Download PDF by Joseph W. Goodman: Introduction To Fourier Optics, Third Edition, Problem

By Joseph W. Goodman

ISBN-10: 0070242542

ISBN-13: 9780070242548

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Extra info for Introduction To Fourier Optics, Third Edition, Problem Solutions

Example text

The Fourier transform of this expression is F {tA (x, y)} = π 2 d 2 2 J1 (πdρ) × 2 cos(πsfX ), πdρ 2 + f 2 . Taking the squared magnitude of this expression, using the identity cos2 θ = where ρ = fX Y 1 2 (1 + cos 2θ), and introducing the scaling parameters appropriate for the optical Fourier transform, we obtain the following expression for the intensity point-spread function (under the assumption that the intensity of the wave at the exit pupil is unity): I(u, v) = |h(u, v)|2 = 2 4  π d  2 16λ2 zi2 J1 √ πd u2 +v 2 λzi √ πd u2 +v 2 λzi We can now answer the specific questions of the problem: 2  1 + cos 2πsu λzi .

Consider a coherent wavefield described by U (x, y; t) = U (x, y) e−j2πνt . The mutual intensity of such a wavefield at points (x1 , y1 ) and (x2 , y2 ) is given by J(x1 , y1 ; x2 , y2 ) = U (x1 , y1 ; t) U ∗ (x2 , y2 ; t) = U (x1 , y1 ) U ∗ (x2 , y2 ) e−j2πνt ej2πνt = U (x1 , y1 ) U ∗ (x2 , y2 ). From Eq. (6-11) we see that this wavefield is fully coherent. 58 Chapter 7 7-1. The intensity distribution exposing the film is known to be of the form I(x, y) = Io + ∆I(x, y), with the further restriction that ∆I ≪ I0 .

2 4 4 (b) The figure illustrates the finite amplitude transfer function and the object frequency components present. Noting that zi = 2f , the cutoff frequency of the amplitude transfer function is ~ f Amplitude transfer function fi Spectral components of input ~ f fo Figure 6-12: fo = w/2λf. To obtain any variations of intensity in the image, it is necessary that at least two spectral components of the object be passed by the amplitude transfer function. From the figure, this will be the case (assuming f˜ ≤ 2fo ) provided fi ≤ fo , or equivalently provided sin θ ≤ w .

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Introduction To Fourier Optics, Third Edition, Problem Solutions by Joseph W. Goodman


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