Coherence decides whether amplitudes or intensities add

For a deterministic coherent field, propagation through a linear optical system is linear in complex amplitude. In a shift-invariant region the image field is the convolution of the object field with an amplitude impulse response a. The camera then records its modulus squared, up to a calibration factor. This final step is nonlinear in the object field. Consequently there is generally no single intensity PSF that convolves arbitrary coherent object intensities to give their camera image.

Consider two object contributions that produce fields A and B at one detector position. Their detected intensity contains two separate squared amplitudes and a cross term that depends on relative phase. Changing phase can alter the intensity without changing either individual contribution. A phase-only object can thus become visible after propagation or filtering even though its transmitted intensity is uniform in its own plane. Predicting that image requires its phase distribution and the coherent propagation operator, not only its local brightness.

For spatially incoherent object emission, distinct points have zero mutual field correlation on the measurement average. Averaging the squared field sum eliminates cross terms between different points. The resulting image intensity is the convolution of object intensity with the normalized squared modulus of the amplitude impulse response. Temporal narrowband assumptions are still needed if one wants a single wavelength-dependent impulse response; a broadband camera usually adds wavelength-dependent intensity images with spectral weights.

Partial coherence lies between these cases and requires the mutual intensity, a two-position correlation, rather than just one field or one intensity map. The two-point formula retains a cross term multiplied by a complex degree of coherence. Saying that incoherent light never interferes is misleading: a beam can interfere with a sufficiently matched copy of itself. The relevant statement here is that the specific distinct object-point contributions are mutually uncorrelated under the observation average.

For a clear scalar paraxial pupil with numerical aperture NA at wavelength λ, the coherent amplitude transfer function has transverse support up to NA/λ in object-space cycles per metre. The incoherent intensity OTF can extend to 2NA/λ because it is a pupil autocorrelation. These act on different input quantities. Comparing the numbers alone does not prove that incoherent imaging is always superior: contrast, object type, noise, phase sensitivity, and a chosen resolution criterion all affect the practical comparison.

Use the wave lab to test the cross term directly. Reset and set both controls to phase zero degrees and ratio one, predicting field amplitude two. Reset and set phase 180 degrees and ratio one, predicting zero. At a quarter-cycle difference the amplitude is the square root of two. The plotted sum is a field, not measured intensity; square its amplitude to make the optical comparison. This fully coherent pair cannot be changed into incoherent illumination by simply selecting a fixed phase.