Gaussian beam propagation in a Lorentz-violating vacuum in the presence of a semi-transparent mirror

In this paper we study the propagation of structured optical scalar beams in a Lorentz-violating (LV) vacuum parametrized by a constant 4-vector u μ and in the presence of a semi-transparent mirror. The two bosonic degrees of freedom of the electromagnetic field can be described by a LV extension of the massless scalar field theory, whose LV part is characterized by the term (u · ∂ϕ)2. The mirror at a surface Σ is modelled by a delta-type potential in the Lagrange density for the LV scalar field, i.e. λ δ(Σ)ϕ 2, where the parameter λ controls the degree of transparency of the mirror. Using Green’s function techniques, we investigate the propagation of a Gaussian beam in the presence of a mirror which is perpendicular to the propagation direction and for two particular choices of the background 4-vector: parallel and perpendicular to the propagation direction. To quantify the Lorentz-violating effects we introduce the fidelity as a measurement of the closeness of the propagated field distribution with respect to that in the conventional vacuum. In the absence of the mirror (λ = 0) the fidelity is found to be close to one, and hence LV effects are quite small. However in the presence of the mirror, there are regions where the fidelity drops to zero, thus implying that LV effects could be clearly differentiated from the propagation in vacuum. Within the paraxial approximation we determine analytically the LV effects upon the Rayleigh range, the radius of the beam, the Gouy phase and the radius of curvature of the wavefronts. We discuss possible scenarios where our results could apply, by using optically transparent multiferroic materials, which offer unprecedented opportunities to tailor structured beam propagation, as well as to simulate an LV vacuum.


Introduction
The study of light propagation has fuelled exceptional technological applications in optical communication [1,2], optical sensing [3,4] and quantum information processing [5,6]. With the rapid development of new technologies to produce and characterize structured light beams [7][8][9], such as optical manipulation and optically driven micromachines, the study of beam propagation through different media has taken a remarkably upgrowth. In particular, the propagation of light in anisotropic media has attracted great attention due to their multiple uses in many modern optical and electro-optical waveguide devices.
The development of artificially engineered materials has sparked considerable interest in the last decades, since their optical properties can be controlled in a desired fashion. They offer a unique opportunity to test phenomena predicted by nonoptical theories which are difficult to be observed in the lab. As a result, a plethora of new discoveries on laser beams and their propagation properties have been achieved. For example, nematic liquid crystals [10,11] and lead glasses [12,13] offer an opportunity to investigate light propagation in strongly nonlocal nonlinear media [14][15][16]. In a similar fashion, metamaterial implementations have been used to mimic well-known gravitational phenomena that are difficult to observe directly using astronomical data. The transformation optics approach is the appropriate dictionary which allow us to translate cosmic phenomena to optics in the lab [17][18][19][20]. For instance, microstructured optical waveguides around a microsphere have been used to trap light by mimicking gravitational lensing [21,22]. Interestingly, the recent advances in the precision of cavity-based frequency measurements offer a platform to measure directly the effects of the Earth's gravity field upon the propagation of laser beams [23,24].
Symmetry breaking is a ubiquitous property in nature and diverse fields of modern physics. Among various symmetry breaking phenomena, spontaneous symmetry breaking lies at the heart of many fascinating and fundamental properties of nature. It is involved in Higgs physics, [25], double-well Bose-Einstein condensates [26] and topological phases [27]. Optical devices also offer an interesting arena for the study of spontaneous symmetry breaking; they hold great potential for the study of fundamental physics. It has been observed experimentally, for example, in temporal pulses in planar waveguides [28], on beam profiles in a photonic lattice [29] as well as in ultrahigh-Q whispering-gallery microresonator [30]. All this teach us that high precision optical experiments and the engineered optical materials on-demand provide an interesting test-bed to study the breaking of other of the fundamental symmetries of nature: Lorentz symmetry.
Lorentz symmetry is a cornerstone of quantum field theories and general relativity. However, candidate theories for quantum gravity predict observable deviations from known physics, including departures from Lorentz invariance. For example, in the low-energy limit of the bosonic string, some vector and tensor fields can acquire nonzero expectation values, which in turn introduce privileged directions in spacetime, thus breaking Lorentz invariance [31]. Nowadays, investigations concerning Lorentz violation are mostly conducted under the framework of the standard model extension (SME), which is an effective field theory that contains the standard model, general relativity, and all possible operators that break Lorentz invariance [31,32]. The search for Lorentz violating (LV) effects have been developed in different branches of physics. For example, the photon sector of the SME acts in many respects like a nontrivial optical medium [33,34]. Along this vein, in this paper we suggest that the study of structured optical beam propagation in a Lorentz-violating vacuum could provide information on the Lorentz-symmetry breaking scenarios arising from the Planck scale. Since no deviation from Lorentz invariance has been detected yet, from a high-energy perspective it is expected that the background tensor fields are small. However, there are scenarios in which Lorentz-symmetry is naturally broken, and the corresponding LV coefficients acquire a precise meaning; besides, they are not necessarily small. For example, the energy bands in crystalline solids exhibits an effective Lorentz symmetry at low energies, and the breaking of this is manifested as the various electronic phases of the system [35]. Therefore, the electromagnetic response of crystalline solids could be described by the photon sector of the SME, in which the background tensor fields are determined by the structural and electronic properties of the system. Finally, the development of artificially engineered materials also offers a unique opportunity to test LV effects.
In this paper we study the propagation of structured optical beams in a Lorentz-violating vacuum and in the presence of a semi-transparent mirror. The origin of this 4-vector, which breaks Lorentz-symmetry, arises from the CPT-even sector of the SME, which renders a Lorentz-violating electromagnetic field theory parametrized by a constant 4-vector u μ . We show that the two bosonic degrees of freedom of the electromagnetic field can be described by a LV massless scalar field theory which is derivable from the Lagrange density u We also analyse the effects of a semi-transparent mirror upon propagation and investigate if the mirror can help to detect the effects of the Lorentz symmetry breaking, and find that it is indeed the case. The presence of the semi-transparent mirror at a surface Σ is modelled by including a delta-type potential in the Lagrange density for the LV scalar field, i.e. ( ) ld f S , where λ controls the degree of transparency of the mirror [36]. The name semi-transparent mirror arises because the limit λ → ∞ is equivalent to impose Dirichlet boundary conditions on the scalar field at the surface [37,38]. Physically, the degree of transparency is related to the plasma frequency [39]. In order to investigate field propagation we compute the Green's function for different choices of the 4-vector u μ , namely, a spacelike u μ pointing in the parallel and perpendicular directions to field propagation. The mirror is chosen perpendicular to the field propagation also. We obtain a general expression for the propagated field in terms of the angular spectrum of the input mode, which can be applied to arbitrary structured optical beam sources. Here we focus on the propagation of the archetypal model in optics: the Gaussian beam. We first evaluate numerically the field intensity profile and next we obtain analytical expressions in the paraxial regime which are in agreement with the quantitative observations from the numerics. As a measure of the closeness between the field in a LV vacuum and that in conventional vacuum, we introduce the fidelity [40]. In the absence of the mirror, the fidelity is close to unity, thus implying that the LV effects are quite small. However, in the presence of the mirror, the fidelity exhibits regions where it drops to zero in some regions of the parameter space defined by the position of the mirror and the degree of transparency. This means that, for certain choices of these parameters, the field propagated in an LV vacuum behaves strongly different from the field propagated in conventional vacuum, thus giving us a mechanism for disentangling the effects of Lorentz symmetry breaking. This paper is organized as follows. In section 2 we introduce the theoretical model we shall consider, which is the CPT-even sector of the minimal SME in the presence of a semi-transparent mirror. We discuss its main properties and show that a particular choice for the LV tensor coefficient yields a Lorentz-violating massless scalar field theory for each bosonic mode of the electromagnetic field parametrized by a second rank-tensor, which is expressed in terms of a single four vector. In section 3 we derive the Green's function for the LV scalar field in the presence of a semi-transparent mirror considering different configurations for the background four vector. We use these results to study, in section 4, the propagation of a Gaussian beam. We numerically evaluate the expression for the propagated field and then we evaluate it analytically in the paraxial regime. We explore the effect of the semi-transparent mirror and find new interesting results. Finally, in section 5 we summarize our results and discuss possible realizations with optically transparent multiferroic metamaterials.

Theoretical framework
In this paper we consider the electromagnetic field in a Lorentz-symmetry breaking scenario parametrized by a CPT-even constant tensor. The Lorentz-symmetry breaking is achieved by a tensorial coupling in the Lagrangian as follows [33,34]: where J μ is the 4-current that couples to the electromagnetic 4-potential A μ , and F μ ν = ∂ μ A ν − ∂ ν A μ is the electromagnetic field tensor, which satisfies the Bianchi identity ò μ ν α β ∂ μ F αβ = 0, ensuring the U(1) gauge invariance of the action S d The dimensionless coefficients k α β μ ν controls Lorentz and CPT symmetry breakdown. In the simplest scenario this coefficient is constant, so that energy and momentum are conserved [41]. We adopt this assumption here and we will be working in a 3 + 1 dimensional space-time with Minkowski metric η μ ν = ( + , − , − , − ).
Variation of the Lagrange density (1) produces the equation of motion Current conservation can be verified directly by taking the divergence at both sides of equation (2). Certain linear combinations of the LV coefficients k αβμν simplify the analysis of this theory. One useful set can be written in terms of four 3 × 3-matrices κ DE , κ HB , κ DB and κ HE , which can be expressed in terms of the different components of the tensor k αβμν . Explicit expressions for this dependence are presented in [33,34]. It is interesting that, with the definition of these matrices, the field equations (2) can be cast in the form of the standard Maxwell equations in homogeneous anisotropic media, with the modified constitutive relations Therefore, a LV vacuum acts in many respects like a nontrivial optical medium. Now we delve on possible realizations of the theory (1). As discussed, candidate theories for quantum gravity predict observable deviations from known physics, including departures from Lorentz invariance [31,32]. In this scenario, since no violation of this symmetry has been detected yet, the coefficient k αβμν is expected to be small in all earth-based frames. Indeed, different combinations of the coefficient have been stringently constrained by experiments (see [33] and references therein). The study of light propagating in a medium characterized by the constituent relations (3) could shed information about the LV effects arising from the photon sector of the SME. As we know, Lorentz symmetry appears as an effective symmetry of low-energy condensed-matter systems and its breaking manifests as various electronic phases of matter. In this scenario, the coefficients for Lorentz violation are determined by the structural and electronic properties of the material and hence they are not necessarily small. This invites for a nonperturbative study of Lorentz breaking scenarios. In condensed-matter physics the theory (1) can be realized, for example, in naturally existing anisotropic and magnetoelectric matter. Also, they can be realized under demand in artificially created composite materials which exhibit unusual properties which are not found in nature, such as metamaterials and some classes of multiferroics, which are particularly attractive since they allow the simulation of curved spacetime backgrounds. For example, the constituent relations (3) describe the magnetoelectric multiferroic Cr 2 O 3 , in which case the matrices κ DE and κ HB contribute to the conventional permittivityˆ( )  w and permeabilityˆ( ) m w tensors, and the magnetoelectric tensor =arises from the coupling between spins and electric dipoles in the material [42]. All of these matrices are determined by the lattice structure of the material and they are frequency-dependent. The case of multiferroic materials could be specially relevant for the results reported in this paper, since they can be optically transparent and act like a nontrivial vacuum.
A particular choice for the LV tensor in equation (1), namely k αβμν = θò αβμν , appears in the description of the recently discovered topological phases of quantum matter. In this case θ is a nondynamical scalar field coupling whose origin is purely quantum-mechanical and its form depends on the symmetries of the system: it is equal to π for time-reversal invariant topological insulators [43] and it becomes space and time dependent for Weyl semimetals due to the breaking of time-reversal and spatial-inversion symmetries [44].
It is widely known that the two bosonic degrees of freedom of the electromagnetic field have essentially the same dynamics as the one described by two massless scalar fields. This is so because each component of the Maxwell equation (in the Lorentz-gauge) ,A μ = J μ can be seen as a single massless Klein-Gordon equation ,f = J, where f is any of the components of the 4-potential and J the corresponding 4-current source. Here, , = ∂ μ ∂ μ stands for the D'Alembert operator. Along this line, we now ask whether the Lorentz-violating field equations (2) can be described in terms of a Lorentz-violating scalar field. We answer this question in the affirmative for a particular choice of the the coefficient k F ( ) abmn , namely Interestingly, this theory also emerges in other contexts. For example, the electromagnetic field in a locally Minkowski spacetime, which is a Finsler spacetime with vanishing connection and curvature, satisfies also the field equation (2) with the LV coefficient as in equation (4). In this case, the tensor h μ ν is related to the deformation of the spacetime from Minkoski to locally Minkoski [45]. Since Finsler spacetime is intrinsically anisotropic and naturally induces Lorentz symmetry breaking, the tensor h μ ν effectively describes the breaking of such symmetry. On the other hand, the theory (5) come up also in the description of a scalar field in a general stationary space-time background at zero-order approximation, i.e. when the size of the system is small, the metric g μν (x) only need to be expanded to zeroth-order g μν (0) = η μ ν + h μ ν , being h μ ν a constant tensor [46].
Here, there is a subtle point that deserves to be discussed in detail. A given coefficient for Lorentz violation can lead to observable effects only if it cannot be eliminated from the action via field redefinitions or coordinate choices. In the present case, the source-free LV theory (5) can be cast in the form of the Lorentz-symmetric theory albeit in different coordinates and with a field redefinition. Explicitly, under the change of coordinates . However, there are scenarios which preclude the elimination of the LV coefficients by coordinate and field redefinitions. For example, when the theory involves different fields which interact among themselves, LV effects can be eliminated only for one field at a time. For example, the theory of Lorentz-violating scalar and spinor fields interacting through the Yukawa coupling [47] and the radiatively induced quantum corrections in Lorentz-violating quantum electrodynamics [48]. Also, the presence of boundaries which by themselves break Lorentz invariance prevents the elimination of LV effect by a coordinate redefinition. The problem at hand falls in this latter case and hence we discuss it in detail.
Let us consider the free-source LV scalar field theory (5) in the presence of a two-dimensional semitransparent mirror. We take a coordinate system such that the mirror is located perpendicular to the z-axis, located at z = z 0 . This configuration is described by adding the potential term z z where λ 0. Two comments are in order. First, in the limit λ → ∞ , the additional term is equivalent to imposing Dirichlet boundary conditions on the scalar field at z = z 0 and hence the name of semi-transparent mirror for finite λ. Second, the presence of the delta potential precludes the elimination of the LV coefficient by means of a coordinate choice, since while the h μ ν (∂ μ f)(∂ ν f) term can be absorbed by the kinetic term, it reappears in the argument of the Dirac delta. Therefore, the plane z = z 0 in the original coordinates is mapped to a new plane with an orientation which depends on the LV background tensor. The LV scalar field theory (6), with the particular choice h μ ν = u μ u ν , being u μ a dimensionless Lorentzviolating background 4-vector, has been used to study the Casimir effect between two parallel conducting plates embedded in a nontrivial spacetime background [49][50][51][52][53], with the two plates modelled by two delta-like potentials in the limit λ → ∞ . It has also been used to investigate the Casimir effect for perfectly conducting sphere [54] and cylinder [55]. Therefore, any physical configuration in a LV background, with prescribed boundary/initial conditions, must be sensitive to the nontrivial vacuum.
The model we shall consider in this paper is given by equation (6) with the particular choice h μ ν = u μ u ν , and a semitransparent located at the z = z 0 plane. Under these assumptions, the field equation becomes subject to a prescribed initial condition for the scalar field at the z = 0 plane. This model is appropriate to investigate the propagation of optical scalar beams in a Lorentz-violating background in the presence of a semitransparent mirror, which is the main goal of this paper.

Green's function approach
In this section we employ Green's function techniques to investigate the propagation of optical beams. We look for solutions of equation (7) in the form of monochromatic waves, i.e.
where ω is the angular frequency and r, ( ) f w is the frequency-dependent complex amplitude of the field. In the known zoology of structured light, they are represented in the 2D transverse plane to the propagation axis, which we take along the z-direction. As the geometry of the problem suggests, it is convenient to choose a coordinate system in which one of the axes coincide with the direction of the beam propagation. Therefore, upon propagating a distance z, the complex amplitude can be expressed as: w ¢ ¢ l^^i s the corresponding Green's function in the frequency domain, which satisfies the inhomogeneous Helmholtz equation for a pointlike source: We now focus on the calculation of the GF which solves equation (9). The GF we consider has translational invariance in the directions perpendicular to the z axis, that is in the transverse x and y directions, while this invariance is broken in the z direction due to the presence of the semitransparent mirror. Exploiting this symmetry we further introduce the Fourier transform in the transverse coordinates With all of the above, it is found that the reduced GF satisfies the differential equation he solution to equation (11) is simple, but not straightforward. For solving it we employ a method similar to that used for obtaining the GF for the one-dimensional δ-function potential in quantum mechanics, where the free GF is used for integrating the GF equation with δ-interaction. In the problem at hand, to integrate equation (11), we assume that the GF in the absence of the mirror is known, i.e. g z z , u 0 ( ) ( ) ¢ . Using the properties of the Dirac delta, one can easily integrate equation (11) to obtain: , and then substitute into the same equation, yielding This result implies that the reduced GF in the presence of the mirror is fully determined by the reduced GF in its absence. As expected, for λ = 0 one recovers the solution in the absence of the mirror. Besides, taking the limit λ → ∞ and evaluating the result at z z 0 ¢ = one gets that , which corresponds to the Dirichlet condition we usually impose at a perfect mirror.
Following the above discussion, the problem now consists in evaluating the reduced GF in the absence of the mirror, which satisfies the differential equation f u μ is assumed to be small, one can directly solve this equation perturbatively up to some order in u μ . However, one can follow standard strategies of electromagnetism to compute the reduced GF in an analytical fashion, valid to all orders in u μ . The general solution for an arbitrary u μ is simple, but it is not illuminating at all. So, we shall consider two special cases which correspond to the spatial part of u μ being (i) perpendicular and (ii) parallel to the mirror, i.e. (i) u μ = (0, 0, 0, u z ) and (ii) u μ = (0, u x , u y , 0), respectively. In the following we compute the reduced GF in these two cases.
3.1. Case (i) : u μ = (0, 0, 0, u z ) In this case the reduced GF equation (14) simplifies to . The choice between them is dictated by the physical requirement that g u 0 z ( ) must be an outgoing wave as z recedes to infinity, since the source is located at z¢: The coefficients A and B have to be determined by imposing the boundary conditions at the singular point z z = ¢. The final result is The Green's function in coordinate representation is obtained by substituting this result into the ansatz (10) and computing the momentum integrals. Fortunately, in this case one can obtain a closed form expression. The result is G z z e R r r , ; , The corresponding reduced GF in the presence of the semi-transparent mirror is obtained by substituting equation (18) into equation (13). One gets The final expression for the Green's function in coordinate representation cannot be evaluated analytically, but we solve it numerically in the next section.

Case
In this case the reduced GF satisfies he solution to this equation is simple and follows the same lines as above. In this case, the boundary conditions at the singular point z z = ¢ are: Substitution of the reduced GF (22) into equation (13) yields the GF in the presence of a semi-transparent mirror: Once again, the final Green's function in coordinate representation cannot be evaluated in an analytical fashion, but it can be treated numerically.
With the reduced Green's functions (18) and (22) we are able to analyse the propagation of scalar optical beams in a Lorentz violating vacuum for any initial input field. To include the effect of the semi-transparente mirror instead we have to use the reduced GFs (19) and (23).

Timelike case
Now we briefly discuss the timelike case, which is defined by the background four vector u μ = (u 0 , 0, 0, 0). In this case the reduced Green's function equation (14) reduces to We observe that, unlike the spacelike cases considered above, where components of the 4-vector enters into the reduced Green's functions (18) and (22)  Physically, this can be interpreted as a modification in the speed of light, which potentially can be tested in experiments. However, from the mathematical point of view, the beam propagation is carried out such as it happens in conventional vacuum. The consequences of this choice are interesting though evident so that they do not require at this stage further theoretical study. Therefore, hereafter we will primarily focus on the spacelike cases.

Optical beam propagation
Once we have computed the Green's function, we are ready to investigate the propagation of an optical scalar beam in a Lorentz-violating vacuum as described by the constant four-vector u μ , and in the presence of a semitransparent mirror of strength λ.
( ) f w^b e the field on the input plane located at z = 0. The field propagates through a Lorentzviolating medium and find, at z = z 0 , a semi-transparent mirror with degree of transparency given by λ. We now ask for the propagated field at a plane located at z > 0. It is given by equation (8), with the appropriate Green's function. Substituting the Fourier decomposition of the GF, given by equation (10), into equation (8) his expression is ready to be applied to an arbitrary input field. However, the intricate expressions for the Green's function limit the possible analytical evaluation of the integral (25), but we can turn to numerical calculations. In the next section we focus on the propagation of Gaussian beams.
The quantitative study is carried out by evaluating the intensity I r r Since the background field is constant, energy and momentum are conserved. Therefore, the total power P z d I r r will be preserved during propagation, independently of the four-vector u μ .
On the other hand, in order to quantify the effects of the Lorentz-violating background and the semitransparent mirror, we introduce a quantity to measure the closeness of the propagated field with respect to that in conventional vacuum (defined by u μ = 0). To this end we introduce the fidelity, defined by [40] f u u u u u u u u , , In a few words, the fidelity is a measure of the structural resemblance between the propagated fields in Lorentzviolating backgrounds characterized by u μ and u¢ m . By definition, f (λ) (u μ , u μ ) = 1, which means that the overlap between the propagated fields is maximum. With the help of the fidelity we shall investigate the usefulness of the mirror in detecting LV effects.
We close this section with an important comment regarding the propagation of optical beams. In general, the study of light propagation requires vector solutions to Helmholtz equation, thus allowing the identification of the polarization states. For example, the propagation of vector Bessel beam [56] and vector Helmholtz-Gauss beams [57] have been analysed in the Lorentz-symmetric case. It would be of wide interest to include the LV effects in the propagation of vector beams. Here, we focus on the propagation of scalar beams, which is appropriate for linearly polarized light under certain circumstances [58], and we leave the general analysis for future investigations.

Gaussian beam propagation
The unique properties of structured light are now extensively used for many technological applications and are still under current investigation both theoretically and experimentally. Among them, the Gaussian beam (GB) is a fundamental model in optics with a wide variety of applications. It represents a good approximation to the electric field amplitude of the fundamental mode at the output of a standard laser [59], as well as to the electric amplitude in weakly guiding waveguides with cylindrical symmetry [60]. In practice, it is used in innumerable experimental devices, such as laser stabilization cavities [61] and gravitational wave detectors [62]. Recently, GBs have been considered as a test bed for the impact of Earth's gravity on beam propagation [23,24].
A GB can be described as a scalar wave whose wave fronts are predominantly transverse to the main direction of propagation, here taken as the z-axis, with the perpendicular components of the wave vector k ⊥ exhibiting a Gaussian distribution. The amplitude distribution of a Gaussian beam of waist size w 0 at the transmitting aperture located at z = 0 is taken as [63] . This result implies that the transverse profile of the propagated field remains axially symmetric. This integral cannot be evaluated analytically in terms of known special functions and hence we proceed to numerically evaluate it. In the numerical simulations we suppose that we have a He-Ne laser beam (with wavelength of 632.8 nm) that is focused to a spot w 0 = 0.5 mm. Figures 1(a) and (b) shows the field intensity profile I r r normalized in units of I 0 as a function of the dimensionless coordinates r ⊥ /w 0 and z/w 0 for different values of the parameter u z . First, we observe that the Gaussian profile of the beam remains unaffected as it propagates. Second, we can see that the LV parameter u z affects the evolution of the beam radius and beam waist as a function of z, moving away from a focal point. Qualitatively: the beam radius and beam waist increase as a function of u z . Later we back to the discussion of these observations in a quantitative fashion.
In order to quantify the dissimilarity of the propagated field in a Lorentz-violating vacuum with respect to the propagation in conventional vacuum we resort to the fidelity, as defined by equation (27). Using the propagated field distribution (30) we obtain the analytical expression

+ -
Clearly, the fidelity is unity for u z = 0 and drops abruptly to zero as |u z | → 1. If u z is assumed a very small parameter, the fidelity is close to unity and hence Lorentz-violating effects modifies solely slightly the properties of the beam. In order to better understand the effect of Lorentz symmetry violation upon the propagation of the Gaussian beam we recall that its wavefronts are predominantly transverse to the propagation direction. In fact, in most actual implementations |k ⊥ | = |k z | so that the paraxial approximation is adequate. Interestingly, the integral (30) can be evaluated analytically in the paraxial regime. To this end we have to expand the k z ʼs appearing in equation (30) in a Taylor series: [64]. We observe that the amplitude term varies slowly with k ⊥ , so the zeroth order term is sufficient, but the phase term is more sensitive, thus the second order term is necessary. Under this approximation one can perform the integral in equation (30) to obtain: is the radius of the beam at any position z (measured from the focus), z u z z u , a r c t a n = is the Gouy phase and R z u z z u z , 1 is the radius of curvature of the beam's wavefronts at z. Since |u z | < 1, the Rayleigh range in the presence of Lorentz symmetry breaking is always lesser than the vacuum value z k w 0 2 R 0 0 2 ( ) = . This implies that the radius of the beam satisfies: w(z, u z ) > w(z, 0), i.e. the Lorentz-violating term disfavors the focalization of the beam, thus confirming our numerical findings in figures 1(a) and (b).
The Gouy phase, which is a phase advance acquired by a Gaussian beam around the focal region during propagation, plays an salient role in optics [65,66]. For example, it is important for the lateral trapping force at the focus of optical tweezers and leads to phase velocities that exceed the speed of light in vacuum [67] and it has been experimentally observed in terahertz beams [68]. In the problem at hand one finds that the Gouy phase in the presence of LV is smaller than the Gouy phase in free space, i.e. ψ(z, u z ) < ψ(z, 0). Physically, this means that the contribution to the advancement of the phase of the beam becomes slowed down due to the LV term as compared to that in vacuum. Finally, the qualitative conclusions we extracted from figures 1(a) and (b) agree with the fact that the radius of curvature increases due to the Lorentz symmetry breaking, i.e. R(z, u z ) > R(z, 0).
As evinced by the nearly unit fidelity and small changes in the properties of the Gaussian beam during propagation for small values of u z , we have to use alternative methods to detect a signal of Lorentz symmetry breaking with an optical setup as considered in this work. To this end, in the following we shall analyse the effect of the semi-transparent mirror. In this case, the propagated field distribution is given by equation (25), with the reduced Green's function (19) and the angular spectrum k i.f. ( ) f^. We obtain parameter λ. A similar effect is produced when a Gaussian beam propagating in a medium with a given index of refraction impinges upon a planar interface and transmits to a medium with a large index of refraction [69]. As evinced in figures 2(a) and (b), the qualitative behaviour of the field intensity profile, before and after the mirror, is quite similar to the behaviour of a Gaussian beam propagating in vacuum (i.e. without Lorentz symmetry violation). For the sake of looking for significant differences between the propagated field in vacuum (u z = 0) and in a Lorentz breaking background (u z ≠ 0), we now evaluate the fidelity, as defined in equation (27). The expression is cumbersome and not illuminating, so here we only present the numerical results. In figure 2(c) we show the fidelity as a function of λw 0 and z 0 w 0 . The dark regions show us that there are different parameter sets where the fidelity drops to almost zero. This means that, in such regions, the field intensity profile in the presence of a LV background has no resemblance with the corresponding field intensity in vacuum. This result is quite interesting and evinces the usefulness of the mirror in the present analysis: in the absence of the mirror, the fidelity (31) is close to unity for small values of u z , while the presence of the mirror produces regions where the fidelity is near to zero, thus making differentiable the presence of the LV background from the vacuum. f =^normalized in units of I 0 as a function of the dimensionless coordinates x/w 0 , y/w 0 and z/w 0 for a particular orientation of the vector u ⊥ . Since u ⊥ is assumedly small, the effects upon the propagated field are expected to be small as well. Therefore, we take the (unrealistic) large values u x = 0.4 and u y = 0.8 for a better appreciation of the effects. Figures 3(a) (b) shows the cross section of the normalized density profile at  x/w 0 = 0.5 (y/w 0 = 0.5). We observe that the effect of the breaking of Lorentz symmetry is a deflection in the direction of the beam along the vector u ⊥ . Besides, figure 3(c) shows that the cross section at z/w 0 = 0.5. One observes that the propagated field exhibits an elliptical cross-section, thus implying that the initially circular beam profile becomes an elliptical profile due to the effects of Lorentz symmetry violation.
For a better understanding of the general properties of the propagated field described from the numerical results in figure 3, we now evaluate the integral (34) in the paraxial regime. To this end, we rotate our coordinate system around the propagation direction such that u ⊥ coincides with one of the axes, for example the x axis. Power expanding the square roots in equation (34) for k ⊥ = k 0 , keeping the second order in the exponential factor and the zeroth order in the amplitude we get is the radius of the beam at position z and z u z z u , a r c t a n is the Gouy phase. Clearly, this result shows that the beam parameters are different in the x and y directions. While the propagation in the y direction is exactly the same as in standard vacuum (i.e. in the Lorentz symmetric case), the propagation in the x direction becomes LV-dependent. Since |u x | < 1, one can directly verify that z R (u x ) < z R (0), thus implying that radius of the beam increases in the u ⊥ -direction. This explains the broadening of the beam observed in figure 3. Also, it is clear that the Gouy phase is acquired at a different rate in each direction, being larger in the Lorentz-violating direction.
We now investigate the effect of the semi-transparent mirror. In this case, the propagated field is given by: normalized in units of I 0 ) as a function of the dimensionless coordinates x/w 0 , y/w 0 and z/w 0 for u x = 0.4 and u y = 0.8. In figure 4, at left (center) we show the cross section of the normalized density profile at y/w 0 = 0.5 (x/w 0 = 0.5), with the mirror located at z 0 /w 0 = 1 and transparency λw 0 = 1. The inset plots show the same cross sections for the less transparent case with λw 0 = 3. As expected, once the beam is transmitted, it follows a similar behaviour to that in a Lorentz-violating vacuum, with the intensity diminished according to the transparency of the mirror. This becomes clear in the insets, which shows the intensity profile for a less transparent mirror. Figure 4, at right, show the cross section at z/w 0 = 0.5. As expected, the beam preserves its elliptical profile but its intensity diminishes.
As before, by means of the fidelity, one can quantify the dissimilarity between the propagated field in a Lorentz breaking vacuum as compared with that in conventional vacuum. However, in the present case there is a subtlety we have to consider. As shown in equation (34), the momentum along the propagation direction k z can exceeds the natural frequency of the field, thus implying that the effects due to Lorentz symmetry breaking will become appreciable at large distances. This suggests that, in order to look for LV effects at short distances, we must analyse the fidelity at planes transverse to propagation, i.e. at fixed z. One cannot obtain an analytical expression analogous to equation (31), but we can resort to numerical calculations. In figure 5(a) we show the transverse fidelity f (0) (u x , 0) for different values of the distance z (in units of w 0 ). As expected, we observe that the transverse fidelity decays faster as we move along z. In short, the LV effects can be better appreciated as far we are from the input source. As in the case of a LV background pointing along the propagation direction, we now ask if the presence of the mirror could help or not for looking LV effects in the propagated field distribution in the case at hand. To this end, we now evaluate numerically the transverse fidelity at z = w 0 . Figure 5(b) shows a density plot for different values of the transparency λ and the position of the mirror z 0 . As a consistency check we first observe that the fidelity is close to one near to the axes, as expected. Also, we can see the emergence of a dark region where the fidelity drops to smaller values, thus implying that the propagated field in the LV vacuum differs from the propagated field in a conventional vacuum. Therefore, in both cases (i) and (ii), the presence of the mirror plays a fundamental role since it allow us to distinguish Lorentz-violating effects.

Summary and discussion
In this paper we analysed the propagation of optical beams in a Lorentz-violating vacuum characterized by a constant 4-vector u μ and in the presence of a semi-transparent mirror. The origin of this 4-vector, which breaks Lorentz symmetry, can be interpreted in different ways. From the particle physics side, candidate theories for quantum gravity (such as strings) predict the appearance of nonzero expectation values of some vector and tensor fields, thus introducing privileged directions in spacetime. On the other hand, condensed-matter systems naturally break Lorentz invariance. They exhibit an effective Lorentz symmetry at low energies whose breaking manifests as the various electronic phases of the system. In this sense, the nontrivial vacuum arising from the standard model extension can be seen as an anisotropic optical media, and then, optical phenomena could be used as a test bed for the physics at the Planck scale.
The rapid developments of artificially engineered materials on demand have open a new avenue to experimentally test phenomena predicted by nonoptical theories which are difficult to be observed in the lab. For example, by means of metamaterial implementations it has been possible to bring to the laboratory wellknown gravitational phenomena that are difficult to observe directly using astronomical data, such as gravitational lensing. In a similar fashion, nematic liquid crystals and lead glasses have been used to investigate light propagation in strongly nonlocal nonlinear media as well as to study the spontaneous breaking of chiral symmetry. Merging the above ideas, here we suggest that optical analogue systems can be used to mimic the effects caused by the breaking of Lorentz symmetry. In particular, here we investigate the effects of a nontrivial vacuum and a semi-transparent mirror upon the propagation of structured optical beams.
The LV part of the field theory we consider is characterized by the tensor coupling k α β μ ν F αβ F μν , where the dimensionless tensor k α β μ ν has constant components and parametrizes the breakdown of Lorentz invariance. With a particular choice of this tensor, given by equation (4), the two bosonic degrees of freedom of the electromagnetic field are found to satisfy a Lorentz-violating massless Klein-Gordon equation. The presence of the mirror is implemented by means of a delta-type potential in the Lagrangian density, such that the degree of transparency is controlled by a parameter λ. The limit λ → ∞ is equivalent to impose Dirichlet boundary conditions on the field at the surface. Using Green's function techniques we solve the field equation and study the propagation of a Gaussian beam under these conditions. Our numerical results evince how the symmetry breaking parameter affects field propagation. We confirm our finding by computing analytically the propagated field in the paraxial regime. Salient features are, for instance, the modification of the Gaussian beam parameters (i.e. Rayleigh range, radius of the beam, Gouy phase and radius of curvature of the wavefronts) as a function of the LV parameter. Furthermore, we find that the presence of the mirror, the fidelity drops to zero in some regions of the parameter space defined by the position of the mirror and the degree of transparency. This provides us a mechanism for distinguishing the effects of Lorentz symmetry braking.
The study of beam focusing has attracted great interest due to its possible applications, for example, in laser materials processing or surgery. In the latter case, it is highly important to focus a laser beam down to the smallest spot possible to maximize intensity and minimize the heated area. This has motivated the focusing of optical beams by diverse mechanisms. For example, the authors in [70] studied the propagation of a nondiffractive Bessel beam in a nonlinear Kerr medium, and the main finding is that the focusing or defocusing of the beam can be tuned by means of the sign of the nonlinear index coefficient. Here we reach a similar conclusion, with the difference that the propagation is carried out in a linear medium but in the presence of a nontrivial background vector.
We close by commenting on possible realizations of the predicted effects in existing material media. As discussed above, metamaterial implementations represent promising candidates to observe our predictions, since the anisotropy (and hence the background 4-vector) can be changed literally at will. Multifferroic materials fullfill an essential ingredient required to observe optical beam propagation in the media: optical transparency. To the best of our knowledge, the optical transparency of topological materials, which are other potential candidates, has not been investigated yet. However, topological insulators can be understood as a bulk dielectric with a surface Hall effect, and therefore the optical transparent region may correspond to the trivial regions of the bulk dielectric. Of course, this requires a formal study. On the other hand, the optical transparency of light in multiferroic materials, such as Ca 2 CoSi 2 O 7 , Sr 2 CoSi 2 O 7 and Ba 2 CoGe 2 O 7 , has been extensively studied both from the theoretical and experimental sides [71,72]. Magnetoelectric multifferoics are simultaneously both ferromagnetic and ferroelectric, and the dynamical or optical magnetoelectricity appears due to the coupling of the coexisting order parameters. When the optical magnetoelectric effect is strong enough, the aforementioned materials become fully transparent for a given propagation direction, while they absorb light travelling in the opposite direction. Therefore, they represent a good test bed for the predictions presented in this paper. The next obstacle is related to the geometry of the material. We must guarantee that the material exhibits an axially symmetric magnetoelectric tensor. This is of course a great experimental challenge. Along this line, recent studies report cylindrically symmetric optical transparent magnetoelectric materials composed by magnetostrictive Fe 72.5 Si 12.5 B 15 microwires and piezoelectric poly(vinylidene fluoride-trifluoroethylene) [73]. All in all, our predictions are theoretical for the time being, but we expect that with the recent advances in optical transparent magnetoelectric materials, a properly engineered material (on demand) may simulate the properties of the Lorentz-violating vacuum considered in this work, where the reported properties could be observed.