L32 · Heat changes the optical path

Thermal Lensing & High-Power Scaling

Turn absorbed pump into heat, stress and optical phase. Record a probe camera, fit a compensator and compare beam quality with a separate cavity stability model.

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 $\text{—}$

Separate cavity round-trip half trace $\text{—}$

Separate cavity stability $\text{—}$

Separate 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

Does better cooling also fix the beam?

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

Heat and capture, then fit the compensator. Camera records remain frozen until another capture; changing compensation preserves the temperature. Cooling shifts absolute temperature, while wider heating or lower heat load reduces the gradient.

An ideal cylindrical gain element, four millimeters thick, cooled only at its side. Absorbed pump power and the heat fraction set deposited heat; Gaussian or top-hat radial heating is uniform along the axis. Constant illustrative coefficients and a radial finite-volume heat equation give transient and steady temperatures. Free-boundary isotropic plane-stress slices give radial and hoop stress and relative axial bulging. This is a reduced mechanical model, not a three-dimensional finite-element solution. Thermo-optic index change plus the consistent plane-stress expansion give scalar optical path. A fixed weak Gaussian probe passes once through it and a chosen compensator; a two-dimensional complex Fourier transform produces pixels at a physical lens back focal plane. The camera refines its grid and rejects unresolved phase sampling. Camera pixels and their center line share this field; the cold image is a separate reference. The stress-optic depolarization estimate is a separate polarization diagnostic and is not applied to the scalar camera. A separate symmetric cavity uses the fitted thermal lens only. There is no active laser efficiency, gain, damage or fracture prediction. Axial cooling, temperature-dependent coefficients and vector optical propagation are omitted. At large temperature rise the constant-material extrapolation is illustrative. Distances and crystal size are enlarged schematically.

Start with the essentials

Focus question: Does better cooling also fix the beam?
One-sentence intuition: Compare temperature, phase, recorded focal intensity and the limits of a compensator.

Core mathematical model

Deposit and remove heat

\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 boundary carries heat into coolant; a finite heat-transfer coefficient also raises the entire temperature.

A steady Gaussian gradient

\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}

At fixed heat and conductivity the steady radial gradient is independent of the side cooling coefficient.

Connect heat to 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}

Free-boundary isotropic plane-stress slices are a deliberate reduced approximation.

Build a consistent scalar optical path

\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 index change and relative end expansion; fitting defocus leaves a residual aberration.

Acquire the physical 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}

Pixels and their center line are calculated from the same field. An ideal quadratic compensator cannot remove higher-order phase.

Separate two diagnostics

\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}

Stress-induced polarization loss and paraxial cavity stability are separate references, not camera reconstruction or laser efficiency.

Common difficulties

Cold coolant eliminates the lens

Typical misconceptionA low edge temperature implies a flat wavefront.

Better mental modelThe gradient is set by deposited heat and conductivity as well as pump geometry.

The camera is laser efficiency

Typical misconceptionCore collection equals total active laser conversion efficiency.

Better mental modelIt measures a fixed scalar weak probe; gain dynamics and laser extraction are not modeled.

Run the experiment

01
Heat the element
Advance and capture, then compare the high-heat and wide-pump presets.
What to observe: The same heat spread over a wider area creates a weaker phase gradient.
02
Test cooling honestly
Compare side cooling at equal pump and heat fraction.
What to observe: Temperature shifts while the constant-conductivity steady lens remains almost unchanged.
03
Compensate and reacquire
Fit the compensator after heating and compare the acquired pixels with the cold reference.
What to observe: Defocus improves; nonquadratic aberration and stress depolarization remain.