Title of Invention: SYSTEMS AND METHODS FOR ACOUSTIC MODE CONVERSION Abstract: The present disclosure provides systems and methods associated with mode conversion for ultrasound and acoustic radiation devices. A mode converting structure (holographic metamaterial) is formed with a distribution of acoustic material properties selected to convert an acoustic pressure pattern from a first mode to a second mode to attain a target radiation pattern that is different from the input radiation pattern. A solution to a holographic equation provides a sufficiently accurate approximation of a distribution of acoustic material properties to form a mode converting device. One or more optimization algorithms can be used to improve the efficiency of the mode conversion and generation of the acoustic mode converter. Detailed Description: If an Application Data Sheet (ADS) has been filed on the filing date of this application, it is incorporated by reference herein. Any applications claimed on the ADS for priority under 35 U.S.C. §§ 119, 120, 121, or 365(c), and any and all parent, grandparent, great-grandparent, etc., applications of such applications are also incorporated by reference, including any priority claims made in those applications and any material incorporated by reference, to the extent such subject matter is not inconsistent herewith. CROSS-REFERENCE TO RELATED APPLICATIONS The present application claims the benefit of the earliest available effective filing date(s) from the following listed application(s) (the “Priority Applications”), if any, listed below (e.g., claims earliest available priority dates for other than provisional patent applications or claims benefits under 35 U.S.C. § 119(e) for provisional patent applications, for any and all parent, grandparent, great-grandparent, etc., applications of the Priority Application(s)). In addition, the present application is related to the “Related Applications,” if any, listed below. Priority Applications NONE. Related Applications If the listings of applications provided above are inconsistent with the listings provided via an ADS, it is the intent of the Applicant to claim priority to each application that appears in the Priority Applications section of the ADS and to each application that appears in the Priority Applications section of this application. All subject matter of the Priority Applications and the Related Applications and of any and all parent, grandparent, great-grandparent, etc., applications of the Priority Applications and the Related Applications, including any priority claims, is incorporated herein by reference to the extent such subject matter is not inconsistent herewith. TECHNICAL FIELD This disclosure relates to mode converting structures for acoustic radiation devices. A mode converting structure may include a volumetric distribution of acoustic material properties to covert an input acoustic mode to an output acoustic mode, for a finite frequency range. BRIEF DESCRIPTION OF THE DRAWINGS The patent or application file contains at least one drawing executed in color. Copies of this patent or patent application publication with color drawing(s) will be provided by the Office upon request and payment of the necessary fee. FIG. 1A illustrates an example of a volumetric holographic medium showing real values of a volumetric distribution of refractive indexes for increasing the directionality of a two-dimensionally isotropic line source. FIG. 1B illustrates an example of a holographic solution showing imaginary values of a volumetric distribution of refractive indexes for increasing the directionality of a two-dimensionally isotropic line source. FIG. 2A illustrates an example of a field distribution of a line source surrounded by a holographic metamaterial (mode converting structure) that has a distribution of refractive indexes approximating the distribution shown in FIGS. 1A and 1B. FIG. 2B illustrates an example of a far-field radiation pattern of the line source in FIG. 2A surrounded by the mode converting structure having the distribution of refractive indexes shown in FIGS. 1A and 1B. FIG. 3 illustrates an example of a volumetric distribution of refractive indexes for converting a radiation pattern of a two-dimensional point dipole antenna into a directional beam. FIG. 4A illustrates an example of a field distribution of a two-dimensional point-dipole source surrounded by a mode converting structure that has a distribution of refractive indexes approximating the distribution shown in FIG. 3. FIG. 4B illustrates an example of a far-field radiation pattern of the two-dimensional point dipole source in FIG. 4A surrounded by the mode converting structure having the distribution of refractive indexes shown in FIG. 3. FIG. 5 illustrates an example of an optimized volumetric distribution of refractive indexes generated by optimizing the solution shown in FIG. 1A. FIG. 6A illustrates an example of a field distribution of the two-dimensionally isotropic line source surrounded by a mode converting structure that has a distribution of refractive indexes approximating the distribution shown in FIG. 5. FIG. 6B illustrates an example of a far-field radiation pattern of the two-dimensionally isotropic line source in FIG. 6A surrounded by the mode converting structure having the distribution of refractive indexes shown in FIG. 5. FIG. 7 illustrates an example of an optimized volumetric distribution of refractive indexes generated by optimizing the solution shown in FIG. 3. FIG. 8A illustrates an example of a field distribution of the two-dimensional point dipole source surrounded by a mode converting structure that has a distribution of refractive indexes approximating the distribution shown in FIG. 7. FIG. 8B illustrates an example of a far-field radiation pattern of the two-dimensional point dipole source in FIG. 8A surrounded by the mode converting structure having the distribution of refractive indexes shown in FIG. 7. FIG. 9 illustrates an example of an acoustic mode converting structure converting acoustic radiation from a first mode to a second mode. FIG. 10A illustrates an example of an acoustic pressure intensity pattern of a circular ultrasonic horn enhanced with an optimized binary mode converting structure. FIG. 10B illustrates an example of a mode converting structure optimized with a binary volumetric distribution of refractive indexes configured to be inserted into the ultrasonic horn. FIG. 10C illustrates the binary optimized mode converting structure inserted into the ultrasonic horn. FIG. 11A illustrates a representation of discretization of a volumetric distribution of acoustic material properties into a plurality of discrete acoustic material properties. FIG. 11B illustrates a close up view of a portion of FIG. 11A. FIG. 11C illustrates a representation of a possible embodiment of a cylindrical mode converting structure with individual voxels assigned discrete acoustic material properties. FIG. 12 illustrates a representation of the effective distribution of acoustic material properties of the mode converting structure for voxels with sub-wavelength dimensions. FIG. 13 illustrates one embodiment of a representation of a mode converting structure within a first waveguide configured to convert ultrasonic energy from a first mode to a second mode. FIG. 14 illustrates a simulated embodiment of ultrasonic energy with the first waveguide in a first mode converted by the mode converting structure into a second mode within the second waveguide. DETAILED DESCRIPTION According to various embodiments, systems, apparatuses, and methods are described herein that relate to mode converting structures configured to modify field patterns of electromagnetic radiation (EMR) devices and/or pressure fields of acoustic radiation (AR). For electromagnetic radiation, a mode converting structure may be generated with a volumetric distribution of refractive indexes that can be used to convert an electromagnetic field from a first mode as generated by an original EMR device to a second mode. For acoustic radiation, a mode converting structure may be generated with a volumetric distribution of acoustic material properties that affect the refractive indexes of acoustic waves (such as properties associated with linear elasticity tensor and dynamic density tensor). The acoustic mode converting structure may convert pressure fields from a first mode as generated by an AR device to a second mode. Some of the embodiments of mode converting structures described herein are equally applicable to AR devices and EMR device. Other embodiments and descriptions may be more suitable for one type of radiation than the other. For example, embodiments describing a distribution of dielectric constants or relating to electrical permittivity may be more suitable for EMR devices. On the other hand, embodiments describing a distribution of acoustic material properties may be more suitable for AR devices. Nevertheless, both are described herein because in many instances elements and configurations of the various embodiments are interchangeable. In particular, the notion of refractive index is applicable both to EMR and AR devices. The mathematical basis for using refractive index descriptions includes the Wave Equation, Helmholtz Equation, and ray approximations (ray acoustics, ray optics, etc.). In some embodiments, a mode converting structure may be formed that provides a first target mode conversion for EMR device and a second target mode conversion for an AR device. Furthermore, the wavelength of an AR device may coincide with the wavelength of an EMR device based on the methods described herein. For example, the second mode may have a narrower beamwidth, a higher directional gain, lower far-field sidelobes, and/or a more uniform radiation profile in the radiative near-field. In some embodiments, the mode converting structure may modify the pressure field to compensate or negate the effects of a re-radiating, reflective, or refractive object in the near-field or far-field of the radiation device (EMR or AR). An ultrasonic AR device may generate ultrasonic radiation corresponding to an acoustic near-field distribution of a loudspeaker, electrostatic loudspeaker, a piezoelectric transducer, a MEMS transducer, a magnetostrictive transducer, an electromagnetic acoustic transducer (EMAT), an electrodynamic loudspeaker, and a passive radiator. A mode converting structure may receive a pressure field pattern on an input side of the mode converting structure and output a pressure field pattern on an output side corresponding to an output acoustic radiation in a second mode. The output acoustic radiation in the second mode may correspond to a different acoustic near-field distribution. The distribution of refractive indexes, dielectric constants or acoustic material properties of a mode converting structure may be selected to modify a field or pressure pattern of the radiating device for a finite frequency range and convert the radiation from a first mode to a second mode. In various embodiments an EMR mode converting structure, the mode converting structure may be idealized as a graded-permittivity structure having a continuous distribution of dielectric constants, such that there are no abrupt changes in permittivity across the structure. Given a finite range of wavelengths, a discretized piecewise-continuous approximation of the graded-permittivity structure may be electromagnetically equivalent for a given bandwidth. Similarly, a graded-density or other graded-acoustic-material-property structure may be generated and/or used to generate a discretized piecewise-continuous approximation of the mode converting structure having the volumetric distribution of acoustic material properties. Thus, in various embodiments, the mode converting structure may be divided into a plurality of sub-wavelength voxels. That is, the mode converting structure may be conceptually thought of as comprising a plurality of voxels (three-dimensional pixels) whose largest dimension is smaller than a wavelength within the relevant bandwidth. For example, each voxel may have a maximum dimension that is less than half of a wavelength (e.g., the smallest wavelength) within a predetermined frequency range. The mode converting structure may comprise metamaterials and be referred to as a holographic metamaterial device useful to modify the near-field and/or far-field of an EMR or AR device for a particular frequency range. In some embodiments, the voxels may be cubes, parallelepipeds, tetrahedrons, pyramids, prisms, various regular polyhedrons, or other polyhedrons. In some embodiments, a voxel may have one or two dimensions that are sub-wavelength while the other dimension(s) are larger than a wavelength. The mode converting structure may be flat or curved and may be ridged or flexible (or have flexible portions at least). A mode converting structure may have a non-uniform thickness and/or may fill voids to create a uniform thickness with non-converting material. In various embodiments, a combination of voxel shapes and/or sizes may be used. Moreover, voxels may be shaped and/or sized such that little or no space, gaps, or voids exist between voxels. Alternatively, voxels may be arranged such that gaps or voids of various sizes and/or shapes exist. In some embodiments, the gaps or voids may be ignored and/or negligible in calculating the volumetric acoustic material properties. Alternatively, the gaps or voids may be assigned one or more acoustic material properties corresponding to a vacuum or to air or another fluid (liquid or gas) that fills the gaps or voids. Examples of materials include immersion liquids, mineral oil, synthetic oil, and polymer oil. Other examples of materials may include: one or more metamaterials having varying acoustic material properties, aperiodic photonic crystals, composite materials, porous materials, foam materials, layered composite materials, stratified composite materials, fiber-bundle materials, micro-rod materials, nano-rod materials, Because of the effect temperature and pressure may have on acoustic material properties, in some embodiments, mode converting structures may be rated for specific temperature and/or pressure ranges or combinations. In some embodiments, standard ambient temperature and pressure (SATP) may be used as 25 degrees Celsius and 100 kPa of pressure. In various embodiments, a mode converting structure may be selected to include one or more materials have unique acoustic material properties to attain the target volumetric distribution of acoustic material properties. Examples of acoustic material properties include, but are not limited to, linear elasticity tensor, dynamic density tensor, bulk modulus, elastic modulus, and density. Acoustic material properties of a particular material may vary depending on the finite frequency range for which the ultrasonic mode converter is configured to function. Throughout this disclosure, discussions of discretizing the mode converting structure may mean different things in various embodiments. For example, in some embodiments, the conceptual discretization of the mode converting structure may be used for optimization algorithms, while the mode converting structure may not be physically discretized. In some embodiments, the discretization of the mode converting structure may be a physical discretization of the mode converting structure. Physical discretization may be useful for manufacturing simplification (e.g., for three-dimensional printing of a mode converting structure). Thus, allusions to discretizing, dividing into voxels, and the like should be understood in the context of either manufacturing or optimization, and sometimes both. In the absence of explicit context, discussions should be construed as implying manufacturing and optimization individually, and as implying the possibility of a discretization for both manufacturing and optimization. A manufactured mode converting structure may be positioned relative to a radiation device (EMR or AR), may be removable, and/or may be configured as a retrofit solution for an existing antenna system. The shape and dimensions of the mode converting structure may be adapted based on the radiation device used. In various embodiments, an EMR device may include, by way of example but not limitation, a radio frequency antenna, an optical radiation transmitter, and an optical radiation receiver, and/or an electro-optical EMR device configured to convert between electric current and optical radiation or vice versa. In various embodiments, an AR device may include, by way of example but not limitation, an acoustic transducer, an acoustic transmitter, an acoustic receiver, an ultrasound transducer, an infrasound transducer, an audible acoustic radiation device, a vibration device, a sonic radiation device, and/or an optical radiation antenna configured to convert between acoustic vibration and electric current. An EMR mode converting structure may be used to modify field distributions of the EMR device. An AR mode converting structure may be used to modify a pressure field distribution of the EMR device. For instance, a mode converting structure may be used to modify the field or pressure distribution of a low-directivity antenna to correspond to that of a higher-directivity, narrow-beamwidth antenna. The mode converting structure allows for the beam pattern of a radiation device to be modified without any modification, or at least any significant modification, to radiation device, radiation feed, or the environment of the radiation device. Thus, the mode converting structures disclosed herein allow for a retrofit modification of existing radiation devices. As specific examples, a mode converting structure may improve the directivity of an AR device that normally has a directivity between 2, 5, 10, 15, or 20+ dBi. In other embodiments, an acoustic mode converting structure may be configured to narrow a far-field beamwidth of a main or side lobe, create maximum pressure fields in certain locations, or create pressure nulls in certain locations. The mode converting structure may be configured with a volumetric distribution of acoustic material properties to modify the far-field radiation pattern to compensate for a re-radiating object positioned in the reactive/Fresnel or radiative/Fraunhofer near-field of the AR device. The following specific examples use radio frequency (RF) antennas as an example of EMR devices generally. However, it is appreciated that many of the same concepts, embodiments, and general functionality of the systems and methods described herein are equally applicable to other frequency ranges of EMR, including those utilizing low-frequency RF, microwave, millimeter-wave, Terahertz, far and mid-infrared, near infrared, visible light, ultraviolet, x-rays, gamma rays, and so forth. It is appreciated that the sizes, refractive index values, materials, and other variables may be adjusted based on the particular spectrum in use. Furthermore, it is appreciated that many of the same concepts can be applied to, for example, transducers instead of antennas and ultrasound instead of electromagnetic energy. Many antennas, such as electrically small dipoles, monopoles, and loop antennas are fundamentally limited in their directionality. Horn-shaped antennas can have better directionality, so long as their dimensions are not significantly sub-wavelength. At any rate, once fabricated and installed, fixed-shape, single-feed antennas generally have a fixed radiation pattern and a certain beamwidth, although it may vary based on frequency. Additionally, as described above, a mode converting structure may be divided into a plurality of sub-wavelength voxels that each have a maximum dimension that is less than a wavelength (e.g., three-quarters, half, one-third, one-quarter, one-tenth of a wavelength) for a specific frequency range. As above, each voxel may then be assigned one of a plurality of dielectric constants (or acoustic material properties) to approximate an identified distribution of dielectric constants (or acoustic material properties). A mode converting structure may be used for free-space transmission and/or receiving. Alternatively or additionally, a mode converting may be utilized to convert acoustic or electromagnetic energy from a first mode to a second mode within a first waveguide, between two waveguides, between a waveguide and free space, and/or between free space and a waveguide. For example, the distribution of acoustic material properties may convert acoustic energy within the waveguide at a first mode to a second mode. As another example, the distribution of acoustic material properties may be configured to convert acoustic energy within a first waveguide from a first mode to a second mode for transmission through one or more additional waveguides. Similarly, the distribution of dielectric constants may convert electromagnetic energy within the waveguide at a first mode to a second mode. As another example, the dielectric structure may be configured to convert electromagnetic energy within a first waveguide from a first mode to a second mode for transmission through one or more additional waveguides. Again, in each of the embodiments described herein, the embodiments, modifications, adaptations, equations, algorithms, and/or other variations may be adapted for use in free-space applications, in transmitting antennas, in receiving antennas, within a waveguide, between two different waveguides, from a transmission line to free space, from a first transmission line to a second transmission line, between a waveguide transmission line to a non-waveguide transmission line, from a non-waveguide transmission line to a waveguide transmission line, for any of a wide variety of frequencies and bandwidths, and/or in combinations and permutations of any of the above. Thus, embodiments in which radiation devices, antennas, and free-space applications are used as the example application for a mode converting device are equally applicable to embodiments relating to waveguides and other transmission lines, even if not explicitly stated herein. As described above, mode converting structures may be used to transform the near-field and/or far-field of a fixed radiation device without necessarily modifying the radiation device, installation, and/or surrounding environment. According to various embodiments, a holographic solution may be used to determine a volumetric distribution of dielectric constants that can provide a desired field transformation and mode conversion for free-space applications and waveguide/transmission line applications alike. Similarly, a holographic solution may be used to determine a volumetric distribution of acoustic material properties that can provide a desired pressure transformation and mode conversion for free-space applications and waveguide/transmission line applications alike. For example, a mode converting structure may include a volumetric distribution of primary acoustic refractive indices. A primary acoustic refractive index, np, may correspond to the speed of primary wave, or the p-sound. Specifically, np=c0/cp, where c0 is the speed of sound in the ambient fluid and cp is the speed of p-sound in that medium. In such an embodiment, a volumetric distribution of primary acoustic refractive indices can be determined using Equation 1 below, or a variation thereof: n2(x, y, z)−1=βpin·p*goal/|pgoal|2 Equation 1 In Equation 1, n2(x, y, z) represents a volumetric distribution of primary acoustic refractive indices in an x, y, z coordinate system. In many embodiments described herein, a Cartesian coordinate system is used as a default example; however, any of a wide variety of coordinate systems are suitable, including cylindrical, polar, barycentric, trilinear, and other coordinate systems. In fact, in some embodiments, alternative coordinate systems may be preferable to simplify calculations and/or facilitate manufacturing. For instance, a cylindrical coordinate system may be useful for a manufacturing technique in which the volumetric distribution of dielectric constants corresponds to a uniform rotation of a two-dimensional planar cross section around an axis of revolution. In Equation 1, β represents a normalization constant and pin represents an input pressure field distribution of acoustic radiation from (1) an AR device on the surface of the mode converting structure relative to the x, y, z coordinate system, (2) the input pressure field distribution of radiation within a first waveguide or first waveguide portion (e.g., an acoustic transmission medium) relative to the x, y, z coordinate system, or (3) the input pressure field distribution of the radiation from free space (e.g., air) into a waveguide or other transmission medium, relative to the x, y, z coordinate system. Subsequent descriptions of Equation 1 with application to AR devices, are equally applicable to waveguide, various transmission mediums, and free-space (e.g., air) applications. Field pgoal represents the “goal” or selected, desired, target output pressure field distribution of radiation from the mode converting structure relative to the x, y, z coordinate system. The calculated distribution of dielectric constants may be approximated by conceptually dividing the mode converting structure into a plurality of voxels. Each voxel can then be assigned an acoustic refractive index. In some embodiments, each voxel may be assigned a spatial average value corresponding to the average calculated acoustic refractive index for the volume of the voxel. In embodiments in which the discretization is binary, ternary, or N-ary, each voxel may be assigned an acoustic refractive index from a selection of N discrete acoustic refractive indices, where N is an integer greater than 1 (2 for binary, 3 for ternary, and so forth). As a specific example, a region having a size that is distinguishable at the frequency of the ultrasonic radiation may contain multiple voxels. If the region should have, on average, an acoustic refractive index of 0.486, this may be satisfied by conceptually dividing the region into 100 voxels and using a first material with an acoustic refractive index of 0.9, to fill 31 of those voxels and a second material with an acoustic refractive index of 0.3, to fill the other 69 voxels. Thus, the average acoustic refractive index of the region will approximate 0.486. Similar approximations can be made using any number of materials having any number of acoustic refractive index. In some embodiments, frequency-dependent metamaterials having effective acoustic refractive indices that are more than 1, less than 1, and even less than 0 may be utilized. In various embodiments, the volumetric distribution of acoustic refractive indices may be substantially homogenous in one spatial dimension of the coordinate system, such that the volumetric distribution of the mode converting structure is effectively two-dimensional even though it is physically a three-dimensional object. For instance, the volumetric distribution may correspond to a uniform extrusion of a planar two-dimensional distribution perpendicular to the plane. In various embodiments, an acoustic mode converting device may include one or more transparent materials. Examples of various possible materials include, but are not limited to transparent inorganic glasses, transparent polymers, glass polymers, thermoplastics, inorganic glasses, doped inorganic glasses, nanoparticle embedded glasses, nanoparticle embedded polymers, polymers with varying polymerization numbers, etc. In some embodiments, a mode converting structure may take the form of a functional or even structural element. For example, a transparent mode converting structure may be used as a window for a car or building. Opaque mode converting structures may be used as wall coverings, doors, flooring, mirrors, etc. As previously noted, many of the embodiments of mode converting structures described herein are applicable to AR devices and EMR device. Of course, because the properties of electromagnetism and acoustics are different, different variables and modifications to the equations may be appropriate. However, examples described in terms of acoustic radiation are often applicable to electromagnetic radiation, and vice versa. The following table illustrates some analogies and similarities of acoustics and electromagnetism. Table 1 below is found in “Acoustic Metamaterial Design and Applications” by Shu Zhang, University of Illinois at Urbana-Champaign, 2010, which is hereby incorporated by reference in its entirety. TABLE 1 Acoustics Electromagnetism (TMz) Analogy ∂ P ∂ x = - i   ωρ x  u x ∂ E z ∂ x = - i   ωμ y  H y ∂ P ∂ y = - i   ωρ y  u y ∂ E z ∂ y = i   ωμ x  H x ∂ u x ∂ x + ∂ u y ∂ y = - i   ωβP ∂ H y ∂ x - ∂ H x ∂ y = - i   ωɛ z  E z Acoustic pressure P Electric field Ez −Ez ↔ P Particle velocity ux uy Magnetic field Hx, Hy Hy ↔ −ux Hx ↔ uy Dynamic density ρx ρy Permeability μx, μy ρx ↔ μy ρy ↔ μx Dynamic Permittivity ϵz ϵz ↔ β compressibility β FIG. 1A illustrates an example of a holographic solution showing real values of a volumetric distribution of dielectric constants 100 using a modified version of Equation 1 above for electric fields: εhol(x, y, z)−1=βEgoal·E*in/|Ein|2 Equation 1.1 The illustrated volumetric distribution of dielectric constants is calculated for an idealized two-dimensionally isotropic line source EMR device. A mode converting structure (i.e., a holographic metamaterial) with a corresponding distribution of dielectric constants could be used as a cover for the line source EMR device to increase the directionality of the line source EMR device. Similar to FIG. 1A, Equation 1 above may be used to determine a distribution of acoustic refractive indices. Alternatively, a modified version of Equation 1 may be used to determine a volumetric distribution of a combination of materials having varying linear elasticity tensors, dynamic density tensors, bulk moduli, elastic moduli, and densities. FIG. 1B illustrates the imaginary values of the volumetric distribution of dielectric constants 150 for the same line source EMR device described in conjunction with FIG. 1A above. The “goal” or “target” electromagnetic field used to generate FIGS. 1A and 1B is a plane wave with infinite directivity. The example, although idealized, illustrates one method for generating a mode converting structure for converting the electromagnetic field generated by an EMR device to a second mode with improved radiation characteristics. In practice, the finite aperture of the holographic metamaterial domain limits the actual directivity that can be attained. To account for the aperture effect, another option would be to use a Gaussian beam whose waist is equal to or small than the diameter of the holographic metamaterial domain. FIG. 2A illustrates an example of a field distribution 200 of the line source EMR device surrounded by a holographic metamaterial (mode converting structure) that has a distribution of dielectric constants approximating the distributions (real and imaginary) shown in FIGS. 1A and 1B. FIG. 2B illustrates an example of a far-field radiation pattern 250 of the line source EMR device surrounded by the mode converting structure having the distribution of dielectric constants (real and imaginary) shown in FIGS. 1A and 1B. The Equation 1.1 for electric fields and permittivity values may be used to identify a distribution of dielectric constants with complex permittivity values in all four quadrants of the complex variable plane, including the half-plane corresponding to active-gain medium, and possibly the quadrant corresponding to a passive, negative-permittivity medium. In such embodiments, active-gain permittivity values and negative permittivity values may be attainable using metamaterials. For instance, the distribution of dielectric constants can be discretized into sub-wavelength voxels each being assigned a particular permittivity value. Some of the voxels may be assigned permittivity values that can be implemented with traditional low-loss dielectrics, while other voxels may be assigned permittivity values (active-gain and negative) that can be implemented with metamaterials. In some situations, it may be desirable to utilize low-loss dielectrics in which ε′>1 and ε″«1. Such materials may be referred to as non-superluminal low-loss dielectrics (NSLLDs). Some material may only be considered NSLLD for specific frequency bands. Accordingly, the materials used to generate a mode converting structure may depend highly on the specific frequencies and bandwidths utilized by a particular EMR device. For acoustic mode converting structures, an alternative to Equation 1 may be used as: n2(x, y, z)−1=β|pgoal+pin|2/|pin|2 Equation 2 In Equation 2 above, n2(x, y, z) represents a volumetric distribution of acoustic refractive indices in an x, y, z coordinate system. Again, any coordinate system may be used that is suitable for the calculation of the distribution of acoustic refractive indices and/or is useful for mapping a manufacturing process. Constant β represents a non-zero normalization constant and field pin represents an input pressure field distribution of acoustic radiation from an AR device, within a waveguide or other transmission medium, and/or on a coupling between a waveguide and free space on the surface of the mode converting structure relative to the x, y, z coordinate system. Field pgoal represents the “goal” or selected/desired/target output field distribution of acoustic radiation from the mode converting structure relative to the x, y, z coordinate system. Solving the equations above and/or other equations described herein, may be performed using an optimization algorithm in which the acoustic refractive indices are treated as optimizable variables. The real and/or imaginary parts of the acoustic refractive indices may be treated as independently optimizable variables, or complex values may be selected and used as the optimizable variables. Any of a wide variety of optimization algorithms may be used, including those (1) in which a cost function is determined for each modification or group of modifications, (2) in which a gradient of a cost function based on partial derivatives is made with respect to each of the optimizable variables, and (3) in which a sensitivity vector is calculated using an adjoint solution-based sensitivity calculation. In some embodiments, a constrained optimization algorithm may be used in which the acoustic refractive indices are treated as optimization variables constrained to have real parts greater than or equal to approximately N and imaginary parts equal to approximately M, where N and M are real numbers. In other embodiments, a guess- and-check approach may be used in which an initial guess is used to solve the holographic solution using any one of the equations described herein. Non-exhaustive examples of specific optimization algorithms are described in greater detail below. FIG. 3 returns to an example of an electromagnetic radiation mode converting structure and illustrates an example of a volumetric distribution of dielectric constants 300 for converting a radiation pattern of a two-dimensional point dipole antenna into a directional beam. The distribution of dielectric constants illustrated in FIG. 3 can be found using an electromagnetic variation of Equation 2 above with the target or goal output field set as a plane wave: εhol(x, y, z)−1=β|Egoal+Ein|2/|Ein|2 Equation 2.1 For the purposes of this calculation, the plane wave would be indistinguishable from a finite-width beam, given the finite diameter of the metamaterial domain used. FIG. 4A illustrates an example of a field distribution 400 of a two-dimensional point-dipole antenna surrounded by a mode converting structure that has a distribution of dielectric constants approximating the distribution shown in FIG. 3. FIG. 4B illustrates an example of a far-field radiation pattern 450 of the two-dimensional point dipole antenna in FIG. 4A surrounded by the mode converting structure having the distribution of dielectric constants shown in FIG. 3. Equations 1 and 2 above provide adequate solutions to finding distributions of acoustic refractive indices for generating mode converting structures. However, further optimization may improve the efficiency of the mode conversion and compensate for the finite metamaterial domain. The metamaterial domain may be conceptually split into a plurality of voxels, where each voxel is approximately less than one-half wavelength (e.g., one-tenth of a wavelength). Each voxel may be conceptually populated with a spatial average of the continuous acoustic refractive index found using Equation 1 or 2 above. This discrete distribution of dielectric constants may be used as an initial guess in an optimization algorithm. The optimization algorithm may treat the real and imaginary values of the acoustic refractive index in each voxel as independent control variables. Alternatively, the complex (or real) value in each voxel may be treated as an independent value. Any of a wide variety of optimization algorithms may be used. For example, a small perturbation to one of the control variables may be made, and then the forward wave propagation problem may be solved to determine the effect of the perturbation. This may be referred to as a cost function optimization in which the cost function is the difference between the target or goal field and the field produced by the current state of the optimization variables. The finite difference in the goal, divided by the small perturbation value of the control variable, may be referred to as the finite-difference estimate of the cost function partial derivative. After computing all of the partial derivatives with respect to all control variables, the combined vector may be referred to as the “gradient” of the cost function, also known as “a sensitivity vector.” In other embodiments or as an alternative in the same embodiments, an adjoint method may be used that is based on the analytical derivatives of the equation describing the forward problem. The adjoint method may be used to produce the entire sensitivity vector after solving just one auxiliary problem known as the adjoint problem, whose computational complexity is the same as the complexity of one forward problem of the same size. In some applications, this may reduce the amount of computation per optimization step by a factor of N, where N is the number of control variables. Once a sensitivity vector is obtained, an iteration of a standard Newton, damped Newton, conjugate-gradient, or any other gradient-based iterative nonlinear solver may be used to determine the next configuration. In some embodiments the optimization algorithms may use heuristics as part of the optimization process. This may be useful in embodiments where the control variables are non-differentiable. In those embodiments it may be difficult to determine the gradient of the sensitivity vector or the analytic derivative. For example, in some embodiments, the optimization algorithm may use iterative heuristic optimization techniques such as, particle swarm optimization (PSO) or genetic optimization to determine an optimal solution. In one embodiment the iterative heuristic optimization may begin by generating a family of possible solutions. The family of possible solutions may contain thousands or millions of possible optimal solutions. Each, of the possible solutions may be generated according to a predetermined representation of the optimal solution domain. In other embodiments the predetermined solutions may be generated randomly. The predetermined representation may include properties of the optimal solution (e.g., transmission mode, manufacturing constraints, acoustic material properties, boundary conditions, or optimization variables). With each iteration of the heuristic optimization process the possible solutions may be evaluated using a fitness function. Solutions that are “more fit”, as determined by the fitness function, may be stochastically selected to continue on to the next iteration. Each, iteration may add new possible solutions or remove less fit solutions. In addition, with each successive iteration properties of each solution may be modified, altered, mutated, exchanged, updated, or changed in any way or combination of ways useful in determining an optimal solution. The algorithm may iterate a finite number of times or it may iterate until an acceptable solution is reached. An optimization algorithm may be utilized until a predetermined termination tolerance(s) is met. A termination condition can be imposed on some norm of the sensitivity vector, in which case the optimization algorithm is guaranteed to converge. A termination condition can be imposed as an inequality on the scalar value of the cost function, in which case the algorithm may fail to meet the imposed condition. In various embodiments, it may be useful to apply a termination condition to a sensitivity vector, and to take the final value of the optimization cost function as an output of the algorithm instead of an input to the algorithm. For applications that require the final value of the cost function to be below a certain tolerance, the optimization loop that failed to produce such an outcome can be repeated with a different initial guess. Each of Equations 1 and 2 define a family of initial guesses, each of which can be used to initiate a different optimization loop. Such loops are entirely independent and can be computed in parallel, using distributed computing. Using an electromagnetic variation as an example, FIG. 5 illustrates the optimization 500 of the real part of the dielectric constant distribution shown in FIG. 1A using the adjoint sensitivity method and a conjugate-gradient nonlinear solver. In the illustrated optimized dielectric constant distribution, discretized values from FIG. 1A are used as an initial guess. The optimization algorithm converged to the solution shown in FIG. 5. FIG. 6A illustrates an example of a field distribution 600 of the two-dimensionally isotropic line source EMR device surrounded by a mode converting structure that has a distribution of dielectric constants approximating the optimized distribution shown in FIG. 5. FIG. 6B illustrates an example of a far-field radiation pattern 650 of the two-dimensionally isotropic line source EMR device surrounded by the mode converting structure having the optimized distribution of dielectric constants shown in FIG. 5. Comparison of FIG. 6A with FIG. 2A and FIG. 6B with FIG. 2B show the improvement in mode conversion efficiency of the optimized solution. FIG. 7 illustrates the optimization 700 of the dielectric constant distribution shown in FIG. 3 using the adjoint sensitivity method and a conjugate-gradient nonlinear solver. In the illustrated optimized dielectric constant distribution, discretized values from FIG. 3 are used as an initial guess. The optimization algorithm converged to the solution shown in FIG. 7. FIG. 8A illustrates an example of a field distribution 800 of the two-dimensional point dipole antenna surrounded by a mode converting structure that has a distribution of dielectric constants approximating the optimized distribution shown in FIG. 7. FIG. 8B illustrates an example of a far-field radiation pattern 850 of the two-dimensional point dipole antenna in FIG. 8A surrounded by the mode converting structure having the optimized distribution of dielectric constants shown in FIG. 7. Comparison of FIG. 8A with FIG. 4A and FIG. 8B with FIG. 4B show the improvement in mode conversion efficiency of the optimized solution. The conceptual voxels described above are assigned a discrete permittivity value; however, the total number of unique values is unlimited as each one may be any real (or potentially complex) value. In some embodiments, it may be useful to limit the total number of unique values. Thus, instead of assigning each voxel a value based on the spatial average, εav, of the continuous distribution over that region, each voxel could be assigned, as a binary example, one of two values, ε1 or ε2. For instance, each voxel may either be assigned a permittivity value of “1” or “X”, where 1 represents a vacuum and X represents a permittivity value greater than 1. Such a binary discretization may be thought of as similar to gray-scale imaging where only white and black dithering is used. The Boolean decision to assign each voxel to either ε1 or ε2 may be based on whether εav is above or below a threshold value. A similar binary discretization approach may be used for acoustic embodiments, in which a first material having a first acoustic material property value may be used as one type of voxel and a second material having a different acoustic material property value may be used as another type of voxel. The result may be considered a piecewise-constant distribution of dielectric constants or acoustic material properties. So long as the feature sizes of each voxel are sufficiently small (e.g., sub-wavelength), the mode converting structure may be electromagnetically or acoustically equivalent to a continuous distribution for a given bandwidth of EMR or AR. In various embodiments, the piecewise-constant distribution of dielectric constants or acoustic material properties may be binary, ternary, or quaternary in nature, or otherwise limited to a specific number of unique permittivity values. Thus, in some embodiments, Equation 1 and/or 2 may be used to determine a continuous distribution of acoustic material properties. Optimization algorithms may then be employed using discretized average acoustic material property values (e.g., acoustic refractive indices. A mode converting structure may then be manufactured using the discretized distribution of optimized average acoustic material properties. In some embodiments, Equation 1 and/or 2 may be used to determine a continuous distribution of acoustic material properties. Optimization algorithms may then be employed using discretized average acoustic material properties. The discretized average acoustic material properties may then be discretized into N values for an N-ary discretization (where N is 2 for binary discretization, 3 for ternary discretization, and so forth). A mode converting structure may then be manufactured using the discretized distribution of optimized N-ary acoustic material properties. For example, an N-material three-dimensional printer may be used to deposit a material with one of the N acoustic material properties in each respective voxel. In embodiments in which the piecewise-constant distribution is used in the optimization algorithms, it may be desirable to preserve the ability to use real-valued control variables while still accounting for the N-ary nature of the structures being optimized. An algebraic transformation may be used to map the real-valued control variable to the N-ary-valued dielectric constants. An example of such a transformation for a binary piecewise-constant distribution with permittivity values ε1 and ε2 is as follows: ε(x, y, z)=ε1+(ε2−ε1)θδ(l(x, y, z)) Transformation 1 In Transformation 1, l(x, y, z) is a real-valued function of coordinates with values bounded to the [−1; 1] interval (called the level-set function), and θδ(l) is a smoothed Heaviside function, which, by definition, is equal to zero for 1<−δ, unity for l>δ, and is continuous with its first (and possibly second) derivatives for all l. The value of the smoothing parameter δ may be chosen as 0.1; however, this value can be selected differently based on the specific application to achieve more accurate results. The transformation allows optimization algorithms designed for continuous, real-valued control variables to be used for N-ary discretized approximations by using near-N-ary values as realistic approximations to N-ary values. After optimization has been performed, the values may be converted back into discretized N-ary values based on whether each optimized value is above or below one or more threshold values, where the number of threshold values is equal to N−1. In acoustic embodiments in which the piecewise-constant distribution of acoustic material properties is used in the optimization algorithms, it may again be desirable to preserve the ability to use real-valued control variables, while still accounting for the N-ary nature of the structures being optimized. A similar algebraic transformation may be used to map the real-valued control variable to the N-ary-valued acoustic material properties. An example of such a transformation for acoustic refractive indices for a binary piecewise-constant distribution with dynamic compressibility b1 and b2 is as follows: n2(x, y, z)=b1+(b2−b1)θδ(l(x, y, z)) Transformation 1.1 In Transformation 1.1, l(x, y, z) is again a real-valued function of coordinates with values bounded to the [−1; 1] interval, and θδ(l) is a smoothed Heaviside function. In some embodiments, the value of the smoothing parameter δ is chosen as 0.1. Again, this value can be selected differently based on the specific application to achieve more accurate results. FIG. 9 illustrates an example 900 of an acoustic mode converting structure 920 converting input acoustic radiation 910 from a first mode and outputting acoustic radiation 930 in a second mode. FIG. 10A illustrates an example of a radiation intensity pattern 1000 of a circular horn 1060 enhanced with an optimized binary mode converting structure 1070 (i.e., holographic metamaterial). FIG. 10B illustrates a representation of a mode converting structure 1070 optimized with a binary volumetric distribution of dielectric constants or acoustic material properties configured to be inserted into the horn 1060. The binary volumetric distribution of dielectric constants is illustrated as various grayscale patterns to show that average values over any given region may be a factor of the ratio of voxels assigned ε1 or b1 (shown as white) and ε2 or b2 (shown as black). It can be appreciated that for a ternary or other N-ary embodiment, additional colors might be used to represent the various possible discretization alternatives and approximations. FIG. 10C illustrates the binary optimized mode converting structure 1070 inserted into the horn 1060. As illustrated, the mode converting structure 1070 may be specifically manufactured (i.e., a volumetric boundary may be imposed) so that it limits the total width to no wider than the maximum width of the horn 1060. In the illustrated embodiment, the volumetric boundary allows the mode converting structure 970 to protrude from the horn by a small amount. The equations above describe an x, y, z coordinate system. Many possible variations of Equations 1 and 2 are possible and may be utilized in combination with the discretization and optimization techniques described herein. A variation of Equation 1 that may be used is provided below: n2hol−1=βpgoal·p*in/|pin|2 Equation 3 In Equation 3 above, n2hol represents a volumetric distribution of acoustic material properties in any of a wide variety of three-dimensional coordinate systems. Similar to Equation 1, β represents a normalization constant and pin represents an input pressure field distribution on a mode converting waveguide junction (free space or other transmission medium) or from a device on the surface of the mode converting structure relative to the three-dimensional coordinate system. pgoal represents a selected or desired output pressure field distribution from the mode converting structure relative to the three-dimensional coordinate system. In some embodiments, a variation of Equation 2 may be used as provided below: n2hol−1=β|pgoal+pin|2/|pin|2 Equation 4 In Equation 4 above, n2hol represents a volumetric distribution of dielectric constants in any of a wide variety of three-dimensional coordinate systems. Similar to Equation 1, β represents a normalization constant and pin represents an input pressure field distribution on a mode converting waveguide junction, at a waveguide termination, or from an AR device on the surface of the mode converting structure relative to the three-dimensional coordinate system. pgoal represents a selected or desired output pressure field distribution from the mode converting structure relative to the three-dimensional coordinate system. In any of the embodiments described herein, values for n2hol below a minimum threshold value may be set to a predetermined minimum value. Similarly, values for n2hol above a maximum threshold value may be set to a predetermined maximum value. In other embodiments, a plurality of discrete values for n2hol may be available and each of the calculated values of n2hol may be assigned one of the available discrete values by rounding down to the nearest available value, rounding up to the nearest available value, and/or assigned to the closest matching value. As previously described, many variations of Equations 1 and 2 may be used to find the holographic solution and calculate the volumetric distribution of dielectric constants. Another example of such an equation is provided below: n2(x, y, z)=α+βpgoal·p*in/|pin|2 Equation 5 In Equation 5 above, n2(x, y, z) represents a volumetric distribution of dielectric constants in an x, y, z coordinate system. α and β represent selectable constants and pin represents an input field distribution of radiation from a AR device on the surface of the mode converting structure relative to the x, y, z coordinate system. pgoal represents the “goal” or selected/desired output field pressure distribution of radiation from the mode converting structure relative to the x, y, z coordinate system. It is appreciated, without exhaustive recitation herein, that any of the variations, embodiments, or methods for solving a holographic solution or equation described herein may be used in conjunction with any of the other variations, embodiments, or methods of any other holographic solution or equation. In Equation 5, a value for α may be selected to optimize impedance matching between the input mode and a mode-converting medium. A value a may be selected to optimize impedance matching
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