User:CraigNGC/subpage FourierOptics

Fourier optics is one of the three major viewpoints for understanding classical optics (the other two being the diffraction integral viewpoint and geometrical optics). The subject area known as "Fourier optics" is also known as the plane wave spectrum technique in the broader context of general electromagnetic theory. This approach stems from the fact that in source-free regions (and virtually all of classical optics pertains to source-free regions), electromagnetic fields may be expressed in terms of a spectrum of propagating and evanescent (a fancy word meaning exponentially decaying) plane waves.

Origin of Plane Wave Spectrum Representation of the Electric Field

The plane wave spectrum concept stems from the homogeneous Electromagnetic wave equation, which itself is a by-product of the basic Maxwell's equations specified to source-free regions. In the frequency domain, the homogeneous Electromagnetic wave equation assumes the form:

$$ \nabla^2 E_u + k^2E_u = 0   $$

where u = x, y, z and k = 2π/λ. We may readily find solutions to this equation in rectangular coordinates by using the principle of separation of variables for partial differential equations. This principle says that in separable orthogonal coordinates, we may construct a so-called product solution to this wave equation of following form:

$$ E_u(x, y, z) = f_x(x) \times f_y(y) \times f_z(z)$$

i.e., a solution which is expressed as the product of a function of x, times a function of y, times a function of z. If we now plug this product solution into the wave equation, where

$$ \nabla^2 E_u = \frac{\partial E_u}{\partial x^2} + \frac{\partial E_u}{\partial y^2} + \frac{\partial E_u}{\partial z^2} $$

we obtain

$$\frac{}{}f_x(x)f_y(y)f_z(z) + f_x(x)f_y(y)f_z(z) + f_x(x)f_y(y)f''_z(z) + k^2f_x(x)f_y(y)f_z(z)=0 $$

which may be rearranged into the form:

$$  \frac{f_x(x)}{f_x(x)}+ \frac{f_y(y)}{f_y(y)} + \frac{f''_z(z)}{f_z(z)} + k^2=0 $$

We may now argue that each of the quotients in the equation above must, of necessity, be constant. For, say the first quotient is not constant, and is a function of x. None of the other terms in the equation has any dependence on the variable x. Therefore, the first term may not have any x-dependence either; it must be constant. Let's call that constant -kx2. Reasoning in a similar way for the y and z quotients, we now obtain three ordinary differential equations for the fx, fy and fz, along with one separation condition:

$$\frac{d^2}{dx^2}f_x(x) + k_x^2 f_x(x)=0$$

$$\frac{d^2}{dy^2}f_y(y) + k_y^2 f_y(y)=0$$

$$\frac{d^2}{dz^2}f_z(z) + k_z^2 f_z(z)=0$$

$$k_x^2+k_y^2+k_z^2=k^2$$

Each of these 3 differential equations has the same solution, a complex exponential, so that the elementary product solution for Eu is:

$$E_u(x,y,z)=e^{j(k_x x + k_y y)} e^{\pm j \sqrt{k^2-k_x^2-k_y^2}z}$$

which represents a propagating or exponentially decaying plane wave solution to the homogeneous wave equation. The - sign is used for a wave propagating/decaying in the +z direction and the + sign is used for a wave propagating/decaying in the -z direction (this follows the engineering time convention, which assumes an ejωt time dependence). This field represents a propagating plane wave when the quantity under the radical is positive, and an exponentially decaying wave when it is negative (in passive media, we always choose the root with a negative imaginary part, to represent decay, not amplification).

A general solution to the Electromagnetic wave equation in rectangular coordinates is formed as a superposition of elementary plane wave solutions as:

$$E_u(x,y,z)=\int\int E_u(k_x,k_y)e^{j(k_x x + k_y y)} e^{\pm j \sqrt{k^2-k_x^2-k_y^2}z} dk_x dk_y$$

This plane wave spectrum representation of the electromagnetic field is the basic foundation of Fourier Optics, because we see that when z=0, the equation above simply becomes a Fourier transform (FT) relationship between the field (in a given plane of an optical system), and the plane wave content of that field. This is an extremely important point to recognize. What this says is that if we take some field distribution in the object plane of an imaging system, then take the FT of that distribution, then we know the plane wave content (i.e., the radiation pattern) of that field distribution in space. Stated another way, the radiation pattern of any planar field distribution is the FT of that distribution. In addition, we may determine the image plane distribution of some object plane distribution by tracing the progress of the individual plane wave components through the imaging system, and then re-assembling them in the image plane, each with its own particular phase.

The separation condition,

$$k_x^2+k_y^2+k_z^2=k^2$$

which so closely resembles the equation for the length of a vector in terms of its rectangular components, suggests the notion of the k-vector, defined (for propagating plane waves) in rectangular coordinates as

$$\mathbf k = k_x \hat \mathbf x + k_y \hat \mathbf y + k_z \hat \mathbf z $$

and in the spherical coordinate system as

$$ k_x = k sin \theta cos \phi $$

$$ k_y = k sin \theta sin \phi $$

$$ k_z = k cos \theta $$

We'll make use of these spherical coordinate system relations in the next section.

Fourier Transforming Property of Lenses

If a transmissive object is placed one focal length in front of a lens, then its Fourier transform will be formed one focal length behind the lens. We may show this using what we now know about the plane wave spectrum representation of the transmittance function in the front focal plane. Consider the figure below (click to enlarge)

in this figure, we imagine the front focal plane transmittance function as being decomposed into a spectrum of plane waves, each prapagating at a different angle with respect to the optic axis of the lens (i.e., the horizontal axis). We'll consider one plane wave component, propagating at angle θ with respect to the optic axis. We'll assume θ is small (paraxial approximation), so that

$$k_x/k \cong sin \theta \cong \theta$$

and

$$k_z/k \cong cos \theta \cong 1-\theta^2 $$

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

$$ e^{j k f cos \theta} $$

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

$$ e^{j k f / cos \theta} $$

and the sum of the two phase lengths is 2f for paraxial plane wavess. Each plane wave component of the field in the front focal plane appears as a 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.

4F Correlator

One of the primry applications of Fourier Optics is in the mathematical operation of correlation. This has historically been done with a device known as a 4F correlator, shown in the figure below (click to enlarge).

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 (kx, ky) domain. 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 FT of that function is then formed 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(kx,ky) x G(kx,ky). This product now lies in the "input plane" of the second lens, 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.

Applications
Fourier optics is used in the field of optical information processing, the staple of which is the classical 4F processor.

The Fourier transform properties of a lens provide numerous applications in optical signal processing such as spatial filtering, optical correlation and computer generated holograms.

Fourier optical theory is used in interferometers, optical tweezers, atom traps, and quantum computing. Concepts of Fourier optics are used to reconstruct the phase of light intensity in the spatial frequency plane (see adaptive-additive algorithm).