Study of Ultrafast Polarization and Carrier Dynamics in Semiconductor Nanostructures: a THz Spectroscopy Approach

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1 Study of Ultrafast Polarization and Carrier Dynamics in Semiconductor Nanostructures: a THz Spectroscopy Approach Inaugural-Dissertation zur Erlangung des Doktorgrades der Fakultät für Mathematik und Physik der Albert-Ludwigs-Universität Freiburg im Breisgau vorgelegt von Dmitry Turchinovich aus St.Petersburg im Mai 2004

2 Dekan: Prof. Dr. Rolf Schneider Leiter der Arbeit: HD Dr. Peter Uhd Jepsen Referent: HD Dr. Peter Uhd Jepsen Korreferent: Prof. Dr. Hellmut Haberland Tag der Verkündigung des Prüfungsergebnisses:

3 Publications in Refereed Journals Part of the work presented in this thesis is based on the following publications in refereed journals: D.Turchinovich, A.Kammoun, P.Knobloch, T.Dobbertin, and M.Koch Flexible all-plastic mirrors for the THz range Appl. Phys. A 74, 291 (2002) D.Turchinovich, K.Pierz, and P.Uhd Jepsen InAs/GaAs quantum dots as efficient free carrier deep traps phys. stat. sol. (c) 0, 1556 (2003) D.Turchinovich, P.Uhd Jepsen, B.S.Monozon, M.Koch, S.Lahmann, U.Rossow, and A.Hangleiter Ultrafast polarization dynamics in biased quantum wells under strong femtosecond optical excitation Phys. Rev. B 68, (R) (2003) D.Turchinovich, B.S.Monozon, M.Koch, S.Lahmann, A.Hangleiter, and P.Uhd Jepsen Ultrafast polarization dynamics in optically excited biased quantum wells Proceedings of SPIE 5354, 151 (2004) D.Turchinovich, P.Uhd Jepsen, and B.S.Monozon Biased semiconductor quantum wells under ultrafast optical excitation: theoretical model of dynamical screening submitted to Phys. Rev. B (2004) Publications not included in this thesis: D.Turchinovich, P.Knobloch, G.Luessem, and M.Koch THz time-domain spectroscopy on 4-(trans-4-pentylcyclohexyl)-benzonitril Proceedings of SPIE 4463, 65 (2001) Patents: T.Dobbertin, P.Knobloch, D.Turchinovich, and M.Koch Patent DE A1 Optisches Bauelement (Plastic dielectric mirrors for THz frequency range) (2002) 3

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5 Preface Most of the results presented in this Thesis were obtained at the Department of Molecular and Optical Physics, University of Freiburg, in the group of HD Dr. Peter Uhd Jepsen. The results presented in the Section 2.4 of the Chapter 2 of this Thesis were obtained during the author s stay at the Institut für Hochfrequenztechnik, Technical University of Braunschweig, in the group of Prof. Dr. Martin Koch. 5

6 Preface 6

7 Contents Introduction 9 1 Principles of generation and detection of electromagnetic transients in nonlinear crystals Maxwell s equations. Wave equation and its solution in the far field Linear and nonlinear contributions to polarization Optical rectification and second harmonic generation Linear electrooptic effect and phase retardation Conclusions Terahertz time-domain spectroscopy. Design and characterization of the experimental setup Experimental THz-TDS setup driven by a femtosecond amplifier Femtosecond Ti:Sapphire amplifier system THz spectrometer based on ZnTe generation and detection crystals Propagation of an electromagnetic wave packet through the medium. THz spectral analysis Simulation of the experimental THz pulses. Phase matching and absorption considerations Plastic dielectric mirrors designed for the THz range Conclusions Ultrafast polarization dynamics in the biased semiconductor quantum wells under ultrafast optical excitation. Dynamical screening effect Optical transitions in a biased quantum well. Quantum-confined Stark effect Samples Experiment THz emission spectroscopy Time-integrated photoluminescence spectroscopy Theory

8 Contents Model of dynamical screening Theoretical results and discussion Conclusions Trapping of the free carriers in InAs/GaAs quantum dot structures Contribution of the free carriers into the dielectric function. Drude conductivity model Samples Experiment and discussion Extraction of the spectral information Conclusions Bibliography 113 Acknowledgements 119 8

9 Introduction Ultrafast carrier and polarization dynamics in semiconductors is one of the most exciting areas in modern physics. Contemporary electronics and optoelectronics is nearly entirely based on the advanced semiconductor technology. Since in our information-oriented society there is an ever-growing demand for the higher bandwidth in the telecom systems and for the higher clock rate in the computers, the ultrafast phenomena in semiconductors start to play a crucial role in the development of new devices. In the modern telecommunications systems the information is already transferred through the optical fibers at the terabit/s rate (although not yet using a single laser source), and the computers are operating at the several gigahertz clock rate. From the point of view of the fundamental physics, experimental and theoretical investigations of the processes occurring on the femtosecond and picosecond timescale are also of a great interest, since it allows us to understand the mechanisms of such extremely important processes as the transport in strong fields, relaxation, recombination and trapping of the charge carriers, formation of the excitons, modification of the semiconductor band structure due to the optical excitation and many others. The ultrafast phenomena in semiconductors is a booming research area for already two decades, since the femtosecond lasers came into the market. The experimental techniques such as degenerate and non-degenerate pump-probe experiments, four-wave mixing, time-resolved photoluminescence and many others are widely used by the semiconductor research community. In the recent decade the terahertz (THz) spectroscopy began to play an everincreasing role in the experiments. The frequency of 1 THz corresponds to one oscillation per picosecond. In the equivalent units 1 THz is: 1 THz 1 ps mev 0.3 mm 33 cm K The THz range is thus bridging the gap between the microwave and the infrared ranges, and is sometimes referred to as far-infrared. One should notice here that the low THz range lies below the room-temperature background of 26 mev = 6.3 THz. The lack of efficient emitters and detectors in the THz frequency range was the main problem for the experimentalists in the past. The main workhorses in the 9

10 Introduction millimeter and submillimeter wave spectroscopy were the continuous-wave sources such as backward-wave oscillators (BWO) [1] or sum-frequency mixed Schottky and Gunn diodes (see e.g. [2] and [3]) used as the emitters, and helium-cooled bolometers used as the detectors. In this case the useful spectroscopy bandwidth typically covered the range between 0.01 and 1.5 THz, and no time-resolved experiments were possible. Another approach to generate THz radiation is a quantum-cascade laser (QCL), first proposed by R.F.Kazarinov and R.A.Suris back in 1971 [4]. For the THz generation the QCL employs the intraband transitions in a biased doped semiconductor heterostructure. The first operating device was demonstrated however only in 1994 by J.Faist and coworkers [5], and despite the impressive progress in the QCL development achieved ever since, these devices are not yet able to operate at a room temperature. Also in 1971 a completely new approach to generate the ultrashort electromagnetic transients was proposed by the group of Y.R.Shen: optical rectification of strong picosecond laser pulses in the nonlinear crystals [36]. They demonstrated the THz emission in the range THz. This approach is based on the fundamental nonlinear optical effect and it does not require cooling of the THz generation crystal, although in this first demonstration the helium-cooled bolometer was again used as a detector. The optical rectification effect will be discussed in detail in this Thesis and will be widely employed in the experiments presented here. In the 1980s the group of D.H.Auston at Bell Laboratories pioneered yet another approach to generate and detect THz radiation at room temperatures using ultrashort laser pulses - THz time-domain spectroscopy (THz-TDS). This approach employs the generation of an ultrashort conduction current or polarization spike in the biased intrinsic semiconductor by optical gating of the gap between the biasing electrodes [6]. In the absence of the short laser pulse no current flows across this gap. Once the short laser pulse arrives to the semiconductor surface, it excites free charge carriers, which are accelerated by the applied electric field. The rise time of the photocurrent surge is given by the duration of the excitation laser pulse and can be of the sub-picosecond duration. Such a sharp change in the current results in the radiation of the few-cycle electromagnetic pulse with the duration even shorter than the rise time of the current surge. This electromagnetic transient has a broadband frequency spectrum in the THz range. The photoconductive generation of the THz transients is an alternative to the optical rectification process using the ultrashort laser pulses. In order to detect such an electromagnetic transient at room temperature one can use a photoconductive detection. For the photoconductive detection one uses a semiconductor with short free carrier life time, so that the free carriers excited by an ultrashort laser pulse can contribute to the current for not much longer than the duration of the laser pulse itself. The semiconductor is initially unbiased but 10

11 still has the electrodes grown on its surface. In the absence of the THz electric field the gating of the gap between the electrodes with a laser pulse does not result in any current across this gap. Once both THz pulse and the gating laser pulse are incident on the surface, the optically excited electrons and holes start drifting across the gap accelerated by the THz electric field, which results in the photocurrent across the gap. By delaying the gating laser pulse in respect to the THz pulse one is able to read the electric field strength in the THz pulse, which is proportional to the measured photocurrent, with the time resolution given by the carrier lifetime in the semiconductor and the step in time delay between the THz and gating pulses. This results in the field-resolved detection of an ultrashort electromagnetic transient. An alternative approach to detect the THz transient is to use a transparent electrooptic crystal as a detector. In this case for the THz detection one sends a weak probe ultrashort laser pulse through the crystal. In the absence of the THz electric field in the case of an isotropic crystal this probe pulse will experience no phase retardation. In the presence of the THz field the electrooptic crystal will demonstrate a Pockels effect, which will result in the phase retardation of the laser pulse, given the probe laser pulse is properly polarized. This induced phase retardation is directly proportional to the electric field strength in the THz pulse, and can be measured e.g. by ellipsometry. Again, by delaying the laser pulse in respect to the THz pulse one can thus measure the temporal evolution of the electric field in the incident THz pulse. This technique was first demonstrated in 1996 by the groups of X.-C.Zhang [37] and P. Uhd Jepsen and H.Helm [38] and is called a free-space electrooptic sampling (FEOS). In the experiments presented in this Thesis the THz generation by optical rectification of the strong laser pulses in the nonlinear crystal and the THz detection using a free-space electrooptic sampling was used. Once THz-TDS became a conventional experimental technique, many researchers focussed their effort on the semiconductors studies. The linear spectroscopy on different bulk semiconductors was performed by several authors. The main spectroscopic features in the bulk semiconductors at THz frequencies are caused by the polar lattice vibration (phonons) and by the free carrier absorption. The phonon and free carrier contributions to the dielectric function of the semiconductors were studied for example by the groups of P. Uhd Jepsen [44] and D.Grischkowsky [43], [7]. It was demonstrated that the phonon contribution to the dielectric function can be described by the harmonic oscillator model, and the free-carrier contribution can be described by the Drude model of conductivity. One more approach to study the ultrafast processes in semiconductors is to perform optical-pump THz-probe spectroscopy, when the modification of the semiconductor properties induced by the strong optical pump pulse is probed in the THz range. This gives one an opportunity to monitor the modification of the sample s properties in THz range with sub-ps resolution. Using this technique the 11

12 Introduction time-resolved free-carrier lifetime and mobility in many bulk semiconductors was measured, for example in GaAs [8], [9] and Si [10]. Recently the plasma formation process was observed in bulk GaAs with sub-10 fs resolution in the optical-pump THz-probe spectroscopy experiment [11]. Also the very impressive results on the exciton formation in the semiconductor quantum wells were obtained using opticalpump THz-probe technique. In this case the optical pump either resonantly created an exciton in the 1s ground state, or the uncorrelated electron-hole pairs high in the continuum, and with the THz pulse the intra-excitonic 1s 2p transition was probed. This gave an important information on the exciton formation dynamics [17], [18]. Another type of experiments performed using the THz-TDS technique is the THz emission spectroscopy. In this type of experiments the THz emission from the optically excited sample is measured. The emitted THz signal is a direct fingerprint of an ultrafast polarization and/or conduction current change in the sample induced by an ultrafast optical excitation, and thus gives an important direct information on ultrafast carrier and polarization dynamics in the system of interest. In this manner the strong-field transport in the bulk semiconductors [12], [13] and the ultrafast polarization dynamics in the semiconductor quantum wells were studied [14], [15], [16]. This Thesis is organized as follows: In the Chapter 1 the basic physics - electrodynamics and nonlinear optics, involved in the THz generation and detection is reviewed. In the Chapter 2 we will describe the THz-TDS experiment based on the nonlinear crystals and will perform the characterization of the experimental setup. We will also discuss the performance of the dielectric mirrors for the THz range designed by us. In the Chapter 3 we present the experimental and theoretical studies on the ultrafast polarization dynamics in the biased semiconductor quantum wells under ultrafast optical excitation. We will demonstrate a new effect of the dynamical screening of the electric field inside the quantum well by that of the polarized carriers created by the ultrafast optical pulse, and the consequences of such a screening on the key optical and electronic properties of the system. The strongly internally biased InGaN/GaN quantum well structure is used as a sample in this work. In the Chapter 4 our studies on the ultrafast free-carrier trapping into the quan- 12

13 tum dots will be shown on the example of the InAs/GaAs Stranski-Krastanow - grown quantum dot structure. We will demonstrate the significant reduction of the free-carrier lifetime in such a structure compared to the bulk GaAs crystal, and the decrease in the free-carrier mobility. The possible explanations for these effects will be also discussed in this Chapter. 13

14 Introduction 14

15 Chapter 1 Principles of generation and detection of electromagnetic transients in nonlinear crystals. In this chapter we will review the fundamental physics involved in the generation and detection of the short electromagnetic transients in nonlinear crystals, central for the context of this Thesis. We start with the Maxwell s equation in vacuum and medium and then arrive at the wave equation. The solution of the wave equation will give us an important result: the far-field temporal shape of an electromagnetic transient induced by the time-varying sources - conduction current and polarization. Then we will consider the linear and nonlinear contributions to the optically induced polarization, and will discuss the two very important nonlinear optics effects: optical rectification and second harmonic generation. Optical rectification of ultrashort laser pulses in nonlinear crystals will be employed in this work to generate the THz electromagnetic transients. Finally, we will describe the linear electrooptic effect in nonlinear crystals, and the phase retardation of the incident optical electromagnetic wave induced by it. This effect will be used in detection of the THz transients by means of the timeresolved optical probing of the electrooptic effect induced by the THz electric field in the nonlinear crystal. 15

16 Electrodynamics 1.1 Maxwell s equations. Wave equation and its solution in the far field. Electromagnetic phenomena are governed by the Maxwell s equations. In vacuum they have the form of E = ρ ε 0, (1.1) E = B t, (1.2) B = 0, (1.3) B = µ 0 ε 0 E t + µ 0J, (1.4) where E is an electric field strength, B - magnetic induction, J - current density, ρ - charge density, µ 0 and ε 0 are the magnetic and electric permeabilities of vacuum, and t - time. µ 0 and ε 0 are related to the speed of light c = ε 0 µ 0 1. E, B, and J are the vectors. Using the identity ( B) = 0 on the eq. (1.4), and taking the derivative of the eq. (1.1) we arrive to the charge conservation law ρ t = J. (1.5) In a medium the charge density ρ can be represented as a sum of the charges added externally ρ ext (e.g. by the incoming current), and the polarized charges ρ pol, related to the polarization density P as ρ = ρ ext + ρ pol, (1.6) ρ pol = P. (1.7) The current J can be represented as a sum the conduction J cond and displacement J disp currents, and the latter can be related to the polarization density P as J = J cond + J disp, (1.8) J disp = P t. (1.9) Now we will introduce the material-related parameters: electric displacement D and magnetic field strength H D = εe = E + P ε 0 = (1 + χ ε 0 ), (1.10) 16

17 Physical principles H = B µ 0, (1.11) where ε is the dielectric constant and χ is a dielectric susceptibility. The polarization density is P = (ε 1)ε 0 E = χe. (1.12) Now we rewrite the eqs. ( ) taking into account the new material parameters and eqs. ( ) [19, 20] D = ρ ext ε 0, (1.13) E = B t, (1.14) B = 0 (1.15) D H = J cond + ε 0 t. (1.16) The eqs. ( ) are called material Maxwell s equations. Here the conduction current J cond and the polarization P (included into displacement D) are the only time-varying parameters, describing the response of the medium to the electromagnetic field. The eqs. ( ) can be combined into 2 E 1 c 2 2 E t 2 = µ 0 ( Jcond t ) + 2 P. (1.17) t 2 This equation describes the propagation of an electromagnetic wave and is hence a wave equation. The only two time-varying source terms are again the conduction current J cond and the polarization P. We are interested in the far-field on-axis solution to the wave equation 1.17, which relates the temporal shape of an electromagnetic signal emitted by a slab of material with a time-varying spatially uniform conduction current J cond and/or polarization P at the axis normal to the slab and at the considerable distance form it. Such a solution have been presented by several authors [21], [22], [23], [24] in the form E rad (t) µ ( ) 0 S Jcond (t) + 2 P(t). (1.18) 4π z t t 2 where S is the emitting area and z is a distance from the emitter surface to the detection point. This result is very important for us, since it allows for the reconstruction of the polarization an carrier dynamics in an electromagnetic signal emitter, assuming that the radiated electric field is properly detected. 17

18 Nonlinear Optics 1.2 Linear and nonlinear contributions to polarization. In the linear case the polarization P can be expressed as (see 1.12) P (r, t) = + χ (1) (r r, t t )E(r, t )dr dt, (1.19) where χ (1) is a linear susceptibility, representing the 2 nd rank tensor. Let us assume that the electric field has a form of a monochromatic plane wave E(r, t) = E(k, ω) = E 0 (k, ω)exp(ik r iωt), (1.20) where r is a coordinate, k is a wavenumber, ω - angular frequency, and E 0 - electric field amplitude. The Fourier transform of eq. (1.20) gives us P (r, t) = P (k, ω) = χ (1) (k, ω)e(k, ω), (1.21) and the dielectric constant is then ε(k, ω) = 1 + χ(1) ε 0 (k, ω). (1.22) In the nonlinear case the eq. (1.21) can be expanded into the series of power of E: P (r, t) = χ (1) (r r, t t )E(r, t )dr dt χ (2) (r r 1, t t 1 ; r r 2, t t 2 ) E(r 1, t 1 )E(r 2, t 2 )dr 1 dt 1 dr 2 dt , (1.23) where χ (i) is an i th order nonlinear susceptibility, representing the tensor of (i + 1) th rank. If E can be expressed as a group of plane monochromatic waves E(r, t) = i E(k, ω), (1.24) then the Fourier transform of eq. (1.23) will give [25] P (k, ω) = P (1) (k, ω) + P (2) (k, ω) + P (3) (k, ω) +... (1.25) 18

19 Physical principles with P (1) (k, ω) = χ (1) (k, ω)e(k, ω), P (2) (k, ω) = χ (2) (k = k i + k j, ω = ω i + ω j )E(k i, ω i )E(k j, ω j ), (1.26) P (3) (k, ω) = χ (3) (k = k i + k j + k k, ω = ω i + ω j + ω k ) E(k i, ω i )E(k j, ω j )E(k k, ω k ). From the eqs. (1.26) it is clearly seen that the higher order nonlinear susceptibilities provide mixing of the frequencies of the different monochromatic waves, given the matching of their wavenumbers. Now we will rewrite the eqs. (1.26) into the conventional tensor form (keeping the wavenumber matching in mind): P (1) i (ω) = χ (1) ij (ω)e j(ω), P (2) i (ω = ω j + ω k ) = χ (2) ijk (ω)e j(ω)e k (ω), (1.27) P (3) i (ω = ω j + ω k + ω l ) = χ (3) ijkl (ω)e j(ω)e k (ω)e l (ω). The ratio of the two successive polarization terms inside the medium in the series (1.25) scales roughly as P (n+1) P (n) = χ (n+1) E χ (n) E (1.28) E mat where E mat is the inherent electric field inside the medium, which is typically on the order of 10 8 V/cm, and therefore χ (n+1) /χ (n) Emat 1 [26]. Thus the higherorder contributions to the nonlinear polarization will always be much weaker than the lower-order ones. The form of the χ (n) tensor is also strongly dependent on the internal symmetry of the medium. In the substances with inversion symmetry χ (2n) = 0, where n = 1, 2, 3,.., and therefore the nonlinear polarization contains only odd-order terms. This can be proved by the following explanation: A medium with inversion symmetry contains a regular lattice of points such that inversion (replacing each atom at coordinate r, relative to the point, with the one with r) leaves the crystal structure unchanged (invariant) [27]. Let us now induce the even-number higher-order polarization P (2n) r = χ (2n) E (2n) in such a medium along the direction r. The inversion symmetry implies that the polarization P (2n) r = χ (2n) E (2n) along the direction r should be equal to P (2n) r in its absolute value, but it is also obvious that it should have an opposite sign. This is only possible, given the even power of the electric field, if χ (2n) = 0. 19

20 Nonlinear Optics 1.3 Optical rectification and second harmonic generation. Here we will focus ourselves on the nonlinear 2 nd order polarization induced by an optical excitation. Let us assume that we have two plane electromagnetic waves E 1 (t) = A 1 (t)cos(ω 1 t) and E 2 (t) = A 2 (t)cos(ω 2 t), where A 1,2 (t) are the envelopes and ω 1,2 are the carrier frequencies. Then the 2 nd order nonlinear polarization P (2) from (1.27) will have a form P (2) (ω = ω 1 + ω 2 ) = = χ (2) ijk (ω)a 1(t)A 2 (t)cos(ω 1 t)cos(ω 2 t) = (1.29) = 1 2 χ(2) ijk (ω)a 1(t)A 2 (t)cos[(ω 1 ω 2 )t] + cos[(ω 1 + ω 2 )t] = = P (2) (2) (ω) + P (ω). P (2) (ω) now is a sum of two terms P (2) (2) (ω) and P Σ (ω) containing the difference and sum frequencies. Let us now assume that both incident electromagnetic waves are identical, i.e. A 1 = A 2 = A and ω 1 = ω 2 = ω. Then (1.29) can be reduced to Σ P (2) (2ω = ω + ω) = 1 2 χ(2) ijk (2ω)A2 (t) χ(2) ijk (2ω)A2 (t)cos(2ωt) = (1.30) = P (2) (2) (0) + P Σ (2ω). As a result we have two non-zero polarization terms, both dependent on electric field amplitude. The term P (2) is completely independent of the carrier frequency and describes the effect called optical rectification. The term P (2) Σ contains oscillations at the doubled carrier frequency, and describes an effect called second harmonic generation (SHG). The carrier-frequency-independent 2 nd order polarization term P (2) is proportional to the square of the electromagnetic wave amplitude, i.e. directly proportional to the incident wave intensity, and in general is time-dependent. In the case of an unmodulated electromagnetic wave we have A(t) = const. Therefore in this case the optical rectification effect will result in the permanent polarization inside a medium. In the case of a modulated electromagnetic wave, the term P (2) will be timedependent, and therefore, according to the wave equation (1.17) will result in an electromagnetic radiation with the far-field time-varying electric field strength defined by the eq. (1.18). This property will be used to generate the THz electromagnetic signals in nonlinear crystals, and is a central theme in this work. It should be noted here, that not all the nonlinear crystals have both good SHG and optical rectification efficiencies. Indeed, the co-propagation of the three 20

21 Physical principles signals: carrier, second harmonic, and an optically rectified signal, requires good phase matching in a very wide frequency range over a considerable optical path inside the crystal, which is not the case for most dielectrics. Usually the crystals possessing the high SHG efficiency show very small optical rectification efficiency and vice versa. 1.4 Linear electrooptic effect and phase retardation. If a low-frequency or a dc electric field E 0 (Ω 0) is applied to a medium, the optical dielectric constant of the medium ε(ω, E 0 ) will be a function of E 0. For a small E 0, ε(ω, E 0 ) can be expanded into series of the power of E 0 [25]: ε(ω, E 0 ) = ε (1) + ε (2) (ω + Ω)E 0 + ε (3) (ω + 2Ω)E 0 E (1.31) Since D = εe + P ε 0 and P = χ (1) E + χ (2) E , the nonlinear terms of optical dielectric constant will be ε (2) (ω + Ω) = 1 ε 0 χ (2) (ω + Ω), ε (3) (ω + 2Ω) = 1 ε 0 χ (3) (ω + 2Ω, ) (1.32)... In a medium with no inversion symmetry the electrooptic effect will be dominated by the linear term ε (2) E 0. This effect is called a linear electrooptic effect, or Pockels effect. The quadratic electric field term ε (3) E 0 E 0 will exist in any type of medium, and is called the Kerr effect. Now we will consider the linear electrooptic effect in a crystalline medium. The spatial dispersion of a refractive index n = ε of the crystal without an applied electric field is described by a refractive index ellipsoid x 2 n 2 x + y2 n 2 y + z2 n 2 z = 1, (1.33) where the directions x, y and z are the principal dielectric axes, i.e. the axes along which D and P are collinear. Application of an external electric field E 0 (x, y, z) will modify the dielectric constant tensor. It will induce the birefringence in the crystal where the propagating electromagnetic waves polarized along and perpendicularly to the external 21

22 Nonlinear Optics field experienced no anisotropy in the absence of the applied field, or modify the inherent birefringence in case of an originally anisotropic crystal. The effect of the applied electric field on the refractive index ellipsoid can be expressed by the changes in the constants 1/n 2 x, 1/n 2 y, 1/n 2 z of the index ellipsoid. Following the conventions of [27], we show the new (deformed) index ellipsoid in the form of ( ) n 2 4 ( ) ( ) 1 1 x 2 + n 2 1 n ( ) 2 1 yz + 2 n 2 2 xz ( ) 1 y 2 + n ( ) 2 1 n 2 z xy = 1 (1.34) 6 which now contains the mixed terms. If one chooses the axes x, y and z parallel to the principal dielectric axes of the crystal, then without an applied electric field eq. (1.34) will reduce to the eq. (1.33). Then ( ) 1 n 2 1 = 1 ; E 0 n =0 2 x ( ) 1 n 2 2 = 1 ; E 0 n =0 2 y ( ) 1 n 2 3 = 1, E 0 n =0 2 z and the mixed terms are ( ) 1 n 2 4 E 0 =0 = ( ) 1 n 2 5 E 0 =0 = ( ) 1 n 2 6 E 0 =0 = 0. The linear changes in the coefficients ( 1, where i = 1...6, induced by the applied n )i 2 electric field E 0 (E x, E y, E z ) are defined by ( ) 1 = n 2 i 3 r ij E j, (1.35) j=1 where i = and j = 1, 2, 3 denotes the projections of the applied electric field E 0 on the axes x, y, z. The eq. (1.35) expressed in the matrix form will read: 22

23 Physical principles Centrosymmetric Cubic, 43m 3m e.g.: Si e.g.: GaAs, ZnT e e.g.: LiT ao 3, LiNbO r 22 r r 22 r 13 All elements zero r 33 r r r 41 0 r r 41 r Table 1.1: Electrooptic tensors for the crystals of different symmetries [27]. ( 1 n 2 )1 ( 1 n 2 )2 ( 1 n 2 )3 ( 1 n 2 )4 ( 1 n 2 )5 ( 1 n 2 )6 = r 11 r 12 r 13 r 21 r 22 r 23 r 31 r 32 r 33 r 41 r 42 r 43 r 51 r 52 r 53 r 61 r 62 r 63 E 1 E 2 E 3 (1.36) The tensor r ij is called an electrooptic tensor, and its form depends on the internal symmetry of the material. In the centrosymmetric materials all the elements r ij = 0 since the nonlinear second order susceptibility in such materials is χ (2) = 0. In order to calculate the propagation of the electromagnetic wave through the electrooptic material in the presence of applied electric field, one has to transform the the deformed index ellipsoid (1.34) to the principal axes of original ellipsoid (1.33). Then the deformed ellipsoid will not contain mixed terms, but the original indices n x, n y, n z E0 =0 will depend on n x, n y, n z E0, E 1, E 2, E 3, and the components of r ij. In general, after this transformation the principle axes of the deformed index ellipsoid x, y, z E0 will not coincide with the principle axes of the original index ellipsoid x, y, z E0 =0. In the following we will denote the axes of the original index ellipsoid x 0, y 0, z 0, and the new axes of the deformed ellipsoid x, y, z. In Table 1.1 we present the forms of the electrooptic tensors for the crystals of different symmetries: centrosymmetric, cubic (43m), and 3m crystals. Now we will perform the calculations for the refractive index change and the orientations for the new axes x, y, z for the 110 -oriented ZnTe crystal of the thickness d. ZnTe is a cubic crystal of the 43m symmetry class. In the absence 23

24 Nonlinear Optics <110> y y 0 x 0 (110) 45 0 x E 0 E opt d z 0 = z Figure 1.1: Orientation of the principal axes of the original (x 0, y 0, z 0 ) and the deformed (x, y, z) index ellipsoids due to the linear electrooptic effect in 110 -oriented ZnTe crystal. The electric field E 0 is applied along the z = z 0 axis. E opt shows the plane of polarization of an incident plane wave. The electric field E 0 is applied along the z = z 0 axis. of an applied electric field the refractive indices n x,y,z = n o are equal in all three directions x 0, y 0, and z 0. Let us assume that the electric field E 0 is oriented parallel to the z 0 axis of the crystal (see Fig. 1.1). The deformed index ellipsoid, as it follows from (1.34, 1.35, 1.36) and the Table 1.1 will have the form of x 2 n y2 + z2 + 2r n 2 0 n 2 41 E 0 xy = 1 (1.37) 0 As a result of the applied electric field, the axis z = z 0 did not transform, but the axes x and y, previously both pointing out of the crystal surface, now rotated by 45 0 about the z axis so, that x is lying in the plane of the crystal surface (110), and y is normal to the plane of the crystal surface (and collinear with the direction 110 ). The refractive indices along the new axes will be n x = n n3 0r 41 E 0 n y = n n3 0r 41 E 0 (1.38) n z = n 0 Let us now send a plane electromagnetic wave in the direction normal to the crystal surface so, that it is polarized at 45 0 in respect to the z axis (its polarization 24

25 Physical principles is marked as E opt in the Fig. 1.1), and therefore to the direction of the applied electric field E 0. Since the polarization of the incident electromagnetic wave has zero projection on the y axis, it will be only influenced by the x and z components of the refractive index n x and n z, which are now different from each other. The incident electromagnetic wave will thus experience a phase retardation, as its x and z components will now propagate through the crystal with different velocities. The total phase difference between these two components is φ = ω(n x n z )d c = 1 ωn 3 or 41 E 0 d 2 c (1.39) where ω is an angular frequency of the incident electromagnetic wave. The phase retardation in the nonlinear crystal induced by the low-frequency electric field is generally used in the detection of the THz transients. This is done by means of time-resolved sampling of the phase retardation of an ultrashort optical probe laser pulse, which is induced by the electric field in the incident THz transient. 1.5 Conclusions In this Chapter we have discussed the physical principles of generation and detection of electromagnetic transients in nonlinear crystals. In the Section 1.1 we demonstrated, that the electromagnetic signal is induced by the time-varying conduction current and polarization (eq. 1.17), and that the electric field of this signal in the far-field will be proportional to the first temporal derivative of the conduction current and the second temporal derivative of polarization (eq. 1.18). We are thus able to predict the temporal shape of the electromagnetic signal induced by the known time-varying sources, and also we are able to reconstruct current and polarization dynamics of the unknown radiation source, given the emitted electromagnetic signal is properly measured. In the Section 1.2 the linear and nonlinear contributions to the optically induced polarization in nonlinear crystals were discussed, and in the Section 1.3 we have considered the two most important 2 nd order nonlinear effects: the polarization induced in the nonlinear crystal at the doubled carrier frequency (SHG), and the polarization, which temporal dependency is only given by the intensity profile of the incident electromagnetic signal, but which is completely independent of its carrier frequency - optical rectification signal. According to the main results of the Section 1.1, the time-varying SHG and optical rectification signals will result in the electromagnetic radiation. This, in particular, gives us an opportunity to convert the modulated optical signals with high carrier frequency into the low frequency electromagnetic radiation using the optical rectification effect. If the 25

26 Nonlinear Optics intensity of the optical signal will time-vary on sub-picosecond time scale, the emitted electromagnetic transient will belong to the THz range. In the Section 1.4 we have demonstrated, that the electric field applied to the nonlinear crystal will result in the modification of its optical properties, which may cause the phase retardation of an incident electromagnetic wave, given the orientation of polarization of this wave is chosen properly. If the applied electric field is delivered to the nonlinear crystal by a THz transient, we will be able to detect the temporal dependency of the electric field of this transient using the time-resolved optical probing of the phase retardation, induced by the slow-varying electric field in the THz transient. 26

27 Chapter 2 Terahertz time-domain spectroscopy. Design and characterization of the experimental setup. In this Chapter we will discuss the experimental aspects of the THz generation and detection in nonlinear crystals with ultrashort laser pulses using a terahertz time-domain spectroscopy (THz-TDS) approach. At first we will consider the design of the THz setup and the femtosecond Ti:Sapphire amplifier system which is used to drive the THz setup. We will perform the setup characterization and outline the phenomena responsible for the setup specifications, such as THz frequency bandwidth. We will consider the propagation of the THz transient through a slab of a dielectric, and discuss the procedure used for the spectral data analysis. We will also perform the numerical simulations of the emitted and detected THz pulses, addressing such issues as the THz absorption and the phase matching in the generation and detection crystals. We will also discuss the fundamental issue of energy flow in THz generation process. We will consider the THz-TDS experiment performed on the CdTe crystal at room temperature and will discuss the frequency resolution an the reliability of the calculated dielectric spectra. Finally we will discuss the design and characterization of the polymer-based dielectric mirrors for the THz range. 27

28 THz-TDS 2.1 Experimental THz-TDS setup driven by a femtosecond amplifier Femtosecond Ti:Sapphire amplifier system. In this work we use a THz setup driven by a commercial femtosecond Ti:Sapphire (Ti:Sa) amplifier Clark MXR CPA 1000 system [28]. The amplifier system consists of the following main parts: i) Kerr-lens modelocked Ti:Sa oscillator delivering 100 fs - long laser pulses with the central wavelength tunable around 800 nm at the 100 MHz repetition rate, with an average cw power of 0.3 mw. Given the above characteristics and the beam diameter of 2 mm one can estimate the single pulse energy of 3 nj/pulse, the average pulse power of 30 kw, peak intensity of 0.95 MW/cm 2, and taking into account [26] I(t) = 1 2 ε 0cẼ2 (t) (2.1) where I is intensity and Ẽ(t) is an average envelope electric field strength of the pulse, the peak electric field strength in the pulse of 18.9 kv/cm 2. This Ti:Sa oscillator is pumped with a commercial diode-pumped Nd:YVO 4 frequencydoubled cw VERDI T M [29] laser, with an output of 2.5 W at 532 nm. ii) Grating-based pulse stretcher, which chirps the 100 fs transform-limited oscillator pulse to the approximately 80 ps length conserving the pulse bandwidth. This pulse stretching is used in order to significantly decrease the peak intensity of the laser pulse before seeding it into the amplifier, since high intensity will lead to the amplifier crystal damage. iii) Ti:Sa regenerative amplifier, pumped by the frequency doubled commercial Nd:YAG laser delivering the 10 ns long pump pulses with a repetition rate of 1 khz and an average cw power of 9.1 W at 532 nm. Excited by this pump pulse the Ti:Sa crystal of the regenerative amplifier turns into a gain medium for a period of approximately 70 ns, which, together with a cavity of an amplifier results in the lasing within the gain spectrum bandwidth of a Ti:Sa crystal. The synchronized Pockels cell lets the seed pulses from a stretcher in with a 5 ns time window and a repetition rate of 1 khz. This provides for only one stretched pulse to enter an amplifier in a 1 ms period. This stretched pulse travels inside the amplifier cavity, thus gaining energy from the excited Ti:Sa amplifier crystal. Once it is amplified, the synchronized Pockels cell lets it out of the amplifier and it enters the compressor. 28

29 Experimental setup 5 J H A J? D A H + F HAII H + F HAII H 5JHAJ?DAH, A = O JA A I? F A & & # = J? 0 = F = JA.=H=@=O H J= J H = " F = JA F =HE AH / H = J E C / H = J E C 5AA@>A= ) F * A = H A F A JEJE H = JA 2? A I?A JD E BE F =HE AH 6 E 5=+ HOIJ= ) F EBEA H JA AI? F A A A II I? E = J H BI I E = J H 6 E 5 =?HOIJ= F H E I I 8-4, 6 1 BHAG K K> A@H ; 8 2 # 9? M #! BHAG K K> ; ) / #! 0 2 ' 9 A IA Figure 2.1: The schematic of a Ti:Sa regenerative amplifier. Image is adapted from the original drawing by Bernd v. Issendorf [31]. 29

30 THz-TDS iv) Grating-based pulse compressor. It gives a negative chirp to the amplified, but still strongly chirped pulse, thus after a round trip compressing it into a 100 fs transform-limited pulse with a cw average power of up to 0.85 W and the repetition rate of 1 khz. Thus, after the amplification we have 100 fs transform-limited pulses tuned around 800 nm with the energy of 0.85 mj/pulse, the average pulse power of 8.5 GW, the peak intensity of 45 GW/cm 2 (at 5 mm beam diameter), and, according to the eq. (2.1), the peak electric field strength of 1.3 MV/cm, delivered with the repetition rate of 1 khz. The pulse intensity stability of the system is within 3%. The schematic of the amplifier system is shown in the Fig In the Fig. 2.2 we show the intensity spectrum and the intensity autocorrelation trace of the Ti:Sa amplifier system tuned around 805 nm. In order to represent the autocorrelation on the true time scale, the time axis of the measured autocorrelation signal should be corrected. The intensity autocorrelation is given by S(t ) + I(t)I(t t )dt (2.2) If one assumes a Gaussian temporal shape of the laser pulse, the autocorrelation signal will be S(t ) + e t2 e (t t ) 2 dt = + e 2t2 e t (t 2t) dt (2.3) and therefore the autocorrelation trace will still have a Gaussian shape, but its full width at half maximum will be 2 times longer. Thus in order to arrive to a true time scale the time axis should be divided by 2. For the correction factors for various temporal shapes of the pulses one can consult [30] THz spectrometer based on ZnTe generation and detection crystals. Here we will describe the THz time-domain spectrometer based on optical rectification of the 100 fs laser pulses with the central wavelength of approximately 800 nm in the ZnTe crystal and detection of the THz pulses in another ZnTe crystal using free-space electrooptic sampling (FEOS) technique. Both emitter and detector ZnTe crystals are 110 -oriented and 1 mm - thick. ZnTe is one of the most popular THz emitters and detectors used with near-infrared laser sources since being a wide bandgap semiconductor (E g (ZnT e) = 2.28 ev) it is basically transparent at the laser pump wavelength, but shows good nonlinear properties. At 800 nm it has both large 2 nd order nonlinear susceptibility χ (2) = esu [32] and electrooptic coefficient r 41 = 4.04 pm/v [34]. In the Fig. 2.3 the schematic of a THz time domain spectrometer is shown. The 100 fs laser pulse with the central 30

31 Experimental setup Intensity [arb. units] Gaussian fit FWHM = 14 nm a) Wavelength [nm] Intensity autocorrelation Gaussian fit corrected time axis FWHM = 113 fs b) Time [fs] Figure 2.2: a) Spectrum of the regenerative amplifier output: measured data (circles) and a Gaussian fit (solid line) b) Intensity autocorrelation on the corrected time axis: measured data (squares) and a Gaussian fit (solid line). For the time axis correction see text. 31

32 F K F > A = ( L E I E > A H A = H 1 A = O 6 A J A? J E > A = ( & 6 0 C A A H = J E > A = ( & BH 6 E 5 = = F EBEA H THz-TDS 6 A J A? J H - B 16 EHH H 2 E = C A B J D A A EI I E I F J 5 ) , - B 16 EHH H 6 A A E J J A H + D F F A H Figure 2.3: Schematic of the THz setup. OEM - off-axis ellipsoidal mirror with θ in = θ out = 90 0, ITO mirror - indium-tin oxide coated quartz slab, VDL - variable delay line, POL - polarizer, the pump beam is another delayed fraction of the laser beam, which can be used for optical pump - THz probe experiments. 32

33 Experimental setup READ: (V A V B ) ZnTe < 110 > VWP PD A PD B PBS x NIR probe THz x y z 45 0 x 0 y y 0 z = z 0 E THz Figure 2.4: Electrooptic detection unit based on Pockels effect in a oriented ZnTe crystal. VWP - variable wave plate, PBS - polarizing beam spitter, PD A and PD B - photodiodes A and B, V A and V B are the voltages produced by PD A and PD B. Inset: principal axes of original (x 0, y 0, z 0 ) and distorted (x, y, z) refractive index ellipsoid. wavelength of around 800 nm excites a nonlinear polarization spike in the emitter ZnTe crystal, which produces the electromagnetic pulse with the frequency spectrum in the THz range. This process is described above by the equations (1.30, 1.17, and 1.18). The THz pulse is guided to the detector ZnTe crystal by the offaxis ellipsoidal (OEM) and plane mirrors. The OEMs are made of aluminum, and the plane mirrors are made of a thick quartz slab coated with conductive indiumtin oxide (ITO). These ITO mirrors allow for high transmission of light at optical wavelength, but are highly reflective in the THz range. In our setup we use a 1 : 1 imaging optical scheme, so that the emission spot of the emitter ZnTe crystal is imaged 1 : 1 onto the plane in the middle of the spectrometer (sample position), which is then imaged further onto the plane of the detector crystal. This, together with an emission spot of several millimeters in diameter (i.e. larger than the emitted THz wavelength), provides the frequency-independent THz radiation spots at the emitter, sample and detector. Once the THz pulse is transmitted to the surface of the detector ZnTe crystal, its electric field induces a birefringence in this crystal. This birefringence is probed by the phase retardation (see Eq. 1.39) of a very weak 800 nm laser pulse with a pre-aligned polarization. This probe laser pulse is temporally delayed with respect to the pump laser used for the THz emission using the variable delay line (VDL). We are therefore able to measure the phase retardation (which is proportional to 33

34 THz-TDS the electric field in the THz pulse) with a time resolution, given by the width of the THz detection probe pulse and the steps in time delay. Taking into account that both THz pump and THz probe laser pulses originate from the same laser beam and normally have the same duration, we arrive to the fundamental limitation of THz-TDS method: one can not measure the THz signals which are shorter in duration than the probe laser pulse itself. Therefore in a conventional THz- TDS setup the bandwidth of the measured THz pulse can not exceed that of the detection laser pulse. The schematic of the detection unit is shown in the Fig The principal axes of the crystal in the absence of THz electric field are oriented as follows: z 0 axis is collinear with z, and x 0 and y 0 belong to the x y plane. THz pulse is polarized along z 0 axis, and the probe laser pulse is polarized at 45 0 in respect to the THz pulse polarization plane. In the absence of THz electric field the probe pulse will be transmitted through the crystal without any modification, and after propagating through the variable wave plate (VWP) its polarization will become circular. This will provide the equal light intensity incident on the photodiodes PD A and PD B, and therefore the difference in their voltages will be zero. Once the electric field of the THz pulse is applied to the crystal, the principal axes transform as shown in the Fig. 2.4, and according to the Eq the x and y components of the probe laser pulse will be influenced by different refractive indices, which will result in the phase retardation between these components. After propagation through the VWP this phase-retarded optical probe pulse will become elliptically polarized, which will result in different light intensities incident on PD A and PD B. The difference in the voltages produced by PD A and PD B will be proportional to the induced phase retardation, and therefore (see Eq. 1.39) to the electric field strength in the THz pulse. Varying the delay between the THz pulse and optical probe one can therefore temporally sample the electric field in the THz pulse. Taking into account the Eq. 1.39, the measured electric field in the THz pulse in case of the FEOS in cubic electrooptic crystals can be estimated as E [T Hz] = 2 V c V max ωn 3 0r 41 d, (2.4) where V is the difference in the voltages between the photodiodes PD A and PD B. V max is the voltage produced by each of the photodiodes when they are illuminated with the probe beam. In case of no THz field V = V max V max = 0. n 0 and r 41 are the refractive index and the electrooptic coefficient of the electrooptic crystal at the frequency of the probe beam ω, respectively. d is the thickness of the crystal. The electrooptic signal V in our experiment was read by a commercial lock-in amplifier SR 830 DSP [35] at the frequency 125 Hz corresponding to 1/8 of the Ti:Sa amplifier repetition rate. The THz generation beam was modulated at this frequency with a chopper (see Fig. 2.3). For the ZnTe crystal probed at 800 nm n 0 = 3.22 [33] and r 41 = 4.04 pm/v 34

35 Experimental setup [34]. In case of 1 mm thickness, the electric field in the THz pulse is at least E [T Hz] = V [V/cm]. (2.5) V max THz generation by optical rectification of ultrashort laser pulses in nonlinear crystals was first demonstrated by the group of Y.R.Shen in 1971 [36]. The FEOS detection scheme was first introduced by the groups of X.-C.Zhang [37] and P.Uhd Jepsen and H.Helm [38] in Both of these techniques are widely used ever since and allow for coherent detection of the temporal evolution of the electric field in the ultrashort transients. Recently the generation and detection of THz pulses with the peak electric field strength on the order of 1 MV/cm was demonstrated using the optical rectification generation and FEOS detection schemes [39]. In the Fig. 2.5 the free space THz pulse generated and detected in our ZnTebased spectrometer and its amplitude and phase spectra are shown. This pulse was generated by the 800 nm central wavelength, 100 fs long laser pulse with the fluence of approximately 0.5 mj/cm 2. The electric field strength in the THz pulse was calculated according to Eq The THz pulse has a relatively complicated shape, with a main single-cycle oscillation positioned at around 2.6 ps time delay, which is followed by decaying anharmonic oscillations. The maximum signal-to-noise (S/N) ratio of this pulse is approximately This THz pulse has a useful bandwidth in the range THz, which is considerably smaller than the bandwidth of the 100 fs excitation laser pulse. The temporal shape and the bandwidth of a THz pulse produced by the optical rectification in an optically transparent nonlinear crystal is determined by the temporal shape and bandwidth of the excitation pulse as well as by the phase mismatch between the optical and THz signals co-propagating through the crystal, and the THz absorption in the crystal. These effects, as well as the method of simulation of the THz pulses will be described below. As was mentioned above (see Section 1.2) the 2 nd -order nonlinear susceptibility χ (2), responsible for the optical rectification, has the spacial anisotropy, as it mixes the polarizations induced in the different directions. It is analogous to the directional variance of the electrooptic effect, described in Section 1.4. Hence the induced polarization, and therefore the emitted THz signal should be dependent on the orientation of the ZnTe crystal in respect to the polarization of the excitation laser pulse. In the Fig. 2.6,a we present the THz pulses generated in the 110 -oriented ZnTe crystal rotated about its normal 110, at the various angles between the orientation of the z-axis and the polarization of the pump beam. All the pulses have identical shape, but not the absolute strength and phase. Depending on the crystal rotation angle some of the pulses have positive, and some of them have negative polarity. We will describe their polarity by the phase of the 1 st peak. The strength and the phase are governed by the magnitude of χ (2) and the direction of induced 2 nd -order nonlinear polarization. In the Fig. 2.7 the orientation of the crystal in respect to the pump beam polarization is shown. The induced polarization P = χ (2) E i E j is a mixed term of 35