ULTRAFAST COHERENT SPECTROSCOPIES AND LASER SOURCES BASED ON THIRD-ORDER NONLINEAR OPTICAL PHENOMENA

This document is composed of four different research efforts. In the first chapter of this dissertation, I present my experimental work on seeded Yb-doped Q-switched fiber laser based on stimulated Brillouin scattering (SBS). I began by experimenting with different types of fiber laser cavity configurations in order to find an ideal cavity that would successfully generate the free-running stochastic nonlinear SBS mirror as well as to reinforce the injected light mode. Experimentally, the ideal configuration involved a diffraction grating as the cavity back-mirror in the Littrow configuration with added cavity losses to enhance the SBS effect. Single SBS pulses exhibited the cascaded SBS effect and pulse width (FWHM) typically less than 2 ns. Continuous sine wave modulated light, continuous triangular wave modulated light and a dynamic pulsed halfsine modulated light injections successfully synchronized the SBS repetition rate, and to some degree the pulse-to-pulse intensity, to the waveform frequency. The injected waveforms effectively seeded the SBS Q-switching mechanism and stabilized the laser repetition rate in a novel way. Output spectrum showed the injected waveforms nearly, but not completely, suppressed the free-running cavity modes due to fiber terminations and the output pulse wavelength was slightly blue-shifted compared to the injection spectrum. This was a curious result and is under further investigation. In the second part of this work, I show experimentation regarding the resolution of fine spectral features within several Raman active vibrational modes in potassium titanyl phosphate (KTP) crystal. Measurements are performed using a femtosecond timedomain coherent anti-Stokes Raman scattering spectroscopy technique that is capable of delivering equivalent spectral resolution of 0.1 cm. The Raman spectra retrieved from my measurements show several spectral components corresponding to vibrations of different symmetry with distinctly different damping rates. In particular, linewidths for unassigned optical phonon mode triplet centered at around 820 cm are found to be 7.5±0.2 cm, 9.1±0.3 cm, and 11.2±0.3 cm. Results of these experiments will ultimately help to design an all-solid-state source for sub-optical-wavelength waveform generation that is based on stimulated Raman scattering. In the third section of this document, I demonstrate and analyze a series of experiments in traditional and soft condensed matter using coherent optical spectroscopy and microscopy with ultrafast time resolution. I will show capabilities of resolving both real and imaginary parts of the third-order nonlinearity in the vicinity of Raman resonances from a medium probed within microscopic volumes with an equivalent spectral resolution of better than 0.1 cm. I can differentiate between vibrations of various types within unit cells of crystals, as well as perform targeted probes of areas within biological tissue. Vibrations within TiO6 octahedron and the ones for the Ti-O-P intergroup were studied in potassium titanyl phosphate crystal to reveal multiline structure within targeted phonon modes with closely spaced vibrations having distinctly different damping rates (~0.5 ps-1 versus ~1.1 ps-1). I also detected 1.7-2.6 ps-1 decay of C-C stretching vibrations in fat tissue and compared that with the corresponding vibration in oil. I lastly demonstrate an effective microspectroscopy technique by tracing the dispersion of second order nonlinear susceptibility ( ( ) 2 χ ) in a monolayer tungsten diselenide (WSe2). The ( ) 2 χ dispersion obtained with better than 3 meV photon energy resolution showed peak value being within 6.3-8.4×10 m/V range. I estimate the fundamental bandgap to be at 2.2 eV. Sub-structure in the ( ) 2 χ dispersion reveals a contribution to the nonlinearity due to exciton transitions with exciton binding energy estimated to be at 0.7 eV.

generation that is based on stimulated Raman scattering.
In the third section of this document, I demonstrate and analyze a series of experiments in traditional and soft condensed matter using coherent optical spectroscopy and microscopy with ultrafast time resolution. I will show capabilities of resolving both real and imaginary parts of the third-order nonlinearity in the vicinity of Raman resonances from a medium probed within microscopic volumes with an equivalent spectral resolution of better than 0.1 cm -1 . I can differentiate between vibrations of various types within unit cells of crystals, as well as perform targeted probes of areas within biological tissue. Vibrations within TiO6 octahedron and the ones for the Ti-O-P intergroup were studied in potassium titanyl phosphate crystal to reveal multiline structure within targeted phonon modes with closely spaced vibrations having distinctly different damping rates (~0.5 ps-1 versus ~1.1 ps-1). I also detected 1.7-2.6 ps-1 decay of C-C stretching vibrations in fat tissue and compared that with the corresponding vibration in oil.           (1) and (2)   represents instrument function that allows to measure dephasing times (T 2 ) shorter than 150 fs. Solid curves are simulated CARS signals that make use Equations (3) and (4) and assuming single exponential decay for G(t) with 150fs (black) and 250 fs (red) dephasing times………………………………………………………………………...……………73 Corresponding spectrum of the resonant third order nonlinearity (χ (3) (ω)) obtained using experimental CARS transient and applying Equation (5). Imaginary part of χ (3) is shown in blue open circles. Solid black line is the best fit to the spectrum assuming three components shown in separate colors and with parameters provided in the text; (c) Real part of χ (3) (ω)……………………………………………………………………………..76 between the driving E 1 ,E 2 pulsed fields and the probe field E 3 . Vibration at 1072 cm -1 has been targeted; (b) CARS image the same area when the probe pulse was delayed by 600 fs; (c) Time-domain CARS signal obtained from the ~20 µm diameter fat area located at the center of the mouse tissue for the image shown in Fig. 3.5(a). Red line represents the best fit to the data obtained by using formulae (3)- (4) and varying the corresponding line parameters (see text); (d) Real (black circles) and imaginary (blue circles) parts of χ (3) (ω) xv obtained using the experimental CARS transient shown in part (c) and Equation (5). Red line is theoretical fit that makes use two-component Lorentzian function (Δν 1,2 =4.1,6.2 cm -1 , 28 cm -1 separation between components, A 1 /A 2 =13:4)………………………...….78       [2]. SBS based Q-switched (SBS-QS) fiber lasers have displayed a novel ability to compress Q-switched pulses to less than a few ns pulse width (FWHM) as well as high-peak-power, often in excess of 10 kW, which are several orders of magnitude narrower and greater, respectively, than pulses generated by conventional Q-switched fiber lasers [1,[3][4][5][6][7][8]. The addition of Yb-doped double-clad fiber (DCF) to SBS-QS fiber laser models has been pivotal in the advancement of high-power fiber lasers. Cladding pumping due to Yb-doped DCF technology allows for high-power broad stripe, relatively cheap laser diodes to be coupled into active fiber with superior launching efficiency. High-intensity light confined within the single-mode, Yb-doped core due to cladding pumping allows for SBS-QS fiber lasers to easily reach power thresholds of various nonlinear effects, such as low-threshold SBS. The combination of cladding pumping and Q-switching allows for high-energy, high-peak-power, short pulse emission in regular or irregular self-pulsing regimes [7][8][9][10][11].
These types of fiber lasers have proved to be very beneficial in the field of medicine, industrial processing, range finding, and remote sensing, as well as in applications such as LIDAR [4].
Of course, utilizing SBS as a Q-switching mechanism comes with a few considerable drawbacks. In single-mode fiber (SMF), SBS is due to stochastic thermal noise fluctuations within the medium density of the fiber [12] with the driving optical field being random superposition of laser modes in the cavity. The noise source constituting these thermal fluctuations in the medium density, known as the Langevin noise source, is assumed to be a Gaussian random variable with zero mean and is δcorrelated in space and time. In short, output laser characteristics in SBS-QS fiber lasers, unless externally controlled, are strongly affected by an initial distribution of laser modes in the cavity and by thermal noise. Because of this characteristic, the output pulse train of SBS-QS fiber lasers has considerable repetition rate instability and corresponding pulse-to-pulse intensity fluctuations. It has been shown experimentally that the spacing between adjacent Stokes pulses can vary greatly. In some cases, the variance of the repetition rate in SBS-QS fiber lasers was found to be ~20% of the period [14]. Additionally, the intensity of the Stokes output has been shown to exhibit 100% temporal fluctuations in the sense that the normalized standard deviation of the pulse intensity is equal to unity [13].
A considerable amount of experimental and theoretical efforts have been undertaken in order to enhance SBS-QS fiber laser output as well as to overcome its glaring deficiencies. Enhancements of the actual fiber cavity design itself have been numerous. Fiber Bragg gratings (FBG) spliced onto the back end of active fiber cavities have been used in order to introduce narrow spectrum feedback in an effort to more efficiently generate SBS (in the case of an Yb-doped DCF core, the back-reflected light being approximately 1.06 µm wavelength) [8,11,16,18]. Often in experimental setups, several tens of meters of passive fiber are spliced to the active fiber in order to enhance the distributed backscattering SBS effect or, in some cases, the passive fiber itself served as a back mirror utilizing narrow-spectrum Rayleigh scattered light to retro-reflect back into the active fiber [1,4,6,[16][17]. There have also been studies using a 10/90 coupler in the form of a Rayleigh-stimulated Brillouin scattering ring interferometer. The ring interferometer acts as a spectrally-selective resonating energy source which enhances a cavity mode until it reaches SBS threshold, successfully generate nanosecond pulses, much shorter than the round-trip time of the fiber resonator length [14,18]. External Qswitching mechanisms, typically in the form of acousto-optic modulators (AOM) or Cr 4+: YAG saturable absorbers (SA), have been added to the all-fiber cavity design as well in order to enhance SBS giant pulse peak-power and to stabilize the repetition rate by forcing SBS initiation by increasing HR feedback once the fiber is inverted to a high level [3-6, 15, 19]. In addition to adjustments in laser cavity design, SBS-QS repetition rate has been stabilized experimentally using a pulsed pump laser diodes [7,20] and, in the case most relevant to our current work, by externally injecting square-wave modulated light into the fiber cavity [14]. The latter study showed great success in developing a novel method of gain control in SBS-QS fiber lasers, drastically improving repetition rate variance from about 20% of the period to about 1% of the period [14].
For sufficiently intense incident light, with a spectral width within a few tens of MHz, a nonlinear SBS mirror can be efficiently generated in single-mode fiber. The SBS mirror produces fast transient backscattered Stokes feedback and can convert more than 80% of incident intensity into the narrow Stokes pulse intensity [23].
Our current study aims to improve upon previous experimental stabilization efforts by injecting periodic waveforms into the fiber cavity that will govern the nonlinear SBS mirror and the Q-switching effect. The externally seeded optical mode with optimized parameters (i.e. amplitude and bandwidth) is expected to significantly suppress the stochastic nature of laser generation and initiation of the SBS effect that is behind the Q-switching mechanism in this type of laser. In other words, the optimized optical mode seeded is expected to control initial laser dynamics that yields in a stable output pulsed train synchronized to the frequency of the seeded waveform in each cycle of Q-switched pulse generation. Figure 1.1 (a) shows the theoretical interpretation of our current experimental investigation. The 1064nm waveform with frequency ω L (orange), is injected into the core of highly inverted Yb-doped DCF. The waveform is amplified as it traverses the inverted fiber until its intensity reaches the SBS threshold. At this point, the nonlinear SBS mirror is generated where the acoustic sound wave (grey) has driven frequency Ω and the backscattered Stokes wave (red) has a downshifted frequency (ω L -Ω). If the active fiber is pumped above transparency level (for small signal gain) and there is sufficient cavity feedback, a waveform single passing through the gain fiber with sufficient length can at the same time generate a Stokes mode/pulse that can in turn generate an efficient nonlinear SBS mirror. This can further create a subsequent Stokes ordered pulse propagating in the opposite direction. This process, known as the cascaded SBS effect, is depicted in Figure 1.1 (b) and typically generates Stokes orders until the population inversion is essentially depleted.
The experimental setup is shown in Figure 1 We began by experimenting with the type of back mirror in the fiber laser configuration in order to get a free-running stochastic SBS lasing effect which very clearly showed characteristic repetition rate and intensity fluctuations in the fiber laser.
Initially, we choose to use an HR mirror as the back mirror of the cavity (Figure 1.3).
With a low pump power of 433 mW, the oscilloscope trace in (c)), the output spectrum shows a broad spectral cluster for CW lasing from approximately 1073nm to 1081nm with a largest gain at approximately 1075nm.
We then decided to block the HR mirror in the cavity completely, allowing the two perpendicularly cleaved fiber terminations with corresponding 4% Fresnel reflection as the only source of cavity feedback. The oscilloscope traces with this bare-fiber cavity configuration with varying pump power is shown in Figure 1  shows the output spectrum for pump power 954 mW, less than threshold. In this image there are a few spectral components where the largest gain corresponds to approximately 1072 nm. As the pump power increases past threshold in Figure 1.6 (c), the output spectrum shows several different spectral components with relatively high gain values but the maximum gain value is still approximately corresponding to 1072 nm. This image shows that SBS pulsing exhibits unstable wavelength lasing patterns that our injection experiment will ultimately stabilize.
Because the highest gain during free-running SBS lasing in the bare-fiber cavity configuration is at approximately 1072 nm, we encountered an experimental difficulty.
The central wavelength of our injected light was approximately 1064.6 nm, as shown in problem was resolved because ytterbium is homogenously broadened and therefore, if we inject with sufficiently high intensity, other free-running SBS modes will be completely suppressed. To test this idea, we injected CW light with varying loss and detected the output spectrum using a partial mirror as the cavity back mirror. In Figure 1.8 (a), the CW injection had a very high loss and the injected 1064nm CW light is slightly more intense than the free-running SBS lasing modes. As the CW injection loss decreases (as shown in Figure 1. 8 (b) and (c)), the CW injection spectrum almost completely suppresses the free-running modes. This is an indication that even though we cannot line up the wavelength of the injection and the wavelength of the largest gain, if we inject with sufficient intensity we can still drown out the other free-running SBS modes.
In order to maximize the effect of the injection, we installed a diffraction grating that will provide wavelength dependent feedback (analogous to a fiber Bragg grating).
The grating is 1200 lines/mm and is used in the Littrow configuration at an angle that favors the diffracted mode being at a wavelength that matches as precisely as possible the wavelengths of the seeded optical mode. Figure 1.9 shows the oscilloscope traces for the Littrow configuration with varying pump power. Figure 1.9 (a) shows self-pulsing relaxation oscillations with a low pump power of 824 mW. As the pump power was increased from 824 mW to 1.08 W (Figure 1.9 (b)), the relaxation oscillation mode grows with no indication of its development into short-pulse mode due to SBS Q-switching.
The laser dynamics are entirely affected by CW mode generation that depletes the optical gain. As the pump power is increased further to 1.47 W then to 1.84 W, shown in Figure   1.9 (c) and (d), the relaxation oscillation self-pulsing is further amplified and the repetition rate increases further and some SBS pulses are seen. The transient dynamics of SBS are not seen clearly for this configuration so a variable attenuator was added to the fiber cavity to introduce cavity losses and enhance the SBS lasing effect. Figure 1. 10 (a) shows two distinct linear relationships for the output power as a function of pump power.
The output spectrum, shown in Figure    we see relaxation oscillation self-pulsing and as the pump power increases to 1.08 W in part (b), those relaxation oscillation pulses clearly increases in amplitude and repetition rate. As the pump power increases past SBS threshold (~1.3 W), as shown in Figure 1.11 (c), the SBS mode is favored (i.e. experiencing more gain than CW) and several different Stokes pulses are evident. As pump power increases further to 1.84 W, as shown in the pump power exceeds threshold, we see a significantly larger slope corresponding to SBS lasing. In Figure 1.12 (b) and (c), the output spectrum is shown with a broad span and a narrow span, respectively, for a pump power greater than threshold. We see in these spectra the tuned wavelength reinforced by the diffraction grating in the Littrow configuration at ~1066 nm which is significantly less intense than the free-running SBS modes as a result of the added cavity losses. Stokes pulses evident and part (d) shows that the Stokes pulse has <4 ns pulse width (FWHM). This is a curious result, as we were expecting a pulse width <2 ns. We believe this is because of the fluctuating pulse measurements due to the stochastic SBS pulse generation.
The first type of waveform injection experimented with was a continuous sine wave modulated light with a frequency of 160 kHz. Figure 1.14 (a) and (b) shows the oscilloscope traces with the sine wave injection (blue) and the Stokes pulses (yellow).
The output pulse train shows the SBS repetition rate is largely synchronized to the frequency of the sine wave modulated light injection. Figure 1.14 (c) and (d) depicts a single SBS pulse with the sine wave injection. Figure 1.14 (c) shows a cascaded SBS effect with 3 or 4 Stokes orders, however it is far less pronounced than the cascaded SBS effect for the free-running SBS Littrow configuration with loss shown in Figure 1.14 (c). identifies two distinct relationships. For pump power less than threshold (~1.1 W), there is a quadratic relationship and then after threshold is achieved, the characteristic linear relationship is observed for SBS lasing. This implies that the sine wave undergoes amplification due to double-passing through the inverted fiber until it has an intensity large enough to seed the SBS mechanism itself. For all pump powers in excess of this value, the sine wave seeds the SBS process excluding thermal noise initiation. Figure   1. 15 (b) and (c) shows the output spectrum. Figure 1. 15 (b) shows that the ~1064 nm spectrum does not completely suppress the modes due to termination at ~1074 nm as we would have hoped. We did not have amplifier equipment to properly amplify the injection enough to completely suppress the approximately 1074 nm modes. Figure 1.15 (c) shows a very curious result. The output spectrum (blue line) has a slight blue-shifted wavelength compared to the injection spectrum (dotted red). This implies the output frequency has increased seemingly indicating anti-Stokes pulse generation instead of Stokes pulse generation. The top of the output pulse in Figure 1.15 (c) also shows evidence of several Stokes orders but the spectrum analyzer has a max resolution of 0.05 nm and we cannot detect the exact shape. We are currently looking into this unusual effect.
The second type of waveform we experimented with was a continuous triangular wave modulated light with a frequency of 130 kHz. cascaded SBS effect with a few Stokes orders but is not as pronounced as the freerunning SBS grating and losses configuration shown in Figure 1.11 (c). Figure 1. 16 (d) shows the pulse width (FWHM) in the presence of the triangular wave injection to be <2 ns.
The output measurements for the continuous triangular wave modulated light injection are shown in Figure 1.17. Figure 1.17 (a) shows the output power as a function of the pump power. We see two very distinct relationships here. For pump power less than threshold (~1.2 W), we see a quadratic relationship and for pump power greater than the threshold, see a linear relationship that is characteristic of SBS lasing. We believe this is due to the triangular wave becoming amplified as it double-passes the inverted fiber until it is sufficiently amplified to seed the SBS mechanism. We also think that the bandwidth of the injected mode is better matched to seed the power into intrinsic cavity modes that are further amplified and serve as an efficient driving field for the SBS effect. Figure 1. 17 (b) shows the injection intensity is nearly large enough to completely subdue the lasing modes at ~1074 nm and is more effective at suppressing these modes than the sine wave injected light. If we had a pulse amplifier we could completely suppress these extraneous modes but this was not available to us. Figure 1.17 (c) gives a very similar curious result as we have seen in the sine wave injection output. The output pulse (blue) has slightly blue-shifted wavelength relative to the injected spectra (dotted red). We would have expected a larger wavelength, as is indicative of a Stokes order, and this result seems to indicate that the output is due to an anti-Stokes reaction. We also see several Stokes orders in the peak of the output pulse but we can only make out 0.05 nm maximum resolution from our spectrum analyzer. We are currently looking into this result. A high spectra resolution Fabry-perot etalon is needed to study this effect.
The last type of waveform that showed success was an inverted half-sine wave with periodic CW lasing between half-sine waves. The half-sine width (frequency) was 160 kHz and CW lasing with a 15 µs period. This dynamic waveform was made so that the stretch of CW lasing was long enough to deplete the population inversion between SBS pulses by cross gain modulation and to suppress spontaneous Brillouin scattering.
The half-sine wave was then made narrow enough to be the exact amount of time it takes for SBS to be generated. This injection was used with the bare-fiber termination cavity configuration prior to the addition of the diffraction grating. By making these adjustments to this dynamic waveform, we were able to synchronize the SBS repetition rate to the waveform frequency, as seen in Figure   injection. Figure 1.19 (a) shows the output power as a function of the pump power. It is obvious that the power-power data have at least two distinct slopes that can be explained as a strong interference between SBS lasing at seeded (1064.6 nm) and lasing wavelength corresponding to higher gain (~1072 nm). The SBS threshold is larger than the threshold values corresponding to the previous two waveforms due to the addition of the grating. , with what appears to be Stokes orders on its peak that cannot be made out due to the maximum resolution of the spectrum analyzer, compared to the injection spectrum (dotted red). This result was unexpected and we are currently looking into it further but is important to note that this effect is not due to the grating tuning as this configuration did not include the grating.
In conclusion, we have displayed a novel method of stabilizing the repetition rate instability by seeding the SBS pulses with a periodic waveform. We chose a cavity configuration with a diffraction grating in the Littrow configuration with cavity losses in order to maximize the effectiveness of the injection and simultaneously enhancing the generation of the SBS pulsing effect. Injections of continuous sine wave modulated light, continuous triangular wave modulated light and half-sine wave pulse modulated light injections successfully synchronized the SBS repetition rate to the waveform frequency and seeded the SBS mechanism for pump power greater than the threshold values. This project is ongoing and requires more funding and equipment in order to investigate further.                   Precise information on fine structure and decay of Raman active modes is essential from both fundamental and device applications point of views. Time-domain studies provide direct information on decay and dephasing processes for vibrational modes and, for solid-state media, provide most valuable information as concerned parametric phonon interaction due to deformation potential anharmonicity. In frequency domain, dispersion of the corresponding nonlinear optical susceptibility is an essential characteristic in order to get an insight into physics of intra-and interatomic groups interactions. In this paper we focus on an important nonlinear optical gain material that is used both as intracavity and external gain material in multi-wavelength laser devices. The attention has recently grown due to possible applications of efficient frequency converters in generating phase-locked frequency combs for attosecond waveform generation. Potassium titanyl orthophosphate KTiOPO 4 (KTP) is a widely known optical material that is particularly attractive for nonlinear optical applications. Because of its high nonlinear optical coefficient and its optical and mechanical stability, the crystal is used in laser sources as an optical frequency converter. Its large electro-optic coefficient, low dielectric constant and ion exchange properties also make it suitable for electro-optic [1] and waveguided laser devices [2]. The crystal was previously shown to be an efficient source for multi-wavelength pulse generation via stimulated Raman scattering (SRS) [3,4] or as a combination of SRS and efficient second order frequency conversion [5].
Renewed interest came with recent SRS experiments on high-frequency crystal vibrations that promised a pathway towards a solid-state sub-optical-cycle waveform source [6][7][8].
In other words, materials with high second and third order nonlinearity associated with several Raman active vibrations at high frequency range are of interest from the standpoint of generating a frequency comb that would ultimately support attosecond waveforms [9]. Knowledge of key properties of lattice vibrations is thus important in the light of the applications of this material as a nonlinear gain (of both second and third order) medium.
KTP's vibrational spectra are quite complex. The spectra consist of about 100 Raman active peaks as a result of the crystal's multiatomic unit cell. The complexity makes it difficult to perform comprehensive and unambiguous phonon line assignment, to precisely measure bandwidth and separation of individual Raman active peaks, as well as to estimate Raman cross-section for each individual phonon line. Even though the material has been known for more than three decades, detailed spectroscopic studies on its Raman active vibrations are relatively scarce [10][11][12]. The performed studies helped to elucidate contributions to Raman and infrared spectra from major atomic units within the primitive cell, as represented by TiO 6 octahedra and PO 4 tethrahedra. Also, important details concerning line assignments and their major characteristics were provided by the studies. However, the information ultimately proved to be contradictory and detailed spectral features of some peaks were not provided by these experiments. In particular, the first comprehensive Raman study of KTP [10] [11]. A study that followed later stated that the Raman line detected at ~830 cm -1 is an intergroup (Ti-O-P) vibration, but provided no details on the detected linewidths and separations for the different peaks [12].
At room temperature, the Raman spectroscopy of KTP has also been investigated from 10 to 1400 cm -1 [13][14] and also studied as a function of high pressure revealing the existence of two additional phase transitions near the critical pressures of 5.5 and 10 GPa [15]. Temperature dependent Raman scattering were studied [16][17] and found that no phonon mode coalesces to central peak near T c and reported it as a sign of damped soft mode [16]. A study on polarized Raman spectra showed strongest phonon line located at 234 cm -1 [18]. It is worth mentioning that no experimental or theoretical study can be found which addresses phonon dispersion properties or mechanisms for phonon line decay. As was mentioned above, there is a motivation for a more detailed characterization of phonon vibrations in the material in the light of a search for an efficient solid-state media for a sub-optical-waveform source. Indeed, the crystal possesses several high-scattering cross-section phonon modes within energy range of 200-1000 cm -1 . The modes are conveniently spaced apart so that generation of a frequency comb, via SRS with intrinsically phase-locked spectral components, would provide multi-octave bandwidth to support sub-femtosecond pulses.
In this work, we present data on the decay of some of the KTP crystal phonon modes within 640 -850 cm -1 . We reveal the fine structure of the vibrations by retrieving the vibrational system's response function and Raman spectra. Our data provide details on the crystal's complex vibrational spectra supported by important quantitative results.
The data obtained for an unassigned vibrational mode at 820 cm -1 supports the conclusion that the modes decay noticeably slower when compared to high-frequency modes originating from vibrations of the main TiO 6 or PO 4 atomic groups. We attempt to explain our linewidth results within the framework of parametric phonon interaction due to the deformation potential anharmonicity.
Time-domain CARS spectroscopy is a valuable tool that enables probing the dynamics of elementary excitations in condensed matter. This technique monitors in time a degree of coherence within the lattice or molecular vibrations created by two ultrashort optical pulses at an earlier moment of time. Tracing the net coherence provides information on characteristic relaxation and dephasing processes. In our studies, we employed three-color CARS geometry with widely tunable 110-150 fs pulses [19,20].
The experimental set up is schematically shown in Figure  MHz. The OPOs utilize high parametric gain periodically poled lithium tantalate (PPSLT) crystals. The OPOs were simultaneously pumped by a split output of a highpower mode-locked Ti:sapphire oscillator tuned to 765 nm. Detailed OPO characteristics and performance were reported in our recent publications [21,22]. The OPOs with pulsed outputs at 970-1020 nm and 1050-1100 nm, served to coherently drive lattice vibrations with energies within 600 -990 cm -1 . Another small part of the Ti:sapphire oscillator was delayed and served as a probe pulse. All of the three pulses were intrinsically synchronized, made to overlap in space, and focused by a high numerical aperture (NA~1.25) objective lens. In the detection arm, we used a high numerical aperture (NA~0.9) condenser followed by a diffraction grating and a set of bandpass filters. This permitted efficient detection of the signal of interest on the background of other signals generated within the focal volume. A photomultiplier tube (PMT) with high gain and quantum efficiency (Hamamatsu model #R10699) was used to detect anti-Stokes signal photons at selected wavelengths. The PMT current output was digitized by a high-speed data acquisition card. Using this experimental arrangement, we can routinely detect CARS signals versus probe pulse delay times within five decades. The corresponding total power on the sample from the three beams does not exceed 15 -20 mW. Other details and characteristics of the set up are described in our most recent work [20]. tensor components [10,11] are involved in Raman mode excitation and scattering processes during CARS.
Lattice dynamics in condensed matter is modeled as time-dependent behavior regarding the expectation value of molecular/atomic displacement amplitude under a driving force. This driving force consists of a pair of pulsed fields with an optical frequency difference matching the energy of vibration quanta [23][24][25]. Quantitatively, the scattering CARS signal at anti-Stokes frequency in the time-domain (S as (t d )) can be expressed as the following: In the above equation, ( ) and ε(t) are normalized time-dependent envelopes for atomic displacement amplitude and probe pulse, respectively. This also implies that ζ 0 represents detected anti-Stokes signal at a zero delay. In Equation (1) In the equation above, g(t) represents the response function of the corresponding vibrational system to δ-pulsed driving fields coupled with normalized time-dependent envelopes for the driving pulse amplitudes.  (1) and (2) are of Fredholm type-I and can be solved using the Fourier transform method [26]. This is ensured by the correlation integral theorem and the fact that spectra and/or envelopes of ε 1 , ε 2 , and ε pr pulses are known and can be measured. In the case when ( ) is a real function, the response function g(t) and its Fourier transform can be ultimately obtained. The condition holds true for many types of vibrational systems that do not involve diffusional phase shifting events. As a consequence, precise spectra and fine features in the vicinity of Raman active vibrations can be effectively resolved. an approach reported earlier by our group, described in Ref.
[27], yields in somewhat distorted spectral data. Thus, equations above need to be solved in order to retrieve Raman spectra along with the dispersion of the real part of the associated resonant third order nonlinearity (χ (3) (ω)). The Fourier transform (S as (ω)) of the measured timedependent CARS signal is a first step in solving the equations. The corresponding result is shown in Figure 2.2 (b). The spectrum is smooth, as high-frequency noise in the timedomain CARS signal has a relatively low spectral power density and is not visible on linear scale. The main characteristic of the spectrum is a broadband and high-intensity pedestal, associated with ultrafast signal rise-time. The pedestal masks a narrower spectral feature. The latter may reflect a slower decay rate due to the phonon decay process mentioned above. Knowing the measured probe pulse spectrum, (I pr (ω)=FT(ε pr 2 (t)), and applying the inverse Fourier transform operation, allows the collection of time-domain data on the coherent displacement amplitude ( ( )) at Further, having ! , ! available from OPO pulse autocorrelation and spectral measurements and a Fourier transform of ( ), one can arrive to resonant third order nonlinearity (χ (3) (ω)) spectra contained in real and imaginary parts of g(ω). for the doublet. The difference in the bandwidths is explained by different damping rates for in-plane and along long axis vibrations within the TiO 6 octahedron [10]. A third component is also pronounced in the spectra with a position shifted to lower energies by 65 cm -1 . This mode has a different symmetry and represents ν 2 (E g ) anti-phase stretching vibration within TiO 6 octahedra. The peak can be better resolved under condition when one of the OPOs is detuned to provide more efficient coherent excitation for the ν 2 (E g ) mode. As a result, the time-dependent CARS signal exhibits a more pronounced quantum beats pattern. Using this arrangement, the spectral bandwidth of the ν 2 (E g ) mode was determined to be 21.3±0.7 cm -1 . The obtained parameters for the main ν 1 (A 1g ) doublet and for the ν 2 (E g ) modes are in good general agreement with the referenced reports [10,11].
We must note, however, that consistent bandwidth and Raman shift data for the doublet components could not be found throughout Raman spectroscopy characterization studies of KTP crystal published in the past [10][11][12][13][14][15][16][17][18][28][29][30][31]. The result of fitting imaginary part of the resonant third order nonlinearity (i.e. Raman spectrum) using Lorentz-shaped multi-peak curves is also shown in Fig mode. This proves the fact that the mode is noticeably weaker in its intensity when compared to the ν 1 (A 1g ) and ν 2 (E g ) modes. Retrieved Raman spectra (Im(χ (3) (ω))) and data for the real part of the third order nonlinearity are shown in Figures 2.3 (g(t)) that was obtained by solving equations (1) and (2)   Applications of nonlinear optics have been widely regarded as powerful tools that are capable of providing quantitative spectral information in condensed matter characterization. These applications span from plasma to solid-state materials and nanostructures, as well as to interfaces and biological media [1,2,3,4,5]. Raman effect based spontaneous and coherent scattering techniques are of special attention due to their selectivity and sensitivity down to a chemical bond level, and that in turn provides access to important physical mechanisms and fundamental interactions.
However, in spontaneous Raman spectroscopy, reliable data on weak resonances is difficult to obtain. In the last decade, coherent Raman spectroscopy methods have been successfully demonstrated in optical microscopy [6,7,8,9,10]. Spectral responses with a resolution of a few cm −1 have been demonstrated in various applications that focused on the characterization of biological matter. It is worth noting that these approaches provide only dispersion for the imaginary part of the resonant optical nonlinearity. The overwhelming majority of studies are in the frequency domain, with a recent focus on novel solid-state materials [11,12,13,14] and molecular/biological media [15,16,17,18]. The coherent Raman based microscopy techniques were recently applied to image biological tissue and cells with sub-micron spatial resolutions capable of resolving the spatial structures of key constituents when their characteristic Raman active vibrations are targeted [19,20]. The spontaneous Raman microscopy has been applied with a greater focus towards the analysis of specific vibrations within cells, tissues and their spectral features. Multi-line spectra were analyzed with respect to relative changes in the intensities and spectral shifts for main phonon lines, depending on growth techniques for solid-state materials as well as scattering geometries with respect to crystal axes [21,22]. For biological media, efforts were focused towards the goal of correlating those with bio-molecular alterations occurring on cellular and subcellular levels [23,24]. Resolving lattice or molecular vibration damping rates Γ (or linewidths, ∆ν = 1/Γ) and line shapes, which are ultimately key parameters in characterizing atomic/molecular bonds, has not been of specific effort primarily due to insufficient detection sensitivity and spectral resolution. The vibration damping rate is strongly influenced by unit cell structure and respective phonon properties in crystals and inter-and intra-molecular interactions in soft media. Therefore, the ability to measure Raman line shapes and sub-structures with high precision is absolutely important from the standpoint of having access to fine inter-and intra-molecular interactions and their physical mechanisms. Spontaneous Raman spectroscopy has a typical resolution of ~3-7 cm −1 and often requires the application of data postprocessing algorithms to retrieve spectra. Thus, for current state-of-the-art spectral domain techniques, the ability to access critical inter-and intra-molecular interactions of key molecular groups within biological specimens is limited. Coherent Raman microscopy, which employs picosecond pulses to attain high peak powers, cannot provide a better spectral resolution due to the pulse bandwidths (~5-20 cm −1 ).
Additional measurement artifacts and sources of imprecision come from a need to mechanically tune laser wavelengths in point-by-point spectral measurements with an obvious adverse impact on precision. Thus, the desired precision in detecting Raman line shapes and bandwidths may not be achieved. Precise information on dispersion of the nonlinear optical susceptibility in Raman active media is essential in order to gain insight into the physics of intra-and inter-atomic and molecular interactions. For solid state, detailed knowledge of Raman active vibrational properties in intrinsic and doped materials, as well as in thin films, is of high value from the standpoint of identifying mechanisms that limit transport parameters which are critical for device applications.
For biological media, this can become a powerful tool in detecting small spectral changes that can be translated to variations in bio-molecular composition and ultimately lead to disease diagnoses on a molecular level.
In this report, we discuss and implement a method that is applied to characterize crystal phonons and Raman active vibrations in soft matter. We show how to obtain high spectral resolution dispersion data for the resonant part of the third-order optical nonlinearity (χ (3)  Third-order nonlinear polarization ( (!) ) is the origin of the scattered signal at shifted frequency in coherent Raman spectroscopy. Here, we will consider a case of coherent anti-Stokes Raman scattering (CARS) with three incident fields involved. The nonlinear polarization in the frequency-domain is then expressed in terms of the thirdorder susceptibility (χ (3) (ω as ; ω 1 , −ω 2 , ω 3 )) and is a nonlinear response of a medium to incident optical fields E 1 ,E 2 , and E 3 : In the above equation, E 1 and E 2 are the complex amplitudes of the driving pulses and E 3 is the complex amplitude of the time-delayed probe pulse. The nonlinear polarization defined in Equation (1) represents the macroscopic change of polarization connected with the coherent material excitation and probing in the CARS process. If one chooses to represent the polarization through the molecular susceptibility tensor (α), the nonlinear polarization expression in frequency-domain can be written as following: N -density of molecules, Q -coherent amplitude of molecular or lattice vibrations at the driving frequency, ω as -anti-Stokes frequency. The interaction between the electromagnetic field and the vibrational mode is described by the molecular susceptibility tensor. These two equivalent definitions of nonlinear polarization in the frequency-domain for the 3-color CARS geometry allows for the definition of the thirdorder susceptibility in terms of the molecular susceptibility tensor and the coherent vibrational amplitude. The macroscopic coherence is the statistical average of molecular displacements (q i ) with Raman frequency (ω R ) and independent phases (ϕ i ). Upon the action of the driving pair of pulses (E 1 , E 2 ), the displacements are forced to accept common phase (ϕ 1 -ϕ 2 ). The difference frequency of the pair (ω 1 -ω 2 ) is tuned to match Raman active vibration (ω R ) in order to achieve maximum value of Q. After the action of the two fields, the coherence decays freely in time as a result of inter-and intraatomic/molecular interactions that lead to changes in the individual phases (pure dephasing) [25,26]. Third wave/pulse E 3 , that can be delayed in time with respect to the driving fields, is scattered on the time-dependent coherence Q(t,) producing a signal at the anti-Stokes frequency (ω as ).
The time-domain coherent amplitude can be represented in terms of an instantaneous response function G(t), that is the characteristic of the ensemble oscillating at Raman frequency (ω R =ω 1 -ω 2 ), coupled to the finite width driving pulses [26]: The above equation is a measure of the coherent excitation of the molecules. The stimulated process excites the vibrational system coherently, i.e. the molecules oscillate in unison within the excited volume. The forced common phase then loses uniformity in time due to various dephasing properties. Time-domain CARS signal (S as (t d )) can be expressed as the following: The κ constant represents combined field coupling coefficients to the molecular/atomic system (κ 12 ) for E 1 ,E 2 fields and E 3 field to the coherent amplitude. The time-domain CARS signal, mathematically defined in Equation (4) extremely short optical pulses, or when G(t) does not change much on a timescale of the pulsewidths, the integrations are not necessary to calculate the signal (S as (t)). Taking into account Equations (1) and (2), a final expression for χ (3) can be straightforwardly derived: Thus, the experimental CARS transient signal (S as (t d )) can be used as a source function to obtain imaginary and real parts of χ (3) with the help of Equation (5). We therefore display a quantitative method for deriving the imaginary and real parts of the third-order nonlinearity utilizing the Fourier transform method for solving Equations (3) and (4)   Three-color pulse CARS images at different time-delays can be generated with a spatial resolution of 380 nm. We were able to trace transient non-resonant CARS signals, generated in the forward direction, within higher than five orders of magnitude in solid glass material while the average power of the combined beams was kept under 30 mW (~0.15 nJ/pulse per beam) at the sample. The non-resonant time-domain signal in glass (see Figure 3.1(c)) represents instrument function showing our capability to detect decay in coherence with a characteristic dephasing time (T 2 ) shorter than 150 fs. This is shown by comparing the experimental signal (blue ovals) with simulated CARS transients    Figure 3.2 (b) can be fit well when the two line components (ω R1 = 706 cm -1 and ω R2 = 768 cm -1 ) are involved. The corresponding response function is ) . The best fit is achieved when the amplitude ratio is 9:5 and the corresponding T 21 , T 22 times of 495±10 fs and 515±10 fs. It is obvious that in this case, due to broad bandwidths of E 1 , E 2pulses (~85 cm -1 ), the two components can be driven with nearly equal strength opposite to the result shown in Figure 3.2 (a). In other words, the result shown in Figure 3.2 (a) corresponds to phonon mode targeting condition where primarily only one phonon mode centered at (i.e. ω 1 -ω 2 ~ 706 cm -1 ) is driven and probed. This results in absence of the beat pattern and decay of the signal that is best fit with the single exponential decay response function presented above. Figures 3.2 (c,d) show the corresponding spectra for χ (3) , both for real and imaginary parts, obtained using the formalism described previously that required solutions for the integral Equations (3) and (4). Solid lines are simulated spectra assuming Lorentzian lineshape and line parameters that have been mentioned above. This highlights the asymmetry seen in the spectra obtained using the experimental data. The best fit to the imaginary part is obtained assuming three components with ratios of 26:15:11, linewidths of 21 cm -1 , 24 cm -1 , 17 cm -1 and line separations of 65 cm -1 and 16 cm -1 , respectively.
Multiple Raman active modes can be driven by E 1 , E 2 -pulses at the same time, given fairly broad bandwidth pulses. When KTP phonon modes centered at ~830 cm −1 are targeted, a complex nature of Raman active vibrations in the crystal is clearly seen in the detected CARS transients. Figure 3.

3(a) shows the time-domain CARS
signal obtained for the case of coherently driven vibrations with the center wavelengths for E 1 , E 2 -pulses adjusted to 967 nm and 1052 nm, respectively. The signal carries a well pronounced modulation pattern that represents a coherent beat signal due to interference of at least two spectral components. This is governed by the exponential decay trend with a time constant that is clearly longer than the one observed for ν 1 (A 1g ) mode. This is also a substantially longer time than the pulsewidths used in the measurements and the width of the instrument function for the set up. We have therefore analyzed the obtained data for this case with the help of the result expressed by Equation (5). The non-resonant background, due to electronic nonlinearity, has a noticeable presence in the data as can be revealed by Fourier analysis of the CARS transient. The clustered phonon modes at ~830 cm −1 that are driven by the broadband pulses have a weaker contribution, highlighting the fact that the modes have a smaller Raman scattering cross-section compared to the ν 1 (A 1g ) and ν 2 (E g ) modes, as well as narrower linewidths. Raman spectra retrieved from the time-domain signal are represented by the imaginary part of χ (3) (ω) and are shown in  cm -1 . This is in fairly good agreement with reported spontaneous Raman scattering data that provided the range of 20.4-29.0 cm -1 and 34.6-39.2 cm -1 for the two [34]. Linewidths (FWHM) for the three components are 9.1±0.4 cm -1 , 7.5±0.6 cm -1 , and 11.2±0.5 cm -1 , respectively. This compares fairly well to ranges of 10.2-12.6 cm -1 , 9.2-10.8 cm -1 , and 14.0-16.4 cm -1 , respectively, that have been reported for the spontaneous Raman spectroscopy measurements. A component amplitude ratio of 23:4:15 can also be retrieved and represents the best fit to the imaginary part of the χ (3) spectra. As was mentioned above, due to a slower decay, phonon linewidths are more than two times narrower when compared to the high frequency modes (ν 1 (A 1g ) and ν 2 (E g )), as shown in the results presented in Figure 3.2. This is explained by the fact that the ν 1,2 modes have overtone decay channels with a higher phonon density of states for resulting lower energy phonons in parametric decay processes.
Thus, lack of the efficient decay channels for ~830 cm -1 mode leads to a significantly lower damping rate for the driven vibrations. We believe that the mode in a real function for Q(t). Namely, t = ! !(!!!! ! ) where ƒ(t) = α ! for t ≪ t v and ƒ t = βt for t > t v [35]. Constants α and β are molecular parameters (e.g. mean velocity, diffusion rate coefficient, etc.). Constant t v is velocity correlation time. The most recent data available in literature for the vibrational mode shows a single peak with a ~7 cm -1 linewidth [36]. The discrepancy is explained by a lack of sensitivity and resolution (~5 cm -1 ) to provide more precise data.
We have applied our method and experimental capabilities to enable quantitative microscopy where, as an example, we characterized biological tissue. We demonstrate imaging of tissue constituents and then perform dephasing time measurements in targeted areas of the tissue by driving and probing in time characteristic Raman active vibrations.  Single point measurement often led to data of poor quality that was the result of tissue damage. This problem will be more closely investigated in our future work. The obtained time-domain CARS signal shows a pronounced beat pattern of at least two spectral components with the corresponding dephasing times of 2.6 and 1.7 ps. The obtained transient has been analyzed by generating theoretical curves to fit the experimental data and the results are summarized in the caption. The corresponding dispersion of χ (3) obtained with the help of Equation (5)   represents instrument function that allows to measure dephasing times (T 2 ) shorter than 150 fs. Solid curves are simulated CARS signals that make use Equations (3) and (4) and assuming single exponential decay for G(t) with 150fs (black) and 250 fs (red) dephasing times.    Large variations in the SHG signal have made this point-by-point measurement insensitive to smaller features in the SHG spectra. In addition, the incident photon energy range was limited so that the near bandgap nonlinear optical response of the material could not be obtained.
In this letter, we report on a microspectroscopy method applied to characterize monolayer WSe 2 within a photon energy range of 2.4-3.2 eV. We detect single-shot second harmonic (SH) spectra from monolayer WSe 2 material with better than 0.3 nm resolution (~3 meV) and low noise (5-6% rms) using broadband femtosecond continuum pulses. Fine sub-structure features that can be detected within the main peak of ( ) 2 χ indicate the impact of near bandgap exciton transitions. We retrieve, with a fairly good precision, the fundamental bandgap and exciton binding energy. The absolute values for refined theoretical models for 2D materials.
The experimental idea is presented in Fig. 4.1(a). Femtosecond continuum pulses with a smooth spectral envelope centered in the near-IR are used to generate SHG signal within an atomically thin semiconductor sample. Spectrum of the second harmonic signal that carries spectral signatures of the sample is then detected with fine (~3 meV) spectral resolution. The laser part of the experimental arrangement, shown in Fig. 4.1(b), generates spectrally smooth shape for the ultra-broad continuum in the near infrared (780-1050 nm) with typical power density of ~50 µW/nm. Chromatic dispersion compensated optics were used to deliver the fundamental beam to the sample within less than 400 nm spotsize. The incident beam can be angle scanned using a galvo-mirror scanner to provide 200×200 µm 2 imaging area. SH signal from the sample was collected through the same objective lens in the backward direction. The SHG signal spectra were resolved using a monochromator (Horiba, Inc. model: iHR320) and a cooled CCD detector (Syncerity-UV/Vis, Horiba Inc.). Single layer WSe 2 flakes were prepared by micromechanical cleavage of bulk WSe 2 crystal on 90 nm SiO 2 on a Si substrate. . We believe that this is due to local field variations affecting ( ) 2 χ at the flake/SiO 2 interface. The SHG spectrum is shown in Fig. 4.3(a) by the blue curve. We have also performed, for comparison purposes, point-by-point wavelength tuning SHG measurements with wavelength tunable Ti:sapphire oscillator (filled circles data in Fig.   4.3(a)). The SHG signal fluctuations are significantly higher (σ=±54%) for this case. We believe that much higher SHG signal variations versus wavelength observed in the pointby-point measurements are due to couple of additional sources of the variation. Namely, changes in the fundamental beam parameters like pulsewidth and spatial mode while the wavelength is tuned. The lowered precision for the SHG spectra were also observed in the referred point-by-point measurements [5] as authors pointed to the uncertainties in the puslewidths while the wavelength was tuned as being the main reason.
The observed increase in the resonant nonlinearity ( )

2D
χ matches well with splitoff band transitions (i.e. B-exciton) if one considers bandstructure parameters for single layer of WSe 2 at K-point such as bandgap ( g E ) and split-off energy ( SO Δ ) [5,6]. A similar effect has been observed in MoS 2 involving a different energy valley that is at Γ point [3].
We have obtained ( )

2D
χ spectra using two approaches. The first one exploits the relationship between the fundamental and SH powers. In the other one, we used a comparative approach when a material with known second order nonlinearity is used. We have chosen thin KTP crystal. The crystal is well characterized from many aspects. The second order effective nonlinearity ( eff d ) is between 1.72-2.01 pm/V [7] for the incoming beam polarization and crystal orientation that we used. By normalizing our WSe 2 SH data to the one obtained from the crystal ( ), we obtain a ratio ( ) that provides dispersion of the absolute value of and is free from measurement artifacts (e.g., ω 2 T , etc.). The result is displayed in Fig. 4.3(b) by blue curve for a comparison. Some difference can be seen at lower photon energies. We note that SHG and fundamental signal have been detected by different spectral devices in the first method. The second method is free from the artifact.
Bloembergen et al [8] were first to consider SHG from thin layers and interfaces.
SHG power has been derived by Merano [9] by considering a real experimental situation of a sheet of 2D material on the top of the layered substrate. Following the approach developed in [9,10] and making use of the results by Boyd et al. [11] SHG pulse peak power from 2D material ( D P 2 2 ) and bulk crystal ( P 2 B ) can be expressed as shown in Supplement [12]. Based on that, the absolute value of ( )

2D
χ expressed in terms the crystal's effective nonlinearity eff d , refractive index ( b n ), numerical aperture of the objective lens ( Θ ) and the measured ratio ( ρ ) is: The above equation is an estimate of the second-order nonlinearity for the atomically thin crystal in terms of the normalization of the two-dimensional second harmonic data compared to the previous data from a reference crystal. Using this approximation eliminates the artifact corresponding to the second-order nonlinearity measurement obtained by the power relationship method.
It is important to note that the two-layered substrate can lead to the enhancement or attenuation of the SHG intensity due to interference effects that depends on the 2) thickness of the SiO 2 layer. This is taken into account in the factor containing complex field reflectivity ( changes by only few percent within the photon energy range that we have. The median peak value of for WSe 2 monolayer at the SH photon energy of 2.76 eV can be estimated at 7.3×10 -19 pm 2 /V. We provide ±15% range owing to several factors such as the range for eff d itself and taking into account signal variations across the flake that were discussed earlier. The value is on the same order of magnitude with the one quoted for the measured MoS 2 sheet nonlinearity in [13] and factor of 2.7 smaller than the one provided by Merano [9] for MoS 2 . If we straightforwardly use solutions provided in [8] and assume that the sample still has bulk refractive index [14] and nonlinearity ( ) The above equation is an estimate for the second-order nonlinearity when the 2D comparison layer is treated as a bulk-like system. In this approach, one must consider the SHG process in bulk media under tight focusing conditions as well as the treatment of the SHG problem in the single-layer sample.
In the formula above, ( ) n f ′ is a factor containing refractive indices of bulk WSe 2 and is coming from solutions for the SH field [17], ζ -factor accounts for the interference effects. Additional details are provided in Supplement [12]. This approach may cause controversy and disagreement. We, however, we would like to provide an estimate just for sake of comparing the material's nonlinearity with the one of other bulk non- 2) centrosymmetric semiconductors. The estimated ( ) 2 2D χ value is in the 932-1233 pm/V range for peak SH photon energy while the off-peak value is about 440 pm/V. The values are comparable with the ones for well known non-centrosymmetric semiconductors (GaAs, CdTe, ZnSe [15]) that are used in parametric devices and frequency converters in the infrared [16]. Comprehensive theoretical treatment and modeling of the second order nonlinearity is based on rigorous approaches outlined in [17,18]. Here we will use an approach based on parabolic bands approximation, accounting for Coulomb effects via exciton continuum states above bandgap in order to estimate dispersion of the absolute value of ( ) 2 χ due to interband transitions first. The expression for ( ) 2 χ along the electric field and induced dipole moment in x-direction can be represented by [18]:  (3) and (4) [6]. We have also considered momentum matrix elements and dephasing rates are k-, and therefore energy, independent. The summation over k-states can be replaced by integral over energy that couples the resonant denominator in Equation (4) and joint density of states factor for the first conduction and top valence bands for . Figure   4.4 shows the results of our calculations when γ was set to 35 meV. It is important to note that the nonlinearity is sensitive to a change of g E . This is shown by comparing two simulations when g E is changed from 2.22 eV (green curve) to 2.15 eV (dash-dotted curve). It is apparent that when the bandgap is set at ~ 2.2 eV, a better match with the experimental data around the peak area is obtained. We believe that band nonparabolicity becomes critical to explain the experimental data at higher photon energies.
On the lower energy side, one finds that the rise in simulated ( ) 2 χ is noticeably sharper when compared to our experimental data. The discrete exciton contributions can be represented mainly by resonant factors that are similar to the term in Equation (4)   well with the ones reported in [5,6]. Exciton binding energy obtained from our data is also in close agreement with the value of 0.6±0.2 eV reported by Wang et al. [5].
In conclusion, we have demonstrated precise measurements of the second order nonlinearity in atomically thin layer of semiconducting material within broad range of photon energies. Using ultra-broadband continuum pulses, we were able to detect fine features in the ( ) 2 χ dispersion with high spectral resolution (<3 meV). The nonlinearity onset is primarily due to monolayer WSe 2 states that couple valence bands, excitonic levels and the continuum states above the first conduction band. Using our data, we estimate peak nonlinearity range for a sheet of WSe 2 at 6.

A.1. Pump laser and seed DBR laser
In the SBS laser experiment there were two different light sources. The first of which was a 5W 14 pin cooled multimode pump laser diode (EM4). This laser diode had a center wavelength of 976 nm, was fiber coupled with a fiber NA of 0.22 and cladding pumped the Yb-fiber in the laser cavity. A laser diode is an electrically pumped semiconductor laser in which a current-carrying p-n junction acts as a gain medium. This type of laser is ideal for several applications where brightness is essential.
The second type of light source in this experiment was a 1064nm 14 pin DBR laser (EM4). This model had a 150mW output power and a laser linewidth of 10 MHz. It contains a cooler, thermistor and monitor detector. A distributed Bragg reflector (DBR) is an InGaAs quantum well laser diode and is ideal in applications where mode stability, low RIN and stable PM properties are needed. We used this laser as a seeder for our fiber laser injection setup.

A.2. Mode-locked Ti:sapphire laser
The primary laser source used in the CARS experiments was the mode-locked Ti:sapphire ultrafast laser (Mira-HP, Coherent) which was tunable from 700 to 1000 nm and had an output power of ~3-3.7 mW. The mode-locked laser was pumped by a green laser with output power 17 W and a repetition rate (frequency) of 76 MHz. The pulsed output light is passed through a two-cavity prism to compensate the pulse. The majority of the beam is split into two equal parts and synchronously pump the two optical parametric oscillators (OPO) and the remainder of the laser emission is used as the probe pulsed light for the coherent anti-Stokes Raman spectroscopy (CARS) experiments.

A.3. Optical parametric oscillators (OPO)
Two OPOs were used simultaneously in the aforementioned publications. The first OPO setup involves an optical resonator containing a stoichiometric lithium tantalite (PPSLT) nonlinear crystal as a nonlinear gain media. The nonlinear crystal allows for quasi-phase matching condition. The OPO generates parametric oscillation at the near-IR pump wavelengths for both continuous wave (CW) and pulsed mode. The nonlinear crystal has 1 mm width, 0.5 mm thickness and 15 mm length. The grating period is varied from 17.5 to 24.8 µm with 0.6 µm differences between consecutive periods. The pump power, delivered by the TiS mode-locked laser, was 1.15 mW and was focused onto the nonlinear crystal by a 76 mm focal length lens. The OPO cavity consists of two concave mirrors and three parallel substrate mirrors. A pair of Brewster cut angle prisms was used to compensate for the dispersion. The optimal distance between prisms (apex to apex), which generates short pulses within the tuning range 960 -1050 nm, was found to be 280 mm. The pulse characteristics are shown in Figure A.1.
The second OPO is based on a periodically pooled lithium niobate (PPLN) nonlinear crystal with a similar setup as the first OPO. This OPO is capable of generating wavelengths ranging from 1050 to 1100 nm.

A.4. Supercontinuum generation
The broadband continuum pulses needed for the SHG setup are a result of the generation of the supercontinuum from a photonics crystal fiber (PCF). Supercontinuum generation produces ultra-broadband optical spectrum when pumped by a high-power laser source. In our experimental configuration, the pump laser source is composed of femtosecond pulses. A photonic crystal, with core diameter 1.2 µm and zero group dispersion at 750 nm, was used in order to produce broadband (450 -1150 nm) continuum. A characteristic spectrum is given in Figure A.2.
The PCF was installed on an XYZ stage (466A XYZ fixture, Newport Corp.), aligning the PCF marked polarization with the polarization of the Ti:sapphire beam. This alignment was completed with the use of a half-wave plate on a rotating mount to change the linear polarization angle and a Glan-Thompson polarizer to make sure the input polarization was vertical. The XYZ stage incorporated a 40x microscopic objective lens to effectively couple light into the fiber. The input power into the PCF did not exceed 50 mW in order to avoid damaging the fiber during the light coupling adjustments. Once the continuum pulse was successfully observed, the input power was increased to 100 mW and the translational stage and steering mirrors were tuned to optimize the effect. At this time, an objective lens (20x, Newport Corp.) was installed and adjusted to collimate the output beam. It is important to note that the properties of the generated continuum pulses are defined by the amount of coupled power rather than the coupling frequency. The supercontinuum generation setup is shown in Figure A.3.

B.1. Chapter 4 Supplement
The electric field for the incident beam at fundamental frequency is represented by the following expression: In this equation, we have used the SI system of units. The intensity is then written as: The second harmonic pulse peak power generated in bulk crystal under tight Gaussian focusing conditions can be represented by the expression that follows from formula (2.58) under (2.99) (p.3609 Ref. [11]): n 2 -refractive index at SH frequency, Q -numerical aperture of the microscope objective. Given the fact that the incident and SH beams are partially reflected off the entrance and exit facets correspondingly, we write: We have neglected the dispersion in the crystal (i.e. n 2 ≈n 1 =n) in order to arrive at Equation (4). Also, we have used w 0 ≈l/pQ for high NA objective.
The solution for the SH field in the 2D layer and the corresponding intensity (I 2 ) can be obtained by using Equation (12) in Ref. [9]. Both beams (SH and the fundamental) have the Gaussian spatial distribution and the field's amplitude can be represented via fundamental pulse peak power using: and 2 0 0 The latter relationship is obtained by integrating Equation (2) over space. The complex factor represented by: accounts for the interference effect in the two-layered substrate for both fundamental and SH beams [9,10].
Following the solution for the electric field at SH generated in 2D material [9] and taking into account Equations (5) and (6), the peak power at SH frequency (P 2 2D ) is: Then, the ratio (ρ) of the two peak powers (2D sheet/bulk) is: And This equation is represented as Equation (2) in the manuscript. It is worth noting that in Equation (10), sheet second order nonlinearity is measured in m 2 /V.
The following work corresponds to the estimate for the nonlinearity [m/V], when the 2D layer is treated as bulk-like system. For a nonlinear interface, the solution for the electric field at SH frequency is given [8] by: n 2 , n 1 stand for the refractive indices (at SH and fundamental frequencies, respectively) of the bulk-like WSe 2 thin layer material with the actual numerical values for different wavelengths (optical frequencies) provided in [14]. Furthermore, we write: n2 , n3 -refractive indices of SiO 2 (subscript 2) and Si (subscript 3) at SH freq, tthickness of SiO 2 layer.

B.2. Seeded SBS Q-switched fiber laser based on SBS procedure
There were four different procedural components in this setup: the SBS laser operation procedure, the injection waveform generation procedure, the injection/fiber core coupling procedure and the procedure for running the experiment itself. In the SBS laser operation procedure the fiber position first needed to be optimized. This was done by free-space coupling the multi-mode pump laser through an aspheric lens and into the core and inner cladding of the active fiber. The fiber end and the aspheric lens were positioned on an XYZ translational stage controlled by three spatial micrometers and had a rotating fiber holder used to match the vertical polarization of the injected light mode with the PANDA fiber slow-axis (see Figure B.1). The pump laser diode was initially set to have a low current (400 mA) and a power meter was put on the opposite side of the fiber measuring how much pump light was propagating through the fiber. The micrometers were adjusted until the power meter measured the maximum throughput power. Because the active fiber was double-clad fiber, the pump light didn't have to couple into the narrow fiber core. Instead, the pump light was coupled into the inner cladding waveguide (125µm diameter) and upon contact with the ytterbium ions in the doped single-mode core, the light was almost immediately absorbed creating population inversion in the active fiber. Cladding-pumping is a critical property of high-power fiber lasers and amplifiers and is shown schematically in Figure B.2 (a). A relevant property of double-clad fiber is that if the fiber termination is perpendicularly cleaved, 4% Fresnel reflection of the outgoing beam will act as feedback into the single-mode fiber core but if the fiber termination is angle cleaved (typically less than 10 degrees) the reflection will propagate into the inner cladding and not back into the fiber core. This property is illustrated in Figure B.2 (b). If an HR mirror or a partially reflecting mirror is used in the cavity configuration, a power meter is setup after the dichroic mirror, the pump laser current is increased to a current large enough to generate lasing but not large enough to initiate SBS pulsing (typically approximately1000mA) and the mirror is tuned until the maximum output power is measured. The XYZ translational stage holding an aspheric lens in front of the fiber-coupled fast photodetector is then tuned to see the laser traces on the oscilloscope.
Once the SBS laser has been optimized, the injection components need to be setup. First, the seed DBR laser is turned on with a moderately high current (220mA) and the emitted light is free-space coupled into the input fiber of the Mach-Zehnder modulator. This input light coupling is optimized with micrometers until a power meter registers a maximum amount of power exiting the modulator output fiber. The modulator had to then warm up for a half-hour or else the exiting light power would drift. Once warmed up, the Mach-Zehnder modulator could operate properly without any drift detected. The arbitrary function generator drove the RF signal into the modulator generating the waveform and the DC bias voltage applied to the modulator was at the quadrature (half max) point for proper waveform injection generation (see Figure B.3).
The output fiber is connected to a fiber-collimator and then the collimator is rotated in its mount to assure that the exiting light is vertically polarized. A two-lens system was constructed with a moving translational stage under the far lens to narrow to the beam waist. This light then reflected off of a two-mirror system to direct the beam path and couple as much optical power possible through the free-space isolator. As the injection light exits the isolator and passes through the 50/50 beamsplitter, the beam path is tuned slightly to make the light incident on a photodetector.
In order to effectively couple the injection light into the fiber core (and not the inner cladding), we first setup and tuned a diffraction grating as the cavity back mirror in the Littrow configuration. Tuning the grating required injecting CW light of 1064 nm wavelength and adjusting the angles of the grating until the output power after the 50/50 beamsplitter was at a maximum level. The grating configuration is schematically depicted in Figure B.4. Once the grating was configured, the injection light beampath and the laser output beampath were overlapped by having both beams pass through two irises separated by approximately two feet of open table space. The pump laser was set to a low current which would generate somewhat inverted fiber but not strong enough to initiate lasing and a 10 kHz sine wave was injected into the fiber. The reflected amplified sine wave was then measured by the fast photodetector and the oscilloscope. The beamsplitter and other mirrors were tuned in order to generate the most amplified sine wave on the oscilloscope as possible and then the pump current was adjusted lower until the sine wave was no longer visible to show that the injection successfully passes through the fiber core and not passing through the inner cladding which would not show an amplification property with the varying pump laser current.
Once the injection waveform light was successfully coupled into the fiber core, we could run the experiment. The SBS laser output was blocked and the pump current was set to its maximum current used in the experiment (typically 3.6A to 4A) and the laser warmed up for 10 minutes before any measurements were taken. After this time, the injection light was blocked and the SBS laser was unblocked and free-running SBS oscilloscope traces were recorded. The injection was then unblocked and the SBS laser output was then measured on the oscilloscope, on the optical spectrum analyzer and the power meter. Typical pulse trains and individual SBS pulses were both measured in this fashion.

B.3. Coherent anti-Stokes Raman spectroscopy (CARS) procedure
In the CARS procedure, two coherent light beams with frequencies ω 1 and ω 2 can be used to drive a Raman vibrational mode at frequency ω R = ω 1 -ω 2 . The two beams are overlapped in time and space to effectively generate this effect. A third pulse of frequency ω 3 with a variable time delay is then injected into the Raman excited medium generating a CARS signal with an up-shifted, anti-Stokes frequency ω 3 + ω R . The experimental configuration is shown in Figure B Special attention had to be paid to making an initial zero time delay for the CARS signal whereby the three light sources had spatial and temporal overlap. In this effort, two irises were placed after the dichroic mirror and the beams were guided through the center of both irises. To setup the correct timing between the beam overlap, the optical path lengths of each beam was first measured taking care to include the thickness of any optic along the beam path and the translational stages for the delay line.
The method of finding the zero-delay point involved the use of a Beta-Barium Borate (BBO) nonlinear optical crystal to observe the sum frequency generation (SFG) signal resulting from the overlap of the input beams. Initially, both OPOs were blocked and the Ti:sapphire laser was focused through the BBO crystal. Then, the crystal was vertically rotated and the output light was the wavelength corresponding to the color purple and the crystal was slightly rotated to optimize the signal. Once this has been accomplished, the Ti:sapphire light was blocked and the two OPOs were unblocked. The crystal is then rotated displaying two light beams resulting from the second harmonic signals of the OPOs at two different crystal positions. The crystal position is then set between the two second harmonic signal spots. The mirror on the translational stage was moved backward and forward in order to detect a mixing signal. If one of the OPOs is blocked, the signal disappears. This implies that there is zero time delay for the two OPOs. At this point, one OPO is blocked and the Ti:sapphire light is unblocked. The same procedure as was used for the alignment of the two OPOs was employed to line up the Ti:sapphire and either OPO light. At this point, the three beams are in zero time delay. To test this outcome, if we block one of the OPO light sources one of the output spots will disappear and if the other OPO is blocked, the other spot will disappear.
During this procedure, the power of OPO-1 was set at 56 mW and 200 mW for OPO-2 thus yielding a combined power of 256 mW after DCM1.

B.4. Second harmonic generation (SHG) microspectroscopy procedure
SHG is the nonlinear process where an input wave composed of a short optical pulse with frequency ω passes through a nonlinear material lacking inversion symmetry and generates a pulse wave with twice the initial optical frequency 2ω. SHG is also known as frequency doubling and is a special type of the nonlinear process sum frequency generation (SFG). A schematic diagram of SHG is given in Figure B.8.
The experimental setup used in the SHG microspectroscopy procedure employed a high-repetition rate femtosecond Ti:sapphire laser with a central wavelength tuned to 750 nm. This light was coupled into a photonic crystal fiber (PCF) with a core diameter of 1.2 µm utilized a 40x objective lens (NA = 0.75) to generate a spectrally stable continuum in the range of ~450 nm to ~1200 nm (UV to IR light spectrum) with a total power of approximately 45 mW. The power of the incident beam coupled into the PCF was approximately 100 mW. The generated continuum was dispersed by a pair of prisms and then a portion of the spectrum, having a fairly smooth envelope in the wavelength range of interest, was selected as the fundamental beam. The beam was angle scanned with a scanning area of approximately 200x200 µm 2 in the image plain of the objective.
Samples in this procedure were positioned using a micrometer driven translational stage.
A schematic diagram of the SHG experimental setup is shown in Figure 4.1 in Chapter 4.
The SHG microspectroscopy measurement procedure began with taking the spectral and power measurements of the incoming fundamental beam before the objective. This was done with a power meter and an optical spectrum analyzer (Anritsu, model MS 9710C). After these measurements, the second-harmonic signal from the sample was collected through the same objective in the backward direction and filtered out with a dichroic mirror and shortpass filter (SPF) before entering a calibrated grating monochromator (Horiba, iHR320). The monochromator had a cooled, sensitive CCD detector (Horiba, model Syncerity-356399). The SHG signal beam was additionally sent to a photomultiplier tube (PMT) to enable imaging of the sample. Data acquisition was performed using a data acquisition card as well as with the monochromator's USB interface. Both experimental components were controlled with the LabVIEW interface software.      with optical frequency ω passes through a nonlinear crystal generating forward-and