Holographic recording is based on interference between two coherent beams: a reference beam and an object beam. The object beam carries information about the wavefront coming from the object, while the reference beam provides a stable phase reference. When the two beams overlap on a recording medium, they produce a fine fringe pattern. That fringe pattern stores both amplitude and phase information indirectly, which is why a hologram can later reconstruct a three-dimensional-looking wavefront.
In the simplest geometry for two plane waves, the most important quantity is the spacing between adjacent interference fringes. If the angle between the two beams is \(\theta\), then the fringe spacing is
\[
d_f=\frac{\lambda}{2\sin(\theta/2)}.
\]
Here \(\lambda\) is the wavelength of the recording light. This formula shows two basic trends immediately. First, shorter wavelength means smaller fringe spacing. Second, larger beam angle also means smaller fringe spacing because \(\sin(\theta/2)\) increases.
For the sample input
\(\theta=30^\circ\)
and
\(\lambda=633\ \text{nm}\),
the half-angle is
\[
\theta/2=15^\circ.
\]
Therefore
\[
d_f=\frac{633\ \text{nm}}{2\sin 15^\circ}.
\]
Since
\[
2\sin 15^\circ \approx 0.5176,
\]
we get
\[
d_f \approx \frac{633}{0.5176}\ \text{nm} \approx 1223\ \text{nm}.
\]
So the fringe spacing is about
\[
d_f \approx 1.22\ \mu\text{m},
\]
which matches the sample output.
This is an extremely fine pattern. In fact, holographic recording media must be able to resolve very small spatial features in order to store the interference fringes properly. Another useful related quantity is the spatial frequency of the fringes:
\[
\nu_g=\frac{1}{d_f}.
\]
If the spacing is expressed in micrometers, this can be converted into lines per millimeter. High-angle or short-wavelength recordings therefore lead to high spatial-frequency holograms.
The local recorded intensity on the plate oscillates from bright to dark as you move across the interference pattern. In a simple equal-amplitude model, the recorded fringe intensity can be written as
\[
I(x)=I_0\cos^2\left(\frac{\pi x}{d_f}\right),
\]
where \(x\) is position along the recording plate and \(I_0\) is a reference intensity scale. This means that every time the plate position changes by one fringe spacing \(d_f\), the pattern repeats.
The calculator uses this relation to preview the local fringe intensity at a chosen point on the hologram. That makes it easy to see whether the selected point corresponds to a bright fringe, a dark fringe, or something in between.
The really important physical idea comes during reconstruction. If the developed hologram is illuminated again with the original reference beam, the recorded fringe pattern acts like a very fine diffraction grating and reconstructs the original object wavefront. In the simplest symmetric preview used here, the reconstructed direction corresponds to the original object-beam direction, which is why the tool shows a reconstruction angle associated with the recording geometry.
This is only a basic preview, not a full holography solver. Real holography can involve spherical wavefronts, off-axis reference beams, amplitude and phase modulation of the medium, thick or thin holograms, and Bragg selectivity in volume holograms. But the foundational recording idea is still the same: coherent beams interfere, and that interference pattern is stored in the recording medium.
The animation in this calculator shows the object beam and reference beam meeting at the plate, the fine fringe strip that is recorded there, and a schematic reconstructed beam when the hologram is read out with the reference beam again. The right-side plot shows how the fringe intensity varies across the plate over several fringe periods.
So the core formulas to remember are
\[
d_f=\frac{\lambda}{2\sin(\theta/2)},
\qquad
\nu_g=\frac{1}{d_f},
\qquad
I(x)=I_0\cos^2\left(\frac{\pi x}{d_f}\right).
\]
These relations explain why holographic recording requires coherent light, why fringe spacing becomes extremely fine, and how a recorded interference pattern can later reconstruct a stored optical wavefront.
