A reference beam makes phase observable

Write the object and reference complex amplitudes at the recording plane as O and R. Their intensity contains the object intensity, the reference intensity, and two conjugate cross terms. These last terms encode the phase difference. Mutual coherence, compatible polarizations, and mechanical stability are necessary during the exposure. A bright reference by itself does not recover phase if the relative phase wanders through many cycles before the detector finishes averaging.

In off-axis holography, tilt a plane-wave reference slightly relative to the object beam. Its known transverse phase ramp shifts the two cross-term spectra to opposite carrier frequencies. A Fourier-domain window can isolate one cross term if the carrier, object bandwidth, and sampling rate leave them separated from each other and the central intensity terms. Too little tilt causes overlap; too much tilt produces fringes finer than the pixels can sample. Neither problem is solved just by increasing reference power.

For a simple thin amplitude hologram, suppose developed transmittance varies linearly with recorded intensity over a limited exposure range. Illuminating that pattern with the original reference gives several field terms. One is proportional to the object field multiplied by reference intensity; another involves the conjugate object and produces a twin image. Other terms form an undiffracted background. Off-axis geometry separates these orders in angle. Real recording materials can be nonlinear, so this elementary transmittance law is a deliberate model.

A digital hologram instead stores camera samples and extracts the complex cross term numerically, then propagates it with a calibrated diffraction model. The recovered field is defined relative to the reference, not an independently measurable absolute phase of the universe. Refocusing can move a reconstruction plane within the information supported by the measurement. It does not restore unrecorded spatial frequencies, remove speckle automatically, or supply complete three-dimensional material properties from one arbitrary scattering scene.

Rather than separate orders spatially, record several exposures while stepping a uniform reference phase. For a stable object and known real reference amplitude r, phases zero, π/2, π, and 3π/2 separately probe the real and imaginary parts of the object field. Subtracting opposite exposures removes both separate intensities. The four measurements therefore yield two quadratures without needing an off-axis carrier, although motion between frames or phase-step calibration errors can corrupt the result.

Use the waves lab as a fully coherent object–reference pair. Reset and set both controls to phase zero degrees and amplitude ratio 0.5; predict amplitude 1.5. Reset again and set phase 180 degrees and ratio 0.5; predict amplitude 0.5. The squared amplitudes differ by two in its arbitrary intensity units. The lab shows signed waveforms rather than camera exposure values and has no hologram, Fourier filtering, or diffraction reconstruction. Its purpose is to test the interference algebra before interpreting a recorded pattern.