= Lippmann fringe spacing
{c}
{title2=$d=\lambda_0/(2n)$}
At <normal incidence>, let $z$ measure distance from the reflector into a uniform transparent layer with <refractive index> $n$. For a <monochromatic wave> of <vacuum> <wavelength> $\lambda_0$, the layer <wavenumber> is $k=2\pi n/\lambda_0$. Represent the incident and returning <electric fields> by complex <wave amplitudes> with time factor $e^{-i\omega t}$ suppressed:
$$
E_i(z)=E_0e^{-ikz},\qquad E_r(z)=rE_0e^{ikz},\qquad r=|r|e^{i\phi}.
$$
The time-averaged squared <electric field>, normalized to the incident contribution, is
$$
\frac{\langle E^2\rangle}{\langle E_i^2\rangle}
=1+|r|^2+2|r|\cos(2kz+\phi).
$$
This electric exposure pattern has maxima separated by
$$
2kd=2\pi,\qquad d=\frac{\pi}{k}=\frac{\lambda_0}{2n}.
$$
The <wave phase> of the reflector shifts the maxima's positions but leaves their spacing unchanged in this uniform-layer model. The electric pattern, rather than the zero net <energy flux> of an ideal <standing wave>, is the relevant exposure proxy. An imperfect mirror reduces its contrast: $(I_{\max}-I_{\min})/(I_{\max}+I_{\min})=2|r|/(1+|r|^2)$, where $I$ denotes this exposure proxy.
Choose $\lambda_0=600\,\mathrm{nm}$ and $n=1.5$. Then the medium <wavelength> is $400\,\mathrm{nm}$ and $d=200\,\mathrm{nm}$. For an ideal phase-reversing reflector, $r=-1$ and the normalized pattern is $4\sin^2(kz)$: its first two maxima are at $z=100$ and $300\,\mathrm{nm}$. These are chosen calculations, not measured properties of a historical <photographic emulsion> or the optical response of <Mercury>.
\Image[/lippmann-standing-wave-depth-fringes.png]
{title=A <standing wave> records depth fringes $200\,\mathrm{nm}$ apart}
{description=Original schematic and calculation for chosen $\lambda_0=600\,\mathrm{nm}$, $n=1.5$ and an ideal phase-reversing mirror. The upper diagram distinguishes recording with a reflector from later reflected-colour readout. The lower plot shows the normalized electric exposure pattern, not a measured developed density profile. The first fringe position depends on reflector <wave phase>; the period does not. Material widths in the apparatus schematic are illustrative.}
{source=lippmann-standing-wave-depth-fringes.py}
For multiple mutually incoherent <wavelengths>, time averaging adds exposure contributions rather than their instantaneous <electric fields>. In a linearized recording model,
$$
D(z)\propto\int S(\lambda_0)q(\lambda_0)
\left[1+|r(\lambda_0)|^2+2|r(\lambda_0)|\cos\left(\frac{4\pi n(\lambda_0)z}{\lambda_0}+\phi(\lambda_0)\right)\right]d\lambda_0.
$$
Here $S$ is incident spectral exposure density, $q$ a wavelength-dependent recording sensitivity and $D$ a model exposure response. An <optical spectrum> therefore writes overlapping depth oscillations, not necessarily visibly separated sheets. This is a useful connection to the <Fourier transform>: approximately, depth is paired with optical spatial <frequency> $2n/\lambda_0$. Finite recording depth, varying sensitivity and nonlinear <photographic processing> prevent treating this expression as an exact inversion rule for an actual plate.
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