3+1 formulation of light modes in nonlinear electrodynamics

Fig. 1. Configuration of the background magnetic field B=B0ẑ, the light propagation direction n̂, and the two polarization vectors δE_⊥ and δE_‖. The vector δE_⊥ is perpendicular to the plane formed by B and n̂, while δE_‖ is in the plane. The light modes in Table III refer to this configuration.

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$Departures of the refractive indices n and the parallel polarization vector’s angle ϕ from their free-space values as functions of the magnetic field strength for PM and HES Lagrangians: (a) n⊥ − 1 and n‖ − 1; (b) (ϕ − π/4)/π. Here, the parameters η1 and η2 of the PM Lagrangian are chosen to match the HES Lagrangian in the weak-field limit: η1/4 = η2/7 = e4/(360π2m4). Bc = m2c2/eℏ = 4.4 × 1013 G. The propagation vector’s angle (θ in Fig. 1) is π/4.$

Fig. 2. Departures of the refractive indices n and the parallel polarization vector’s angle ϕ from their free-space values as functions of the magnetic field strength for PM and HES Lagrangians: (a) n_⊥ − 1 and n_‖ − 1; (b) (ϕ − π/4)/π. Here, the parameters η₁ and η₂ of the PM Lagrangian are chosen to match the HES Lagrangian in the weak-field limit: η₁/4 = η₂/7 = e⁴/(360π²m⁴). B_c = m²c²/eℏ = 4.4 × 10¹³ G. The propagation vector’s angle (θ in Fig. 1) is π/4.

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Table 1. Determination of polarization vectors for a given refractive index. The symbol ∼ between two quantities indicates that the two vectors are equivalent, i.e., differ only by a factor. In this table, the case n = 1 is excluded. In the nondegenerate case, n = 1 leads to the FPC. In the degenerate case, n = 1 can hold when δE ∝Z × X for general configurations as well as the FPC.

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Table 1. Determination of polarization vectors for a given refractive index. The symbol ∼ between two quantities indicates that the two vectors are equivalent, i.e., differ only by a factor. In this table, the case n = 1 is excluded. In the nondegenerate case, n = 1 leads to the FPC. In the degenerate case, n = 1 can hold when δE ∝Z × X for general configurations as well as the FPC.

Nondegenerate case (c ≠ 0)	δE = pX′ + qY′ + rZ′
n (≠ 1) from MN − H² = 0 (33)	X′, Y′, Z′ from (26)
Cases	(p, q, r)
H ≠ 0, M ≠ 0, N ≠ 0	$(H, - M, - (Z \cdot X) H / d^{2} + (Z \cdot Y) M / d^{2})$
	∼ (−N, H, (Z ⋅X)N/d² − (Z ⋅Y)H/d²)
H = 0, M ≠ 0, N = 0	(0, − M, (Z ⋅Y)M/d²) ∼ (0, 1, − (Z ⋅Y)/d²)
H = 0, M = 0, N ≠ 0	(−N, 0, (Z ⋅X)N/d²) ∼ (1, 0, − (Z ⋅X)/d²)
H = 0, M = 0, N = 0	$(p, q, - (Z \cdot X) p / d^{2} - (Z \cdot Y) q / d^{2})$ ;
	p and q are arbitrary.
Degenerate case (c = 0)	δE = pX′ + rZ′
n (≠ 1) from $\tilde{M} \tilde{N} - {\tilde{H}}^{2} = 0$ (38)	X′, Z′ from (36)
Cases	(p, r)
$\tilde{H} \neq 0$ , $\tilde{M} \neq 0$ , $\tilde{N} \neq 0$	$(\tilde{H}, - \tilde{M}) \sim (- \tilde{N}, \tilde{H})$
$\tilde{H} = 0$ , $\tilde{M} \neq 0$ , $\tilde{N} = 0$	$(0, - \tilde{M}) \sim (0, 1)$
$\tilde{H} = 0$ , $\tilde{M} = 0$ , $\tilde{N} \neq 0$	$(- \tilde{N}, 0) \sim (1, 0)$
$\tilde{H} = 0$ , $\tilde{M} = 0$ , $\tilde{N} = 0$	(p, r); p and r are arbitrary.

Table 2. Procedure to find light modes in a nonlinear nondegenerate vacuum. The steps from 7 to 9 should be implemented for each value of n found in step 6. In Sec. IV C, we present the essential formulas used in the procedure in order. A similar procedure can be set up in the degenerate case.

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Table 2. Procedure to find light modes in a nonlinear nondegenerate vacuum. The steps from 7 to 9 should be implemented for each value of n found in step 6. In Sec. IV C, we present the essential formulas used in the procedure in order. A similar procedure can be set up in the degenerate case.

Step	Task	Equations
1	Specify Lagrangian	(16)
2	Calculate a², ab, b², c², and d² without specifying E and B	(24)
3	Specify E, B, and $\hat{n}$ and rule out the FPC
4	Calculate a², ab, b², c², and d² for E and B, and confirm c ≠ 0	(24)
5	Calculate α, β, and γ in terms of a², ab, b², c², and d²	(32)
6	Calculate λ^(±) and n > 0	(31) and (34)
7	Calculate X, Y, Z, V, X′, Y′, and Z′	(23) and (26)
8	Calculate H, M, and N	(28)
9	Determine (p, q, r) and calculate δE	Table I, (25)

Table 3. Refractive indices and polarization vectors for various Lagrangians in a purely magnetic background field: $B = B_{0} \hat{z}$ and $\hat{n} = (\sin θ, 0, \cos θ)$ , as shown in Fig. 1. For HES, a², c², and d² are from (55). The FPC is not included here.

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Table 3. Refractive indices and polarization vectors for various Lagrangians in a purely magnetic background field: $B = B_{0} \hat{z}$ and $\hat{n} = (\sin θ, 0, \cos θ)$ , as shown in Fig. 1. For HES, a², c², and d² are from (55). The FPC is not included here.

Lagrangian	Degeneracy (c = 0)	(Refractive index)²	Polarization vector	Parameters
Post-Maxwellian (PM)	No	$n_{⊥}^{2} \equiv \frac{1}{1 - μ \sin^{2} θ}$	δE_⊥ ≡ (0, 1, 0)	$μ = \frac{2 η_{1} B_{0}^{2}}{1 - η_{1} B_{0}^{2}}$
Post-Maxwellian (PM)	No	$n_{‖}^{2} \equiv \frac{1 + ϵ}{1 + ϵ \cos^{2} θ}$	$δ E_{‖} \equiv (\cos θ, 0, - \frac{1}{1 + ϵ} \sin θ)$	$ϵ = \frac{2 η_{2} B_{0}^{2}}{1 - η_{1} B_{0}^{2}}$
Born–Infeld (BI)	No	$n_{‖}^{2}$	pδE_⊥ + qδE_‖	$ϵ = B_{0}^{2} / T$
Born–Infeld (BI)	No			p, q arbitrary
ModMax (MM)	Yes	$n_{⊥}^{2} (= 1)$	δE_⊥	μ = 0
ModMax (MM)	Yes	$n_{‖}^{2}$	δE_‖	ϵ = e^2g − 1
Heisenberg–Euler–Schwinger (HES)	No	$n_{⊥}^{2}$	δE_⊥	$μ = B_{0}^{2} a^{2} / d^{2}$
Heisenberg–Euler–Schwinger (HES)	No	$n_{‖}^{2}$	δE_‖	$ϵ = B_{0}^{2} c^{2} / d^{2}$

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Chul Min Kim, Sang Pyo Kim. 3+1 formulation of light modes in nonlinear electrodynamics[J]. Matter and Radiation at Extremes, 2025, 10(3): 037201

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Paper Information

Received: Sep. 27, 2024

Accepted: Mar. 3, 2025

Published Online: Jul. 16, 2025

The Author Email:

DOI:10.1063/5.0240870

Topics

laser devices and laser physics

Lasers and Laser Optics

Laser physics

laser manufacturing

Instrumentation, Measurement and Metrology

Table 2. Procedure to find light modes in a nonlinear nondegenerate vacuum. The steps from 7 to 9 should be implemented for each value of n found in step 6. In Sec. IV C, we present the essential formulas used in the procedure in order. A similar procedure can be set up in the degenerate case.

Table 2. Procedure to find light modes in a nonlinear nondegenerate vacuum. The steps from 7 to 9 should be implemented for each value of n found in step 6. In Sec. IV C, we present the essential formulas used in the procedure in order. A similar procedure can be set up in the degenerate case.

Table 3. Refractive indices and polarization vectors for various Lagrangians in a purely magnetic background field: B=B0ẑ and n̂=(sin⁡θ,0,cos⁡θ), as shown in Fig. 1. For HES, a2, c2, and d2 are from (55). The FPC is not included here.

Table 3. Refractive indices and polarization vectors for various Lagrangians in a purely magnetic background field: B=B0ẑ and n̂=(sin⁡θ,0,cos⁡θ), as shown in Fig. 1. For HES, a2, c2, and d2 are from (55). The FPC is not included here.

Table 3. Refractive indices and polarization vectors for various Lagrangians in a purely magnetic background field: $B = B_{0} \hat{z}$ and $\hat{n} = (\sin θ, 0, \cos θ)$ , as shown in Fig. 1. For HES, a², c², and d² are from (55). The FPC is not included here.

Table 3. Refractive indices and polarization vectors for various Lagrangians in a purely magnetic background field: $B = B_{0} \hat{z}$ and $\hat{n} = (\sin θ, 0, \cos θ)$ , as shown in Fig. 1. For HES, a², c², and d² are from (55). The FPC is not included here.