Fourier Optics and Beam Quality

Stub. Full prose lives in STUDY_PLAN.md §Week 3.

Goals

• A lens performs a Fourier transform between its focal planes.
• A spatial filter is a low-pass filter on spatial frequencies.
• The B-integral and self-focusing are the dominant beam-quality threats.
• The Strehl ratio is the dimensionless beam-quality figure of merit.

Master equations

Diffraction-limited focal spot:

$d_{spot} \approx \frac{\lambda f}{D}$

B-integral (accumulated nonlinear phase):

$B = \frac{2\pi}{\lambda}\int n_2 I(z)\, dz \qquad \text{(NIF design budget: } B < 1.8 \text{ rad)}$

Strehl ratio for RMS wavefront error $\sigma$:

$S \approx \exp\!\left[-\left(\frac{2\pi\sigma}{\lambda}\right)^2\right]$

NIF tie-in

The 60-meter Cavity Spatial Filter makes no intuitive sense until you see it as a Fourier device. You need 60 m because (a) the beam is 40 cm wide and (b) you must reach the Fraunhofer regime ($F = a^2/(\lambda L) \lesssim 1$) to perform a clean Fourier transform. The pinhole at focus is the low-pass filter on spatial frequencies — it scrubs out high-k components that would self-focus into damaging hot spots.

Deformable mirrors handle the low-spatial-frequency wavefront errors (thermal distortion in amplifier glass) that the SF pinhole leaves alone — because they're below the cutoff. SF + DM = a two-stage cleanup: SF for high frequencies, DM for low.

Self-check