Why a transverse magnetic field makes the vacuum birefringent
In classical electromagnetism, empty space does not prefer one linear polarization over another. In quantum electrodynamics, however, the vacuum acts like a polarizable medium when it sits inside a strong background magnetic field. Virtual charged particles loop in and out of existence, and that interaction makes the two orthogonal polarizations travel with slightly different phase velocities. The effect is tiny, but it is real in the weak-field approximation that this calculator uses.
This page is built for order-of-magnitude planning. It is not a full beamline simulation, and it does not model cavity enhancement, diffraction, residual gas, mirror coatings, stress birefringence, or detector calibration. What it does give you is a clean way to see how the basic QED scaling works: the birefringence depends on the square of the field, the phase delay grows with path length, and shorter wavelengths accumulate more retardance for the same Δn.
Because the numbers are so small, the most meaningful interpretation is often comparative. If one magnet run gives four times the field, the leading-order signal is sixteen times larger. If you double the interaction length, you double the phase delay. If you keep the geometry the same and shorten the wavelength, the calculated retardance increases in proportion to 1/λ. Those scaling rules are the heart of the calculator.
How to use this QED vacuum birefringence calculator
Start by entering the transverse field strength, the length of the region where the beam and magnetic field overlap, and the probe wavelength in nanometers. The form expects a single pass through a uniform field; if you are thinking about an optical cavity, remember that the calculator still works with the geometric path length and leaves the cavity buildup factor to your own judgment. That makes it useful for comparing magnet designs or checking whether a proposed beamline can plausibly reach the phase sensitivity you need.
For the cleanest interpretation, use the wavelength of the actual probe beam in vacuum rather than a value copied from a material datasheet. If the beam is not exactly perpendicular to the field, or if the magnetic field varies substantially across the interaction region, the simple estimate is still a helpful baseline but no longer a complete description. The calculator also assumes a vacuum path, so any gas pressure, plasma, or stressed optic in the real setup should be treated as an extra systematic that may dominate the QED signal.
When you submit the form, the output area summarizes the two refractive-index shifts, the birefringence between them, the accumulated phase retardance, and the 45-degree ellipticity estimate. The sign and scale are more important than the last digit: in laboratory conditions the effect is usually far below ordinary optical noise, which is exactly why experimenters rely on long paths, high fields, and careful suppression of false birefringence sources.
QED vacuum birefringence formulas used in this calculator
The calculator first converts the magnetic field into the dimensionless ratio below, so you can see how far the setup sits from the QED critical scale.
From that ratio, the leading-order weak-field approximation gives the polarization-dependent index shifts for the two normal modes of light.
The birefringence is the difference between those two shifts, which is the quantity that drives the phase retardance across the path length.
Finally, the calculator turns the birefringence into a relative phase delay, and the 45-degree ellipticity estimate is simply half of that phase delay in this simple model.
Worked example: 10 T field, 1 m path, 500 nm probe
With a 10 T transverse magnetic field, a 1 m single-pass path, and a 500 nm probe wavelength, the calculator predicts a field ratio of about 2.27 × 10-9. Using the weak-field formulas above, that gives n∥ − 1 ≈ 9.27 × 10-22, n⊥ − 1 ≈ 5.30 × 10-22, and Δn ≈ 3.97 × 10-22. The corresponding phase delay is about 4.99 × 10-15 rad, with a 45-degree ellipticity estimate of about 2.50 × 10-15 rad. That is a perfectly sensible result for a laboratory field: the signal is minuscule, but it scales in the expected direction when B, L, or λ change.
If you want a larger value, the most effective lever is usually the magnetic field because the signal scales as B squared. A longer path helps linearly, and a shorter wavelength helps linearly as well, but neither of those can compensate for a field that is too weak. In real experiments, the challenge is not getting a nonzero answer from the model; it is distinguishing that tiny QED phase shift from mirror birefringence, residual gas, and instrument drift.
How to interpret the QED birefringence outputs
The output list is designed to move from the basic polarization shifts to the quantities that matter for polarization optics. The refractive-index shifts tell you how each polarization mode behaves individually, while Δn tells you how far apart the modes are. The phase delay and ellipticity then translate that microscopic split into the sort of signal an experiment would try to observe.
How to read the QED vacuum birefringence outputs
| Quantity |
Meaning |
Practical note |
| n∥ - 1, n⊥ - 1 |
Polarization-specific refractive-index shifts above one. |
The shifts are so small that the full refractive index usually rounds back to 1 in ordinary floating-point output. |
| Δn |
The birefringence between the parallel and perpendicular polarization axes. |
This is the quantity that tells you how strongly the two polarization axes split in the vacuum. |
| Δφ |
The relative phase delay accumulated over the path. |
It scales with L / λ, so longer path lengths and shorter wavelengths both strengthen the signal. |
| 45-degree ellipticity |
The small ellipticity estimate for an input polarization halfway between the two axes. |
Real experiments must separate this phase-delay signal from mirror, gas, alignment, and detector systematics. |
Assumptions and limits of the QED vacuum birefringence model
This calculator intentionally stays close to the simple Euler-Heisenberg picture so that the scaling is transparent. That makes the output easy to use for rough planning, but it also means you should read the result as a leading-order estimate rather than a full numerical simulation of an experiment. If any of the following assumptions are violated, the returned value still tells you the direction of the effect, but not necessarily the exact magnitude you would measure in the lab.
- Weak-field QED regime. The formulas assume B is far below Bc. Near the critical scale, higher-order corrections and a more complete treatment become important.
- Transverse uniform geometry. The light is assumed to cross the field at right angles through a region with roughly constant B. Strong gradients or a tilted path change the simple scaling picture.
- Vacuum path only. The calculation does not include gas, plasma, coatings, mirror stress, or stressed windows. Any of those can produce ordinary birefringence that overwhelms the QED signal.
- Single wavelength. The model uses one monochromatic probe wavelength and does not account for bandwidth, pulses, cavity dispersion, or finite coherence length.
- No detector model. The output estimates the physical signal only. Shot noise, modulation scheme, integration time, and calibration strategy remain your responsibility.
FAQ about QED vacuum birefringence
Can this calculator prove QED vacuum birefringence in a lab setup?
No. It estimates the leading-order vacuum signal from the weak-field Euler-Heisenberg approximation, so it is best used for scaling and planning. Real experiments also need cavity gain, modulation, detector noise control, and checks for gas or optic birefringence.
What if the magnetic field is close to the QED critical field?
The weak-field formula assumes B is far below the QED critical field of about 4.414 × 109 T. As you approach that scale, the calculator's approximation stops being reliable and a higher-order treatment is needed.