Tangential (yz)
Sagittal (xz)
Koechner eq. 5.69: a fold mirror used off-axis has ftangential=(R/2)cosΞΈ and fsagittal=(R/2)/cosΞΈ β the two planes generally see different stability.
Transverse modes (m,n) sit at Ξ½ = FSRΓ[q + (m+Β½)Οt/2Ο + (n+Β½)Οs/2Ο] relative to the longitudinal comb. When Ο/2Ο is a simple fraction p/N, mode families from different q overlap in frequency β the well-known transverse-mode-degeneracy effect (strongest at confocal, Ο=90Β°).
Tangential
Sagittal
Ratio wsagittal/wtangential at the reference plane. A perfectly round beam gives 1.00; folded cavities with fold mirrors are astigmatic and deviate from 1.00 unless compensated.
| Element | z (mm) | wt (mm) | ws (mm) | Area (mmΒ²) |
|---|
β¦ tangential Β· β sagittal. Shaded = stable (0<gβgβ<1 or β1<gβgβ<0). This point uses only the two terminal mirrors β it does NOT move for fold-mirror curvature or thermal lensing. For the real stability of the full cavity, use the "effective gβgβ" numbers below and the header badge, not this point's position.
Mark parameters to optimize with the π― button next to any slider in the element list (left panel), then run.
Solves the self-consistent axis offset (IβM)β»ΒΉΒ·(E,F) induced by a fixed per-round-trip perturbation, then propagates it through the cavity with ordinary ABCD matrices. Tilt and decenter of the selected mirror are applied together (2Γ doubling on reflection for each, matching the existing tilt convention); set either to 0 to isolate the other's effect.
Pattern uses I_mn(x,y) = [H_m(β2x/w_t)]Β² [H_n(β2y/w_s)]Β² exp(β2xΒ²/w_tΒ²β2yΒ²/w_sΒ²), evaluated at the cavity waist (M1 reference plane) using the current wββ, wββ. MΒ² of a pure TEMmn mode is exactly (2m+1) in the tangential direction and (2n+1) in the sagittal direction β a standard, closed-form result of Gaussian-beam theory (not a wave-optics simulation).
z is found from the desired output when in target mode by solving E_out/E_sc = zβ1βln z numerically. Defaults reproduce Koechner's own Nd:YAG worked example (Β§8.1): expect R_optβ0.65, t_pβ11ns, P_pβ9MW.
Scales the cavity's ideal TEMββ waist by MΒ² using the standard real-beam relations ΞΈ_MΒ²=MΒ²Ξ»/(Οwβ) and w(z)=wββ[1+(MΒ²Ξ»z/ΟwβΒ²)Β²]. Use this to predict the actual output spot size downstream when the real beam isn't diffraction-limited.
Ξ· = [2wβw_p/(wβΒ²+w_pΒ²)]Β² β assumes the pump and cavity mode share the same waist location with matched wavefronts. This is a fast estimate, not Koechner's full 3-D overlap integral (eq. 6.71), which additionally needs the pump's MΒ², absorption profile, and focus offset solved numerically.
N_F = aΒ²/(Ξ»L). Large N_F (β«1) β the aperture is many Fresnel zones across, diffraction loss is negligible and the paraxial ABCD treatment is trustworthy. N_F β² 1 β significant diffraction, the simple clipping estimate elsewhere in this tool becomes unreliable.
Independently perturbs every element's ROC, length/thickness, and (for fold mirrors) AOI with Gaussian noise at the Ο set above, then reruns the same round-trip stability/eigenmode solve used everywhere else in this tool for each trial. Angular tilt and lateral decenter of a single mirror are handled separately in the Alignment Sensitivity tab β this tab is for dimensional manufacturing/fabrication tolerances across the whole cavity.
Run the analysis to see results.
setParam(label, key, value) β set a numeric parameter on the element named "label"
getParam(label, key) β read a parameter back
metrics() β {stableT, stableS, w0t, w0s (mm), g1, g2} for the current design
log(...args) β print to this Output panel
elementLabels() β array of current element labels, in order