While all electromagnetic phenomena are described by the Maxwell equations, material interactions, sources, boundaries and receivers differ across the frequency spectrum. This note briefly reviews the electromagnetic theory of diffraction and discusses how it is applied in optical analysis.
The Maxwell equations are
is the electric field, 
is the electric displacement, B is the magnetic
induction, H is the magnetic field, J
is the current density and
is the
charge density.
The fields are further related by the material equations
where P is the polarization of material and M is the magnetization. In most optical materials, M=0 and P is a function of E. The simplest and most common case is
Then
where
and
As a result of the high frequencies and small wavelengths of optical
fields, charge interactions with the optical field involve quantum
mechanical effects which cannot accurately be analyzed by continuous
models. In optical diffraction problems, 
is always neglected. In principle, J is also neglected,
although we discuss below how the current density can be used
formally to add a loss component to the pemitivitty, 
.
We also note that E and D need not be colinear,
meaning that 
may in general be tensor
valued.
Using the material relations, we substitute in the Maxwell equations to find the wave equations
The equations are reduced to a simpler form by the vector identity
From the Maxwell equations we know that
,
where we have assumed for the moment that 
is scalar valued. Thus,
and
A medium in which 
is homogeneous.
Media in which this is not the case include optical fiber, graded
index lenses, and volume holograms.
In isotropic media, the wave equations are
In a typical diffraction problem, one is given the field on a particular boundary or source and one wishes to calculate the field on a different boundary or in a different region. This is a linear problem in the sense that the field produced on the output boundary by a linear superposition of fields on the input boundary is the linear superposition of the output fields produced by the individual input fields. Formally, one might express this situation as follows:
For the diffractive transformation 
from
the input boundary to the output boundary, the transformation
of the field 
satisfies
.Since the diffractive transformation is linear (and in general also shift invariant), Fourier analysis is extremely useful in analyzing diffraction problems.
One solves a diffraction problem using the Maxwell or wave equations and boundary conditions. Note that diffraction is not a linear problem from all perspectives. For example, if one considers the input to be the boundary condition and the output to be the field scattered by the boundary, the output for a linear superposition of inputs is not necessarily a linear superposition of the individual outputs. This situation arises, as an example, when one uses arrays of surface features to modulate a scattered field. The addition of new surface features does not necessarily modulate the scattered field in a linear fashion.
The ease with which nonlinearity can be introduced in the spatial domain contrasts with the difficulty of introducing nonlinearity in the temporal domain. The process of generating the optical field is highly nonlinear but once the field is generated and propagating, effects which couple different optical frequencies are relatively rare. For this reason, the use of Fourier analysis to characterize the temporal and spectral response of optical systems is nearly universal, while the use of Fourier analysis in the spatial domain is somewhat more limited. The linearity of the temporal response is a mixed blessing, however, when one considers the difficulty of accurately characterizing temporal aspects of the field. Spatial distributions and boundary conditions can be measured to extra-ordinary accuracy, but the exact temporal behavior of optical fields is fundamentally unknowable. Analysis of diffraction problems generally focuses on "quasi-monochromatic fields," which one treats as single frequency Fourier components but which ultimately are stochastic.
In the Fourier domain, the wave equation for the electric field takes the form
where * represents a multi-dimensional convolution operator. The possibility of a convolution over temporal frequency is not realized in practice and points out a flaw in the materials response equations. In general, the dielectric response of a linear material is characterized by a linear relationship in the frequency domain and a convolution in the time domain, e.g.
so that
The upper limit of integration at time t is necessary to maintain causality. The time integrated response is consistent with the physical situation, in which the polarization vector describes the microscopic response of the charge in the material to the driving field. One does not expect that this response will be instantaneous in the time domain. This means, as discussed above, temporal aspects of the diffraction problem are linear and there is no need to consider a convolution over temporal frequencies.
The convolution of the perimittivity and the field in the spatial
frequency domain is physically realistic, however. The response
of the field to spatial variations in 
is
local in the spatial domain and global in the spatial frequency
domain, in contrast to temporal variations in 
,
where the response is local in the spectral domain and global
in the time domain. The convolution 
thus
represents the potential nonlinearity of the diffraction problem
with respect to variations in spatial distributions.
Limiting ourselves to homogeneous materials, the Fourier transform of the wave equation is
.
If 
is a scalar, this equation has solutions
only if 
. Allowing for the possibility
that 
is a tensor, solutions correspond
to those values of 
for which the determinant
of the matrix operator
vanish. The equation 
reduces the range
of 
from three dimensions to two dimensions.
The surface defined by this equation is called the wave normal
surface. In isotropic materials (materials in which the perimitivitty
is a scalar), the wave normal surface is a sphere in k-space
of radius 
. In anisotropic materials (crystals),
the wave normal surface splits into two sheets, so that there
is are two solutions for k in almost every direction. Propagation
in ansisotropic materials is considered in detail in Optical
waves in crystals, by Yeh and Yariv.
Each solution for k corresponds to an eigenvector E. The direction of E is the polarization. For the isotropic case, two possible polarizations exist for each k. In the general case, each eigenvector corresponds to a different value of k.
Once the solution for 
has been determined,
one may inverse Fourier transform to find the general solution
The challenge of this representation is that, despite the fact
that k lies in 3D space, the relationship between k
and 
discussed above means that the combination
of k and 
only spaces 3-space and
the Fourier representation need actually involve integration over
three dimensions.
Returning to the diffraction problem, our goal is to use the Fourier representation of the field to find the field on the output boundary given the field on the input boundary.
Suppose as an example that we consider the diffraction problem
sketched above. We are given the field 
on the input plane and we wish to calculate the field 
on the output plane. The planes contain the x and y axes and are
displaced along the z axis. Given the linearity of the diffraction
problem with respect to time, as discussed above, there is no
loss of generality in limiting our analysis to quasi-monochromatic
fields. Once the transformation is known as a function of 
inverse transforming for the temporal response is straightforward.
At a single frequency, the Fourier representation of the field
is
we include only two dimensions in our integration for k in view of the 2D range for k discussed above.
The relationship between k and 
defined by the wave normal surface is called the dispersion function.
Using the dispersion function, we can always determine one of
the four variables 
if the other three
variables are given. Suppose, as an example, that 
is the constrained variable (i.e. suppose that we are free
to select 
and 
arbitrarily). Dropping the time dependence, the phasor field as
a function of space can be expressed
Eq. provides an opening for quick solution of the diffraciton
problem. Suppose that the input plane containing 
is the 
plane. Then, according to Eq.
, the field in this plane is
.
We have dropped the polarization vector for simplicity, the effect
of shifting to scalar fields is discussed briefly below. According
to Eq. , 
is essentially the Fourier transform
of 
. Now suppose that the output plane
is the 
plane. According to Eq. ,
.
Writing these relationships in terms of the Fourier transform
of 
, 
, and defining
and 
yields
and
Comparing 
and 
,
we see that the transformation is linear and shift-invariant with
a transfer function
The impulse response of the diffractive transformation 
is the inverse Fourier transform of 
.
In an isotropic space the dispersion relationship is
where 
. The wave normal surface in free
space is a sphere, as sketched below.
The transfer function for the free space dispersion relationship is
In many optical systems, the transfer function and the impulse
can be simplified by the paraxial approximation. This approximation
states that spatial frequencies in the fields of interest are
such that the angular spread of the diffracted signal about the
axis of propagation is small. The paraxial approximation corresponds
to limiting 
and 
to the purple region on the wave normal surface. Under the paraxial
approximation we assume that over the spatial bandwidth range
of interest 
. This approximation is valid
if the separation, d, between the input and output planes is much
greater than aperture of interest in each plane.
Under the paraxial approximation,
The impulse response for diffraction from 
to 
is the inverse Fourier transform of
the approximate transfer function,
Our goal for the remainder of this note is to evaluate this integral.
The limits of integration are from the maximum to the minimum
spatial frequency along each axis over the small purple circle
on the wave normal sphere. Despite our assumption that 
,
we will assume that at almost all points over the aperture of
interest 
. This assumption allows us to
extend the limits of integration in Eq. to plus and minus infinity.
Completing the squares in the exponents with the substitutions
and 
yields
The use of this impulse response to analyze diffraction and imaging problems is the focus of Wilson and GF. We will build on this kernel throughout the term.
Finally, we briefly return to the question of polarization, which we dropped midway through our discussion. Other than notational simplicity, there was no real reason for us to drop the vector aspect of the field and we could easily revise our Fourier analysis to include vector components. The difficulty with vectors relates to assumptions about boundary conditions and spatial nonlinearity which we have avoided in this note. In computing the transformation from one plane to the next, we have implicitly assumed that the field in the input plane is a knowable object. Unfortunately, this is not generally the case in optical systems. To create known fields, one might try to modulate plane wave inputs using transmission or reflection masks.
The modulation of the field to create spatial distributions is
a very complex subject. As discussed above, spatial modulation
is not a linear process. The simplest approximation to modulation,
known as the Kirkhoff boundary condition, involves the assumption
that the field distribution after a modulator is equal to the
product of a transmission factor for the modulator and the input
field. This approximation often works well, but is not an accurate
reflection of the microscopic interaction between fields and surfaces.
Moving beyond the Kirhoff approximation generally involves numerical
methods. The inclusion of vector polarization effects in these
models is complex but often necessary for accurate results.