= Optical biprism
An <optical biprism> combines two shallow prism regions that deflect parts of an incident beam in different directions. In the familiar Fresnel arrangement, associated with <Augustin-Jean Fresnel>, the regions act like two thin <optical prisms> placed base to base. <Light> from one narrow illuminated slit emerges as if from two separated <virtual images> of that slit. Where the two parts overlap, their <optical coherence> permits <wave interference>.<User's Manual: Bi-Prism Experiment by Raman Scientific Instruments>[PDF pp2–4, apparatus and action of the biprism]{ref}
This supplies two related optical sources without requiring two independent lamps to maintain a stable <phase difference>. A bench apparatus uses a narrow slit, the <optical biprism>, and a movable eyepiece or screen. A micrometer translation measures the spacing of several <interference fringes>. A <convex lens> can image the two virtual sources in either of two conjugate positions; if their measured image separations are $d_1,d_2$, their actual full separation is $d=\sqrt{d_1d_2}$. The inverse magnifications of the two positions explain the geometric mean.<User's Manual: Bi-Prism Experiment by Raman Scientific Instruments>[PDF p2, formula and apparatus; PDF pp4–5, virtual-source interference]{ref}
A thin-prism calculation makes the geometry explicit. Let each region have small wedge angle $\alpha$, <refractive index> $n$ relative to air, and slit-to-prism distance $a$. The paraxial deviation is $\delta\simeq(n-1)\alpha$. Backward extension of the emergent rays locates the virtual sources a distance $a\delta$ above and below the original slit, so
$$
d\simeq2a(n-1)\alpha.
$$
At a screen a further distance $b$ beyond the prism, the source-to-screen distance is $D=a+b$. Near the axis, the <optical path difference> is $dy/D$, giving <interference fringe spacing>
$$
\beta\simeq\frac{\lambda D}{d}.
$$
The symbols here use $d$ for the full source separation; the apparatus manual uses $2d$ for that same quantity.
For an original chosen model, $n=1.50$, $\alpha=1^\circ$, $a=0.10\,\mathrm m$, $b=0.90\,\mathrm m$ and $\lambda=600\,\mathrm{nm}$ give $d\simeq1.745\,\mathrm{mm}$ and $\beta\simeq0.344\,\mathrm{mm}$. The wedge angle must be converted to radians in the small-angle formula.
\Image[/fresnel-biprism-virtual-sources.png]
{title=Two virtual sources and the interference fringes of a thin optical biprism}
{description=Original paraxial model with the stated chosen parameters. Solid lines are selected physical rays; dashed lines extend the emergent rays backward to the <virtual images>. The prism is schematic, and axial and transverse scales differ. The right panel assumes equal coherent amplitudes and shows the central <interference fringes>, normalized to unit maximum.}
A finite slit width, limited <optical coherence>, unequal beam intensities and misalignment can reduce fringe visibility. The <Wollaston prism> instead separates <linear polarization> components using <birefringence>; its two beams do not automatically interfere on an ordinary intensity screen because their polarizations are orthogonal.
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