Concepts
The ideas the rest of these pages build on: where things are, which way they face, how fields are written, and when light from two paths interferes.
Units
| Quantity | Unit |
|---|---|
| Lengths and positions on the bench | mm |
| Angles | degrees, anticlockwise from +x |
| Wavelength | nm, in vacuum |
| Optical power | mW (sources), dBm (fibre amplifiers) |
| Fibre length | m; fibre loss in dB/km |
| Frequencies | Hz in the electronics, MHz for modulators |
| Pulse duration | fs; group delay dispersion in fs² |
Coordinates and angles
The bench is the plane seen from above: x to the right, y up. Every element has a position (x, y) and an angle, and that angle sets its own frame. The element’s local +x points along the angle, and local +y is 90° anticlockwise from it.
What the angle means depends on the element:
| Element | Its angle is |
|---|---|
| Source | the direction the light leaves in |
| Mirror, grating | the direction its working face points (its normal, towards the light) |
| Screen, photodetector, fibre tip | the direction its face points, towards the light arriving |
| Lens | the direction of its optical axis; light travelling that way meets surface 1 first |
| Plane elements (stop, waveplate, polariser, mask, filter, …) | their normal: 0 or 180 for a beam travelling along x |
| Isolator | its forward direction |
To fold a beam travelling in direction \(a\) into direction \(b\), a plane mirror faces \((a + 180° + b)/2\). For example, a beam travelling +x (0°) sent up to +y (90°) needs a mirror at 135°.
Surfaces
Every surface is a curve in its element’s frame, \(x = z_0 + \mathrm{sag}(y)\), over \(\vert y\vert\) up to its semi-aperture. The sag of a conic with aspheric terms is
\[\mathrm{sag}(y) = \frac{c\,y^2}{1 + \sqrt{1 - (1+k)\,c^2 y^2}} + a_2 y^2 + \sum_{j \ge 1} A_j\, y^{2j+2} + \sum_{j \ge 1} \alpha_j\, \vert y\vert ^{j}\]with curvature \(c = 1/R\) and conic constant \(k\): 0 for a sphere, −1 for a paraboloid. The last sum holds odd terms. Its first term, a cone, is how an axicon is made.
Polarisation
Light carries two complex amplitudes:
- s, perpendicular to the bench plane;
- p, in the plane, perpendicular to the ray.
Together they form a Jones vector \((E_s, E_p)\). A polarisation element is a 2 × 2 Jones matrix on that vector.
The p direction follows a parity convention. It is the in-plane normal to the ray, and it reverses at each reflection. A mirror then turns s and p over alike, and at normal incidence \(r_p = r_s\), as it must when s and p cannot be told apart.
Fields and phase
The ray tracer accumulates phase as \(e^{+ikL}\) along each ray’s optical path \(L = \sum n_i \ell_i\). The field engine writes fields as \(e^{-ikz}\), following Siegman. In that convention a Gaussian beam is
\[E(r, z) \propto \exp\!\left(-\frac{i k r^2}{2 q(z)}\right), \qquad q(z) = z + i z_R, \qquad z_R = \frac{\pi w_0^2}{\lambda}.\]The two meet where the field engine takes in the tracer’s Jones vectors, which enter conjugated.
Launches and coherence
Light from one laser is one launch. Wherever two paths of a launch meet, on a screen, a photodiode or in a fibre, their fields add. Separate lasers add as powers, never as fields.
A source can be split from another, like two fibre launches off one coupler. The two are then one launch with a phase and an extra length of fibre between them, so their light interferes.
A laser’s linewidth \(\Delta\nu\) sets its coherence length \(L_c = c/\Delta\nu\). Two paths of one launch whose optical paths differ by \(\Delta L\) interfere with the fringe contrast
\[\vert \gamma(\Delta L)\vert = \begin{cases} \exp\left(-\pi \vert \Delta L\vert / L_c\right) & \text{Lorentzian line (a laser)}\\[4pt] \exp\left(-\dfrac{(\pi \Delta L / L_c)^2}{4 \ln 2}\right) & \text{Gaussian line (an LED, a Doppler-broadened line)} \end{cases}\]and the cross term of each pair of fields is weighted by it. \(\Delta L\) is the difference in group paths, so glass counts by its group index. A coherence length of 0 means a single frequency: every pair counts in full.
A broadband source is given by its bandwidth \(\Delta\lambda\). Its coherence length is \(\lambda^2/\Delta\lambda\), a few wavelengths, so only near-equal paths still make fringes: white-light fringes.
The beam tree
The tracer sends one reference ray down the middle of each Gaussian source. Wherever that ray splits, at a beam splitter, a partly reflecting surface, a grating order or each round trip of a cavity, the beam branches. The branches form a tree. Every Gaussian-beam result is worked out along this tree: beam sizes, the diffracted field, cavity modes, and what reaches fibres and photodiodes.
Tabs and link planes
A project can hold several benches, each on a tab. Light keeps to its own bench. Two link planes that share a pair name are one plane seen from two benches: light crossing one leaves the other, and beams, fields and cavities carry across. In fibre, two feedthroughs do the same, and two electrical feedthroughs carry signals between tabs.
Sweeps
Any number in a panel can be swept: it moves back and forth over a range on a shared clock, and the bench is worked out again at each step. The Sweeps menu also has:
- an optimiser: a golden-section search for one parameter, or Nelder–Mead from a Latin hypercube start for several;
- a tolerance study: Monte Carlo over the swept ranges.