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Fourier optics

The Fourier model for image formation provides a simple and elegant explanation of the limits of resolution of an optical system and of the nature of the image itself. The underlying principle is the same as that used for the analysis of sounds or electronic signals. The whole of sound reproduction theory is based on the principle that any sound, no matter how complex, is made up of sinusoidal (wave-form) components of various amplitudes, frequencies, and phases. The converse, that any sound pattern can be built up from an appropriate spectrum of sinusoids, is equally true. This is called respectively Fourier analysis and Fourier synthesis, after Joseph Fourier (1768-1830), who first worked out the mathematical principles.

The precise optical analogy is that the light beam emitted (reflected or transmitted) by any object, no matter how complex, is identical with the pattern of diffraction that would be transmitted by a spectrum of sinusoidal diffraction gratings of various transmittances, spatial frequencies, and spatial phases, and, for a two-dimensional object, orientations. Similarly, any desired light beam can be constructed by superimposing appropriate sinusoidal gratings on a plane wavefront. One small extension is necessary. Because (unlike a sonic or electrical signal) a transmittance cannot be negative, there is always a ‘d.c.’ component present in an optical grating, equal to half the amplitude of the grating.

The Fourier approach treats imaging as a diffraction phenomenon, and a lens as a device for performing an optical Fourier analysis (in two dimensions) of the sinusoidal functions that together create the shape of the wavefront from the object. The principle is approached by considering a simple sinusoidal grating (Fig. 1). This produces a single pair of diffracted plane wavefronts. If the grating is situated at the front focal plane of the lens, these wavefronts will converge towards the rear focal plane, forming two spots; this can be readily demonstrated using laser light. The electrical field in this plane is the Fourier transform (i.e. the spatial frequency spectrum) of the object grating, and represents—as it must—a single spatial frequency.

When the object is more complicated, the diffraction pattern at the rear focal plane is also more complicated. But it still consists of pairs of spots. Each pair indicates a specific amplitude, spatial frequency, and orientation of the originating sinusoidal grating by (respectively) their intensity, separation, and angle. Indeed, simple objects can be identified from their diffraction patterns. The letter V of (Fig. 2) shows the characteristic pattern of two rectangles at an angle to one another. One thing the visible pattern does not show is the spatial phases of the gratings; but they are nevertheless present in the electric field.

A fundamental property of a Fourier transform is that it obeys the commutative law. This implies that if a Fourier transform is carried out on a Fourier transform, the result is just the original function. So a second lens positioned with its front focal plane in the rear focal plane of the first will produce a replica of the original object at its own rear focal plane (Fig. 3). This, of course will not be news to a photographer: it is part of traditional geometrical optics. What was not generally appreciated, and not actually seen before lasers appeared, was the presence of the diffraction pattern that proved the existence of the optical Fourier transform in the principal focal plane of the lens. (It is, of course, present even with white light, but is smeared out by dispersion and lack of coherence.)

One important outcome of the Fourier principle is that it indicates the absolute limit of resolution of a lens. Because the highest spatial frequencies in the object wavefront are diffracted to the largest angle, they may miss the lens altogether (Fig. 4). Closing the lens aperture down progressively cuts off the higher spatial frequencies, just as applying a high-frequency blocking filter removes the highest sound frequencies from a hi-fi amplifier. The effect was first noticed in the early 1950s, when it was seen that the image of a Sayce grating (a rectangular intensity-profile grating, Fig. 5a) became markedly sinusoidal in profile at the fine-detail end as the aperture of the lens under test was closed down, indicating that the higher spatial frequencies present in the bars were not being recorded (Fig. 5b).

Fig.-1
Fig.-1


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Fig.-2


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Fig.-3


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Fig.-4


Fig.-5a
Fig.-5a


Fig.-5b
Fig.-5b

— Graham Saxby

Bibliography

  • Steward, E. G., Fourier Optics: An Introduction (1989).
  • Saxby, G., The Science of Imaging: An Introduction (2002)
 
 
Wikipedia: Fourier optics

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 or spectral domain 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 (i.e., exponentially decaying) plane waves.

Electromagnetic Wave Propagation

The Wave Equation

Fourier optics begins with the homogeneous, scalar wave equation:

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

where u(r,t) is a real-valued, scalar representation of an electromagnetic wave propagating through free space.

The Helmholtz Equation

If we next assume that the solution of this equation takes a time-harmonic form, or in other words,

u(\mathbf{r},t) = \mathrm{Re} \left\{ \psi(\mathbf{r}) e^{j\omega t} \right\}

and substitute this expression into the wave equation, we derive the time-independent form of the wave equation, also known as the Helmholtz equation:

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

where

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

is the wave number, j is the imaginary unit, and ψ(r) is the time-independent, complex-valued amplitude of the propagating wave.

The paraxial approximation

We can simplify the complex wave amplitude further by a simple change of variable:

\psi(\mathbf{r}) = A(\mathbf{r}) e^{-j \mathbf{k} \cdot \mathbf{r}}

where

\mathbf{k} = k_x \mathbf{x} + k_y \mathbf{y} + k_z\mathbf{z}

is the wave vector, and

k = ||\mathbf{k}|| = \sqrt{k_x^2 + k_y^2 + k_z^2} = {\omega \over c}

is the wave number. Next, using the paraxial approximation, we assume that

k_x^2 + k_y^2 << k_z^2

or equivalently,

\sin \theta \approx \theta

where θ is the angle between the wave vector k and the z-axis.

As a result,

k \approx k_z

and

\psi(\mathbf{r}) \approx A(\mathbf{r}) e^{-jkz}

The paraxial wave equation

Substituting this expression into the Helmholtz equation, we derive the paraxial wave equation:

\nabla_T^2 A - 2jk { \partial A \over \partial z} = 0

where

\nabla_T^2 = {\partial^2 \over \partial x^2} + {\partial^2 \over \partial y^2}

is the transverse Laplacian operator.

Transfer function of wave propagation


Origin of Plane Wave Spectrum Representation of the Electric Field

The plane wave spectrum concept stems from the homogeneous Electromagnetic wave equation (see also Electromagnetic radiation, and 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, the wavenumber of the medium. 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 the following form:

Eu(x,y,z) = fx(x)×fy(y)×fz(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, using the scalar Laplacian (aka, Laplace operator) in rectangular coordinates

\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

f''x(x)fy(y)fz(z) + fx(x)f''y(y)fz(z) + fx(x)fy(y)f''z(z) + k2fx(x)fy(y)fz(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 homogeneous Electromagnetic wave equation in rectangular coordinates is formed as a weighted 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. Stated another way, the radiation pattern of any planar field distribution is the FT of that distribution (see Huygens-Fresnel principle}. 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 k-vector, or wave vector, defined (for propagating plane waves) in rectangular coordinates as

\bold k = k_x \hat \bold x + k_y \hat \bold y + k_z \hat \bold 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 to the right (click to enlarge)

On the Fourier Transforming Property of Lenses
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On the Fourier Transforming Property of Lenses

In this figure, we assume a plane wave incident from the left and imagine the front focal plane transmittance function as being decomposed into a spectrum of plane waves, each propagating 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

\frac{k_x}{k} \cong \sin \theta \cong \theta

and

\frac{k_z}{k} \cong \cos \theta \cong 1 - \frac{\theta^2}{2}

and

\frac{1}{\cos \theta} \cong \frac{1}{1 - \frac{\theta^2}{2}} \cong 1 + \frac{\theta^2}{2}

In the figure, the plane wave phase, moving horizontally from the front focal plane to the lens plane, 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 waves. 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 primary applications of Fourier Optics is in the mathematical operations of cross-correlation and convolution. This has historically been done with a device known as a 4F correlator, shown in the figure below (click to enlarge).

4F Correlator
Enlarge
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 (kx, ky) domain. 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 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).

See also

References

  • Goodman, Joseph (2005). Introduction to Fourier Optics, 3rd ed,, Roberts & Co Publishers. ISBN 0974707724.  or online here
  • Hecht, Eugene (1987). Optics, 2nd ed., Addison Wesley. ISBN 0-201-11609-X. 
  • Wilson, Raymond (1995). Fourier Series and Optical Transform Techniques in Contemporary Optics. Wiley. ISBN 0471303577. 

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Photography Encyclopedia. The Oxford Companion to the Photograph. Copyright © 2005 by Oxford University Press. All rights reserved.  Read more
Wikipedia. This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Fourier optics".  Read more

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