The pump pulses were crossed tac and the probe pulses were

ENCYCLOPEDIA OF
MODERN OPTICS SECOND EDITION

EDITORS IN CHIEF

VOLUME 2

Spectroscopy ▪ Terahertz ▪ Optics of Semiconductors and 2D Materials▪ Nonlinear Optical Spectroscopy ▪ Metamaterials and Plasmonics▪ Lasers, UV Lasers, Random lasers

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EDITORIAL BOARD

v

VOLUME 2

Juraj Darmo and Karl Umerrainer
Ulrich Hlluner, Rupert /-luber, Machillo Kim,

Materials I?ohit. P PrasanlwlI'lar, Dmitly A Ytll"Olslti, and AfHoineue J Taylor 123

Tutorial on Multidimensional Coherent Spectroscopy Marl? Siemens 150

LIST OF CONTRIBUTORS FOR VOLUME 2

11lSl'itute of Phot.Ollics (HId Nmwt.ecll11o!ogy, Milano, Italy

lean-Claude Dids
Universily of New Mexico, lllim'luerque, NM, United States

John Harter
University of Californill, Sallta Barbllrll, CII, Ullited SUlles

Martin R Ilofmann
Ru/l" University 8ocl'lll11l, BociJllm, Germany

Juraj Darmo

Stephan W Koch
Philipps Unil/ersit)' of Marblllg, Marblllg. Ge,."wn)'

ix

Slares NM, Unit.ed SUItes

Bachana Lomsadze
University of Michigtll1, Ann Arbor, MI, United States Cristian Manzoni
Instiwte of Photonics and Nanotechnology, Milano, llilly 5R Meech
University of East Anglia, Norwich, UK
Jerome V Moloney
Universit)' of Ariwlla, Tllcson, AZ, Unit.ed Stales
Calan Moody
Nat.ional Instit.me of Sumdtmis & Technology, BOllldel', CO, United Slales
GA Mourou
Ulliver.lity of Michigan, Ann Arbor, MI, USA
Cururaj V Naik
II ice University, I-Iollston, TX, Ullited Stales
G Timolhy Noe II
Rice Unillersity, I-Iouston, 'Ix, United States
Jennirer P Ogilvie
Unillersity of Michigli11, Anll Arbor, /\;11, United Slates D Pelrosyan
InSI:itme of f-JectwlIic Structure (Inri Laser, Heraidioll, Greece
Rohit P Prasankumar
Cenler for Int.egrated NalIOlecJuwlogies, Los A/amos, NM. Ullit.ed Stal.es
Klaus Reimann
Max 130m Instilllte for Nonlinear Optics and
Short.-PlIlse Spect.1'Oscopy, Berlill, Germany
Marco Santagiustina

Caterina Vozzi
Institllle for Photonics and Nanotechnologies CNR-IFN, Milano, Italy

L Wang
Chitlese Academy of Sciences, Beijil1g, China

W Zawadzki
Polish Academy of Sciences, Warsaw. Poland

John J Zayhowski
Lincoll1 Laboratory, Massachusett.s illst.it.ute of Technology, Lexington, MA, Uniled States

This is the second, updated edition of the Encyclopedia of Modern Optics. There are 197 entries, many of them new or updated, reflecting the enormous progress in the optical sciences and technology and the ever-expanding impact since the publication of the first edition. Some of the new topics are:

E Vauthey, University of Geneva, Geneva, Switzerland

r 2005 Elsevier Ltd. All rights reserved.

Introduction

Over the past two decades, holographic techniques have proved to be valuable tools for investigating the dynamics of chemical processes. The aim of this chapter is to give an overview of the main applications of these techniques for chemical dynamics. The basic principle underlying the formation and the detection of elementary transient holograms, also called transient gratings, is first presented. This is followed by a brief description of a typical experimental setup. The main applications of these techniques to solve chemical problems are then discussed.

where L¼lpu/(2sin ypu) is the fringe spacing, lpu is the pump wavelength, and Ipu is the intensity of one pump pulse.

The modulation of the absorbance, DA, is essentially due to the photoinduced concentration change, DC, of the different chemical species i (excited state, photoproduct, …):

DA lpr� �¼ X ei lpr� �DCi ð5Þ

Fig. 2 Classification of the possible contributions to a transient grating signal.

susceptibility. Electronic OKE occurs in any dielectric material under sufficiently high light intensity. On the other hand, nuclear OKE is mostly observed in liquids and gases and depends strongly on the molecular shape.

Dnt dis related to the temperature-induced change of density. If a fraction of the excitation energy is converted into heat, through a nonradiative transition or an exothermic process, the temperature becomes spatially modulated. This results in a variation of density, hence to a modulation of refractive index with amplitude Dnt d. Most of the temperature dependence of n originates from the density. The temperature-induced variation of n at constant density is much smaller than Dnt d.

dis related to the variation of volume upon population changes. This volume comprises not only the reactant and product Dnv
molecules but also their environment. For example, in the case of a photodissociation, the volume of the product is larger than that of the reactant and a positive volume change can be expected. This will lead to a decrease of the density and to a negative Dnv d.

A typical experimental arrangement for pump–probe transient grating measurements is shown in Fig. 3. The laser output pulses are split in three parts. Two parts of equal intensity are used as pump pulses and are crossed in the sample. In order to ensure time coincidence, one pump pulse travels along an adjustable optical delay line. The third part, which is used for probing, can be frequency converted using a dye laser, a nonlinear crystal, a Raman shifter, or white light continuum generation. The probe pulse is sent along a motorized optical delay line before striking the sample at the Bragg angle. There are several possible beam config-urations for transient grating and the two most used are illustrated in Fig. 4. When the probe and pump pulses are at different wavelengths, they can be in the same plane of incidence as shown in Fig. 4(a). However, if the pump and probe wavelengths are the same, the folded boxcars geometry shown in Fig. 4(b) has to be used. The transient grating technique is background free and the diffracted signal propagates in a well-defined direction. In a pump–probe experiment, the diffracted signal intensity is measured as a function of the time delay between the pump and probe pulses. Such a setup can be used to probe dynamic processes occurring in timescales going from a few fs to a few ns, the time resolution depending essentially on the duration of the pump and probe pulses. For slower processes, the grating dynamics can be probed in real time with a cw laser beam and a fast photodetector.

Fig. 5 Time profile of the diffracted intensity measured with a solution of malachite green after excitation with two pulses with close to counterpropagating geometry.

In principle, the diffracted signal may also contain contributions from the optical Kerr effect, DnK, but we will assume here that this non-resonant and ultrafast response is negligibly small. The density change can originate from both heat releasing processes and volume differences between the products and reactants, the former contribution usually being much larger than the latter. Even if the heat releasing process is instantaneous, the risetime of the density grating is limited by thermal expansion. This expansion is accompanied by the generation of two counterpropagating acoustic waves with wave vectors,~kac ¼ 7ð2p=LÞ~i One can distinguish two density gratings: 1) a diffusive density grating, which reproduces the spatial distribution of temperature and which decays by thermal diffusion; and 2) an acoustic density grating originating from the standing acoustic wave and whose amplitude oscillates at the acoustic frequency uac.

where r, b, Cv, and aac are the density, the volume expansion coefficient, the heat capacity, and the acoustic attenuation constant of the medium, respectively, Q is the amount of heat deposited during the photoinduced process, and DV is the corresponding volume change. As the standing acoustic wave oscillates, its corresponding density grating interferes with the diffusive density grating. Therefore, the total modulation amplitude of the density and thus Dnd exhibits the oscillation at uac. Fig. 5 shows the time profile of the diffracted intensity measured with a solution of malachite green. After excitation to the S1 state, this dye relaxes nonradiatively to the ground state in a few ps. For this measurement, the sample solution was excited with two 30 ps laser pulses at 532 nm crossed with an angle close to 1801. The continuous line is the best fit of Eqs. (9) and (10). The damping of the oscillation is due to acoustic attenuation. After complete damping, the remaining diffracted signal is due to the diffusive density grating only.

where f(t) is a normalized function describing the time evolution of the temperature and/or volume change. In many cases, f(t)¼ exp( � krt), kr being the rate constant of the process responsible for the change. Fig. 6 shows the time profile of the diffracted intensity calculated with Eqs. (9) and (11) for different values of kr.

If several processes take place, the total change of refractive index is the sum of the changes due to the individual processes. In this case, Dnd should be expressed as

The above equations describe the growth of the density phase grating. However, this grating is not permanent and decays through diffusive processes. The phase grating originating from thermal expansion decays via thermal diffusion with a rate constant kth given by

where Dth is the thermal diffusivity. Table 1 shows kth values in acetonitrile for different crossing angles.

For example, Fig. 7(a) shows the energy diagram for a photoinduced electron transfer (ET) reaction between benzophenone (BP) and an electron donor (D) in a polar solvent. Upon excitation at 355 nm,1BP* undergoes intersystem-crossing (ISC) to3BP* with a time constant of about 10 ps. After diffusional encounter with the electron donor, ET takes place and a pair of ions is generated. With this system, the whole energy of the 355 nm photon (E¼3.49 eV) is converted into heat. The different heat releasing processes can be differentiated according to their timescale. The vibrational relaxation to1BP* and the ensuing ISC to 3BP* induces an ultrafast release of 0.49 eV as heat. With donor concentrations of the order of 0.1 M, the heat deposition process due to the electron transfer is typically in the ns range. Finally, the recombination of the ions produces a heat release in the

Similarly, the optoelastic constant of a material, r∂n/∂r, can be determined from the amplitude of the signal (see Eq. (11)). This is done by comparing the signal amplitude of the material under investigation with that obtained with a known standard.

Finally, the thermal diffusivity, Dth, can be easily obtained from the decay of the thermal density phase grating, such as that shown in Fig. 7(c). This technique can be used with a large variety of bulk materials as well as films, surfaces and interfaces.

The temporal variation of DCi can be due either to the dynamics of the species i in the illuminated grating fringes or to processes taking place between the fringes, such as translational diffusion, excitation transport, and charge diffusion. In liquids, the decay of the population grating by translational diffusion is slow and occurs in the microsecond to millisecond timescale, depending on the fringe spacing. As thermal diffusion is typically hundred times faster, Eq. (14) is again valid in this long timescale. Therefore if the population dynamics is very slow, the translational diffusion coefficient of a chemical species can be obtained by measuring the decay of the diffracted intensity as a function of the fringe spacing. This procedure has also been used to determine the temperature of flames. In this case however, the decay of the population grating by translational diffusion occurs typically in the sub-ns timescale.

In the condensed phase, these interfringe processes are of minor importance when measuring the diffracted intensity in the short timescale, i.e., before the formation of the density phase grating. In this case, the transient grating technique is similar to transient absorption, and it thus allows the measurement of population dynamics. However, because holographic detection is background free, it is at least a hundred times more sensitive than transient absorption.

Fig. 9 Transient grating spectra obtained at various time delays after excitation at 355 nm of a solution of chloranil and 0.25 M methylnaphthalene (a): from top to bottom: 60, 100, 180, 260, 330 and 600 ps; (b): from top to bottom: 100, 400, 750, 1100, 1500 ps).

Polarization Selective Transient Grating

In the above applications, the selection between the different contributions to the diffracted signal was essentially made by choosing the probe wavelength and the crossing angle of the pump pulses. However, this approach is not always sufficient. Another important parameter is the polarization of the four waves involved in a transient grating experiment. For example, when measuring population dynamics, the polarization of the probe beam has to be at magic angle (54.71) relatively to that of the

R1111 ¼ R1122 þ R1212 þ R1221 ð16Þ where the subscripts are the Cartesian coordinates. Going from right to left, they design the direction of polarization of the pump, probe, and signal pulses. The remaining elements can be obtained by permutation of these indices (R1122¼R1133¼R2211¼…). In a conventional transient grating experiment, the two pump pulses are at the same frequency and are time coincident and therefore their indices can be interchanged. In this case, R1212¼R1221 and the number of independent tensor elements is further reduced:

R1111 ¼ R1122 þ 2R1212 ð17Þ The tensor R can be decomposed into four tensors according to the origin of the sample response: population and density changes, electronic and nuclear optical Kerr effects:

The table shows that R1212(d)¼0, i.e., the contribution of the density grating can be eliminated with the set of polarization (01,901,01,901), the so-called crossed grating geometry. In this geometry, the two pump pulses have orthogonal polarization, the polarization of the probe pulse is parallel to that of one pump pulse and the polarization component of the signal that is orthogonal to the polarization of the probe pulse is measured. In this geometry, R1212(p) is nonzero as long as there is some polarization anisotropy, (r a 0). In this case, the diffracted intensity is

Idif ðtÞpjR1212ðpÞj2p½DCðtÞ � rðtÞ�2 ð20Þ The crossed grating technique can thus be used to investigate the reorientational dynamics of molecules, through r(t), especially when the dynamics of r is faster than that of DC. For example, Fig. 10 shows the time profile of the diffracted intensity measured in the crossed grating geometry with rhodamine 6G in ethanol. The decay is due to the reorientation of the molecule by rotational diffusion, the excited lifetime of rhodamine being about 4 ns.

Fig. 10 Time profile of the diffracted intensity measured with a solution of rhodamine 6G with crossed grating geometry.

Concluding Remarks

The transient grating techniques offer a large variety of applications for investigating the dynamics of chemical processes. We have only discussed the cases where the pump pulses are time coincident and at the same wavelength. Excitation with pump pulses at different wavelengths results to a moving grating. The well-known CARS spectroscopy is such a moving grating technique. Finally, the three pulse photon echo can be considered as a special case of transient grating where the sample is excited by two pump pulses, which are at the same wavelength but are not time coincident.