= Finite-layer model of Lippmann reflection
{c}
{title2=$\lambda_B=2nd\cos\theta$}
To isolate the readout mechanism, replace the processed <photographic emulsion> by $N$ equally spaced, identical weak scattering planes, with separation $d$. Take the readout <refractive index> $n$ to be real and independent of <wavelength>, and let $\theta$ be the internal <angle> from the plane normal. Adjacent specularly backscattered contributions differ in <wave phase> by $2k_zd=4\pi nd\cos\theta/\lambda_0$. Their <geometric series> gives
$$
r_N=a\sum_{j=0}^{N-1}e^{ij\delta},\qquad
\delta=\frac{4\pi nd\cos\theta}{\lambda_0}-2\pi,\qquad
F_N=\frac{|r_N|^2}{N^2|a|^2}
=\left[\frac{\sin(N\delta/2)}{N\sin(\delta/2)}\right]^2.
$$
Here $a$ is a small plane reflection <wave amplitude>, assumed constant in this illustrative band; $F_N=1$ at $\delta=0$ by continuity. Since incident and reflected waves occupy the same external medium, $|r_N|^2$ is the model <reflectance> when transmission factors are absorbed into $a$. Require $N|a|\ll1$ for the weak-scattering sum to remain self-consistent. Normalizing to $F_N$ discards absolute brightness; it does not predict a highly reflecting historical plate.
The first-order peak obeys
$$
\lambda_B=2nd\cos\theta.
$$
This is <Bragg's law> with an <angle> measured from the normal, rather than the glancing <Bragg angle>. For external viewing <angle> $\alpha$ in air approximated by <refractive index> one, <Snell's law> gives $n\sin\theta=\sin\alpha$, hence
$$
\lambda_B(\alpha)=2nd\sqrt{1-\frac{\sin^2\alpha}{n^2}}.
$$
The expression describes specular readout: illumination must supply the matching incident direction. Tilting a plate under fixed narrow illumination can make the signal disappear as well as change its selected colour.
For the chosen $n=1.5$ and $d=200\,\mathrm{nm}$, calculated peaks are $600\,\mathrm{nm}$ at $\alpha=0$, $565.69\,\mathrm{nm}$ at $30^\circ$, and $529.15\,\mathrm{nm}$ at $45^\circ$. Greater obliquity selects a shorter <wavelength>. Uniform shrinkage to $d'=sd$, with unchanged $n$, multiplies every peak <wavelength> by $s$: a chosen $s=0.95$ moves the normal peak to $570\,\mathrm{nm}$. More generally, $\lambda'_B/\lambda_B=(n'/n)s$ at <normal incidence>. A change of thickness alone is insufficient if processing also changes the <refractive index>.
Finite $N$ gives a finite spectral response. The nearest zeros satisfy $\delta=\pm2\pi/N$, giving $\lambda_\pm=\lambda_B/(1\pm1/N)$. At the normal $600\,\mathrm{nm}$ peak, $N=30$ gives neighbouring zeros at approximately $580.65$ and $620.69\,\mathrm{nm}$; $N=10$ gives $545.45$ and $666.67\,\mathrm{nm}$. Thus greater sampled depth narrows the central band in this model. The small-detuning distance from peak to a nearest zero is approximately $\lambda_B/N$; it is not the full width at half maximum.
\Image[/lippmann-angular-colour-finite-stack.png]
{title=Calculated angular colour shift and finite-stack <reflectance> shape}
{description=Original weak-scattering finite-grating model with chosen $n=1.5$ and $d=200\,\mathrm{nm}$. Left: predicted first-order peak versus external <angle>, with a separate illustrative 5 percent shrinkage calculation at unchanged <refractive index>. Right: normalized $F_N$, comparing $N=10$ and $N=30$ at normal readout and $N=30$ at $45^\circ$ external readout. Every curve is calculated; none is a measured spectrum.}
{source=lippmann-angular-colour-finite-stack.py}
Real processed <photographic emulsions> have a continuous, irregular depth response rather than identical planes. Absorption, wavelength-dependent <refractive index>, unequal scattering strengths, a nonideal exposure reflector, changes during <photographic processing> and multiple <scattering of light> can alter both peak positions and shapes. A transfer calculation with measured material parameters would be needed for quantitative <reflectance>. The present model explains the phase-selection mechanism and the tradeoff between depth and spectral width; it cannot establish faithful reproduction of an arbitrary incident <optical spectrum>. Its useful lesson is that a photograph can record colour through sub-<wavelength> structure, while its eventual appearance still depends on illumination, viewing geometry and processing.
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