Diffraction limit

The diffraction limit is the finest detail an optical system can resolve, and the smallest spot it can focus light to, when the only limit is diffraction. A lens collects only the part of the light that falls within its aperture, and the missing part of the wavefront spreads each image point into a small pattern, the Airy disk for a round lens. The scale of that pattern is the wavelength divided by the numerical aperture (NA): a microscope objective resolves periodic detail down to $\lambda \mathrm{/}(2\text{NA})$, 196 nm for green light at 550 nm and NA 1.4, and a telescope of diameter $D$ separates stars down to an angle of $1.22\lambda \mathrm{/}D$. An instrument whose aberrations are small enough for diffraction alone to set its resolution is called diffraction-limited. The usual test is a Strehl ratio of at least 0.8, which corresponds to an RMS wavefront error of about $\lambda \mathrm{/}14$ (the Maréchal criterion).

The formulas

Three statements of the limit are in common use. They differ in their constants and in what they describe, and all scale as $\lambda \mathrm{/}\text{NA}$.

1. Abbe limit. The finest period of a grating-like object that a microscope objective can image, when the illumination fills the objective's aperture:

$$d=\frac{\lambda }{2\text{NA}}\mathrm{.}$$

With a narrow, on-axis illumination cone it is $\lambda \mathrm{/}\text{NA}$.

2. Rayleigh criterion. Two point sources are just resolved when the center of one Airy pattern falls on the first dark ring of the other:

$$\begin{array}{rl}{\displaystyle d}& {\displaystyle =0.61\frac{\lambda }{\text{NA}}\text{(object plane)},}\\ {\displaystyle \theta }& {\displaystyle =1.22\frac{\lambda }{D}\text{(angle)}\mathrm{.}}\end{array}$$

The Sparrow criterion, where the dip between the two peaks just disappears, gives $0.47\lambda \mathrm{/}\text{NA}$. The Airy Disk Calculator evaluates the Rayleigh, Sparrow and Abbe distances, and the encircled energy, for any f-number or NA.

3. Focused Gaussian beam. The smallest $1\mathrm{/}{e}^2$ waist radius a lens of numerical aperture NA can form is

$$w_0\approx \frac{\lambda }{\pi \text{NA}}\mathrm{.}$$

Why the limit exists

A lens accepts only the plane-wave components of the light that travel within its acceptance cone, at angles up to ${\theta }_{\text{max}}$ from the axis. Fine detail in an object is carried by the steep components: a pattern of period $p$ diffracts light at $\sin \theta =\lambda \mathrm{/}p$, and if that angle lies outside the cone, the detail never reaches the image. In Fourier-optics terms, an aberration-free lens passes transverse spatial frequencies up to $\text{NA}\mathrm{/}\lambda$ with coherent illumination and up to $2\text{NA}\mathrm{/}\lambda$ with incoherent illumination, where the modulation transfer function falls to zero. At a given wavelength, the only way a conventional, far-field system can resolve finer detail is a larger NA, $n\sin {\theta }_{\text{max}}$, obtained with a wider cone or an immersion medium of higher index $n$.

Values

The Abbe limit $\lambda \mathrm{/}(2\text{NA})$ at common wavelengths:

Because $\text{NA}=n\sin {\theta }_{\text{max}}$, the medium in front of the lens caps it. Dry objectives reach 0.95; water immersion ($n$ = 1.33) reaches 1.27; oil immersion ($n$ = 1.518) reaches 1.4 in standard objectives and 1.49 in objectives made for total internal reflection fluorescence. Solid-immersion lenses of high-index material, such as silicon used at infrared wavelengths to inspect chips through their substrate, go beyond any immersion oil.

Lithography

Optical lithography writes its resolution as a half-pitch $k_1\lambda \mathrm{/}\text{NA}$, and $k_1=0.25$ is the diffraction limit for a single exposure. That gives 36 nm for 193 nm immersion scanners (NA 1.35), 10 nm for extreme-ultraviolet (EUV) scanners at 13.5 nm and NA 0.33, and 6 nm for high-NA EUV scanners at NA 0.55; air and every immersion fluid absorb 13.5 nm light, so EUV scanners use mirrors in vacuum, with NA below 1. Production processes run at a $k_1$ somewhat above 0.25, using phase-shift masks, off-axis illumination and optical proximity correction, and print finer pitches by multiple patterning, which splits one layer's pattern over several exposures.

Diffraction-limited laser beams

A laser beam is diffraction-limited when it is an ideal Gaussian beam, whose beam parameter product, waist radius times far-field half-angle, is $\lambda \mathrm{/}\pi$. A real beam's product is larger by the factor $M^2$ (beam quality), and it focuses to a spot $M^2$ times larger through the same lens. The procedure is in measuring M² beam quality.

Fibers and waveguides

The same scaling limits how tightly a waveguide can confine light. A mode confined by an index step of numerical aperture NA cannot be much narrower than about $\lambda \mathrm{/}(\pi \text{NA})$ in radius: about 4 µm at 1550 nm for the NA of about 0.12 set by the 0.36% index step of standard single-mode fiber (Corning's specified 0.14 is measured at the 1% level of the far field), which is why its mode field diameter is 10.4 µm. A silicon strip waveguide, with an index contrast of 3.48 against 1.44, confines light to about 500 × 220 nm; its 220 nm thickness is close to half a wavelength in silicon, $\lambda \mathrm{/}(2n)$ = 0.22 µm at 1550 nm. Plasmonic waveguides confine light more tightly still, at the cost of loss in the metal.

Beyond the diffraction limit

Several techniques resolve detail finer than the diffraction limit. They do not make a lens focus light more tightly; each adds information or physics that conventional far-field imaging lacks.

STED, PALM and STORM reach resolutions of tens of nanometers in cells, and their development was recognized by the 2014 Nobel Prize in Chemistry.

Common questions

What does diffraction-limited mean?

That an optical system's aberrations are small enough for diffraction to be the only thing limiting its resolution: its focused spot is an Airy pattern of the size set by the wavelength and the NA. The usual criterion is a Strehl ratio of 0.8 or more.

What is the diffraction limit of the human eye?

With a 3 mm pupil at 550 nm, $1.22\lambda \mathrm{/}D$ is $2.2\times {10}^{-4}$ rad, or 0.77 arcminute. Normal (20/20) visual acuity corresponds to resolving detail of 1 arcminute, close to that limit; the spacing of the cone cells in the fovea limits the eye at a similar scale.

Is the diffraction limit λ/2?

Only for the highest numerical apertures: $\lambda \mathrm{/}(2\text{NA})$ becomes $\lambda \mathrm{/}2$ at NA 1, and immersion objectives go somewhat below it, to about $\lambda \mathrm{/}2.8$ at NA 1.4. For a camera lens or a low-NA objective, the limit is several wavelengths.

Can the diffraction limit be broken?

Not by a better lens: the limit follows from the wavelength and the range of angles the lens collects. Super-resolution methods resolve finer detail by using fluorescence switching, near-field probes or patterned illumination, as listed above.

References: M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), ch. 8. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2005), for the coherent and incoherent cutoff frequencies. E. Hecht, Optics, 5th ed. (Pearson, 2017), ch. 10. M. G. L. Gustafsson, "Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy," Journal of Microscopy 198, 82 (2000). S. W. Hell and J. Wichmann, "Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy," Optics Letters 19, 780 (1994). L. Schermelleh et al., "Super-resolution microscopy demystified," Nature Cell Biology 21, 72 (2019).