Supervisor:
Prof. Dario POLLI
Co-supervisor:
Benedetta TALONE
Author:
Martina RECCHIA
Student ID: 89954
Academic Year 2018-2019
ical investigation, a very important requirement is to be able to visualize large sample areas. In the previous version of the presented system, a laser-scanning based imaging modality was available, with a Field Of View (250x250 µm2)
lim-ited essentially by the Objective. For this reason, in this thesis work particular attention is given to the implementation of the hardware, i.e. a XY motorized sample stage, and to the development of the software in order to perform also sample-scanning imaging modality. Moreover, the latter enables to test quickly the real imaging capability of the new kind of phenomenon exploited. Indeed, in the laser-scanning based imaging modality, which employs galvanometric mirrors, a perfect alignment of all the optics over all the possible beam position is required to avoid vignetting. Instead, with a XY motorized sample stage the optical sys-tem needs to be aligned only to a fixed laser beam position. Experimental results on test samples (Poly-methyl methacrylate, Polystyrene and fluorescent beads) and biological samples, obtained with sample-scanning based imaging, will be then presented, showing the huge potential of this kind of system.
La necessità di strumenti diagnostici sempre più affidabili e efficienti ha innescato una rinascita nello sviluppo della strumentazione per microscopia ottica. I mi-croscopi convenzionali, basati su tecniche ottiche lineari, presentano una serie disvantaggi: alta energia e bassa profondità di penetrazione della luce UV e VIS utilizzate e imaging a bassa risoluzione a causa dei processi lineari coinvolti. Queste limitazioni possono essere superate con sistemi di imaging basati su pro-cessi ottici non lineari, che impiegano la luce nella gamma spettrale NIR. Questo lavoro di tesi mira a introdurre un microscopio ottico home-made, multimodale e non lineare, che consente di eseguire microscopia a dispersione Raman anti-Stokes Coerente e microscopia a fluorescenza da eccitazione a due fotoni.
Dato l’obiettivo finale di impiegare questo strumento nel campo dell’indagine biologica, un requisito necessario è quello di essere in grado di visualizzare ampie aree di campione. Nella versione precedente del sistema presentato, era disponi-bile la sola modalità di imaging basata sulla scansione laser, la quale è carat-terizzata da campo visivo (250x250 µm2) limitato essenzialmente dall’obiettivo.
Per questo motivo, durante questo lavoro di tesi, si è prestata particolare at-tenzione all’implementazione dell’hardware, ovvero del supporto motorizzato XY su cui viene disposto il campione, e allo sviluppo del relativo software al fine di aver a disposizione anche una modalità di imaging a laser fisso. Inoltre, quest’ultima modalità consente di testare rapidamente la reale capacità di imag-ing di nuove tipologie di fenomeno sfruttate. Infatti, nella modalità di imagimag-ing basata sulla scansione laser, che impiega specchi galvanometrici, è necessario un perfetto allineamento di tutte le ottiche su tutte le possibili posizioni del fascio per evitare la vignettatura. Invece, grazie alla nuova modalità implementata, il sistema ottico richiede di essere allineato solo ad una posizione fissa del fascio laser.
Saranno quindi presentati risultati sperimentali su campioni di prova (poli-metilmetacrilato, polistirene e sfere fluorescenti) e campioni biologici, che mostrano l’enorme potenziale di questo tipo di sistema.
1.4 Basic Principle of Two-Photon Excitation Fluorescence . . . 44
1.4.1 Introduction to the Fluorescence Process . . . 44
1.4.2 Two-Photon Excitation Fluorescence (TPEF) . . . 48
2 Nonlinear Optical Microscopy 50 2.1 Introduction . . . 50
2.2 Two-photon excitation microscopy . . . 51
2.3 Second-Harmonic Generation Microscopy . . . 55
2.3.1 TPEF and SHG Combined Microscopy . . . 57
2.4 Coherent Anti-Stokes Raman Scattering (CARS) Microscopy . . . 58
2.4.1 The CARS Field Distribution under Tight Focusing . . . . 58
2.4.2 Imaging Properties Of Cars Microscopy . . . 59
2.4.3 The CARS Light Source . . . 61
2.4.4 The CARS Laser Scanning Microscope . . . 62
3 The multimodal microscope 64 3.1 Multi-branch fiber-format excitation laser source for CARS and TPEF modalities . . . 64
3.2 Multimodal NLO microscope layout . . . 66
3.2.1 General Software Interface . . . 70
3.2.1.1 Motorized XY-scanning stage software . . . 75
3.2.2 Microscope Resolution . . . 79
4 Results and discussion 82 4.1 Imaging test . . . 82
4.1.1 Sample-scanning vs Laser-scanning based modality . . . . 88
5 Biological Result and discussion 96 5.1 Introduction . . . 96 5.2 Experimental results and qualitative analysis . . . 97 5.3 Test of the large area imaging capability . . . 106
1.3 Field orientation for propagation in birifrangent media. . From[60] 26 1.4 General form of the index ellipsoid. The coordinate axes X, Y and
Z are the optically principal axes. n1 , n2 and n3 are the principal
refractive indices of the anisotropic medium.[56] . . . 27 1.5 Graphical method to obtain no and ne. . . 27
1.6 Phase-matching diagram for second-harmonic generation in a pos-itive uniaxial medium, for the case of type I phase matching. . From [60] . . . 29 1.7 General view of a QPM crystal . From [60] . . . 30 1.8 Second-harmonic intensity as a function of distance into a
nonlin-ear crystal, for k = 0, k =/ 0, and first-order QPM. The dashed line shows the average growth in the SHG intensity under QPM. From [60] . . . 31 1.9 Frequency-domain picture of ultrashort-pulse SHG. Frequencies
from the input pulse combine in pairs to drive second-harmonic frequency components. [60] . . . 32 1.10 Schematic of spontaneous Raman scattering. (A) Incident light
is scattered into Rayleigh ωP, Stokes (ωS = ωP− ωR), and
anti-Stokes components (ωaS = ωP + ωR). (C) Energy diagram for
Stokes Raman scattering and (B) anti-Stokes Raman scattering. The ground state level is labeled g, the vibrational level v, and the intermediate level j. Far from electronic resonance, the intermedi-ate stintermedi-ates j have infinitely short lifetimes and are thus considered virtual states. Solid arrows correspond to applied light; wiggled arrows indicate the emitted radiation. [35] . . . 34
1.11 Conceptual scheme for the four possible CRS signals generation. (a) Two input fields at frequency ωP (pump) and ωS (Stokes)
in-teract with a Raman active χ(3) medium (the bi-atomic molecule),
generating at the output four fields (CARS, SRL, SRG, CSRS). The arrows thickness is proportional to the intensity of the rela-tive field. (b) Conceptual representation in the spectral domain, where the χ(3) interaction generates a modulation of the non-linear
refractive index that modulates the input fields at the beating fre-quency Ω. This interaction creates side-bands on the input fields, that are shifted of the beating frequency Ω. . . 36 1.12 Schematic of the CARS light-matter interaction through
Jablon-ski dia- grams (dashed lines are virtual levels and solid lines real levels). The subscripts p, s and aS stands for pump, Stokes and anti-Stokes, respectively. . . 40 1.13 Schematic of the possible four-wave-mixing light-matter
interac-tions, through Jablonski diagrams (dashed lines are virtual levels and solid lines real levels), leading to the same anti-Stokes radia-tion at frequency ωaS. The subscripts P, S and aS stands for pump,
Stokes and anti-Stokes, respectively. (A) Resonant contribution to coherent anti-Stokes Raman scattering (CARS) process by tar-geted molecules, (B) another four-wave mixing process providing electronic nonresonant contribution to CARS signal, and (C) elec-tronic nonresonant contribution to CARS signal from background (non- target) molecules. From [30]. . . 42 1.14 Spectral shapes of (A) different components of χ(3), (B) the
indi-vidual squared components, (C and D) full |χ(3)(Ω)|2 at different
ratios of resonant and nonresonant contributions. From [30]. . . . 43 1.15 Modified Franck-Condon energy state diagram shows the
electronic-vibronic energy levels and energy transitions of a molecule. The ground singlet state is S0, the first excited singlet state is S1, and the first excited triplet state is T1. Transitions from absorption (A), fluorescence (F), and phosphorescence (P) are indicated by boldfaced lines with arrowheads. The wavy lines indicate various types of non-radiative transitions: internal conversion (thermal relaxation) (IC) and intersystem crossing (S1 to T1 transitions) (ISC). The horizontal lines within each electronic energy level rep-resent the vibronic levels. [35] . . . 45
2.1 MPE simplified optical pathways. In the MPE optical pathways the emission pinhole is removed since the only emitted light reach-ing the sensor is comreach-ing from the currently point scanned volume in the sample. No other fluorescence signal is produced elsewhere. [21] . . . 54 2.2 (a) Lateral and axial view of IPSF and IP SF2 (b) Gaussian
func-tion profile [64] . . . 54 2.3 Schematic of a 2PE microscope with epi-fluorescence and
trans-fluorescence detection. . . 56 2.4 The HRS from two molecules located close together. (a) If the
molecules are parallel, their HRS is in phase and interferes con-structively. (b) If the molecules are anti-parallel, their HRS is out of phase and cancels. . . 57 2.5 A combined scanning TPEF-SHG microscope. A pulsed laser
beam is focused into a thin sample. The TPEF is collected in the backward direction using a dichroic beamsplitter. The SHG is collected in the forward direction after blocking both transmit-ted laser light and fluorescence. The TPEF and SHG signals are acquired simultaneously. . . 58 2.6 Schematic of CARS signal detection in the far field with tight
focusing. . . 59 2.7 CARS microscope setup. (A) The pump and Stokes beam
pa-rameters are optimized in terms of spatial mode and polarization orientation. The beams are spatially overlapped before coupling into the microscope. (B) The CARS microscope shows much simi-larity with the (confocal) scanning microscope. For rapid scanning a beam scanner is employed. Strongest signals are collected at the non-descanned forward and backward detectors. A proper band-pass filter is used to reject all incident radiation while band-passing the anti-Stokes radiation. . . 62
3.1 Multi-branch fiber-format excitation laser source architecture and wavelength output. Schematic of a multi-branched Er:fiber laser source for multimodal nonlinear microscopy. DM, dichroic mirror; PPLN, periodically-poled lithium niobate; EDFA, erbium-doped fiber amplifier; HNLF, highly nonlinear fiber; BS, beam splitter; Si, silicon. Delay line and DM are the beam conditioning devices. 65 3.2 Spectra of narrowband pump (Arm 0) and tunable narrowband
Stokes (Arm 1) beams. The curve in blue indicates the wavelength dependent average power of the Stokes beam reaching the sample. 66 3.3 Sketch of the multimodal NLO laser-scanning microscope. In the
figure are depicted the detection schemes for trasmission CARS/TPEF modalities in forward detection. DM, dichroic mirror; PMT,
photo-multiplier tube. . . 67
3.4 Real picture of the multimodal NLO laser-scanning microscope. Picture of the entire microscope system with in blue indicated the optical and mechanical components shown in Fig. 3.3. The red line shows the optical beam path. The yellow line indicates the optical path followed by the white lamp light from top to bottom, ending on a camera. . . 68
3.5 Bright-field illumination pathway schematic. In yellow it’s repre-sented the white light optical path and in red the laser (pump and Stokes) optical path coming from the laser excitation source. . . . 69
3.6 Mad City Vertical Axis Panel . . . 71
3.7 Galvo Panel . . . 71
3.8 Stokes Panel . . . 72
3.9 Single Point Spectrum window . . . 72
3.10 Full Scanner Panel . . . 72
3.11 Standa Panel . . . 73
3.12 Log Panel . . . 73
3.13 Screenshots of the Matlab software graphic interface controlling the home-made microscope system. First main window. . . 74
3.14 Screenshots of the Matlab software graphic interface controlling the home-made microscope system. Second main window. . . 74
3.15 (a) Setting Standa Position Panel. (b) Bright-field image of a sample of PS (10-µm diameter) beads .The two figures together show how the displacement of the sample, seen by the user and operates thank to the software, occurs. . . 75
3.16 a) Sketch of the relative movement of the laser during the scan, taking the as reference system an inertial reference system with the motorized XY-scanning stage. The green arrows represent the distance over which acceleration and deceleration are performed. The blue arrows represent the component of the ideal scan. Green area is the one over which the laser moves without acquisition of respective signal. DimX and DimY are equal to the parameters chosen by the user for the scan. b) Speed-Time diagram of the laser relative motion. . . 77
sample and (b)PS beads of 10-µm diameter samples. . . 83 4.2 CARS spectrum of PS and PMMA from [26]. . . 83 4.3 Excitation specta of SHEROTM fluorescent particles of ∼10-µ
di-ameter. . . 84 4.4 Emission specta of SHEROTM fluorescent particles of ∼10-µ
diam-eter. . . 84 4.5 Transmission and CARS image of PMMA beads of 6-µm
diame-ter sample. Both images are characdiame-terized by a scanned area of 100×100 µm2, 100×100 pixels and a RDT=1 s. . . 85
4.6 Transmission and CARS image of PMMA beads of 6-µm diame-ter sample. Both images are characdiame-terized by a scanned area of 300×300 µm2, 300×300 pixels and a RDT=3 s. . . 85
4.7 Transmission and CARS image of PMMA beads of 6-µm diame-ter sample. Both images are characdiame-terized by a scanned area of 500×500 µm2, 500×500 pixels and a RDT=2.5 s. . . 85
4.8 Trasmission and CARS image of PS beads of 10-µm diameter sam-ple. Both images are characterized by a scanned area of 100×100 µm2, 100×100 pixels and a RDT=1 s. . . 86
4.9 Transmission and CARS image of PS beads of 10-µm diameter sample. Both images are characterized by a scanned area of 300×300 µm2, 300×300 pixels and a RDT=3 s. . . 86
4.10 Transmission and CARS image of PS beads of 10-µm diameter sample. Both images are characterized by a scanned area of 500×500 µm2, 500×500 pixels and a RDT=2.5 s. . . 86
4.11 Transmission and TPEF image of SHEROTM fluorescent particles
of 10-µm diameter sample. Both images are characterized by a scanned area of 100×100 µm2, 100×100 pixels and a RDT=1 s. . 87
4.12 Transmission and TPEF image of SHEROTM fluorescent particles
of 10-µm diameter sample. Both images are characterized by a scanned area of 300×300 µm2, 300×300 pixels and a RDT=3 s. . 87
4.13 Transmission and TPEF image of SHEROTM fluorescent particles
of 10-µm diameter sample. Both images are characterized by a scanned area of 500×500 µm2, 500×500 pixels and a RDT=2.5 s. 87
4.14 Transmission and CARS image of PMMA beads of 6-µm diameter sample. Both images, taken in laser-scanning modality, are char-acterized by a scanned area of 100×100 µm2, 100×100 pixels and
a RDT=1 s. . . 88 4.15 Different degrees of Vignetting severity . . . 88 4.16 Field Distortion in a Two-way Mirror Deflection System . . . 89 4.17 Singular value intensities for the different singular vales. Much of
the contributions to the the image come from the first 10 singular values. The singular values have been sorted in descent order and normalized to the maximum of theirs. . . 94 4.18 Images obtained by successive truncation of Eq. 4.8. The number
of singular value terms is written in the top-left corner of each image. It is important to note that from the penultimate image the obteined ones are identical to the measured one. . . 95 5.1 Representative micro-CT images of a lumbar vertebrae of a mouse
[61] . . . 98 5.2 Transmission image (300×300 µm2, 300×300 pixels, RDT=3s) of
a longitudinal longitudinal section of a mouse spine. . . 99 5.3 Single Point spectrum related to the point indicated in fig 5.4 . . 99 5.4 Bright-Field image of the region chosen for the analysis. . . 100 5.5 CARS image (300×300 µm2, 300×300 pixels, RDT=3s) of a
lon-gitudinal section of a mouse spine. . . 100 5.6 Transmission image (100×100 µm2, 200×200 pixels, RDT=1s) of
a longitudinal section of a mouse spine. . . 101 5.7 CARS image (100×100 µm2, 200×200 pixels, RDT=1s) of a
lon-gitudinal section of a mouse spine. . . 102 5.8 Typical Raman spectrum of spongy bone tissue showing the major
bands and the corresponding compounds. Background signal has been removed [29] . . . 102 5.9 CARS image (100×100 µm2, 200×200 pixels, RDT=1s) of a
lon-gitudinal section of a mouse spine. . . 103 5.10 CARS image (100×100 µm2, 200×200 pixels, RDT=1s) of a
lon-gitudinal section of a mouse spine. . . 103 5.11 Comparison between Raman spectra of different
formaldehyde-based compounds (see legend). The vertical dotted lines indicate the position of the symmetric n S and anti-symmetric n AS modes of C–H stretching observed in the liquid aqueous solution.[13] Actually,the one on which we must concentrate in this discussion is the only spectrum of the Paraformaldehyde. . . 104 5.12 TPEF image (100×100 µm2, 200×200 pixels, RDT=1s) of a
lon-gitudinal section of a mouse spine. . . 105 5.13 CARS image (100×100 µm2, 200×200 pixels, RDT=1s) of a
lon-gitudinal section of a mouse spine. . . 105 5.14 Transmission image (1×1 mm2, 1000×1000 pixels, RDT=5s) of a
1.1 Summary of Birefringent Phase Matching for Second-Harmonic Generation [60] . . . 29 4.1 8MTF-102LS05, Standa LTD specification . . . 90 4.2 Intervals of time needed to do the measurement specified in the
first column with the different modalities. . . 93 4.3 (a)Relative delay TStanda[s] vs TStanda, ideal[s] ; (b) Relative delay
in most tissues.
Figure 1: TPEF image (1×1 mm2, 1000×1000 pixels, RDT=5s) of a longitudinal section of a mouse spine.
Another disadvantage of the conventional techniques is that they show im-portant limitations due to the need of introducing chemical labels to the studied samples, which could interfere with biological functionalities. Moreover, for a heterogeneous system, as biological ones, one needs spatial resolution to focus on the object of interest, and chemical selectivity to probe particular species. Taking into account of all these negative aspects, it clear the need of a different kind of phenomenon on which based the operation of microscopes.
Coupling of nonlinear optics and scanning microscopy has generated a panel of imaging tools for biology and materials research. Because each nonlinear optical (NLO) imaging modality is sensitive to specific molecules or structures, multi-modal NLO imaging capitalizes the potential of NLO microscopy for studies of complex biological tissues.
The coupling of multiphoton fluorescence, second harmonic generation, and coherent anti-Stokes Raman scattering (CARS) has allowed investigation of a broad range of biological questions thanks to their big advantages.
• In general, exploiting a non-linear phenomenon, it is possible to obtain a better resolution with respect to the one obtained in linear case.
• Chemical imaging by use of inherent molecular vibration signals avoids the photobleaching problem and perturbations to cell functions induced by fluorophore labeling.
• Vibrational microscopy based on the Raman scattering, exploits the vibra-tional modes of molecules to differentiate them. Indeed, each sample has a chemical structure that determines a peculiar vibrational spectrum, which provides a signature that can be exploited for the identification of molecules. In this thesis work, a Multimodal Nonlinear Optical Microscope, with which it is possible to perform Coherent Anti-Stokes Raman scattering (CARS) mi-croscopy and Two-photon excitation fluorescence (TPEF) mimi-croscopy, will be presented.
CARS is a four-wave mixing process in which a Pump beam (ωp) and a Stokes
beam (ωS) interact with a sample to generate a signal at the anti-Stokes frequency
ωAS= 2ωP– ωS. The CARS signal arises from the nonlinear polarization induced
by Pump and Stokes beams due to the third-order susceptibility of the sample. The third-order susceptibility consists of a non-resonant part, which is indepen-dent of ωP – ωS frequency, and a resonant part, which depends on the ωP – ωS
frequency. Because CARS signal can be significantly enhanced by molecular vi-brations when ωP – ωS is close to the vibrational frequency of a chemical bond,
the CARS microscopy allows label-free and chemically selective imaging .
In TPEF, the target molecule absorbs two photons to reach an excited elec-tronic state and then emits a single-fluorescence photon of higher energy than either of the incident photons. Compared to one-photon fluorescence, TPEF microscopy, provides inherent three-dimensional (3-D) resolution with tightly fo-cused excitation. The small excitation volume of TPEF greatly reduces overall specimen photodamage as well as fluorophores photobleaching. Moreover, the use of near infrared (NIR) excitation for TPEF allows deeper penetration into the imaged tissues. Owing to the availability of fluorescent labels, TPEF has become the most widely used NLO imaging modality and a potent tool for biological studies.
Chapter 1 gives a complete overview about the theoretical knowledge needed to understand every physical processes exploited during the presented work. Chapter 2 introduced the main type of nonlinear microscopy, underling their prin-cipal characteristics and state of the art in their implementation. Then, Chapter
Theoretical Foundations
This first chapter provides an overview on the physical processes that underlie the nonlinear biological microscopy.
1.1
Principles of Nonlinear Optics
Nonlinear optics is the study of phenomena that occur as a consequence of the modification of the optical properties of a material system induced by the presence of light. Typically, only laser light is sufficiently intense to modify the optical properties of a material system. Otherwise, for intensities not that high, linear optics has to be used to analyze the phenomena that can occur in the material, for example the processes of transmission, reflection, refraction, superposition and birefringence. Nonlinear optics can be seen as an extension of linear optics and it is needed to explain many phenomena that the latter is not able to explain. [8].
Both linear and nonlinear optical processes can be understood by considering the interaction between the field of the electromagnetic radiation and the charged particles of the sample. Nonlinear phenomena are due to the inability of the dipoles in the optical medium to respond in a linear fashion to the alternating electric field (E-field) associated with the light beam: since the atomic nuclei are too massive and inner-core electrons too tightly bound to respond to the alternating E-field at the frequency of light ( 103 THz), the outer electrons of the
atoms in a material are primarily responsible for the polarization of the optical medium induced by the beam’s E-field, since they start to oscillate and result slightly displaced from their equilibrium positions. An induced electric dipole moment arises:
µ(t) = −e × r(t) (1.1) where e is the electron charge and r(t) its displacement.
The macroscopic polarization results from adding up all the single contribu-tions:
into account:
P(t) = 0[χ(1)E(t) + χ(2)E2(t) + χ(3)E3(t) + ...] ≡ P(1)(t) + P(2)(t) + P(3)(t) + . . . .
(1.4) where χ(2) and χ(3) are the second-order and the third-order nonlinear
suscep-tibilities, respectively. When ultrashort pulses are used,the Nonlinear terms of polarization become preponderant, since they are characterized by an high peak power and consequently by a very strong electric field, . Generally, χ(2), if it is
different from zero, is bigger than χ(3), thus the first effects that can be seen are
the ones related to the second order polarization. It is important to underline also that there are other ways to excited a nonlinear behaviour of the optical medium, and these are based on using optical frequencies near the resonant frequency of the oscillating dipoles.
The most usual method for describing nonlinear optical phenomena is writing the polarization in terms of E(t) and the reason is that a time-varying polar-ization can provide new components of the electromagnetic field. This can be demonstrated starting from the Maxwell equations, by assuming to have a non magnetic medium (M = 0,χ(m) = 0) and that there are no current and charge
density sources (ρ = 0,J = 0) (these hypothesis are justified by the aim of this dissertation): ∇ × E = −∂B ∂t (1.5) ∇ × H = ∂D ∂t (1.6) ∇ · D = 0 (1.7) ∇ · B = 0 (1.8)
From which it is possible to derive the following wave equation for a medium with a refractive index n:
∇2E − 1 c2 ∂2E ∂t2 = µ0 ∂2P ∂t2 (1.9)
where, P can be split into its linear and nonlinear component:
P = P(1)+ PNL. (1.10) Now it’s easy to see that the nonlinear polarization (i.e. the nonlinear response of the medium) acts as a source term in this equation that has the form of a driven wave equation.
1.2
Second Order Nonlinearity
The second-order processes occur in a medium when there is a nonzero contribu-tion from the second-order nonlinear polarizacontribu-tion:
P(2) = 0χ(2)E2, (1.11)
that means a nonzero second-order susceptibility χ(2). The material for which
χ(2) is different from zero are called Non − Centrosymmetric.
Let us consider the second-order nonlinear polarization that arises with input fields at two different frequencies. The overall input electric field is taken as :
E(t) = 1 2(A1e
iω1t+ A
1eiω2t+ c.c) (1.12)
and so, the second-order contribution of the nonlinear polarization turns out to be:
P(2)(t) =0χ
(2)
4 (A
2
1ei2ω1t+ A22ei2ω2t+ A21*e−i2ω1t+ A22*e−i2ω2t+ (1.13)
+ 2A1A2ei2(ω1+ω2)t+ 2A1*A2*e−i2(ω1+ω2)t+ 2A1A2*+ (1.14)
+ 2A2A1*+ 2A1A2*ei2(ω1−ω2)t+ 2A1*A2ei2(ω2−ω1)t). (1.15)
In a full treatment of nonlinear optics, one must consider both the nonlinear polarization and the field as vectors and so the nonlinear susceptibility then takes the form of a tensor χijk(2). Moreover, the latter may be both complex and
frequency dependent.
In general, the second-order nonlinear susceptibility tensor has 27 components (i, j, k = 1:3) that must be known to fully specify the nonlinear polarization, even for a fixed set of frequencies. Fortunately, the nonlinearity tensor exhibits a num-ber of powerful symmetries that can greatly reduce the numnum-ber of independent tensor components. Concerning with nonlinearities associated with bound elec-trons, assuming that all the optical frequencies of interest are well below the electronic absorption resonances of the nonlinear material, and well above any other resonances (such as those associated with lattice vibrations in the nonlin-ear crystal), the following conditions applies:
P (ω1+ ω2) = 20χ(2)E1E2 (SFG), (1.18)
P (ω1 − ω2) = 20χ(2)E1E∗2 (DFG), (1.19)
P (0) = 20χ(2)(E1E∗1+ E2E∗2) (OR). (1.20)
where:
• SHG stands for second − harmonic generation: two photons at a given frequency are absorbed by the material and one at a double frequency is emitted ;
• SFG is sum − frequency generation: two photons at ω1 and at ω2 are
absorbed by the material and one at (ω1 + ω2)is emitted ;
• DFG is difference − frequency generation :two photons at ω1 and at
ω2are sent on the material, the one at ω2is absorbed and one at (ω2− ω1)is
emitted by the material (obviously assuming ω2 bigger than ω1 ). Since, at
the end, is like to have the amplification of the light at ω1, this process is
also called Optical P arametrical Amplification (OPA). Falls in the case of difference − frequency generation also the one for which a photon at ω2 is sent and absorbed by the material, with the consequent emission of
two photons: one at ω1 and one at (ω1− ω2). In this case, this process is
called Spontaneous P arametric Down Conversion (SPDC) ;
• OR is optical rectification: it refers to the development of a DC or low-frequency polarization when intense laser beams propagate through the medium.
At this point, it is important is to determine what are the changes of the field equations, if the second term of the Polarization is taken into account.
It is possible to work in the plane wave approximation and derive the expres-sion of the signal emitted by the sample. By assumption, the incident fields are plane waves with constant amplitude and phase in the transverse plane (x,y), which propagate along the z direction. Consequently, the wave equation can be written as: ∂2E ∂z2 − 1 c2 ∂2E ∂t2 = µ0 ∂2P L ∂t2 + µ0 ∂2P NL ∂t2 (1.21)
In this framework, the linear effect are negligible, so it is possible to consider only the nonlinear term.
Generally, the field and the nonlinear polarization can be expressed as follows: E(z, t) = 1 2e(A(z)eˆ i(kz−ω0t)+ c.c.) (1.22) PNL = 1 2p(pˆ NL(z)e i(kNLz−ω0t)+ c.c.) (1.23)
Inserting Eq. 1.22 and Eq.1.23 in Eq. 1.21 and calculating the partial deriva-tives, exploiting also the Slowling Varing Approximation (SVEA), for which the considered field is characterized by an amplitude that doesn’t change too much over a distance comparable with its wavelength ,what is obtained is the following:
∂A ∂z + 1 vg ∂A ∂t − iGV D 2 ∂A ∂t2 = −iµ0ω0c 2n e −i∆kz pNL(ˆe · ˆp) (1.24)
where vg is the group velocity, GV D is the group velocity dispersion and
∆k = kp− k0.
Let’s consider the T hree W ave Mixing approach, in which we consider two incoming fields which interact with the sample and a third field which corre-sponds to the response of the sample: let’s assume that the photons involved in the interaction with the χ(2) crystal, are such that their frequencies respect the
following relation : (ω1+ ω2) = ω3.
The overall Nonlinear polarization can be written as: PNL = 0χ2[
1
2(A1(z, t)e
i(ω1t−k1z)+A
2(z, t)ei(ω2t−k2z)+A3(z, t)ei(ω3t−k3z))]2 (1.25)
Due to the fact that the square of the bracket must be perform, many fre-quencies are obtained. Let’s assume that the only interactions that turn out to be efficient are the ones that continued to respect the relation mentioned above be-tween the three frequencies. In this way, the preponderant terms of the Nonlinear polarization turn out to be :
PNL(ω3) = 1 40χ 2(A 1A2ei[(ω1+ω2)t−(k1+k2)z]+ c.c.) (1.26) PNL(ω2) = 1 40χ 2 (A3A*1ei[(ω3−ω1)t−(k3−k1)z]+ c.c.) (1.27) PNL(ω1) = 1 40χ 2(A 2*A3ei[(ω3−ω2)t−(k3−k2)z]+ c.c.) (1.28)
For each of this field, it is possible to write the corresponding wave equation using Eq.1.24, obtaining three coupled differential equations, that turn out to be decoupled in the case in which χ(2) is switch off, that means that is just this
term that makes possible the exchange of energy between these fields. Starting with Eq. 1.24, it is possible to consider a further hypothesis based on a physical treatment: 1
vg ∂A
2cni
where the effective nonlinear coefficient defftakes into account the tensor
prod-ucts of the fields and the nonlinear susceptibility tensors as well as the projection of the nonlinear polarization onto the fields . For lossless nonlinear media, it is real.
The three coupled equations that are obtained making all the substitutions are: ∂A1 ∂z + 1 vg1 ∂A1 ∂t = −iγ1A3A2 *e−i∆kz (1.31) ∂A2 ∂z + 1 vg2 ∂A2 ∂t = −iγ2A3A1 *e−i∆kz (1.32) ∂A3 ∂z + 1 vg3 ∂A3 ∂t = −iγ3A1A2e i∆kz (1.33) where ∆k = k3− k1− k2.
Considering a frame reference that travels with the same velocity of vg3, and
so t0 = t − z
vg3, the equations become:
∂A1 ∂z + δ13 ∂A1 ∂t = −iγ1A3A2 *e−i∆kz (1.34) ∂A2 ∂z + δ23 ∂A2 ∂t = −iγ2A3A1 *e−i∆kz (1.35) ∂A3 ∂z = −iγ3A1A2e i∆kz (1.36) where δ13 = vg11 − vg31 , δ23= vg21 − vg31 .
Moreover, considering also monochromatic waves characterized by temporal derivaties equal to zero:
∂A1 ∂z = −iγ1A3A2 *e−i∆kz (1.37) ∂A2 ∂z = −iγ2A3A1 *e−i∆kz (1.38)
Figure 1.1: (a) Geometry of second-harmonic generation. (b) Energy-level diagram describing second-harmonic generation. From [35]
∂A3
∂z = −iγ3A1A2e
i∆kz (1.39)
Starting from these equations, it is possible to analyze the different Second Order Processes.
1.2.1
Second-Harmonic Generation (SHG)
The process of second-harmonic generation is shown in Fig. (1.1).
A laser beam at frequency ω illuminates a nonlinear optical material and a beam of light at frequency 2ω is created. The transfer of energy from the input field to the output field can be visualized in terms of the energy-level diagram shown on the right-hand side of the figure. The process can be seen as one in which two photons from the input beam are lost and one photon in the output beam is created.
In order to analyze the process from the analitical point of view, let’s consider a monochromatic input field at frequency ω0 directed onto a nonlinear crystal of
length L. This is called fundamental field and induces a nonlinear polarization at the second-harmonic frequency 2ω0. The intensity of the radiation emitted
in the harmonic generation process can be predicted by means of a propagation calculation.
Starting from the Eqs. 1.37, 1.38 and 1.39, considering now ω1 and ω2
degen-erate and equal to ω0 and the other one equal to 2ω0, two partial equations are
obtained: ∂Aω0 ∂z = −iγω0 A* ω0 √ 2 A√ω0 2e −i∆kz (1.40) ∂A2ω0 ∂z = −iγ2ω0 Aω0 2 2 e i∆kz (1.41)
where the difference in the propagation constants, ∆k = k2ω0−2kω0, is referred
to as the phase mismatch.
Under the assumption that Aω0is not appreciably modified by the nonlinear
I2ω0 = |A2ω0| = γ 2ω0I ω0
∆kz12 4 (1.43) The intensity reaches a maximum at distance lc = ∆kπ , that is known as
Coherent Length, and then diminishes to zero, repeating sinusoidally with pe-riod 2lc. lc indicates the maximum thickness of the nonlinear crystal that can
contribute to generating a second harmonic. Often, it is as short as tens of mi-crometers.
From Eq. 1.43 , it is possible to see how SHG conversion efficiency, I2ω0
Iω0 , is
proportional to the input intensity and to the square of the propagation length. It is important to recall that this result is valid only in the low- conversion-efficiency limit.
Effect of Phase Matching
The condition ∆k = 0 is known as the condition of P erfect P hase Matching and is a requirement for efficient generation of second-harmonic radiation. When this condition is fulfilled, the term sin(∆kz12)
∆kz12 in Eq. 1.43 is equal to unity, since it
corresponds to a sinc function of argument ∆kz1
2. Thus the intensity becomes
I2ω0 = γ 2 2ω0I 2 ω0 z2 4 (1.44)
It is important to remember that in any situation where the second-harmonic reaches an amplitude that is significant compared to the input field amplitude, pump depletion effects must be considered through coupled nonlinear (Eq. 1.41 and 1.40 ) and so 1.44 is no longer valid.
Figure 1.2 shows the variation of second-harmonic intensity as a function of distance into the nonlinear crystal, both with and without phase matching.
Before seeing how the condition can be experimentally reached, let’s see its physical meaning. The material is made by molecules and atoms and when an external field comes onto the medium, they start to oscillate, emitting radiation. The situation is the same of having a lot of emitters. By considering the medium divided into slices, once the fundamental field, at ω0, arrives onto once slice, it
start to emits the second-harmonic , at 2ω0, that propagates toward the output.
Figure 1.2: Effect of phase matching on continuous-wave second-harmonic generation: (a) second- harmonic intensity in the non-depleted pump approximation, as a function of distance into the nonlinear crystal, both with and without phase matching; (b) output intensity as a function of phase mismatch. [60]
each slice, must interfere constructively. A constructive interference is obteined when the two field at ω0 and 2ω0 have the same phase velocity
vpω0 = vp2ω0 (1.45) ω0 k(ω0) = 2ω0 k(2ω0) (1.46) 2k(ω0) = k(2ω0) (1.47)
Remembering that the wave vector is given by k(ω) = n(ω)ω
c , it is easy to
see how the perfect phase matching condition can not be reached without some precautions, due to the fact that there are not dispersionless media.
2n(ω0)ω0 c =
n(2ω0)2ω0
c (1.48)
n(ω0) = n(2ω0) (1.49)
Phase Matching in Birefringent Media
The phase mismatch in second-harmonic generation arises due to the variation of the refractive index with frequency. The most widespread method for achieving phase matching makes use of birefringent media (media with anisotropic linear properties).
The idea is that the refractive index’s dependence on the polarization and on the propagation direction in a birefringent nonlinear crystal is an additional degree of freedom that can be exploited to compensate for the variation with frequency. Fortunately, many of the media that have large second-order nonlin-earities also possesses a linear birefringence that can be used for phase matching. Let’s consider a nonmagnetic media, i.e. B = µ0H. A media is said to
be isotropic when its optical characteristic are complitely indipendent from the direction. Vice versa, if this property is not present, the media is said to be
Going more in detail, let’s now explore the character of plane-wave solutions of Maxwell’s equations in anisotropic media. Exploiting the complex form of the fields, that for the electric one can be written as
E(r, t) = Re{ ˜Eei(ωt−k·r)} (1.52) where, for a plane-wave, ˜E is a constant; applying Maxwell’s curl equations, 1.5 and 1.6, for a current-free region , one can yield:
k × ˜E = µ0ω ˜Hand k × ˜H = −ω ˜D (1.53)
from which it is possible to see that ˜H is perpendicular to both ˜E and ˜D as well as to the propagation vector k. ˜D and k are also perpendicular. These conditions also hold for waves in isotropic media. However, unlike the isotropic plane-wave case, ˜E need not be perpendicular to k. The orientation of the field vectors is sketched in Fig. 1.3 where k is chosen to lie in the y-z plane and H is taken to be polarized along ˆx. The E and D fields, as well as the propagation vector k and the direction of energy flow (E×H ), all lie in a plane perpendicular to H. In this sketch x, y, and z refer to crystallographic axes of the medium.
θdenotes the angle that the propagation vector k makes with the z axis. Phase matching may be achieved under suitable conditions by selecting θ properly. It is important to note a further peculiarity that arises in anisotropic media: the direction of energy flow, E × H, need not be parallel to k but can be separated by a walk − off angle, ρ, with respect to k ; ρ is also the angular separation between the E and D fields.
In order to understand how phase matching can be achieved, it is necessary to go more deep in the property of anisotropic crystal. If the media is free of leaks, the matrix turns out to be real and symmetrical. All the symmetric tensors can always be diagonalized by choosing a suitable reference system, which in this case turns out to be the one characterized by the so-called crystallographic axes, with respect to which the crystal is found to enjoy symmetry properties. Chosing those axis the matrix has the following form:
= 11 0 0 0 22 0 0 0 22 (1.54)
Figure 1.3: Field orientation for propagation in birifrangent media. . From[60]
As it is known, the elements of the matrix are linked with the refractive indices (for the chosen coordinate system nii =
q
ii
0), for which is possible to
derive the following relation: X2 n112 + Y 2 n222 + Z 2 n332 = 1 (1.55)
The Eq. 1.40 represent an ellipsoid, the so-called the index ellipsoid ( Fig. 1.4). This one allows to identify the refractive index of an anisotropic medium depending on the direction of light propagation. Thus, based on the specific direction of propagation, each wave, that enters the crystal,will see a different refractive index.
The shape of the ellipsoid depends on the properties of the crystal. For the purposes of the discussion, let’s consider a crystal with an axis of symmetry (n11 = n22 = n0 6= n33 = ne): the crystal is called uniaxial, positive or negative,
depending on the sign of the difference of the indices ne−no; the axis of symmetry,
i.e. the one along which the index seen is ne = n33, is called extraordinary axis
; the other one, along which no = n11 = n22, is called ordinary axis .
Once the wave plane is determined on the graph given by a Cartesian coordi-nate system determined by the crystallographic axes (plane passing through the origin and perpendicular k), it is necessary to consider the intersection of this with the index ellipsoid. In this way an ellipse is obtained, as shown in Fig. 1.5, whose axes identify the directions in which the incoming wave is decomposed and their length is equal to the index of refraction in that direction of polarization.
Any wave that enters the material can be decomposed, according to the afore-mentioned axes, into two waves, ordinary (o-wave) and extraordinary (o-wave), which will see different refraction indices of the material, respectively no and ne,
Figure 1.4: General form of the index ellipsoid. The coordinate axes X, Y and Z are the optically principal axes. n1 , n2 and n3 are the principal refractive indices of the
anisotropic medium.[56]
depending on their propagation directions. no turns out to be constant, while ne
will depend on the θ angle.
Thus, for the ordinary wave the value propagation constant is simply given by k = ωn0
c , while the extraordinary wave the value propagation constant is given
by k = ωne(θ) c where 1 ne2(θ) = cos 2(θ) no2(θ) +sin 2(θ) no2(θ) (1.56) The effective index of refraction for the extraordinary wave depends on the propagation angle with respect to the z-axis (ne(θ)) and varies smoothly between
no for θ = 0° and ne for θ = 90°.
We are now in a position to understand birifringent phase matching .
Let’s first consider positive uniaxial media. There are two schemes to achieve phase matching:
1. An input e-wave at the fundamental frequency drives an o-wave at the second-harmonic frequency. Since the electric field associated with the input e-wave appears twice, we abbreviate this interaction e + e → o. Phase matching occurs when
no(2ω) = ne(ωo, θ) (1.57)
This condition can be achieved by adjusting the propagation angle θ in the nonlinear crystal, which tunes the effective index ne(ωo, θ) for the e-wave
to the needed value. This procedure is known as angle tuning. Substitut-ing Eq. 1.56 into Eq. 1.57 yields the followSubstitut-ing expression for the phase-matching angle θp: sin2(θp) = 1 no2(2ωo) − 1 no2(ωo) . 1 ne2(ωo) − 1 no2(ωo) (1.58) For a phase-matching angle to exist, the refractive indices must satisfy
ne(ωo) ≥ no(2ωo) (1.59)
i.e. the crystal must have sufficient birifrangence. This scheme in which two identically polarized fundamental waves interact to produce the second harmonic is termed type I phase matching and is sketched in 1.6
2. One input o-wave and one input e-wave, both at ωo, interact to drive an
o-wave at 2ωo (o + e → o). This scheme, where both an o-wave and e-wave
are present at the input, is known as type II phase matching . As above, phase matching is achieved via angle tuning. The condition for type II phase matching is:
2ωono(2ωo) c = ωono(ωo) c + ωone(ωo, θ) c (1.60) which reduces to 2no(2ωo) = no(ωo) + ne(ωo, θ) (1.61)
Figure 1.6: Phase-matching diagram for second-harmonic generation in a positive uniaxial medium, for the case of type I phase matching. . From [60]
Interaction
Type Phase-Matching Condition Birefringence Requirement Positive uniaxial (ne > no) Type I e + e → o no(2ω0) = ne(2ω0, θ) ne(ω0) ≥ no(2ω0) Type II o + e → o 2no(2ωo) = no(ω0) + ne(ω0, θ) ne(ω0) + no(ω0) ≥ 2no(2ω0) Negative uniaxial ( (no > ne) Type I o + o → e ne(2ωo, θ) = no(ω0) no(ω0) ≥ ne(2ω0) Type II o + e → e 2ne(2ωo, θ) = no(ω0) + ne(ω0, θ) no(ω0) + ne(ω0) ≥ 2ne(2ω0)
Table 1.1: Summary of Birefringent Phase Matching for Second-Harmonic Generation [60]
This equation has a real solution when
ne(ωo) ≥ no(2ωo) (1.62)
Therefore, greater birefringence is required compared to type I phase match-ing.
To be complete, the phase-matching conditions for both positive and negative uniaxial crystals are summarized in Table 1.1.
An important special case is the one of phase matching for θp = 90°, known
as noncritical phase matching. This case is desirable since there is no spatial walk-off. When 90° phase matching is possible, temperature tuning (i.e., exploit-ing the temperature dependence of the refractive indices) is often used for fine adjustments of the birefringence while maintaining a 90° propagation geometry.
Figure 1.7: General view of a QPM crystal . From [60]
Quasi-Phase Matching
An alternative to birefringent phase matching is the so-called quasi − phase matching, QPM. In this technique a phase mismatch ∆k between the funda-mental and second harmonic fields can be compensated for by using a nonlinear crystal with a periodic structure such that the sign of χ(2)changes every coherence
length lcof the crystal. Such a periodic structure could be constructed by slicing
a crystal into many thin slabs, each having a width equal to lc and then placing
the slabs back together in a manner such that each slab is rotated 180° relative to its neighbors. Since materials with nonzero second-order susceptibilities χ(2) lack
inversion symmetry, the resulting crystal will have a second-order susceptibility that changes sign each coherence length of the crystal,(Figure 1.7).
The coherence length of the crystal is the length over which the direction of energy flow is from the fundamental field to the second harmonic field. Thus, just as the direction of energy flow in the crystal is about to switch so as to cause attenuation of the second-harmonic field, since the relative phase between the field and the polarization at 2ω0 has slipped by π, the sign of χ(2) changes.
This change in sign restores the proper phase relation between the dipoles and the harmonic field and so ensures continued amplification of the second-harmonic field. So the QPM crystal is built such that sign changes in the nonlinear coefficient occur when this π phase slippage has just accumulated, which has the effect of resetting the relative phase back to zero. As a result, the contributions to the second-harmonic field produced by each of the regions of material of length lc all add up between them (Fig. 1.8). Averaged over a crystal length L lc the
second-harmonic field grows linearly with crystal length, and the intensity grows quadratically, although still less rapidly than the case of perfect phase matching. Since the small coherence length associated with the phase mismatch in many crystals makes the construction of a QPM crystal somewhat impractical, instead, external fields can be used to perform a periodic poling of a nonlinear ferroelectric material or a non-linear polymer. This permanent periodic poling produces the alternating sign of χ(2) needed for QPM. A common structure of this type is
commonly referred to as periodically poled lithium niobate, or PPLN.
1.2.2
Ultrashort-Pulse SHG
Qualitatively, when it is treated the second-harmonic generation starting from short pulses, what happens is that frequencies from the input pulse combine
Figure 1.8: Second-harmonic intensity as a function of distance into a nonlinear crys-tal, for k = 0, k =/ 0, and first-order QPM. The dashed line shows the average growth in the SHG intensity under QPM. From [60]
in various pairs (i.e., nominally, ω0 and Ω − ω0) to drive frequency components
(i.e. Ω) of the nonlinear polarization, and hence of the second-harmonic field. Energy conservation is satisfied automatically. This interaction is sketched from a frequency-domain perspective in Fig. 1.9.
Let’s analyze new effects that occur with ultrashort pulses, considering one with a certain bandwidth and centered at frequency ω0, that satisfies the type I
phase matching (since only one input wave is considered): 2k(ω0) = k(2ω0
What is important to understand is how the phase matching conditions changes if we consider ω0+ ∆ω, istead of ω0.
To identify the dominant effects, let’s expand the phase mismatch to the first order in optical frequency:
∆k = k2ω(2ω0) − 2kω(ω0) + ∂k2ω ∂ω 2ω0 − ∂kω ∂ω 2ω0 (Ω − 2ω0) (1.63)
If we assume phase matching for input frequency ω0 the first two terms go
away.
Identifying the derivatives with respect to ω as the inverse group velocities of the input and second-harmonic waves evaluated at frequencies ω0 and 2ω0,
respectively, we have ∆k(Ω) = vg-1(2ω0) − vg-1(ω0) (Ω − 2ω0) = ∆(vg-1)(Ω − 2ω0) (1.64)
Note that phase matching does not imply group velocity matching.
Within the limits of the first-order expansion, the phase mismatch is a function of the second-harmonic frequency only. Its magnitude is related to the mismatch in inverse group velocities.
Figure 1.9: Frequency-domain picture of ultrashort-pulse SHG. Frequencies from the input pulse combine in pairs to drive second-harmonic frequency components. [60]
• The nonlinear polarization driving the second harmonic is proportional to the autoconvolution of the fundamental field complex spectrum. Therefore, for bandwidth-limited pulses, the spectral width of the nonlinear polariza-tion is slightly broader than that of the input spectrum.
• Not all of the second-harmonic frequencies that could potentially be driven by the nonlinear polarization are phase matched. The effect is that of a filter acting solely on the second-harmonic field and not on the fundamental, with an effective filter bandwidth, ∆υ = 0.88
|∆(vg-1)|L, that is inversely proportional
to the difference in group delay through the crystal for the fundamental and second-harmonic waves. Therefore, in cases where there is substantial group velocity mismatch (GVM), the second-harmonic spectrum can be significantly narrower than the input spectrum.
• All of the frequencies in the input spectrum can participate in the genera-tion of phase- matched second-harmonic frequencies. Hence, a high-energy conversion efficiency into the second harmonic is possible even in the pres-ence of group velocity mismatch leading to the filtering effect cited above. These results are strictly valid only for a first-order expansion in the frequency variables.
As final consideration, it is important to give an estimate of the final duration of the second harmonic pulse. The second harmonic travels with a lower group velocity than the fundamental. Considering the crystal as given by several slices, in each of which a contribution to the second harmonic is generated,the second
pulse at 2ω0 arrive at the end of the crystal.
1.3
The Coherent Raman Scattering process
1.3.1
Introduction
It was shown how media that present centrosymmetry do not have non-linear quadratic, quartic, and so on, optical properties. The nonlinearity of odd order is instead always present, even if of a smaller entity. In the case of cubic phenomena, the interaction of four fields within the material induced by the cubic suscepti-bility will occur and there are many more processes that take place through the same.
Given the purpose of the discussion, it is possible to concentrate only on the Raman scattering related phenomenon (a spontaneous inelastic scattering event induced by molecular vibrations) that makes able to understand the Coherent Raman Scattering (CRS) , term that includes a family of label-free, non-invasive, non-linear optical techniques in which, typically, two input light fields induce a Raman-active vibrational transition in the material, which acts as the imaging contrast mechanism for microscopy applications [2].
These molecular vibrational modes, thanks to which it is possible to have Raman scattering phenomenon, are referred to vibrational motions of both in-dividual chemical bonds and chemical groups, thus form a contrast mechanism to not only identify molecules but also to exploit structural dynamics and in-ter/intra molecular interactions. The Raman spectrum of a specific molecule is the representation of the unique combination of chemical bonds that make this molecule different from others, thus ensuring the univocal identification of the molecule itself.
CRS and spontaneous Raman scattering are strictly related, but there are important differences between the two optical techniques. In spontaneous Raman scattering, the light-induced vibrational motions in the sample are incoherent while in CRS the vibrational modes are excited in a coherent manner. So, CRS offers a way to enhance the Raman scattering effect by means of a coherent non-linear polarization that emits a coherent Raman signal, which is stronger than
Figure 1.10: Schematic of spontaneous Raman scattering. (A) Incident light is scattered into Rayleigh ωP, Stokes (ωS = ωP − ωR), and anti-Stokes components
(ωaS = ωP+ ωR). (C) Energy diagram for Stokes Raman scattering and (B)
anti-Stokes Raman scattering. The ground state level is labeled g, the vibrational level v, and the intermediate level j. Far from electronic resonance, the intermediate states j have infinitely short lifetimes and are thus considered virtual states. Solid arrows correspond to applied light; wiggled arrows indicate the emitted radiation. [35]
the spontaneous Raman one of several orders of magnitude, thus making possible high-speed label-free imaging of living systems.
Spontaneous Raman scattering was firstly discovered in 1928 by Raman, who observed that when monochromatic light of frequency ωPis incident on molecules,
the scattered light contains an array of different colors. The strongest compo-nent has the same frequency as the incident light, but weaker contributions with shifted frequencies ωP± ωR are also observed. The first contribution is referred
to as the Rayleigh component whereas the shifted contributions are now known as the Raman scattered components, with the red-shifted frequencies called the Stokes components and the blue-shifted frequencies called the anti − Stokes components. Raman quickly realized that the frequencies ωR correspond to
char-acteristic frequencies of molecular motions, and that the spectrum obtained can be used to identify and characterize molecules.
Figure 1.10 displays the energy diagrams of spontaneous Raman scattering. Typically, ωP lies in the visible or near-infrared range of the spectrum ( 1.5 − 3.7
103 THz, or 500 − 1250nm), while the nuclear vibrational motions ωR are in
the mid- to far-infrared range (75 ÷ 470 THz, or 4÷ 25 µm). Because the fre-quency of the applied light exceeds by far the nuclear vibrational frequencies, the nuclei are unable to respond instantaneously to the electromagnetic field. On the contrary, the electrons follow the oscillating field adiabatically. The Raman effect is therefore understood as a field-induced modulation of the electron cloud, which exhibits shifted frequency components due to the presence of nuclear nor-mal modes. A disadvantage of the Raman technique is the intrinsic weakness of the scattering process. Only 1 part out of 106 of the incident radiation will be
scattered into the Stokes frequency when propagating through 1 cm of a typical Raman active medium. This feebleness severely complicates several applications, including vibrational imaging of biological samples. Here nonlinear methods come to the rescue.
CRS is a third-order non-linear optical process that belongs to the class of four-wave mixing, which means that the simultaneous interaction of three input
frequency detuning ωP- ωS matches a characteristic vibrational frequency Ω of
the molecules under study ( ωP- ωS =Ω), four possible CRS interaction schemes
take place [32], each one generating a new field which is the result of a four-wave mixing process:
• the coherent anti − Stokes Raman scattering (CARS) field at frequency ωaS = ωP +( ωP − ωS),
• the coherent Stokes Raman scattering (CSRS) field at frequency ωcsrs=
ωS − ( ωP − ωS),
• the stimulated Raman gain (SRG) at frequency ωS and the stimulated
Raman loss (SRL) at frequency ωP. The SRG and SRL processes are part
of the stimulated Raman scattering (SRS) interaction.
In Fig.1.11 a conceptual scheme of the CRS non-linear interaction process is depicted, where the two input fields, pump and Stokes fields (Fig.1.11(a)), generate a non-linear refractive index modulation of the medium at the beating frequency Ω, equal to ωP − ωS, that can interact with them, leading to the
creation of new side-bands (Fig. 1.11(b)). These new side-bands are exactly the output fields derived from the CRS interaction (CARS, SRG, SRL, CSRS), as it is shown in Fig. 1.11(b).
This section of the chapter will focus on the analysis of the physics underlying the Raman Scattering process, giving a briefly explanation of the Spontaneus Raman scattering, to move then to CRS and CARS.
The classical description of Raman scattering and CARS takes the vibrational motion explicitly into account, in the form of a harmonic oscillator. Although the classical model provides an intuitive picture of the process, a quantum mechan-ical description is required to predict the intensities of the Raman and CARS vibrational signals.
1.3.2
Classical model of vibrational motions
To better understand the CRS processes, it is necessary to highlight the rela-tionship between the electron displacement and the electric field of the electro-magnetic radiation. The light-matter interaction can be described assuming to
Figure 1.11: Conceptual scheme for the four possible CRS signals generation. (a) Two input fields at frequency ωP(pump) and ωS (Stokes) interact with a Raman active
χ(3) medium (the bi-atomic molecule), generating at the output four fields (CARS,
SRL, SRG, CSRS). The arrows thickness is proportional to the intensity of the relative field. (b) Conceptual representation in the spectral domain, where the χ(3) interaction
generates a modulation of the non-linear refractive index that modulates the input fields at the beating frequency Ω. This interaction creates side-bands on the input fields, that are shifted of the beating frequency Ω.
work in the adiabatic approximation, which states that the nuclei, having a mass which is approximately 2000 times bigger than the electron mass, cannot follow the oscillation of the electric field, while the electrons, being lighter, will oscillate and it is possible to associate to them an adiabatic electron potential which de-pends on the nuclear coordinates. Thus, the electronic polarizability α(t), which links the electric dipole moment to the driving electric field, will be influenced by the nuclear modes.
By the assuming an input field that is a monochromatic plane wave, of the form E(t) = Ae−iωpt+ cc , omitting vectorial notation for simplicity, the
magni-tude of the electric dipole moment is :
µ(t) = α(t)E(t) (1.66) When no nuclei modes are present, the polarizability can be approximated as a constant, α0, but in the presence of nuclear modes, the electronic polarizability
can be expanded in a Taylor series in the generalized nuclear coordinate Q: α(t) = α0+
∂α ∂Q
Q(t) + ... (1.67) The nuclear motion along Q can be assumed to be that of a classical harmonic oscillator:
Q(t) = 2Q0cos(ωυt + φ) = Q0
h
the quantity ∂α
∂Q, which describes how the polarizability varies along Q. Raman
scattering therefore occurs only in molecules in which the applied field brings about a polarizibility change along the nuclear mode. The condition of polariz-ibility change forms the basis of the selection rules for Raman spectroscopy, which depend on the symmetry of the nuclear mode in the molecule.
The strength of the Raman signal is commonly expressed in terms of the differential scattering cross-section, defined as the amount of scattered intensity within the elementary solid angle, divided by the incident intensity [35]:
σdiff ≡ ∂σ ∂Ω = ω ¯IRr2 ωRI¯ (1.70) where ¯I(¯IR) is defined as the cycle-averaged Poynting vector of the incoming
(scattered) light.
The differential cross-section can be related to square modules of the polar-izibility change (i.e., the first-order term in the Taylor expansion of the polariz-ibility): σdiff∝ ∂α ∂Q 2 (1.71) The total intensity of the Raman scattered light is then found by integrat-ing over the solid angle and by summintegrat-ing the scatterintegrat-ing contributions of all N molecules incoherently:
Itot = N I
Z
σdiffdΩ (1.72)
Thus, the spontaneous Raman signal scales linearly with the incident light in-tensity I, with the number of scattering molecules and with the Raman scattering cross-section, directly proportional to
∂α ∂Q 2 .
The classical model provides an empirical connection between the Raman scattering cross-section and the observed intensities of the Raman lines, but it is unable to predict the Raman transition strengths. To take the scattering interactions explicitly into consideration, a quantum mechanical description is necessary.
The classical description of the CRS process provides an intuitive picture of the molecular motions driven by two external fields. So with respect to spontaneous Raman, where there is a single incident field impinging onto the sample, here two incident fields induce oscillations in the molecular electron cloud. These oscillations form an effective force along the vibrational degree of freedom, which drives the nuclear vibrational modes. The nuclear modes will spatially coherently modulate the refractive properties of the material, such that a third field arriving onto the sample will experience this modulation and give rise to the Stokes and the anti-Stokes signal.
The electric field of two incoming beams always called Pump and Stokes can be written as:
E1(t) = A1(t)e−iω1t+ cc (1.73)
E2(t) = A2(t)e−iω2t+ cc (1.74)
Assuming that the vibrational mode of the nucleus is that of a damped har-monic oscillator, with resonant frequency ωυ , that the frequency of Pump, ω1,
and the one of the Stokes, ω2, are far from the vibrational frequency of the
os-cillator, that means that the nuclei cannot be driven by the oscillation of the incident fields, while the electrons can follow them, and that the intensity of the two fields are large enough, then non linear processes can occur such that the electrons start to oscillate at the frequency Ω = ω1− ω2.
This oscillation is due to the time-varying force experienced by the nuclear vibrational mode, which arises thanks to the fields, that can be written as:
F (t) =∂α ∂Q 0 [A1A∗2e−iΩt+ cc] (1.75)
The equation of motion for the molecular vibration along Q is expressed as: d2Q(t) dt2 + 2γ dQ(t) dt + ωυQ(t) = F (t) m (1.76)
where γ is the damping constant, m is the reduced mass of the nuclear oscil-lator and ωυ is the resonance frequency of the harmonic oscillator.
A solution to Eq.1.76 can be found by adopting the trial solution of the form Q(t) = Q(Ω)e−iωt+ cc (1.77) ,
that lead to the following equation for the amplitude of the vibration : Q(ωυ) = 1 m ∂α ∂Q 0 A1A∗2 ω2 υ− Ω2 − 2iΩγ (1.78) Notice that the nuclear mode is efficiently driven when the difference frequency between the Pump and the Stokes fields, Ω, approaches the resonance frequency ωυ of the vibration and depends on the amplitude of the Pump and Stokes field
and on the term ∂α ∂Q
ω1+ Ω is the coherent anti-Stokes Raman scattering frequency.
Combining Eq.1.78 and Eq.1.79 with the general form of polarization, P (t) = P (ω)e−iωt, it is possible to find the amplitude of the different polarization com-ponent. Focusing on the anti-Stokes one:
P (ωaS) = N m ∂α ∂Q 2 0 A21A∗2 ω2 υ− Ω2− 2iΩγ = 60χNL(Ω)A21A∗2 (1.81)
where the non linear susceptibility is defined as [14]: χNL = N 60m ∂α ∂Q 2 0 1 ω2 υ− Ω2− 2iΩγ = 60χNL(Ω)A21A∗2 (1.82)
The induced polarization in the material produces four signal at four different frequencies, two of them are at the fundamental frequencies of the Pump and the Stokes, while the other two are new frequencies which correspond to the Stokes and the anti-Stokes coherent Raman signals.
1.3.3
Coherent anti-Stokes Raman scattering
Coherent anti-Stokes Raman scattering (CARS) is the result of the frequency mixing between two pump photons, at frequency ωP, and one Stokes photon, at
frequency ωS, to generate one anti-Stokes photon, at frequency ωaS= 2ωP-ωS, as
it is shown in Fig.1.12.
In fact, by impinging on a Raman active medium with two fields, the Pump at frequency ωP and the Stokes at frequency ωS, whose frequency detuning Ω=ωP
-ωS equals the resonance frequency of the vibrational mode of the medium ωυ
(Ω = ωυ), a non-linear third-order polarization at the anti-Stokes frequency ωaS
is generated which can irradiate a field at the same optical frequency. This non-linear polarization is the result of two first interactions made by one Pump photon and a Stokes photon through a virtual level (dashed line in Fig. 1.12) to generate a coherent state that is the superposition of the ground and first excited vibrational level. A third interaction with another Pump photon carries the system to the final virtual state where, through the third-order non-linear polarization, is emitted the anti-Stokes photon.
Figure 1.12: Schematic of the CARS light-matter interaction through Jablonski dia-grams (dashed lines are virtual levels and solid lines real levels). The subscripts p, s and aS stands for pump, Stokes and anti-Stokes, respectively.
From an analytical point of view, exploiting the four wave-mixing approach and considering the degenerate case where the first and third fields coincide with the Pump (E1=E3=E1) at frequency ωP and the second one coincides with the
Stokes (E2=ES) at frequency ωS, the radiating field (EaS) oscillates at the
anti-Stokes frequency ωaS = 2ωP− ωS, has a wavevector kaS = 2kP− kS and turns to
be, after a propagation of length L in the Raman active medium : EaS(L) ∝ −iχ(3)(Ω)EP2ES*Lsinc
∆kL 2
ei∆kL2 (1.83)
where ∆k = kaS + kS− 2kP is the wave vector mismatch . The wave vector
phase-match sets, on the one hand, the maximum interaction length for effi-cient CARS conversion and, on the other hand, the direction where the CARS field will be emitted ( k is a vector). Under the assumption of tight focusing condition, which is the typical situation in microscopy experiments using high numerical-aperture objectives, the interaction length L becomes small and the phase-matching condition relaxes (∆k = 0), thus we can neglect ∆kL and set it to zero.
Since during detection, what is measured is the intensity, which is proportional to the squared modulus of the field, it is possible to write the intensity of the anti-Stokes signal as:
IaS(L) ∝ χ (3)(Ω) 2 IP2IS*L2sinc2 ∆kL 2 ei∆kL2 (1.84)
From Eq.1.84 it is possible to note:
• a quadratic dependence on the interaction length L
• a quadratic dependence on the pump intensity and linear on the Stokes intensity
(Fig. 1.13A). Another combination of the same input frequencies is also possible. This new combination leads to another FWM phenomenon (Fig. 1.13B), which is coming purely from the electronic contribution and does not involve the real vibrational levels of the investigated molecules. This nonresonant process consti-tutes the nonresonant contribution χ(3)NR. In addition, the background molecules
(the molecules other than the targeted ones), which are not in resonance with the input frequency combination, may also contribute to nonresonant part χ(3)NR
through their electronic response.
In short, nonresonant contribution χ(3)NR is coming from pure electronic
re-sponse away from the resonance. The non-resonant contribution, which is typ-ically referred to as non-resonant background (NRB), arises from the fact that other four-wave mixing processes, in addition to the resonant CARS process, compete simultaneously to generate a signal at the anti-Stokes frequency ωaS.
While CARS process is mediated by Raman vibrational resonances, the other processes, originating from the four-wave-mixing (FWM) mechanism, can ex-change energy via the third-order non-linear susceptibility of the molecules even in absence of real Raman vibrational transitions (since the energy and momen-tum conservation is preserved). For example, in a biological samples, typically, there are not only the target molecules but also others present, like the solvents molecules which usually do not share the same vibrational resonances as the stud-ied ones. Therefore, these non-target molecules cannot contribute to the resonant term of the susceptibility but they can give electronic non-resonant contributions, by interacting with the incoming fields.
Electronic response is instantaneous because electrons being lighter in mass can follow the fields adiabatically. Hence χ(3)NR is spectrally flat and can be
treated as constant in a small frequency range (Fig. 1.14A). In general, it is possible to write to overall third-order non-linear susceptibility, since χ(3)R is a
complex quantity that can be separated into its real and imaginary parts, as: χ(3)(Ω) = <(χ(3)R(Ω)) + i=(χ(3)R(Ω)) + (χ(3)NR) (1.85) where, according to Eq.1.82,
<(χ(3)R(Ω)) ∝ N ωυ− Ω
![Figure 1.3: Field orientation for propagation in birifrangent media. . From[60]](https://thumb-eu.123doks.com/thumbv2/123dokorg/7499490.104388/28.892.220.722.149.498/figure-field-orientation-propagation-birifrangent-media.webp)
