Model laser‑driven stimulated emission using flexible physical inputs quickly and clearly here. Choose Einstein or cross‑section methods, then download reports in CSV and PDF.
This calculator provides two common, practical forms for stimulated emission rate.
Illustrative values for comparison only.
| Method | Key Inputs | Per-Atom Rate (s⁻¹) | Notes |
|---|---|---|---|
| Einstein | B21=1.0e-14, u=3.0e-6 J/m³ | 3.0e-20 | Low energy density regime |
| Einstein | B21=2.5e-12, u=1.2e-3 J/m³ | 3.0e-15 | Stronger radiation field |
| Cross-section | σ=3.0e-20 m², Φ=5.0e24 | 1.5e5 | Flux entered directly |
| Cross-section | σ=1.0e-21 m², I=2.0e4 W/m², λ=1064 nm | 1.07e2 | Flux derived from intensity |
| Cross-section | σ=8.0e-21 m², I=1.0e5 W/m², λ=1550 nm | 6.25e2 | Telecom wavelength example |
Stimulated emission rate sets how quickly an excited population can be converted into coherent photons. In gain media, higher rates generally support higher small‑signal gain and faster amplification, provided population inversion exists. This calculator helps compare regimes using either Einstein coefficients or cross‑sections.
The Einstein form uses Rst = B21·u, where u is the radiation energy density at the transition. For weak fields, u may be very small, giving rates far below 1 s⁻¹ per atom. For strongly pumped cavities, the same transition can move many orders of magnitude higher.
A common practical relation is Rst = σ·Φ, with σ in m² and photon flux Φ in photons·m⁻²·s⁻¹. Typical stimulated emission cross‑sections for solid‑state and fiber systems are often around 10⁻²¹ to 10⁻¹⁹ m², depending on line shape, doping, and wavelength.
If you know intensity, the calculator derives Φ = I / (h·c/λ). At 1064 nm, photon energy is about 1.87×10⁻¹⁹ J, so an intensity of 10⁴ W/m² corresponds to roughly 5×10²² photons·m⁻²·s⁻¹. This makes unit handling critical for correct rates.
Per‑atom Rst becomes volumetric using RV = N2·Rst. Excited‑state densities vary widely: low‑doped media might be 10²¹–10²³ m⁻³, while highly inverted systems can approach 10²⁴ m⁻³ in localized regions.
Multiply RV by active volume to estimate total stimulated emissions per second. For example, RV=10²⁸ s⁻¹·m⁻³ in a 10⁻⁶ m³ mode volume yields about 10²² events/s. This is a rate metric, not output power; extraction and losses still matter.
Very large Rst values imply rapid depletion of inversion under strong fields. In real devices, this links to gain saturation and steady‑state balance between pumping and stimulated emission. Use the calculator to test “what‑if” changes in σ, Φ, and N2 to see sensitivity.
Confirm that σ and Φ are referenced to the same wavelength and polarization conditions. For intensity‑derived flux, ensure λ is entered correctly (nm vs µm) and check intensity units (W/m² vs mW/cm²). Exporting CSV/PDF helps document consistent assumptions across design iterations.
It reports stimulated emission rate per excited atom in s⁻¹. If you provide N2, it also outputs volumetric rate, and with volume it estimates total stimulated events per second.
Use Einstein if you already have B21 and energy density u at the transition. Use the cross‑section method if your material data provides σ and you can estimate photon flux from intensity or measurements.
Energy density u can be tiny in free space or weak fields, making B21·u very small. Rates can jump in cavities or strongly driven systems where u at the transition frequency is much larger.
It assumes monochromatic light at wavelength λ and uses photon energy h·c/λ. If your source has broad spectrum, pulsed structure, or strong spatial variation, treat the result as an effective average.
Many gain materials fall around 10⁻²¹ to 10⁻¹⁹ m² near strong transitions, but values vary by line shape, temperature, doping, and polarization. Always use σ data matched to your operating wavelength.
No. It counts stimulated transitions per second, not energy leaving the cavity. To estimate power, you would multiply extracted photon rate by photon energy and include coupling efficiency and losses.
σ, Φ, N2, and volume span many orders of magnitude, so unit conversion errors amplify quickly. Double‑check nm vs µm, cm² vs m², and cm³ vs m³ before trusting exported reports.
Use this tool to compare optical gain scenarios safely.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.