1. Introduction

We present a physically motivated wavevector formulation that discretizes propagating electromagnetic (EM) waves within bulk nonreciprocal magneto-optical (MO) materials and examine the resulting nonlocal states within the material. This discretization results in a bulk material response that creates a complex amplitude superposition of states. Here we demonstrate that applying a single solution to the classical MO inhomogeneous wave equation generates a natural discretization of the solution. The procedure described here utilizes a physically motivated discretization based on analysis of MO beam splitting [1,2]. We demonstrate that beam-splitting in bulk MO materials yields a multiplicity of polarization states characterized by complex amplitudes, depending on the reflection angle of the beam splitter. Presentation of these results will be broken down into three sections presented in following order: the classical discretization of the EM wave, physically motivated state superpositions with complex amplitudes, and computational verification and comparisons.

MO materials induce nonreciprocal propagation of light and have been shown to exhibit polarization-based beam splitting [1–3]. This nonreciprocal beam splitting occurs through the separation of propagation directions for different optical polarization states relative to the MO material’s magnetization direction. The physical separation of the polarization states, or magneto-optic birefringence, creates states elliptically-polarized perpendicular to the magnetization direction while simultaneously generating states linearly-polarized parallel with the magnetization. Both birefringent states are activated through internal reflection within the bulk MO material, each having different refractive indices. These unique refractive indices and simultaneous activation result in the reflection of two distinct outputs from a single incident input within the bulk MO material. This double reflection/refraction within circularly-birefringent media such as chiral and MO materials has been studied previously [4–7]. However, the previous works do not address the emergence of a multiplicity of discretized states within each given polarization pair.

Such discretized multiplicity is due to MO materials containing a unique and special property: The refractive indices of the linearly- and elliptically- polarized states are not only pairwise different from each other but are also distinct for different angles of reflection/refraction. Hence, there is an angular-dependent continuum infinity of refractive indices for the reflected light in these materials. MO materials maintain their nonreciprocal functionality upon beam separation allowing for the creation of nonreciprocal photonic devices such as beam steerers, isolators, and routers [8–11].

We have previously presented an analysis and experimental verification of the nonreciprocal beam splitting in bismuth-substituted iron garnet [1]. Within [1] it is demonstrated that MO nonreciprocity induces double and triple reflection/refraction. This multiple reflection/refraction emerges as the separation of magneto-optically birefringent states, with either elliptical or linear polarization. Triple refraction occurs when the incident optical beam propagates away from the MO magnetization direction. The case of triple refraction was experimentally verified with the observation that increasing the beam separation increases individual beam width for each state. A diagram showing the double refraction and quadruple reflection is displayed as Fig. 1.

 

Fig. 1. Shown in this diagram is the beam splitting effect for input light propagating along the magnetization direction with optical components having refractive indices ${n_ + }$ or ${n_ - }$. Following the phase matching condition at the interface between the media, the reflected beams propagate at different angles hence ${\theta _{i, \pm }} \ne {\theta _{r, \pm }}$. The transmitted beams also follow this same principle. The COMSOL simulation was constructed following the diagram shown here with ${\theta _{i, \pm }} = 45^\circ $. Within COMSOL the incident gaussian beam is either left- or right-circularly polarized.

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We demonstrate here that EM waves undergoing this classical splitting produce a unique set of discretized superposition states. Each such superposition state is shown to individually contain unique complex amplitudes determining the contribution of the polarization states. This superposition of states with complex amplitudes may be akin to a typical quantum state, potentially giving rise to quantum mechanical effects within a classical mechanical architecture. In this formulation, a single incident input will split into two states becoming a superposition of elliptical and linear states with complex amplitudes based on the direction of propagation relative to the MO magnetization direction. By adjusting the direction of propagation, the EM field distribution in the elliptical and linear states will change. This will in turn cause the values of the complex amplitudes governing each state to change. We postulate that this will cause single photons to undergo a change in probability amplitudes for each state by adjusting the direction of propagation. Recently it has been proposed that probabilistic computers may be a more achievable alternative to quantum computers [12–14]. Similar to quantum computers, probabilistic computers use the probability of a state or operation for computation. The primary difference being that quantum computers use complex amplitudes which can be positive or negative, real or complex while probabilistic computers simply use the likelihood of an outcome. The unique adjustment of the EM field distribution and state probabilities has the potential to be used as a probabilistic operation.

The complex amplitudes are dictated by the boundary conditions and Maxwell equations of the MO material. Thus, upon internal reflection there is a finite set of allowable complex amplitudes which generates a finite set of modes of reflection. These modes all contain unique complex amplitudes for the elliptical and linear polarization states and interfere with each other as the superposition of all physical modes propagates as an ensemble following reflection. Notice that the generation of this mode multiplicity is not due to confinement or resonance effects. Rather, it arises from the variance in refractive index for different beam-propagation directions in MO materials.

Similarly, this suggests that a single photon that experiences this kind of nonreciprocal double reflection/refraction transforms into spatially separated states, each propagating with different refractive indices. The photon would undergo self-interference as it propagates through the MO material. This phenomenological photon self-interference and nonlocality may be further studied and applied in a controllable way [15–17]. Nonlocality may present many applications especially pertaining to quantum optics [18,19].

2. Discretizing the EM wave

The inhomogeneous wave equation in magneto-optic media is discussed in detail in [1]. This equation is derived from the classical Maxwell’s equations in the presence of a MO material permittivity tensor. The MO material permittivity tensor is shown below as Eq. (1).

Following the application of the MO permittivity tensor to the Maxwell equations the wave equation for MO material takes the form of Eq. (2), shown below.

This wave equation may be solved by examining its axial extremes which results in the existence of two different states within the bulk MO material. One said state exhibits linear polarization in the direction of the material’s magnetization, labeled as the z-axis. While the second state is elliptically-polarized in the xy-plane (x- and y-axes being perpendicular to the magnetization). The existence of these two states may be experimentally verified by the methods explained in [1]. The elliptical state also contains two different variants depending on incident helicity. These two variants become the standard circularly-polarized states for the case of propagation along the magnetization direction.

The inhomogeneous wave equation in a magneto-optic material produces several solutions with different distributions in the polarization states (elliptical and linear) and several propagation constants for each of those polarization states. This fact is the result of magneto-optical materials supporting a continuum of different refractive indices in different propagation directions. To analyze this, we assume a plane wave solution to the inhomogeneous wave equation, shown in Eq. (3):

This reformulated wave equation (Eq. (3)) may be combined with the boundary conditions for EM wave reflection/refraction to solve for the wavevector $\boldsymbol{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over k} }$ and the electric field values for any propagation direction. Within the Supplementary we present a derivation showcasing that the inhomogeneous wave equation may indeed be solved for with a single wavevector $\boldsymbol{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over k} }$. This derivation reveals that one single wavevector $\boldsymbol{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over k} }$ will still result in the expected axial extremes used to solve for the inhomogeneous wave equation. Notice that the wave equation contains coupling terms between different vectors components. Numerical methods may be applied to solve for all variables of an optical beam that is internally reflected within a bulk MO material. The primary, and simplest, example is that the EM wave is initially propagating along the magnetization (along the z-axis) and is then internally reflected in the material in some different direction. We postulate that upon reflection or refraction not along the magnetization direction, any single photon will enter into a superposition of these various states. As the boundary conditions and Maxwell’s equation enforce the simultaneous presence of both elliptical and linear states.

Since the refractive index in MO materials depends on the direction of propagation and the direction of the optical electric field vectors relative to the magnetization, these electric field components will propagate with different refractive indices and in separate directions. Notice that the magnetization couples the x- and y-components resulting in elliptically-polarized states. This induces the beam splitting effect or double reflection/refraction [1–3] in bulk MO materials. However, here the discretized EM wave is introduced with a single wavevector $\boldsymbol{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over k} }$. This will force the reflected EM wave to propagate with a single wavevector. To numerically solve the given set of equations we apply Wolfram Mathematica as it contains a powerful numerical solving function that allows for the evaluation of a set of equations. The solution for both the wavevector $\boldsymbol{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over k} }$ and the corresponding electric fields may then be obtained for any given direction of propagation relative to the magnetization. The specific MO material used for these calculations was bismuth-substituted iron garnet. However, similar results compared to those presented here should be expected for any bulk MO material that is characterized by a (gyrotropic) permittivity tensor. Numerical solutions at the boundary for the bulk MO interface and air for the reflected light with a 532 nm wavelength are shown in Table 1. Although light at the wavelength of 532 nm is highly absorbed within bismuth-substituted iron garnets, the material possesses a larger magneto-optic gyrotropy than in the infrared, and hence a more pronounced magneto-optical response. For the purposes of this work the absorption was excluded, to analyze the effects of the wavevector discretization.

Table 1. Mathematica Boundary Condition Solutions

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The values shown in Table 1 represent the physical modes found for $45^\circ $ reflection relative to interfacial normal of the MO material interface with air. Where the incident light propagates along the magnetization direction, with an initial right circular polarization. While Table 1 only represents the physical modes for $45^\circ $ reflection, the full set of values, including nonphysical modes, for $45^\circ $ reflection and for other reflection angles are given in the Supplementary. The number of modes depends on the reflection angle and the wavelength of the incident light. Larger permittivity tensor g values will yield more potential modes, while smaller g values will yield less modes. As an example, we found 16 potential modes for 532 nm light reflected at $45^\circ $, while we found 12 modes for 1550 nm light at the same angle. Each individual physical mode is activated in reflection and together constitute the full reflected wave. However, each mode contributes more or less to the outgoing wave. This contribution or weight for each mode may be determined experimentally or using simulations as explored here. Following the results shown in Table 1 it may be seen that there are some modes that appear to contain gain and loss. We postulate that instead this is in fact the flow of energy from one mode to another. Separate propagation of each individual mode would then have either positive or negative energy flow. However, since all physical modes are activated upon reflection the resulting superposition of all modes will obey the conservation of energy. For each single solution to the Maxwell equations and the EM boundary conditions there exist physical and nonphysical modes, much like for the case of waveguides and fiber optics. These physical and nonphysical modes are present for all angles of reflection and wavelengths. The physicality of a single mode may be determined using experimental or simulation comparison of results and rare cases such as zero/infinite electric field and/or wavevector values.

These results, including the example of $45^\circ $-reflection shown in Table 1, indicate that all physical wavevector $\boldsymbol{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over k} }$ values must contain real and imaginary parts. Although only the result for $45^\circ $ reflection is presented in Table 1, every solution for the wavevector $\boldsymbol{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over k} }$ with any angle of reflection contains a real and an imaginary part. Examples for other reflection directions are presented in the Supplementary. The real part of each discretized wavevector $\boldsymbol{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over k} }$ is not experimentally observed. In fact, the single optical field represented by this wavevector contains a superposition of states that propagate with experimentally measurable real wavevectors that interfere with each other.

3. Physically motivated state superpositions

Based on the solutions found above for the wavevector $\boldsymbol{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over k} }$, we make the ansatz that the wavevector $\boldsymbol{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over k} }$ is a superposition of propagation constants for the elliptically- and linearly-polarized states: $\boldsymbol{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over {k_m}} } = {a_m}{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over {k_E}} } + {b_m}{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over {k_L}} } $. Here the wavevector $\boldsymbol{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over {k_m}} } $ has the subscript m, denoting which physical mode composes the state superposition. Since the wavevector $\boldsymbol{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over {k_m}} } $ contains both a real and imaginary component for all propagation orientations with respect to the MO magnetization, the amplitudes ${a_m}$ and ${b_m}$ must be complex. Meanwhile, the ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over {k_E}} } \; and\; {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over {k_L}} } $ wavevectors have experimentally measured real values. With this ansatz, a review of Eq. (3) reveals unique results. Within Eq. (3) the x- and y-components of the electric field describe the elliptical polarization, while the z-component describes the linear polarization state. Moreover, as shown in Eq. (3), there is a codependence of all electric field components with each other, while there are also cross terms in the wavevector $\boldsymbol{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over {k_m}} } $. This codependence and cross terms under the above ansatz indicate that the elliptically-polarized state and the linearly-polarized state make up a unique superposition state. Under this ansatz we assume that the complex amplitudes ${a_m}$ and ${b_m}$ take the forms shown in Eq. (4).

Under these definitions, the discretized EM wave propagates as a superposition of the elliptically- and linearly-polarized states. However, since we have found from Table 1 that there exists a discretized set of physical modes defining the wavevector $\boldsymbol{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over {k_m}} } $ at the boundary for each direction of propagation, there must be multiple solutions to Eq. (5).

The measurable wavevectors ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over {k_E}} } $ and ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over {k_L}} } $, shown in Eq. (6), are real and constant for a given propagation direction relative to the magnetization, as shown in [1].

Both equations result in a real observable value for propagation in any direction relative to the magnetization. This results in the discretization of the wavevector $\boldsymbol{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over {k_m}} } $ for a given single inhomogeneous wave equation solution. Here the wavevector $\boldsymbol{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over {k_m}} } $ may be defined as the discretized state vector describing the propagation of the superposition of elliptical and linear states. This discretization state vector has a finite set of values, as shown in the example of $45^\circ $ reflection in Table 1. Each physical wavevector contains its own unique complex coefficients for the known elliptical polarization ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over {k_E}} } $ and linear polarization ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over {k_L}} } $ wavevectors. The latter, ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over {k_E}} } $ and ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over {k_L}} } $, are the measurable real wavevectors, which are deterministically defined based on the phase matching condition discussed in detail in [1]. The presence of two different polarization states, propagating in different directions, is very suggestive. For the case of a single photon launched into this system, we postulate that it will exist in a superposition of those two polarization states propagating in different directions. This is something that needs to be verified experimentally. Furthermore, the superposition of states with unique probabilities of occurrence may be used to create a probabilistic computer. A probabilistic computer has been recently proposed as a more achievable alternative to a quantum computer [12–14].

Moreover, a classically reflected optical beam will contain a superposition of all physical modes. The probability amplitude for each polarization state will change over propagation length due to coupling and interference of the physical modes with each other. Each mode having its own initial complex amplitudes, determined by the wavevector $\boldsymbol{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over {k_m}} } $ evaluated at the boundary. These theoretical results are verified below using computational methods at different wavelengths.

4. Computational results and comparison

To verify the proposed theory for the superposition of states, computational methods may be applied to simulate the discretized EM wave propagation. For this purpose, we use COMSOL to simulate reflection within a commonly used MO material [20–22], bismuth-substituted iron garnet (BiIG). This is a ferrimagnetic material [22], that maintains its magnetization state after poling and does not require a constantly applied external magnetic field to preserve it [9,10,23]. Different bismuth substitution levels may be used to adjust the Faraday rotation [20–22]. Within the COMSOL program we applied the MO material permittivity tensor, described in Eq. (1). This permittivity tensor induces a coupling of the EM field x- and y-components normal to the magnetization, forcing the propagating EM wave to gyrate in the xy-plane. The parametric values used in the simulations are $\sqrt \varepsilon \approx 2.57$ and $g \approx 0.037$ for the propagation wavelength of 532 nm. Similarly, the values used for the propagating wavelength of 1550 nm was $\sqrt \varepsilon \approx 2.31$ and $g \approx 0.002$. The COMSOL 3D simulation consists of a BiIG rectangular prism of the dimensions of $9\; \mu m\; \times 3\; \mu m\; \times 1.5\; \mu m$ with one end of the prism cut at the angle of $45^\circ $. The angle of $45^\circ $ is the optimal test, as in reflection we should observe the EM states’ refractive indices approaching those for the Cotton-Mouton effect. Prior to the $45^\circ $ interface a right circularly polarized gaussian beam is sent through the BiIG material in the same direction as the material magnetization to reflect/refract at the interface. In reflection, the gaussian beam travels nearly along the x-axis within the material. While in refraction the light is transmitted into air outside of the material. A full picture of the proposed design is presented in Fig. 1, while the incident angle is ${\theta _{i, \pm }} = 45^\circ $.

After the light internally reflects within the BiIG material the Mathematica solutions, shown in Table 1, predict that the reflected light undergoes self-interference with each physical individual mode interfering with the other modes. This results in the modes experiencing oscillation in amplitude while propagating along the direction of propagation, in this case near perpendicular to the material magnetization. Their superposition results in the propagation of a wavefront that does not propagate at the same speed in different directions with respect to the MO magnetization. This is due to each individual mode consisting of a unique superposition of elliptical and linear polarization states; these states have separate and unique refractive indices as seen with double reflection/refraction. The difference in superposition of different refractive indices results in an individual speed of propagation for each individual mode. This produces an interference pattern due to the superposition of all physical modes in 3-dimensional space, causing the overall wavefront to broaden as experimentally observed, referred to as beam broadening in [1]. The Mathematica predictions along with the COMSOL verification are shown in Fig. 2 for the wavelength of 532 nm. Normally light at 532 nm is highly absorbed within bismuth-substituted iron garnets. However, we did not apply any ohmic losses in our calculations and instead assumed no loss for the comparison of the Mathematica and COMSOL results. For verification that the described phenomena are valid for wavelengths outside of the visible we also computed the results for 1550 nm wavelength. Within BiIG material, light at 1550 nm experiences low loss while the wavevector discretization we predict is still present. However, the magneto-optical effects are greatly reduced in comparison to the visible. These results for light at 1550 nm wavelength are shown in the Supplementary.

 

Fig. 2. Overlayed Mathematica and COMSOL results for comparison and verification with 532 nm light. The Mathematica results are the numerical predictions for the electric field containing the superposition of all allowable modes after $45^\circ $ reflection. The COMSOL results are the measured electric fields that propagates along the project paths for the polarization states. With the input of a 532 nm gaussian beam reflected at a $45^\circ $ interface with air. The results are as follows (a), (b), (c) for comparison of the electric field in the x, y, z-directions respectively. (d) COMSOL 3D graphic displays the magnitude of the Poynting vector for the $45^\circ $ reflection of a 532 nm gaussian beam.

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Within Fig. 2 (a-c) we plot the Mathematica numerical solution for the electric field for all physical modes given a $45^\circ $ reflection of an incident 532 nm plane wave. Also overlayed, within Fig. 2 (a-c) we show the COMSOL results for the same described case with an incident 532 nm gaussian beam. The incident gaussian beam propagates along the magnetization, reflects at $45^\circ $ and propagates nearly perpendicular to the magnetization. Following the results shown in Fig. 2 (a-c), it can be seen that the Mathematica predictions closely resemble the COMSOL simulation results. This confirms the proposed theory of the existence of the superposition of the physical wavevector $\boldsymbol{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over {k_m}} } $ modes, each mode with its own unique superposition of the elliptical and linear polarization states.

Here it is important to note the differences and challenges in the Mathematica and COMSOL verification. Since each physical mode has a unique probability of occurring the superposition of all modes will have different weighted contributions for each solution in Mathematica. To resolve this within Mathematica we apply a weighting function ${W_m}$ for each physical mode. A full description of the weight function along with a short derivation for its calculation is shown in the Supplementary. Here let us note that the relative weights for each mode changes as the light propagates away from the boundary. This suggests that there is an energy transference between the modes as we previously postulated. The relative found weights for each physical mode for 532 nm light one wavelength from the boundary are as follows: ${W_1} \approx{-} 0.258 + 0.089\textrm{i}$, ${W_2} \approx{-} 0.258 + 0.089\textrm{i}$, ${W_3} \approx{-} 0.381 + 0.704\textrm{i}$, ${W_4} \approx{-} 0.188 + 0.124\textrm{i}$, ${W_5} \approx{-} 0.352 + 0.078\textrm{i}$, ${W_6} \approx{-} 0.168 + 0.039\textrm{i}$. Notice here that the weighting functions describing the superposition of the physical modes are complex. This superposition may be akin to a quantum superposition with complex amplitudes. Hence, it may be possible to use such superposition states for quantum mechanically based device. This would be possible while utilizing classical entanglement, which is a recently reported exciting phenomenon [24–26]. The respective weights for each mode may be verified experimentally, while the comparison of the COMSOL and Mathematica results reveals the general solution for $45^\circ $ reflection. Another important note is that COMSOL solves for the reflected states’ phase intrinsically, while the phase for each state is included in the electric field $\boldsymbol{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over {E_m}} } $ and wavevector $\boldsymbol{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over {k_m}} } $ solutions in Mathematica. This will shift the Mathematica results slightly when compared to the COMSOL results.

Another confirmation is the COMSOL 3D graphic, shown in Fig. 2(d). This graphic shows the 3-D magnitude of the Poynting vector in the plane of propagation (xz-plane in COMSOL). Within this graphic there is an interference pattern proceeding after internal reflection in the bulk MO material. This interference pattern expands with propagation leading to the beam broadening phenomenon. The interference pattern itself is a result of the superposition and interference of the different physical wavevector $\boldsymbol{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over {k_m}} } $ modes. These modes all start with individual initial values directly following reflection, but after propagating a long enough distance the modes will compile into a single broadening wavefront. Therefore, the graphic, shown in Fig. 2 (d), reveals the superposition of the individual wavevector $\boldsymbol{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over {k_m}} } $ modes, and since the different wavevectors propagate with separate and different speeds the superposition of all physical wavevectors results in wavefront broadening.

The numerical and simulation results shown here predict that when a circularly-polarized photon is initially propagating along the magnetization direction and is incident on the boundary of the MO material, the photon will reflect into a quantized set of pairs. Each pair propagates with its own refractive index for each of the two contained photon states, either elliptically-polarized or linearly-polarized. Each pair constitutes a separate EM wave mode propagating in a separate and unique direction. The reflected state pairs propagate with discretized refractive indices and are coupled pairwise. Different EM wave mode solutions will interfere with each other resulting in the interference pattern of the total EM fields found in Fig. 2. The complete result will be a photon composed of elliptically- and linearly-polarized states with unique probabilities depending on the mode. These two states will propagate with separate refractive indices and in unique directions. Resulting in a nonlocalized single photon.

5. Conclusion

Within the context of this article, we have presented a theory of discretized EM waves in MO materials. These discretized waves consist of a finite number of modes emerging in bulk media, and not as a result of confinement or resonance effects. Each mode contains a known and unique superposition of elliptically- and linearly-polarized states. The theory predicts that these modes occur with a definite measurable probability. This results in a combined superposition of physical modes, yielding a unique and predictable interference pattern.