Second harmonic generation applied to biomimetic interfaces

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Thesis

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Second harmonic generation applied to biomimetic interfaces

LICARI, Giuseppe Léonardo

Abstract

Interface properties dramatically differ from those of the bulk phase. Specific techniques need to be used to study interfaces and, here, surface second harmonic generation was employed.

This process is allowed at interfaces but not in the bulk. We have investigated biomimetic interfaces using Yellow Oxazole (YO) cyanine dyes and a dithienothiophene (DTT-1) push-pull molecule. YO compounds are well-known DNA probes and we showed that they could be used to study interfaces in the presence or not of DNA. DTT-1 was used as a mechanosensitive probe for detecting order and surface pressure at interfaces, extending its mechanism to nonlinear spectroscopy. Computer simulations were performed in parallel to the experiments in order to have a better microscopic view of the systems. The work presented in this thesis suggests that both classes of compound may be applied in the future as biological probes using second harmonic generation.

LICARI, Giuseppe Léonardo. Second harmonic generation applied to biomimetic interfaces . Thèse de doctorat : Univ. Genève, 2018, no. Sc. 5198

DOI : 10.13097/archive-ouverte/unige:103771 URN : urn:nbn:ch:unige-1037712

Available at:

http://archive-ouverte.unige.ch/unige:103771

Disclaimer: layout of this document may differ from the published version.

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U

NIVERSITÉ DE

G

ENÈVE

Section de chimie et biochimie Départment de chimie physique

F

ACULTÉ DES

S

CIENCES

Professeur E. Vauthey

Second Harmonic Generation Applied to Biomimetic Interfaces

THÈSE

présentée à la Faculté des sciences de l’Université de Genève pour obtenir le grade de Docteur és sciences, mention chimie

par

Giuseppe Leonardo L

ICARI de

Marsala (Italie)

Thèse N°5198

G

ENÈVE

Atelier ReproMail

2018

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Dedico questa tesi ai miei genitori, Anna e Gaspare.

Mi hanno insegnato che con la semplicità si possono fare grandi cose.

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The most beautiful thing we can experience is the mysterious.

It is the source of all true art and science.

Albert Einstein

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T

ABLE OF

C

ONTENTS

Page

Abstract ix

Résumé xi

Acknowledgments xiii

List of Tables xv

List of Figures xvii

1 General Introduction 1

1.1 Liquid Interfaces . . . 2

1.1.1 Interfaces in Biology . . . 2

1.2 Interface-Selective Techniques . . . 3

1.3 Aims of the Thesis . . . 5

2 Concepts Behind the Experiments 7 2.1 Linear Spectroscopy . . . 8

2.1.1 Light-Matter Interaction . . . 8

2.1.2 Absorption . . . 10

2.1.3 Fluorescence . . . 10

2.1.4 Radiative and Non-Radiative Rate Constants . . . 11

2.2 Nonlinear Spectroscopy . . . 12

2.2.1 Second Harmonic Generation . . . 13

2.2.2 Classical Harmonic and Anharmonic Oscillators . . . 15

2.2.3 Symmetry Properties of the Susceptibility . . . 16

2.2.4 The Three-Layers Model . . . 18

2.2.5 Susceptibility, Hyperpolarizability and Orientation . . . 19

2.2.6 Surface Second Harmonic Generation Setup . . . 24

3 Concepts Behind the Simulations 27

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TABLE OF CONTENTS

3.1 Quantum Mechanics . . . 28

3.1.1 The Schrödinger Equation and Variational Principle . . . 28

3.1.2 The Hartree-Fock Approximation . . . 30

3.1.3 Density Functional Theory . . . 32

3.1.4 Computing Properties . . . 37

3.1.5 Limitations . . . 39

3.2 Molecular Dynamics . . . 40

3.2.1 The Basic Machinery . . . 41

3.2.2 The Energy Potentials . . . 44

3.2.3 Computing Properties . . . 46

3.2.4 Limitations . . . 49

4 Fluorescent DNA Probes at Liquid/Liquid Interfaces 51 4.1 Introduction . . . 52

4.2 Results and Discussion . . . 53

4.2.1 SSHG Spectra . . . 53

4.2.2 Polarization-Resolved SSHG and Orientation at the Interface . . . 57

4.2.3 Quantum-Chemical Calculations . . . 62

4.2.4 Time-Resolved SSHG . . . 66

4.3 Conclusions . . . 69

5 Combining Second Harmonic with Simulations 71 5.1 Introduction . . . 72

5.2 Results and Discussion: TOSAC-3 at Biomimetic Interfaces . . . 74

5.2.1 Affinity of the Dye for the Interfaces . . . 74

5.2.2 Molecular Orientation at the Interfaces . . . 79

5.2.3 Conclusions . . . 88

5.3 Results and Discussion: Effect of DNA . . . 89

5.3.1 Disruption of Aggregates . . . 89

5.3.2 Reorientation and Induced Chirality . . . 90

5.3.3 A Model of the DNA Complex . . . 94

5.3.4 Impact of DNA on the Dynamics at the Interface . . . 97

5.3.5 Conclusions . . . 98

6 A Mechanosensitive Probe for Biomimetic Membranes 101 6.1 Introduction . . . 102

6.2 Results and Discussion . . . 104

6.2.1 Interfacial Spectra of the Mechanosensor . . . 104

6.2.2 Origin of the Mechanosensitivity . . . 106 vi

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TABLE OF CONTENTS

6.2.3 Average Interfacial Orientation of the Dye . . . 110

6.2.4 Ultrafast Excited-State Dynamics at the Interface . . . 114

6.2.5 Imaging in Giant Unilamellar Vesicles . . . 116

6.3 Conclusions . . . 118

General Conclusions 121

A Experimental Details 125

B List of Publications 137

C List of Symbols and Abbreviations 139

Bibliography 145

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A

BSTRACT

I

nterfaces are of fundamental interest in many areas of research. Their properties can dramatically differ from those in the bulk phase due to the anisotropy of forces experienced by interfacial molecules. In particular, biological interfaces, such as cellular membranes, play a major role in signaling, transport and catalysis, among many others. Although the study of interfaces is of utmost importance, the research in this field has suffered from the scarcity of techniques that specifically measure an interfacial signal. Surface Second Harmonic Generation (SSHG) is a technique where light at a given frequency is converted into an electromagnetic wave at twice this frequency. This process is not allowed in symmetric media but it is operative at interfaces where the symmetry is broken, providing therefore an effective interface-specificity.

In this thesis, we have investigated biomimetic interfaces using SSHG. Two classes of com- pound were considered. Firstly, the cyanine dyes belonging to the the Yellow Oxazole (YO) family were studied at interfaces. These compounds are well-known biological probes and can detect DNA at very small concentrations. The aim of this study was to find out if these dyes could be used as interfacial probes for biological applications. We found that the aggregation behavior of YO dyes can drastically change upon adsorption at the interface and that their dynamics is also highly environment-dependent. The molecular orientation underwent also substantial variations upon changing the characteristics of the interface. The addition of DNA had a large impact on the interfacial response of a specific cyanine dye, TOSAC-3, and revealed also the rising of a chiral signal. Overall, the data suggest that these compounds may be used to probe nucleic acid at biological-like interfaces and to explore chirality in supramolecular interfacial architectures.

Secondly, a dithienothiophene push-pull molecule was used as a mechanosensitive probe for detecting order and surface pressure at interfaces. The innovative mechanism of this probe was already applied in fluorescence and, here, it was extended to nonlinear spectroscopy. We found

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ABSTRACT

that both the SSHG spectrum and the orientation are good reporter of interfacial fluidity, with the latter being more sensitive over the former. The dynamics of the mechanosensor was shown to slow down in a constrained environment, and, since a similar effect was observed for the cyanine TOSAC-3, may represent a general property of molecules undergoing large amplitude motion in the excited state.

Computer simulations were performed in parallel to the experiments in order to have a better microscopic view of the molecular systems under study. Both electronic structure and molecular dynamics simulations provided useful insights in the static and dynamic properties of the probes.

The combination of experiments and simulations gave overall a clearer picture and facilitated the interpretation of the data.

The work presented in this thesis suggests that both classes of compound may be applied in the future as biological probes for interfaces in combination with nonlinear techniques such as second harmonic generation.

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R

ÉSUMÉ

L

es interfaces présentent un intérêt considérable dans de nombreux domaines de recherche. Leurs propriétés peuvent différer notablement de celles du cœur des ma- tériaux en raison de l’anisotropie des forces subies par les molécules aux interfaces.

En particulier, les interfaces biologiques, telles que les membranes cellulaires, jouent un rôle critique, entre autres, dans la signalisation, le transport et la catalyse. Bien que l’étude des interfaces soit de grande importance, la recherche dans ce domaine a été ralentie dû au nombre réduit de techniques mesurant spécifiquement un signal à l’interface. La Génération de Seconde Harmonique (GSH) est une technique où la lumière, à une fréquence fondamentale, est convertie en une onde électromagnétique au double de la fréquence fondamentale. Ce processus n’est pas permis dans des milieux symétriques mais est opérationnel aux interfaces où la symétrie est réduite, fournissant ainsi une spécificité d’interface élevée.

Dans cette thèse, nous avons étudié des interfaces biomimétiques à l’aide de la GSH. Deux classes de composés ont été considérées. Premièrement, des cyanines, appartenant à la famille des oxazoles jaunes (Yellow Oxazole, YO), ont été étudiés aux interfaces. Ces composés sont des sondes biologiques bien connues pouvant détecter l’ADN à de très faibles concentrations. Le but de cette étude était de comprendre si ces composés pouvaient être utilisés comme sonde des inter- faces pour des applications biologiques. Nous avons observé que le comportement d’agrégation des colorants YO peut radicalement changer lors de l’adsorption à l’interface et que leur dynamique est également fortement dépendante de l’environnement. L’orientation moléculaire peut aussi subir des variations substantielles en changeant les caractéristiques de l’interface. L’addition d’ADN a eu un grand impact sur la réponse d’une molécule spécifique, TOSAC-3, et a également révélé l’apparence d’un signal chiral. Les données suggérent que ces composés pourraient être utilisés pour examiner des molécules d’acide nucléique à des interfaces biomimétiques et pour

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RÉSUMÉ

explorer la chiralité dans des architectures supramoléculaires aux interfaces.

Deuxièmement, une molécule "push-pull" appartenant à la famille des dithiénothiophènes a été utilisée comme sonde mécano-sensible pour détecter l’ordre et la pression de surface aux différentes interfaces. Le mécanisme original de cette sonde a été appliqué en fluorescence et, ici, a été étendu à la spectroscopie non linéaire. Les résultats obtenus montrent que le spectre GSH et l’orientation sont tous deux de bons indicateurs de la fluidité de la membrane, l’orientation étant plus sensible que le spectre. La dynamique de cette sonde, tout comme TOSAC-3, a été ralentie dans un environnement contraint, ce qui peut représenter une propriété générale des molécules subissant un large réarrangement de la structure dans l’état excité.

Des simulations numériques ont été effectuées parallèlement aux expériences afin d’avoir une meilleure vision microscopique des systèmes moléculaires étudiés. Des simulations de structure électronique et de dynamique moléculaire ont fourni des informations utiles sur les propriétés statiques et dynamiques des sondes. La combinaison d’expériences et de simulations a donné une image plus claire et a facilité l’interprétation des données.

Le travail présenté dans cette thèse suggère que les deux classes de composés peuvent être appliquées à l’avenir comme sondes biologiques pour les interfaces en combinaison avec des techniques non linéaires, telles que la génération de second harmonique.

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A

CKNOWLEDGMENTS

T

he work described in this thesis has been accomplished at the University of Geneva in the department of Physical Chemistry. The first person I would like to acknowledge is Prof. Eric Vauthey, my thesis supervisor. He accepted me in the group and with his enthusiasm and encouragement he has stimulated my curiosity and love for research during these last few years. He was always available for discussion and opened to any idea. I couldn’t have asked for a better supervisor.

I want to thank Prof. Kaori Sugihara (Université de Genève) and Prof. Sylvie Roke (École Polytechnique Féderale de Lausanne) for accepting the job of reading and evaluating this thesis and participating as members of the jury during my PhD defense.

During my PhD, I met many people in our research group, and will always have nice memories of them. Thanks to them, I was able to continue with my work. We spent also a lot of time together outside the university. The days where things were not working in the lab would have been harsh without them around. Therefore, I would like to thank immensely all the present and past members of our group. In particular, I am very thankful to Sophie Jacquemet. She was always there to help with bureaucratic papers but she was also always available just for a friendly chat.

I want to thank Didier Frauchiger, whose wonderful technical skills are of an essential value for the whole group. I am glad I could count on his help.

I would like to express my gratitude to all the people I have collaborated with. I had the chance to work with very competent and experienced people. They have taught me a lot of things and their contribution has enriched substantially my work.

I would like to thank all my friends, especially the ones I met in Geneva. With them I shared beautiful moments during my doctoral period. I would have enjoyed much less the time in Geneva without their presence.

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ACKNOWLEDGMENTS

An acknowledgment goes also to my accordion and to music in general. Music has been for me a very powerful weapon to fight against the moments of stress and difficulty. My accordion is for me as a friend, to which I can talk deliberately and from which I receive a comforting response, in the form of acoustic vibrations.

My family has always supported the decisions I made in my life. They have watched over me and shared their wise thoughts. I thank them for their suggestions, comprehension and love.

Thanks in particular to my nieces and nephews, the most beautiful gift I could have asked for from my sisters. Their joy and vivacity have filled many (otherwise boring) days of my life and have always reminded me that we should not forget the kids inside all of us.

I would like to thank Delphine. Many times she had stayed next to me while I was working, just to keep me company. Many times she understood the things to do to help me completing the PhD adventure. I hope she realizes how much she means to me, even if I am not a man of many sweet words.

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L

IST OF

T

ABLES

TABLE Page

2.1 Coefficients determined from refractive indices and angle of incidence. . . 20

4.1 Analysis of the polarization-resolved SSHG data measured with YOYO-1 and YOSAC-1. 58 4.2 DFT excitation energies and oscillator strengths of YO, H- and J-dimers. . . 63

5.1 Results from the analysis of TOSAC-3 polarization-resolved SSHG data. . . 80

5.2 Comparison of the experimental and simulated interfacial TOSAC-3 orientations. . . 81

5.3 Parameters describing the TDM orientation in the molecular frame. . . 83

5.4 Analysis of the polarization-resolved SSHG data of cyanine dyes in the presence of DNA. 91 5.5 Time constants derived from the fit to the TR-SSHG data of TOSAC-3. . . 97

6.1 Time constants obtained from FLUPS and TA data. . . 109

6.2 Analysis of the polarization-resolved SSHG data of DTT-1. . . 112

6.3 Time constants derived from the fit to the TR-SSHG data of DTT-1. . . 114

A.1 Number of molecules contained in each MD simulation. . . 131

A.2 Parameters extracted from the fit to the DFT potentials. . . 133

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L

IST OF

F

IGURES

FIGURE Page

2.1 Jablonski diagram depicting spectroscopic processes. . . 11

2.2 Energy level scheme illustrating the possible resonances in SHG. . . 14

2.3 Dependence of the linear and second order susceptibility as a function of frequency. . 16

2.4 Three layer model geometry for surface second harmonic generation. . . 19

2.5 Euler angles and reference frames. . . 21

2.6 Dependence of the orientationDparameter on angle distributions. . . 23

2.7 TR-SSHG setup used for the work in this thesis. . . 24

4.1 Schematic representation of the excitonic interaction. . . 52

4.2 Structure of the fluorescent DNA probes. . . 53

4.3 Electronic absorption spectra of YOYO-1 in various solvents. . . 54

4.4 Concentration dependence of the SSHG spectra of different cyanines. . . 54

4.5 Absorption and SSHG spectra of YOSAC-3. . . 56

4.6 SSHG spectra of YOYO-1 at different polarization geometries. . . 57

4.7 SSHG spectra of YOSAC-1 at different polarization geometries. . . 58

4.8 Polarization profiles measured with YOYO-1 at different concentrations. . . 59

4.9 Polarization profiles measured with YOSAC-1 at different concentrations. . . 60

4.10 Tilt angle as a function of YOYO-1 and YOSAC-1 concentration. . . 61

4.11 Electronic transitions and hyperpolarizabilities of YO, H- and J-dimers. . . 63

4.12 Frontier molecular orbitals of YO, H- and J-dimers. . . 64

4.13 Frequency-dependent hyperpolarizability tensor for the YO, H-dimer and J-dimer. . . 65

4.14 TR-SSHG profiles measured with YOYO-1, YOSAC-1 and YOPRO-1 at the interface. 67 5.1 Structures of TOSAC-3 and phospholipids. . . 73

5.2 Experimental adsorption isotherms of TOSAC-3 at interfaces. . . 74

5.3 Examples of unit cells used for the MD simulations. . . 76

5.4 Potential of mean force profiles of TOSAC-3 at interfaces. . . 76

5.5 Density profiles obtained from simulations for TOSAC-3. . . 78

5.6 Polarization-resolved SSHG data of TOSAC-3 without or with monolayer. . . 79 xvii

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LIST OFFIGURES

5.7 Orientation parameterDas a function of TOSAC-3 concentration. . . 80

5.8 Experimental and simulated absorption spectrum of TOSAC-3. . . 82

5.9 Orientation anglesµandand corresponding distribution widths from MD. . . 83

5.10 Variation of the skewed Gaussianbparameter as a function of the surface concentration. 84 5.11 MD snapshots illustrating typical interfacial orientations of the dye. . . 85

5.12 Comparison of the simulated and the tilt angle distributions deduced from experiments. 85 5.13 MD density profiles of simulation molecules at interfaces. . . 87

5.14 SSHG spectra of TOSAC-3 at the interface without and with DNA. . . 90

5.15 Polarization-resolved SSHG measurements of cyanine dyes without and with DNA. . 92

5.16 Effect of DNA on TOSAC-3 and YOPRO-1 orientations. . . 94

5.17 Normalized density profiles obtained from simulations with DNA/TOSAC-3 complexes. 95 5.18 Distances and angle distributions for TOSAC-3 and DNA in different complexes. . . . 96

5.19 TR-SSHG data recorded with TOSAC-3 at the interface without and with DNA. . . . 97

6.1 Structure of DTT-1, DPPC and DOPC. . . 103

6.2 SSHG spectra of DTT-1 at various interfaces and mean molecular areas. . . 104

6.3 Molecular orbitals of the mechanosensitive dye. . . 105

6.4 Simulated absorption spectra and potential energy surfaces of DTT-1a. . . 106

6.5 Steady-state and time-resolved fluorescence measurements of DTT-1. . . 108

6.6 Total intrinsic hyperpolarizability of DTT-1a as a function of the dihedral angle. . . . 109

6.8 Example of frequency-dependent hyperpolarizability tensor calculated for DTT-1a. . . 110

6.7 Polarization profiles measured with DTT-1 in different experimental conditions. . . . 111

6.9 Mean tilt angle of DTT-1 at the interfaces as function of the probe wavelength. . . 113

6.10 TR-SSHG profiles measured with DTT-1 at the interfaces. . . 114

6.11 Transient absorption measurement of DTT-1. . . 116

6.12 Two-photon excitation fluorescence microscopy of a GUV. . . 117

A.1 Reflectance of silver and aluminum mirrors used as references. . . 127

A.2 Linear dependence of the square root of the SSHG signal with the power of the probe. 128 A.3 Optimized structure of TOSAC-3 with characteristic atoms highlighted. . . 133

A.4 Characteristic distributions of dihedral angles of TOSAC-3. . . 134

A.5 Area-normalized distribution of the thioester dihedral angle of TOSAC-3. . . 134

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C

HAPTER

1

G

ENERAL

I

NTRODUCTION

L

iquid interfaces play a fundamental role in various different areas of science. The chemical and physical properties of liquid interfaces belong to an interdisciplinary research area, and thus there are investigations covering several surface-related topics, such as electric potential, ion solvation, adsorption, extraction or catalysis. In this chapter, a general view of research at surfaces is presented, with a specific focus on biological interfaces. The main approaches for interfacial investigations will be also introduced. The aim of this thesis will be discussed in the last section. The introduction to the techniques and the computational methods used in this thesis is left to chapters 2 and 3. The results of the work are described in chapters 4-6 while experimental and simulation details are listed in Appendix A.

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CHAPTER 1. GENERAL INTRODUCTION

1.1 Liquid Interfaces

At a phase boundary, molecules experience anisotropic forces that can cause a specific arrange- ment of other species or of the solvent. For example, an accumulation of charged surfactants at a boundary can give rise to an electric field that drives the adsorption of other molecules by Coulombic interactions.1Two immiscible liquids may appear as two independent and isolated chemical systems but a microscopic observation of the phenomena taking place at the interface would reveal a multitude of dynamic processes occurring on a short timescale, such as transfer of mass or energy. Moreover, the bulk properties can be well-defined by macroscopic quantities such as viscosity or polarity. These properties are difficult to define at an interface since the region of interest is now confined to few nanometers. One way to extract useful information about the in- terface is to use a dye as a local probe whose properties change according to the environment. For example, the triphenylmethane dye malachite green has been used as a probe of local viscosity2 or the excited-state lifetime of eosin B was shown to depend on the H-bond donating ability of the solvent.3,4In this approach, it is very important to understand the behavior of the probe already in the bulk phase before interpreting the results at the interface.

1.1.1 Interfaces in Biology

Membranes play a key role in cells since they separate the interior from the extracellular space.

Besides their structural and protective role, membranes control the flux of molecules inside the cell, promote signaling and serve as a surface for adsorption of several biostructures.5,6Models of membranes have been of great interest in order to investigate their properties. In particular, the interface between two immiscible liquids has been widely used and many fundamental processes at this interface such as photosynthesis, biocatalysis, ion pumping, and others have been addressed.7The investigation of biointerfaces has relevance for biomaterials and biomedical applications. Replacing parts of living systems requires a deep knowledge of the interactions at the interface between the biomaterials.8,9In drug delivery, it is crucial to study the effect of coatings in the release of the active molecule.10Moreover, the activity of some drugs is associated with cell membrane recognition and, therefore, the mechanism of action can be understood only after an appropriate investigation of the membrane-drug interactions.11

Biological interfaces comprise many components such as phospholipids, cholesterol and proteins. Various films and assemblies have been introduced in order to mimic biological archi- tectures. Among many, Langmuir films and liposomes are of some note. The latter are spherical vesicles that contain at least one lipid bilayer. They can mimic the compartmentalization, cur- vature and fluidity of cell membranes.12The different possible sizes of a liposome have given rise to the classification in Small, Large and Giant Unilamellar Vesicles (SUV, LUV and GUV respectively). MultiLamellar Vesicles (MLV) and multivesicular liposomes are also encountered when dealing with this membrane model. A Langmuir monolayer instead consists of a surfactant

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1.2. INTERFACE-SELECTIVE TECHNIQUES

spread on the water surface and packed together by mechanical barriers to obtain a compact film with a certain surface pressure. The simpler Langmuir films have many advantages over liposomes. For example, many experimental variables such as surface pressure, composition, tem- perature, pH and domain formation can be easily controlled in monolayers. Moreover, monolayer investigations usually require small amount of material, which reduces the costs for experiments involving expensive proteins. The characterization of these films can give valuable information on the orientation, penetration, interaction and hydration of molecules at the interface.9For example, the conformation of peptides has been observed to affect the interaction with lipids in the membrane.13The presence of lipid monolayers at an ITIES (Interface Between Two Immiscible Electrolyte Solution) has showed a great effect on the rate of ion transfer across the interface.7The dehydration and orientation of phospholipid headgroups was found to depend on the monolayer surface pressure and on the presence of cations such as Ca2+.14

1.2 Interface-Selective Techniques

From the previous discussion one realizes that the study of interfaces is crucial for both fun- damental investigations and technological applications. However, there are some experimental limitations in this field of research. The main problem resides in the "buried" nature of the interface. The molecules that compose the interfacial boundary are considerably fewer than those composing the surrounding bulk. Therefore, unless an interface-specific technique is employed, the signal generated by the bulk will always overwhelm the response from the interface. To date, this problem has been tackled using techniques that are extremely selective to the interfacial environment and that can exclude the signal coming from the bulk medium. Here we focus the attention on optical techniques, which probably represent the most widely used to this purpose.

Total Internal Reflection (TIR) fluorescence confines the beam close to the interface in order to reduce the contribution from the bulk and is especially suited for phenomena at liquid/solid interfaces.15The fluorophore is placed in the bulk phase with the lower refractive index and it is excited by the evanescent field generated by a laser beam traveling in TIR geometry through a high-refractive-index material. The evanescent wave has a penetration depth in the order of few hundreds of nanometers, determined by the wavelength of the light, and, therefore, the fluorescence comes only from molecules in the region close by the interface. TIR fluorescence is also used in microscopy to selectively excite surface-bound fluorophores in specimens.16The drawback of this technique is the requirement of an emitting species. On the other hand, in attenuated TIR (or Attenuated Total Reflectance, ATR)17spectroscopy one measures infrared spectra of any molecule close to the interface within the evanescent wave penetration depth. This technique does not suffer from the restriction to use a fluorophore. Moreover, it can be easily implemented in a standard infrared spectrometer. Using this technique it is also possible to determine the composition of multilayered samples.

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CHAPTER 1. GENERAL INTRODUCTION

Another surface technique, related to ATR, is InfraRed Reflection-Absorption Spectroscopy (IRRAS), also known as Grazing-Angle InfraRed Spectroscopy (GAIRS).18Mid-infrared light with a defined polarization state impinges onto the surface at a certain incidence angle. The reflected light is detected at an angle equal to the incidence angle and provides vibrational frequencies and intensities of the adsorbed molecules. IRRAS can be used to identify different functional groups and their orientation relative to the surface. It is often applied to thin films on reflective materials such as self-assembled monolayers deposited on metallic surfaces but it has also been applied for lipid/protein films at air/water interfaces.

Signal enhancement due to surface plasmon resonance is used in Surface Enhanced Raman Scattering (SERS) spectroscopy in order to study the surface of metallic nanoparticles. With this technique, the intensity of the Raman bands can be increased up to the level of detecting single molecules.19The plasmon resonance is due to the amplification of the electric field of the incident radiation by the surrounding surface of the metallic nanoparticle. The interaction between the adsorbate and the metal surface can also lead to a signal enhancement because of a change in the molecular polarizability. Furthermore, Raman is an inherently label-free and noninvasive technique and it has been used also in microscopy to visualize membrane-associated processes without staining.20

All these techniques, however, are not able to selectively probe molecules in close proximity of the interfacial region, which has only a thickness of the order of 1-2 nm. Another approach that mitigates this issue consists of studying an intrinsic property of the interface, which cannot therefore be observed in the bulk phase. This is the case for any even order nonlinear optical technique such as Second Harmonic Generation (SHG) and Sum Frequency Generation (SFG), which rely on the second order nonlinear susceptibility ˜¬(2). These processes are forbidden in centrosymmetric media in the electric-dipole approximation but are allowed in asymmetric environments such as an interface between two liquids. In SFG, two incident electric fields at frequency!1and!2overlap at the interface and produce a third field with!1+!2frequency.21,22 One of the two fundamental fields is typically in the infrared region and is broadband, giving access to the vibrational spectrum of interfacial molecules. However, vibrational transitions have usually smaller oscillatory strengths than electronic transitions, therefore vibrational SFG is in general less sensitive than electronic SFG or SHG. In this thesis, we have employed electronic SHG spectroscopy in which the two incident photons possess the same frequency (!1=!2), being therefore a particular case of sum frequency generation. The main advantage of this technique over SFG resides in the use of only one optical beam to generate the nonlinear response, which simplifies the experimental setup. Moreover, the electronic resonance increases the second har- monic signal and allows the detection of molecules at relatively low interfacial concentrations.

SHG and SFG have also been employed for the study of nonplanar surfaces, such as micro- and nanoparticles in suspensions.23,24The scattered SFG or SHG signal can be collected at different angles with respect to the incident light and can give information on the adsorption of molecules

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1.3. AIMS OF THE THESIS

on the particles and on the size and shape of the particles themselves. The SHG experiments discussed in this work have been performed in a reflective geometry at liquid/liquid interfaces and will be more extensively described in the next chapter.

Although performing experiments is always the best way to get insights on a specific phe- nomenon, there are cases where no experimental techniques can be used to interrogate the sample or to extract a particular piece of information. In such a case, one can resort to computa- tional methods. There are all sort of simulation tools which may be used to predict properties associated to molecules. In reference to biomimetic interfaces, the method of choice is without a doubt Molecular Dynamics (MD).25Experimental methods give information averaged over a large number of molecules and within a relatively large time window. Conversely, MD provides atomic spatial resolution and fast observation time, capable of following molecular vibrations.

MD has been widely used to investigate lipid bilayers and their interaction with other molecules.

The simulations should be always compared to experiments in order to validate the model, which then can be used to predict other properties. Molecular dynamics can give access to information that cannot be readily obtained at interfaces following an experimental approach.

1.3 Aims of the Thesis

The work presented in this thesis is aimed at studying molecular probes that could be used to gain insights into specific properties of biomimetic interfaces. The main spectroscopic technique that was applied throughout is Surface SHG (SSHG). This nonlinear optical technique and its capabilities are discussed more in detail in chapter 2. What is important to mention here is that it provides a selective investigation of only a few interfacial layers without interference from the upper and lower bulk phases. Two simple models for membranes were employed: liquid/liquid and liquid/phospholipid monolayer/liquid interfaces, the latter being more closely related to biological environments. Although these models may appear a naive representation of the true cell environment, the data presented later on, especially in chapter 6, give evidence that there are similarities between the behavior of the molecular probes in these models and in more realistic system such as in unilamellar vesicles.

Additional tools that were used in this work are theoretical simulations. They were crucial in supporting the experimental findings and giving the fine details of the microscopic environment under investigation. We employed Quantum Mechanical (QM) and MD approaches, both of which contributed giving different point of views of the system. In fact, QM can be used to calculate molecular properties for a small ensemble of atoms only. For example, transition energies, nonlin- ear response tensors and potential energy surfaces were predicted at a high level of theory but for the isolated molecular probes. On the other hand, MD is a relatively cheap computational method that can be employed for a large molecular model (hundreds of thousand of atoms). Thus, we used MD to simulate on a fully atomistic scale the interfaces investigated here, giving access

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CHAPTER 1. GENERAL INTRODUCTION

to a detailed microscopic picture and to an appropriate comparison with the experimental data.

The probes used here have a substantial hydrophobic character which promotes their ad- sorption at the interface. Two dye families will be discussed in the following chapters. First, Yellow Oxazole (YO) and Thiazole Orange (TO) cyanine dyes will be presented in chapter 4 and 5.

They are well-known for their sensitivity to DNA and are currently used as fluorescent probes in biology. Our goal was to use their affinity towards interfaces and DNA in order to employ them as interfacial sensors for nucleic acids. Second, in chapter 6, we discuss a novel mechanosensitive dye which can discriminate between ordered and disordered membrane phases. The SSHG in- vestigations at biomimetic interfaces described in this chapter show that its mechanism is valid also in the nonlinear regime and opens promising avenues of possible applications in nonlinear optical microscopy.

Although SSHG has been demonstrated several decades ago, the amount of available data especially on excited-state dynamics at liquid interface is still scarce. Clearly more systematic studies are required before a more comprehensive picture of static and photoinduced interfacial phenomena can be drawn. Here, in all the investigations, we attempted to exploit the capabilities of the SSHG technique in order to extract all available information, such as interfacial spectra, molecular orientations, adsorption isotherms and dynamics. The common aim was to include all these insights into a coherent and meaningful framework.

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C

HAPTER

2

C

ONCEPTS

B

EHIND THE

E

XPERIMENTS

M

olecular spectroscopy is the branch of chemistry that studies the interaction between light and matter. The electromagnetic spectrum extends over several orders of magni- tude in frequency with energies varying between feV and MeV (microwaves to gamma rays). Such a large light palette allows for different type of measurements and investigations, all of which can provide a specific piece of the puzzle. This chapter presents some basic notions in linear and nonlinear spectroscopy necessary to follow the discussions in chapters 4-6. The attention will be focused on surface second harmonic generation, which is the main experimental technique used throughout in this thesis. The basic theory, models and experimental setup are examined and linked to the work described in the following chapters.

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CHAPTER 2. CONCEPTS BEHIND THE EXPERIMENTS

2.1 Linear Spectroscopy

2.1.1 Light-Matter Interaction

Light is an electromagnetic wave consisting of an electric~E(~r,t) [V/m] and a magneticB(~~r,t) [T or V·s/m2] field oscillating perpendicularly to each other.26,27The properties of light in classical wave theory are described by the Maxwell equations, which in the "macroscopic" formulation and in vacuum read:

r·~D=Ω (Gauss’s law for electric field), (2.1a) r·~B=0 (Gauss’s law for magnetic field), (2.1b) r £E~=°@~B

@t (Ampere’s law), (2.1c)

r £~H=~J+@~D

@t (Faraday’s law), (2.1d)

where~D,~H,Ωand~Jare the electric displacement [C/m2], the magnetic field intensity [A/m], the volume charge density [C/m3] and the current density [A/m2], respectively (the space and time dependence is dropped for simplicity and the units are expressed in the International System of Units). Thepolarizationandmagnetizationare the electric and magnetic dipole moments induced by an external electromagnetic field in a macroscopic medium and are related to equations 2.1 by the following relations:

~D="0~E+~P, (2.2a)

B~=µ0H~+M,~ (2.2b)

where"0[C/V·m] andµ0[T·m/A] are the vacuum permittivity and permeability. The propagation of light through a medium is completely determined by the response of the polarization and of the magnetization. Focusing the attention on the electric component of the light and assuming a linear functional relation between the electric displacement and the electric field in the case of weak fields, equation 2.2a can be rewritten as

D~="0~E+"0¬˜(1)~E="0(1+¬˜(1))~E="~˜E. (2.3)

˜

¬(1)is the linear optical susceptibility, a complex quantity, and ˜"is the absolute permittivity given by the product of"0and the relative permittivity of the medium (˜"r=1+¬˜(1)). The polarization~P is important in order to determine optical properties such as absorption, emission or scattering. A similar reasoning brings to the magnetic susceptibility and the permeability of the material ( ˜µ).

Light propagating in an uniform isotropic medium nonconducting and free from charges is 8

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2.1. LINEAR SPECTROSCOPY

governed by a simplified version of the Maxwell equations, namely

r·~E=0, (2.4a)

r·~B=0, (2.4b)

r £~E=°@~B

@t, (2.4c)

r £~B=˜µ@~E

@t. (2.4d)

Since the electric field of the light generates the perpendicular magnetic field, and vice versa, equation 2.4c can be rearranged into a differential equation where the dependence on~Bis removed:

r £≥ r £~E¥

=r £

°@~B

@t

!

@

@t

≥r £~B¥

@

@t

˜

"˜µ@~E

@t

!

. (2.5)

Sincer £(r £E)=~ r(r·~E)° r2~Eand considering equation 2.4a one obtains r2~E=˜µ@2E~

@t2. (2.6)

This is the fundamental wave equation for the light traveling through a medium and it has as a possible solution the harmonic plane wave

~E=Asin(~k·~r°!t)=1

2E~0ei(~k·~r°!t)+c.c., (2.7) whereE~0andA~are the amplitudes of the wave (E~0=A/i),~ ~kis the propagation wavevector and!is the angular frequency. The modulus of the wavevector is related to the wavelength,

|~k| =, whereas the angular frequency is related to the frequency and the period of the wave,

!=2º∫=T.

From equation 2.6 one can extract the phase velocity of the wave in a medium,

˜

v¡=(˜"˜µ)°1/2. (2.8)

In vacuum, the phase velocity coincides with the speed of lightc, which is defined as∏/T. When light propagates through a dielectric medium, the phase velocity in the material is reduced with respect to free space and it is related to the electric susceptibility as in

˜

n=v˜¡(vacuum)

˜

v¡(material)=≥ 1+¬˜(1)¥°1/2

. (2.9)

The quantity ˜nis the complex refractive index which is usually decomposed in its real and imaginary parts

˜

n=n+iK, (2.10)

the refractive indexnand the attenuation indexK. The former as the effect to modify the phase velocity of light in a material. The latter quantifies the amplitude variation of the wave when

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CHAPTER 2. CONCEPTS BEHIND THE EXPERIMENTS

interacting with the medium and it is related to the absorption process. The real and imaginary parts of the complex refractive index are related to each other by a Kramers-Kronig relation.28 The energy of an electromagnetic wave is shared equally between its electric and magnetic fields. The flow of electromagnetic energy associated with a traveling wave is represented by the Poynting vector (~S[W/m2])

~S=c2"0E£~ ~B. (2.11)

The irradiance (or intensity,I), i.e. the average energy per unit area per unit time, derived from the Poynting vector can be expressed as

I=c"0

2E20, (2.12)

which relates the radiant flux density with the square of the amplitude of the electric field.

2.1.2 Absorption

In general, a molecule can respond to the incoming light in two ways. It can "scatter" the light without modifying its frequency. Otherwise, if the light energy matches the difference between two molecular quantum levels, it can absorb the electromagnetic radiation and be promoted to a higher energy level. Therefore, a beam passing through a solution containing absorbing molecules will loose part of its intensity. The Beer-Lambert law describes how a monochromatic source of light with irradianceI0incident on a cell of thicknessldecreases exponentially when it interacts with an absorbers of concentrationc

I=I010°≤cl=I010°A. (2.13)

The quantityAis termed the absorbance or optical density of the sample (A=≤cl) whereas≤is the molar extinction coefficient. The concentration is usually expressed in molarity [mol/L or M]

andlin cm. Since the absorbance is a dimensionless quantity,must be expressed as M°1cm°1. A plot ofA, or≤, as a function of the incident radiation frequency, or∏, is called an absorption spectrum. The attenuation indexKin equation 2.10 is connected to the absorbance:

K=ln10A∏

4ºl . (2.14)

2.1.3 Fluorescence

After absorption of light, the energy accumulated by the absorber is dissipated through various channels (Figure 2.1). The population of high vibrational levels (vn) of an excited state (e.g. S1) can undergo rapid vibrational relaxation to the lowest vibrational level (v0). The nonradiative transition from a higher to a lower electronic state, namely internal conversion, can also take place and occurs within picoseconds. All these processes are fast and precede an additional possible dissipation channel: fluorescence.

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2.1. LINEAR SPECTROSCOPY

S0 S1 S2

Internal conversion Vibrational relaxation

Fluorescence

Absorption

Intersystem crossing

Phosphorescence

T1

FIGURE2.1.Jablonski diagram depicting some non-radiative (dashed arrows) and ra- diative (solid arrows) processes of excited closed shell molecules.

A molecule in the lowest vibrational level of its first excited state (Sv10) can relax to a vibrationally-excited ground-state level (Sv0n) by emission of light. The radia- tive process is then followed by thermal equilibration of the population to thev0of the ground state. This gives rise to the peculiar mirror image characteristic of ab- sorption and emission spectra. This is particularly true for molecules that do not experience significant varia- tion of the structure upon excitation, which assures a similar spacing of the vibrational levels. Fluorescence takes place typically from the lowest excited electronic state (Kasha’s rule) because internal conversion and vi- brational cooling are usually faster then radiative pro- cesses.

The energy dissipation through vibrational relax- ation or internal conversion has also an additional effect on the fluorescence spectrum. Since the emission of elec-

tromagnetic radiation occurs always at lower energies than the excitation light, the fluorescence spectrum is shifted to lower energy with respect to the absorption. The difference between posi- tions of the band maxima is denoted as the Stokes shift.

Other dissipation channels can contribute to the relaxation of molecules in their excited state. For example a closed-shell organic molecule in its singlet first excited state can undergo an intersystem crossing toward a triplet state (T1, Figure 2.1) where one electron has reversed its spin (change of spin multiplicity). Emission from a triplet state is also possible (phosphorescence) but occurs on a longer time scale since it originates from a spin-forbidden transition.

2.1.4 Radiative and Non-Radiative Rate Constants

There are two types of fluorescence, spontaneous and stimulated. The former has been mentioned in section 2.1.3. Stimulated emission occurs when a strong light beam interacts with the molecules in their electronic excited state and induces radiation at the same frequency, phase and direction of the incoming light. This process requires the energy of the field to match exactly a transition in the molecule and it is the basis of the laser.

The rate for spontaneous emission for atoms is described by dNex

dt =°AE·Nex, (2.15)

whereNexis the excited-state population andAEis the Einstein coefficient for fluorescence. The fluorescence lifetime is related toAE:

AE=1

ø. (2.16)

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CHAPTER 2. CONCEPTS BEHIND THE EXPERIMENTS

Stimulated emission and absorption share instead the same rate constant, namely the Einstein coefficientBE. The rates for upward (absorption) and downward (stimulated emission) transitions are given by

dNex

dt =BE·Ω(∫)·Nex, (2.17a)

dNex

dt =°BE·Ω(∫)·Nex, (2.17b) withΩ(∫) equals to the external radiation density.

The molecular equivalent ofAEis the radiative rate constantkrad. All other deactivation pathways of the first excited state are also associated with rate constants (e.g.kICfor internal conversion). The overall rate at which the S1state decays to the ground state is a summation over all the single rate constants (kS1=P

iki). The radiative rate constant can be measured for example by following the decrease of spontaneous emission with time, which gives an exponential decay as in

Ifl(t)=Ifl010°kS1t. (2.18) The S1state decays monoexponentially to the ground state through a radiative pathway. In other cases, the decay can be multiexponential due to e.g. the presence of multiple species. Fluorescence lifetime measurements can be performed for example by Time-Correlated Single Photon Counting (TC-SPC) or by broadband fluorescence upconversion, whose details are discussed in Appendix A.

The fate of the excited-state population after the excitation with electromagnetic radiation can be investigated through various other time-resolved spectroscopic techniques, which can measure dynamics in the femtosecond to microsecond time range. The perturbation needed to bring the system into the excited state must be shorter than the rate of the process of interest. Therefore, short optical pulses generated by state-of-the-art lasers are required. For instance, Transient Absorption (TA) can have a time resolution in the order of a few femtoseconds and can allow for the detection of emissive, non-emissive and dark states. However, TA and broadband fluorescence upconversion do not belong anymore to the realm of linear spectroscopy. The principles behind these techniques are to be seek in nonlinear optics.

2.2 Nonlinear Spectroscopy

Electromagnetic fields with high irradiance are able to modify the optical properties of a material.

The consequences of strong electromagnetic fields interacting with matter is the domain of nonlinear optics, which first began with the discovery of second harmonic generation by Franken et alin 1961, just one year later the discovery of the first laser.28

In section 2.1.1 we saw that the induced polarization~Pof a material depends on the strength

~Eof an applied optical field as

~P="0¬˜(1)~E. (2.19) 12

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2.2. NONLINEAR SPECTROSCOPY

In nonlinear optics, the polarization is not anymore linear with the field and can be described by a power series in the field strength:

~P="0( ˜¬(1)E+~ ¬˜(2)~E2+¬˜(3)E~3+...)¥~P(1)+~P(2)+~P(3)+... . (2.20) Here we introduced the second- and third-order nonlinear optical susceptibility, ˜¬(2)and ˜¬(3) respectively, which give rise to several nonlinear processes. Nonlinear susceptibilities ˜¬(n)have units of [m/V]n°1.

Since we are interested in nonlinear optical phenomena of second order, only the second term of the polarization expansionP~in equation 2.20 is necessary here. In its most generalized form, the second-order polarizationP~(2)is expressed as a multipole series expansion:29

(2)=~P(2)°~r·Q$(2)+µ0

i!~r £M~(2)+... , (2.21) where quadrupole and magnetic field terms can contribute to the total polarization. However, for the molecules studied in this thesis one can safely neglect the last two contributions and apply the electric dipole approximation. The electric displacement in equation 2.2a can then be expressed as

D~="0˜"r·~E+P~(2), (2.22) which can be used in the Maxwell equations 2.1 in order to derive the wave equation in the nonlinear regime:

r £≥ r £E~¥

° r2~E+"0µ0"˜r@2~E

@t2+µ0@2P~

@t2=0. (2.23)

The solution of this equation29provides the conditions at which there is an efficient generation of a second order nonlinear response. In particular, the intensity of the generated field scales as

I(2)(2!)/l2

√sin(¢~k l/2)

¢~k l/2

!2

I2(!), (2.24)

wherelrepresents the interaction length, i.e. the path over which the fundamental beam is in phase with the nonlinear signal, and¢~k=~k(2!)°2~k(!)) is the phase mismatch, namely the phase relation between the nonlinear induced polarization and the propagating electric field in the material. In case of perfect phase matching,¢~k=0, the conversion efficiency towards the nonlinear frequency is the highest. The coherence length is defined as

lc=2º

¢k (2.25)

and corresponds to the material length after which the converted power periodically cancels out.

2.2.1 Second Harmonic Generation

The second harmonic generation phenomenon rises from the interaction of a laser field at fre- quency!with a nonlinear medium, resulting in the generation of light at frequency 2!. ˜¬(2)

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CHAPTER 2. CONCEPTS BEHIND THE EXPERIMENTS

ω

ω

non-resonant one-photon resonant

two-photon

resonant fully resonant

FIGURE2.2.Energy level scheme illustrating the possible resonances in SHG. Stationary and virtual states are represented by solid and dotted horizontal lines respectively.

is usually small with respect to the linear susceptibility ( ˜¬(2)/ ˜¬(1)ª10°12m/V) but it increases significantly when one of the fields is in resonance with a transition of the material (Figure 2.2).

The plot of the SHG intensity as a function of the frequency of the incident laser beam repre- sent a SHG spectrum and reflects one- and two-photon resonances with the nonlinear material.

Polarization-resolved data instead can be recorded by measuring the SHG intensity as a function of the fundamental light polarization. The analysis of these data, described more thoroughly in section 2.2.5, can provide an accurate description of the orientational behavior of adsorbates at surfaces. The SHG signal can also be collected varying the concentration of the species giving rise to the nonlinear response. In this case, one can investigate the adsorption efficiency and estimate the Gibbs free energy associated to the surface adsorption. Finally, if an additional pump beam tuned at an excitation frequency of the chromophore is present, Time-Resolved (TR) SHG experiments can be carried out by delaying the pump with respect to the fundamental probe beam.

SHG can be generated only from non-centrosymmetric media. In the electric dipole approxi- mation this can be easily shown by considering that in a centrosymmetric medium the inversion of the propagation wavevector~kdoes not affect the sign or the magnitude of the susceptibility ˜¬(2). However, it does change the sign of the incident and second harmonic fields (~E(!) andP~(2)(2!), respectively). The second order polarization in equation 2.20 can thus be written as:

°~P(2)(2!)="0¬˜(2)h

°~E(!)ih

°~E(!)i

="0¬˜(2)E~2(!). (2.26)

Comparison with the original equation implies that~P(2)(2!)=°~P(2)(2!), which can be satisfied only in case~P(2)(2!)=0. This condition means that ˜¬(2)=0, since the incident field is assumed to be different from zero. At non-centrosymmetric media, such as interfaces, the symmetry is broken and the~P(2)(2!)=°~P(2)(2!) equality does not necessarily hold, allowing surface second harmonic generation.

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2.2. NONLINEAR SPECTROSCOPY

2.2.2 Classical Harmonic and Anharmonic Oscillators

The classical description based on the Lorentz oscillator model can explain and give access to the general properties of linear and nonlinear optical phenomena without employing more sophisticated quantum mechanical approaches. In this model a molecule is composed by electrons of massmand charge°eand nuclei hold together by forces that behave according to Hooke’s law. The potentialU(x) acting between the electron and the nucleus is represented by a series expansion of the distancexbetween the two particles:

U(x)=1 2kx2+1

3ax3+1

4bx4+... , (2.27)

withk,aandbconstants. The restoring force is the derivative of this potential F=°dU(x)

dx =°(kx+ax2+bx3+...). (2.28) A friction forceis also introduced in order to avoid singular points at resonance frequencies.

An incident electromagnetic waveE(t) perturbs the electron-nucleus system by Coulomb forces.

Considering all the previous terms, the equation of motion can be written then as md2x

dt2+2m∞dx

dt+kx+ax2+bx3+...=°eE(t), (2.29) where the first term represents the acceleration endured by the electron.

In the harmonic approximation only the first term of the force in equation 2.28 is considered and it is valid only for small external fields, which cause small displacements of the electron from its equilibrium position. Solving equation 2.29 requires a perturbative approach, with the solution given by the following series in the displacementx:

x(t)=X1 i=1

ix(i)(t). (2.30)

The equation for the first order inusing∏E(t) instead ofE(t) in equation 2.29 reads now d2x(1)

dt2 +2∞dx(1) dt +k

mx(1)=°e

mE(t), (2.31)

which has as solution

x(1)(t)=° e 2m

E(!)e°i!t

£!20°!2°2i∞!§°c.c. (2.32) Here the external electric field was expressed using the complex notation:

E(t)=1 2

hE(!)e°i!t+E§(!)ei!ti

. (2.33)

Moreover, the resonance frequency!0=p

k/mof an un-driven and un-damped harmonic oscillator was used. Assuming a large number of electron-nucleus systems per unit volume,N, one can obtain the total polarization induced in the medium from equation 2.32:

P(1)(t)=°Nex(1)(t)=Ne2 2m

E(!)e°i!t

£!20°!2°2i∞!§+c.c. (2.34)

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CHAPTER 2. CONCEPTS BEHIND THE EXPERIMENTS

frequency ω0 ω0/2

A

B

FIGURE2.3.Dependence of the (A) linear and (B) second order susceptibility as a function of the incident frequency. The real and imaginary parts are depicted in dotted and solid lines respectively.

From a comparison of equation 2.19 and 2.34, the linear susceptibility can thus be written as

˜

¬(1)(!)=Ne2

"0m

£ 1

!20°!2°2i∞!§. (2.35) Figure 2.3A shows the dependence of the real and imaginary part of ˜¬(1)as a function of the external electric field nearby the resonance at!0.

For an intense external field one can include an additional nonlinear term for the electron displace- ment from equation 2.28 and the equation of motion now becomes

d2x dt2+2∞dx

dt+k mx+a

mx2=°e

mE(t). (2.36) This equation can be solved again with the perturba- tion technique. The solution at the second order in

reads x(2)(t)=°ae2

m2

[E(!)]2e°2i!t

£!20°4!2°4i∞!§£!20°!2°2i∞!§2°c.c.

(2.37)

Similarly as for equation 2.34 one can extract the macroscopic second order polarization P(2)(t)=°Nex(2)(t)=Nae3

4m2

[E(!)]2e°2i!t

£!20°4!2°4i∞!§£

!20°!2°2i∞!§2+c.c. (2.38) Thus, comparing with the second order term in equation 2.20, the macroscopic nonlinear suscep- tibility is written as

˜

¬(2)(2!;!,!)=Nae3

"0m2

£ 1

!20°4!2°4i∞!§£

!20°!2°2i∞!§2. (2.39) In the last equation, we observe that there are two resonance frequencies, at!0and at!0/2 (Figure 2.3B). The parameteracharacterizes the strength of the nonlinearity. The classical description of a dumped and driven anharmonic oscillator is able thus to explain the origin of the optical nonlinearity. Moreover, if the external field is composed by two different frequencies, the anharmonic oscillator explains also the rise of other phenomena such as sum or difference frequency generation.

2.2.3 Symmetry Properties of the Susceptibility

Symmetry arguments can be used to find the independent and nonzero elements of the second order susceptibility tensor, which is usually composed by several elements.28In section 2.2.1, the

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2.2. NONLINEAR SPECTROSCOPY

effect of theinversion symmetrywas already shown to cancel ˜¬(2)and forbid second harmonic generation in a centrosymmetric medium. Here, other types of relevant symmetry relations will be mentioned. Let us consider the following expression for the nonlinear polarization in the general case of any second order process:

~P(2)(!n+!m)="0¬˜(2)i jk(!n+!m,!n,!m)~Ej(!n)~Ek(!m). (2.40) Intrinsic permutation symmetry. The interchange ofnandmandjandkindices in equation 2.40 results in the expression ˜¬(2)ik j(!n+!m,!m,!n)~Ek(!m)~Ej(!n) which produces numerically the same nonlinear polarization~P(2)(!n+!m). This means practically that the order of the two incoming beams in the product~Ej(!n)~Ek(!m) does not matter. Therefore the following equality for the elements of the tensor can be written:

˜

¬(2)i jk(!n+!m,!n,!m)=¬˜(2)ik j(!n+!m,!m,!n). (2.41) Overall permutation symmetry. Far from a resonance of the medium, the susceptibility tensor is purely real. This can be easily seen from the anharmonic oscillator model in equation 2.39 when|!°!0| >>∞. In this condition (lossless media), all frequencies of the susceptibility (!n+!m,!nand!m) can be freely interchanged while simultaneously permutating the Cartesian indices (i,jandk). This has the consequences that

¬(2)i jk(!n+!m,!n,!m)=¬(2)jki!n,!m!n°!m). (2.42) Kleinman symmetry. The nonlinear susceptibility is essentially independent of frequency when the fundamental frequencies are much smaller than the lowest resonance in the medium.

This is shown in equation 2.39 for the frequency!approaching zero. Since in this case we are far from resonance the overall permutation symmetry must also be valid. However, considering that the susceptibility does not depend on the frequency we can permute the Cartesian indices without the need of permuting the frequency indices. Thus, the following condition must be fulfilled:

¬(2)i jk(!n+!m,!n,!m)=¬(2)jki(!n+!m,!n,!m)=

¬(2)ki j(!n+!m,!n,!m)=¬(2)jik(!n+!m,!n,!m)=

¬(2)k ji(!n+!m,!n,!m)=¬(2)ik j(!n+!m,!n,!m).

(2.43)

The Kleinman symmetry is very useful because can greatly reduce the number of independent nonzero susceptibility elements.

Spatial symmetry. If we consider for instance a crystal where theXandYdirections are equivalent and different from theZdirection, a rotation of e.g. 90±around theZaxis would leave the crystal structure unaltered. In such a case, the optical response in either theXor theYdirection should be the same and, for example, the¬(2)ZX Xand¬(2)ZY Yelements should be identical. Therefore, it appears that crystal symmetry and group theory may be used in order

17

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