← Questions and answers (section 09)

Distances between optical parts: calculator

A collimated laser beam barely changes over a bench, so most distances between flat optics (mirrors, wave plates, cubes) are free: they are set by the size of the mounts, the table holes and the apertures. Only a few distances are fixed by optics: the lens distances in the double-pass cat's eye, the lens spacing of a telescope, and the length difference between the two arms. The calculators below use standard Gaussian-beam optics (ABCD matrices) and the data-sheet values of our parts; every formula is shown next to its result.

0Which distances are fixed and which are free

Distance	Set by	Our value	How exact
AOM → cat's-eye lens	lens focal length f	100 mm	not critical for our small RF scans (calculator 3 gives the walk)
cat's-eye lens → retro mirror	f (mirror at the focus)	100 mm	the critical one: an offset δ makes the return beam converge (calculator 3)
telescope lens → lens (beam expander)	f₁ + f₂	50 + 100 = 150 mm	set by the output beam staying the same size far away (calculator 2)
collection lens → photodiode	the spot must fit the chip	about 40–50 mm	loose: the spot only has to fit 3.6 × 3.6 mm with margin
control path − probe path (split → cell)	laser linewidth	within about 0.5 m	calculator 4
between flat optics (mirrors, plates, cubes, isolator)	mount size, table holes, apertures	free	the beam must fit every aperture (calculator 2); mounts must not touch
the two alignment irises	accuracy wanted	as far apart as possible	calculator 5

1Fibre collimator → beam size

Wavelength (nm)Rb D1, 794.98 nm in vacuum Fibre mode field diameter (µm)PM780-HP: 5.3 ± 1.0 µm at 850 nm (ask the laser maker for the actual fibre) Collimator focal length f (mm)planned CFC11A-B (BOM L05): 11 mm Beam diameter wanted (mm)for the "focal length needed" line

Result	Value	Formula

What this shows for our plan. With the planned 11 mm collimator and a PM780 fibre the beam is about 2.1 mm (Thorlabs' own table gives 2.1 mm for this collimator), not the 1.5 mm the equipment list assumes. That is close to the AOM's 2 mm active aperture (calculator 2 shows about 93 % passing). A collimator of about 7.5–8 mm focal length gives about 1.5 mm; a 2:1 telescope in front of the AOMs does the same. Still to be decided; measure the real beam first.

2Beam along a path: size at every part, clipping, lenses

Example start with the beam of calculator 1 or start diameter mm (waist at the start)

Part	Position	Beam there	Passes

Edit the path (distances, lenses, apertures)

Type	Value	Label

Transmission of a centred Gaussian beam (1/e² radius w): round aperture of radius a: 1 − exp(−2a²/w²); rectangle: erf(√2 a_x/w) · erf(√2 a_y/w). A round aperture of radius 1.5 w passes 98.9 %, so "aperture at least 1.5 × the beam diameter" is the usual rule. "Max beam" is the largest beam diameter the maker allows (the isolator), not a clear aperture. Distances in the examples are typical, not measured: type in your own.

3Double-pass AOM (cat's eye): the distances that matter

RF centre frequency (MHz) Acoustic velocity (mm/µs)TeO₂ longitudinal, G&H 3080-125 drawing Cat's-eye lens f (mm)AC254-100-B (BOM O08) Beam diameter in the AOM (mm) AOM → lens distance d: Mirror offset from the focus δ: RF scan of the probe (± kHz)spec: ±200 kHz around 79.965 MHz Distance from the AOM to the cell (mm) PBS → AOM (mm)

Result	Value	Formula

Why d = f and the mirror at f. The ABCD matrix of the unfolded arm (AOM → d → lens → f + δ → mirror → back) has B = 2(f − d) and C = 2δ/f² for small offsets. B = 0 (d = f) means the returning beam comes back through the same point of the AOM whatever the RF frequency; C = 0 (δ = 0) means it comes back collimated. For our ±200 kHz scan B hardly matters (sub-micrometre walk), so the lens-to-mirror distance is the one to get right: put the mirror where the returning beam has the same size far away as the incoming one (or where the 0th and 1st orders are sharpest on a card). Double-pass practice: Donley et al., Rev. Sci. Instrum. 76, 063112 (2005).

4Control and probe path lengths

Laser linewidth Δν (MHz)Eagleyard BFY12: 0.6 typ, 1.0 MHz at 15 mW Control: split PBS → cell (mm)a double-pass arm counts twice Probe: split PBS → cell (mm)

Result	Value	Formula

Both beams come from one laser, so its phase noise cancels in the two-photon (EIT) resonance only if both arrive at the same time. A delay τ leaves a relative phase variance 2πΔντ (a Lorentzian, white-frequency-noise laser, as in delayed self-heterodyne theory); the noisy part is spread over about a megahertz, far wider than the kilohertz EIT line, so it only lowers the contrast. Real DFB lasers also have slower (1/f) frequency noise; measure the arm lengths on the bench with a tape.

5Alignment accuracy with two irises

Centring error on each iris e (mm)assumption: by eye on a closed iris with the IR card Distance between the irises L (mm) Length of the bench after them (mm) Iris → optic for the back-reflection check (mm)

Result	Value	Formula

Geometry only: the further apart the irises, the smaller the angle error. See the alignment question in section 09 for the method.

6What these calculators do not do

Thin lenses and perfect Gaussian beams only: no aberrations, no lens thickness, no diffraction rings from a clipped beam (the transmission is a first-order estimate).
The AOM's diffraction efficiency for a given beam size is not modelled: ask the maker for the figure at 795 nm and at our beam size.
A free-space laser (for example the Yanban option) has an elliptical, astigmatic beam: measure its diameters with the camera and type them in as the start diameter.
The saturated-absorption (SAS) geometry and the mechanical size of each mount are not calculated: check the mount drawings.

7Sources of the formulas and numbers

Gaussian beams and ABCD matrices: H. Kogelnik and T. Li, "Laser beams and resonators", Appl. Opt. 5, 1550 (1966); waist after a lens: S. A. Self, Appl. Opt. 22, 658 (1983).
Collimator beam diameter 4λf/(π MFD) and the maximum waist distance: Thorlabs fibre collimator pages (CFC-11X-B sheet); PM780-HP mode field diameter: Thorlabs PM fibre table.
AOM: Bragg angle λf/(2v), separation λf/v, rise time ≈ 0.64 D/v (transit time of the sound across the beam); G&H 3080-125 outline drawing 97-01598-01 rev D (4.2 mm/µs, 2.5 × 2 mm aperture, 23/34/65 ns for 125/200/400 µm beams).
Double-pass AOM: E. A. Donley et al., Rev. Sci. Instrum. 76, 063112 (2005) (NIST PDF).
Phase noise of a delayed copy of a laser: T. Okoshi, K. Kikuchi and A. Nakayama, Electron. Lett. 16, 630 (1980); L. E. Richter et al., IEEE J. Quantum Electron. 22, 2070 (1986).
Every function of this page is checked against these values by a test script (32 checks: Thorlabs examples, G&H rise times, ray traces and a numerical integration).