The plate was exposed to light and its own reflection
Lippmann pressed a reflective backing against a fine-grained photosensitive layer. Light coming in and light reflected back interfered throughout the depth of the emulsion, so the exposure varied with depth as well as across the image. Development kept this fine layered structure, and the layers could later reflect selected colours.
The invention lay in this recording step. Many thin films show interference colours, but none of them has recorded an image. A screen that computes a hue from a wavelength also skips the step in which light physically wrote the colour into the plate.
Sources: [1]
Layer spacing is set by the wavelength inside the emulsion
At normal incidence in a uniform medium, neighbouring intensity maxima of the standing wave sit half a wavelength apart, with the wavelength measured in that medium. The ideal spacing is the vacuum wavelength divided by twice the refractive index. The phase change on reflection moves the first maximum but leaves the spacing unchanged.
Light with a broad spectrum writes many such patterns on top of one another. The developed plate has a finite thickness and its own material response, and it holds no list of separate red, green and blue values. The colour seen later also depends on the viewing light and the viewing angle.
Sources: [1]
Recording and viewing follow different light paths
In the historical account, light passed through the glass and the emulsion before reaching the reflective backing during exposure. Later, under white light, the developed layers reflect a selected spectrum back to the viewer. A diagram that still sends the viewing light into the mercury backing mixes up the two stages.
Tilting the plate changes the optical path between layers and can shift the colour you see. Development, shrinkage and the backing reflection also affect how faithfully the spectrum comes back. A convincing colour does not prove that the original spectrum was recovered exactly.
Modern experiments measure how the colours distort
Plate thickness and backing reflection can distort the reproduced spectrum, as the EPFL authors showed with an experimentally tested model. That qualifies the common claim that interference photography preserves every colour perfectly. Their work studies how a plate responds and how a spectrum can be reconstructed; it does not retell the history of the 1891 invention.
This article shows no digitized spectrum from an original Lippmann plate. The spacing formula covers single-wavelength light. Reconstructing a broad spectrum faithfully needs a model of both the optics and the material response.
Sources: [2]
Start with thin-film colours, then add exposure
The linked model shows how optical thickness picks out the reflected colour. Use it to predict how the colour changes with angle. Then go back to the historical account and find the two steps the model lacks: exposure and development. Those steps are what separate a recorded photograph from a film that was already there.
Sources: [1]