1. A system has a one dimensional pupil function given by P(x) = Rect() given an...

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1. A system has a one dimensional pupil function given by P(x) = Rect() given an aperture of radius \(a). The object is assumed to be coherent, monochromatic, with a wavelength, A, a distance \(d) from the pupil. For simplicity, we will do this problem in 1D. We want to find the impulse response at the focal plane of this system. First, remind yourself of how position in the pupil at relates to a spatial frequency fr in the focal plane by drawing a ray from the pupil to the focus and the phase fronts that are normal to this ray spaced at A. Use this relation to transform the pupil function as a function of x to the frequency transfer function which is a function of spatial frequency fr. Finally, Fourier transform this function to find the impulse response in terms of horizontal position, x, and any other given quantities: a, d, lambda. Normalize the impulse response to have a maximum amplitude of unity. 1. A system has a one dimensional pupil function given by P(x) = Rect() given an aperture of radius \(a). The object is assumed to be coherent, monochromatic, with a wavelength, A, a distance \(d) from the pupil. For simplicity, we will do this problem in 1D. We want to find the impulse response at the focal plane of this system. First, remind yourself of how position in the pupil at relates to a spatial frequency fr in the focal plane by drawing a ray from the pupil to the focus and the phase fronts that are normal to this ray spaced at A. Use this relation to transform the pupil function as a function of x to the frequency transfer function which is a function of spatial frequency fr. Finally, Fourier transform this function to find the impulse response in terms of horizontal position, x, and any other given quantities: a, d, lambda. Normalize the impulse response to have a maximum amplitude of unity.