Thin Phase Object The Phase Grating Approximation

This theory goes back to Alpress and Sanders [23], Cowley and Iijima [24], and Hutchison [25].

After the interaction of an electron with wavelength A with an object with local potential V(x, y, z) the wavelength of the electron varies as A':

-j2meE

where E is the electrostatic acceleration potential and m and e are the mass and charge of the electron.

Since the electron travels perpendicular to the x, y plane in the z direction, the phase shift change dx(x, y, z), traveling through dz, is given as dX(x, y, z) = 2w dz(1/A' - 1/A)

and with a = w/(AE) and V/E ^ 1 one obtains d(x, y, z) V/(E)dz = V(x, y, z) dz

The total phase shift traveling through a thickness q0 (q0 is on the order of 0.1 nm) is given by r q0 rq0

X(x, y) = / dX(x, y,z) = a V(x, y, z) dz = aVp(x, y)

0 0

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