I'm trying to figure out how to simplify the problem where laser pulses are focused to a small spot in bulk glass. The waist of the beam is about 20 microns.

At the wavelength used there is only two-photon absorbtion, so all of the energy is absorbed at the focal point. After some amount of laser pulses this point inside the glass becomes hot and dissipates heat to the surrounding glass.

Basically I'd like to calculate the temperature distribution near the focal point. My guess is to use the cylindrical shape of heat source, but I don't know the right differential equation to model it in matlab.

Can anybody suggest the correct approach?


1 Answer 1


The 'go to' partial differential equation here is surely the Heat equation (Fourier), here in one dimension:

$$\frac{\partial T}{\partial t}=\kappa \frac{\partial^2T}{\partial x^2}+\frac{\dot{Q}(x,t)}{c_p\rho }$$

It can be easily expanded into three dimensions or expressed in polar, spherical or cylindrical coordinates.

It's not clear from your question whether these 'pulses' are really just short (or intermittent) bursts or continuous forms of irradiation. That would have to determined to define the source term $\frac{\dot{Q}(x,t)}{c_p\rho }$.

If you have some idea of the temperature in the 'spot' after a 'pulse', then you can drop the source term and use that temperature as an initial condition. The equation should then be easily separable:

Say for two dimensions, $T(x,y,t)=X(x)Y(y)\Theta(t)$. Insert into the 2D PDE to obtain three ODEs for $X$, $Y$ and $\Theta$.

Including a sketch of the geometry of your problem in your question would help decide the 'optimum' coordinate system to use. I'm quite interested in helping out with this problem but it does lack some information to take this any further.

  • $\begingroup$ Pulsed nanosecond laser at 355 nm was used in the experiment. The source in the equation is really desired. $\endgroup$ Commented Jul 21, 2016 at 15:48
  • $\begingroup$ Assuming little or no scattering the source term should be determinable from the absorbtivity of the glass (Lambert-Beer law?). The light absorbed is converted to heat. Can you describe the geometry better? "Bulk glass": very thick? Very wide? $\endgroup$
    – Gert
    Commented Jul 21, 2016 at 15:54
  • $\begingroup$ Right, in real experiment sample is 1.5 mm thick. Absorption coefficient α at this wavelength is less than 0.5 cm^-1. But as I sad pick power laser intensity is very high (gigawats/cm^2). So the initial temperature determination is actually separate problem $\endgroup$ Commented Jul 21, 2016 at 16:04
  • $\begingroup$ Looks like a quasi-2D problem, with temperature gradient in the $r$ (distance from laser beam) direction and $\partial T/\partial z \approx 0$. So now a source term is needed. $\dot{Q}$ for $r\approx 0$. And if the glass gets quite hot, a sink term for convective losses will also be needed. $\endgroup$
    – Gert
    Commented Jul 21, 2016 at 16:12
  • $\begingroup$ The model would be quite similar to: syvum.com/cgi/online/serve.cgi/heat/heat1003.html but here the source term is avoided by using boundary conditions (I appreciate that won't work for you) and accounts for convective losses. No doubt Bessel Functions would also pop up in your model. $\endgroup$
    – Gert
    Commented Jul 21, 2016 at 16:44

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