4754
OPTICS LETTERS / Vol. 38, No. 22 / November 15, 2013
Dispersion of the resonant nonlinear optical susceptibility obtained with femtosecond time-domain coherent anti-Stokes Raman scattering Shan Yang1 and Feruz Ganikhanov2,* 1
Department of Mechanical and Aerospace Engineering, Case Western Reserve University, Cleveland, Ohio 44106, USA 2
Department of Physics, University of Rhode Island, Kingston, Rhode Island 02881, USA *Corresponding author:
[email protected] Received September 12, 2013; accepted October 10, 2013; posted October 15, 2013 (Doc. ID 197586); published November 12, 2013
We propose and experimentally demonstrate a method that is capable of resolving both real and imaginary parts of third-order nonlinearity (χ 3 ) in the vicinity of Raman resonances. Dispersion of χ 3 can be obtained from a medium probed within microscopic volumes with a spectral resolution of better than 0.10 cm−1 . © 2013 Optical Society of America OCIS codes: (070.4340) Nonlinear optical signal processing; (190.7110) Ultrafast nonlinear optics; (300.6500) Spectroscopy, time-resolved; (320.7130) Ultrafast processes in condensed matter, including semiconductors. http://dx.doi.org/10.1364/OL.38.004754
Precise information on dispersion of the nonlinear optical susceptibility in Raman active media is essential in order to get an insight into physics of intra- and intermolecular interactions. Though fairly high spectral resolution (few cm−1 ) can be attained in spontaneous Raman spectroscopy, the method suffers from low detection sensitivity and long integration times in order to produce reliable data. Coherent Raman spectroscopy and microscopy have been shown to be extremely useful in spectroscopic studies [1–4]. Ultimate spectral resolutions (∼40 MHz) can be achieved with cw lasers [5] and ultrabroad spectral range can be covered using unique laser sources in multiplex coherent anti-Stokes Raman scattering (CARS) experiments [6]. Coherent Raman spectroscopy methods adapted for microscopy [7–10] have also demonstrated the capability to obtain spectra. However, the approaches provide only dispersion for the imaginary part of the resonant optical nonlinearity. In this Letter, we demonstrate a method that is capable of producing high spectral resolution dispersion data, for both real and imaginary parts, of the resonant third-order nonlinearity of a medium probed within microscopic volumes. Time-domain CARS transients traced with femtosecond pulses within orders of magnitude in the signal decay can lead to resolution of fine spectral features in χ 3 dispersion that cannot be reliably detected by Raman-based spectroscopy techniques. Third-order nonlinear polarization (P 3 ) is a source of detected signals in coherent Raman spectroscopy. The polarization is normally expressed in terms of the third-order susceptibility (χ 3 ωas ; ω1 ; −ω2 ; ω3 ) and interacting optical fields with amplitudes E 1 , E 2 , and E 3 as P 3 ωas 6χ 3 ωas ; ω1 ; −ω2 ; ω3 E 1 E 2 E 3 :
(1)
Following the molecular polarizability tensor (α) approach [11], the same polarization can be represented as 1 ∂α P 3 ωas N Qω1 − ω2 E 3 ; 2 ∂q
where N is the molecular density, and Qω1 − ω2 stands for the macroscopic molecular or lattice vibration coherent amplitude stimulated in a Raman active medium by a pair of waves with E 1 - and E 2 amplitudes at ω1 and ω2 frequencies, respectively. The macroscopic amplitude is a statistical average of amplitudes of the individual molecular displacements (qi ) within the interaction volume. Comparing the above equations, one can conclude that if the coherent amplitude (Q) can be measured at each ω1 , ω2 frequency pair setting, then the dispersion of χ 3 is also known. Relating formulae (1) and (2) to each other relies on an approximation when interaction of the molecular system with external fields is represented by a model in which α is expanded only up to the first order in qi . Such approximation is justified in the vicinity of Raman resonances since other terms that represent higher-order tensors and nonlinear expansion for α result in negligible contributions. The coherent anti-Stokes signal Sωas ∼ jP 3 j2 can be viewed as a result of scattering of a probe wave with amplitude E 3 on the coherence built by the excitation pair of pulses in the material. The excitation and probe events in CARS can be treated independently [12,13]. The coherent amplitude decays freely due to various dephasing and decay processes caused by intra- and inter-molecular interactions. Anti-Stokes signal intensity versus time delay (td ) of the probe pulse E 3 t with respect to the excitation pair can be found using the following equations: Z∞ ∂E z; t S as td ∝ κas E 3 t − td Qz; t; jE as tj2 dt; as ∂z −∞ 3a
(2)
0146-9592/13/224754-04$15.00/0
Qt
N 1X ∂q qi t; i γ i iγ x qi κ q E 1 tE 2 t: ∂t N i1
(3b) In the equations, constants κ as and κq stand for probe and excitation field coupling to the molecular system. © 2013 Optical Society of America
November 15, 2013 / Vol. 38, No. 22 / OPTICS LETTERS
4755
The individual amplitude qi is a complex function and thus can be characterized by its amplitude and phase parts qi t q0 tejφt . The amplitude can be affected by intra-molecular interactions with other molecular or lattice degrees of freedom. The phase term is impacted by external (inter-molecular) interactions with surrounding bath by means of collisions and velocity drifts. The time-dependent solution for Qt is, in general, a complex function that takes into account various amplitude and phase relaxation processes. In the case of extremely short optical pulses tp ≪ 1∕γ i ; 1∕γ x , the integrations that are necessary to calculate S as t can be lifted and a solution for the signal is expressed in a simpler formula: S as td ζjQtd j2 I 3 l2 ;
(4)
where I 3 ∼ jE 3 j2 , l is the interaction length, and ζ is a factor accounting for various laser beam parameters. It is obvious that a finite integral exists for the absolute value of Qt. Furthermore, Qt 0 at t < 0 and thus the Fourier transform of Qt at ω ω1 − ω2 represents the coherent amplitude (ω1 − ω2 ) in Eq. (2). For cases when Qt is a real function, the coherent amplitude can be expressed in terms of the measured time-domain CARS signal Eq. (4) yields in Qtd ∼ Std 1∕2 . Further, taking into account Eqs. (1) and (2), we can arrive to the following expression for χ 3 : 1 ∂α χ ωas ; ω1 ; −ω2 ; ω3 N κq 12 ∂q 3
Z
∞ 0
S as t 1∕2 −iωt e dt: S as 0 (5)
Here, S as 0 is CARS signal at zero delay. Thus, both the imaginary and real parts of χ 3 can be obtained in one measurement. The coherent amplitude’s properties as a function of time and the fact that P 3 is linearly dependent on Q allow the use of Kramers–Kronig relations for χ 3 obtained by our method. The classical model was followed here since it provides an effective theoretical outline for the method and conveniently relates the CARS signal amplitude to the measurable Raman line cross section [12]. The density matrix approach to calculating time-dependent nonlinear polarization for an isolated vibration [14] should also result in the relationship above given the fact that the time evolution of the nondiagonal elements follows the same dynamics as Qt. We should note that in the case of chirped pulses, Qt is a complex function and the extraction of dispersion data from CARS transients is not possible. In our experiments we used synchronously pumped optical parametric oscillators (OPOs) driven by a mode-locked Ti:sapphire laser running at 76 MHz (Fig. 1). Details and characteristics for the laser sources are provided in our recent work [15,16]. Each OPO can be independently tuned to a central wavelength within the 950–1250 nm range in order to satisfy the resonance condition for the Raman active mode with vibration frequency Ων . The OPO pulses had a typical bandwidth of ∼80–100 cm−1 determining the frequency range for the dispersion data that can be obtained using Eq. (5). A delayed replica of the pump laser pulse at 780 nm served as
Fig. 1. Three-color femtosecond CARS is enabled by two independently tunable OPOs that provided broadband pulses at central frequencies of ω1 and ω2 in order to phase in vibrations within subfemtoliter excitation volume. Decay of coherent amplitude Qt is probed by a pulse at ωpr that is a small part of the Ti:sapphire oscillator output. Fourier transform of the square root of the time-domain p CARS signal measured with high dynamic range S as td provides dispersion of χ 3 .
a probe. The three time-synchronized pulses were precompensated for the dispersion from steering optics and the high numerical aperture objective lens. The anti-Stokes signal was spectrally filtered using diffraction grating and bandpass filters to minimize the background noise floor. Typically, the CARS signal that was generated within approximately 0.2 fL probe volume could be traced within five orders of magnitude versus the delay time (td ) of the probe pulse. Examples of optical phonon mode decay in solid-state material made of high-quality crystals are considered first. Due to crystal potential anharmonicity, the primary route of decay of the coherent amplitude is a threephonon parametric interaction process that often results in the mode decaying into phonons of lower energy. Thus, for qi t in the equation set (3), there is no phase-shifting process and the dynamics is governed by time-dependent amplitude q0 . The coherent amplitude Qt is a real function and the χ 3 dispersion can be calculated directly from Eq. (5). Calculations and experimental data show that the effect of the instrument function to S as td is negligible at time delays td 2tp , where tp represents the excitation pulsewidths. Figure 2(a) shows the experimental CARS transient obtained from potassium titanyl phosphate (KTP) crystal when an optical phonon mode corresponding to one of the fundamental vibrations of distorted TiO6 octahedra at ΩR ∼ 21 THz (or 700 cm−1 ) [17] is excited and probed. The investigated phonon split into lower energy phonons
4756
OPTICS LETTERS / Vol. 38, No. 22 / November 15, 2013
Fig. 2. (a) Experimental time-domain CARS signal (open circles) from KTP crystal. Solid red and blue curves represent the calculated CARS signal with decay times of 495 and 505 fs, respectively. (b) Dispersion of the resonant χ 3 (open circles) using Fourier transform of the experimental time-domain data. Solid red line represents the calculated two-peak Lorentzian lineshape with the following parameters: the ratio for the excited doublet line amplitudes is 21.50, and lower and higher frequency peak linewidths (ΔνL1;2 ) are 21.00 and 24.20 cm−1 , respectively. Inset shows differences in spectra that would correspond to the different time-domain signal decay times indicated above. An order of magnitude difference seen for the CARS signals (td > 2 ps) corresponds to less than 15% change in magnitude of the spectra.
via a homogenous process and therefore γ i Γ, resulting in exponential decay of the coherent amplitude Qt at rate Γ. The exponential decay time of T 2∕Γ 495 fs for the corresponding time-domain CARS signal can be measured with excellent precision. Figure 2(b) shows the result of the Fourier transform of the experimental CARS transient. By comparing the result with the Lorentzian function [red line in Fig. 2(b)], one can see that the real and imaginary parts of χ 3 versus ω1 − ω2 exhibit a nearly perfect case of a homogenously broadened line with a linewidth ΔνL Γ∕πc 21.00 0.20 cm−1 . The spectral resolution, precision, and sensitivity, which are important achievements in this approach, are demonstrated using the following considerations. The calculated time-domain CARS signal using a slightly different decay time constant (T 505 fs) shows a noticeable discrepancy (factor of 6) with the experimental data at a delay time of about 2.5 ps. In the spectral domain, this translates to the calculated curve shown in the blue line in the inset of Fig. 2. Differences between the two pieces of spectral data do not exceed 15% in magnitude within areas closer to the line wings. Spectral resolution better than 0.15 cm−1 and wavelength
stability better than 0.03% would be required to detect those changes in a spontaneous Raman experiment. A small peak at ∼760 cm−1 is also seen in the spectrum. This is due to a weak excitation of the second line in the Raman doublet corresponding to slightly different vibration frequencies in the axial and planar directions of TiO6 [17]. The time-domain signal shows a pronounced quantum beats pattern even though the second line is barely within the effective excitation spectrum of the E 1 and E 2 driving fields. Figure 3(a) shows the CARS transient for the case in which both Raman lines in the doublet are more efficiently excited compared to the case of Fig. 2 by slightly tuning the central wavelength of one of the OPOs. As expected, the time-domain CARS signal exhibits a quantum beat pattern with stronger modulation depth as a result of interference of the two components in the doublet. The spectral data obtained using formula (5) lead to the conclusion that the two lines are separated by ∼60 cm−1 . The corresponding linewidths for the second component of the doublet are ΔνL2 24.20 0.30 cm−1 . This indicates a faster decay of that vibration due to a possibly higher density of states for lower energy phonons. Raman spectra of oil are of great interest for different fields of research. Unlike solid state, molecules can move
Fig. 3. (a) Experimental time-domain CARS signal (open circles) from KTP crystal for the case in which pulses at ω1 and ω2 were tuned to efficiently excite both lines in the Raman doublet. (b) Dispersion of the resonant χ 3 ω1 − ω2 obtained from time-domain data using Eq. (5). Solid red line represents the calculated Lorentzian lineshape with the following parameters: the ratio for the doublet line amplitudes corresponds to 5.40, and the lower and higher frequency peak linewidths (ΔνL1;2 ) are 21.00 and 24.20 cm−1 . The solid red line on part (a) is the calculated CARS signal corresponding to the same amplitude ratio as mentioned above, while the phonon decay times are T 1;2 2∕πcΔvL1;2 , and the peak position difference is 60 cm−1 .
November 15, 2013 / Vol. 38, No. 22 / OPTICS LETTERS
Fig. 4. (a) Experimental time-domain CARS signal (open circles) from oil. The ω1 and ω2 pulses were tuned to excite vibrations near 870 cm−1 (C-C bending mode). (b) Dispersion of the resonant χ 3 ω1 − ω2 obtained from time-domain data using Eq. (5). Solid red line represents the calculated Lorentzian lineshape with Γ∕πc 3.82 0.08 cm−1 . The inset in part (a) shows the transient when the wavelength of one of the OPOs is slightly detuned so that the neighboring Raman line that is spaced apart by ∼21 cm−1 can be efficiently excited, thus resulting in the quantum beat pattern.
freely and the coherent amplitude can decay due to dephasing caused by both collisions and translational motions. Collisions result in random phase shifts in qi while the phase is changing linearly in time in between due to the Doppler effect. The dephasing rate due to collisions is uniform and leads to γ i Γc . The dephasing due to the translational motion is accounted for by setting γ χ Ων νi t in Eq. (3b). For Gaussian distribution of molecular velocities, the ensemble averaging in Eq. (3b) results in a real function for Qt. Namely, Qt q0 e−Γt−f t , where f t αt2 for t ≪ tv and f t βt for t > tv [18]. The q0 amplitude is time-independent since intramolecular energy redistribution can be neglected for vibrations within lower level states. Constants α and β account for vibration and molecular parameters (e.g., Ωv , mean squared velocity, diffusion coefficient, etc.). Constant tv stands for the velocity’s correlation time, and it is directly proportional to the diffusion coefficient and inversely proportional to the mean squared velocity. Figure 4 shows the CARS transient obtained from the oil sample when the two excitation pulses (E 1 and E 2 ) were wavelength tuned to the target Raman active mode near 865 cm−1 corresponding to a weak C-C bending vibration. For the case of oil, tv ∼ 1 fs at room temperature. Time-domain experimental data [Fig. 4(a)] show characteristic decay time of 1.4 ps that is much shorter than the estimated diffusional dephasing time T D 2∕β ∼ 20 ms.
4757
Thus, we make an obvious conclusion that the dephasing is governed by collisions with corresponding rate Γc 0.26 ps−1 . By performing Fourier transform of the CARS transient, we can find that the line shown on Fig. 4(b) fits well into the Lorentzian profile with linewidth ΔνL 3.82 0.08 cm−1 . The detection sensitivity offered by time-domain CARS can clearly reveal unresolved features for the weak Raman mode in oil, as can be seen from the data presented in the inset of Fig. 4(a). The most recent data available in the literature for the vibrational mode do not show a resolved twopeak structure with narrow linewidth (
