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Laser and optics calculators

108 engineering calculators covering pulse energy, beam propagation, fibre, detectors, ranging and LiDAR. Every one is multi-directional: choose the variable you want solved and fill in the rest.

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Laser Fundamentals 12 calculators

Peak power, fluence, divergence and beam quality - the relationships everything else is built on.

Pulse Peak Power Instantaneous peak power from pulse energy and pulse width. P_peak = E / τ Average Power Continuous average power from repetition rate and energy per pulse. P_avg = E · f_rep Fluence Laser energy per unit area - the value that sets material damage thresholds. F = E / Area Intensity Instantaneous laser power per unit area. I = P / Area Gaussian Beam Divergence Far-field half-angle divergence at the diffraction limit. θ = λ / (π · w₀) Rayleigh Length Distance over which the beam expands from its waist until the spot area doubles. z_R = π · w₀² / λ Spot Size at Distance z Beam radius at a distance z from the waist. w(z) = w₀ · √[1 + (z/zR)²] Focused Spot Diameter Theoretical focal spot diameter of a laser beam through a lens. d ≈ (1.27 · λ · f) / D Single-Photon Energy Convert between wavelength and quantum energy in eV. E = 1239.8 / λ Duty Cycle & Thermal Load Fraction of time the laser is on, from pulse width and repetition rate. Duty = τ · f_rep Pulse Spatial Length Physical distance an ultrashort pulse occupies in space. L = c · τ Beam Quality Factor M2 How far a real beam deviates from a theoretical Gaussian beam. M² = (π·w₀·θ) / λ

Swept-Source & Chaos (FDML) 10 calculators

Swept-source OCT and laser chaos: cavity synchronisation, k-space sampling and coherence.

FDML Cavity Length & Sync Locking the sweep frequency to the round-trip time of light in the cavity. f_sweep = c / (n_eff · L) OCT Axial Resolution FDML-OCT axial resolution set by the sweep bandwidth. δz = (2ln2/π) · (λ₀²/Δλ) Sweep Bandwidth Conversion Convert a wavelength sweep span in nm into the equivalent frequency bandwidth in THz. Δν ≈ (c/λ₀²) · Δλ k-Space Sampling Linearized resampling step in wavenumber space. Δk = 2πΔλ / λ₀² ADC Imaging Depth Limit Maximum alias-free imaging depth imposed by the ADC sample rate. z_max = (c/4n) · (f_s/f_sweep) Chaos Feedback Delay External feedback delay tau set by the length of the feedback loop. τ = n_eff · L_delay / c Instantaneous Coherence Length Coherence performance of a swept laser judged from its instantaneous linewidth. L_c = λ₀² / Δλ_inst Relative Intensity Noise (RIN) Statistical description of laser output power fluctuation. RIN = <δP²> / (P² · Δf) FDML Sync Score Score how well the current sweep frequency matches the physical cavity length. Score: Sync(f_sw vs c/nL) Laser Phase Noise Phase noise power spectral density at a given offset frequency. L(f) = 10 · log10(S_phi(f)/2)

Pulses & Energy 10 calculators

Pulse energy, average power, duty cycle and nonlinear pulse propagation in fiber.

Peak Power Relation between pulse energy, pulse width and peak power. P_peak = E / τ Average Power Relation between average power, energy per pulse and repetition rate. P_avg = E × f_rep Duty Cycle Ratio of laser on-time to the full period. D = τ × f_rep Fluence Energy per unit area. F = E / Area Intensity Power per unit area. I = P / Area Sum-Frequency Generation (SFG) Nonlinear optical process: 1/l3 = 1/l1 + 1/l2 ω₃ = ω₁ + ω₂ Nonlinear Phase (SPM B-integral) Estimate self-phase modulation (B-integral) of ultrashort or high-power light in fiber. φ_NL = (2π/λ) · n₂ · I · L_eff Soliton Parameters Fundamental soliton (N=1) condition and related length scales in fiber, assuming a sech-squared shape. N² = (γP₀T₀²) / |β₂| Dispersion Broadening Width of a Gaussian pulse after propagating through a dispersive medium. τ(L) = τ₀ · [1 + (L/L_D)²]^0.5 Transform Limit Theoretical shortest pulse width from the time-bandwidth product. Δν · Δτ = K (Sech²: 0.315)

Q-Switched Lasers 2 calculators

Q-switch output energy and the longitudinal modes of the resonator.

Q-Switch Laser Design Simulate the output of actively or passively Q-switched lasers, using the Degnan model. E_out ≈ η·P_pump·τ·(1-e^(-1/fτ)) Longitudinal Mode Spacing Frequency and wavelength spacing of a resonator's longitudinal modes; the center wavelength strongly affects the wavelength spacing. Δν = c / 2nL, Δλ = λ² / 2nL

Fiber & Lasers 10 calculators

Numerical aperture, mode field diameter, coupling loss, amplifier gain and resonator stability.

Fiber Numerical Aperture (NA) Convert between NA and acceptance angle (single-mode about 0.13, multimode about 0.22). NA = sin(θ_half) V-Number & Modes Normalized frequency V of a fiber, and the single-mode versus multimode cutoff condition. V = (2πa/λ) · NA Mode Field Diameter (MFD) Marcuse approximation of the mode field diameter 2w of a step-index fiber. w/a ≈ 0.65 + 1.6/V^1.5 + 2.8/V^6 Fiber Coupling Efficiency Fiber coupling loss caused by transverse offset of a Gaussian beam. η = exp(-2(x/w)²) Dispersion Pulse Broadening Pulse broadening in time caused by chromatic dispersion. Δτ = |D| · L · Δλ Fiber Amplifier Gain Gain saturation behaviour of a fiber amplifier such as an EDFA or YDFA. G = G₀ / (1 + P_in / P_sat) Fiber Laser Output CW fiber laser output power from slope efficiency. P_out = η · (P_pump - P_th) Resonator Stability Two-mirror stability criterion using g parameters. Stable when 0 < g1*g2 < 1. g = 1 - L/R Two-Mirror Cavity Waist Waist radius and its position inside a stable two-mirror cavity. w₀² = (λL/π)·[g1g2(1-g1g2)/(g1+g2-2g1g2)²]^0.5 Thermal Lensing Thermally induced focal length of a laser crystal under high pump power. f_th ≈ (K · A_pump) / (P_heat · dn/dT)

Spectrum & Utilities 9 calculators

Unit and spectrum conversions, gratings, coatings, etalons and thermal drift.

Spectrum Converter Convert between wavelength, frequency and wavenumber. f = c/λ, ν̃ = 1/λ Angle Converter High-precision conversion across degrees, radians, milliradians, arcminutes and arcseconds. θ_rad = θ_deg · π / 180 Optical Density (Filtering) Convert between transmittance T and optical density OD. OD = -log₁₀(T) Grating Equation Relation between incidence and diffraction angles of a diffraction grating. d(sinα + sinβ) = mλ Wavelength Thermal Drift Shift of the laser center wavelength caused by a change in ambient temperature. Δλ = α · ΔT Doppler Shift Frequency offset produced by a target's radial velocity. Δf = 2 · v · cosθ / λ Single-Layer AR Coating Optimum refractive index and thickness (QWOT) of a single-layer antireflection coating. n₁ = √(n₀nₛ), d = λ₀ / 4n₁ Bragg Mirror Peak reflectance of a periodic high-reflectance coating stack. R = [(n₀(n_H)²ᴺ - nₛ(n_L)²ᴺ) / (...)]² Fabry-Perot Etalon Free spectral range (FSR) and finesse of a Fabry-Perot cavity. F = π√R / (1-R), FSR = c / 2nL

Beam & Optics 14 calculators

Spot sizes, refraction, polarization, diffraction limits and centering tolerances.

Spot Size vs Distance Beam size at a given distance, from the divergence angle. W(z) ≈ W₀ + 2 · z · tan(θ/2) Focused Spot Spot size of a lens-focused beam at the diffraction limit. 2w₀ = (4·λ·f) / (π·D) Snell's Law Refraction angle at the interface between two media. n₁ · sin(θ₁) = n₂ · sin(θ₂) Brewster's Angle Angle of incidence at which P-polarized light is not reflected. θ_B = arctan(n₂ / n₁) Critical Angle (TIR) Total internal reflection angle for light travelling from a denser to a rarer medium. θ_c = arcsin(n₂ / n₁) Fresnel Reflection Surface reflectance and transmittance at normal incidence. R = ((n₂-n₁)/(n₂+n₁))² Malus's Law Intensity of polarized light after passing through an analyser. I = I₀ · cos²(θ) Extinction Ratio A measure of polarization purity. ER(dB) = 10 · log₁₀(P_max/P_min) Surface Tilt & Translation Relation between the center-of-curvature translation of a spherical surface and the surface tilt it causes. θ = arctan(Δ / R) Centering: Transmission to Reflection Convert transmitted centering error into reflected centering error or surface tilt. T = (n - 1) · R ISO 10110 Centering ISO 4/sigma(L): convert wedge angle into edge thickness difference (ETD). ETD ≈ D · tan(σ) Angular Resolution (Rayleigh) Rayleigh criterion: the smallest angle at which an optical system resolves two point sources. θ = 1.22 · λ / D Airy Disk Diameter Diffraction spot diameter formed at the image plane of an ideal optical system. d = 2.44 · λ · N Microscope Resolution (Abbe) Lateral microscope resolution, set by numerical aperture and medium refractive index. δ = 0.5 · λ / NA

Ranging & LiDAR 12 calculators

Time-of-flight, phase and FMCW ranging, the link budget, resolution and atmospheric loss.

Pulse Time of Flight (TOF) Target distance from the measured pulse round-trip time. D = (c · Δt) / 2 Phase-Shift Ranging Distance from the phase difference of a modulated wave. D = (c / 4πf) · Δφ Triangulation Ranging Short-range target displacement or absolute distance from geometry. D = (L · f) / x LiDAR Link Budget Estimate the optical power captured at the receiver. Pr = Pt · (ρA/πR²) · η Range Resolution Ability to separate two adjacent targets. ΔR = cτ / 2 Unambiguous Range Maximum detection range set by the pulse repetition frequency (PRF). R_max = c / (2 · PRF) LiDAR Signal-to-Noise Ratio Assess detection quality. SNR = Ps / √(Pb + Pd) Doppler Shift Frequency offset produced by a target's radial velocity. v = (λ · Δf) / 2 LiDAR Far-Field Spot Beam diameter at long range. S ≈ D_exit + R · θ Angular Resolution (Scanning) Sets the lateral resolution of a scanning system. Δx = R · Δθ Atmospheric Attenuation Round-trip atmospheric transmission estimate. T = exp(-2αR) FMCW Ranging Target distance from the beat frequency. R = (c · fb · T) / (2 · B)

Permittivity & Dielectrics 5 calculators

How a material's permittivity sets capacitance, refractive index and wave velocity.

Permittivity to Capacitance Compute the capacitance of a parallel-plate capacitor, or back out the material permittivity. C = ε₀ · εr · A / d E-Field & Displacement Vector Electric field strength and electric displacement inside a dielectric. D = ε₀ · εr · E Permittivity to Refractive Index Classical conversion between permittivity and optical refractive index for non-magnetic materials. n ≈ √εr Wave Velocity in a Medium Propagation speed of an electromagnetic wave in a material of a given permittivity. v = c / √εr Wavelength in a Medium Wavelength inside a medium relative to the vacuum wavelength. λm = λ₀ / √εr

APD Avalanche Photodiodes 7 calculators

Gain, bias, temperature compensation and bandwidth of an APD receiver.

APD RC Rise Time RC rise time set by junction capacitance, which itself depends on permittivity. tr = 2.2 · R · C(εr) APD Output Photocurrent Total output current from incident power, base responsivity and avalanche gain. Iph = Pin · Re · (M/100) APD Sample Voltage Output voltage amplitude developed across the sampling resistor. Vout = (Iph + Id) · Ri APD Gain-Voltage Model Empirical fit of gain against reverse voltage, for breakdown trend analysis. M = 1 / [1 - (Vr/Vbr)ⁿ] APD Temperature Compensation Bias voltage shift needed to hold gain constant as ambient temperature drifts. Vr(T) = Vref + δ · (T - Tref) APD Bandwidth Minus 3 dB bandwidth and rise time set by junction capacitance and load resistance. fbw = 1/(2πRC), tr = 0.35/fbw APD Dark-Current Dissipation Static power the APD itself dissipates while reverse biased. Pdark = Id · Vr

SPAD Single-Photon Detection 11 calculators

Photon counting, dead time, afterpulsing, crosstalk and dark-count behaviour.

Photon Energy & Count Convert between average optical power and the number of photons arriving within an integration window. E_ph = hc/λ, N = P·t / E_ph Detection Efficiency & Count Rate Count rate from photon flux, photon detection efficiency and optical path loss. R = Φ · PDE · 10^(−L/10) Dark Counts & SNR Photon-counting signal-to-noise ratio from signal counts, dark counts and integration time. SNR = S / √(S + D) Dead Time & Count Correction Correct between measured and true count rate for SPAD dead time, in both directions. M = T / (1 + T·τd) Timing Jitter & Range Resolution Range resolution from detector timing jitter, or the jitter needed to hit a target resolution. ΔR = c · σt / 2 SiPM Array Saturation Number of fired microcells for a given incident photon number and PDE, i.e. SiPM saturation. N_out = N_pix · [1 − e^(−PDE·N_ph / N_pix)] Afterpulsing Afterpulse probability as a function of hold-off time and trap lifetime. P_ap = P₀ · e^(−t_hold / τ) Array Crosstalk Total crosstalk probability across the neighbouring cells of an SPAD array. P_tot = [1 − (1 − p_ct)^n] Gated-Mode Analysis Effective background noise in gated operation, from gate width, repetition rate and raw noise. N_eff = N_raw · (Width · Freq) DCR vs Temperature Dark count rate against temperature, using the doubling-temperature coefficient. DCR(T) = DCR₀ · 2^((T−T₀)/k) PDE Spectral Curve Gaussian model of photon detection efficiency against wavelength around the peak wavelength. PDE(λ) = PDE_max · e^(−½((λ−λ₀)/σ)²)

Beam Combining & Control 6 calculators

Polarization combining, splitters, DOE fan-out and combining efficiency.

PBS Polarization Combining A polarizing beamsplitter combines two orthogonally polarized beams; compute the combined output power. P_out = P₁ + P₂ × η_pbs Fiber Optic Splitter A fiber-coupled splitter divides the input into two ports by ratio; compute the power at each port. P₁ = P_in × R, P₂ = P_in × (1-R) × η DOE Diffractive Splitting A diffractive optical element splits an incident beam into N equal-intensity orders; compute the intensity of each order. I_n = I₀ × η_DOE / N_orders Laser Array Combining Total output power of several lasers combined after each one's own coupling efficiency. P_total = Σ(P_i × η_i) Beamsplitter Power Budget Conservation between reflectance, transmittance and absorption of a beamsplitter. R + T + A = 1 Laser Power Superposition Incoherent sum of several independent lasers after their coupling efficiencies. P_combined = Σ P_i × η_coupling

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