1. Introduction

The visual perception of the human eye is particularly well suited for the recognition of interfaces, i.e. the sharp transition between bright and dark areas, for general structure and features identification. Taking advantage of this observation, various microscopy techniques were developed to highlight frontiers between areas of different refractive indices such as dark-field (DF), differential-interference-contrast (DIC) [1,2] or the Nobel price honored phase-contrast (PC) microscopy [3]. These techniques date back to the beginning of microscopy itself (DF [4]) or are aged almost a century (PC, DIC) but still belong to the standard configurations of basic state-of-art light microscopes contributing to numerous scientific breakthroughs.

In the context of nonlinear microscopy, the second-(SHG) [5,6] and third-harmonic generation (THG) contrast mechanisms [7,8] are well known for their ability to highlight concentration gradients and interfaces. More precisely, SHG requires the absence of centro-symmetry within the illuminated area, and is otherwise forbidden as for all $\chi ^{(2)}$ processes. This property is evidently given for any matter interface placed within the focal volume. As a $\chi ^{(3)}$ process, THG is permitted in homogeneous media but requires a rarely given anomalous dispersion of the refractive index, i.e. $n_{\lambda }\geq n_{3\lambda }$ [7]. Conversely, if the sample is structured axially or transversely, THG radiation can be detected under normal dispersion conditions [7]. SHG and THG microscopy are straightforward to implement experimentally, since they require only one laser beam, and can be detected simultaneously [9]. However, the SHG signal at interfaces is weak and SHG microscopy has been mostly applied to non-centro-symmetric bulk samples such as collagen [6,10]. THG microscopy is usually implemented with an excitation laser in the 1200-1500 nm range to enable for the detection of a visible THG radiation wavelength at $\lambda /3$. Practically, efficient SHG and THG microscopy require pulsed fs-laser sources at power levels that are often close to the sample damage threshold. Other three and four wave mixing processes, such as sum frequency microscopy [11] (SFM, $\chi ^{(2)}$) and third-order sum frequency microscopy [12] (TSFM, $\chi ^{(3)}$), have also been implemented to probe molecular vibrational modes using 2 different excitation wavelengths. In this case, a mid-IR beam is tuned to be in resonance with a certain vibrational level to introduce chemical sensitivity and amplify the otherwise non-resonant SFM or TSFM wave-mixing. The second NIR-beam is used to blue-shift the resulting radiation via a $\chi ^{(2)}$ or $\chi ^{(3)}$ process into the detection range of commonly used single-element detectors (PMTs).

Similar to SFM and TSFM, coherent anti-Stokes Raman scattering (CARS) microscopy [13] enables to image molecular vibrational bonds but uses only near-IR beams that are less prone to be absorbed by water. In non-degenerated CARS, the pump, probe and Stoke beams with angular frequency $\omega _{p1}$, $\omega _{p2}$ and $\omega _{S}$, respectively, are combined to address a molecular vibration $\Omega =\omega _{p1}-\omega _{S}$ generating anti-Stokes radiation at $\omega _{as}=\omega _{p1}+\omega _{p2}-\omega _{S}$. Note that most of the work reported so far in CARS microscopy uses a 2 colours degenerated scheme where a single beam acts as the pump and probe, $\omega _{p1}=\omega _{p2}$. CARS is not in general an interface sensitive contrast mechanism but probes a wide range of the sample $K$-support [13,14]. With some modifications in the excitation or detection path, however, CARS microscopy becomes edge sensitive. Improving the image contrast Lin et al. used an annular mask in the detection path to suppress the anti-Stokes radiation from the homogeneous part of the sample propagating predominantly along the principle optical axis [15]. Further, Gachet et al. investigated the radiation pattern of chemical interfaces [16,17] and suggested a scheme where the anti-Stokes far-field radiation is measured for different angles to suppress the non-resonant background by a subtraction of signals [18–21]. Instead of filtering or analyzing the detection path, Krishnamachari et al. investigated the impact of phase modulation pattern within the excitation path also referred to as focus engineering [22–24]. It was shown that tuning the phase profile of the excitation allows to highlight chemical interfaces in x-,y- or z-direction [22]. The fidelity of detection path methods [15–17] decreases, however, with increasing scattering strength of the material affecting the propagation direction of the anti-Stokes radiation. On the contrary, focus engineering methods using phase-masks [22–24] are less sensitive to the impact of scattering of the anti-Stokes photons but any optical density difference in the excitation path before the focus can cancel any interface sensitivity limiting both approaches to thin or optically homogeneous samples. Thus, the developed interface highlighting CARS approaches can not yet fully compete with SHG or THG microscopy.

Here, we present an alternative approach to highlight chemical interfaces by shaping the k-space profile (or equivalently the amplitude profile of the excitation pupil of the microscope objective) of the 3 contributing pump, probe and Stokes beams. Due to this specific illumination geometry the radiation of any homogeneous sample ($\textbf {K}$(0,0,0)=1 else $\textbf {K}$=0, where $\textbf {K}$ denotes the sample spatial wavevectors) is suppressed, i.e. does not find any phase matched combination of the incident excitation k-vectors of the pump ($\textbf {k}_{p1}$), Stokes ($\textbf {k}_{S}$) and probe ($\textbf {k}_{p2}$) beams. We call this approach dark-field (DF-) illuminated CARS as being sensitive to interfaces only. Further, any phase variation arising from different optical path length for the pump, probe and Stokes beam does not affect its sensitivity towards interfaces. Within this paper, we investigate the object’s spatial frequency support $\textbf {K}$ that is probed for various excitation pupil function arrangements on grounds of the Ewald sphere of the anti-Stokes radiation and the effective excitation wave vector given by the sum of the pump plus probe minus the Stokes wave-vector $\textbf {k}_{\textrm {eff}}=\textbf {k}_{p1}+\textbf {k}_{p2}-\textbf {k}_{S}$. Then, we implement a vectorial model using the Debye-Wolff integral and dynadic Green function to model a proposed experiment.

2. Proposed experimental realization

Figure 1 displays one hypothetical realization of a DF-CARS experimental setup following our previous coherent Raman implementations [25,26]. A powerful laser source is partially used as Stokes beam and is frequency-doubled to pump two independently tunable optic parametric oscillators (OPO) that serve as pump and probe beam to drive a 3-color CARS process. For the 2-color version only one OPO is required. All 3(2) beams are superimposed in space and time just as in a regular laser scanning CARS scheme and the anti-Stokes radiation is detected in forward direction. As the only novelty, some razor edges are added to truncate the pump, probe and Stokes beams independently in the lateral direction (see also insets in Fig. 1). To avoid any blurring of the apertures while the laser propagate, the razor edges are imaged by a 2 lens 4f-combination onto the laser scanning mirrors which themselves are imaged by the tube and scan lens to the back-focal plane of the objective lens (pupil aperture). Evidently, the shaping of the pupil transfer could be also achieved by spatial light modulators to gain in flexibility. Simpler but less flexible, dielectric optical filter placed at the back-aperture of the objective lens could be used as a 3$^{\textrm {rd}}$ alternative to generate the desired amplitude modulation of the pupil function. As displayed in the lower part of Fig. 1, different 2- or 3-color masks placed in the back-aperture of the excitation objective (amplitude pupil transfer functions) are useful to probe different K-vector distributions of the object as we show below.

 

Fig. 1. Pupil filter and proposed experimental setup: a) 2- and 3-color DF-CARS pupil filters are displayed highlighting interfaces along the x-, y- or z-axis. b) 1 pump laser, 2 optical parametric oscillators, 3 razor edges, 4 4f-telescope, 5 laser scanning microscope, 6 photo-electron multiplier tube. The right inset shows the beam preparation magnified. Note that the pump, probe and Stokes beam do not overlap at the entrance of the objective lens but leave a gap in-between them to avoid probing K(0,0,0) with any illumination k-vector combination.

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3. Object spatial frequency support

In the plane wave regime, momentum conservation for the object spatial frequency $\textbf {K}$ is given by $\textbf {K}=\textbf {k}_{p1}+\textbf {k}_{p2}-\textbf {k}_{S}-\textbf {k}_{aS}$ where $\textbf {k}_{p1}$, $\textbf {k}_{p2}$, $\textbf {k}_{S}$ and $\textbf {k}_{aS}$ are the wave-vectors of the pump-, probe-, Stokes- and anti-Stokes radiation. The derivation of this results can be found in [27]. Furthermore, the effective excitation k-vectors can be derived from the pump, probe and Stokes k-vector as $\textbf {k}_{\textrm {eff}}=\textbf {k}_{p1}+\textbf {k}_{p2}-\textbf {k}_{S}$ and the probed sample K-vector can be simply derived as $\textbf {K}=\textbf {k}_{\textrm {eff}}-\textbf {k}_{aS}$, a situation equivalent to the linear scattering case. For the latter, only the incident angle of the excitation k-vector can be tuned but not the its absolute value. In non-linear CARS microscopy, however, $\textbf {k}_{\textrm {eff}}$ can now take different angles and absolute values (|$\textbf {k}_{\textrm {eff}}$|) depending on how $\textbf {k}_{p1}$, $\textbf {k}_{p2}$ and $\textbf {k}_{S}$ are arranged. This flexibility of $\textbf {k}_{\textrm {eff}}$ is central to perform DF-CARS.

Focused beams can be considered as the coherent superposition of a continuum of plane waves with their k-vectors located on a cap of a sphere - see also Eq. (1). Truncating the transmission of the excitation pupil function (back aperture of the objective lens) also truncates the k-vectors cap of the sphere. Note that we neglect at this heuristic level polarization effects that would lead to an alteration of the cap’s amplitude for high incident angles for the x-,y- and z-polarization contributions. However, polarization effects will be fully taken into account in the numerical section below.

In the case of CARS microscopy, we have to consider all possible combinations of incident pump, probe, Stokes and outgoing anti-Stokes k-vectors (plane waves) to find the object $\textbf {K}$-space that is probed. Finding all combinations is mathematically equivalent to the convolution operation of these pump, probe and Stokes cap of spheres or arcs in 2D (where the k-vectors point to). This convolutions is performed step-wise in Fig. 2 schematically (pump: green, Stokes: red, probe: yellow, anti-Stokes: blue). For illustration purposes, we neglected the full convolution, i.e. take one cap of a sphere as a brush and use it to retrace the other cap, but we selected only 3 points (left, center, right) of each arc to estimate the final object spatial frequency support.

 

Fig. 2. Spatial frequency support in conventionally illuminated CARS-microscopy (a) and DF-CARS (b, c, d, e). The gray ellipse outlines the probed area in the object K-space depending on the illumination k-space configuration for the pump $\textbf {k}_{p1}$ (green), Stokes $\textbf {k}_{S}$ (red), probe $\textbf {k}_{p2}$ (yellow) beams. The anti-Stokes radiation $\textbf {k}_{aS}$ (blue) is detected in forward direction. Configuration a, b, c are 3 colors cases; configurations d, e are 2 colors cases. Note that the gray area for DF-CARS does not overlap with the K(0,0,0).

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In Fig. 2(a), the result from Hashimoto et al. [14] of conventional laser scanning microscope (LSM) CARS is sketched. As obvious from Fig. 2(a), LSM-CARS probes an object spatial frequency support around K(0,0,0) (which also explains why a homogeneous sample gives a CARS signal). Interestingly, the probed object K-support is larger than for linear scattering microscopy and elongated along the K$_x$ axis. As a result, LSM-CARS provides a better resolution in lateral (K$_x$) direction than in axial (K$_z$) direction and is generally better resolved than linear scattering LSM assuming that the same excitation wavelength are applied. From the construction displayed in Fig. 2(a) it also appears that there is no ’missing cone’ in CARS as compared to linear scattering [28], a property that is beneficial to CARS tomography and ptychography reconstructions [27]. Flipping the direction of the anti-Stokes vectors in Fig. 2(a) would result the object support for Epi-CARS [29] which is, therefore, centered around K$_z\approx$ 2k$_{aS}$. Unfortunately, Epi-CARS images are often dominated by forward directed CARS photons that were back-scattered by the sample or object holder.

Having understood the significance of the CARS transfer functions (i.e. the CARS probed spatial frequency support $\textbf {K}$ of the object), our proposed DF-CARS schemes derives from selecting suitable wave-vector combinations of the incident pump, probe and Stokes beams so that no anti-Stokes radiation is detected in forward direction for the object frequency $\textbf {K}$(0,0,0). Figure 2(b) sketches the situation depicted in Fig. 1(a) for mask 1. The distributions of $\textbf {k}_{p1}$ (pump:green) and $\textbf {k}_{S}$ (Stokes: red) are symmetric with respect to the z-axis and while the center of the $\textbf {k}_{p2}$ distribution (probe: yellow) is normal to the sample - see also Fig. 3. Figure 2(b) reveals that in this case only K(K$_x$,K$_y$,K$_z$) with K$_x$>0 are probed so that interfaces along the x-axis featuring significant contributions for K$_x$>0 will generate a strong anti-Stokes signal in forward-direction. Obviously, turning the pupil mask by 90$^{\circ }$ (mask 2, Fig. 1(a)) probes interfaces along the perpendicular direction with K$_y$>0. To achieve contrast for z-interfaces with K$_z$<0, mask pattern 3 in Fig. 1(a) is suitable with the probe and pump wave-vectors entering from opposite sides while the Stokes beam fills the well-isolated center part of the excitation pupil. As evident from Fig. 2(c) the probed object spatial frequency support $\textbf {K}$ stays centered in K$_x$ (and K$_y$) but is shifted in -K$_z$ direction.

 

Fig. 3. Coordinate system and declaration of variables. $x_p$, $y_p$, $\theta$ and $\phi$ are the Cartesian and polar coordinates of the pupil, respectively. $\textbf {r}(x,y,z)$ and $\textbf {R}(R,\Theta ,\Phi )$ are the coordinates of the object and far-field, respectively.

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Sensitivity towards chemical interfaces in x-, y- and z-direction can be also obtained by using the same wave-vector for the pump and probe beam (degenerated CARS). The design of the incident pump, probe and Stokes wave-vectors can be found in Fig. 1(a) for the masks 4-6 while the corresponding probed object spatial frequency supports $\textbf {K}$ are sketched in Figs. 2(d) and (e). Mask 4 and 5 probe the x- and y-interfaces, respectively, while mask 6 probes the z-interfaces. Note also that mask 6 (Fig. 2(e)) probes also a significant larger area with an extended spread in K$_x$ for K$_z$<0. The heuristic description of DF-CARS as presented above provides a simple geometrical view to determine the object spectrum being probed by various color-dependent pupil function filter. The anticipated interface sensitivity of the Mask 1-6 in Fig. 1(a) shall be supported by numerical calculations in the following.

4. Numerical calculations

We revisit now the DF-CARS schemes numerically using the Debye-Wolff integral and dynadic Green function approach presented in [29–31]. For the sake of completeness, the most important equations and variables are introduced in the following. An overview for the declaration of variables can be found in Fig. 3. We assume that a x-polarized plane wave enters the objective lens so that the focal field distribution is given by:

The integrated anti-Stokes radiation power measured in the far-field is provided by:

5. Simulation results

For the simulation, we assume an excitation objective lens with air immersion and a numerical aperture (NA) of 0.95 which is equivalent to a $\theta _{max}=72^{\circ }$. For the collection of anti-Stokes radiation, we set $\Theta _{max}=30^{\circ }$. The pupil amplitude mask $M$ defines the distribution of incidences for each excitation color. As an example for mask 1 shown in Fig. 1, the pupil mask for $\lambda _{p1}$ is set to:

As a reference sample, we selected a binary 2D spoke pattern [32] defined as $N=1/2+\operatorname {sgn} [1/2\cdot \cos (5\pi \varphi )]$ with $\varphi$ being the angle between the x-axis (z-axis) and the sample position vector $\textbf {r}$. The results of the calculations for x-, y- and z-interface sensitive excitation can an be found in Fig. 4. It is clearly apparent that only the edges of the spoke pattern give signal while the homogeneous parts of the object return negligible anti-Stokes radiation in forward direction. The closer the cloud of probed object frequencies is located to K(0,0,0) and the wider its spread in K$_y$ direction, for the x-interfaces sensitive configuration (Figs. 4(b)and (f)), the better interfaces are detected along the diagonal xy-direction featuring significant contributions among those K with |K$_x$|>0 and |K$_y$|>0. As a weakness of our approach for visualizing the probed object K-support, we only estimate the area that finds at minimum one combination of k-vectors. Evidently, some object spatial frequencies are probe more efficiently than others. This observation - together with the large spread of the probed object frequency cloud in Fig. 2(e) (probing well large areas with K$_x$>0 and K$_x$<0) - explains why some xz-interfaces return even stronger signal than pure z-interfaces in Fig. 4(h). If arbitrary oriented interfaces are to be detected, a consecutive measurement using several masks, e.g. 1-3 or 4-6 from Fig. 1(a), would have to be applied. Experimentally, this could be achieved by rotating the image of the knife-edges using dove prism [33] or by rotation of color-dependent dielectric filters (designed as in Fig. 1(a)) which are placed at the back-focal plane of the objective lens. A weighted summation of x-, y- and z-interfaces sensitive DF-CARS image would yield the full chemical edge and interface map of the sample in 3D.

 

Fig. 4. Synthetic DF-CARS images of a binary spoke pattern: a) ground truth e) conventional LSM-CARS contrast b) mask 1 (see Fig. 1); 3-color CARS; $k_x$-sensitive c) mask 2; 3-color CARS; $k_y$-sensitive d) mask 3; 3-color CARS; $k_z$-sensitive f) mask 4; 2-color CARS; $k_x$-sensitive g) mask 5; 2-color CARS; $k_y$-sensitive h) mask 6; 2-color; $k_z$-sensitive. The scale-bar equals 20 ${\mathrm{\mu}}$m.

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To exemplify the impact of DF-CARS in a biomedical research context, we selected as a second sample structure the scheme of an eukaryotic cell as displayed in Fig. 5(a). As before, we assume that the image brightness is proportional to the anti-Stokes scatterer density. Note that the cell scheme serves just for illustration and has no relation with a real cellular lipid distribution, furthermore the same ground truth image is considered in the (x,y) and (x,z) planes. The simulated imaging results displayed in Fig. 5 nicely outline the potential of DF-CARS for highlighting cell organelles which would be otherwise hidden under large amounts of signal and noise arising from the homogeneous scatterer distribution parts of the cell.

 

Fig. 5. Synthetic DF-CARS images of a eukaryotic cell scheme: a) ground truth; the images brightness is proportional to the scatterer density. e) conventional laser scanning CARS image. Rest) DF-CARS images corresponding to excitation pupil filter displayed at the top right corner. The multiplication factor on b-d and f-h) indicates the scaling to the [0,1] color scale. The blue scale-bar equals 20 ${\mathrm{\mu}}$m.

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Depending on the illumination geometry and object structure, the DF-CARS signal intensity is relatively weak compared to conventional laser scanning CARS microscopy. For the faint cell structure displayed in Fig. 5(a), the DF-CARS intensity will be 2-4 orders of magnitude weaker than for the conventional filling of the objective lens’ back aperture. For this estimation, it was assumed that the focal field for each color is normalized to its strongest focal field polarization state (e.g. E$_{\textrm {x,norm}}$=E$_{\textrm {x}}$/|E$_{\textrm {x}}$|, E$_{\textrm {y,norm}}$=E$_{\textrm {y}}$/|E$_{\textrm {x}}$| and E$_{\textrm {z,norm}}$=E$_{\textrm {z}}$/|E$_{\textrm {x}}$| for |E$_{\textrm {x}}$|>|E$_{\textrm {y}}$| and |E$_{\textrm {x}}$|>|E$_{\textrm {z}}$|). Thus, pixel dwell times in DF-CARS microscopy will range from hundreds of ${\mathrm{\mu}}$s to ms compared to hundreds of ns [34] in conventional CARS microscopy.

We want to close this section with a discussion on the impact of the non-resonant background and linear scattering. DF-CARS suppresses the anti-Stokes radiation of the homogeneous non-resonant background arising from the electronic part of the non-linear susceptibility $\chi ^{(3)}_{aS,el}$. Yet, it is sensitive to off-resonance CARS signal arising from any interfaces of discontinuities arising in $\chi ^{(3)}_{aS,el}$ spatial distributions. Off resonance DF-CARS images under mask 3 or 6 (see Fig. 1) will just look as THG images with the advantage that the excitation and radiation wavelengths can be selected at will. The presence of linear scattering, however, may damp the contrast of DF-CARS. Opposed to THG, it is possible for linear scattering to change the propagation direction of the excitation beams to obtain momentum conservation for K(0,0,0) in DF-CARS microscopy. Furthermore, the blue Ewald arc (cap of sphere in 3D) sketched in Fig. 2 is just the forward collected part of the full Ewald circle. If any part of the full circle crosses K(0,0,0), anti-Stokes radiation is generated which could be deflected by linear scattering into the forward direction. Thus, DF-CARS demands measures to reduce scattering events such as the application of NIR-excitation.

6. Conclusion

We investigated a 2- or 3-color laser scanning CARS microscopy scheme, named dark-field CARS (DF-CARS), able to highlight chemical interfaces. In our proposed implementation, DF-CARS uses color dependent pupil amplitude mask for the pump, Stokes and probe beams. Depending on the masks configuration we predicted a chemical interface sensitivity along the x-, y- or z-direction using the concept of the effective excitation wave-vector $k_{\textrm {eff}}=k_{p1}+k_{p2}-k_{S}$ and the Ewald sphere representation of the out-going anti-Stokes radiation. DF-CARS highlights interfaces as it prevents momentum conservation from the homogeneous part of the sample K(0,0,0). Compared to other nonlinear interface sensitive methods such as SHG or THG microscopy, DF-CARS provides molecular specificity. Furthermore, the DF-CARS excitation and radiation wavelength are spectrally closer to each other as compared to SHG or THG bringing more flexibility and facilitating tissue penetration. Finally, off-resonance DF-CARS provides similar information as THG microscopy but its larger wavelength tunability either allows to gain in spatial resolution or to target spectral windows with minimum or maximum absorption.