PHYS 408 HOMEWORK 8 SOLUTIONS

Thus we cannot infer any information about nx y from

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Unformatted text preview: assage through the a thin cell with √ n index n(x, y ), the image just beyond the cell is, within an overall constant phase, I0 eik0 d0√(x,y) for cell thickness d0 and illumination wavenumber k0 outside the cell (in free space). The corresponding intensity is | I0 eik0 d0 n(x,y) |2 = I0 , which is completely independent of n(x, y ). Thus we cannot infer any information about n(x, y ) from the brightfield image. Note, however, the thin-film transmission formula does rely on some assumptions, notably those relating to thin-film transmission in section 2.4 of Saleh and Teich. For the sake of comparison with the forthcoming phase contrast imaging analysis, note that one useful way to quantify the approximations in this brightfield imaging analysis is calculation of the intensity of the Taylor expansion of the transmited field. One will find (as described in the lab manual) that the correction to the constant term is second order in the phase contrast, i.e. there are no first-order corrections to the intensity. This is useful conceptually because is motivates us to search for masks which admit first-order (thus higher in magnitude than second-order) phase contrast terms. See section 1.8 of the lab manual, especially page 13, for a more detailed solution, but without numbers specific to this homework set. The image function is E0 exp(ik0 d0 n(x, y )), for k0 = 2π /(0.600 µm) and d0 = 5 µm. Thus the input image is an exponential of a product of rectangular functions which is easier to handle (for our purposes here) if we factor it as follows, for n0 = 1.5000 and noting k0 d0 ￿ 1, and subsequently Taylor expand: f (x, y ) = E0 exp(ik0 d0 n(x, y )) = E0 exp(ik0 d0 n0 ) exp (iθ(x, y )) ≈ E0 exp(ik0 d0 n0 ) (1 + iθ(x, y )) ￿ ￿ ￿x￿ y with θ(x, y ) = 0.0001 · k0 d0 rect rect ∆x ∆y (19) p<mask (x, y ) ∝ F{f (x, y )}|￿ ⊥ =(k/f )￿ ⊥ ∝ δ (x)δ (y ) + iF{θ(x, y )}|￿ ⊥ =(k/f )￿ ⊥ k x k x (21) (20) and ∆x = ∆y = 20 µm. The image in the fourier pl...
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