The wave equation

Fourier optics begins with the homogeneous, scalar wave equation (valid in source-free regions):

$${\displaystyle \left(\nabla ^{2}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial {t}^{2}}}\right)u(\mathbf {r} ,t)=0.}$$

where ${\displaystyle c}$ is the speed of light and u(r,t) is a real-valued Cartesian component of an electromagnetic wave propagating through a free space (e.g., u(r, t) = E_i(r, t) for i = x, y, or z where E_i is the i-axis component of an electric field E in the Cartesian coordinate system).

Sinusoidal steady state

If light of a fixed frequency in time/wavelength/color (as from a single-mode laser) is assumed, then, based on the engineering time convention, which assumes an ${\displaystyle e^{i\omega t}}$ time dependence in wave solutions at the angular frequency ${\displaystyle \omega =2\pi f}$ with ${\displaystyle f=1/\tau }$ where ${\displaystyle \tau }$ is a time period of the waves, the time-harmonic form of the optical field is given as

$${\displaystyle u(\mathbf {r} ,t)=\operatorname {Re} \left\{\psi (\mathbf {r} )e^{i\omega t}\right\}.}$$

where ${\displaystyle i}$ is the imaginary unit, ${\displaystyle \operatorname {Re} \left\{x\right\}}$ is the operator taking the real part of ${\displaystyle x}$,

$${\displaystyle \omega =2\pi f}$$

is the angular frequency (in radians per unit time) of light waves, and

$${\displaystyle \psi (\mathbf {r} )=a(\mathbf {r} )e^{i\phi (\mathbf {r} )}}$$

is, in general, a complex quantity, with separate amplitude ${\displaystyle a}$ in non-negative real number and phase ${\displaystyle \phi }$.

The Helmholtz equation

Substituting this expression into the scalar wave equation above yields the time-independent form of the wave equation,

$${\displaystyle \operatorname {Re} \left\{\left(\nabla ^{2}+k^{2}\right)\psi (\mathbf {r} )\right\}=0}$$

where

$${\displaystyle k={\omega \over c}={2\pi \over \lambda }}$$

with the wavelength ${\displaystyle \lambda }$ in vacuum, is the wave number (also called propagation constant), ${\displaystyle \psi (\mathbf {r} )}$ is the spatial part of a complex-valued Cartesian component of an electromagnetic wave. Note that the propagation constant ${\displaystyle k}$ and the angular frequency ${\displaystyle \omega }$ are linearly related to one another, a typical characteristic of transverse electromagnetic (TEM) waves in homogeneous media.

Since the originally desired real-valued solution ${\displaystyle u(\mathbf {r} ,t)}$ of the scalar wave equation can be simply obtained by taking the real part of ${\displaystyle \psi (\mathbf {r} )e^{i\omega t}}$, solving the following equation, known as the Helmholtz equation, is mostly concerned as treating a complex-valued function is often much easier than treating the corresponding real-valued function.

$${\displaystyle \left(\nabla ^{2}+k^{2}\right)\psi (\mathbf {r} )=0.}$$

Solving the Helmholtz equation

Solutions to the Helmholtz equation in the Cartesian coordinate system may readily be found via the principle of separation of variables for partial differential equations. This principle says that in separable orthogonal coordinates, an elementary product solution to this wave equation may be constructed of the following form:

$${\displaystyle \psi (x,y,z)=f_{x}(x)f_{y}(y)f_{z}(z)}$$

i.e., as the product of a function of x, times a function of y, times a function of z. If this elementary product solution is substituted into the wave equation, using the scalar Laplacian in the Cartesian coordinates system

$${\displaystyle \nabla ^{2}\psi ={\frac {\partial ^{2}\psi }{\partial x^{2}}}+{\frac {\partial ^{2}\psi }{\partial y^{2}}}+{\frac {\partial ^{2}\psi }{\partial z^{2}}},}$$

then the following equation for the 3 individual functions is obtained

$${\displaystyle f_{x}''(x)f_{y}(y)f_{z}(z)+f_{x}(x)f_{y}''(y)f_{z}(z)+f_{x}(x)f_{y}(y)f_{z}''(z)+k^{2}f_{x}(x)f_{y}(y)f_{z}(z)=0}$$

which is readily rearranged into the form:

$${\displaystyle {\frac {f_{x}''(x)}{f_{x}(x)}}+{\frac {f_{y}''(y)}{f_{y}(y)}}+{\frac {f_{z}''(z)}{f_{z}(z)}}+k^{2}=0}$$

It may now be argued that each quotient in the equation above must, of necessity, be constant. To justify this, let's say that the first quotient is not a constant, and is a function of x. Since none of the other terms in the equation has any dependence on the variable x, so the first term also must not have any x-dependence; it must be a constant. (If the first term is a function of x, then there is no way to make the left hand side of this equation be zero.) This constant is denoted as -k_x². Reasoning in a similar way for the y and z quotients, three ordinary differential equations are obtained for the f_x, f_y and f_z, along with one separation condition:

$${\displaystyle {\begin{aligned}{\frac {d^{2}}{dx^{2}}}f_{x}(x)+k_{x}^{2}f_{x}(x)&=0\\[1pt]{\frac {d^{2}}{dy^{2}}}f_{y}(y)+k_{y}^{2}f_{y}(y)&=0\\[1pt]{\frac {d^{2}}{dz^{2}}}f_{z}(z)+k_{z}^{2}f_{z}(z)&=0\\[3pt]k_{x}^{2}+k_{y}^{2}+k_{z}^{2}&=k^{2}\end{aligned}}}$$

Each of these 3 differential equations has the same solution form: sines, cosines or complex exponentials. We'll go with the complex exponential as ${\displaystyle \psi (x,y,z)}$ to be a complex function. As a result, the elementary product solution ${\displaystyle \psi (x,y,z)}$ is

$${\displaystyle {\begin{aligned}\psi (x,y,z)&=Ae^{ik_{x}x}e^{ik_{y}y}e^{ik_{z}z}\\&=Ae^{i(k_{x}x+k_{y}y)}e^{ik_{z}z}\\&=Ae^{i(k_{x}x+k_{y}y)}e^{\pm iz{\sqrt {k^{2}-k_{x}^{2}-k_{y}^{2}}}}\end{aligned}}}$$

with a generally complex number ${\displaystyle A}$. This solution is the spatial part of a complex-valued Cartesian component (e.g., ${\displaystyle E_{x}}$, ${\displaystyle E_{y}}$, or ${\displaystyle E_{z}}$ as the electric field component along each axis in the Cartesian coordinate system) of a propagating plane wave. ${\displaystyle k_{i}}$ (${\displaystyle i=x}$, ${\displaystyle y}$, or ${\displaystyle z}$) is a real number here since waves in a source-free medium has been assumed so each plane wave is not decayed or amplified as it propagates in the medium. The negative sign of ${\displaystyle k_{i}}$ (${\displaystyle i=x}$, ${\displaystyle y}$, or ${\displaystyle z}$) in a wave vector ${\displaystyle \mathbf {k} =k_{x}{\hat {\mathbf {x} }}+k_{y}{\hat {\mathbf {y} }}+k_{z}{\hat {\mathbf {z} }}}$ (where ${\textstyle \left|\mathbf {k} \right|=k={\omega \over c}={2\pi \over \lambda }}$) means that the wave propagation direction vector has a positive ${\displaystyle i}$ (${\displaystyle i=x}$, ${\displaystyle y}$, or ${\displaystyle z}$)-component, while the positive sign of ${\displaystyle k_{i}}$ means a negative ${\displaystyle i}$ (${\displaystyle i=x}$, ${\displaystyle y}$, or ${\displaystyle z}$)-component of that vector.

Product solutions to the Helmholtz equation are also readily obtained in cylindrical and spherical coordinates, yielding cylindrical and spherical harmonics (with the remaining separable coordinate systems being used much less frequently).

The complete solution: the superposition integral

A general solution to the homogeneous electromagnetic wave equation at a fixed time frequency ${\displaystyle f}$ in the Cartesian coordinate system may be formed as a weighted superposition of all possible elementary plane wave solutions as

$${\displaystyle \psi (x,y,z)=\int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }\Psi _{0}(k_{x},k_{y})~e^{i(k_{x}x+k_{y}y)}~e^{\pm iz{\sqrt {k^{2}-k_{x}^{2}-k_{y}^{2}}}}~dk_{x}dk_{y}}$$

(2.1)

with the constraints of ${\displaystyle k_{x}^{2}+k_{y}^{2}+k_{z}^{2}=k^{2}}$, each ${\displaystyle k_{i}}$ as a real number, and ${\displaystyle k={\omega \over c}={2\pi \over \lambda }}$ where ${\displaystyle \omega =2\pi f}$. In this superposition, ${\displaystyle \Psi _{0}(k_{x},k_{y})}$ is the weight factor or the amplitude of the plane wave component with the wave vector ${\displaystyle (k_{x},k_{y},k_{z})}$ where ${\displaystyle k_{z}}$ is determined in terms of ${\displaystyle k_{x}}$ and ${\displaystyle k_{y}}$ by the mentioned constraint.

Next, let

$${\displaystyle \psi _{0}(x,y)=\psi (x,y,z)|_{z=0}.}$$

Then:

$${\displaystyle \psi _{0}(x,y)=\int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }\Psi _{0}(k_{x},k_{y})~e^{i(k_{x}x+k_{y}y)}~dk_{x}dk_{y}}$$

The plane wave spectrum representation of a general electromagnetic field (e.g., a spherical wave) in the equation (2.1) is the basic foundation of Fourier optics (this point cannot be emphasized strongly enough), because at z = 0, the equation simply becomes a Fourier transform (FT) relationship between the field and its plane wave contents (hence the name, Fourier optics).

Thus:

$${\displaystyle \Psi _{0}(k_{x},k_{y})={\mathcal {F}}\{\psi _{0}(x,y)\}}$$

and

$${\displaystyle \psi _{0}(x,y)={\mathcal {F}}^{-1}\{\Psi _{0}(k_{x},k_{y})\}}$$

All spatial dependence of each plane wave component is described explicitly by an exponential function. The coefficient of the exponential is a function of only two components of the wave vector for each plane wave (since other remained component can be determined via the above mentioned constraints), for example ${\displaystyle k_{x}}$ and ${\displaystyle k_{y}}$, just as in ordinary Fourier analysis and Fourier transforms.

Connection between Fourier optics and imaging resolution

Let's consider an imaging system where the z-axis is the optical axis of the system and the object plane (to be imaged on the image plane of the system) is the plane at ${\displaystyle z=0}$. On the object plane, the spatial part of a complex-valued Cartesian component of a wave is, as shown above, ${\textstyle \psi _{0}(x,y)=\int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }\Psi _{0}(k_{x},k_{y})~e^{i(k_{x}x+k_{y}y)}~dk_{x}dk_{y}}$ with the constraints of ${\displaystyle k_{x}^{2}+k_{y}^{2}+k_{z}^{2}=k^{2}}$, each ${\displaystyle k_{i}}$ as a real number, and ${\displaystyle k={\omega \over c}={2\pi \over \lambda }}$ where ${\displaystyle \omega =2\pi f}$. The imaging is the reconstruction of a wave on the object plane (having information about a pattern on the object plane to be imaged) on the image plane via the proper wave propagation from the object to the image planes, (E.g., think about the imaging of an image in an aerial space.) and the wave on the object plane, that fully follows the pattern to be imaged, is in principle, described by the unconstrained inverse Fourier transform ${\textstyle \psi _{0,{\text{unc}}}(x,y)=\int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }\Psi _{0}(k_{x},k_{y})~e^{i(k_{x}x+k_{y}y)}~dk_{x}dk_{y}}$ where ${\displaystyle k_{i}}$ takes an infinite range of real numbers. It means that, for a given light frequency, only a part of the full feature of the pattern can be imaged because of the above-mentioned constraints on ${\displaystyle k_{i}}$; (1) a fine feature which representation in the inverse Fourier transform requires spatial frequencies ${\textstyle {\frac {k_{T}}{2\pi }}}$, where ${\displaystyle k_{T}}$ are transverse wave numbers satisfying ${\displaystyle k_{T}^{2}=k_{x}^{2}+k_{y}^{2}\geq k^{2}}$, can not be fully imaged since waves with such ${\displaystyle k_{T}}$ do not exist for the given light of ${\displaystyle k}$ (This phenomenon is known as the diffraction limit.), and (2) spatial frequencies with ${\displaystyle k_{T}<k}$ but close to ${\displaystyle k}$ so higher wave outgoing angles with respect to the optical axis, requires a high NA (Numerical Aperture) imaging system that is expensive and difficult to build. For (1), even if complex-valued longitudinal wavenumbers ${\displaystyle k_{z}}$ are allowed (by an unknown interaction between light and the object plane pattern that is usually a solid material), ${\displaystyle k_{z}}$ give rise to light decay along the ${\displaystyle z}$ axis (Light amplification along the ${\displaystyle z}$ axis does not physically make sense if there is no amplification material between the object and image planes, and this is an usual case.) so waves with such ${\displaystyle k_{z}}$ may not reach the image plane that is usually sufficiently far way from the object plane.

In connection with photolithography of electronic components, these (1) and (2) are the reasons why light of a higher frequency (smaller wavelength, thus larger magnitude of ${\displaystyle k}$) or a higher NA imaging system is required to image finer features of integrated circuits on a photoresist on a wafer. As a result, machines realizing such an optical lithography have become more and more complex and expensive, significantly increasing the cost of the electronic component production.

Paraxial wave propagation (optic axis assumed as z axis)

A solution to the Helmholtz equation as the spatial part of a complex-valued Cartesian component of a single frequency wave is assumed to take the form:

$${\displaystyle \psi (\mathbf {r} )=A(\mathbf {r} )e^{-i\mathbf {k} \cdot \mathbf {r} }}$$

where ${\displaystyle \mathbf {k} }$ is the wave vector, and

$${\displaystyle \mathbf {k} \cdot \mathbf {r} =k_{x}\mathbf {x} +k_{y}\mathbf {y} +k_{z}\mathbf {z} }$$

and

$${\displaystyle k=\|\mathbf {k} \|={\sqrt {k_{x}^{2}+k_{y}^{2}+k_{z}^{2}}}={\omega \over c}}$$

is the wave number. Next, use the paraxial approximation, that is a small-angle approximation such that

$${\displaystyle k_{x}^{2}+k_{y}^{2}\ll k_{z}^{2}}$$

so, up to the second order approximation of trigonometric functions (that is, taking only up to the second term in the Taylor series expansion of each trigonometric function),

$${\displaystyle {\begin{aligned}\sin \theta &\approx \theta \\\tan \theta &\approx \theta \\\cos \theta &\approx 1-\theta ^{2}/2\end{aligned}}}$$

where ${\displaystyle \theta }$ is the angle (in radian) between the wave vector k and the z-axis as the optical axis of an optical system under discussion.

As a result,

$${\displaystyle k_{z}=k\cos \theta \approx k(1-\theta ^{2}/2)}$$

and

$${\displaystyle \psi (\mathbf {r} )\approx A(\mathbf {r} )e^{-i(k_{x}x+k_{y}y)}e^{ikz\theta ^{2}/2}e^{-ikz}}$$

The paraxial wave equation

Substituting this expression into the Helmholtz equation, the paraxial wave equation is derived:

$${\displaystyle \nabla _{T}^{2}A-2ik{\partial A \over \partial z}=0}$$

where

$${\displaystyle \nabla _{T}^{2}=\nabla ^{2}-{\partial ^{2} \over \partial z^{2}}={\partial ^{2} \over \partial x^{2}}+{\partial ^{2} \over \partial y^{2}}}$$

is the transverse Laplace operator in the Cartesian coordinates system. In the derivation of the paraxial wave equation, the following approximations are used.

• ${\displaystyle \theta }$ is small (${\displaystyle \theta \ll 1}$) so a term with ${\displaystyle \theta ^{2}}$ is ignored.
• Terms with ${\displaystyle 2k_{x}}$ and ${\displaystyle 2k_{y}}$ are much smaller than a term with ${\displaystyle 2k}$ (or ${\displaystyle 2k_{z}}$) so these two terms are ignored.
• ${\textstyle \left|{\frac {\partial ^{2}}{\partial z^{2}}}A(\mathbf {r} )\right|\ll \left|k{\frac {\partial }{\partial z}}A(\mathbf {r} )\right|}$ so a term with ${\textstyle {\frac {\partial ^{2}}{\partial z^{2}}}A(\mathbf {r} )}$ is ignored. It is the slowly varying envelope approximation, means that the amplitude or envelope of a wave ${\displaystyle A(\mathbf {r} )}$ is slowly varying compared with the major period of the wave ${\textstyle \lambda ={\frac {2\pi }{k}}}$.

Spatial versus angular bandwidth

The equation (2.2) above is critical to making the connection between spatial bandwidth (on the one hand) and angular bandwidth (on the other), in the far field. Note that the term "far field" usually means we're talking about a converging or diverging spherical wave with a pretty well defined phase center. The connection between spatial and angular bandwidth in the far field is essential in understanding the low pass filtering property of thin lenses. See the section 6.1.3 for the condition defining the far field region.

Once the concept of angular bandwidth is understood, the optical scientist can "jump back and forth" between the spatial and spectral domains to quickly gain insights which would ordinarily not be so readily available just through spatial domain or ray optics considerations alone. For example, any source bandwidth which lies past the edge angle to the first lens (This edge angle sets the bandwidth of the optical system.) will not be captured by the system to be processed.

As a side note, electromagnetics scientists have devised an alternative means to calculate an electric field in a far zone which does not involve stationary phase integration. They have devised a concept known as "fictitious magnetic currents" usually denoted by M, and defined as

$${\displaystyle \mathbf {M} ~=~2\mathbf {E} ^{\text{aper}}\times \mathbf {\hat {z}} .}$$

In this equation, it is assumed that the unit vector in the z-direction points into the half-space where the far field calculations will be made. These equivalent magnetic currents are obtained using equivalence principles which, in the case of an infinite planar interface, allow any electric currents J to be "imaged away" while the fictitious magnetic currents are obtained from twice the aperture electric field (see Scott [1998]). Then the radiated electric field is calculated from the magnetic currents using an equation similar to the equation for the magnetic field radiated by an electric current. In this way, a vector equation is obtained for the radiated electric field in terms of the aperture electric field, and the derivation requires no use of stationary phase ideas.

Eigenfunction (natural mode) solutions: background and overview

In the case of differential equations, as in the case of matrix equations, whenever the right-hand side of an equation is zero (For example, a forcing function, forcing vector, or the source of a force is zero.), the equation may still admit a non-trivial solution, known in applied mathematics as an eigenfunction solution, in physics as a "natural mode" solution, and in electrical circuit theory as the "zero-input response." This is a concept that spans a wide range of physical disciplines. Common physical examples of resonant natural modes would include the resonant vibrational modes of stringed instruments (1D), percussion instruments (2D) or the former Tacoma Narrows Bridge (3D). Examples of propagating natural modes would include waveguide modes, optical fiber modes, solitons and Bloch waves. an Infinite homogeneous media admits the rectangular, circular and spherical harmonic solutions to the Helmholtz equation, depending on the coordinate system under consideration. The propagating plane waves that we'll study in this article are perhaps the simplest type of propagating waves found in any type of media.

There is a striking similarity between the Helmholtz equation (2.3) above, which may be written

$${\displaystyle \left(\nabla ^{2}+k^{2}\right)f=0,}$$

and the usual equation form for the eigenvalues / eigenvectors of a square matrix A,

$${\displaystyle \left(\mathbf {A} -\lambda \mathbf {I} \right)\mathbf {x} =0,}$$

particularly since both the scalar Laplacian ${\displaystyle \nabla ^{2}}$ and the matrix A are linear operators on their respective functions / vector spaces. (The minus sign in this matrix equation is, for all intents and purposes, immaterial. However, the plus sign in the Helmholtz equation is significant.) It is perhaps worthwhile to note that the eigenfunction solutions / eigenvector solutions to the Helmholtz equation / the matrix equation, often yield an orthogonal set of the eigenfunctions / the eigenvectors which span (i.e., form a basis set for) the function space / vector space under consideration. The interested reader may investigate other functional linear operators (so for different equations than the Helmholtz equation) which give rise to different kinds of orthogonal eigenfunctions such as Legendre polynomials, Chebyshev polynomials and Hermite polynomials.

In the matrix equation case in which A is a square matrix, eigenvalues ${\displaystyle \lambda }$ may be found by setting the determinant of the matrix equal to zero, i.e. finding where the matrix has no inverse. (Such a square matrix is said to be singular.) Finite matrices have only a finite number of eigenvalues/eigenvectors, whereas linear operators can have a countably infinite number of eigenvalues/eigenfunctions (in confined regions) or uncountably infinite (continuous) spectra of solutions, as in unbounded regions.

In certain physics applications such as in the computation of bands in a periodic volume, it is often a case that the elements of a matrix will be very complicated functions of frequency and wavenumber, and the matrix will be non-singular (I.e., it has the inverse matrix.) for most combinations of frequency and wavenumber, but will also be singular (I.e., it does not have the inverse matrix.) for certain specific combinations. By finding which combinations of frequency and wavenumber drive the determinant of the matrix to zero, the propagation characteristics of the medium may be determined. Relations of this type, between frequency and wavenumber, are known as dispersion relations and some physical systems may admit many different kinds of dispersion relations. An example from electromagnetics is an ordinary waveguide, which may admit numerous dispersion relations, each associated with a unique propagation mode of the waveguide. Each propagation mode of the waveguide is known as an eigenfunction solution (or eigenmode solution) to Maxwell's equations in the waveguide. Free space also admits eigenmode (natural mode) solutions (known more commonly as plane waves), but with the distinction that for any given frequency, free space admits a continuous modal spectrum, whereas waveguides have a discrete mode spectrum. In this case, the dispersion relation is linear, as in section 1.3.

K-space

For a given ${\displaystyle k}$ such as ${\displaystyle k={\omega \over c}={2\pi \over \lambda }}$ for a homogeneous vacuum space, the separation condition,

$${\displaystyle k_{x}^{2}+k_{y}^{2}+k_{z}^{2}=k^{2}}$$

which is identical to the equation for the Euclidean metric in a three-dimensional configuration space, suggests the notion of a k-vector in a three-dimensional "k-space", defined (for propagating plane waves) in rectangular coordinates as:

$${\displaystyle \mathbf {k} ~=~k_{x}\mathbf {\hat {x}} +k_{y}\mathbf {\hat {y}} +k_{z}\mathbf {\hat {z}} }$$

and in the spherical coordinate system as

• ${\textstyle k_{x}=k~\sin \theta ~\cos \phi }$
• ${\textstyle k_{y}=k~\sin \theta ~\sin \phi }$
• ${\textstyle k_{z}=k~\cos \theta ~}$

Use will be made of these spherical coordinate system relations in the next section.

The notion of k-space is central to many disciplines in engineering and physics, especially in the study of periodic volumes, such as in crystallography and the band theory of semiconductor materials.

The two-dimensional Fourier transform

A spectrum analysis equation (calculating the spectrum of a function ${\displaystyle u(x,y)}$):

$${\displaystyle U(k_{x},k_{y})=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }u(x,y)e^{-i(k_{x}x+k_{y}y)}dxdy}$$

A synthesis equation (reconstructing the function ${\displaystyle u(x,y)}$ from its spectrum):

$${\displaystyle u(x,y)={\frac {1}{(2\pi )^{2}}}\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }U(k_{x},k_{y})e^{i(k_{x}x+k_{y}y)}dk_{x}dk_{y}}$$

The normalizing factor of ${\displaystyle {1}/{(2\pi )^{2}}}$ is present whenever angular frequency (radians) is used, but not when ordinary frequency (cycles) is used.

Input plane

The input plane is defined as the locus of all points such that z = 0. The input image f is therefore

$${\displaystyle f(x,y)=U(x,y,z){\big |}_{z=0}}$$

Output plane

The output plane is defined as the locus of all points such that z = d. The output image g is therefore

$${\displaystyle g(x,y)=U(x,y,z){\big |}_{z=d}}$$

The 2D convolution of input function against the impulse response function

$${\displaystyle g(x,y)~=~h(x,y)*f(x,y)}$$

i.e.,

$${\displaystyle g(x,y)=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }h(x-x',y-y')~f(x',y')~dx'dy'}$$

(4.1)

The alert reader will note that the integral above tacitly assumes that the impulse response is NOT a function of the position (x',y') of the impulse of light in the input plane (if this were not the case, this type of convolution would not be possible). This property is known as shift invariance (Scott [1998]). No optical system is perfectly shift invariant: as the ideal, mathematical point of light is scanned away from the optic axis, aberrations will eventually degrade the impulse response (known as a coma in focused imaging systems). However, high quality optical systems are often "shift invariant enough" over certain regions of the input plane that we may regard the impulse response as being a function of only the difference between input and output plane coordinates, and thereby use the equation above with impunity.

Also, this equation assumes unit magnification. If magnification is present, then eqn. (4.1) becomes

$${\displaystyle g(x,y)=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }h_{M}(x-Mx',y-My')~f(x',y')~dx'dy'}$$

(4.2)

which basically translates the impulse response function, h_M(), from x′ to x = Mx′. In eqn. (4.2), h_M will be a magnified version of the impulse response function h of a similar, unmagnified system, so that h_M(x,y) = h(x/M,y/M).

Derivation of the convolution equation

The extension to two dimensions is trivial, except for the difference that causality exists in the time domain, but not in the spatial domain. Causality means that the impulse response h(t − t′) of an electrical system, due to an impulse applied at time t', must of necessity be zero for all times t such that t − t′ < 0.

Obtaining the convolution representation of the system response requires representing the input signal as a weighted superposition over a train of impulse functions by using the sifting property of Dirac delta functions.

$${\displaystyle f(t)=\int _{-\infty }^{\infty }\delta (t-t')f(t')dt'}$$

It is then presumed that the system under consideration is linear, that is to say that the output of the system due to two different inputs (possibly at two different times) is the sum of the individual outputs of the system to the two inputs, when introduced individually. Thus the optical system may contain no nonlinear materials nor active devices (except possibly, extremely linear active devices). The output of the system, for a single delta function input is defined as the impulse response of the system, h(t − t′). And, by our linearity assumption (i.e., that the output of system to a pulse train input is the sum of the outputs due to each individual pulse), we can now say that the general input function f(t) produces the output:

$${\displaystyle g(t)=\int _{-\infty }^{\infty }h(t-t')f(t')dt'}$$

where h(t − t′) is the (impulse) response of the linear system to the delta function input δ(t − t′), applied at time t'. This is where the convolution equation above comes from. The convolution equation is useful because it is often much easier to find the response of a system to a delta function input - and then perform the convolution above to find the response to an arbitrary input - than it is to try to find the response to the arbitrary input directly. Also, the impulse response (in either time or frequency domains) usually yields insight to relevant figures of merit of the system. In the case of most lenses, the point spread function (PSF) is a pretty common figure of merit for evaluation purposes.

The same logic is used in connection with the Huygens–Fresnel principle, or Stratton-Chu formulation, wherein the "impulse response" is referred to as the Green's function of the system. So the spatial domain operation of a linear optical system is analogous in this way to the Huygens–Fresnel principle.

System transfer function

If the last equation above is Fourier transformed, it becomes:

$${\displaystyle G(\omega )~=~H(\omega )\cdot F(\omega )}$$

where

• ${\displaystyle G(\omega )}$ is the spectrum of the output signal
• ${\displaystyle H(\omega )}$ is the system transfer function
• ${\displaystyle F(\omega )}$ is the spectrum of the input signal

In like fashion, eqn. (4.1) may be Fourier transformed to yield:

$${\displaystyle G(k_{x},k_{y})~=~H(k_{x},k_{y})\cdot F(k_{x},k_{y})}$$

The system transfer function, ${\displaystyle H(\omega )}$. In optical imaging this function is better known as the optical transfer function (Goodman).

Once again it may be noted from the discussion on the Abbe sine condition, that this equation assumes unit magnification.

This equation takes on its real meaning when the Fourier transform, ${\displaystyle ~G(k_{x},k_{y})}$ is associated with the coefficient of the plane wave whose transverse wavenumbers are ${\displaystyle ~(k_{x},k_{y})}$. Thus, the input-plane plane wave spectrum is transformed into the output-plane plane wave spectrum through the multiplicative action of the system transfer function. It is at this stage of understanding that the previous background on the plane wave spectrum becomes invaluable to the conceptualization of Fourier optical systems.

Fourier transforming property of lenses

If a transmissive object is placed at one focal length in front of a lens, then its Fourier transform will be formed at one focal length behind the lens. Consider the figure to the right (click to enlarge)

On the Fourier transforming property of lenses

In this figure, a plane wave incident from the left is assumed. The transmittance function in the front focal plane (i.e., Plane 1) spatially modulates the incident plane wave in magnitude and phase, like on the left-hand side of eqn. (2.1) (specified to z = 0), and in so doing, produces a spectrum of plane waves corresponding to the FT of the transmittance function, like on the right-hand side of eqn. (2.1) (for z > 0). The various plane wave components propagate at different tilt angles with respect to the optic axis of the lens (i.e., the horizontal axis). The finer the features in the transparency, the broader the angular bandwidth of the plane wave spectrum. We'll consider one such plane wave component, propagating at angle θ with respect to the optic axis. It is assumed that θ is small (paraxial approximation), so that

$${\displaystyle {\frac {k_{x}}{k}}=\sin \theta \simeq \theta }$$

and

$${\displaystyle {\frac {k_{z}}{k}}=\cos \theta \simeq 1-{\frac {1}{2}}\theta ^{2}}$$

and

$${\displaystyle {\frac {1}{\cos \theta }}\simeq {\frac {1}{1-{\frac {1}{2}}\theta ^{2}}}\simeq 1+{\frac {1}{2}}\theta ^{2}}$$

In the figure, the plane wave phase, moving horizontally from the front focal plane to the lens plane, is

$${\displaystyle e^{ikf\cos \theta }}$$

and the spherical wave phase from the lens to the spot in the back focal plane is:

$${\displaystyle e^{ikf/\cos \theta }}$$

and the sum of the two path lengths is f (1 + θ²/2 + 1 − θ²/2) = 2f; i.e., it is a constant value, independent of tilt angle, θ, for paraxial plane waves. Each paraxial plane wave component of the field in the front focal plane appears as a point spread function spot in the back focal plane, with an intensity and phase equal to the intensity and phase of the original plane wave component in the front focal plane. In other words, the field in the back focal plane is the Fourier transform of the field in the front focal plane.

All FT components are computed simultaneously - in parallel - at the speed of light. As an example, light travels at a speed of roughly 1 ft (0.30 m) per nanosecond, so if a lens has a 1 ft (0.30 m) focal length, an entire 2D FT can be computed in about 2 ns (2 × 10⁻⁹ seconds). If the focal length is 1 in, then the time is under 200 ps. No electronic computer can compete with these kinds of numbers or perhaps ever hope to, although supercomputers may actually prove faster than optics, as improbable as that may seem. However, their speed is obtained by combining numerous computers which, individually, are still slower than optics. The disadvantage of the optical FT is that, as the derivation shows, the FT relationship only holds for paraxial plane waves, so this FT "computer" is inherently bandlimited. On the other hand, since the wavelength of visible light is so minute in relation to even the smallest visible feature dimensions in the image i.e.,

$${\displaystyle k^{2}\gg k_{x}^{2}+k_{y}^{2}}$$

(for all k_x, k_y within the spatial bandwidth of the image, so that k_z is nearly equal to k), the paraxial approximation is not terribly limiting in practice. And, of course, this is an analog - not a digital - computer, so precision is limited. Also, phase can be challenging to extract; often it is inferred interferometrically.

Optical processing is especially useful in real time applications where rapid processing of massive amounts of 2D data is required, particularly in relation to pattern recognition.

Hardware implementation of the system transfer function: The 4F correlator

The theory on optical transfer functions presented in the section 5 is somewhat abstract. However, there is one very well known device which implements the system transfer function H in hardware using only 2 identical lenses and a transparency plate - the 4F correlator. Although one important application of this device would certainly be to implement the mathematical operations of cross-correlation and convolution, this device - 4 focal lengths long - actually serves a wide variety of image processing operations that go well beyond what its name implies. A diagram of a typical 4F correlator is shown in the figure below (click to enlarge). This device may be readily understood by combining the plane wave spectrum representation of the electric field (section 1.5) with the Fourier transforming property of quadratic lenses (section 6.1) to yield the optical image processing operations described in the section 5.

4F Correlator

The 4F correlator is based on the convolution theorem from Fourier transform theory, which states that convolution in the spatial (x,y) domain is equivalent to direct multiplication in the spatial frequency (k_x, k_y) domain (aka: spectral domain). Once again, a plane wave is assumed incident from the left and a transparency containing one 2D function, f(x,y), is placed in the input plane of the correlator, located one focal length in front of the first lens. The transparency spatially modulates the incident plane wave in magnitude and phase, like on the left-hand side of eqn. (2.1), and in so doing, produces a spectrum of plane waves corresponding to the FT of the transmittance function, like on the right-hand side of eqn. (2.1). That spectrum is then formed as an "image" one focal length behind the first lens, as shown. A transmission mask containing the FT of the second function, g(x,y), is placed in this same plane, one focal length behind the first lens, causing the transmission through the mask to be equal to the product, F(k_x,k_y) × G(k_x,k_y). This product now lies in the "input plane" of the second lens (one focal length in front), so that the FT of this product (i.e., the convolution of f(x,y) and g(x,y)), is formed in the back focal plane of the second lens.

If an ideal, mathematical point source of light is placed on-axis in the input plane of the first lens, then there will be a uniform, collimated field produced in the output plane of the first lens. When this uniform, collimated field is multiplied by the FT plane mask, and then Fourier transformed by the second lens, the output plane field (which in this case is the impulse response of the correlator) is just our correlating function, g(x,y). In practical applications, g(x,y) will be some type of feature which must be identified and located within the input plane field (see Scott [1998]). In military applications, this feature may be a tank, ship or airplane which must be quickly identified within some more complex scene.

The 4F correlator is an excellent device for illustrating the "systems" aspects of optical instruments, alluded to in the section 5 above. The FT plane mask function, G(k_x,k_y) is the system transfer function of the correlator, which we'd in general denote as H(k_x,k_y), and it is the FT of the impulse response function of the correlator, h(x,y) which is just our correlating function g(x,y). And, as mentioned above, the impulse response of the correlator is just a picture of the feature we're trying to find in the input image. In the 4F correlator, the system transfer function H(k_x,k_y) is directly multiplied against the spectrum F(k_x,k_y) of the input function, to produce the spectrum of the output function. This is how electrical signal processing systems operate on 1D temporal signals.

Image Restoration

Image blurring by a point spread function is studied extensively in optical information processing, one way to alleviate the blurring is to adopt Wiener Filter. For example, assume that ${\displaystyle o(x,y)}$ is the intensity distribution from an incoherent object, ${\displaystyle i(x,y)}$ is the intensity distribution of its image which is blurred by a space-invariant point-spread function ${\displaystyle s(x,y)}$ and a noise ${\displaystyle n(x,y)}$ introduced in the detection process:

$${\displaystyle i(x,y)=o(x,y)\otimes s(x,y)+n(x,y)}$$

The goal of image restoration is to find a linear restoration filter that minimize the mean-squared error between the true distribution and the estimation ${\displaystyle {\hat {o}}(x,y)}$. That is, to minimize

$${\displaystyle \varepsilon ^{2}=|o-{\hat {o}}|^{2}}$$

The solution of this optimization problem is Wiener filter:

$${\displaystyle H\left(f_{X},f_{Y}\right)={\frac {S^{*}\left(f_{X},f_{Y}\right)}{\left|S\left(f_{X},f_{Y}\right)\right|^{2}+{\frac {\Phi _{n}\left(f_{X},f_{Y}\right)}{\Phi _{o}\left(f_{X},f_{Y}\right)}}}},}$$

where ${\displaystyle S\left(f_{X},f_{Y}\right)}$, ${\displaystyle \Phi _{o}\left(f_{X},f_{Y}\right)}$, ${\displaystyle \Phi _{n}\left(f_{X},f_{Y}\right)}$ are the power spectral densities of the point-spread function, the object and the noise.

The recording geometry

Ragnarsson proposed a method to realize Wiener restoration filters optically by holographic technique like setup shown in the figure.^[2][3] The derivation of the function of the setup is described as follows.

Assume there is a transparency as the recording plane and an impulse emitted from a point source S. The wave of impulse is collimated by lens L1, forming a distribution equal to the impulse response ${\displaystyle h}$. Then the distribution ${\displaystyle h}$ is then split into two parts:

1. The upper portion is first focused (i.e., Fourier transformed) by a lens L2 to a spot in the front focal plan of lens L3, forming a virtual point source generating a spherical wave. The wave is then collimated by lens L3 and produces a tilted plane wave with the form ${\displaystyle r_{0}e^{-j2\pi \alpha y}}$ at the recording plane.
2. The lower portion is directly collimated by lens L3, yielding an amplitude distribution ${\displaystyle {\frac {1}{\lambda f}}H{\left({\frac {x_{2}}{\lambda f}},{\frac {y_{2}}{\lambda f}}\right)}}$.

Therefore, the total intensity distribution is

$${\displaystyle {\begin{aligned}{\mathcal {I}}\left(x_{2},y_{2}\right)&=\left|r_{o}\exp \left(-j2\pi \alpha y_{2}\right)+{\frac {1}{\lambda f}}H\left({\frac {x_{2}}{\lambda f}},{\frac {y_{2}}{\lambda f}}\right)\right|^{2}\\&=r_{o}^{2}+{\frac {1}{\lambda ^{2}f^{2}}}\left|H\left({\frac {x_{2}}{\lambda f}},{\frac {y_{2}}{\lambda f}}\right)\right|^{2}+{\frac {r_{o}}{\lambda f}}H\left({\frac {x_{2}}{\lambda f}},{\frac {y_{2}}{\lambda f}}\right)\exp \left(j2\pi \alpha y_{2}\right)+{\frac {r_{o}}{\lambda f}}H^{*}\left({\frac {x_{2}}{\lambda f}},{\frac {y_{2}}{\lambda f}}\right)\exp \left(-j2\pi \alpha y_{2}\right)\end{aligned}}}$$

Assume ${\displaystyle H}$ has an amplitude distribution ${\displaystyle A}$ and a phase distribution ${\displaystyle \psi }$ such that

$${\displaystyle H\left({\frac {x_{2}}{\lambda f}},{\frac {y_{2}}{\lambda f}}\right)=S\left({\frac {x_{2}}{\lambda f}},{\frac {y_{2}}{\lambda f}}\right)\exp \left[j\psi \left({\frac {x_{2}}{\lambda f}},{\frac {y_{2}}{\lambda f}}\right)\right],}$$

then we can rewrite intensity as follows:

$${\displaystyle I\left(x_{2},y_{2}\right)=r_{o}^{2}+{\frac {1}{\lambda ^{2}f^{2}}}S^{2}\left({\frac {x_{2}}{\lambda f}},{\frac {y_{2}}{\lambda f}}\right)+{\frac {2r_{o}}{\lambda f}}S\left({\frac {x_{2}}{\lambda f}},{\frac {y_{2}}{\lambda f}}\right)\cos \left[2\pi \alpha y_{2}+\psi \left({\frac {x_{2}}{\lambda f}},{\frac {y_{2}}{\lambda f}}\right)\right].}$$

Note that for the point at the origin of the film plane (${\displaystyle (x_{2},y_{2})=\mathbf {0} }$), the recorded wave from the lower portion should be much stronger than that from the upper portion because the wave passing through the lower path is focused, which leads to the relationship ${\textstyle r_{0}\ll {\frac {1}{\lambda f}}}$.

In Ragnarsson' s work, this method is based on the following postulates:

1. Assume there is a transparency, with its amplitude transmittance ${\displaystyle t}$ proportional to ${\displaystyle s(x,y)}$, that has recorded the known impulse response of the blurred system.
2. The maximum phase ${\displaystyle \phi }$ shift introduced by the filter is much smaller than ${\displaystyle 2\pi }$ radians so that ${\displaystyle e^{j\phi }\approx 1+j\phi }$.
3. The phase shift of the transparency after bleaching is linearly proportional to the silver density ${\displaystyle D}$ present before bleaching.
4. The density is linearly proportional to the logarithm of exposure
   $${\displaystyle I=r_{o}^{2}+{\frac {1}{\lambda ^{2}f^{2}}}S^{2}.}$$

   
5. The average exposure ${\displaystyle {\bar {I}}}$ is much stronger than varying exposure
   $${\displaystyle \Delta I\left({\frac {x_{2}}{\lambda f}},{\frac {y_{2}}{\lambda f}}\right)={\frac {2r_{o}}{\lambda f}}S\left({\frac {x_{2}}{\lambda f}},{\frac {y_{2}}{\lambda f}}\right)\cos \left[2\pi \alpha y_{2}+\psi \left({\frac {x_{2}}{\lambda f}},{\frac {y_{2}}{\lambda f}}\right)\right].}$$

   

By these postulates, we have the following relationship:

$${\displaystyle \Delta t\propto \Delta \phi \propto \Delta D\propto \Delta (\log I)\approx {\frac {\Delta I}{I}}.}$$

Finally, we get a amplitude transmittance with the form of a Wiener filter:

$${\displaystyle \Delta t\propto {\frac {S^{\ast }}{r_{0}^{2}\lambda ^{2}f^{2}+\left\vert S\right\vert ^{2}}}.}$$

Functional decomposition and eigenfunctions

The twin subjects of eigenfunction expansions and functional decomposition, both briefly alluded to here, are not completely independent. The eigenfunction expansions to certain linear operators defined over a given domain, will often yield a countably infinite set of orthogonal functions which will span that domain. Depending on the operator and the dimensionality (and shape, and boundary conditions) of its domain, many different types of functional decompositions are, in principle, possible.