PHYS 408 HOMEWORK 8 SOLUTIONS

The phase shift for other fourier components over

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Unformatted text preview: ane just before the mask is the fourier transform of the input image, in real space, as proved in the previous homework: (22) Introduce a “dot” of dielectric at x = y = 0 in the fourier plane such that the DC term – i.e. the kx = ky = 0 term – undergoes a phase shift of φ relative to all other fourier components. If this dielectric has refractive index nmask and thickness dmask , the phase shift for light passing through it is k0 nmask dmask . The phase shift for other fourier components over that same distance (through free space, or some other material of assumed index at k0 equal to 1) is k0 dmask . Thus we want k0 (nmask − 1)dmask = φ, which can be satisfied by at least one combination of nmask and dmask . The image just beyone the mask, output image, and output intensity are p>mask (x, y ) gout (x, y ) I4f ∝ ∝ ∝ eiφ δ (x)δ (y ) + iF{θ(x, y )}|￿ ⊥ =(k/f )￿ ⊥ k x eiφ + iθ(−x, −y ) ￿ iφ ￿ ￿e + iθ(−x, −y )￿2 ≈ 1 + 2 sin(φ)θ(−x, −y ) (23) (24) (25) (26) where the θ2 was neglected, as it was in part a, the case of brightfield image. Are we better off than in the case of brightfield imaging? Yes, we can have a the non-zero first-order phase contrast term we were motivated to generate. We can maximize the image contrast by letting φ = π /2, and will assume this value for φ here onwards. The non-DC terms are still small compared to the DC term, but we’re still better off than in the case of brightfield imaging, where – within the approximations of our model – phase contrast was not directly imagable. In real-life, you may still be able to see object features in the brightfield, but phase contrast microscopy can greatly enhance visibility. Below is a Matlab script that numerically handles this example of phase contrast imaging. Note that the analytical expression for the output intensity derived above agrees with the numerical part to better than a few tenths of a percent, so the numerical work here is overkill, although still good practice. We would not expect the analytical work to be remarkably difficult here for the reason the output image well-represents the actual object. This makes phase contrast imaging that much more remarkable, i.e. that such a ’simple’ tweak can yield a great improvement in visibility. 3 3.3 matlab script clear...
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