Fourier optics is the study of classical
opticsOptics is the branch of physics which studies the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...
using techniques involving
Fourier transformIn mathematics, Fourier analysis is a subject area which grew out of the study of Fourier series. The subject began with trying to understand when it was possible to represent general functions by sums of simpler trigonometric functions...
s and can be seen as an extension of the
Huygens-Fresnel principleThe Huygens–Fresnel principle is a method of analysis applied to problems of wave propagation...
. The underlying idea is that any "wide" wave which moves forward (like an ocean wave coming towards the shore) can actually be thought of as an infinite amount of "wave points", all of which could (when they collide with something in the way) move relatively independently of each other. The theorem basically says square objects can be made by combining an infinite amount of curved objects. If a wave is far enough away from something that it can be simplified to a square block moving forward, a
Fraunhofer diffractionIn optics, Fraunhofer diffraction , or far-field diffraction, is a form of wave diffraction that occurs when field waves are passed through an aperture or slit causing only the size of an observed aperture image to change due to the far-field location of observation and the increasingly planar...
would be created. When the wave is close enough that more attention must be paid to the individual wave points and the wave can only be simplified to a round ball instead of a square block, a
Fresnel diffractionIn optics, Fresnel diffraction or near-field diffraction is a process of diffraction that occurs when a wave passes through an aperture and diffracts in the near field, causing any diffraction pattern observed to differ in size and shape, depending on the distance between the aperture and the...
would be created. These mathematical simplifications and calculations are the realm of Fourier analysis and synthesis -- together, they can describe what happens when light passes through various slits or mirrors curved one way or the other, or is fully or partially reflected. Fourier optics forms much of the theory behind
image processing techniquesIn electrical engineering and computer science, image processing is any form of signal processing for which the input is an image, such as photographs or frames of video; the output of image processing can be either an image or a set of characteristics or parameters related to the image...
, as well as finding applications where information needs to be extracted from optical sources such as in
quantum opticsQuantum optics is a field of research in physics, dealing with the application of quantum mechanics to phenomena involving light and its interactions with matter.- History of quantum optics :...
. To put it in a slightly more complex way, similar to the concept of
frequency and
time used in traditional Fourier transform theory, Fourier optics makes use of the
spatial frequencyIn mathematics, physics, and engineering, spatial frequency is a characteristic of any structure that is periodic across position in space. The spatial frequency is a measure of how often the structure repeats per unit of distance. The SI unit of spatial frequency is cycles per meter...
domain (
kx,
ky) as the conjugate of the spatial (
x,
y) domain. Terms and concepts such as transform theory, spectrum, bandwidth, window functions and sampling from one dimensional
signal processingSignal processing is an area of electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time to perform useful operations on those signals...
are commonly used.
Propagation of light through a homogeneous, source-free medium
Light can be described as a waveform propagating through free space (vacuum) or a material medium (such as air or glass). Mathematically, the amplitude of the wave is represented by a scalar wave function
u that depends on both space and time:
where
represents position in three dimensional space, and
t represents time.
The Wave Equation in the Time Domain
Fourier optics begins with the homogeneous, scalar
wave equationThe wave equation is an important second-order linear partial differential equation of waves, such as sound waves, light waves and water waves. It arises in fields such as acoustics, electromagnetics, and fluid dynamics...
(valid in source-free regions):
where
u(
r,
t) is a
real numberIn mathematics, the real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real...
, Cartesian component of an electromagnetic wave propagating through free space.
The Helmholtz Equation in the Frequency Domain
If it is assumed that the solution of this equation takes a time-
harmonicIn acoustics and telecommunication, a harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency. For example, if the fundamental frequency is...
form:
where
and this expression is substituted into the wave equation, the time-independent form of the wave equation may be derived, also known as the
Helmholtz equationThe Helmholtz equation, named for Hermann von Helmholtz, is the elliptic partial differential equationwhere ∇2 is the Laplacian, k is the wavenumber, and A is the amplitude.-Motivation and uses:...
:
where
is the wave number,
j is the
imaginary unitIn mathematics, physics, and engineering, the imaginary unit is denoted by i or the Latin j or the Greek iota...
, and ψ(
r) is the time-independent,
complex-valuedA complex number, in mathematics, is a number comprising a real number and an imaginary number; it can be written in the form a + bi, where a and b are real numbers, and i is the standard imaginary unit, having the property that i
2 = −1...
component of the propagating wave. Note that the propagation constant, k, and the frequency, , are linearly related to one another, a typical characteristic of transverse electromagnetic (TEM) waves.
Paraxial plane waves (Optic axis is assumed z-directed)
As will be shown rigorously in the following section, an elementary product solution to this equation takes the form:
where
is the
wave vectorA wave vector is a vector representation of a wave. The wave vector has magnitude indicating wavenumber , and the direction of the vector indicates the direction of wave propagation....
, and
is the wave number. Next, using the
paraxial approximationIn geometric optics, the paraxial approximation is an approximation used in ray tracing of light through an optical system .A paraxial ray is a ray which makes a small angle to the optical axis of the system, and lies close to the axis throughout the system...
, it is assumed that
or equivalently,
where θ is the angle between the wave vector
k and the z-axis.
As a result,
and
The paraxial wave equation
Substituting this expression into the Helmholtz equation, the paraxial wave equation is derived:
where
is the transverse Laplacian operator, shown here in Cartesian coordinates.
The plane wave spectrum: the foundation of Fourier optics
The
plane waveIn the physics of wave propagation, a plane wave is a constant-frequency wave whose wavefronts are infinite parallel planes of constant amplitude normal to the phase velocity vector....
spectrum concept is the basic foundation of Fourier Optics. The plane wave spectrum is a continuous spectrum of
uniform plane waves, and there is one plane wave component in the spectrum for every tangent point on the far-field phase front. The amplitude of that plane wave component would be the amplitude of the optical field at that tangent point. Again, this is true only in the far field, defined as: Range = 2 D
2 / λ where D is the maximum linear extent of the optical sources and λ is the wavelength (Scott [1998]). The plane wave spectrum is often regarded as being discrete for certain types of periodic gratings, though in reality, the spectra from gratings are continuous as well, since no physical device can have the infinite extent required to produce a true line spectrum.
The plane wave component propagating parallel to the optic axis is analogous to the DC component of an electrical signal. Bandwidth in electrical signals relates to the difference between the highest and lowest frequencies present in the spectrum of the signal. For
optical systems, bandwidth is a measure of how far a plane wave is tilted away from the optic axis, so for this reason, this type of bandwidth is often referred to as angular bandwidth or spatial bandwidth. It takes more frequency bandwidth to produce a short pulse in an electrical circuit, and more angular (or, spatial frequency) bandwidth to produce a sharp spot in an optical system (see discussion related to
Point spread functionThe point spread function describes the response of an imaging system to a point source or point object. A more general term for the PSF is a system's impulse response, the PSF being the impulse response of a focused optical system. The PSF in many contexts can be thought of as the extended blob...
).
The plane wave spectrum arises naturally as the
eigenfunctionIn mathematics, an eigenfunction of a linear operator, A, defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. More precisely, one has...
solution to the homogeneous
electromagnetic wave equationThe electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. The homogeneous form of the equation, written in terms of either the electric field E or the magnetic field B, takes the...
in rectangular coordinates (see also
Electromagnetic radiationElectromagnetic radiation is a ubiquitous phenomenon that takes the form of self-propagating waves in a vacuum or in matter. It consists of electric and magnetic field components which oscillate in phase perpendicular to each other and perpendicular to the direction of energy propagation...
, which derives the wave equation from Maxwell's equations in source-free media, or Scott [1998]). In the
frequency domainIn electronics, control systems engineering, and statistics frequency domain is a term used to describe the analysis of mathematical functions or signals with respect to frequency, rather than time....
, the homogeneous electromagnetic wave equation in the frequency domain, or the
Helmholtz equationThe Helmholtz equation, named for Hermann von Helmholtz, is the elliptic partial differential equationwhere ∇2 is the Laplacian, k is the wavenumber, and A is the amplitude.-Motivation and uses:...
, assumes the form:
where
u =
x,
y,
z and
k = 2π/λ, the
wavenumberWavenumber in most physical sciences is a wave property inversely related to wavelength, having SI units of reciprocal meters . Wavenumber is the spatial analog of frequency, that is, it is the measurement of the number of wavelengths per unit distance, or more commonly times that, or the number...
of the medium.
EigenfunctionIn mathematics, an eigenfunction of a linear operator, A, defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. More precisely, one has...
solutions
In the case of differential equations, as in the case of matrix equations, whenever the right-hand side of an equation is zero (i.e., the forcing function / forcing vector is zero), the equation may still admit a non-trivial solution, known as an
eigenfunctionIn mathematics, an eigenfunction of a linear operator, A, defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. More precisely, one has...
solution. In matrix algebra, this occurs when the determinant of the matrix is zero, i.e when the matrix has no inverse. There is a similarity between eqn. (2.0) above, which may be written
and the usual equation for the
eigenvalues/eigenvectorsIn mathematics, eigenvalue, eigenvector, and eigenspace are related concepts in the field of linear algebra. Linear algebra studies linear transformations, which are represented by matrices acting on vectors. Eigenvalues, eigenvectors and eigenspaces are properties of a matrix...
of a square matrix,
A,
Both the scalar Laplacian, : and the matrix,
A are linear operators. 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, it is often the case that the elements of a matrix will be functions of frequency and wavenumber, and the matrix will be non-singular for most combinations of frequency and wavenumber, but will also be singular for certain other 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 the ordinary waveguide, which may admit numerous dispersion relations, each associated with a unique mode of the waveguide. Each propagation mode of the waveguide is known as an
eigenfunctionIn mathematics, an eigenfunction of a linear operator, A, defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. More precisely, one has...
solution (or eigenmode solution) to Maxwell's equations in the waveguide. Free space also admits eigenmode solutions, but with the distinction that for any given frequency, free space admits a continuous modal spectrum, whereas waveguides have a discrete mode spectrum.
Separation of variables
Solutions to the homogeneous wave equation (2.0) may be found in rectangular coordinates by using the principle of
separation of variablesIn mathematics, separation of variables is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation....
for partial differential equations. This principle says that in separable
orthogonal coordinatesIn mathematics, orthogonal coordinates are defined as a set of d coordinates q = in which the coordinate surfaces all meet at right angles...
, an
elementary product solution may be constructed to this wave equation of the following form:
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 this
elementary product solution is plugged into the wave equation, using the scalar Laplacian (aka,
Laplace operatorIn mathematics and physics, the Laplace operator or Laplacian, named after Pierre-Simon de Laplace, is a differential operator, specifically an important case of an elliptic operator, with many applications. It is denoted by the symbols Δ, ∇2, or ∇·∇. In physics, it is used in the...
) in rectangular coordinates
this is obtained
which may be rearranged into the form:
It may now be argued 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. The constant is referred to as -
kx². Reasoning in a similar way for the
y and
z quotients, three ordinary differential equations are obtained for the
fx,
fy and
fz, along with one
separation condition:
Each of these 3 differential equations has the same solution, a complex exponential, so that the elementary product solution for
Eu is:
-
-
-
-
which represents a propagating or exponentially decaying uniform 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 e
jωt 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, the root with a non-positive imaginary part must always be chosen, to represent uniform propagation or decay, but not amplification).
The superposition integral
A general solution to the homogeneous electromagnetic wave equation in rectangular coordinates is formed as a weighted superposition of all possible elementary plane wave solutions as:
where the integrals extend from minus infinity to infinity.
This plane wave spectrum representation of the electromagnetic field is the basic foundation of Fourier Optics (this point cannot be emphasized strongly enough), because when
z=0, the equation above simply becomes a
Fourier transform (FT) relationship between the field and its plane wave content (hence the name, "Fourier optics").
All spatial dependence of the individual plane wave components is described explicitly via the exponential functions. The coefficients of the exponentials are only functions of spatial wavenumber
kx,
ky, just as in ordinary Fourier analysis and
Fourier transformIn mathematics, Fourier analysis is a subject area which grew out of the study of Fourier series. The subject began with trying to understand when it was possible to represent general functions by sums of simpler trigonometric functions...
s.
Free space as a low-pass filter
When
the plane waves are evanescent (decaying), so that any spatial frequency content in an object plane transparency which is finer that one wavelength will not be transferred over to the image plane, simply because the plane waves corresponding to that content cannot propagate. In connection with lithography of electronic components, this phenomenon is known as the diffraction limit and is the reason why light of progressively higher frequency (smaller wavelength) is required for etching progressively finer features in integrated circuits.
The far field approximation and the concept of angular bandwidth
The equation above may be evaluated asymptotically in the far field (using
stationary phase approximationIn mathematics, the stationary phase approximation is a basic principle of asymptotic analysis, applying to oscillatory integralstaken over n-dimensional space Rn where the i = √−1. Here f and g are real-valued smooth functions. The role of g is to ensure convergence; that...
) to show that the field at the point (
x,
y,
z) is indeed due solely to the plane wave component (
kx,
ky,
kz) which propagates parallel to the vector (
x,
y,
z), and whose plane is tangent to the phasefront at (
x,
y,
z). The mathematical details of this process may be found in Scott [1998] or Scott [1990]. The result of performing a stationary phase integration on the expression above is the following expression,
which clearly indicates that the field at (x,y,z) is directly proportional to the spectral component in the direction of (x,y,z), where,
and
Stated another way, the radiation pattern of any planar field distribution is the FT of that source distribution (see
Huygens-Fresnel principleThe Huygens–Fresnel principle is a method of analysis applied to problems of wave propagation...
, wherein the same equation is developed using a
Green's functionIn mathematics, a Green's function is a type of function used to solve inhomogeneous differential equations subject to boundary conditions. The term is also used in physics, specifically in quantum field theory, electrodynamics and statistical field theory, to refer to various types of correlation...
approach). Note that this is NOT a plane wave, as many might think. The radial dependence is a spherical wave - both in magnitude and phase - whose local amplitude is the FT of the source plane distribution at that far field angle. The plane wave spectrum has nothing to do with saying that the field behaves something like a plane wave for far distances.
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 section 5.1.3 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.
K-Space
The separation condition,
which so closely resembles the equation for the Euclidian norm of a vector in terms of its rectangular components, suggests the notion of k-vector, or
wave vectorA wave vector is a vector representation of a wave. The wave vector has magnitude indicating wavenumber , and the direction of the vector indicates the direction of wave propagation....
, defined (for propagating plane waves) in rectangular coordinates as
and in the
spherical coordinate systemIn mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance from a fixed origin, the elevation angle of that point from a fixed plane, and the azimuth angle of its orthogonal...
as
We'll make use of these spherical coordinate system relations in the next section.
The two-dimensional Fourier transform pairs
Analysis Equation:
Synthesis Equation:
note: the normalizing factor of : is present whenever angular frequency (radians) is used, but not when ordinary frequency (cycles) is used.
Optical systems: General overview and analogy with electrical signal processing systems
An optical system consists of an input plane, and output plane, and a set of components that transforms the image
f formed at the input into a different image
g formed at the output. The output image is related to the input image by convolving the input image with the optical impulse response,
h (known as the
point-spread function, for focused optical systems). The impulse response uniquely defines the input-output behavior of the optical system. By convention, the optic axis of the system is taken as the
z-axis. As a result, the two images and the impulse response are all functions of the transverse coordinates,
x and
y.
The impulse response of an optical imaging system is the output plane field which is produced when an ideal mathematical point source of light is placed in the input plane (usually on-axis). In practice, it is not necessary to have an ideal point source in order to determine an exact impulse response. This is because any source bandwidth which lies outside the bandwidth of the system won't matter anyway, so therefore it's not necessary in determining the impulse response. The source only needs to have as much (angular) bandwidth as the optical system.
Optical systems typically fall into one of two different categories. The first is the ordinary focused optical imaging system, wherein the input plane is called the object plane and the output plane is called the image plane. The field in the image plane is desired to be a high-quality reproduction of the field in the object plane. In this case, the impulse response of the optical system is desired to approximate a 2D delta function, at the same location (or a scaled location) in the output plane corresponding to the location of the impulse in the input plane. The
actual impulse response typically resembles an Airy function, whose radius is on the order of the wavelength of the light used. In this case, the impulse response is typically referred to as a
point spread functionThe point spread function describes the response of an imaging system to a point source or point object. A more general term for the PSF is a system's impulse response, the PSF being the impulse response of a focused optical system. The PSF in many contexts can be thought of as the extended blob...
, since the mathematical point of light in the object plane has been spread out into an Airy function in the image plane.
The second type is the optical image processing system, in which a significant feature in the input plane field is to be located and isolated. In this case, the impulse response of the system is desired to be a close replica (picture) of that feature which is being searched for in the in the input plane field, so that a convolution of the feature against the input plane field will produce a bright spot at the feature location in the output plane.
Input plane
The input plane is defined as the locus of all points such that
z = 0. The input image
f is therefore
Output plane
The out plane is defined as the locus of all points such that
z =
d. The output image
g is therefore
The 2D convolution of input function against the impulse response function
i.e.,
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
comaIn optics , the coma in an optical system refers to aberration inherent to certain optical designs or due to imperfection in the lens or other components which results in off-axis point sources such as stars appearing distorted. Specifically, coma is defined as a variation in magnification over...
in focused imaging systems). However, high quality optical systems are often "shift invariant enough" over certain regions of the input plane that the equation above can be used with impunity for certain calculations.
Also, this equation assumes unit magnification. If magnification is present, then eqn. (4.1) becomes
which basically translates the impulse response function, h
M, from x' to x=Mx'. In (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
causalityCausality is the relationship between an event and a second event , where the second event is a direct consequence of the first....
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 esponse requires representing the input signal as a weighted superposition over a train of impulse functions by using the
sifting property of
Dirac delta functionThe Dirac delta or Dirac's delta is a mathematical construct introduced by theoretical physicist Paul Dirac. Informally, it is a generalized function representing an infinitely sharp peak bounding unit area: a 'function' δ that has the value zero everywhere except at x = 0 where its value is...
s.
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 is then simplified to a single delta function input, which would be the
impulse response of the system, h(t - t'). Thus, the output of the linear system to a general input function
f(
t) is shown to be:
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 and 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 principleThe Huygens–Fresnel principle is a method of analysis applied to problems of wave propagation...
, or Stratton-Chu formulation, wherein the "impulse response" is referred to as the
Green's functionIn mathematics, a Green's function is a type of function used to solve inhomogeneous differential equations subject to boundary conditions. The term is also used in physics, specifically in quantum field theory, electrodynamics and statistical field theory, to refer to various types of correlation...
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:
where
is the spectrum of the output signal
is the system transfer function
is the spectrum of the input signal
In like fashion, (4.1) may be Fourier transformed to yield:
Once again it may be noted from the discussion on the
Abbe sine conditionThe Abbe sine condition is a condition that must be fulfilled by a lens or other optical system in order for it to produce sharp images of off-axis as well as on-axis objects...
, that this equation assumes unit magnification.
This equation takes on its real meaning when the Fourier transform, : is associated with the coefficient of the plane wave whose transverse wavenumbers are: :. 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.
Applications of Fourier optics
Fourier optics is used in the field of optical information processing, the staple of which is the classical 4F processor.
The
Fourier transformIn mathematics, Fourier analysis is a subject area which grew out of the study of Fourier series. The subject began with trying to understand when it was possible to represent general functions by sums of simpler trigonometric functions...
properties of a
lensA lens is an optical device with perfect or approximate axial symmetry which transmits and refracts light, converging or diverging the beam. A simple lens is a lens consisting of a single optical element...
provide numerous applications in optical signal processing such as spatial filtering, optical correlation and computer generated holograms.
Fourier optical theory is used in
interferometryInterferometry is the technique of diagnosing the properties of two or more waves by studying the pattern of interference created by their superposition. The instrument used to interfere the waves together is called an interferometer...
,
optical tweezersAn optical tweezer is a scientific instrument that uses a focused laser beam to provide an attractive or repulsive force , depending on the refractive index mismatch to physically hold and move microscopic dielectric objects...
,
atom trapsA magnetic trap uses a magnetic gradient to trap neutral particles with a magnetic moment. Although such traps have been employed for many purposes in physics research, they are most well-known for being the last stage in cooling atoms to achieve Bose-Einstein condensation...
, and quantum computing. Concepts of Fourier optics are used to reconstruct the
phaseThe phase of an oscillation or wave is the fraction of a complete cycle corresponding to an offset in the displacement from a specified reference point at time t = 0. Phase is a frequency domain or Fourier transform domain concept, and as such, can be readily understood in terms of simple harmonic...
of light intensity in the spatial frequency plane (see
adaptive-additive algorithmIn the studies of Fourier optics, sound synthesis, stellar interferometry, optical tweezers, and diffractive optical elements it is often important to know the spatial frequency phase of an observed wave source. In order to reconstruct this phase the Adaptive-Additive Algorithm , which derives...
).
Fourier transforming property of lenses
If a transmissive object is placed one focal length in front of a
lensA lens is an optical device with perfect or approximate axial symmetry which transmits and refracts light, converging or diverging the beam. A simple lens is a lens consisting of a single optical element...
, then its
Fourier transformIn mathematics, Fourier analysis is a subject area which grew out of the study of Fourier series. The subject began with trying to understand when it was possible to represent general functions by sums of simpler trigonometric functions...
will be formed one focal length behind the lens. Consider the figure to the right (click to enlarge)
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 approximationIn geometric optics, the paraxial approximation is an approximation used in ray tracing of light through an optical system .A paraxial ray is a ray which makes a small angle to the optical axis of the system, and lies close to the axis throughout the system...
), so that
and
and
In the figure, the
plane wave phase, moving horizontally from the front focal plane to the lens plane, is
and the
spherical wave phase from the lens to the spot in the back focal plane is:
and the sum of the two path lengths is
f (1 + θ
2/2 + 1 - θ
2/2) = 2
f 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 functionThe point spread function describes the response of an imaging system to a point source or point object. A more general term for the PSF is a system's impulse response, the PSF being the impulse response of a focused optical system. The PSF in many contexts can be thought of as the extended blob...
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 transformIn mathematics, Fourier analysis is a subject area which grew out of the study of Fourier series. The subject began with trying to understand when it was possible to represent general functions by sums of simpler trigonometric functions...
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 . / ns, so if a lens has a . focal length, an entire 2D FT can be computed in about 2 ns (2 x 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 new supercomputers such as the petaflop IBM Roadrunner 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.,
(for all
kx,
ky within the spatial bandwidth of the image, so that
kz 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.
Object truncation and Gibbs phenomenon
Spatially modulated electric field, shown on the left-hand side of eqn. (2.1), typically only occupies a finite (usually rectangular) aperture in the x,y plane. The rectangular aperture function acts like a 2D square-top pulse function, where the field is assumed to be zero outside this 2D rectangle. The spatial domain integrals for calculating the FT coefficients on the right-hand side of eqn. (2.1) are truncated at the boundary of this aperture. This step truncation can introduce inaccuracies in both theoretical calculations and measured values of the plane wave coefficients on the RHS of eqn. (2.1).
Whenever a function is discontinuously truncated in one FT domain, broadening and rippling are introduced in the other FT domain. A perfect example from optics is in connection with the
Point spread functionThe point spread function describes the response of an imaging system to a point source or point object. A more general term for the PSF is a system's impulse response, the PSF being the impulse response of a focused optical system. The PSF in many contexts can be thought of as the extended blob...
, which for on-axis plane wave illumination of a quadratic lens (with circular aperture), is an Airy function,
J1(
x)/
x. Literally, the point source has been "spread out" (with ripples added), to form the Airy point spread function (as the result of truncation of the plane wave spectrum by the finite aperture of the lens). This source of error is known as
Gibbs phenomenonIn mathematics, the Gibbs phenomenon, named after the American physicist J. Willard Gibbs, is the peculiar manner in which the Fourier series of a piecewise continuously differentiable periodic function behaves at a jump discontinuity: the nth partial sum of the Fourier series has large...
and it may be mitigated by simply ensuring that all significant content lies near the center of the transparency, or through the use of
window functionIn signal processing, a window function is a function that is zero-valued outside of some chosen interval. For instance, a function that is constant inside the interval and zero elsewhere is called a rectangular window, which describes the shape of its graphical representation...
s which smoothly taper the field to zero at the frame boundaries. By the convolution theorem, the FT of an arbitrary transparency function - multiplied (or truncated) by an aperture function - is equal to the FT of the non-truncated transparency function convolved against the FT of the aperture function, which in this case becomes a type of "Greens function" or "impulse response function" in the spectral domain. Therefore, the image of a circular lens is equal to the object plane function convolved against the Airy function (the FT of a circular aperture function is
J1(
x)/
x and the FT of a rectangular aperture function is a product of sinc functions, sin
x/
x).
Fourier analysis and functional decomposition
Even though the input transparency only occupies a finite portion of the
x-
y plane (Plane 1), the uniform plane waves comprising the plane wave spectrum occupy the
entire x-
y plane, which is why (for this purpose) only the longitudinal plane wave phase (in the
z-direction, from Plane 1 to Plane 2) must be considered, and not the phase transverse to the
z-direction. It is of course, very tempting to think that if a plane wave emanating from the finite aperture of the transparency is tilted too far from horizontal, it will somehow "miss" the lens altogether but again, since the uniform plane wave extends infinitely far in all directions in the transverse (
x-
y) plane, the planar wave components cannot miss the lens.
This issue brings up perhaps the predominant difficulty with Fourier analysis, namely that the input plane function, defined over a finite support (i.e., over its own finite aperture), is being approximated with other functions (sinusiods) which have infinite support (
i.
e., they are defined over the entire infinite
x-
y plane). This is unbelievably inefficient computationally, and is the principal reason why
waveletLoosely speaking, a wavelet is a wave-like oscillation with an amplitude that starts out at zero, increases, and then decreases back to zero. It can typically be visualized as a "brief oscillation" like one might see recorded by a seismograph or heart monitor. Generally, wavelets are purposefully...
s were conceived, that is to represent a function (defiined on a finite interval or area) in terms of oscillatory functions which are also defined over finite intervals or areas. Thus, instead of getting the frequency content of the entire image all at once (along with the frequency content of the entire rest of the
x-
y plane, over which the image has zero value), the result is instead the frequency content of different parts of the image, which is usually much simpler. Unfortunately, wavelets in the
x-
y plane don't correspond to any known type of propagating wave function, in the same way that Fourier's sinusoids (in the
x-
y plane) correspond to plane wave functions in three dimensions. However the FTs of most wavelets are well known and could possibly be shown to be equivalent to some useful type of propagating field.
On the other hand,
Sinc functionIn mathematics, the sinc function, denoted by sinc and sometimes as Sa, has two definitions. In digital signal processing and information theory, the normalized sinc function is commonly defined by...
s and Airy functions - which are not only the point spread functions of rectangular and circular apertures, respectively, but are also cardinal functions commonly used for functional decomposition in interpolation/sampling theory [Scott 1990] -
do correspond to converging or diverging spherical waves, and therefore could potentially be implemented as a whole new functional decomposition of the object plane function, thereby leading to another point of view similar in nature to Fourier optics. This would basically be the same as conventional ray optics, but with diffraction effects included. In this case, each point spread function would be a type of "smooth pixel," in much the same way that a soliton on a fiber is a "smooth pulse."
Perhaps a lens figure-of-merit in this "point spread function" viewpoint would be to ask how well a lens transforms an Airy function in the object plane into an Airy function in the image plane, as a function of radial distance from the optic axis, or as a function of the size of the object plane Airy function. This is kind of like the
Point spread functionThe point spread function describes the response of an imaging system to a point source or point object. A more general term for the PSF is a system's impulse response, the PSF being the impulse response of a focused optical system. The PSF in many contexts can be thought of as the extended blob...
, except now we're really looking at it as a kind of input-to-output plane transfer function (like MTF), and not so much in absolute terms, relative to a perfect point. Similarly, Gaussian wavelets, which would correspond to the waist of a propagating Gaussian beam, could also potentially be used in still another functional decomposition of the object plane field.
Far-field range and the 2D2 / λ criterion
In the figure above, illustrating the Fourier transforming property of lenses, the lens is in the near field of the object plane transparency, therefore the object plane field may be regarded as being a superposition of plane waves, each one of which propagates to the lens. This is defined via the far-field criterion as: Range = 2
D2 / λ where
D is the maximum linear extent of the optical sources and λ is the wavelength (Scott [1998]).
D of the transparency is on the order of cm (10
-2 m) and the wavelength of light is on the order of 10
-6 m, therefore
D/λ is on the order of 10
4. This times
D is on the order of 10
2 m, or hundreds of meters. On the other hand the far field distance from a PSF spot is on the order of λ. This is because D for the spot is on the order of λ, so that
D/λ is on the order of unity. One times
D (i.e., λ) is on the order of λ (10
-6 m).
Since the lens is in the far field of any PSF spot, the field incident on the lens from the spot may be regarded as being an asymptotic spherical wave, as in eqn. (2.2), not as a plane wave spectrum, as in eqn. (2.1). On the other hand, the lens is in the near field of the entire input plane transparency, therefore eqn. (2.1) - the full plane wave spectrum - accurately represents the field incident on the lens from that larger, extended source.
Lens as a low-pass filter
A lens is basically a low-pass plane wave filter (see
Low-pass filterA low-pass filter is a filter that passes low-frequency signals but attenuates signals with frequencies higher than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a high-cut filter, or treble cut filter when used in...
). Consider a "small" light source located on-axis in the object plane of the lens. It is assumed that the source is small enough that, by the far-field criterion, the lens is in the far field of the "small" source. Then, the field radiated by the small source is a spherical wave which is modulated by the FT of the source distribution, as in eqn. (2.2), Then, the lens passes - from the object plane over onto the image plane - only that portion of the radiated spherical wave which lies inside the edge angle of the lens. In this far-field case, truncation of the radiated spherical wave is equivalent to truncation of the plane wave spectrum of the small source. So, the plane wave components in this far-field spherical wave, which lie beyond the edge angle of the lens, are not captured by the lens and are not transferred over to the image plane. Note: this logic is valid only for small sources, such that the lens is in the far field region of the source, according to the 2
D2 / λ criterion mentioned previously. If an object plane transparency is imagined as a summation over small sources (as in the Whittaker-Shannon interpolation formula, Scott [1990]), each of which has its spectrum truncated in this fashion, then every point of the entire object plane transparency suffers the same effects of this low pass filtering.
Loss of the high (spatial) frequency content causes blurring and loss of sharpness (see discussion related to
Point spread functionThe point spread function describes the response of an imaging system to a point source or point object. A more general term for the PSF is a system's impulse response, the PSF being the impulse response of a focused optical system. The PSF in many contexts can be thought of as the extended blob...
). Bandwidth truncation causes a (fictitious, mathematical, ideal) point source in the object plane to be blurred (or, spread out) in the image plane, giving rise to the term, "point spread function." Whenever bandwidth is expanded or contracted, image size is typically contracted or expanded accordingly, in such a way that the space-bandwidth product remains constant, by Heisenberg's principle (Scott [1998] and
Abbe sine conditionThe Abbe sine condition is a condition that must be fulfilled by a lens or other optical system in order for it to produce sharp images of off-axis as well as on-axis objects...
).
Coherence and Fourier transforming
While working in the frequency domain, with an assumed e
jωt (engineering) time dependence, coherent (laser) light is implicitly assumed, which has a delta function dependence in the frequency domain. Light at different (delta function) frequencies will "spray" tha plane wave spectrum out at different angles, and as a result these plane wave components will be focused at different places in the output plane. The Fourier transforming property of lenses works best with coherent light, unless there is some special reason to combine light of different frequencies, to achieve some special purpose.
4F Correlator
When the plane wave spectrum representation of the electric field (
section 2) is combined with the Fourier transforming property of quadratic lenses (
section 5.1), it leads naturally to the development of numerous 2D image processing devices (
this section). One of the primary applications of Fourier Optics is in the mathematical operations of
cross-correlationIn signal processing, cross-correlation is a measure of similarity of two waveforms as a function of a time-lag applied to one of them. This is also known as a sliding dot product or inner-product. It is commonly used to search a long duration signal for a shorter, known feature...
and
convolutionIn mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...
, and these have 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 theoremIn mathematics, the convolution theorem states that under suitableconditions the Fourier transform of a convolution is the pointwise product of Fourier transforms. In other words, convolution in one domain equals point-wise multiplication in the other domain...
from
Fourier transformIn mathematics, Fourier analysis is a subject area which grew out of the study of Fourier series. The subject began with trying to understand when it was possible to represent general functions by sums of simpler trigonometric functions...
theory, which states that
convolutionIn mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...
in the spatial (
x,
y) domain is equivalent to direct multiplication in the spatial frequency (
kx,
ky) 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(
kx,
ky) x
G(
kx,
ky). 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
convolutionIn mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...
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
section 4 above. The FT plane mask function,
G(
kx,
ky) is the system transfer function of the correlator, which we'd in general denote as
H(
kx,
ky), 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(
kx,
ky) is directly multiplied against the spectrum
F(
kx,
ky) of the input function, to produce the spectrum of the output function. This is how electrical signal processing systems operate on 1D temporal signals.
Afterword: Plane Wave Spectrum Within the Broader Context of Functional Decomposition
Electrical fields are really just particular types of mathematical functions and, as such, may often be represented in many different ways. In the
Huygens-FresnelThe Huygens–Fresnel principle is a method of analysis applied to problems of wave propagation...
or Stratton-Chu viewpoints, the electric field is represented as a superposition of point sources, each one of which gives rise to a
Green's functionIn mathematics, a Green's function is a type of function used to solve inhomogeneous differential equations subject to boundary conditions. The term is also used in physics, specifically in quantum field theory, electrodynamics and statistical field theory, to refer to various types of correlation...
field. The total field is then the weighted sum of all of the individual Greens function fields. That seems to be the most natural way of viewing the electric field for most people - no doubt because most of us have, at one time or another, drawn out the circles with protractor and paper, much the same way Thomas Young did in his class paper on the
Double-slit experimentIn quantum mechanics, the double-slit experiment demonstrates the inseparability of the wave and particle natures of light and other quantum particles. A coherent light source illuminates a thin plate with two parallel slits cut in it, and the light passing through the slits strikes a screen...
. However, it is by no means the only way to represent the electric field, which may also be represented as a spectrum of sinusoidally varying plane waves. In addition, Frits Zernike proposed still another
functional decompositionFunctional decomposition refers broadly to the process of resolving a functional relationship into its constituent parts in such a way that the original function can be reconstructed from those parts by function composition...
based on his
Zernike polynomialsIn mathematics, the Zernike polynomials are a sequence of polynomials that are orthogonal on the unit disk. Named after Frits Zernike, they play an important role in geometrical optics. -Definitions:There are even and odd Zernike polynomials...
, defined on the unit disc. The third-order (and lower) Zernike polynomials correspond to the normal lens aberrations. And still another functional decomposition could be made in terms of
Sinc functionIn mathematics, the sinc function, denoted by sinc and sometimes as Sa, has two definitions. In digital signal processing and information theory, the normalized sinc function is commonly defined by...
s and Airy functions, as in the Whittaker-Shannon interpolation formula and the Nyquist-Shannon sampling theorem. All of these functional decompositions have utility in different circumstances. The optical scientist having access to these various different representational forms has available a richer insight to the nature of these marvelous fields and their properties. Embrace these different ways of looking at the field, rather than viewing them as being in any way conflicting or contradictory.
Functional Decomposition and Eigenfunctions
The twin subjects of
eigenfunctionIn mathematics, an eigenfunction of a linear operator, A, defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. More precisely, one has...
expansions and
functional decompositionFunctional decomposition refers broadly to the process of resolving a functional relationship into its constituent parts in such a way that the original function can be reconstructed from those parts by function composition...
, both briefly alluded to in this Wikipedia article, 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 functionsIn mathematics, two functions and are called orthogonal if their inner product is zero. Whether or not two particular functions are orthogonal depends on how their inner product has been defined. A typical definition of an inner product for functions is...
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.
See also
- Abbe sine condition
The Abbe sine condition is a condition that must be fulfilled by a lens or other optical system in order for it to produce sharp images of off-axis as well as on-axis objects...
- Huygens-Fresnel Principle
The Huygens–Fresnel principle is a method of analysis applied to problems of wave propagation...
- Point spread function
The point spread function describes the response of an imaging system to a point source or point object. A more general term for the PSF is a system's impulse response, the PSF being the impulse response of a focused optical system. The PSF in many contexts can be thought of as the extended blob...
- Phase contrast microscopy
Phase contrast microscopy is an optical microscopy illumination technique in which small phase shifts in the light passing through a transparent specimen are converted into amplitude or contrast changes in the image....
- Fraunhofer diffraction
In optics, Fraunhofer diffraction , or far-field diffraction, is a form of wave diffraction that occurs when field waves are passed through an aperture or slit causing only the size of an observed aperture image to change due to the far-field location of observation and the increasingly planar...
- Fresnel diffraction
In optics, Fresnel diffraction or near-field diffraction is a process of diffraction that occurs when a wave passes through an aperture and diffracts in the near field, causing any diffraction pattern observed to differ in size and shape, depending on the distance between the aperture and the...
- Adaptive-additive algorithm
In the studies of Fourier optics, sound synthesis, stellar interferometry, optical tweezers, and diffractive optical elements it is often important to know the spatial frequency phase of an observed wave source. In order to reconstruct this phase the Adaptive-Additive Algorithm , which derives...