Resolution extension through the moiré effect. If an unknown sample structure (a) is multiplied by a known regular illumination pattern (b), a beat pattern (moiré fringes) will appear (c). The moiré fringes occur at the spatial difference frequencies between the pattern frequency and each spatial frequency component of the sample structure and can be coarse enough to observe through the microscope even if the original unknown pattern is unresolvable. Otherwise-unobservable sample information can be deduced from the fringes and computationally restored.

Structured-illumination concept. (a) The set of sample spatial frequencies that can be observed by the conventional microscope defines a circular observable region of radius k₀ in frequency space. (b) If the excitation light contains a spatial frequency k₁, a new set of information becomes visible in the form of moiré fringes (hatched circle). This region has the same shape as the normal observable region but is centered at k₁. The maximum spatial frequency that can be detected (in this direction) is k₀ + k₁.

Generation of harmonics by nonlinear fluorescence. (a) The nonlinear dependence of the fluorescent emission rate on the illumination intensity in the saturation regime. (b) The emission pattern resulting from sinusoidally patterned illumination with peak pulse energy densities of (from the bottom to top curve) 0.25, 1, 4, 16, and 64 times the saturation threshold. (c) Amplitude of each Fourier component of such emission patterns as a function of the illumination pulse energy (log scale). Equivalently, the curves indicate the fraction of the image signal that stems from each harmonic of the illumination pattern. As the illumination energy increases, more and more harmonics come into play (lower curves). The calculations used the scalar model of light and were for steady-state illumination (a) and for the pulse length of 0.64 ns used in the experiments and a realistic fluorescent lifetime of 3.5 ns (b and c).

Resolution extension by nonlinear structured illumination. (a) The region of frequency space that is observable by conventional microscopy (compare with Fig. 2a). (b) An example of a sinusoidal illumination pattern. (c) That illumination pattern has three frequency components: one at the origin (black), representing the average intensity, and two at ±k₁, representing the modulation (dark gray). These are also the frequency components of the effective excitation under linear (i.e., nonsaturating) structured illumination. Under conditions of saturation, or other nonlinear effects, a theoretically infinite number of additional components appear in the effective excitation; the three lowest harmonics are shown here (light gray). (d) Observable regions for conventional microscopy (black), linear structured illumination (dark gray), and nonlinear structured-illumination microscopy (light gray) based on those three lowest harmonics. (e) Corresponding observable regions if the procedure is repeated with other pattern orientations. The much larger region of observable spatial frequencies (e) compared with that shown in a makes it possible to reconstruct the sample with correspondingly increased spatial resolution.

Verification of harmonic production through saturation. (a) Profiles through images of a thin fluorescent layer illuminated by a Gaussian laser beam modulated by sinusoidal stripes with a period of 2.5 μm, with a peak energy density per pulse of 0.58 mJ/cm² (bottom curve) and 37 mJ/cm² (top curve). The bottom curve approximately follows the sinusoidal illumination pattern, because the peak energy density is below the saturation regime. In the top curve, the higher pulse energy causes fluorescence to saturate near the peaks of the pattern, leading to an asymmetric curve form with broad peaks and sharp valleys. (That the observed valleys do not reach zero is expected because of blurring by the observation optics and does not necessarily imply that the actual intensity minima are nonzero.) (b) Fourier transforms corresponding to the profiles shown in a showing five detectable harmonics in the high-energy pattern (arrows). Only the lowest harmonic is detectable in the low-energy pattern (arrowhead). The vertical axis is logarithmic (base 10); the curves in b have been separated vertically for clarity.

A field of 50-nm fluorescent beads, imaged by conventional microscopy (a), conventional microscopy plus filtering (b), linear structured illumination (c), and saturated structured illumination using illumination pulses with 5.3 mJ/cm² energy density, taking into account three harmonic orders in the processing (d). Because no scanning is necessary, a wide field can be imaged simultaneously.

Lateral profiles through images of an isolated 50-nm red fluorescent bead, acquired by SSIM (solid trace) and conventional microscopy (dashed trace). (The structure along the baseline in the SSIM profile is caused by noise.) The average FWHM of 100 such SSIM profiles through different beads was 59 nm, a dramatic reduction compared with the 265-nm width for unprocessed conventional microscopy.