Development of a multi channel acquisition scheme for a broadband Stimulated Raman Scattering microscopy system

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Facoltà di scienze matematiche, fisiche e naturali

Corso di laurea magistrale in Fisica Medica

Development of a multi channel

acquisition scheme for a broadband

Stimulated Raman Scattering

microscopy system

Candidato

Luca Genchi

Relatori

Prof. Carlo Liberale

Prof. Francesco Fuso

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We are like dwarfs sitting on the shoulders of giants. We see more, and things that are more distant, than they did, not because our sight is superior or because we are taller than they, but because they raise us up, and by their great stature add to ours. - John of Salisbury

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Abstract

Coherent Raman scattering microscopy has emerged as a high-speed, high sensitivity vibrational imaging platform. Stimulated Raman scattering (SRS) microscopes, in particular, with their subcellular resolution, combined with the capability to collect precise vibrational signatures, are allowing to perform quantitative chemical imaging even in live cells. Being intrinsically label-free, vibrational spectroscopic microscopes allow the measurement of biological samples without perturbing the function of molecules.

The SRS microscope implemented at the Vibrational Imaging laboratory (Vibra Lab) at King Abdullah University of Science and Technology (KAUST) uses spectral shaping of a femtosecond laser through a fast and narrowband Acousto Optical Tunable Filter (AOTF), which allows scanning all the way from the Raman fingerprint region to the CH-stretch region without the need of any change in the optical setup, therefore ensuring high acquisition speed. The high enough spectral resolution (7 cm−1) of the AOTF allows to properly resolve Raman peaks in the highly crowded fingerprint region.

The aim of this thesis work, carried out during an internship at KAUST, is to characterize and improve the signal to noise ratio measurement of the Vibra Lab SRS microscope. In this thesis, a brief introduction to vibrational spectroscopic imaging is provided in chapter 1, pointing out critical aspects and strengths of coherent Raman microscopy and introducing biological applications of this technique. A more detailed description of the setup and discussion of its performance is given in chapter 3. In chapter 4, the AOTF is utilized to develop a multi channel acquisition scheme using Hadamard reconstruction method. In chapter 5, applications of this setup are discussed. At the end, in chapter 6, I summarize the work done and introduce future developments.

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Chapter 1

Introduction to Vibrational

spectroscopic imaging

Vibrational spectroscopy has been largely used to study molecules both in gas and condensed phase, as well as at interfaces. The idea of using the spectrum of biomolecules to act as a natural contrast medium for imaging of living systems is opening new scenarios for the understanding of cellular machinery and giving new possibilities for the molecule-based diagnosis. Vibrational spectroscopic imaging devices, thanks to their sensitivity and capability to analyze samples without labels, are supposed to become fundamental clinical tools for disease diagnosis and therapy effectiveness evaluation in the future [1–4].

The transition from spectroscopy to spectroscopic imaging, however, is not only a combi-nation of spectrometry and microscopy. After a brief introduction about the advantages of using spectroscopic microscopes, in this chapter I will discuss critical aspects of coherent Ra-man scattering (CRS) systems, focusing the attention on stimulated RaRa-man scattering (SRS) microscopes and the state of the art of the research at the moment.

1.1 Advantages of using spectroscopic microscopes

The most commonly used microscopy techniques to image biological samples are based on fluorescent molecular tags, as green fluorescent proteins that facilitate the imaging of protein dynamics in living cells, and the recent development of super-resolution fluorescence microscopy methods has allowed the study of cellular structures at a nanometric scale. The labeling ap-proach used in these techniques, however, presents some possible side effects and limitations:

1. labels may perturb the function of a biological molecule;

2. delivery of labels to a target is sometimes difficult, especially under in vivo conditions; 3. potential toxicity sometimes prevents the use of labels in human subjects.

Vibrational spectroscopic imaging overcomes these barriers and it’s really suited to in vivo applications [1, 3–6]. Molecules can be recognized thanks to their distinctive vibrational sig-nature produced by quantized vibrations of chemical bonds. These vibrational spectra can be recorded using infrared (IR) absorption or Raman scattering. IR imaging techniques, however, have limited spatial resolution due to the long wavelength utilized [3], and also have limited in

vivo applications due to strong absorption by water and the induced heat in the samples.

Raman spectroscopy, which uses visible or near IR light for excitation, can also be used to obtain a chemical map of the sample, with a reasonable spatial resolution. Raman spectrometers have been extensively used for analysis of cells and tissues. There are already commercially available microscopes based on the so called spontaneous Raman process, which usually requires an acquisition time on the order of a second to measure the Raman spectrum at a given position of a sample, implying that to acquire images of 100 × 100 pixels a relevant portion of an hour is

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required. Spontaneous Raman microscopes are thus not suitable to study dynamical processes of biological samples because of their limited acquisition rate.

Instead, Coherent Raman scattering microscopes, which use the so called stimulated Raman process, are orders of magnitude (4 − 6) faster than spontaneous Raman microscopes [7, 8], at the cost of a substantial increase in complexity of the apparatus. Indeed very few systems exist at a commercial level, due to the not yet mature state of the engineering of the systems and the ongoing proposal of new configurations.

1.2 Coherent Raman Scattering microscopy

In 1928, C.V. Raman and his collaborator K.S. Krishnan described what they called a new type

of secondary radiation [9], being the inelastic Raman scattering. Because of the small cross

section, such discovery had little practical use until the invention of the laser. Lasers, giving a bright light source, not only increased the signal of spontaneous Raman scattering, but also facilitated the investigation of coherent Raman scattering [10]. Coherent anti-Stokes Raman scattering (CARS) was first used to produce an image in 1982 [11], but because of technical difficulties was not pursued again until 1999 [12]. Thanks to the emergence of commercially available near-infrared pulsed laser, coherent Raman imaging has grown from that day and many different acquisition schemes have been developed. In 2008, Freudiger et al. [13] presented the first biomedical imaging application using a heterodyne detected SRS microscope.

As discussed in detail in chapter 2, Coherent Raman Scattering microscopy is a class of third-order nonlinear microscopy techniques for label-free, non-invasive and non-destructive imaging that overcomes the frame rate limitation of spontaneous Raman microscopes. It has thus emerged as a high-speed, high sensitivity vibrational imaging platform. Coherent Raman microscopes use two synchronized and frequency detuned laser beams, the Pump (ωp) and the

Stokes (ωs). When the difference between the frequencies of the fields matches the frequency of a

Raman-active molecular vibration, the molecules in the focal volume oscillate in phase and four different processes can occur at the same time: coherent anti-Stokes Raman scattering (CARS), coherent Stokes Raman scattering, stimulated Raman gain (SRG) and stimulated Raman loss (SRL); the last two are strictly connected to each other, so they are usually addressed as stimulated Raman scattering (SRS). All these techniques have been used for spectroscopy of biological samples. In particular, SRS is interesting because is free of nonresonant background [13–15] (present in CARS as discussed in chapter 2) and provides vibrational spectral profiles that are nearly identical to spontaneous Raman spectroscopy [8, 15], giving access to the wide catalog of spectra already existing in the literature; last, but most important, the SRS intensity is linearly dependent on molecular concentration [3,8], so it can be used as quantitative method for chemical imaging [15–17]. These reasons currently make SRS the leading CRS microscopy technique for biomedical applications.

1.2.1 Biomedical applications of Raman microscopes

In biological samples, approximately 90% of the relevant Raman peaks are found in the

fin-gerprint spectral region, from ∼500cm−1 _{to ∼1800cm}−1_{, while the other 10% is found in the}

higher energy CH-stretch region, from ∼2700cm−1 to ∼3300cm−1 [3].

CRS microscopes provide information on the biochemical composition of tissues generating image contrast using the Raman active vibrational frequency of a given chemical compound. Moreover, the nonlinear CRS signal is generated only at the focus where the excitation inten-sities are the highest. This leads to 3D sectioning capability [12], which is essential for imaging tissues or cell structures. This allows label-free visualisation for a number of biomedical appli-cations, here briefly listed for some specific examples.

Neurological applications

The sheats of neuronal axons are made of myelin, a dense and lipid-rich electrical insulator, which gives a strong Raman response in the CH-stretch region. Demyelination plays a large role in neurodegenerative diseases, and can be studied using CRS microscopes [18–20].

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Cancer studies

Recently, lipid droplets have been identified as distinctive markers in colorectal cancer [21] and in general as key player in cancer cells and tissues, especially in cancer stem cells [22], increasing the already existing interest in imaging using CH-stretch region of spectra [23], being CH bond present in such droplets.

Drug interactions

The fast acquisition speed, combined with good photostability and lack of phototoxicity asso-ciated with CRS, permits real-time imaging of drug distribution within cells with a resolution that allows precise intracellular registration [2, 5, 13, 24].

SRS, in particular, being linearly dependent on Raman scatterers concentration [8], allows quantitative imaging and provides a unique opportunity to understand drug distribution within individual cells by delivering single cell pharmacokinetic and pharmacodynamic read-outs [24].

Intraoperative histology

Recently, intraoperative application of SRS has been demonstrated [25]. SRS microscopy can locate tumor infiltration in areas that appeared normal by eye, which suggests that this tool could be applied during surgery. Using two peaks in the CH-stretch, infact, Ji et al. [25] demonstrated ex vivo and in vivo imaging of healthy and tumorous brain tissue.

SRS microscopy has also been demonstrated as an equivalent diagnostic tool of traditional hematoxylin and eosin (H&E) staining [3, 4, 8, 25, 26], being though faster and without intro-ducing artifacts due to sample preparation.

1.2.2 Performance of CRS microscopes

Using the fact that a cell is a spatially and temporally organized dynamic system, recording a fingerprint spectrum at every pixel of a microscopy image with sufficient speed and performing multivariate analysis on the resulting data set, a given molecule could be located and monitored in real time in a living system, giving information about its functionalities. However, this is not easy to achieve at this point because of many technical difficulties leading to several questions: is it possible to increase the speed of spectral acquisition to microsecond scale in order to capture the dynamics of biomolecules in a living cell? Is it possible to achieve enough sensitivity to detect target molecules in a tissue environment without harming the cells? Is it possible to reach a nano scale resolution in order to enable imaging of biological structures? And last, but most important, is it possible to detect a spectrum from a centimeter deep tissue for in vivo diagnosis?

These are now still open questions [4] about vibrational spectroscopic imaging, although some advances have been done in these directions [3]. Even if there are still some doubts about the optimal configuration for coherent Raman scattering, it is clear and uncontroversial that this approach offers the best sensitivity and the highest image acquisition speed for vibrational spectroscopic imaging [1].

The majority of CRS microscopes acquire spectral information in a time of approximately 10µs per cm−1and can generate good quality signal over a particular and limited spectral range. The fastest narrowband methods are able to perform acquisitions in a time down to ∼100ns per cm−1, but they can acquire signals only in the CH-stretch range, the strongest spectral region. Boht CARS and SRS show similar detection limits, between tens of µmol per liter to tens of mmol per liter [3].

It is important to underline that, even if the theoretical background of coherent Raman scattering is well known [8], fundamentals of photodamage are still lacking, thus generally limiting performance of microscopes through conservative light power. A better understanding of photodamage process could give an improvement in signal generation and detection.

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CARS microscopes

Different coherent Raman imaging techniques exhibit a wide range in performance. Narrow-band CARS, for example, acquire images with a short pixel dwell time (∼ 0.2µs) [27] but only stimulates narrow regions of the spectrum, so they lack in chemical information. Using nar-rowband excitations, multi-colour images can be formed from several co-localized images with different spectral content. It is important to underline that narrowband CARS microscopy can acquire at video rates thanks to an high signal to noise ratio (SNR), as achievable in the CH-stretch region, but does not show the same performance at lower wavenumbers, in the fingerprint region [3].

Other techniques, such as broadband CARS [28], allow performing spectral acquisition from fingerprint to CH-stretch regions, at the expense of a longer (∼ 10ms) pixel dwell time [29].

The main limitation of CARS microscopes is the presence of the nonresonant background; this limitation becomes more significant for biological samples in the more populated fingerprint region of Raman spectra, making raw CARS signal unusable [3]. These limitations are overcome by SRS microscopes.

SRS microscopes

Contrary to narrowband CARS microscopes, narrowband SRS has been used to demonstrate high SNR across the entire Raman spectrum [30]. They are intrinsically free of nonresonant background, present in CARS, as will be explained in chapter 2; nevertheless, unless complicated setups are used [14], they may still present some background signal arising from third harmonic generation (THG) and cross phase modulation (XPM) [17]. SRS microscopes usually utilize MHz lock-in detection in order to reduce the laser intensity noise to the shot noise limit [31,32]. The disadvantage of using a lock-in detection scheme is that it is not simple to implement an array of phase-synchronized detectors, limiting thus the possibility to have a broadband SRS acquisition scheme. Moreover, due to sample damage constraints, spectroscopic SRS techniques utilize average laser power similar to that typically used by narrowband SRS, but divided over a broader spectral range [32].

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Chapter 2

Theoretical treatment of Raman

scattering

Coherent Raman Scattering (CRS) is a special class of light-matter interactions, whose response contains information about molecular vibrations resonant with the difference in frequencies of two incident light fields ω1, ω2. It is thus possible to probe low frequencies nuclear vibrations of

molecules and material using optical light fields, tuning the Ω = ω1− ω2into the characteristic

vibrational mode ωv.

Coherent Raman techniques are related to spontaneous Raman scattering. Spontaneous and coherent both allow vibrational spectroscopic examination of molecules with visible and infrared light sources, but CRS provides much stronger vibrationally sensitive signals [7]. Co-herent Raman scattering not only gives a much improved signal level compared to spontaneous Raman [8, 10, 33], making possible to have fast scanning capabilities [1, 3, 13], but also offers a more detailed control of the Raman response, making possible to selectively probe both the electronic and vibrational response properties of the sample [8]; CRS also provides a more detailed information about molecules orientation inside samples [34, 35].

In this chapter I briefly analyze nonlinear optical processes and basics of Raman interactions, discussing more in detail the coherent Raman scattering (CRS) process. A classical model [8] of interactions is given, which is useful to get qualitative but not necessarily inaccurate information about the Raman processes; successively a more accurate, semi-classical model [8,36] is provided and finally stimulated Raman scattering is discussed using a quantum approach [8, 10, 33, 37].

2.1 Nonlinear optical processes

It is possible to describe both linear and nonlinear optical effects as resulting from interaction of the electric field component of electromagnetic radiation with charged particles in materials and molecules. Using electromagnetic waves it is thus possible to apply a periodic perturbation to a sample, producing vibrations in the electric structure of the target. Electromagnetic waves in the visible and near-infrared oscillate with a frequency of ν ∼ 103_{THz, too high for the heavy}

nuclei to follow adiabatically the oscillation but not for the lightweight electrons. The driving fields thus slightly displace electrons from their equilibrium positions, inducing an electric dipole moment ~µ parallel to the field. Projecting onto electric field direction we have

µ(t) = −e · r(t) (2.1)

with e electric charge of the electron, r(t) displacement. The latter depends on how strong the electron is bound to the nucleus, being bigger for weakly bound electrons and smaller for tightly bound ones. When the displacement is not too large, i.e. electrons are close to nucleis, it is possible to approximate the electron binding potential as an harmonic one. The macroscopic polarization is obtained adding up all the N electric dipoles in the unit volume

~

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Material χ(3) _{λ (nm)} _Reference

Water 1.3 × 10−14 1060 [38] Ethanol 1.3 × 10−14 1060 [38] Vegetable oil 1.3 × 10−14 1060 [38] Silica 1.3 × 10−14 1060 [39]

Table 2.1: Magnitude of χ(3)_{measured with third-harmonic generation measurements at the indicated}

excitation wavelength.

and, assuming the applied electric field is weak compared to the field that binds the electrons, we have

~

P (t) = 0χ ~E(t) (2.3)

with 0 electric permittivity in vacuum, χ susceptibility of the material.

As already stated, eq. 2.3 holds only in weak fields approximation. For stronger electric fields, electrons are farther displaced from equilibrium positions and the binding potential can no longer be approximated as harmonic; thus, we have to add corrections to the polarization. Assuming that the anharmonic contributions are small compared to the harmonic effect, it is possible to express the electron displacement, and thus the polarization, as a power series in the field, obtaining, along the direction of the field

P (t) = 0χ(1)E(t) + χ(2)E2(t) + χ(3)E3(t) + . . . =

= P(1)(t) + P(2)(t) + P(3)(t) + . . . (2.4) with χ(n)_{, P}(n) _{respectively n-th order of susceptibility and polarization.}

The coherent Raman effects arise from the third-order contribution to the polarization P(3)

of materials, so their magnitude is given by the strength of the triple product of incoming fields and the amplitude of the third-order susceptibility χ(3)_{. To understand the effects of}

nonlinearity to the material it can be useful to compare the amplitude of the linear susceptibility

χ(1)_{, close to the unity in condensed phase, to the amplitude of the third-order χ}(3) _{of different}

materials, listed in table 2.1. The nonlinearity of electronic motion becomes more significant when the magnitude of the external applied field is comparable to the atomic field that binds electrons, Ea ≈ 2 · 107 esu, meaning that the intensity of the laser beam should be I ∼ 1014

W cm−2; it is thus clear that, being the typical laser intensities used in Raman spectroscopy on the order of ∼ 1010 W cm−2, the approximation used to state eq. 2.4 holds [8]. The magnitude of χ(3)_{, however, grows larger when the electron displacement is enhanced, such}

as under electronically resonant conditions; in this case, the third-order nonlinear response is stronger. Nuclear resonances can also affect the electronic nonlinear susceptibility. The adiabatic potential that electrons feel, in fact, depends on positions and movements of the nucleis; the electronic polarizability is thus perturbed by lattice vibrations.

With CRS, we obtain chemical information from the sample by probing nuclear resonances.

2.2 Classical model of Raman scattering

2.2.1 Spontaneous Raman scattering

Although visible and infrared electromagnetic waves move electrons, their oscillatory motion contains information about position and movement of nuclei. This happens because the elec-tronic potential depends on nuclear coordinates, thus the elecelec-tronic polarizability is perturbed by the presence of nuclear modes. To show this dependence, it is useful to use the relation between the electric dipole moment ~µ and the polarizability α(t), under the assumption that

the driving frequency is far from any electronic resonance of the system

~

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and, using the displacement Q of nuclei from their positions in absence of external fields, express the polarizability with a Taylor expansion near Q = 0

α(t) = α0+ ∂α ∂Q ₀ Q(t) + . . . (2.6)

with α0constant polarizability, obtained far from any nuclear resonance. The first order of the

expansion expresses the coupling between nuclear and electron coordinates. We can assume an harmonic motion of the nuclei

Q(t) = 2Q0cos(ωvt + φ) = Q0ei(ωvt+φ)+ e−i(ωvt+φ)

(2.7) with Q0amplitude, ωv nuclear resonance frequency, φ phase of the nuclear mode vibration.

Introducing the oscillating field E(t) = Ae−iω1t_{+ c.c. in eq. 2.5, the dipole moment along}

the field direction becomes

µe(t) = α0+ ∂α ∂Q ₀ 2Q0cos(ωvt + φ) · Ae−iω1t_{+ c.c.} = = α0Ae−iω1t+ A ∂α ∂Q ₀ Q0

e−i(ω1−ωv)t+iφ_{+ e}−i(ω1+ωv)t−iφ

+ c.c. (2.8) which oscillates at several frequencies. The first term of eq. 2.8 is the elastic Rayleigh scattering that doesn’t change the vibration frequency; the two terms in the square brackets represent the anelastic Raman scattering with a shift in the vibration frequency, respectively ω1− ωv

(Stokes-shifted) and ω1+ ωv (anti-Stokes-shifted). It is important to note that the anelastic

terms depend directly on the amplitude of ∂α ∂Q

₀ and so on the nuclear motions.

Generation of the spontaneous Raman scattering signal

As stated in eq. 2.8, in the classical model of Raman scattering the harmonic nuclear motion creates frequency-shifted components in the electric dipole. The amplitude of Stokes and anti-Stokes vibrations are thus proportional to the magnitude of the electric field radiated by the dipole at corresponding frequencies. In this section the analysis is made for the Stokes frequency

ωs = ω1− ωv. The procedure is the same, with obvious substitutions, for the anti-Stokes

component.

Using the far field approximation, it is possible to express the amplitude of the electric field along ~r emitted from the oscillating dipole in 2.8 as

E(ωs) = ω2 s 4π0c2 |µ(ωs)| eikr r sin θ (2.9)

with θ, r observation angle and distance, k wave vector of the radiated field. The outgoing energy flux is given by the modulus of the Poynting vector

S(ωs) =

0c

2 |E(ωs)|

2 _(2.10)

and the total energy irradiated from the dipole is obtained integrating over the solid angle

I(ωs) = ω4s 12π0c3 Q20|A|2 ∂α ∂Q 2 (2.11) According to the classical model here shown, the total energy irradiated from the dipole is thus proportional to the intensity of the incoming radiation Iin = |A|2, to ω4s and shows a

dependence on_∂Q∂α

2

which is experimentally assumed as a cross section σ of the process. It is also important to note that the spontaneous Raman emission is incoherent because nuclear vibrations of different molecules are uncorrelated, so each molecule i carries his own phase φi;

consequently, the intensity of the total Raman emission is proportional to N times eq. 2.11, being N the total number of Raman scatterers.

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Using a light source with intensity I0, the total scattered Stokes-shifted light from a sample

with molecular density n and length z can be written as

I(ωs) = nzσ(ωs)I0 (2.12)

The classical model provides an intuitive interpretation of the light-matter interaction, but fails when it tries to describe the resonant polarizability and is not capable to predict the relationship between Stokes and anti-Stokes intensities. Before providing a semi-classical de-scription of the process using a quantum formalism for matter and a classical one for waves, given in section 2.3, a classical description of CRS is given.

2.2.2 Coherent Raman scattering

The classical model of the coherent Raman scattering provides an interpretation of the interac-tion as caused by driven nuclear oscillainterac-tions in the sample. This derivainterac-tion is given for a single harmonic nuclear mode per molecule, but can be easily generalized. The treatment of the CRS phenomenon can thus be divided in two steps:

1. two incoming beams produce oscillations in the molecular electron cloud. These vibrations create a force field along the vibrational degree of freedom, driving nuclear modes; 2. the nuclear motion creates a coherent spatial modulation of the refracting properties of

the material. This modification is the reason for the scattering fields and creates the Raman shift.

Driven Raman mode

As made in section 2.2.1, it is possible to sketch the nuclear motion as an oscillation in the

Q coordinate. The vibrational motion will be given by a damped harmonic motion with a

resonance frequency ωv. The system is hit by two incoming beams E1, E2 respectively with

frequencies ω1, ω2 far from the resonance ωv of the material (ω1 > ω2 >> ωv). Electrons

surrounding the nuclei follow adiabatically the field and, if the intensities of fields are high enough, nonlinear effects can occur and electrons can move at combination of frequencies, including the difference one Ω = ω1− ω2. There is thus an effective force on the oscillator

F (t) = ∂α ∂Q ₀ A1A∗2e−iΩt+ c.c. (2.13) being A1, A2respectively the intensities of E1and E2. What equation 2.13 says is that, because

the electronic and nuclear motions are related through a nonzero _∂Q∂α₀, the modulated electron cloud creates a force that oscillates at Ω frequency and affects nuclear modes. The nuclear motion is thus determined by

d2Q dt2 + 2γ dQ dt + ωvQ(t) = F (t) m (2.14)

with γ damping coefficient and m reduced mass of the oscillator. Solving the above equation we find Q(t) = Q(Ω)e−iΩt+ c.c. Q(Ω) = 1 m ∂α ∂Q ₀ A1A∗2 ω2 v− Ω − 2iΩγ (2.15) and the resonance effect is now clear, being the amplitude of the oscillatory motion the largest when the difference frequency Ω matches the oscillator’s resonance one ωv.

Probe modulation

Because of the nuclear motion in eq. 2.15, the applied electric fields E1, E2 experience an

altered electronic polarizability when they propagate in the material. Using equations 2.2, 2.5 and 2.6 we have P (t) = N α0+ ∂α ∂Q 0 Q(t) E1(t) + E2(t) (2.16)

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The first term in square brackets of eq. 2.16 gives the linear polarization PL(t) of the material,

while the second one describes the contribution to the third-order nonlinear polarization PN L

due to the driven nuclear modes. Substituting eq. 2.15 and using the expression of the fields

E1= A1e−iω1t+ c.c., E2= A2e−iω2t+ c.c., we obtain

PN L(t) = P (ω1)e−iω1t+ P (ω2)e−iω2t+ P (ωcs)e−iωcs+ P (ωas)e−iωast+ c.c. (2.17)

with ωcs = 2ω2− ω1 called coherent Stokes frequency, ωas = 2ω1− ω2 anti-Stokes frequency.

Defining the nonlinear susceptibility

χN L(Ω) = n 6m0  ∂α ∂Q ₀ 2 ₁ ω2 v− Ω2− 2iΩγ (2.18) we can explicitate all the nonlinear polarizations of equation 2.17

P (ω1) = 60χN L(Ω)|A2|2A1 (2.19) P (ω2) = 6∗0χN L(Ω)|A1|2A2 (2.20) P (ωcs) = 6∗0χN L(Ω)A22A ∗ 1 (2.21) P (ωas) = 60χN L(Ω)A21A∗2 (2.22)

These are, respectively, responsible for Stimulated Raman Loss (SRL) eq. 2.19, Stimulated Raman Gain (SRG) eq. 2.20, Coherent Stokes Raman Scattering (CSRS) eq. 2.21, Coherent Anti-Stokes Raman Scattering (CARS) eq. 2.22. They all depend on the magnitude of the same χN L in eq. 2.18, so they are all comparable in magnitude, being produced by the same

nuclear vibration at ωv. It is important to underline at this point that, even if the four CRS

effects have similar amplitude, the actual detected signal for different CRS techniques may not have similar strength.

Energy flux in CRS

In coherent Raman processes, energy contained in the incoming fields is redirected in two different ways

dissipative processes: when the total energy contained in all the light beams is lower after

passing through the sample. In this case, the material absorbs energy;

parametric processes: new light fields are generated without energy exchange with the

mate-rial, that acts merely as a mediator. The total energy of light fields can be thus conserved. Let’s start the discussion considering the fields generated with new frequencies ωcs and ωas.

The nonlinear field in the anti-Stokes frequency is given by

Eas= Aase−iωast (2.23)

with corresponding intensity

I(ωas) =

0c

2 |Aas|

2 _(2.24)

that, if no fields with ωasare present before the interaction in the sample, can only be generated

by the nonlinear polarization oscillating at ωas; the magnitude Aas is thus proportional to

P (ωas), and using eq. 2.22 we have

I(ωas) ∝ χN L 2 I₁2I2 (2.25)

with I1, I2 intensities of the fields oscillating at ω1and ω2 respectively. The same discussion is

valid for the coherent Stokes

I(ωcs) ∝ χN L 2 I₂2I1 (2.26)

Intensities of eq. 2.25 and 2.26 are both detected as homodyne signals, i.e. proportional to the modulus square of the nonlinear polarization. In this case, the energy of both beams is

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extracted from the incident beams oscillating at ω1and ω2through a parametric process, so no

excitation occurs in the sample.

It is possible to introduce an additional field, commonly called local oscillator, oscillating at

ωasor ωcswith a well defined phase relation with the nonlinear polarization in the material. In

this case, there is more than one source of radiation at the signal frequency and the intensity can be written as I(ωas) = 0c 2 E_as(3)+ E_asloc 2 ∝ E_as(3) 2 +E_asloc 2 + E_as(3)∗ E_asloc+ E_as(3) E_asloc∗ (2.27)

with Eas(3), Easlocelectric fields of the third-order polarization and the local oscillator respectively.

The terms in the square brackets represent the heterodyne mixing that depends on both fields. It is possible to express the heterodyne intensity

Ihet(ωas) = 2Alocas

Re(E_as(3)) cos φ + Im(E_as(3)) sin φ

= = 2α

Re(χN L) cos(φ − φp) + Im(χN L) sin(φ − φp)

(2.28)

with α = |Aloc

asA21A2|, Alocas amplitude of the local oscillator, φ phase difference between E

(3)

as and

the real Eloc

as, φp phase difference between the radiated field E

(3)

as and the induced polarization

P (ωas). Under resonance conditions, however, the nonlinear susceptibility in eq. 2.18 becomes

purely imaginary, thus the total intensity in the anti-Stokes channel is given by

I(ωas) ∝ E_as(3) 2 +E_asloc 2 + 2α Im(χN L) sin(φ − φp) (2.29)

that clearly depends on the phase difference ∆φ = φ − φp. A more detailed discussion needs

calculation of the phase difference using propagation factors [8] but is beyond the scope of this section. Here we can briefly discuss interesting cases arising from eq. 2.29. When ∆φ = 0, the heterodyne terms is zero and the total intensity is given by the sum of the homodyne anti-Stokes term and the local oscillator intensity. When ∆φ = −π/2, being the Im(χN L) > 0 as visible

in eq. 2.18, the total intensity is less than the sum of anti-Stokes and local oscillator terms; in this condition, the CARS process is not purely parametric anymore as dissipative contribution arises.

What we learned so far is that the presence of a phase coherent local oscillator can affect the sensitivity of the measurement, allowing to probe parametric or dissipative processes. The discussion given above is useful to understand the SRL and SRG processes. In the former, the signal is probed at ω1 frequency so the P (ω1) acts as source of E

(3)

1 field, that interferes with

the external field E1, acting here as local oscillator. The total intensity detected at ω1 will be

I(ω1) = 0c 2 E (3) 1 + E1 2 ∝ E (3) 1 2 +E1 2 + 2I1I2

Re(χN L) cos(∆φ) + Im(χN L) sin(∆φ)

(2.30) On a detector in the far field the phase shift will be ∆φ = −π/2 [8], i.e. the real part of the material response is π/2 phase retarded compared to E1, the intensity becomes

I(ω1) ∝ E (3) 1 2 +E1 2 − 2I1I2Im(χN L) (2.31)

The loss of intensity due to the last term of eq.2.31 arises due to the destructive interference between the driving and induced fields. This attenuation is mediated by the dissipative part of the interaction, being proportional to the imaginary part of the χN L.

The same description given for the SRL can be done for the SRG. The incoming field at ω2

acts as a local oscillator and creates interference. Using a ∆φ = π/2 and χ∗_{N L}= −χN Lat the

resonance, we have I(ω2) ∝ E (3) 2 2 +E2 2 + 2I1I2Im(χN L) (2.32)

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Figure 2.1: Input and output spectra of SRS and CARS.

From eq. 2.32 we can see that the intensity at ω2 experiences a growth; this gain is due to

constructive interference between the driving and induced fields. A sketch of the effect of SRL and SRG on the power of interacting field is shown in figure 2.1.

It is possible to selectively detect the heterodyne portion of signal, thus the resulting SRL and SRG signals are directly proportional to the dissipative part of coherent Raman interaction

Ihet(ω1) = −2I1I2Im(χN L) (2.33)

Ihet(ω2) = +2I1I2Im(χN L) (2.34)

2.2.3 SRL and SRG signals

Equations 2.33 and 2.34 tell us that, if we use an heterodyne detection, the signal detected in SRL and SRG microscopes are directly proportional to the intensities of the applied fields. When these measurements are normalized to the intensity of the applied field, one obtains

SRL = I het_(ω 1) I1 ∝ −2I2Im(χN L) (2.35) SRG = I het_(ω 2) I2 ∝ +2I1Im(χN L) (2.36)

Both SRL in 2.35 and SRG in 2.36 don’t depend on the intensity of the probed field (Pump for SRL, Stokes for SRG) but are directly proportional to the intensity of the other field applied (Stokes for SRL, Pump for SRG).

2.3 Semi-classical model of Raman scattering

The picture given by the classical model in the previous section used an harmonic oscillator, driven at a frequency ωv, to justify the modulation that affects the amplitude of incoming

fields E1, E2. SRL and SRG are produced by destructive and constructive interferences, and

subsequent loss (eq. 2.31) and gain (eq. 2.32) are thus proportional to imaginary part of the nonlinear susceptibility. The classical model, however, doesn’t take into account the quantized nature of nuclear oscillations. A more rigorous treatment could be provided using quantum mechanics to describe the matter, still maintaining a classical description of fields.

It is possible to describe the state of the material in terms of molecular wavefunctions, written as a superposition of molecular eigenstates

ψ(~r, t) =X

i

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with ~r coordinate vector of both electrons and nuclei, cn projection of ψ along eigenstates.

Introducing ˆH0, the Hamiltonian of the system in the absence of any external field, it is possible

to write the time evolution of the system

i~dψ_dt = ˆH0ψ (2.38)

and being ψi eigenstates of the Hamiltonian, their evolution is given by

ψi(~r, t) = ai(~r)e−iωit (2.39)

with ai(~r) the spatially varying part of the wavefunction. When an external field is applied,

the Hamiltonian of the system is given by ˆ

H = ˆH0+ ˆV (t) (2.40)

where the interaction term is given by ˆ

V (t) = −ˆ~µ · ~E(t) (2.41)

thus both electrons and nuclei are set in motion by the electric field they experience at time t. The electric dipole operator of eq. 2.41 is given by

ˆ

µ =X

j

ejˆrj (2.42)

with the sum running over electrons and nucleis.

Once the wavefunction of the system is found with the new Hamiltonian, it is possible to calculate the polarization as the expectation value of the dipole operator

P (t) = N < ˆµ(t) >= N hψ(r, t)| ˆµ |ψ(r, t)i (2.43)

with N number density in the volume V .

In order to write the polarization of eq.2.43, we need to find the wavefunction of the system with the Hamiltonian of eq. 2.40. In principle we could use perturbation theory and expand the wavefunction to the n-th order. This method, however, is not able to properly account for broadening mechanisms of spectroscopic features due to coupling to other degrees of freedom. A more common way to describe the phenomenon is using the density matrix formalism.

2.3.1 Density Matrix

Using the eigenstates |ni of the unperturbed system, we can define the density matrix operator as

ˆ

ρ ≡ |ψ(t)i hψ(t)| =X

nm

ρnm(t) |ni hm| (2.44)

that depends on the matrix elements ρnm= hn| ˆρ |mi and the operator |ni hm|, called coherence.

Diagonal elements of the matrix ρnn give the probability of the system to be in |ni, while the

off diagonal elements ρnm, n 6= m, imply that the system is found in a coherent superposition

of eigenstates |ni and |mi and represent the amplitude of the coherence. The time evolution of the density operator can be written as

d ˆρ dt = − i ~ _ˆ H, ˆρ (2.45) with the square brackets indicating the commutator operation_ˆ

H, ˆρ = ˆH ˆρ − ˆρ ˆH. We can now

express the dipole operator as function of the density one

< ˆµ(t) >=X

nm

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It is possible to find the density matrix of the system with a perturbation expansion of ˆρ(t)

in powers of the electric field ˆ

ρ(t) = ρ(0)(t) + ρ(1)(t) + ρ(2)(t) + ρ(3)(t) + . . . (2.47)

with ρ(n)_{(t) representing the n-order contribution in the electric field. The n = 0 order}

repre-sents the unperturbed density matrix at thermal equilibrium

ρ(0)(t) = ρ(−∞) = e

− ˆH/kBT

Tr e− ˆH/kBT

(2.48)

with kBBoltzmann’s constant. The successive terms have way more complicated expressions [8],

so it can be useful to introduce a more compact form that utilizes the Liouville space operators H, V(t). These operators act on an ordinary operator ˆA

H ˆA ≡_ˆ

H, ˆA

V(t) ˆA ≡_ˆ

V (t), ˆA

(2.49) It is thus possible to rewrite eq. 2.45 as

d ˆρ

dt = −

i

~H ˆ

ρ (2.50)

As already stated using the classical model, the coherent Raman interaction is a third-order process; we are thus looking for ρ(3)_{. The calculation of this term is beyond the scope of this}

section, I will use the expression predicted by perturbation theory [40]

ρ(3)(t) = −i ~ 3Z ∞ 0 dτ3 Z ∞ 0 dτ2 Z ∞ 0 dτ1 × G(τ3)V(t − τ3)G(τ2)V(t − τ3− τ2)G(τ1)V(t − τ3− τ2− τ1)ρ(−∞) (2.51)

where the variables τi run over the interval between the application of a light field incident

at ti−1 and the one incident at ti; the Liouville space Green’s function G(t) describes the

propagation of the material system when no light fields are present, and is given by

G(τ ) ≡ θ(τ )e−iHτ /~ (2.52)

with θ(τ ) the Heavyside step function. The interpretation of eq. 2.51 is that the system starts from a thermal equilibrium condition ρ(−∞) and is then perturbed by successive light fields, as described by V operator. Between different interactions, the system evolves according to the Green’s function G.

2.3.2 Response function and third-order susceptibility

We can use eq. 2.51 to evaluate the polarization 2.43. In particular, we are interested in the third order expansion

P(3)(t) = N Tr ˆµρ(3)(t) (2.53) that we can obtain as

P_i(3)(t) = NX jkl Z ∞ 0 dτ3 Z ∞ 0 dτ2 Z ∞ 0 dτ1R (3) ijkl(τ3, τ2, τ1) × Ej(t − τ3)Ek(t − τ3− τ2)El(t − τ3− τ2− τ1) (2.54)

where the indices i,j,k,l indicate the polarization orientation in cartesian coordinates and R(3)_ijkl is the third-order response function

R(3)_ijkl(τ3, τ2, τ1) =  i ~ 3 ˆ µiG(τ3)ˆµLjG(τ2)ˆµLkG(τ1)ˆµLlρ(−∞) (2.55)

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with ˆµL _{representing the Liouville space version of the electric dipole operator ˆ}_{µ. The response}

function in eq. 2.55 describes the time-ordered response of the material to the light fields, and it is needed when time-resolved coherent Raman spectroscopy experiments are performed. For what it concerns CRS microscopy, at the moment, the equation is not useful because the response is rarely time-resolved in fast imaging applications. It is more interesting to point out the dependence on the magnitude of the polarization at different vibrational frequencies, because this is more practically related to imaging experiments [35].

In order to get the nonlinear polarization in the frequency domain, it is useful to assume that light sources are spectrally narrow; with this assumption, we can write the nonlinear polarization that oscillates at the signal frequency ω4= ω1+ ω2+ ω3

P_i(3)= Pi(ω4)e−iω4t+ c.c. (2.56)

and using the fact that Pi(t) is a real function, the relation Pi∗(ω4) = P (3)

i (−ω4) holds. The

amplitude of the nonlinear polarization at frequency ω4is given by

Pi(ω4) = N

X

jkl

R(3)_jkl(ω4, ω1+ ω2, ω1)Ej(ω1)Ek(ω2)El(ω3) (2.57)

with R(3)_jkl frequency domain response function given by

R(3)_jkl(ω4, ω1+ ω2, ω1) = −1 ~ 3 ˆ µiG(ω4)ˆµLjG(ω1+ ω2)ˆµLkG(ω1)ˆµl−Lρ(−∞) (2.58) In eq. 2.58, G is the frequency domain Green’s function

G(ω) = −i Z ∞

0

dtG(t)eiωt (2.59)

Green’s function in eq. 2.59 gives the frequency information contained in the density matrix during a propagation period, while the response function in eq. 2.58 gives the evolution of the system in response to incoming fields in terms of molecular coherences. In particular, the second propagator is the one that gives the Raman sensitive signal.

It is convenient to describe system response to a particular combination of fields using the third-order susceptibility χ(3)_ijkl, obtained summing over all field permutation of the response function R(3)_ijkl. The χ(3)_ijklis thus given by several terms, but can be simplified [4] into

χ(3) = χ(3)_{N R}+ χ

(3)

R

∆ − iΓ (2.60)

where χ(3)_{N R} is the electronic contribution that is vibrationally nonresonant, χ(3)_R is related to the Raman scattering cross section, Γ is the Raman line width, ∆ = ωv− (ω1− ω2) is the

detuning from a vibrational transition at ωv. The reason why the χ(3) has been written as in

eq. 2.60 is that in CRS microscopy we are typically concerned with vibrational resonances of non-absorbing molecules, thus it is more practical to interpret the results of an experiment with this compact expression that highlights only the relevant vibrational resonances, grouping all non resonant terms in χ(3)_{N R}.

The spectral phase behaviour of χ(3)_{is an important aspect of CRS, especially in CARS and}

CSRS. The resonant nonlinear susceptibility, in fact, creates a field that interferes in a different way with the nonresonant field on each side of the resonance, creating a distortion of spectra. This is the reason why CARS and CSRS spectra are not directly comparable with spontaneous Raman ones. In fact, the spectrum S(Ω) for CARS and CSRS experiments, as function of the difference frequency Ω = ω1− ω2, is given by

S(Ω) ∝χ(3)(Ω) 2 =χ (3) N R 2 +χ (3)(Ω) R 2 + 2χ(3)_{N R}Re χ(3)_R (Ω) (2.61) In contrast, SRS experiments use an heterodyne detection and give a spectrum directly proportional to the imaginary part of χ(3)(Ω)

S(Ω) ∝ ±2 Im χ(3)(Ω)

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with the sign depending on the technique used, being positive for SRG and negative for SRL. It is now clear the first advantage of SRS techniques; being the measured spectrum directly proportional to the imaginary part of χ(3)_{, it is directly comparable with spontaneous Raman}

spectra that, as we will see in 2.4.1, are proportional to the imaginary part of χ(3) _{as well.}

Spatial polarization properties of χ(3)

The χ(3)_{is calculated taking into account all the combinations of fields in the response function}

of eq. 2.58. As already stated, this function depends on the polarization states of the incident fields i,j,k,l. We can consider them to be linear and expressed in cartesian coordinates. The third-order susceptibility is thus a fourth-rank tensor with 34 _{= 81 separate elements; the}

number of nonzero and independent elements depends on the spatial symmetry of the material itself. For biological samples, for example, the medium is predominantly aqueous and isotropic, thus the number of nonzero elements reduces to 21; for ordered materials, including membrane structures and crystalline materials, this assumption of isotropy doesn’t hold and the number of nonzero and independent elements can be different [8].

2.4 Quantum model of Raman Scattering

An approach considering the quantized nature of photons must be taken into account when modeling the response from only a few Raman scatterers, which produce signals in the single photon regime. A full quantum description, that highlights the quantized nature of fields, provides a quantitatively more accurate description of the energy exchange between fields and molecules.

There are two different approaches that can be used to build the quantum model of Raman scattering. The first one, more similar to the semiclassical model of section 2.3, models the material evolution in terms of the density matrix operator defined in 2.3.1, providing accurate estimates of Raman cross section. The second approach is based on Kramers-Heisenberg model for transition rates and highlights the energy flux from the molecular perspective.

In the quantum model, the optical electrical field is described by a wavefunction |ψFi, and

the expectation value of the optical electric field is calculated using the operator ˆ

E(~r, t) = ˆEs(~r, t) + ˆEs†(~r, t) (2.63)

with the two terms given by ˆ Es(~r, t) = ~ωs 20V 1/2 ˆ ase−i(ωst− ~ks·~r) (2.64) ˆ Es†(~r, t) = ~ωs 20V 1/2 ˆ a†sei(ωst− ~ks·~r) (2.65)

with ˆas, ˆa†s annihilation and creation operators for mode s, V quantization volume of photon

mode s [8]. The annihilation and creation operators act on a mode with ns photons as follow

ˆ

as|ψFs(n)i = (ns− 1)1/2|ψFs(n − 1)i (2.66)

ˆ

a†s|ψFs(n)i = n1/2s |ψsF(n + 1)i (2.67)

The system Hamiltonian now is composed by three terms ˆ

H = ˆH0+ ˆHF+ ˆHint (2.68)

with ˆH0 unperturbed Hamiltonian, ˆHF contribution from the field degrees of freedom, ˆHint

interaction between the field and the material. The last two terms are given by ˆ

HF =

X

s

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ˆ Hint= ˆEs(~r, t) ˆV†(~r) + ˆE†s(~r, t) ˆV (~r) (2.70) where ˆV is ˆ V (~r) = N X α=1 δ(~r − ~rα) X a,b>a µab|ai hb| (2.71)

with the α index running over all molecules, supposed identical for simplicity. |ai and |bi are the wavefunctions of different energy levels.

The most important difference between the semiclassical approach of section 2.3 and the quantum model is given by eq. 2.70 that produces a change in both the material and the field degrees of freedom when an interaction occurs.

The photon number in a mode s is given by the expectation value of the operator ˆ

Ns= ˆa†sˆas (2.72)

that, when applied to the wavefunction |ψs

F(n)i, has ns, number of photons in mode s, as

eigenvalue. The signal detected in this mode is thus given by

Ss=

dNˆs

dt (2.73)

The signal of eq. 2.73 has an intuitive form, being just a representation of the change of number of photons with frequency ωs. This can be calculated solving the density matrix of the

total system ˆρtot(t), that includes both material and field. The signal of eq.2.73 can be written

as Ss= d dtTr _ˆ Nsρˆtot(t) = − 2 ~Im TrEˆs(~r, t) ˆV† (2.74) Another way to describe the interaction is focusing on the transitions between states in the material. When field and material are coupled, a transition between states correspond to a change in the fields. This approach is only useful to calculate the dissipative signals. The transition rate between state |ai and |ni, a generic excited state, is given by the Fermi’s golden rule Ra→n= 2π ~2 X n hn| ˆHint|ai 2 δ(ωn− ωa) (2.75)

that can be extended to multi-photon interaction using the Kramers-Heisenberg form [36]. For what it concerns this treatment, the most interesting thing is that material transitions are directly related to the change in the number of photons, it is thus possible to write

Ss∝ ±Ra→n (2.76)

where the sign is positive when a photon in the s mode is absorbed, negative if it is emitted.

2.4.1 Spontaneous Raman scattering

The spontaneous Raman process, as already stated in section 2.2.1, involves a strong driving field at ω1frequency. In this condition, the incoming field can be treated as classical, while the

field emitted at frequency ω2 is treated with a quantum description; the density matrix of ω2

mode is initially a vacuum state, so the lowest order density matrix operator that contributes to the signal S2 is ˆρ

(3)

tot, that involves two interactions with ω1 and two interactions with ω2.

A Jablonski diagram of the process is sketched in picture 2.2. The final emission is at ω2 =

ω1− (ω1− ω2), thus the density matrix of the ω2field has raised from the vacuum state, |0i h0|,

to the one photon state |1i h1|, the radiated photon. Using eq. 2.74 it is possible to write the signal S2= 1 ~4 X k2 ~ω2 20V 2 ImR(ω2, ω2− ω1, −ω1) E1 2 (2.77) with the sum running over all ω2 modes with wave vector k2 in the volume V . Equation 2.77

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Figure 2.2: Energy diagram of spontaneous Raman scatter-ing.

but the detected signal is linearly proportional to the intensity of the incident field, I1= |E1|2.

It is possible to express the strength of the Raman signal using the differential Raman cross

section σ(ω2) = 1 9 ω1ω23 π22 0~2c4 ImR(ω2, ω2− ω1, −ω1) E1 2 ≈ C1Imχ(3)(Ω) (2.78) with the 1/9 factor resulting from an average over molecular orientation, C1 a proportionality

constant. Equation 2.78 confirms what was revealed in advance in section 2.3.2 through eq. 2.62, when SRS spectra were said to be directly comparable with the spontaneous Raman ones, being both proportional to the imaginary part of χ(3)_.

Before the interaction with the field takes place, the molecule is in the ground state |ai. After the Raman process occurs, the molecule is in the vibrationally excited state |bi. The molecule has thus gained an energy ~(ω1− ω2) = ~ωv. Using the Kramers-Heisenberg version

of the rate eq. 2.75, we can write the transition rate as

Ra→b∝ ω1ω2 V2 E1 2 X n µbnµan ω1− ωna+ iγ 2 δ(ω1− ω2− ωba) (2.79)

with E1expectation value of the field amplitude with ω1frequency. It is now possible to deduce

the differential cross section at the Raman resonance

σ(ω2) ∝

ω1ω32

c4 |α|

2 _(2.80)

with α transition amplitude, assuming the form of a single pathway Raman transition polariz-ability α =X n µbnµna ω1− ωna+ iγ (2.81)

2.4.2 Stimulated Raman scattering

An analysis similar to the one presented for the spontaneous Raman scattering can be given for the stimulated Raman scattering (SRS) process. In this case we need to consider two different field modes, ω1 and ω2, already occupied by photons. During the SRS process one photon of

the ω1 mode is absorbed and one of ωs mode is emitted, giving rise respectively to the signal

Raman loss (SRL) and to the signal Raman gain (SRG), while the molecule is excited from the ground state |ai to the vibrationally excited state |bi. The energy scheme is exactly the same as for the spontaneous Raman process, shown in figure 2.2. SRS process is thus dissipative and can be written using a Kramers-Heisenberg form [8]; ignoring contributions from electronic resonances and using ωba= ω1− ω2, we get the rate

Ra→b∝ ω1ω2 V2 E1 2 E2 2 X n µbnµan ω1− ωna+ iγ 2 δ(ω1− ω2− ωba) (2.82)

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Comparing the transition rate in eq. 2.82 with the one of the spontaneous Raman scattering displayed in eq. 2.79, we have an increase of a factor |E2|2. This is caused by the presence

of the ω2 mode before the interaction. The enhanced transition rate produces an enhanced

rate of photon loss in the ω1 mode and an enhanced gain in the ω2 mode; this implies that,

as long as the photon flux is well above the shot-noise limit, the SRL and SRG optical signals are many orders of magnitude higher than the corresponding signal arising from spontaneous Raman scattering.

Homodyne signals are not dissipative, so their rate can not be written using the Kramers-Heisenberg form. A description of heterodyne detected CARS signal could be done [8], but it’s beyond the scope of this treatment.

2.5 CRS microscopy techniques

Once the classical, semiclassical and quantum models of the Raman process have been presented, it is useful to summarize the information given and underline the most important aspects concerning CRS microscopy [4].

Coherent Raman Scattering is one of the nonlinear optical processes called four wave mixing. In CRS microscopy, two laser sources with frequencies ωs, called Stokes, and ωp, called Pump,

are used. When these waves are both spatially and temporally overlapped into a nonlinear medium, four CRS processes occur at the same time:

CARS Coherent Anti-Stokes Raman Scattering, at frequency 2ωp− ωs; CSRS Coherent Stokes Raman Scattering, at frequency 2ωs− ωp; SRG Signal Raman Gain, at frequency ωs;

SRL Signal Raman Loss, at frequency ωp.

There is also another process, called Raman-induced Kerr Effect (RIKE) [41] where a new polarization component is produced [8].

The signal of the CRS is created by the third order nonlinear polarization oscillating at frequency ωsign

P(3)(ωsign) = 0Dχ(3)(ωsign; ω0, ωp, −ωs)E0EpEs∗ (2.83)

with E0 electric field oscillating at ω0 frequency, D indicating the number of possible

permu-tations of the fields; D = 6 for three different fields, D = 3 for ω0= ωs or ω0= ωp. The χ(3)

contains a resonant part χ(3)_R and a real non resonant one χ(3)_{N R}:

χ(3) = χ(3)_{N R}+ χ

(3)

R

∆ − iΓ (2.84)

where χ(3)_R is related to the Raman scattering cross section, Γ is the Raman line width, ∆ = Ω − (ωp− ωs) is the detuning with respect to a vibrational transition at frequency Ω.

If we take the imaginary part of the χ(3) we have Imχ(3)₌ χ

(3)

R Γ

∆2_{+ Γ}2 (2.85)

Equation 2.85 shows a Lorentzian profile. Utilizing the slowly varying amplitude

approxi-mation, it is possible to write the CRS field

ECRS(ωsign) = iχ(3)(ωsign; ω0, ωp, −ωs)E0EpEs∗ (2.86)

Focusing the discussion onto SRL, in which ω0= ωsand ωsign= ωp, the field is given by

ESRL(ωp) = iχ

(3)

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with Elo local oscillator field. The total intensity is thus given by a mixture of the SRL field

and the local oscillator one

ISRL= Eloc+ ESRL 2 (2.88) Utilizing an heterodyne detection, obtained inducing a modulation in the Pump using a modulated Stokes wave, the laser Pump intensity is dominated by the interference term between the SRL field and the local oscillator Ep

∆I(ωp) = −2 Imχ

(3)

SRLIpIs (2.89)

where the negative sign indicates a loss of intensity.

Similar to the SRL, the SRG shows the same spectral profile as spontaneous Raman and a change of intensity given by

∆I(ωs) = 2 Imχ

(3)

SRLIpIs (2.90)

Basically the combination of SRL and SRG transfers an energy ~(ωp− ωs) to the sample,

resulting in a true vibrational excitation.

2.5.1 Other nonlinear optical phenomena

Several different optical processes can occur, generated by the second-order χ(2)_{and third-order}

χ(3) _{susceptibility of materials. A list of phenomena can be easily found in literature [40, 42].}

Here some of the third-order phenomena will be briefly discussed; biological samples, being mostly centrosymmetric, have negligible second-order susceptibility [40]. Some of the third order effects don’t affect CRS measurements, so they are not discussed in this section.

Optical Kerr effect

The third order contribution to the nonlinear polarization can be written as

P(3)(t) = 0χ(3)E˜3(t) (2.91)

and in the general case in which ˜E is given by several frequency components, the polarization

assume the complicated expressions stated in the previous sections. Considering for simplicity a monochromatic field with frequency ω, the polarization into the medium can be written as [42]

P (ω) = 0χ(1)+

3χ(3) 8 |E(ω)|

2_{E(ω) =}

0χef fE(ω) (2.92)

Assuming that the nonlinear contribution is small compared to the linear one, it is possible to write the refractive index of the material as

nef f =p1 + χef f ≈ n0+ n2I(ω) (2.93)

with I intensity of the field oscillating at ω, n2given by

n2=

3 2n2

00c

χ(3) (2.94)

The intensity dependence of the refractive index in eq. 2.93 produces many interesting non-linear effects. For what it concerns microscopy, the most important are self-phase modulation (SPM) and cross-phase modulation (XPM).

Self phase modulation (SPM)

When an optical pulse propagates into a third-order nonlinear medium, the optical Kerr effect produces a time-dependent change in the refractive index experienced by the beam, as stated in eq. 2.93. For this reason, a light beam experiences a change of the refractive index according to its intensity and therefore has a self phase modulation (SPM). An energy diagram of the process is shown in figure 2.3.

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ω

Figure 2.3: Jablonski diagram of the SPM. Being a third-order effect, self phase modulation is a four-wave process that produce a phase delay in the propagating beam.

ω

_p

ω

_p

ω

_s

ω

_s

Figure 2.4: Jablonski diagram of the XPM. Cross phase modulation is a four-wave process that pro-duce a phase delay in the propagating beam.

However, SPM doesn’t affect SRS measurements; for example, performing a SRL measure-ment, the information is obtained as a difference between the intensity of the Pump beam when the Stokes is present and when it’s not; being the SPM always present, it is cancelled doing the difference between the two Pump intensities. An energy diagram of the process is shown in figure 2.3.

Cross phase modulation (XPM)

Cross phase modulation is a nonlinear phase shift of an optical field induced by another field with different characteristics, such as the wavelength. Considering an electric field given by two different components

E = E1e−iω1t+ E2e−iω2t+ c.c. (2.96)

the refractive index experienced by the field at ω1is given by

n ∝ n0+ n2 |E1|2+ 2|E2|2) (2.97)

In equation 2.97 all components of polarization at frequencies other than ω1and ω2are not

considered. The first term of the equation is given by the SPM, while the second one is the XPM. The energy diagram of the process is shown in figure 2.4.

The important feature of XPM in microscopy is that it creates a modulation of the divergence angle of the beam, due to the above said nonlinear modulation of the refractive index in the focal volume, giving rise to artifacts on CRS spectra when not all angular spectrum of the beam, i.e. its full Numerical Aperture, is collected, as will be analyzed in section 3.3.1.

Development of a multi channel acquisition scheme for a broadband Stimulated Raman Scattering microscopy system

Facoltà di scienze matematiche, fisiche e naturali

Corso di laurea magistrale in Fisica Medica