Axion–Photon Conversion Probability Calculator
What this calculator estimates
Axion searches are often described as a hunt for incredibly faint signals, but the practical question is usually more specific: if an axion enters a magnetic region, how likely is it to emerge as a photon that your experiment could then detect? This calculator estimates that single-pass conversion probability with the standard simplified vacuum mixing formula used in many introductory axion–photon discussions. It is most useful when you want a quick order-of-magnitude feel for how magnetic field strength, magnet length, coupling, axion mass, and photon energy push the probability up or down.
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 tell them apart quickly.
What each input means in plain language
Every field in the form maps directly onto a term in the oscillation 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.
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.
The formula used by the calculator
The page uses the usual vacuum conversion expression for axion–photon mixing in a uniform magnetic field. Written in compact form, the probability is
with the phase-mismatch parameter
There are two messages hidden in that expression. First, when qL is very small, the sinc-like factor is close to 1, so the probability behaves approximately like (gBL/2)2. In that coherent limit, stronger fields and longer magnets help in a very direct way. Second, when qL becomes large, the sine term oscillates and the ratio sin(qL/2)/(qL/2) suppresses the probability. That is the mathematical signature of axion and photon waves slipping out of phase.
The script on this page follows that logic exactly. It converts the magnetic field from tesla to GeV2, the length from meters to GeV−1, and both mass and energy from eV to GeV. Those conversions are necessary because the compact mixing formula is most naturally expressed in natural units. The output probability is therefore dimensionless, while qL acts as a regime flag: values below about 1 suggest coherence, and values far above 1 indicate noticeable phase washout.
If you prefer to think abstractly, the calculator is still just a function of several inputs. The general input–output view is preserved below because it is a useful reminder that the page is applying a specific physics model to a small, clearly labeled set of parameters.
For this specific calculator, the weights and nonlinear behavior are not arbitrary bookkeeping. They correspond to real physical scaling: the coupling multiplies the amplitude, the field and length set the mixing opportunity, and the oscillatory factor decides how much of that opportunity survives coherence loss.
Worked 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.
How to interpret the result without overreading it
The 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.
The output line includes three pieces of information. P is the conversion probability. qL is the coherence indicator. Regime labels the case as coherent or incoherent using the simple threshold built into the script. If the result is coherent, increasing B, L, or gaγ usually helps in the intuitive way. If the result is incoherent, a stronger magnet alone may not rescue the scenario; improving energy or reducing the mass-induced phase mismatch can matter more.
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.
Assumptions and model boundaries
This 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 number like a physicist, not like a headline
Small 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.
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.
Saving Your 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.