Biaxial Gaussian Beams, Hermite–Gaussian Beams, and Laguerre–Gaussian Vortex Beams in Isotropy-Broken Materials

Abstract

We have developed the paraxial approximation for electromagnetic fields in arbitrary isotropy-broken media in terms of the ray–wave tilt and the curvature of materials’ Fresnel wave surfaces. We have obtained solutions of the paraxial equation in the form of biaxial Gaussian beams, which is a novel class of electromagnetic field distributions in generic isotropy-broken materials. Such beams have been previously observed experimentally and numerically in hyperbolic metamaterials but have evaded theoretical analysis in the literature up to now. Biaxial Gaussian beams have two axes: one in the direction of the Abraham momentum, corresponding to the ray propagation, and another in the direction of the Minkowski momentum, corresponding to the wave propagation, in agreement with the recent theory of refraction, ray–wave tilt, and hidden momentum [Durach, 2024]. We show that the curvature of the wavefronts in the biaxial Gaussian beams correspond to the curvature of the Fresnel wave surface at the central wave vector of the beam. We obtain the higher-order modes of the biaxial beams, including the biaxial Hermite–Gaussian and Laguerre–Gaussian vortex beams, which opens avenues toward studies of the optical angular momentum (OAM) in isotropy-broken media, including generic anisotropic and bianisotropic materials.

1. Introduction

In this paper, which is an extension of Ref. [1], we continue to conceptualize and describe isotropy-broken materials as a broad class of electromagnetic media, which do not feature isotropy. They are characterized by Fresnel wave surfaces without spherical symmetry and by non-transverse electromagnetic fields. The introduction of isotropy-broken media was inspired by the previous progress in studies of anisotropic and bianisotropic metamaterials [2,3,4,5,6,7].

It is well known from the work of Fermat and Huygens in the XVII century that rays are directed perpendicular to the wavefronts in isotropic media. Correspondingly, when considering a paraxial beam of rays, the wavefronts propagate close-to-parallel with the beam axis. The non-parallel to beam propagation of wavefronts in uniaxial media has been discussed [8,9] and numerically visualized in beams refracted into hyperbolic metamaterials [10,11], but an analytical theory of this effect is missing from the literature. This effect corresponds to a general phenomenon of ray–wave tilt due to non-parallel Abraham and Minkowski momenta and the presence of tangential components of hidden momentum and bound charge waves in isotropy-broken materials [1]. In this manuscript, we obtain the paraxial equation for isotropy-broken media and express its solutions as biaxial Gaussian beams, with one axis corresponding to the direction of the ray and another to wavefront propagation.

The Fresnel wave surfaces of isotropic media are spherically symmetric with negative curvatures inversely proportional to the wavelength $\lambda$ [12]. Correspondingly, Gaussian beams in isotropic media are expected to have converging wavefronts before foci and diverging wavefronts after, which is the case in all conventional media with positive indices of refraction. Surprisingly, it was demonstrated that negative-index isotropic media feature converging wavefronts after passing the focal plane [13]. This effect has also been numerically observed in the emission of sources into hyperbolic metamaterials [14,15], but has also evaded theoretical analysis in the literature. In this manuscript, we directly relate the curvature of the Fresnel wave surface to the curvature of the wavefronts in biaxial Gaussian beams. For example, open-topology Fresnel wave surfaces of multihyperbolic media [16,17,18,19,20,21,22,23,24,25] feature a positive curvature, which results in Gaussian beams with converging wavefronts after foci.

The orbital angular momentum (OAM) of light in isotropic media has occupied a central role in photonics recently with potential applications in areas such as communications, sensing, and optical manipulation [26,27,28,29]. Nevertheless, there are no studies on the OAM of electromagnetic fields inside isotropy-broken materials. We obtain high-order modes in the form of biaxial Hermite–Gaussian and Laguerre–Gaussian vortex beams with a topological charge. This result opens a new frontier for investigations into the OAM in isotropy-broken materials.

2. Materials and Methods

Consider a plane wave with wave vector $\mathit{k}=\left(k_x,k_y,k_z\right)$ and frequency $\omega =k_{0}c$ propagating through an isotropy-broken material described by bianisotropic constitutive relations:

$$\left(\begin{array}{c}\mathit{D}\\ \mathit{B}\end{array}\right)=\widehat{M}\left(\begin{array}{c}\mathit{E}\\ \mathit{H}\end{array}\right)=\left(\begin{array}{cc}\widehat{ϵ}& \widehat{X}\\ \widehat{Y}& \widehat{\mu }\end{array}\right)\left(\begin{array}{c}\mathit{E}\\ \mathit{H}\end{array}\right),$$

(1)

The properties of such a wave are described using the characteristic matrix method. The z-component of the wave vector and the transverse field satisfy the following eigenproblem:

$$\widehat{\Delta }\left(k_x,k_y\right)\cdot {\left(E_x,E_y,H_x,H_y\right)}^T=\left({\displaystyle \frac{k_z}{k_0}}\right){\left(E_x,E_y,H_x,H_y\right)}^T$$

(2)

The characteristic matrix of an isotropy-broken material given by Equation (1) is [23,25]

$$\widehat{\Delta }\left(k_x,k_y\right)=\widehat{\Delta }={\widehat{P}}^{-1}\cdot \left({\widehat{M}}_{\parallel ,\parallel }-\left({\widehat{M}}_{\parallel ,z}+{\widehat{q}}^T\right)\cdot {\widehat{M}}_{z,z}^{-1}\cdot \left({\widehat{M}}_{z,\parallel }+\widehat{q}\right)\right)$$

(3)

where additional matrices are defined in terms of vector $\mathit{q}=\mathit{k}/k_0$:

$${\widehat{M}}_{\parallel ,\parallel }=\left(\begin{array}{cccc}{ϵ}_{11}& {ϵ}_{12}& X_{11}& X_{12}\\ {ϵ}_{21}& {ϵ}_{22}& X_{21}& X_{22}\\ Y_{11}& Y_{12}& {\mu }_{11}& {\mu }_{12}\\ Y_{21}& Y_{22}& {\mu }_{21}& {\mu }_{22}\end{array}\right),{\widehat{M}}_{\parallel ,z}=\left(\begin{array}{cc}{ϵ}_{13}& X_{13}\\ {ϵ}_{23}& X_{23}\\ Y_{13}& {\mu }_{13}\\ Y_{23}& {\mu }_{23}\end{array}\right),$$

$$\widehat{P}={\widehat{P}}^{-1}=\left(\begin{array}{cccc}0& 0& 0& -1\\ 0& 0& 1& 0\\ 0& 1& 0& 0\\ -1& 0& 0& 0\end{array}\right),\widehat{q}=\left(\begin{array}{cccc}0& 0& -q_y& q_x\\ q_y& -q_x& 0& 0\end{array}\right)$$

The longitudinal fields satisfy

$${\left(E_z,H_z\right)}^T=-{\widehat{M}}_z^{-1}\cdot \left({\widehat{M}}_{z\parallel }+\widehat{q}\right)\cdot {\left(E_x,E_y,H_x,H_y\right)}^T$$

(4)

The characteristic equation for the eigenvalue problem of Equation (2) is a quartic equation, which corresponds to the Fresnel wave surface of medium given by Equation (1):

$$\mathcal{H}\left(\mathit{k},k_0\right)=q_z^4-\mathrm{t}\mathrm{r}\left(\widehat{\Delta }\left({\mathit{q}}_{\perp }\right)\right)q_{\mathrm{z}}^3-\xi \left({\mathit{q}}_{\perp }\right)q_{\mathrm{z}}^2-\zeta \left({\mathit{q}}_{\perp }\right)q_z+\det \left(\widehat{\Delta }\left({\mathit{q}}_{\perp }\right)\right)=0$$

(5)

where $\xi ={\displaystyle \frac{1}{2}}\left(\mathrm{t}\mathrm{r}\left({\widehat{\Delta }}^2\right)-\mathrm{t}\mathrm{r}{\left(\widehat{\Delta }\right)}^2\right)$, $\zeta ={\displaystyle \frac{1}{6}}\left(2\mathrm{t}\mathrm{r}\left({\widehat{\Delta }}^3\right)-3\mathrm{t}\mathrm{r}\left({\widehat{\Delta }}^2\right)\mathrm{t}\mathrm{r}\left(\widehat{\Delta }\right)+\mathrm{t}\mathrm{r}{\left(\widehat{\Delta }\right)}^3\right)$.

At given ${\mathit{k}}_{\perp }=(k_x,k_y)$ Equation (5) has 4 roots $k_z=k_z^{\left(i\right)}$, $i=1,\dots ,4$ corresponding to the points $\left(k_x,k_y,k_z^{\left(i\right)}\right)$ at the Fresnel wave surface. For plane wave propagating paraxially to the z-axis,

$$k_z^{\left(i\right)}\left({\mathit{k}}_{\perp }\right)=k_z^{\left(i\right)}\left(0\right)+{\mathit{k}}_{\perp }^T\cdot \mathit{t}+{\displaystyle \frac{1}{2!}}{\mathit{k}}_{\perp }^T\cdot \widehat{H}\cdot {\mathit{k}}_{\perp }+\dots$$

(6)

where $\mathit{t}=\left({\partial }_{k_x}\left[k_z^{\left(i\right)}\left({\mathit{k}}_{\perp }\right)\right],{\partial }_{k_y}\left[k_z^{\left(i\right)}\left({\mathit{k}}_{\perp }\right)\right]\right)$ and ${\widehat{H}}_{ij}={\displaystyle \frac{{\partial }^2{k}_z^{\left(i\right)}\left({\mathit{k}}_{\perp }\right)}{\partial k_i\partial k_j}}$ are the gradient and Hessian matrix, respectively. Similarly, the eigenvectors of Equation (3) can be expressed as

$${\Gamma }_{\perp }^{\left(i\right)}\left({\mathit{k}}_{\perp }\right)={\Gamma }_{\perp }^{\left(i\right)}\left(\mathbf{0}\right)+\widehat{J}\cdot {\mathit{k}}_{\perp }+\dots$$

(7)

where $\widehat{J}=\left({\displaystyle \frac{\partial {\Gamma }_{\perp }^{\left(i\right)}}{\partial k_x}},{\displaystyle \frac{\partial {\Gamma }_{\perp }^{\left(i\right)}}{\partial k_y}}\right)$ is the Jacobian matrix.

We consider a Gaussian beam with a focus in the $z=0$ plane. Its transverse field ${\Gamma }_{\perp }^{\left(i\right)}={\left(E_x,E_y,H_x,H_y\right)}^T$ can be expressed as

$${\Gamma }_{\perp }^{\left(i\right)}\left({\mathit{r}}_{\perp },z\right)=w_0\int {\displaystyle \frac{d{\mathit{k}}_{\perp }}{{\left(2\pi \right)}^2}}{e}^{i{\mathit{k}}_{\perp }{\mathit{r}}_{\perp }}{e}^{-{\displaystyle \frac{k_{\perp }^2{w}^2}{4}}}{\Gamma }_{\perp }^{\left(i\right)}\left({\mathit{k}}_{\perp }\right)\cdot \exp \left(i{k}_z^{\left(i\right)}\left({\mathit{k}}_{\perp }\right)z\right)$$

(8)

Note that Equation (8) represents a superposition of plane wave solutions of Maxwell equations in an arbitrary isotropy-broken medium with characteristic relations of Equation (1) and Fresnel wave surface of Equation (5). The superposition principle is one of the basis tenets of electromagnetism [30]. Substituting the paraxial approximation of Equations (6) and (7) into Equation (8), we obtain

$${\Gamma }_{\perp }^{\left(i\right)}\left({\mathit{r}}_{\perp },z\right)={\Gamma }_{\perp }^{\left(i\right)}\left(\mathbf{0}\right)\cdot e^{i{k}_z^{\left(i\right)}\left(0\right)z}\cdot G(\mathit{r})-i\cdot e^{i{k}_z^{\left(i\right)}\left(0\right)z}\left(\widehat{J}\cdot {\mathbf{\nabla }}_{r_{\perp }}\right)\cdot G(\mathit{r})$$

where the second term is considered small. The function $G\left(\mathit{r}\right)$ is given by

$$G\left(\mathit{r}\right)=w_0\int {\displaystyle \frac{d{\mathit{k}}_{\perp }}{{\left(2\pi \right)}^2}}{e}^{i{\mathit{k}}_{\perp }\mathit{\rho }}\cdot \exp \left({\mathit{k}}_{\perp }^T\cdot \widehat{\alpha }\left(z\right)\cdot {\mathit{k}}_{\perp }\right),\hspace{1em}\widehat{\alpha }\left(z\right)=-{\displaystyle \frac{w_0^2}{4}}\widehat{1}+{\displaystyle \frac{i}{2!}}\widehat{H}z$$

(9)

and $\mathit{\rho }\left(z\right)=\left(x+t_{x}z,y+t_{y}z\right)$. Evaluating the integral in Equation (9), we obtain the desired expression for the biaxial Gaussian beam profile in the arbitrary isotropy-broken material as

$$G\left(\mathit{r}\right)={\displaystyle \frac{1}{4\pi }}{\displaystyle \frac{w_0}{\sqrt{\det \left(\widehat{\alpha }\left(z\right)\right)}}}\exp \left({\displaystyle \frac{1}{4}}\mathit{\rho }\left(z\right)\cdot {\widehat{\alpha }\left(z\right)}^{-1}\cdot {\mathit{\rho }}^T\left(z\right)\right)$$

(10)

3. Results

3.1. Paraxial Equation and Biaxial Gaussian Beams in Isotropy-Broken Materials

Note that function $G\left(\mathit{r}\right)$ satisfies the generalized paraxial equation in isotropy-broken media:

$$i{\displaystyle \frac{\partial G\left(\mathit{r}\right)}{\partial z}}-{\displaystyle \frac{1}{2!}}{\nabla }_{\mathit{\rho }}\cdot \left(\widehat{H}{\nabla }_{\mathit{\rho }}G\left(\mathit{r}\right)\right)=0$$

(11)

The expressions in Equations (10) and (11) become especially transparent in the coordinate system, where the Hessian matrix of the Fresnel wave surface $\widehat{H}$ is diagonal $\widehat{H}=r_x\widehat{\mathit{x}}\widehat{\mathit{x}}+r_y\widehat{\mathit{y}}\widehat{\mathit{y}}$, where coefficients $r$ characterize the curvatures of the Fresnel wave surface in the direction of the wavefront propagation. In these notations, Equation (11) turns into

$$i{\displaystyle \frac{\partial G\left(\mathit{r}\right)}{\partial z}}-{\displaystyle \frac{r_x}{2!}}{\displaystyle \frac{{\partial }^{2}G\left(\mathit{r}\right)}{{\partial }^2{\rho }_x}}-{\displaystyle \frac{r_y}{2!}}{\displaystyle \frac{{\partial }^{2}G\left(\mathit{r}\right)}{{\partial }^2{\rho }_y}}=0$$

(12)

while the solution Equation (10) is expressed as

$$G\left(\mathit{r}\right)=g\left(x,t_x,r_x\right)g\left(y,t_y,r_y\right),$$

(13)

$$g\left(x,t_x,r_x\right)={\displaystyle \frac{w_0^{1/2}}{2{\pi }^{1/2}}}{\displaystyle \frac{1}{\sqrt{\left(\frac{w_0^2}{4}-\frac{i}{2!}{r}_{x}z\right)}}}\exp \left(-{\displaystyle \frac{1}{4}}{\displaystyle \frac{{\left(x+t_{x}z\right)}^2}{\frac{w_0^2}{4}-\frac{i}{2!}{r}_{x}z}}\right)$$

(14)

We can rewrite Equation (14) as

$$g\left(x,t_x,r_x\right)={\displaystyle \frac{1}{{\pi }^{1/2}}}{\displaystyle \frac{1}{\sqrt{w_x\left(z\right)}}}\exp \left(-{\displaystyle \frac{{\left(x+t_{x}z\right)}^2}{w_x^2\left(z\right)}}\right)\exp \left(i\left\{{\displaystyle \frac{{k_z\left(x+t_{x}z\right)}^2}{2R_x\left(z\right)}}-{\displaystyle \frac{1}{2}}{\psi }_{G_x}\right\}\right),$$

(15)

where the Gouy phase, beam radius, and wavefront radius of curvature in the x-direction are, respectively,

$${\psi }_{G_x}=\arctan \left({\displaystyle \frac{z}{z_x}}\right),\hspace{1em}{w}_x\left(z\right)=w_0\sqrt{1+{\left({\displaystyle \frac{z}{z_x}}\right)}^2},\hspace{1em}{R}_x\left(z\right)=-r_x{k}_{z}z\left(1+{\left({\displaystyle \frac{z_x}{z}}\right)}^2\right).$$

(16)

The Rayleigh length in the x-direction is $z_x=-{\displaystyle \frac{w_0^2}{2r_x}}$.

As we demonstrate in Figure 1 and as can be seen from Equation (16), all of the parameters of the beam depend on the curvature of the Fresnel wave surface. Specifically, the Rayleigh length $z_x$ and, correspondingly, the confocal parameter $2z_x$ are inversely proportional to the curvature of the Fresnel wave surface. Fresnel wave surfaces of isotropic media have a spherical symmetry and their curvatures are negative $r_x<0$. In this case, the wavefront radius of curvature is positive $R_x>0$, as can be seen in Figure 1a. If the curvature of the Fresnel wave surface is positive $r_x>0$, the wavefront radius of curvature is negative $R_x<0$, corresponding to converging wavefronts after passing the focal plane, as shown in Figure 1b,c. Converging wavefronts in Gaussian beams have been previously discussed in relation to isotropic negative index materials [13], but never for isotropy-broken media. Please note that if $r_x>0$, the sign of the accumulated Gouy phase is negative.

The equation for constant phases is

$${\displaystyle \frac{{k_z\left(x+t_{x}z\right)}^2}{2R_x\left(z\right)}}+{\displaystyle \frac{{k_z\left(y+t_{y}z\right)}^2}{2R_y\left(z\right)}}-{\displaystyle \frac{1}{2}}{\psi }_{G_x}-{\displaystyle \frac{1}{2}}{\psi }_{G_y}+k_{z}z=C_{phase}$$

(17)

Note that since, generally speaking, $r_x$ and $r_y$ are different at Fresnel wave surfaces in isotropy-broken media, the Rayleigh lengths $z_x$ and $z_y$, beam radii $w_x\left(z\right)$ and $w_y\left(z\right)$, and wavefront curvatures $R_x\left(z\right)$ and $R_y\left(z\right)$ are different as well. Correspondingly, the resulting Gaussian beams feature both intensity and phase astigmatism. This astigmatism is illustrated in Figure 1d,e for different signs of $r_x$ and $r_y$. While in all cases, a difference in the magnitude of $r_x$ and $r_y$ leads to an elliptical intensity spot, constant phase curves become hyperbolic if $r_x{r}_y<0$.

From Equations (10)–(17), we see that wavefronts propagate at an angle to the beam, the phenomenon recently described by Durach as ray–wave tilt in isotropy-broken media in Ref. [1]. Ray–wave tilt can be seen as the differential aberration of the ray and wave sources when seen in the material rest frame [1]. This is illustrated in Figure 1c, where wavefronts propagate at a 45-degree angle to the beam, which possesses different ray and wave axes and represents a biaxial Gaussian beam.

In accordance with Ref. [1], wavefronts propagate in the direction of the time-averaged Minkowski momentum, whose density is given by ${\mathit{g}}_{Min}={\displaystyle \frac{1}{4\pi c}}\mathrm{R}\mathrm{e}\left\{{\mathit{D}}^{\ast }\times \mathit{B}\right\}$, i.e., along the z-axis in the notations above, while the beam propagates in the direction of the Abraham momentum whose time-averaged density is ${\mathit{g}}_{Abr}={\displaystyle \frac{1}{4\pi c}}\mathrm{R}\mathrm{e}\left\{{\mathit{E}}^{\ast }\times \mathit{H}\right\}$. The surfaces of the constant field strength form hyperboloids along the axis $x=-t_{x}z, y=-t_{y}z$, i.e., in the direction of the Abraham momentum:

$${\displaystyle \frac{{\left(x+t_{x}z\right)}^2}{w_0^2\left\{1+{\left(\frac{z}{z_x}\right)}^2\right\}}}+{\displaystyle \frac{{\left(y+t_{y}z\right)}^2}{w_0^2\left\{1+{\left(\frac{z}{z_y}\right)}^2\right\}}}=C_{fstr}$$

(18)

The ray–wave tilt corresponds to the components of the hidden momentum $\Delta \mathit{g}={\mathit{g}}_{Abr}-{\mathit{g}}_{Min}$, which is transverse to the wavefront propagation direction [1]:

$$\overline{{\mathsf{\Delta }\mathit{g}}_{\perp }}=-{\displaystyle \frac{k_0}{c{k}^2}}\left(\mathrm{R}\mathrm{e}\left\{i{\rho }_{be}{\mathit{D}}^{\ast }+i{\rho }_{bm}{\mathit{B}}^{\ast }\right\}\right)$$

(19)

According to Equation (19) the ray–wave tilt is related to the propagation of electric and magnetic bound charge waves, which are carried by the beam in the isotropy-broken media [1]. To illustrate this, we find the bound charges by investigating the longitudinal fields using Equation (4):

$$\begin{gathered} {\left(E_z,H_z\right)}^T=\left\{-{\widehat{M}}_z^{-1}\cdot {\widehat{M}}_{z\parallel }+i{\widehat{M}}_z^{-1}\cdot \left(k_0^{-1}\widehat{\partial }\right)\right\}\cdot {\left(E_x,E_y,H_x,H_y\right)}^T, \\ \widehat{\partial }=\left(\begin{array}{cccc}0& 0& -{\partial }_{{\rho }_y}& {\partial }_{{\rho }_x}\\ {\partial }_{{\rho }_y}& -{\partial }_{{\rho }_x}& 0& 0\end{array}\right) \end{gathered}$$

$$\widehat{\partial }\left(\begin{array}{c}{\mathit{E}}_{\perp }\\ {\mathit{H}}_{\perp }\end{array}\right)=\left(\begin{array}{cccc}0& 0& -{\partial }_{{\rho }_y}& {\partial }_{{\rho }_x}\\ {\partial }_{{\rho }_y}& -{\partial }_{{\rho }_x}& 0& 0\end{array}\right)\cdot {\left(E_x,E_y,H_x,H_y\right)}^T$$

Using the fact that the transverse gradient of $G\left(\mathit{r}\right)$ satisfies

$${\nabla }_{\mathit{\rho }}G\left(\mathit{r}\right)={\displaystyle \frac{1}{2}}{\widehat{\alpha }}^{-1}\mathit{\rho }G\left(r\right)=-{\displaystyle \frac{2}{w_0^2}}\left({\displaystyle \frac{\left(x+t_{x}z\right)}{1+i\frac{z}{z_x}}},{\displaystyle \frac{\left(y+t_{y}z\right)}{1+i\frac{z}{z_y}}}\right)G\left(r\right)$$

we rewrite

$$\widehat{\partial }\left(\begin{array}{c}{\mathit{E}}_{\perp }\\ {\mathit{H}}_{\perp }\end{array}\right)=-{\displaystyle \frac{2}{w_0^2}}\left(\begin{array}{cccc}0& 0& -{\displaystyle \frac{\left(y+t_{y}z\right)}{1+i\frac{z}{z_y}}}& {\displaystyle \frac{\left(x+t_{x}z\right)}{1+i\frac{z}{z_x}}}\\ {\displaystyle \frac{\left(y+t_{y}z\right)}{1+i\frac{z}{z_y}}}& -{\displaystyle \frac{\left(x+t_{x}z\right)}{1+i\frac{z}{z_x}}}& 0& 0\end{array}\right)\left(\begin{array}{c}{\mathit{E}}_{\perp }\\ {\mathit{H}}_{\perp }\end{array}\right)=\widehat{P}(\mathit{\rho })\left(\begin{array}{c}{\mathit{E}}_{\perp }\\ {\mathit{H}}_{\perp }\end{array}\right)$$

Following this, the full field vectors of the biaxial Gaussian beam are

$$\begin{gathered} \left(\begin{array}{c}\mathit{E}\\ \mathit{H}\end{array}\right)=G\left(\mathit{r}\right)\exp \left(i{k}_{z}z\right)\left\{\left(\begin{array}{c}{\mathit{E}}_0\\ {\mathit{H}}_0\end{array}\right)+\widehat{\mathit{z}}{\widehat{M}}_z^{-1}\cdot \left(i{k}_0^{-1}\widehat{P}\right)\left(\begin{array}{c}{\mathit{E}}_{\perp 0}\\ {\mathit{H}}_{\perp 0}\end{array}\right)\right\} \\ =G\left(\mathit{r}\right)\exp \left(i{k}_{z}z\right)\left\{\left(\begin{array}{c}{\mathit{E}}_0\\ {\mathit{H}}_0\end{array}\right)+\widehat{\mathit{z}}\left(\begin{array}{c}\delta E_z\left(\mathit{r}\right)\\ \delta H_z\left(\mathit{r}\right)\end{array}\right)\right\} \end{gathered}$$

(20)

The bound charges can be found as

$${4\pi \rho }_e=-4\pi \nabla \cdot \mathit{P}=\nabla \cdot \mathit{E}=\nabla \left\{G\left(\mathit{r}\right)\exp \left(i{k}_{z}z\right)\right\}\cdot {\mathit{E}}_0+{\partial }_z\left\{G\left(\mathit{r}\right)\exp \left(i{k}_{z}z\right)\delta E_z\left(\mathit{r}\right)\right\}$$

$${4\pi \rho }_m=-4\pi \nabla \cdot \mathit{M}=\nabla \cdot \mathit{H}=\nabla \left\{G\left(\mathit{r}\right)\exp \left(i{k}_{z}z\right)\right\}\cdot {\mathit{H}}_0+{\partial }_z\left\{G\left(\mathit{r}\right)\exp \left(i{k}_{z}z\right)\delta H_z\left(\mathit{r}\right)\right\}$$

To demonstrate the propagation of biaxial Gaussian beams through a generic bianisotropic medium, in Figure 2, we consider a material with the Fresnel wave surface $\mathcal{H}=0$, as shown in Figure 2a, and with the material parameter matrix $\widehat{M}$ color-coded in Figure 2b [see also Equation (1)]. It is a tetra-hyperbolic material. We consider a central plane wave with $n=1.915$ propagating along the z-axis, i.e., ${\mathit{g}}_{Min}\propto \widehat{\mathit{z}}$ shown in Figure 2a as a black dashed vector. The direction of the Abraham momentum and the normal to the Fresnel wave surface corresponds to $k_0{t}_x=2.21$ at the central wave vector. The curvature of the Fresnel wave surface is $k_0{r}_x=12.16$.

In Figure 2c, we plot $\mathrm{R}\mathrm{e}\left\{g\left(x,t_x,r_x\right)e^{i{k}_{z}z}\right\}$ for a beam of width ${k_{0}w}_0=20$ centered on this wave. As can be seen in Figure 2c, the ray is directed along the Abraham momentum, while the phase propagates along the Minkowski momentum. The wavefronts are converging after passing the focus of the beam, which corresponds to the positive curvature of the Fresnel wave surface at the central wave vector, which can be seen in Figure 2a.

In Figure 2d,e, we show the electric and magnetic bound charge waves, which propagate along with the fields of the beam. Interestingly, while the bound magnetic charge has a diminished magnitude at the center of the beam, the bound electric charge is shifted towards the x-axis with respect to the fields.

3.2. Higher-Order Biaxial Hermite–Gaussian Beams in Isotropy-Broken Media

To obtain the higher-order biaxial Hermite–Gaussian modes in the isotropy-broken media from the fundamental mode in Equations (14)–(16), we utilize the following operator:

$${\left({\displaystyle \frac{\partial }{\partial {\rho }_x}}+\alpha {\rho }_x\right)}^n=\exp \left(-{\displaystyle \frac{\alpha {\rho }_x^2}{2}}\right){\left({\displaystyle \frac{\partial }{\partial {\rho }_x}}\right)}^n\exp \left({\displaystyle \frac{\alpha {\rho }_x^2}{2}}\right),\hspace{1em}\alpha =-{\displaystyle \frac{2e^{i{\psi }_{G_x}}}{w_0{w}_x\left(z\right)}}$$

The resulting higher-order mode after excluding constants is expressed as

$$\begin{gathered} g_n\left({\rho }_x\right)=g\left({\rho }_x\right)e^{-in{\psi }_{G_x}}{H}_n\left({\displaystyle \frac{\sqrt{2}{\rho }_x}{w_x\left(z\right)}}\right),\mathrm{o}\mathrm{r} \\ {g}_n\left({\rho }_x\right)={\displaystyle \frac{1}{{\pi }^{1/2}}}{\displaystyle \frac{1}{\sqrt{w_x\left(z\right)}}}\exp \left(-{\displaystyle \frac{{\left(x+t_{x}z\right)}^2}{w_x^2\left(z\right)}}\right)H_n\left({\displaystyle \frac{\sqrt{2}{\rho }_x}{w_x\left(z\right)}}\right)\exp \left(i\left\{{\displaystyle \frac{{k_z\left(x+t_{x}z\right)}^2}{2R_x\left(z\right)}}-N{\psi }_{G_x}\right\}\right) \end{gathered}$$

(21)

where the Hermite polynomials are $H_n\left(\xi \right)={\left(-1\right)}^n\exp \left({\xi }^2\right){\left({\displaystyle \frac{d}{d\xi }}\right)}^n\exp \left(-{\xi }^2\right)$ and $N=\left(n+{\displaystyle \frac{1}{2}}\right)$.

In Figure 3, we plot the biaxial Hermite–Gaussian beams for the same parameters as in Figure 1c for $n=1$ and $n=2$. We observe that the transverse pattern of the field is perpendicular to the wavefront propagation direction, while the wavefronts are converging after passing the focal plane due to the positive curvature of the Fresnel wave surface.

3.3. Laguerre–Gaussian Vortex Beams in Isotropy-Broken Media

Vortex beams have attracted a lot of attention in recent years due to their ability to carry and transfer OAM to materials. Consider a fundamental astigmatic biaxial Gaussian beam with equal Rayleigh lengths $z_x=z_y=z_R$, such that $w_x^2/w_y^2=r_x/r_y$:

$$G(\mathit{r})={\displaystyle \frac{1}{\left(z-i{z}_R\right)}}\exp \left(-i{\displaystyle \frac{\left(\frac{{\rho }_x^2}{r_x}+\frac{{\rho }_y^2}{r_y}\right)}{2\left(z-i{z}_R\right)}}\right)$$

From the fundamental mode, we obtain higher-order biaxial Laguerre–Gaussian vortex beams with radial indices $p\ge 0$ and azimuthal indices $l$:

$$G_{pl}(\mathit{r})={\left({\displaystyle \frac{z+i{z}_R}{z-i{z}_R}}\right)}^p{\left({\displaystyle \frac{1}{z-i{z}_R}}\right)}^{\left|l\right|+1}{\left({\displaystyle \frac{{\rho }_x}{\sqrt{r_x}}}-{\displaystyle \frac{i{\rho }_y}{\sqrt{r_y}}}\right)}^{\left|l\right|}\exp \left(-i{\displaystyle \frac{\left(\frac{{\rho }_x^2}{r_x}+\frac{{\rho }_y^2}{r_y}\right)}{2\left(z-i{z}_R\right)}}\right)L_p^{\left|l\right|}\left[-z_R{\displaystyle \frac{\left(\frac{{\rho }_x^2}{r_x}+\frac{{\rho }_y^2}{r_y}\right)}{z^2+z_R^2}}\right]$$

In Figure 4, we plot biaxial Laguerre–Gaussian vortex beams in isotropy-broken medium. We see that the optical vortices are wrapped around the ray axis such that the transverse pattern is perpendicular to the wavefront propagation direction. The wavefronts are converging after passing the focal plane due to the positive curvature of the Fresnel wave surface at the central wave vector.

We introduce the following parameters:

$${\rho }_L=\sqrt{-z_R\left({\displaystyle \frac{{\rho }_x^2}{r_x}}+{\displaystyle \frac{{\rho }_y^2}{r_y}}\right)},{\phi }_L=\arctan \left({\displaystyle \frac{{\rho }_y\sqrt{r_x}}{{\rho }_x\sqrt{r_y}}}\right),$$

(22)

$$w_L\left(z\right)=\sqrt{2}\sqrt{z^2+z_R^2},{\psi }_L=\arctan \left({\displaystyle \frac{z}{z_R}}\right),R_L\left(z\right)=z\left(1+{\left({\displaystyle \frac{z_R}{z}}\right)}^2\right),\hspace{1em}N=2p+\left|l\right|$$

(23)

Using a normalizing constant $C_{pl}$, we express the biaxial Laguerre–Gaussian vortex beams as

$$G_{pl}={\displaystyle \frac{C_{pl}}{w_L\left(z\right)}}{\left({\displaystyle \frac{{\rho }_L\sqrt{2}}{w_L\left(z\right)}}\right)}^{\left|l\right|+1}{L}_p^{\left|l\right|}\left[{\displaystyle \frac{2{\rho }_L^2}{w_L^2\left(z\right)}}\right]\exp \left(-{\displaystyle \frac{{\rho }_L^2}{w_L\left(z\right)}}\right)\exp \left({\displaystyle \frac{i{\rho }_L^2}{2z_R{R}_L\left(z\right)}}+i\left(N+1\right){\psi }_L\right)e^{-i\left|l\right|\varphi }$$

(24)

4. Discussion

The presence of Gaussian beams is ubiquitous in optics, and their applications range from ultrafast dynamics in quantum materials [31], to superluminal laser plasma filaments [32], to ultrafast photonics [33]. In this paper, we pioneer the consideration of Gaussian beams in isotropy-broken media, which includes non-parallel Abraham and Minkowski momenta and the first introduction of OAM vortex beams in isotropy-broken media, which opens a new frontier for OAM studies. We thank the anonymous reviewer who brought to our attention that isotropic media can propagate titled Gaussian beams [34]. It is interesting to note that these solutions can be obtained in terms of Lorentz transformation into a moving frame [35]. The consideration of media in moving frames is known to break isotropy and even cause transformations of the hyperbolic topological classes [1].

To conclude, we considered electromagnetic fields in isotropy-broken media in paraxial approximation. We derived the paraxial equation and obtained three classes of its solutions, namely, biaxial Gaussian, Hermite–Gaussian, and Laguerre–Gaussian beams. The parameters of the beams were directly related to the properties of the Fresnel wave surface at the central wave vector of the beam.