Excitation of plasmon-polaritons in graphene via non-linear mixing of optical waves
Abstract
Em 2015, plasmões de superficie (SPPs) foram pela primeira vez excitados em grafeno por um processo não linear (NL) de diferença de frequências, causado pela interação de dois feixes óticos [1]. Cálculos teóricos, usando teoria de perturbações, preveem uma resposta muito mais fraca do que a necessária para explicar os dados experimentais. Experiências e resultados teóricos recentes revelam a importância dos SPPs na relaxação de portadores de carga em grafeno e apontam para a possibilidade da sua amplificação em condições fora do equilíbrio, por exemplo, quando uma folha de grafeno é incidida por feixes óticos muito intensos. No entanto, o mecanismo por trás da sua geração por diferença de frequências ainda não é totalmente compreendido. Nesta tese, calculamos as condutividades óticas não lineares relevantes ao processo e mostramos que as observações experimentais da Ref. [1] têm a sua origem em processos físicos para além da geração coerente de SPPs por diferença de frequências, descrita pela abordagem da teoria de perturbação da matriz de densidade. Também estudamos a dinâmica dos portadores de carga no grafeno quando estes se encontram fora do equilíbrio e apresentamos o nosso modelo de eletrões quentes para explicar a sua influência na geração de SPPs. Os resultados experimentais podem ser compreendidos se para além do processo de diferença de frequências de segunda ordem, considerarmos também a amplificação do campo SPP devido ao relaxamento interbanda de eletrões quentes. Para descrever esta amplificação na presença da geração por diferença de frequências, escrevemos uma equação de evolução para o número de ocupação de SPPs. Os nossos cálculos mostram que estes dois processos, ocorrendo em simultâneo, podem aumentar fortemente a taxa de geração não-linear de plasmões de superfície. Finalmente, o trabalho experimental, realizado para gerar SPPs através do mecanismo teorica mente considerado, também é apresentado. O principal desafio foi a contruir um esquema de deteção sensível o suficiente para observar a excitação dos SPPs. Várias técnicas, tais como deteção sensível de fase (lock-in amplifier) e deteção balanceada, foram exploradas. O esquema ótico foi construído, mas a potência do laser precisa ser estabilizada e consequentemente ainda não tivemos sucesso. No entanto, os métodos e resultados apresentados na parte experimental desta tese são uma referência importante para continuação do trabalho no futuro.
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Rui Jorge Pinto Dias Excitation of plasmon-polaritons in Graphene via non-linear mixing of optical waves setembro de 2023 UMinho | 2023 Rui Jorge Pinto Dias Excitation of plasmon-polaritons in Graphene via non-linear mixing of optical waves Universidade do Minho Escola de Ciências
Rui Jorge Pinto Dias Excitation of plasmon-polaritons in Graphene via non-linear mixing of optical waves Dissertação de Mestrado Mestrado em Física Trabalho efetuado sob a orientação do Professor Doutor Mikhail Igorevich Vasilevskiy Professor Doutor José Carlos Viana Gomes Universidade do Minho Escola de Ciências setembro de 2023
Copyright and Terms of Use for Third Party Work This dissertation reports on academic work that can be used by third parties as long as the internationally accepted standards and good practices are respected concerning copyright and related rights. This work can thereafter be used under the terms established in the license below. Readers needing authorization conditions not provided for in the indicated licensing should contact the author through the RepositóriUM of the University of Minho. License granted to users of this work: https://creativecommons.org/licenses/by/4.0/ ii
Agradecimentos Eu gostaria de começar por agradecer aos meus orientadores, Professores Mikhail Vasilevskiy e José Carlos Viana Gomes, por me terem orientado nesta dissertação de mestrado e pelas enumeras discussões que tivemos. O trabalho que aqui apresento não existiria sem o seu apoio. Aprendi muito com eles e apesar da enorme dificuldade do projeto, sinto-me feliz por o ter aceitado. Gostaria de também agradecer ao meu colega Diogo Cunha, pela sua ajuda na aplicação do modelo de eletrões quentes aos dados experimentais e por rever os cálculos numéricos, e ao Professor Michael Belsley e ao Dr. Peter Schellenberg, pela ajuda no laboratório e pelas discussões tivemos. Em especial, quero agradecer ao Dr. Manuel Rodrigues pela enorme ajuda no laboratório. Sempre se mostrou disponível e paciente comigo e salvou-me muitas vezes da minha própria asneira. Por fim gostaria de agradecer aos meus pais, à minha irmã e aos meus amigos por me sempre aturarem, mesmo nos meus dias mais melancólicos. iii
Statement of integrity I hereby declare having conducted this academic work with integrity. I confirm that I have not used plagiarism or any form of undue use of information or falsification of results along the process leading to its elaboration. I further declare that I have fully acknowledged the Code of Ethical Conduct of the University of Minho. University of Minho, Braga, September 2023 Rui Jorge Pinto Dias iv
Geração ótica não linear de plasmões de superfície em grafeno Resumo Em 2015, plasmões de superficie (SPPs) foram pela primeira vez excitados em grafeno por um processo não linear (NL) de diferença de frequências, causado pela interação de dois feixes óticos [1]. Cálculos teóricos, usando teoria de perturbações, preveem uma resposta muito mais fraca do que a necessária para explicar os dados experimentais. Experiências e resultados teóricos recentes revelam a importância dos SPPs na relaxação de portadores de carga em grafeno e apontam para a possibilidade da sua amplificação em condições fora do equilíbrio, por exemplo, quando uma folha de grafeno é incidida por feixes óticos muito intensos. No entanto, o mecanismo por trás da sua geração por diferença de frequências ainda não é totalmente compreendido. Nesta tese, calculamos as condutividades óticas não lineares relevantes ao processo e mostramos que as observações experimentais da Ref. [1] têm a sua origem em processos físicos para além da geração coerente de SPPs por diferença de frequências, descrita pela abordagem da teoria de perturbação da matriz de densidade. Também estudamos a dinâmica dos portadores de carga no grafeno quando estes se encontram fora do equilíbrio e apresentamos o nosso modelo de eletrões quentes para explicar a sua influência na geração de SPPs. Os resultados experimentais podem ser compreendidos se para além do processo de diferença de frequências de segunda ordem, considerarmos também a amplificação do campo SPP devido ao relaxamento interbanda de eletrões quentes. Para descrever esta amplificação na presença da geração por diferença de frequências, escrevemos uma equação de evolução para o número de ocupação de SPPs. Os nossos cálculos mostram que estes dois processos, ocorrendo em simultâneo, podem aumentar fortemente a taxa de geração não-linear de plasmões de superfície. Finalmente, o trabalho experimental, realizado para gerar SPPs através do mecanismo teoricamente considerado, também é apresentado. O principal desafio foi a contruir um esquema de deteção sensível o suficiente para observar a excitação dos SPPs. Várias técnicas, tais como deteção sensível de fase (lock-in amplifier) e deteção balanceada, foram exploradas. O esquema ótico foi construído, mas a potência do laser precisa ser estabilizada e consequentemente ainda não tivemos sucesso. No entanto, os métodos e resultados apresentados na parte experimental desta tese são uma referência importante para continuação do trabalho no futuro. Palavras-chave: plasmónica em grafeno; condutividade não linear; eletrões quentes; “laser” de plasmões v
Excitation of plasmon-polaritons in graphene via non-linear mixing of optical waves Abstract Several years have passed since surface plasmons-polaritons (SPPs) were excited for the first time in graphene by a frequency-difference nonlinear (NL) process caused by the interaction of two optical beams [1, 2]. Theoretical calculations, based on standard perturbation theory predict a much weaker frequency-difference response than it is required to explain the experimental data. Recent experimental results, along with theoretical calculations, demonstrate the importance of SPPs in the relaxation dynamics of non-equilibrium carriers in graphene and point out the possibility of their amplification in non-equilibrium conditions, e.g. under strong optical pumping. Nonetheless, the complete mechanism behind their generation by frequency-difference still remains to be explained. In this thesis, we calculate the relevant NL optical conductivities and show that, indeed, the experimental observations of Ref. [1] must have originated from physical processes beyond the coherent frequency-difference generation of SPPs described by the density-matrix perturbation theory approach. We study the dynamics of non-equilibrium carriers in graphene and present our hot electron model to explain their influence on the SPP generation. This model is used to understand the interplay between the SPPs and non-equilibrium carriers. We suggest that the experimental results of Ref. [1] can be understood by considering, in addition to the second-order frequency-difference process, the amplification of the SPP field via interband relaxation of hot electrons. We propose a master equation to describe this amplification in the presence of the frequency-difference generation. Our calculations show that these two processes, occurring simultaneously, can strongly enhance the rate of resonant generation of surface plasmons by two adjusted optical beams. Finally, experimental work was carried out to generate SPPs via the theoretically considered mechanism. The main challenge appeared to be their detection. Several techniques, such as phase sensitive (lock-in amplifier) and balanced detection, have been explored. The optical setup has been built, but the laser power still needs to be stabilized, so we have not been successful yet. Nevertheless, the methods and results presented in the experimental part of this thesis can be an important benchmark for successes in the future. Keywords: graphene plasmonics; nonlinear conductivity; hot electrons; plasmon lasing vi
Contents 1 Introduction 10 1.1 Generation of graphene SPPs by two optical beams . . . . . . . . . . . . . . . . . . . 10 1.2 Objectives and structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3 Units............................................. 14 2 Graphene plasmonics 16 2.1 Fundamentals........................................ 16 2.1.1 Hybridization of SPPs with substrate excitations . . . . . . . . . . . . . . . . 18 2.2 Electron-plasmon kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.1 Graphene’s electronic dispersion relation . . . . . . . . . . . . . . . . . . . . . 19 2.2.2 Electronic transitions via interaction with plasmons . . . . . . . . . . . . . . . 20 3 Nonlinear generation of SPPs in graphene 22 3.1 DF-field enhancement at the plasmon resonance . . . . . . . . . . . . . . . . . . . . 22 3.2 Optical response in graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.3 DF second-order conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.4 Reflection coefficients of the optical beams . . . . . . . . . . . . . . . . . . . . . . . 30 3.4.1 How to detect the optically generated SPPs? . . . . . . . . . . . . . . . . . . 32 3.5 Concludingremarks .................................... 33 4 Hot carriers in graphene 35 4.1 Carrier dynamics in graphene upon ultrafast optical excitation . . . . . . . . . . . . 35 4.2 Hotelectronmodel..................................... 38 4.2.1 Non-thermalized carriers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.2.2 Thermalized carriers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.2.3 Non-equilibrium conductivity due to hot carriers . . . . . . . . . . . . . . . . 45 4.2.4 Differential reflectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.3 Numerical results and concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . 47 vii
1.1 Generation of graphene SPPs by two optical beams SiO2 EI b EI a θa θb qa qbq ESP x z Figure 1.1: Schematics of all-optical generation of surface plasmons via a frequency-difference NL process of interaction of two optical beams, aand b, represented by their electric fields Ea,b and their incident angles θa,b.Esp represents the SPP field. qj(j=a, b)are the in-plane momentum of the incoming beams and q=qb−qais the SPP momentum. respectively. The DF process is expected to lead to the generation of SPPs of frequency ω=ωb−ωa. In spite of being centrosymmetric, with a second-order optical response that should vanish by symmetry [12], graphene can still display a nonzero second order optical conductivity if one uses oblique optical beams with nonzero and unequal in-plane components of the wavevector, thus effectively breaking graphene’s symmetry. This idea was experimentally implemented by Constant et al. [1] and investigated further in a subsequent article [13]1. These experiments used graphene on top of SiO2(quartz) and two 100 fs p-polarized laser pulses, a pump (pulse b), and a probe (pulse a), to excite SPPs, as shown in Fig. 1.1. They fixed the probe wavelength at λa= 615 nm, while the pump wavelength, λb, was varied between 615 and 545 nm. By recording the differential reflectivity of the probe, under the condition of energy and in-plane momentum matching for SPPs, they observed a resonance associated with SPP generation (see Figs. 1.2 and 1.3) with the frequency and momentum: ω=ωb−ωaq=qb−qa, (1.1) where ωj= 2πc/λjand qj= (ωj/c) sin θjexare the in-plane momentum of the pulses. The differential reflectivity is defined as: ∆R RΦb =R−R0 RΦb , (1.2) where Ris the reflectance of the probe when the pump is present, R0is the reflectance in the absence 1All-optical generation of SPPs was also reported in another publication [2]. However, this article does not contain sufficient details and is not clear enough, in my view. Therefore, it will not be discussed further in this thesis. 11
I Introduction 2 0 2 Delay time (ps) 0.00 0.02 0.04 0.06 R / R (mJ cm 2) Figure 1.2: Experimental results from Constant et al. [1], taken from figure 2 of the article. For this case, the incident angles were θb=−35◦and θa= 75◦.Left: SPP dispersion relation (black lines), for a damping rate γp≈2π×1.6ps−1and an effective Fermi level of 500 meV, and DF scan (qb−qa, ωb−ωa) as performed in the experiment (blue line). Optical phonons of the substrate are represented by the red dashed lines. Right: Measured differential reflectivity as a function of the time delay between probe and pump. The black curve shows a typical non-resonant measurement when the difference frequency between pump and probe (61.2 THz) does not match with a surface plasmon state. The red curve shows an additional symmetric contribution to the recorded reflection signal when the difference frequency matches the frequency of a SPP (23.8 THz), highlighted in the left figure by the red circle. of the pump and Φbis the pump’s fluence. The graphene sample used in their experiments was ndoped, with a natural Fermi level of 300 meV. They weakly focused the beams on graphene, with a spot size of approximately 300 µm in radius. The pump pulse fluence used was Φb∼0.1−0.2mJ cm−2while the probe fluence was two orders of magnitude smaller, Φa∼10−2Φb. They performed the experiment for three different geometries (different angles of incidence). The resonance observed in the differential reflectivity was interpreted to be caused by an exchange of energy between pump and probe due to excitation of the SPP field. In other words, pump and probe by DF created SPPs, which in their turn, interacted with the pump, also by a DF process, and generated photons at the frequency of the probe. Consequently, the observed resonance involves an interaction between three fields, therefore it is a third order process, composed of two intermediate second order processes. To model the SPP dispersion, Constant et al. used an effective Fermi level of 500 meV, which they justified as a consequence of photodoping, expected under illumination by ultrafast pulses [14], and introduced an effective formula for the second order conductivity: σ(2)(ω) = iσ(2) ω ωa , (1.3) where σ(2)is a fitting parameter. They accounted for the coupling between graphene SPPs and substrate (quartz) optical phonons according to Ref. [15]. Their main results are summarized in 12
1.1 Generation of graphene SPPs by two optical beams 210123 Delay time (ps) 0.00 0.01 0.02 0.03 0.04 0.05 0.06 R / Rb (mJ cm 2) 21012 Delay time (ps) Figure 1.3: Same as in Fig. 1.2, but with non-resonant and resonant signals plotted separately for better visualization. Geometry θbθaλb(nm) ν(THz) ∆R RΦbmJ cm−2σ(2)(fAmV−2) 1) 45◦35◦607 7.0 -0.0097 24 2) 40◦20◦597 12 -0.025 75 3) -35◦75◦587 23.8 0.062 180 Table 1.1: Main experimental results of Ref. [1] for the three different geometries. νis the DF frequency at which a SPP resonance was observed. σ(2)was extracted using the model of Ref. [1]. For all geometries λa= 615 nm, Φb∼0.1−0.2mJ cm−2and Φa∼10−2Φb. To calculate the SPP dispersion curve, they assumed a damping rate of γp≈2π×1.6ps−1and a effective Fermi level of 500 meV. This table was taken from article [16], where the authors of Ref. [1] further analyzed their results. Table 1.1. Even though the relation of the observed resonant peak in the differential reflectance to the generation of graphene surface plasmons looks doubtless, as we will see in the next Sections, the results in Table 1.1 deviate a lot from what had been predicted [9]. First, the resonant signal appears on top of a broader asymmetric band, which is also observed outside of the plasmon resonance. This asymmetric temporal variation of the differential reflectance was tentatively attributed to hotelectron effects [13] and it is not well understood. Secondly, theoretical calculations of the secondorder conductivity for the scheme in Fig. 1.1, using perturbation theory [9, 16, 17], predict a response orders of magnitude smaller than estimated by the experiment. Thus, one specific objective of this thesis was to understand the real reason behind the unexpectedly large signal observed by Constant et al. 13
I Introduction 1.2 Objectives and structure of the thesis As mentioned above, the main goal of this Master project was to understand the feasibility of generation of useful surface plasmons in graphene by combining two optical beams. The work presented here is divided into two parts. In the first part we study and model theoretically the experimental observations of Refs. [1, 13], namely, the origin of both non-resonant and resonant signals. The second part describes the experimental work done in an attempt to reproduce the results of Refs. [1, 13] and further study the phenomenon. The thesis consists of seven chapters and eleven appendices. In the next chapter (Chapter 2), fundamental (linear) properties of graphene plasmons and surface plasmon-polaritons are presented. In Chapter 3, the DF nonlinear response of graphene is calculated. The obtained formulae for the nonlinear conductivity are used to model the differential reflectivity. The results are incompatible with the observations in Refs. [1, 13]. In search for an answer, the dynamics of non-equilibrium carriers in graphene is studied and a hot electron model, to describe this carriers, is presented. These are the subjects of Chapter 4. The fifth chapter deals with SPPs in non-equilibrium graphene and proposes a new mechanism, SPP lasing, to explain the experimental observations. Chapter 6 presents the carried out experimental work. The use of an optical parametric amplifier to control the frequency difference between beams is discussed as well as the best approach to detect the generation of SPPs. The unsolved technical issues are addressed at the end. Finally, Chapter 7 sums up the study. 1.3 Units The system of units used in this work was the Gaussian one (CGS). However, the nature of the problem is better understood and the calculations are more easily performed in non-system units µm / ps−1/ meV. The reason is that in these units the Fermi velocity in graphene is: vF= 1 µm ps−1. (1.4) To make things easier for the reader, I present here the necessary fundamental constants in such units. Namely, Plank’s constant and the velocity of light are: ℏ= 0.658 meV ps , c = 300 µm ps−1. (1.5) The fine structure constant is: αf=e2 ℏc=1 137 (1.6) and this gives us an easy way of determining the electron’s charge ein this units. Finally, as useful conversion relations, we have 1W= 6.24 ·109meV ps−1(1.7) 14
1.3 Units and: W cm−2= 62.4meV ps−1µm−2(1.8) As a convention, the Greek letter νis frequency and ωthe radial frequency: ω= 2πν ; (1.9) νis measured in THz, while ωin ps−1. 15
Chapter 2 Graphene plasmonics In this chapter, we introduce basic concepts of the electronic, optical, and plasmonic properties of graphene, which serve as background for the detailed discussions in the following chapters. Details of the calculations can be found in the Appendix. 2.1 Fundamentals SPPs are evanescent electromagnetic (EM) waves coupled to the free electron plasma oscillations, that propagate along the surface of a conductor [6, 7] and decay exponentially away from it. In our case, the conductor is a graphene sheet placed between two dielectrics, for example air and SiO2. In graphene, we can distinguish between two types of SPPs [6, 7]: transverse magnetic (TM) SPPs and transverse electric (TE) ones. We are only interested in the TM-SPPs and will not consider TE-SPPs1. To introduce the necessary concepts, consider a graphene sheet between two dielectrics, with dielectric constants ε1and ε2, referring respectively to the top and bottom dielectric, as shown in Fig. 2.1. For simplicity, assume monochromatic solutions of Maxwell’s equations. For an SPP with 1Since, these excitations are composed of electrons coupled to a EM field, the word polariton should be used when addressing them. However, since in this thesis, there is no room for confusion, for simplicity I might call SPPs, surface plasmons or simply plasmons. x z Medium 2 Medium 1 Graphene Figure 2.1: Illustration of a single graphene sheet (blue line) between two insulators with dielectric constants ε1and ε2. 16
2.1 Fundamentals a frequency ωand a wavevector q=qex, propagating along the x-direction, its EM field has the form: Esp = (E2,x,0, E2,z)eiqxeiκ2ze−iωt z < 0 (E1,x,0, E1,z)eiqxe−iκ1ze−iωt z > 0 , Hsp = H2eyeiqxeiκ2ze−iωt z < 0 H1eyeiqxe−iκ1ze−iωt z > 0 . (2.1) Inserting the fields (2.1) into Maxwell’s equations (see Appendix A for details) we obtain for κmand field components: κ2 m=q2−ω2 cεm, (2.2) Em,z =i(−1)m+1 q κm Em,x . (2.3) From the boundary conditions including the continuity of the tangential component of the electric field and discontinuity of the normal component, Ej,z, when there is a surface charge present, assuming linear optical response of graphene, we find the SPP dispersion equation: D(ω, q) = ε2 κ2 +ε1 κ1 +4πi ωσ(1)(ω, q)=0. (2.4) Its solution yields the SPP dispersion relation ω(q). Here σ(1)(ω, q)is graphene’s non-local linear conductivity. The solutions to (2.4) are real or complex, depending on whether the real part of the conductivity vanishes or not. Since we shall include losses in the conductivity, its real part will be non-zero, so the solutions will be complex. We can either look for solutions with a complex wavevector q: q(ω) = Req(ω) + iImq(ω)(2.5) or with a complex frequency: ω(q) = Reω(q) + iImω(q)(2.6) The choice depends on the physical situation. If we have a continuous excitation of SPPs, it is better to consider qcomplex. On the other hand, if SPPs are excited by pulsed radiation, it is better to take ωcomplex. We shall use both choices, so it will be specified whenever necessary. An example of the plasmonic dispersion curve is shown in Fig. 2.2. At the frequencies of interest (low SPP frequencies), σ(1)(ω, q)is well approximated by the local conductivity σ(1)(ω). There are two contributions to σ(1)(ω)coming from intraband and interband optical transitions: σ(1)(ω) = σintra(ω) + σinter(ω), (2.7) where σintra is the Drude contribution, describing intraband transitions, and σinter denotes the con17
II Graphene plasmonics 0 20 40 60 Re(q) ( m 1) 0 10 20 30 40 (THz) 0 20 40 60 Re(q) ( m 1) 10 1 100 Im(q) ( m 1) Figure 2.2: (right) Plasmon dispersion curve ω(q)(blue line), calculated using both the Drude and interband conductivities. In this calculation qwas chosen to be complex. The light line is shown in green and the black line represents the electronic dispersion, below which we have the Landau Damping region (shaded region). (left) Losses. The calculation parameters were: Fermi energy EF= 300 meV, ε1= 1 and ε2= 2.4, electron momentum relaxation rate γp= 2π×2.6 ps−1, interband relaxation rate γinter = 0.1ps−1(added to to avoid bad numerical behavior) and a temperature T= 300 k. tribution of interband transitions. For the Drude term (see Appendix B.2) at equilibrium: σintra(ω) = 4σ0i π kBT ℏ(ω+iγp)ln n1 + eEF/kBT1 + e−EF/kBTo , (2.8) while for the interband term we have: σinter(ω) = 2σ0 πZ+∞ 0 dε (fv(ε)−fc(ε)) 1 ℏγinter −i(2ε+ℏω)+1 ℏγinter +i(2ε−ℏω). (2.9) Here EFis graphene’s Fermi energy with respect to the Dirac point, γpis the electron momentum relaxation rate, γinter is the interband scattering rate and σ0=πe2/(2h). For the plasmon frequencies of interest, the Drude term is much bigger than the interband one. Thus unless stated otherwise, only the Drude term will be used to describe SPPs in graphene. 2.1.1 Hybridization of SPPs with substrate excitations If graphene is placed on top of a polar substrate, such as SiO2, plasmons will couple to optical phonons of the substrate, leading to new hybrid modes called surface plasmon-phonon polaritons [7] (for simplicity we also refer to them as SPPs). The coupling between optical phonons and SPPs is easily introduced by considering a frequency dependent dielectric function for the substrate of the form [7]: ε2(ω) = ε∞+X j fjω2 TO,j ω2 TO,j −ω2−iωγTO,j , (2.10) 18
2.2 Electron-plasmon kinematics where ωTO,j are the phonon frequencies, γTO,j the damping rates, fjthe oscillatory weights and ε∞, high-frequency dielectric constant. This expression works well in THz and the mid-IR regimes. For SiO2, from Ref. [15], the high-frequency dielectric constant is ε∞= 2.4while the optical phonon frequencies and oscillatory strengths are: ωTO = 2π(13.44,23.75,33.84) ps−1(2.11) and f= (0.7514,0.1503,0.6011) , (2.12) respectively. The damping rates are: γTO = 2π(0.80,1.27,1.27) ps−1. (2.13) An example of the resulting dispersion relation is plotted in Fig. 1.2. 2.2 Electron-plasmon kinematics 2.2.1 Graphene’s electronic dispersion relation For small energies, most electronic processes in graphene can be described by approximating the electronic dispersion relation as linear near the Dirac point [7]: εs,k=sℏvFk, (2.14) with s=−1(+1) for the valence (conduction) band, vF= 106m/s the Fermi velocity and kis the electron’s wavevector. This approximation is known as the Dirac cone approximation. Equation (2.14) is symmetric with respect to zero energy and displays particle-hole symmetry. In this approximation, electrons are well described by an effective Hamiltonian: H0=vFσ·p, (2.15) where σ= (σx, σy)with σjdenoting the j-th Pauli Matrix and pthe momentum operator. The eigenstates of Hamiltonian (2.15) are: |k, s⟩=χs(k)|k⟩, (2.16) where: |k⟩=eik·r √S, χs(k) = 1 √2"e−iθ( k) s#. (2.17) Here, Sis the graphene’s area and θ(k)is the angle the electron momentum does with the x-axis. 19
II Graphene plasmonics εk SPP EFEF X a) b) c) Figure 2.3: Electronic transitions by emission or absorption of a plasmon (SPP), represented by yellow arrows. The electrons are depicted by the blue circles and holes by the white ones. a) Intraband transition with absorption of a plasmon, in equilibrium, with Fermi energy EF. b) Interband transition forbidden by Pauli blocking. In equilibrium, plasmons with low energy cannot promote electrons to the conduction band, since the states are already occupied. c) Out of equilibrium, with population inversion, these transitions are no longer Pauli blocked. 2.2.2 Electronic transitions via interaction with plasmons Since SPPs possess a certain dispersion relation and their EM field can be quantified, they can be treated as quasiparticles. In the subsequent Sections, it will be important to understand the role of plasmons in electronic transitions, so here we study the kinematics of electron-plasmon interactions. An electron can be scattered by a plasmon and either jump to a state in the same band (intraband transition) or to the other band (interband transition), with creation or destruction of an electronhole pair. Let us consider first the case of intraband transition, shown in Fig. 2.3 a). If the electron is scattered from a state |ki, s⟩to a state |k, s⟩by absorption of a plasmon with energy-momentum “coordinates” (q, ωq), energy and momentum conservation conditions read: εk−εki=ℏωq k−ki=q . (2.18) By choosing a reference frame such that q=qex, we find that the possible electron momenta, which are solutions to system (2.18), lie in the hyperbola: (kx−q/2)2 a2−k2 y b2=k2 0, (2.19) 20
3.3 DF second-order conductivity |2'⟩ kx SP SP EF |1⟩ |2⟩ |3⟩ |3'⟩ |1'⟩ |2'⟩ kx εk photon b SP EF |1⟩ |2⟩ |3⟩ |3'⟩ |1'⟩ Figure 3.1: (Left) Transitions and states considered in the calculation of the conductivity for DF generation at frequency ωand ωa: Channel 1 (right) and Channel 2 (left). (Right) For the process of generation of pump photons (ωb), the arrows are inverted. between the states: |1⟩=|−1,k1⟩,|2⟩=|1,k2⟩,|3⟩=|1,k3⟩, (3.16) and the intraband transition in the conduction band, and channel 2 involving the states |1′⟩=|−1,k1′⟩,|2′⟩=|−1,k2′⟩,|3′⟩=|1,k3′⟩, (3.17) with the intraband transition in the valence band. For the case of DF generation of the field oscillating with the frequency ω=ωb−ωaand momentum q=qb−qa, we have Ω= (ω, q)and Ωa,b =ωa,b,qa,b. The only relevant component of the conductivity tensor, for the experimental arrangement of Fig. 1.1 is the one with i=j=k=x, so the general expression (3.14) takes the form: σ(2) xxx (Ω,Ωb,−Ωa) = e3 ℏ2ωbωaSX 123 vx 23,−qvx 31,qbvx 12,−qa ω32 −ω−iγ32 f(ε1)−f(ε3) ω31 −ωb−iγ31 −f(ε2)−f(ε1) ω12 +ωa−iγ12 +e3 ℏ2ωbωaSX 1′2′3′ vx 1′2′,−qvx 3′1′,qbvx 2′3′,−qa ω2′1′−ω−iγ2′1′f(ε3′)−f(ε2′) ω2′3′+ωa−iγ2′3′−f(ε1′)−f(ε3′) ω3′1′−ωb−iγ3′1′, (3.18) where we retained only resonant terms. Evaluation of the matrix elements of the modulated velocity operator (3.15) can be done in different ways, the most straightforward is considering the velocity operator v=vFσ(see Appendix B.3 for details). In principle, in the velocity gauge one has to calculate them with the full Hamiltonian including the interaction [21, 22]. However, in our case, the interaction operator commutes with eir·qand we can use the unperturbed Hamiltonian (without 27
III Nonlinear generation of SPPs in graphene A). The matrix elements contain Kronecker δ-s that cancel out two summations in Eq. (3.18) and yield the conservation of momentum q=qb−qa. To compute Eq. (3.18), we expand the matrix elements vj mn,q, the energy differences, ωmn, and the Fermi–Dirac distributions to the first order in the wavevectors qjand q. For example, for a state |n⟩=|1,k′⟩with k′=k+q, f(εn) = fc(εk′)≈fc(εk) + ∂εkfc(εk)qj·∇kεk. (3.19) We denote by fc(ε)the distribution function for the electrons in the conduction band and by fv(ε) = fc(−ε)the distribution for electrons in the valence band. Additionally, for simplicity, we set equal all phenomenological damping constants, γnm =γ. With these approximations, considering both channels, the zero-th order terms in momentum cancel out, and the lowest order term in the conductivity is proportional to q, thus yielding a finite result. At zero temperature, the following analytical expression is obtained: σ(2) xxx (Ω,Ωb,−Ωa)≈δq,qb−qa e3v2 FEF ℏ3ωbωaπ2 2ωF−ωa (2ωF−ωa)2+γ2qA(ω, q, γ), (3.20) where ωF=EF/ℏis the Fermi frequency and A(ω, q, γ) = π(ω+iγ) (vFq)4 (vFq)2+ 2(ω+iγ)2 s1−vFq ω+iγ 2 −1 (3.21) is an auxiliary function, which is independent of qin the limit q→0: lim q→0A(ω, q, γ) = −π 4 1 ω+iγ . (3.22) Figure 3.2 shows the behavior of the real and imaginary parts of σ(2) xxx (Ω,Ωb,−Ωa)as functions of ωand q, for two values of the Fermi energy. The results obtained by the analytical formula (3.20) are confirmed by direct numerical evaluation of Eq. (3.18). In the left panel, for a fixed value of q, the real part of the conductivity peaks at ω≈vFq. An important feature of Eq. (3.20) for the second-order conductivity σ(2) xxx (Ω,Ωb,−Ωa)is that it goes to zero as q→0, as it should for a centrosymmetric material. This physically meaningful feature was lacking in the results of Ref. [11]. The reason for this probably is the problem with intraband matrix elements for the roperator, first pointed out by Blount [20]. The analytical formula derived in Ref. [16], although showing the correct limiting behavior for q→0, is somewhat different from ours and requires more work. It is important to emphasize that the points in Fig. 3.2 represent the results of direct numerical evaluation of Eq. (3.18), performed without approximations such as expansions in terms of wavevectors, so these spectra must be correct within the model and approach used. According to Eq. (3.8), it is the imaginary part of the conductivity that is important to describe 28
3.3 DF second-order conductivity 0 25 50 75 100 125 150 175 ( ps 1 ) 6 4 2 0 2 (2) (cgs units) Re (2) Im (2) 0246810 q ( m 1) 1 0 1 2 3 4 5 (2) (cgs units) Re (2) Im (2) Figure 3.2: Real and imaginary parts of σ(2) xxx (Ω,Ωb,−Ωa)plotted vs ω(left) and q(right). The solid and dashed lines were calculated using expression (3.20) with Fermi levels EF= 300 and 500 meV, respectively. The triangles and squares are numerical calculations of expression (3.18). In the right panel, θI bwas fixed at 40◦, while in the left plot, ωwas fixed at 94ps−1. In both cases ωa= 3065ps−1,γ= 10.6meV and θI a= 20◦. the SPP generation process, so from Fig. 3.2, Imσ(2) xxx (Ω,Ωb,−Ωa)≈3esu ≈0.1fA m V−2 for ω= 50 ps−1. Even though it is larger than the value calculated in Ref. [16], it is below the experimentally estimated value in Table 1.1 by some 3 orders of magnitude. Similar calculations can be carried out for the conductivities oscillating with the frequencies ωaand ωb, representing the DF generation of photons aand sum frequency generation of photons b, respectively. Diagrammatically, the generation of photons bis the same as for SPP generation, but with the arrows inverted, Fig. 3.1 (right). Analytical expressions, obtained with the same approximations as Eq. (3.20), are: σ(2) xxx (Ωb,Ωa,Ω)≈ −δq,qb−qa e3v2 FEF ℏ3ωaωπ2 q 2ωF−ωa−iγ A(ω, q, γ), (3.23) σxxx (Ωa,Ωb,−Ω)≈δq,qb−qa e3v2 FEF ℏ3ωbωπ2 q 2ωF−ωa−iγ A(ω, q, −γ). (3.24) We notice that in limit γ→0, 1 ωa|σxxx (Ωa,Ωb,−Ω)|=1 ωb|σ(2) xxx (Ωb,Ωa,Ω)|=1 ω|σxxx (Ω,Ωb,−Ωa)|, (3.25) that is our expressions exhibit the symmetry expected in lossless media. 29
III Nonlinear generation of SPPs in graphene 3.4 Reflection coefficients of the optical beams The reflection coefficients of the optical beams are affected by the the SPP generation and this is how the latter was detected in Ref. [1]. Here we present the reflection and transmission coefficients and calculate them using the results of the previous Section. The model presented in this Section is similar to the one used by Constant et al. [1]. There are two differences: 1. We will not take into account the coupling between SPPs and optical phonons. This only affects the position of the resonance, not its order of magnitude. So, for the time being, we confine ourselves by this simpler approach; 2. We shall use expressions (3.20), (3.23) and (3.24) for the nonlinear conductivity, not the model (1.3), introduced by Constant et al. [1]; We consider a situation similar to the experiment, with graphene cladded by two dielectrics with dielectric constants ε1= 1 and ε2= 2.4and two p-polarized optical waves impinging on it (see Fig. 3.3). The reflection and transmission coefficients, rjand tj, respectively, are defined as usual: rj=ER j/EI j, tj=ET j/EI j, (3.26) with j=a, b.EI jis the incident field and ER(T) jis the reflected (transmitted) field. The calculation is carried out in a standard way [23]. Graphene’s current must include a NL term, such as in Eq. (3.7), so, for fields aand b, the nonlinear currents are: j(2) x(ωa, qa) = 1 2σ(2) xxx (Ωa,Ωb,−Ω)Eb,x(z= 0)E∗ SP,x(z= 0) (3.27) and: j(2) x(ωb, qb) = 1 2σ(2) xxx (Ωb,Ωa,Ω)Ea,x(z= 0)ESP,x(z= 0) , (3.28) respectively, with ESP the SPP field generated by DF mixing, as given in Eq. (3.8) and described in Section 3.1. To put it in another way, we are describing the differential reflectivity, recorded in the experiment, as as a measure of exchange of energy between beams band a, that is mediated by the plasmon field. From the boundary conditions, the transmission coefficients are (see Appendix C for details): ta=t(0) a 1−2t(0) a|t(0) b|2π2iσ(2) xxx (Ωa,Ωb,−Ω)σ(2)∗ xxx (Ω,Ωb,−Ωa)|EI b|2cos2θT b √ε1cωD∗(ω, q)cos θT a!, (3.29) tb=t(0) b 1+2t(0) b|t(0) a|2π2iσ(2) xxx (Ωb,Ωa,Ω)σ(2) xxx (Ω,Ωb,−Ωa)|EI a|2cos2θT a √ε1cωD(ω, q)cos θT b!, (3.30) 30
3.4 Reflection coefficients of the optical beams x z Medium 2 Medium 1 Graphene kjIkjR θjI θjT kjT Figure 3.3: Scheme of p-polarized waves propagating from medium 1 to medium 2, with graphene at the interface z= 0. The incident angle for wave jis θI j, while the transmitted is θT j.kI jand kT jare the incident and transmitted wavevectors. where t(0) jdenotes the transmission coefficient in the absence of the second pulse, t(0) j=2(ε1/kI jz) sec θT jcos θI j (ε1/kI jz)+(ε2/kT jz) + 4πi ωj σ(1)(ωj) , (3.31) kI(T) jz is the z-component of the incident (transmitted) wave and D(ω, q)is the SPP dispersion relation (2.4). The corresponding reflection coefficients are given by: rj= 1 −sec θI jcos θT jtj(j=a, b), (3.32) with θI jand θT jthe incidence and transmission angles. Notice that at the SPP resonance condition ReD(ω, q)=0, and the reflection is enhanced. From Eqs. (3.29) and (3.30), we see that the transfer of energy between the two optical fields is resonant under the condition of DF phase matching between the incident beams and the SPP frequency and momentum. This process, in spite of being due to the second-order response of graphene, is in fact of the third order in optical fields. Figure 3.4 presents the spectral variation of the differences in the reflectance, ∆R=R−R0, transmittance, ∆T=T−T0, and absorbance, ∆A=A−A0(the quantities with the subscript 0correspond to the absence of beam b), for the situation presented in Supplementary Material of Ref. [1] where θI a= 20◦and θI b= 40◦. All these spectra show the SPP resonance at ωSP = 80 ps−1. The reflectance of the probe beam in Fig. 3.4 shows a dip at resonance, but it could be a peak if θI awere larger than the Brewster angle (situation corresponding to Fig. 2 of Ref. [1]) because ∆Ra≈(r(0) a∆r∗ a+r(0)∗ a∆ra),r(0) ais nearly real and changes its sign at the Brewster angle, while 31
III Nonlinear generation of SPPs in graphene 50 100 150 ( ps 1 ) 0.2 0.1 0.0 0.1 0.2 ( A / A 0) 109 Probe Pump Figure 3.4: Differential reflectance (left, solid lines), transmittance (left, dashed lines) and absorbance (right) of the pump (blue curves) and probe (green curves) beams as functions of ω. Here the Fermi energy EF= 500 meV, θI a= 20◦and θI b= 40◦, while other parameters are the same as in Fig. 3.2. The plasmon resonance is observed at approximately ωSP = 80 ps−1. We assumed the same intensities for both optical beams, Ia=Ib= 1016 erg cm−2s−1, in order to facilitate the comparison. its variation owing to the NL effect, ∆ra, does not. The chosen value of the intensity of the pump, Ib= 1016 erg cm−2s−1, approximately corresponds to the fluence of 0.1 mJ/cm2and pulse duration of 0.1 ps, mentioned in Ref. [1]. From figures S1 of Supplementary Material of Ref. [1] we can find the depth of the resonant minimum of ∆R/R0≈1.6·10−3, while our calculations predict a value of the order of 10−10 at the SP resonance. Since ∆Ris quadratic in the second-order conductivity, our calculated result for it is lower than the value extracted from the experiment in Ref. [1] by roughly 3 orders of magnitude, as already mentioned before in Section 3.3. 3.4.1 How to detect the optically generated SPPs? As pointed out in the previous Section, the differential reflectance method [1, 13] relies on a process that is of the third order with respect to the optical fields. Considering the optical beams as plane waves, it seems that the variation of the reflection coefficients of these beams is small and consequently hard to detect, unless very high intensities of the pump beam are used. Then other (non-electronic) types of nonlinearity can arise, for instance, due to photothermal effects [16], which is less interesting from the point of view of applications. Thus, it might be preferable to try to detect the optically generated surface plasmons using another route. Time-resolved THz spectroscopy is a powerful technique in the spectral range relevant to graphene SPPs [24] and it could help detecting the flux of plasmons escaping from the optical spots where they were generated. Below we shall evaluate this flux. Let us assume for simplicity that the optical beams are focused by cylindrical lenses, so that the system is uniform along the y-direction. As a first approximation, we neglect the uncertainty of the 32
3.5 Concluding remarks in-plane components of the wave vectors qaand qb, as if they were plane waves. In this situation, the energy flux in the x-direction associated with the generated SPPs, per unit length along y, is (see Appendix D for details): J=qω 16πX m=1,2 εm κ3 m|Em,x(ω, q)|2∝ |D(ω, q)|−2. (3.33) At the plasmon resonance, the evanescent field is strongly enhanced and the energy flux increases. Since SPPs are dissipative because of the Ohmic losses, we can neglect the inverse process of optical photon generation with the propagation of SPPs. Thus, the flux, Eq. (3.33), is responsible for the removal of energy from the optical beams. If the incident energy fluxes are J(I) a,J(I) b(per unit length along y) and if both reflected and transmitted beams aand bare measured, it should be possible to detect a variation of the total energy of both optical beams since, at the plasmon resonance, it should be diminished by Jbecause of the energy conservation, J(I) a+J(I) b=J(R) a+J(T) a+J(R) b+J(T) b+La+Lb+J, (3.34) where Lstands for losses associated with the transmission and reflection of the optical beams at graphene-covered surface, which are related to Reσ(1)(ωa,b)and, therefore, can be evaluated. Detection of a positive difference between the left-hand side and the parenthesis in the right-hand side of Eq. (3.34) will indicate the SPP generation, J > 0. Another possibility is to try to detect the SPP flux by converting it into propagating THz light, as it was done in the first demonstrations of graphene surface plasmons [25, 26]. The advantage in this approach, is that the variation is a secondorder process with respect to the optical fields, consequently, it should require lower intensities of the optical beams. 3.5 Concluding remarks It is clear that the calculated value of the second-order conductivity of optically pumped graphene is too low (by approximately 3 orders of magnitude) to explain the experimental results of Ref. [1]. This should be expected since, in the spectral range of interest, the electronic transition for SPP emission, shown in Fig. 3.1, has a very low probability to happen, because intraband transitions via SPP emission/absorption are only phase matched for plasmons with large momenta (see Section 2.2). Thus, there must be an important effect we did not include in the model or the perturbative approach is not valid because of the high intensity of the pump pulse. The answer seems to lie in the incoherent part of measured differential reflectance, non-resonant curve in Fig. 1.2. This signal is probably due to out-of-equilibrium electrons. In a recent work [27], it was observed that this hot electrons play a crucial role in the efficiency of terahertz highharmonic generations. Therefore, it may be that the presence of the incoherent part in the measured differential reflectance signal cannot be ignored. In fact, theoretical works [28, 29] suggest that out-of33
III Nonlinear generation of SPPs in graphene equilibrium graphene has the necessary conditions for plasmon amplification and this could account for the huge signal experimentally observed. So, we have to find out how these hot electrons behave and how they can participate in the generation of SPPs. Unfortunately, in the presence of hot electrons, the use of a perturbative approach to describe the electronic nonlinearities and the nonlinear optical coupling to plasmons is no longer valid in graphene [30]. An alternative could be the use of a hot-electron model to describe the non-equilibrium electrons generated with strong, ultrafast light pulses. In the next two chapters, we explore these ideas. 34
Chapter 4 Hot carriers in graphene As discussed in the previous Section, the signal observed by Constant et al. [1] cannot be explained in terms of the standard perturbation theory. Although the idea that the observed signal was due to generation of SPPs by the DF process is not ruled out, the observed resonance is too strong to be attributed to a non-linear process alone. A clue to this enigma, I believe, lies in the asymmetric line shape of the background signal in Fig. 1.2, measured when pump and probe are out of resonance. This signal probably comes from the relaxation dynamics of the out-of-equilibrium electrons [31, 32] excited by pump. At high pump intensity, the shape of the Dirac cone promotes the formation of an inverted population state [33] between holes and electrons. It has been proposed [29, 34] and experimentally shown [35] that graphene SPPs play a role in the dynamics of these hot carriers and vice-versa. In particular, theoretical calculations suggest that the SPP field can experience gain via stimulated plasmon emission in photoinverted graphene via electron-hole recombination [29, 36]. In order to understand how plasmons can experience gain, we must first understand the dynamics of hot carriers in graphene. Thus, the subject of this chapter is the study of the dynamics of hot carriers in graphene. We organize the chapter as follows: in Section 4.1 the dynamics of hot carriers in graphene is discussed, with its several stages and relaxation channels that have been observed. In Section 4.2, we introduce a model based on microscopic rate equations that describes this dynamics. Finally, our calculated results are compared with experiment, in Section 4.3. 4.1 Carrier dynamics in graphene upon ultrafast optical excitation We start by discussing the ultrafast dynamics of non-equilibrium electrons in graphene [37], when hot carriers are excited optically with an ultrafast laser pulse, usually a femtosecond pulse. We focus on interband excitation, when the incident pump photon energy ℏωbis high enough to promote electrons to the conduction band, that is, when ℏωb>2EF, with EFbeing the Fermi level. When an ultrafast laser pulse impinges on graphene, electrons from the valence band are excited to the conduction band. Two populations of non-thermalized particles are created: the excited 35
IV Hot carriers in graphene electrons in the conduction band and the respective holes in the valence band. These non-thermalized carriers may block the promotion of new electrons to the conduction band. Also, they evolve to different thermal distributions, with their own temperature. Such electrons and holes, which are in equilibrium between themselves but not with the crystal lattice are called "hot". The true equilibrium of such hot carriers is eventually achieved. The dynamics of these hot carriers in graphene is nowadays understood to occur via several stages and decay channels [37] (see Fig. 4.1). In the first stage, the promptly excited carriers with a non-Fermi-like distribution primarily undergo carrier-carrier scattering on a 10-fs time scale [38, 39] and two population of hot carriers, one of electrons and another of holes, are formed. These populations can be described as Fermi-like distributions, with different chemical potentials but an equal temperature, different from that of the lattice [30]. For longer timescales, the relaxation of hot carriers towards their equilibrium state is accomplished via several stages and decay channels, such as optical phonon emission [31], Auger recombination and hot plasmon emission [34, 35, 40], on a timescale of hundreds of femtoseconds, as well as direct and disorder-assisted acoustic phonon emission on picoseconds scale. Thermalization with the lattice is achieved before the electrons and holes have reached equilibrium between themselves (i.e, the electrons and holes quasi-Fermi levels become equal, Fe=Fh=EF). Re-establishing the equilibrium concentrations of electrons and holes is the slowest process in graphene. 36
4.2 Hot electron model To describe the evolution of thermalized carriers, we write the balance equations for ntand pt: ˙nt=nnt τee −nt−n0 τr , (4.34) ˙pt=pnt τee −pt−p0 τr , (4.35) where n0and p0are the equilibrium electron and hole concentrations1. On the other hand, the instantaneous thermalized electron concentration can be related to a quasi Fermi level Fe(t)and electronic temperature T(t): nt(t) = gsgv SX k fe(εk, Fe(t), T(t)) = 2 πkBT(t) ℏvF2 Γ (2) F1Fe(t) kBT(t), (4.36) where gsand gvare the spin and valley degeneracies for graphene and Fn(z)is the Fermi integral: Fn(z) = 1 Γ(n+ 1) Z+∞ 0 dx xn 1 + exp (x−z), (4.37) with Γ(n)being the Euler gamma function. Equations (4.34) and (4.36) establish a relation between Fe(t)and T(t). To completely determine these two functions we need a second equation. At the smaller timescale ∼τε, the thermalized electrons, having been scattered by phonons, impurities, and other lattice imperfections, give in their energy, εt, to the lattice. We write the energy relaxation for electrons also in the form of a balance equation: ˙εt=εnt τee −εt−ε′ 0 τε , (4.38) where εnt is the energy of the non-thermalized electrons. This energy is: εnt =Z+∞ 0 dεερ(ε)g(ε, t) = exp −ℏ2ω2 b 8∆2 Eρℏωb 2∆2 ε˜g(t) + √2π∆ερℏωb 2ℏωb 2+2∆2 ε ℏωb˜g(t). (4.39) Since 2∆ε≪ℏωb, using Eq. (4.28), εnt approximates to: εnt ≈ℏωb 2nnt(t)(4.40) 1The second term in Eqs. (4.34) and (4.35) represents e-h recombination. This recombination is a nonlinear two-particle process [30] that is proportional to the product of the electron and hole densities. This recombination rate should be written as αrec (ntpt−n0p0)with αrec called the recombination coefficient. However, at relatively low excitation levels (which may be the case in our situation, at the latest stage of the relaxation process when the interband recombination becomes important) we may introduce a recombination time by writing the recombination coefficient as αrec =1 τr(n0+p0). In this approximation, we can decouple the balance equations for electrons and holes and write them as Eq. (4.34) and (4.35). 43
IV Hot carriers in graphene and Eq. (4.38) may be simplified: ˙εt≈ℏωb 2 nnt(t) τee −εt−ε′ 0(t) τε , (4.41) where ε′ 0(t)is the energy of the electron distribution when thermalized with the lattice at temperature T0: ε′ 0(t) = gsgv SX k εkfe(εk, Fe(t), T0) = 2 π (kBT0)3 (ℏvF)2Γ (3) F2Fe(t) kBT0. (4.42) Equation (4.41) can be solved using Green’s function method: εt=e−t/τεZt −∞ dt′et′/τεℏωb 2 nnt(t′) τee +ε′ 0(t′) τε. (4.43) We define: ˜ε(t) = ℏωb 2 e−t/τε τee Zt −∞ dt′et′/τεnnt(t′)(4.44) and since Fe(t)is a slowly varying function compared to 1/τε, we approximate: e−t/τεZt −∞ dt′et′/τεε′ 0(t′) τε≈e−t/τεε′ 0(t)Zt −∞ dt′et′/τε τε =ε′ 0(t). (4.45) Equation (4.43) assumes the form: εt(t) = ˜ε(t) + ε′ 0(t). (4.46) On the other hand, the energy of the thermalized electrons is: εt=2 π (kBT(t))3 (ℏvF)2Γ (3) F2Fe(t) kBT(t), (4.47) so equating Eqs. (4.46) and (4.47), the second relation for Fe(t)and T(t)is: 2 π (kBT(t))3 (ℏvF)2Γ (3) F2Fe(t) kBT(t)= ˜ε(t) + ε′ 0(t). (4.48) It remains to determine the chemical potential for the hole distribution. We could write similar balance equations for holes as we did for electrons. However, since charge must be conserved, we have: p0+n(t) = p(t) + n0. (4.49) This allows us to determine the hole concentration at any given time. Separating contributions from 44
4.2 Hot electron model thermalized and non-thermalized carriers, we have: pt(t) = p0+nnt(t) + nt(t)−pnt(t)−n0. (4.50) Remembering that pnt =nnt, Eq. (4.50) becomes: pt(t) = p0+nt(t)−n0. (4.51) Knowing pt(t), we can calculate the hole’s chemical potential Fh(t)using the relation: pt(t) = 2 πkBT(t) ℏvF2 Γ (2) F1−Fh(t) kBT(t). (4.52) 4.2.3 Non-equilibrium conductivity due to hot carriers Now that we know how the quasi-Fermi levels and the electronic temperature evolve with time, we can study the non-equilibrium transient conductivity. We assume that, to first order, the optical non-equilibrium interband conductivity of graphene can be described by the usual formula (see Eq. (B.33)): σneq(ω, t) = 4σ0 πℏ(γinter −iω)Z+∞ 0 dε 1−f(e)(ε, t)−f(h)(ε, t) [ℏγinter +i(2ε−ℏω)] [ℏγinter −i(2ε+ℏω)] , (4.53) with γinter meaning the interband scattering rate. Here f(e)and f(h)are the full distribution functions for the electrons and holes with contributions from thermalized and non-thermalized carriers: f(e,h)(ε, t) = ge,h(ε, t) + fe,h (ε, Fe,h(t)) , (4.54) with ge,h given in Eq. (4.21). The extension of the usual formula for the interband conductivity (B.33) to a time dependent version (4.53) is valid as long as, at the frequency of observation ω, the field oscillates several times before the physical conditions change. In other words, this approach works if the two time scales, 1/ω and the time it takes for chemical potentials and temperature to change, are sufficiently different: ωτee >1. (4.55) Here, we take τee as the time of change of the system, since it is the smallest characteristic time parameter. Because it is expected that τee lies in the time range 10 −30 fs [38], at the optical frequencies of interest (λa≤615 nm) this condition is satisfied. As shown in Appendix E, to the first order, the differential reflectivity only depends in the real part of the transient conductivity. Also, it is expected that ℏγinter ≪kBT[30] so the real part of 45
IV Hot carriers in graphene expression (4.53) may be written as: Reσinter(ω, t) = Reσnt(ω, t) + Reσt(ω, t), (4.56) with σt(ω, t)coming from the thermalized carriers: Reσt(ω, t) = σ0 sinh ℏω−(Fe(t)−Fh(t)) 2kBT cosh Fh(t) + Fe(t) 2kBT+ cosh ℏω−(Fe(t)−Fh(t)) 2kBT(4.57) and σnt(ω, t)from the non-thermalized carriers2: Reσnt(ω, t) = −σ0gℏω 2, t. (4.58) 4.2.4 Differential reflectivity The experimental results we are interested in understanding were obtained by means of pumpprobe spectroscopy and measure the changes in the reflectance of graphene. Let us call δtthe time delay between the pulses band a. The differential reflectivity is given by (see Appendix E for derivation): ∆Ra Ra =−cos2θa cos αaˆ t(0) a2 Φaˆr(0) a 4π cσ0Z∞ −∞ dt′Ait′ 2f(e)ℏωa 2, t′+δt+f(h)ℏωa 2, t′+δt . (4.59) Here Ai(t′)is the incident pulse’s amplitude: Ai(t) = aiexp −t2 τ2 a, (4.60) where aiis a constant and τais the pulse time width. Φais the probe’s fluence: Φa=Zdt |Ai(t)|2(4.61) and ˆ t(0) aand ˆr(0) aare the usual linear transmission and reflection coefficients. Finally, αaand θaare the incident and refracted angles, respectively. 2We notice that the "slowly varying amplitude" approximation is still valid here because the relevant frequency in Eq. (4.58) is ωband also satisfies ωτee >1. 46
4.3 Numerical results and concluding remarks 0.0 2.5 5.0 7.5 10.0 t (ps) 200 100 0 100 200 300 Fe , h 500 1000 1500 2000 2500 T (K) -0.25 0 0.25 0.75 1.25 t (ps) 0.06 0.05 0.04 0.03 0.02 0.01 0.00 0.01 R / R 0 b (mJ 1 cm2) Figure 4.2: Left: Quasi-Fermi levels and temperature as functions of the time delay between pump and probe, t, as determined by the model. Fe,Fhand Tare represented by the blue, green and red curves, respectively. Right: Differential reflectivity normalized to the pump fluence. The black line with dots are the experimental results taken from Ref. [1]. The blue line is the calculated result from our model with τee = 10 fs, τε= 0.4ps and τr= 1 ps. The red dashed line shows the calculation (blue line) but with a correction factor f= 0.45 for the amplitude and a shift in time of t0= 0.1ps. 4.3 Numerical results and concluding remarks We can use our model to simulate the non-resonant signal measured by Constant et al. [1], as shown in Figs. 4.2 and 4.3 (see Appendix F for numerical implementation). The time parameters, τee,τεand τrused were: τee = 10 fs , τε= 0.4ps , τr= 1 ps . (4.62) These are not the parameters that best fit our model to the experimental results. Taking τee = 1 fs, leads to a better agreement with the experimental results. However, since it is generally accepted that τee ∼10 −30 fs [38], this value looks unrealistic. The rest of the parameters are the same as in the experiment: the pump and probe incident angles are αb= 40◦and αa= 20◦, respectively; pump and probe have equal frequencies, ωb=ωa= 3064.97 ps−1(λa= 615 nm); the substrate is SiO2with a refractive index n2=√ε2= 1.4601 at frequency ωaand the pump fluence is Φb= 0.26 mJ cm−2. The calculated dependence deviates from the observations in two ways: it yields a signal bigger by approximately a factor of two and the peak of the calculation appears approximately 0.1 ps earlier than the measurement. The reasons for these differences could be: 1. Information concerning experimental details that we have from Ref. [1] may be incomplete, for instance, we assume that both pulses are Gaussian and have the same duration. However, if it is not so, the spectral distribution of non-thermalized carriers will be different. We also checked that assuming a non-Gaussian pump would change the temporal distribution of hot 47
IV Hot carriers in graphene carriers. Both effects would affect the differential reflectance dependence on the probe pulse delay. 2. In the work [1], the beams were focused using a single lens. This technical choice implies that in order to have an angle of incidence different from zero, the beams cannot be centered in the lens, and this introduces spatial and temporal chirp in the pulses [41] that perhaps cannot be ignored. 3. The data of Ref. [1] is noisy, which makes the fit difficult to perform. In fact, the data shows a weird feature: when ωa=ωb, it is expected that the peak of the signal to be positioned at ≈0ps, since it is when most electrons are pumped to the conduction band thus blocking the probe more intensively. This is not the case in the data of Ref. [1] (see black curve in Fig. 4.2). For the peak of the signal to appear later, the non-thermalized electrons must thermalize so that the population of thermalized electrons grows enough to block the probe at frequency ωa=ωb. This is possible if the non-thermalized electrons thermalize over the timescale of 1 fs. To conclude this chapter, the hot electron model yields a good understanding of the processes taking place in graphene under fast optical pumping. They have to be taken into account when considering all-optical generation of surface plasmons. 1000 0 1000 (meV) 0.0 0.2 0.4 0.6 0.8 1.0 f( ) EF -0.0 ps 0.3 ps 1.0 ps 0 200 400 0.0 0.5 1.0 EF 0 2000 4000 ( ps 1) 1.0 0.5 0.0 0.5 1.0 Re ( neq /0 ) -0.0 ps 0.3 ps 1.0 ps Figure 4.3: Left: Hot carrier distribution for different delay times, determined using the the Fermi levels and temperature in Fig. 4.2. The inset shows the equilibrium Fermi-Dirac distribution. Right: Transient conductivity (4.56). 48
Chapter 5 Plasmon lasing 5.1 Plasmon emission as a new electronic relaxation channel As discussed in Chapter 2, SPPs are collective charge oscillations, mediated by long-range Coulomb interactions. But, in a situation out of equilibrium, is this possible? The answer is yes and experimental evidence for optically generated non-equilibrium plasmons was given by Ni et al. [42]. In their work, using infrared pump–probe nano-spectroscopy, SPPs were activated and detected using femtosecond optical pulses in a specimen of intrinsic graphene that otherwise lacked infrared plasmonic response at equilibrium. To excite SPPs, a focused IR probe beam was used to illuminate a metalized tip of an atomic force microscope and generate a strong evanescent electric field [6]. At equilibrium, no response was observed. However, upon ultrafast optical excitation using a fs pump pulse, Ni et al. observed charge oscillations, consistent with graphene SPPs. The amplitude of this oscillations depended on the delay between pulses: at observation times bigger than 2 ps, when the amount of charge carriers started to be small, the features of the oscillations disappeared. Thus, we may conclude that the out-of equilibrium carriers can hold SPPs or, more correctly, hot SPPs. Nevertheless, this does not reveal how important are SPPs as a relaxation channel and if they strongly couple to electron-hole pairs. The strong interplay of plasmons and eletron/hole excitations in graphene was revealed by Aaron et al. [43, 44]. Using angle-resolved photoemission spectroscopy, they observed a renormalization of the Dirac cone due to coupling of SPPs with holes. Later, Kim et al. [35, 45] experimentally showed that hot SPPs participate in the relaxation dynamics as a decay channel for the excited carriers. By impinging a graphene device with a 100 fs pulse at 850 nm and collecting the resulting emission spectra using a Fourier-transform infrared microscope, they found that the higher the Fermi level1of graphene, the stronger the infrared emission is. This behavior is the opposite of thermal emission, which should reduce for increasing Fermi energy. The observed response is consistent with plasmon emission, as the phase space for plasmonic emission increases with Fermi energy [45]. Some of the plasmons, emitted by the hot electrons, scatter into photons, leading to the observed infrared emission. To enhance conversion of SPPs into thermal radiation, 1They controlled the Fermi energy by electrostatic gating. 49
V Plasmon lasing + Figure 5.1: Schematics of the plasmon lasing in photoinverted graphene. A particular SPP mode is generated by the DF process (yellow arrow). This mode lives in the SPP reservoir and will stimulate electrons and holes to recombine with emission of a plasmon of the same mode. As a consequence the SPP population grows and the field is amplified. gold nanodisks were added to the sample. The role of this nanodisks was to scatter the SPP field into photons that could be measured and effectively enhance their presence. Detailed analysis showed a plasmon emission rate of a stunning five orders of magnitude higher than the thermal emission [45]. These results clearly show that we cannot ignore the hot carriers created by the pump. Furthermore, in photoinverted graphene, SPPs may experience gain, through electron-hole recombination [10, 29], thus suggesting that the reason behind the signal observed in Ref. [1] was SPP lasing: a particular SPP mode is selected by the DF process and, due to the non-equilibrium electron-hole population, this mode is amplified, giving rise to a signal several orders of magnitude higher than the predicted by the perturbation theory (see Fig. 5.1). In this chapter we introduce a model, based on the master equation formalism, to describe this process. In Section 5.2, we explain how the SPP dispersion curve is calculated in non-equilibrium graphene. In Section 5.3, we write the master equation to describe the evolution of the SPP population. Section 5.4 deals with the problem of including DF generation in the master equation. The change in the differential reflectivity, due to the generation of plasmons, is estimated in Section 5.5. Finally, we make some concluding remarks in Section 5.6. 5.2 Non-equilibrium plasmon dispersion relation and gain Previous theoretical works [29, 36] showed that photoinverted graphene has the right conditions for SPP gain, e.g an increase of their non-equilibrium population. In this Section we explore this effect. Consider a graphene sheet in a photoinverted state, with electron and hole distributions characterized by the chemical potentials Feand Fh, respectively, at a temperature T. Interband transitions, with the energies below ∼(Fe+Fh), have low probability of being blocked and, as discussed in Section 2.2, such electron-hole pairs can recombine via plasmon emission. In this regime, the interband 50
5.2 Non-equilibrium plasmon dispersion relation and gain Loss Gain Figure 5.2: Plasmon dispersion, ωsp(q) + iγsp(q), calculated at zero temperature with both intraband and interband conductivities. The parameters of the calculation were Fe=−Fh= 300 meV, γinter = 0.01 ps−1to avoid bad numerical behavior, ε1= 1 and ε2= 2.4as usual. The blue curve shows the solutions of the Eq. (5.4) for a momentum relaxation rate γp= 0.01 ps−1, while the orange curve shows the dispersion for a much more realistic value of γp= 2π×2.6ps−1. The γsp(q)plot clearly shows that for certain frequencies it is expected plasmonic gain, even for finite γp. conductivity, for a sufficiently low frequency ω, is negative (see Fig. 4.3) and the optical gain is possible. Consequently, the SPP field instead of being damped, is amplified (and off damped). This idea is easily understood if we consider a particular case of intrinsic photoinverted graphene, Fe=−Fh>0, at zero temperature. In these conditions, expressions (B.26) for the intraband conductivity and (B.33) for the interband conductivity yield the following analytical expressions: σintra(ω) = 4σ0 π Fe+|Fh| ℏγp−iℏω(5.1) and σinter(ω) = σ01 + 1 πarctan ℏω−2|Fh| ℏγinter +1 πarctan ℏω−2Fe ℏγinter −1 πarctan ℏω+ 2|Fh| ℏγinter −1 πarctan ℏω+ 2Fe ℏγinter −iσ0 2πln (ℏγinter)2+ (2|Fh|+ℏω)2 (ℏγinter)2+ (2|Fh|−ℏω)2!−iσ0 2πln (ℏγinter)2+ (2Fe+ℏω)2 (ℏγinter)2+ (2Fe−ℏω)2!, (5.2) respectively. Here γpis the momentum relaxation rate introduced in Section 4.2 and γinter is the interband scattering rate. Notice that the interband conductivity is negative for small ωand γinter: lim ω→0lim γinter→0σinter(ω) = −σ0. (5.3) 51
V Plasmon lasing Gain Loss Figure 5.3: Plasmon dispersion, ωsp(q) + iγsp(q), for the same conditions as in Fig. 5.2, but calculated at finite temperature T= 2000 k, also with both intraband and interband conductivities. Even at finite temperatures, plasmon gain is possible. As before, the plasmon dispersion relation is determined by the poles, ω(q), of Eq. (2.4): D(ω, q) = ε1 κ1 +ε2 κ2 +4πi ωσ(1)(ω)=0. (5.4) However, σ(1)(ω)now is the transient conductivity with both contributions (5.1) and (5.2). For γp=γinter = 0 (no losses) and small ω, in the equilibrium regime, the conductivity is purely imaginary and the solutions of the dispersion equation, ω(q), are real. However, in photoinverted graphene the real part of the interband conductivity is non-zero for small ωand the solutions to Eq. (5.4) become complex, ωsp(q) + iγsp(q). For Reσinter <0,γsp(q)is positive and the SPP field instead of being damped, is amplified with time. Figure 5.2 shows the plasmon dispersion curve obtained using σ(1) modeled with Eqs. (5.1) and (5.2) for different γp. Figure 5.3 shows calculated results for temperature T= 0, obtained using Eqs. (B.28) and (4.57) for the intraband and interband conductivity, respectively. Even at high temperatures, plasmon gain seems possible. These results are in good agreement with what was expected, the plasmon channel of the electronic relaxation is likely in the 100-fs time scale. Figures 5.2 and 5.3 show a damping rate of the order ∼10 ps−1that corresponds to a time scale of ∼100 fs. Using the hot electron model from Section 4.2 we can predict the SPP dispersion for different time delay t. For the same conditions as in Fig. 4.2, we plot the SPP dispersion in Fig. 5.4. 52
5.4 DF-generation of plasmons The matrix element in (5.26) for the process in Fig. 5.6 is: ⟨F|ˆ Hint|I⟩=√nb√na+ 1pnq+ 1M(q,qa,qb)(5.28) and the emission rate of plasmons is: Γsp =2π ℏ(nq+ 1) (na+ 1) nb|M(q,qa,qb)|2δ(ℏωb−ℏωa−ℏωq), (5.29) where M(q,qa,qb)will be specified below. The photon modes in Eq. (5.29) are not specified. At the surface, for a particular frequency, three type of modes are present, corresponding to incident, reflected and transmitted photons. These modes have equal in-plane momentum qjbut the z-component of the wavevector and the polarization vector are different. One could say that the optical field at the interface z= 0 is in a superposition of incident, reflected and transmitted states. The interaction, which causes reflection/refraction of photons on the surface and leads to the three kind of photons, is not included in the Hamiltonian (5.24) and diagrams of Fig. 5.6. We include this interaction phenomenologically. For example, in the emission rate of plasmons (5.29) both reflected and transmitted photons contribute. In fact, the full field at the interface contributes and stimulates the emission of plasmons. Thus, ignoring absorption in graphene, the photon occupation numbers naand nbin Eq. (5.29) should be equal to the number of incident photons. For the emission rate of photons athe situation is different. Here, one has to distinguish between emission in reflection or emission in transmission, since the emission in a particular direction will be stimulated differently by the presence of the beam. The same will be true for the inverse process, where a plasmon and a photon aare absorbed and a photon bis emitted. 5.4.1 Electron-photon and electron-plasmon interaction The SPP generation is mediated by electrons that are hidden in the matrix element M(q,qa,qb) in Hamiltonian (5.24). The full Hamiltonian describes graphene’s electrons, the radiation fields a and b, and the plasmon field: ˆ H=ˆ H0+ˆ He-ph +ˆ He−pl , (5.30) where ˆ H0is graphene’s + fields’s + plasmons’s bare Hamiltonian that is diagonal, in the appropriate basis. ˆ He−pl is the interaction between electron and plasmon as determined in Eq. (5.9) and ˆ He-ph is the electron-photon interaction. We write this term as in Section (3.3): He-ph =e cvFX j σxAx j(t, x, z = 0) , (5.31) 59
V Plasmon lasing so in the second quantization4: ˆ He-ph =e cvFX jX kjX s′,s X k Ax kjc† s′,k+qjcs,kˆ bkjFx s′,s(k+qj,k)e−iωkjt +e cvFX jX kjX s′,s X k Ax kjc† s′,k−qjcs,kˆ bkjFx s′,s(k−qj,k)eiωkjt, (5.32) where: Ax kj=s2πℏc2 η2V ωkj ekj·ex. (5.33) Here, ηis the index of refraction. For the incident and reflected fields η= 1, while for the transmitted field ηis the index of refraction of the substrate. For future calculations it is better to distinguish between aand bfields, so we rewrite this term as: ˆ He−ph =ˆ Va+ˆ Vb, (5.34) with: ˆ Vj=e cvFX kjX s′,s X k Ax kjc† s′,k+qjcs,kˆ bkjFx s′,s(k+qj,k)e−iωkjt +e cvFX kjX s′,s X k Ax kjc† s′,k−qjcs,kˆ bkjFx s′,s(k−qj,k)eiωkjt. (5.35) The sum over kjruns over incident, reflected and transmitted fields. The process of interest involves the annihilation of a photon, the creation of another photon and the creation of a plasmon. This is a third-order process in which ˆ He−ph acts twice and ˆ He-pl acts once. The Hamiltonian needs to be transformed into a form that contains an effective photon– plasmon interaction [47]. As given above, the Hamiltonian is written in terms of a basis set that spans the Hilbert space of the system (electrons, plasmons, and photons), that diagonalizes ˆ H0. The basis consists of states: |m⟩=|nb⟩|na⟩|nq⟩|..., Nk,s, ...⟩, (5.36) where Nk,s is the occupation number for the electron state |k, s⟩. We can change to a new basis set of states |˜m⟩=U|m⟩via unitary transformation U. In this new basis, the Hamiltonian matrix elements are: ⟨˜n|ˆ H|˜m⟩=⟨n|˜ H|m⟩, (5.37) where ˜ H=U†ˆ HU. Hence, consider the canonical transformation U=e−Swhere Sis an operator such that S†=−S.˜ Hhas the same eigenvalues as ˆ Hand its eigenstates are obtained by the 4See Appendix (G.1) for quantization of the optical field. 60
5.4 DF-generation of plasmons operator eSacting on the corresponding eigenstates of ˆ H. Expanding e±S,˜ Hreads: ˜ H=ˆ H0+ˆ Hint +hS, ˆ H0i+hS, ˆ Hinti+1 2! hS, hS, ˆ H0ii+1 2! hS, hS, ˆ Hintii +1 3! hS, hS, hS, ˆ H0iii+1 3! hS, hS, hS, ˆ Hintiii+... , (5.38) where ˆ Hint =ˆ Va+ˆ Vb+ˆ He−pl. The operator Sis chosen such that: hS, ˆ H0i=−ˆ Hint . (5.39) This choice simplifies Eq. (5.38) to: ˜ H=ˆ H0+1 2hS, ˆ Hinti+1 3hS, hS, ˆ Hintii+... =ˆ H0+ˆ H2+ˆ H3+... (5.40) Consider two eigenstates |I⟩and |F⟩of ˆ H0. The choice given by Eq. (5.39) establishes: ⟨F|hS, ˆ H0i|I⟩=−⟨F|ˆ Hint|I⟩≡⟨F|S|I⟩=⟨F|ˆ Hint|I⟩ EF−EI . (5.41) This shows that Sis proportional to ˆ Hint. Since our interest is in third-order interaction, we consider the third term ˆ H3in Eq. (5.40): ˆ H3=1 3hS, hS, ˆ Hintii=1 3S2ˆ Hint −2Sˆ HintS+ˆ HintS2, (5.42) that using identity (5.41) gives the transition element MF I : MFI =1 3⟨F|ˆ H3|I⟩ =1 3X m,n ⟨F|ˆ Hint|m⟩⟨m|ˆ Hint|n⟩⟨n|ˆ Hint|I⟩ (EF−Em) (En−EI) (Em−En){3En−3Em+EF−EI}. (5.43) In the Fermi’s golden rule, energy is conserved, so the initial and final states have the same energy, EF=EI, and we obtain: MFI =X m,n ⟨F|ˆ Hint|m⟩⟨m|ˆ Hint|n⟩⟨n|ˆ Hint|I⟩ (EF−Em) (En−EI). (5.44) We now substitute the expression for ˆ Hint while keeping only the terms that correspond to ph →pl 61
V Plasmon lasing →ph processes with different photons: MFI =X m,n ⟨F|ˆ Va|m⟩⟨m|ˆ Vb|n⟩⟨n|ˆ He−pl|I⟩ (EF−Em) (En−EI)+X m,n ⟨F|ˆ Va|m⟩⟨m|ˆ He−pl|n⟩⟨n|ˆ Vb|I⟩ (EF−Em) (En−EI) +X m,n ⟨F|ˆ Vb|m⟩⟨m|ˆ Va|n⟩⟨n|ˆ He−pl|I⟩ (EF−Em) (En−EI)+X m,n ⟨F|ˆ Vb|m⟩⟨m|ˆ He−pl|n⟩⟨n|ˆ Va|I⟩ (EF−Em) (En−EI) +X m,n ⟨F|ˆ He−pl|m⟩⟨m|ˆ Va|n⟩⟨n|ˆ Vb|I⟩ (EF−Em) (En−EI)+X m,n ⟨F|ˆ He−pl|m⟩⟨m|ˆ Vb|n⟩⟨n|ˆ Va|I⟩ (EF−Em) (En−EI). (5.45) We can look at the terms in Eq. (5.45) as diagrams of the form shown in Fig. 5.6 and use Feynman rules to calculate the matrix element5. The rules are: 1. With each internal fermion line, associate a wavevector k; 2. Conserve wavevector at each vertex; 3. Assign at each vertex the respective interaction matrix element: (a) plasmon vertex e cvFAx∗ q(0)pnq+ 1s2πc LqωqS, (5.46) for plasmon emission and e cvFAx q(0)√nqs2πc LqωqS, (5.47) for plasmon absorption; (b) photon vertex e cvFpnj+ 1 X kjX kX s,s′ Ax∗ kjFx s′,s(k−qj,k), (5.48) for photon emission and e cvF√njX kjX kX s,s′ Ax kjFx s′,s(k+qj,k), (5.49) for photon absorption; 4. Multiply by the factor (−1)F+n, where F is the number of closed fermion loops and nthe number of vertices6; 5. To each vertex assign the respective fermion occupation number; 5Only the first two are shown, the others are obtained by permutations of the vertices. 6This can be seen explicitly if we say that the initial state is: |..., N1, ..., N2′, ...., N2, ...., N3⟩= (−1)p|N1, N2′, N2, N3, ...⟩(5.50) 62
5.4 DF-generation of plasmons 6. Choose an initial state. Assign the propagator to the first fermion line: 1 En−EI , (5.55) where Enis the energy of the intermediate state and EIthe energy of the initial state; 7. Choose an initial state. Assign the propagator to the second fermion line: 1 EF−Em , (5.56) where Emis the energy of the intermediate state and EFis the energy of the final state. 5.4.2 Plasmon emission/absorption In a non-equilibrium situation, we have electrons in the conduction band and holes in the valence band, so the initial state for the process can be any state filled with an electron. For example, in the diagram in Fig. 5.6 we could have: 1→3→2→1or 2→1→3→2or 3→1→2→3. (5.57) All these permutations and different channels are encoded in the sums in Eq. (5.45). Consider Channel I from Fig. 5.6, with plasmon emission by the electronic transition in the conduction band. This process is composed of interactions with permutation: 1 −→ b(absorption) 3 −→ pl(emission) 2 −→ a(emission) 1 −→ . (5.58) We discard terms that do not respect this permutation. The initial electronic state is: |i⟩=|N1, N2′, N2, N3, ...⟩. (5.59) As discussed above, the photon acan be emitted in reflection or in transmission. For SPP emission, both cases contribute, so one should consider the total field. If absorption by graphene is ignored, the photon modes relevant for the emission of SPPs are the incident modes. Using the Feynman Thus, the matrix elements introduce for example: ⟨3|ˆ Vb|I, 1⟩ → (−1)2p⟨N1−1, N2′, N2, N3+ 1|c† 3c1|N1, N2′, N2, N3⟩= (−1)2p(1 −N3)N1(−1)N1−1+N2′+N2(5.51) ⟨2|ˆ He−pl|3⟩ → (−1)2p⟨N1−1, N2′, N2+2, N3|c† 2c3|N1−1, N2′, N2, N3+1⟩= (−1)2p(1 −N2) (−1)N1−1+N2′+N2(−1)N1−1+N2′ (5.52) ⟨F, 1|ˆ Va|2⟩ → (−1)2p⟨N1, N2′, N2, N3|c† 1c2|N1−1, N2′, N2+ 2, N3⟩= (−1)2p(−1)N1−1+N2′(5.53) so when multiplied the sign is positive: ⟨F, 1|ˆ Va|2⟩⟨2|ˆ He−pl|3⟩⟨3|ˆ Vb|I, 1⟩ → (1 −N2) (1 −N3)N1(5.54) 63
V Plasmon lasing rules, the contributions from Channel I to the matrix element MFI are: C1=e cvF3Ax kbAx∗ q(0)Ax∗ ka√nkbpnq+ 1pnka+ 1s2πc LqωqSδq,qb−qa ×X k3X k2 δk3,k1+qbδk1+qa,k2F−1,1(k1,k2)F1,1(k2,k3)F1,−1(k3,k1) ×N1(1 −N3) (1 −N2) (ℏωa−εk1−εk2) (εk3+εk1−ℏωb)+N3(1 −N2) (1 −N1) (εk3+εk1−ℏωb) (ℏωq+εk2−εk3) +N2(1 −N1) (1 −N3) (ℏωq+εk2−εk3) (ℏωa−εk1−εk2). (5.60) The sums over kjare dropped because only the incident fields contribute to generation of SPPs. For Channel II, the process is of the form: 1 −→ b(absorption) 3 −→ a(emission) 2′ −→ pl (emission) 1 −→ , (5.61) so the contribution from this channel is: C2=e cvF3Ax kbAx∗ q(0)Ax∗ ka√nkbpnq+ 1pnka+ 1s2πc LqωqSδq,qb−qa ×X k2′X k1′ δk3−qb,k1δk2′,k3−qaF−1,−1(k1,k2′)F−1,1(k2′,k3)F1,−1(k3,k1) × N1(1 −N3) (1 −N2′) ℏωq+εk′ 2−εk1(εk3+εk1−ℏωb) +N3(1 −N2′) (1 −N1) (εk3+εk1−ℏωb)ℏωa−εk′ 2−εk3 +N2′(1 −N1) (1 −N3) ℏωa−εk′ 2−εk3ℏωq+εk2′−εk1 . (5.62) Finally, we average over quasi-equilibrium electrons to yield the quasi-FD functions. For example: N1(1 −N3) (1 −N2)→fv(k1) (1 −fc(k3)) (1 −fc(k2)) , (5.63) The elements Ax kjin Eqs. (5.60) and (5.62) are given in Eq. (5.33). As explained in the Appendix (G.1), the normalization factor: 2πℏc2 ωjη2V1/2 (5.64) corresponds to one photon in a volume V. If we have a beam of cross section Sand the laser is placed at a distance Rfrom the sample, then Vis the volume of the beam: V=S ×R. (5.65) 64
5.4 DF-generation of plasmons If the laser power is Pand it generates photons with wavevector k≈kj, the number of photons in this volume is: nkj=P ℏωkj ∆t, (5.66) where: ∆t=R cn(5.67) is the time that it takes a photon to reach the surface where it is removed from the beam (either reflected or absorbed). Using expressions (G.35) and (G.37), for the electric and magnetic fields, the space average of the Poynting vector operator, ˆ P(r, t) = c 4πˆ E׈ H, (5.68) is: ˆ P=c 4πX k2πℏωk V η nˆ b† kˆ bk+ˆ b† kˆ bkoˆ k. The expectation value for the state |nkj⟩is: Dˆ PE=ℏcωj 2V η 2nkj+ 1ˆ kj. (5.69) As nkj≫1: Dˆ PE≈ℏcωj V η nkjˆ kj=Ijˆ kj. (5.70) Here Ijis the intensity: Ij=ℏcωj V η nkj. (5.71) Therefore, the final expression for the matrix element MF I in the case of emission is: M(+) FI =C1+C2 =s2πℏc2 V ωbs2πℏc2 V ωa cos αbcos αa√nkbpnq+ 1pnka+ 1rℏc Sδq,qb−qaMk ≈2πc ηωbωarℏcωbnkb Vrℏcωanka Vpnq+ 1 cos αbcos αarℏc Sδq,qb−qaMk =2πc ωbωapIbIapnq+ 1 cos αbcos αarℏc Sδq,qb−qaMk, (5.72) where αjis the angle of incidence of field j,Iaand Ibare the intensities of the incident optical fields 65
V Plasmon lasing aand band: Mk=e cvF3s2πc Lqωq X k3X k2 δk3,k+qbδk+qa,k2F−1,1(k,k2)F1,1(k2,k3)F1,−1(k3,k) fv(εk) (1 −fc(εk3)) (1 −fc(εk2)) (ℏωa−εk−εk2) (εk3+εk−ℏωb)+fc(εk3) (1 −fc(εk2)) (1 −fv(εk)) (εk3+εk−ℏωb) (ℏωq+εk2−εk3) +fc(εk2) (1 −fv(εk)) (1 −fc(εk3)) (εk2−εk3+ℏωq) (ℏωa−εk−εk2) +X k2′X k1′ δk−qb,k1′δk2′,k−qaF−1,−1(k1′,k2′)F−1,1(k2′,k)F1,−1(k,k1′) fv(εk1′) (1 −fc(εk)) 1−fv(εk2′) ℏωq+εk′ 2−εk1′εk+εk1′−ℏωb+fc(εk)1−fv(εk2′)1−fv(εk1′) εk+εk1′−ℏωbℏωa−εk′ 2−εk +fv(εk2′)1−fv(εk1′)(1 −fc(εk)) ℏωa−εk′ 2−εkℏωq+εk2′−εk1′ . (5.73) For the situation where a plasmon and a photon aare absorbed and a photon bis emitted, the result is similar to Eq. (5.72) as expected, since the electronic transitions are the same (only the direction of the arrows is different, but since this is third order process, the minus sign cancels out). The only true difference is the occupation number for the fields aand band the plasmon field: in this case, plasmons are absorbed and photons of frequency ωbare emitted to the reflected field and transmitted fields, so the final occupation number of plasmons is nqinstead of nq+ 1. For the photons, if absorption is ignored the total field at the interface should, again, be equal to the incident field so the the occupation numbers to be consider are again naand nband the matrix MF I in the case of absorption is M(−) FI =2πc ωbωapIbIa√nqcos αbcos αarℏc Sδq,qb−qaMk. (5.74) If the reader is not convinced that indeed this is the case, consider the sum of both reflected and transmitted photons of frequency ωb: A(T),x kbqnk(T) b +1+A(R),x kbqnk(R) b + 1 2 , (5.75) where A(j),x kbis the field amplitude in Eq. (5.33) with j=R, T and nk(T) b and nk(R) b are the occupation 66
5.4 DF-generation of plasmons numbers for transmitted and reflected photons. Using Eq. (5.71), this sum can be written: A(T),x kbqnk(T) b +1+A(R),x kbqnk(R) b + 1 = s2πℏc2 η2V ωbqnk(T) b + 1 cos θb+s2πℏc2 V ωb cos αbqnk(R) b + 1 ≈s2πc ω2 b sI(T) j ηcos θb+qI(T) jcos αb =s2πc ω2 b Ibcos αb sT η cos θb cos αb +√R!. Ignoring absorption in graphene, the reflection and transmission coefficients satisfy Eq.(C.15): rj= 1 −sec αbcos θbtj, (5.76) so that the sum becomes: A(T),x kbqnk(T) b +1+A(R),x kbqnk(R) b + 1 = s2πc ω2 b Ibcos αb, (5.77) that is the sum of transmitted and reflected photons, in zero-th order, equals the incident photons. 5.4.3 DF plasmon emission rate We now have all the necessary tools to calculate the frequency difference emission rate of a SPP with energy-momentum coordinates (ωq,q). Using Eq. (5.72), the emission rate of plasmons, Γ(+) sp , becomes: Γ(+) sp =(2πc)3 ω2 bω2 a IbIa(nq+ 1) cos2αbcos2αaMDF (q,qa,qb)δ(ℏωb−ℏωa−ℏωq), (5.78) where: MDF (q,qa,qb) = δq,qb−qa 1 SX k|Mk|2. (5.79) In Eq. (5.79), the summation is over the electronic sates. The rate for plasmon absorption is obtained using Eq. (5.74): Γ(−) sp =(2πc)3 ω2 bω2 a IbIanqcos2αbcos2αaMDF (q,qa,qb)δ(ℏωb−ℏωa−ℏωq). (5.80) The DF-generation rate in the master equation is: ΓDF (q) = Γ(+) sp −Γ(−) sp =(2πc)3 ω2 bω2 a IbIacos2αbcos2αaMDF (q,qa,qb)δ(ℏωb−ℏωa−ℏωq). (5.81) 67
V Plasmon lasing Figure 5.7: ΓDF (ps−1) as a function of time delay tbetween the pump and probe and the frequency difference ω. The dynamical evolution of hot carriers is described using the hot electron model from Section 4.2. The calculation was performed for the same conditions and parameters as in Figs. 4.2 and 5.5. The red dashed line is the time-dependent SPP dispersion curve calculated using only the intraband conductivity (B.28). To introduce losses and the corresponding broadening we replace the Dirac δ-function by a Lorentzian: δ(ℏωb−ℏωa−ℏωq)→1 πℏ γloss(q) (ωb−ωa−ωq)2+γ2 loss(q), (5.82) where γloss =−γsp(q), and write: ΓDF (q) = (2πc)3 πℏω2 bω2 a IbIacos2αbcos2αaMDF (q,qa,qb)γloss(q) (ω−ωq)2+γ2 loss(q). (5.83) γsp(q)is obtained by considering complex solutions, ωsp =ωq+iγsp, for the dispersion equation and ω=ωb−ωa. In Fig. 5.7 we plot ΓDF as a function of ωand time. To obtain Fig. 5.7, the integral in Eq. (5.79) was calculated numerically. The calculation is quite time-consuming and was performed in the following way: the electron dynamics was determined using the hot electron model for ω= 0 and it was used to calculate ΓDF for all other frequencies. To put it in another way, the situation simulated was the one with a pre-pump at ω= 0 and then the frequency of field bwas varied, ωb=ωa+ω. This is a somewhat different from the Constant et al. experiment, where the field band the pump are the same. The latter would require the simulation of the electron dynamics for each ω. The final form of the Eq. (5.5) for the SPP population is: dnq dt = Γ1(q)+Γ2(q)nq+ ΓDF(q)−γloss(q)nq−neq q. (5.84) 68
6.3 Testing sensitivity Amplifier Ti-Sa Laser TOPAS-C Sample Detection Scheme beam a beam b Delay Line Figure 6.3: Optical setup for generation of SPPs in Graphene using the amplifier and TOPAS-C. The Ti:sapphire laser seeds the amplifier and the output pulse is used to excite SPPs and seed the TOPAS-C. diate experiment to determine how sensitive the set up mounted in the Femtosecond Laboratory of the CF-UM-UP could be. The designed experiment is a simple sum frequency (SF) experiment [12] (see Fig. 6.4). Two 84 fs Ti:sapphire laser pulses, aand b, at wavelength λ0= 794 nm are focused on a BBO crystal, using a spherical lens with 2.5 cm of focal length. The beams are focused at equal angles θa=θb= 20◦. A SF-signal is generated at wavelength, λs= 397 nm, and propagates at angle θ= 0◦. To control the phase matching between fields, the BBO crystal was mounted on a rotating stage to allow control of the angle Ψ. The crystal used was bought from Eksma Optics [53]. The goal of the experiment was to measure the power depletion of beam a. This measurement was carried out using a fast photodetector and a lock-in amplifier, whose principle is explained in Appendix I. Although BBO is a very efficient medium for SF-generation, the power depletion, ∆Pa, should be small compared to the incident power, PI a. For the geometry shown in Fig. 6.4, we may estimate this power depletion. For PI a≈PI b≈10 mW, the measured power of the SF-signal was Ps≈60 µW. From the Manley–Rowe relations (see Appendix H for details), the power depletion should then be of the order: ∆Pa∼Ps 2= 30µW . (6.6) This corresponds to a power variation ∆Pa/P I a∼10−3. This experiment has two main advantages: 1. The SF-signal is visible by eye, making the alignment of the setup easy. 2. The SF-signal is proportional to the product of the powers of the incident pulses (see Appendix H): Ps∝PaPb. (6.7) This means that if both Paand Pbare lowered by a factor of 10, the SF-power and the power depletion will drop by 100, thus allowing us to easily test the sensitivity of the fast photodetector + lock-in amplifier system. 75
VI Experimental generation of SPPs z y Ψ Side View y x z θa θb Top view SF signal Delay Line Chopper Lens BBO Photodiode+Lock-in amplifier Figure 6.4: Sum frequency generation in BBO crystal. The Ti:sapphire laser pulses are the red arrows and the SF signal is the blue arrow. θband θaare the incident angles and Ψis the crystal’s rotation angle. (Top) Geometry of the problem, side and top view of the crystal. (Bot) Experimental setup. A detailed description of the BBO as well as a semi-qualitative calculation of the SF-generation rate in the context of this experiment are given in Appendix (H). The photodetector used has a fast response, with a rise time of 1 ns [54]. To estimate the intrinsic noise of the photodiode, we used a tektronix oscilloscope with a 500 MHz bandwidth to acquire the data fastly. Since the pulses had a time width of approximately 100 fs, what we saw in the oscilloscope was the photodiode’s RC response convoluted with that of the oscilloscope. We acquired in total 32967 pulses, at sampling rate of 20 GHz, and integrated the signal of each pulse. The obtained average value was: A= 1.49 ±0.02 mV . (6.8) Using this value, we estimate the noise of the measurement using the photodetector to be around 1%. For a small signal with a magnitude of 10−5, this corresponds to a signal to noise ratio (SNR) of 1: SNR = 20 log10 10−5 10−2=−60 dB . (6.9) Hence, in order to measure such small signals, the lock-in must be of capable of removing at least 60 dB of noise. This is possible, if in the lock-in amplifier we set a time constant of 1 second and a 60 dB/decade filter. 1The definition of the dB unit is given in Appendix (I.1). 76
6.3 Testing sensitivity 150 100 50 0 50 100 150 t (fs) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Signal ( W) (14.9,11.8) mW 150 100 50 0 50 100 150 t (fs) 0.0 0.2 0.4 0.6 0.8 1.0 Signal ( W) (6.99,2.705) mW 150 100 50 0 50 100 150 t (fs) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Signal ( W) (3,2.14) mW 150 100 50 0 50 100 150 t (fs) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Signal ( W) Figure 6.5: Measured pump depletion signal. These results were obtained in the early runs. At each run, the power is cut down. The legends in the images are the incident power of beams a and b,(PI a, PI b). The last image is a combination of the 3 runs, for comparison. As it can be seen, the oscillations appear on top of the pump depletion and seem to be random. Furthermore, they do not go down with PI aPI b. 6.3.1 Results The first results obtained are shown in Fig. 6.5. On top of the pump depletion signal, we observe oscillations that do not decrease with the incident power, PI aPI b, contrary to what was expected. These oscillations seem to be linear and to some degree random in nature since they do not match for different runs. Although we do not know exactly the cause, these oscillations might be modulated scattered light that somehow finds its way to the photodetector and interferes with the main beam. The scattering mechanism must have its origin in an interaction between the beams, because if the beam ais blocked, these oscillations disappear, so it cannot just be the light scattered from the surface (see Fig. 6.6). To make sure that the oscillations were not caused by bad alignment, the optical setup was realigned. The oscillations disappeared and the results obtained are shown in Fig. 6.7. The observed 77
VI Experimental generation of SPPs 150 100 50 0 50 100 150 t (fs) 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 Signal ( W) (12.8,10.22) mW 150 100 50 0 50 100 150 t (fs) 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 Signal ( W) Figure 6.6: (Left) Pump depletion. (Right) Pump depletion (same as in the left) (blue) plus the result when beam ais blocked (red). This result shows that the oscillations cannot be attributed only to simple scatter of light from beam b. behavior is as expected. The results were taken during the same day. For the second situation, the total power was reduced approximately by a factor of 10. We were able to detect signal variations down to: ∆Pa PI a≈0.08 ·10−3 1.85 ∼10−5, (6.10) as desired. Nonetheless, it is clear that there are considerable fluctuations. In principle, the results can be improved by a more careful alignment and by increasing the time constant and filter roll-off in the lock-in, so the sensitivity of the measurement can be improved if needed. 200 100 0 100 200 t (fs) 0 1 2 3 4 5 Signal ( W) (15.5,9.26) mW 150 100 50 0 50 100 150 t (fs) 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 Signal ( W) (1.85,1.11) mW Figure 6.7: Measured pump depletion signal. The legend shows the incident power of each beam (PI a, PI b). The situation on the right has the total power reduced by a factor of 8.369 relatively to the situation on the left. 78
6.4 Aligning TOPAS-C 0.0 0.5 1.0 1.5 2.0 t (ps) 0.006 0.005 0.004 0.003 0.002 0.001 0.000 R / R 0 0.0 0.5 1.0 1.5 2.0 t (ps) 0.020 0.015 0.010 0.005 0.000 R / R 0 b (mJ cm 2) Figure 6.8: Expected signal fluctuations in the differential reflectivity. (Left) Differential reflectivities calculated using the hot electron model of Sec. 4.2. The parameters are the same as in Fig. 4.2. The curves were calculated for different fluence fΦb. The blue line was calculated with a correction factor f= 1, while the red and green lines for f= 1.1and f= 1.4, corresponding to fluctuations of 10 and 40%, respectively. (Right) Same as in the left plot but the differential reflectivity is normalized by the pump fluence. 6.4 Aligning TOPAS-C After concluding that the lock-in amplifier could be used to detect small changes in the reflectivity due to the presence of SPPs, we went on to aligning the setup in Fig. 6.3, including the TOPAS-C2. The main difficulties encountered in setting the TOPAS-C were finding the correct position of the lenses to focus on the nonlinear crystals as well as the right crystal and beam angles for phase matching. Since TOPAS-C is built in a very compact way, these tasks are challenging. The easiest way to perform the alignment is to remove most optical components and then reinstall them, one by one to ensure that the beam goes in a straight line and is centered in the lenses. Even with this approach, problems appear because the lenses are not normal ones: depending on the lens rotational angle, the focal point changes. Moreover, the input pump must have a good spatial and temporal profile to ensure maximum efficiency and stability. Therefore, the optics before the TOPAS-C, especially the amplifier, must also be correctly aligned. In the end, the TOPAS-C was not properly working. The output pulses had a nice spectral shape, but the output power was very unstable; power fluctuations up to 40% in the course of one second were observed. We looked for the problem but have not found it. Additional alignment of the TOPAS-C did not lead to any considerable improvement. The problem must come from the amplifier. The power fluctuations of the pump are a serious problem because the hot electron signal is nonlinear and consequently it also fluctuates. This fluctuations are not removed by the lock-in, since they are intrinsic to the signal. The solution is monitoring the output power of TOPAS-C, 2Aligning the TOPAS-C alone took us a month. 79
VI Experimental generation of SPPs pulse by pulse, and normalize the signal to the pump fluence, pulse by pulse (see Fig. 6.8). This is not possible with the lock-in. Moreover, the lock-in was tested in an experiment using a laser at 80 MHz rate. Will it work as well at a 1 KHz rate? In principle, the lock-in should still be capable of removing the same amount of noise as determined before, but it will take more time to stabilize since the sampling rate is much lower. Consequently, the detection scheme needs to be changed. 6.5 Balanced detection In spite of knowing that the problem was in the amplifier, it was unclear where it came from and fixing it could take months. As a consequence, we decided that normalizing the signal to the pump (TOPAS-C) fluence was the best approach. For this, we needed a way of both measuring the probe and monitor the pump power, pulse by pulse. This can be done using two photodetectors, one for the probe and another for the pump. The signals from the photodetectors are integrated separately using two boxcars and are divided, pulse by pulse, posteriorly in the computer. The used setup is shown in Fig. 6.9. The operation principle and functioning of the boxcar is explained in Appendix K. Since the experiment required increased SNR, the probe was measured using a balanced photodetector, which has the ability to cancel common noise and detect small signal fluctuations in a large DC signal. This detector uses a reference and a beam with the signal to be measured, in our case the reflection from the graphene sample. The two photodiodes in the balanced photodetector are identical and connected in such a way that their photocurrents cancel. The output of the balanced pair is zero until there is some difference in the intensity of one of the beams, causing the pair to become unbalanced and a net signal to appear in the output. Consequently, common-mode noise that is present in both the reference and signal beams is cancelled out and does not appear as part of the signal. On the other hand, any imbalance between the photocurrents generated by the reference and signal detectors is amplified and is seen as the received signal. 6.5.1 Use of the boxcar SR250 To integrate the pulses, the boxcar used was the SR250 [55], which is an analog model, with associated limitations [56]. This model does not have graphical user interface, so the SR245 is used to communicate via PC. The SR245 can communicate simultaneously with two SR250, and can store in its memory 3711 integrals (1855 from each boxcar). If one wants to acquire more data, the information in the SR245 needs to be exported to the computer and its memory emptied. To communicate with the SR245, a communication interface with computer is needed. The easiest to use is the GPIB, but for an unknown reason, when data is scanned, the return values between channels (fast photodiode and balanced photodetector) are not synchronized. In other words, the two boxcars are not interrogated at the same time, so while one boxcar sends a value only after a pulse was integrated, the other would answer earlier or later, and return a value for 80
6.5 Balanced detection Sample Probe Pump Balanced photodetector Monitor photodetector PC SR250 SR250 SR245 Figure 6.9: Second approach to detect generation of SPPs. The changes in the probe are measured using a balanced photodetector. The pump power is monitored by a fast photodetector. Two boxcar averagers SR250 integrate the signals and send them to the SR245 computer interface module. This module sends the data to the computed where the integrated signals are divided. the moment when no pulse was present. As a consequence, the channels are not correlated and the signal cannot be normalized. In the end, we had to use the RS232 interface, which is older and less friendly than the GPIB. For a baud rate of 9600, this interface worked fine when the data was downloaded value by value. This takes a long time; for example, for a scan of 1800 pulses in each boxcar, which corresponds to 1.8 seconds, it takes approximately 1 min to fully download the data to the computer. A much faster option is to download the full data from the SR245 in a 2 byte binary format. However, a timeout error appeared and the download always failed. We do not know the origin of this error, but it is something to be fixed to optimize data acquisition. 6.5.2 Testing the balanced photodetector We used the Nirvana auto-balanced photoreceivers, Model 2007 [57]. According to the detector’s data sheet, this model is capable of reducing common noise in 50 dB at frequencies from DC to 125 kHz. The device measures the difference in power between the reference and signal photodiodes: Sig =Psig −Pref , (6.11) where Psig and Pref are the incident powers on the signal and reference photodiodes, respectively. The device has shown some problems in the past and we were not confident if it would perform well. To test the device, we set up the experiment in Fig. 6.10. The goal was to verify Malus’s law, using the unstable TOPAS-C, and measure the balanced photodiode output, Sig(θ), where θis the polarizer’s angle. At each θ, 500 pulses were integrated. The polarizer was mounted in the signal beam’s path. Figure 6.11 a) shows how the measured voltage fluctuates with each pulse, for a fast photodiode 81
VI Experimental generation of SPPs SR250 TOPAS_C Balanced photodetector SR250 SR245 PC Polarizer sig ref Fast photodetector Figure 6.10: Balanced Photodetection Setup. and for the balanced photodetector. At a first look, it seams that the balanced photodetection yields a much more stable measurement, effectively removing common noise. The noise in the fast photodetector is given by the standard deviation of the measurement. With this detector, the measured mean, µ1, and the standard deviation, σ1, were: µ1≈390 µW , σ1≈40 µW . (6.12) The ratio between the signal and full background is: σ1 µ1≈0.1 = 10% . (6.13) This means that the fast photodiode can detect signal variations down to 10% in the TOPAS-C. Smaller changes are completely washed out by the noise. To estimate the rejection in common noise by the balanced photodetector, we used the results in Fig. 6.11 b). The plot shows Sig(θ)(blue dots with error bars) and the fit function (black line): Sig(θ) = Pref rcos2θ−1, (6.14) where Pref and rare fitting parameters. ris the ratio between the incident powers, r=Psig (θ= 0) Pref . (6.15) The fitted values are: r= 1.04 , Pref = 187.4µW , with negligible uncertainty. Close to the balanced state (θ= 10.5◦), the standard deviation of the 82
6.5 Balanced detection 0 200 400 nº pulse 0 200 400 600 Power ( W ) 10 0 10 (degrees) 5 0 5 Power ( W ) Figure 6.11: (Left) Measured pulse power from TOPAS-C. In total, 500 pulses were acquired. The orange dots are the power measured with the fast photodiode and the blue dots, with the balanced photoreceivers. (Right) Signal measured using the balanced photodetector for different angles θof the polarizer. At each θ, 500 pulses were acquired. The blue dots with error bars are experimental data while the black line is resulting fit. The horizontal dashed red line shows the perfect balanced region. measurement is σ2= 0.5µW. Thus, the ratio between the signal and full background is: σ2 V0≈3·10−3= 0.3% . (6.16) The balanced photodetectors improved the noise-background ratio from 10% of the full scale down to 0.3%. We estimate the rejected common noise (RCN) to be: RCN = 20 log10 σ2 Pref −20 log10 σ1 µ1≈32 dB . (6.17) This is lower than what is expected to be the full capacity of the Nirvana auto-balanced photoreceiver. In principle, the rejection of noise could be improved by injecting the beams using optical fibers, instead of free space incidence. There is a chance that the detector is broken. In fact, the device shows a weird behavior: when more light goes to the signal photodiode than to the reference photodiode, the signal blows up and goes to a DC value. To get it working again, the power ratio needs to be readjusted and the beams have to be blocked, so that the detector has time to completely discharge. Only then it responds again. This behavior is unexpected and its origin is unknown to us. Is the balanced detector + boxcar averager system sensitive enough to detect SPPs? It is difficult to answer this question, but it is possible to estimate the sensitivity of the measurement in a best case scenario. To detect SPPs with a SNR = 10, the detection must be sensitive to changes of the order of: Psig −Pref Pref ∼10−5(6.18) 83
VI Experimental generation of SPPs in magnitude. For this to be possible, noise must be lowered to: 20 log10 10−5−20 log10 σ2 Pref ≈ −50 dB . (6.19) Considering laser noise only and assuming that it behaves as a white noise, to remove 50 dB of noise, the number of acquired pulses Nmust be at least (see Appendix K): N=σ2/Pref 10−52 = 104. (6.20) 6.6 Problems to be solved A lot of work still needs to be done in order to be capable of generating and detecting SPPs. According to the estimates from the previous section, to detect changes of the order of 10−5, we need to acquire about 104pulses. We have performed performed the experiment in these conditions, but never saw a signal that corresponded to either SPP generation or hot electrons. Clearly, more pulses need to be integrated to improve the statistics. The problem is that it takes approximately 6 minutes to acquire 104pulses, so, if for example we want to increase the acquisition number by a factor of 5, the acquisition time would increase to 30 minutes and the experiment would take too much time. Thus, data acquisition must be improved and to do this, the timeout error mentioned in Subsection 6.5.1 must be fixed. Moreover, the amplifier needs to be fixed: the origin of the instability must be found and corrected. This will change the conditions and will require a fine tune of TOPAS-C, which will have to be calibrated again. Finally, we must find an easy way of knowing if the pulses arrive at the sample at the same time. Using a NL crystal, such as BBO, this is easily done by finding the SF-signal, generated when the pulses are superimposed in time. However, this approach only works for small incident angles. For larger angles, the phase match condition is never satisfied. 84
B.2 Linear optical response of graphene grouping terms of the same order in λ, we find for zeroth order: ∂tρ(0) nm =−i ℏ[ˆ H0,ˆρ(0)]nm −γnm ρ(0) nm −ρ(eq) nm (B.6) and for N-th order: ∂tρ(N) nm =−i ℏ[ˆ H0,ˆρ(N)]nm −i ℏ[ˆ Vint,ˆρ(N−1)]nm −γnmρ(N) nm , N > 0. (B.7) The commutator [ˆ H0,ˆρ(N)]nm is easy to calculate since we know unperturbed eigenfunction and eigenvalues: [ˆ H0,ˆρ(N)]nm = (εn−εm)ρ(N) nm . Equation (B.6) becomes: ∂tρ(0) nm =−i ℏ(εn−εm)ρ(0) nm −γnm ρ(0) nm −ρ(eq) nm . (B.8) This equation describes the time evolution of the system in the absence of any external field, so it has as solution the system in equilibrium: ρ(0) nm =ρ(eq) nm =δnmf(εn). (B.9) Equation (B.7) becomes: ∂tρ(N) nm =−(iωnm +γnm)ρ(N) nm −i ℏ[ˆ Vint,ˆρ(N−1)]nm , (B.10) with ωnm = (εn−εm)/ℏand has the solution: ρ(N) nm (t) = ρ(N) nm (−∞)−i ℏZt −∞ dt′[ˆ Vint,ˆρ(N−1)]nme(iωnm+γnm)(t′−t). (B.11) B.2 Linear optical response of graphene Here we shall derive graphene’s linear optical conductivity in the long-wavelength limit. To avoid any divergence, the calculation is done in the length gauge1[21, 22]. Consider a graphene sheet that is being impinged by an electric field of the form: Ex(t) = E0 2e−iωt +c.c . (B.12) 1Actually, the calculation can also be done in the velocity gauge, but it requires considering electronic states that depend on the effective momentum (ℏk+eA/c) and is less straightforward. 91
II Optical response of graphene In the length gauge, the interaction term is written: ˆ Vint =exEx(t). (B.13) To determine the linear response, Eq. (B.11) must be solved for N= 1: ρ(1) nm(t) = −i ℏ(f(εm)−f(εn)) Zt −∞ dt′Vnm(t′)e(iωnm+γnm)(t′−t). (B.14) The evaluation of Vnm is a bit tricky because the operator xis not well defined in momentum space. From Refs. [7, 20, 21]: ⟨k1, s1|x|k2, s2⟩=(2π)2 iA δs1,s2 ∂ ∂k2,x δ(k1−k2)−(2π)2 A sin θ1 2k1 (1 −δs1,s2)δ(k1−k2) =Fs1,s2(k1,k2). (B.15) Consequently, the matrix elements Vnm are: Vnm(t) = eFsn,sm(kn,km)E0 2e−iωt +eFsn,sm(kn,km)E∗ 0 2eiωt (B.16) and Eq. (B.14) gives: ρ(1) nm(t) = −ie ℏ(f(εm)−f(εn)) Fsn,sm(kn,km)E0 2 e−iωt γnm +i(ωnm −ω)+E∗ 0 2 eiωt γnm +i(ωnm +ω). (B.17) The x-component of the current operator is: ˆ jx=−evFσx, (B.18) so the averaging yields: jx=Trˆ jxˆρ(1) =ie2vF ℏX sνkνX sn,kn (f(εν)−f(εn)) Fsn,sν(kn,kν)E0 2 e−iωt γnν +i(ωnν −ω)+E∗ 0 2 eiωt γnν +i(ωnν +ω)σx νn =1 2j(ω)e−iωt +1 2j∗(ω)eiωt . (B.19) By definition the conductivity is: jx(ω) = σ(ω)E0(ω)(B.20) 92
B.2 Linear optical response of graphene so: σ(ω) = ie2vF ℏX sνkνX sn,kn (f(εν)−f(εn)) Fsn,sν(kn,kν)σx νn γnν +i(ωnν −ω). (B.21) We now consider separately, interband and intraband contributions. B.2.1 Intraband conductivity The intraband contributions correspond to terms with sν=sn=±1. Equation (B.21) yields: σintra(ω) = ie2vF ℏX snX kνkn (f(εsn(kν)) −f(εsn(kn))) Fsn,sn(kn,kν)σx snsn(kν,kn) γnν +i(ωsn,sn(kn,kν)−ω). (B.22) We simplify the problem and write the same damping parameter γp(electron momentum relaxation rate) for all states. Furthermore, the matrix element Fsn,sn(kn,kν)for intraband transitions is simply: Fsn,sn(kn,kν) = −(2π)2 iA δsn,snδ(kn−kν)∂ ∂kn,x , (B.23) so Eq. (B.22) becomes: σintra(ω) = −e2vF ℏAX k∂ ∂kx fc(ε(k))σx 1,1(k,k) γ−iω +∂ ∂kx fv(ε(k))σx −1,−1(k,k) γ−iω . (B.24) Replacing the sum by integration, Eq. (B.24) simplifies to : σintra(ω) = −e2 2πh 4π ℏγ−iℏωZ∞ 0 dεε ∂ ∂εfc(ε)−∂ ∂εfv(ε). Introducing: σ0=πe2 2h, (B.25) allows us to write σintra(ω)as: σintra(ω) = −4σ0 π 1 ℏγ−iℏωZ∞ 0 dεε ∂ ∂εfc(ε)−∂ ∂εfv(ε). (B.26) The integral has analytical solution, if fcand fvare Fermi like distributions. Consider the more general situation of quasi-equilibrium: fc=1 1 + exp ε−Fe kBT, fv=1 1 + exp −ε+Fh kBT, (B.27) 93
II Optical response of graphene where Feand Fhthe electron and hole chemical potentials, respectively and Tthe temperature. In this regime the integral in (B.26) yields the analytical expression: σintra(ω) = 4σ0i π kBT ℏ(ω+iγp)ln n1 + eFe/kBT1 + e−Fh/kBTo . (B.28) The equilibrium regime is easily obtained by setting Fe=Fh=EF. B.2.2 Interband conductivity The interband contributions correspond to the terms with sν=−sn=±1. For this case we set γnm =γinter. Equation (B.21) yields: σinter(ω) = ie2vF ℏX kν,kn (fc(εkν)−fv(εkn)) F−1,1(kn,kν)σx 1,−1(kν,kn) γinter +i(ω−1,1(kn,kν)−ω) +ie2vF ℏX kν,kn (fv(εkν)−fc(εkn)) F1,−1(kn,kν)σx −1,1(kν,kn) γinter +i(ω1,−1(kn,kν)−ω). (B.29) Since: F−1,1(kn,kν) = F1,−1(kn,kν) = −(2π)2 A sin θn 2kn δ(kn−kν)(B.30) and: ω1,−1(kn,kν) = −ω−1,1(kn,kν) = vF(kn+kν), (B.31) Eq. (B.29) simplifies to: σinter(ω) = e2vF ℏX k sin2θ 2kfv(εk)−fc(εk) γinter −i(2vFk+ω)+fv(εk)−fc(εk) γinter +i(2vFk−ω). (B.32) Replacing the sum by integration: σinter(ω) = 2σ0 πZ∞ 0 dε (fv(ε)−fc(ε)) 1 ℏγinter −i(2ε+ℏω)+1 ℏγinter +i(2ε−ℏω). (B.33) In equilibrium, at zero temperature Eq. (B.33) yields: σinter(ω) = σ01 + 1 πarctan ℏω−2EF ℏγinter −1 πarctan ℏω+ 2EF ℏγinter −iσ0 2πln (ℏγinter)2+ (2EF+ℏω)2 (ℏγinter)2+ (2EF−ℏω)2!. 94
B.3 Second order conductivity In quasi-equilibrium, taking the zero-temperature limit, a simple expression is also obtained: σinter(ω) = σ01 + 1 πarctan ℏω−2Fe ℏγinter −1 πarctan ℏω+ 2Fe ℏγinter +σ01 πarctan ℏω−2|Fh| ℏγinter −1 πarctan ℏω+ 2|Fh| ℏγinter −iσ0 2πln (ℏγinter)2+ (2|Fh|+ℏω)2 (ℏγinter)2+ (2|Fh|−ℏω)2!−iσ0 2πln (ℏγinter)2+ (2Fe+ℏω)2 (ℏγinter)2+ (2Fe−ℏω)2!. (B.34) This is similar to the equilibrium result, but with the additional contribution from the holes. Nonetheless, there is a crucial difference. In the limit γinter →0(no interband losses) and ω→0, σinter(ω)yields a negative value: lim γinter,ω→0σinter(ω) = −σ0. This is different from the equilibrium situation, where the conductivity should yield zero. The fact that the real part of the interband conductivity is negative implies that the solution for ωand qfrom the SPP dispersion equation are complex. Physically, it represents gain, i.e. generation of photons. In real situations ℏγinter ≪kBTso, in the limit γinter →0, Eq. (B.33) has the analytical form: σinter(ω) = σ0 sinh ℏω−(Fe−Fh) 2kBT cosh Fh+Fe 2kBT+ cosh ℏω−(Fe−Fh) 2kBT. (B.35) B.3 Second order conductivity In this section we derive the expressions (3.20), (3.23) and (3.24) for DF conductivity. We start by determining the general expression (3.14) for second order conductivity. The calculation is performed in the velocity gauge to account for effects of spatial dispersion: ˆ H=vFσ·p+e cA(t, r), (B.36) where A(t, r)is the vector potential. The interaction term is: ˆ Vint =evF cσ·A(t, r). (B.37) We expand A(t, r)in plane waves: A(t, r) = 1 2X qA(ωq,q)ei(q·r−ωqt)+c.c. (B.38) 95
II Optical response of graphene For each frequency ωq, there is a wave-vector qassociated to it. Since the field is real, its Fourier components satisfy the relation A(ωq,q) = A∗(−ωq,−q). We work in the gauge where the electrostatic potential is zero and ∇·A= 0. Then, by definition, the electric field is E(t) = −(1/c)∂tA(t) and the relation between the Fourier components is: E(ωq,q) = iωq cA(ωq,q). (B.39) The calculation of the second-order response in graphene is performed using the density matrix formalism. To determine the second order conductivity, Eq. (B.11) needs to be solved for N= 2. Using Eq. (B.17) we find: ρ(2) nm(t) = e 2ℏ2X lX qX p f(εl)−f(εn) ωqωp vj nlpvi lmq ωnl −ωp−iγnl Ej(ωp,p)Ei(ωq,q) ωnm −(ωp+ωq)−iγnm e−i(ωp+ωq)t −e 2ℏ2X lX qX p f(εm)−f(εl) ωqωp vi nlqvj lmp ωlm −ωp−iγlm Ej(ωp,p)Ei(ωq,q) ωnm −(ωp+ωq)−iγnm e−i(ωp+ωq)t, (B.40) where vj mnq=vF⟨m|σjeir·q|n⟩are the matrix elements of the spatially modulated velocity operator. In Eq. (B.40) the Einstein summation convention is used. The single particle current operator is: J(r0, t) = −evδ(r−r0) = −e SvX u eiu(r−r0), (B.41) with Sbeing the area of the graphene sheet and r0the particle’s position. The second order contribution to the expectation value of the this operator is: ⟨Jk⟩=Trρ(2)Jk(r0, t) =e3 (2ℏ)2SX nml X qX pX u f(εm)−f(εl) ωpωq vi nlqvj lmpvk mnu ωlm −ωp−iγlm Ej(ωp,p)Ei(ωq,q) ωnm −(ωp+ωq)−iγnm e−i(ωp+ωq)te−iu·r0 −e3 (2ℏ)2SX nml X qX pX u f(εl)−f(εn) ωpωq vj nlpvi lmqvk mnu ωnl −ωp−iγnl Ej(ωp,p)Ei(ωq,q) ωnm −(ωp+ωq)−iγnm e−i(ωp+ωq)te−iu·r0. (B.42) 96
B.3 Second order conductivity The matrix elements vi nlqvj lmpvk mnuimpose momentum conservation relations that cancel out the summation over u, that is u=−(q+p). Therefore, expression (B.42) is rewritten as: ⟨Jk⟩=(evF)3 4ℏ2SX nml X qX p f(εm)−f(εl) ωqωp σi nlqσj lmpσk mn−(q+p) ωlm −ωp−iγlm Ej(ωp,p)Ei(ωq,q) ωnm −(ωp+ωq)−iγnm e−i(ωp+ωq)tei(q+p)·r0 −(evF)3 4ℏ2SX nml X qX p f(εl)−f(εn) ωqωp σj nlpσi lmqσk mn−(q+p) ωnl −ωp−iγnl Ej(ωp,p)Ei(ωq,q) ωnm −(ωp+ωq)−iγnm e−i(ωp+ωq)tei(q+p)·r0. (B.43) Notice that Ej(ωp,p)is the total field. It is more convenient to write the incoming fields aand b explicitly: Ej(ωp,p) = Eb,j(ωp,p) + Ea,j(ωp,p). (B.44) Making this substitution in (B.43), terms corresponding to interaction of the field with itself will appear. We are not interested in these, so we truncate them and rewrite (B.43) as: ⟨Jk⟩=e3 4ℏ2SX nml X qX p f(εm)−f(εl) ωqωp vi nlqvj lmpvk mn−(q+p) ωlm −ωp−iγlm Ea,j(ωp,p)Eb,i(ωq,q) ωnm −(ωp+ωq)−iγnm e−i(ωp+ωq)tei(q+p)·r0 +e3 4ℏ2SX nml X qX p f(εm)−f(εl) ωqωp vi nlqvj lmpvk mn−(q+p) ωlm −ωp−iγlm Eb,j(ωp,p)Ea,i(ωq,q) ωnm −(ωp+ωq)−iγnm e−i(ωp+ωq)tei(q+p)·r0 −e3 4ℏ2SX nml X qX p f(εl)−f(εn) ωqωp vj nlpvi lmqvk mn−(q+p) ωnl −ωp−iγnl Ea,j(ωp,p)Eb,i(ωq,q) ωnm −(ωp+ωq)−iγnm e−i(ωp+ωq)tei(q+p)·r0 −e3 4ℏ2SX nml X qX p f(εl)−f(εn) ωqωp vj nlpvi lmqvk mn−(q+p) ωnl −ωp−iγnl Eb,j(ωp,p)Ea,i(ωq,q) ωnm −(ωp+ωq)−iγnm e−i(ωp+ωq)tei(q+p)·r0. (B.45) The second order conductivity is defined as: Jk(ωp+ωq,p+q) = 1 2σ(2) kji(Ω,Ωp,Ωq)Eb,j(ωp,p)Ea,i(ωq,q), (B.46) where the vectors Ωp= (ωp,p)and Ω= (ωp+ωq,p+q)are used to simplify the notation. From comparison with (B.45), we conclude that the second order conductivity is: 97
II Optical response of graphene σ(2) jki (Ω,Ωp,Ωq) = e3 ℏ2ωqωpSX nml vk mn−(q+p)vj lmpvi nlq ωlm −ωp−iγlm f(εm)−f(εl) ωnm −(ωp+ωq)−iγnm +e3 ℏ2ωqωpSX nml vk mn−(q+p)vj nlpvi lmq ωlm −ωq−iγlm f(εm)−f(εl) ωnm −(ωp+ωq)−iγnm −e3 ℏ2ωqωpSX nml vk mn−(q+p)vj nlpvi lmq ωnl −ωp−iγnl f(εl)−f(εn) ωnm −(ωp+ωq)−iγnm −e3 ℏ2ωqωpSX nml vk mn−(q+p)vj lmpvi nlq ωnl −ωq−iγnl f(εl)−f(εn) ωnm −(ωp+ωq)−iγnm . (B.47) In general, σjki (Ω,Ωp,Ωq)is a function of ωp,ωq,qand pas well as ωp+ωqand q+p. B.3.1 Derivation of the second-order conductivities Beginning with the general expression (B.47) for the 2-nd order conductivity, we derive explicit formulae for the processes of generation of a frequency difference excitation out of two incident photons, ω=ωa−ωb, and for the inverse processes contributing to the reflected optical beams. We use the notation Ω= (ω, q),Ωa= (ωa,qa)and Ωb= (ωb,qb). The recipe of the calculation of all three different cases is the same : in the summation, only the two channels in Fig. 3.1 are considered and the elements ωnm,vj nmpand the Fermi-Dirac distributions are expanded to the first order in the momenta, qjand q. With these approximations, a cancellation of the zeroth order contribution, between the channels, is observed, and only the first order contributes to the conductivity. We start with the calculation of σ(2) xxx (Ω,Ωb,−Ωa). Inserting in expression (B.47) the states in Fig. 3.1 we obtain: σ(2) xxx (Ω,Ωb,−Ωa) = e3 ℏ2ωbωaSX 123 vx 23,−qvx 31,qbvx 12,−qa ω32 −ω−iγ32 f(ε1)−f(ε3) ω31 −ωb−iγ31 −f(ε2)−f(ε1) ω12 +ωa−iγ12 +e3 ℏ2ωbωaSX 1′2′3′ vx 1′2′,−qvx 3′1′,qbvx 2′3′,−qa ω2′1′−ω−iγ2′1′f(ε3′)−f(ε2′) ω2′3′+ωa−iγ2′3′−f(ε1′)−f(ε3′) ω3′1′−ωb−iγ3′1′. (B.48) We call the first line in (B.48), channel 1, and the second line, channel 2. Let us focus on channel 1. Expansion of the spatially modulated velocity matrix elements and of the elements ωnm yields, respectively: vx 23,−qvx 31,qbvx 12,−qa≈δq,qb−qaδk3,k1+qbδk1+qa,k2v3 Fcos θ1sin2θ1(B.49) and ω31 ≈2vFk1+vFqbcos θ1,ω12 ≈ −2vFk1−vFqacos θ1and ω32 ≈vFqcos θ1.The expansion of 98
B.3 Second order conductivity the Fermi-Dirac distribution gives: fc(εk+p)≈fc(εk) + ∂εkfc(εk)ℏvFpcos θ. (B.50) fc(ε)is the occupation function of conduction states and fv(ε) = fc(−ε)of the valence states. With these approximations, setting k1=kand θ1=θ, we can write for Channel 1: C1≈δq,qb−qa (evF)3 ℏ2ωbωaSX k cos θsin2θ ω32 −ω−iγ fv(εk)−fc(εk) ω31 −ωb−iγ −fc(εk)−fv(εk) ω12 +ωa−iγ +δq,qb−qa (evF)3ℏvF ℏ2ωbωaSX k cos θsin2θ ω32 −ω−iγ ∂εkfc(εk)−qbcos θ ω31 −ωb−iγ −qacos θ ω12 +ωa−iγ . Changing the summation to integration and then performing a change of variable, ε=ℏvFk, X k→gS (2π)2ZkC 0 dkk Z2π 0 dθ →gS (ℏvF)2(2π)2ZεC 0 dεε Z2π 0 dθ . (B.51) Channel 1 contribution is written: C1=δq,qb−qa e3vF ℏ4ωbωaπ2ZεC 0 dεε Z2π 0 dθ cos θsin2θ vFqcos θ−ω−iγ fv(ε)−fc(ε) 2ε/ℏ+vFqbcos θ−ωb−iγ −δq,qb−qa e3vF ℏ4ωbωaπ2ZεC 0 dεε Z2π 0 dθ cos θsin2θ vFqcos θ−ω−iγ fv(ε)−fc(ε) 2ε/ℏ+vFqacos θ−ωa+iγ −δq,qb−qa e3v2 F ℏ3ωbωaπ2ZεC 0 dεε Z2π 0 dθ cos θsin2θ vFqcos θ−ω−iγ ∂εfc(ε)qbcos θ 2ε/ℏ+vFqbcos θ−ωb−iγ +δq,qb−qa e3v2 F ℏ3ωbωaπ2ZεC 0 dεε Z2π 0 dθ cos θsin2θ vFqcos θ−ω−iγ ∂εfc(ε)qacos θ 2ε/ℏ+vFqacos θ−ωa+iγ . (B.52) Here gis the degeneracy factor equal to 4 and kcis an upper limit that has to be specified. If we require that the circle in k-space has the same area as the hexagon (the first BZ), then: kc=s8π 3√3 1 a(B.53) 99
II Optical response of graphene and for a= 1.42 Å [7], εc≈104meV. At zero temperature, Eq. (B.52) simplifies to: C1=δq,qb−qa e3vF ℏ4ωbωaπ2ZεC EF dεε Z2π 0 dθ cos θsin2θ vFqcos θ−ω−iγ 1 2ε/ℏ+vFqbcos θ1−ωb−iγ −δq,qb−qa e3vF ℏ4ωbωaπ2ZεC EF dεε Z2π 0 dθ cos θsin2θ vFqcos θ−ω−iγ 1 2ε/ℏ+vFqacos θ1−ωa+iγ +δq,qb−qa e3v2 FEF ℏ3ωbωaπ2Z2π 0 dθ cos2θsin2θ vFqcos θ−ω−iγ qb 2ωF+vFqbcos θ−ωb−iγ −δq,qb−qa e3v2 FEF ℏ3ωbωaπ2Z2π 0 dθ cos2θsin2θ vFqcos θ−ω−iγ qa 2ωF+vFqacos θ−ωa+iγ . (B.54) Now let us turn to Channel 2. For this channel: vx 1′2′,−qvx 3′1′,qbvx 2′3′,−qa≈ −δq,qb−qaδk2′,k3−qaδk3−qb,k1v3 Fcos θ3sin2θ3(B.55) and ω2′3≈ −2vFk3+vFqacos θ3,ω31 ≈2vFk3−vFqbcos θ3and ω2′1=ω2′3+ω31 ≈ −vFqcos θ3. As before, the Fermi-Dirac function is also expanded, in this case, fv(ε). However, the first order contribution vanishes at equilibrium. Setting k3=k, θ3=θand going to integration, assuming zero temperature situation, C2is written: C2≈ −δq,qb−qa e3vF ℏ4ωbωaπ2ZEC EF dεε Z2π 0 dθ cos θsin2θ −vFqcos θ−ω−iγ 1 2ε/ℏ−vFqacos θ−ωa+iγ +δq,qb−qa e3vF ℏ4ωbωaπ2ZEC EF dεε Z2π 0 dθ cos θsin2θ −vFqcos θ−ω−iγ 1 2ε/ℏ−vFqbcos θ−ωb−iγ . (B.56) Notice that the two lines in expression (B.56) for channel 2 cancel out the first two lines in for channel (B.54), when we sum the channels, thus avoiding any divergent behavior. This observation is made clear if we perform the change of variable θ→θ+πin (B.56) and use the periodicity of the integrand to rewrite the limits of integration in θto be 0 and 2π, again. With this modification, the final expression for the conductivity is: σ(2) xxx (Ω,Ωb,−Ωa)≈δq,qb−qa e3v2 FEF ℏ3ωbωaπ2Z2π 0 dθ cos2θsin2θ vFqcos θ−ω−iγ qb 2ωF+vFqbcos θ−ωb−iγ −δq,qb−qa e3v2 FEF ℏ3ωbωaπ2Z2π 0 dθ cos2θsin2θ vFqcos θ−ω−iγ qa 2ωF+vFqacos θ−ωa+iγ ≈δq,qb−qa e3v2 FEF ℏ3ωbωaπ2 2ωF−ωa (2ωF−ωa)2+γ2qA(ω, q, γ), (B.57) 100
Appendix D Energy flux of SPPs Here we calculate the energy flux, associated with the generated graphene plasmons, that propagates along the surface, in the direction of the wavevector of the plasmons. The energy flux density is given by the Stokes vector: S=Rec 8πE×H∗, (D.1) with Hbeing the magnetic field. For SPPs, this vector has the form: SSP = (S2,x,0, S2,z)e2κ2zz < 0 (S1,x,0, S1,z)e−2κ1zz > 0 . (D.2) Its x-component is (m= 1,2): Sm,x =−c 8πReEm,zH∗ m,y. (D.3) From Maxwell’s equations, the electric and magnetic field components of SPPs satisfy, Hm,y(ω, q) = −(−1)m+1iωεm cκm Em,x . (D.4) Using this relation together with Eq. (2.3), we are able to write the energy flux density in terms of the x-component of the electric field: Sm,x =qωεm 8πκ2 m|Em,x|2. (D.5) The Em,x component is determined by Eq. (3.8). Hence, the energy flux density, per unit length along y, is: J=Z+∞ −∞ dzSx=qω 16πX m=1,2 εm κ3 m|Em,x(ω, q)|2. (D.6) 107
Appendix E Differential reflectance in the presence of hot electrons As a result of the evolution of the out-of-equilibrium carriers’ population, the non-equilibrium optical conductivity, σneq, is assumed to vary slowly with time, so that it may be described as a function of both time and frequency, σneq(ω, t) = 4σ0 πℏ(γ−iω)Z+∞ 0 dε 1−f(e)(ε, t)−f(h)(ε, t) [ℏγ+i(2ε−ℏω)] [ℏγ−i(2ε+ℏω)] . (E.1) Here, γis the interband scattering rate, f(e)(ε, t)and f(h)(ε, t)are the electron and hole distributions with contributions from thermalized and non-thermalized carriers. In general, the current density can be written as jx(t) = Z∞ −∞ dt′σt, t′Ext′, (E.2) where Ex(t′)is the x-component of the probe’s electric field at z= 0 and σt, t′=Z∞ −∞ dω 2πσneq ω, t′exp −iω t−t′ =4σ0 πZ∞ −∞ dω 2πℏ(γinter −iω)Z∞ 0 dε 1−f(e)(ε, t′)−f(h)(ε, t′) (ℏγ−i(2ε+ℏω)) (ℏγ+i(2ε−ℏω)) exp −iω t−t′ . (E.3) The integral over ωin Eq. (E.3) can be solved by analytic continuation and using the residue theorem. The poles are located at ω=±2ε/ℏ−iγ , (E.4) i.e, both of them lie within the lower half-plan. For t < t′the integral vanishes, which traduces the causality principle, and the contour must be closed in the lower half plane. Employing the residue 108
theorem, Eq. (E.3) simplifies to: σt, t′=4σ0 πℏΘt−t′e−γ(t−t′)Z∞ 0 dε 1−f(e)(ε, t′)−f(h)(ε, t′)cos 2ε ℏt−t′, (E.5) where the Heaviside function Θ (t−t′)expresses the causality principle. It is worth noting that the electric field in Eq. (E.2) corresponds to the transmitted field at z= 0. With this in mind, let us proceed to define the fields. The electric fields at z= 0 can be written as follows: E(i) x(r, t) = Ai(t) cos αei(sin αkax−cos αkaz−ωat), (E.6) E(r) x(r, t) = Ar(t) cos αei(sin αkax+cos αkaz−ωat), (E.7) E(t) x(r, t) = At(t) cos θei(nsin θkax−ncos θkaz−ωat), (E.8) where ka=ωa/c and ωais the wavevector and frequency of the probe beam, αand θare the incident and transmitted angles of the probe, nis the index of refraction of the substrate and Ai,Arand At are the incident, reflected and transmitted amplitudes of the fields. This amplitudes will be defined as Gaussian functions: Ai(r,t)(t) = ai(r,t)exp −t2 τ2 a, (E.9) where τais the duration time of the probe beam. Applying Maxwell’s equations enables us to compute the corresponding magnetic fields. This results in the following expressions: H(i) y(r, t) = ωa kaccos αE(i) x(r, t), (E.10) H(r) y(r, t) = −ωa kaccos αE(r) x(r, t), (E.11) H(t) y(r, t) = nωa kaccos θE(t) x(r, t). (E.12) The reflection and transmission coefficients are determined using the following boundary conditions: E(i) x+E(r) x=E(t) x, (E.13) H(i) y+H(r) y=H(t) y+4π cjx(x, t), (E.14) or in terms of the field amplitudes, Aicos α+Arcos α=Atcos θ, (E.15) Ai−Ar=nAt+4π cjx(x, t)e−i(sin αkax−ωat). (E.16) 109
V Differential reflectance in the presence of hot electrons In the situation without pumping, the current is written simply as: jx(x, t) = σeq (ωa)E(t) x(x, t), (E.17) where σeq(ωa)is the equilibrium optical conductivity. At the frequency ωa,σeq(ωa)≈σ0. As a consequence, Eq. (E.16) yields: Ai−Ar=nAt+4π cσ0Atcos θ. (E.18) Using (E.15) and (E.18), the familiar expression for the linear transmission coefficient is obtained, Ar= cos θ−ncos α+4π cσ0cos θcos α cos θ+ncos α+4π cσ0cos θcos α Ai=t(0) aAi. (E.19) In the case involving pumping, jx(x, t)is determined by Eq. (E.2) and the conductivity is given in Eq. (E.5). Therefore, multiplying Eq. (E.16) by cos αand summing it with Eq. (E.15) yields: 2Aicos α= (cos θ+ncos α)At+4π cjx(x, t) cos αe−i(sin αkax−ωat)(E.20) = (cos θ+ncos α)At +16σ0 cℏcos αcos θ Z∞ −∞ dt′Z∞ 0 dεΘt−t′e(iωa−γ)(t−t′)1−f(e)(ε, t′)−f(h)(ε, t′)cos 2ε ℏt−t′Att′. Applying a Fourier-transform in the variable t, Eq. (E.20) becomes: 2Aicos α= (ncos α+ cos θ)At +16σ0 cℏcos αcos θ(E.21) Z∞ −∞ dt′Z∞ t′ dt Z∞ 0 dεe(iωa−γ)(t−t′)eiωt 1−f(e)(ε, t′)−f(h)(ε, t′)cos 2ε ℏt−t′Att′, where: Ai(r,t)(ω) = ZdteiωtAi(r,t)(t). (E.22) The integral over the variable tin (E.21) yields: Z∞ t′ dte(iωa−γ)(t−t′)eiωt cos 2ε ℏt−t′=ieit′ω(iγ +ω+ωa) (iγ +ω+ωa−2ε/ℏ) (iγ +ω+ωa+ 2ε/ℏ)(E.23) 110
and consequently, 2Aicos α= (ncos α+ cos θ)At +16σ0 cℏcos αcos θZ∞ −∞ dt′Z∞ 0 dε 1−f(e)(ε, t′)−f(h)(ε, t′)Att′ i(iγ +ω+ωa)eit′ω (iγ +ω+ωa−2ε/ℏ) (iγ +ω+ωa+ 2ε/ℏ), (E.24) or in terms of the non-equilibrium conductivity (E.1): 2Aicos α= (ncos α+ cos θ)At+4π ccos αcos θZ∞ −∞ dt′σneq(ω+ωa, t′)Att′eit′ω. (E.25) The last term of the equation represents a convolution of both pulses, since the non-equilibrium conductivity is a consequence of the pump beam. If there is a delay δtbetween pump and probe, the conductivity is simply changed to, σneq ω, t′→σneq ω, t′+δt. (E.26) Eq. (E.25) can solved by iteration in ∆σ(ω+ωa, t′+δt) = σneq (ω+ωa, t′+δt)−σ0, 2Aicos α=ncos α+ cos θ+4π cσ0cos αcos θAt+4π ccos αcos θZ∞ −∞ dt′∆σω+ωa, t′+δtAtt′eiωt′. (E.27) The reflected and transmitted field amplitudes are expanded: A(r,t)=A(0) (r,t)+A(1) (r,t)+... (E.28) where A(j) (r,t)denotes the j-th iteration. The zeroth iteration consists in neglecting the second term in the RHS of (E.27) and yields Eq. (E.19) as expected. The first order of the transmitted amplitude is: A(1) t=−4π c cos αcos θt(0) ncos α+ cos θ+4π cσ0cos αcos θZ∞ −∞ dt′∆σω+ωa, t′+δtAit′eit′ω. (E.29) 111
V Differential reflectance in the presence of hot electrons Using Eq. (E.15), the reflected field amplitude is determined to be: Ar=cos θ cos αAt−Ai. (E.30) Therefore, the expression for the iteration of zeroth-order is A(0) r=− ncos α−cos θ+4π ccos αcos θσ0 ncos α+ cos θ+4π ccos αcos θσ0 Ai=−ˆr(0) aAi(E.31) and for the first-order, A(1) r=−4π c cos2θt(0) ncos α+ cos θ+4π cσ0cos αcos θZ∞ −∞ dt′∆σω+ωa, t′+δtAit′eit′ω. (E.32) The reflectivity of the whole structure, under pumping, is defined to be Ra(δt) = R∞ −∞ dω |Ar(ω)|2 R∞ −∞ dω |Ai(ω)|2. (E.33) Substituting the expansion for Ar, up to first-order in ∆σ, in Ra(δt)yields: Ra(δt)≈R(0) a+cos2θ cos αˆ t(0)2ˆr(0) a 2πΦa 4π cReZ∞ −∞ dω Z∞ −∞ dt′∆σω+ωa, t′+δtAi(ω)Ait′eit′ω, (E.34) where Φa=Zdt |Ai(t)|2=Z∞ −∞ dω 2π|Ai(ω)|2=a2 irπ 2τa(E.35) and R(0) a=ˆr(0) a 2. Thus the differential reflectivity is: ∆Ra Ra =Ra−R(0) a R(0) a =cos2θ cos αˆ t(0)2 2πΦaˆr(0) a 4π cReZ∞ −∞ dω Z∞ −∞ dt′∆σω+ωa, t′+δtAi(ω)Ait′eit′ω. (E.36) Ai(ω)is a Gaussian function centered at ω= 0. Since the pulse spectral width is much smaller than ωa, we make the approximation, ∆σω+ωa, t′+δt≈∆σωa, t′+δt. (E.37) Then, Eq. (E.36) simplifies to: ∆Ra Ra≈cos2θ cos αˆ t(0) a2 Φaˆr(0) a 4π cZ∞ −∞ dt′Re∆σωa, t′+δtAit′ 2. (E.38) 112
∆Ra/Raonly depends on the real of σneq (ωa, t′+δt). In the limit γinter →0, using Sokhotski– Plemelj theorem, the real part of the conductivity is: Reσneq ωa, t′+δt=σ01−f(e)ℏωa 2, t′+δt−f(h)ℏωa 2, t′+δt (E.39) and consequently, ∆Ra Ra =−cos2θ cos αˆ t(0) a2 Φaˆr(0) a 4π cσ0Z∞ −∞ dt′Ait′ 2f(e)ℏωa 2, t′+δt+f(h)ℏωa 2, t′+δt . (E.40) 113
Appendix F Numerical implementation of the hot electron model The equations describing the model presented in Sec. 4.2 cannot be solved analytically. The most accurate and easy way to solve the differential equations is using the forth order Runge-Kutta method. If implemented without care, the calculation can take some time, so it is useful to provide some details of the implemented algorithm. Since the pump is a 100-fs pulse, the initial time is chosen as t0= 0.4ps. The initial conditions were: Fermi energy EF=Fe=Fh= 300 meV and temperature T0= 300 K. To solve the differential equations, a step h1= 0.02 fs was used to solve the nonlinear Eqs. (4.36), (4.48) and (4.52) at each h2≈0.3fs. The algorithm used was: 1. Apply Runge-kutta 4th order scheme with step h1to solve Eqs. (4.31) and (4.34) from time tito tf=ti+h2; 2. At time tf=ti+h2, calculate the integral (4.44): ˜ε(tf) = I0+ℏωb 2 e−tf/τε τee Ztf ti dt′et′/τεnnt(t′)(F.1) and solve Eqs. (4.36), (4.48) and (4.52). I0is the value of the integral at the initial moment, which is zero in the beginning; 3. Update the values of time, tito tf, and corresponding initial conditions of the integral: I0→˜ε(tf)(F.2) as well as the quasi-Fermi levels and temperature to Fe(tf),Fh(tf)and T(tf). Also update the initial non-thermalized and thermalized carrier concentrations; This procedure is repeated until tfequals the final time of the measurement chosen to be 2 ps. 114
Appendix G Quantization of the SPP electromagnetic field In this section, using the classical expression for the energy, the electromagnetic field, in a dispersive lossless dielectric medium, is quantized following Ref. [58]. For electric and magnetic fields with a harmonic time dependence: E(r, t) = Eω(r)e−iωt +c.c , (G.1) B(r, t) = Bω(r)e−iωt +c.c , (G.2) the time-averaged classical electromagnetic energy in the presence of a dispersive, lossless dielectric is given by: UEM(ω) = 1 4πZd3rE∗ ω(r)∂ ∂ω [ωεr(r, ω)] Eω(r) + |Bω|2, (G.3) where εr(r, ω)is the relative dielectric function. The idea is to take the above equation as the quantum mechanical energy of a EM field eigenmode with frequency ω. We work in the Weyl gauge, in which the scalar potential is set to zero: ϕ= 0 , (G.4) such that the electric and magnetic fields are obtained only from the vector potential A: E(r, t) = −1 c ∂A(r, t) ∂t ,B(r, t) = ∇×A(r, t). (G.5) In our case of interest, graphene is between two dielectrics, ε1and ε2, that are homogeneous. Thus, A(r, t)is expanded by making explicit the momentum in the xy-plane and since plasmons follow a 115
VII Quantization of the SPP electromagnetic field dispersion relation ωq=ω(q), the expansion is written as: A(r, t) = 1 √SX q αqe−iωqteiq·rAq(z) + c.c , (G.6) so that qnot only is the plasmon momentum but also represents a mode ωq. The coefficients αq are dimensionless. The mode functions Aqhave units [A]·[distance]and Shas units of area. From Maxwell’s Eqs. (A.3) and (A.4), we obtain the wave equation for E: ∇×∇×Eq(r) = ω2 c2εr(r, ωq)Eq(r). (G.7) Using expansion (G.6) in the wave equation, the eigenvalue equation for Aq(z)is obtained: Dq×Dq×Aq(z) = ω2 q c2εr(r, ωq)Aq(z), (G.8) where the operator Dqdefined as: Dq=iq+ez∂z. (G.9) Equation (G.8) is just Ampere’s law in the dielectric medium for the mode-function Aq(z). Next, we assume that the total time-averaged energy for the vector potential is given by: UEM =X q UEM(ωq)|αq|2. (G.10) The quantization of the theory is done by transforming the amplitudes αqinto quantum mechanical operators: αq→s2πc Lqωq ˆaq, α∗ q→s2πc Lqωq ˆa† q, (G.11) with commutation relations: hˆaq,ˆa† pi=δq,p. (G.12) Lqis the called the mode-length. This term is determined by demanding that the quantum mechanical Hamiltonian obtained from Eq. (G.10): ˆ H=X q 2πc 2Lqωq UEM(ωq)ˆa† qˆaq+ ˆaqˆa† q(G.13) coincides with the Hamiltonian for a collection of harmonic oscillators: ˆ H=1 2X q ℏωqˆa† qˆaq+ ˆaqˆa† q. (G.14) 116
H.2 Crystal and lab frames components. For the wavevector kI b, using Eq. (H.4), the eigenmodes are are found to be: ko b=ω cno(sin αo,cos αosin θ, cos αocos θ)(H.18) and: ke b=ω cne(θ, αe) (sin αe,cos αesin θ, cos αecos θ), (H.19) with: 1 n2 e(θ, αe)=cos2αecos2θ n2 o +cos2αesin2θ n2 e +sin2αe n2 e . (H.20) The respective eigen-polarizations are: ˆ do=1 pcos2αosin2θ+ sin2αo (−cos αosin θ, sin αo,0) , (H.21) ˆ de=1 2 1 psin2αecos2θ+ sin2θsin (2αe) cos θ, cos2αesin (2θ),−2 sin2αe−2 cos2αesin2θ. (H.22) For beam a, the results are the same but with −αeand −αo. Thus, the fields aand binside the crystal are: Da=−sin θ toDI a pcos2αosin2θ+ sin2αo eiko b·re−iωt ˆ do(−αo)(H.23) +cos θsin α teDI a pcos2θsin2α+ sin2θeike a·re−iωt ˆ de(−αe), Db=−sin θ toDI b pcos2αosin2θ+ sin2αo eiko b·re−iωt ˆ do(αo)(H.24) −cos θsin α teDI b pcos2θsin2α+ sin2θeike b·re−iωt ˆ de(αe), where toand teare the transmission coefficients for ordinary and extraordinary polarizations, respectively. The respective electric fields are given by E=ηD: Ea=−sin θ toEI aeiko b·re−iωt n2 opcos2αosin2θ+ sin2αo ˆ do(−αo)(H.25) +cos θsin α teEI aeike a·re−iωt n2 e(θ, αe)pcos2θsin2α+ sin2θ ˆ de(−αe) + cos θsin α teEI aeike a·re−iωt ¯n2 e(θ, αe)pcos2θsin2α+ sin2θ ˆ ka, 123
VIII Description of the experiment with BBO Eb=−sin θ toEI beiko b·re−iωt n2 opcos2αosin2θ+ sin2αo ˆ do(αo)(H.26) −cos θsin α teEI beike b·re−iωt n2 e(θ, αe)pcos2θsin2α+ sin2θ ˆ de(αe)−cos θsin α teEI beike b·re−iωt ¯n2 e(θ, αe)pcos2θsin2α+ sin2θ ˆ kb, where 1 ¯n2 e(θ, αe)=n2 e−n2 ocos αcos θpcos2θsin2α+ sin2θ n2 en2 o . (H.27) The angles αoand αe, as well as the transmission coefficients, are determined by the boundary conditions. H.3 Sum frequency generation We start the calculation of the sum frequency process with the non-linear wave equation [12]: ∇×∇×Es−ω2 s c2ε(ωs)Es=4πω2 s c2PNL , (H.28) with Esdenoting the field at sum frequency ωs=ωa+ωband PNL =χ(2) :EaEb. (H.29) In practice, Eq. (H.28) is very difficult to solve. Nonetheless, we can make it easier if we approximate the fields by plane waves and remember that sum frequency generation is only appreciable when there is momentum conservation, that is, when ∆k=ka+kb−ksis equal to zero. This is only true for the situation when ordinary components of fields aand binteract to produce a sum frequency field with extraordinary polarization: o+o→e. (H.30) Thus, only the ordinary components of fields aand bare of relevance when writing PNL in Eq. (H.29). Taking into account only the ordinary polarizations, PNL is written: PNL = 2 0 0 0 0 d31 −d22 −d22 d22 0d31 0 0 d31 d31 d33 0 0 0 EaxEbx EayEby 0 0 0 EaxEby +EayEbx = 2 −d22 (EaxEby +EayEbx) d22 (EayEby −EaxEbx) d31 (EaxEbx +EayEby) . (H.31) The z-component of the fields is not considered because the ordinary eigenmode only has xand 124
H.3 Sum frequency generation y-components. Explicitly, PNL is: PNL = 2 0 −d22 sin2θ n2 osin2θcos2α+ sin2α d31 sin4θ−sin2θtan2α n4 ocos2αsin2θ+ tan2α2 t2 o n4 o EI aEI bei(ko a+ko b)·r = 4ps t2 o n4 o EI aEI bei(ko a+ko b)·r. (H.32) From the experiment, the SF field appears at αs= 0, so its eigenmode is: ˆ de(αs= 0) = (0,cos θ, −sin θ), (H.33) the polarization is: PNL = 4deff t2 o n4 o EI aEI bei(ko a+ko b)·rˆ de(αs= 0) + 4dk t2 o n4 o EI aEI bei(ko a+ko b)·rˆ ks, (H.34) where ˆ ks=ˆ ka+ˆ kb= (0,sin θ, cos θ)(H.35) is the direction of propagation of the SF signal and deff =ps·ˆ de(αs= 0) , dk=ps·ˆ ks. (H.36) Finally, we can write the non-linear wave Eq. (H.28) as: ∇×∇×Es−ω2 s c2ε(ωs)Es=16πω2 s c2deff t2 o n4 o EI aEI bei(ko a+ko b)·rˆ de(αs= 0) +16πω2 s c2dk t2 o n4 o EI aEI bei(ko a+ko b)·rˆ ks. (H.37) This is still complicated to solve. However, we invoke the fact that the polarization cannot radiate an electromagnetic wave along its direction of propagation. In fact, from Gauss law: ˆ ks·Ds= 0 ≡ˆ ks·ε(ω)Es= 4πˆ ks·PNL , (H.38) so the condition for the ˆ kscomponent of the polarization is automatically satisfied. Thus, the radiated field only has an extraordinary component and the wave equation reduces to a scalar one: d2Es dξ2+ω2 c2n2 e(θ, αs= 0, ωs)Es=−16πω2 s c2deff t2 o n4 o EI aEI bei(ko a+ko b)·r, (H.39) 125
VIII Description of the experiment with BBO with ξdenoting the variable along ˆ ks. We now write the SF-field as Es=As(ξ)eiks·ξ, (H.40) with Ajbeing a slowly varying amplitude. Also, since the depletion of beams aand bis small, we approximate EI aand EI bas constants in Eq. (H.37). This then gives: d2As dξ2+ 2iks dAs dξ +ω2 s c2n2 e(θ, αs= 0, ωs)−k2 sAs(ξ) = −16πω2 s c2deff t2 o n4 o EI aEI bei∆kξ , (H.41) with (for ωa=ωb): ∆k= ∆k·ˆ ks= 2ω cno(ω) cos αo−2ω cne(θ, αs= 0,2ω). (H.42) Since, the crystal is mounted on a rotating stage, we can control θand tune it so that ∆k= 0. Hence from now on, we assume the condition of perfect phase match, ∆k= 0. Using the slowly varying amplitude approximation, Eq. (H.41) becomes: dAs dξ ≈i8πω2 s ksc2deff t2 o n4 o EI aEI b, (H.43) which has solution: As=i8πω2 s ksc2Ddeff t2 o n4 o EI aEI b, (H.44) where Dis an effective interaction distance. This allows us to conclude that the SF-field is of the form: As∝EI aEI b. (H.45) Since we have pulses with finite bandwidth, As(ω) = iβ Zdω′EI b(ω−ω′)EI a(ω′), (H.46) where βthe constant of proportionality, β=i8πω2 s ksc2Ddeff t2 o n4 o , (H.47) that can be determined by fitting to the experimental results. From the Manley–Rowe relations (assuming no losses) [12], the intensity variation of beam ais: ∆Ia(t) = −1 2Is(t). (H.48) 126
H.3 Sum frequency generation Here Is∝ |As(t)|2is the intensity of the SF-field. Therefore, the power depletion is: ∆Pa(t)≈ −1 2Ps(t)∝PI aPI b. (H.49) 127
Appendix I Lock-in amplifier Lock-in amplifiers are used to detect and measure very small AC signals, all the way down to a few nanovolts [61, 62]. Accurate measurements may be made even when a small signal is obscured by noise sources many thousands of times larger. This device uses a technique known as phase-sensitive detection to extract signals in a defined frequency band around a specific reference frequency. Signals at frequencies outside the frequency band are rejected and do not affect the measurement. In the experiment, the process under study is excited at a fixed reference frequency νrand the lock-in detects the response from the experiment at that frequency. In our case, the pump beam (beam b) is modulated by an optical chopper, that fixes νr. The lock-in then uses a phase-lockedloop (PLL) to generate the reference signal. The PLL locks the internal reference oscillator to the external reference, ωr= 2πνr, resulting in a reference sine wave at ωrwith a fixed phase shift θref. Since the PLL actively tracks the external reference, changes in ωrdo not affect the measurement. To understand how the lock-in works, let us, for the sake of simplicity, consider the following example. Imagine that the optical chopper modulates the beam as a sine wave1. Then, the signal is of the form Vsig sin (ωrt+θsig), where Vsig is the signal amplitude. The lock-in amplifier generates its own sine wave, VLsin (ωLt+θref), and then multiplies it by the signal using a phase sensitive detector (PSD). The output of the PSD is the product of two sine waves: Vpsd =VLVsig sin (ωrt+θsig) sin (ωLt+θref) =VLVsig 2cos [(ωr−ωL)t+θsig −θref] −VLVsig 2cos [(ωr+ωL)t+θsig +θref], one at the difference frequency ωr−ωLand the other at the sum frequency ωr+ωL. If the PSD output is passed through an ideal low pass filter, the AC signals are removed. Then, in general, there is nothing left. However, if ωrequals ωL, the difference frequency component will be a DC 1In reality, the chopper modulates the beam as a square wave. This is not a problem, since the lock-in’s reference can be locked to the square wave’s first harmonic. 128
signal. In this case, the filtered PSD output will be: Vpsd =VLVsig 2cos (θsig −θref). (I.1) This is a very nice signal because it is a DC signal proportional to the signal amplitude. This is possible because the PPL locks ωLequal to ωr. It also ensures that the phase between the signals does not change with time, otherwise cos (θsig −θref)would change and Vpsd would not be a DC signal. The phase dependency θ=θsig −θref can be eliminated by adding a second PSD. If the second PSD multiplies the signal with the reference oscillator shifted by 90◦, its low pass filtered output will be: Vpsd2=VLVsig 2sin θ. (I.2) The first output is usually called x-component while the second, the y-component: x=Vsig cos θ y =Vsig sin θ(I.3) and they represent the signal as a vector relative to the lock in reference oscillator. By computing the magnitude Rof the signal vector, the phase dependency is removed: R=px2+y2=Vsig , (I.4) where Rmeasures the signal amplitude and does not depend upon the phase between the signal and lock in reference. In practice, things are a bit more complicated. Since ideal filters don’t exist, some noise will always be present at the output of the PSD. Real filters2have a bandwidth of frequencies that are allowed to pass. Frequencies outside this bandwidth are heavily attenuated. Mathematically, filters are well described in the frequency domain. The relationship between the incoming signal Vin(ω) and the filtered signal Vout(ω), in the frequency domain, is given by: Vout(ω) = H(ω)Vin(ω), (I.5) where H(ω)is called the transfer function of the filter. For low-pass filters, H(ω)is well approximated by: H(ω) = 1 1 + iωτ , (I.6) where τ=RC is the filter time constant and Rand Care the resistance and capacitance of the RC circuit, that is, the low-pass filter. The cut-off frequency fcis defined as the frequency at which the signal power is reduced by −3 dB or one half. The attenuation grows ten times every tenfold frequency increase above fc=f−3dB. This equals 20 dB/decade corresponding to an amplitude 2In the case of the Lock-in, these filters are low-pass filters. 129
IX Lock-in amplifier BW 2st filter BW 1st filter Figure I.1: Attenuation 20 log10 Vout/Vint of frequencies for a low pass filter. First order (blue line) and second order (orange line) low pass filters. Increasing the order of the filter decreases the bandwidth (BW) and improves the roll-off. reduction by a factor of 2 every doubling of the frequency. For the filter described by H(ω), the cut-off frequency is: fc=1 2πτ . (I.7) Compared to the idealized filter, the first-order filter has a fairly poor roll-off behavior. To increase the roll-off steepness it is common to cascade several of these filters. For every filter added the filter order is increased by 1. Since the output of one filter becomes the input to the following one, we can simply multiply their transfer functions. The transfer function of an n-th order filter is: Hn(ω) = 1 1 + iωτ n . (I.8) In Fig. I.1 the behavior of a first and second order filters is shown. We now have a complete picture of the principles behind lock-in amplifier. After mixing the input signal with the reference signal, the input signal spectrum is shifted by the demodulation frequency ωr. Low-pass filtering further transforms the spectrum through a multiplication by the filter transfer function Hn(ω). The demodulated signal contains all frequency components around the reference frequency, weighted by the filter response: Vout(ω) = Hn(ω)Vin(ω−ωr). (I.9) This equation clearly shows that demodulation behaves like a bandpass filter in that it picks out the frequency spectrum centered at νrand extending on each side by f−3dB. The choice of the time constant (or cut-off frequency) and filter’s roll-off is usually a trade-off between the quality of the results and the time that takes to do the experiment. The bigger the filter’s time constant and order, the longer it takes for the lock-in’s output to settle. Although, a 130
I.1 The dB unit bigger time constant and filter order means more stable results, this is not a synonym of better results, since the experimental conditions change over time. In our case, the laser power fluctuates over the day and if the experiment takes too long, this fluctuations will start to appear in the final results. Therefore, the order and time constant are chosen to give the most stable outputs in the lowest time possible. The best combination is found by trial. A time constant of 1 second and a 60 dB/decade are enough to get good results, and this was the combination used to measure the pump depletion. I.1 The dB unit The dB is a logarithmic way of describing a ratio. The ratio may be power, sound pressure, voltage or intensity or several other things. In electrical circuits, dissipated power is typically proportional to the square of voltage or current when the impedance is constant. Taking voltage as an example, this leads to the equation for power gain level LG: LG= 20 log10 Vout Vin dB , (I.10) where Vout is the root-mean-square (rms) output voltage and Vint is the rms input voltage. 131
Appendix J TOPAS-C TOPAS-C is an optical parametric amplifier (OPA) [63]. This device exploits the second-order nonlinearity of nonlinear crystals such as BBO to transfer energy from a fixed frequency pump pulse to a variable frequency signal pulse, whose frequency is easily tuned. The principle of an OPA is shown in Fig. J.1: in a nonlinear crystal, energy is transferred from a high frequency and high intensity pump beam, at frequency ω3, to a lower frequency and lower intensity signal beam, at frequency ω1, which is amplified. Additionally, a third beam, called the idler beam, at frequency ω2, is created to fulfill energy conservation. In principle, the signal frequency to be amplified can vary from 0to pump frequency ω3, and correspondingly the idler can vary from ω3to 0. This mechanism is similar to DF generation, except for the strength of the interacting fields: DF generation arises when the fields at ω3and ω1have comparable intensities, while OPA occurs when the field at ω1is much weaker. The OPA process is efficient when, in addition to energy conservation, the phase-matching condition: ∆k=k1+k2−k3= 0 (J.1) is satisfied, where kiis the wave vector of the i-th beam. The TOPAS-C provides an optical amplifier with continuously variable frequency (determined by the phase-matching condition) and represents an easy way of changing the output frequency of an otherwise fixed femtosecond laser system. In our case, the pump is the output from the amplifier at λ3= 800 nm and the output from TOPAS-C is to be changed between 800 nm and 729 nm in ω3 ω1 ω2 Figure J.1: Optical parametric amplification: the pump at frequency ω3excites the system that relaxes via stimulated emission of photons at frequency ω1, thus amplifying the signal. 132
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