**5.1 Methods of improving spatial resolution of CARS microscopy**

As well known, there is a theoretical limitation of the spatial resolution for any far-field optical microscopes because of the existence of light diffraction. Ernst Abbe defined the diffraction limit as [124]:

$$d = \frac{0.61\lambda}{n\sin\phi} \cong \frac{\lambda}{2NA} \,\, \,\,\tag{5.1}$$

where *d* is the resolvable minimum size, λ is the wavelength of incident light, n is the refraction index of the medium being imaged in, φ is the aperture angle of the lens, and NA is the numerical aperture of the optical lens. It is obvious that for an optical microscope, d is the theoretical limit of spatial resolution. The samples' spatial features, smaller than approximately half the wavelength of the used light, would never be able to be resolved.

In recent years, in order to meet the requirements on the study of life science and material science, ones have found several ways to overcome the optical diffraction limit and obtained sub-diffraction limited spatial resolution theoretically. In fluorescence microscopy, the success of the resolution enhancement techniques relies on the ability to control the emissive properties of fluorophores with a proper optical beam. The most important developments for breaking through the diffraction barrier are sub-diffraction-limited resolution fluorescent imaging techniques, such as photo activated localization microscopy (PALM) [125], stochastic optical reconstruction microscopy (STORM) [126], and stimulated emission depletion (STED) microscopy [127, 128], which have opened up notable prospect for sub-cellular structure and bio-molecular movement and interaction imaging.

As one of label-free nonlinear imaging techniques, the spatial resolution of CARS microscopy is higher (about 300nm lateral resolution) than the one of traditional linear optical microscopy, but it is still a diffraction-limited imaging technique. Today, how to achieve a sub-diffractionlimited CARS microscopy has become one of attractive topics all over the world. Compared with developments of the fluorescence nanoscopy, the method for breaking through the diffraction limitation in CARS microscopy is still under theoretical research.

In 2009, Beeker et al. firstly presented a way to obtain a sub-diffraction-limited CARS microscopy in theory [129]. With the density matrix theoretical calculations, they found that

Ultra-Broadband Time-Resolved Coherent Anti-Stokes Raman Scattering

theory.

method [131].

molecular vibrational mode.

broadband T-CARS microscopy.

**5.2 Phonon depletion CARS microscopy[132]**

Spectroscopy and Microscopy with Photonic Crystal Fiber Generated Supercontinuum 197

Kim M. Hajek et al presented a theoretical analysis and simulation of a wide-field CARS microscopy with sub-diffraction-limited resolution in 2010 [131]. The configuration and a simulation result are shown in figure 21. In this method, two coherent pump beams were used and interfered in the sample plane, forming a standing wave with variable phase. The numerical simulation showed that a super-resolved image with three times better lateral resolution could be obtained by image processing method of standing-wave frequency

Fig. 21. (a) Schematic of stand-wave wide-filed CARS microscopy, (b) phase-matching condition, (c) simulation result of the point spread function (PSF) of the proposed

All above discussed sub-diffraction-limited CARS microscopy open up the possibility of achieving sub-diffraction-limited CARS microscopy. Unfortunately, these approaches can only be used in the single frequency CARS microscopy based on the signal of a single

As discussed above, successful methods of breaking through the theoretical diffraction limitation of CARS microscopy depend on the controllable emissive properties of the useful signals in the focus. But the above suggested methods for breaking through the diffraction limited resolution can only be used for dealing with the single bond signal. By researching the CARS process with quantum optics theory, we presented our method for breaking through the diffraction limitation, unlike the above methods, which is effective for ultra-

In our theoretical model, all incident laser fields, generated signal field and the material system are all described with quantum mechanics theory. In the CARS process, the first light-matter

the molecular vibrational coherence in CARS can be strongly suppressed by using an annular mid-infrared laser to control the pre-population of the corresponding vibrational state. The energy level diagram is shown in figure 19. Thereby the emission of generated CARS signals in the annular area of point spread function could be significantly suppressed and the spatial resolution can be improved considerably.

Fig. 19. Energy level diagram for CARS extended with an additional level 4 . Energy level 1 - 4 are ground state, vibrational state, excited vibrational state and control state, respectively. ωP, ωpr, ωS, and ωctrl are the frequencies of pump, probe, Stokes and control laser [129].

Alexei Nikolaenko et al provided their CARS interferometric theory in the same year [130]. In this theoretical research, a stabilized, phase-adjustable interferometer was used to achieve nearly complete interference between the local oscillator and the pump- and Stokes-induced CARS radiation. The schematic of the CARS interferometry setup is shown in figure 20. Their theoretical analysis showed that the energy loss in the anti-Stokes channel is accompanied by an energy gain in the pump and Stokes channels. This implied that the CARS interferometry provided a controllable switching mechanism of anti-Stokes radiation from the focal volume, which might be a possible technique for improving the spatial resolution of the CARS microscopy.

Fig. 20. Schematic of the CARS interferometry. ES, error signal; LO, unit for generating local oscillator; WP, wedged plate; BS, 50-50 beam splitter; DM, dichroic mirror; FB, optical feedback signal; MO, microscope objective; BPF, bandpass filter; Cond., condenser; PMT, photomultiplier [130].

the molecular vibrational coherence in CARS can be strongly suppressed by using an annular mid-infrared laser to control the pre-population of the corresponding vibrational state. The energy level diagram is shown in figure 19. Thereby the emission of generated CARS signals in the annular area of point spread function could be significantly suppressed

Fig. 19. Energy level diagram for CARS extended with an additional level 4 . Energy level 1 - 4 are ground state, vibrational state, excited vibrational state and control state,

respectively. ωP, ωpr, ωS, and ωctrl are the frequencies of pump, probe, Stokes and control laser [129].

Alexei Nikolaenko et al provided their CARS interferometric theory in the same year [130]. In this theoretical research, a stabilized, phase-adjustable interferometer was used to achieve nearly complete interference between the local oscillator and the pump- and Stokes-induced CARS radiation. The schematic of the CARS interferometry setup is shown in figure 20. Their theoretical analysis showed that the energy loss in the anti-Stokes channel is accompanied by an energy gain in the pump and Stokes channels. This implied that the CARS interferometry provided a controllable switching mechanism of anti-Stokes radiation from the focal volume, which might be a possible technique for improving the spatial

Fig. 20. Schematic of the CARS interferometry. ES, error signal; LO, unit for generating local oscillator; WP, wedged plate; BS, 50-50 beam splitter; DM, dichroic mirror; FB, optical feedback signal; MO, microscope objective; BPF, bandpass filter; Cond., condenser; PMT,

and the spatial resolution can be improved considerably.

resolution of the CARS microscopy.

photomultiplier [130].

Kim M. Hajek et al presented a theoretical analysis and simulation of a wide-field CARS microscopy with sub-diffraction-limited resolution in 2010 [131]. The configuration and a simulation result are shown in figure 21. In this method, two coherent pump beams were used and interfered in the sample plane, forming a standing wave with variable phase. The numerical simulation showed that a super-resolved image with three times better lateral resolution could be obtained by image processing method of standing-wave frequency theory.

Fig. 21. (a) Schematic of stand-wave wide-filed CARS microscopy, (b) phase-matching condition, (c) simulation result of the point spread function (PSF) of the proposed method [131].

All above discussed sub-diffraction-limited CARS microscopy open up the possibility of achieving sub-diffraction-limited CARS microscopy. Unfortunately, these approaches can only be used in the single frequency CARS microscopy based on the signal of a single molecular vibrational mode.
