Reference
Formula reference
This is the set of equations I use for laser process development, optical instrument design, vacuum systems, and production quality. It is organized like a laser-optics handbook, with every symbol and unit defined. Where a formula has a live calculator on this site, the entry links to it.
Photon energy & spectroscopy
Photon energy#
Energy carried by a single photon of a given wavelength or frequency. Calculator →
\[ E = h\nu = \frac{hc}{\lambda}, \qquad E\,[\text{eV}] \approx \frac{1239.842}{\lambda\,[\text{nm}]} \]
\(E\) photon energy (J or eV) · \(h\) Planck constant · \(\nu\) frequency (Hz) · \(c\) speed of light · \(\lambda\) vacuum wavelength
Wavenumber and frequency#
The spectroscopic wavenumber is proportional to energy, so level spacings add linearly. Calculator →
\[ \tilde\nu = \frac{1}{\lambda} = \frac{\nu}{c}, \qquad \tilde\nu\,[\text{cm}^{-1}] = \frac{10^{7}}{\lambda\,[\text{nm}]}, \qquad 1\ \text{cm}^{-1} \leftrightarrow 29.979\ \text{GHz} \]
\(\tilde\nu\) wavenumber (conventionally cm⁻¹) · \(\nu\) frequency (Hz) · \(\lambda\) vacuum wavelength
Molar and thermal energy equivalents#
Converts a per-particle energy to a per-mole energy and to the temperature at which \(k_BT\) equals it. Calculator →
\[ \begin{gathered} E_m = N_A E, \qquad T_\text{eq} = \frac{E}{k_B} \\[4pt] 1\ \text{eV} = 96.485\ \tfrac{\text{kJ}}{\text{mol}} = 8065.54\ \text{cm}^{-1} = 11\,604.5\ \text{K} \end{gathered} \]
\(E_m\) molar energy (J/mol) · \(N_A\) Avogadro constant · \(k_B\) Boltzmann constant · \(T_\text{eq}\) temperature equivalent (K) · 1 kcal = 4.184 kJ (thermochemical)
Photon flux and photons per pulse#
Number of photons per second in a beam of power \(P\), or per pulse of energy \(E_p\). Calculator →
\[ \Phi = \frac{P}{h\nu} = \frac{P\lambda}{hc} \approx 5.034\times10^{15}\,P\,[\text{W}]\;\lambda\,[\text{nm}]\ \text{s}^{-1}, \qquad N_\text{pulse} = \frac{E_p\lambda}{hc} \]
\(\Phi\) photon flux (photons/s) · \(P\) optical power (W) · \(E_p\) pulse energy (J) · \(\lambda\) vacuum wavelength
Photodetector responsivity#
Photocurrent per watt of incident light for a detector with quantum efficiency \(\eta\). Calculator →
\[ \mathcal{R} = \frac{I_\text{ph}}{P} = \eta\,\frac{e\lambda}{hc} \approx \eta\,\frac{\lambda\,[\text{nm}]}{1239.84}\ \ \text{A/W} \]
\(\mathcal{R}\) responsivity (A/W) · \(I_\text{ph}\) photocurrent (A) · \(\eta\) external quantum efficiency (electrons per incident photon, 0–1 without gain) · \(e\) elementary charge
Planck's law (spectral radiance)#
Spectral radiance of an ideal blackbody at temperature \(T\). Calculator →
\[ B_\lambda(\lambda,T) = \frac{2hc^2}{\lambda^5}\,\frac{1}{\exp\!\left(\dfrac{hc}{\lambda k_B T}\right) - 1} \]
\(B_\lambda\) spectral radiance (W·m⁻²·sr⁻¹ per m of wavelength) · \(T\) temperature (K). Spectral exitance of a Lambertian surface is \(M_\lambda = \pi B_\lambda\). Real (grey) emitters: multiply by emissivity \(\varepsilon(\lambda)\).
Wien's displacement law#
Wavelength of peak spectral radiance (per unit wavelength) of a blackbody. Calculator →
\[ \lambda_\text{max}\,T = b = 2.897\,771\,955\times10^{-3}\ \text{m}\cdot\text{K} \]
\(\lambda_\text{max}\) peak of \(B_\lambda\) · \(b\) Wien constant. The peak of the per-frequency spectrum \(B_\nu\) lies elsewhere: \(\nu_\text{max} \approx 5.879\times10^{10}\ \text{Hz/K}\times T\).
Stefan–Boltzmann law#
Total power radiated per unit area, integrated over all wavelengths and the hemisphere. Calculator →
\[ M = \varepsilon\,\sigma\,T^4, \qquad \sigma = 5.670\,374\,419\times10^{-8}\ \text{W}\,\text{m}^{-2}\,\text{K}^{-4} \]
\(M\) radiant exitance (W/m²) · \(\varepsilon\) total hemispherical emissivity (1 for a blackbody) · \(\sigma\) Stefan–Boltzmann constant · \(T\) temperature (K)
Doppler (thermal) linewidth#
Gaussian FWHM of a spectral line broadened by the Maxwell–Boltzmann velocity distribution. Calculator →
\[ \frac{\Delta\nu_D}{\nu_0} = \frac{\Delta\lambda_D}{\lambda_0} = \sqrt{\frac{8\,k_B T\ln 2}{m c^2}} \approx 7.162\times10^{-7}\sqrt{\frac{T\,[\text{K}]}{M\,[\text{u}]}} \]
\(\Delta\nu_D,\ \Delta\lambda_D\) Doppler FWHM · \(\nu_0,\ \lambda_0\) line center · \(m\) emitter mass (kg) · \(M\) mass in atomic mass units · \(T\) kinetic temperature (K)
Natural (lifetime) linewidth#
Lorentzian FWHM set by the finite lifetime of the excited state. Calculator →
\[ \Delta\nu_N = \frac{1}{2\pi\tau}, \qquad \Delta\tilde\nu_N\,[\text{cm}^{-1}] \approx \frac{5.309}{\tau\,[\text{ps}]} \]
\(\Delta\nu_N\) Lorentzian FWHM (Hz) · \(\tau\) upper-state lifetime (s). If both levels decay, \(1/\tau = 1/\tau_\text{upper} + 1/\tau_\text{lower}\).
Voigt linewidth (Olivero–Longbothum)#
FWHM of a Gaussian convolved with a Lorentzian, such as Doppler plus natural or pressure broadening. Calculator →
\[ f_V \approx 0.5346\,f_L + \sqrt{0.2166\,f_L^2 + f_G^2} \]
\(f_V\) Voigt FWHM · \(f_L\) Lorentzian FWHM · \(f_G\) Gaussian FWHM (same units). Empirical fit, accurate to about 0.02 %.
Grating equation#
Angle at which a ruled or holographic grating diffracts wavelength \(\lambda\) into order \(m\). Calculator →
\[ m\lambda = d\,(\sin\alpha + \sin\beta), \qquad \text{Littrow } (\alpha = \beta = \theta_L):\ \ m\lambda = 2d\sin\theta_L \]
\(m\) diffraction order (integer) · \(d\) groove spacing (1/groove density) · \(\alpha,\ \beta\) angles of incidence and diffraction measured from the grating normal; they have the same sign when on the same side of the normal
Angular and linear dispersion#
How fast the diffraction angle, and the position at a spectrometer's focal plane, change with wavelength. Calculator →
\[ \frac{d\beta}{d\lambda} = \frac{m}{d\cos\beta}, \qquad \frac{d\lambda}{dx} = \frac{d\cos\beta}{m\,f} \]
\(d\beta/d\lambda\) angular dispersion (rad/m) · \(d\lambda/dx\) reciprocal linear dispersion (e.g. nm/mm) at the focal plane of a camera optic of focal length \(f\), detector normal to the diffracted beam
Resolving power and spectral bandpass#
The theoretical resolution of a grating, and the practical bandpass set by the slit. Calculator →
\[ R = \frac{\lambda}{\Delta\lambda} = mN, \qquad \Delta\lambda_\text{bp} \approx w_s\,\frac{d\lambda}{dx} \]
\(R\) resolving power · \(N\) number of illuminated grooves · \(\Delta\lambda_\text{bp}\) bandpass (FWHM) · \(w_s\) slit width, assuming unit magnification and a slit image wider than one pixel
Gaussian beams & focusing
Gaussian intensity profile#
Transverse profile of a TEM₀₀ beam, its peak irradiance, and the FWHM ↔ 1/e² conversion. Calculator →
\[ I(r) = I_0\,e^{-2r^2/w^2}, \qquad I_0 = \frac{2P}{\pi w^2}, \qquad d_\text{FWHM} = \sqrt{2\ln 2}\;w = 0.5887\,(2w) \]
\(w\) 1/e² intensity radius · \(I_0\) on-axis irradiance (W/m²) · \(P\) total power · 86.5 % of the power lies within \(r \le w\)
Focused spot diameter (with truncation)#
1/e² diameter at the focus of a lens, including beam quality and clipping by the clear aperture. Calculator →
\[ d_F = \mathrm{APO}(T)\,\frac{\lambda f' M^2}{D_b}, \qquad T = \frac{D_b}{D_t} \]
\(d_F\) focused 1/e² diameter · \(D_b\) input 1/e² beam diameter at the lens · \(D_t\) clear aperture (lens, scanner mirror, or iris) · \(f'\) focal length · \(M^2\) beam quality · APO apodization factor: \(4/\pi \approx 1.273\) for \(T \lesssim 0.5\), 1.45 at \(T = 0.7\), 1.83 at \(T = 1\)
Focused waist radius (untruncated)#
Radius form of the focusing formula for a collimated beam with its waist near the lens. Calculator →
\[ w_0 = \frac{M^2\lambda f'}{\pi w_L} \]
\(w_0\) focused waist radius (1/e²) · \(w_L\) input 1/e² radius at the lens · valid when the input Rayleigh range is much longer than \(f'\), the lens is aberration-free, and there is no clipping
Far-field divergence#
Asymptotic half-angle of the expanding beam far from the waist. Calculator →
\[ \theta = \frac{M^2\lambda}{\pi w_0} \]
\(\theta\) far-field half-angle divergence (rad, 1/e²); full angle is \(2\theta\) · \(w_0\) waist radius
Rayleigh range#
Distance from the waist at which the beam area has doubled (radius × √2). Calculator →
\[ z_R = \frac{\pi w_0^2}{M^2\lambda} = \frac{w_0}{\theta} \]
\(z_R\) Rayleigh range (m) · confocal parameter \(b = 2z_R\) · \(w_0\) waist radius · \(\theta\) half-angle divergence
Beam radius vs. distance#
Hyperbolic envelope of a propagating Gaussian (or embedded-Gaussian) beam. Calculator →
\[ w(z) = w_0\sqrt{1+\left(\frac{z}{z_R}\right)^2} \]
\(w(z)\) 1/e² radius at axial distance \(z\) from the waist · for \(z \gg z_R\), \(w \approx \theta z\)
Wavefront curvature and Gouy phase#
Radius of curvature of the phase fronts and the extra on-axis phase picked up through focus. Calculator →
\[ R(z) = z\left[1+\left(\frac{z_R}{z}\right)^2\right], \qquad \psi(z) = \arctan\frac{z}{z_R} \]
\(R\) wavefront radius (flat at the waist, minimum \(2z_R\) at \(z = z_R\)) · \(\psi\) Gouy phase of the fundamental mode (total shift \(\pi\) through focus; Hermite–Gaussian mode \(nm\) acquires \((n+m+1)\psi\))
Beam parameter product#
Waist radius × divergence half-angle: an invariant of the beam through ideal optics. Calculator →
\[ \mathrm{BPP} = w_0\,\theta = \frac{M^2\lambda}{\pi} \]
BPP usually quoted in mm·mrad · a diffraction-limited beam at 1064 nm has BPP = 0.339 mm·mrad · for multimode fiber delivery, \(\mathrm{BPP} \approx (d_\text{core}/2)\cdot\mathrm{NA}\)
Depth of focus for a spot-size tolerance#
Total axial range over which the beam radius stays within a fraction \(\varepsilon\) of the waist. Calculator →
\[ \mathrm{DOF} = 2\,z_R\sqrt{(1+\varepsilon)^2 - 1} \]
DOF total length (±half on either side of focus) · \(\varepsilon\) allowed fractional growth of the radius (0.05 for 5 %) · \(\varepsilon = \sqrt2 - 1\) recovers the full \(\pm z_R\)
Beam expander#
Two-lens telescope: diameter scales up by \(M\), divergence scales down by \(M\). Calculator →
\[ M = \left|\frac{f_2}{f_1}\right| = \frac{D_\text{out}}{D_\text{in}}, \qquad \theta_\text{out} = \frac{\theta_\text{in}}{M}, \qquad L = f_1 + f_2 \]
\(M\) magnification · \(f_1,\ f_2\) input and output focal lengths; Keplerian: both positive (internal focus); Galilean: \(f_1 < 0\) (no internal focus, shorter) · \(L\) thin-lens spacing for collimated output
F-theta scan lens#
Image height is linear in scan angle, so galvo commands map linearly to position. Calculator →
\[ y = f\,\theta, \qquad \text{field} = 2f\theta_\text{max}, \qquad \theta = 2\,\theta_\text{mech} \]
\(y\) spot position from the optical axis · \(f\) f-theta focal length · \(\theta\) optical scan angle (rad), twice the mechanical mirror rotation \(\theta_\text{mech}\) · spot size from the focused-spot formula with \(D_b\) at the scan mirrors
Gaussian transmission through a circular aperture#
Fraction of power passing a centered circular aperture or iris. Calculator →
\[ \frac{P_t}{P} = 1 - \exp\!\left(-\frac{2a^2}{w^2}\right) \]
\(a\) aperture radius · \(w\) 1/e² beam radius at the aperture · \(a = w\): 86.5 %; \(a = 1.5w\): 98.9 %; \(a = 2w\): 99.97 %
Fiber imaging: spot size and NA#
A multimode fiber end face imaged onto the work by a collimator \(f_1\) and focusing lens \(f_2\).
\[ d_\text{spot} = d_\text{core}\,\frac{f_2}{f_1}, \qquad \mathrm{NA} = \sin\alpha = \sqrt{n_\text{core}^2 - n_\text{clad}^2}, \qquad \sin\alpha' = \frac{f_1}{f_2}\sin\alpha \]
\(d_\text{core}\) fiber core diameter · \(\alpha\) output cone half-angle in air · \(\alpha'\) focusing cone half-angle · \(n_\text{core},\ n_\text{clad}\) refractive indices · the imaged spot is roughly top-hat, not Gaussian
Interface optics & filters
Snell's law#
Refraction at a planar interface between two media. Calculator →
\[ n_1\sin\theta_1 = n_2\sin\theta_2 \]
\(n_1,\ n_2\) refractive indices · \(\theta_1,\ \theta_2\) angles of incidence and refraction from the surface normal
Fresnel reflectance (s and p)#
Power reflectance of an uncoated interface for each polarization. Calculator →
\[ \begin{gathered} R_s = \left|\frac{n_1\cos\theta_i - n_2\cos\theta_t}{n_1\cos\theta_i + n_2\cos\theta_t}\right|^2 \\[4pt] R_p = \left|\frac{n_1\cos\theta_t - n_2\cos\theta_i}{n_1\cos\theta_t + n_2\cos\theta_i}\right|^2 \end{gathered} \]
\(s\) (TE) polarization ⟂ plane of incidence, \(p\) (TM) ∥ · \(\theta_t\) from Snell's law · lossless dielectrics: \(T = 1 - R\) · unpolarized: \(R = (R_s + R_p)/2\)
Normal-incidence reflectance#
Reflection loss per uncoated surface at 0°. Calculator →
\[ R_0 = \left(\frac{n_1 - n_2}{n_1 + n_2}\right)^2 \]
\(R_0\) reflectance (0–1) · glass with \(n = 1.5\) in air: 4.0 % per surface
Brewster angle#
Angle of incidence at which p-polarized reflectance goes to zero. Calculator →
\[ \tan\theta_B = \frac{n_2}{n_1} \]
\(\theta_B\) Brewster angle · fused silica (\(n \approx 1.45\) at 1064 nm) from air: \(\theta_B \approx 55.4^\circ\)
Critical angle (total internal reflection)#
Beyond this internal angle, light going into a lower-index medium is totally reflected. Calculator →
\[ \sin\theta_c = \frac{n_2}{n_1}, \qquad n_1 > n_2 \]
\(\theta_c\) critical angle · glass (\(n = 1.5\)) to air: 41.8°
Transmission of an uncoated window#
Non-absorbing plate in air, summing incoherent multiple reflections between its two faces. Calculator →
\[ T_\text{plate} = \frac{(1-R)^2}{1-R^2} = \frac{1-R}{1+R} \;\overset{\theta = 0}{=}\; \frac{2n}{n^2+1} \]
\(R\) single-surface reflectance · \(n\) plate index · valid for thick plates or broadband light (no etalon fringes) · \(n = 1.5\): \(T = 92.3\,\%\)
Optical density, transmission, and decibels#
Logarithmic attenuation of filters, eyewear, and attenuators. Calculator →
\[ \mathrm{OD} = -\log_{10} T, \qquad T = 10^{-\mathrm{OD}}, \qquad \text{loss}\,[\text{dB}] = 10\,\mathrm{OD}, \qquad \mathrm{OD}_\text{stack} = \sum_i \mathrm{OD}_i \]
\(T\) transmittance (0–1) · stacking rule ignores interreflections between filters (tilt reflective ND filters to avoid them)
Beer–Lambert absorption#
Exponential attenuation of light inside an absorbing medium. Calculator →
\[ T_\text{int} = e^{-\alpha\ell} = 10^{-A}, \qquad A = \varepsilon c\ell = \frac{\alpha\ell}{\ln 10}, \qquad \delta = \frac{1}{\alpha} \]
\(T_\text{int}\) internal transmittance (excludes surface reflections) · \(\alpha\) absorption coefficient (m⁻¹) · \(\ell\) path length · \(A\) absorbance (decadic) · \(\varepsilon\) molar absorptivity (L·mol⁻¹·cm⁻¹) · \(c\) concentration (mol/L) · \(\delta\) optical penetration depth
Laser processing & damage
Pulse energy and duty cycle#
Relations between average power, repetition rate, and pulse duration. Calculator →
\[ E_p = \frac{P_\text{avg}}{f_\text{rep}}, \qquad D = \tau f_\text{rep}, \qquad T_\text{rep} = \frac{1}{f_\text{rep}} \]
\(E_p\) pulse energy (J) · \(P_\text{avg}\) average power (W) · \(f_\text{rep}\) repetition rate (Hz) · \(D\) duty cycle · \(\tau\) pulse duration (FWHM) · \(T_\text{rep}\) pulse period
Peak power and pulse-shape factor#
Peak power of a pulse from its energy and FWHM duration. Calculator →
\[ P_\text{peak} = k\,\frac{E_p}{\tau}, \qquad k = \begin{cases} 1 & \text{rectangular} \\ 2\sqrt{\ln 2/\pi} \approx 0.939 & \text{Gaussian} \\ \ln\!\left(1+\sqrt2\right) \approx 0.881 & \text{sech}^2 \end{cases} \]
\(P_\text{peak}\) peak power (W) · \(\tau\) FWHM pulse duration · \(k\) temporal shape factor (ratio of peak power to \(E_p/\tau\))
Peak and average fluence#
Energy per unit area for a Gaussian spot. Ablation thresholds are usually quoted as peak fluence. Calculator →
\[ F_0 = \frac{2E_p}{\pi w_0^2}, \qquad \bar F = \frac{E_p}{\pi w_0^2} = \frac{F_0}{2} \]
\(F_0\) on-axis (peak) fluence (J/m², usually quoted in J/cm²) · \(\bar F\) average over the 1/e² area · \(w_0\) 1/e² radius at the work surface
Peak irradiance#
Instantaneous on-axis intensity. It governs nonlinear absorption and plasma formation. Calculator →
\[ I_0 = \frac{2P_\text{peak}}{\pi w_0^2} = \frac{F_0\,k}{\tau} \]
\(I_0\) peak irradiance (W/m², usually quoted in W/cm²) · \(P_\text{peak}\) peak power · \(k\) pulse-shape factor · \(\tau\) FWHM duration
Ablated diameter and threshold (Liu method)#
Crater diameters at several pulse energies give both the beam radius and the threshold fluence. Calculator →
\[ D^2 = 2w_0^2\,\ln\!\left(\frac{F_0}{F_\text{th}}\right) = 2w_0^2\,\ln\!\left(\frac{E_p}{E_\text{th}}\right) \]
\(D\) ablated (crater) diameter · plot \(D^2\) against \(\ln E_p\): slope \(= 2w_0^2\), x-intercept \(= \ln E_\text{th}\), then \(F_\text{th} = 2E_\text{th}/(\pi w_0^2)\) · requires a Gaussian spatial profile
Ablation depth per pulse#
Logarithmic depth law for absorption-limited removal above threshold. Calculator →
\[ L = \delta\,\ln\!\left(\frac{F_0}{F_\text{th}}\right), \qquad F_0 > F_\text{th} \]
\(L\) depth per pulse (on axis) · \(\delta\) effective penetration depth: optical (\(1/\alpha\)) for ultrashort pulses at low fluence, the electron heat-diffusion length at high fluence, and an empirical fit parameter for ns pulses, where heat diffusion dominates · breaks down when plasma shielding sets in at high \(F_0/F_\text{th}\)
Multi-pulse incubation#
Threshold fluence falls as defects accumulate from repeated pulses on the same site. Calculator →
\[ F_\text{th}(N) = F_\text{th}(1)\,N^{\,S-1} \]
\(N\) pulses per site · \(S\) incubation coefficient (\(S = 1\): no incubation; many metals \(S \approx 0.8\text{–}0.9\)) · empirical power law, usually saturating at large \(N\)
Pulse spacing and overlap#
Center-to-center spacing of pulses along a scan line and the fractional overlap of adjacent spots. Calculator →
\[ \Delta x = \frac{v}{f_\text{rep}}, \qquad \mathrm{OL} = 1 - \frac{\Delta x}{D}, \qquad N_\text{geo} = \frac{D\,f_\text{rep}}{v} \]
\(v\) scan speed · \(D\) spot diameter: state whether this is the 1/e² diameter or the ablated diameter · \(\mathrm{OL} < 0\) means gaps between spots · \(N_\text{geo}\) pulses landing within one spot diameter
Hatch (line) overlap and area rate#
Line-to-line overlap for raster or hatch filling, and area processed per unit time. Calculator →
\[ \mathrm{OL}_h = 1 - \frac{h}{D}, \qquad \dot A = v\,h \]
\(h\) hatch distance (line pitch) · \(\dot A\) area coverage rate per pass (m²/s), excluding jump and acceleration time
Effective number of pulses (Gaussian)#
Accumulated fluence at a point divided by the single-pulse peak fluence, for a line (1D) or a hatched area (2D). Calculator →
\[ N_\text{eff,1D} = \sqrt{\frac{\pi}{2}}\,\frac{w_0\,f_\text{rep}}{v}, \qquad N_\text{eff,2D} = \frac{\pi}{2}\,\frac{w_0^2\,f_\text{rep}}{v\,h} \]
\(w_0\) 1/e² radius · derived by summing Gaussian fluence over pulses; accurate when \(\Delta x,\ h \lesssim w_0\) · accumulated peak fluence \(\approx N_\text{eff}\,F_0\) (ignores incubation and heat accumulation)
LIDT pulse-duration scaling#
Empirical rule for converting a specified fluence damage threshold to a different pulse duration. Calculator →
\[ \mathrm{LIDT}(\tau) \approx \mathrm{LIDT}_\text{spec}\,\sqrt{\frac{\tau}{\tau_\text{spec}}} \]
LIDT as fluence (J/cm²) · \(\tau\) pulse duration · holds for thermally dominated damage, roughly 10 ps to 100 ns, at the same wavelength, beam size, and test protocol (1-on-1 vs S-on-1) · not valid for femtosecond pulses · wavelength scaling depends on the coating, so prefer data measured at your wavelength
Thermal diffusion length#
How far heat spreads during a pulse or dwell time. It sets the scale of the heat-affected zone. Calculator →
\[ L_\text{th} = 2\sqrt{\kappa\,\tau}, \qquad \kappa = \frac{k}{\rho\,c_p} \]
\(\kappa\) thermal diffusivity (m²/s) · \(k\) thermal conductivity · \(\rho\) density · \(c_p\) specific heat · \(\tau\) pulse duration or interaction time · conventions vary by a factor of 2 (\(\sqrt{\kappa\tau}\) vs \(2\sqrt{\kappa\tau}\)) · stainless steel (\(\kappa \approx 4\) mm²/s) at 10 ns: \(L_\text{th} \approx 0.4\) µm
Thermal relaxation and heat accumulation#
An order-of-magnitude criterion for when heat from successive pulses starts to build up. Calculator →
\[ \tau_\text{th} \approx \frac{d^2}{4\kappa}, \qquad \text{accumulation when } f_\text{rep} \gtrsim \frac{1}{\tau_\text{th}} \]
\(d\) heated spot diameter · \(\kappa\) thermal diffusivity · 30 µm spot on stainless steel: \(\tau_\text{th} \approx 56\) µs (≈ 18 kHz) · overlap shortens the effective time between pulses at a site
Vacuum
Kinetic-theory results assume an ideal gas in equilibrium. Practical forms are for air at 20 °C, where \(\bar v \approx 463\) m/s. Scale them by \(\sqrt{T/M}\) for other gases (multiply by about 2.7 for He and 3.8 for H₂). For unit changes, use the pressure converter; 1 Torr = 1.333 mbar = 133.3 Pa.
Gas number density#
Molecules per unit volume from the ideal-gas law. Calculator →
\[ n = \frac{p}{k_B T} \]
\(n\) number density (m⁻³) · \(p\) pressure (Pa) · \(T\) temperature (K) · at 20 °C: \(2.47\times10^{20}\) m⁻³ per Pa, or \(3.3\times10^{16}\) cm⁻³ at 1 Torr
Mean molecular speed#
Average speed of the Maxwell–Boltzmann distribution. It appears in every molecular-flow conductance. Calculator →
\[ \bar v = \sqrt{\frac{8k_B T}{\pi m}}, \qquad v_\text{rms} = \sqrt{\frac{3k_BT}{m}}, \qquad v_p = \sqrt{\frac{2k_BT}{m}} \]
\(\bar v\) mean speed · \(v_\text{rms}\) root-mean-square speed · \(v_p\) most probable speed · \(m\) molecular mass (kg) · air at 20 °C: \(\bar v = 463\) m/s
Mean free path#
Average distance a molecule travels between collisions with other gas molecules. Calculator →
\[ \lambda_\text{mfp} = \frac{k_B T}{\sqrt2\,\pi d_m^2\,p}, \qquad \lambda_\text{mfp}\,[\text{cm}] \approx \frac{5.0\times10^{-3}}{p\,[\text{Torr}]}\ \ (\text{air}) \]
\(d_m\) molecular (kinetic) diameter, about 0.37 nm for air · \(p\) pressure · rule of thumb for air at room temperature: 5 cm at 1 mTorr, 50 m at 10⁻⁶ Torr
Knudsen number and flow regime#
Ratio of the mean free path to the characteristic size of the duct. It decides which conductance formula applies. Calculator →
\[ \mathrm{Kn} = \frac{\lambda_\text{mfp}}{D} \]
\(D\) tube diameter or characteristic dimension · \(\mathrm{Kn} < 0.01\): viscous (continuum) flow · 0.01–0.5: transitional · \(\mathrm{Kn} > 0.5\): molecular flow (some texts use \(\mathrm{Kn} > 1\))
Impingement rate and monolayer time#
Molecular flux striking a surface, and how long it takes to deposit one monolayer. Calculator →
\[ \Phi = \frac{p}{\sqrt{2\pi m k_B T}} = \frac{n\bar v}{4}, \qquad t_\text{ML} \approx \frac{N_s}{s\,\Phi} \]
\(\Phi\) impingement rate (m⁻²·s⁻¹) · \(N_s\) surface site density, about 10¹⁹ m⁻² · \(s\) sticking coefficient · N₂ at 10⁻⁶ Torr and 20 °C: \(t_\text{ML} \approx 2.6\) s (with \(s = 1\))
Throughput, pumping speed, and conductance#
Gas throughput is conserved along a pumping line in steady state. Calculator →
\[ Q = p\,S = C\,(p_1 - p_2) \]
\(Q\) throughput (Pa·m³/s or mbar·L/s) · \(S\) pumping speed at the point where \(p\) is measured (m³/s or L/s) · \(C\) conductance of the element between pressures \(p_1\) and \(p_2\)
Orifice conductance (molecular flow)#
The maximum conductance an opening of area \(A\) can have. Its practical form is the upper limit for any port. Calculator →
\[ C_\text{orifice} = \frac{\bar v}{4}\,A \approx 11.6\ \frac{\text{L}}{\text{s}} \times A\,[\text{cm}^2] \ \ (\text{air}) \]
\(A\) open area · thin aperture, molecular regime · air at 20 °C
Long-tube conductance (molecular flow)#
Knudsen's result for a long round tube. Note the steep dependence on diameter. Calculator →
\[ C_\text{tube} = \frac{\pi}{12}\,\bar v\,\frac{d^3}{L} \approx 12.1\,\frac{d^3\,[\text{cm}^3]}{L\,[\text{cm}]}\ \ \text{L/s} \ \ (\text{air}) \]
\(d\) inner diameter · \(L\) length, with \(L \gg d\) (for short tubes use the orifice conductance × Clausing transmission probability) · independent of pressure
Long-tube conductance (viscous flow)#
Poiseuille conductance for laminar flow. It grows with mean pressure, so roughing lines are rarely limiting at high pressure. Calculator →
\[ C_\text{visc} = \frac{\pi d^4\,\bar p}{128\,\eta L} \approx 180\,\frac{d^4\,[\text{cm}^4]\;\bar p\,[\text{Torr}]}{L\,[\text{cm}]}\ \ \text{L/s} \ \ (\text{air}) \]
\(\bar p\) mean pressure in the tube · \(\eta\) dynamic viscosity (air at 20 °C: \(1.81\times10^{-5}\) Pa·s) · laminar, fully developed flow, \(L \gg d\)
Series conductance and effective pumping speed#
Conductances in series add like resistors in parallel. The pump's speed at the chamber is always less than its rated speed. Calculator →
\[ \frac{1}{C_\text{tot}} = \sum_i \frac{1}{C_i}, \qquad S_\text{eff} = \frac{S\,C}{S + C} \]
\(S\) pump speed at its inlet · \(C\) total conductance between pump and chamber · \(S_\text{eff}\) speed delivered to the chamber · if \(C = S\), half the pump is wasted
Pump-down time and ultimate pressure#
Exponential evacuation of a volume at constant speed, and the base pressure set by the gas load. Calculator →
\[ t = \frac{V}{S_\text{eff}}\,\ln\frac{p_0}{p}, \qquad p_\text{ult} = \frac{Q_\text{gas}}{S_\text{eff}} \]
\(V\) chamber volume · \(p_0\) starting pressure · \(Q_\text{gas}\) total gas load (outgassing + permeation + leaks) · valid in the roughing range, where \(S_\text{eff}\) is constant and the volume gas dominates; below roughly 10⁻²–10⁻³ mbar (system-dependent), desorption from surfaces controls the pump-down
Machine vision
Thin-lens imaging and magnification#
Object and image conjugates for a lens of focal length \(f\). Calculator →
\[ \frac{1}{f} = \frac{1}{s_o} + \frac{1}{s_i}, \qquad m = \frac{s_i}{s_o} = \frac{f}{s_o - f} \]
\(s_o\) object distance (approximately working distance plus the offset to the front principal plane) · \(s_i\) image distance · \(m\) magnitude of the (inverting) magnification
Field of view and pixel footprint#
Object-side field and the size of one pixel projected onto the part. Calculator →
\[ \mathrm{FOV} = \frac{\text{sensor size}}{m}, \qquad p_\text{obj} = \frac{p_\text{px}}{m} \]
sensor width or height · \(p_\text{px}\) pixel pitch · \(p_\text{obj}\) pixel footprint on the object · rule of thumb: at least 2 px per line pair (Nyquist) and 3–5 px across a defect for robust detection
Working f-number and object-side NA#
The effective f-number at finite conjugates is larger than the number on the lens barrel. Calculator →
\[ N_w = N\,(1+m), \qquad \mathrm{NA}_\text{obj} \approx \frac{m}{2N_w} \]
\(N\) lens f-number (infinity-focus) · \(N_w\) working (image-side) f-number · assumes a pupil magnification near 1 · paraxial
Depth of field (object side)#
Axial range of the part that stays within an acceptable blur on the sensor. Calculator →
\[ \mathrm{DOF} \approx \frac{2N c\,(1+m)}{m^2} = \frac{2 N_w c}{m^2} \]
\(c\) acceptable blur-circle diameter at the sensor (often 1–2 pixels) · \(N\) f-number · \(m\) magnification · valid when DOF is much smaller than the working distance; diffraction blur adds when \(c\) approaches \(2.44\lambda N_w\)
Diffraction limit: Airy disk and Rayleigh resolution#
Smallest blur a perfect lens can form. Compare it with the pixel pitch before buying more megapixels. Calculator →
\[ d_\text{Airy} = 2.44\,\lambda N_w, \qquad r_\text{obj} = \frac{0.61\,\lambda}{\mathrm{NA}_\text{obj}} \]
\(d_\text{Airy}\) diameter to the first dark ring at the sensor · \(r_\text{obj}\) Rayleigh two-point resolution on the object · \(\lambda\) illumination wavelength
Motion blur, exposure, and line rate#
Blur from part motion during the exposure, and the line rate a line-scan camera needs for square pixels. Calculator →
\[ B = \frac{v\,t_\text{exp}}{p_\text{obj}}, \qquad t_\text{exp,max} = \frac{B_\text{max}\,p_\text{obj}}{v}, \qquad f_\text{line} = \frac{v}{p_\text{obj}} \]
\(B\) blur in pixels · \(v\) part speed · \(t_\text{exp}\) exposure or strobe duration · typical \(B_\text{max}\) = 0.5–1 px · \(f_\text{line}\) line rate (lines/s) for a square object pixel
Quality & SPC
Process capability: Cp and Cpk#
Short-term (within-subgroup) capability. Cpk penalizes an off-center mean. Calculator →
\[ C_p = \frac{\mathrm{USL}-\mathrm{LSL}}{6\hat\sigma_w}, \qquad C_{pk} = \min\!\left(\frac{\mathrm{USL}-\bar x}{3\hat\sigma_w},\ \frac{\bar x-\mathrm{LSL}}{3\hat\sigma_w}\right) \]
USL, LSL specification limits · \(\bar x\) process mean · \(\hat\sigma_w\) within-subgroup standard deviation, \(\bar R/d_2\) or \(\overline{MR}/1.128\) · requires a stable (in-control), approximately normal process
Process performance: Pp and Ppk#
Long-term (overall) performance using the total sample standard deviation. Calculator →
\[ P_p = \frac{\mathrm{USL}-\mathrm{LSL}}{6s}, \qquad P_{pk} = \min\!\left(\frac{\mathrm{USL}-\bar x}{3s},\ \frac{\bar x-\mathrm{LSL}}{3s}\right) \]
\(s = \sqrt{\sum (x_i-\bar x)^2/(n-1)}\) overall sample standard deviation · \(P_{pk}\) noticeably below \(C_{pk}\) points to between-subgroup variation (drift, shifts)
Taguchi capability Cpm#
Capability index that penalizes deviation from the target, not just from the center of the specification. Calculator →
\[ C_{pm} = \frac{\mathrm{USL}-\mathrm{LSL}}{6\sqrt{s^2 + (\bar x - T)^2}} \]
\(T\) target value · \(s\) standard deviation · \(\bar x\) mean
Expected fraction out of specification#
Nonconforming fraction predicted by a normal model. Calculator →
\[ p_\text{out} = \Phi\!\left(\frac{\mathrm{LSL}-\mu}{\sigma}\right) + 1 - \Phi\!\left(\frac{\mathrm{USL}-\mu}{\sigma}\right), \qquad \text{ppm} = 10^6\,p_\text{out} \]
\(\Phi\) standard normal CDF · \(\mu,\ \sigma\) process mean and standard deviation · a one-sided \(C_{pk} = 1.33\) gives about 32 ppm · tail estimates depend strongly on the normality assumption
X̄–R control limits#
Shewhart three-sigma limits for subgroup means and ranges. Calculator →
\[ \mathrm{UCL}_{\bar x},\ \mathrm{LCL}_{\bar x} = \bar{\bar x} \pm A_2\bar R, \qquad \mathrm{UCL}_R = D_4\bar R, \qquad \mathrm{LCL}_R = D_3\bar R \]
\(\bar{\bar x}\) grand mean · \(\bar R\) average subgroup range · constants depend on the subgroup size \(n\); for \(n = 5\): \(A_2 = 0.577\), \(D_3 = 0\), \(D_4 = 2.114\), \(d_2 = 2.326\)
Individuals and moving range (I–MR) limits#
Control limits when each measurement is its own subgroup, which is common for slow or expensive measurements. Calculator →
\[ \mathrm{UCL},\ \mathrm{LCL} = \bar x \pm 2.66\,\overline{MR}, \qquad \mathrm{UCL}_{MR} = 3.267\,\overline{MR} \]
\(\overline{MR}\) average moving range of consecutive points · \(2.66 = 3/d_2\) with \(d_2 = 1.128\) for \(n = 2\) · \(3.267 = D_4\) for \(n = 2\)
DPMO and sigma level#
Defect rate normalized by opportunities, and the conventional Six Sigma level. Calculator →
\[ \mathrm{DPMO} = \frac{10^6\times\text{defects}}{\text{units}\times\text{opportunities}}, \qquad \sigma_\text{level} = \Phi^{-1}\!\left(1 - \frac{\mathrm{DPMO}}{10^6}\right) + 1.5 \]
\(\Phi^{-1}\) inverse standard normal CDF · the +1.5 term is the conventional long-term mean-shift allowance; 3.4 DPMO ↔ 6σ, 6210 DPMO ↔ 4σ
Rolled throughput yield#
Probability that a unit passes every process step right the first time. Calculator →
\[ \mathrm{RTY} = \prod_i Y_i, \qquad Y_i \approx e^{-\mathrm{DPU}_i} \]
\(Y_i\) first-pass yield of step \(i\) · \(\mathrm{DPU}_i\) defects per unit at step \(i\) (the exponential form assumes Poisson-distributed defects)
Overall equipment effectiveness (OEE)#
Fraction of planned production time that yields good parts at the ideal rate. Calculator →
\[ \begin{gathered} \mathrm{OEE} = A \times P \times Q \\[4pt] A = \frac{\text{run time}}{\text{planned time}}, \quad P = \frac{t_\text{ideal}\times\text{total count}}{\text{run time}}, \quad Q = \frac{\text{good count}}{\text{total count}} \end{gathered} \]
\(A\) availability · \(P\) performance (speed) · \(Q\) quality rate · \(t_\text{ideal}\) ideal cycle time per part · equivalently \(\mathrm{OEE} = t_\text{ideal}\times\text{good count}/\text{planned time}\)
Takt time#
The production pace required to meet customer demand. Calculator →
\[ T_\text{takt} = \frac{\text{net available time}}{\text{customer demand}} \]
net available time = shift time minus planned breaks and meetings · every station's cycle time must be ≤ \(T_\text{takt}\) (in practice ≤ 85–95 % of takt to absorb losses)
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