## Abstract

We propose a scheme for achieving widefield coherent anti-Stokes Raman scattering (CARS) microscopy images with sub-diffraction-limited resolution. This approach adds structured illumination to the widefield CARS configuration [Applied Physics Letters **84**, 816 (2004)]. By capturing a number of images at different phases of the standing wave pattern, an image with up to three times the resolution of the original can be constructed. We develop a theoretical treatment of this system and perform numerical simulations for a typical CARS system, which indicate that resolutions around 120 nm are obtainable with the present scheme. As an imaging system, this method combines the advantages of sub-diffraction-limited resolution, endogenous contrast generation, and a wide field of view.

©2010 Optical Society of America

## 1. Introduction

The importance of optical microscopy to biological research is universally recognized. Ideally, an imaging microscope should combine attributes including sub-diffraction-limited resolution for effective imaging of structures down to molecular scales, a field of view comparable to that of the ubiquitous confocal fluorescence microscope, and high endogenous contrast. While fluorescence microscopy relies on assumptions that the fluorescent tag is attached to the target species of interest, and that it does not perturb the behavior or chemistry of the system studied, endogenous contrast mechanisms carry no such inherent risks.

At best, most existing microscopy techniques offer only two of the above ideal attributes. A wide range of methods exist for achieving lateral resolution below the diffraction limit, the vast majority of which rely on fluorescent tagging of target species. Some techniques are, in fact, heavily dependent on the properties of photoswitchable fluorophores, namely photoactivated localization microscopy (PALM) [1,2] and stochastic optical reconstruction microscopy (STORM) [3]. Controlling the structure of excitation light fields, on the other hand, is key to stimulated emission depletion (STED) microscopy [4], and the various structured illumination methods (also known as patterned excitation microscopy) [5–7]. Typically achievable resolutions range between 16 and 70 nm for STED [8,9], and 50-120 nm for structured illumination microscopy [5–7].

Over the last ten or so years, coherent anti-Stokes Raman scattering (CARS) microscopy [10] has grown in popularity as an imaging technique offering intrinsic contrast from the target species, with no need to tag. This advantage comes at the price of meeting energy and phase-matching conditions for this four-wave-mixing process, of which the latter is the more onerous. In consequence, almost all CARS implementations involve a collinear phase-matching geometry, with signal generated from a point, which is then scanned to build up an image of the sample. The usefulness of CARS microscopy is limited, however, by the achievable spatial resolution, typically around 300 nm [11–13].

Very recently, several schemes have been proposed for reducing the collinear excitation spot to nanometer size, as is done in STED, to generate sub-diffraction-limited CARS images [14,15]. No experimental implementation of these schemes yet exists. More fundamentally, reducing the spot size leads necessarily to a trade-off between resolution and field of view, in order to maintain reasonable image acquisition timescales.

A much more promising route to super-resolved CARS microscopy is via the alternative widefield phase-matching geometry [16,17]. We propose using structured illumination methods with the widefield geometry to generate images with sub-diffraction-limited resolution, intrinsic contrast, and an approximately 50 μm field of view: a combination of all three ideal microscopy attributes.

Super-resolution is achieved by illuminating the sample with a phase-variable transverse volumetric standing wave and recording a series of images at different phases, which are then combined to yield a single sub-diffraction-limited image. The image processing is similar to structured illumination fluorescence microscopy [18], but more closely related to its coherent illumination counterpart [19].

## 2. Proposed scheme

A schematic diagram of the energy levels and excitation/signal beams involved in the CARS interaction is given in Fig. 1 . We recall that CARS is a four-wave-mixing process, involving excitation of the target species with a pump field ${\omega}_{\text{p}}$ and a Stokes field ${\omega}_{\text{S}}$, where the energy difference, $\Omega ={\omega}_{\text{p}}-{\omega}_{\text{S}}$, is tuned to a vibrational transition in the molecule. A third probe beam, here degenerate in frequency with ${\omega}_{\text{p}}$, incident on the molecule will be scattered by this virtual grating, generating an anti-Stokes field at frequency ${\omega}_{\text{as}}=2{\omega}_{\text{p}}-{\omega}_{\text{S}}$.

The fields must also satisfy a phase-matching condition $\left|\Delta \text{k}\right|\xb7l<<\pi $, where the phase mismatch is ${\text{\Delta k=k}}_{\text{as}}\text{-}\left({\text{2k}}_{\text{p}}{\text{-k}}_{\text{S}}\right)$, and *l* is the interaction length, i.e. the length over which a CARS signal will be generated. In the widefield geometry [16], $\left|\Delta \text{k}\right|$ is minimized by delivering the Stokes beam through the microscope objective, while the two pump beams impinge on the sample from below, at high angle, and from opposite directions to one another. The three laser beams are all located in a single plane, and overlap in the sample plane, which is at right angles to that containing the beams. CARS signal from an area of around 50 μm diameter is emitted anti-parallel to the incoming Stokes beam, and is imaged through the microscope objective onto a detector (camera). The widefield phase-matching configuration is shown in Fig. 2
, with the inset illustrating that $\left|\Delta \text{k}\right|$ is minimized.

As shown in Fig. 2, the fact that the pump beams counter-propagate in this phase-matching geometry makes them ideally suited to generating a phase-variable standing-wave field, as used in structured illumination microscopy. In brief, a structured excitation field acts to increase the spatial-frequency passband of the microscope, thus resulting in increased (or super-) resolution. The microscope passband is determined by the objective numerical aperture (NA) and the wavevector of the imaged light, as shown (in purple) in the centre of the frequency spectrum in Fig. 3 . In frequency space, the standing-wave field is represented by two shift frequencies $\pm \left({\text{k}}_{\text{px}}\right)/\pi $, where ${\text{k}}_{\text{px}}$ is the projected magnitude of the pump beam wavevector in the sample plane. Regions in frequency space, of the same size as the microscope passband, are convolved with the shift frequencies and appear in the image. These regions can be separated and moved back to their original frequency-space locations (shown in green in Fig. 3) by acquiring a number of images at different phases of the standing wave [7,18,19].

In the system proposed here, the standing wave frequency cannot be chosen arbitrarily, but is set by the species under investigation (which determines the energy condition) and the widefield geometry. It is not limited, however, by the microscope passband, unlike most structured illumination methods which image the standing wave onto the sample through the objective lens [5,6,18]. Thus we are free to choose the NA of our objective lens to produce the greatest possible resolution increase compatible with the energy and phase-matching conditions. As seen in Fig. 3, by choosing an objective with $\text{NA}={\text{k}}_{\text{px}}/{\text{k}}_{\text{as}}$, the original microscope passband just overlaps the shifted regions in the spatial-frequency spectrum; this corresponds to the condition for maximum increase of the effective (i.e. super-resolved) microscope passband. In this case, the passband is increased to three times its original size, resulting in a resolution three times the diffraction limit of the original system.

This is a greater resolution increase than the two times improvement obtainable with simple structured illumination fluorescence microscopy [7,18]. It is attributable both to the nonlinear nature of the CARS process, in which the pump beam term is squared, and to the fact that the standing wave frequency is not limited by the microscope passband. Note that an additional advantage is that the illumination wavelength is longer than that of the anti-Stokes signal beam.

Some increase in axial resolution should also result from this system, due to the inherent sectioning effect of using structured illumination [20]; this is not modeled in the present paper.

#### 2.1 Theoretical treatment

The following treatment sets out the theoretical basis for the claims outlined above. We take the equation for the intensity at the image plane of an in-focus thin object generated by coherent light [21]

where $F\left[g(m,n)\right]$ denotes the Fourier transform of the function*g*(

*m,n*), $c(m,n)$is the coherent transfer function (CTF) for an arbitrary microscope objective as given in Gu [21], and $o(x\text{'},y\text{'})$is the object function.

We take this last to be the CARS polarization

where*a*is some multiplicative constant, ${\chi}^{(3)}$ is the third-order susceptibility of the target transition in the species of interest,

*N*(

*x,y*) is the spatial distribution of target molecules across the sample, and

*E*and

_{p}*E*refer to the pump and Stokes fields respectively. Note that to show explicitly the dependence of

_{S}*o(x,y)*on the target species spatial distribution, ${\chi}^{(3)}$ in Eq. (2) is effectively the susceptibility per molecule. As constants, both

*a*and ${\chi}^{(3)}$ will henceforth be neglected in this analysis.

For simplicity, the present treatment assumes perfect phase-matching, and takes the Stokes field to be a plane wave in the negative z direction, with no transverse beam variation, and unity amplitude ${E}_{s}^{*}(x,y)=\text{exp}\left(i{\text{k}}_{\text{S}}z\right)$. (The imaged beam travels in the positive z direction up the objective lens.) We have also neglected the rapidly oscillating factor in $\text{exp}\left(i\omega \text{t}\right)$ in the expressions for the various fields, as these terms average to unity over detection timescales.

Furthermore, we consider a one-dimensional standing wave for the pump field, as shown in Fig. 2, such that ${E}_{p}(x,y)=\text{exp}\left[i{\text{(k}}_{\text{px}}x+{\text{k}}_{\text{pz}}z+\varphi /2)\right]+\text{exp}\left[i{\text{(-k}}_{\text{px}}x+{\text{k}}_{\text{pz}}z-\varphi /2)\right]$, with *ϕ* the phase difference between the two pump beams. As with the Stokes field, we take the beams to have unity amplitude and no transverse beam structure for added simplicity.

Substituting for the fields into Eq. (2), applying the convolution theorem, and performing the relevant Fourier transforms, we find the Fourier transform of the object function $F\left[o\left(x\text{'},y\text{'}\right)\right]=O(m,n)$to be

*N*(

*x,y*). Note that in Eq. (3) we have neglected the global phase factor which results from the combination of the z components of the pump and Stokes beams, as it goes to unity when the intensity is measured [19].

Equation (3) clearly shows the shift in spatial frequency which results from using structured illumination, with the spatial distribution of target molecules appearing three times, in two cases shifted by an amount proportional to the spatial frequency of the standing wave.

Substituting Eq. (3) into Eq. (1) and applying the convolution and shift theorems of Fourier transforms, we obtain an expression for the intensity at the detector

In Eq. (4) we see that the frequency shift has passed into the microscope coherent transfer function, or in other words, the shift affects the effective microscope passband. Rather than simply showing the molecular distribution at spatial frequencies contained in $c\left(m,n\right)$, the intensity at the detector contains contributions from two additional spatial frequency regions. As illustrated previously in Fig. 3, the centre of the transfer function has been duplicated at shifts of $\pm {\text{k}}_{\text{px}}/\pi $, meaning that additional spatial frequencies present in $N(m,n)$ appear in the image. Hence, for a correct choice of objective NA, the super-resolving system is capable of passing spatial frequencies in a region three times the usual system passband.

## 3. Image processing

An image processing scheme is necessary in order to generate the super-resolved image from a series of images at different phases of the standing wave. Due to the coherent nature of the CARS signal, this scheme varies substantially from those used in fluorescent structured illumination microscopy [7,18]. It is analogous to that reported in [19], but modified by the nonlinearity of CARS.

We know that the super-resolved image will be constructed from a number of images at different phases of the standing wave. Rearranging Eq. (4) into sinusoidal and cosinusoidal components, we obtain

As there are five terms in Eq. (5), a minimum of five images at appropriate phases of the standing wave are necessary to isolate the coefficients A-E. The simplest means is to use images at phases equally spaced around the unit circle; if the frequency of the standing wave is known, A-E can then be isolated as shown in Eq. (12)-(16) in Appendix 1. Note that we have made no assumptions about the distribution of target molecules *N(x,y)* in arriving at this point.

The super-resolved image is to be constructed from a combination of the coefficients A-E. To determine the appropriate combination, we compare the information contained in these coefficients to the image intensity that would be obtained from a system illuminated by unstructured light but with an increased passband in one dimension (see Eq. (17) in Appendix 2). Effectively, our super-resolved technique mimics such an ‘ideal’ system. This approach was previously used by Littleton *et al.* [19].

We find that combining the coefficients as

## 4. Results and discussion

To demonstrate the increase in resolution obtainable with the proposed super-resolved CARS system (superCARS), we perform numerical simulations in one dimension (the super-resolved dimension) of the system point-spread function (PSF). The full-width half maximum (FWHM) of the PSF is a common measure of resolution. The simulations are adapted from the method of Ref [19], and model the system for the simplified conditions used in the above theoretical development (such as plane-wave beams, etc.).

In order to give concrete typical resolution values, we simulate the ubiquitous CARS test case of the 3050 cm^{−1} transition in polystyrene, and use a typical Stokes beam wavelength of 1064 nm. All other parameters (such as the pump beam wavelength, phase-matching angle, etc.) are set by the energy and phase-matching conditions. We choose the simulated objective NA to give the maximum resolution increase (i.e. three times) as $\text{NA}={\text{k}}_{\text{px}}/{\text{k}}_{\text{as}}$, which in this case is equal to 0.78 for an air objective.

For comparison, we use the same parameters to simulate a standard widefield CARS system, with the same microscope passband (NA = 0.78), and the ‘ideal’ comparison system discussed above ($\text{NA=3}\times \text{0 .78}$). The PSFs for all three cases are displayed in Fig. 4 . While a standard CARS system would give a lateral resolution of 362 nm, the super-resolved system has a resolution of 122 nm, as does the equivalent ‘ideal’ comparison case. Clearly, the proposed superCARS system has a resolution increase of three times over standard CARS imaging on the same system.

A resolution increase of three times is the maximum possible with the superCARS method; for the parameters used here, this only occurs for an NA of 0.78. Use of a higher NA objective lens (as is usual) would permit a larger total passband, in absolute terms, leading to higher resolution overall. The resolution increase compared to standard CARS imaging, however, would be less, although would still be at least a factor of two. Use of an objective lens with an NA very much higher than the equality condition ($\text{NA}={\text{k}}_{\text{px}}/{\text{k}}_{\text{as}}$) would involve additional complexity in the image processing, as it implies overlap of the three frequency-space regions contributing to the image. When $\text{NA}\approx {\text{k}}_{\text{px}}/{\text{k}}_{\text{as}}$, these regions are distinct; this is the case discussed in this paper.

A further demonstration of the improvement of the proposed method over standard CARS imaging is provided by Fig. 5 . This depicts the simulated intensity distribution obtained for a sample of randomly distributed 100 nm diameter polystyrene beads under super-resolved and standard CARS imaging. As before, the figure shows the situation in one dimension only, such that the ‘beads’ appear as top-hat functions. The superCARS system resolves many beads where standard CARS would fail to do so, although resolution will also ultimately be limited by the signal-to-noise ratio of the system.

## 5. Conclusion

We have proposed a scheme for adding super-resolution capability to widefield CARS microscopy, as a means to obtaining widefield microscopy images with sub-diffraction-limited resolution and endogenous contrast. It is the configuration of the two pump beams which enables super-resolved imaging. The beams are made to be coherent with one another and to interfere in the sample plane, forming a standing wave whose phase may be varied. At least five images at standing-wave phases equally spaced around the unit circle are combined, with knowledge of the standing-wave frequency, to construct a single super-resolved image. A theoretical treatment and computer simulations show that a three times lateral resolution increase is possible with this system, compared to the same system in standard CARS mode. For the typical CARS test species, polystyrene, and typical illumination wavelengths, this corresponds to a resolution of 122 nm. Higher resolutions may be obtainable using higher NA objective lenses than in our simulations, at the price, however, of increased complexity in the image processing. The proposed scheme constitutes an important step in the development of optical microscopy tools for biological research.

## 6. Appendix 1

The coefficient expressions from Eq. (5) are as follows

where ${F}^{*}\left[g(m,n)\right]$ represents the complex conjugate of the Fourier transform of *g(m,n)*.

The various coefficients may be isolated from images at phases spaced equally around the unit circle as shown below

where *n* is the total number of equally spaced images used and $n\ge 5$.

## 7. Appendix 2

We develop an expression for the image intensity arising from a CARS system without structured illumination, but with three times the microscope passband.

Eq. (17) follows from Eq. (1), where we have used *N(x,y)* as the object function, and $\text{c'}(m,n)=\text{c}(m,n)+\text{c}(m+{\text{k}}_{\text{px}}/\pi ,n)+\text{c}(m-{\text{k}}_{\text{px}}/\pi ,n)$ as the coherent transfer function. We have performed Fourier transforms, and applied the convolution theorem as appropriate.

It is immediately clear from comparison with Eq. (8) and (10) that the second term in curly brackets in Eq. (17) is equal to ${\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\text{B}$, while the third term is equal to D. The first term in Eq. (17) resembles A of Eq. (7), with different weighting, however, of the constituent terms.

Noting that the microscope CTF is real and even in spatial frequency space, and that the spatial distribution of target molecules *N(x,y)* is real, we find that $F\left[\text{c}(m\pm {\text{k}}_{\text{px}}/\pi ,n)N(m,n)\right]={F}^{*}\left[\text{c}(m\mp {\text{k}}_{\text{px}}/\pi ,n)N(m,n)\right]$. Hence, ${\left|F\left[\text{c}(m+{\text{k}}_{\text{px}}/\pi ,n)N(m,n)\right]\right|}^{2}={\left|F\left[\text{c}(m-{\text{k}}_{\text{px}}/\pi ,n)N(m,n)\right]\right|}^{2}$. We thus deduce that the first term in curly brackets in Eq. (17) is equivalent to ${\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\text{A+}{\scriptscriptstyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\sqrt{{\text{D}}^{2}+{\text{E}}^{2}}$. Consequently, ${I}_{\text{super-resolved}}(x,y)={\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\text{A+}{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\text{B+D+}{\scriptscriptstyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\sqrt{{\text{D}}^{2}+{\text{E}}^{2}}={I}_{\text{ideal}}(x,y)$, as given in Eq. (6).

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