^{†}Corresponding author. Email: huangyongs@gmail.com
^{*}Project supported by the National Basic Research Program of China (Grant No. 2011CB808104) and the National Natural Science Foundation of China (Grant No. 11105233).
Contrary to the superposition principle, it is well known that photorefraction exists in the vacuum with the presence of a strong static field, a laser field, or a rotational magnetic field. Different from the classical optical crystals, the refractive index also depends on the phase of the strong electromagnetic field. We obtain the phase and direction dependence of the refractive index of a probe wave incident in the strong field of a circularpolarized plane wave by solving the Maxwell equations corrected by the effective Lagrangian. It may provide a valuable theoretical basis to calculate the polarization evolution of waves in the strong electromagnetic circumstances of pulsar or neutron stars.
In a vacuum, a photon can always metamorphose into an electron and a positron and then annihilate each other becoming a photon again within a short time mandated by the uncertainty principle. The process keeps on repeating itself. The electron and positron can also interact by exchanging a photon or more photons. Therefore, the vacuum is polarized and magnetized by strong fields and exhibits nonlinear optical characteristics.^{[1– 13]} In a static or rotational magnetic field, the vacuum leads to birefringence^{[4, 5, 7, 8, 14]} for the probe waves. Until now, the experimental results^{[5, 6]} are so close to the birefringence predicted by the existing theories.^{[4, 7, 8]} Heyl and Shaviv^{[5]} calculated the direction dependence of the refractive index and the polarization evolution of a probe wave in a strong rotational magnetic field. Based on the Maxwell equations, accompanied with the rotational magnetic field B_{r}, a strong rotational electric field E_{r} exists. Due to the electric field E_{r}, the phase velocity, the group velocity, and the refractive index of the probe wave depend on the angle between the wave vector and E_{r} × B_{r} and the phase of the strong electromagnetic field B_{r}. To check the light– light diffraction experimentally, King, Piazza, and Keitel^{[16, 17]} proposed a matterless doubleslit scenario by using headon collisions of a probe laser field with two ultraintense laser beams. Kryuchkyan and Hatsagortsyan^{[9]} predicted the “ brag scattering” rate of a probe wave by spatially modulated strong electromagnetic fields. The phase and direction dependence of the refractive index, the relative permittivity, and the relative permeability of a probe wave in a strong rotation electromagnetic field have not been discussed in detail, but they are significant for the calculation of the polarization evolution, which maybe helpful for observations to distinguish emissions from a pulsar and to deduce the structure of the magnetosphere.^{[15, 18– 20]}
The photon polarization tensor is a powerful tool for describing the vacuum polarization in a homogeneity magnetic field, ^{[21, 22]} an electric field, ^{[22]} or an electromagnetic field.^{[23, 24]} With the photon polarization tensor, researchers investigated the angle dependence of vacuum birefringence^{[25]} in a homogeneous magnetic field, the amplitude for the scattering of a photon^{[26, 27]} by an intense laser field, and the light diffraction.^{[28]} However, the phase and direction dependence of photorefraction in a circularpolarized plane wave has not been discussed in detail.
Without using the photon polarization tensor, there is another simple and effective method to obtain the refractive index from the dispersion relationship by solving the Maxwell equations corrected by the effective Lagrangian 𝔏 _{eff}^{[29]} in a homogeneous^{[4, 7, 8]} or slowlyvarying field. By using this method, in this paper, the refractive index of a probe wave with an arbitrary incident angle in the presence of a rotational strong electric and a rotational strong magnetic field is studied. With the obtained dispersion relationship of the probe wave, the phase and direction dependence of the refractive index, the permittivity, and the permeability are calculated in detail. It may provide a valuable theoretical basis to calculate the polarization evolution of waves in the strong electromagnetic circumstances of pulsar^{[30]} or neutron stars^{[19, 20, 31, 32]} and to understand the atmospheric emission of neutron stars.^{[32– 37]}
The external field is assumed to be a lowfrequency strong circularpolarized electromagnetic wave,
where ϕ _{Ω } = Ω t – k_{r}z, k_{r} = Ω /c, the vacuum light velocity is c = 3.0 × 10^{8} m/s, and cB_{0} ≲ 10^{18} V/m, which is the critical Schwinger field of electron– positron pair production in the vacuum.
Now assume that the wave vector of the probe wave is k_{p} = (k sin(θ )cos(ϕ ), k sin(θ )sin(ϕ ), k cos(θ )) in the rotation coordinate system (Ê _{r}, B̂ _{r}, ẑ ), where θ and ϕ are the angle between k_{p} and ẑ and the angle between the projection of k_{p} on the Ê _{r}– B̂ _{r} plane and Ê _{r}, respectively.
Figure 1 shows four cases as follows. (i) There is no interaction between the parallel incident probe wave and the rotational magnetic and electric field. (ii) The refractive indices of the parallel polarized probe wave and the perpendicular polarized probe wave are 1+ 14ξ and 1 + 8ξ , respectively, with
where ϕ _{0} − π /2 is the angle between the projection of k_{p} on the x̂ – ŷ plane and the x axis. Set
in the rotational coordinate system. Therefore, (x̂ _{p}, ŷ _{p}, k̂ _{p} = k_{p}/k_{p}) is a new unit orthogonal basis. After some algebra, we obtain
in the laboratory system (x̂ , ŷ , ẑ ). Therefore, the electromagnetic field of the probe wave satisfies
where the components in the rotational system (x̂ _{p}, ŷ _{p}, k̂ _{p}) and the components in the laboratory system (x̂ , ŷ , ẑ ) satisfy the rotational transformation,
With E_{3} = 0, we obtain
With the effective Lagrangian 𝔏 _{eff}, ^{[29]} the nonclassical electric displacement field D_{p} = D − ε _{0}E_{r} and magnetic field strength
where the tensors ϵ ^{r}, μ ^{– 1, r} in the rotational system and those in the laboratory system satisfy ϵ ^{r} = ϵ ^{L}R and μ ^{– 1, r} = μ ^{– 1, L}R. The ϵ ^{r} and μ ^{– 1, r} are expressed as
where n = ck/ω is the refractive index.
With the wave assumption and the linearization of the Maxwell equations corrected by the effective Lagrangian 𝔏 _{eff}, the dispersion relationship is obtained after some straightforward algebra
It contains three special cases for θ = 0, θ = π , and θ = π /2.
If θ = 0, the dispersion relationship becomes two secondorder equations with roots of 1, − 1− 14ξ for the parallel polarization and roots of 1, − 1− 8ξ for the perpendicular polarization. The − 1− 14ξ and − 1− 8ξ correspond to the backpropagation waves. Therefore, there is no interaction between the forwardpropagating wave and the rotational field.
If θ = π , the dispersion relationship has roots of – 1, 1 + 14ξ for the parallel polarization and roots of – 1, 1 + 8ξ for the perpendicular polarization. The refractive indices are the same as those in the static strong magnetic field.^{[4, 7]}
If θ = π /2, the dispersion relationship becomes
therefore, the refractive index is
Figure 2 shows the direction dependence of the refractive indices. There are four modes for an arbitrary incident angle. The n_{1} satisfies n_{1} ≤ − 1 and decreases with increasing θ for θ ∈ [0, π /2) and increases for θ ∈ (π /2, π ]. The n_{2} satisfies n_{2} < − 1 for θ ∈ [0, π /2) and n_{2} > 1 for θ ∈ (π /2, π ]. The n_{1} and n_{2} cannot reach a finite value at θ = π /2. The n_{3} satisfies n_{3} ≤ 1 for θ ∈ [0, π /2) and n_{3} > − 1 for θ ∈ (π /2, π ], and becomes zero at θ = π /2. The n_{4} increases with increasing θ for ϕ ≤ 3π /4 and reaches the maximum value at θ ≈ 3π /5 for ϕ > 3π /4. For θ = π /2,
Figures 3 and 4 show the phase dependence of the refractive indices. The n_{1} satisfies n_{1} ≤ − 1 and reaches the maximum value at the axis of symmetry, ϕ = π /2. The n_{2} reaches the minimum (maximum) value at the symmetry axis for θ < π /2 (θ > π /2). The n_{3} satisfies n_{3} ≤ 1 and reaches the minimum value at the axis of symmetry for θ < π /2. For θ > π /2, n_{3} ≥ − 1 and reaches the maximum value at the axis of symmetry for θ < π /2. The n_{4} reaches the minimum value at the symmetry axis.
With the refractive indices and Eqs. (9) and (10), the phase dependence of the permittivity ε ^{L} and the permeability μ ^{L} corresponding to the four refractive indices can be obtained. Since the four components,
We have obtained the dispersion relationship of a probe wave propagating in a strong circularpolarized electromagnetic field with an arbitrary incident angle. Three special cases for θ = 0, π /2, π have been discussed in detail. The phase and direction dependence of the refractive index, the permittivity, and the permeability in the laboratory system have been obtained. The relationships have some symmetries. For the perpendicular incident probe wave with θ = π /2, there is only one permissible mode with the refractive index
Until now, in the laboratory, the tightly focused strong circularpolarized laser pulse is the only way to generate the rotating electromagnetic field. However, if the probe wave is not propagating parallel to the focused pulse, the optical path is about several micrometers. Comparing the probe wave propagating through the strong field of the pulse and a probe wave with the same frequency propagating in a vacuum without any field, the phase shift of interference fringes, Δ ϕ _{p}, could be used to reveal the refractive index in the strong field. For a strong laser with an intensity of 10^{20} W/cm^{2}, wavelength of 1 μ m and the probe wave with a wavelength of 1 μ m, we have  B_{r} ≈ 10^{5} T and Δ ϕ _{p} ≈ 10^{− 13}. Therefore, it needs quite a precise measurement that we cannot realize currently.
We are grateful to Felix Karbstein for many interesting and enlightening discussions.
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