Coherent nonlinear single-molecule microscopy

We investigate a nonlinear localization microscopy method based on Rabi oscillations of single emitters. We demonstrate the fundamental working principle of this technique using a cryogenic far-field experiment in which subwavelength features smaller than $\lambda /10$ are obtained. Using Monte Carlo simulations, we show the superior localization accuracy of this method under realistic conditions and a potential for higher acquisition speed or a lower number of required photons as compared to conventional linear schemes. The method can be adapted to other emitters than molecules and allows for the localization of several emitters at different distances to a single measurement pixel.

Figure 1

(a) Experimental setup with two cascaded acousto-optic modulators (AOMs) to obtain short laser pulses with a high signal to background ratio. (b) The optical setup inside the cryostat with a solid-immersion lens (SIL). (c) The level scheme of a single molecule.

Figure 2

Intensity dependence of the response from a single molecule in temporally integrated detection. Pulse duration was 4 ns, repetition rate was 700 kHz, and the intensity was integrated over 100 ms. The raw experimental data is shown in blue, whereas a theoretical fit is shown in red. Figure adapted from Ref. [12].

Figure 3

Experimental raster-scanned images of a single molecule without (a) and with (b) pulsed excitation (4-ns pulses, 700-kHz repetition rate). The effectively larger PSF of the illumination spot due to the field dependence is visible. The rings around the central spot have widths of $\approx 40$ nm.

Figure 4

Simulated raster-scanned image of a single molecule and line scans. (a) The usual linear optical response of a single molecule. (b) If the excitation light is pulsed and the pulse width is shorter than the singlet $T_1$ time of the molecule, several rings appear, representing Rabi oscillations. [(c) and (d)] Cross sections of (a) and (b).

Figure 5

Basic derivation of pixelated images for Monte Carlo estimations. (a) Calculated PSF in the absence of amplitude damping and detector noise (analytical). (b) Numerically estimated pixel distribution for 1000 detected photons. (c) Line cuts for pixel distribution and fitted PSF.

Figure 6

Localization accuracy for different numbers of detected photons. Gaussian fitting (upper red curve) and coherent Rabi imaging microscopy (lower blue curve). Both curves scale as $1/\sqrt{N}$, as expected for pure shot-noise limited localization performance (background noise is neglected in these calculations). The images used for fitting are shown above, labeled with the number of detected photons. (Upper row) CORIM; (lower row) assumed Gaussian PSF. For details of this simulation see text.

Figure 7

Localization accuracy for Gaussian localization (red curve) and for coherent Rabi imaging microscopy (blue curve) as a function of the corresponding pulse area (top abscissa). With increasing pulse strength, the Gaussian PSF widens first and reduces the localization accuracy by increasing the effective spot size. At an intensity of about $2\pi$, the localization gets better than linear imaging. The oscillations for higher field strength are explained by opening and closing the inner spot of the PSF. The images used for fitting are shown above, labeled with the value of $I_0$, which effectively changes the pulse area. (Upper row) CORIM; (lower row) assumed Gaussian PSF. See text for details.

Figure 8

Localization accuracy for different pixel sizes. The coherent Rabi imaging microscopy (blue curve) has a higher stability against fitting. Note that background noise is neglected such that only pixelation noise is present. Example images for fitting are shown above with the corresponding pixel size in nm. (Upper row) CORIM; (lower row) Gaussian PSF. For details of this simulation see text.

Figure 9

(a) Simulated image of two molecules located at a distance of 20 nm experiencing the same effective pulse area with the same illumination strength. The concentric rings with subwavelength features can be easily differentiated. (b) (Blue) A line cut through the center of the PSF; (red, dashed) the Gaussian intensity profile of the illuminated area.

Figure 10

(a) Simulated image of four molecules (red dots), located at at small displacements from the measurement point, indicated by the green cross. (b) Electric field amplitude at the green cross as a function of the field strength. (c) Fourier transform of the field amplitude shown in (b). The four emitters are well resolved although the field variation at only a single pixel is taken into account. If the inhomogeneous broadening of the Rabi response is negligible, the distance of all emitters to the measurement spot can be determined.