The mathematical details of this process may be found in Scott [] or Scott []. The result of performing a stationary phase integration on the expression above is the following expression,. Stated another way, the radiation pattern of any planar field distribution is the FT of that source distribution see Huygens—Fresnel principle , wherein the same equation is developed using a Green's function approach. Note that this is NOT a plane wave. The plane wave spectrum has nothing to do with saying that the field behaves something like a plane wave for far distances.

Equation 2. 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.

### Scattering wave energy propagation in a random isotropic scattering medium: 1. Theory

See section 5. 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 for calculating the far zone electric field which does not involve stationary phase integration. They have devised a concept known as "fictitious magnetic currents" usually denoted by M , and defined as. 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 []. 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. Fourier optics is somewhat different from ordinary ray optics typically used in the analysis and design of focused imaging systems such as cameras, telescopes and microscopes. Ray optics is the very first type of optics most of us encounter in our lives; it's simple to conceptualize and understand, and works very well in gaining a baseline understanding of common optical devices.

Unfortunately, ray optics does not explain the operation of Fourier optical systems, which are in general not focused systems. Ray optics is a subset of wave optics in the jargon, it is "the asymptotic zero-wavelength limit" of wave optics and therefore has limited applicability. We have to know when it is valid and when it is not - and this is one of those times when it is not.

For our current task, we must expand our understanding of optical phenomena to encompass wave optics, in which the optical field is seen as a solution to Maxwell's equations. This more general wave optics accurately explains the operation of Fourier optics devices.

In this section, we won't go all the way back to Maxwell's equations, but will start instead with the homogeneous Helmholtz equation valid in source-free media , which is one level of refinement up from Maxwell's equations Scott []. From this equation, we'll show how infinite uniform plane waves comprise one field solution out of many possible in free space. These uniform plane waves form the basis for understanding Fourier optics.

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The plane wave 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. 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.

As in the case of electrical signals, bandwidth is a measure of how finely detailed an image is; the finer the detail, the greater the bandwidth required to represent it. 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 also relates to spatial frequency content spatial bandwidth , but it also has a secondary meaning. It also measures how far from the optic axis the corresponding plane waves are tilted, and so this type of bandwidth is often referred to also as angular 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 function.

The plane wave spectrum arises naturally as the eigenfunction or "natural mode" solution to the homogeneous electromagnetic wave equation in rectangular coordinates see also Electromagnetic radiation , which derives the wave equation from Maxwell's equations in source-free media, or Scott []. In the case of differential equations, as in the case of matrix equations, whenever the right-hand side of an equation is zero i. 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.

## Fractional Fourier analysis of elastic wave scattering in inhomogeneous materials

Infinite homogeneous media admit the rectangular, circular and spherical harmonic solutions to the Helmholtz equation, depending on the coordinate system under consideration. The propagating plane waves we'll study in this article are perhaps the simplest type of propagating waves found in any type of media. The interested reader may investigate other functional linear operators which give rise to different kinds of orthogonal eigenfunctions such as Legendre polynomials , Chebyshev polynomials and Hermite polynomials.

In certain physics applications such as in the computation of bands in a periodic volume , it is often the case that the elements of a matrix will be very complicated 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 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 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 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. 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. 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 optical 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 since it cannot even be captured by the optical system , so therefore it's not necessary in determining the impulse response. The source only needs to have at least 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 linearly 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 function , 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 input plane field, so that a convolution of the impulse response an image of the desired feature against the input plane field will produce a bright spot at the feature location in the output plane.

It is this latter type of optical image processing system that is the subject of this section. Section 5. The input image f is therefore. The output image g is therefore. 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 []. 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. The extension to two dimensions is trivial, except for the difference that causality exists in the time domain, but not in the spatial domain.

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 shifting property of Dirac delta functions. 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. 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. 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. 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 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 interferometry , 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. 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. 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.

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The various plane wave components propagate at different tilt angles with respect to the optic axis of the lens i. The finer the features in the transparency, the broader the angular bandwidth of the plane wave spectrum. In the figure, the plane wave phase, moving horizontally from the front focal plane to the lens plane, is. 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. If the focal length is 1 in. 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. 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. Use the link below to share a full-text version of this article with your friends and colleagues.

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In this paper we provide a complete formulation of scattered wave energy propagation in a random isotropic scattering medium. First, we formulate the scattered wave energy equation by extending the stationary energy transport theory studied by Wu to the time dependent case. The iterative solution of this equation gives us a general expression of temporal variation of scattered energy density at arbitrary source and receiver locations as a Neumann series expansion characterized by powers of the scattering coefficient.

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Solving the scattered wave energy equation using a Fourier transform technique, we obtain a compact integral solution for the temporal decay of scattered wave energy which includes all multiple scattering contributions and can be easily computed numerically. Examples of this solution are presented and compared with that of the single scattering, energy flux, and diffusion models. We then discuss the energy conservation for our system by starting with our fundamental scattered wave energy equation and then demonstrating that our formulas satisfy the energy conservation when the contributions from all orders of scattering are summed up.

We also generalize our scattered wave energy equations to the case of nonuniformly distributed isotropic scattering and absorption coefficients. To solve these equations, feasible numerical procedures, such as a Monte Carlo simulation scheme, are suggested. Our Monte Carlo approach to solve the wave energy equation is different from previous works Gusev and Abubakirov, ; Hoshiba, based on the ray theoretical approach.

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## Wave Scattering Theory

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