= Holographic wavefront reconstruction
{title2=$I=|R+O|^2$}
Take stationary scalar <monochromatic waves> of the same <frequency> and <polarization of an electromagnetic wave>, with time convention $e^{-i\omega t}$. Write their <complex amplitudes> at the recording plane as $R=A_R e^{i\phi_R}$ and $O=A_O e^{i\phi_O}$, absorbing the common conversion to <irradiance> into the units. An exposure short enough to avoid fringe motion but long compared with an optical period records
$$
I=|R+O|^2=|R|^2+|O|^2+R^*O+RO^*
=A_R^2+A_O^2+2A_RA_O\cos(\phi_O-\phi_R).
$$
The two cross terms are where relative <wave phase> enters the measured <light intensity>. Blocking the <holographic reference wave> leaves $|O|^2$ and removes them. Loss of mutual <optical coherence>, incompatible <polarization of an electromagnetic wave> or motion that averages the relative <wave phase> similarly reduces fringe contrast. These are useful controls for distinguishing a <hologram> from an ordinary brightness record. One local cosine value does not uniquely determine a <wave phase>; known reference structure and separation of the cross terms supply the additional information in the example below.
Model a thin weakly modulated recording by its <amplitude transmittance>
$$
t=t_0+\beta I,\qquad |\beta I|\ll|t_0|.
$$
Here $t_0$ and $\beta$ are real constants in a local linear response to exposure at a fixed exposure time. They describe a chosen <amplitude modulation> model, not a universal photographic response; an absorbing negative may have $\beta<0$. Keep $0\leq t\leq1$ for this passive real-transmission model. Ignore material thickness, saturation, shrinking, clipping and changes of <wavelength>. Illuminating the processed record with the original $R$ gives, immediately after the plate,
$$
Rt=\underbrace{\bigl[t_0+\beta(|R|^2+|O|^2)\bigr]R}_{\text{zero-order background}}
+\underbrace{\beta|R|^2O}_{\text{desired object field}}
+\underbrace{\beta R^2O^*}_{\text{conjugate field}}.
$$
This follows by multiplying the preceding identity by $R$; no <wave phase> detector has been assumed. If $|R|$ is uniform, the desired term reproduces $O$ up to a constant. A nonuniform $|R|^2$ weights the recovered field. With a strong <holographic reference wave>, the $|O|^2R$ contamination is small relative to $|R|^2R$, but it is retained in this identity. Subsequent <diffraction> propagates each term: reproduction at the plate is the starting boundary field, not yet a focused image.
\Image[/holography-recording-and-reconstruction-geometry.png]
{title=Recording <light intensity> and reconstructing <wavefronts>: original functional geometry for <in-line holography>}
The schematic uses a <plane wave> reference and one <optical point source> to separate the functions. The top row records $R+O$; the bottom removes the object and illuminates the developed <hologram> with $R$. Dashed extensions identify the desired <virtual image>; the conjugate component converges toward a <real image>. All three outgoing components coexist. The coloured paths distinguish terms, not different <wavelengths>; distances and strengths are schematic. This functional drawing is not a reconstruction of the historical apparatus.
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