Optical pumping of intersubband polaritons

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Dipartimento di Fisica ”Enrico Fermi”

TESI DI LAUREA MAGISTRALE IN FISICA

Optical pumping of

intersubband polaritons

CANDIDATO RELATORE

Filippo Velli

Alessandro Tredicucci

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Introduction

After Einstein’s insights on the photoelectric effect and the breakthrough of quantum mechanics, the treatment of light interacting with matter, which has always sparked curiosity throughout history, has radically changed due to the concept of wave-particle duality and the discovery of the fundamental quantum of light, the photon. Attributing a corpuscular nature to light has revealed a connection with the material ”atomic” counterpart not apparent in the clas-sical description of electromagnetism. One of the more striking effects of this deeper understanding of light-matter interaction is without doubt the polariton, a mixed, part photon part material excitation, quasi particle first proposed in a seminal paper by Hopfield [1].

Under normal circumstances the interaction of the electromagnetic field with local material oscillators, such as atomic transitions, can be well described by the Fermi golden rule in the perturbative theory of quantum mechanics [2]. A photon emitted or scattered by an atom can in fact occupy a macroscopic quantity of free space modes, leading to a regime of irreversible interaction dubbed weak light-matter coupling. The same holds true if viceversa after the absorption of a photon the material oscillator dephases due to interaction with the environment. However if a single mode of the electromagnetic field interacts with a resonant material excitation, in an ideal system where the dephasing processes are absent a reversible exchange of energy can occur, a regime called strong light-matter coupling. From a quantum mechanical point of view, the strongly coupled system is described by new eigenstates, which are a linear superposition of the light and matter eigenstates of the originally uncoupled Hamiltonians. These new states are defined as polaritons. Another way to view them is as a consequence of the avoided crossing principle, arising when the dispersion of the light mode intersects the resonance of the matter oscillator.

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Polaritons in semiconductor microcavities, at low densities, have been ob-served experimentally for the first time in 1992 [3], with excitons, hydrogenoid bound states between electron-hole pairs in quantum wells, playing the role of the matter excitations. Since excitons obey the same Bose-Einstein statistics as photons, a variety of discoveries have subsequently been made for exciton-polariton systems. Some examples include Bose-Einstein condensation [4], ob-servation of polariton superfluidity [5], and most importantly the development of bosonic lasing of polaritons [6]. This lasing phenomenon shares similarities to conventional lasing in that it exhibits a ”threshold behaviour”, however it has a completely different physical origin and does not rely con the concept of population inversion [7].

More recently efforts have been made to transfer the experimental findings of exciton-polariton systems to intersubband polaritons, theoretically predicted in 1997 [8] and observed experimentally in 2003 [9]. They rely on the intersubband transition in semiconductor quantum wells as the matter excitation, which has been the basis of major technological achievements such as the quantum cascade laser [10, 11]. As such the first important difference is the spectral region of potential applications, namely the THz and mid infrared.

This thesis is inserted in the MIR BOSE research project collaboration, which has the ultimate goal of developing an instersubband polariton device exhibiting bosonic lasing. As such we have benefitted from the possibility to utilize a de-vice fabricated in Paris in order to perform our experimental measurements. The thesis is structured as follows. In Chapter 1 we have given a theoretical overview of intersubband polaritons, focusing also on the class of optical resonators which support the cavity mode. The mechanisms of intersubband polariton scatter-ing, specifically with resonant LO-phonons, which are fundamental for bosonic lasing, have also been discussed. We have then performed a characterization of the polaritonic sample, both using semiclassical numerical simulations and by performing angle resolved reflectance spectroscopy measurements, which are pre-sented in Chapter 2. Chapter 3 is dedicated instead to the experimental setup of our primary objective, optical pumping of intersubband polaritons using a high power CO2laser. We have given a technical description of the laser, which works

in a pulsed regime, and have measured the pump beam characteristics in order to obtain an estimate of the average incident power. We have also described the critical alignment process and the operating principles of the interferometer we

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have utilized. In Chapter 4, after describing the adopted measuring methods (both lock-in amplifier and boxcar averager), we present and discuss the main experimental results of the thesis, the detection of radiation re emitted by the polaritonic sample compatible with resonant polariton scattering processes. We give physical interpretation to different measuring parameters, and offer possible improvements moving forward.

Before concluding, we would briefly like to familiarize the reader with the mid infrared region of the electromagnetic spectrum. In the literature different energy units are frequently used because of the highly interdisciplinary nature of polaritonics. We have chosen to adopt the meV for the first two chapters, since they are most commonly used in theoretical simulations, and instead the wavenumber in cm−1 for the last two describing the experimental results. They are in fact the standard units of Fourier transform spectroscopy, the technique we have used to obtain the spectra. When needed we have referenced both units. A conversion table is shown below.

Range subrange [eV] [nm] Infrared (IR) NIR 0.886 - 0.413 1400 - 3000 MIR 413 - 24.8 meV 3 - 50 µm FIR 24.8 - 1.24 meV 50 µm - 1 mm Terahertz (THz) 124 - 1.24 meV 10 µm - 1 mm [cm−1] [THz] Infrared (IR) NIR 7140 - 3330 214 - 100 MIR 3330 - 200 100 - 6.0 FIR 200 - 10 6.0 - 0.3 Terahertz (THz) 1000 - 10 30 - 0.3

Quick conversion units: the spectral band of our thesis covers approximately the range 50 ÷ 150 meV in the Mid Infrared.

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Chapter 1

Overview of intersubband

polaritons

In this chapter we will try to provide an insight into intersubband (ISB) polari-ton systems, first realized in [9], by summarising the most important theoretical aspects. Special care will be given to concepts that are relevant for the experi-mental implementations which are the main subject of this thesis.

1.1 Intersubband transitions in quantum wells

Semiconductor quantum wells (QWs) are the centerpiece of a variety of physical systems and devices. Their versatility stems from their almost ideal, textbook-like quantum behaviour: different semiconductor III-V compounds can be grown to form heterojunctions at the interfaces, where offsets for the electrons in both the conduction and valence bands create barriers or wells, depending on the materials used. This leads to one dimensional size-quantization in the growth direction (z), so that charge carriers in the heterolayers lose their bulk properties and behave as if their motion were effectively two dimensional.

A complete treatment of the physics of these systems can be found in [12]. The description is based on the envelope function model, in which periodic Bloch functions of the bulk semiconductors are used as a basis set to expand the heterojunction wavefunctions around a high-symmetry point of the reciprocal lattice:

ψ(r) =X

l,k

Fl,keik·rul,k(r)

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where l is the number of bands chosen for the expansion. Since the heterostruc-ture is translationally invariant in the layer plane the envelope function depends only on z. Thus choosing the Γ(k = 0) point we obtain

ψ(r) =X

l

χl(z)eikk·rkul(r),

where χ(z) is the 1D envelope function and kk ≡ (kx, ky). While the Bloch

functions vary rapidly with the lattice periodicity, the envelope functions vary more slowly with the dimension of the layers.

In order to find the dispersion relation of these states in a first rough approx-imation it is sometimes sufficient to use a single parabolic S conduction band as model for the expansion (l = 1), with the effective mass dependent on the well and barrier materials. Proceeding similarly to the k · p method [13] the problem reduces to the eigenvalue equation

" EC(z) − ~ 2 2 ∂ ∂z 1 m∗_(z) ∂ ∂z + ~2k2_k 2m∗_(z) # χ(z) = Eχ(z), (1.1)

where EC(z) is the conduction band energy profile for k=0. The solutions in this

approximation of for example AlGaAs/GaAs quantum wells (depicted in Fig. 1.1) are the lowest lying bound states with boundary conditions χ(z),_m∗1_(z)dχ_dz

both continuous at the interfaces and energies

En(kk) = En(0) +

~2k2k

2m∗.

It can then be said that these states form energy levels called subbands which are identified by the quantum number of the corresponding envelope function and that have identical parabolic in-plane dispersions. For this reason the joint density-of-states of a transition between consecutive subbands is δ like, similarly to simple atomic transitions. An important point to make is that the spac-ing between these energy levels is not determined by the energy gap between the conductance and valence bands of the semiconductor material, but by the thickness of the wells. Thus the intersubband transition energy is a controllable quantity which can be engineered during the growth process.

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Figure 1.1: Conduction band edge profile of AlGaAs/GaAs quantum wells with the two energy levels and associated envelope functions, solutions of Eq. 1.1. The in plane dispersion relations of E1 and E2subbands are represented underneath.

The dipole matrix element of the intersubband transition can be written as

hχ_i|e · p|χ_fi = δ_k,k0e ·

Z

χ∗_i(z)pzχf(z)dz, (1.2)

which yields vertical optical transitions and the e k z selection rule corresponding to a transverse magnetic (TM) polarized electromagnetic wave [12].

1.2 Confinement of light in semiconductor

microcav-ities

The term microcavity indicates an optical resonator which has dimensions close to or below the wavelength of light. Confinement is achieved either with reflec-tion off a single interface, such as a metallic surface, or with microstructures periodically patterned on the scale of the resonant wavelength [14]. Sometimes a combination of these two approaches is used for different spatial directions in the same microcavity. Semiconductor microcavities such as AlGaAs/GaAs multilayers thus not only confine electrons, but also naturally provide confine-ment for light either via their geometry (the Bragg mirror), or simply via the difference in refractive index with a bounding medium. With an optically active material embedded in the microcavity, the interaction between light and matter

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can be enhanced or suppressed, such as in the famous Purcell effect. This is due to the inherent discreteness of the optical modes and their related density of states.

Many different geometries for microcavities exist, however we want to focus on the class of dielectric slab waveguides. What distinguishes them is that the light confinement is restricted to one direction, which leads to a conservation law of the in plane wave vector component, thus allowing coupling with a single pho-tonic mode at a fixed energy. This property is fundamental to reach the strong light matter coupling regime, detailed in the next section. A general introduc-tion to these stacks of dielectric films can be found in many textbooks such as [15], whose notation we follow. To satisfy the need to provide greater confine-ment of light in thinner active regions, not obtainable with dielectric waveguides, we will then introduce the metal-insulator-metal cavity which also has a planar slab geometry. A sketch of both dielectric and MIM resonators, embedding a semiconductor heterostructure, is shown in Fig. 1.2.

metal metal dielectric dielectric GaAs AlGaAs x z

Figure 1.2: Schematic of microcavity planar slab resonators on which most of intersub-band polariton systems rely. Left : Dielectric slab waveguide. Right : Metal-insulator-metal (MIM) waveguide.

The confinement of optical modes in planar slab waveguides is restricted to the z direction where they have a finite dimension and varying dielectric permit-tivity function ε(z) = n2(z) corresponding to the layers, while the propagation takes place in the translationally invariant (x, y) plane. Choosing x as the prop-agation direction, Maxwell’s equations have solutions of the form

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which are subject to the continuity of the tangential components at the dielec-tric interfaces. The wavevector in plane component β is continous while kz is

”quantized” for each mode identified by the mode number m. In this case the wave equation reduces to the Helmholtz equation

∂2 ∂z2 + ω_m2 c2 ε(z) − β 2 A(x, z) = 0 (1.3)

where A(x, z) is any cartesian component of the fields.

The condition for confined modes is that the fields must fall off exponentially ∝ e−kzz _{outside of the guiding center layer while being sinusoidal inside. This}

implies that the refractive index of the guiding region n2 is greater than the

index of the external bounding region n1 (n2 > n1), assuming a symmetrical

waveguide. Furthermore analyzing (1.3) the condition on the in plane propaga-tion wave vector is n1ω/c < β < n2ω/c since ∂2A/∂z2 must be of opposite sign

at the interfaces.

Introducing the effective index of refraction defined as neff= cβ/ωm, we can

ob-serve then that a confined mode travels through an effective medium with index neff which varies from n1 to n2 according to the dispersion of the mode, when

the thickness d of the guiding layer is fixed.

The complete solutions of the wave equation, which can be solved numeri-cally, comprise both TE and TM guided modes. The fields of TM modes, which obey the selection rule of intersubband transitions, are:

Hy(x, z, t) = Hm(z)ei(ωmt−βx) Ez(x, z, t) = β ωεHm(z)e i(ωmt−βx) Ex(x, z, t) = − i ωε ∂Hy ∂z . (1.4)

One way to achieve stronger coupling between the cavity mode and the in-tersubband transition is to increase the confinement of the mode by sandwiching the semiconductor QWs between two metal layers. These metal-insulator-metal planar waveguides (MIM), studied extensively in microwave systems [16], sup-port both a purely transverse TEM mode and the fundamental TM0 mode,

which has no cutoff frequency and no in-plane electric field component. Many photonic devices in the far infrared region have also employed the higher

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con-finement properties of metal-dielectric interfaces for example taking advantage of surface plasmon modes [17]. In general, as metals are highly reflective in the infrared (they behave as almost perfect electric conductors), the challenge is to find an efficient way of coupling external light into the waveguide. An idea to solve this is to pattern the top surface with a periodic 1D metallic grating with ribbons orthogonal to the x direction, which has the effect of folding the disper-sion of the modes inside the Brillouin zone, as in Fig. 1.3. This allows external incident light to couple with the waveguide modes within the otherwise empty air light cone, by satisfying the condition

kx+ 2π a m = β, (1.5)

where a is the grating period and m is an integer.

kx ω kx ω air semicond uctor mod e β #/a a

Figure 1.3: Folding scheme of a metal insulator metal waveguide, with a periodic 1D patterning on the top surface. The design has been developed in [18].

After the first studies in the THz region with patch gratings [19], a dis-persive MIM resonator has been experimentally realized which exhibits strong polaritonic coupling in the mid-IR [20]. The simulations and device presented in chapter two have this geometry. A detailed summary of the different waveguides and optical resonators of intersubband polariton devices can be found in [18].

1.3 Quantum picture of intersubband polaritons

We will now focus on outlining the theoretical framework of intersubband po-laritons, first developed by Ciuti and Carusotto [21]. Starting from a

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Hopfield-15

like Hamiltonian [1] they discuss the different coupling regimes determined by the vacuum Rabi frequency ΩR and the intersubband transition frequency ω12.

When the latter is resonant with the cavity mode and the vacuum Rabi fre-quency exceeds both the transition and mode linewidths (relaxation rates), the light-matter interaction is in the strong coupling regime, leading to polaritonic eigenstates1.

Afterwards we will briefly treat the scattering mechanisms for ISB polaritons focusing specifically on scattering induced by the interaction with longitudinal optical phonons. This process enables the possibility of final-state stimulation and a quantum degenerate regime with the effective lasing of polaritons in an optical pumping configuration [22, 7].

1.3.1 Second quantization Hamiltonian and creation of polari-ton states

The Hamiltonian for a system with intersubband transitions coupled to photonic cavity modes can be written in the interaction picture within the second quan-tization formalism. We will consider a series of nQW identical doped quantum

wells embedded in a planar cavity: the first subband is populated at low tem-perature by the two-dimensional electron gas of density N2DEG and transitions

of energy ~ω12 exist between the first and the second subband.

The creation operator of a photon in the fundamental cavity mode with TM polarization and in-plane wave vector k is a†_k. Since the subband dispersions are parallel, all the electron states in the first subband have basically the same transition energy and are identically coupled to the cavity mode. Thus the state arising from the absorption of a photon should be a symmetrical superposition of all possible electronic excitations. This is called the bright intersubband ex-citation given by the creation operator

b†_k = 1 pnQWN2DEGS nQW X j=1 X |q|<kF c(j)†_2,q+kc(j)_1,q, (1.6)

where the superposition is also over the nQW quantum wells, and c

(j) 1,q, c

(j)† 2,q+k

1_{They also introduce the so called ultra-strong coupling (USC) regime when Ω}

Ris a large fraction of ω12, attainable however only for intersubband systems with very large carrier densi-ties and in the far infrared, which is not our case. However we will follow the full general case up to a point as it is instructive.

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are the fermionic operators which respectively annihilate an electron in the first subband and create an electron in the second subband for a given quantum well. kF is the Fermi wave vector of the two-dimensional electron gas so that in the

ground state |F i electronic states in the first subband with k < kF are occupied,

empty otherwise.

In a dilute limit the intersubband excitation operators are approximately bosonic and obey the corresponding commutation rules, i.e.

1 S

X

k

hb†_kbki N2DEG =⇒ [bk, b†_k0] ' δk,k0. (1.7)

Starting from the coupled light-matter Hamiltonian in the Coulomb gauge, where the interaction comes from the minimal coupling hA · pi, we can then write the Hopfield-like Hamiltonian in second quantization

H = H0+ Hres+ Hanti

which is divided into three distinguishable terms

H0 = X k ~ωc,k a†_kak+ 1 2 +X k ~ω12b†_kbk, (1.8) Hres= ~ X k n iΩR,k(a†kbk− akb†k) + Dk(a†kak+ aka†k) o , (1.9) Hanti = ~ X k n iΩR,k(akb−k− a†_kb†−k) + Dk(aka−k+ a†_ka†−k) o . (1.10)

H0 is the uncoupled energy of the bare cavity photon and intersubband

polarization fields, which depend on their respective number operators.

Hres is the resonant part of the light-matter interaction which depends on the

vacuum Rabi energy ~ΩR,k. The linear term is the simultaneous annihilation

of a photon and creation of an intersubband excitation or vice versa with the same in-plane wave vector. Dk ' Ω2R,k/ω12 for a single transition is instead a

higher order term which contains only photon operators since it derives from the squared vector potential part of the minimal coupling, and has only the effect of giving a blue shift to the polariton energy.

Hanticonsists of the counter rotating terms which are neglected when making the

rotating wave approximation: they correspond to the simultaneous destruction or creation of two excitations with opposite in-plane wave vector.

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The vacuum Rabi energy, which is the coupling constant corresponding to one photon, depends on the characteristics of the intersubband microcavity system. For our case with a single TM mode the frequency is [21, 23]

ΩR,k = 2πe2ω12 ε∞m0ωc,kLeffcav N2DEGneffQWf12 1/2 , (1.11)

where Leff_cav is the effective thickness of the cavity photon mode and neff_QW is the effective number of quantum wells which are identically coupled to the mode. The oscillator strength f12 depends on the dipole moment and frequency of the

transition.

A more recent study in the electrical dipole gauge [24] takes into account the well known depolarization shift effect in semiconductor quantum wells, which is essentially a collective screening effect of the cavity mode by the intersubband carriers [25]. Without going into too much detail of the theoretical calculations, they find that the intersubband transition frequency is renormalised as

ω12= q ω2 12+ ωP2(1 − f ∗ 12fw),

where ωP is the plasma frequency and the factor fw is the ratio between an

effective length Leffwhich indicates the region of space with active intersubband

excitations and the effective thickness of the cavity. For double metal cavities with high confinement however fw → 1 so the depolarization shift decreases

substantially. Further analyzing fw from an experimental standpoint, for an

ar-bitrary multi-quantum well structure embedded in a resonator with well (barrier) thickness Lw(Lb), the formula as first reported in [23] reads

fw ≡

Lw

Lb+ Lw

Γ, (1.12)

where Γ is the fraction of cavity mode volume which couples to the intersubband transition.

We are now ready to find the polariton eigenstates by exactly diagonalizing the Hamiltonian through a Bogoliubov transformation. Introducing the lower polariton (LP) and upper polariton (UP) annihilation operators

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where j ∈ (LP,UP), we must solve the eigenvalue equation

Mkvj,k = ωj,kvj,k. (1.14)

vj,k = (wj,k, xj,k, yj,k, zj,k)T is the vector of the Hopfield coefficients (also known

as mixing fractions) that obey the normalization condition

|w_j,k|2_{+ |x}

j,k|2− |yj,k|2− |zj,k|2 = 1 (1.15)

imposed by the Bose commutation rule for the polariton operators. They cor-respond to the ”photonic” and ”material excitation” fractions of the polariton quasi-particles. The Hopfield-like matrix corresponding to our Hamiltonian is

Mk =        ωc,k+ 2Dk −iΩR,k −2Dk −iΩR,k iΩR,k ω12 −iΩR,k 0 2Dk −iΩR,k −ωc,k− 2Dk −iΩR,k −iΩ_R,k 0 iΩR,k −ω12        . (1.16)

The eigenvalues of Mk are (±ωLP,k, ±ωUP,k) which give rise to the two

branches in the dispersion relation of the polaritons. These eigenvalues are strongly dependent on the ratio ΩR,k/ω12shown in Fig. 1.4 at resonance ωc,k=

ω12.

We will focus on the ordinary strong coupling limit ΩR,k/ω12 1 which is

more relevant in the mid-IR region, while referring to [21] for a detailed discus-sion of the USC regime when the ratio is not negligible compared to 1. In our case the polariton operator can be approximated as

pj,k' wj,kak+ xj,kbk (1.17)

with |wj,k|2+ |xj,k|2 = 1, meaning that the annihilation operator is a

superposi-tion of only the photon and intersubband annihilasuperposi-tion operators with the same in-plane wavevector, thus ignoring antiresonant terms. The eigenvalue problem then reduces to the upper left two by two sub-matrix (without the Dk term) and

the solutions are

ω_LP(UP),k=

ω12+ ωc,k∓

q

(ωc,k− ω12)2+ 4Ω2_R,k

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Figure 1.4: Theoretical graphs showcasing strong coupling. Left : Theoretical disper-sion relation obtained from Eq. 1.18 using ΩR,k/ω12= 0.02 . The hopfield coefficients

are also plotted for the upper polariton. Right : splitting as function of the coupling strength, adapted from [21].

Both the polariton frequencies and the corresponding mixing fractions are plot-ted in Fig. 1.4 as a function of the cavity mode frequency.

At resonance, where ω12 = ωc and we have dropped the wave vector index,

the typical anticrossing behavior in the dispersion is clearly visible. It is very similar to the equivalent dressed state picture for single atoms strongly coupled with light.

The eigenvalues then correspond to

ωLP(UP),res' ω12∓ ΩR. (1.19)

and the mixing fractions are |wLP,res|2 ' |xLP,res|2 ' 1/2. The minimum Rabi

splitting ΩRthus occurs at resonance in a symmetrical position in the dispersion.

As the coupling strength increases and anti-resonant terms become important, asymmetries appear both in the polariton splitting and in the mixing fractions, as illustrated in Fig. 1.4.

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1.3.2 Stimulated scattering and expected lasing of ISB polari-tons

A very useful property of bosonic particles is that the scattering rate from an initial to a final state is enhanced (stimulated) proportionally to the occupation number in the final state plus 1 [7]. This property was used in exciton-polariton systems2 _{to develop a different type of lasing, which does not rely on population}

inversion but on final state stimulation [6].

If the scattering time toward the final state is shorter than the total lifetime of said state (taking into account both radiative and non radiative losses), then the population build-up leads to a quantum degenerate regime. A two dimensional gas of particles or quasi-particles is said to exhibit quantum degeneracy if the mean separation between the particles is inferior to their so called De Broglie wavelength λDB = 2π/k, where k is the particle wave vector, leading to an

overlap of the wavefunctions [26]. This condition is given by √

nλDB < 1, (1.20)

where n is the quasi particle 2D density. Assuming the two dimensional gas to be thermalized and the quasi particles obey a free particle dispersion modified by the effective mass m∗, the energy can be written as ~2k2/2m∗= kBT , where

kB is the Boltzmann constant and T is the temperature [7]. It follows then that

the condition of Eq. 1.20 becomes n kBT

≥ 2m

∗

h2 . (1.21)

Polariton condensation can therefore be met at much higher temperatures than for excitons, since the light component reduces by orders of magnitude the effective mass. The other important parameter is the polariton density. In exciton-polariton systems the upper limit is given by Mott transition density, since the exciton is a two particle bound state and can be broken by collective screening effects. For intersubband polaritons there is no strict upper limit as long as (1.7) is satisfied by providing enough electronic doping in the

quan-2_{Exciton-polaritons share many similarities with intersubband polaritons and have been} the guideline in the field of intersubband polaritonic research, since they are also bosonic quasi particles formed by a photon strongly coupled to a material excitation: the exciton specifically a bound state between an electron-hole pair which are attracted by a coulomb field.

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tum wells. Furthermore maximising the doping concentration also impacts the estimated output power of a polariton laser

Pout= ~ω

τrad

nS, (1.22)

where τrad is the radiative lifetime of the final state, n is the final state density

and S is the device surface.

For these reasons a roadmap to the development of an optically pumped ISB polariton LED device with potential lasing capabilities was proposed [27]. A MIM waveguide device in the mid-IR region with the energy minimum at k = 0 in the LP dispersion was found to be the most promising system. They exhibit quality factors of the order Qcav,tot ' 20 as the lower limit (at 10 µm

τcav,tot = 0.1 ps), and have the possibility of tailoring the polaritonic lifetimes

by accurately choosing the photon and ISB excitation fractions via 1 τLP-rad,tot = |wLP| 2 τcav-rad,tot +1 − |wLP| 2 τISB-rad,tot , (1.23)

where τISB-rad,tot ' (90 ns, 1 ps) is the radiative and total lifetime of the

inter-subband excitation.

A diagram of the dispersion of the system is depicted in Fig. 1.5. The chosen scattering mechanism to populate the final state is polariton scattering induced the emission of an LO-phonon, first studied in [22]. The coupling to bulk phonons is introduced in the Hamiltonian via the Frohlich interaction, and by evaluating the matrix element of the transition they find a scattering rate

Γn2,n1 ₌ n2(n1+ 1) τ21 = n2 S(n1+ 1)|wLP| 2_{(1 − |w} LP|2) ωLO ΓLO 4e2LQWf ρ~ , (1.24)

where n2 is the occupancy of the UP starting state with wave vector k, n1 is

the occupancy of the LP final k = 0 state, S the sample surface, ωLO/ΓLO' 100

for GaAs phonons, ρ is the relative dielectric constant due to phonons, LQW

is the quantum well length and f ∝ LQW is a form factor typically around 0.1.

The factor (n1+ 1) takes into account final state stimulation.

In order to have buildup of the occupation number in the final state the spontaneus scattering rate Γn2,0_{must be compared with the polariton damping}

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E

k

ISB Cavity LP UP

τ

21

τ

1rad

τ

1ΝR

τ

2NR

n

2

n

1

ℏω

pump

ℏω

LO

Figure 1.5: Sketch of the dispersion of an optically pumped polaritonic device relying on a MIM cavity and LO phonon scattering, relevant for operation as bosonic laser. The populations and various loss channels are indicated, as well as the states of the system.

model based on Fig. 1.5 for the pump and signal occupation numbers dn2 dt = AIpumpS ~ωpump − n2(n1+ 1) τ21 − n2 τ2NR dn1 dt = n2(n1+ 1) τ21 − n1 τ1tot , (1.25)

where A is the absorption coefficient at the pump frequency and angle and Ipump

is the optical pump intensity.

Imposing steady state solutions and for n1 n2, from the second equation

we obtain a ”threshold” pump polariton density nthr₂ ' τ21/τ1tot, where by

threshold we mean the minimum value of n2 for the buildup of the final state

occupation number n1. Ignoring the second term in the first equation, relevant

only above the eventual stimulation threshold, we obtain a threshold pumping intensity

I_pumpthr = ~ωpump Aτ2NR

nthr₂

S . (1.26)

Remembering (1.24) and inserting the approximate values for the lifetimes this yields an intensity of the order of 7 × 104 W/cm2, which poses a target goal for the development of an intersubband polariton laser.

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Chapter 2

Polaritonic device simulation

and analysis

The design of a polaritonic light emitting device requires control over many different physical properties of the system. The lower polariton dispersion must exhibit an energy minimum at the center of the brillouin zone [27]; the doping of the MQW active region must be tailored to achieve the desired coupling strength between the instersubband transitions and the cavity modes; finally the damping rates of the system and specifically the radiative loss channel of the optical resonator must be tuned to allow efficient coupling with the external world.

This last problem cannot be addressed in the non-dissipative quantum treat-ment presented in chapter one, though it is fundatreat-mental to predict and interpret the measureable quantities (reflectance, luminescence etc.) of an optical experi-ment such as the one subject of this thesis. Therefore in this chapter we employ a semiclassical approach [28] to perform simulations of our polaritonic device [18]. They are a useful tool to investigate the optical responses of the system in the linear response regime regardless of their microscopic origin and will serve as a term of comparison to the linear spectroscopy measurements we performed to characterize the polaritonic sample, also presented later in the chapter. In this way we can double-check if the sample meets the design parameters, excluding fabrication inconsistencies.1

1

We were not involved in the fabrication process of our polaritonic sample, which was conducted in Paris within the MIR-BOSE collaboration. The multi QW active region was grown via molecular beam epitaxy and the metal-insulator-metal resonator geometry was obtained

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2.1 RCWA simulations in the effective medium

ap-proach

To find the optical response of an intersubband polaritonic device via classical electromagnetism, the active multi QW region must be modeled with a linear and local response theory. The effective medium approach, detailed in [28] for intersubband systems, treats the multi quantum well structure as an anisotropic dispersive dielectric medium with resonances and dissipation mechanisms con-tained in the real and imaginary terms of the dielectric function.

The in-plane component of the dielectric tensor is Drude-like and does not usually vary significantly from bulk values, since it represents the various intra-subband relaxation processes, including scattering with impurities and phonons. The intersubband transition active component is instead Lorentzian like and can be expressed as εz(ω) = ε∞ 1 − f0 ε2_∞ ε2 w ω_P2 ω12− ω2− iωΓ12 −1 , (2.1)

where Γ12/2 is the damping rate of the intersubband transition, f0 is the

oscil-lator strength approximated to one for the two subband quantum well and εw is

the dielectric constant of the well material. The plasma frequency depends on the dopant volume concentration and reads:

ωP = s e2_n 2D ε0ε∞m∗(Lb+ Lw) , (2.2)

where Lb and Lw are the barrier and well thicknesses and n2D is the nominal

dopant sheet density. An important point is that the background dielectric permittivity ε∞is a weighted average of the barrier and well permittivities taking

into account their respective thicknesses. For our system GaAs/Al0.3Ga0.7As ,

the two permittivities are very close to each other ∞ = [Lbεb + Lwεw]/(Lb +

Lw) ' 10.5 .

This model proves to be very practical since in the numerical method we will now present values for the resonance and the damping rate can be plugged in ”by hand” after obtaining them with for example transmittance measurements of the bare quantum wells (Γ12corresponds to the experimental FWHM at resonance).

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Rigorous coupled wave analysis (RCWA) is a numerical method which solves Maxwell’s equations for a multilayer dielectric structure with incoming plane waves as boundary conditions. It utilizes the scattering matrix treatment to propagate the waves through the structure, thus simulating an experimental optical measurement [29]. The layers can be periodically patterned in one or two dimensions, however we will limit ourselves to the 1D case, corresponding to the grating that provides the pattern for the top layer of our MIM waveguide.

Figure 2.1: Sketch of polaritonic device to highlight the geometry of the metal-insulator-metal resonator. The field vectors indicate a TM polarized mode. Adapted from [20].

The code we are using [30] takes as inputs the geometrical parameters of the system, namely the thicknesses d of each layer, the pattern period a and the filling factor f for every layer, which corresponds to the duty cycle percentage of the metal grating. The other parameters are of course the incoming wave vector k0 and angle of incidence θ, and the dielectric functions of the layers. All

the optical constants and the in plane component of the dielectric tensor, which includes optical phonon contributions, are taken from [31]; the dielectric function of gold is defined instead according to [32]. The outputs are the reflectance, transmittance and layer by layer absorbance, which obey the energy conservation normalization condition R + T + A = 1. The code also optionally computes the complex field components on an x-z grid to help localize their presence in the waveguide. For every simulation, only the intersubband active TM polarization is considered.

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will use as an instructive example the parameters of our polaritonic sample detailed in the following section. The four layers are air/ metal/ dielectric/ metal corresponding to the superstrate, patterned gold, MQW and bottom gold layer (there is no transmittance since the thickness of the bottom gold contact layer is larger than the penetration depth of the field). The dispersion relation can be reconstructed by running simulations which span both the incidence angle θ (and thus the wave vector in-plane component kx = k0sin θ) and the frequency

ω. Choosing the frequency bandwith between 72 and 155 meV and angular range from 5 to 65 degrees, we have performed simulations tuning the parameters to highlight the differences in the optical response, shown in Fig. 2.2. We start by considering the bare cavity dispersion (left panel of Fig. 2.2), which is obtained when the semiconductor layer has no doping and is treated as a passive dielectric effective medium.

Three photonic bands, which stem from the folded TM0 mode, are clearly

visible as reflectance dips. They start from around ∼ 80 meV when the first crossing of the air light cone, represented as the dash-dotted line, occurs. In the absence of a periodic patterning the double metal cavity TM0 mode dispersion,

which reads

ω = √kxc ε∞

, (2.3)

is a line in the ω-kx plane with slope 1/

√

∞ ∼ 1/3.2 times the air light line.

The result is therefore consistent with the folding mechanism of Eq. (1.5) within the brillouin zone of boundary kxπ/a = 1, where β is now k0

√

ε∞. In other

words, an incident photon can enter the waveguide when its in-plane wavevector component matches the cavity modal propagation constant by a multiple of the reciprocal lattice wave-vector, and dwell in it as a waveguide photon until it is either absorbed by the various loss channels or re-radiated into free space modes of matching wave vector. The presence of photonic gaps between the bands is justified since the filling factor f perturbs the MIM waveguide modes, as explained in the photonic crystal formalism [33]. The keen eye can also spot a feature above the third band which stems from the folded air light line inside the brillouin zone (this is also visible in the other panels).

By populating the MQW subbands via electronic doping, the ISB transition ( ω12 ' 120 meV, represented by the horizontal dashed line, and Γ12 ' 14

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the resonant middle photonic band splits, exhibiting the typical anticrossing behavior of strong coupling apparent in both the middle and right panel of Fig. 2.2. The upper and lower branch coincide respectively with the upper and lower polariton states, and in fact if the intersubband damping rate Γ12is set to zero,

substituting the z component of the dielectric tensor (2.1) into the dispersion relation (2.3) leads to approximate semiclassical solutions for the polaritonic dispersion [18].

Figure 2.2: RCWA simulations of a double metal resonator with a MQW effective medium middle layer. The reflectance is plotted as a colormap in function of the energy and in plane wave vector. Left panel: bare cavity dispersion with a passive effective medium. The first three photonic bands of the waveguide TM0 mode are visible.

Mid-dle/Right panel: polaritonic splitting in the dispersion with increasing doping concen-tration.

The two panels differ in the value of the nominal sheet doping of the quan-tum wells: for the middle one n2D = 2.2 × 1012cm−2, while for the right one

n2D = 4.4 × 1012cm−2. The distance in energy between the two branches (i.e.

the Rabi splitting) is proportional to the square root of the doping, and thus increases linearly with the plasma frequency. At resonance, where the mini-mum anti-crossing occurs, from Eq. (2.2) we can estimate 2ΩR = ωP

√ fw .

ωP = 28 meV, which is compatible with the energy distance extracted from the

simulated dispersions ∼ 25 meV.

We also note that the polariton linewidths are not as sensitive to the doping as one could expect. It is well known that intersubband transition linewidths

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suffer from large inhomogeneous broadening (a multiple emitter effect) due to the thickness irregularities of the quantum wells [35]. Recently however it has been demonstrated that as long as the linewidth of the bare ISB transition does not exceed the coupling constant ΩR, intersubband polaritons are immune

to inhomogeneous broadening and are thus only affected by the homogeneous part [34]. Since ISB systems are characterized prevalently by their large coupling constants this effect is often visible and turns out useful to ensure better efficiency in optical pumping of polaritons.

2.2 Characterization via angle resolved reflectance

spectroscopy

We proceed now to describe the polaritonic device we will be using to conduct the main experiment of this thesis. which was courteously sent to us from Paris within the MIR BOSE research collaboration. The device relies on a dispersive, metal-insulator-metal resonator geometry as sketched in Fig. 2.1. The top surface is lithographically patterned with a 0.1 µm thick 1D metallic grating of period a = 4.4 µm and filling factor f = 78% as detailed in [20]. The bottom gold contact layer has instead a thickness of about 1 µm. The middle active region consists of a GaAs/AlGaAs multiple quantum well system: 36 period-repetitions of 8.3 nm thick GaAs quantum wells separated by 20 nm thick Al0.33Ga0.67As barriers. They are grown via molecular beam epitaxy on

an undoped GaAs substrate, and Si-δ doping is introduced at the center of the barriers to keep the donor impurities as far away from the heterojunctions as possible [36]. The nominal doping sheet density for our sample, built from wafer HM3872, is n2D = 4.4×1012cm−2. Prior to fabrication of the resonator, the bare

intersubband transition was probed in transmission both at room temperature and 78 K yielding a transition frequency of respectively 121 and 124 meV [34].

Since the surface area is relatively small (2.5 × 2.5 mm) and the patterning is very easily susceptible to handling damage, we mounted the sample on a copper block using conductive high-vacuum glue. Figure 2.3 shows an optical image of the surface pattern and a photo of the mounted device. Whenever necessary we have used isopropyl alcohol and vented dry air to clean the surface.

To characterize the device and verify the design parameters the polaritonic absorption must be probed at different in-plane wave vectors. We have therefore

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Figure 2.3: Optical image taken with a laser writer of a portion of the sample surface. The pattern of period a = 4.4 µm formed by gold stripes etched with electron beam lithography is clearly discernible. On the right is a photo of the device mounted on the copper sample holder.

performed angle-resolved reflectance spectroscopy using a commercial fourier-transform infrared spectrometer, commonly referred to as FTIR. The available one we used is a Nicolet Magna 860 equipped with a Globar infrared thermal source, a KBr beam splitter and a deuterated triglycine sulfate (DTGS) detector operating at room temperature. This setup guarantees a good broadband cov-erage of our spectral region of interest between 80 and 140 meV (∼ 9÷15 µm), which is also conveniently free of air absorption lines, allowing non-vacuum op-eration.

Because in the double metal waveguide there is no transmission through the back plate optical signals have to be acquired in a reflection configuration, which poses a problem since the FTIR spectrometer generally works in transmission. To solve it we built a reflection unit, pictured in Fig. 2.4, that fits the mea-surement chamber: the device is mounted on a x-y translator in order to center the surface area with the focal point of the spectrometer’s IR beam, while three mirrors are used to re-align the beam path with the detector. Two are silver coated plane mirrors, of which one is kept fixed while the other can be positioned in a number of discrete points that cover a good angular range, and the third is a 90◦ gold coated parabolic mirror, chosen to correct the divergence of the beam caused by the increase in path length.

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Prior to every measurement the entire system, with the sample rotated at the correct incidence angle, is externally aligned using a diode laser pointer fixed at the same height of the beam in the chamber. Great care must be taken to select the zero-order reflection since especially in the visible range the light scattering on the grating produces many orders of diffraction. By inserting a polarizer in front of the sample (a Thorlabs wire grid polarizer on silicon substrate with wavelength range 7-15 µm ), we can select the correct TM polarization that couples with the waveguide, in which case the electric field is contained in the plane of incidence and perpendicular to the grating. The Globar infrared source in fact emits approximately as a black body and thus has a uniform distribution over all polarization states. For each angle we then performed the following measurements: first we acquire a set of spectra of the polaritonic device and then another set for a gold reference sample with approximately the same surface area for normalization. Averaging to increase the signal to noise ratio and then taking the ratio of the two yields the reflectance spectra plotted in Fig. 2.5 (top panel). Sample holder Polarizer M1 M2 M3

Figure 2.4: Photograph of the reflection unit positioned in the FTIR measurement chamber. Propagation of the light beam is approximately indicated with the white dotted lines, and is from left to right.

Three absorption dips are clearly visible in each spectrum. The lowest energy one corresponds to the first photonic band of the waveguide, while the other two

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correspond for increasing energy respectively to the lower and upper polariton branches, which indicates the system is indeed in the strong coupling regime. The dispersive character of the resonator is evidenced by energy shift of the minimums over the whole angle range (∼ 15 meV for the UP and ∼ 5 meV for the LP). One thing to note is that the contrast of the dips is not very large. This has an experimental explanation: while we did try to tighten the aperture of the spectrometer’s beam and center the sample as well as possible at the beam waist, it is very probable that a fraction of the reflected light comes from the surrounding area, including the copper block, thus reducing the overall percentage absorbed by the sample. In addition at room temperature the dominant non-radiative losses lead to a reduction of the polariton lifetimes and thus of an increase in their linewidth. We also plotted the extracted polaritonic minimums over an RCWA simulation (this time in function of the angle) with the design parameters of Fig. 2.2 (middle panel). The two are in very good agreement, thus validating the estimated Rabi energy ΩR ∼ 12.5 meV given in

the previous section.

Of course we could not probe the reflectance at normal incidence in this geometry but we fully expect from the progression of the dispersion to observe an energy minimum at kk = 0, enabling a potential buildup of polaritons at

lower temperatures. We also expect to see at cryogenic temperatures a small blue shift of the dispersion of both polaritonic branches due to the typical redshift for increasing temperatures of the intersubband transition frequency. This is caused by two factors: the bandgap becoming smaller with increasing temperature and the filling of states at larger in-plane wave vectors caused by thermal fluctuations in combination with the slight non parabolicity of the subbands [37]. Taking this into account and running simulations with the shifted intersubband energy (ω12 ∼ 124 meV), we estimated the wave vector of the UP distanced in energy

by one LO phonon from the energy minimum (ωLO= 294 cm−1∼ 36 meV [38]).

The corresponding pump energy and incidence angle result in

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Figure 2.5: Observation of intersubband polaritons in the double metal cavity po-laritonic device. Top panel: reflectance measurements at different angles using FTIR spectroscopy. The resolution of the acquired spectra is 8 cm−1 ∼ 1 meV. The dashed lines are a guide to the eye and indicate the polaritonic dispersion. Bottom panel: Ex-tracted energy minima overlayed on a RCWA simulation with the parameters of Fig. 2.2

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Chapter 3

Experimental setup

To achieve significant optical pumping of intersubband polaritons it is necessary to choose a high power laser source for the incident pump beam. This involves utilising specific optical elements to manipulate the alignment and beam proper-ties. We have thus conducted the measurements of our main experimental work in professor Carelli’s lab at the University of Pisa, who generously allowed us to use his test bench and a CO2 laser previously home designed and built to

conduct spectroscopy in the mid to far IR region [39].

Our goal is to measure the radiation scattered from the polaritonic device, in order to be able to detect the component deriving from the radiative decay of the ISB polaritons. Current theoretical predictions suggest this decay must be orders of magnitude smaller than the pump beam, even in the best case scenario of polariton lasing phenomena. It is thus important to optimize the experimen-tal conditions in order to meet the many different requirements, ranging from cryogenic compatible operation to a detection scheme with rotating degrees of freedom, all with the highest possible signal to noise ratio. In this chapter we will therefore illustrate the entire experimental apparatus, shown in the diagram below (Fig. 3.1). First we will characterize the CO2 laser source and afterwards

describe the entire detection setup and the alignment process. Lastly we will briefly give an overview of the working principles of the compact spectrometer we have used for our measurements.

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MCT M1 M2 M3 CO2 cryostat interferometer f#2 pol 1 pol 2 f#1

Figure 3.1: Diagram of the experimental apparatus for optical pumping of intersubband polaritons. All the optical elements are labeled; the pump beam is colored in red and the expected re emission is colored in green. Sketch is not to scale.

3.1 CO

2

laser characteristics

The CO2 laser at our disposal is a high power waveguide gas laser which

oper-ates in a pulsed mode, previously built primarily to serve as an optical pump source for far infrared gas lasers. Since it was last employed in research a few ago, a major part of this thesis has been to restore and check the correct func-tioning of the laser. We have also refurbished some parts in order to obtain good performances, including the output coupler mirror and components of both the vacuum and cooling circuits. After briefly outlining the essential theory, we will give a technical overview of the laser and its characteristics.

The laser emission in a CO2 molecule takes between roto-vibrational energy

levels close to the ground state, giving the system a high efficiency [40]. The other components of the gas mixture, which serves as the active medium when excited by an electrical discharge, are N2and He. N2functions as a metastable excitation

channel, where a large part of the accumulated energy is transferred to the CO2

upper laser level (001) through collisions, providing the population inversion with respect to the lower symmetrical vibrational levels (100) and (0200). The rapid decay of these levels to the ground state (000) is guaranteed by inelastic collisions, which however provoke heat accumulation in the gas mixture. To avoid a detrimental rise in temperature, heat is quickly dispersed by continuous reflow of the mixture and more importantly by the He component, which being a light gas accelerates diffusion towards the waveguide walls.

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smaller energy separations and identified by the angular momentum quantum number J . Because of its structure, the CO2 molecule can exist only in totally

symmetrical states. The two transitions (001) → (100) and (001) → (020_{0) are}

therefore between groups of lines which obey the selection rules ∆J = ±1: they are called the P band (∆J = 1) and R band (∆J = −1). The laser emission can thus span four bands including a P and R band centered at 1064 cm−1 and a P and R band centered at 964 cm−1. Experimentally in each band the most intense emission is for J = 20. This is convenient because our predicted pump energy of Eq. 2.4 practically coincides with the 9 R(20)=1078.6 cm−1 transition. However to obtain laser emission corresponding only to the desired line a dispersive element such as a metallic grating must be inserted in the resonator [39]. 3.1.1 Technical description grating mirror electrodes ground

Figure 3.2: Photograph of the CO2 laser body. The blue and red arrows correspond

respectively to the flow of the cooling liquid and of the gas mixture. The two separate discharges along the waveguide are clearly discernible in purple. The micrometer to turn the grating is placed at the top of the chamber, while the screws to align the front mirror are behind the metal optical plate.

A photo of the laser is shown in Fig. 3.2. The structure consists of a circular hollow dielectric waveguide 75 cm long and 3 mm in diameter. It is a glass Pyrex capillary, chosen for the high surface quality and good thermal properties. Two electrodes are inserted in the middle which generate two separate but simultane-ous electrical discharges toward the grounded endpoints of the waveguide. Each

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branch is sleeved in a secondary glass casing in which the cooling liquid (water and Ethylene Glycol) flows to absorb excess heat and keep the gas temperature under control. An external thermostat regulates the working temperature of the liquid, usually fixed around −4◦C. The gas mixture (we have used a mix with 9% carbon dioxide, 10% nitrogen and the rest helium) is fed into the capillary and kept in flux by a rotative pump.

The optical resonator is formed by an output coupler mirror made of ZnSe with nominal reflectivity R = 90% installed in front of the cavity, while at the back a blazed metal diffraction grating patterned with 150 grooves/mm is used in the Littrow configuration. The grating is placed in a chamber equipped with an external micrometer screw which allows to vary its inclination, in order to provide selectivity over all the emission lines.

The blazed grating allows to achieve maximum optical power on the first diffraction order. The diffraction angles are determined by the grating equation

d(sin θi+ sin θo) = mλ, (3.1)

where d is the groove spacing, θi and θo are the incidence and diffraction angles

respectively and λ is the wavelength. In the Littrow configuration, the incidence and diffraction angles are identical, i.e. the diffracted beam is back-reflected in the direction of the incident beam. Imposing θi = θo and m = 1 we obtain

2d sin θi = λ. (3.2)

Knowing the groove spacing and the center wavelength of the CO2 band the

inclination angle of the grating is thus ' 42◦. Varying the inclination around that angle provides the selective reflectivity which allows to change the laser lines. To check experimentally on which line the laser is working we have used a CO2 spectrum analyzer, also based on a grating and a calibrated thermal

sensitive ruler.

Since the grating polarizes the radiation in the plane orthogonal to the grooves, or parallel to the optical table, the modes that propagate in the waveg-uide are hybrid EH1m modes with a linearly polarized electric field [41]. For the

fundamental mode E11, the intensity as function of the inner radius is given by

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end of the waveguide these modes couple to the free space gaussian modes, and only the EH11 mode couples with a good efficiency to the lowest order TEM00

mode. It is therefore crucial to obtain a good beam to carefully align the waveg-uide, grating and mirror: the waveguide can be roughly straightened by acting on three screws that regulate the two supports, while the mirror orientation can be corrected with four finer screws that turn the steel plate on which it is mounted. Additionally, a piezoelectric ceramic (PZT) with a high voltage con-troller can be used to correct the distance of the mirror from the waveguide up to 10 µm and thus the frequency tunability of the cavity. For a laser resonator indeed the separation between two successive TEM modes (called free spectral range) is

∆ν = c

2nL, (3.3)

where n is the index of refraction and L the length of the cavity; this yields in our case a FSR of approximately 200 MHz. Varying the piezo thus allows to obtain the best power output around two successive longitudinal modes.

The entire mode selection process must be carried out painstakingly in or-der to guarantee a beam that can be effectively focused. We can examine the resulting beam shape using sets of thermal image plates with different sensitiv-ities illuminated by an ultraviolet lamp. With the help of a HeNe alignment laser, collinear with the CO2 after refracting through a Brewster window, we

place the thermal plate at a sufficient distance to distinguish if the free space mode is approximately a low order gaussian. The areas where the laser impinges on the plate are in fact visualized as dark spots because of the higher surface temperature.

Lastly we must discuss briefly the laser’s power supply, which pilots the discharge in the gas mixture. It works in a pulsed regime, generating two in-dependent discharges, and it is of a capacitive type. The charge and discharge circuits are controlled by insulated gate bipolar transistors, which work with large currents and voltages up to the kV range, while guaranteeing a high oper-ation frequency. They act as fast switches when the gate is driven with voltage impulses of ' 15 V. The discharge can thus be tuned between 450 and 950 Hz with an impulse width from 10 to 50 µs. While the power supply has its own impulse generator, it suffers from very large jitters, making it impossible to use it as reference for lock-in measurements. We have thus used an external trigger

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to pilot the gate, using the pulse waveform of a Rigol DG4102 wavefunction generator, with 643 Hz frequency and 1 % duty cycle. With this setup we have obtained a maximum average power of 1.1 W at the source, i.e. not counting mirror losses, measured with a Scientech 365 power energy meter.

3.1.2 Beam diameter at the focal point using a knife-edge tech-nique

Our objective is to obtain the highest possible incident intensity on the polari-tonic sample since, as we have seen, polaritons start accumulating only at high pump powers. This intensity is of course function of the CO2 laser pulse peak

power, but also of the surface area of the sample on which we are able to focus it. It is therefore important to determine the beam’s diameter and quality at the focal point so to have a better estimate of the intensity.

We have experimented with different configurations for the guiding and fo-cusing optics, before settling on the one shown in Fig. 3.1, composed of three mirrors. Mirror M1, placed at a distance of about 2 m from the laser on the other side of the optical table, is a spherical concave solid copper mirror, with a curvature R= 90 cm and a large surface area to collect the divergent beam. The beam is sent at a slight angle on to a second plane mirror M2, also solid copper, which serves to center the beam at 90◦ with respect to the final off-axis parabolical gold mirror M3 (Thorlabs 25 mm diameter, 150 mm reflected focal length) which focuses it on the sample. The radiative losses of the beam over this path (from the laser source to the final focal point) amount to about 15 %. We have used knife-edge scan measurements, a technique frequently em-ployed for infrared laser sources [42], to reconstruct the beam’s characteristics around the focal point. In a knife-edge scan a very sharp blade, mounted on a three axis translation stage, is placed at a certain z position in the xy plane with the edge oriented in either direction, (let’s suppose x). The power of the beam portion transmitted through the knife-edge is then recorded for different values of the blade’s x position. Assuming the beam’s 2D intensity distribution to be separable in the two directions I(x, y) = Ix(x) × Iy(y), which is reasonable

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39 can be written as P (x) = Z x −∞ Z ∞ −∞ I(x0, y)dx0dy = Z ∞ −∞ Iy(y)dy · Z x −∞ Ix(x0)dx0 ∝ Z x −∞ Ix(x0)dx0. (3.4)

This means that P (x) is nothing more than the integration of Ix(x). Therefore

by taking the derivative we can obtain the intensity distribution function along the x direction, mathematically Ix(x) ∝ dP/dx.

The process is then repeated for different z values around the expected focus position and the same for the other direction. In the two panels of Fig. 3.3 we have plotted acquired data points {P } and the resulting curves of the x and y scans respectively. During the scan uncertainties exist due to power fluctuations or mechanical inaccuracy of the translation step size, especially toward the peak in the slope. These will be amplified by the post scan process of numerically taking the derivative to reconstruct the beam intensity distribution. If the dis-tribution is close to a gaussian, we can reduce the errors by fitting directly these sets of data employing the error function via the expression

P (x) = A + B · erf √ 2(x − x0) wx ! , (3.5)

where A, B, x0 are free parameters and wx coincides with the fitted 1/e2 radius

of the gaussian beam. Another possibility is to use a smoothing algorithm for the discrete data points {P } [42]; this can be especially useful as a term of comparison or if the intensity profile is very irregular and far from a gaussian. We have thus tried it by adding points by linear interpolation and then using a fourth order Savitzky-Golay smoothing method, yielding similar results. In Fig. 3.4 the resulting intensity distribution for the y data sets are shown as example. The open circles represent the derivatives of the raw knife-edge data, the blue curve of the error function fitting and the red the smoothing method.

In order to determine the beam waist at the focal point we can use the values of the 1/e2 beam radii {wx,y(z)} extracted from the fitting process. How the

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mathe-Figure 3.3: Raw knife-edge data at different z positions. The total power on the detector is around 650 mW. Left panel : scans in the y direction. Right panel : scans in the x direction. matically by wx,y(z) = w0x,y v u u t1 + " (z − z0x,yλMx,y2 πw2_0x,y #2 , (3.6)

where w0x,y is the beam waist, z0x,y the position of the beam waist, λ the

wavelength and M2

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Figure 3.4: Intensity distributions Iy at different positions around the focus. The open

circles are from the raw knife-edge data, while the blue and red curves are from the error function fitting and the smoothing method.

the x, y direction scans respectively. Inserting the wavelength corresponding to our laser line, λ = 9.27 µm, and leaving the others as free parameters, by fitting we obtain the curves shown in Fig. 3.5. The caustic shape is typical for near gaussian modes. The resulting beam waists, extracted from the fit parameters, are w0x= 0.45 mm and w0y = 0.39 mm. The evident astigmatism of the beam

is the result of a combination of two factors. An actual astigmatism of the CO2

beam, caused by a not perfect alignment of the output coupler mirror and the cavity, and an apparent astigmatism caused by the centering of the beam on the surface of the 90◦ parabolical mirror.

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We can now discuss the estimated peak intensity of the beam which impinges on the surface of the sample in our optical pumping configuration. The surface area S can be approximated by πw02, where we have taken an average of the

radii in the two directions, yielding S ≈ 5 × 10−3 cm2. With the highest average power at the sample position of 950 mW (after mirror losses), we have an average intensity of I = P/S ≈ 190 W/cm2. Taking into account a pulse repetition rate of 643 Hz and a duty cycle of ≈ 1% the peak intensity over one pulse width of τ ≈ 15µs is Ipeak ≈ 20 kW/cm2.

Figure 3.5: Change of the 1/e2_{beam radii in the x and y directions as function of the}

position. The dots are the extracted values from the error function fit, while the curve is the result of fitting via Eq. 3.6.

It is important to note that this value is the relevant one for our experiment, since from Eq. 1.24 the intersubband polariton-phonon scattering time scales are around the µs order. We can observe that it is not far off from the expected lasing threshold discussed at the end of chapter 1, however a few considerations must be taken into account. Firstly the coupling efficiency between the free space mode and the intersubband active guided mode is never unity, since only a portion of the incident light is absorbed through the grating. This means the peak intensity we have achieved is an overestimate of the actual pump values in the upper polariton state. The second consideration regards laser damage thresholds on the GaAs quantum wells. It has been reported in the literature

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that laser induced damage threshold on the surface of bulk GaAs in the mid infrared may vary from 100 to 250 kW/cm2 depending on the wavelength and on the duration of the incident pulses [43, 44]. Most of the damage is caused by local surface heating effects (carrier diffusion and Auger recombination). It is not clear how these values may change for a more complex structure such as the one of our device, where the gold layer provides a very high confinement at the interface, however it is a factor that must definitely be taken into account when trying to further push the ceiling of the pump intensity.

3.2 Angle resolved pump and detection setup

As explained in chapter 2, the primary condition to observe resonant scattering of intersubband polaritons is to correctly select the incidence angle of the laser beam in order to match the in plane wave vector of the upper polariton injection state. Additionally we must be able to collect scattered light at a set angle of detection with high accuracy. This provides the main requirement that our experimental setup has rotating degrees of freedom both for the detection unit and for the cryostat containing the sample.

We have utilized a large ring-shaped optical plate which is graded in degrees and free to turn as a base on which to mount all the components of the detection unit. At the center of the ring we have fixed another rotatable circular plate graded in degrees to serve as the base for the cryostat mount. In this way the the sample can be aligned perfectly in the center of rotation of the detector. A photo of the workbench is shown in Fig. 3.6.

The preparation of the sample consists first in the cleaning process with isopropyl alcohol gently and vented dry air. Afterwards we screw the base plate into a copper sample holder. To assure good thermal conductivity we have inserted thin Indium foils as a middle contact layer. The sample holder can then be fixed to the tip of the cold finger following the same procedure, with the orientation of the grating orthogonal to the pump beam’s polarization (TM polarization rules). The cryostat we have used is an Oxford Instruments Optistat connected with a Mercury iTC temperature controller. To start the cooling procedure a high-vacuum pump brings the pressure in the sample chamber down to E-5 mbar; we then pour the liquid nitrogen in a specific storage chamber and refuel every 30 min, since it is not supplied via a static flow. The cold

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Ring base plate

interferometer

detector sample chamber

Figure 3.6: Photograph of the experimental bench. The dotted red line indicates the pump beam path, while the blue indicates the alignment direction of the detection unit, normal to the sample.

finger is equipped with a Lakeshore DT-670 silicon diode temperature sensor, calibrated between 1.4 and 500 Kelvin. We have substituted one of the sample chamber windows with a 5.1 cm ZnSe (zinc selenide) window, which offers a large transmission band between 3÷18 µm, and has low absorption in the visible red spectrum. This is especially useful for alignment purposes when using the HeNe laser. Additionally, placing the sample as close to the window as possible is crucial to allow a wide angular aperture, since both the injected and collected light pass through it.

Referring back to Fig. 3.1, the detection unit comprises of five optical ele-ments plus the detector: two lenses for collimation purposes and the interfer-ometer sandwiched between two polarizers, which are necessary for the mea-surement. The working principles of the interferometer are detailed in the next section. The detector is an MCT (mercury-cadmium-telluride) photoconductive detector 1 that requires liquid nitrogen cooling. It has an active area of 0.25

Optical pumping of intersubband polaritons

Dipartimento di Fisica ”Enrico Fermi”