QED Vacuum Birefringence Calculator
Vacuum as an Optical Medium
In classical electromagnetism, the vacuum is a perfectly linear medium: light of any polarization travels at the same speed, and the refractive index is exactly one. Quantum electrodynamics (QED) introduces a richer picture. Virtual electron–positron pairs briefly appear and disappear, allowing the vacuum to behave like a very weak, nonlinear optical material when exposed to intense electromagnetic fields.
One key prediction of QED is vacuum birefringence. When a strong, static magnetic field is present, the quantum vacuum acquires a tiny difference in refractive index for light polarized parallel and perpendicular to the field. As light propagates through the field region, these two polarization components accumulate slightly different phases, making the vacuum act like an extremely weak wave plate.
This calculator implements the leading-order QED prediction for this effect using the Euler–Heisenberg effective Lagrangian. Given a transverse magnetic field strength, a path length, and a probe wavelength, the tool estimates:
- the refractive indices for parallel and perpendicular polarizations, n∥ and n⊥,
- the birefringence Δn = n∥ − n⊥,
- the relative phase shift Δφ between the two polarization components, and
- the induced polarization rotation for light initially polarized at 45° to the magnetic field.
The effect is extraordinarily small in typical laboratory conditions, but it is conceptually important and experimentally relevant in high-precision optics and in the study of extreme astrophysical environments such as magnetars.
QED Vacuum Birefringence Formulae
The strength scale that governs QED nonlinearities in a magnetic field is the critical field
numerically Bc ≈ 4.414 × 109 T. For magnetic fields much weaker than this critical value (B ≪ Bc) and for light propagating perpendicular to a uniform, static magnetic field, the Euler–Heisenberg effective Lagrangian predicts two slightly different refractive indices:
n∥ = 1 + \( \dfrac{7}{90} \dfrac{\alpha}{\pi} \Big( \dfrac{B}{Bc} \Big)^2 \), for light polarized parallel to the magnetic field,
n⊥ = 1 + \( \dfrac{2}{45} \dfrac{\alpha}{\pi} \Big( \dfrac{B}{Bc} \Big)^2 \), for light polarized perpendicular to the magnetic field,
where α is the fine-structure constant.
The birefringence is then
Δn = n∥ − n⊥ ≈ \( \dfrac{1}{30} \dfrac{\alpha}{\pi} \Big( \dfrac{B}{Bc} \Big)^2 \).
Over a path length L and at vacuum wavelength λ, the relative phase shift between the two polarization components is
Δφ = .
In this calculator, you provide B (tesla), L (meters), and λ (nanometers). Internally, the wavelength is converted to meters before evaluating the phase shift. The resulting Δn is dimensionless, while Δφ is returned in radians.
How to Use This Calculator
The form above is designed to be simple while still covering realistic experiment and observation scenarios. To use it effectively:
- Enter the transverse magnetic field strength B (T). Use the field component perpendicular to the light propagation direction. Typical laboratory dipole or superconducting magnets reach from about 1 T to several tens of tesla. Leave the value at 10 T for a representative strong lab field.
- Specify the path length L (m). This is the physical distance that light travels within the region where the magnetic field is present and approximately uniform. In real experiments, an optical cavity can increase the effective path length by many orders of magnitude compared with the magnet length.
- Set the wavelength λ (nm). Provide the vacuum wavelength of your probe light in nanometers. Visible light lies roughly between 400 nm and 700 nm. Infrared and ultraviolet wavelengths can also be used as long as the underlying approximations remain valid.
- Run the calculation. After entering the inputs, use the compute action to obtain Δn, Δφ, and the corresponding polarization rotation for an initial 45° linear polarization.
A short descriptive text near the results area explains each reported quantity and its units. You can adjust the parameters to explore how the effect scales with magnetic field strength, path length, and wavelength.
Interpreting the Results
The numbers produced by this calculator are typically extremely small, especially for laboratory magnetic fields. It is therefore useful to understand the approximate size of each quantity and how it should be interpreted.
- Birefringence Δn. This is the difference in refractive index between parallel and perpendicular polarizations. For B on the order of 10 T, Δn is usually around 10−23, far beyond direct index measurements but still physically meaningful.
- Phase shift Δφ (radians). The phase difference accumulates as light propagates. Even with a tiny Δn, long path lengths and short wavelengths can lead to a detectable Δφ. For comparison, a phase shift of 2π rad corresponds to one full optical cycle.
- Polarization rotation. For linearly polarized light initially at 45° to the field direction, the difference in phase between the two orthogonal polarization components manifests as a rotation of the polarization ellipse. In the small-angle limit used here, the rotation angle is approximately Δφ/2.
In practical experiments, polarization rotations of order 10−9 rad are already very challenging to measure. The raw predictions from this calculator for typical lab fields are often many orders of magnitude smaller, which explains why resonance enhancement, modulation techniques, and long integration times are crucial.
Below is a compact comparison of example scenarios that illustrate how dramatically the effect changes as you vary B, L, and λ.
| Regime | B (T) | L (m) | λ (nm) | Approx. Δφ (rad) | Approx. rotation (rad) |
|---|---|---|---|---|---|
| Laboratory, modest magnet | 2.5 | 1 | 1064 | ∼ 3 × 10−17 | ∼ 1.5 × 10−17 |
| Laboratory, strong magnet & long path | 10 | 10 | 500 | ∼ 3 × 10−14 | ∼ 1.5 × 10−14 |
| Extreme field, magnetar-like | 105 | 104 | 1 | ∼ 3 × 10−3 | ∼ 1.5 × 10−3 |
These values are order-of-magnitude examples; the calculator will compute more precise numbers using the same underlying formulas. The key message is that in realistic laboratory fields the phase shift is incredibly small, while near magnetars the effect can become much more pronounced.
Laboratory and Astrophysical Applications
In the laboratory, QED vacuum birefringence has motivated a series of ultra-precise optical experiments. Because the signal scales as B2 and as L/λ, common strategies include:
- using the strongest available superconducting or pulsed magnets,
- employing high-finesse optical cavities to effectively multiply the path length by several orders of magnitude,
- modulating the magnetic field or the polarization to shift the signal to a narrow frequency band, and
- using heterodyne or interferometric detection schemes to reach extremely small rotation sensitivities.
Experiments such as PVLAS are designed around precisely this type of enhancement. A cavity with finesse F can increase the effective path length by a factor on the order of F/π compared with a single pass through the magnet region, bringing the predicted rotation closer to current detection thresholds. For more detailed optical design, you may want to combine this tool with specialized optics calculators such as cavity finesse or beam divergence estimators.
In astrophysics, neutron stars with ultra-strong magnetic fields—magnetars—offer a natural environment where QED effects in the vacuum become significant. Surface magnetic fields can approach or exceed 1010 T, close to or above the critical field. For X-ray photons with very short wavelengths and path lengths comparable to stellar radii, the predicted phase shifts and polarization changes are large enough to be relevant for observational polarimetry.
While the simple formulas implemented in this calculator are strictly valid only in the weak-field approximation, they still provide useful order-of-magnitude guidance for how sensitive X-ray polarimeters must be and how QED effects compare with plasma-induced birefringence in such environments.
Assumptions, Validity Range, and Limitations
The formulas used here come from the leading-order Euler–Heisenberg effective Lagrangian and rely on several important physical and numerical assumptions. You should keep these in mind when interpreting the results.
- Weak-field regime. The expressions assume B ≪ Bc. For fields approaching or exceeding the critical field, higher-order QED corrections and more complete treatments become necessary. In such regimes, this calculator should be regarded only as providing rough estimates.
- Propagation geometry. The derivation is for light propagating perpendicular to a uniform, static magnetic field, with polarizations defined parallel and perpendicular to the field direction. For arbitrary geometries or mixed electromagnetic fields, the simple formulas may not apply directly.
- Uniform, static field. The model assumes that the magnetic field is homogeneous over the entire path length and does not vary in time. Real magnets have fringe fields, gradients, and possibly time-dependent behavior; these are neglected here.
- Vacuum conditions. Only the QED vacuum contribution is included. Any birefringence from residual gas, plasma, optical components, or material media is ignored. In many practical setups, material effects can be orders of magnitude larger than the vacuum contribution.
- Leading-order QED only. Higher-order loop corrections and dispersive effects are not included. The fine-structure constant is treated as a constant, and no energy dependence of the effective couplings is modeled.
- Monochromatic, coherent light. The calculation assumes a single wavelength and coherent propagation. Effects such as finite bandwidth, coherence length, and temporal pulse structure are not modeled.
- No experimental systematics. Systematic effects such as mirror birefringence, stress-induced birefringence in optics, cavity misalignment, detector noise, and thermal drifts are entirely neglected. In real experiments, these often dominate the error budget.
Because of these assumptions, the calculator is best used as a conceptual and design aid rather than as a substitute for full numerical modeling or detailed experimental simulations. When applying the results to real systems, always cross-check with more comprehensive treatments that include geometry, plasma effects, and higher-order corrections where appropriate.