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Chapter 3. Wave optics
other point (x b ). Finally, we divide by ¯
h to give the phase, thus
obtaining the wave function (3.186). Incidentally, this approach is
accurate in the sense that it implicitly includes all orders af aberrations.
3.2.3 Quantum interference effects in electromagnetic potentials
In the preceding sections we investigated wave-optical interference
which occurs when single free-particle amplitudes corresponding
to alternative paths of motion add coherently. We now extend
this discussion to the case where electromagnetic potentials are
present. We consider a hypothetical monochromatic point source
of electrons at axial coordinate z a , which coincides with the front
focal plane of a lens at axial coordinate z L1 . This is shown schematically in Figure 3.5. A screen with two slits is located directly behind the lens. The slits are illuminated by a monochromatic plane
wave in the paraxial approximation. A second lens at axial coordinate z L2 produces a diffraction pattern on a viewing screen located
at axial coordinate z b , which is assumed to coincide with the back
focal plane of the second lens. The solid lines correspond to the
classical rays for the two alternative paths. Bright fringes appear
where the amplitudes corresponding to two alternative paths add
constructively. This occurs where the optical path lengths differ
by an integral number of wavelengths.
Next we assume a magnetic flux which is entirely confined to the
cross-hatched circle, where the lines of flux are oriented perpendicular to the plane of the figure. Such a flux can be produced
in principle by a very long solenoid with very fine, closely spaced
windings, where the axis of the solenoid is also oriented perpendicular to the plane of the figure. We assume that the magnetic field
Chapter 3. Wave optics
other point (x b ). Finally, we divide by ¯
h to give the phase, thus
obtaining the wave function (3.186). Incidentally, this approach is
accurate in the sense that it implicitly includes all orders af aberrations.
3.2.3 Quantum interference effects in electromagnetic potentials
In the preceding sections we investigated wave-optical interference
which occurs when single free-particle amplitudes corresponding
to alternative paths of motion add coherently. We now extend
this discussion to the case where electromagnetic potentials are
present. We consider a hypothetical monochromatic point source
of electrons at axial coordinate z a , which coincides with the front
focal plane of a lens at axial coordinate z L1 . This is shown schematically in Figure 3.5. A screen with two slits is located directly behind the lens. The slits are illuminated by a monochromatic plane
wave in the paraxial approximation. A second lens at axial coordinate z L2 produces a diffraction pattern on a viewing screen located
at axial coordinate z b , which is assumed to coincide with the back
focal plane of the second lens. The solid lines correspond to the
classical rays for the two alternative paths. Bright fringes appear
where the amplitudes corresponding to two alternative paths add
constructively. This occurs where the optical path lengths differ
by an integral number of wavelengths.
Next we assume a magnetic flux which is entirely confined to the
cross-hatched circle, where the lines of flux are oriented perpendicular to the plane of the figure. Such a flux can be produced
in principle by a very long solenoid with very fine, closely spaced
windings, where the axis of the solenoid is also oriented perpendicular to the plane of the figure. We assume that the magnetic field