Photonic magneto-Stark effect based on gradiently modulated magneto-optical medium

Wanguo Liu; Fenghua Liu; Hua Zhao

doi:10.1364/OE.26.000835

1. Introduction

The analogy between an electronic system with a given potential and a photonic system with a modulated permittivity is a common topic during recent decades, which has opened up many ingenious ways to control the light propagation. For example, photonic crystal [1–3], constructed with periodically arranged dielectric media, can produce photonic band gaps and act as a well-known counterpart of crystal. The consequent researches, such as photonic transition [4–6], nonreciprocal propagation [7–10], and effective magnetic field theory for photons [11–13], provide some ingenious methods to control the evolution of photonic states and guide the energy flow to a desirable direction.

The direction-dependent or nonreciprocal propagation is originated from the existence of magnetism, which breaks the time-reversal symmetry [14–17]. Unlike the electrons, photons are neutral particles which cannot directly interact with a magnetic field, and its magnetic response can be only observed by dynamically modulating the background permittivity [11, 18] or using magneto-optical (MO) medium [7–9,12]. Previous studies have shown that when a time-harmonic permittivity is introduced, the effective gauge potential, Lorentz force, and Aharonov-Bohm effect [11,18], can be successfully created for photons. On the other hand, analogously to the electronic system with an applied magnetic field [17], the nonreciprocal propagation of light can be realized by simultaneously employing the MO activity and band gap effect [7–9], that is, magnetic photonic crystals, which also includes some two-dimensional structures [10]. These studies proposed some special ways to realize the effective magnetic field for photons, and provided possibilities to observe their magnetic responses.

Most of the photonic systems mentioned above are designed periodically, and the background permittivity distribution is usually stepped with the variation of coordinates. Also, the studies about nonreciprocity mainly focus on the paired counter-incident waves [8–10]. However, there are very few studies discussing the magnetic responses of photonic states under a general permittivity function. If we use a non-periodic but a non-constant permittivity tensor, what will happen? Besides, in a symmetry-breaking system, the angle-dependent properties of wave have also not been paid much attention. That is, instead of the counter-incident cases, what if we rotate wave’s propagation direction by an arbitrary angle? Actually, for electronic systems, some of the magnetic responses are independent of the periodic potential, and most of them are angle-dependent. A typical example is magneto-Stark (MS) effect, which is related to the oppositely directed Lorentz force on the electron and hole of an electric dipole [19–22]. The MS shift of energy spectrum is linearly related to the applied magnetic field and in-plane wave vector, and it can shift the ground state of the magnetoexciton from a zero in-plane center-of-mass momentum to a finite momentum [21], or measure the giant dipole moment in electronic systems [22]. This reminds us to think about a similar question in photonic systems: is it possible to create an effective magnetic field which renders the eigenvalue of photonic states present the character of MS shift? If so, what condition does the distribution of permittivity satisfy?

We present this work as follows: In Sec. II, we give a brief review of electronic MS effect and construct the characteristic equation for photons in a spatially modulated MO medium. We derive the general condition when the photonic MS (PMS) effect occurs. In Sec. III, we present the PMS effect for plane wave and cylindrical wave as specific examples. Sec. IV summarizes the paper.

2. General formula and condition for photonic magneto-Stark effect

To begin our discussion, let us first review the concept of MS effect [19–22]. If an electric dipole is placed in an applied uniform magnetic field, except for the ordinary Zeeman effect and the diamagnetic shift, there also exists a third magnetic perturbation. This term is related to the Lorentz force produced by exciton’s center-of-mass motion in the magnetic field. Assume the exciton moves in x-y plane, the perturbation Hamiltonian can be generally expressed as follows [19]

H_{S} = \frac{ℏ}{M} p {(B \times k)}_{z}

Here

M

is the exciton mass,

p

is the scalar dipole moment of the electron-hole pair,

B

is the applied magnetic field,

k

is the wave vector, and

ℏ

is the reduced Plank constant. Under an in-plane applied magnetic field, an exciton with an in-plane momentum produces a z-direction Lorentz force and leads to an energy shift. The MS effect described by Eq. (1) is similar to the hydrogenic Stark effect induced by an applied electric field [23]. Therefore, one can regard

E_{e f f} = - \frac{ℏ}{M} {(B \times k)}_{z}

as a quasi-field induced by the dipole’s motion in a magnetic field. Not surprisingly, the MS effect and Stark effect can coexist, offering a way to measure the quasi-field of MS effect [19]. For an exciton with zero wave vector, the Stark splitting should be only proportional to the applied electric field

E

and there is no MS effect. While for an exciton in motion,

E_{e f f}

can be measured by adjusting

E

until the energy splitting vanishes. In such a case, the effects of magnetic field and electric field reach a balance, which means the Stark splitting offsets the MS splitting and

E_{e f f} = - E

is satisfied.

But is it possible to create a MS effect for photons? Photons are neutral particles with zero charge, and “dipole moment” of photons has no definition. Unlike the electron-hole excitons, they cannot interact with an applied magnetic field, and then, how to create such a quasi-field?

To deal with this problem, the first step is to construct the characteristic equation for a photonic state in a general case. As a counterpart of stationary Schrödinger equation which describes the quantum eigenstates, we consider the EM wave equation and study how a spatially modulated permittivity affects a photonic state and its eigenvalue. The medium studied here is non-uniform and anisotropic, and the wave equation for magnetic components $H$ takes the general form

\nabla \times (ε^{- 1} \nabla \times H) = k^{2} H

Note that the MS Hamiltonian in Eq. (1) is time-reversal asymmetric because of the appearance of magnetic field, and therefore we use MO medium to break the time-reversal symmetry. Assume the magnetization is along z axis (transverse magnetic (TM) polarization), the permittivity tensor

ε

of a uniaxial MO crystal and its inversion are given as follows

ε = (\begin{matrix} ε_{/ /} & i η & 0 \\ - i η & ε_{/ /} & 0 \\ 0 & 0 & ε_{⊥} \end{matrix}), ε^{- 1} = (\begin{matrix} \frac{ε_{/ /}}{ε_{/ /}^{2} - η^{2}} & \frac{- i η}{ε_{/ /}^{2} - η^{2}} & 0 \\ \frac{i η}{ε_{/ /}^{2} - η^{2}} & \frac{ε_{/ /}}{ε_{/ /}^{2} - η^{2}} & 0 \\ 0 & 0 & \frac{1}{ε_{⊥}} \end{matrix}) \approx (\begin{matrix} \frac{1}{ε_{/ /}} & \frac{- i η}{ε_{/ /}^{2}} & 0 \\ \frac{i η}{ε_{/ /}^{2}} & \frac{1}{ε_{/ /}} & 0 \\ 0 & 0 & \frac{1}{ε_{⊥}} \end{matrix})

where

η

is the gyrotropic factor which is proportional to the applied magnetic field, and

ε_{/ /}, ε_{⊥}

denote the anisotropic components of non-magnetic permittivity tensor. Since

η

is small compared with the diagonal element of

ε

, we neglect its nonlinear terms in

ε^{- 1}

[24]. The out-of-plane magnetic components have no contribution to the MO effect. Thus we only need to consider

H_{z}

polarization where the wave is bounded in the propagation (x-y) plane. Substituting Eq. (3) into Eq. (2), we derive the following equation with a straightforward calculation (Note that

\frac{\partial}{\partial z} = 0

for a two-dimensional structure)

\frac{\partial}{\partial x} (\frac{i η}{ε_{/ /}^{2}} \frac{\partial H_{z}}{\partial y} - \frac{1}{ε_{/ /}} \frac{\partial H_{z}}{\partial x}) - \frac{\partial}{\partial y} (\frac{i η}{ε_{/ /}^{2}} \frac{\partial H_{z}}{\partial x} + \frac{1}{ε_{/ /}} \frac{\partial H_{z}}{\partial y}) = k^{2} H_{z}

Equation (4), in general, describes a TM wave propagating along an arbitrary direction in x-y plane. When the permittivity tensor is piecewise constant, this eigenvalue problem is well-known and mostly seen in the design of magneto-optical waveguide [25,26]. While here we consider a more general case that the permittivity tensor is a spatially modulated function,

ε = ε (x, y)

. This function can be periodic or non-periodic, and continuous or with countable discontinuous points. For convenience, we introduce relative gyrotropic ratio

α (x, y)

, defined as

α = \frac{η}{ε_{/ /}^{2}}

. Hereinafter we assume the eigenstate

H_{z}

has already been normalized. We rewrite Eq. (4) as follows

(\hat{L} + {\hat{L}}_{S}) H_{z} = k^{2} H_{z} s

where the linear operators

\hat{L}

and

{\hat{L}}_{S}

are given as

\hat{L} = - \frac{\partial}{\partial x} (\frac{1}{ε_{/ /}} \frac{\partial}{\partial x}) - \frac{\partial}{\partial y} (\frac{1}{ε_{/ /}} \frac{\partial}{\partial y})

{\hat{L}}_{S} = i (\frac{\partial α}{\partial x} \frac{\partial}{\partial y} - \frac{\partial α}{\partial y} \frac{\partial}{\partial x})

It can be seen that

{\hat{L}}_{S}

describes the magnetic perturbation. If one discards

{\hat{L}}_{S}

in Eq. (5), it can be seen

\hat{L} H_{z} = k^{2} H_{z}

describes a stationary state of

H_{z}

polarization with no magnetization. This form is already familiar in two-dimensional photonic lattice [27], and its eigenstates are already known. Actually, this form is also applicable for non-periodic cases.

However, with the existence of MO activity, ${\hat{L}}_{S}$ contains physically significant information. As is known, when an EM field is quantized, its energy is carried by a series of photons, and each of their momentum is proportional to the wave vector. According to the standard quatization procedure, the coordinate and momentum should be transformed into operators, $r \to \hat{r}$ and $ℏ k \to ℏ \hat{k} = - i ℏ \nabla$ (Note that the notation $\hat{A}$ here means operator rather than the unit vector). To execute this procedure similarly, we transform ${\hat{L}}_{S}$ in Eq. (7) as a form of cross product

{\hat{L}}_{S} = - {(\nabla α \times \hat{k})}_{z}

where

\nabla = (\begin{matrix} \frac{\partial}{\partial x} & \frac{\partial}{\partial y} \end{matrix})

is the two-dimensional gradient operator.

\hat{k}

is the wave-vector operator. When it acts on its eigenstate, it is equivalent to multiply this state by its wave vector. If one takes

\hat{L}

as a counterpart of a particle’s Hamiltonian in a stationary potential, it is interesting to see the similarity between

H_{S}

in Eq. (1) and

{\hat{L}}_{S}

in Eq. (8). The gradient of relative gyrotropic ratio creates a perturbation term, whose component perpendicular to the propagation direction is responsible for the shift in the energy spectrum. Because of their similarity, the vector field

\nabla α

can be viewed as a counterpart of the magnetic field, which imposes a Lorentz force for photons with wave vector

k

. If the spatial distribution of

α

is properly designed, this term can act as a PMS effect. We can furthermore estimate this effective magnetic field. By comparing the dimension of Eq. (1) and Eq. (8) one can formally get the expression of effective magnetic field

B_{e f f} (p) = - \frac{ℏ}{2 p} \nabla α

Analogously to Eq. (1), the PMS Hamiltonian can be written as

\frac{ℏ}{m} p {(B_{e f f} \times k)}_{z}

, where

m

is the motion mass of a photon. It should be pointed out the expression of

B_{e f f}

contains an uncertain dipole moment

p

, which seems elusive because photon’s dipole has no definition. Actually, this can be interpreted as follows: We do not need to segregate

B

from

p

in Eq. (1), because the MS effect is decided by the product of magnetic field and dipole moment. That is to say, at the point of MS effect, the consequence of

B

acting on

p

is equivalent to that of

C B

acting on

p / C

, where

C

is an arbitrary factor. Therefore, the physical significance of Eq. (9) is not that it creates a magnetic field in reality, but should be that the PMS shift caused by

\nabla α

is equivalent to the consequence of

B_{e f f}

imposing on electron-hole pair with dipole moment

p

. Note that only MO medium with constant gyrotropic ratio is not sufficient to create PMS effect, since

\nabla α = 0

and the effective magnetic field vanishes. Actually, although a nonzero

α

breaks the time-reversal symmetry, such a case satisfies space-inversion symmetry [7].

The direct consequence for perturbation ${\hat{L}}_{S}$ is to eliminate the degeneracy. Consider the simplest case that $\nabla α$ and $ε_{/ /}$ are nonzero constants. One of the eigenstates of unperturbed wave equation $\hat{L} H_{z} = k^{2} H_{z}$ is plane wave $| k 〉 = H_{0} e^{i k \cdot r}$ . Photonic states with equal $| k |$ are 360-degree degenerate with eigenvalue $k^{2}$ (the solid line shown in Fig. 1). On the other hand, $| k 〉$ is also an eigenstate of ${\hat{L}}_{S}$ with eigenvalue $- {(\nabla α \times k)}_{z}$ . Except for the case $\nabla α / / k$ , the eigenvalue of every perturbed state shifts by $| \nabla α | | k | \sin θ$ , which has a one-to-one relationship with their open angle, resulting in different eigenvalues for the new eigenstates (the dashed line shown in Fig. 1). The PMS effect reaches the maximum when $\nabla α ⊥ k$ . Therefore, ${\hat{L}}_{S}$ removes the degeneracy of these states. Once the direction of light is not parallel to the gradient of $α$ ( $θ \neq 0$ and $θ \neq π$ ), the photonic MS effect can take place. Actually, even if $\nabla α$ is not constant, the same argument holds if its direction is fixed, only the perturbed eigenstate may not be strictly solved.

Fig. 1 PMS splitting under the presence of spatially modulated MO ratio, where $k_{0}$ is the wave vector in vacuum. The pure-color background means a constant $α$ , while the gradient one means $α$ is gradiently modulated. When $\nabla α = 0$ (even if $α \neq 0$ ), the eigenvalue of wave vector is 360-degree degenerate. When $\nabla α$ is a non-zero constant, the degeneracy is eliminated almost everywhere except for $\nabla α / / k$ (time-reversal symmetry). The PMS splitting is proportional $| k |$ and isometric between states $| k 〉$ and $| - k 〉$ . It reaches the maximum when $θ = π / 2$ and $θ = 3 π / 2$ ( $\nabla α ⊥ k$ ).

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Like the nonreciprocity in electronic systems can be explained from the MS effect [22], here we should point out that the nonreciprocal propagation of light in previous studies can be completely explained from the perspective of PMS effect. A special example is the one-dimensional nonreciprocal photonic crystal [8,9], in which the PMS effect breaks time-reversal symmetry and leads to nonreciprocal response, since the eigenvalues of photonic states $| k 〉$ and $| - k 〉$ will present an isometric splitting with respect to the unperturbed eigenstates, as shown in Fig. 1. The time-reversal symmetry only remains at the direction $\nabla α / / k$ , under which the PMS splitting goes to zero and nonreciprocity vanishes. This is exactly the reason why the nonreciprocal propagation in magnetic photonic crystal should be realized under the oblique incidence [8,9]. Therefore, the nonreciprocity of paired counter-incident waves is just a special case of PMS effect. Of course, if the gradient of gyrotropic ratio is perpendicular to the propagation direction, the nonreciprocity can be also realized under normal incidence, such as some surface plasmonic structures [28–30].

3. Examples: Photonic magneto-Stark effect for plane wave and cylindrical wave

Here we consider the simplest case in which the gradient of gyrotropic ratio is constant. To ensure the magnetization does not increase to infinity, the modulating area should be limited in a certain range. Therefore, we choose the non-magnetic wavelength of plane wave as $λ = 560 n m$ , and set the distribution function of gyrotropic ratio as

α (x, y) = {\begin{matrix} C \frac{x}{λ}, | \frac{x}{λ} | < 20 \\ 0, | \frac{x}{λ} | \geq 20 \end{matrix}, \begin{matrix} Modulating Aera \\ Non - magnetic Aera \end{matrix}

We choose the nondimensional coefficient $C = 1.5 \times 10^{- 2}$ and depict this distribution function in Fig. 2. Here, in order to see how large the modulating area is, we use $λ$ as a scale for the coordinate (It does not mean $\nabla α$ depends on $λ$ ). Such a distribution allows the gyrotropic ratio to vary from −0.3 to 0.3, and this is achievable for most of the MO media if one sets the applied field uniformly changes with spatial coordinates. Changing the gradient can be realized by shrinking or expanding the modulating area only, and it does not need to change the maximum and minimum of the gyrotropic ratio. Also, the distribution function ensures the width of MO medium is 40 times wider than wavelength, which is very convenient for experimental realization. This function indicates that $\nabla α$ (i.e. the direction of effective magnetic field) is along the x axis. We use Lumerical Mode Solution, a commercially available eigenvalue problem solver, to perform the 2D electromagnetic analysis. Before and after applying a z-direction magnetic field, the propagation of a plane wave along different directions are simulated and shown in Fig. 3. As expected, when the propagation direction is parallel to $\nabla α$ , the effective magnetic field has no influence on the eigenvalue and the wavelength keeps unchanged. While for the non-parallel cases, the wavelength is shifted by the effective magnetic field. The most interesting is, if we imagine the wave crest and trough are oppositely “polarized”, the PMS effect is very figurative: It seems that a fictitious “Lorentz force” is simultaneously “pulling” or “pushing” the wave at the crest and trough, and makes the wavelength be “squeezed” ( $0 < θ < π$ ) or “stretched” ( $π < θ < 2 π$ ). This is quite similar with MS effect, in which the oppositely directed Lorentz force acts on the electric dipole (electron-hole pair) and shift the energy eigenvalue [19,22], and its direction is reversed when $θ$ exceeds $π$ . Note that the shape of plane wave holds since it is a same eigensatate of operators $\hat{L}$ and ${\hat{L}}_{S}$ . These results are totally in accordance with the above prediction. Obviously, it also works when rotating the vector field $\nabla α$ and keeping the propagation direction of wave be invariant, and this allows us to use MO activity to modulate the wavelength.

Fig. 2 Distribution function of gyrotropic ratio $α$ (Eq. (10). $C = 1.5 \times 10^{- 2}$ .

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Fig. 3 Effective magnetic field acts on waves propagating along different directions. $\nabla α$ is determined from Eq. (10) (Fig. 2), where $C = 1.5 \times 10^{- 2}$ . For the left columns, the medium is non-magnetic, and for the right columns, we use MO medium with gradient gyrotropic ratio along x axis. (a) $θ = 0$ , (b) $θ = \frac{π}{4}$ , (c) $θ = \frac{π}{2}$ , (d) $θ = \frac{3 π}{4}$ , (e) $θ = π$ , (f) $θ = \frac{5 π}{4}$ , (g) $θ = \frac{3 π}{2}$ , (h) $θ = \frac{7 π}{4}$ . For $θ = 0$ and $θ = π$ , the gradient of gyrotropic ratio is parallel to the wave vector, and the PMS effect does not occur. For $0 < θ < π$ , the effective magnetic field “squeezes” the wave, while for $π < θ < 2 π$ , it “stretches” the wave. It seems that a “Lorentz force”, produced by the effective magnetic field, is simultaneously “pulling” or “pushing” the wave at the crest and trough.

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What about the case for a cylindrical wave? Of course it is also an eigenstate for operator $\hat{L}$ , but operator ${\hat{L}}_{S}$ is non-diagonal in its representation. According to the perturbation theory, the activity of ${\hat{L}}_{S}$ should be described by the following perturbation matrix [30]

L_{S, k k^{'}} = 〈 k | {\hat{L}}_{S} | k^{'} 〉 = \iint ψ_{k}^{*} (r) {(- \nabla α (r) \times \hat{k} ψ_{k^{'}} (r))}_{z} d^{2} r

Here

ψ_{k} (r) = \frac{H_{0}}{\sqrt{r}} e^{i k r}

is cylindrical wave. Actually, Eq. (10) is applicable for the most general case with arbitrary eigenstate and arbitrary distribution of

α

. For non-degenerate case (the cylindrical wave belongs to this case), Eq. (10) becomes the average value of

{\hat{L}}_{S}

under the eigenstate

ψ_{k} (r)

, and the first order correction of eigenstate is [31]

ψ_{k}^{(1)} (r) = \sum_{k^{'} \neq k} \frac{L_{S, k k^{'}}}{k^{2} - {k^{'}}^{2}} ψ_{k^{'}} (r)

Usually, the integral in Eq. (10) cannot be analytically calculated, and this means the accurate eigenstate cannot be obtained. But it can be predicted that the shape of cylindrical wave will not remain because it is no more a common eigenstate of $\hat{L}$ and ${\hat{L}}_{S}$ . This is in contrast with the plane-wave case, in which the wave keeps its original shape. The numerical results of cylindrical wave propagating in MO medium are depicted in Fig. 4. The wavelength is still set as $λ = 560 n m$ . As seen, when $α$ keeps constant (even if $α \neq 0$ ), the shape of cylindrical wave is totally the same with one in the non-magnetic homogeneous medium (Fig. 4(a)). While $α$ varies along x axis, the wave presents strong direction-dependent character. Similarly to the plane-wave cases, the wavelength of cylindrical wave can be also “squeezed” or “stretched” along different directions (Fig. 4(b)). Continuing increasing the gradient will reshape the wave and lead to the energy flow is almost prohibited along -y direction (Fig. 4(c)). If we changes the direction of $\nabla α$ , the energy flow can be guided to the opposite direction (Figs. 4(d) - 4(f)). Therefore, if properly designed, we can artificially prohibit the energy flow of cylindrical wave from dispersing, and converge it into a desirable direction. This is very convenient for the design of direction-dependent optical devices for cylindrical wave, which is hard to realize in one-dimensional one-way structures. For example, we can let ${(\nabla α)}_{x} > 0$ in upper-half plane, and let ${(\nabla α)}_{x} < 0$ in lower-half plane. This renders the wavefront which is near to the source is restricted in a finite width (see in Fig. 4(f), this area is approximately expressed as $| y | < 2 λ$ ). If we increase $| \nabla α |$ , the energy flow can be almost prohibited at y direction, and the cylindrical wave is approximately transformed into a plane wave propagating along x direction (Fig. 4(g)). This greatly reduces the energy dissipation of cylindrical wave, providing a method to reshape the wavefront and control the evolution of photonic modes.

Fig. 4 Characteristic profiles for cylindrical wave propagating in MO media with different gyrotropic ratio functions. For gradiently modulated The permittivity functions for (b) ~(f) are chosen as Eq. (10). (a) $α = c o n s t$ , (b) $C = 0.5 \times 10^{- 2}$ , (c) $C = 2 \times 10^{- 2}$ . (d) $C = - 0.5 \times 10^{- 2}$ , (e) $C = - 2 \times 10^{- 2}$ , (f) $C = 0.5 \times 10^{- 2}$ for $y > 0$ and $C = - 0.5 \times 10^{- 2}$ for $y < 0$ , (g) $C = 2 \times 10^{- 2}$ for $y > 0$ and $C = - 2 \times 10^{- 2}$ for $y < 0$ . When $α$ is constant, the MO activity has no influence on the wave shape. With the increase of $\nabla α$ , the wavelength is strongly direction-dependent.

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4. Conclusions

In summary, we have shown that, in a MO medium, the gradient of gyrotropic ratio can create an effective magnetic field, which induces a PMS effect. As a counterpart of the electronic MS effect, the PMS effect can be figuratively understood as the activity of “Lorentz force” created by the effective magnetic field, which can adjust the wavelength by shifting the eigenvalue of photonic system. We proved that the PMS effect vanishes once the wave vector is parallel to the effective magnetic field, and reaches the maximum when they are perpendicular. Furthermore, it extends the concept of nonreciprocity from a pair of counter-incident waves to waves propagating along arbitrary different directions. Although we only discussed the PMS effect for plane wave and cylindrical wave under a constant gradient as examples, this theory is undoubtedly applicable for the general cases. We can expect that, through some complicated but ingenious designs of the background permittivity, one can reform the wave into a desirable shape, and control the anisotropic propagation of light. It should be emphasized that the effective magnetic field exists once the permittivity tensor varies along the spatial coordinates, and it does not need to be periodically modulated, which gives a convenient way for the design of optical systems. The PMS effect provides a new mechanism to modulate the wavelength and control the evolution of photonic states.

Funding

The author acknowledges the financial support from the Youth Talent Scientific Project of Education Department of Guizhou Province (Granted No. KY[2017]337), Talent Project of Qiannan Normal University for Nationalities (Granted No. Qnsyrc201720) and the Foundation of Scientific Innovative Research Team of Education Department of Guizhou Province (Granted No. 201329).

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Photonic magneto-Stark effect based on gradiently modulated magneto-optical medium

Abstract

1. Introduction

2. General formula and condition for photonic magneto-Stark effect

3. Examples: Photonic magneto-Stark effect for plane wave and cylindrical wave

4. Conclusions

Funding

References and links

Cited By

Figures (4)

Equations (12)

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