Axion–Photon Conversion Calculator for Vacuum Mixing and qL Coherence
How axion–photon conversion behaves in this calculator
This axion–photon conversion calculator estimates the single-pass probability that an axion becomes a photon while crossing a uniform magnetic field. It uses the simplified vacuum mixing expression that appears in many introductory discussions, so it is best treated as a physics sketch rather than a full detector model. Even so, it gives a useful feel for how magnet strength, magnet length, coupling, axion mass, and photon energy compete with one another.
The key physical idea is coherence. A stronger magnetic field helps because it mixes the axion and photon states more strongly. A longer magnetic region helps because the mixed state has more distance over which to build up amplitude. But those benefits are not unlimited. If the axion mass is heavy enough, or the photon energy is low enough, the axion and photon waves drift out of phase along the path. Once that phase mismatch grows, the oscillatory part of the formula suppresses the conversion even if the magnet itself is excellent. That is why the page reports not only the probability P but also the dimensionless quantity qL, which tells you whether you are still in the coherent regime.
In that sense, this calculator is not just a number generator. It is a compact way to think about a design tradeoff. If your result is tiny because gaγ is tiny, then the problem is weak coupling. If it is tiny because qL is large, the problem is phase mismatch. Those are different physical bottlenecks, and this tool helps you separate them quickly.
How the axion–photon inputs affect the result
Every field in the axion–photon form maps directly onto one term in the mixing formula. A good way to read them is to ask whether each one strengthens the mixing, lengthens the interaction, or spoils coherence:
- Magnetic Field B (T): the strength of the external magnetic field in tesla. In the simplest picture, a larger transverse field gives the axion and photon a stronger chance to mix.
- Region Length L (m): the length of the magnetic region in meters. A longer path usually helps, but only if coherence is preserved across that length.
- Coupling gaγ (GeV−1): the axion–photon coupling constant. This is the interaction strength you are testing or illustrating, often taken from a theory benchmark or sensitivity study.
- Axion Mass ma (eV): the axion mass in electronvolts. Larger mass tends to increase phase mismatch and can push the system out of the coherent regime.
- Photon Energy E (eV): the photon energy in electronvolts. Higher photon energy reduces the phase mismatch term q, which can restore coherence for a fixed mass and length.
As a rule of thumb, B and L try to raise the probability, ma tries to push qL upward, and E can recover coherence by shrinking q = ma2/(2E). That means two scenarios with identical magnets can still behave very differently if their mass and energy scales are not the same.
If you are entering values from a paper or proposal, pay special attention to units. The code converts the engineering units you type into natural units before applying the formula. That means a number that is off by even one power of ten in the coupling or mass can dominate the final answer. The output is dimensionless, so what matters most is whether the scale makes physical sense compared with other scenarios you test.
Axion–photon mixing formula and the coherence term
The page uses the standard vacuum conversion expression for axion–photon mixing in a uniform magnetic field. Written in compact form, the probability is
with the phase-mismatch parameter
The formula is intentionally compact, but the behavior it describes is not arbitrary. The coupling, field, and length enter as a product because they determine the size of the mixing amplitude, while the sine-over-argument factor encodes whether the axion and photon stay in step across the magnet.
Nothing in this page is being averaged with a generic score or a weighted sum. The result comes from the actual oscillation terms, so the probability changes for physical reasons: stronger mixing, a longer path, or less phase mismatch. That also means the calculator is especially useful when you want to compare two setups with one variable held fixed and another nudged up or down.
Because the expression is written in natural units, the code converts tesla, meters, electronvolts, and GeV−1 before it multiplies them. That conversion step is easy to overlook, but it is exactly why a modest-looking change in input can shift the answer by many orders of magnitude. In practice, qL is the more diagnostic quantity if you want to know whether the scenario is coherent or washed out.
The calculator does not collapse those dependencies into a made-up score. It keeps the oscillation picture intact, which is why the output is useful for sanity-checking magnet designs, comparing benchmark masses, and deciding whether better energy alignment would matter more than a small change in field strength.
Worked axion–photon example
Suppose you enter an illustrative vacuum benchmark such as B = 9 T, L = 12 m, gaγ = 1×10−18 GeV−1, ma = 0.000001 eV (that is, 1 μeV), and E = 2 eV. In this case the mass is light enough and the energy high enough that the calculator returns an extremely small qL value, roughly 1.5×10−5. That means the oscillatory suppression term is essentially 1, so the scenario is coherent. The probability then comes out close to 2.9×10−5 in this simplified model.
Now keep the same field, length, and coupling, but raise the axion mass. Because q contains ma2, phase mismatch rises very quickly. If you make the axion hundreds of times heavier, qL is no longer tiny, and the sinc-like term begins to matter. That means two scenarios can have the same magnetic hardware and the same coupling yet produce sharply different probabilities simply because one remains coherent and the other does not. This is exactly the kind of comparison the calculator is meant to support.
You can also read the example the other way around: if a heavier benchmark drives qL upward, then the hardware is no longer the only story. The photon energy and axion mass are part of the tuning problem too, and that is why the calculator keeps both the probability and the coherence indicator visible together.
Reading the axion–photon probability and qL together
The axion–photon probability shown here is a per-axion conversion estimate, not a complete detector yield. Real experiments still need flux, geometry, acceptance, backgrounds, and instrument response. So the number is most useful as a comparative indicator: if you change one variable and the probability rises by orders of magnitude, you have learned something important about the sensitivity of the setup, even before you attach that number to an event rate.
Because qL is the quickest coherence check, it often explains why two setups with similar-looking magnets give very different probabilities. A small qL means the sinc2 factor is near its maximum; a large qL means phase mismatch is doing most of the suppression.
The copy button is useful for that comparative workflow. Run a baseline case, copy it, then change just one variable and compute again. Because the output includes both the probability and qL, you can tell whether a change improved the amplitude side of the formula, the coherence side, or both. That makes the page useful for classroom exercises, proposal back-of-the-envelope checks, and conversations where you want a quick numerical anchor before opening a more detailed code base.
One subtle but useful interpretation point is that E does not act like a direct amplitude booster in the same way B and L do. Instead, higher photon energy helps by shrinking q = ma2/(2E). In other words, energy helps most when coherence is the bottleneck. Once you are already in the small-qL regime, pushing the energy higher yields diminishing conceptual benefit in this simplified formula because the suppression term is already near its maximum.
Also note a sanity check that is worth remembering: probabilities should be physically between 0 and 1. If the simplified expression on this page gives you a value above 1, that does not mean the universe is violating probability. It means your chosen inputs have pushed the approximation outside the range where it should be interpreted literally, or one of the units is inconsistent. In practice, the coupling value is often the first place to double-check.
Model limits for this axion–photon calculator
This axion–photon calculator intentionally stays simple, so it makes several assumptions that are reasonable for a first pass but incomplete for precision work:
- Vacuum mixing: it does not include an effective photon mass from a plasma or buffer gas.
- Uniform field and length: it treats the magnetic region as a single, clean segment with one field strength and one path length.
- No absorption or detector effects: the output is not an observed count rate.
- No cavity or resonant enhancement: it is a direct single-pass estimate, not a full haloscope or resonator model.
- Simple coherence threshold: the page labels regimes using the size of qL, which is a helpful guide but not a complete experimental classification.
Those limits do not make the calculator less valuable; they simply define what kind of question it answers well. It is excellent for quick comparisons, classroom explanation, sanity checks in a magnet design discussion, and rough intuition building for helioscope or light-shining-through-a-wall style scenarios. It is not a substitute for a full propagation code, an experiment-specific likelihood, or a published exclusion analysis.
If you want to use the tool well, a good workflow is to run at least three scenarios: a conservative case, a baseline case, and an optimistic case. Keep the coupling fixed if you are testing hardware changes, or keep the hardware fixed if you are comparing theory benchmarks. Then watch how the balance between the (gBL/2)2 factor and the coherence term changes. That habit turns the calculator from a one-off answer into a small decision aid.
Reading the axion–photon number like a physicist, not like a headline
Small axion–photon probabilities can look discouraging when you see them in isolation, but that is not the right way to read this page. In axion phenomenology, many meaningful scenarios involve probabilities that are tiny on everyday scales. What matters is how the number moves when you change a parameter deliberately. If doubling the magnetic field quadruples the probability while the regime stays coherent, that confirms the expected scaling. If extending the magnet no longer helps because qL has become large, you have learned that coherence, not hardware length, is your present limit.
When you compare two axion benchmarks or two magnet designs, copy the result strings into a notebook and look at how P and qL move together. That is often more informative than staring at a single large or tiny number, because the probability tells you what happened while the coherence term tells you why.
Saving an axion–photon result
Use the copy button to store the exact output string for notes, email, or a lab notebook. A copied result is most helpful when you pair it with the input values and a short sentence describing the scenario, such as whether you were testing a stronger magnet, a lighter axion benchmark, or a higher photon energy. That way the number stays interpretable later instead of becoming an isolated probability with no context.
If you are tracking several runs, the copied P and qL line makes it easy to compare coherent and incoherent cases at a glance. That small habit is often the difference between a result you can reuse and a result you have to recompute from scratch.