Mirror
A mirror reflects from its coated face. It can be plane, spherical or any conic, cut off-axis from its parent surface, partly transmitting, and chirped, adding a group delay dispersion to each reflection.
The model
Shape
The mirror is a surface in its own frame (see Concepts). Its working face points along the mirror’s angle, and its sag is the conic
\[\mathrm{sag}(y) = \frac{c\,(y + y_0)^2}{1 + \sqrt{1 - (1+k)\,c^2 (y + y_0)^2}} - \mathrm{sag}(y_0), \qquad c = \frac{1}{R},\]over \(\vert y\vert\) up to half its length. \(y_0\) is the off-axis offset: the aperture slides that far along the parent curve, which is how an off-axis parabola is made (\(k = -1\)). \(R\) is positive when the mirror is concave towards the light it faces, so its focal length is \(R/2\). Leave \(R\) blank for a plane mirror.
Reflection
Rays and beams reflect by the law of reflection, \(\hat d_r = \hat d - 2(\hat d\cdot\hat n)\hat n\), at the exact point and normal on the surface. A mirror reflects a share \(R_m\) of the power (the reflectance, %) in amplitude \(\pm\sqrt{R_m}\), both polarisations alike.
The sign matters for interference. From its coated face a mirror turns the field over, half a wave. From behind, a partly transmitting mirror reflects the other way up, as a lossless one must by the Stokes relations:
\[t\,t' - r\,r' = 1, \qquad r' = -r.\]With the same sign both ways a cavity would not conserve energy, and its reflected light would not cancel on resonance. The rest of the power, \(1 - R_m\), is transmitted as a branch of its own. A mirror reflecting 100 % absorbs light reaching its back.
For a beam, the reflection is a ray-transfer matrix in the plane, with the surface’s local curvature, and one out of it, found from rays traced through the real surface in three dimensions. Off-axis incidence then gives the tangential and sagittal focal lengths
\[f_t = \frac{R\cos\theta}{2}, \qquad f_s = \frac{R}{2\cos\theta},\]so a curved mirror used off axis makes a beam astigmatic, as it does in the lab.
Tip
A mirror (as any free-space part) can be tipped out of the bench plane by its Tip \(\tau\) (degrees, up positive), as the vertical knob of a kinematic mount tips it. Its normal is then
\[\hat n = (\cos\phi\cos\tau,\ \sin\phi\cos\tau,\ \sin\tau),\]with \(\phi\) its Facing. Rays meet it in 3D, within its round aperture, and reflect by \(\hat d' = \hat d - 2(\hat d\cdot\hat n)\,\hat n\). Square on, a tip \(\tau\) sends the light back rising at \(2\tau\). A tipped 45° fold also turns the polarisation, as a real one does: s and p are taken from the tipped plane of incidence (see heights). Two mirrors tipped \(45°\) one over the other, the upper at its own height and tipped \(-45°\), make a periscope.
Chirped mirrors
A mirror can add a group delay dispersion \(\mathrm{GDD}\) (fs²) to each reflection: a spectral phase \(\tfrac12\,\mathrm{GDD}\,(\omega - \omega_0)^2\) on the light reflected from its face. A negative GDD compresses a positively chirped pulse. See Short pulses.
Settings
| Field | Default | Meaning |
|---|---|---|
length | 25 | mm: the mirror’s width across the bench. |
reflect | 100 | % reflected, 0 to 100 (the rest is transmitted). |
offset | 0 | mm: an off-axis segment of the parent surface, this far from its vertex. |
k | 0 | conic constant (0 sphere, −1 paraboloid). |
R | null | mm radius of curvature, null for flat. Positive: concave towards the light it faces (focuses, f = R/2); negative: convex. |
gdd | 0 | fs² added to each reflection (a chirped mirror; negative compresses a pulse); 0 for an ordinary mirror. |
mapScale | "lin" | "lin" or "log": the irradiance map of the light arriving (its Diffraction tab, as a screen’s). |
vectorFocus | "auto" | "auto", "on" or "off": the vector focus model for steep beams arriving. |
mtfFreq | 50 | lp/mm at which its Diffraction tab quotes the MTF. |
From the tool's own tables, file format version 1: the names are those in a .jekray file.
Readouts
A mirror reads the light arriving at its face as a screen does. Its panel has tabs for:
- the Rays that arrive;
- the Diffraction pattern: the irradiance map, point spread function, MTF and Strehl ratio of the field;
- the Pulse, for a pulsed source.
The light that leaves after it is not changed by reading it.
Assumptions
- The reflectance is the same for s and p, and does not depend on wavelength or angle. For a coating that does, use a lens surface with a thin-film stack, or a dichroic filter.
- The mirror has no thickness: transmitted light passes straight through, without a substrate’s refraction.
- The surface is a perfect conic: no surface figure error or roughness.
Related
- Lens, for refracting surfaces and thin-film coatings.
- Free-space optics: cavities: two or more mirrors facing each other make a cavity, found and solved automatically.