1. Introduction

Much attention has been given to coherent anti-Stokes Raman scattering (CARS) microscopy since its revival in 1999 [1], after it had been previously proposed by Duncan et al. in 1982 [2]. The technique allows three-dimensional imaging with chemical selectivity and is far more sensitive than confocal Raman microscopy. In a classical CARS microscopy set-up, a sample is raster-scanned by two frequency-shifted laser beams in order to spatially map a given vibrational mode. The contrast arises from the four-wave-mixing signal enhancement when the laser frequency difference matches the frequency of the probed vibrational mode.

CARS is at the same time a coherent and a resonant process, meaning that it differs in nature from one or two-photon excited fluorescence (that are resonant and incoherent) and from processes as nonresonant second (SHG) or third harmonic generation (THG) (that are only coherent). Both aspects express in CARS image features: (i) the collected signal is proportional to the square of the coherent sum over the fields emitted within the excited volume; (ii) the collected signal is spectrally dependent as the technique probes specific vibrational resonances.

The coherent nature of CARS accounts for the forward to backward signal ratio [3, 4]. It has also been visually revealed in an experiment where the image of two close objects scanned along the axial direction was modulated by an interference pattern [5]. In another experiment using broadband pulses, Oron et al. have shown it was possible to control spectral interference in order to obtain background-free vibrational spectra [6].

Since the first CARS imaging experiments, dark fringes have been observed at the inter- face between objects such as polystyrene [3, 4, 7, 8] or melanine [9] beads or living yeast cells [10, 11] and water or agarose. Xie and coworkers first proposed a destructive interference mechanism between the object and its surrounding medium to explain this effect [3, 4]. It was then realized that refractive index mismatch between the two media can distort the beams foci and thus alterate the CARS signal in this special region [9]. This interpretation is particularly relevant as polystyrene and melanine refractive indices (n=1.6 and 1.7 respectively) differ significantly from water and agarose’s (n=1.33 for both). Djaker et al. [12] have studied, in the case of polystyrene beads, how such refractive effects distort forward and backward CARS emission. In the case of an interface, Greve et al. [7] have brought an evidence that supports the refractive origin of the signal distorsion.

However, it should be emphasized that CARS is a dispersive process. The respective CARS emissions from two resonant and non resonant objects are thus out of phase. When the interface between these two media is scanned, this phase difference accounts theoretically for destructive interferences. By detuning the excitation, the phase difference changes and a modification of the interface contrast is expected. It is the first time, at our knowledge, that such effects are investigated and clearly demonstrated. This paper will be divided between a theoretical and an experimental part. First, we will describe CARS effect as a resonant process. Considering an isolated Raman line, the spectral dispersion of a CARS resonance will recalled using a “circle-model”, as introduced by Druet et al. [13]. From the previous analysis, and in order to study the phase difference effects between an object and its surrounding, a naive analytical model, and then, a rigorous vectorial model will be introduced. Then, experimental work realized with polystyrene beads immersed in a nonresonant medium will be presented. Relaxing the refractive index mismatch, we will show how varying the phase difference between the two media affects the local image contrast, clearly demonstrating interference effects when interfaces are imaged. Finally, before concluding, these results will be discussed in the context of CARS microscopy.

2. CARS as a resonant process

2.1. Introduction

As a third-order nonlinear process, CARS generation is governed by the χ ⁽³⁾ susceptibility. Contrarly to parametric nonlinear processes, such as second harmonic (SHG) or third harmonic generation (THG), the CARS process is resonant as it probes vibrational states. Druet et al. [13] showed that useful information may be extracted from CARS spectroscopy when exciting beams experience electronic absorption from the investigated medium. However, this technique is not really suited for biological imaging, as it may easily damage the sample. For this reason, our analysis will be restricted to the case where no electronic resonance from the exciting beams occurs.

The χ ⁽³⁾ susceptibility is the superposition of resonant and nonresonant parts χ ⁽³⁾ _R and χ ⁽³⁾ _NR, that respectively account for the vibrational resonance and the instantaneous response of electrons in the medium [14]. χ ⁽³⁾ can be thus expressed as

Assuming a single Raman line, the first term takes a Lorentzian spectral lineshape given by

In the last expression, a is the the CARS resonance oscillator strength (a<0), ω_p and ws are the respective angular frequencies of the pump and Stokes electromagnetic fields E _p and E _s, Ω_R and G are respectively the angular frequency and the half width at half maximum (HWHM) of the associated Raman line. Moreover, the term χ ⁽³⁾ _NR being nonresonant, it can be safely assumed as real [15, 16]. This superposition of resonant and nonresonant terms is responsible for the well-known CARS spectrum asymmetry around resonances [14].

2.2. χ⁽³⁾ behaviour near a resonance

The CARS signal is generated either by the object or by its surrounding so that we introduce the terms χ ⁽³⁾ _O and χ ⁽³⁾ _S (the subscripts O and S standing respectively for the object and its surrounding). As the object is imaged near a CARS resonance, according to the previous subsection, χ ⁽³⁾ _O can be decomposed into its resonant and nonresonant parts following

The surrounding is assumed to be purely nonresonant. χ ⁽³⁾ _O and χ ⁽³⁾ _S are respectively complex and real numbers. A typical experimental CARS spectrum gives only access to |χ ⁽³⁾ _O|². Thus, it is more convenient to express χ ⁽³⁾ _O under its polar form, following the so-called “circle model”, first introduced by Druet et al. [13], and then developed by Fleming and Johnson [17]. We introduce the following notations:

The parameter δω refers to the angular frequency difference between the pump and Stokes beams, ζ represents the normalized δω detuning to the vibrational resonance and then, η normalizes the χ ⁽³⁾ _O nonresonant part amplitude with respect to the resonance strength a. Under a cartesian form, χ ⁽³⁾ _O can be written as

and then under the more convenient polar form as

with

When the CARS resonance is scanned, χ ⁽³⁾ _O describes in the complex plane a circle which center C and radius r are given by

Due to the χ ⁽³⁾ _O,NR nonresonant term, this circle is not centered on the origin 0. On Fig. 1 are depicted (a) the CARS spectra and (b) the χ ⁽³⁾ _O phase as a function of the normalized Raman resonance detuning ζ, and (c) the χ ⁽³⁾ _O susceptibility in the complex plane for three different values of the parameter η. For a very strong CARS resonance (η≪1), this circle is close to a pure Raman resonance. In this case, the susceptibility phase varies over a wide range (that tends towards [0;π]) when the resonance is scanned. On the contrary, for a very weak CARS resonance (η≫1), this range gets very small. Among noticeable features, (i) the resonance (P) and the spectral dip (D) are characterized by the same phase ϕ, (ii) the modulus ρ (and hence the CARS intensity) equals the same when the phase ϕ reaches a maximum (PM) and off resonance (OR).

 

Fig. 1. Theoretical CARS spectra of an isolated Raman line (a), representation of the χ ⁽³⁾ _O tensor phase as a function of the normalized Raman resonance detuning (b) and representation of χ ⁽³⁾ _O in the complex plane (c), for different values of the η parameter. OR 1: Off- Raman resonance; P: Peak CARS resonance; RP: Raman Peak resonance; PM: phase maximum; D: CARS spectral dip; OR 2: Off-Raman resonance.

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Experimentally, the value of the parameter η can be found by measuring the ratio of the maximum (P) to the minimum (D) of CARS signal. Noting this ratio R _P/D, some calculations lead to

The value of this parameter can be also extracted by measuring the Raman shifts associated with the CARS peak (P) and dip (D) and connecting them with information collected by spontaneous Raman spectroscopy [14].

2.3. CARS imaging near an interface

In a typical CARS imaging experiment, an object, surrounded by a nonresonant medium, is scanned. The laser frequency detuning δω is assumed to be here in the vicinity of a vibrational resonance Ω_R of the object. For any point r of the space, the nonlinear polarization induced by the pump and the Stokes fields E_p and E_s is expressed by [18]

where ω_as is the angular frequency of the generated anti-Stokes field E _as, the symbols * and : are used respectively for the complex conjugation and the tensorial product.

This local polarization is responsible for anti-Stokes field emission. The phase of this field accounts for the exciting fields (according to Eq. 10) and χ ⁽³⁾ tensor phases. Following the previous subsection, the object and surrounding third-order susceptibilities are respectively given by the superposition of the terms χ ⁽³⁾ _O,R and χ ⁽³⁾ _O,NR and by χ ⁽³⁾ _S. When the object is scanned, the arising CARS intensity I_CARS is given by the square of the summation over the locally generated anti-Stokes fields. When the object and its surrounding are two homogenous media, it scales with either |χ ⁽³⁾ _O|² or |χ ⁽³⁾ _S|², following the object or surrounding excitation. The situation is more complex when the exciting volume covers the interface between the object and the surrounding. Due to the varying χ ⁽³⁾ tensor phase, the anti-Stokes fields arising from the object and the surrounding are out of phase. The resulting CARS signal is thus affected as the output of an interferometer on which the phase is modified.

As an example, our analysis will focus on the interface between a 6 μm diameter bead and its surrounding. The bead vibrational resonance is defined by η=1.49. Given the bead curvature and the exciting beams complex shape, the CARS signal building is not obvious. Subsequently, two models will be developed to study this interference effect. In order to give a physical insight of this phenomenon, a very simple analytical one-dimensional model will be first developed. Then, a rigorous analysis, including as well the three-dimensional shape of the bead as the vectorial nature of the exciting pump and Stokes beams, will be presented.

2.3.1. One-dimensional model

In this basic model, the bead and its surrounding are assumed to be semi-infinite media separated by a plane interface (see Fig. 2(a–b)). The bead and the surrounding lie respectively in the lower (x<0) and the upper (x≥0) spaces. On the one hand, each medium is simply considered by the amplitude ρ and the phase ϕ of its χ ⁽³⁾ tensor (respectively χ ⁽³⁾ _O and χ ⁽³⁾ _S, following the previous notations). For the bead, ρ=ρ _O and ϕ=ϕ_O and for the liquid (seen as a nonresonant medium), ρ=ρ_S and ϕ=0. On the other hand, the excitation field is taken as constant (1/λ) on a window of width λ. The nonlinearity map and the excitation field can be simply compacted under the functions m and g defined by

For convenience, the dependency of ρ_O and ϕ_O on ζ and η will not be recalled. The interface is scanned along the x-axis so that the arising CARS signal is given by the square of the one-dimensional convolution of the nonlinearity map and the excitation field following

A straight analysis of I_CARS on the [-λ/2;+λ/2] range reveals the existence of a signal minimum, and hence of a spatial dip, if the condition

is fulfilled. On Fig. 2(a), the Eq. 12 is plotted for several values of the normalized Raman detuning ζ. The resonance is defined by η=1.49. This value was chosen on the bases of the 1003 cm^-1 polystyrene resonance (shown on Fig. 4) and assuming a single Raman line. Moreover, ρ_S is kept constant and equals 1. The value taken by the couple (ρ_O;ϕ_O) as a function of the Raman detuning is given by the model developed in the subsection 2.2. The CARS signal is a monotonic function of the scan position x excepted when the phase maximum (PM) is reached, for which the condition given by Eq. 13 is fulfilled. The resulting spatial dip is the expression of the phase difference induced between the object and its surrounding. This effect is fully assessed when the contrasts obtained when the phase maximum is reached (blue curve) and off-resonance (green curve) are compared (see Fig. 2(b)). For these two Raman detunings, the ρ_O value holds but the phase ϕ_O is shifted.

According to Eq. 12, for the particular scan position x=0, the CARS intensity is similar to the output intensity of a two-path interferometer. As a result, the closer ρ_O and ρ_S, and the higher ϕ_O, the deeper the spatial dip.

2.3.2. Three-dimensional vectorial model

From the previous model, the main features of interference effects when scanning an interface have been extracted. A more rigorous description of the physical situation is obtained when (i) the Gaussian and vectorial nature of the incident pump and Stokes exciting fields, (ii) the microscope features and (iii) the nature and the three-dimensional shape of the bead are taken into account. The full description of the model used in this subsection can be found in Ref. [20]. To sum it up, following the framework developed by Richards and Wolf [21], exciting pump and Stokes beams are assumed to be Gaussian and are described as a superposition of plane waves that are focused through a high numerical aperture (NA) microscope objective. Moreover, they are linearly polarized along the same axis and propagate colinearly. The resultant electric fields E _p and E _s, considered as vectorial, are then computed in the vicinity of the focal plane. They induce local Hertzian dipoles in the excitation volume, which orientation, phase and strenght are determined by the mean of Eq. 10. These dipoles act as sources for CARS radiation, which far-field radiation pattern is finally computed. This approach was first introduced in the context of CARS microscopy by Volkmer et al. [3]. Nevertheless, no simplification on the nonlinear polarization direction is made here. The bead is an isotropic medium whose Raman depolarization ratio was assumed to equal 0.33 as this parameter has only little effect on the CARS radiation pattern [22]. Moreover, the refractive index mismatch between the bead and the liquid was neglected.

 

Fig. 2. Theoretical scans of an interface between an object (Obj.) and its nonresonant surrounding (Sur.) for different Raman detunings: black: peak (P); red: dip (D); blue: phase maximum (PM); green: off-resonance (OR). The object resonance is defined by η=1.49. (a)–(b) 1D model: the interface separates two infinite media. The scan position is normalized with respect to the excitation spatial width λ. (c)–(d) 3D model: the interface separates a 6 µm diameter bead from its surrounding. The CARS intensity is normalized with respect to its value in the surrounding.

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Using this three-dimensional model, the theoretical scans obtained when the CARS signal is excited by a 1.2 NA water immersedmicroscope objective and collected in the forward direction with a 0.5 NA condenser are traced on Fig. 2(c). The interface is scanned along the diameter of the equatorial plane of the bead perpendicularly to the optical axis. The pump and Stokes beams are linearly polarized in the scan direction and their respective wavelengths equal 730 nm and 787 nm. Similarly to the results given by the one-dimensional model, the CARS intensity is a monotonic function of the scan position excepted when the polystyrene χ ⁽³⁾ tensor phase is maximum. In this particular case, the dip previously predicted can be found again (blue curve on Fig. 2(d)) and its depth is appreciably the same. Of course, the CARS intensity is no longer a parabolic function of the scan position as the excitation volume is more complex than a gate and the interface is curved. Nevertheless, the one-dimensional model developed in section 2.3.1 seems well-suited when expecting qualitative results.

3. Experimental work

3.1. CARS microscopy set-up

A classical set-up, first proposed in Ref. [23], and previously described in Ref. [12], is used (see Fig. 3). Pump and Stokes pulse trains are delivered by two picosecond tunable mode-locked lasers (Coherent Mira 900, 76 MHz, 3 ps), pumped by a Nd:Vanadate laser (Coherent Verdi). The lasers are electronically synchronized (Coherent SynchroLock System) and are externally pulse-picked (APE Pulse Picker) to reduce their rate down to 3.8 MHz. The pump and Stokes beams are expanded, spatially recombined, injected into a commercial inverted microscope (Zeiss Axiovert 200 M) and focused in the sample through a high-NA microscope objective (Zeiss C-Apochromat, 1.2 NA in water). The sample is raster-scanned with a XYZ piezo stage (Physike Instrument) and the generated forward (F) and backward (E) anti-Stokes signals are detected with two avalanche photodiodes (Perkin Elmer SPCM-AQR-14) used in photon counting mode. This set-up provides lateral and axial resolutions estimated as 750 nm and 2 µm.

 

Fig. 3. CARS microscopy set-up. F: filter; BS: beam splitter; BC: beam combiner; L_E and L_F : lenses; C: condenser (N_A=0.5).

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3.2. Experiments on polystyrene beads

To observe interference effects at interface, 6.2 µm diameter polystyrene beads (Sigma-Aldrich, n≈1.6) embedded in aqueous solution (Cargille Labs, Cargille Immersion Liquid Code OHZB, n=1.556 at 25°C) were imaged. The pump wavelength was fixed to 730.3 nm and the Stokes wavelength was scanned between 784.4 nm and 793.9 nm in order to spectrally probe the aromatic stretching mode of the polystyrene (Raman shift at 1003 cm^-1[19]). This is the strongest Raman peak for this medium. A second peak lies at 1034 cm^-1 so that the considered resonance is not spectraly isolated. However, the latter is less intense than the former. Its measured CARS spectrum is shown on Fig. 4. The two expected peaks are clearly shown in the same time as a spectral dip lying around 1013 cm^-1. The aqueous solution was chosen in such a way that (i) its refractive index matches the polytyrene index and (ii) it does not possess any resonance around 1000 cm^-1. We have not found a liquid matching exactly the polystyrene refractive index (n=1.6). The most suitable one exhibits a 4×10^-2 refractive index mismatch and is free from vibrational modes around 1000 cm^-1. This last point has been experimentally checked (see Fig. 4). Thus, the liquid can be assumed to be a purely nonresonant medium.

The commercially available bead solution was diluted in water and spread on standard microscope slides (170 µm thick). The slides were then dried in a vacuum chamber for 24 hours and left in the experiment room for a few days. Just before experiment, the aqueous solution was deposited on each slide. The whole was finally recovered by another microscope slide. This protocol avoids (i) the formation of bead aggregates and (ii) the presence of residual water around the beads. Such residues change the local refractive index around beads and are responsible for increased refractive effects when their interfaces are imaged.

A single bead was scanned, in its equatorial plane, perpendicularly to the optical axis, for several Raman shifts around the 1003 cm^-1 polystyrene resonance (see Fig. 5(a,b)). In particular, one-dimensional scans along the bead diameter were realized in this plane (dashed line on Fig. 5(a,b)). The results are synthetized on Fig. 5(c,d). All these scans were normalized with respect to the aqueous solution CARS signal. First, the higher contrast is found for the 1003 cm^-1 polystyrene resonance. Second, the image contrast is inverted when the aqueous solution signal is higher than the polystyrene bead. This is true for scans acquired around the polystyrene spectral dip (1013 cm^-1 and 1018 cm^-1). Finally, two dips at the interfaces between the bead and the aqueous solution are always visible, independently of the Raman shift. They are caused by the small remaining refractive index mismatch between the bead and the aqueous solution. When the experiment was carried out with beads which drying was uncompleted, higher dip amplitudes were observed, highlighting this refractive origin. Anyway, on the considered spectral range, this refractive contribution was assumed constant.

 

Fig. 4. Experimental CARS spectra of a 6 µm diameter polystyrene bead (red) and aqueous solution used experimentally (blue). The pump wavelength is fixed to 730.3 nm. The pump and Stokes powers equal 500 µW and 300 µW respectively.

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The manifestation of interference is revealed by comparing the four bead scans taken (i) between 1007 cm^-1 and 1013 cm^-1, (ii) at 1024 cm^-1, (iii) 1035 cm^-1 and (iv) 1097 cm^-1 (Fig. 5(d)). This last Raman shift is located far after the CARS resonance and the associated normalized CARS intensity at the center of the bead is taken as the reference. In this context, the three other Raman shifts (between 1007 cm^-1 and 1013 cm^-1, 1024 cm^-1 and 1035 cm^-1) were chosen such as they exhibit the same CARS intensity at the center of the bead. The bead χ ⁽³⁾ _O tensor phase maximum was assigned to the first one as it lies between the 1003 cm^-1 peak and the 1013 cm^-1 dip. Due to the limited resolution (0.3 nm) of our spectrometer and the high derivative of the polystyrene CARS intensity with the Raman detuning in this spectral range, this spectral position could not be measured accurately. The Stokes laser wavelength was finely adjusted such as the CARS intensity at the center of the bead equalled the reference (1097 cm^-1). The two other Raman shifts were taken around the 1034 cm^-1 peak. According to Fig. 5(d), for these four Raman shifts, the center of the bead and the aqueous solution exhibit a close CARS intensity. As a result, the “interferometer” constitued by the bead and the aqueous solution is quasi-balanced and the effect of the bead χ ⁽³⁾ _O tensor phase on the image contrast is expected to be straigthforward. On both bead/aqueous solution interfaces, the dip amplitude increases when the Raman shift approaches the phase maximum (PM), consistently with our theoretical model (see Fig. 2(d)). In particular, the dip amplitude difference is appreciable when the off-resonance (OR) and phase maximum (PM) (see Fig. 5(e)) signals are compared. This difference is seen on both left and right dips. Indeed, the χ ⁽³⁾ _O tensor phase is expected to increase when the Raman shift δω decreases, provoking destructive interference between anti-Stokes fields emitted by the bead and the aqueous solution. The right and left dips asymmetry was observed for several beads and was caused by a slight misalignment of the pump and Stokes beams with the optical axis.

 

Fig. 5. Two- and one-dimensional scan of a 6.2 µm diameter polystyrene bead embedded in aqueous solution around the 1003 cm^-1 polystyrene resonance. The pump and Stokes powers both equal 500 µW. Bead images (a) on-resonance and (b) off-resonance. The one-dimensional scans are performed along the dashed white lines and are all normalized with respect to the aqueous solution CARS intensity. The pump and Stokes beams linear polarizations are indicated by the white arrows. (c) One-dimensional scans performed along the dashed lines for various Raman resonance detuning and (d) for phase maximum (green), around the second peak (red and blue) and off-resonance (black) only. (e) Spectral positions corresponding to the scans depicted on (c) and associated normalized dip amplitude (bright grey: left dip; dark grey: right dip).

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4. Discussion and conclusion

4.1. Discussion

The theoretical and experimental results shown in this paper present a new aspect of the coherent nature of CARS microscopy. Of course, the coherent consruction of CARS signal is known from the very beginning [24]. However, there is a parallel between (i) the way pulse-shaping techniques extracts Raman spectra from high nonresonant background [6], (ii) the modulation of epi-detected CARS contrast when two thin membranes close to each other are scanned [5] and (iii) the way the Raman detuning affects the contrast of an interface image. In these three cases, a formal analogy is found with a two-path interfometer. The roles of the two interfering “paths” are played by (i) the resonant and nonresonant parts of the CARS signal arising from a homogenous medium, (ii) the signals generated respectively by the two membranes and, (iii) in our experiment, the signals generated respectively by the object and the surrounding. The phase difference between the two “paths” is operated by (i) changing the exciting pulse spectral phase thanks to a spatial light modulator (SLM), (ii) separating the membranes from each other and, (iii) in our case, detuning the excitation from the object Raman resonance. In the first case, the effect is essentially spectral, whereas, in the second one, it is purely spatial. In our work, the observed contrast modification arises from the conjunction of both kind of effects. In the more general context of nonlinear coherent microscopy, which SHG and THG takes also part, the studied effect is really specific to the class of resonant processes such as CARS.

Despite all the care taken experimentally, the refractive index mismatch remains the main responsible for dips appearing at the interface between an objet and its surrounding. Note that this experiment brings another evidence of the refractive origin of these dips in previously mentioned papers. Such effect could be attenuated by imaging infinitely thin biological membranes or organelles in solvent instead of two bulk media. Moreover, our theoretical approach has neglected the excitation beams spectral width. Consequently, the experimentally observed effect is spectrally averaged. To circumvent this drawback, multiplex CARS [25, 26] seems to be the ideal tool. Provided a suited spectral resolution, the multiple interface scans can be done in a “single shot”experiment. Finally, a simple Raman resonance model was assumed, allowing a simple procedure to retrieve the χ ⁽³⁾ tensor phase. However, the interesting “fingerprint” region is spectrally congestioned and the phase retrieval is far more complex. In this perspective, the numerical phase retrieval method proposed by Vartiainen et al. [27] seems promising.

4.2. Conclusion

To conclude, we have shown that CARS contrast at an interface between resonant and nonresonant media can be very different depending on the pump and Stokes field spectral detuning. More specifically when this detuning drives the vibrational resonance with the maximum phase difference (as compared to off-resonance), a spatial dip appears in the CARS image when the interface is scanned. This effect is another evidence of the coherent and resonant nature of the CARS contrast mechanism. This effect could be interestingly used to find the best constrast condition when small biological objects with weak Raman resonances immersed in solvents with high nonresonant level are imaged.