An interference pattern reveals a relationship between alternatives that meet in the same measurement. Its spacing, position and contrast constrain that relationship. Interpreting the pattern also requires knowing the separate intensities, the detector response and the stability of the preparation.
Add amplitudes before intensities
For two coherent fields arriving with the same polarization, the detected intensity is proportional to the squared magnitude of their summed amplitude. Expanding that square gives the separate intensities and a term that depends on their relative phase. This additional term produces the bright and dark fringes.
A useful two-beam description, allowing partial coherence, is
I(φ) = I₁ + I₂ + 2√(I₁I₂) |γ| cos(φ + arg γ).
Here I₁ and I₂ are the separately measured intensities, φ is a controlled phase difference and γ is a normalized mutual-coherence factor for the specified averaging and polarization arrangement. Its magnitude lies between zero and one. Stable, fully coherent beams have |γ| = 1 in this description.
The equation is a model of an optical comparison. It does not by itself identify the cause of a reduced coherence factor. Random phase changes during an exposure, unresolved polarization differences and other forms of averaging can produce similar reductions in the observed contrast.
The same contrast can have two causes
Define visibility by V = (Imax − Imin)/(Imax + Imin), assuming the phase scan reaches the maximum and minimum. The preceding expression gives V = 2√(I₁I₂)|γ|/(I₁ + I₂). Visibility depends on both beam balance and coherence.
Consider two invented arrangements in the same intensity units. In A, the intensities are 2.5 and 2.5 and |γ| = 0.8. In B, they are 4 and 1 and |γ| = 1. Both have total intensity five and interference amplitude four. With matching phase offsets, both produce I(φ) = 5 + 4 cos φ, ranging from one to nine, with visibility 0.8.
The entire ideal phase scan is identical, although the explanations differ. A has balanced beams with reduced mutual coherence. B has fully coherent beams with unequal intensities. A photograph of their fringes cannot resolve this ambiguity on its own.
Measure the alternatives separately
Blocking each arm in turn, while preserving the other arm's preparation, distinguishes the examples: A gives 2.5 in either arm, whereas B gives four and one. Combining those measurements with the phase scan permits an estimate of |γ| within the assumed model.
This test relies on experimental control. If blocking an arm changes the source or feedback in the other arm, the separate measurements may not describe the two-arm arrangement. Background subtraction, detector linearity and phase drift also belong in the measurement account.
A richer observation can therefore resolve an ambiguity without changing the original data. The new information comes from another operation with a different dependence on the unknowns. This is the same principle used in image reconstruction and in the distinction between calibration and prediction.
One detection and many trials
In an experiment with individually registered photons, a single detection is a localized outcome. The interference distribution is inferred from repeated trials with a controlled preparation. A point on the detector does not display a complete fringe pattern.
The corresponding quantum calculation combines probability amplitudes for alternatives that remain coherent in the measurement. If another system retains distinguishable path information, the accessible interference can decrease. The quantitative effect depends on the physical correlations and on which outcomes are retained. The earlier articles on coherence and quantum markers examine that mechanism in detail.
These facts constrain a theory of the experiment, but a shared pattern need not choose uniquely among interpretations that predict the same probabilities. An interpretation becomes empirically distinguishable when it changes an observable distribution under a specified preparation and measurement.
Connect the pattern to completion
The book's quantitative interference record retains off-diagonal information because later phase comparisons can reveal it. Two preparations with the same path populations need not be equivalent under all future tests.
For SMM/DTT, that is a concrete requirement on completion: discarding a distinction that changes a later fringe would discard observable information. A proposed model must recover the relevant amplitudes, composition rule and detection probabilities together. Naming a generative process does not replace this calculation.
The balanced and unbalanced examples above add a complementary lesson. Even a complete fringe scan can leave model parameters undetermined. A good explanation states both what the pattern establishes and which additional observations are needed to identify its cause.
Sources and further reading
- Feynman, Leighton and Sands, Quantum behavior, for amplitude addition and the two-alternative interference term.
- Mohamed Haj Yousef, A quantitative interference record and Coherence and memory. The equal-pattern numerical comparison is a teaching example developed here.