Present day methodologies for mathematical simulation and computational experiment are generally implemented in electromagnetics through the solution of boundary-value (frequency domain) problems and initial boundary-value (time domain) problems for Maxwell’s equations. Most of the results of this theory concerning open resonators have been obtained by the frequency-domain methods. At the same time, a rich variety of applied problems (analysis of complex electrodynamic structures for the devices of vacuum and solid-state electronics, model synthesis of open dispersive structures for resonant quasi-optics, antenna engineering, and high-power electronics, etc.) can be efficiently solved with the help of more universal time-domain algorithms.
The fact that frequency domain approaches are somewhat limited in such problems is the motivation for this study. Moreover, presently known remedies to the various theoretical difficulties in the theory of non-stationary electromagnetic fields are not always satisfactory for practitioners. Such remedies affect the quality of some model problems and limit the capability of time-domain methods for studying transient and stationary processes. One such difficulty is the appropriate and efficient truncation of the computational domain in so-called open problems, i.e. problems where the computational domain is infinite along one or more spatial coordinates. Also, a number of questions occur when solving far-field problems, and problems involving extended sources or sources located in the far-zone.
In the present work, we address these difficulties for the case of - and -waves in axially-symmetrical open compact resonators with waveguide feed lines. Sections 2 and 3 are devoted to problem definition. In Sections 4 and 5, we derive exact absorbing conditions for outgoing pulsed waves that enable the replacement of an open problem with an equivalent closed one. In Section 6, we obtain the analytical representation for operators that link the near- and far-field impulsive fields for compact axially-symmetrical structures and consider solutions that allow the use of extended or distant sources. In Section 7, we place some accessory results required for numerical implementation of the approach under consideration. All analytical results are presented in a form that is suitable for using in the finite-difference method on a finite-sized grid and thus is amenable for software implementation. We develop here the approach initiated in the works by Maikov et al. (1986) and Sirenko et al. (2007) and based on the construction of the exact conditions allowing one to reduce an open problem to an equivalent closed one with a bounded domain of analysis. The derived closed problem can then be solved numerically using the standard finite-difference method (Taflove & Hagness, 2000).
In contrast to other well-known approximate methods involving truncation of the computational domain (using, for example, Absorbing Boundary Conditions or Perfectly Matched Layers), our constructed solution is exact, and may be computationally implemented in a way that avoids the problem of unpredictable behavior of computational errors for large observation times. The impact of this approach is most significant in cases of resonant wave scattering, where it results in reliable numerical data.
2. Formulation of the initial boundary-value problem
In Fig. 1, the cross-section of a model for an open axially-symmetrical () resonant structure is shown, where are cylindrical and are spherical coordinates. By we denote perfectly conducting surfaces obtained by rotating the curve about the -axis; is a similarly defined surface across which the relative
permittivity and specific conductivity change step-wise; these quantities are piecewise constant inside and take free space values outside. Here,; is the impedance of free space;, and are the electric and magnetic constants of vacuum.
The two-dimensional initial boundary-value problem describing the pulsed axially-symmetrical - () and - () wave distribution in open structures of this kind is given by
where and are the electric and magnetic field vectors; for -waves and for -waves (Sirenko et al., 2007). The SI system of units is used. The variable which being the product of the real time by the velocity of light in free space has the dimension of length. The operators, will be described in Section 2 and provide an ideal model for fields emitted and absorbed by the waveguides.
The domain of analysis is the part of the half-plane bounded by the contours together with the artificial boundaries (input and output ports) in the virtual waveguides,. The regions and (free space), such that, are separated by the virtual boundary.
The functions, , , , and which are finite in the closure of are supposed to satisfy the hypotheses of the theorem on the unique solvability of problem (1) in the Sobolev space, where is the observation time (Ladyzhenskaya, 1985). The ‘current’ and ‘instantaneous’ sources given by the functions and, as well as all scattering elements given by the functions, and by the contours and are located in the region. In axially-symmetrical problems, at points such that, only or fields components are nonzero. Hence it follows that;, in (1).
3. Exact absorbing conditions for virtual boundaries in input-output waveguides
in (1) give the exact absorbing conditions for the outgoing pulsed waves and traveling into the virtual waveguides and, respectively (Sirenko et al., 2007). is the pulsed wave that excites the axially-symmetrical structure from the circular or coaxial circular waveguide. It is assumed that by the time this wave has not yet reached the boundary.
By using conditions (2), we simplify substantially the model simulating an actual electrodynamic structure: the -domains are excluded from consideration while the operators describe wave transformation on the boundaries that separate regular feeding waveguides from the radiating unit. The operators are constructed such that a wave incident on from the region passes into the virtual domain as if into a regular waveguide – without deformations or reflections. In other words, it is absorbed completely by the boundary. Therefore, we call the boundary conditions (2) as well as the other conditions of this kind ‘exact absorbing conditions’.
In the book (Sirenko et al., 2007), one can find six possible versions of the operators for virtual boundaries in the cross-sections of circular or coaxial-circular waveguides. We pick out two of them (one for the nonlocal conditions and one for the local conditions) and, taking into consideration the location of the boundaries in our problem (in the plane for the boundary and in the plane for) as well as the traveling direction for the waves outgoing through these boundaries (towards for and towards for), write (2) in the form:
(nonlocal absorbing conditions) and
(local absorbing conditions). The initial boundary-value problems involved in (5) and (6) with respect to the auxiliary functions must be supplemented with the following boundary conditions for all times:
(on the boundaries and of the region for a circular waveguide) and
(on the boundaries and of the region for a coaxial waveguide).
In (3) to (8) the following designations are used: is the Bessel function, and are the radii of the waveguide and of its inner conductor respectively (evidently, if only is a coaxial waveguide), and are the sets of transverse functions and transverse eigenvalues for the waveguide.
Analytical representations for and are well-known and for -waves take the form:
For -waves we have:
Here are the Neumann functions. The basis functions satisfy boundary conditions at the ends of the appropriate intervals (or) and the following equalities hold
in the circular or coaxial waveguide, respectively.
4. Exact radiation conditions for outgoing spherical waves and exact absorbing conditions for the artificial boundary in free space
When constructing the exact absorbing condition for the wave crossing the artificial spherical boundary, we will follow the sequence of transformations widely used in the theory of hyperbolic equations (e.g., Borisov, 1996) – incomplete separation of variables in initial boundary-value problems for telegraph or wave equations, integral transformations in the problems for one-dimensional Klein-Gordon equations, solution of the auxiliary boundary-value problems for ordinary differential equations, and inverse integral transforms.
In the domain, where the field propagates freely up to infinity as, the 2-D initial boundary-value problem (1) in spherical coordinates takes the form
Let us represent the solution as. Separation of variables in (14) results in a homogeneous Sturm-Liouville problem with respect to the function
and the following initial boundary-value problem for:
Let us solve the Sturm-Liouville problem (15) with respect to and. Change of variables, yields the following boundary-value problem for:
With for each equation (17) has two nontrivial linearly independent solutions in the form of the associated Legendre functions and. Taking into account the behavior of these functions in the vicinity of their singular points (Bateman & Erdelyi, 1953), we obtain
Here is a complete orthonormal (with weight function) system of functions in the space and provides nontrivial solutions to (15). Therefore, the solution of initial boundary-value problem (14) can be represented as
where the space-time amplitudes are the solutions to problems (16) for.
Now subject it to the integral transform
where the kernel satisfies the equation (Korn & Korn, 1961)
Since the ‘signal’ propagates with a finite velocity, for any we can always point a distance such that the signal has not yet reached it, that is, for these and we have. Then we can rewrite equation (23) in the form
From (24) the simple differential equation for the transforms of the functions follows:
In this equation, the values and entering into are not defined yet. With and, we have
The last integral is the Hankel transform (Korn & Korn, 1961), which is inverse to itself, and
and the symbol ‘’ denotes derivatives with respect to the whole argument.If is a fundamental solution of the operator (i.e.,, where is the Dirac delta function), then the solution to the equation can be written as a convolution (Vladimirov, 1971). For we have, and then
Let us denote
Then from (Gradshteyn & Ryzhik, 2000) we have for
where and are the Legendre functions of the first and second kind, respectively. For, we can rewrite this formula as.
where and denotes a Legendre polynomial. Considering that
(Janke et al., 1960), and, we can derive
Finally, taking into account the relation, we have from (33)
By using (19), we arrive at the desired radiation condition:
By passing to the limit in (40), we obtain
Formula (41) represents the exact absorbing condition on the artificial boundary. This condition is spoken of as exact because any outgoing wave described by the initial problem (1) satisfies this condition. Every outgoing wave passes through the boundary without distortions, as if it is absorbed by the domain or its boundary. That is why this condition is said to be absorbing.
5. On the equivalence of the initial problem and the problem with a bounded domain of analysis
We have constructed the following closed initial boundary-value problem
where the operator is given by (41). It is equivalent to the open initial problem (1). This statement can be proved by following the technique developed in (Ladyzhenskaya, 1985). The initial and the modified problems are equivalent if and only if any solution of the initial problem is a solution to problem (42) and at the same time, any solution of the modified problem is the solution to problem (1). (In the -domain, the solution to the modified problem is constructed with the help of (40).) The solution of the initial problem is unique and it is evidently the solution to the modified problem according construction. In this case, if the solution of (42) is unique, it will be a solution to (1). Assume that problem (42) has two different solutions and. Then the function is also the generalized solution to (42) for. This means that for any function that is zero at, the following equality holds:
Here, and are the space-time cylinder over the domain and its lateral surface; and are the cosines of the angles between the outer normal to the surface and - and -axes, respectively; the element of the end surface of the cylinder equals.
By making the following suitable choice of function,
it is possible to show that every term in (43) is nonnegative (Mikhailov, 1976) and therefore is equal to zero for all and, which means that the solution to the problem (42) is unique. This proves the equivalency of the two problems.
6. Far-field zone problem. Extended and remote sources
As we have already mentioned, in contrast to approximate methods based on the use of the Absorbing Boundary Conditions or Perfectly Matched Layers, our approach to the effective truncation of the computational domain is rigorous, which is to say that the original open problem and the modified closed problem are equivalent. This allows one, in particular, to monitor a computational error and obtain reliable information about resonant wave scattering. It is noteworthy that within the limits of this rigorous approach we also obtain, without any additional effort, the solution to the far-field zone problem, namely, of finding the field at arbitrary point in from the magnitudes of on any arc, , lying entirely in and retaining all characteristics of the arc. Thus in the case considered here, equation (39) defines the diagonal operator such that it operates on the space of amplitudes of the outgoing wave (19) according the rule
and allows one to follow all variations of these amplitudes in an arbitrary region of. The operator
given by (40), in turn, enables the variations of the field, , to be followed.
It is obvious that the efficiency of the numerical algorithm based on (42) reduces if the support of the function and/or the functions and is extended substantially or removed far from the region where the scatterers are located. The arising problem (the far-field zone problem or the problem of extended and remote sources) can be resolved by the following straightforward way.
Let us consider the problem
which differs from the problem (1) only in that the sources and, are located out of the domain enveloping all the scatterers (Fig. 1). The supports of the functions, , and can be arbitrary large (and even unbounded) and are located in at any finite distance from the domain.
Let the relevant sources generate a field in the half-plane. In other words, let the function be a solution of the following Cauchy problem:
and determines there the pulsed electromagnetic wave crossing the artificial boundary in one direction only, namely, from into.
or, in the operator notations, , – the exact absorbing condition allowing one to replace open problem (47) with the equivalent closed problem
7. Determination of the incident fields
To implement the algorithms based on the solution of the closed problems (42), (51), the values of the functions and as well as their normal derivatives on the boundaries and are required (see formulas (3), (5), (50)). Let us start from the function. In the feeding waveguide, the field incoming on the boundary can be represented (Sirenko et al., 2007) as
Here (see also Section 3), only in the case of -waves and only for a coaxial waveguide. In all other cases. On the boundary, the wave can be given by a set of its amplitudes. The choice of the functions, which are nonzero on the finite interval, is arbitrary to a large degree and depends generally upon the conditions of a numerical experiment. As for the set, which determines the derivative of the functiоn on, it should be selected with consideration for the causality principle. Each pair is determined by the pulsed eigenmode propagating in the waveguide in the sense of increasing. This condition is met if the functions and are related by the following equation (Sirenko et al., 2007):
The function generated by the sources, , and is the solution to the Cauchy problem (48). Let us separate the transverse variable in this problem and represent its solution in the form (Korn & Korn, 1961):
In order to find the functions, one has to invert the following Cauchy problems for one-dimensional Klein-Gordon equations:
Here, , и are the amplitude coefficients in the integral presentations (54) for the functions, , and.
In this paper, a problem of efficient truncation of the computational domain in finite-difference methods is discussed for axially-symmetrical open electrodynamic structures. The original problem describing electromagnetic wave scattering on a compact axially-symmetric structure with feeding waveguides is an initial boundary-value problem formulated in an unbounded domain. The exact absorbing conditions have been derived for a spherical artificial boundary enveloping all sources and scatterers in order to truncate the computational domain and replace the original open problem by an equivalent closed one. The constructed solution has been generalized to the case of extended and remote field sources. The analytical representation for the operators converting the near-zone fields into the far-zone fields has been also derived.
We would like to make the following observation about our approach.
In our description, the waveguide serves as a feeding waveguide. However, both of the waveguides can be feeding or serve to withdraw the energy; also both of them may be absent in the structure.
The choice of the parameters and determining (see Section 4) affects substantially the final analytical expression for the exact absorbing condition on the spherical boundary. When constructing boundary conditions (41), (50), we assumed that and. In (Sirenko et al., 2007), for a similar situation, the exact absorbing conditions for outgoing pulsed waves were constructed with the assumption that and. With such and, equation (21) is the Weber-Orr transform (Bateman & Erdelyi, 1953). However, the final formulas corresponding to (39), (40) for this case turn into identities as, which present a considerable challenge for using them as absorbing conditions. In addition, the analytical expressions with the use of Weber-Orr transform are rather complicated to implement numerically.
The function (see Section 7) can be found in spherical coordinates as well. In this situation, we arrive (see Section 4) at the expansions like (19) with the amplitude coefficients determined by the Cauchy problems
where, , and are the amplitude coefficients for the functions, , and.
The standard discretization of the closed problems (42), (51) by the finite difference method using a uniform rectangular mesh attached to coordinates leads to explicit computational schemes with uniquely defined mesh functions. The approximation error is, where is the mesh width in spatial coordinates, for or for is the mesh width in time variable;, , and. The range of the integers, , and depends both on the size of the domains and on the length of the interval of the observation time. The condition providing uniform boundedness of the approximate solutions with decreasing and is met (see, for example, formula (1,50) in (Sirenko et al., 2007)). Hence the finite-difference computational schemes are stable, and the mesh functions converge to the solutions of the original problems (42), (51).
As opposed to the well-known approximate boundary conditions standardly utilized by finite-difference methods, the conditions derived in this paper are exact by construction and do not introduce an additional error into the finite-difference algorithm. This advantage is especially valuable in resonant situations, where numerical simulation requires large running time and the computational errors may grow unpredictably if an open problem is replaced by an insufficiently accurate closed problem.