This sheet accompanies Lecture 3 (CPA lasers) and goes deeper into focal-spot physics and peak-intensity diagnostics. It complements Exercise 2 (which already introduces sunlight focusing and crude ATLAS estimates from Lecture 2).
Part A — Gaussian beam (pulse) in space and time
Familiarize yourself with the Gaussian models used throughout the course. For a linearly polarized beam propagating along \(x\), a common intensity model is \[ I(x,y,z,t) = I_0\, \frac{w_0^2}{w(z)^2}\, \exp\!\left(-\frac{2r^2}{w(z)^2}\right)\, \exp\!\left(-\frac{4t^2}{\tau_G^2}\right), \] with \(r^2 = y^2 + z^2\), \(w(z) = w_0\sqrt{1 + (z/z_R)^2}\), \(z_R = \pi w_0^2/\lambda\), and \(\tau_G\) the \(1/e^2\) half-duration.
- Show that the FWHM of the intensity profile in the focal plane (\(z=0\)) is \(d_{\mathrm{FWHM}} = w_0\sqrt{2\ln 2}\).
- Derive the relation between pulse energy \(E_{\mathrm{pulse}}\) and peak intensity \(I_0\) for a separable Gaussian in three spatial dimensions and time (integrate over \(x,y,z,t\); the beam is taken as collimated over the short interaction volume, so \(w(z)\approx w_0\) suffices).
- A transform-limited pulse has \(\tau_G \approx 0.44\,\tau_{\mathrm{FWHM}}\) for the standard intensity FWHM duration \(\tau_{\mathrm{FWHM}}\). Rewrite your result from (2) in terms of \(\tau_{\mathrm{FWHM}}\) and \(d_{\mathrm{FWHM}}\).
Part B — Final OAP: what sets the smallest spot?
At CPA facilities the last element before the target is usually an off-axis parabolic mirror (OAP). For a beam of diameter \(D\) incident on a focussing optic of focal length \(f\), the f-number is \(f/\# = f/D\).
- For a Gaussian beam with waist \(w_{\mathrm{in}}\) incident on an ideal thin lens (or OAP in the paraxial limit), the focussed waist is \(w_0 \approx \lambda f/(\pi w_{\mathrm{in}})\). If the near-field beam fills the aperture (\(w_{\mathrm{in}} \approx D/2\)), express \(w_0\) and \(d_{\mathrm{FWHM}}\) in terms of \(\lambda\), \(f\), and \(D\).
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Estimate the Airy-disk diameter for a circular aperture,
\(d_{\mathrm{Airy}}\approx 1.22\,\lambda\,f/D\), and compare it numerically
(in units of \(\lambda\,f/D\)) with your Gaussian \(d_{\mathrm{FWHM}}\) from (4).
Then distinguish two different “limits” that both scale like \(\lambda f/D\)
or \(f/D\), but for different physics:
- Geometric image size (Exercise 2, Part A, sunlight): an extended incoherent source (the Sun) is imaged by similar triangles; the focal spot size is set by the source’s angular diameter, not by diffraction.
- Diffraction limit: a spatially coherent wave filling the aperture cannot be focussed smaller than \(\sim\lambda\,f/D\) (Airy / Gaussian waist).
- LION uses \(f/\# \approx 5\) with \(d_{\mathrm{FWHM}} \approx 5\,\mu\mathrm{m}\); LUX pursues \(f/\# \approx 1\). For the same \(\lambda\) and delivered power, how do you expect peak intensity to scale? What practical limits (wavefront quality, aberrations, damage) might prevent reaching the ideal diffraction limit?
Part C — Focal spot distribution
Use the attenuated ATLAS 3000 focus image exercise_focal_spot.png and analyse it from scratch (Python or MATLAB).
- Open the image with an appropriate software (e.g. Python or MATLAB) and visualize it.
- Plot a one-dimensional line-out (also called profile) in the horizontal direction (the dimension with more pixels). A line-out means the pixel values at a fixed \(y\)-position — a one-dimensional data trace. The line-out should go through the maximum of the image.
- Fit a Gaussian function to the line-out and determine the Full-Width at Half Maximum (FWHM) width in units of pixels.
- The pixel size of the image corresponds to \(0.53\,\mu\mathrm{m}\) in the focal plane. Assume that the spot is circular and that the laser pulse contains an energy of \(10\,\mathrm{J}\). The FWHM duration of the laser pulse is \(30\,\mathrm{fs}\). Calculate the peak intensity (you may use your Gaussian relations from Part A).