Raman spectra acquisition of plasmonic-trapped particles

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Raman spectra acquisition

of plasmonic-trapped

particles

by

Valeria Rodr´ıguez Fajardo

A dissertation submitted in partial fulfillment of the

requirements for the degree of

MASTER OF SCIENCE IN OPTICS

at the

Instituto Nacional de Astrof´ısica, ´

Optica y

Electr´onica

September 2010

Tonantzintla, Puebla, M´exico

Advisors:

Dr. Rub´en Ramos Garc´ıa, INAOE

Dr. Romain Quidant, ICFO

c

INAOE 2010

The author hereby grants to INAOE permission to

reproduce and to distribute copies of this thesis

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Abstract

Through the use of plasmons it is possible to confine particles in small areas for long time. Raman spectroscopy can be used as a powerful identification tool, due to the uniqueness of the spectrum for each material. In this way, if we were able to implement both techniques in a single setup, it would be possible to perform non-invasive analysis of single particles. With this in mind, this thesis seeks to build such single experimental setup. In order to carry out our goal, we divide the work in three parts. First we construct setups for the two methods independently, and after this, we combine them. In all experiments we used as 3.55 µm-diameter polystyrene beads. In the setup for Raman spectroscopy we use a 532 nm diode laser to excite the sample. We first took the spectrum of a layer of polystyrene beads and compare it with an accepted standard. We conclude the setup was working properly, as the spectra were very similar. We were able to acquire the spectrum of a single polystyrene bead and we carried out a scan of it. This was done with the aim of determining the dependance of the signal’s intensity on the position of the bead. We observed the polystyrene Raman signal for 5 µm along two axis on the sample’s plane, and for 20 µm along the optical axis. In the plasmonic-trapping setup, we use a Kretschmann-like geometry, in which the metal surface was patterned with a gold disks grid, illuminated under Total Internal Reflexion (TIR) by a 785 nm Ti:Sapphire laser. We studied the trapping process for both p- and s-polarized incident light. We could see the trapping with p-polarization was very effective, whereas with s-polarization it was not. We also analyze the behavior of a trapped particle during 8 min. To do so, we determined its trajectory, and calculated the histograms of the bead’s position in the parallel (x) and perpendicular (y)

directions to the in-plane incident wave-vector, as well as, the potential energy in which the bead was immersed in. We found the particle has a preferable position, shifted respect the center of the gold disk and that the potentials were asymmetric for x and symmetric for y.

The trapping was of plasmonic nature. Finally, we took the Raman spectra of three trapped beads for different excitation powers. By increasing the integration time of the spectrometer,

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we could observe in each case the peaks 1001.4 cm−1_{, 2904.5 cm}−1 _{and 3054.3 cm}−1 _{of the} polystyrene, with very similar SNR’s. The combination of both techniques implemented, opens new perspectives on the study of single particles through Raman spectroscopy, although it requires more refinement.

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Acknowledgments

I would like to show my gratitude to the persons who help me to pursue this work. First, to my supervisors, Rub´en Ramos-Garc´ıa and Romain Quidant, who supported and guided me. I also thank to Maurizio Riguini, who assisted me at the initial stage of the lab work. I am grateful to the institutions INAOE and CONACyT, which provided me with both a very good place to study and economic aid. I would like to show my gratitude also to ICFO, where I found an encouraging place to carried out my lab work. I am heartily thankful to my parents, Gloria and Jorge, and Carmelo, who had support me in this long trip. Lastly, I offer my regards to all of those who supported me in any respect during the completion of the project.

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1 Introduction

Raman Spectroscopy is based in the Raman effect, which is a non-linear process. This effect was observed for the first time in 1927 by the indian physicist Chandrasekhara Raman [1, 2]. This process is characterized by a change in the energy of the scattered photons by a sample respect that of the incident ones, due to the interaction with the vibrational modes of the sample. As consequence, the wavelength spectrum is conformed by sharp peaks, whose wavelength difference with respect to the incoming beam is related to the energies of those vibrational modes. Since this modes are unique for each material under specific physical conditions, the Raman spectra is like a fingerprint. Furthermore, Raman spectroscopy is very versatile, allowing the use of samples almost without previous preparation, that can be solid, liquid or gaseous and providing the opportunity to carry outin vivo measurements. Nevertheless, in general the

signal is very weak and one should be very careful to avoid photo-damage.

The uniqueness of the Raman spectra and the versatility of the method, make this a tech-nique with an outstanding number of applications in many different fields. We are especially interested in single particle Raman spectroscopy, since it enables the identification of chemical compositions in individual particles as well as the monitoring of dynamical processes, in such a small scales. In order to achieve this, it is necessary to confine the particle in the space, that can be reached through the use of electromagnetic fields. The first work in this respect was done in 1984 by Thurn and Kiefer[3]. They obtained Raman spectra from micrometer-sized particles trapped in stable optical potential produced by radiation pressure. With the popular-ization of optical tweezers the use of this technique has been continuously increased. In 1998 Ajito reported on the obtaining of Raman spectra of single toluene microdroplets [4]. Later, in 1999, Ajito and Morita published an article on the simultaneous imaging and Raman spectrum

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acquisition of p-cresol contained in a single toluene microdroplet [5]. In 2001 Ajito and Torim-itsu proposed, what they called, aRaman tweezers microscope, in which they synthesized their

previous works [6].

Single particle Raman spectroscopy has a wide range of applications in medicine and food industry. Mainly in illness diagnosis and the identification of pathogen organisms. A lot of endeavor has been devoted to this topic. Xie and coworkers have written several articles on the acquisition of Raman spectra of optically trapped cells[7, 8] and identification of single living microorganism [9] and chromosomes[10]. In addition, one of the most impressive capabilities of this method consist in the possibility of monitoring processes in real time. For instance, in 2003 Xie and Li reported on the follow-through the changes in the biochemical composition due to heat denaturation of single microorganism [11] and the year after Xieet al. presented

the spectra taken from a single living cell in real-time [12]. Another example is the monitoring of the changes in the molecular conformation due to mechanical stress. This was carried out by Raoet al. in 2009, who took Raman spectra of human blood cells under such mechanical

deformation [13]. A very good review article on the subject was written in 2007 by Petrov[14]. Plasmons have been known since 1957, when Ritchie[15] observed its first experimental evidence. Plasmons are collective oscillations of the free electrons in metals. Our work deals with Surface Plamons Polaritons (SPP), in which the oscillations are confined to the surface

of a metal in a metal-isolator interface. Already in 1991 Sambles[16] wrote a review article, in which he described the SPP excitation methods. The enhancement of the electric field in the vicinity of such metal surface has found a variety of application for the SPP’s, for example in integrated optics, sensing or manipulation of particles. A review of some of its applications was written by Barnes et al. [17].

In particular, we are interested in plasmonic trapping. The evanescent field generated by the plasmons gives intrinsically in-plane confinement only. To achieve the confinement in the plane it is necessary to implement an additional method. This can be accomplished by using two courter-propagating beams, creating a standing wave. In recent days, Wang demonstrated the trapping of a single fluorescent bead by two counter-propagating SPP’s in a gold stripe [18]. Plasmonic-trapping presents some advantages over conventional optical tweezers. For example, this technique enables the parallel manipulation of multiple particles, since a strong focalization is not needed. In 2005 Garc´es-Chav´ez et al. [19] report the simultaneous manipulation of

thousand of particles by using two counter-propagating beams, in such a way they not only trapped the particles, but controlled its speed, direction and position. Another remarkable

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advantage is that this technique is not restricted by the diffraction limit, and hence it allows to trap nanometric particles.

Another way to accomplish the trapping consist in patterning the metal surface. This was proposed in 2004 by Quidant et al. [20]. They reported the simulation and measurement of

the near-field electrical intensity above a squared gold pads grid, and found intensity gradients that might be used to simultaneously trap multiple particles. Extensive efforts have been done to improve this technique since it has promising applications, for example, the implementation of lab-on-a-chip devices. The measurement of the forces on a particle immersed in a plasmonic evanescent field was carried out in 2006 by Volpe et al. [21] and latter in 2007 Righini et al.

demonstrated experimentally the simultaneous trapping of particles on a grid of gold disks, as well as, its utilization to sort particles of different sizes. The year after Righiniet al. analyzed

quantitatively the behavior of a trapped particle [22] and published a review on manipulation of particles via SPP, in which they presented simulations of the force maps above a single gold disks, as well as experimental results on trapping and sorting of particles [23]. The same year Grigorenko et al. succeeded in the trapping of a single sub-micrometer particle [24].

As we have seen, on one side, single particle Raman spectroscopy method has many diverse and amazing applications and, on the other side, plasmonic-trapping does not suffer of some of the limitations of conventional optical tweezers. Both techniques present advantages such as, being non invasive, as well as, its suitability to study samples in suspension. Therefore can be used to perform in vivo analysis for long periods. In this work we pursue to implement an

experimental setup in which both techniques are join together. However, we focus on the initial development of the system and, consequently, we used only inert test samples.

In the search of our goal, we first worked independently on each technique and then im-plemented them together. We present this thesis in the same manner. We start with Raman spectroscopy of single particles. Chapter 2 deals with the basic theory behind the Raman spec-troscopy, including, of course, the description of the Raman effect. In chapter 3, we follow with the detailed description of the constructed setup, as well as the obtainment and processing of data. Finally, we present the obtained results in chapter 4. There, the results are focus in getting a better insight of the setup, rather than the specific shape of the spectra. Similarly, for the plasmonic-trapping, we begin with the theoretical principles in chapter 5. The explanation of the setup, containing all the relevant specificities is given in chapter 6 and in chapter 7 we present the results. In chapter 8 we describe the implemented setup and discuss our results when combining both methods. Finally, we discuss the conclusions in chapter 9.

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References

[1] C. V. Raman and K. S. Krishnan. A new type of secondary radiation. Nature,

121(3048):501–502, March 1928.

[2] C. V. Raman. A change of wavelength in light scattering. Nature, 121(3051):619–620,

April 1928.

[3] R. Thurn and W. Kiefer. Raman -microsampling technique applying levitation by radiation pressure. Applied spectroscopy, 38(1):78–83, 1984.

[4] K. Ajito. Combined near-infrared raman microprobe and laser trapping system: appli-cation to the anaysis of a single organic microdroplet in water. Applied spectroscopy,

52(3):339–342, 1998.

[5] K. Ajito and M. Morita. Imaging and spectroscopic analysis of single microdroplets con-taining p-cresol using the near-infrared laser tweezers/raman microprobe system. Surface science, 427-428:141–146, 1999.

[6] K. Ajito and K. Torimitsu. Near-infrared raman spectroscopy of single particles. Trends in analytical chemistry, 20(5):255–262, 2001.

[7] C. Xie, M. A. Dinno, and Y. Li. Near-infrared raman spectroscopy of single optically trapped biological cells. Optics Letters, 27(4):249–251, 2002.

[8] C. Xie and Y. Li. Confocal micro-raman spectroscopy of single biological cells using op-tical trapping and shifted excitation difference techniques. Journal of applied physics,

93(5):2982–2986, 2003.

[9] C. Xie, D. Chen, and Y. Li. Raman sorting and identification of single living micro-organism with optical tweezers. Optics Letters, 30(14):1800–1802, 2005.

[10] J. F. Ojeda, C. Xie, Y. Li, F. E. Bertrand, J. Wiley, and T. J. McConnell. Chromoso-mal analysis and identification based on optical tweezers and raman spectroscopy. Optics Express, 14(12):5385–5293, 2006.

[11] C. Xie and Y. Li. Study of dynamical process of heat denaturation in optically trapped sigle microorganism by near-infrared spectroscopy. Journal of applied optics, 94(9):6138–6142,

2003.

[12] C. Xie, C. Goodman, M. A. Dinno, and Y. Li. Real-time raman spectroscopy of optically trapped living cells and organelles. Optics Express, 12(25):6208–6214, 2004.

[13] S. Rao, S. B´alint, B. Cossins, V. Guallar, and D. Petrov. Raman study of mechanically induced oxygenation state transtition of red blood cells using otical tweezers. Biophysical journal, 96(1):209–216, 2009.

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[14] D. Petrov. Raman spectroscopy of opticaly trapped particles. Journal of Optics A,

9(8):S139–S156, 2007.

[15] R. H. Ritchie. Plasma losses by fast electrons in thin films. Physical Review, 106(5):874–

881, 1957.

[16] J. R. Sambles, G. W. Bradbery, and F. Yang. Optical excitation of surface plasmons: an introduction. Contemporary Physics, 32(3):173–183, 1991.

[17] W. L. Barnes, A. Dereux, and T. W. Ebbesen. Surface plasmon subwavelength optics.

Nature, 424:824–830, 2003.

[18] K. Wang, E. Schonbrun, P. Steinvurzel, and K. B. Crozier. Scannable plasmonic trapping using a gold stripe. Nano Letters, 10(9):3506–3511, 2010.

[19] V. Garc´es-Ch´avez, G. C. Spalding, and K. Dholakia. Near-field optical manipulation by using evanescent waves and surface plasmon polaritons. Proceedings of SPIE, 5930:51301B,

2005.

[20] R. Quidant, G. Badenes, S. Cheylan, R. Alcubilla, J. Weeber, and C. Girard. Sub-wavelength pattering of the optical near-field. Optics Express, 12(2):282–287, 2004.

[21] G. Volpe, R. Quidant, G. Badenes, and D. Petrov. Surface plasmon radiation forces.

Physical Review Letters, 96(23):238101, 2006.

[22] M. Righini, G. Volpe, C. Girard, D. Petrov, and R. Quidant. Surface plasmon optical tweezers: Tunable optical manipulation in the femtonewton range.Physical Review Letters,

100(18):186804, 2008.

[23] M. Righini, C. Girard, and R. Quidant. Light-induced manipulation with surface plasmons.

Journal of Optics A, 10(9), 2008.

[24] A. N. Grigorenko, N. W. Roberts, M. R. Dickinson, and Y. Zhang. Nanometric optical tweezers based on nanostructured substrates. Nature Photonics, 2(6):365–370, 2008.

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2 Basic Principles of Raman

Spectroscopy

In this chapter we will introduce the basic concepts behind the Raman Effect. Starting with the notion of scattering and vibrational normal modes, followed by the phenomenological description of the Raman scattering. Then we will present the formalism for the power radiated by an electric dipole and continue with the classical description of the Raman effect. At the end of this section, we will comment on the limitations of the classical approach and, in the section after, we will mention briefly the quantum or semi-classical description. We will also describe the intensities of the Raman peaks, which depends on the irradiance of the incident light, sample properties and experimental setup. We will finish by referring to the selection rules for the Raman transitions.

A deeper introduction to this topic is presented by Le Ru[3] in his book “Principles of SERS” and a complete description of the phenomenon for molecules can be found in the book “The Raman effect” by Long[4]. A general overview, as well as some conventions on nomenclature, symbols and units are found in an article by Schrader and Moore[5]. This article is part of a series of articles published by the IUPAC about spectrochemical analysis.

2.1 General description

We will start with the concept of scattering. Figure 2.1 is a schematic representation of a general scattering event. In it is depicted that, as result of the interaction of the incident light and the medium, radiation is emitted in all directions. Moreover, this radiation could also have different energy or state of polarization than the incident light.

When the emitted radiation has the same energy as the incident light, the process is called

linear orelastic. On the contrary, if the emitted energy is different it is named asnon-linear or inelastic scattering. There are several domains in linear scattering depending on the relation

between the wavelength of the incident light and the size of the scatter object. We will deal with the Rayleigh domain, which describes phenomenon when the scattering particle is very small

compared to the incident light’s wavelength (aλ). In the same way, there are various kinds

of non-linear scattering, depending on the nature of the process which causes the scattered light energy shift respect to the excitation light. Obviously we are interested in the Raman effect, which is related to the vibrational modes of the molecules of the scattering medium.

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Figure 2.1: Schematic for the general scattering process.

Raman scattering can be classified as Stokes or anti-Stokes. One refers to the scattering as Stokes when the energy of the scattered light is smaller that the energy of the incident light and anti-Stokes in the opposite case.

The condition for the medium to scatter light in different directions respect to the incident light is that it must present inhomogeneities in at least one of its optical properties[6]. If the medium is homogeneous, for each scattering differential volume always exist another one, whose associated scattered light interferes destructively with that of the first volume. Therefore if we are dealing with an completely homogeneous medium, the scattering takes place only in the forward direction.

2.1.1 Molecule vibrations

The vibrational behavior of a molecule can be described using its vibrational modes. The vibra-tions are deformavibra-tions of the molecule due to the incessant relative movement of its constituent atoms respect to their equilibrium positions. The special case in which all the atoms oscil-late with the same frequency is calledvibrational normal mode (vibrational mode for short). A

molecule with N (2≤N) atoms in the tridimensional space has 3N (3N−1 for linear molecules)

vibrational modes which form a complete base. This set includes rotational or translational movements of the molecule as a whole. This means that any vibration of the molecule can be expressed as a linear combination of these normal modes. Hence one can define the scalar

normal coordinates Qk as the coefficients of the expansion and expresses the vibration in these

terms. This scalar coordinate system simplifies the mathematical treatment of the phenomenon. When the molecule vibrates, its atoms’ nucleus move along with its electrons, causing the wave-functions of the electrons to be deformed. The calculation of this deformed wave functions

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is extremely complicated. However, it can be overcome by taking into account that, since the nuclei are much heavier than the electrons, in general, the electron velocities are much greater than the nuclei velocities. Therefore it can be assumed that the electrons are in its ground state at each nuclei configuration. Mathematically, this means that the total wave-function can be written as the product of two independent functions: one which describes the electronic structure and another which describes the vibrational behavior. This is known as the adiabatic or Born-Oppenheimer approximation[7]. In this way, since the energies for the vibrational modes of the molecule are small compared to the electronics energies, the first can be seen as small perturbations of the later and then as a substructure of it.

2.1.2 Basic description of Raman effect

Figure 2.2 shows the Jablonsky diagram of the scattering processes. The electronic energy levels are represented by bold lines and the vibrational/rotational energy levels by gray lines. In all three cases a photon with energy Ei =~ω induces a transition to a virtual level (dashed

line in the figure) afterwards the molecule decays rapidly. In the first two cases the molecule is in its ground state at the beginning of the process. When the molecule decays to its initial state (Figure 2.2a), the energy of the scattered photon is the same of the incident photon. The process is elastic and it is called Rayleigh scattering. On the other hand, if the molecule decays to a higher energy level (Figure 2.2b) the scattered photon has less energyES =_~ωS than the

incident photon, or, equivalently its wavelength is larger. In the last case (Figure 2.2c) the system is initially in an excited energy level and decays to its ground state. Here, the scattered photon has higher energy EAS = ~ωAS (its wavelength is shorter) than the incident photon.

The last two processes are inelastic and are known as Stokes and anti-Stokes Raman scattering, respectively.

(a) Rayleigh (b) Raman stokes (c) Raman anti-stokes Figure 2.2: Jablonski diagrams of the Rayleigh and Raman Scattering.

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It is important to notice that, first, the induced transition is not necessarily to any real electronic level but to a virtual one. When the electron does reaches an electronic level, the

effect is called resonance Raman scattering (RRS). Second, the diagrams on figure 2.2 only

describe phenomenologically the Raman effect and do not give information of the probability for each process. Nevertheless, we know the averaged occupancy number for a state k,hnki, is

described by the Fermi-Dirac Statistics[8],

hnki= 1 eKB TEk + 1

, (2.1)

where Ek is the energy of the vibrational energy level, and KBT is the thermal energy. This

expression tell us that the probability of an electron to be in an excited state decreases as the energy of this level increases. It also implies that such probability increments as the temperature raises. Therefore, we can conclude without a further analysis that the probability for anti-Stokes Raman Scattering is smaller than for Rayleigh or Stokes scattering.

2.1.3 Raman spectrum

For a theoretical treatment of Raman scattering it is convenient to work with the angular frequency ω. However, the experimental Raman spectra are presented usually in units of

Raman shift ∆ERS, which is defined by

∆ERS =Ei−ES,AS, (2.2)

with Ei and ES,AS the energies of the incident and scattered (Stokes or anti-Stokes) photons

respectively. Then, having in mind the relationship between these energies described before, the Raman shift is positive for Stokes scattering and negative for anti-Stokes. This quantity re-presents the energy difference between the initial and final states of the electron in the molecule levels and therefore the energy of the vibrational mode involved in the process. The most commonly used units to express the Raman shift are relative wave-numbers, ∆˜k, in cm−1_.

A line in the spectrum represents a specific vibrational mode and its raman shift corresponds to the energy or wave-number (depending on the used units). This arises from the fact that the difference in energy between the scattered and the incident light matches the energy of the vibrational mode involved in the transition.

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are completely defined. However this is physically impossible as is established by the Heisenberg uncertainly principle. This principle relates the lifetime and the energy of the system’s level and states that the product of the uncertainties of these two quantities must remain finite. Therefore if the uncertainty in the energy is zero the lifetime should be infinite, which is not physically possible. Consequently the Raman “line” is broadened in a real spectrum.

2.2 Classical Theory

In this section we will treat the basic theory of Raman scattering from a classical point of view. We will start with the electric dipole emission and follow with an introduction to the concept of polarizability, which, as we will see, is deeply related to Raman scattering. At the end of this section we will point out the limitations of this approach. It is important to clarify that we will assume the incident light does not modify the optical properties of the material.

2.2.1 Electric dipole emission

When dealing with sources whose size is considerably smaller than the wavelength it emits, the electromagnetic radiation in the far field can be written as a sum of multipolar terms. In general it is enough to take only the first term of the series: the electric dipolar term. Raman scattering is one of such cases. In some special cases it is also necessary to take into account the quadrupole electric and the magnetic dipole fields. The electric dipole can be written as the product of two functions purely spatial and temporal, this is,

P(t) = pe−iωt (2.3)

where p is the electric dipole moment and it has been assumed a harmonic time dependence. In particular, we are interested in the intensity distribution of the light emitted by the electric dipole. We will consider the simplest case, in which the dipole is in free space, because it is sufficient for our purposes. The electric and magnetic fields corresponding to the electric dipole term are given by[9]

H(r) = ck₄ 2

π(n×p) eikr

r

1− 1

ikr

E(r) = 1 4π0

k2(n×p)×ne ikr

r + [3n(n·p)−p]

1

r3 −

ik r2

eikr

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where r is the position vector of the observation point, r = |r|, n is an unitary vector in the

direction of r. Equations (2.4) are exact for any r. With this fields, the intensity I (The

time-averaged power radiated per unit solid angle) can be calculated using

I = dPrad

dΩ =

1 2Re

r2_n_·_E_×_H∗

. (2.5)

As result we have

I = dPrad

dΩ =

c2 32π2

r

µ0

0

k4|(n×p)×n|2, (2.6)

where c, µ0 and 0 are the light’s velocity, the permeability and the permittivity in vacuum respectively. The state of polarization of the radiated energy depends on the vector (n×p)×n. In case all the components are in phase, the expression takes the particular form

I = dPrad

dΩ =

c2 32π2

r

µ0

0

k4|p|2sin2θ, (2.7)

withθ the angle between the dipole axis and the vectorr. This equation tell us that the power

radiated by a dipole depends on the angle respect to the dipole axis. Also, having in mind that the frequency and the magnitude of the wave-vector are related as

|k|=k = ω

c, (2.8)

such expression implies that the power radiated is proportional to the fourth power of the frequency of oscillation ω.

2.2.2 Polarizability and induced electric dipole

When a electromagnetic wave is incident on a molecule, induces a change in the molecule’s electron cloud. Since the size of the molecule is considerably smaller than the wavelength of the radiation (for the visible spectrum), the electric field can be assumed uniform along the molecule at any time. Hence the electromagnetic radiation induces an electric dipole Pind and

the radiation emitted by the molecule can be described using the dipole formalism.

When the intensity of the light is not too large too modify the medium properties, the induced electric dipole moment is linear with the electric field, this is,

p=↔

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which is time-dependent. ↔

α is a second rank tensor called polarizability.

The polarizability accounts for how strong the electric field polarizes the molecule. It should be noticed that, since the polarizability is a tensor, the induced dipole is not necessarily collinear with the incident field.

When the molecule is not free to rotate and its center of mass is fixed in the space, this is, the atoms are free to vibrate about its equilibrium positions but without net translation of the molecule, each component of the polarizability tensor, αρ,σ, can be expanded in a Taylor series

in terms of the normal coordinates Qk as

αρ,σ = (αρ,σ)0+

X k ∂αρ,σ ∂Qk 0

Qk+1

2

X

k,`

∂2_α

ρ,σ ∂Qk∂Q`

0

QkQ`+..., (2.10)

where the subscript 0 indicate the term should be evaluated at the equilibrium position, theQi

are the normal coordinates and the summations are over all normal coordinates. In the electric harmonic approximation we take only the first two terms and neglect the higher. In this way, each polarizability component is rewritten as

αρ,σ = (αρ,σ)0+

X k ∂αρ,σ ∂Qk 0

Qk = (αρ,σ)0+

X

k

(α0_ρ,σ)kQk, (2.11)

where we have defined the derived polarizability tensor ↔α0 with components

(α0_ρ,σ)k =

∂αρ,σ ∂Qk 0 . (2.12)

As the last two equations are valid for all components, the polarizability tensor is given by

↔

α=↔α0+

X

k

↔

αk, (2.13)

where

↔

αk=

↔

α0_kQk. (2.14)

In the harmonic approximation the normal coordinate Qk takes the form

Qk =Qk0cos(ωkt+δk) (2.15)

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for the vibrational mode k (Equation 2.14) is given by

↔

αk=

↔

α0_kQk0cos(ωkt+δk). (2.16)

In addition, assuming the incident electric field dependence on time is also harmonic, this is,

E(t) =E0cos(ω0t), (2.17)

with E0 a constant vector, the induced dipole moment (Equation (2.9)) corresponding to the

vibrational mode k is

p(k)₍_t_{) =} ↔

α0_kQk0 ·E0cos(ωkt+δk) cos(ω0t). (2.18)

By using the trigonometric identity

cosαcosβ = 1

2[cos(α+β) + cos(α−β)], (2.19)

we finally obtain

p(k)₍_t_{) =} 1 2Qk0

↔

α0_k·E0cos [(ω0+ωk)t+δk] +

1 2Qk0

↔

α0_k·E0cos [(ω0−ωk)t+δk]. (2.20)

This equation represents the behavior of the electric dipole induced by an electromagnetic field for a molecular vibrational modek. Each term in the equation describes itself an electric dipole,

and therefore the field induces two dipoles oscillating with frequencies ω0±ωk. The phase term δk is different for each vibrational mode and evidences the incoherent nature of the scattering.

The two terms in equation (2.20) are the Stokes and anti-Stokes Raman scattering, respec-tively. We rewrite this equation to highlight this fact as

p(k)₍

t) =p(_Stokesk) (ω0+ωk) +p

(k)

anti−Stokes(ω0 −ωk) (2.21)

with

p(k)

Stokes(ω0+ωk) =

1 2Qk0

↔

α0_k·E0cos [(ω0+ωk)t+δk],

p(k)

anti−Stokes(ω0−ωk) = 1 2Qk0

↔

α0_k·E0cos [(ω0−ωk)t+δk].

(2.22)

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this is,

p=pRayleigh(ω0) +

X

k

p(k)

Stokes(ω0+ωk) +

X

k

p(k)

anti−Stokes(ω0−ωk) (2.23)

where the Rayleigh induced dipole pRayleigh(ω0) is

pRayleigh(ω0) =

↔

α0·E0cos(ω0t), (2.24)

and the Stokes and anti-Stokes induced dipoles are given by equation (2.22). Equation (2.23) is the total induced dipole of a molecule immersed in an oscillating electric field. It implies that the molecule emits not only at the same frequency of the incident light, but also at a series of frequencies related to the vibrational normal modes of the molecule. This vibrations-related frequencies of emission constitute the Raman spectrum. Since the vibrational modes of a molecule are unique, so is its Raman spectrum and thus it can be used as a molecular fingerprint. Given that the coefficients in equations (2.22) are equal, the equation (2.23) erroneously predicts that the probabilities for the two processes are the same.

Equation (2.23) succeed in the prediction of the molecules’ frequencies of emission but fails in the prediction of the probabilities of each process. This implies a better model is needed in order to describe the phenomenon in a more reliable manner. The limitations of the classical formalism are further mentioned in section 2.2.3 and a very brief introduction to a more complete model, the quantum approach, is given is section 2.3.

2.2.3 Limitations of the classical approach

The classical description of Raman scattering is good enough when the main objective is the identification based on the uniqueness of the spectrum. However if a deeper understanding is required, the theory fails in several aspects: ,

• the predicted intensities are not correct,

• it does not describe the resonance Raman scattering,

• it does not account for the molecule’s energy levels due to rotations.

In addition, it has also two important limitations related to the procedure used to determine the induced electric dipole. It was assumed the previous knowledge of the vibrational normal modes of the molecule and the polarizability tensor. Such assumptions are not critical for simple molecules but for more complicated molecules, neither the calculus of its vibrational modes is trivial, nor the qualitative attempt to assign a polarizability tensor which describes its

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behavior. Furthermore, in the classical approach the polarizability is not directly related to the structure of the molecule and hence the interpretation of the spectra is limited as it describe qualitative but not quantitatively the phenomenon.

2.3 Basis of the semi-classical or quantum approach

Herein the molecule is treated under the quantum-mechanical formalism, but the light is still treated from a classical point of view. The electromagnetic field is considered as a perturbation of the molecule’s states and therefore the time-dependent perturbation theory is used. The dipolar approximation is still assumed.

We are interested in the transitions between the energy levels of the molecule due to the presence of the radiation. Similarly to the induced electric dipole moment used in the classical approach, we now use the electric dipole moment operator ˆµ. The transition between a initial

state |Ψiiand a final state |Ψfi is described by the transition amplitudeMf,i(t)

Mf,i(t) =hΨf(t)|µˆ|Ψi(t)i. (2.25)

which is related to the probability of transitionPi→f via the square modulus, this is,

Pi→f =|Mf,i|2, (2.26)

which has physical meaning.

The total perturbed function is expressed as a series of nth-order perturbed wave-functions as

|Ψ(t)i=|Ψ(0)(t)i+|Ψ(1)(t)i+...+|Ψ(n)(t)i+..., (2.27)

but we take only the first two terms, this is, the unperturbed and the first order perturbation,

|Ψ(t)i=|Ψ(0)(t)i+|Ψ(1)(t)i. (2.28)

Substituting this in equation (2.25) and splitting the terms we obtain

Mf,i(t) =hΨ

(0)

f (t)|µˆ|Ψ

(0)

i (t)i+hΨ

(0)

f (t)|µˆ|Ψ

(1)

i (t)i

+hΨ(1)

f (t)|µˆ|Ψ

(0)

i (t)i+hΨ

(1)

f (t)|µˆ|Ψ

(1)

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But the time-dependent perturbed wave-function to the nth order Ψ(n) _{can be expressed in} terms of the unperturbed wave-functions Ψ(0) _as

|Ψ(n)₍_t_)i₌X

k

C_k(n)|Ψ(0)_k (t)i, (2.30)

where the coefficientsC_k(n) depend on the electric field to the nth power. Therefore the last term

in (2.29) is quadratic in the electric field. By considering that we are in the linear approximation we neglect this term and the transition amplitude is written as

Mf,i(t) = hΨ

(0)

f |µˆ|Ψ

(0)

i i+hΨ

(0)

f |µˆ|Ψ

(1)

i i+hΨ

(1)

f |µˆ|Ψ

(0)

i i. (2.31)

The first term of the right hand side of this equation depend on only unperturbed wave-functions and is not related to the incident field, hence is not of our interest. We have then that the transition amplitude to the first order perturbation and linear with the electric field is given by

M(1)

f,i(t) =hΨ

(0)

f (t)|µˆ|Ψ

(1)

i (t)i+hΨ

(1)

f (t)|µˆ|Ψ

(0)

i (t)i. (2.32)

It can be shown that this expression can be rewritten as[10]

M(1)

f,i(t) =

X

k

_µ

f k(µki·E0)

ωk−ωi −ω +

µ_ki(µ_{f k}·E0)

ωk−ωf +ω

ei(ωf−ωi−ω)t

2~

+X

k

_µ

f k(µki·E0)

ωk−ωi+ω

+µki(µf k·E0)

ωk−ωf −ω

ei(ωf−ωi+ω)t

2~ (2.33)

with µ_nm = hψ(0)n |µˆ|ψ(0)m i. Such functions |ψi are the spatial component of the unperturbed

wave-functions, this is,

|Ψ(0)_i₌_|_ψ(0)_i_e−iωt

. (2.34)

This expression is again the Born-Oppenheimer approximation, as we have the wave functions are separable in spatial an temporal functions.

The first term in the right hand side of equation (2.33) describes both Rayleigh and Raman scattering. On one hand, when f = i the final and initial states are the same and therefore

corresponds to Rayleigh scattering. On the other hand, when f 6=i the initial and final states

are different and thus represent Raman scattering. The second term represents a two photon transition, which is not related to Raman scattering and hence we will omit it. In this way, the

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transition amplitude for Rayleigh and Raman scattering to first order in the time-dependent perturbation formalism is given by

M(1)

f,i(t) =

X

k

_µ

f k(µki·E0)

ωk−ωi−ω

+ µki(µf k·E0)

ωk−ωf +ω

ei(ωf−ωi−ω)t

2~ , (2.35)

which defines the Rayleigh and Raman scattering occurrence probabilities through equation (2.26). Equation (2.35) reflects that the scattering is a two step process: first a photon is absorbed to an intermediate statek, after which the molecule emits another photon. The state i corresponds with the electronic level mentioned in section 2.1.2, whereas the statek with the

virtual level and the state f with the final level (whether electronic or vibrational).

From equation (2.35) we can define the polarizability in such a way it satisfies

Mf,i=hΨf|

↔

α|Ψii ·E, (2.36)

obtaining the components of the polarizability tensor as

(αρ,σ)f i =

1

~

X

k

_µ

ρ,f kµσ,ki ωk−ωi−ω

+ µρ,kiµσ,f k ωk−ωf +ω

. (2.37)

This equation rules the behavior of the transitions induced under the presence of a electromag-netic field. This formalism predicts the resonance Raman scattering, as when (ωk−ωi) → ω

the first term diverges. In reality the denominator should have a damping term, which ensures the term to remain finite. This term is also responsible to the finite bandwidth of the Raman lines.

2.4 Intensities of the Raman peaks

In a real experiment the spectrum consist of a series of peaks with different intensities. The positions of the peaks are determined by the energies of the vibrational modes of the molecule. The intensities are governed not only by the incident power and sample physical properties but also by the specific experimental setup. To reach this conclusion it is enough to consider the very simple model of a dipole in free space and therefore it is not necessary to introduce more complicated geometries that only will entangle the calculus. In this section we will present the calculus of the peak intensities under such simple model. The detailed procedure can be found

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in the book of Long[4], section 5.5.2.

In general the experiments do not measure the intensity emitted by the scattering medium in all directions but in a specific one. Having in mind this, we will calculate the intensities for two different common illumination-observation geometries. We begin with the geometry of the figure 2.3 in which the measurement is done in a direction perpendicular to the incident light beam. Here the axes represent the laboratory’s reference system, with its origin in the molecule and the z-axis defined in such a way that coincides with the propagation axis of the incident beam. The unitary vectors ˆk0 and ˆks represent the directions of the incident and scattered

radiation and together they define a plane called the scatter plane. In this particular geometry

the scatter plane coincides with theyz-plane. Such plane is useful to name the vectors in terms

of parallel or perpendicular to this plane.

In most experiments, including ours, the excitation source used is linearly polarized, thus our analysis will be done only for incident light with this specific state of polarization. It is important to notice that, in general, this assumption is not necessary. We know the induced electric dipole momentpand the incident electric fieldE0are related through the polarizability

tensor ↔

α as

p=↔

αt·E0 = 

  

αxx αxy αxz

αyx αyy αyz

αzx αzy αzz

       

E0x

E0y

E0z



  

, (2.38)

where the subscriptt stress out the polarizability tensor is different for each possible transition.

We start by calculating the scattered light’s intensity due to incident radiation polarized in the

y-axis, this is, the incident electric field is on the scatter plane and E0 = E

k

0 = E0yˆj. From

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equation (2.38), the induced dipole moment for one molecule becomes

pS =αxyE0yˆi+αyyE0yˆj+αzyE0yˆk (2.39)

withˆi,ˆjand ˆk the unitary vectors of the reference system.

As we are interested in the intensity measurement from they-axis, the induced dipole along

this axis has no effect on the intensity observed. This can be seen from the angle dependence of the radiated power by a dipole (Equation (2.7)). We first focus on the perpendicular electric dipole p⊥

s =pxˆi. Substituting in equation (2.7) the induced dipole in the x-axis we obtain

I(90°,E

k

0,p

⊥

S)

t =

c2 32π2

r

µ0

0

k4_s|αxyE0yˆi|2 =

1 32π2

0c3

ω4_sα2_xyE₀2_y, (2.40)

where it has been taken into account that the observation line and the dipole axis are perpen-dicular (θ = 90°) and we have used equation (2.8) and

c2 = 1 µ00

. (2.41)

The intensity It in equation (2.40) has three terms as superscript: the first one, θ, represents

the angle at which the measurement is done respect the wave vector of the incident light ; the second one accounts for the polarization of the incident radiation and; the third one is the induced dipole whose contribution is calculated.

The intensity measurements are generally made for not one but a collection of N aleatory

oriented molecules. Hence the intensity is given by

I(90°,E

k

0,p

⊥

S)

t =

Nt

32π2 0c3

ω_s4hα2_xyiE₀2_y (2.42)

whereNtis the number of molecules which contribute to the specific transitiontand the

angle-brackets refer to the average over all directions for the molecule’s orientation. Implicitly it has been used the fact that the scattering is incoherent, as the molecules emit with aleatory phases. Therefore the total intensity is just the product of the intensity for one molecule times the number of molecules.

Analogously, the intensity associated with the parallel dipole pk

s is

I(90°,E

k

0,p

k

S)

t =

Nt

32π2 0c3

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The averages hα2_zyi and hα_xy2 i in equations (2.42) and (2.43) can be expressed in terms of the

tensor invariants β and γ. β is called the mean polarizability and accounts for the degree of

symmetry of the vibrational mode. In a similar way γ is the anisotropy and is related, as its

name reads, to the vibrational mode’s anisotropy. These quantities are given by

β = 1

3(αxx+αyy+αzz) (2.44)

and

γ2 = 1

2

|αxx−αyy|2+|αyy−αzz|2+|αzz−αxx|2

+3 4

|αxy −αyx|2+|αxz −αzx|2+|αyz−αzy|2

, (2.45)

which are independent on the system of reference used and that is why they are so useful. Detailed description and calculus of the tensor invariants can be found in appendix A14 by Long[4].

In terms of the tensor invariants we have

hα2_nmi= γ

2

15 (2.46)

with n, m=x, y, z and n 6=m. We have then that the intensity of equation (2.43) is

I(90°,E

k

0,p

k

S)

t =I

(90°,Ek₀,p⊥

S)

t =

Nt

32π2 0c3

ω_s4γ

2

15E02y. (2.47)

Although this expression is completely valid, it is convenient to have the scattered intensity in terms of the irradianceI0 of the incident light. We use the irradiance assuming the incident radiation is a plane wave, thus

I0 = 1₂c0|E0|2. (2.48)

Using this, we have

I(90°,E

k

0,p

k

S)

t =I

(90°,Ek₀,p⊥

S)

t =

Nt

16π22 0c4

ω4_sγ

2

15I0. (2.49)

Finally, we calculate the total intensity due to parallel and perpendicular induced dipoles as

I(90°,E

k

0,p⊥S+p

k

S)

t =

Nt

8π22 0c4

ω_s4γ

2

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Following the same procedure it is found the intensity for the parallel induced dipole psk

when the incident electric field is perpendicular to the scatter plane E0 =E⊥0 =E0xˆiis

I(90°,E

⊥

0,p

k

S)

t =

Nt

16π22 0c4

ω_s4γ

2

15I0, (2.51)

and the perpendicular by

I(90°,E

⊥

0,p

⊥

S)

t =

Nt

16π22 0c4

ω_s445β

2_{+ 4}_γ2

45 I0. (2.52)

The total intensity is then

I(90°,E

⊥

0,p

⊥

S+p

k

S)

t =

Nt

16π22 0c4

ω_s445β

2_{+ 7}_γ2

45 I0. (2.53)

Equations (2.50) and (2.53) evidence the fact mentioned in the beginning of this section, when it was established that the measured intensities for the Raman peaks depend on the experimental setup. In particular for the geometry sketched in figure 2.3, it can be seen that even if the incident light travels the same path and the measurement is done in the same way, if the incident light’s polarization is different the intensity also will be.

Assuming a forward- or back-scattering measurement, the intensities can be calculated as before. In this case results

I(0°or180°,E

⊥

0,p⊥S+p

k

S)

t =I

(0°or180°,Ek₀,p⊥

S+p

k

S)

t =

Nt

16π22 0c4

ω4_s45β

2_{+ 7}_γ2

45 I0. (2.54)

Here the measured intensities are the same for both polarizations of the incident field. This behavior arises from the fact that incident and scattered wave vectors are collinear and hence the scatter plane is not well defined. Hence all the directions are equivalent and the intensity measurements equal. It is also important to notice that equations (2.53) and (2.54) coincide.

The principal statement of this section is that Raman peaks’ intensities are not always the same, depending not only on the incident power and sample but also on the experimental setup. We were able to shown this for the case of a dipole in free space, which is the simplest case we could consider. When increasing the complexity of the studied system, this feature would be even more pronounced. As the polarizability tensor is different for each vibrational mode, this means the relative intensities of the peaks can change for different experiments. The position of the peaks, on the contrary, must remain the same.

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2.5 Selection Rules

In this section we are interested in the conditions for the transition to be active. In order to determined those it is necessary to study in detail molecular vibrations and symmetry, but this is beyond the scope of this document and we will only mention the principal results. A deeper treatment can be found in, for example, the books of Diem[11] and Smith and Dent[12].

A change in the polarizability of the molecule is the fundamental condition for the transition in order to be Raman active.

In the semi-classical approach the selection rules for Raman scattering are given by the equation (2.25), as this expression can not vanish. With this in mind, is convenient to define a polarizability operator ˆα with components

(ˆαρ,σ)f i =

1

~

X

k

ˆ

µρ|kihk|ˆµσ ωk−ωi−ω +

ˆ

µσ|kihk|ˆµρ ωk−ωf −ω

, (2.55)

where again |ki is the intermediate state and i and f represent the initial and final states,

respectively. From this equation we can determine the components of the polarizability tensor as

(αρ,σ)f i =hf|(ˆαρ,σ)f i|ii=

Z

ψ∗_fαˆρ,σψidτ (2.56)

from where it can be seen the symmetry ofψi,ψf and ˆαρ,σ specify the selection rules for Raman

scattering. In the classical formalism, the rule for the transition to be allowed is in the truncated Taylor series for the polarizability tensor (Equation (2.11)). What makes a transition active is the change in the polarizability due to the interaction with the incident radiation, this is, the derived polarizability (Equation (2.12)) must be non zero.

At this point it is necessary to mention the infrared spectroscopy, as it is also of vibrational nature. The basic selection rule for this effect is that the interaction must cause the molecule’s dipole to change. The two techniques are complementary because its selection rules are different and therefore a single technique can not be used to determine all vibrational levels of a molecule. A good comparison between them is found in the book of Ferraroet al.[13]. In particular, if the

molecule has a center of inversion, a specific vibrational mode is either Raman or infrared active but no both. This is known asrule of mutual exclusion and in this case the two techniques are

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References

[1] C. V. Raman and K. S. Krishnan. A new type of secondary radiation. Nature,

121(3048):501–502, March 1928.

[2] C. V. Raman. A change of wavelength in light scattering. Nature, 121(3051):619–620,

April 1928.

[3] E. Le-Ru and P. Etchegoin. Principles of SERS, chapter 2. Elsevier B.V., 2009.

[4] D. A. Long. The Raman Effect. John Wiley & Sons Ltd., 2002.

[5] B. Schrader and D. S. Moore. Laserbased molecular spectroscopy for chemical analysis -Raman scattering process. Pure & Applied Chemistry, 69(7):1451–1468, 1997.

[6] R. W. Boyd. Nonlinear Optics, pages 393–394. Academic Press, third edition, 2008.

[7] J. M. Ziman. Principles of the theory of solids, pages 200–203. Cambridge University

Press, second edition, 1972.

[8] M. Toda, R. Kubo, and N. Saito. Statistical Physics I. Equilibrium Statistical Mechanics,

pages 78–82. Springer-Verlag, 1983.

[9] J. D. Jackson.Classical Electrodynamics, pages 407–413. Jhon Wiley & Sons, third edition,

1999.

[10] P. F. Bernath. Spectra of Atoms and Molecules, pages 298–305. Oxford University Press,

second edition, 2005.

[11] M. Diem. Introduction to modern vibrational spectroscopy. Jhon Wiley & Sons, 1993.

[12] E. Smith and G. Dent. Modern Raman Microscopy - A Practical Approach, pages 80–86.

John Wiley & Sons Ltd., 2005.

[13] J. Ferraro, K. Nakamoto, and C. W. Brown. Introductory Raman Spectroscopy, pages

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3 Raman Spectroscopy Setup

In this chapter we describe the experimental setup employed to obtain Raman spectra, as well as, the procedures used to characterize the setup. During all the experiments polystyrene beads were used as the particles to study. This election was done based on the fact the spectrum for this material is well known.

In the first section we describe the setup used to acquire Raman spectra, followed by the details of its construction and specifications of each part. In the next section we will present the alignment of the setup and we will define the excitation and detection areas in the experiment. We will finish this chapter with the description of the data acquisition and processing.

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3.1 Description of the setup

In this section we will present and explain the experimental setup used to acquire Raman spectra. After presenting a general overview of how it works, we will mention all the relevant details of each part and the specifications of the elements used.

3.1.1 General description

Figure 3.1 shows the schematic of the experimental setup implemented for Raman spectroscopy. The arrows to the side of some elements accounts for the freedom degrees of the object they are attached to. We can subdivide the experiment in three parts: i) excitation of the sample, ii) detection of the scattered light and iii) observation of the sample’s surface. The beam’s colors representation is given as follows: the sample is excited by a green diode laser (green color), the signal detected from the sample is red-shifted (orange color) and for the observation a white lamp is used (gray color).

Excitation

The excitation light coming out of a 532 nm diode-laser (top-left in the figure) is guided by the mirror M1 and a dichroic mirror DM towards an 50X microscope objective. Then the light goes through a beamsplitter BS which has no role at this stage. The excitation part of the setup is closed by the 50X objective, which focus down the light on the sample. The specific point on the sample to be studied can be chosen thanks to a special sample holder that was designed.

Detection

The sample scatters the light in all directions but just the back-scattering light is collected by the same 50X objective. This scattered signal has both Raman and Rayleigh components and is collected by the same 50X objective and travels along the same path as the green light. Then, the dichroic mirror DM reflects the Rayleigh component and transmits the Raman signal. Again, the beamsplitter BS has no role. Two more mirrors direct the signal towards a 4X microscope objective, which focus it down into a fiber end. Before the 4X objective there is a filter to remove the remaining Rayleigh compound of the signal. The fiber is connected to a spectrometer.

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Figure 3.1: Schematic of the Experimental Setup for Raman Spectroscopy.

Observation

Finally, to observe the sample, it is illuminated from the bottom with a white-light lamp. Such light is collected also with the 50X objective and reflected by the beamsplitter BS to a two lenses system which image the sample onto the CCD directly. In order to magnify the object, we use a negative lens after a positive one, in such a way the former was within the focal distance of the latter.

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3.1.2 Details and specifications

In the previous section we presented the general description about how the setup works. In this section we give a detailed description of each part and the specifications for the optical elements used.

Once we have the general feeling about how the setup works, this section gives a detailed description of each part, as well as, the specifications for the optical elements used.

• The laser output power was 4.8 mW but, due to Fresnel losses the total incident power at

the sample was around 3.8 mW.

• All the mirrors M’s used were silver coated (Thorlabs product reference PF10-03-P01).

• We know from chapter 2, that the scattered light has Rayleigh and Raman components, although we are interested only in the second one. Since the first one is considerably more intense, it must be removed in order to avoid both the spectrometer saturation and the concealing of the signal of interest. The dichroic mirror reflects the light at 532 nm but transmits higher wavelengths. Hence such mirror had two different but complementary uses. On one hand, as was mentioned in the past section, it reflects the excitation light towards the sample. And on the other hand, filters out the signal which is collected with the 50X objective. The filtering occurs because the DM reflects the Rayleigh component of the signal and transmits the red-shifted Raman wavelengths. It must be noticed that we cannot measure the anti-Stokes lines, since the filter removes the blue-shifted Raman frequencies. The dichroic mirror DM used was a Semrock Razor Egde product reference LPD01-532RS-25.

• The height from the sample of the 50X microscope objective could be adjusted by means of a linear translation stage with resolution of 10µm. This allows us to collimate the signal from the sample. The objective used was a LMPlanFI Long Working-Distace Semi-Apochromat byOlympus with numerical aperture 0.5.

• All the spectra were taken for polystyrene 3.55µm-diameter beads frommicroParticles GmbH

product reference PS-R-3.6. The beads were immersed in water, and a small drop of this solution was placed on a cover-glass without any cover. The sample was used after the water was evaporated. The initial concentration of particles was 1% but for our purposes we dilute it successively until reaching the optimal density on the cover-glass. The criterion to decide this was subjective. We cannot calculated the density of particles on the cover-glass, as we do not know neither the final concentration, nor the total area on the cover-glass where the beads are spread.

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• The sample was placed on a special holder designed specifically for this purpose, that will be described on section 8.1.4. This holder was attached to 3D-stage with resolution of 1µm.

• Since the Raman signal is so low compared to the Rayleigh, it is necessary to use an additional filter FD to remove most of the remaining undesired signal. The filter used was a Semrock EdgeBasic product reference BLP01-532R-25.

• The signal was sent to the spectrometer via an optical fiber. To couple a 4X microscope objective focus the signal on the entrance of the fiber. In order to do this the objective was mounted on a linear translations stage and the fiber on a XY translation mount. The used fiber was a 62.5µm-diameter Thorlabs Graded Index Multimode product reference M31L02.

• The spectra were taken using a Shamrock Imaging spectrograph product reference SR303i

with a CCD camera Andor product reference iDus DU401A and softwareAndor Solis. The

spectrometer was calibrated using the 632.8 nm laser line of a He-Ne laser.

• The beamsplitter BS was introduced only with the idea of observing the sample. With this in mind, the beamsplitter used reflects only the 8% and transmits the 92%. An uncoated pellicle beamsplitter by Thorlabs product reference BP108 was used.

• The sample was imaged onto the CCD by using two lenses L1 and L2. Since the separation between the sample and the 50X microscope objective is fixed, a focused image is achieved by moving the lenses. Taking into account the white beam is collimated after the 50X objective, it is necessary to use two lenses to image the sample. The first one focuses the beam at a short distance and the second magnifies it to produce a clear image. The used lenses were a 75 mm focal distance coated plano-convex and a −25.4 mm focal length coated bi-concave,

both byThorlabs product references LA1608-B and LD2297-B-N-ST11 respectively. As part

of the signal was also reflected by the beamsplitter, it was necessary to use a filter FO to remove it from the image.

• The image was taken using a Watec Monochromatic Camera product reference WAT-902H

and with the software Vision Assistant 8.6 from National Instruments. The camera had no

lenses and we project the image directly onto the CCD.

• The hemispherical prism HSP shown in the figure has no role in the experiment but was used with the idea of not to change the setup once the plasmonic-trapping is implemented.

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3.2 Additional aspects

3.2.1 Alignment

The alignment of the setup is crucial in order to obtain a good Signal-to-Noise Ratio (SNR), because the signal in the spectrometer depends on the coupling into the fiber and this is very subtle. The coupling is easier to accomplish if the signal beam before the 4X objective is collimated. If this condition is satisfied, the beam is focused at the working distance from the objective and the end of the fiber can be placed in such point, giving a suitable starting point for the optimization of the coupling. This collimated signal beam is achieved by adjusting the height respect to the sample of the 50X objective. Another very important advantage of having the signal collimated is that it is driven to the coupling objective with no losses.

The alignment was done using an auxiliary laser beam which was located under the prism. First this beam was oriented vertically without any optical element. This was done employing geometrical tools, like measuring the distance between the breadboard and the beam. Then one by one each optical element in the path was inserted in place and positioned in such a way that the auxiliary beam was still vertical. Once this beam was aligned, the excitation green beam was adjusted, with the help of the mirror M1 and the dichroic mirror, until the two beams coincide and therefore the green beam was also vertical. Initially a bare cover-glass was located in the sample space, so the reflection was very high, making the alignment easier. Since the glass was horizontally oriented, the reflected beam after the 50X objective was vertical and its path coincides with the auxiliary beam.

By using the mirror M2, the reflected beam was directed horizontally, whereafter the beam was deviated vertically with mirror M3. As before, this was done by geometrical means and without any optical elements. After the beam was properly oriented, the 4X microscope objec-tive and the mount for the fiber were placed accordingly with the previous alignment.

Until now the alignment was based in geometrical considerations, which is good enough to start. However, in order to obtain a good spectrum it is necessary to perform a finer alignment. The power before and after the fiber was measured without the filter FD and the coupling optimized changing the 4X objective’s height and the XY position of the fiber’s mount until at least the 70% of the power is transmitted through the fiber. At this point of the alignment is also convenient to change the height of the 50X objective. After this optimization has been carried out, the filter FD is located in its position and the fiber connected to the spectrometer.

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should be made. For this a sample must be placed on. This sample has to contain enough particles to generate a strong signal. If the previous alignment has been done properly, the signal, although weak, should be visible. The goal is then to improve this signal. This is done first by changing the 50X objective’s height and then the coupling to the fiber. This process is repeated iteratively until reaching a satisfactory SNR. It is possible to be unable to find this point, in such case, the alignment should be started from the beginning.

3.2.2 Excitation and detection areas

With the aim of getting a strong signal there are three important parameters in the experiment: the size of the particle to be analyzed, and the detection and excitation areas. Figure 3.2 shows images of this parameters.

The size of the studied particle is fixed for each experiment, although it can vary from sample to sample. Figure 3.2a shows an isolated polystyrene bead, which was our particle in study. Here the particles density was very low, as the goal of the experiment was to obtain the Raman spectrum for single particles.

The measured signal is only the light that is coupled into the fiber, this is, the fiber restricts the points on the sample from which the signal is obtained. This means that the fiber in the setup behaves as a pinhole, and hence the setup is treated as a confocal one. The detection area AD is precisely the area on the sample whose signal is detected. Figure 3.2b shows the

brightened area on the sample when the fiber is illuminated from its other extreme with a white light lamp, this is, the detection area. The size of such area depends on the relationship between the collection (50X) and coupling (4X) objectives.

The third important parameter is the excitation area, understood as the area where the

(a) Polystyrene bead. (b) Detection area. (c) Excitation area. Figure 3.2: Important parameters in the experiment.

Raman spectra acquisition of plasmonic-trapped particles