Two-Photon Excited Fluorescence Depolarisation

Transcription

1 Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 552 Two-Photon Excited Fluorescence Depolarisation Experimental and Theoretical Development LINUS RYDERFORS ACTA UNIVERSITATIS UPSALIENSIS UPPSALA 2008 ISSN ISBN urn:nbn:se:uu:diva-9285

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3 List of Papers This thesis is a summary based on the following papers, which can be found as reprints at the end of the volume. The papers are referred to in the text by their Roman numerals I-V. I) Two-photon excited fluorescence depolarisation experiments: II. The proper response function for analysing TCSPC data Linus Ryderfors, Emad Mukhtar, Lennart B.-Å. Johansson Chemical Physics Letters 2005, Vol. 411, II) Two-Photon Excited Fluorescence and Molecular Reorientations in Liquid Solutions Linus Ryderfors, Emad Mukhtar, Lennart B.-Å. Johansson Journal of Fluorescence 2007, Vol. 17, III) The Symmetry of Two-Photon Excited States as Determined by Time- Resolved Fluorescence Depolarization Experiments Linus Ryderfors, Emad Mukhtar, Lennart B.-Å. Johansson Journal of Physical Chemistry A 2007, Vol. 111, IV) Excited State Symmetry and Reorientation Dynamics of Perylenes in Liquid Solutions: Time-Resolved Fluorescence Depolarisation Studies Using One- and Two-Photon Excitation Linus Ryderfors, Emad Mukhtar, Lennart B.-Å. Johansson Journal of Physical Chemistry A 2008, Vol. 112, V) Two-Photon Excited Fluorescence Depolarisation and Electronic Energy Migration within Donor-Donor Pairs Linus Ryderfors, Oleg Opanasyuk, Emad Mukhtar and Lennart B.-Å. Johansson Manuscript

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5 Contents 1. Introduction Fundamental concepts One- and multi-photon optical processes Absorption probabilities Fluorescence spectroscopy Time-resolved fluorescence Fluorescence depolarisation Photoselection Fluorescence depolarisation experiments Molecular reorientation dynamics Molecular information from fluorescence depolarisation Fluorescence polarisation upon OPE and TPE Donor-donor energy migration DDEM Electronic energy transport upon TPE Experimental methodology UV-VIS spectroscopic properties of perylenes Discussion of results Improving the response function for analysing TPE fluorescence depolarisation experiments Extension of the work on the non-linear IRF Theoretical model for TPE fluorescence depolarisation in liquid solution Rotational dynamics and the symmetry of two-photon excited states as determined by time-resolved fluorescence depolarisation Global analysis scheme The case of perylenes in liquid solution Future studies inspired by Papers III-IV Two-photon excited fluorescence depolarisation and electronic energy migration within donor-donor pairs Preliminary experiments on a DDEM system Conclusions and future outlook Sammanfattning på svenska...63

6 8. Acknowledgement List of abbreviations References...67

7 1. Introduction Fluorescence spectroscopy and microscopy are indispensible research tools in the natural sciences(1-4). The properties of the fluorescence light are employed for the exploration of photophysical and photochemical processes of electronically excited molecules. Judicious applications of fluorescence as a tool in chemistry and biology give access to structure, dynamics, interaction and sensing of molecular complexes. Fluorescence is also widely used for the visualisation of processes in cells and biological tissues. Time-resolved fluorescence studies have been particularly informative for monitoring the excited state dynamics, for example in the study of electronic energy transport between chromophoric molecules, which is employed to obtain information on protein systems concerning their structure and conformational dynamics as well as the formation of protein aggregates. After excitation with polarised light, the fluorescence can be partially polarised. The polarisation dependence of the fluorescence is monitored in socalled fluorescence depolarisation experiments. These experiments have been utilized for many decades to study the reorientation motion of molecules in solution, solubilised in anisotropic media like liquid crystalline phases, and used as extrinsic or intrinsic probes in biomacromolecular systems(5-14). The vast majority of experimental work on fluorescence spectroscopy, and in particular fluorescence depolarisation spectroscopy, has been carried out using one-photon excitation (OPE) in the visible or ultraviolet spectral region, where many aromatic molecules have their absorption. Recently, two-photon excitation (TPE) of fluorescence has started to attract considerable attention. TPE occurs when a molecule is excited by simultaneously absorbing two photons of half the transition energy in a single quantum event. Compared to OPE, TPE is a rather unlikely process; if a good one-photon absorbing molecule in daylight would on average most definitely absorb a photon every second, a good two-photon absorbing molecule would only be two-photon excited once every thousand years or so. However, the development in ultrafast pulsed laser systems has given access to very high light intensities that make TPE techniques accessible. Most widely used nowadays are titanium-sapphire lasers, typically operated in the near-ir spectral regime around 800 nm. Compared to conventional OPE, TPE introduces new features that scientists can exploit for their benefit. The high photon density required for the 7

8 effect to occur means that the excitation volume is mainly located in the focal waist region. In strongly scattering or absorbing media, such as biologically interesting systems like skin tissue, cells, and model membranes, the near-ir excitation light employed in TPE gives a reduced scattering background and a larger penetration depth. Consequently, TPE of fluorescence has been of great utility in e.g. biomolecular imaging and fluorescence correlation spectroscopy(16-25) in terms of increased spatial and temporal resolution, with better three-dimensional depth profile imaging, and reduced photobleaching and phototoxicity outside the focal volume. TPE further allows the possibility to excite two dyes with the same laser for dual colour measurements. A further benefit of TPE(26-30) is access to higher excitation or ionization energies without the use of a UV light source. Due to the fact that TPE has different spectroscopic selection rules than OPE, TPE spectroscopy has played an important rule in elucidating the nature of electronic states of molecules. The TPE signal also exhibits a more extensive dependence on the polarisation of the excitation source than OPE. For example, the two-photon absorptivity of an isotropic sample depends on the polarisation of the exciting light. In addition, as predicted by theory and confirmed in primary studies, the twophoton interaction can create a sharper angular distribution of excited state molecules, which leads to a higher initial degree of polarisation of the fluorescence(31-34). Despite these promising prospects, relatively few fluorescence depolarisation studies involving TPE have been reported thus far, see e.g. (35-39) and references cited in (29). One idea has been to use TPE in order to increase the dynamic range of the fluorescence depolarisation studies. Furthermore, TPE creates the opportunity to set up several fluorescence depolarisation experiments that can provide independent information, for example using linearly and circularly polarised excitation light. A global analysis scheme of these experiments could give access to studies of complex rotational dynamics, as well as provide an empirical assignment procedure for the nature of the excited vibronic states involved in the TPE process. One reason for the scarcity of experimental studies is the lack of accessible theoretical models relating TPE fluorescence depolarisation to molecular properties such as reorientation. Nevertheless, the presence of molecular reorientations is a necessary consideration for many molecular systems, including biomolecular or biological systems in their natural environment, where TPE induced fluorescence has its greatest potentials. In my thesis, I chose to investigate from a fundamental point of view TPE fluorescence depolarisation from molecules in solution. Part of the work encompasses experimental considerations concerning the extraction of meaningful data from TPE time-resolved fluorescence. It turned out that further development of theoretical models, relating the TPE fluorescence depo- 8

9 larisation to molecular reorientations in liquid solution, was needed to interpret experimental data. In order to achieve molecular information, we proceeded by exploring a new analysis scheme for experimental data sets, involving excitation light of linear or circular polarisation, respectively. As a step towards structure and dynamics determination for more complex biomolecular systems, this thesis presents, what is to the best of my knowledge, the first treatment of TPE electronic energy migration between identical chromophores during molecular reorientations. 9

10 2. Fundamental concepts The aim of this section is to introduce the fundamental principles of TPE fluorescence depolarisation spectroscopy using single-photon timing experiments. A substantial part of the work refers to the development of physical models for interpretation of TPE fluorescence depolarisation experiments. Therefore, the theoretical framework relevant for this is presented. The chapter ends with a section introducing electronic energy migration between identical molecules One- and multi-photon optical processes Optical spectroscopy involves the interaction between matter and electromagnetic radiation in the near UV and visible part of the spectrum. The common theoretical treatment of the interaction is based on time-dependent perturbation theory, treated e.g. in(40). This leads to a series of perturbation terms, with the order n of dependence on the electromagnetic field varying from 1 to infinity. The first term in the perturbation expansion involves the interaction of the electrons with one photon that can be either absorbed or emitted, as shown by the Feynman diagrams in Figure 1a-b. This corresponds to the linear response regime of optical spectroscopy. Non-linear optical spectroscopy involves the interaction of matter with more than one photon. Non-linear effects are well-established and used in many fundamental and applied areas of science and technology(41, 42). The Feynman diagrams of some common non-linear optical interactions are illustrated in Figure 1c-f. Second order non-linear processes are e.g. two-photon absorption, Rayleigh scattering and Raman scattering. Three-photon processes are e.g. three-photon absorption, sum-frequency generation, second harmonic generation and hyper-raman scattering. The quantum process can be resonant or non-resonant. Resonance enhancement of the (non)-linear optical process of order n occurs if the photon energy E, which is solely determined by the frequency of light via Planck's quantization law E =, matches intermediate transitions, and if the energy of n photons matches the overall energy change in the molecule. Nonresonant transitions are often referred to as virtual transitions since they are energy non-conserving. 10

11 Figure 1. Feynman diagrams of some spectroscopic processes taking an electron between state g> and ƒ>. a) one-photon absorption; b) one-photon emission; c) two-photon absorption. d) two-photon scattering; Rayleigh scattering ( 1 = 2 ) and Raman scattering ( 1 2 ); e) three-photon absorption; f) three-photon scattering, e.g. second-harmonic generation, sum-frequency generation and hyper- Rayleigh scattering ( = 3 ); and hyper-raman scattering ( ). The intermediate states n> and m> can be off-resonance, involving so called virtual states. Figure adapted after Peticolas, W. L., Ann. Rev. Phys. Chem.,

12 Absorption probabilities The probability that a molecule absorbs energy from the electromagnetic field depends on properties of the molecule as well as the wavelength and the polarisation of the light. In this work, the theoretical description is limited to the electric dipole regime. The coupling term in the interaction Hamiltonian is then the scalar product of the electric field of light with the electric dipole moment of the electrons in the molecule. The fundamental probability equations for a transition between two electronic states assuming delta-pulses of excitation follow from Fermi s golden rule with respect to proper time-ordered Feynman diagrams, cf. Figure 1. The transition probability amplitude for OPE P is given 1 by; ex ex ex 1 P 2 Here f rg ˆ is the absorption electric dipole moment vector connecting the ground g and excited ƒ electronic states, ex is a unit vector defining the polarisation of the light. The two-photon excitation involves two photons. The probability amplitude P 2 ex is proportional to the absolute square of a certain sum ex, T fg ex (43, 44), i.e.; Where Tfg ex, ex P T, 2 2 ex fg ex ex is defined by; ˆ ˆ ˆ ˆ ex nr g f rn ex ex f rn nr g ex Tfg ex, ex (3) n ng in ng in T, is the sum over all possible products of two successive timeordered fg ex electric ex dipole interactions via intermediate electronic states n, starting in the ground state g and leading to a population of the final excited state f, weighted by the difference between the transition frequency ng from the ground state to the state n and the frequency of the first photon. n r ˆ g and f r ˆ n are the transition dipoles. The resonance condition for the twophoton absorption process is that the sum of the energy of the two photons matches the energy difference between the ground and final states. The intermediate states need not be resonant, but in the case of a resonant intermediate state n the homogenous linewidth n of the state must be taken into consideration in the denominator(44). Quantum mechanical interference effects arise, since that in order to obtain the probability, the square has to be (1) (2) 12

13 taken of the sum over all intermediate states. In this work, the TPE is obtained using a single laser beam, involving two identical photons. Under this 2 condition, the following equation for P can be obtained; 2 2 ex extex P ex (4) With this definition, T ~ is a molecular quantity obeying the transform properties of a second rank tensor. In this work it will hereafter be referred to as the two-photon absorption tensor T ~. An element (T) ab of a 3x3 element Cartesian representation of T ~ must consequently be defined according to(44): T ab n ˆ ˆ ˆ nr ˆ a g f rb n f rb n nra g i i ng n ng n. (5) It can be seen from Eq. 5 that in the case of two photons with identical frequencies, T ~ is symmetric. In the case of intermediate states far from onephoton resonances, it can always be represented using a real basis set such that all components in the Cartesian representation of T ~ are real(45) The role of symmetry Symmetry considerations provide important simplifications in spectroscopy(46). The mathematical language is provided by the theory of group representations. Spectroscopic selection rules restrict the variety of matrix elements between different irreducible representations of electronic states according to the symmetry of the perturbing Hamiltonian. It follows from Eq. 5 that electric dipole allowed two-photon transitions are governed by the dipole product operators that transform under x 2, y 2, z 2, xy, xz or yz symmetry. The one-photon transitions, cf. Eq. 1, transform under x, y or z symmetry. The allowed transitions have symmetry such that the transition electric dipole vector or tensor T ~ connecting the states g> and ƒ> contain the totally symmetric irreducible representation in the molecular point group. Vibronic coupling(47), for example, can relieve the constraints imposed by the selection rules. For a further discussion on symmetry breaking in TPE, see also(48). For molecules with inversion symmetry, the selection rules mean that one-photon and two-photon transitions probe vibronic states that are complementary with respect to parity. This property has been very useful in molecular spectroscopy(26, 27, 30). Moreover, the same property also applies to vibrational spectroscopy; one-photon infrared absorption and Raman spectroscopy, involving two-photon transitions as was shown in Figure 1d, are also complementary. 13

14 2.2. Fluorescence spectroscopy The properties of the fluorescence emission provide a powerful tool for monitoring various photophysical and photochemical events occurring while the photoexcited molecule relaxes to the ground state(1-3). The fluorescence light is also widely used in biological and analytical applications, e.g. in sensing experiments and imaging. The dependence of the fluorescence signal on concentration of the fluorescent molecules, quencher concentration and ambient conditions such as solvent properties, e.g. viscosity, polarity and ph, as well as temperature and pressure has been used to study molecular properties and interactions. Quantities that the scientist can investigate are mainly; the fluorescence intensity and lifetime; the excitation and emission energy spectra; and the polarisation of the emitted light. The merits of the fluorescence process are in particular; its sensitivity, with single-photon and single-molecule detection limits; its often non-destructive nature; and its wide temporal range, from femtoseconds to microseconds. The development of new fluorescence probes, like the green fluorescent protein and organic molecules with increased absorption cross-section, fluorescence quantum yield and photostability, contribute to make fluorescence spectroscopy increasingly useful in diverse applications in science and technology. Figure 2 presents a schematic diagram for the absorption and fluorescence transitions between the ground and first excited singlet electronic states S 0 S 1 of a molecule. The vertical arrows between the vibronic manifolds of the electronic states in mean that the molecule is assumed to be undergoing Franck-Condon transitions, in which the nuclei are not moving during the radiative transition. The overlap of nuclear wavefunctions in the two electronic states determines the relative intensity of the vibronic transitions. Typically, the photo-excited molecule rapidly ends up in the lowest vibrational levels of S 1. The molecule then loses the remaining excess energy to return to the ground state S 0 via a radiative transition, the fluorescence, or intersystem crossing to a triplet state, which may result in phosphorescence, or by internal conversion. 14

15 Figure 2. Potential energy level diagrams illustrating the radiative electronic excitation and relaxation pathways between vibronic manifolds of the ground S 0 and excited S 1 singlet states. The energy of the emission between the lowest vibrational level of S 1 to the vibrational manifolds of the ground state is consequently less than the energy of the corresponding exciting transition from S 0. This means that in OPE, the fluorescence spectrum is a red shifted with respect to the absorption spectrum. In TPE, the energy for the excitation is provided by the sum of two photons. This leads to the rather unusual result that the fluorescence is instead often blue shifted with respect to the excitation light. It is known that emitting fluorophores behave like radiating dipoles. The probability that an excited molecule will emit a photon that will be transmitted through a certain polarisation direction em of a polariser can be given by an expression analogous to Eq. 1; 2 1 P (6) em em em 15

16 Where em is the emission transition moment vector connecting the emitting state S 1 and the ground state S 0. Due to the different electron configuration in the S 1 state with respect to the ground state, the S 1 state can also have a different equilibrium geometry. Since the emission occurs from the relaxed vibronic state S 1, em can in general be different from ex even for the S 0 S 1 transition. Hitherto, fluorescence probes have shown very similar fluorescence spectra independent on whether it was excited via a one- or two-photon process(2). This confirms that the emission is occurring from the same electronic state independent on the mode of excitation. Under ideal conditions, vide infra, the TPE fluorescence intensity F measured from a sample exposed to a laser with the excitation power density exc is given by the following equation(29): f 2 F molexc2l. (7) 2 Here f, mol and l are the detection efficiency, the fluorescence quantum yield, the number density of absorbing molecules and the optical path length, respectively. The two-photon absorption cross-section 2 is given in the units of cm 4 s or often GM, where 1 GM = cm 4 s. The unit is named in honour of Göppert-Mayer who made the first quantum mechanical treatment of twophoton absorption(49). 2 is proportional to the isotropic orientation average of the TPE probability amplitude(44), cf. Eq. 5, and depends on the wavelength of the exciting light. Contrary to the one-photon absorption crosssection, 2 also depends on the polarisation of the exciting light. Eq. 7 predicts that the TPE fluorescence is proportional to the square of the incident photon flux. The focusing of an intense laser beam necessary in TPE experiments, may lead to a number of artifacts such as self-focusing of the beam, saturation, photobleaching and stimulated emission. The validity of Eq. 7 can be experimentally verified by means of a log-log plot of the excitation and emission light intensities. If the excitation is taking place by means of TPE and in the absence of the above mentioned artifacts, the gradient in such a plot should be 2. The quadratic dependence was verified in all the work on TPE induced fluorescence presented in this thesis, and is exemplified e.g. in Figure 1 in Paper I Time-resolved fluorescence Time-resolved measurements of fluorescence, to determine the lifetime and the depolarisation, have proved particularly powerful in elucidating the dynamics and interactions of excited state molecules as related to physical, chemical as well as biological questions(2). Pulsed light sources, such as 16

17 light emitting diodes, flash lamps, laser systems or even synchrotrons, make it possible to monitor the fluorescence intensity as a function of time after the excitation. The fluorescence intensity from a sample following a deltapulse of excitation is proportional to the number of molecules in the emitting state. In many cases, only intramolecular processes compete with the fluorescence. The decay law is then simply a monoexponential decay function: F t Aexp t (8) Where A is the pre-exponential factor. In the exponent, t is time and is the fluorescence lifetime, which is a measure of how long the molecule remains in the excited electronic state on average. For aromatic molecules, is usually in the order of nanoseconds Time-correlated single photon counting In this work, the detection is mainly carried out using single-photon timing detection, also called time-correlated single photon counting(50, 51), TCSPC. This technique is generally accepted as the method of choice for studying time-resolved fluorescence in the time domain. It combines an ideal blend of sensitivity, time-resolution and dynamic range with known statistics of the data which is valuable for a quantitative evaluation of the experiment with respect to the kinetic models(29). TCSPC is a stochastic method; the decay curves are built up as probability histograms correlating a single excitation event with a single emission event one by one. The distribution of data obeys Poisson statistics. When a femtosecond laser is used as excitation source, the timing electronics limits the time-resolution of TCSPC. The current state of the art set-up employs microchannel plate photomultipliers and pico-timing discriminators. In this case the time-resolution can be down to tens of picoseconds, as determined by the width of the instrumental response function (IRF), which describes the response of the detection system response to the exciting light pulse. Due to the finite response time of the detection system, the observed fluorescence decay curves F(t) appear somewhat distorted. The obtained kinetic traces are analysed as the convolution between the IRF, irf(t), and the true decay function u(t) of the kinetics: t FC t u t s irf s ds u t irf t. (9) 0 Here the calculated decay is called F C (t). The convolution operator is denoted by, and t, s and denote time. is a variable time-shift parameter. The IRF is usually collected by measuring scattered excitation light. Intensity 17

18 decays are typically described by a sum of exponential functions with differing decay times i : t A exp t u (10) i i i For biological systems or single molecules a distribution of decay times is sometimes more appropriate(52). Data evaluation of an experimental decay with respect to a kinetic model is usually made by statistical significance tests with respect to the goodness of fit function 2 : 2 2 n2 n 2 FC ti F ti 2 2 wr ti (11) in1 t in1 i In Eq. 11, the fitting is performed from the channel n 1 to n 2. According to Poisson statistics relevant for TCSPC, the standard deviation obey the relation 2 ti Fti. Graphical representations of the weighted residuals wr(t i ) and the autocorrelation functions of these residuals are also useful. An example of a curve fitted TCSPC experiment is given in Figure 3: Figure 3. Example of a TCSPC experiment, showing the curve-fitted fluorescence lifetime decay histograms F(t) and the IRF. The upper two panels show graphs of the weighted residuals wr(t) of the fit and the autocorrelation function of these residuals cr(i). 18

19 The fluorescence decay parameters in the fittings are usually obtained using iterative reconvolution algorithms with respect to the experimental IRF, cf. Eq. 9, thus assessing the decay laws indirectly. The data optimisation is carried out with respect to minimisation of 2 using different search algorithms. For multiexponential decays, gradient or steepest descent analysis like nonlinear least-squares fitting(53) are appropriate, and this is also mainly used in this work. For complex or noisy data, maximum entropy modelling(54) or evolutionary stochastical methods like genetic algorithms(55) can be used Fluorescence depolarisation Photoselection Consider a liquid solution of chromophoric molecules exposed to a short pulse of polarised light, as is pictured in Figure 4. Figure 4. Photoselection of molecules in solution. Initially, the molecules are randomly oriented. Mainly aligned molecules are excited by the polarised light at time t = 0, as represented by the bold lines. Molecular reorientation then tends to randomise the excited state orientation distribution. Being in solution, the molecules have a random orientation. If the light has the right wavelength, some molecules will absorb energy from the light field and be transferred to an excited electronic state. As was shown in the Section 2.1.1, molecules oriented with the transition electric dipole moments parallel to the polarisation vector of the excitation light will preferentially be excited. This process is known as photoselection. The resulting orientation distribution of the electronically excited molecules is anisotropic, meaning that the molecules have a preferred orientation in space. When time proceeds these molecules will return to the ground state e.g. by emitting a photon. In the meantime, molecular reorientations will tend to randomise the initial distribution of excited state molecules again. 19

20 Photoselection in OPE and TPE The excited state orientation distribution formed by photoselection following OPE of randomly oriented ground-state molecules are pictured in Figure 5a. Figure 5. Photoselection of molecules in solution in the case of (a) OPE, and (b) TPE; the orbital-like shapes represents the excited state orientation distributions obtained using linearly polarised excitation light. In OPE there is a cos 2 dependence of the molecular orientation probability with respect to the angle to the polarisation vector of the light. In TPE, a dominating single TPE tensor element is assumed. This results in a cos 4 distribution. The probability for OPE of a molecule is given by Eq. 1, i.e. it is proportional to the cos 2, where is the angle the molecular absorption transition dipole moment vector makes to the electric field vector of light ex. The orientation distribution of the electronically excited state molecules for a randomly oriented sample will be symmetrically distributed along ex, with a preferential direction along this axis according to the cos 2 distribution. In TPE, the photoselection process depends on the molecular TPE transition tensor T, which is given by Eq. 5. If there is a single dominant tensor element governing the transition, this means that both electric dipole interactions are directed along a single specific direction in the molecule. This leads to a sharper excited state distribution than in OPE; a cos 4 - distribution around the polarisation vector of the light ex, see Figure 5b. If there is no such dominating tensor element, the TPE photoselection can lead to excited state orientation distribution with very different shapes(48), not necessarily sharper than in OPE Fluorescence depolarisation experiments The polarisation of the spontaneous emission of a photon from an excited molecule is related to the molecular orientation via the emission transition dipole moment em. Consequently, the fluorescence emitted from a sample excited with polarised light can be partially polarised. In Figure 6, a sche- 20

21 matic illustration of the fluorescence depolarisation experiments is presented: Figure 6. Fluorescence depolarisation experiments. The polarised excitation light impinges along the X L -axis. The fluorescence intensities F t or F t transmitted through parallel and perpendicular settings of the emission polariser are detected. The fluorescence depolarisation is constructed by comparing the intensity of the fluorescence light polarised parallel and perpendicular to the excitation polarisation. The polarised fluorescence experiments in Figure 6 are described with reference to the Cartesian coordinate system L fixed in the laboratory. The excitation light is usually linearly polarised, say along the Z L -axis and is impinging on the sample along the X L -axis. The fluorescence is detected at right angle along the Y L -axis, further down so paraxial conditions can be assumed. The fluorescence is detected through a polariser, either at parallel settings with respect to the excitation polarisation, F (t), i.e. along the Z L axis, or at perpendicular polarisation,, along the X L -axis. Optical filters, as used in this work, or a monochromator, are used to select the wavelength of the observed emitted light. The polarised fluorescence decays are most useful when combined into a sum curve S(t) and a difference curve D(t), i.e. t F t 2GF t F t F t S 3 (12) D t F t GF t (13) The sum curve, S(t), depends solely on the fluorescence relaxation and is directly proportional to the isotropic emission intensity F(t), i.e. free from rotational reorientation effects. This can also be measured with so called 21

22 magic angle settings of the polarisers, e.g. when the emission polariser is oriented at 54.7º to the symmetry Z L -axis(56). The difference curve, D(t), contains information about the reorientation. The fluorescence anisotropy is defined by the well-known ratio(57): r t t D t S F F t t GF 2GF t t. (14) In Eqs , G is a correction factor that accounts for the relative transmission efficiency of the detection system for parallel and perpendicular polarised light. In this work, the anisotropy decay parameters were obtained by analysing globally the S(t) and D(t) decay curves constructed from the polarised intensity histograms(2) according to: 2 3 S t F t GF t F t Dt F t GF t rtst (15) The analysis of the anisotropy decay parameters in r(t) can be made with an iterative reconvolution analysis of the S(t) and D(t) decay curves, analogous to the procedure used for obtaining fluorescence lifetimes in Eq Molecular reorientation dynamics The fluorescence anisotropy r(t) corresponds to orientation correlation functions which describe the reorienting motions of an ensemble of electronically excited molecules. The correlation is between the orientation of the molecule initially excited and when emitting a photon at a time t later. In order to describe the molecular orientations a laboratory frame (X L, Y L, Z L ) and a molecule fixed frame (X D, Y D, Z D ) are introduced, as is illustrated in Figure 7: 22

23 Figure 7. Orientational transformations in fluorescence depolarisation; from a molecule fixed coordinate frame D to a laboratory fixed frame L at times s=0 and s=t. A theoretical expression for the fluorescence anisotropy r(t) can be obtained from constructing the anisotropy quotient Eq. 14 with respect to a description of the polarised fluorescence intensity decays F (t) and F t. Due to the different physical origins behind the excitation probability in OPE and TPE, these intensities will have a different form. Formally, the polarised intensities can be written: ex, em 1,2 0 1 ex L em L F t P P. (16) ex, em Here the angle brackets... stand for a statistical orientation average 0 over the Euler angles L and L which describe the orientation of the molecule with respect to the L coordinate frame at times s = 0 and s = t. The subscripts ex and em indicate the excitation and emission polarisations used in the experiment. The absorption and emission probabilities depend on the projection of the transition dipole vectors or the TPE transition tensor on the light polarisations via Eqs. 1, 5, 7. The polarisations ex and em are conveniently expressed in the L-frame, whereas the molecular properties are most conveniently described with respect to the molecule fixed D-frame, which is reorienting with respect to the L-frame. This means that in order to describe the polarised fluorescence intensities, it is necessary to perform the orientational transformations DL at the times s = 0 and s = t. The fluorescence anisotropy can be derived in a general setting in terms of time-dependent orientation correlation functions. Various microscopic 23

24 and macroscopic order and reorientation motions can be considered. The orientation transformations between different coordinate systems are smoothly carried out within the formalism of spherical tensors, which has been exploited by a number of authors(9, 58-62). The irreducible tensor components are related to the Cartesian components by means of a unitary transformation. The irreducible tensor components transform in a particularly straight-forward manner under rotations using the Wigner rotation matrix elements D p,q ()(63). This means that solving Eq. 16 corresponds to j evaluating expressions of the form; j* 0 j' ; p,q DL p',q' DL C t D D t (17) Evaluation of the orientation average correlation functions C(t) is often carried out via a Greens s function method according to; 0 0 * 0 j' 0 C t d d f D j D G t (18) 0 DL, DL DL DL DL p,q DL p',q' DL DL DL, In Eq. 18, the orientational distribution density of molecules before excitation is denoted by f DL 0. Since the molecules prior excitation are isotropically oriented, the normalised distribution density obeys f DL 0 = 1/ The Green s function G DL DL, t is the conditional orientational probability that a molecule with an initial orientation 0 L has an orientation L at time t later. It is subject to the initial condition: 0 0, 0 G t (19) DL DL DL DL 0 The analytical form of G DL DL, t depends on the model for reorientation, but it can usually be expanded in terms of Wigner rotation matrix elements(9). As a consequence, only correlation functions with j=j and p=p enter in Eq. 17, which greatly reduces the number of correlation functions that need to be computed as compared to macroscopically ordered systems(62). This is particularly useful in calculating the TPE induced fluorescence anisotropy which involves more terms(64, 65) Molecular information from fluorescence depolarisation The fluorescence depolarisation is conveniently quantified in the fluorescence anisotropy r(t), which can be obtained from experiments by measuring the polarised fluorescence intensities as was shown in Figure 6. The termi- 24

25 nology and the molecular information available from these experiments are summarised in the following section. The steady-state fluorescence anisotropy, r ss, is probably the most straightforward quantity to determine in a fluorescence depolarisation experiment since the emission is not discriminated with respect to time; t F t t r r dt F dt (20) ss 0 0 r ss is related to the time average of the angular correlation between the transition dipoles of absorption and emission during the fluorescence lifetime. The fundamental anisotropy, here denoted by r 0f, is the anisotropy in the absence of reorientation. r 0f gives information about the effective angle between the emission dipole moment direction and the absorption transition dipole moment direction in the molecular frame(66): r 0f 2 2 3cos (21) The angle is related to information about the symmetry of the excited vibronic states, since the transitions are expected to be polarised with respect to certain symmetry axes of the molecules. The limiting anisotropy r ss for fluorescent molecules immersed in a rigid solution, i.e. frozen or with high viscosity, should provide r 0f (67). A time-resolved fluorescence depolarisation experiment however, is often performed in order to obtain information about the reorientation dynamics of the fluorescent molecules. The measurement window is always limited by the lifetime of the excited state. This means that the dynamics information is only directly accessible if the reorientation, as described by an effective reorientation correlation time, is taking place on the timescale of the fluorescence lifetime, see Figure 8. 25

26 Figure 8. Schematic illustration showing the experimental conditions for a fluorescence depolarisation experiment of a fluorophore in solution. The fluorescence anisotropy r(t) is often analysed according to multiexponential decays interpreted in terms of local motion, mixed populations or different rotational diffusion rates around different molecular axes. A molecular interpretation of the fluorescence anisotropy requires a physical model for the molecular reorientations. The rotational diffusion model assumes that the molecules are reorienting due to Brownian diffusion, i.e. stochastically, as the result of many small angular steps due to collisions with neighbouring solvent molecules(68, 69). The solvent is treated as a homogenous medium, characterized only by its bulk properties, meaning the viscosity and temperature. The model is strictly valid only for large molecules immersed in a continuous solvent composed of smaller molecules. For a rigid rotor in solution, the fluorescence anisotropy r(t) will in the general case decay according to a sum of five exponentials(66, 70, 71); 26

27 t r t r i exp i (22) i 0, 1, 2 Where the pre-exponents r i depend on the molecular transition dipole directions. The rotational correlation times i are related to the rotational diffusion constants along the principal diffusion axes of the molecule. According to a modified hydrodynamic formalism(72), a single rotation correlation time can be related to solvent properties according to: V f C (23) kt B where, V, k B, T are the viscosity, the hydrodynamic volume, the Boltzmann constant and the temperature respectively. ƒ is a shape factor and C is a factor accounting for various hydrodynamic boundary conditions describing the solute-solvent interactions, e.g. stick or slip conditions. A plot of versus /T in a temperature or viscosity interval is a common test for the validity of the hydrodynamic model, and can also yield the hydrodynamic volume V. The rotational jumps model(73) assumes that the molecules reorient in random uncorrelated angular jumps. The time between the jumps are assumed to be Poisson distributed around an average value and the molecule is fixed between two successive jumps. In the high collision limit, the rotational jumps model for fluorescence anisotropy has the same analytical form as that of the rotational diffusion model(6). The intial anisotropy r(0) measured in an experiment is often lower than the fundamental anisotropy r 0f (1, 67). If the fluorescence is detected by means of TCSPC, the time resolution is usually nanoseconds to tens of picoseconds at the best. The apparent deficit in the initial anisotropy can be due to unresolved contributions of ultrafast processes such as dielectric relaxation, vibrational relaxation and also torsional vibrations of the fluorophore around its equilibrium geometry as well as fast inertial rotational motions (so called librations) in the solvent free volume or the solvation shell. By comparing r(0) with r 0f, information about the angular magnitude of these fast unresolved reorientations can be obtained indirectly(74). If the anisotropy reaches a plateau value different from zero, this means that the excited molecules are not completely randomised during the lifetime of the excited state. This value is often referred to as the residual anisotropy r. A non-zero r is commonly observed for fluorescence probes in macroscopically ordered structures and lipid membranes, or when attached to macromolecules like proteins or other biomolecules. This behaviour means that there are some rotational restrictions, which are indicative of local anisot- 27

28 ropic ordering. The residual anisotropy gives information on molecular order parameters(9) Fluorescence polarisation upon OPE and TPE The initially created orientation distribution of fluorescent molecules differs in the OPE and the TPE fluorescence experiments because of the different physical origins behind the probability of excitation, as was illustrated in Figure 5. The sharper photoselection for TPE and multi-photon excitation can result in a higher initial fluorescence anisotropy, as expressed in the following generalisation of Eq. 21(75): r i 0 2 2i 3cos i 1 (24) 2i 3 2 where i is the number of photons involved in the excitation process. The maximal anisotropy is for parallel absorption and emission dipoles, which amounts to r 1 (0) = 0.4 in OPE, r 2 (0) = 0.57 in TPE and r 3 (0) = 0.67 in threephoton excitation. As previously mentioned, a wide range of anisotropy values can in general be expected in the multi- and TPE experiments, even for the S 0 S 1 transition(48). There even exists TPE tensors T ~ which yield r = 0.61(64, 76). In an OPE experiment, two independent measurements of polarised fluorescence can be made, for example the perpendicular decay curves detected in the fluorescence anisotropy experiment, cf. Figure 6. The general expression for the intensity of a three-photon interaction with randomly oriented immobile molecules contains 15 independent parameters(77). In TPE of fluorescence with two identical exciting photons, four independent parameters can be determined which are related to molecular quantities(44, 78). A convenient choice is the fluorescence excited using linearly and circularly polarised light and detected by emission at parallel and perpendicular polariser settings(44). With circularly polarised excitation, the parallel direction best refers to the propagation direction of the beam, which is the symmetry axis of the photoselected ensemble of excited molecules(79). From the polarised decay curves, the fluorescence anisotropy using linear and circular excitation polarisation can be constructed, r l (t) and r c (t), respectively, cf. Eq. 14. The two set of experiments measures signals that depend on independent combinations of the components of the TPE tensor T ~. This opens the possibility of making a global analysis of the experiments which could also improve the possibility of analysing multiple exponential decays, which is an admittedly difficult task(2). Furthermore, in the case of angleresolved fluorescence anisotropy experiments, OPE can give information 28

29 regarding 2 nd and 4 th rank order parameters(62, 80). TPE can in principle provide additional information about 6 th rank order parameters(44, 64, 65). Another useful experimental quantity that can be determined in TPE is the ratio for isotropic emission using linearly and circularly polarised light, TP : TP c t l t S S (25) In this work, TP is called the two-photon polarisation ratio. This quantity is directly proportional to the ratio of the two-photon absorptivities upon circular and linear excitation light. In order to measure TP, the emission is detected at magic-angle polariser settings(44, 79). It should be noted that the one-photon absorptivity in solution is independent of excitation polarisation. This means that no information is available by measuring the corresponding OPE quantity, which is unity Donor-donor energy migration DDEM Donor-donor energy migration, DDEM, refers to the reversible non-radiative energy transfer between two photophysically identical molecules D A and D B, one of which is initially excited (*); D * A B A * B D D D. (26) DDEM is related to the irreversible donor-acceptor energy transfer, DAET, occurring between distinct donor and acceptor molecules. The latter process is also referred to as Förster resonance energy transfer, FRET. The electronic energy transport can be detected by measuring the fluorescence from the sample. Electronic energy transfer is commonly used in biophysics to determine distances between extrinsic or intrinsic chromophoric groups which are localised in macromolecule or supramacromolecular systems(1, 2, 81). The range of distances available from such studies is within Å, which is comparable to the size of many proteins or the thickness of a biological membrane. This means that the technique complements the atomic resolution available with the widely used X-ray and NMR spectroscopic methods. Advantages with fluorescence based techniques like DAET and DDEM are mainly the sensitivity and the potential to provide structural and dynamic information even from complex macromolecular systems. Unlike DAET, DDEM does not require specific labeling of the proteins, which is in practice difficult to perform. In DAET/DDEM, the energy transfer is mediated by the long-range dipoledipole coupling. Two weakly interacting donor and acceptor groups can be 29

30 linked to a rigid macromolecule, which then acts as a spacer. An example of a molecular system that fulfills the requirements of a DDEM system consists of a bisteroid to which two anthryl groups have been covalently linked(82), see Figure 9. H 3 C O CH 3 O CH 3 H 3 C OH H 3 C OH CH 2 P O O P CH 2 O O Figure 9. The chemical structure of bis(9-anthryl phosphonate) bisteroid (AnBsAn) is displayed together with the configuration angles. R is the distance between the two anthryl donor groups. The rigid structure of the bisteroid keeps the communicating anthryl groups at a well-defined distance, but they may undergo local reorienting motions. The extended Förster theory, EFT(82, 83) is the most complete and applicable description available for the electronic interaction between donordonor or donor-acceptor pairs. The excitation probabilities for the primary (p) and the secondary (s) excited donor group is given by: 30

31 p s p 1 2 t 1exp 2 t' dt' F t 1 2 t 1exp 2 t' dt' F t s t t F t D t 0 D t D (27) 0 In Eqs. 27 the angle brackets... stand for a stochastic average, implying that the excitation probabilities are dictated by the stochastic long range dipole-dipole coupling(84). F D (t) is the fluorescence relaxation of the donor molecule in the absence of energy migration. The stochastic rate of energy migration is described by (t), and it depends on the spectral overlap of the emission and absorption bands, as well as the distance and the mutual orientation of the two donor molecules. Therefore, the rate also depends on the molecular reorientations. For sufficiently small time intervals when the molecules can be regarded as stationary, the rate can be described by the classical Förster theory(85). Moreover, Eqs. 27 show that the total excitation probability due to energy migration is conserved. Hereby the observed fluorescence intensity becomes directly proportional to F D (t). Thus, unlike in DAET, the photophysics relaxation of the donors becomes insensitive to the DDEM process. However, since the DDEM process predominantly takes place between differently oriented fluorophores, it causes fluorescence depolarisation. An analysis of the fluorescence anisotropy of a DDEM system can be used for providing information about the distance of the two interacting chromophores as well as internal configuration angles(82, 84). In the presence of molecular reorientations, the expressions are rather complex and molecular dynamics simulations are required for the analysis of data. In this context Brownian dynamics and Monte-Carlo simulations can be used(86-88). Information from experiments on a mono- and a bis-chromophoric system, cf. Figure 9, are analysed in a global manner(88). The recovery of molecular parameters from data with a reasonable accuracy is a non-trivial problem(83, 87) Electronic energy transport upon TPE TPE-FRET has been used for instance in biological imaging applications on living cells and for bioassays(89-93). TPE-FRET has been shown to give potentially higher energy transfer efficiency, less background, and a deeper imaging depth than the confocal OPE-FRET. A few applications of TPE-DDEM exist, where the aggregation of fluorophore labeled proteins have been studied(94). However, no theory is yet available that describes electronic energy 31

32 transport upon TPE in the presence of molecular reorientation. The Paper V presents a unified theoretical description of OPE and TPE fluorescence depolarisation and DDEM in the presence of molecular reorientations. Preliminary experiments on a model system, cf. Figure 9, are also reported in this thesis. 32

33 3. Experimental methodology The experimental set-up of the TCSPC experiment used in this work for obtaining TPE and OPE polarised fluorescence decays is pictured in Figure 10. Figure 10. A schematic representation of the experimental set-up of the TCSPC experiment used in this work for measuring the TPE polarised fluorescence decays. 33

34 The excitation source for the TPE experiment was a 200 khz femtosecond laser amplifier system(coherent radiation), operating at 800 nm. The system consists of an 18 W cw diode laser pumped titanium-sapphire mode locked laser (Mira), which is used to seed a titanium-sapphire regenerative amplifier (RegA 9000). The central operating wavelength is conveniently tunable from 760 nm to 840 nm. The average power after the amplifier was up to 1 W at 200 khz, thus producing pulses of up to 5 J pulse energy. The pulse width was determined by the autocorrelation trace to be around 200 fs. For the OPE experiments, the laser output was frequency doubled by directing the beam into a Coherent 9400 OPA optical parametric amplifier which is equipped with a -barium borate crystal (BBO) for this purpose. The horizontally polarised output light from the laser amplifier was rotated to vertical polarisation by means of two planar mirrors. The polarisation of the exciting beam was defined by passing the beam through a Glan Taylor prism polariser, and could further be controlled by a /4-plate for obtaining circularly polarised light. A reflective variable neutral density filter was used to control the laser intensity with minimal influence on the laser pulse shape. In order to achieve the high photon flux densities necessary for TPE the laser beam was focused by using a 50 mm focal-length microscope objective, which produces a focal spot size of around 30 m. The time-resolved fluorescence decay was measured using the standard procedure of the TCSPC experiments, outlined in the Section The laser beam was focused into the central part of a 10x10 mm fused silica cuvette containing the sample. The cuvette holder was thermostatted within 1 ºC. The emitted light was collected at a right angle through a 2 mm diameter pinhole spatial masking in order to confine the observation to the region of maximum TPE fluorescence. Due to the localised nature of the two-photon effect, this procedure causes only a small reduction of the collection efficiency of the fluorescence light but should minimise artifacts resulting from e.g. time spread across the excitation beam, second-harmonics generated at and within the walls of the cuvette, scattered excitation light, and interference of possible emission from the cuvette walls(34). The emitted light was further collimated with a quartz lens of 50 mm focal length, led through optical filters and passed through a sheet polariser mounted in a motorised holder, before finally impinging on the detector. Filters are superior for isolating the wavelength of the fluorescence in a TPE experiment compared to monochromators; filters have a higher transmittance efficiency, which is also relatively independent on light polarisation, i.e the G-factor is near unity, cf. Eqs A bandpass filter was used to isolate the emission. In addition, a long-pass filter was also used, since it is necessary in order to eliminate excitation light contamination of the decays in the OPE experiments. The long-pass filter also eliminates the possibility that second harmonic light may influence the decay curve in the TPE experiments. 34

35 In order to achieve the condition that mimics isotropic emission, which is needed for measurements of the fluorescence lifetime and the TPE polarisation ratio TP, the emission polariser was set to the magic angle (i.e., 54.7 to the vertical for the linear excitation experiment and 35.3 to the vertical for circularly polarised excitation light). In the fluorescence depolarisation experiments, the collection was automatically halted every 90 seconds and the emission polariser rotated between the parallel and perpendicular settings. This procedure minimises the influences of long time drifts in the performance of the laser, which is particularly important in TPE fluorescence experiments; the signal intensity is proportional to the square of the laser intensity for TPE fluorescence. The detector consisted of a microchannel plate photomultiplier (MCP-PMT, Hamamatsu R3809U-51), cooled to -40 ºC. The output from the photomultiplier was amplified with a fast rise time preamplifier (Ortec VT120) before being shaped in a pico-timing discriminator (Ortec 9307). This signal provides the stop pulse to the time-to-amplitude converter (TAC, Ortec 566), which registers the time separation of the excitation and emission events in TCSPC. The start pulses were generated by a silicon photodiode monitoring a fraction of the excitation laser beam. These pulses were also sent through a constant fraction discriminator (Tennelec 454), before triggering the start pulse of the TAC. The analogue output voltage signals of the TAC were digitised and stored in a multi-channel pulse-height analyser (MCA, IBH consultants). In order to achieve reasonable statistical precision, a typical experiment was run until counts were collected in the peak channel for the magic angle or D(t) curve. The IRF for analysing the TPE experiments was measured using the nonlinear signal obtained by replacing the sample with a suspension of colloidal gold nanoparticles, cf. Paper I. The width of the IRF was typically 40 ps full width at half maximum. The fluorescence anisotropy and lifetime decay functions were obtained from the polarised decay curves using this IRF, according to the iterative reconvolution procedure outlined in the Sections and A modified Levenberg-Marquardt algorithm for non-linear least squares fits was mainly used in this work to perform the data fitting. The quality of the fits was determined by its reduced 2 -value and the Durbin-Watson parameter. Inspection of graphs of the weighted residuals and autocorrelation function of these residuals was also used. In the analysis of DDEM in Paper V, a genetic algorithm was also employed for the data fitting. 35

36 4. UV-VIS spectroscopic properties of perylenes Different perylenes are mainly used in this work to investigate TPE fluorescence depolarisation. The molecular structure and the absorption and emission spectra of perylene is shown in Figure 11. Perylene belongs to the class of the polycyclic aromatic hydrocarbons. It is an effectively planar molecule with a rotor shape approximating that of a disc. The spectroscopic properties of perylene are well-studied, both in the gaseous and condensed phase. The absorption spectrum of perylene in the UV-VIS region exhibits two main electronic transitions with resolved vibronic structure, centered around 255 nm and 435 nm. These transitions are orthogonally polarised. The S 0 S 1 transitions for perylene in solution exhibit mirror image symmetry and a small Stokes shift of about 200 cm -1. This suggests that the nuclear configuration is very similar in the ground and first excited singlet states, and furthermore that the photophysical pathways of excess vibrational energy relaxation proceed smoothly to the lowest vibrational level of S 1. 1 Normalised absorption/emission 0, Wavelength, nm Figure 11. The molecular structure of perylene is shown together with the absorption and emission spectra of the S 0 S 1 transition of perylene in squalane. 36

37 The absorption and emission spectra are fairly independent of solvent polarity. The fluorescence lifetime of perylene in solution is around 5 ns, with a quantum yield for fluorescence of about 0.9. The fundamental anisotropy of perylene is r 0f = 0.37 and quite independent on excitation and emission wavelength across the lowest energy absorption band(67, 95). This r 0f value is close to the theoretical maximum of 0.4 and suggests that the absorption and emission oscillators are nearly collinear. Rotational dynamics studies of perylene in different solutions have been studied in a number of laboratories. In most cases, two exponentials were resolved in the anisotropy decay, see for exemple (7, ), i.e. r t r exp t r exp t (28) This behaviour of the anisotropy has been interpreted in terms of rotational diffusion constants D and D corresponding to rotational motions spinning in the plane of the molecule and tumbling out of the molecular plane; 1 2 4D 1 0 6D 2D (29) The in-plane rotation D has been estimated to be 6-20 times faster than D. Consequently two pre-exponentials r 2 and r 0 in the TPE fluorescence anisotropy decays can potentially be resolved and give independent information about transition moment directions. The emission oscillator is expected to be parallel to the molecular long axis X D. This restricts the analysis of TPE fluorescence anisotropy to the component of the TPE transition tensor T ~. Perylene should therefore constitute a useful molecular probe for an investigation of reorientation dynamics following TPE. We have further studied two structural analogues of perylene, namely 1,7- diazaperylene (DPe) (101) and 2,5,8,11-tetra-tert-butylperylene (TBPe) (74). N N Figure 12. The molecular structures of DPe (left) and TBPe (right). 37

38 DPe and TBPe are expected to exhibit slightly different reorientation dynamics compared to perylene, whereas the OPE S 0 S 1 transition dipoles are still largely long axis in-plane polarised. The aromatic core of DPe has two nitrogen atoms that allow hydrogen bonding to protic solvents. In non-polar solvents it behaves very similar to perylene. In polar solvents, there is a larger red shift between the absorption and fluorescence spectra and broader vibronic peaks than for perylene, indicating specific solvent interactions likely due to the nitrogen atoms. TBPe on the other hand has bulky tert-butyl groups in the corners of the perylene skeleton. The spectra show a less pronounced vibrational structure. DPe in protic solvents as well as TBPe are expected to have an impeded reorientational freedom in the plane of the molecule as compared to perylene and consequently two less well-separated rotational correlation times. As a means to assess the validity of the simple models used to describe the reorienting motions, experiments were performed in different solvents and within a small temperature interval of 7 21 ºC. In order to resolve the reorientation dynamics, solvents with a quite high viscosity were used. One aim of the studies was to use the time-resolved fluorescence depolarisation for the quantitative determination of the TPE tensor components of the three perylenes for the transition involving two identical 800 nm photons provided by the titanium-sapphire laser system. Point group symmetry arguments enable a fruitful starting point for this discussion, as briefly discussed in the Section The S 0 S 1 transition is assumed to be an inplane * transition. The molecular symmetry is with respect to the valence electrons well approximated by a D 2h point group for perylene and TBPe. DPe has a somewhat lower symmetry, C 2h. Planar aromatic molecules are expected to have mixed two-photon transitions, meaning that diagonal and off-diagonal excitation pathways in the TPE tensor could contribute. The existence of a point of inversion symmetry in the molecular structure for the perylenes implies that the two-photon S 0 S 1 transition is parity forbidden. Nevertheless, transitions are allowed by e.g. intensity borrowing from allowed electronic states via vibronic coupling to ungerade vibrations. Such complex excitation pathways make the task of determining absorption transition tensor components challenging. The fluorescence TPE spectrum between 570 nm and 770 nm of perylene in solution was measured by Yu et al. (102). The one-photon absorption and the TPE spectra look markedly different, as can be expected from the different selection rule for OPE and TPE. The S 1 state appeared only weakly in their TPE spectrum and reveals a peak around 755 nm based on a false origin. The value of the polarisation ratio TP approaching 1.5 above 755 nm indicates that the TPE transition tensor is mainly composed by off-diagonal elements. 38

39 5. Discussion of results 5.1. Improving the response function for analysing TPE fluorescence depolarisation experiments Paper I discusses experimental development for making meaningful timeresolved experiments using TCSPC detection of TPE fluorescence. Research in this field was initiated by David Birch s laboratory in the mid 90 s, using similar equipment as in the work presented in this thesis. Still when analysing the fluorescence decay data from our lab, severe misfits were obtained. The standard deconvolution procedure, cf. Eq. 9, inherited from the OPE experiments uses an IRF collected from the Rayleigh scattering RS, of the excitation light from a cuvette filled with a dilute aqueous suspension of colloidal silica (Ludox). It was then suggested that the analysis of nonlinearly excited fluorescence decays requires a response function obtained from a non-linear optical process. Such an IRF could be obtained from a water suspension of colloidal gold nanoparticles. These samples were reported to exhibit a strong hyper-rayleigh scattering HRS(103) at half the laser wavelength, cf. Figure 1f. Indeed, this procedure yielded better analyses of TPE fluorescence decay data(104). Paper I describes the consequences of using IRFs obtained from the RS and the HRS of colloidal gold nanoparticles in order to analyse the TPE fluorescence population and anisotropy decays. The model system consisted of the scintillator POPOP, 1,4-bis-(5-phenyl-oxoazol-2yl)-benzene, dissolved in ethanol and n-octanol liquid solutions. The experiments were accompanied by synthetically generated TCSPC data of model decay laws. We were interested to discover the effect of using the different IRFs on the result of the fluorescence anisotropy decay parameters that are obtained from analysing the polarised decay histograms. Being defined as a difference, the anisotropy has a lower signal to noise ratio and is probably more sensitive to systematic errors than the fluorescence lifetime decays that were investigated before(104). Typical non-linear and linear Rayleigh scattering IRFs from colloidal gold nanoparticles as measured in the laboratory are pictured in Figure 13. The histogram profile of the non-linear IRF was somewhat narrower (29 ps compared to 35 ps approximated full width at half maximum), it had faster fall and rise times. The most obvious feature (cf. around 16 ns in Figure 3a in 39

40 Paper I), is the absence of the leakage pulse from the cavity dumper in the laser amplifier Non-linear IRF Rayleigh scattering Counts Time, ps Figure 13. Linear and non-linear instrumental response functions IRFs obtained from a suspension of colloidal gold nanoparticles. Approximated full width at half maximums are 29 ps for the non-linear IRF and 35 ps for the linear RS. Solely rotations are responsible for the fluorescence depolarisation of POPOP in solution. The anisotropy could be well described by an exponential decay function with a single rotational correlation time. The excited state population of POPOP in ethanol is completely randomised within a nanosecond, (= ns), which must be considered fast with respect to the timeresolution offered by TCSPC. Indeed, by constructing 2 -surfaces, cf. Figure 14, we showed that the analyses using the non-linear IRF gave a more welldefined initial anisotropy. 40

41 1,5 1,4 Rayleigh scattering Hyper-Rayleigh scattering 2 / 2 min 1,3 1,2 1,1 1 0,45 0,50 0,55 0,60 0,65 0,70 Anisotropy, r(0) Figure graphs monitoring the superior precision of the HRS-IRF ()over the RS-IRF () in retrieving the initial anisotropy r(0) of POPOP in ethanol. Indeed, r(0) = 0.51 ± 0.03 for the HRS-IRF vs < r(0) < 0.7 for the RS- IRF. The value obtained using the HRS-IRF also matched the value r(0)= 0.52 obtained in n-octanol, where the anisotropy decayed slowly enough to be well-resolved ( = 1.6 ns). The experimental findings were supported by reanalysing the synthetic TCSPC data. Single exponential decay laws r(t) = 0.51exp(-t/) and F(t) = exp(-t/1.33 ns) mimicking POPOP were used to construct the polarised fluorescence decay curves by convolution with the experimental HRS-IRF, and addition of Gaussian noise, cf. (104). The reanalyses showed that the RS-IRF could not reproduce the correct anisotropy decay laws when the rotational correlation time was shorter than the fluorescence lifetime = 1.33 ns. This means that the two IRFs are not compatible with each-other, which could be suspected from a visual inspection of their histogram profiles, cf. Figure 13 and Figure 2 in Paper I. In conclusion, Paper I demonstrates that the non-linear IRF yields significantly better statistics for retrieving the anisotropy decay parameters in a TPE fluorescence depolarisation experiment. Using the proper IRF is probably even more important when it comes to resolving complex anisotropy decay laws, involving for example multi-exponential decays. 41

42 Extension of the work on the non-linear IRF Linear and non-linear optical signals from nanosized noble metal colloidal suspensions have received a lot of attention lately, see e.g. (15, ). The origin of the non-linear signals emitted from the noble metal nanoparticles is still debated, but the effects probably involve surface plasmon oscillations. These are the result of the collective excitation of conduction band electrons near the surface of the nanoparticle. As is discussed in the cited references, the noble metal nanoparticles and their aggregates exhibit unique optical, chemical and physical properties that display a sensitive dependence on the size, geometry and surface environment of the nanoparticles, see e.g. (115, 118, 120, 124). Notably they can strongly concentrate an external electric field locally which can magnify the optical signals. These features make them exceptional candidates for e.g. chemical characterization, biosensing and imaging. It should be particularly promising for the enhancement of non-linear effects that are inherently weak. We noted early, as mentioned in my diploma work(125), that the signal obtained from exposing the gold nanoparticle suspension to the focused femtosecond 800 nm laser was not merely at the second harmonic wavelength, but over most of the visible region of the spectrum. Indeed, we have now studied multi-photon absorption induced luminescence (MAIL) from a number of different gold colloid suspensions, including spheres of radius 3.5 nm, 5 nm and 15 nm spheres, and rods of some different aspect ratios. Using a 400 nm laser, no luminescence is detected. However, when exposed to the focused 800 nm femtosecond laser, a whitish light in the focal region of the sample is even visible to the eye. The signal has a non-linear intensity dependence on the excitation light. The emitted light was recently characterised using streak-camera detection operating in the synchroscan mode (Hamamatsu C5680). These experiments are still unpublished, but the features agree with published data(111, 113, 126, 127). The emission has a broad spectral peak, which appears to be prompt in time. In Figure 15 are pictured the extinction spectrum and the MAIL spectrum obtained from a suspension of 15 nm gold spheres. 42

43 1 1 0,4 MAIL intensity 0,8 0,6 0,4 0,2 signal Time, ps 0,3 0,2 0,1 "Absorbance" in 10 mm cuvette Wavelength, nm Figure 15. Extinction spectrum and MAIL spectrum from a suspension of gold nanoparticle spheres of 15 nm radius. The MAIL spectrum is integrated with respect to time. Inset: The time-resolved MAIL signal at 480 nm (thin line) compared to the Rayleigh scattering at 800nm (bold line). The full widths at half maximum of these signals are 9.4 ps and 12 ps, respectively. The MAIL spectrum reported in Figure 15 is uncorrected for inner filter effects, the detector spectral response and the emission filter transmission profile (a 2 mm Schott BG42 filter). Back-calculating the filter absorption broadens the spectrum slightly and shifts it about 20 nm to lower wavelengths. The MAIL spectrum is similar for gold nanoparticles of different size and shape, having a broad emission ranging from 350 nm to 620 nm. The centre of gravity wavelength is around 480 nm, varying slightly with the geometry of the particles. We note the lack of an HRS peak around 400 nm. Probably, the signal observed with the TCSPC detection system reported in Paper I at the second harmonic wavelength was from the slope of the broad spectrum shown in Figure 15. Furthermore, the MAIL signal appears instantaneous over the whole spectrum. The time profile at the centre wavelength is shown in the inset of Figure 15, together with the RS signal measured at 800 nm, which represents the time-resolution of the experiment. The two profiles have a very similar appearance. The full width at half maximum is 9.4 ps for the MAIL and 12 ps for the RS. Femtosecond studies of the gold nanoparticle MAIL suggest that the luminescence is occurring within 50 femtoseconds of the laser pulse excitation(128). With this new information, we can reconsider the origin as to why a nonlinear IRF is more appropriate to analyse the fluorescence decays which are obtained using a non-linear excitation process. The validity of the convolu- 43

44 tion integral Eq. 9 requires that the IRF and the fluorescence are measured under the same experimental conditions of distortions(56). I propose that the MAIL from the gold nanoparticle suspension provides an ideal IRF for analysing time-resolved multi-photon excited fluorescence decays. Indeed, the results and conclusions from the previous publications(71, 104) on using a non-linear IRF to analyse TPE fluorescence decays still hold, albeit the origin of the signal may be different from what was first assumed. The gold MAIL is prompt in time over a broad spectral range, covering at least nm. In fact, the emission wavelength of gold nanoparticles is widely tunable by changing the shape and size of the nanocluster(115, 118, 119). Consequently, the gold MAIL allows the experimentalist to monitor the IRF at the same wavelength as the fluorescence, thereby avoiding the colour effect the photomultiplier(50). In order to correct for the wavelength dependence of the IRF in a single-photon timing experiments, it is common to use the reference convolution method(56). This procedure requires singleexponential references(129), which has to be measured under identical instrumental settings as used for the sample. These references are to date not systematically catalogued for TPE fluorescence. A second advantage is based on the geometrical considerations. Both the gold MAIL and the TPE fluorescence, being non-linear signals with respect to the excitation photon flux, are occurring from a localised region in the focal volume. Moreover, by observing the IRF and the fluorescence at the same wavelength, the same optical filters or monochromator settings can be used to isolate the emission. Consequently, the gold MAIL and the TPE fluorescence are taking the same optical path from the excitation volume towards the detector, thereby illuminating the same area on the photomultiplier. Furthermore, unlike scattering, the MAIL is not completely polarised. An emission anisotropy of r ss = 0.27 using linearly polarised excitation light was measured for the system reported in Figure 15. The detector response can be polarisation sensitive. If the anisotropy is determined from analysing directly the experimental polarised decay curves instead of the constructed sum and difference curves, an IRF with the corresponding polarisation settings can easily be measured with the gold MAIL. 44

45 5.2. Theoretical model for TPE fluorescence depolarisation in liquid solution Paper II is a theoretical paper calculating the time-resolved fluorescence anisotropy obtained upon TPE, using a single exciting beam, of a general anisotropic rigid rotor immersed in liquid solution. The reorienting motions considered were rotational Brownian diffusion as well as unresolved ultrafast restricted reorientations, i.e. librations. The spherical tensor approach was applied to calculate the anisotropy. This idea is not new, and has been used both to obtain OPE(9, 58, 59) and TPE fluorescence anisotropy(44, 64, 65, 70). As was mentioned before, the spherical tensor formalism is very useful for calculating orientation averages since they transform under coordinate change and rotations in a straightforward manner, using the properties of Wigner rotation group matrices(63). It was probably a necessary theoretical language for the development of models for polarised TPE spectroscopy in the presence of molecular reorientation, which was not considered in the direction cosine approach used in the 70 s by McClain(78). Wan and Johnson(44, 70) made the first calculation of two-photon excited fluorescence anisotropy decay in the rotational diffusion regime. However, the general results are not simplified with respect to the physical parameters involved. The other references(64, 65) focus on the properties of TPE fluorescence depolarisation in terms of general orientation correlation functions. The focus in the latter references is on molecularly ordered systems, which require a more involved description, cf. the discussion following Eqs Also, without a physical model for the molecular reorientation, these expressions are not straightforwardly applied in the quantified analysis of experimental data. In the paper II, the expressions for the TPE fluorescence anisotropy following linear and circular excitation light are rederived. Only the terms relevant for a macroscopically isotropic system, like molecules in liquid solution were considered. The resulting expression for the anisotropy in the rotational diffusion regime is given in the article. The analytical form is a sum of five exponentials, just like in OPE(66), cf. Eq. 22. The general expression for the anisotropy was brought to its simplest form in molecular Cartesian coordinates in the diffusion frame, which are familiar to experimentalists. The expression is a function of 12 variables, namely the six components of the symmetrical two-photon absorption tensor components relevant for TPE of two identical photons, the three components of the transition dipole moment for emission, and the three components of the principal diffusion axes system of the molecule. The final form of the general anisotropy decay shows a strong analogy to the corresponding OPE expression derived by Chuang and Eisenthal(66). It is not surprising that the fluorescence anisotropy has the same analytical form in both OPE and TPE. This comes from the fact that fluorescence in solution detects only orientation correlation functions of 45

46 irreducible second rank components. This point is further discussed in the context of Paper V where a proper irreducible representation of the twophoton interaction is presented. The complete expression for the two-photon polarisation ratio TP is also derived. This is an isotropic quantity that is a function of the absorption tensor components only. It was also pointed out how TP can be measured in a straight-forward manner by using magic angle settings of the polarisers. We further allowed the possibility of the diffusing molecules to undergo ultrafast unresolved restricted reorienting motions, i.e. librations. In the separation of time-scales, the transition dipole moment directions in the diffusion frame are hence taken from a distribution of angles(6). The polarisation ratio TP is not affected by librations, since it is an isotropic quantity. However, the librations cause a decrease in the anisotropy pre-exponents r i. This is manifested by an apparent decrease of the initial anisotropy, as is illustrated in the figures presented in the article. As an example, Figure 16 pictures the anisotropy decays upon linearly and circularly polarised TPE of a molecule where the transition is governed by a tensor that consists of a purely molecular T XX polarised component parallel to the emission transition dipole. The anisotropy is plotted for some different maximal libration half-angles MD describing the angular freedom of the molecular frame M with respect to the diffusing frame D. As can be seen in Figure 16, the effect is most visible around time t = 0. Figure 16. Two-photon excited fluorescence anisotropy in the presence of fast unresolved reorientation, i.e. librations. The anisotropy using linearly {r l (t) 0} and circularly {r c (t) 0} polarised light is predicted for different Eulerian maximal halfangles of fast unresolved reorientation. The molecule is an oblate with the diffusion tensor components D Z = 10D X = 10D Y. The emission transition dipole is directed along the molecular X-axis and in the TPE tensor only T XX is non-zero. 46

47 In the case of the perylenes treated in this thesis, the ultrafast reorienting motions should occur mainly in the plane of the molecule. Under this assumption, the pre-exponent r 2 in the anisotropy decay, cf. Eq. 28, will decrease by the factor ƒ: sin f (30) MD MD MD where MD is the maximal angular displacement around an axis perpendicular to the molecular plane. The pre-exponent r 0 is unaffected. This result means that the effect on the TPE anisotropy decay is identical to the OPE expression(74). Such a model is used in OPE for explaining the observed deficit of the initial anisotropy from the fundamental anisotropy, or from values expected from symmetry considerations of the molecules. We are not aware of any previous treatment in the literature of the influence of fast unresolved restricted reorientation on TPE fluorescence anisotropy. 47

48 5.3. Rotational dynamics and the symmetry of twophoton excited states as determined by timeresolved fluorescence depolarisation Global analysis scheme Paper III outlines a procedure for obtaining the relevant molecular quantities from time-resolved polarised TPE fluorescence experiments in liquid solution. The analysis is based on the theoretical models developed in the Paper II. In order to assign the relative magnitude of the TPE tensor components, it is not enough to measure merely the anisotropy measured using linearly polarised excitation light. Several measurements with independent information about tensor components are needed. The idea presented in Paper III is to analyse globally the results from three set of experiments using linear and circular polarised excitation light, namely the two anisotropies r l (t) and r c (t) and the TPE polarisation ratio TP. The procedure can be illustrated with the following flow scheme; Figure 17. A flow scheme overview for the global analysis of several OPE and TPE fluorescence depolarisation experiments which have been applied to determine the two-photon absorption tensor components (T XX, T XY, T YY ). See text for further details. 48

49 In order to calculate the tensor components, the three set of experiments are analysed in the space of tensor components directly, assuming that the same tensor T ~ describes all the three set of experiments. The anisotropy decays are assumed to be described by a bi-exponential decay, cf. Eq. 28, and possibly also influenced by ultrafast unresolved restricted reorientations. As long as the emitting states in OPE and TPE experiments are identical, the reorienting motions should be the same. Therefore, the rotational correlation times 2 and 0 determined from the TPE experiments were verified to match those in the OPE anisotropy decay. For the analysis of the TPE experiments, the magnitude of the ultrafast restricted reorientations are taken from the OPE experiments. The maximal libration half-angles for these unresolved motions was readily determined from a comparison of the OPE fundamental anisotropy and the measured initial anisotropy, vide infra. The libration angles and the TPE tensor components correspond to values of the pre-exponents in the anisotropy decays and the TPE polarisation ratio TP according to equations described in Paper II. Furthermore, the rotational correlation times 2 and 0 were globally linked in the analyses of r l (t) and r c (t). The resulting TPE tensor components were determined by searching for the best statistical fit according to the minimization of the global reduced 2 - value of the r l (t) and r c (t) set of experiments. The obtained tensor components were rejected if they didn t satisfy the experimentally determined value of TP within uncertainty (±0.03) The case of perylenes in liquid solution Perylene, DPe and TBPe were studied by the different OPE and TPE fluorescence polarisation experiments outlined in Figure 17. Laser light of 400 nm was used for the OPE and 800 nm for the TPE. In order to assess the rotation dynamics, the perylenes were investigated at different temperatures and in different solvents, which are viscous enough so that the anisotropy decays are well-resolved using the TCSPC detection technique. These experiments are presented in Paper IV. Fluorescence depolarisation experiments for the S 0 S 1 transitions using linearly polarised OPE and TPE are illustrated in Figure 18. The graphs in this figure show data following linearly polarised excitation light for perylene in 1,2-propanediol at 287 K. The TPE experiments were also performed with circularly polarised excitation light, as illustrated in Figure 19, and the TPE polarisation ratio was determined. The anisotropy data obtained for the different perylenes are presented in Table 1 and Figure

50 Figure 18. Fluorescence depolarisation data obtained from OPE and TPE of perylene in 1,2-propanediol at 287 K. The solid line in the TPE difference curve D(t) (upper panel) shows the best fit of the theoretical model with data, see text. The constructed anisotropy decays r(t) are given in the lower panel, with TPE giving a higher anisotropy. The instrumental response function is indicated in the centre graph below the D(t) decay. The best fits obtained for the OPE and TPE anisotropies are given by: l OPE l TPE 0.215exp 0.44 ns 0.1exp 2.7 ns exp 0.44 ns0.143exp 2.7 ns. r t t t r t t t 50

51 Figure 19. Fluorescence depolarisation data obtained from TPE using linearly (l) and circularly (c) polarised light of perylene in 1,2-propanediol at 287 K. The solid line in the difference curve D(t) (upper panel) shows the best global fit of the theoretical model with data, see text. The constructed anisotropy decays r l (t) > 0 and r c (t) < 0 are given in the lower panel. The best fits are: 0.223exp 0.43 ns0.143exp 2.7 ns exp 0.43 ns0.097exp 2.7 ns. l r t t t c r t t t 51

52 Rotational correlation times [ns] ,1 0,2 0,3 0,4 0,5 /T [cp K -1 ] Rotational correlation times [ns] ,1 0,2 0,3 0,4 0,5 /T [cp K -1 ] Rotational correlation times [ns] ,1 0,2 0,3 0,4 0,5 /T [cp K -1 ] Figure 20. Rotational dynamics of perylene (upper) DPe (centre) and TBPe (lower) graph: rotational correlation times 2 (empty symbols) and 0 (whole symbols) versus / T in solutions of 1,2-propanediol -, n-octanol - and squalane-. 52

53 Table 1. r(0)-values for perylene (Pe), 1,7-diazaperylene (DPe) and 2,5,8,11-tetratert-butylperylene (TBPe) in different solvents. Linear (l) and circular (c) excitation light was used for TPE at 800 nm. Linearly polarised 400 nm light was used in the OPE experiments. TP is the polarisation ratio for isotropic emission. The maximum angle for fast unresolved restricted reorientation MD is also given. As can be seen in Figure 18, the OPE and TPE fluorescence anisotropy decays could be described by the same two rotational correlation times, confirming that the reorientation dynamics was similar in the OPE and TPE cases. This also verifies that focusing the intense 800 nm beam causes negligible local heating in the solution. There was no temperature dependence of the initial anisotropy values reported in Table 1. The linear dependence of rotational decorrelation times for the three perylenes upon / T in Figure 20 shows that the reorientation motions are in qualitative agreement with hydrodynamic theory, cf. Eq. 23. On the otherhand, there was a clear solvent dependence of the initial anisotropy r(0) and the TP. As can be seen from Table 1, a low initial anisotropy correlated with a large solvent free volume; therefore it was attributed to ultrafast restricted reorientation. For the example perylene in 1,2- propanediol, the OPE initial anisotropy r(0 )= 0.31 was obtained. This result is significantly lower than the expected r 0f = The corresponding maximal libration angles, MD = 23º, could be calculated from Eq. 30. The same reorientation motions are expected to be present also in the TPE case. So if one would calculate the TPE tensor components from the anisotropy preexponents obtained directly from curve-fitting of the experimental curves, one would end up with the wrong values. For perylene in 1,2-propanediol, the TPE measured initial anisotropy value of r(0)=0.37 and MD =23º give an estimate of r 0f = 0.41 for the TPE fundamental anisotropy. For the other systems, the anisotropy is higher in the TPE case, ranging from r(0)=0.47 for DPe in glycerol. These anisotropy values are significantly lower than the r = 0.57 expected for a pure long-axis polarised TPE transition. It is also significantly lower than what has been obtained from several other common 53

54 dyes(34, 48). The low initial anisotropy value indicates that the TPE is in fact a mixed transition. This was not unexpected since S 0 S 1 in TPE is parity forbidden. In the Paper III, it is shown how to calculate the anisotropy for a mixed transition using the additive property of the anisotropy(1). This behaviour could straight-forwardly be incorporated in the global analysis scheme pictured in Figure 17. The resulting analysis demonstrated that the TPE process can be described by an in-plane two dimensional T, for which the components exhibit mixed vibronic character. The relative tensor components which were compatible with the experimental results within the experimental uncertainty were mapped out, see Figure 21. Figure 21. Domains of relative tensor element values T XX / T YY and T XY / T YY corresponding to a successful description of the TPE polarisation experiments. Results are shown for 1,7-diazaperylene (upper three domains) and perylene (lower three domains). The solvents are common for the two molecules and indicated by the filling pattern; 1,2-propanediol (horizontal); n-octanol (vertical); squalane (opaque);. The best fits for each solution correspond to values in the middle of the respective domains along the line with a fixed value for T XX / T XY. The obtained values of the relative tensor components from the global analysis give information about the symmetry of the vibronic states involved in the transition(45). Both A g - and B 1g -transitions involving diagonal and offdiagonal tensor components respectively are contributing, with the diagonal A g transitions dominating. A Herzberg-Teller analysis of vibronic coupling relevant for TPE(47, 130), revealed that the TPE transition can be explained by vibronic coupling with vibrational modes of at least two different symmetries. Since T XX -components are dominating, the first excited electronic state 54

55 of perylene likely acts as the main intermediate state in the TPE process at the studied wavelength. An important finding was that the TPE tensor appeared to be solvent dependent. As was shown in Figure 21, the range of values of the tensor components obtained for both perylene and DPe in squalane and the polar solvents overlap only in the boundary region Future studies inspired by Papers III-IV The analyses in Papers III-IV show that the TPE transition induced with an 800 nm laser in perylene is of mixed character with mainly components along the molecular long axis. On the other hand, according to the spectrum of the TPE polarisation ratio measured by Yu et al. at wavelengths slightly below 800nm(102), TP = 1.5 which corresponds to r l (0) = 1/7. Neither of these values is in agreement with the values presented in Papers III-IV, cf. Table 1. The two different studies indicate that there should be a sensitive dependence on wavelength for the TPE polarisation parameters TP and r(t) in the region corresponding to the parity forbidden S 0 S 1 TPE transition. An experimental study could assess the excitation wavelength and solvent dependence of the fluorescence anisotropy and the TPE polarisation ratio. Such a study could also provide a test for recently developed methods in quantum chemical calculations(131, 132). It would further be interesting to assess the picosecond dynamics of the anisotropy decays in solution. Such a study could aim to discover the origins of the ultrafast reorientations of the perylenes in solution that were predicted from the solvent dependence on the polarisation parameters presented in Paper IV, cf. Table 1. Such experiments have been performed using OPE fluorescence detected by means of optical gating, using e.g. the fluorescence up-conversion technique(98, 99). Femtosecond components have been discovered that supposedly make up for the deficit of the initial anisotropy for perylene from 0.4. Librations(98) and vibronic coupling(99) have been suggested as the main reasons for the ultrafast component of the anisotropy decay. A femtosecond study that addresses the dependence on solvent and excitation energy of the fluorescence anisotropy is required. Unfortunately, today s detection techniques for femtosecond dynamics are not sensitive enough to perform TPE depolarisation experiments with a reasonable quality of data. If they will be in the future, the procedure presented in this thesis of analysing globally experiments using different polarisation set-ups of the excitation and emission light will be rewarding. Still, in this context, new models for how the many complicated picosecond processes present in spectroscopy in the condensed phase influence the reorientation dynamics need to be developed. 55

56 5.4. Two-photon excited fluorescence depolarisation and electronic energy migration within donordonor pairs Paper V presents a unified theoretical description for OPE and TPE fluorescence depolarisation in the presence of electronic energy migration within pairs of identical chromophores, so called donor-donor energy migration, DDEM, cf. Section 2.4. The weak dipole-dipole interaction characteristic of Förster electronic energy transfer involves a donor in a vibronically relaxed S 1 state. As long as the emitting state is independent on whether the excitation is occurring via OPE or TPE, the interaction part in the EFT is identical. This means that TPE is readily incorporated in the EFT once a description of the TPE fluorescence anisotropy of the chromophore in the absence of energy transfer is available. In analogy to the established results for OPE-EFT(84), the fluorescence anisotropy r (n) (t) for the DDEM system in the case of OPE (n = 1) and TPE (n = 2) respectively can be written: 56 JJ IJ 1 (31) r (t) r (t) (t) r (t) (t) (n) 1 (n) p 1 (1) p 2 2 J=A,B IJ where p (t) is the excitation probability of the primarily excited donor defined in Eqs. 27. r (n) JJ (t) is the fluorescence anisotropy of the donor in the (1) absence of energy migration. The anisotropy contributions r IJ (t) originate from the energy migration from the lowest excited electronic state of a donor to a donor in its electronic ground state. The expression for this anisotropy term is given by: (1) IJ 0f 2 I J r (t) r P ˆ ( 0 ) ˆ (t) (32) where P 2 denotes the second order Legendre polynomial, and ˆ I and ˆ J are unit vector of the electronic transition dipole moments of donors A and B. Paper V contains a new description of the TPE fluorescence anisotropy, cf. (2) r JJ (t) in Eq. 31. The theoretical system under consideration involves several orientation transformations due to the local motions of the donor molecule on the rigid macromolecule, which in turn is assumed to have a macroscopically isotropic orientation in the solution, cf. Figure 9. Due to second L L 0 T 0 order terms with respect to TPE tensor components T in the expressions for S(t) and D(t), cf. Paper II, the incorporation of several orientation transformations was at first sight a tedious commitment. In OPE on the other hand, the corresponding calculations involve only first order terms and are much more straightforward. However, the following contraction of TT 0 0 into irreducible product tensors is very useful:

57 C C 0 0 s C0,2,4 sc TT C C 2 2 C 2 2 C s 2C1 TpTs-p pc p ps s s (33) where the 2x3 matrices are 3-j symbols(63). In fact, the contraction in Eq. 33 unifies the description of OPE and TPE fluorescence anisotropy. Coordinate C transformation of s is carried out using rank C Wigner rotation matrices. Then the orientation average over the macroscopically isotropic distribution of macromolecules means that only second rank components with C = 2 contributes to D(t) and zeroth rank components with C = 0 contribute to S(t). As is shown in Paper V, the fluorescence anisotropy obtained using linearly polarised excitation light can be written; r (2) (t) D ( )D ( ) 2TrTT 6 M 1 (2)* 0 (2) (2) qm MD qn MD m m -n 3 q,m,n (0) TrT 0 n (34) The angle brackets... are the orientation correlation functions relating the orientation of the local molecular frame M with respect to the macromolecule fixed frame D. T q and M q denote second rank irreducible tensor components in the M frame of the two-photon absorption tensor (T ) and the emission dipole tensor ( M ), respectively. The explicit contractions (0) (2) m and m are given in Paper V. The OPE anisotropy of the same system has a very similar expression: 3 n qm DL qn DL m -n 1 (35) (1) (2)* 0 (2) r (t) D ( )D (,t) A M 5 q,m,n where A q denote second rank irreducible tensor components of the onephoton absorption dipole tensor ( A ). By comparing Eqs , it is seen that the OPE and TPE anisotropy indeed obey the same transformation properties; apart from the numerator, which is isotropic, upon exchanging 2 the second rank contributions A m for 2TrTT m 6m, these are the same expressions. In this context, it can also be noted that the irreducible product C tensor components m would have allowed a simpler description of the TPE anisotropy in Paper II. Especially, the orientational transformations between the diffusing and the molecular frames would have been facilitated in this description. Symmetry considerations limit the number of non-vanishing orientation correlation functions that are needed to calculate the anisotropy in (34). In 57

58 the literature, one often restricts the calculations to molecules possessing cylindrical symmetry. In this case, only correlation functions with m = n are non-zero in Eq. 34. However, the chromophoric molecules often used in DDEM studies have lower symmetry than cylindrical. Consequently, more correlation functions need to be accounted for(133). Paper V gives the explicit expression for the anisotropies of molecules belonging to the point groups D 2h, D 2 and C 2v. The commonly used derivatives of the molecules BODIPY, perylene and anthracene all belong to these point groups. Upon comparing the resulting theoretical expressions for r (1) (t) and r (2) (t), it can be seen that the anisotropy decays should be parallel in the case of an ideal TPE tensor possessing only a component along the absorption dipole vector : ( 12, ) 12, q0 0 q0 q r (t) r V (t) (36) Assuming that, the initial anisotropies are r (1) (0) = 2/5 and r (2) (0) = 4/7. However, since r (2) (0) is often much lower than 4/7 and T ~ might contain several non-zero elements, more orientation correlation functions probably need to be computed in order to describe the reorientation dynamics in TPE. The simulation of such orientation correlation functions was research in progress at time of the submission for the present thesis Preliminary experiments on a DDEM system Indeed, we have also carried out experiments on a DDEM system (unpublished data). The system studied was pictured in Figure 9. One or two anthryl groups have been covalently linked to a bisteroid in a symmetric manner. Fluorescence depolarisation measurements were preformed on the monoform (AnBs) and the bisform (AnBsAn) dissolved in 1,2-propanediol. OPE at 390 nm and TPE at 780 nm should involve the S 0 S 1 transitions(82). The photophysics decay of AnBs and AnBsAn are very similar, whereas the anisotropy decay of AnBsAn has a more rapid decay, due to the presence of DDEM, see Figure

59 Figure 22. OPE (left), and TPE (right) fluorescence decay experiments of AnBs and AnBsAn in 1,2-propanediol at 283 K. The upper graphs illustrate the photophysics and the weighted residuals of the curve fitted decays. The lower graphs illustrate the fluorescence depolarisation experiments; the middle plot shows the curve fitted difference curves D(t), the lower plots show the constructed anisotropy decays r(t) and the upper graphs show the weighted residual plots of the curve-fitted D(t). The anisotropy is modeled by a bi-exponential decay with a plateau. As can be seen in Figure 22, the photophysics decay of the fluorescence are similar in OPE and TPE. The TPE fluorescence signal was shown to be proportional to the square of the laser intensity by constructing log-log plots. The obtained quadratic dependence confirmed the TPE process. The r 2 (t) is only slightly higher than r 1 (t) which, together with TP = 0.60 for AnBs and TP = 0.63 for AnBsAn, suggests the presence of several non-zero TPE tensor components. This also indicates that more orientational correlation functions should contribute in TPE compared to OPE. The low residual anisotropy value r correlates with a wide local orientation distribution of the anthryl groups. The analysis of the anisotropy decays of the bischromophoric system in terms of the DDEM model is described using Brownian dynamics simulations 59

60 of the rotational dynamics and the electronic interactions, which are modeled by the EFT(84, 87). The fitting is carried out in the Fourier domain with a genetic algorithm. Using a global analysis scheme for the AnBs and AnBsAn system upon OPE at 390 nm, the following map of configuration parameters compatible with the experimental data is obtained: Figure 23. Results of fitting of fluorescence depolarisation data of AnBsAn in 1,2- propanediol at 10 o C with respect to configuration parameters defined in Fig 9. The precision in the configuration angles is very low due to the low local ordering. Judging from the fit, the value of R/R 0, where R 0 is Förster radius, is within in the range Substitution of the known value for the Förster radius R 0 = 27 Å gives R = Å, which is in reasonable agreement with the estimated value from the molecular structure(82). 60