191
3.3. Diffraction
following to two definitive texts by Born and Wolf [11], and by
Goodman [36] for detailed and comprehensive discussion.
3.3.1 The Fresnel–Kirchhoff relation
In mathematical terms, the central problem in diffraction theory is
to calculate the amplitude u(x), given specified, known boundary
conditions. For a free particle this is a solution to the Helmholtz
equation, given by
(v
2 + k
2 ) u(x) = 0,
(3.230)
where k is a constant, and u(x) is the spatial part of the wave
function. This is precisely the scalar wave equation applicable to
light, in which case k = ω/c, and c is the speed of light. Allowing
for this, we therefore anticipate that the results to follow are otherwise equally valid for a photon and a charged particle. In this
section, we describe a Green’s function approach originally derived
by Sommerfeld [85] for light optics to achieve this. This methodology is known as the Rayleigh–Sommerfeld solution. The reader
is referred to the text by Goodman [36] for a comprehensive discussion, including the interesting historical attempts to correctly
understand this problem.
For the present purpose, we assume the particle propagates freely,
in the absence of electric and magnetic fields. We begin by stating
a very general result, which will prove to be useful. We assume two
arbitrary, complex functions U (x) and V (x), where these functions
are finite and differentiable over an arbitrary, closed volume τ . We
form the quantity U v
2 V − V v
2 U , and integrate this over the
volume τ . It follows that
U v
2 V − V v
2 U dτ = v · [ U vV − V vU ] dτ. (3.231)
τ
τ
3.3. Diffraction
following to two definitive texts by Born and Wolf [11], and by
Goodman [36] for detailed and comprehensive discussion.
3.3.1 The Fresnel–Kirchhoff relation
In mathematical terms, the central problem in diffraction theory is
to calculate the amplitude u(x), given specified, known boundary
conditions. For a free particle this is a solution to the Helmholtz
equation, given by
(v
2 + k
2 ) u(x) = 0,
(3.230)
where k is a constant, and u(x) is the spatial part of the wave
function. This is precisely the scalar wave equation applicable to
light, in which case k = ω/c, and c is the speed of light. Allowing
for this, we therefore anticipate that the results to follow are otherwise equally valid for a photon and a charged particle. In this
section, we describe a Green’s function approach originally derived
by Sommerfeld [85] for light optics to achieve this. This methodology is known as the Rayleigh–Sommerfeld solution. The reader
is referred to the text by Goodman [36] for a comprehensive discussion, including the interesting historical attempts to correctly
understand this problem.
For the present purpose, we assume the particle propagates freely,
in the absence of electric and magnetic fields. We begin by stating
a very general result, which will prove to be useful. We assume two
arbitrary, complex functions U (x) and V (x), where these functions
are finite and differentiable over an arbitrary, closed volume τ . We
form the quantity U v
2 V − V v
2 U , and integrate this over the
volume τ . It follows that
U v
2 V − V v
2 U dτ = v · [ U vV − V vU ] dτ. (3.231)
τ
τ