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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 thinﬁlm transmission formula does rely on some assumptions, notably those relating to thinﬁlm 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 ﬁrstorder corrections to the intensity. This is useful conceptually because is motivates us to search
for masks which admit ﬁrstorder (thus higher in magnitude than secondorder) 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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 Fall '11
 KirkW.Madison
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