We present a theoretical investigation of coherent anti-Stokes Raman
scattering (CARS) that is modulated by periodically depleting the ground state
population through Rabi oscillations driven by an additional control laser. We
find that such a process generates optical sidebands in the CARS spectrum and
that the frequency of the sidebands depends on the intensity of the control
laser light field. We show that analyzing the sideband frequency upon scanning
the beams across the sample allows one to spatially resolve emitter positions
where a spatial resolution of 65 nm, which is well below the diffraction-limit,
can be obtained

Carsten Cleff

Carsten Fallnich

Chris J. Lee

H. H. Szeto

Herman L. Offerhaus

Jennifer L. Herek

Klaus-Jochen Boller

P. W. Milonni

Petra Groß

Willem P. Beeker

English

arXiv

We present a theoretical investigation of coherent anti-Stokes Raman scattering (CARS) that is modulated by periodically depleting the ground-state population through Rabi oscillations driven by an additional control laser. We find that such a process generates optical sidebands in the CARS spectrum and that the frequency of the sidebands depends on the intensity of the control laser light field. We show that analyzing the sideband frequency upon scanning the beams across the sample allows one to spatially resolve emitter positions where a spatial resolution of 65 nm, which is well below the diffraction limit, can be obtained

Beeker, Willem P.

Lee, Chris J.

Boller, Klaus-Jochen

Groß, Petra

Cleff, Carsten

Fallnich, Carsten

Offerhaus, Herman L.

Herek, Jennifer L.

NARCIS 

Spatially dependent Rabi oscillations: An approach to sub-diffraction-limited coherent anti-Stokes Raman-scattering microscopy

University of Twente Research Information

PHYSICAL REVIEW A 81, 012507 (2010)Spatially dependent Rabi oscillations: An approach to sub-diffraction-limited coherent anti-StokesRaman-scattering microscopyWillem P. Beeker, Chris J. Lee, and Klaus-Jochen Boller*Laser Physics & Nonlinear Optics Group, MESA+ Research Institute for Nanotechnology, University of Twente,†P.O. Box 217, Enschede NL-7500 AE, The NetherlandsPetra Groß, Carsten Cleff, and Carsten FallnichInstitut für Angewandte Physik, Westfälische Wilhelms-Universität, D-48149 Münster, GermanyHerman L. Offerhaus and Jennifer L. HerekOptical Sciences Group, MESA+ Research Institute for Nanotechnology, University of Twente, P.O. Box 217,Enschede NL-7500 AE, The Netherlands(Received 13 November 2009; published 25 January 2010)We present a theoretical investigation of coherent anti-Stokes Raman scattering (CARS) that is modulatedby periodically depleting the ground-state population through Rabi oscillations driven by an additional controllaser. We find that such a process generates optical sidebands in the CARS spectrum and that the frequencyof the sidebands depends on the intensity of the control laser light field. We show that analyzing the sidebandfrequency upon scanning the beams across the sample allows one to spatially resolve emitter positions where aspatial resolution of 65 nm, which is well below the diffraction limit, can be obtained.DOI: 10.1103/PhysRevA.81.012507 PACS number(s): 31.15.ap, 87.64.−t, 78.20.Bh, 78.47.jhI. INTRODUCTIONOptical microscopy is one of the key techniques used to an-alyze biological processes at systemic, cellular, and subcellularlevels. However, current optical microscopy techniques haveinsufficient resolution to observe interactions at the resolutiondesired by cell biologists—that is, at the molecular or singlefunctional group level.Current near-field techniques are sub-diffraction-limited,such as scanning near-field optical microscopy [1], butthey suffer from the disadvantage that they are limited tosurfaces. Two notable far-field techniques have achieved sub-diffraction-limited resolution: stimulated emission depletion(STED) microscopy [2] and stochastic optical reconstructionmicroscopy (STORM) [3,4]. Both techniques require that thesample be labeled with fluorescent dyes. In addition, STEDrequires that the intensity of the sample be rather high toresonantly saturate the electronic transitions of the label,hastening photobleaching and potential cytotoxicity [5,6].STORM, on the other hand, uses very low light intensities, butrequires very special labels that can be intentionally switchedbetween dark and light states. Furthermore, STORM imagestake a substantial amount of time to acquire [3], obstructinglive cell imaging.Ideally, biologists would prefer to employ label-free imag-ing techniques, such as coherent anti-Stokes Raman scattering(CARS) microscopy, which replaces labeling by detectingthe chemically specific vibrational modes of the molecules.Furthermore, the nonresonant nature of the excitation pro-cess limits photodamage [7]. Techniques to achieve sub-diffraction-limited CARS imaging are the subject of activeinvestigation. Linear techniques to suppress CARS emission*k.j.boller@utwente.nl†URL: http://lpno.tnw.utwente.nl/have been the subject of both experimental and theoreticalinvestigations [8]. However, linear techniques cannot be usedto obtain sub-diffraction-limited resolution. In previous work,we investigated a saturation process, analogous to STED, thatprevents the buildup of the vibrational coherence required forCARS emission [9]. However, the saturation was presumedto be fully incoherent, via a combination of long lifetimesand short dephasing times, leading us to explore alternativeapproaches that also include coherent effects.Here, we use a density matrix approach [10] to model theCARS emission process. We consider CARS emission in afour-level system (see Fig. 1) in combination with an additionallight field that couples an additional level |4〉, called the controllevel, to the ground-state level |1〉. A mechanism that leads tosub-diffraction-limited CARS microscopy has been identifiedfor this system: for the case that the lifetime of |4〉 and thedephasing of the |1〉-|4〉 transition are long compared to theinverse transition rate imposed by the additional laser, Rabioscillations between |1〉 and |4〉 induce a Rabi splitting of theCARS emission. The Rabi splitting is intensity dependent,which can be used to resolve features within a diffraction-limited volume.II. THEORETICAL FRAMEWORKA detailed description of the level scheme, density matrixequations, and light fields has been given elsewhere [9]. Insummary, the four-level system (see Fig. 1) has a groundstate (|1〉), a vibrational state (|2〉), an excited electronic state(|3〉), and a control state (|4〉). Transitions between the groundand excited states, the vibrational and excited states, and theground and control states are dipole allowed, while all othertransitions are dipole forbidden. The medium is irradiated withfour pulsed light fields, two of which (pump, ωp; Stokes, ωS)are two-photon resonant with the |1〉-|3〉 transition, drivingthe population into |2〉 and inducing a coherence between1050-2947/2010/81(1)/012507(4) 012507-1 ©2010 The American Physical SocietyWILLEM P. BEEKER et al. PHYSICAL REVIEW A 81, 012507 (2010)|1|4|3|2prsp carscFIG. 1. Energy level diagram for the CARS process with anadditional level |4〉. Level |1〉 is the ground level and initially fullyoccupied. Level |2〉 is a vibrational level of the medium, level |3〉 isthe excited level, and level |4〉 is the control state, which has a lowdecoherence rate. The arrows indicate possible transitions inducedby the incident laser fields, which are far detuned from the transitionfrequency (ωp , ωS , and ωpr), or on resonance (ωc). Through coherentanti-Stokes Raman scattering (CARS), the medium emits light at thefrequency, ωCARS, while the additional control field, ωc, drives Rabioscillations at the |1〉-|4〉 transition.|1〉 and |2〉 (we refer to this as the vibrational coherence).The vibrational coherence induces two sidebands on the probelight field (ωpr), one of which is known as the CARS emissionfrequency (ωp − ωS + ωpr = ωcars).The additional laser (called the control laser ωc), resonantwith the |1〉-|4〉 transition, influences the CARS emission. Tounderstand the control laser’s effect, we calculate the temporalevolution of the laser pulse envelopes and the envelopes ofthe density matrix elements, along with their radiating fields,as described in detail in Ref. [9]. We study CARS emissionfor the case where |4〉 has a decoherence rate that is slowcompared to the duration of the laser pulses, resulting in Rabioscillations between |1〉 and |4〉. Although |4〉 is considered tobe a vibrational state in this calculation such as in [11] or [12],any dipole allowed state with the requisite decoherence ratecan be used.For generality, we choose energy levels, detunings, andpulse durations that are typical for CARS emission processesfrom molecules. The |1〉-|3〉 transition angular frequency is setto 4700 THz (approx. 400 nm or 25 000 cm−1), and the |1〉-|2〉frequency is set to 314 THz (approx. 6 µm or 1667 cm−1).Likewise, we choose a |1〉-|4〉 transition frequency of 565 THz(approx. 3.33 µm or 3000 cm−1).The combination of the pump laser wavelength as 800 nmand Stokes as 923 nm is taken to provide two-photon resonancewith the |1〉-|2〉 transition. The probe laser wavelength is takenas 600 nm. The total lifetime of state |3〉 and the lifetime ofstate |4〉 are taken to be on the order of nanoseconds [13], whilethe decoherence rates between states involved in the CARSemission process are of the order of picoseconds (see, e.g.,[13–15]). All the laser pulse durations τ are set to 7 ps (1/e2)except for the control laser field, which is continuous duringthe interval of CARS emission. The numerical calculationsextend over 30 ps in steps of 0.1 fs. The amount of output datawas set to 70 000 points for fast computation and causes onlya small amount of discretization noise.III. RESULTSWe performed a series of calculations of the emissionspectrum around the CARS frequency. In these calculationsFIG. 2. (a) Emission spectra containing the CARS emission peakfor increasing values, on a logarithmic scale, of the induced Rabifrequency oscillations between states |1〉 and |4〉. The amount ofsplitting of the sidebands is equal to R . (b) Emission spectracontaining the CARS emission peak for increasing values, on alogarithmic scale, of the decoherence rate of 14 with R = 4 THzfor each spectrum. Note that the amplitude of the sidebands decreasesand disappears around where R equals the decoherence rate. Theincrease in Rayleigh scattering of the incident light fields for the lowerset of decoherence rates leads to an increase in background signal atthe CARS emission wavelength.ωc was taken to be resonant with the |1〉-|4〉 transition(the detuning 14 = 0), while the decoherence rate of thistransition was set to 14 = 0.1 THz. The amplitude of thecontrol field and, hence, the Rabi frequency (R) were variedin steps. The results are shown in Fig. 2 where the spectraaround the CARS frequency, as emitted by the medium, fora number of values of R are given. A single emission peakis seen at low Rabi frequencies and indicates standard CARSemission, as expected. It can be seen that, for increasing R ,the spectra show two symmetrically placed sidebands. Fromour calculations, we find that these sidebands show a spacingfrom the CARS carrier frequency that coincides with R , givenby [10]R =√E2cµ214h̄2+ 214. (1)012507-2SPATIALLY DEPENDENT RABI OSCILLATIONS: AN . . . PHYSICAL REVIEW A 81, 012507 (2010)Here Ec =√(2Ic/ε0) is the electric field of the controllaser, Ic is the control intensity, and µ14 is the dipole momentat the control transition. From this, we conclude that theRabi oscillation periodically depletes the ground state, which,in turn, modulates the vibrational coherence. As a result,the CARS emission becomes amplitude modulated whichspectrally shows up as two Rabi sidebands. The splitting ofseveral THz is large enough to be distinguishable in a realCARS experiment using standard spectrum analyzers.To obtain a noticeable Rabi splitting in the CARS spectrum,it is critical that the frequency of the Rabi oscillations exceedsthe decoherence rate 14. This can be seen in Fig. 2(b), wherethe spectrum of the CARS emission, modulated with R =4 THz, is shown for increasing values of the decoherence rate.Up to decoherence rates which correspond to sub-picoseconddecoherence times, the 4 THz modulation is still detectable inFig. 2(b). Obtaining such high Rabi frequencies is important,because the decoherence times in liquids are very short,typically on the order of 5 ps [11].The induced sidebands can be used to obtain positionalinformation by noting from Eq. (1) that the Rabi splitting ofthe sidebands depends on the intensity of the control laser.Furthermore, the geometrical alignment of the dipole momentµ14 may be obtained by rotation of the polarization angleof the control laser beam, providing additional informationon the structure of the sample. Considering a single emitterilluminated by a Gaussian beam, we note that the radialdistance of the emitter from the center of the control beamcan be obtained by measuring the exact frequency of thesidebands, which can be called a Rabi labeling of the emitterposition. Such a space-dependent Rabi frequency is a well-known effect [16]; however, it is usually considered to causeundesired broadening and is eliminated, e.g., in pump-probeexperiments, by using a much larger pump beam as comparedto the probe beam [17].The absolute position of an emitter location can be calcu-lated by trilateration. We illustrate this with a two-dimensionalexample; however, a similar approach also applies to threedimensions. Consider an emitter located at the Cartesiancoordinates x1 and y1. The observed CARS emission spectrum,with its sideband frequency depending on the local intensityof the control laser, does not determine these coordinates butinstead provides the distance from the control beam center,r1 =√x21 + y21 , which is a ring of possible locations centeredaround the control beam. By scanning the control laser fora known distance, dx, along the x axis, the radial locationof the emitter is changed to (x1 − dx, y1). As a result, theemitter modulation frequency will change, revealing a newdistance, r2 =√(x1 − dx)2 + y21 . The expressions for r1 andr2 can then be solved for x1, and a subsequent scan along y bya known dy determines y1 as well. Turning our attention to thespatial resolution of such a positioning scheme, we show thatthe scheme can provide CARS with sub-diffraction-limitedresolution.We note that the control laser, resonant with the |1〉-|4〉transition and focused to a diffraction-limited Gaussian-shapedintensity profile, centered on the other laser beams, illumi-nating a distribution of emitters, will generate a distributionof Rabi sidebands on the CARS signal. If the control laserFIG. 3. (Color online) Spatial resolution as a function of the Rabifrequency and radial location of the emitter.has a peak envelope field strength of E0 and a 1/e-spot-sizeof w0, then the range of observed Rabi frequencies withinthe diffraction-limited spot is max = |µ14E0/h̄| in the centerto min = 1/e|µ14E0/h̄| at radial distance w0. The radialresolution is then determined by how accurately the Rabi-splitting frequency can be measured in the CARS spectra.Given a Rabi frequency, , corresponding to a radial distancefrom the center of the focus of r , and a frequency measurementaccuracy of d, corresponding to a radial resolution of dr , itcan be shown thatdr = w202r + dr ln( − d). (2)It can be seen from the transcendental Eq. (2) that drbecomes smaller than the diffraction limit (smaller than w0)and that higher Rabi frequencies lead to higher resolution,while emitters located away from the center of the controlbeam are resolved better than emitters close to the center. Theresolution is expected to be maximal at the steepest intensityslope of the control beam; therefore, by using a Gaussian-shaped control beam, the highest resolution is expected tobe found off center. This is illustrated in Fig. 3 where wehave plotted the Rabi-labeling resolution as a function ofboth Rabi frequency and radial location, for d = 3 cm−1(which is a typical value for the linewidth of the CARSpeak in a spectrum). This figure shows that considerableimprovement over diffraction-limited CARS microscopy ispossible.To make the improvement explicit, we consider a specificexample for a microscope with a numerical aperture of 1.2and laser wavelengths that were used in our calculations. Inthis case, the diffraction-limited resolution is 171 nm. If thecontrol laser (λ = 3.3 µm), focused to a diffraction-limitedspot, induces 100 cm−1 Rabi oscillations, then at a radialdistance of 1.2 µm the resolution is 65 nm. We estimatefrom the data presented in [11] that the laser intensityrequired to generate a Rabi frequency of 100 cm−1 is about500 MW/cm2, which is comparable to the intensities usedin standard CARS experiments and sufficiently low to avoidmultiphoton excitation [12]. However, it should be noted that012507-3WILLEM P. BEEKER et al. PHYSICAL REVIEW A 81, 012507 (2010)such a value will depend on the molecular system understudy.IV. CONCLUSIONSIn conclusion, we have demonstrated a route to obtainingsub-diffraction-limited CARS signals. By resonantly driv-ing the excitation of a control state with relatively longdecoherence times, intensity and thus spatially dependentsidebands in the CARS emission spectrum can be generated.By accurately measuring this Rabi splitting in the CARSspectrum, objects can be resolved to a resolution below thediffraction limit. Using typical numbers, we show that aresolution in the order of 65 nm may be achievable. 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Spatially dependent Rabi oscillations:An approach to sub-diffraction-limited coherent anti-Stokes Raman-scattering microscopy

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