Unformatted text preview: (x, y )} ⊥ =(k/f ) ⊥
k
x 1 (8) 2 optical switching The input function is
f (x, y ) = Aδ (x − α)δ (y − α)f orα = 0.5 cm (9) Propagate this through half of the 4f system, to just before the mask. Here, the function will now take the form –
according to a result from the previous homework – the fourier transform of f (x, y ) mapped onto real space, a result further
analyzed in problem 1 above. The result: p< (x, y ) =
=
=
−ie2ikf
λf
−ie2ikf
λf
−ie2ikf
λf The desired output function is
F{f (x, y )} ⊥ =(k/f ) ⊥
k
x
Ae−ikx α e−iky α ⊥ =(k/f ) ⊥
k
x Ae−i(kα/f )(x+y) g (x, y ) = Aδ (x − α)δ (y − α) (10)
(11)
(12) (13) Propagate this “backwards” to just after the mask, which can be done by using the result from problem 1: p> (x, y ) =
−ie2ikf
λf F −1 {f (x, y )} ⊥ =(k/f ) ⊥
k
x
−ie2ikf +ikx α +iky α
=
Ae
e
⊥ =(k/f ) ⊥
k
x
λf
2ikf
−ie
=
Ae+i(kα/f )(x+y)
λf (14)
(15)
(16) Now that we have explicit forms for the function just before and just after the mask, we can write down an explicit form
for the mask itself: m(x, y ) = p> (x, y )
e+i(kα/f )(x+y)
= −i(kα/f )(x+y) = e+2i(kα/f )(x+y)
p< (x, y )
e (17) Recalling that the transmission of a thin ﬁlm of is just t(x, y ) ≈ eikd0 eik(n−1)d(x,y) , we can – within an overall phase which
can be adjusted by adjusting the maximum thickness d0 – achieve a mask with the derived transmission above with a thin
ﬁlm of thickness according to
d(x, y ) = α
(x + y )
2(n − 1)f (18) This is just a prism with increasing thickness in both x and y , which can be fabricated directly (or be composed of a
succession of a prism in x and a prism in y ). 2 3 phase contrast imaging 3.1 brightﬁeld imaging 3.2 phase contrast imaging, phase masking √
See section 1.8 of the lab manual, especially page 13. Plane wave I0 eikz illumination, p...
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 Fall '11
 KirkW.Madison
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