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Diffraction

What an image tells you about the camera (or telescope or your eye)

Diffraction causes star pattern around bright lights
Nova exploding pusher target
  • The first target I designed for the Nova laser!
  • Photo taken by a National Geographic photographer with an ordinary (nice) camera, not a Nova diagnostic.
  • The ten pointed stars have nothing to do with the fact that Nova had ten laser beams.
  • Instead, this camera had a five sided lens iris, visible in the reflections below and to the right of the target.
  • I don't understand many things about this image (e.g.- the red color of the chamber port behind the target), but the ten pointed stars are caused by diffraction from the pentagonal lens iris.
Five and eight bladed irises
You can guess telescope from its diffraction pattern

Hubble telescope image

Hubble telescope image

Webb telescope image

Webb telescope image

(Beware, however, that published telescope images are usually "cleaned up" to more accurately present the object in the image.)

Diffraction spikes are 2D Fourier transform of pupil

Hubble telescope pupil shows supports for secondary mirror

Hubble telescope pupil Hubble telescope pupil FT

Webb telescope pupil shows secondary supports plus hexagonal segment edges

Webb telescope pupil Webb telescope pupil FT
  • Straight edges produce perpendicular diffraction spikes
  • Hubble: Spikes are from secondary mirror struts
  • Webb: Six spikes from hex segment edges, horizontal spikes from vertical mirror strut
  • Pentagonal lens iris: Ten spikes!
What does a lens have to do with Fourier transforms?
  • An approximate (small angle) modern formulation of Huygen's Principle.
  • Imagine a field of stars at various angles $\theta$ relative to optical axis.
  • Each star angle $\theta$ corresponds to a given transverse wavenumber $k_x$ in the plane of the primary mirror (or lens). Wave amplitude is brightness.
  • The lens (or mirror) maps $\theta \approx k_x/k$ in the lens plane into position $x$ in the focal plane according to $\theta=x/f$, where $f$ is focal length.
  • Thus each position $x$ in the image plane gets only the light with a particular transverse wavenumber $k_x$ in the lens plane. A lens Fourier transforms incident light waves!
lens plane image plane x f f θ θ kz kx k = 2π/λ
How do Fourier transforms cause diffraction spikes?
  • Sharp edges of pupil or shadow of support struts contain high spatial frequencies - large $k_x$ in lens plane means large extent in $x$ in image plane.
  • Fourier transform of a top hat (like strut shadow) is a sinc function ($\sin x/x$), which you can use to make rough models of diffraction spikes.
  • Spike width $\Delta x$ and period of bands in image is directly proportional to light wavelength $\lambda$ and inversely proportional to shadow width $D$ in lens plane $\Delta\theta = \Delta x/f = \lambda/D$.
  • Note that spike amplitude a given distance from center is roughly independent of the shadow width - only band period changes.
lens plane D image plane
Diffraction has many uses in modern optical design
  • Diffraction spikes are a nuisance (for scientific instruments), but the effect can be extremely useful too!
  • Diffraction gratings are used to separate colors, bend beams, and amplify very short pulse laser beams.
  • Most integrated optics work by diffraction - so transmitting your phone conversations often relies on diffractive optics.
  • NIF laser has diffractive phase plate in its final optics to produce a smooth laser spot on the target.
  • If you want to play with Fourier transforms or scientific programming in general, I highly recommend learning SciPy, which you can get from Anaconda.