ) as conjugate variables. The classical example is the circular aperture with diameter D of a (refractor) telescope for which the resulting field at focus f0 can be expressed by the closed function with J1(p) the first order Bessel function of argument p, whose first zero is at 1.22, resulting in the Airy’s disk radius. Therefore, the resolving power (in radians) of the instrument will be given by since the minimum separation possible corresponds to the radius of the diffraction disk. Once the total received power is proportional to the square of the electric field, Eq. (3) also predicts that light intensity at the focus is inversely proportional to the square of the f0/D ratio Numerical Integration of Fresnel-Kirchhoff Integral Except for the somewhat artificial case of a squared aperture (Klein, 1970), other cases can only be analyzed by numerical simulation of Eq. (2). Let us denote u = xA and v = yA the coordinates on the aperture planes. If (x) represents Heaviside’s function (Abramowitz & Stegun, 1972) such that (x) = 1 for x > 0, and (x) = 0 otherwise, an aperture stop described by a main radius of R1 and a small hole of radius R2 (R2 < R1) may be represented by the two parameter function: A refractor aperture case can be simulated using Eq. (5) in (2) and making R2 = 0. On the other hand, a Maksutov0 1 0 [ ( / 2)sin ] ( ) E 2 sin ikf A J k D e E iD f     (3)