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.

Frozen active-scaling compensation $\text{—}$

Active output at current temperature $\text{—}$

Active efficiency at current temperature $\text{—}$

Unextracted fluorescence power $\text{—}$

Rejected optical power $\text{—}$

Active power balance residual $\text{—}$

Pump utilization by the cavity mode $\text{—}$

Calculated steady scaling points $\text{—}$

Elapsed heating time $\text{—}$

Deposited heat power $\text{—}$

Central temperature rise $\text{—}$

Outer-cell temperature rise $\text{—}$

Heat carried to coolant $\text{—}$

Stored thermal energy $\text{—}$

Probe-weighted thermal lens power $\text{—}$

Residual path after best quadratic fit $\text{—}$

Relative bulge per face $\text{—}$

Largest hoop-minus-radial stress $\text{—}$

Stress-polarization loss estimate $\text{—}$

Uncompensated cavity half trace $\text{—}$

Uncompensated cavity stability $\text{—}$

Uncompensated paraxial mode radius $\text{—}$

Captured camera pixels $\text{—}$

Captured on-axis ratio to cold probe $\text{—}$

Captured power within fifty micrometers $\text{—}$

Captured compensator power $\text{—}$

Explore the mechanism

Can better cooling alone fix the beam of a high-power laser?

Absorbed pump power

P_{\rm abs}

60\,\mathrm{W}

Deposited heat fraction

\eta_h

0.25

Pump radius

w_p

0.8\,\mathrm{mm}

Pump profile: Gaussian / top-hat

b

0

Element outer radius

R

2\,\mathrm{mm}

Side heat-transfer coefficient

h

5000\,\mathrm{\frac{W}{m^2K}}

Thermal conductivity

\kappa

14\,\mathrm{\frac{W}{mK}}

Diverging compensation power

C

0\,\mathrm{m^{-1}}

Detail: crystal / coolant / camera

j

0

The page captures a simulated probe after thirty seconds of heating. Parameter changes recapture the image; changing compensation keeps the current temperature. Cooling shifts the absolute temperature, while a wider heat source or a lower heat load reduces the gradient. The camera uses a fixed logarithmic display relative to the cold peak; power readouts retain linear units.

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.

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 misconceptionIf the edge is cold, the wavefront must be flat.

Better mental modelThe gradient depends on the deposited heat, the conductivity and the pump geometry.

The camera measures laser efficiency

Typical misconceptionThe power collected in the camera core equals the total conversion efficiency of the laser.

Better mental modelThe 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

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.
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.
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.
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.