Preston Bohm
optics information theory

Diffraction Limited Resolution

Resolving two points across the spectrum d = 0.61 λ / NA
IR NIR vis UV DUV EUV soft X X-ray e⁻
Probe
Visible
λ = 550 nm (green light)
Diffraction-limited spot
353 nm
can resolve: small organelles
Two points, 300 nm apart
NOT RESOLVED
NA (fixed objective)
0.95
Image plane: two point sources 300 nm apart, blurred by the point-spread function (field of view 1.2 µm)
Intensity along the line joining them; a dip appears only once the spot is smaller than the gap
Slide from infrared to electrons. The two point sources never move; only the wavelength of the probe changes. The blur is the point-spread function, whose width is set by the diffraction-limited spot size d = 0.61 λ/NA. Visible light (≈550 nm) cannot separate features 300 nm apart, and the two blobs merge into one. Shorten the wavelength and the spot collapses: by deep UV the pair splits, and an electron's picometre wavelength resolves them with room to spare. That is the whole reason electron microscopes exist. Assumption: the numerical aperture is held fixed at NA = 0.95 for every probe, so the spot size depends only on wavelength. That keeps the demo clean but is not physical at the short-wavelength end. Real X-ray and electron optics work at tiny numerical apertures (NA of order 0.01) and suffer lens aberrations, so their achieved resolution is far coarser than 0.61 λ/NA with NA = 0.95 would suggest. Read the short-wavelength spots here as the diffraction floor, not the number an instrument actually delivers.

Point the best optical microscope ever built at two specks 200 nm apart and you see one smudge. No polishing or magnification fixes it: the obstruction is the light, not the lens. Waves through any finite aperture spread out, setting a hard floor on the finest detail an instrument can resolve. That floor is fixed by one number: the wavelength.

Why a lens cannot focus to a point

A perfect lens collecting a plane wave produces not a point but an Airy pattern, a bright central disk ringed by faint halos, because the aperture truncates the wave and a truncated wave diffracts. The width of that disk is the point-spread function (PSF), the image of an ideal point, and every image you record is the true scene convolved with it. When two points sit closer than the PSF is wide, their blurs merge and the information that there were two of them is gone.

The angular radius of the Airy disk for a circular aperture of diameter D is

\[ \sin\theta \;=\; 1.22\,\frac{\lambda}{D} \]

Rayleigh's criterion calls two points "just resolved" when the peak of one Airy disk falls on the first dark ring of the other. For a microscope with numerical aperture \( \mathrm{NA} = n\sin\alpha \), Abbe's 1873 result gives the smallest resolvable separation

\[ d \;=\; \frac{0.61\,\lambda}{\mathrm{NA}} \qquad\Longleftrightarrow\qquad d_{\min} \approx \frac{\lambda}{2\,\mathrm{NA}}. \]

Both say the same thing: resolution scales with wavelength. NA cannot exceed the refractive index of the medium, so even oil-immersion optics stall near \( \mathrm{NA}\approx 1.4 \). The only lever with orders of magnitude left is \( \lambda \).

The wavelength ladder, from IR to electrons

So the history of microscopy is a march down the wavelength scale. Visible light bottoms out near 200 nm, fine for cells but useless for a virus. Ultraviolet and deep-UV push lithography and biology further; chip fabs moved to 193 nm DUV and then 13.5 nm extreme ultraviolet. X-rays reach ångström wavelengths and image atomic planes directly.

Then comes the trick that breaks the optical scale: a particle of momentum p has a de Broglie wavelength

\[ \lambda \;=\; \frac{h}{p} \;=\; \frac{h}{\sqrt{2 m_e e V}}\,, \]

so an electron accelerated through a modest voltage carries a wavelength thousands of times shorter than visible light. At 100 kV the relativistically corrected value is about 3.7 pm, smaller than an atom. That is the electron microscope's working principle: not better lenses, but a far shorter wave. The interactive above walks the ladder, with spot size d tracking \( \lambda \) at fixed NA so you can watch the two points fuse and split.

The caveat that keeps electron microscopists employed

Wavelength sets the floor, not the achieved resolution. An electron's 3.7 pm wavelength does not buy 3.7 pm images: magnetic round lenses suffer unavoidable spherical and chromatic aberration, so usable apertures are tiny (NA of order 0.01) and resolution lands near 0.05–0.1 nm, not picometres. Aberration correction, which earned a share of the 2017 chemistry prize, claws back what the short wavelength promised. The diffraction limit sets what is possible; engineering decides how close you get.

The logic runs in reverse for what we cannot shrink. Radio astronomers, stuck with metre and centimetre waves, recover resolution the only other way the equations allow: making D enormous, synthesizing apertures the size of the Earth. Shrink \( \lambda \) or grow D, you fight the same ratio. Diffraction never goes away; you only choose which side of \( \lambda/D \) to push.

References

  1. E. Abbe, "Beiträge zur Theorie des Mikroskops und der mikroskopischen Wahrnehmung," Archiv für mikroskopische Anatomie, vol. 9, no. 1, pp. 413–468, 1873, doi: 10.1007/BF02956173.
  2. Lord Rayleigh, "Investigations in optics, with special reference to the spectroscope," Philosophical Magazine, vol. 8, no. 49, pp. 261–274, 1879, doi: 10.1080/14786447908639684.
  3. M. Born and E. Wolf, Principles of Optics, 7th ed. Cambridge, U.K.: Cambridge Univ. Press, 1999.
  4. L. de Broglie, "Recherches sur la théorie des quanta," Ph.D. dissertation, Univ. of Paris, Paris, France, 1924.
  5. D. B. Williams and C. B. Carter, Transmission Electron Microscopy: A Textbook for Materials Science, 2nd ed. New York, NY, USA: Springer, 2009, doi: 10.1007/978-0-387-76501-3.