Intensity data leave phase ambiguities

Let a coherent object field be u and its Fourier transform be U, using the cycles-per-length convention. The measured far-field intensity is proportional to the squared modulus of U after known propagation factors are removed. Taking the inverse Fourier transform of this intensity produces the autocorrelation of u. Thus the data contain pairwise difference information. They are not equivalent to direct access to the complex Fourier transform, whose inverse would immediately supply the field.

Autocorrelation connects an object with a conjugated shifted copy of itself, integrating over their overlap. It can identify support extents and repeated separations, yet it does not assign those separations uniquely to individual features. With finite sampled data one must distinguish circular DFT autocorrelation from the aperiodic correlation of a compact object padded with zeros. Insufficient padding mixes opposite boundaries. A reconstruction that ignores this distinction can match its own numerical model while misinterpreting the physical experiment.

Multiplying the whole object by one constant unit-modulus phase rotates every Fourier coefficient but leaves all intensities unchanged. Translating the object multiplies its transform by a frequency-dependent unit-modulus phase and also leaves the intensity unchanged. Conjugating and reversing the object produces the conjugate transform, again preserving the measured magnitude. A support condition or an external position reference may rule out some translations, but an intensity-only experiment does not determine a global field phase.

These familiar ambiguities are not the entire uniqueness problem. One-dimensional Fourier magnitude can have further nontrivial ambiguities even with finite support. Higher-dimensional support, positivity, oversampling, masks, or diversity measurements can improve identifiability under explicit assumptions, but no universal recovery guarantee follows from the word oversampling alone. An opaque or complex-valued specimen may violate nonnegativity assumptions used successfully for other objects. A prior is useful only when its physical justification is stated.

A basic alternating procedure Fourier transforms an object estimate, replaces its modulus by the measured modulus while retaining its current phase, inverse transforms, then enforces object-domain constraints. Those constraints might set values outside a known support to zero or impose a justified real nonnegative object. The sets are generally nonconvex, so iterates can stagnate or settle at different consistent solutions. A decreasing residual demonstrates improved agreement with the specified data model, not proof that a unique physical object has been found.

The vectors instrument isolates magnitude-only information. Reset and set both controls to angle zero degrees and magnitude one; predict components one and zero. Reset again and set angle 90 degrees and magnitude one; predict components zero and one. The magnitude stays fixed while the vector changes. This is an analogy for one complex coefficient with missing phase. The instrument displays both components and does not perform Fourier propagation, impose support, or run a phase-retrieval algorithm.