Temporal Faraday and Other Magneto-Optic Effects

Abstract

We consider temporal optical effects in the presence of static fields, and more generally in anisotropic optical media, such as magnetized materials. Magneto-optical effects are due not just to phase shifts between the different eigenmodes, as in static media, but also to temporal variations in the frequency and mode amplitudes. Faraday rotations, Cotton–Mouton effects and other polarimetric processes due to static magnetic or electric fields are discussed. Examples of magneto-plasmas are compared with those in nonlinear Kerr media. These temporal processes could be of general interest in plasma physics and photonics.

1. Introduction

Temporal effects in optical media have received considerable attention in recent years [1,2,3], and the area of spacetime optics has shown considerable growth (see the reviews [4,5]). Several different temporal processes and configurations have been studied, including time refraction and time reflection [6,7], temporal beam-splitters [8,9], photon acceleration [10,11], spatiotemporal Bragg gratings [12] and the related problems of time crystals [13,14,15], light amplification [16,17], temporal quantum optics [18,19,20,21] with direct extensions to electron QED [22], time-varying metamaterials [23,24,25], and negative refraction in epsilon near-zero systems [26,27]. Quite recently, the case of Faraday rotations induced by a temporal interface was studied [28].

In this paper, we return to the temporal Faraday effect and examine the more general problem of temporal magneto-optics. It is known that time-domain magneto-optics has been considered in the past, especially in condensed matter physics [29,30,31]. But here, special attention is given to the frequency and amplitude variations that occur at the temporal interfaces. We also show that Faraday rotation cannot be observed at a single temporal interface as recently suggested [28], but that a sequence of two opposite temporal interfaces is necessary. We consider not only Faraday rotation, but also a Cotton–Mouton type of ellipticity, induced by temporal processes in birefringent media. Polarization rotation and induced ellipticity result from three different temporal processes, which are wave amplification, frequency shifts and a phase difference between the two eigenmodes propagating in the medium. In static media, only the phase difference will remain, due to different wavenumbers.

The contents of this paper are as follows. First, in Section 2, we consider a single temporal interface that we describe as a time-refraction process. We show that, in this case, no Faraday rotation can be observed. In Section 3, we examine the restored frequency shifts that result from two consecutive temporal interfaces with opposite signs, and calculate the resulting Faraday rotation, for transmitted and reflected waves. Notice that this rotation is fundamentally distinct from that occurring in a static medium, because it depends on the temporal and not on the spatial properties of the medium. It can therefore be observed in different physical geometries. In Section 4, we consider two specific physical situations. The first is wave propagation in a non-stationary plasma. We calculate the temporal Faraday effect that can be observed for wave propagation along the direction of a static magnetic field ${\mathbf{B}}_0$. Similar effects persist for oblique propagation, and the Cotton–Mouton effect is discussed for perpendicular propagation. The other example is related to a nonlinear optical medium, which exhibits the well-known phenomenon of Kerr birefringence, but now considered in a time-varying medium. This example is analogous to the Cotton–Mouton effect in a non-stationary plasma. In Section 5, we examine the more general situation where the sharp temporal interfaces are replaced with arbitrary temporal changes in the optical medium. Temporal magneto-optical effects can still be observed, as long as the temporal changes are temporally symmetric, and bring the medium to its initial state. Finally, in Section 6, we state our conclusions.

2. Single Temporal Interface

We consider an optical medium with an axis with symmetry, for instance, a static magnetic field ${\mathbf{B}}_0$. We assume a wave propagating along a direction, with wavevector ${\mathbf{k}}_0$ and frequency ${\omega }_0$, before entering the anisotropic medium, as described by the electric field

$${\mathbf{E}}_0(\mathbf{r},t)=\sum _{\pm }{\mathbf{E}}_{0\pm }\left(t\right)exp(i{\mathbf{k}}_0·\mathbf{r})+c.c.$$

(1)

where $(\pm )$ represent two orthogonal modes (not only right- and left-hand circular polarization, but also linear orthogonal modes, according to the physical situation), and

$${\mathbf{E}}_{0\pm }\left(t\right)={\mathbf{E}}_{0\pm }exp(-i{\omega }_{0}t)+{{\mathbf{E}}^{\prime }}_{0\pm }^{*}exp\left(i{\omega }_{0}t\right),$$

(2)

where the amplitudes ${\mathbf{E}}_{0\pm }$ and ${{\mathbf{E}}^{\prime }}_{0\pm }$ correspond to modes with the same wavenumber $k_0$ and frequency ${\omega }_0$, but propagating in opposite directions. Regarding the initial conditions, at $t=t_0$, we assume linear field polarization along a direction defined by the unit vector $\mathbf{e}$, such that

$${\mathbf{E}}_{0\pm }=E_0\mathbf{e},{{\mathbf{E}}^{\prime }}_{0\pm }=E_0^{\prime }\mathbf{e}.$$

(3)

To be more specific, let us assume propagation along the z-axis, using ${\mathbf{k}}_0=k_0{\mathbf{e}}_z$, parallel to the static magnetic field ${\mathbf{B}}_0=B_0{\mathbf{e}}_z$. For linear polarization at an angle ${\theta }_0$ from the x-axis, we can write the unit polarization vector as

$$\mathbf{e}={\displaystyle \frac{1}{\sqrt{2}}}\left[exp(-i{\theta }_0){\mathbf{e}}_{+}+exp(+i{\theta }_0){\mathbf{e}}_{-}\right],$$

(4)

where ${\mathbf{e}}_{\pm }$ represents the left- and right-hand unit vectors, with ${\mathbf{e}}_{\pm }=\left({\mathbf{e}}_x\pm i{\mathbf{e}}_y\right)/\sqrt{2}$. We then assume a sudden change in the refractive index of the medium $n(\omega ,\mathbf{k})$, occurring at $t=t_0$, such that, for invariant ${\mathbf{k}}_0$, we have a frequency shift determined by

$$n\left(\omega \right)=\left\{\begin{array}{c}{n}_0\left(\omega \right),(t<t_0)\hfill \\ n_{\pm }\left(\omega \right),(t\ge t_0)\hfill \end{array}\right.,\omega \left(t\right)=\left\{\begin{array}{c}{\omega }_0,(t<t_0)\hfill \\ {\omega }_{\pm }=(n_0/n_{\pm }){\omega }_0,(t\ge t_0)\hfill \end{array}\right.,$$

(5)

It is well known that, in order to satisfy the validity of Maxwell’s equation for all times, the fields have to satisfy the continuity conditions

$$\mathbf{D}(\mathbf{r},t_0-\delta )=\mathbf{D}(\mathbf{r},t_0+\delta ),\mathbf{B}(\mathbf{r},t_0-\delta )=\mathbf{B}(\mathbf{r},t_0+\delta ),$$

(6)

for $\delta \to 0$, where the displacement and magnetic fields are related to the wave electric field by the relations

$$\mathbf{D}=\overline{\overline{ϵ}}·\mathbf{E},(\nabla \times \mathbf{E})=-{\displaystyle \frac{\partial \mathbf{B}}{\partial t}},$$

(7)

and $\overline{\overline{ϵ}}$ is the dielectric tensor. Explicit expressions of this tensor can be found in textbooks, such as [32] for plasmas, and [33] for anisotropic media. This allows us to write for each wavevector component ${\mathbf{k}}_0$ of the spectrum, valid for $t=t_0$, the following wave amplitude relations:

$$n_0^2\left(E_0+E_0^{\prime }\right)=n_{\pm }^2\left(E_1+E_1^{\prime }\right),$$

(8)

and

$${\displaystyle \frac{1}{{\omega }_0}}\left(E_0-E_0^{\prime }\right)={\displaystyle \frac{1}{{\omega }_{\pm }}}\left(E_1-E_1^{\prime }\right).$$

(9)

It is now useful to define the parameters

$${\alpha }_{\pm }={\displaystyle \frac{n_0}{n_{\pm }}}={\displaystyle \frac{{\omega }_{\pm }}{{\omega }_0}},$$

(10)

which allow us to write the amplitude relations

$${\alpha }_{\pm }^2\left(E_0+E_0^{\prime }\right)=E_1+E_1^{\prime },{\alpha }_{\pm }\left(E_0-E_0^{\prime }\right)=E_1-E_1^{\prime }.$$

(11)

We can then obtain the transmission and reflection coefficients for each polarization state, $T_{\pm }$ and $R_{\pm }$, as

$$T_{\pm }\equiv {\left({\displaystyle \frac{E_1}{E_0}}\right)}_{\pm }={\displaystyle \frac{{\alpha }_{\pm }}{2}}{\left[({\alpha }_{\pm }+1)+({\alpha }_{\pm }-1){\left({\displaystyle \frac{E_0^{\prime }}{E_0}}\right)}_{\pm }\right]}_{\pm },$$

(12)

and

$$R_{\pm }\equiv {\left({\displaystyle \frac{E_1^{\prime }}{E_0}}\right)}_{\pm }={\displaystyle \frac{{\alpha }_{\pm }}{2}}{\left[({\alpha }_{\pm }-1)-({\alpha }_{\pm }+1){\left({\displaystyle \frac{E_0^{\prime }}{E_0}}\right)}_{\pm }\right]}_{\pm }.$$

(13)

They are sometimes called the temporal Fresnel formulae, and correspond to a straightforward generalization of previous results [8,20]. We notice that they depend on the ratio between the initial amplitudes for propagating and counter-propagating modes, $(E_0^{\prime }/E_0)$, which clearly indicates that this is a linear four-wave mixing process. This contrasts with the case of a static boundary, which satisfies different boundary conditions and where the corresponding Fresnel formulae are independent of $E_0^{\prime }$. We also notice that, at this point, we cannot talk about a total field polarization state, because for $t\ge t_0$, each mode oscillates with different frequencies, ${\omega }_{+}\ne {\omega }_{-}$. We simply have two independent modes, with two different polarizations and frequencies, where the total field associated with the wavevector ${\mathbf{k}}_0$ is determined by the sum of these two modes:

$${\mathbf{E}}_1(\mathbf{r},t)=\sum _{\pm }{\mathbf{E}}_{1\pm }\left(t\right)exp(i{\mathbf{k}}_0·\mathbf{r})+c.c..$$

(14)

with

$${\mathbf{E}}_{1\pm }\left(t\right)={\mathbf{E}}_{1\pm }exp(-i{\omega }_{\pm }t)+{{\mathbf{E}}^{\prime }}_{1\pm }^{*}exp\left(i{\omega }_{\pm }t\right),$$

(15)

where ${\omega }_{\pm }=k_{0}c/n_{\pm }$.

3. Restored Frequency

In order to consider possible polarimetric effects on a single wave mode, we need a process that can restore the initial frequency, ${\omega }_0$. This can be achieved using a second temporal boundary with the opposite sign, occurring at some later time $t_1=t_0+\tau$, where the refractive index returns to its initial value: $n(\omega ,t\ge t_1)=n_0(\omega ,t<t_0)$. These two consecutive temporal boundaries define what could be called a temporal beam-splitter [8].

It is particularly useful to derive the final transmission and reflection coefficients, $T_{2\pm }$ and $R_{2\pm }$, resulting from this temporal beam-splitter in the presence of birefringence effects. For this purpose, we take the particular case of $E_0^{\prime }=0$. Using Equations (12) and (13), and following the procedure outlined in reference [8], we can determine the fields at $t=t_1$, as given by

$$T_{2\pm }\equiv {\left({\displaystyle \frac{E_2}{E_0}}\right)}_{\pm }={\displaystyle \frac{{\alpha }_{\pm }{\beta }_{\pm }}{4}}\left\{({\alpha }_{\pm }+1)({\beta }_{\pm }+1)e^{-i({\omega }_{\pm }-{\omega }_0)\tau }+({\alpha }_{\pm }-1)({\beta }_{\pm }-1)e^{i({\omega }_{\pm }+{\omega }_0)\tau }\right\},$$

(16)

and

$$R_{2\pm }\equiv {\left({\displaystyle \frac{E_2^{\prime }}{E_0}}\right)}_{\pm }={\displaystyle \frac{{\alpha }_{\pm }{\beta }_{\pm }}{4}}\left\{({\alpha }_{\pm }-1)({\beta }_{\pm }+1)e^{i({\omega }_{\pm }-{\omega }_0)\tau }+({\alpha }_{\pm }+1)({\beta }_{\pm }-1)e^{-i({\omega }_{\pm }+{\omega }_0)\tau }\right\},$$

(17)

where ${\beta }_{\pm }=1/{\alpha }_{\pm }$. This simplifies to

$$T_{2\pm }\equiv {\left({\displaystyle \frac{E_2}{E_0}}\right)}_{\pm }=\left[cos\left({\omega }_{\pm }\tau \right)-{\displaystyle \frac{2i}{{\alpha }_{\pm }}}({\alpha }_{\pm }^2+1)sin\left({\omega }_{\pm }\tau \right)\right]e^{i{\omega }_0\tau },$$

(18)

and

$$R_{2\pm }\equiv {\left({\displaystyle \frac{E_2^{\prime }}{E_0}}\right)}_{\pm }={\displaystyle \frac{2i}{{\alpha }_{\pm }}}({\alpha }_{\pm }^2-1)sin\left({\omega }_{\pm }\tau \right)e^{-i{\omega }_0\tau },$$

(19)

This means that the total field of the mode $({\omega }_0,{\mathbf{k}}_0)$, coming out of the temporal beam-splitter, is

$${\mathbf{E}}_0(\mathbf{r},t\ge t_0+\tau )=E_0\sum _{\pm }\left[T_{2\pm }\left(\tau \right){\mathbf{e}}_{\pm }{e}^{i{\mathbf{k}}_0·\mathbf{r}-i\omega t}+R_{2\pm }^{*}\left(\tau \right){\mathbf{e}}_{\pm }{e}^{-i{\mathbf{k}}_0·\mathbf{r}+i\omega t}\right]+c.c..$$

(20)

This is to be compared with the initial field (1) and (2). In particular, it means that the total transmitted field is determined by

$${\mathbf{E}}_2\left(t\right)=E_0\left(T_{2+}{\mathbf{e}}_{+}+T_{2-}{\mathbf{e}}_{-}\right),$$

(21)

where we have used a simplified notation, which has to be compared with

$${\mathbf{E}}_0\left(t\right)=E_0\left({\mathbf{e}}_{+}+{\mathbf{e}}_{-}\right),$$

(22)

This shows that three different effects were introduced by the temporal beam-splitter: (i) field amplification, due to the factors ${\alpha }_{\pm }$; (ii) rotation of the polarization direction; (iii) the creation of ellipticity. The first effect already exists in isotropic media. The other two are due to anisotropy, and will be illustrated next. To understand them, we write Equation (18) in the form

$$T_{2\pm }=\left|T_{2\pm }\right|exp\left(i{\phi }_{\pm }\right),$$

(23)

with

$$\left|T_{2\pm }\right|=\sqrt{{cos}^2\left({\omega }_{\pm }\tau \right)+{\displaystyle \frac{1}{4{\alpha }_{\pm }^2}}{\left({\alpha }_{\pm }^2+1\right)}^2{sin}^2\left({\omega }_{\pm }\tau \right)},$$

(24)

and

$$tan{\phi }_{\pm }={\displaystyle \frac{\left({\alpha }_{\pm }^2+1\right)}{2{\alpha }_{\pm }}}tan\left({\omega }_{\pm }\tau \right).$$

(25)

For the trivial case of ${\alpha }_{\pm }=1$, we are reduced to $|T_{2\pm }|=1$ and ${\phi }_{\pm }={\omega }_{\pm }\tau$, as expected. Similarly, for the reflected wave, we would obtain

$${{\mathbf{E}}^{\prime }}_2\left(t\right)=E_0\left(R_{2+}{\mathbf{e}}_{+}+R_{2-}{\mathbf{e}}_{-}\right),$$

(26)

to be compared with ${{\mathbf{E}}^{\prime }}_0\left(t\right)=0$. The amplitudes of the above transmission and reflection coefficients oscillate as a function of the temporal width $\tau$, as in the case of the usual spatial beam-splitter. This clearly shows the existence of temporal interference, different for each mode, as illustrated in Figure 1. The above results are applied to a specific physical situation in the next section, where temporal processes in the presence of static magnetic and electric fields will be considered.

4. Specific Examples

4.1. Magnetized Plasmas

As a first example, let us consider a plasma in a static magnetic field. We assume that the plasma is suddenly created at time $t=t_0=0$, and vanishes at time $t=\tau$. We assume the most interesting case of parallel propagation, which means wave propagation in the direction of the static magnetic field, and take $n_0=1$ for $t<0$. Alternatively, we could assume the sudden creation of a (quasi) static magnetic field in a pre-created plasma, but the physical conditions would be more difficult to define. The eigenmodes ± can therefore be identified with right- and left-hand circularly polarized modes [32]. This leads to

$${\alpha }_{\pm }={\displaystyle \frac{1}{\sqrt{{ϵ}_{\pm }\left(\omega \right)}}},{ϵ}_{\pm }\left(\omega \right)=1-{\displaystyle \frac{{\omega }_p^2}{\omega (\omega \pm {\omega }_c)}},$$

(27)

where ${\omega }_p$ is the electron plasma frequency, and ${\omega }_c=e{B}_0/m_e$ is the electron cyclotron frequency. For a given initial frequency $\omega (t<0)={\omega }_0>({\omega }_p,{\omega }_c)$, we obtian

$${\omega }_{\pm }={\displaystyle \frac{{\omega }_0}{n_{\pm }}}={\omega }_0{\alpha }_{\pm },$$

(28)

These new frequencies are therefore determined by the equation

$${\omega }_{\pm }^2-{\displaystyle \frac{{\omega }_p^2}{{\omega }_{\pm }({\omega }_{\pm }\pm {\omega }_c)}}-{\omega }_0^2=0.$$

(29)

We can see that the frequency shift can be very strong when we approach the resonant frequency for one of the modes, ${\omega }_{-}\to {\omega }_c$. Strong effects can also be expected near a cutoff, when $n_{\pm }\to 0$. In both situations, we can have ${\omega }_{\pm }\gg {\omega }_0$.

Here, we disregard such critical situations and use, for illustration, the case of very high frequencies, such that ${\omega }_0\gg ({\omega }_p,{\omega }_c)$. In this case, we expect very small frequency shifts, $\mathrm{\Delta }{\omega }_{\pm }\ll {\omega }_0$, as determined by the approximate solution

$$\mathrm{\Delta }{\omega }_{\pm }\equiv ({\omega }_{\pm }-{\omega }_0)=\pm {\displaystyle \frac{1}{2}}{\displaystyle \frac{{\omega }_p^2}{{\omega }_0^2}}{\omega }_c.$$

(30)

We can then calculate the final rotation of the electric polarization state, valid for $t\ge \tau$, as determined by the above expressions for the transmission and reflection coefficients. This leads to

$$T_{2\pm }={\left[1-{\displaystyle \frac{1}{2}}\left(1\mp {\displaystyle \frac{{\omega }_p^2{\omega }_c}{4{\omega }_0^3}}\right){sin}^2\left({\omega }_{\pm }\tau \right)\right]}^{1/2}exp\left(i{\phi }_{\pm }\right),$$

(31)

where we have used the phases ${\phi }_{\pm }$, given by

$$tan{\phi }_{\pm }=\left(1\pm {\displaystyle \frac{{\omega }_p^2{\omega }_c}{2{\omega }_0^3}}\right)tan\left({\omega }_{\pm }\tau \right),$$

(32)

and

$$R_{2\pm }={\displaystyle \frac{{\omega }_p^2{\omega }_c}{2{\omega }_0^3}}sin\left({\omega }_{\pm }\tau \right)exp(\mp i\pi /2).$$

(33)

This is corroborated by the numerical results shown in Figure 2, where the initial polarization state is assumed to be linear. We can then state that a temporal beam-splitter in a plasma produces an effect similar to the well-known Faraday effect, which results from the presence of a static magnetic field $B_0$ parallel to the propagation direction. The significant difference with respect to the spatial Faraday process is due to the changes in the wave mode frequency and amplitude. This difference is directly associated with the energy non-conservation of the temporal process.

In the same figure, we also present the electric fields $E_{+}\left(t\right)$ and $E_{-}\left(t\right)$, as obtained from numerical solutions of the wave equation in the time-varying medium. In order to obtain these solutions, we replaced the sharp temporal changes in the medium with smooth changes using tangent-hyperbolic functions. The difference between the two modes in terms of amplitudes and frequencies is clearly shown. During the time interval $\tau$, the two different modes have circular polarizations in opposite directions and belong to two distinct spectral components. For this reason, we can only talk of polarization changes for $t>\tau$, when the wave emerges from the temporal beam-splitter with the initial frequency ${\omega }_0$.

This magneto-optic effect is not limited to parallel propagation, because two orthogonal polarization states exist in a plasma for high frequency waves propagating along any direction with respect to the static magnetic field. In particular, for wave propagation in a direction perpendicular to ${\mathbf{B}}_0$, the two eigenmodes correspond to linear polarization, parallel and perpendicular to the static field. In this case, we should replace the previous notation $(+,-)$ with a more appropriate notation $(\Vert ,\perp )$. Using, once more, the cold plasma model, we can specify these new modes based on their dielectric functions [32].

$${ϵ}_{\Vert }\left(\omega \right)=1-{\displaystyle \frac{{\omega }_p^2}{{\omega }^2}},{ϵ}_{\perp }\left(\omega \right)=1-{\displaystyle \frac{{\omega }_p^2}{{\omega }^2}}{\displaystyle \frac{{\omega }^2-{\omega }_p^2}{({\omega }^2-{\omega }_{uh}^2)}},$$

(34)

where ${\omega }_{uh}={({\omega }_p^2+{\omega }_c^2)}^{1/2}$ is the upper-hybrid frequency. The frequency shifts observed inside the temporal beam-splitter are now given by

$${\omega }_{\Vert ,\perp }={\displaystyle \frac{{\omega }_0}{n_{\Vert ,\perp }}}={\omega }_0{\alpha }_{\Vert ,\perp },$$

(35)

which is formally identical to Equation (28). This would lead to a change in ellipticity, a characteristic feature of the Cotton–Mouton effect. There is a small problem related to the field continuity relations. A longitudinal field component is needed to define the complete state of the perpendicular wave mode, which is absent before the first temporal boundary, as well as after the second one. However, the longitudinal field component is immediately established in the plasma, in response to the transverse component, because of the negligible electron inertia [28].

4.2. Nonlinear Kerr Medium

Another important example is related to a nonlinear optical medium, and not a plasma. We consider a generic Kerr medium, where the optical birefringence is associated with a static electric field ${\mathbf{E}}_0$, instead of a static magnetic field. This case is similar to the above, with two linearly polarized eigenmodes, but where now the static field modifies the refractive index of the parallel polarization, not the perpendicular one. Instead of Equation (34), we then have [33]

$${ϵ}_{\Vert }\left(\omega \right)=1+{\chi }^{\left(1\right)}\left(\omega \right)+{\lambda }_0\kappa {\left|E_0\right|}^2,{ϵ}_{\perp }\left(\omega \right)=n_0^2\equiv 1+{\chi }^{\left(1\right)}\left(\omega \right),$$

(36)

where ${\lambda }_0=2\pi /k_0$ is the wavelength, ${\chi }^{\left(1\right)}$ is the linear susceptibility, and $\kappa$ is a characteristic constant of the optical material. An increase in the refractive index will induce a down-shift in the parallel frequency mode ${\omega }_{\Vert }$, as given by

$${\omega }_{\Vert }\simeq {\omega }_0\left(1-{\displaystyle \frac{1}{2}}{\lambda }_0\kappa {\left|E_0\right|}^2\right),{\omega }_{\perp }={\omega }_0.$$

(37)

This leads to a change in the polarization ellipticity, as shown by the solutions of the wave equation, represented in Figure 3. For the initial state of linear polarization, we obtain, at the end of the temporal beam-splitter, elliptic polarization, very similar to what would occur in the Cotton–Mouton effect.

5. Slow Time Variations

It is well known that the temporal discontinuities discussed above are only valid for temporal changes that occur on a time scale much shorter than the wave period. This is, in general, very difficult to achieve. We therefore need to extend our discussion to the case of arbitrary temporal variations in the medium. This can be described by a sequence of infinitesimal time steps, from t to $t+dt$[34]. Let us then assume that the refractive index of the medium is described by a continuous function of time, $n\left(t\right)$, to be specified. In analogy with Equation (1), let us consider the total electric field associated with a given wavevector component $\mathbf{k}$:

$$\mathbf{E}(\mathbf{r},t)=\sum _{\pm }{\mathbf{E}}_{\pm }\left(t\right)exp(i\mathbf{k}·\mathbf{r})+c.c..$$

(38)

with

$${\mathbf{E}}_{\pm }\left(t\right)={\mathbf{A}}_{\pm }exp[-i{\varphi }_{\pm }\left(t\right)]+{{\mathbf{A}}^{\prime }}_{\pm }^{*}exp\left[i{\varphi }_{\pm }\left(t\right)\right].$$

(39)

But, in contrast with Equation (2), where the field amplitudes were assumed to be constant, the new field amplitudes ${\mathbf{A}}_{\pm }=A_{\pm }{\mathbf{e}}_{\pm }$ and ${\mathbf{A}}_{\pm }=A_{\pm }{\mathbf{e}}_{\pm }$ are now slowly varying functions of time, and evolve on a time scale much longer than the wave period, and the phase functions are defined by

$${\varphi }_{\pm }\left(t\right)={\int }_{t_0}^t{\omega }_{\pm }\left(t^{\prime }\right)d{t}^{\prime },$$

(40)

where $t_0$ is the initial time, and ${\omega }_{\pm }\left(t\right)$ are the time-varying mode frequencies. Applying the approach of [34] for each mode, it is then possible to show that the field amplitudes evolve as

$${\displaystyle \frac{d{A}_{\pm }}{dt}}=-{\displaystyle \frac{{\mathrm{\Omega }}_{\pm }}{2}}\left\{3A_{\pm }+A_{\pm }^{\prime }exp\left[+2i{\varphi }_{\pm }\left(t\right)\right]\right\},$$

(41)

and

$${\displaystyle \frac{d{A}_{\pm }^{\prime }}{dt}}=-{\displaystyle \frac{{\mathrm{\Omega }}_{\pm }}{2}}\left\{3A_{\pm }^{\prime }+A_{\pm }exp\left[-2i{\varphi }_{\pm }\left(t\right)\right]\right\},$$

(42)

with

$${\mathrm{\Omega }}_{\pm }={\displaystyle \frac{1}{n_{\pm }}}{\displaystyle \frac{d{n}_{\pm }}{dt}}.$$

(43)

These equations can be formally solved [5]. But, for the present purposes, it is useful to consider the plausible case of very weak reflected waves, where we have $\left|A_{\pm }^{\prime }\left(t\right)\right|\ll \left|A_{\pm }\left(t\right)\right|$, and we can use the approximate expressions

$${\displaystyle \frac{d{A}_{\pm }}{dt}}=-{\displaystyle \frac{3}{2}}{\mathrm{\Omega }}_{\pm }{A}_{\pm },$$

(44)

and

$${\displaystyle \frac{d{A}_{\pm }^{\prime }}{dt}}=-{\displaystyle \frac{1}{2}}{\mathrm{\Omega }}_{\pm }{A}_{\pm }exp\left[-2i{\varphi }_{\pm }\left(t\right)\right],$$

(45)

This leads to simple expressions for the transmission and reflection coefficients, as given by

$$T_{\pm }\left(t\right)\equiv {\displaystyle \frac{A_{\pm }\left(t\right)}{A_{\pm }\left(t_0\right)}}=n^{-3/2}\left(t\right),$$

(46)

and

$$R_{\pm }\left(t\right)\equiv {\displaystyle \frac{A_{\pm }^{\prime }\left(t\right)}{A_{\pm }\left(t_0\right)}}\simeq -{\displaystyle \frac{1}{2}}{\int }_{t_0}^t{\mathrm{\Omega }}_{\pm }\left(t^{\prime }\right)exp\left[-2i{\varphi }_{\pm }\left(t^{\prime }\right)\right]d{t}^{\prime }.$$

(47)

As an illustrative example, we consider the case of a magneto-plasma, assuming a constant magnetic field, $B_0$, and a plasma density evolving as as Gaussian function, with a typical duration of ${\mathrm{\Delta }}_t\gg 1/\omega \left(t_0\right)$. We can write

$${\omega }_p^2\left(t\right)={\omega }_{p0}^{2}exp\left(-{\displaystyle \frac{t^2}{{\mathrm{\Delta }}_t^2}}\right),$$

(48)

where ${\omega }_{p0}$ is the maximum value of the plasma frequency. For parallel propagation, this leads to approximate values of the circularly polarized eigenmode frequencies, determined by

$$n_{\pm }\left(t\right)={\displaystyle \frac{{\omega }_0}{{\omega }_{\pm }\left(t\right)}}\simeq 1\mp {\displaystyle \frac{{\nu }_p}{2}}exp\left(-{\displaystyle \frac{t^2}{{\mathrm{\Delta }}_t^2}}\right),{\nu }_p={\displaystyle \frac{{\omega }_{p0}^2{\omega }_c}{{\omega }_0^2}},$$

(49)

where we use the initial wave frequency ${\omega }_0=\omega (t_0\to -\infty )$, and the auxiliary frequency ${\nu }_p\ll {\omega }_0$. This leads to

$${\varphi }_{\pm }\left(t\right)={\omega }_{0}t\pm {\displaystyle \frac{{\nu }_p}{4}}\sqrt{\pi }\left[1+Erf\left(t\right)\right].$$

(50)

These expressions can be used to calculate the coefficients (46) and (47). Let us focus on the transmission coefficient. We have

$$T_{\pm }\left(t\right)\simeq 1\mp {\displaystyle \frac{3}{2}}{\displaystyle \frac{{\nu }_p}{{\mathrm{\Delta }}_t^2}}{\int }_{t_0}^t{t}^{\prime }exp\left(-{\displaystyle \frac{{t^{\prime }}^2}{{\mathrm{\Delta }}_t^2}}\right)d{t}^{\prime }=1\pm {\displaystyle \frac{3}{2}}{\nu }_{p}exp\left(-{\displaystyle \frac{t^2}{{\mathrm{\Delta }}_t^2}}\right),$$

(51)

We can see that, for $t\to \infty$, the asymptotic value of $T_{\pm }$ tends to one, and the reflection coefficient becomes zero. However, even in the unfavorable case of $T_{\pm }=1$, temporal Faraday rotation still exists because of the frequency difference between the two modes. The total phase shift resulting from such a difference is given by

$$\theta \left(t\right)\equiv {\varphi }_{+}-{\varphi }_{-}={\displaystyle \frac{{\nu }_p}{2}}\sqrt{\pi }\left[1+Erf\left(t\right)\right].$$

(52)

This is illustrated in Figure 4. Also represented are the wave solutions of the field amplitudes, $E_{+}\left(t\right)$ and $E_{-}\left(t\right)$, for a given wavenumber k. We take into account the temporal variation in the refractive index, which introduces a first-order term in the wave equation, proportional to $\partial \mathbf{E}/\partial t$.

If, instead of circularly polarized eigenmodes, we have two linearly polarized modes, as in the nonlinear Kerr effect, the temporal transmission coefficients in this approximation will be $T_{\perp }=1$, and

$$T_{\Vert }=1-{\displaystyle \frac{3}{2}}{\int }^t{\mathrm{\Omega }}_{\Vert }\left(t^{\prime }\right)d{t}^{\prime },{\mathrm{\Omega }}_{\Vert }\simeq -{\displaystyle \frac{\sqrt{\pi }}{4}}{\chi }_{NL}{\displaystyle \frac{t}{{\mathrm{\Delta }}_t^2}}exp\left(-{\displaystyle \frac{t^2}{{\mathrm{\Delta }}_t^2}}\right),$$

(53)

where ${\chi }_{NL}={\lambda }_0\kappa {\left|E_0\right|}^2$, and $E_0$ is the static electric field. Again, this leads to $T_{\Vert }(t\to \infty )=1$. Still, in this case, a final frequency shift between the two linear modes remains, due to the different frequencies, leading to a final change in ellipticity. This means that, in all cases, with or without temporal amplification, a temporal beam-splitter with an arbitrary temporal profile always leads to magneto-optical effects.

6. Conclusions

In this paper, we have studied the magneto-optical effects, such as Faraday rotation and similar processes, resulting from reversible temporal perturbations of a birefringent medium. These perturbations were described as temporal beam-splitters, using sharp temporal boundaries and by slowly varying the boundaries described by a Gaussian function. The cases of a cold magnetized plasma, and nonlinear Kerr media, were examined in detail.

We have shown that, in addition to the phase difference between eigenmodes, two other processes directly associated with the temporal variations in the medium contribute to these magneto-optical effects. One is the amplification resulting from the influence of temporal boundaries, when the transmission coefficients can be significantly larger than one. The other is the mode frequency shifts due to non-conservation of the total wave energy. As we have seen, these effects are particularly important for sharp temporal boundaries. Notice that in static media, only the phase difference between the two independent eigenmodes would remain, but now due to a difference in the wavenumbers. In the case of a Faraday rotation, the eigenmodes correspond to left- and right-hand polarization, propagating along the static magnetic field. In the case of Cotton–Mouton and nonlinear Kerr effects, the two eigenmodes are linearly polarized and propagate across the static magnetic or electric fields. The effects discussed here are similar, but not identical, to those occurring in a static medium, and can be observed in different physical geometries where space boundaries are replaced by temporal ones.

We were able to demonstrate that these temporal effects can be observed in a variety of physical situations and in different anisotropic media, such as a plasma or a nonlinear Kerr optical medium, where the anisotropy is due to the presence of a static electric or magnetic field. They could eventually be used to develop new configurations in spacetime optics, as well as in diagnostics of ultrashort laser pulses.