Ground-state depletion for subdiffraction-limited spatial resolution in coherent anti-Stokes Raman scattering microscopy

We theoretically investigate ground-state depletion for subdiffraction-limited spatial resolution in coherent anti-Stokes Raman scattering (CARS) microscopy. We propose a scheme based on ground-state depopulation, which is achieved via a control laser light field incident prior to the CARS excitation light fields. This ground-state depopulation results in a reduced CARS signal generation. With an appropriate choice of spatial beam profiles, the scheme can be used to increase the spatial resolution. Based on the density matrix formalism we calculate the CARS signal generation and find a CARS signal suppression by 75$\%$ due to ground-state depletion with a single control light field and by using two control light fields the CARS signal suppression can be enhanced to 94$\%$. Additional control light fields will enhance the CARS suppression even further. In case of a single control light field we calculate resulting CARS images using a computer-generated test image including quantum and detector noise and show that the background from the limited CARS suppression can be removed by calculating difference images, yielding subdiffraction-limited resolution where the resolution achievable depends only on the intensity used.

Figure 1

Energy diagram of the CARS process. The ground state $|1\rangle$ is initially fully occupied. State $|2\rangle$ is a Raman-active vibrational state and $|3\rangle$ is the upper electronic state acting as control state. An additional control state is included by adding an infrared-active vibrational state $|4\rangle$. A control light field resonant with the vibrational state $|4\rangle$ or the electronic level $|3\rangle$ is used to reduce the ground-state population and, thereby, to suppress the CARS signal generation.

Figure 2

(a) Population of the states $|1\rangle$ (black, ${\rho }_{11}$), $|2\rangle$ (blue, ${\rho }_{22}$), $|3\rangle$ (red, ${\rho }_{33}$), and $|4\rangle$ (green, ${\rho }_{44}$) as a function of the control laser intensity $I_{c,el}$ in resonance with the $|1\rangle$–$|3\rangle$ transition. (b) The population of the levels as a function of $I_{c,vib}$ in resonance with $|1\rangle$–$|4\rangle$ transition. (c) CARS intensity as a function of the control laser intensity $I_c$ resonant with the $|1\rangle$–$|3\rangle$ (black solid curve) or $|1\rangle$–$|4\rangle$ (red dashed curve) transition.

Figure 3

CARS intensity as a function of the control intensity for three cases. (a) Two simultaneously irradiated control light fields, one resonant with the transition to the electronic state ($|1\rangle –|3\rangle$) and one resonant with the transition to the vibrational state ($|1\rangle –|4\rangle$) leading to a suppression of the CARS signal by 90$\%$ (black solid curve). (b) A subsequent irradiation of two control light fields, the first one resonant with the $|1\rangle –|3\rangle$ transition and the second one resonant with the $|1\rangle –|4\rangle$ transition, irradiated 20 ps delayed to each other resulted in a ground-state depletion of 75$\%$ and a CARS signal suppression of 94$\%$ (red dotted curve). (c) For comparison a single control light field resonant with the $|1\rangle –|3\rangle$ transition, suppressing the CARS signal by 75$\%$ (blue dashed curve).

Figure 4

(a) Population difference between the vibrational state $|4\rangle$ and the Raman state $|2\rangle$ for different intensities of the control beam as a function of the nonradiative transition rate $R_{42}$ between the vibrational state $|4\rangle$ and the Raman state $|2\rangle$; (b) resulting depletion of the CARS signal. (c) Normalized CARS intensity as a function of the control light field intensity for a nonradiative transition rate $R_{42}$ of 0 THz (black solid curve) and $0.1$ THz (red dashed curve).

Figure 5

(a) Normalized CARS intensity as a function of the control light field intensity for coherence lifetimes ${\Gamma }_{41}$ of the vibrational state $|4\rangle$ of 50, 100, 200, 400, 1000, and 5000 fs. (b) The saturation intensity of the CARS signal suppression decreases with increasing coherence lifetime ${\Gamma }_{41}$.

Figure 6

Cross sections of the spatial beam profiles of the pump (red), Stokes (blue), probe (green), and the two control (black) light fields. The inset shows the phase plate used to create the donut-shaped beam profile of the control light fields.

Figure 7

(a) By irradiating a control light field with a donut-shaped spatial profile the CARS excitation profile could be narrowed significantly below the diffraction limit. The achieved width depended directly on the used control light field intensity. The limited suppression of 75$\%$ leaves a significant diffraction-limited background. (d) Using two subsequent control light fields the CARS excitation profiles were similarly narrowed but the background was significantly lower. Two-dimensional illustrations are shown in (b) and (e) for an excitation without control light field and in (c) and (f) for the narrowest excitation profile of each case [(a) and (d) red curve].

Figure 8

FWHM (black solid line) of the complete excitation profile (corresponding to the achievable lateral resolution) and FWHM of the pedestal (blue dashed line) of the CARS excitation profiles with a single control light field [Fig. 7a] as function of the donut peak intensity of the control light field. The width of the pedestal saturates at the diffraction-limit of a CARS excitation which will create a diffraction-limited background below a subdiffraction-limited CARS image.

Figure 9

(a) Test sample simulating fine structure of e.g. brain tissue. By calculating the convolution of the test image with the excitation profile, a CARS image was obtained for the case that (b) no control light field was applied and the case (c) that a control light field was applied that resulted in an excitation profile with a FWHM of 18 nm shown in Fig. 7c. While in (b) no subdiffraction-limited features are resolved in (c) the fine structure of the test sample is visible on top of the diffraction-limited background. The background can be removed by calculating a difference image between (b) and (c). Due to the coherent nature of the CARS process (d) a difference image from intensity images generates aberrations in the boundary area in contrast to (g) a difference image from the electric field amplitude. As experimental images always contain noise both cases have been considered with (d) and (g) quantum noise only (maximum photon number of ${10}^5$) and an additional detector noise of (e) and (h) 0.58$\%$ rms and (f) and (i) 2.31$\%$ rms.