An optical setup for chiroptical spectroscopy with surface waves

POLITECNICO DI MILANO

Scuola di Ingegneria Industriale e dell’Informazione

Corso di Laurea Magistrale in Engineering Physics

Dipartimento di Fisica

AN OPTICAL SETUP FOR

CHIROPTICAL SPECTROSCOPY

WITH SURFACE WAVES

Supervisor: Prof. Paolo BIAGIONI

Co-supervisor: Prof. Maria Elisabetta BRENNA

Master thesis by:

Erika MOGNI - Mat. 898688

Academic year 2018/2019

POLITECNICO DI MILANO

Scuola di Ingegneria Industriale e dell’Informazione

Corso di Laurea Magistrale in Engineering Physics

Dipartimento di Fisica

SIMULATION OF OPTICAL MODULATION

IN AlGaAs NANOSTRUCTURES DUE TO

THERMAL EFFECTS

Supervisor: Prof. Paolo BIAGIONI

Master thesis by:

Francesco RUSCONI - Mat. 858626

Contents

Abstract ... 11

Riassunto ... 12

1. Circular dichroism ... 13

1.1 Circular dichroism measurement ... 13

1.2 Superchirality ... 15

1.3 Photonic crystals and chiral surface waves ... 15

2. The experimental setup ... 22

2.1 Second Harmonic Generation (SHG) ... 22

2.2 Experimental setup ... 24

2.3 Alignment criticalities ... 27

3. Preliminary experimental data ... 31

3.1 Gold surface plasmon polariton ... 31

3.1.1 Surface plasmon polaritons ... 31

3.1.2 Properties of surface plasmon polaritons ... 35

3.1.3 Excitation of surface plasmon polaritons ... 35

3.1.4 Experimental data ... 37

3.2 Circular dichroism baseline of the BK7 glass ... 39

3.2.1 Characterization of the prism ... 39

3.2.2 Circular dichroism baseline ... 41

4. Circular dichroism measurements ... 45

4.1 Setup characterization ... 45

4.1.1 The CD setup ... 45

4.1.2 Circular dichroism baseline of the empty cuvette ... 46

4.1.3 Circular dichroism baseline with CHCl3 ... 47

4.2 Test for molecular CD ... 49

4.2.1 The molecule ... 49

4.2.2 Circular dichroism measurements ... 50

5. Surface waves on a 1DPC ... 54

Conclusions ... 63

Appendix: LabView acquisition program ... 65

Acknowledgments ... 66

List of figures

Fig 1.1 Schematic representation of a 1DPC. ... 16 Fig 1.2 Dispersion relations for: a) Uniform system: all layers with the same electric constant with arbitrarily

assigned periodicity of !. b) Low contrast of dielectric constants: 13 to 12. There is a small gap in frequency between the upper and the lower branches of the lines. No allowed modes in the crystal has a frequency in this gap. c) High contrast of dielectric constants: 13 to 1. As the contrast increases, also the band gap becomes wider. (Taken from Ref. [6]). ... 17

Fig 1.3 Band structure of a multilayer film. Blue indicates TM modes polarized so that the electric field points

in the x direction. Red indicates TE modes polarized on the yz plane. It is evident that in ordinary conditions dispersion relations do not overlap because they show different slopes. (Taken from Ref. [6]). ... 18

Fig 1.4 Schematic representation of a 1DPC terminated with a defect able to excite both TE and TM modes.

[5] ... 19

Fig. 1.5 1DPC band structures for TE and TM illumination, and their superposition. The black solid and

dashed lines represent the light line in water and in the "#$%= 1.53 incident medium, respectively. The colored

lines are the dispersion relations for the Bloch surface modes. (Taken from Ref. [5]). ... 20

Fig 1.6 Superimposed reflection maps for TE and TM illuminations. TE results to be narrow while the TM

appears only as a halo. (Taken from Ref. [5]). ... 21

Fig 2.1 Scheme of second harmonic generation. Two incoming photons at frequency +, produce an output

photon of frequency 2+,. ... 23

Fig 2.2 Optical arrangement for second harmonic generation. ... 23 Fig. 2.3 Schematic of the experimental setup for angular spectra acquisition. (PEM=photoelastic modulator).

... 24

Fig 2.4 Schematic representation of a rhombohedral calcite crystal. The optic axis passes through corner H

and point I on side BF [8]. ... 24

Fig 2.5 Schematic of a Glan-Taylor polarizer. ... 25 Fig 2.6 PEM modulation of light. If the peaks of retardation reach one fourth of the wavelength of light, they

correspond to right and left circularly polarized light. ... 26

Fig. 2.7 Schematic representation of the configuration of the two semi-cylindrical BK7 glasses to limit the

focusing effect. ... 26

Fig. 2.8 Top view (a) and side view (b) of the rotational part of the system. Different colors indicate different

elements of the setup associated to degrees of freedom of the sample stage. ... 27

Fig 2.9 a) If the alignment is good, the incoming light does not suffer refraction and hits the center of rotation

of the semi-cylindrical sample. b) Also, when a rotation occurs, the beam is not deviated and hits the center of rotation of the BK7 glass. ... 28

Fig 2.10 Off-axis laser misalignment example. a) When the BK7 glass is at 0° with respect to the incoming

beam, light is refracted with a certain angle. b) When the semi-cylinder rotates, the incident light hit a different point of the prism and the transmitted beam does not hit the center of rotation anymore. ... 28

Fig 2.11 Off-axis sample misalignment example. a) When the BK7 glass is perpendicular to the incoming laser

rotates, the light is again not deviated and points the center of rotation of the glass, but in this case a different point of the semi-cylinder reflects the light. ... 29

Fig 3.1 Real and imaginary part of the dielectric constant for gold using the Drude-Sommerfeld free-electron

model. The real part is the solid line, the imaginary part is the dashed line. Note that different scales are used for real and imaginary parts. (Taken from Ref. [9]). ... 32

Fig 3.2 Model and experimental dielectric function of gold. Open circle: experimental values taken from [10].

Squares: model of the dielectric function considering the contribution of both free-electron and of single interband transition. a) Imaginary part. b) Real part. (Taken from Ref. [9]). ... 33

Fig 3.3 Interface between two media with dielectric functions ., and ./. The interface is defined at 0 = 0 in a

Cartesian coordinate system. We consider only p-polarized light as we are looking for homogeneous solutions that decay exponentially away from the interface. ... 34

Fig 3.4 Experimental setups to excite surface plasmon polaritons by means of evanescent waves created at the

interface with a medium with " > 1. a) Otto configuration: evanescent wave at glass/air interface; SPP at air/metal interface. b) Kretschmann configuration: evanescent wave at glass/metal interface; SPP at metal/air interface. ... 36

Fig 3.5 Excitation of SPP in the Kretschmann configuration. The reflectivity of the exciting beam is plotted as

function of the angle of incidence and curves for different metal thicknesses (in nm) are shown. Gold is taken into account but also a trace for silver is plotted for comparison. Minima correspond to SPP excitations. (Taken from Ref. [9]). ... 37

Fig 3.6 Excitation of SPP in Kretschmann configuration: experimental results. The reflectivity is plotted as

function of the angle of incidence. Different curves represent measurements taken by slightly changing the position of the center of rotation of the semi-cylindrical prism by the use of a micrometer. Error bars are within the dimensions of the dots. ... 38

Fig 3.7 Representation of angles involved in the Snell’s law. Light passing from a medium with refractive index

", to a medium with refractive index "/ is transmitted following this rule. ... 39

Fig 3.8 Total internal reflection graphs showing the intensity of the reflected beam as a function of the angle

of incidence. These graphs show how the position of the glass on the sample holder, and so the position of the center of rotation, can affect the measurement. Error bars are referred to 2 angular scans. ... 40

Fig 3.9 Reflectivity spectrum of the BK7 glass after a first grossly cleaning of its surface. Error bars are referred

to 2 angular scans. ... 40

Fig 3.10 Total internal reflection spectra for TE (a) and TM (b) polarized light. Error bars are referred to 10

angular scans. ... 41

Fig 3.11 Circular dichroism baseline of the BK7 semi-cylindrical glass. The amplitude is plotted as a function

of the angle of incidence of the illuminating light. Different curves are related to different positions of the sample with respect to the center of rotation. Error bars of the yellow line are referred to 10 angular scans. Error bars of the pink line are referred to 5 scans while error bars of the purple line are referred to 3 angular scans. ... 42

Fig 3.12 Circular dichroism amplitude signal of the BK7 semi-cylindrical glass for TE and TM polarizations of

light. The amplitude signal is plotted as function of the angle of incidence of the illuminating beam. Error bars are referred to 10 angular scans. ... 43

Fig 3.13 Circular dichroism phase spectrum of the BK7 semi-cylindrical glass. The phase is represented as

function of the angle of incidence of the illuminating beam. Both TE and TM polarizations of the incident light are represented. Error bars are referred to 10 angular scans. ... 43

Fig 4.1 Experimental setup for CD transmission measurements on the solution with molecules and CHCl3. 45

Fig 4.2 Circular dichroism amplitude signal of the setup without any sample between the PEM and the

detector. Error bars are referred to 10 measurements, 2 seconds each. ... 46

Fig 4.3 Circular dichroism amplitude signal of the empty cuvette. Error bars are referred to 10 measurements,

2 seconds each. ... 47

Fig 4.4 Transmission spectrum of CHCl3. ... 48

Fig 4.5 Circular dichroism amplitude signal of the solvent CHCl3 contained in the cuvette. Error bars are referred to 10 measurements, 2 seconds each. ... 49

Fig 4.6 a) 2D representation of the o-MR-(R)-PEA molecule. b) 3D representation of the o-MR-(R)-PEA

molecule. ... 50

Fig 4.7 a) Simulation of the absorbance spectrum of the o-MR-PEA molecule. b) Simulation of the CD signal

of the o-MR-PEA molecule. ... 50

Fig 4.8 Absorption spectrum of a solution 1.9 µM of o-MR-(R)-PEA molecule in CHCl3. ... 51

Fig 4.9 a) Red line: absorbance spectrum of the o-MR-PEA molecule without taking into account the

interaction with the solvent. Blue line: absorbance spectrum of the o-MR-PEA molecule taking into account the interaction with the solvent. b) Red line: simulation of circular dichroism signal of the o-MR-PEA molecule without taking into account the interaction with the solvent. Blue line: simulation of circular dichroism signal of the o-MR-PEA molecule taking into account the interaction with the solvent. ... 51

Fig 4.10 CD amplitude spectrum of a solution 1.9 µM of o-MR-(R)-PEA molecule in CHCl3. Error bars are referred to 10 measurements, 2 seconds each. ... 52

Fig 4.11 CD phase spectrum of a solution 1.9 µM of o-MR-(R)-PEA molecule in CHCl3. Error bars are referred to 10 measurements, 2 seconds each. ... 53

Fig 5.1 Schematic representation of the structure of the one-dimensional photonic crystal. Materials are

indicated and also layers with different thicknesses are highlighted. ... 54

Fig 5.2 Excitation of TM (pink) and TE (blue) mode of surface waves of the one-dimensional photonic crystal.

From a) to f) curves are acquired at wavelengths from 380 nm to 500 nm with a step of 20 nm. The TE modes are sharper and narrower, while the TM mode is barely visible at 380, 400 and 420 nm while it is no more recognizable for longer wavelengths. Error bars of the TM modes are referred to 8 angular scans while error bars of the TE modes are referred to 2 angular scans. ... 56

Fig 5.3 a) Comparison of the TE mode of surface waves at different wavelengths. The mode shifts to smaller

angles as the wavelength increases. b) Zoom of spectrum a) in the region between 50° and 77° to underline the shift of the TE mode of the surface waves. Error bars are referred to 2 angular scans. ... 57

Fig 5.4 Comparison between the TE mode of surface waves of the crystal #DM1 and #DM2. For the #DM2

crystal error bars are referred to 2 angular scans while for crystal #DM1 error bars are referred to 4 angular scans. ... 58

Fig 5.5 TE mode of the surface waves of the crystal DM1. By looking around the peak with a fine angular

measurement it is possible to conclude that the half width at half maximum of the peak is roughly 0.3°. Error bars are referred to 8 angular scans. ... 59

Fig 5.6 Comparison between TM mode of surface waves of the #DM2 and the #DM1 crystals. The peak is

broad and poorly visible, but it is possible to recognize the shift between the two peaks. Error bars are referred to 8 angular scans. ... 60

Fig 5.7 Simulation of the reflectivity intensity for TE and TM modes of surface waves of the multilayer with

refractive indexes found in literature. ... 60

Fig 5.8 Simulation of the intensity of TE and TM modes of surface waves of the multilayer with refractive

indexes found in literature. Blue dots are experimental point from our measurement of the TE mode of crystal #DM2. ... 61

List of tables

Tab 3.1 Measurement parameters for the excitation of gold SPP. Start and end angles define the angular

interval of the measurements. The scan is done by acquiring one point every delta angle for a time equal to the acquisition time. After each rotation the system waits for a time equal to the dead time before starting the new acquisition while after the end of each scan the system waits for a time equal to the wait time before starting a new scan. ... 37

Tab 3.2 Measurement parameter for circular dichroism baseline of BK7 glass. ... 42 Tab 4.1 Measurement parameters for the acquisition of the circular dichroism signal of the empty cuvette. 46 Tab 4.2 Measurement parameters for the acquisition of circular dichroism signal of molecules of the cuvette

full of CHCl3. ... 48

Tab 4.3 Measurement parameters for the acquisition of circular dichroism signal of the solution 1.9 µM of

o-MR-(R)-PEA molecule in CHCl3. ... 52

Tab 5.1 Thicknesses set in accordance with information about refractive indexes in literature. ... 55 Tab 5.2 Thicknesses adjusted in order to match refractive indexes used in simulations. Quantities in italics

are those changed for the second multilayer growth. ... 55

Tab. 5.3 Measurement parameters for the excitation of TE or TM modes of surface waves of the

one-dimensional photonic crystal. ... 55

Tab 5.4 Measurement parameters for the analysis of the crystal #DM1. ... 58 Tab 5.5 Measurement parameters for the investigations of the width of the TE mode of the surface wave in

Abstract

Abstract

Chirality is a geometric symmetry property which describes objects that cannot be superimposed onto their mirror image. This characteristic emerges only when an interaction with a chiral environment occurs, otherwise two different enantiomers are identical in their physical and chemical properties. Enhanced techniques for chiral detection are in high demand since many drugs interact differently with the human body based on their chirality, which can be evaluated by circular dichroism (CD) spectroscopy. Since in general CD signals are very weak, enhancing the sensitivity of CD measurements is very appealing for basic applications in chemistry and molecular physics and, in the long run, for life-science as well. The idea of this work is characterizing a new experimental setup that will be used to enhance the CD signal by using a combination of photonic crystals and surface waves, both in the blue and UV region. The goal of the commissioning is to confirm that the setup is capable of performing angle-resolved CD measurements (as required for the photonic crystal) with the required angular accuracy, angular reproducibility, and with a low CD baseline.

The first test that is performed is on a thin film of gold with the aim to excite a surface plasmon polariton. This will be the confirmation that angular measurements can be achieved, and it is possible to determine which degree of angular precision the system owns.

Then, since the one-dimensional photonic crystal used for enhancing the CD signal will be deposited on a BK7 glass substrate, the complete characterization of the CD response of a semi-cylindrical BK7 glass used for the coupling to the photonic crystal is performed. In this way it is possible both to confirm the capability of the setup in recovering CD signals and to quantify the CD baseline contribution of the substrate during circular dichroism measurements.

Another investigation is done on chiral molecules synthetized by a collaboration with the Department of Chemistry, Materials and Chemical Engineering of Politecnico di Milano. The aim of this analysis is to quantify the magnitude of the CD signals of molecules that would be compared with the enhanced signal once the complete setup is mounted and ready to perform CD measurements with the photonic crystal.

Finally, the full characterization of a one-dimensional photonic crystal especially designed for the application is performed. This investigation is run to check the capability of the system to excite TE and TM modes of surface waves of the multilayer, the basic ingredient to obtain chiroptical enhancement.

During all the measurements, the alignment of the setup turns out to out to be a fundamental and critical parameter for all the investigations.

Experimental results on the gold surface plasmon polaritons show the good reproducibility of the optical setup while angular spectra of the one-dimensional photonic crystal show a good accuracy of the system since it can detect a surface mode with a full width at half maximum of about 0.3°. Circular dichroism measurements give an order of magnitude of the CD amplitude baseline of the BK7 glass, between 0.5 and 4 mdeg (still viable to improvements), and of the CD signals of molecules that will be deposited on the photonic crystal. For those we collect a baseline, measured with a cuvette, of about 0.3 mdeg, while the maximum CD signal is about 1 mdeg for a 1.9 µM solution. The results confirm that the system and the samples, in their current configuration, already represent a viable candidate for a first demonstration of enhanced CD with superchiral surface waves.

Riassunto

Riassunto

La chiralità è una proprietà geometrica di simmetria che descrive oggetti che non possono essere sovrapposti alla loro immagine speculare. Questa caratteristica emerge solo quando avviene un'interazione con un ambiente chirale, altrimenti i due enantiomeri sono identici nelle loro proprietà fisiche e chimiche. Le tecniche per migliorare il segnale chirale sono richieste poiché molti farmaci interagiscono con il corpo umano in base alla loro chiralità, che può essere valutata mediante spettroscopia di dicroismo circolare (CD). In generale, i segnali CD sono molto deboli, quindi aumentare la sensibilità delle misure di dicroismo circolare è interessante per applicazioni in chimica e fisica molecolare. L'idea di questo lavoro è caratterizzare una nuova configurazione sperimentale che sarà utilizzata per migliorare il segnale CD usando una combinazione di cristalli fotonici e onde di superficie, entrambe nella regione del blu e dell’UV. L’obiettivo dello studio è di provare che il setup sia in grado di svolgere misure angolari di CD (come richiesto per i cristalli fotonici) con l’accuratezza e la riproducibilità angolare richieste e con un basso contributo di CD.

Il primo test viene eseguito su un sottile film d'oro con l’obiettivo di eccitare un plasmone di superficie. Questa misura testa la capacità di svolgere scansioni angolari e di determinare quale grado di precisione possiede il sistema.

Successivamente, poiché il cristallo fotonico monodimensionale utilizzato per migliorare il segnale CD verrà depositato su un substrato di vetro BK7, viene eseguita la caratterizzazione completa della risposta CD di un vetro BK7 semicilindrico usato per l’accoppiamento con il cristallo fotonico. In questo modo è possibile confermare la capacità del setup nel misurare segnali CD e quantificare il contributo di fondo che tale vetro introduce nelle misure di dicroismo circolare.

Un'altra indagine viene condotta sulle molecole chirali sintetizzate con la collaborazione del Dipartimento di Chimica, Materiali e Ingegneria Chimica del Politecnico di Milano. Lo scopo di questa analisi è quello di quantificare l’ampiezza del segnale di dicroismo circolare in modo da poterlo comparare al segnale aumentato una volta che l’apparato sperimentale sarà completo e pronto per misure CD con il cristallo fotonico. Infine, viene eseguita la completa caratterizzazione di un cristallo fotonico monodimensionale progettato unicamente per questa applicazione. Questa misura viene svolta per testare la capacità del sistema di eccitare i modi TE e TM delle onde superficiali del multistrato, l'ingrediente di base per ottenere l’aumento del segnale chiroottico. Durante tutte le misure, l’allineamento del setup si è dimostrato essere un parametro tanto fondamentale quanto critico per la correttezza delle analisi. I risultati ottenuti con il plasmone di superficie dell’oro dimostrano una buona riproducibilità del setup mentre gli spettri angolari del cristallo fotonico monodimensionale evidenziano una buona accuratezza del sistema che è in grado di risolvere un modo di superficie con una larghezza di riga di circa 0.3°. Le misure di dicroismo circolare forniscono un ordine di grandezza dell’ampiezza del fondo del vetro BK7, tra circa 0.5 e 4 mdeg, e del segnale CD delle molecole che saranno depositate sul cristallo fotonico. Per queste ultime il fondo, misurato in cuvette, ha un valore di circa 0.3 mdeg mentre il massimo segnale di CD raggiunge un valore di circa 1 mdeg per una soluzione 1.9 µM. I risultati confermano che il sistema e i campioni, nella configurazione attuale, rappresentano un valido candidato per una prima dimostrazione di aumento del segnale CD con onde superchirali di superficie.

Chapter 1 Circular dichroism

Chapter 1

Circular dichroism

In this chapter, the concept of circular dichroism (CD) is introduced. Then, superchirality is explained as an important improvement for CD spectroscopy. Finally, one-dimensional photonic crystals are described as novel platforms created to enhance the circular dichroism signal.

1.1 Circular dichroism measurement

Chirality is the property of a three-dimensional object to be geometrically distinct from its mirror image. This characteristic emerges only when an interaction with a chiral environment occurs, otherwise two different enantiomers are identical in their physical and chemical properties. The ability to detect molecular chirality is of great importance since e.g. DNA and proteins, the building blocks of life, are chiral and most drugs interact differently with the human body depending on their chirality. For this reason, assessing the chiral configuration and conformation of molecules is appealing not only for physics research, but also for biology, medicine and pharmacology.

The chiral properties of molecules can be studied with chiroptical spectroscopy techniques [1].

Circular dichroism (CD) spectroscopy is one of the most used techniques to investigate different enantiomers; it measures the different absorption between left (+) and right (−) circularly polarized light (CPL). The fractional difference in absorption is typically measured combining the switching between the two polarizations and a detection scheme using a lock-in amplifier.

A chiral molecule subjected to a monochromatic electromagnetic field generates an electric dipole moment p and a magnetic dipole moment m given by:

4

5 = 6789 − :;<=9, >5 = ?7=9 − :;<89.

Quantities with a tilde are complex, e.g. 67 = 6@_{+ :6}@@_{: 67 is the electric polarizability, ?7 is the magnetic}

susceptibility and ;< is the isotropic mixed electric-magnetic dipole polarizability. 89 and =9 are the local fields at the molecule. Taking the real part of each side of the equations physical quantities can be obtained. The rate of excitation of a molecule can be written as

B±₌D /E6

@@_F89F/_{+ ?}@@_F=9F/_{G ± ;}@@_{+ IJK89}∗_{⋅ =9N,}

where + is the frequency of the electromagnetic field and ± refers to different CPL. The term ?@@_F=9F/ _associated

to the magnetic dipole moment can be neglected as it is negligibly small compared to 6@@_F89F/_{. We use the}

identity + IJK89∗_{⋅ =9N = =̇ ⋅ 8 − 8̇ ⋅ =, where 8 and = indicates real part of the field components, getting}

B±₌ /

Chapter 1 Circular dichroism

where XY is the permittivity in the free space, ST=P_ZQF89F /

is the time-average electric energy density and C is the optical chirality, defined as

V ≡PQ /8 ⋅ ∇ × ^ + , /__Q= ⋅ ∇ × = = − P_QD / IJK89 ∗_{⋅ =9N,}

where `Y is the permeability of the free space. For a monochromatic circular polarized light plane wave, the

dissymmetry factor a, that measures the fractional difference in absorption between left and right CPL, is defined as

a ≡ 2b_bc_cdb_fbe_e.

By substituting the expression for the absorption rate, we get a general expression of the dissymmetry factor: a = − Eg_ihh_hhG E_Dk/j

lG.

The chiral asymmetry in the rate of excitation of a small molecule is therefore proportional to the product of the chiral properties of the matter and the chiral properties of the electromagnetic field [2].

The circular dichroism signal is defined by the differential absorptivity between left and right handedness ∆n: = nf_{− n}d_{= pqa E}re

rcG, where n indicates the absorptivity while I is the light intensity.

This relation is found starting from the Beer’s law describing the attenuation of light intensity in a medium I = IY10dPstuvw,

where X is the molar extinction coefficient, p is the path length, and xyzs is the concentration of the chiral

analyte in moles per liter.

From that expression, one can define the absorptivity as n = Xpx = pqa ErQ

rG.

Another way to measure the circular dichroism is by using the ellipticity {, defined as {(rad) =√red√rc

√re_f√rc.

Using the Beer’s law, it is possible to express this quantity using the absorptivity as {(rad) ≈ ∆nÅ$ (,Y)_Z ,

which is valid for small values of ∆n.

It is possible also convert this parameter into millidegrees, obtaining {(mdeg) ≈ 32982 ⋅ ∆n.

This last unit is commonly used for circular dichroism spectra. For standard CD spectrometers the common limit of detection is around 1 mdeg.

For most small molecules, the absorption cross section for left- and right-circularly polarized light differ by less than one part per thousand. The weakness of CD is a consequence of the small size of most molecules with respect to the wavelength of light: the circularly polarized field undergoes just a perceptible twist over a distance of molecular dimensions. This twist provides only a weak perturbation to the overall rate of excitation [3].

Looking at the dissymmetry factor expression, one can therefore notice that there are two different contributions: the first, Eg_ihh_hhG , that depends only on the properties of the molecule, and the second, E_Dk/j

lG, that depends only on the characteristics of the electromagnetic field. Substantial effort has been devoted to

Chapter 1 Circular dichroism

calculating CD spectra for a variety of molecule at multiple levels of theory and to designing molecules that show large optical dissymmetry at particular wavelengths. These treatments focused on the molecular aspects of CD, relying on circularly polarized waves as source of excitation. With the advent of near field optics, plasmonics, and metamaterials, scientists engineer electromagnetic fields that are far more involved than circularly polarized light and so they act on the second term of the dissymmetry factor [3].

1.2 Superchirality

In the last few years some novel approaches have been proposed to enhance the CD signal by modifying the chiral properties of the probing electromagnetic field through the control of the associated optical chirality C, which determines the degree of asymmetry in the absorption rate of a chiral molecule between left and right circularly polarized light in the dipolar approximation. Circularly polarized plane waves are a common example of a chiral electromagnetic field with their optical chirality for a unitary intensity waves expressed as

Vjáà± = ±PQ_/D.

In this framework, a field distribution where V is beyond Vjáà± value is typically called “superchiral”.

The aim is to design an ideal chiral sensing platform, that should have the following characteristics:

• It can generate optical chiralities of both handedness by reversing the polarization state of the incident field;

• It is able to provide uniform superchiral optical fields, |V| > FVjáà± F, over large areas;

• It is able to work at wavelengths extending from the UV to the IR depending on the platform design, with a particular attention to the high energy range of spectrum, below 400 nm, where most electronic molecular transitions occur.

Plasmonic nanostructures have been widely proposed and employed as available solutions for the enhancement of the optical chiral response of biomolecules. However, chiral plasmonic sensing does not meet all the above criteria simultaneously. In particular, superchiral optical fields are spatially confined to the so called plasmonic “hot spots” and large homogeneous optical chiralities are usually obtained only for chiral plasmonic nanostructures, thus negating the possibility of handedness switching upon reversal of the incident polarization state. Indeed, the highest importance of molecular spectroscopy in the blue and UV energy range, which is not accessible with standard Au plasmonics, is pushing for the adoption of novel plasmonics materials such as aluminum, which can operate in the blue and near-UV range.

Uniform surface-enhanced fields for sensing, which are not found in substrates based on nanoparticles, are instead obtained when surface plasmon polaritons or Bloch surface waves are employed [4].

1.3 Photonic crystals and chiral surface waves

Bloch surface waves are surface waves supported by a semi-infinite one-dimensional photonic crystal (1DPC) and exist both as transverse electric (TE) and transverse magnetic (TM) modes [5].

The simplest possible infinite photonic crystal consists of alternating layers of materials with different dielectric constants: a multilayer film, as illustrated in Fig 1.1.

Chapter 1 Circular dichroism

Fig 1.1 Schematic representation of a 1DPC.

The material is periodic in the z direction, and homogeneous in the xy plane. This allows us to classify the modes using the wave vector in the plane ä∥, the wave vector in the z direction åç and the band number n. The

wave vectors specify how the mode transforms under translation operators, and the band number increases with frequency.

We can write the modes in the Bloch form:

éè,ë_í,ä_∥(ì) = îïä∥⋅ñîïëíçóè,ë_í,ä_∥(0).

The function u(z) is periodic, with the property ó(0) = ó(0 + ò) whenever R is an integral multiple of the spatial periodicity !. Because the crystal has continuous translational symmetry in the xy plane, the wave vector ä∥ can assume any value. However, the wave vector åç can be restricted to a finite interval, the

one-dimensional Brillouin zone, because the crystal has discrete translational symmetry in the z direction. If the primitive lattice vector is !ôö then the primitive reciprocal lattice vector is /õ_ú åç and the Brillouin zone is

−õ_ú< åç≤õ_ú. Consider waves that propagate entirely in the z direction, crossing the dielectric sheets at normal

incidence. In this case, ä∥= 0 and only the wave vector component åç is important. Without possibility of

confusion, we can then abbreviate åç by å. In Fig 1.2 we plot dispersion relations +è(å) for three different

Chapter 1 Circular dichroism

Fig 1.2 Dispersion relations for: a) Uniform system: all layers with the same electric constant with arbitrarily assigned periodicity of !. b) Low contrast of dielectric constants: 13 to 12. There is a small gap in frequency between the upper and the lower branches of the lines. No allowed modes in the crystal has a frequency in this gap. c) High contrast of dielectric constants: 13 to 1. As the contrast increases, also the band gap becomes wider. (Taken from Ref. [6]).

The left-hand plot is for a system in which all of the layers have the same dielectric constant; the medium is actually uniform in all three directions. A periodicity of ! is arbitrarily assigned to this system. In a homogeneous medium, the speed of light is reduced by the index of refraction. The modes lie along the light line, given by

+(å) =të

ß .

The center plot is for a nearly-homogeneous medium with alternating dielectric constants of 13 and 12. It looks very similar to the homogeneous case with one important difference: there is a small gap in frequency between the upper and lower branches of the lines. There is no allowed mode in the crystal that has a frequency within this gap, regardless of å. Such a gap is called photonic band gap.

The right-hand plot is for a structure with a much higher dielectric contrast of 13 to 1: it is clear that the gap widens considerably as the dielectric contrast is increased.

The bands above and below the gap can be distinguished by where the energy of their modes is concentrated: in the high-e regions, where low frequency modes concentrate their energy, or in the low-e regions, where high frequency modes concentrate their energy.

No electromagnetic modes are allowed to have frequencies in the gap. If we send a light wave with a frequency in the photonic band gap onto the face of the crystal from outside no purely real wave vector exists for any mode at that frequency. Instead, the wave vector is complex. The wave amplitude decays exponentially into the crystal. Stating that there are no states in the photonic band gap, means that there are no extended states like the mode given by the Bloch form éè,ëí,ä∥(ì) = îïä∥⋅ñîïëíçóè,ëí,ä∥(0). Instead, modes are evanescent, decaying exponentially:

é(ì) = î†äô_ó(0)îd°ç_.

They are just like the Bloch modes but with a complex wave vector å + :¢. The imaginary component of the wave vector causes the decay on a length scale of ,

°. Although evanescent modes are solutions of the eigenvalue November 15, 2007 Time: 01:44pm chapter04.tex

46 CHAPTER 4 0.00 0.05 0.10 0.15 0.20 0.25 0.30 -0.5 -0.25 0 0.25 0.5 GaAs/Air Multilayer

Wave vector ka/2π

Photonic Band Gap

n= 1 n= 2 0.00 0.05 0.10 0.15 0.20 0.25 0.30 -0.5 -0.25 0 0.25 0.5

Wave vector ka/2π

Photonic Band Gap

GaAs/GaAlAs Multilayer n= 1 n= 2 0.00 0.05 0.10 0.15 0.20 0.25 0.30 -0.5 -0.25 0 0.25 0.5

Wave vector ka/2π

GaAs Bulk Fr equenc y ω a/2 π c

Figure 2: The photonic band structures for on-axis propagation, as computed for three

different multilayer films. In all three cases, each layer has a width 0.5a. Left: every layer has the same dielectric constantε= 13. Center: layers alternate betweenεof 13 and 12. Right: layers alternate betweenεof 13 and 1.

The Physical Origin of Photonic Band Gaps

For now, consider waves that propagate entirely in the z direction, crossing

the sheets of dielectric at normal incidence. In this case, k∥ = 0 and only the

wave vector component kzis important. Without possibility of confusion, we can

abbreviate kzby k.

In figure 2, we plot ωn(k) for three different multilayer films. The left-hand

plot is for a system in which all of the layers have the same dielectric constant; the medium is actually uniform in all three directions. The center plot is for a structure with alternating dielectric constants of 13 and 12, and the right-hand plot is for a structure with a much higher dielectric contrast of 13 to 1.2

The left-hand plot is for a homogeneous dielectric medium for which we have arbitrarily assigned a periodicity of a. But we already know that in a homogeneous medium, the speed of light is reduced by the index of refraction. The modes lie along the light line (as in the subsectionIndex guidingof chapter 3), given by

ω(k)_{= √}ck

ε. (2)

2 _{We use these particular values because the static dielectric constant of gallium arsenide (GaAs) is}

about 13, and for gallium aluminum arsenide (GaAlAs) it is about 12, as reported in Sze (1981). These and similar materials are commonly used in devices. Air has a dielectric constant very nearly equal to 1.

Chapter 1 Circular dichroism

problem, they diverge as z goes to ±∞ (depending on the sign of ¢). Consequently, there is no physical way to excite them within an idealized crystal of infinite extent. However, a defect or an edge in an otherwise perfect crystal can terminate this exponential growth and thereby sustain an evanescent mode. If one or more evanescent modes are compatible with the structure and symmetry of a given crystal defect, we can excite a localized mode within the photonic band gap. One-dimensional photonic crystals can localize states only in one dimension and the state is confined to a given plane [6]. Bloch surface waves are already used for surface sensing and spectroscopic applications [4-7] as they can show a large field enhancement, propagate through long distances and exist for both polarizations (TE and TM). The idea is to use Bloch surface waves also for CD measurements but, in order to do that, surface waves with a high optical chirality are required. This condition can be fulfilled if the dispersion relations of the two different surface wave modes overlap. This does not happen in ordinary conditions, as pointed out in Fig. 1.3, where it is evident that the two dispersion relations show different slopes, and therefore the TE and TM waves cannot be excited simultaneously, a fundamental condition for the creation of chiral surface waves (CSWs).

Fig 1.3 Band structure of a multilayer film. Blue indicates TM modes polarized so that the electric field points in the x direction. Red

indicates TE modes polarized on the yz plane. It is evident that in ordinary conditions dispersion relations do not overlap because they show different slopes. (Taken from Ref. [6]).

To address this requirement, a novel chiral platform based on the combination of a 1DPC with a properly engineered anisotropic surface defect was recently proposed. Electrodynamic calculations show that such a platform, which is well within the modern fabrication capabilities, supports CSWs originating from the coherent superposition of TE and TM surface modes, providing homogeneous and switchable superchiral fields over arbitrarily large areas and wide spectral ranges. Additionally, the use of standard dielectric materials allows moving the operation wavelength toward the high-energy blue and UV end of the spectrum. The design of the chiral sensing platform is based on two fundamental considerations regarding the optical properties of a semi-infinite 1DPC terminated with a surface defect. Specifically:

• 1DPCs support both TE and TM surface modes;

• The slopes of the respective dispersion relations are a function of the effective refractive index of the defect.

Chapter 1 Circular dichroism

19 It follows naturally that, for a properly designed structure terminated with an optically anisotropic defect, the TE and TM dispersion relations can superimpose in the (å∥, +) space within the band structure forbidden

region. This eventually enables the coherent excitation of both TE and TM modes over a wide spectral range, thus obtaining a chiral surface wave upon the introduction of a õ_/ phase shift between them. A practical implementation of the proposed design is illustrated in Fig 1.4.

Fig 1.4 Schematic representation of a 1DPC terminated with a defect able to excite both TE and TM modes. [5]

The semi-infinite 1DPC consists of alternating high ("§) and low ("à) refractive index materials. For the

structure proposed by Giovanni Pellegrini et al. in Ref. [5] they choose Ta2O5 ("•= 2.06 + 0.001:) and SiO2

("ß= 1.454 + 0.0001:), which are a standard choice for 1DPC fabrication, and suitable for the generation of

both TE and TM surface modes. The 1DPC is then terminated with an optical anisotropic defect consisting in an additional Ta2O5/ SiO2 multilayer with a period much smaller than the principal period. This defect acts as an effective medium characterized by the diagonal dielectric tensor:

ε™™ = ´

ε∥ 0 0

0 ε∥ 0

0 0 ε¨

≠,

where ε∥= εÆÆ = εØØ, ε¨= ε∞∞ and, since ε∥≠ ε¨, a uniaxial birefringence is produced. This means that different

polarizations now propagate with different velocities, as they see different refractive indexes.

The TE and TM band structure for the semi-infinite 1DPC, calculated with the MIT Photonic-Bands package adopting the supercell method, are reported in Fig 1.5.

RAPID COMMUNICATIONS

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FIG. 1. A schematic representation of the 1DPC sensing

plat-form, with the incoming TE and TM linearly polarized waves

represented at the bottom, and the corresponding TE and TM surface

modes at the top. Crystal and defect periods are defined as d

1DPC

and

d

def

, respectively, while d

H,L

stands for the thickness of the high and

low refractive index materials in the 1DPC.

of the chiral sensing platform is based on two fundamental

considerations regarding the optical properties of a

semi-infinite 1DPC terminated with a surface defect. Specifically,

(i) 1DPCs support both TE and TM surface modes, and

(ii) the slopes of the respective dispersion relations are a

function of the effective refractive index of the defect [

24

,

27

].

It follows naturally that, for a properly designed structure

terminated with an optically anisotropic defect, the TE and

TM dispersion relations can superimpose in the (k

_∥

,ω) space

within the band structure forbidden region, where k

_∥

is the

wave vector component parallel to the crystal surface. This

eventually enables the coherent excitation of both TE and TM

modes in a wide spectral range, thus obtaining a chiral surface

wave upon the introduction of a π/2 phase shift between them.

A practical implementation of the proposed design is

illustrated in Fig.

1

. The semi-infinite 1DPC consists of

alternating high (n

H

) and low (n

L

) refractive index materials.

For our structure we choose Ta

2

O

5

(n

H

= 2.06 + 0.001i) and

SiO

₂

(n

_L

_{= 1.454 + 0.0001i), which are a standard choice for}

1DPC fabrication, and allow for the generation of both TE and

TM surface modes [

28

–

30

]. Material dispersion is ignored

for simplicity, yet it can be straightforwardly included in the

design process if needed. The 1DPC is then terminated with

an optically anisotropic defect characterized by the diagonal

dielectric tensor:

¯¯ε =

⎛

⎜

⎝

ε

_xx

0

0

0

ε

_xx

0

0

0

ε

_zz

⎞

⎟

⎠.

(2)

In order to obtain the necessary uniaxial birefringence, we

exploit form anisotropy and design the defect as an additional

Ta

₂

O

₅

/SiO

₂

multilayer with a period d

_def

much smaller than

the principal period d

_1DPC

. The diagonal elements of the tensor

are then obtained by standard Maxwell homogenization and

expressed as [

31

]

ε

_xx

_{= f}

_def

ε

_H

_{+ (1 − f}

_def

) ε

_L

,

(3)

ε

_zz

₌

'

_f

def

ε

H

+

(1 − f

def

)

ε

L

(

− 1

,

(4)

FIG. 2. (a) 1DPC band structures for TE and TM illumination,

and their superposition. The black continuous and dashed lines

represent the light line in water and in the n

inc

= 1.53 incident

medium, respectively. (b) Superimposed reflection maps for TE and

TM illuminations. The thin dashed lines indicate a spectral slice taken

at constant incident angle θ

c

. Inset: z profile of the 1DPC dielectric

constants. (c) Spectral slices of the reflectivity map for TE and TM

illuminations at different incident angles θ

c

.

where ε

_H,L

_{= n}

2

H,L

, f

def

is the filling factor of the high

refractive index material, and the upper surface of the defect

is in contact with water. All the geometrical parameters of the

1DPC are detailed in the Supplemental Material (SM) [

32

].

The TE and TM band structure for the semi-infinite 1DPC,

calculated with the MIT Photonic-Bands package adopting

the supercell method [

33

], are reported in Fig.

2(a)

. The 1DPC

supports both TE and TM surface modes, highlighted by the

lines within the forbidden regions of the band structure (white

areas). The shaded regions correspond to allowed optical

modes extended inside the 1DPC. The same modes can either

be extended or exponentially decaying in the upper semispace

depending on their position above or below the light line

241402-2

TE TM Reflected beam 1DPC Defect TE TM nL nH

Chapter 1 Circular dichroism

20

Fig. 1.5 1DPC band structures for TE and TM illumination, and their superposition. The black solid and dashed lines represent the light

line in water and in the "inc= 1.53 incident medium, respectively. The colored lines are the dispersion relations for the Bloch surface

modes. (Taken from Ref. [5]).

The semi-infinite 1DPC supports both TE and TM surface modes, highlighted by the lines within the forbidden regions of the band structure, indicated as white area. The shaded regions correspond to allowed optical modes extended inside the 1DPC. The same modes can either be extended or exponentially decaying in the upper semi-space depending on their position above or below the light line + = µå∥, where µ =_è t

∂∑∏π∫ is the speed of light in water and "ªºΩæø= 1.33. By superimposing the two band structures, as displayed in the right panel of

Fig. 1.5, the optically anisotropic defect allows us to obtain overlapping TE and TM surface mode dispersion relations in the (å∥, +) space. This achievement offers the unique possibility to excite surface modes with

arbitrarily polarization states in a wide spectral range, in direct contrast to both standard surface plasmon polaritons (which exist only as TM modes) and isolated Bloch surface waves. The same phenomenon is modeled for an actual 1DPC with a finite number of periods, illuminated in the Kretschmann configuration for proper momentum matching. The finite 1DPC, consisting of two crystal periods plus the anisotropic surface defect, is illuminated from an incident medium with refractive index "#$%= 1.53. The reflection maps for both

TE and TM illuminations in Fig 1.6 display the surface modes as a narrow dark band (TE mode), superimposed to a lighter and broader band (TM mode), in agreement with the overlapping observed for the semi-infinite 1DPC, demonstrating how the adopted design approach is sound even for a realistic finite size structure. The total internal reflection onset is also clearly visible around the { ≃ 60° incident angle as a sharp variation in the reflected intensity.

RAPID COMMUNICATIONS

GIOVANNI PELLEGRINI et al.

PHYSICAL REVIEW B 95, 241402(R) (2017)

FIG. 1. A schematic representation of the 1DPC sensing

plat-form, with the incoming TE and TM linearly polarized waves

represented at the bottom, and the corresponding TE and TM surface

modes at the top. Crystal and defect periods are defined as d

1DPC

and

d

def

, respectively, while d

H,L

stands for the thickness of the high and

low refractive index materials in the 1DPC.

of the chiral sensing platform is based on two fundamental

considerations regarding the optical properties of a

semi-infinite 1DPC terminated with a surface defect. Specifically,

(i) 1DPCs support both TE and TM surface modes, and

(ii) the slopes of the respective dispersion relations are a

function of the effective refractive index of the defect [

24

,

27

].

It follows naturally that, for a properly designed structure

terminated with an optically anisotropic defect, the TE and

TM dispersion relations can superimpose in the (k

_∥

,ω) space

within the band structure forbidden region, where k

_∥

is the

wave vector component parallel to the crystal surface. This

eventually enables the coherent excitation of both TE and TM

modes in a wide spectral range, thus obtaining a chiral surface

wave upon the introduction of a π/2 phase shift between them.

A practical implementation of the proposed design is

illustrated in Fig.

1

. The semi-infinite 1DPC consists of

alternating high (n

H

) and low (n

L

) refractive index materials.

For our structure we choose Ta

2

O

5

(n

H

= 2.06 + 0.001i) and

SiO

2

(n

L

= 1.454 + 0.0001i), which are a standard choice for

1DPC fabrication, and allow for the generation of both TE and

TM surface modes [

28

–

30

]. Material dispersion is ignored

for simplicity, yet it can be straightforwardly included in the

design process if needed. The 1DPC is then terminated with

an optically anisotropic defect characterized by the diagonal

dielectric tensor:

¯¯ε =

⎛

⎜

⎝

ε

xx

0

0

0

ε

xx

0

0

0

ε

zz

⎞

⎟

⎠.

(2)

In order to obtain the necessary uniaxial birefringence, we

exploit form anisotropy and design the defect as an additional

Ta

2

O

5

/SiO

2

multilayer with a period d

def

much smaller than

the principal period d

1DPC

. The diagonal elements of the tensor

are then obtained by standard Maxwell homogenization and

expressed as [

31

]

ε

xx

= f

def

ε

H

+ (1 − f

def

) ε

L

,

(3)

ε

zz

=

'

f

def

ε

H

+

(1 − f

def

)

ε

L

(

− 1

,

(4)

FIG. 2. (a) 1DPC band structures for TE and TM illumination,

and their superposition. The black continuous and dashed lines

represent the light line in water and in the n

inc

= 1.53 incident

medium, respectively. (b) Superimposed reflection maps for TE and

TM illuminations. The thin dashed lines indicate a spectral slice taken

at constant incident angle θ

c

. Inset: z profile of the 1DPC dielectric

constants. (c) Spectral slices of the reflectivity map for TE and TM

illuminations at different incident angles θ

c

.

where ε

H,L

= n

2_H,L

, f

def

is the filling factor of the high

refractive index material, and the upper surface of the defect

is in contact with water. All the geometrical parameters of the

1DPC are detailed in the Supplemental Material (SM) [

32

].

The TE and TM band structure for the semi-infinite 1DPC,

calculated with the MIT Photonic-Bands package adopting

the supercell method [

33

], are reported in Fig.

2(a)

. The 1DPC

supports both TE and TM surface modes, highlighted by the

lines within the forbidden regions of the band structure (white

areas). The shaded regions correspond to allowed optical

modes extended inside the 1DPC. The same modes can either

be extended or exponentially decaying in the upper semispace

depending on their position above or below the light line

Chapter 1 Circular dichroism

21

Fig 1.6 Superimposed reflection maps for TE and TM illuminations. TE results to be narrow while the TM appears only as a halo. (Taken

from Ref. [5]).

The coherent superposition of the TE and TM surface modes with the appropriate relative phase ¬t results in

a sharp optical chirality enhancement well above one order of magnitude in the whole analyzed spectral range and an enhancement of more than 2 orders of magnitude in the CD signal [5]. Superchirality is only achieved if both TE and TM surface modes are simultaneously launched, while the ordinary excitation of TE and TM evanescent waves is not sufficient to generate sizable superchirality. The obtained chiral field meets all the criteria for an ideal sensing platform. First, the handedness of the optical chirality can be readily switched by alternating between left and right incident elliptical polarization states. Second, the in-plane translational invariance of the system implies that the chirality enhancement is obtained over arbitrarily large areas, where the only realistic constraints are imposed by limitations in the illumination and fabrication processes. Third, the platform can operate in a wide spectral range in the blue and UV part of the spectrum, opening up the possibility to perform surface enhanced CD spectroscopy. Finally, it is worth noting that the obtained surface wave is intrinsically chiral, i.e., the optical chirality does not originate from the interference between the incident and local fields, which is often the case for most plasmonic platforms.

RAPID COMMUNICATIONS

GIOVANNI PELLEGRINI et al.

PHYSICAL REVIEW B 95, 241402(R) (2017)

FIG. 1. A schematic representation of the 1DPC sensing

plat-form, with the incoming TE and TM linearly polarized waves

represented at the bottom, and the corresponding TE and TM surface

modes at the top. Crystal and defect periods are defined as d

1DPC

and

d

def

, respectively, while d

H,L

stands for the thickness of the high and

low refractive index materials in the 1DPC.

of the chiral sensing platform is based on two fundamental

considerations regarding the optical properties of a

semi-infinite 1DPC terminated with a surface defect. Specifically,

(i) 1DPCs support both TE and TM surface modes, and

(ii) the slopes of the respective dispersion relations are a

function of the effective refractive index of the defect [

24

,

27

].

It follows naturally that, for a properly designed structure

terminated with an optically anisotropic defect, the TE and

TM dispersion relations can superimpose in the (k

_∥

,ω) space

within the band structure forbidden region, where k

_∥

is the

wave vector component parallel to the crystal surface. This

eventually enables the coherent excitation of both TE and TM

modes in a wide spectral range, thus obtaining a chiral surface

wave upon the introduction of a π/2 phase shift between them.

A practical implementation of the proposed design is

illustrated in Fig.

1

. The semi-infinite 1DPC consists of

alternating high (n

H

) and low (n

L

) refractive index materials.

For our structure we choose Ta

2

O

5

(n

H

= 2.06 + 0.001i) and

SiO

2

(n

L

= 1.454 + 0.0001i), which are a standard choice for

1DPC fabrication, and allow for the generation of both TE and

TM surface modes [

28

–

30

]. Material dispersion is ignored

for simplicity, yet it can be straightforwardly included in the

design process if needed. The 1DPC is then terminated with

an optically anisotropic defect characterized by the diagonal

dielectric tensor:

¯¯ε =

⎛

⎜

⎝

ε

xx

0

0

0

ε

xx

0

0

0

ε

zz

⎞

⎟

⎠.

(2)

In order to obtain the necessary uniaxial birefringence, we

exploit form anisotropy and design the defect as an additional

Ta

2

O

5

/SiO

2

multilayer with a period d

def

much smaller than

the principal period d

1DPC

. The diagonal elements of the tensor

are then obtained by standard Maxwell homogenization and

expressed as [

31

]

ε

xx

= f

def

ε

H

+ (1 − f

def

) ε

L

,

(3)

ε

zz

=

'

f

def

ε

H

+

(1 − f

def

)

ε

L

(

− 1

,

(4)

FIG. 2. (a) 1DPC band structures for TE and TM illumination,

and their superposition. The black continuous and dashed lines

represent the light line in water and in the n

inc

= 1.53 incident

medium, respectively. (b) Superimposed reflection maps for TE and

TM illuminations. The thin dashed lines indicate a spectral slice taken

at constant incident angle θ

c

. Inset: z profile of the 1DPC dielectric

constants. (c) Spectral slices of the reflectivity map for TE and TM

illuminations at different incident angles θ

c

.

where ε

H,L

= n

2_H,L

, f

def

is the filling factor of the high

refractive index material, and the upper surface of the defect

is in contact with water. All the geometrical parameters of the

1DPC are detailed in the Supplemental Material (SM) [

32

].

The TE and TM band structure for the semi-infinite 1DPC,

calculated with the MIT Photonic-Bands package adopting

the supercell method [

33

], are reported in Fig.

2(a)

. The 1DPC

supports both TE and TM surface modes, highlighted by the

lines within the forbidden regions of the band structure (white

areas). The shaded regions correspond to allowed optical

modes extended inside the 1DPC. The same modes can either

be extended or exponentially decaying in the upper semispace

depending on their position above or below the light line

Chapter 2 The experimental setup

Chapter 2

The experimental setup

In this chapter the experimental setup for the characterization of the photonic crystals is described. The first two sections are dedicated to the optical arrangement used to generate the second harmonic from the beam of a Ti:Sapphire laser and for the acquisitions of angular spectra. The last section is focused on the description of the setup alignment, that comes out to be a critical parameter during measurements.

2.1 Second Harmonic Generation (SHG)

Since we want to investigate a broad range of the wavelength spectrum, a widely tunable coherent white light source is required but, since it is not available in our laboratory, a high power automated femtosecond Ti:Sapphire laser is used. This laser can work between 690 nm and 1040 nm, wavelengths longer than those we want to investigate, roughly between 400 nm and 500 nm, so we set an optical arrangement in order to have second harmonic generation (SHG).

When electromagnetic radiation interacts with matter, the response can be linear or nonlinear, depending on the relevance of non-linear terms of the interaction. Assuming that atoms in matter can be modeled as harmonic oscillators it is possible to write a linear relation between the displacement and the driving force. In this case, the field emitted will have the same frequency as the original radiation. The polarization P(t) can be written as:

√(ƒ) = XY?(,)8(ƒ),

where XY is the permittivity in vacuum and ?(,) is the linear susceptibility, that is in general a tensor relating

components of the electric field to components of the polarization vector, of the considered medium. This relation is valid as long as the intensity of the illuminating radiation is low; when its intensity increases, the oscillators experience larger displacements and nonlinear terms become more important. In this case we have to consider also higher order terms of the susceptibility and so the relation between P(t) and E(t) is not linear anymore. Now all terms that influence the oscillations must be taken into account:

√(ƒ) = ∑ Xè Y?(è)8è(ƒ),

where ?(è) _{represents the susceptibility of the n}th _{order of the process, which is associated to the n}th _{power of} the irradiated electric field. The susceptibility ? contains the characteristics of the material and brings them into the equation.

At a physical level, in the process of harmonic generation the nth _{term of the nonlinear relation can be}

associated to n incoming photons with frequency + that produce one photon of frequency "+ as output, keeping the conservation of energy and satisfying the selection rules for parity and angular momentum. Focusing on the second harmonic generation, if two incoming photons at frequency + excite a non-centrosymmetric crystal, a photon at frequency 2+ would be the output of the process, as shown in Fig 2.1.

Chapter 2 The experimental setup

Fig 2.1 Scheme of second harmonic generation. Two incoming photons at frequency +, produce an output photon of frequency 2+,.

In order to generate the second harmonic of the Ti:Sapphire laser we use a barium borate (BBO) crystal, arranged in the setup as depicted in Fig 2.2.

Fig 2.2 Optical arrangement for second harmonic generation.

The BBO crystal is set inside a telescope made by two lenses with focal length of 30 mm and 50 mm, respectively, then the light composed by the fundamental and the second harmonic wavelength hits a dichroic mirror that transmits the fundamental component and reflects the doubled-frequency one. Then light passes through bandpass filters in order to have a clean beam at the second-harmonic wavelength. The last step is the coupling with a monomodal optical fiber that allows bringing the light to the second part of the experimental setup where angular spectra are acquired.

Laser source BBO crystal Dichroic mirror Bandpass filter Fiber coupling Beam stopper

Chapter 2 The experimental setup

2.2 Experimental setup

Angle-resolved CD spectra are taken using the following experimental setup:

Fig. 2.3 Schematic of the experimental setup for angular spectra acquisition. (PEM=photoelastic modulator).

The laser beam comes from the fiber coupled with the second harmonic generation part of the setup, previously described.

The beam is collimated by a lens and passes through a Glan-Taylor polarizer that allows to set the linear polarization of the incoming beam. This element belongs to the family of Glan-type prisms, i.e. the optical axis lies in the plane of the entrance face [8]. The Glan-Taylor polarizer is constituted by two right angle prisms of calcite divided by an air gap. Crystals of calcite can be easily cleaved in order to produce rhombs of the form shown in Fig. 2.4.

Fig 2.4 Schematic representation of a rhombohedral calcite crystal. The optic axis passes through corner H and point I on side BF [8].

A B C D E F G H I Optic axis Glan-Taylor

polarizer Rotational stages

Detector Lock-In Amplifier Amplitude Phase PC Intensity ! θ1 θ2 Fiber output PEM

Chapter 2 The experimental setup

The optical axis HI is the direction in the crystal along which two sets of refracted waves travel at the same velocity. Any plane, such as DBHF, which contains the optical axis and is perpendicular to the two opposite faces of the rhomb ABCD and EFGH is called principal section. The rhomb is then cut into two right angle elements that will make the optical element.

The Glan-Taylor polarizer has the principal section perpendicular to the plane of the cut, and it is constituted by two right angle prisms divided by an air gap. This separation occurs along the major side of the two elements. In this kind of polarizer, only p-polarized light can be transmitted, while s-polarized light undergoes total internal reflection. The transmission of the p-polarized light is highly efficient, as the angle of incidence at the gap can be reasonably close to the Brewster’s angle. The transmitted beam is completely polarized, while the reflected one is not, as depicted in Fig 2.5.

Fig 2.5 Schematic of a Glan-Taylor polarizer.

The Glan-Taylor polarizer is mounted on a goniometer support that allows for the rotation of the two prisms. By rotating the two prisms one can switch the polarization of the transmitted beam, from transverse magnetic (TM) to transverse electric (TE). This is possible because the beam at the fiber output is in general elliptically polarized.

Once the light polarization is set, the beam goes through a photo-elastic modulator (PEM) by Hinds Instruments, which model is PEM-90. This instrument is used for modulating at a fixed frequency the polarization of a beam of light. Its operation is based on the photoelasticity phenomenon. The basic design of the photo-elastic modulator consists of a piezoelectric transducer and a crystal of transparent material. The transducer is tuned to the resonance frequency of the optical element, that is at about 50 kHz. A current is then sent through the transducer to vibrate the crystal through stretching and compressing which changes the birefringence of the transparent material. In fact, as a solid material becomes birefringent after mechanical compression, also an optical element under stress can exploit this phenomenon. This means that field components parallel and perpendicular to the modulator axis travel at slightly different velocities. The perpendicular component then either “leads” or “lags” the parallel component after passing through the PEM. The phase difference created between the two components is called retardation and oscillates as a function of time. An important condition occurs when the peak of the retardations reaches exactly one-fourth of the wavelength E∆_ZG of light. When this condition occurs, at the peak, the polarization vector traces a right handed

Chapter 2 The experimental setup

spiral about the optical axis. Such light is called “right circularly polarized”. The polarization oscillates then between right and left circular polarized light at the modulation frequency, with other polarizations in between as shown in Fig 2.6.

Fig 2.6 PEM modulation of light. If the peaks of retardation reach one fourth of the wavelength of light, they correspond to right and left

circularly polarized light.

The photo elastic modulator is aligned with an inclination of 45° of the optical axis with respect to the Glan-Taylor polarizer so that the modulation between the two circular polarizations occurs at the modulator frequency.

After this element the beam reaches, as depicted in Fig. 2.3, a semi-cylindrical BK7 glass that we use as a coupling element to excite Bloch surface waves. Since a semi-cylindrical glass acts as a lens, the light is focused inside the crystal and then broadens, affecting the angular precision of the measurement. For this reason, we add another BK7 semi-cylinder to the system, positioned between the PEM and the sample holder as shown in Fig. 2.7.

Fig. 2.7 Schematic representation of the configuration of the two semi-cylindrical BK7 glasses to limit the focusing effect.

In this way we limit the focusing effect and the incidence angle for the excitation of the Bloch surface wave is well defined inside the BK7 substrate.

The sample is mounted on an OptoSigma rotational stage (model OSMS-60YAW) that allows to strike the glass with different angles.

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4

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−λ₄

Linear Right circular Linear Left circular Linear

First glass

Second glass