---
title: "Thermal Lensing & High-Power Scaling Interactive Lab | Sandbox Physics"
description: "Can better cooling alone fix the beam of a high-power laser? Not on its own."
canonical_url: "https://sandboxphysics.com/labs/thermal-lensing/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/thermal-lensing/"
last_modified: "2026-10-27"
---

L32 · Heat changes the optical path

# Thermal Lensing & High-Power Scaling

**Can better cooling alone fix the beam of a high-power laser?** Not on its own. The pump power absorbed in the crystal sets up a temperature gradient, which bends the light like a lens and adds stress. Beam quality depends on the resulting phase profile across the beam, not just on the peak temperature. A compensating optic can correct only part of the distortion.

Follow the absorbed pump as it becomes heat, stress and optical phase. Record a probe beam on a camera, fit a compensating lens and compare the beam quality with a separate cavity stability model.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

The element is a cylinder four millimetres long, cooled on its side, and a set fraction of the absorbed pump becomes heat. Radial finite volumes give the temperature as it changes in time. Constant, illustrative material coefficients then give a simplified plane stress, the bulging of the faces and a scalar optical path. The weak-probe camera follows one complex field through the heated crystal and the compensator. Its cold image is a separate reference, and a capture is refused when the sampling cannot resolve the field. The running laser is calculated separately, in a symmetric 30-centimetre cavity with mirrors of 50-centimetre curvature and a thermal lens plus compensator at its center. The same temperature sets the Gaussian mode, the share of the pump it uses, the power rejected for residual wavefront error and the stress-polarization loss. A quasi-steady four-level equation with saturated gain then gives the circulating and extracted power. The absorbed pump splits into the set heat, useful output, rejected optical power and unextracted fluorescence. The thermal coupling runs one way: heat affects the laser, while the heat fraction and saturation intensity stay fixed. Temperature-dependent gain, heating that depends on extraction, multimode recovery, fracture and damage are left out. When the Gaussian design is unstable, its single-mode output is zero; this does not predict that all real laser emission stops. The scaling curves are calculated steady states, not efficiencies derived from the camera or calibrated hardware performance. The apparatus on screen is still the probe bench.

Laser foundations · Light, gain and feedback

[L01 · Compare sources](https://sandboxphysics.com/labs/laser-source-comparison/) [L02 · Count transitions](https://sandboxphysics.com/labs/light-matter-transitions/) [L03 · Build inversion](https://sandboxphysics.com/labs/population-inversion/) [L04 · Amplify once](https://sandboxphysics.com/labs/single-pass-amplifier/) [L05 · Add feedback](https://sandboxphysics.com/labs/passive-optical-resonator/) [L06 · Start from noise](https://sandboxphysics.com/labs/laser-noise-startup/) [L07 · Build and debug](https://sandboxphysics.com/labs/build-debug-laser/) [L08 · Measure and diagnose](https://sandboxphysics.com/labs/laser-diagnostic-bench/) [L09 · Select axial modes](https://sandboxphysics.com/labs/gain-longitudinal-modes/) [L10 · Confine the beam](https://sandboxphysics.com/labs/stable-resonator-geometry/) [L11 · Select spatial modes](https://sandboxphysics.com/labs/transverse-laser-modes/) [L12 · Propagate and measure](https://sandboxphysics.com/labs/gaussian-beam-quality/) [L13 · Align and match](https://sandboxphysics.com/labs/alignment-mode-matching/) [L14 · Optimize extraction](https://sandboxphysics.com/labs/output-coupler-optimization/) [L15 · Drive the dynamics](https://sandboxphysics.com/labs/laser-dynamics-relaxation/) [L16 · Store and release](https://sandboxphysics.com/labs/q-switched-giant-pulse/) [L17 · Lock phases into pulses](https://sandboxphysics.com/labs/mode-locking-frequency-comb/) [L18 · Calibrate the comb](https://sandboxphysics.com/labs/optical-frequency-comb/) [L19 · Inject carriers](https://sandboxphysics.com/labs/semiconductor-laser-diode/) [L20 · Compare gain media](https://sandboxphysics.com/labs/laser-medium-zoo/) [L21 · Distributed gain](https://sandboxphysics.com/labs/fiber-fbg-laser/) [L22 · Choose feedback](https://sandboxphysics.com/labs/laser-feedback-architectures/) [L23 · Balance ultrafast pulses](https://sandboxphysics.com/labs/ti-sapphire-oscillator/) [L24 · Stretch and compress](https://sandboxphysics.com/labs/chirped-pulse-amplification/) [L25 · Trap and extract](https://sandboxphysics.com/labs/regenerative-multipass-amplifier/) [L26 · Convert frequencies](https://sandboxphysics.com/labs/nonlinear-frequency-conversion/) [L27 · Scatter and cascade](https://sandboxphysics.com/labs/raman-brillouin-lasers/) [L28 · Synchronize a slave](https://sandboxphysics.com/labs/injection-locking/) [L29 · Lock to a reference](https://sandboxphysics.com/labs/pdh-frequency-lock/) [L30 · Measure phase noise](https://sandboxphysics.com/labs/laser-linewidth-phase-noise/) [L31 · Scatter for feedback](https://sandboxphysics.com/labs/random-laser/) [L32 · Diagnose thermal scaling](https://sandboxphysics.com/labs/thermal-lensing/) [L33 · Combine coherent fields](https://sandboxphysics.com/labs/coherent-beam-combining/) [L34 · Sustain a Kerr pulse](https://sandboxphysics.com/labs/microresonator-kerr-comb/) [L35 · Return a delayed field](https://sandboxphysics.com/labs/external-feedback-chaos/) [L36 · Scan without mode hops](https://sandboxphysics.com/labs/tunable-laser-scan/)

Physical basis: [MIT: coherence and spectrum](https://ocw.mit.edu/courses/res-6-006-video-demonstrations-in-lasers-and-optics-spring-2008/resources/coherence-length-and-source-spectrum/) · [Rate equations](https://ocw.mit.edu/courses/6-977-ultrafast-optics-spring-2005/resources/chapter2/) · [Pumping and gain media (§7.2)](https://ocw.mit.edu/courses/6-974-fundamentals-of-photonics-quantum-electronics-spring-2006/bf3bdeb971819ddc9c68719d28bb2745_classnotes.pdf) · [Laser dynamics](https://ocw.mit.edu/courses/6-977-ultrafast-optics-spring-2005/resources/chapter4/)

Physics tutorial

## Cooling and beam quality answer different questions

**Background** A pump can heat a crystal more strongly at its center than at its edge.

**Why it matters** Absorbed pump power turns into heat, stress and a change in optical phase. You can record the effect on a probe beam with a camera, fit a compensating lens and compare the beam quality with a separate model of cavity stability.

Start with the essentials

- **Focus question**: Can better cooling alone fix the beam of a high-power laser?
- **One-sentence intuition**: Cooling lowers the temperature, but the lens comes from the temperature gradient, and a compensator can only remove its quadratic part.

### Core mathematical model

#### Heat goes in and flows out

$$
\begin{aligned}P_h&=\eta_hP_{\rm abs}\\\rho c\,\partial_tT&=\kappa\nabla_r^2T+q(r)\\-\kappa T\prime(R)&=h[T(R)-T_c]\end{aligned}
$$

The side surface carries heat into the coolant. Because the heat-transfer coefficient is finite, the whole crystal also sits at a higher temperature.

#### Steady gradient for a Gaussian pump

$$
\begin{aligned}T\prime(r)&=-\frac{P_h(1-e^{-2r^2/w_p^2})}{2\pi\kappa Lr\,Z}\\Z&=1-e^{-2R^2/w_p^2}\end{aligned}
$$

For a fixed heat load and conductivity, the steady radial gradient does not depend on how well the side is cooled.

#### Temperature creates stress

$$
\begin{aligned}A(r)&=\frac{2}{r^2}\int_0^rT(s)s\,ds\\\sigma_r&=\frac{E\alpha_T}{2}[A(R)-A(r)]\\\sigma_\theta&=E\alpha_T\!\left[\frac{A(R)+A(r)}2-T(r)\right]\end{aligned}
$$

The Lab treats the crystal as isotropic slices in plane stress with a free outer surface. This is a deliberate simplification.

#### Optical path through the heated crystal

$$
\begin{aligned}\Delta\mathcal{L}&=L\chi[T(r)-T(0)]\\\chi&=n_T+(n-1)(1+\nu)\alpha_T\\\mathcal{L}_{\rm fit}&=b-\tfrac12\Phi r^2\end{aligned}
$$

The optical path includes the change in refractive index and the bulging of the end faces relative to the center. Fitting a pure defocus still leaves some aberration behind.

#### Image in the focal plane

$$
\begin{aligned}E_{\rm out}&=e^{-r^2/w_b^2}e^{ik(\Delta\mathcal{L}+Cr^2/2)}\\E_f&\propto\mathcal{F}_{2D}[E_{\rm out}]\\I_f&=|E_f|^2\end{aligned}
$$

The camera pixels and the line through their center come from the same calculated field. An ideal quadratic compensator cannot remove higher-order phase.

#### Two separate checks: depolarization and cavity stability

$$
\begin{aligned}\delta(r)&=kLC_s(\sigma_\theta-\sigma_r)\\\eta_\perp&=\tfrac12\langle\sin^2(\delta/2)\rangle\\\left|\frac{A+D}{2}\right|&<1\end{aligned}
$$

The polarization loss from stress and the stability of a paraxial cavity are separate reference calculations. Neither is derived from the camera, and neither is the laser efficiency.

#### Gain equals loss in the running laser

$$
\begin{aligned}P_u&=(1-\eta_h)P_{\rm abs}\,\mathcal O\\g(P_c)&=P_u/(P_s+P_c)\\P_c&=\max(0,P_u/\Lambda-P_s)\end{aligned}
$$

A calculated four-level Gaussian cavity uses the heated mode and the share of the pump it uses. The heat fraction is fixed in advance, so heat affects the laser but the laser does not change the heat.

#### Where the absorbed pump power goes

$$
\begin{aligned}P_{\rm out}&=\Lambda_{\rm ext}P_c\\\eta_{\rm opt}&=P_{\rm out}/P_{\rm abs}\\P_{\rm abs}&=P_h+P_{\rm out}\\&\quad+P_{\rm rejected}+P_{\rm fl}\end{aligned}
$$

Residual phase error, polarization rejection and changes of the mode all reduce the laser output. The share of probe power the camera collects in its core is still a separate probe measure.

### Common difficulties

#### Cold coolant eliminates the lens

**Typical misconception** If the edge is cold, the wavefront must be flat.

**Better mental model** The gradient depends on the deposited heat, the conductivity and the pump geometry.

#### The camera measures laser efficiency

**Typical misconception** The power collected in the camera core equals the total conversion efficiency of the laser.

**Better mental model** The camera sees a weak, fixed scalar probe beam. The laser efficiency is calculated separately from the heated cavity mode and a quasi-steady balance of gain and loss; it is not inferred from the camera.

### Run the experiment

1. 01

   #### Heat the element

   Click Advance & capture, then compare the High heat / tight pump and Widen the pump presets.

   **What to observe:** Spreading the same heat over a wider area gives a weaker phase gradient.

2. 02

   #### Test what cooling does

   Keep the pump and heat fraction the same and compare different side cooling.

   **What to observe:** The temperature shifts, but with constant conductivity the steady lens hardly changes.

3. 03

   #### Compensate and capture again

   After heating, fit the compensator and compare the captured pixels with the cold reference.

   **What to observe:** Defocus improves, but aberration beyond the quadratic part and stress depolarization remain.

4. 04

   #### Calculate active scaling

   Click Calculate active scaling, then widen the pump or reduce the deposited heat.

   **What to observe:** Compare the hot and cold output, the optical efficiency of the laser and the separate probe collection measure.

Physical conclusion

Look at temperature, phase and the recorded focal spot separately, and remember what a compensator cannot fix.

Waves & optics · Starter path

## From one ray to interference measurement

Start with refraction, find where the ray picture stops working, then use interference to measure tiny distances.

[Start here → **Optical Boundary Explorer** Turn the incident angle and watch reflected and transmitted power trade off. Then switch to the critical-angle scene. Observe · 11 min](https://sandboxphysics.com/labs/optical-boundary-explorer/)

1. [1 Optical Boundary Explorer](https://sandboxphysics.com/labs/optical-boundary-explorer/)
2. [2 Rainbow Caustic Observatory](https://sandboxphysics.com/labs/rainbow-laboratory/)
3. [3 Camera Obscura](https://sandboxphysics.com/labs/camera-obscura/)
4. [4 Coherent Wave Composer](https://sandboxphysics.com/labs/coherent-wave-composer/)
5. [5 Young Double-Slit Bench](https://sandboxphysics.com/labs/young-double-slit-bench/)
6. [6 Michelson Precision Lab](https://sandboxphysics.com/labs/michelson-precision-lab/)

[View the full path →](https://sandboxphysics.com/topics/waves-and-optics/#starter-path)