AlGaAs on-chip device emitting a light beam carrying spin and orbital angular momentum

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POLITECNICO DI MILANO

School of Industrial and Information Engineering

Master Degree in Physics Engineering

AlGaAs on-chip device emitting a

light beam carrying Spin and

Orbital Angular Momentum

Supervisor:

Prof. Roberto Osellame

Cosupervisor: Prof. Sara Ducci

Master degree thesis of:

Giorgio Maltese

Matr. 819242

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Acknowledgments

I am grateful to Prof. Roberto Osellame from Politecnico di Milano who accepted to be the relator of this master research project, carried out at Matériaux et Phénomènes Quantiques Laboratory in Paris. I would also thank the Prof. Claudia Dallera, who supported me during all my second year of master study in Paris.

My deep gratitude goes to Prof. Sara Ducci, which gave me the opportunity to work in her group on a very interesting project. Her warm encouragement never let me down and her guidance helped me to stay focused even in hard times.

I am also thankful to the researcher Yacine Halioua for sharing with me his experience and knowledge. Besides, he taught me how to approach complex problems, such as the design of a completely new device.

I would like to thank the clean room engineers Stephan Suffit, Christophe Manquest and Pascal Filloux from MPQ and Michael Rosticher from École Normale Superiore. They introduced me to the clean room and taught me how to use many fabrication equipments.

Thanks also to the phd students Claire Autebert, which showed me how to work efficiently in a clean room, and Guillaume Boucher, who taught me how to present scientific data in a professional manner. Besides, I would like to thank the postdocs Pierre Allain and Yao Qifeng for many inspiring discussions about physics.

I also take this opportunity to express my gratitude to all the other members of MPQ for their incessant support and friendship. Thanks to them, I had a very good time during the whole year of master research project. Finally, I would like to thank both my Italian and French classmates for the great years we spent together. At last, a special thanks goes to my girlfriend, all my friends and my family for their endless support.

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1 Abstract

We have developed an on-chip device based on aluminium gallium arsenide (AlGaAs) which spins and twists a linearly polarized (LP) beam at the telecom wavelength (λ =

1.55µm). At the device output, the beam carries both Spin Angular Momentum (SAM) and Orbital Angular Momentum (OAM).

The conversion from LP to SAM/OAM occurs through an asymmetric waveguide. The modal dispersion adds a phase delay∆φ=π/2 between the fundamental TE and TM

modes, generating a circularly polarized beam carrying SAM. Since the light is tightly focused as it propagates along the waveguide, part of its SAM is converted into OAM of first order, carried by the longitudinal component of the electric field.

Our device has proved to be monomode and has comfortable fabrication tolerances, thanks to its geometry and material composition. We observed that the induced TE-TM phase delay varies in a range of∆φ ≈ 90◦±30◦ by finely tuning the wavelength. This additional feature agrees with a simple model of birefringent and not loss-less Fabry-Perot.

The emitted OAM is interesting for coding quantum states on a new basis, which could lead to faster and safer quantum communication protocols on chip. In addition, the integrated device could be monolithically integrated in more complex quantum photonics circuits. As instance, thanks to the direct band gap AlGaAs, it could be integrated in lasers, other than devices based on χ2 non linear optics, like entangled photons sources.

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2 Estratto

Abbiamo sviluppato un dispositivo integrato basato sulla piattaforma arseniuro di gal-lio e alluminio (AlGaAs) in grado di mettere in torsione e rotazione un fascio di luce linearmente polarizzato (LP) alla lunghezza d’onda telecom (λ =1.55µm). All’uscita del dispositivo, il fascio trasporta sia Momento Angolare di Spin (Spin Angular Mo-mentum, SAM) che Momento Angolare Orbitale (Orbital Angular MoMo-mentum, OAM). La conversione da LP a SAM/OAM avviene grazie ad una guida d’onda asimmetrica. La dispersione modale induce un ritardo di fase di∆φ=π/2 tra i modi fondamentali

quasi-trasverso elettrico (TE) e quasi-trasverso magnetico (TM), generando un fascio circolarmente polarizzato che trasporta SAM. Dato che la luce è fortemente confinata mentre si propaga lungo la guida d’onda, parte dal suo SAM è convertito in OAM del primo ordine, trasportato dalla componente longitudinale del campo elettrico.

Il nostro dispositivo ha dimostrato di lavorare in condizioni di singolo-modo ed offrire comode tolleranze di fabbricazione, grazie alla sua geometria e composizione. Inoltre abbiamo osservato che il ritardo di fase indotto tra TE e TM può essere scelto in un in-tervallo∆φ≈90◦±30◦variando in modo fine la lunghezza d’onda della luce entrante. Questa caratteristica aggiuntiva è spiegabile con un semplice modello di Fabry-Perot con una cavità birefringente e con perdite.

L’OAM emesso può rappresentare una base aggiuntiva per codificare stati quantistici, per protocolli di comunicazione su chip più veloci e sicuri. Inoltre, il dispositivo può essere integrato in più complessi circuiti fotonici per il calcolo quantistico. Per esem-pio, grazie all’intervallo diretto dell’AlGaAs tra le bande di valenza e conduzione, può essere integrato in ssorgenti laser o in dispositivi basati sull’ottica non lineare in χ2, come sorgenti di fotoni in entanglement.

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3 Introduction

At the beginning of the previous century Albert Einstein unveiled the quantum nature of light by predicting its wave-particle duality. Sixty years later, the development of laser has allowed to produce coherent and monochromatic light beams, which could be manipulated through several kind of optical devices. An important example is optical fibers, which have revoluzionized classical communication by guiding laser light at low losses. More recently optics has become a strategic choice for quantum information as well. Quantum information is the field that studies how information is encoded, manipulated and detected in quantum systems. As instance, one important field of quantum information is quantum communication, whose protocols are much more secure than the corresponding classical ones. The most common is quantum key distribution (QKD), based on the impossibility to copy secret keys, encoded in quantum states, without corrupting them.

Nowadays our challenge is to implement the concepts of quantum information and communications in integrated photonics devices, extremely compact and massively parallel, directly compatible with the existing telecom network and operating at room temperature. Among the different platforms, we can mention glass written by fem-toseconds laser, which have already demonstrated complex circuits (e.g. beam split-ters, interferomesplit-ters, 3D waveguides) and doped glass for active devices (e.g. ampli-fiers, lasers) [16]. On the other hand, if a higher miniaturization is needed, semicon-ductors materials such as silicon (Si) and aluminum gallium arsenide (AlGaAs) may be chosen due to their high refractive index contrast. The main advantage of Si is its compatibility with the spread CMOS technology, but its indirect band gap makes the fabrication of electrically driven sources difficult. Furthermore its lattice structure does not permit to generate photon pairs via non linear phenomena of the second or-der, so people need to rely on the third order non linear effects, which are smaller. An important example of second order non linear effect is spontaneous parametric down conversion (SPDC). In SPDC one incoming photon, named pump photon, generates two photons, called signal and idler, who share its energy. In order to have an efficient conversion from the pump photons to the signal/idler photons, the total momentum must be conserved. In a wave-picture this condition is called phase-matching. AlGaAs is a particularly suitable platform to perform SPDC: the symmetry of its lattice struc-ture is compatible with second order non linear conversions, while its refractive index range ( n ∈ [2.98, 3.37]at telecom λ= 1.55µm) is wide enough to allow a fine phase-matching engineering. Moreover, AlGaAs presents a direct band gap that allows a

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3 Introduction

radiative recombination, hence lasers.

For the above reasons, we decided to design an integrated device for the generation of Orbital Angular Momentum (OAM) based on AlGaAs. The long term objective is to monolithically integrate this new miniature component to the photon pairs sources already developed in the team.

3.1 AlGaAs photon pairs sources

AlGaAs integrated quantum sources, based on SPDC, have been demonstrated at the MPQ laboratory (Laboratoire Matériaux et Phénomènes Quantiques) by the team led by Sara Ducci, at the University Paris Diderot.

Among them, we can mention an electrical pumped source of photons pairs [13], illustrated in fig. 3.1, and a transversely optically pumped source of counter propa-gating twin photons, rapresented in fig. 3.3, conceived to engineer the frequency state of the photon pairs with great versatility [53].

3.1.1 Electrically pumped integrated twin photon source

Figure 3.1: Electrical pumped AlGaAs source of polarisation entangled photons de-veloped at MPQ. On the left, the schematic of the device: an electrically pumped quantum well generates pump photons which give raise to SPDC. Polarisation-entangled photons at telecom wavelength λ = 1.57µm are emitted. On the right, image of the device obtained by Scanning Electron Microscope (SEM).

The electrically driven device is a multilayer AlGaAs waveguide grown by Molecular Beam Epitaxy (MBE). It is based on an AlGaAs heterostructure containing a quantum well lasing at 785nm; the laser photons can be converted by type II internal SPDC to generate photon pairs in the telecom range. As image 3.1 shows, the quantum well is inserted in the core of the structure. The lasing mode is the Transverse Electric Bragg

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3 Introduction

mode, as two Bragg mirrors provide the vertical confinement by photonic band gap. The twin photons are emitted in the fundamental TE00and TM00modes, confined by total internal reflection. TEBragg, TE00 and TM00 lateral confinement is obtained by total internal reflection at the AlGaAs-air interface by the 4µm wide ridge, obtained through a wet etching along the (011) crystalline axis.

Figure 3.2: On the left, the far-field profile of the Bragg mode emitted by the laser. On the right, the time-correlation histogram of TE/TM photon pairs at 1.57µm, at room temperature (T=25◦C).

The spatial intensity distribution of the laser beam is observed in the near and far field. Fig 3.2 on the left shows the far field profile of the laser beam recorded on a screen. The generation of photon pairs is demonstrated through temporal correlation measurements (Fig. 3.2, on the right) The internal generation efficiency of the device is µ ≈ 7 · 10−11 _{pairs/injected electron in the stimulated regime, above the lasing} threshold. The SPDC efficiency is µSPDC10−9 pairs/pump photon, comparable with the SPDC of passive devices based on the same phase-matching scheme [33].

3.1.2 Transversally pumped integrated twin photon source

A transverse pump configuration (fig. 3.3 on the left) is used to engineer the pho-ton pairs frequency state. A pump field (λ = 775µm) impinges on top of an AlGaAs waveguide with an angle θ generating two counterpropagating, orthogonally polarised twin photons (λ≈1.55nm) through SPDC. The pumping field is confined in a micro-cavity consisting of two Bragg reflectors in order to enhance the conversion efficiency [53]. The source has an efficiency of about 10−11pairs per pump photon and a spectral linewidth of 0.3nm for a 1mm long sample.

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3 Introduction

Figure 3.3: On the left, schematic of the optically pumped AlGaAs source of twin pho-tons developed at MPQ. A pump beam impinges on top of the waveguide and via SPDC produces two possible twin photons states. Phase-matching is satisfied automatically along the waveguide direction, while a periodic modulation of the waveguide core is required in the epitaxial direction. On the right, signal (red line) and idler (blue line) wavelength depends on the incoming angle θ, due to momentum conservation. The pump beam incidence angle ±θ corresponds the frequency of the degenerate photon

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3 Introduction

can be generated. In particular, by pumping the device with the angles correspond-ing to frequency degeneracy, the produced state presents a high level of polarization entanglement showing a violation of Bell inequalities and a fidelity of 83% to the Bell state (fig. 3.4).

Figure 3.4: Density matrix reconstruction of the Bell state produced by the optically pumped twin photon source.

3.2 AlGaAs device to generate a light beam carrying OAM

The above mentioned devices have been conceived to generate quantum states in which photons are entangled in frequency and polarization. However, light offers an additional physical quantity to play with. As Allen showed in 1992 [3], light beams carry an orbital angular momentum (OAM), a degree of freedom that allows the gen-eration of hyper-entangled quantum states [5].

In the past fifteen years, OAM light beams have been investigated in free space quantum optics and quantum communication experiments [44, 69]. These twisting beams are interesting because of their potential infinite number of states, in contrast to the only two orthogonal states offered by polarization. As qu-bits(0, 1)can be encoded in the polarization state of photons [34], photons with OAM can be the carrier of

qu-dits(a, b, c, d, e..), a superposition of d different quantum states [20]. Since the unit of information is much richer, the number of channels for the propagating information increases while more complex secret keys can be used. As a result, faster and safer communications are possible.

Besides, OAM of light has found applications in many other fields as well, like particle micromanipulation [63], imaging [22], free space communication system [72] and even astronomy [70]. It has also contributed to pioneering fundamental physics, leading to the discovery of rotational Doppler frequency shift [14] other than the for-malization of angular uncertainty relations [21] and Poincarè hyper-spheres [55].

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3 Introduction

Nowadays, one of the challenges of OAM optics is to generate and propagates OAM light beams in integrated chips. Some of the main difficulties in guiding OAM light beams in waveguides are the rotational symmetry requirements and the intermodal coupling. Still, starting from a recently published proposal on a SOI device [40], I have designed, fabricated and characterized a monomode and ideally lossless

de-vice, based on AlGaAs, to generate a light beam carrying OAM. The main goal for the future is to start from this integrated device to generate hyper-entangled OAM quantum states. The idea is to have a new miniature component to generate OAM, which could be also monolithically integrated to the other devices already developed in the team.

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4 Spin and Orbital Angular Momentum of

light

4.1 Angular momentum of light

In 1619 Kepler observed that comets’ tail always points away from the Sun and ex-plained it by introducing for the first time the concept of radiation pressure of light [36]. To explain light radiation pressure, in 1862 Maxwell introduced the idea of linear momentum of light and in 1900 Lebedev proved it experimentally [39]. Nowadays it is well known that light carriers a linear momentum p, directed along its direction of propagation k and with an amplitude of p= h

λ per photon, as states by the de Broglie’s relation. It is less known that, in addition to p, a light beam can also posses an angular momentum L, directed along the direction of propagation of light as well. To visualise it, we may consider a light beam propagating along x and a particle situated in the transverse plane at the position r from the beam axis (fig. 4.1).

If the beam carries an angular momentum, it would exert on the particle an impulse given by the classical definition L=r×p.

This relation can be satisfied by two different kind of light beams. The first one (fig. 4.1, left) is a circularly polarized beam and carries a spin angular momentum (SAM, L=Ls), which makes the particle spins around its axis. The other one (fig. 4.1,

right), has a helicoidal phase structure and carries orbital angular momentum (OAM,

L = Lo ), which forces the particle to rotate in a circular orbit, following the beam

phase structure.

According to the definition of angular momentum, SAM and OAM can exist only if the light beam has azimuthal momentum in the transverse plane: pθ 6=0. Thus plane waves, whose momentum is collinear with the direction of propagation due to the phase structure consisting of by infinite parallel planes, cannot carry any angular mo-mentum. Yet, in practice plane waves do not exist: light beams are collimated in free space or confined as they propagate within the waveguide core. Furthermore, they are always being observed and measured with system of finite apertures. Therefore any light beam can, in principle, carries an angular momentum. Such angular momentum is either SAM, due to a circular state of the beam polarization, or OAM, related to the beam spatial phase structure.

The previous condition on pθ implies also that E and B have a projection in the direction of propagation, Ex 6= 0∨Hx 6=0, as a propagating e.m. field is formed by

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4 Spin and Orbital Angular Momentum of light

Figure 4.1: Comparison between light beams carrying SAM (left) and OAM (right), as they interact with a particle. SAM, circularly polarized, makes the particle spin, while OAM, due to a helicoidal phase structure, makes the particle rotate around the light propagation axis.

(k, E, H)orthogonal each others. In the case of OAM, all the requirements on E, B and

pθ are taken into account by the Poynting vector P, which indicates the density flux of energy of the beam and is by definition parallel to the momentum: P= E×B. As fig. 4.2 on the right illustrates, in OAM beams the Poynting vector skews around the beam axis during propagation. At a given transverse plane, a circular flux of energy would be observed as a result of pθ. On the other hand, in the case of plane waves the Poynting vector is always parallel to the direction of propagation, so no OAM can be carried (fig. 4.2, left).

Figure 4.2: Comparison between the wavefronts of a plane wave (left) and a beam carrying OAM (right). The former one has constant phase surfaces, while the latter one posses a helicoidal phase structure, as highlighted by the skewing Poynting vector.

From this conclusion it is easy to understand the definition of angular momentum of light: J = e0R r× (E×B)d3r. The same expression can also be derived from the Noether’s theorem [10, 12], which returns an AM of light divided into two separate terms: J=e0R (E×A)d3r+e0∑

R

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The first contribution refers to the possibility of a light beam to spin and carry spin angular momentum (SAM), while the second one to twist and carry orbital angular momentum (OAM).

4.2 Spin Angular Momentum

SAM is carried by circularly polarized (CP) light beams, in which the polarization vector rotates as the light propagates (fig. 4.1 (a)), as Poynting demonstrated at the beginning of 1900 [43]. Indeed, CP light beams with a radially decreasing intensity profile satisfies Ls = r×p 6= 0 [29] and a CP photon carries an unit of SAM: Ls = ¯h [61]. The sign of Ls reflects the CP rotation verse, also known as handedness. From the point of view of the detector, oriented towards the light source, right handed CP beams have Ls= ¯h, while left handed Ls= −¯h.

A circular polarization state of the beam can be described as the superposition of two orthogonal linearly polarized states, delayed each other by a π

2 phase shift (fig. 4.3 ). Therefore CP beams can be obtained from orthogonal LP beams originating from the same source, as occurs in a quarter-wave plate (QW). A LP beam impinges on the QW at 45◦ with the optic axis, equally exciting two orthogonally polarized states. Due to birefringence, the orthogonal modes propagate through the QW with different effective refractive index and thus velocity, accumulating a phase delay. By properly choosing the QW plate length, their relative delay can be tuned exactly to π

2. As we will see in the next section, an analogous mechanism can be used to generate a light beam carrying an orbital angular momentum of first order.

(a) (b)

Figure 4.3: SAM light beam, also known as CP beam. As (a) shows, each polarization vector spins during propagation. (b) A CP beam can be decomposed into a TE and a TM beam, relatively delayed by a π

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4.3 Orbital Angular Momentum

Orbital angular momentum (OAM) of light was introduced for the first time by Darwin in 1932 in order to explain quadrupole transitions in atoms [15]. In a quadrupole transition the absorption of one photon increases the atomic angular momentum by 2¯h, so it cannot be induced solely by a CP photon carrying SAM (Ls = ¯h). More generally, multipole transitions involving∆L=γ¯h can be explained only by supposing

that photon has an OAM of Lo = (γ−1)¯h in addition to SAM. However, the form of

light beams carrying OAM was unknown until 1992, when Allen demonstrated that Laguerre-Gaussian (LG) modes carry a well-defined value of OAM [3]:

u(r, φ, z) = C LG l p w(z) r√2 w(z) !|l| exp − r 2 w2₍_z₎ L|pl| 2r2 w2₍_z₎ exp −ik r 2 2R(z)

exp(ilφ)exp[i(2p+ |l| +1)ζ(z)] (4.1)

Such modes posses a helicoidal phase structure, expressed by the term exp(ilφ), where l is the azimuthal mode index. As the beam propagates, this helicoidal phase structure rotates so that each photon carries an OAM of Lo = ±l¯h, with l is the azimutal index of the LG modes, also named orbital order.

Both OAM and SAM are quantized in unit of ¯h because in both cases the discretization of AM arises from the beam invariance to rotation around its axis. However, we underline that OAM has nothing to do with polarization and so with SAM. Rather, OAM arises from the spatial distribution of the beam and for such reason it is an additional degre of freedom with respect to polarization. Conversely to SAM, which allows only the two orthogonal states right (Ls = ¯h) and left handed CP (Ls = −¯h), OAM has a set of infinite set of eigenstates L = ±l¯h and thus a single OAM photon can transmit a much richer information.

The helicoidal phase structures explains the reason for the discretization of OAM in unit of ¯h. Since either E or H exist out of plane, p has an azimuthal component

p` = ¯hk0cos(θ) = ¯hk0_2πrlλ , where θ is the inclination of the phase front, and thus Poynting vector, with respect to the beam axis.

The effect of this skewing energy flux is straightforward: if an OAM were shone on you, it would exert a torque, while a common light beam would just push you away.

If we record the propagating phase structure at a fixed transverse plane, we would obtain a helicoidal phase profile. As fig. 4.4 shows, the phase changes only az-imuthally by 2lπ, where l is the LG azimutal index that defines the number of helices per wavelength and thus the order of carried OAM. As l increases, the phase struc-ture evolves from parallel planes (l = 0), to a helicoidal surface (l = ±1) and two intertwined helicoidal surfaces (l = ±2).

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Figure 4.4: Phase structure, phase and intensity profiles of the first two orders OAM light: l∈ [2,−2]. In l=0 case, the phase structure is made by disconnected surfaces (e.g. parallel plane in plane waves) and no OAM is carried. If l= ±1, the phase structure is a single helical surface, whose step length is given by the wavelength λ. For higher order modes l > 1, the wavefront is composed by l intertwined helices, whose step length is λ

l. The helical phase structure rotation verse reflects the sign of l.

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A direct consequence of the helicoidal phase profile is a donut shaped intensity profile. Indeed, at the beam axis the phase has a singularity [75, 59] and thus it cannot be defined, as for the the time zone at the centre of the North Pole [77]. As result, the intensity is zero at the center of the beam, where there is a dark vortex called optical vortex. The optical vortex is defined according to its topological charge l, equal to the OAM order, and does not carry any energy and thus OAM. If we consider the whole phase structure of the beam, the phase singularity coincides with the line of the beam propagation axis. Therefore, the beam energy and thus momentum (both linear and orbital) is not carried at the beam axis, where there is no light. We can see in fig. 4.4, on the right, that the intensity increases radially from the beam center, with a polynomial growth as predicted by eq. 4.1.

4.3.1 HG to LG transformation

There are several ways to generate an OAM beam of light. The most common (fig. 4.5) are based on spiral phase plates [9, 71, 51, 68], forked diffractive gratings and computer generated holograms [8, 30, 7, 67] and q-plates [45, 35].

(a) (b) (c)

Figure 4.5: Most common devices to generate OAM: (a) a spiral phase plate [26], (b) A diffracting grating (c) a q-plate [46].

However, these methods cannot be implemented in integrated optics, given the limi-tation imposed by growth, patterning and etching fabrication steps. For such purpose there is only one suitable method, named cylindrical mode conversion of Hermite-Gausssian (HG) to Laguerre-Gaussian (LG) beams. Incidentally, this method was also the first one proposed by Alllen in 1992 [3].

An OAM beam can be decomposed in a specific number of HG beams, described by eq. 4.2, each one with a properly phase delay.

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4 Spin and Orbital Angular Momentum of light un(x, z) = 2 π 1/4 1 2n_n!w 0 1/2 q0 q(z) 1/2 q0 q∗₀ q∗(z) q(z) n/2 Hn √ 2x w(z) ! exp −i kx 2 2q(z) (4.2) The previous transformation is possible because both HG and LG form a complete set of solutions for the Helmholtz equation, in cartesian and cylindrical coordinates respectively. In particular, Hermite polynomials are related to Laguerre polynomials [1] by eq. 4.3: m+n

∑

k = (2i)kP_k(n−k,m−k)(0)H(n+m−k,0)H(0,k) =2(n+m)(−1)n(x−iy))(m−n)L (m−n) n (x2+y2) (4.3) Since we are interested in generating an OAM of first order, we consider the case of first order Hermit polynomials, for which m = 1 and n=0. The previous relation reduces to:

H1,0(x) −iH0,1(y) =2(x−iy)L10(x2+y2) (4.4) Given the definition of LG (eq. 4.1) and HG modes (eq. 4.2), we can rewrite eq. 4.4 in terms of corresponding modes. The resulting HG into LG conversion for first order OAM is summarized by: eq. 4.5.

LG(0,1) =HG1,0+iHG0,1= HG1,0+ei

π

2_HG

0,1 (4.5)

According to the derived conversion, illustrated in fig. 4.6, a LG mode carrying an OAM of the first order (l = 1) can be obtained by superposing one HG(1,0) mode,

first order along the horizontal direction, and one HG(0,1) mode, first order along the

vertical direction, with the latter one delayed by a π

2 phase shift. If the relative phase delay is−π

2, the resulting LG beam would carry an OAM of l = −1 instead of l=1.

4.3.2 Generation of an OAM light beam in a waveguide

The HG to LG conversion, also known as π

2-converter after the required phase delay, can be implemented in free space by a pair of cylindrical lenses [3]. Still, this kind of conversion is not commonly used to generate OAM light beams as first order HG modes cannot be easily emitted by laser sources. On the other hand, in a waveguide the HG to LG conversion can be implemented, as Liang has recently demonstrated for a hybrid SOI/plasmonic waveguide [40]. If the core of the waveguide is chosen small enough, the two fundamental quasi-TE and quasi-TM modes present a not neg-ligible electric field along the direction of propagation. The two Ex components are

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Figure 4.6: HG first order modes HG(1,0)and HG(0,1), delayed each other by a π₂ phase

shift, superpose to give a LG light beam that carries OAM l=1.

Ex = 1 iβ∇TET

which for quasi-TE and quasi-TM can be rewritten as: Ex,TE ≈i dETE_y dy Ex,TM ≈idE TM z dz (4.6) These two modes are equally excited by an input beam linearly polarized (LP) at 45◦ and π

2 phase shifted by a plasmonic layer of Cu. In this way a light beam carrying OAM_l=1 can be generated.

However, despite the brilliant idea to generate OAM through the longitudinal elec-tric field component of the modes, Liang’s SOI/hybrid waveguide has not been fab-ricated. The main reason is probably related to the complexity of the waveguide structure. In addition to the usual fabrication steps, SiO2 thermal oxidation and Cu deposition are required, preceded by the corresponding alignment steps. Besides, as the generated OAM depends strongly on the birefringence of the hybrid SOI/delay section, there are some demanding tolerances in the waveguide geometry. As a result, only a phase shift of π

2 ±π8 is achievable, while an exact phase shift of π2 is mandatory to generate a well-defined OAM light beam.

Conversely, I have conceived an original design of a AlGaAs-based waveguide (chapter 7), which is much less demanding in terms of fabrication tolerances and

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steps (chapter 8). Once I have developed its fabrication process in the clean room, I have fabricated it (chapter 9) and characterized it (chapter 10).

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5 Numerical Simulation: Finite-difference

Time-domain

5.1 The Finite-difference Time-domain simulation method

5.1.1 Introduction of FDTD method

The design and optimization of the photonics device presented in this thesis is based on the Finite-Difference Domain (FDTD) method. The Finite-Difference

Time-Domain (FDTD)method is a discretization of the Maxwell’s equations, whose

tem-poral and spatial derivatives are approximated by the central difference. Given its flexibility, speed and accuracy, the FDTD simulation method is applied not only for electromagnetism applications, such as photonics circuits, antennas, scattering and wave propagations problem, but also to biomedicine and biophotonics [23, 18, 60], photovoltaics [50, 17] and also radiation environement issues [25] and earthquake de-tection [62].

FDTD was conceived by Kane Yee in 1966 [78] but its notoriety started to spread ten years later, thanks to advances in the calculation power of the computer proces-sors [42]. Nowadays, computer massive parallelization allows to solve FDTD grids of incredibly high dimension, such as 3000×3000×3000 nodes. Furthermore, the FDTD algorithm is fast if we consider the number of equations that it solves. Given N total number of degree’s of freedom, only O(N) floating-point operations are needed for each iteration in time. On the other hand, a FDTD simulation has the disadvantage that the whole space needs to be discretized, while the electric and magnetic fields are calculated in every point of the grid. This restrition is a waste of computation resources for homogeneous media, such as the open space. As instance, if we simu-late a perfectly conducting sphere which admits an electric field only on its surface, it would be necessary to calculate the electric field in its volume. These unnecessary calculations become important especially in 3D simulations. Indeed, simulations in x, y and z can be too demanding: as the size of the structure to simulate grows linearly, the number of degrees of freedom N and the corresponding operations to solve grow cubically.

Concerning the meshing of the simulation space, the original FDTD algorithm pro-posed by Yee discretizes and calculates the Maxwell equation on a rigid orthogonal grid, with a constant step in x, y and z. However, such grid constitues a limitation for

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5 Numerical Simulation: Finite-difference Time-domain

the modelling of complex geometries, like cilindrical antennas, micro disks or objects presenting some parts with very small features. To simulate such detailed structure, a grid refinement is required. However, as the number of degrees of freedom N trans-lates into a long calculation time ∝O(N), a global mesh with very small spacing (and thus time steps, as we will show in the paragraph about stability and numerical dis-perion) is not feasible.

A first method to solve to this issue is to use a sub-cell model [32], as proposed by Simpson and Holland in 1981. Also sub-griding is an efficace way to resolve geometric structures which necessitate a local refinement. Instead of improving the whole global mesh, a very fine local mesh is added to describe accurately the object to simulate.

Among the important innovations to the FDTD algorithm, we also highlight the Per-fectly Matched Layer (PML) absobing medium, proposed by J. Berenger in 1994 [11]. PML allows to model efficacely unbounded problems for several kind of media: lossy, inhomogeneous, anisotropic, dispersive and even non linear ones. The only price to pay is to extend the mesh and thus calculate additional nodes in the Perfectly Matched Layer.

Concerning the nature of the electromagnetic radiation that can be simulated, FDTD is a broad-band simulation. This means that a single simulation allows to study a broad frequency response. Such advtantage may become a disadvantage if a very narrow band response is desired or if the material constants, like the refractive index n(λ), are

not known for all the frequency of the source spectrum.

In the following paragraphs we discuss in details the principles of the FDTD method that we have briefly introduced. We start from the core of the FDTD algorithm, based on the central difference approximation and the discretization of the Maxwell’s equa-tions. Afterwards the problem of stability and numerical dispersion of the FDTD method is presented. We conclude by illustrating the most common light sources and boundary condition, such as PML.

5.1.2 The central difference approximation

The discretization Finite Difference underlying FDTD is based on the central difference method. Given a function f(x)which belongs to C∞, f(x)can be approximated around the point x, within a small distance∆x, by the Taylor series expansion 5.1:

f(x± ∆x 2 ) = f(x) ± ∂ f(x) ∂x ∆x 2 + ∂2f(x) ∂x2 ∆x 2 2₁ 2! ± ∂3f(x) ∂x3 ∆x 2 3₁ 3!+.. (5.1) By subtracting the expansions on both positive and negative sides of x, we can express the first-order derivative of f(x) with respect to x by the central difference approximation:

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5 Numerical Simulation: Finite-difference Time-domain ∂ f(x) ∂x ≈ f(x+∆x₂ ) − f(x−∆x₂ ) ∆x +O(∆x2) (5.2) where O(∆x2) =∆x2 ∂3f(x) ∂x3 1 24.

As eq. 5.2 shows, the main advantage of the central difference approximation is that the error on the approximation of ∂ f(x)

∂x , diminuishes as ∆x

2 _as _{∆x decreases. For} example, if ∆x reduces by a factor of 10, the error O(∆x2) is 100 times smaller. The amplitude of such error depends on the smoothness of the function f , as O(∆x2) ∝

∂3f(x)

∂x3 . Therefore f functions which vary abruptly or rapidly are approximated with the same error value of smoother functions only if a smaller∆x is chosen. It follows that in the case of electromagnetic waves the size of ∆x depends on the simulated

radiation wavelength, with smaller∆x needed for higher frequency signals.

5.1.3 Maxwell’s Equations

A time-dependent current or charge density generates an electromagnetic field. The evolution in space and time of such electromagnetic field is described by the four Maxwell equations. In particular, FDTD is based on Faraday’s and Amepere’s laws. Faraday’s law (eq. 5.3) dictates that both a change in time of the magnetic flux density

B and magnetization M induce the accumulation of a voltage across a closed circuit

∇ ×E. On the contrary, Ampere’s law (eq. 5.4) affirms that both a change in time of the electric flux density D and a current J generate a magnetic flux intesity∇ ×H

about the area they pierce.

∇ ×E= −∂B ∂t −M (5.3) ∇ ×B=µ0 J+ ∂D ∂t (5.4) The electric and magnetic flux densities D and B are related to the corresponding electric and magnetic intensities E and H by the consitutive relations of the material. In linear and isotropic media with permittivity e = e0er and permeability µ = µ0µr, where e0 and µ0 are the permittivity and permeability of free-space, while e0 and µ0 the relative permittivity and permeability of the material:

D=eE (5.5)

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5.1.4 Yee’s algorithm

FDTD is based on Yee’s algorithm, presented for the first time by Kane S. Yee in 1966.

Yee’s algorithm discretizes the space and time derivatives in Maxwell’s equations for both electric and magnetic field, following the central difference approximation. Firstly, the continous tridimensional space is discretized by a regular cubical grid. Such grid in x, y, z is formed by nodes which can be indicated by the integers i, j, k as:

(x, y, z)_i,j,k = (i∆x, j∆y, k∆z). As in Maxwell’s equation the time derivatives of E and

B must be approximated too, the time is discretized analogously: t= n∆t. Il follows

that at any node and at a given discrete instant, a spatially and time varying function f(x, y, z, t)can be espressed as:

f(x, y, z, t) = f(i∆x, j∆y, k∆z, n∆t) = f_i,j,kn (5.7) Let’s consider now the case of electromagnetism, in which the two functions f(x, y, z, t)

of eq. 5.7 are the electric E(x, y, z, t)and magnetic fields H(x, y, z, t), linked through the Mawell’x equations previously illustrated.

The idea of Yee’s algorithm is to discretize the curl in both Faraday’s (eq. 5.3) and Am-pere’s laws (eq. 5.4). As figure 5.1 (a) shows, along every axis of the grid the electrical field and the magnetic field are alternated. Also, if E is projected along the tangential direction, H is projected along the perpendicular one and viceversa.

The reason is that such disposition approximates the Faraday’s law curl (eq. 5.3) on each face of the grid cube, as fig. 5.1 (a) illustrates. Furthermore, Yee’s grid is the graphical rapresentation of the central difference approximation method. As instance, let’s consider the Faraday’s law along the x direction. We can rewrite it as:

µδHx δt = δEy δz − δEz δy −Mx (5.8)

By approximating its space and time derivatives through the central difference the-orem, we obtain: µ Hn_x+1 i,j+1_{2 ,}k+1₂ −H n x_i,j₊1 2 ,k+12 ∆t = En+12 y_i,j₊1 2 ,k+1 −En+12 y_i,j₊1 2 ,k ∆z − En+12 z_i,j₊_1,k₊1 2 −En+12 y_i,j,k₊1 2 ∆y −M n+1₂ x_i,j₊1 2 ,k+12 (5.9) Eq. 5.9 is rapresented by Yee’s grid: H is located at the center of each cube face, between two equidistant E vectors. Therefore, as in central difference approximation, the spatial derivatives of E in the Faraday’s equation are approximated with an error smaller than the second order. Since electric and magnetic fields are also alternated in time, their time derivative is accurate at the second order as well. As a result, the whole vectorial Faraday’s equation, along x, y, z, can be approximated with a second-order accuracy in both space and time.

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Figure 5.1: Yee’s algorithm grids. On the top, the Faraday’s grid. On the bottom, the Ampere’s grid. [27]

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(fig. 5.1 on the bottom). On each grid cell face, the magnetic field H, tangential to the face edges, encloses the perpendicular E, placed at the face center. Therefore H performs the curl operation, as required by Ampere’s equation. The discretization of Ampere’s equation is similar to the Faraday’s one. As instance, the x projection of Ampere’s law is given by:

e En+12 x_i₊1 2 ,j,k −En−12 x_i₊1 2 ,j,k ∆t = H_zn i+1_{2 ,}j+1_{2 ,}k−H n z_i₊1 2 ,j−12 ,k ∆y − Hn_y i+1_{2 ,}j,k+1₂ −H n y_i₊1 2 ,j,k−12 ∆z −Jxn_i₊1 2 ,j,k (5.10) Now let’s consider both Faraday’s and Ampere’s grids. As fig. 5.2 illustrates, the two grids are staggered to form the Yee’s lattice. The edges of Ampere’s grid inter-secate the centers of Faraday’s grid faces and viceversa. The interlacing of the two grids allows the electric and magnetic field vectors to be consistent, as Ampère’s and Faraday’s equations are satisfied by the same electromagnetic field. Other than rapre-senting correctly the curl operation and the central difference approximation method, Yee’s lattice is a natural choice for describing the evolution in time of H and E. As Faraday’s equation (5.3) suggests, the time derivative of H depends on the curl of

E. Thus, at a given instant n+1, the value of the magnetic field Hn+1 _{is calculated} from both the previous values of the magnetic field Hnand the discretized curl of the electric field E, in a recursive scheme summarized by eq. 5.11.

H_xn+1 i,j+1 2 ,k+12 = H_xn i,j+1 2 ,k+12 +∆t µ    En+12 y_i,j₊1 2 ,k+1 −En+12 y_i,j₊1 2 ,k ∆z − En+12 z_i,j₊_1,k₊1 2 −En+12 z_i,j,k₊1 2 ∆y −M n+1 2 x_i,j₊1 2 ,k+12    (5.11) The same logic is followed to calculate the other two magnetic field components along y and z: Hn+1 y_i₊1 2 ,j,k+12 , Hn+1 z_i₊1 2 ,j+12 ,k

. Given that the Faraday’s law and the Ampère’s law are specular, the electric field projections En+12

x_i₊1 2 ,j,k , En+12 y_i,j₊1 2 ,k and En+12 z_i,j,k₊1 2 are calcu-lated similarly, starting from the already stored values of the E components and the numerical curl of the surrounding H field.

At this point we may note that H and E are staggered not only in space, but also in time. As eq. 5.11 shows, E is calculated in semi-integers time instants 1₂+n, while

Hin integer time instants 1+n. So E is updated in the middle of two consecutive H updates and viceversa. This method, known as leap-frog scheme, has the advantage to reduce the equations to solve at the same time from six to three. Also, it can be shown that leads to a zero numerical dispertion error for a wave propagation. On the other side, the leap-frog method becomes unstable if the time-step∆t is higher than a given value, limiting the upper value of∆t, as we will discuss in the following stability paragraph.

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Figure 5.2: Yee lattice: Faraday’s and Ampere’s grids are staggered. [27] .

So far we have derived the FDTD algorithm following the original work by Yee, which started from the differential form of Faraday’s (eq. 5.3) and Ampére’s equations (eq. 5.4) to obtain their discretized expressions (eq. 5.9 and 5.10 for the x-component of H and E).

To have a more view on the principle underlying FDTD, we briefly show how FDTD equations can be derived starting from the Maxwell Faraday’s and Ampere’s equations in the integral form instead of the differential one. In the simple case of a linear, isotropic and lossless material, the Faraday’s law affirms:

d dtµ Z Z ΣB ·dS= I ∂Σ E ·d` − Z Z ΣM ·dS (5.12)

where Σ is a face on the Faraday’s Yee’s cube (fig.5.1). FDTD algorithm is derived straightforward by making the following approximations on E and H:

1. Along eachΣ edge, E has a constant tangential component 2. Over the wholeΣ face, H has a constant normal component

As instance, let’s suppose that Σ is the face normal to x axis, then eq. 5.12 can be rewritten as: δt δtµH n+1 2 x_i,j₊1 2 ,k+12 ∆y∆z= − En+12 y_i,j₊1 2 ,k ∆y+En+12 z_i,j₊_1,k₊1 2 ∆z−En+12 y_i,j₊1 2 ,k+1 ∆y−En+12 z_i,j,k₊1 2 ∆z −Mn+12 x_i,j₊1 2 ,k+12 ∆y∆z (5.13)

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As last step, we approximate the time derivative in eq. 5.13 via the central differ-ence, simplifying on both side of the equation∆y∆z. We find out that, starting from the Faraday’s integral equation, we obtain the samefirst equation of FDTD algorithm (eq. 5.9), as previously derived starting from the corresponding Faraday’s differential equation.

5.1.5 Stability and numerical disperion

Due to the leap-frog time evolution, FDTD algorithm can be not stable under given conditions. It can be demonstrated ([27], p. 51) that the leap-frog method is stable only if the time-step∆t is upper bounded:

∆t< 1 c· 1 q 1 ∆x2 +_∆y12 + _∆z12 (5.14) In a cubic grid (∆=∆x,y,z), the upper limit of∆t expressed by eq. 5.14 becomes:

∆t< ∆

c 1

√

3 (5.15)

Therefore a tridimensional FDTD simulation is stable only if the wave propagates by a fraction of a cell length∆ for each time step ∆t. In particular, respect to the grid, such distance corresponds to one third of the cubic cell diagonal.

To minimize the numerical dispersion error, the choice of∆t is usually: ∆t= ∆

c 0.99

√

3 (5.16)

Concerning the dispersion, it can can shown that FDTD is inherently disperive. By letting a discretized plane wave (K, E0e−i(K·r)eiωt; H0e−i(K·r)eiωt)propagate in the FDTD grid - which corresponds to solve the six discrete FDTD Maxwell’s equations (like eq. 5.9,5.10) - we find out that the the wave propagation must satisfied the fol-lowing relation: 1 c2 1 ∆tsin2 ω∆t 2 = 1 ∆xsin2 ek_x∆x 2 ! + 1 ∆ysin2 ek_y∆y 2 ! + 1 ∆zsin2 ek_z∆z 2 ! (5.17) Eq. 5.17 is known as dispersion relation as it quantifies how much a wave is dis-persed as it propagates in a medium. In the ideal case of no discretization (or infinitely accurate discretization),∆x, ∆y , ∆z=0 and thus∆t=0 (eq. 5.15). Therefore eq. 5.17 becomes:

ω2

c2 = (k 2

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Eq. 5.18 describes the dispersion relation of a wave in the continuous space and it implies:

k= 2π

λ (5.19)

However, in FDTD ∆x, ∆y, ∆z, and ∆t are necessarily finite. By solving eq. 5.17 iterativaly - as instance with Newton’s method - it can be shown that the numerical wavevector ek differs from the exact wavevector k of eq. 5.19. As a consequence, the wave accumulates a phase error as it propagates through the discrete space:

e= 180

◦

π

·(ek−k) (5.20)

e

k depends on the wave propagation direction, so that the phase error e is anisotropic. Along a specific direction of propagation, the phase error increases as the spatial sam-pling becomes less fine. As instance, lets consider ∆x = λ

20 = 0.05λ, ∆y = 0.04λ and ∆z = 0.03λ. Then, phase error is highest along x and smallest along z, as fig. 5.3 illustrates.

Figure 5.3: Phase error in xyz space, for ∆x = 0.05λ, ∆y = 0.04λ, ∆z = 0.03λ and ∆t= ∆_c0.99√

3 [27]

We also notice that the phase error reduces quadratically with the decrease of the mesh size: e∝ _∆12, due to the second order accuracy of the central difference

approxi-mation.

However, ∆ cannot be choosen arbitrarily small: in a tridimensional simulation the number of calculations grows cubically as∆ decreases. A compromise between the phase error accuracy and the calculation time is necessary. Usually, a phase error smaller than 1◦ per wavelength (e < 1_λ◦) is tollerated, so that a sampling of∆ = ∆x =∆y = ∆z= λ

20 is chosen. Anyway, we notice that the phase error accumulates as the wave propagates.

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For very long optical structure, to minimize the global phase error, a fine discretization along the direction of propagation should be adopted.

5.1.6 FDTD sources

FDTD’s goal is to reproduce real physics problems by solving Maxwell equations. Thus, it is also important to simulate real physical sources. The first property of a source to consider is its bandwidth, namely the frequency band of the electromagnetic fields that is going to propagate in the FDTD space. According to ∆ = λ

20 previsouly discussed, for stability reason the spatial and thus temporal discretization sampling depends on the smallest wavelength, or highest frequency. The limitation in band-width translates into a limitation in time. Indeed, physical sources have a transient behaviour, which is taken into account in FDTD simulation, inherently time-dependent.

Gaussian and derived gaussian souces

A first attempt to simulate a time and band-limited source is to use a Gaussian pulse, with a given time delay t0 and pulse width tw:

g(t) =e

−(t−_t0)2

t2w _(5.21)

The Gaussian pulse semi-bandwidth is fbw = _πt1_w, as its Fourier transform is: G(f) =tw

√

πe(π f)

2_t2

w_e−j2π f t0 _(5.22)

The pulse magnitude is highest in f =0 and decreases rapidly with the frequency: at twice the semi-bandwidth G(2 fbw) < G₁₀₀(0). As convention, the highest frequency is chosen as fhigh = πt2w, so that the spatial FDTD discretization is given by∆=

λ

20 = cπt2·20w. The gaussian has maximum energy in f =0. A simulation may required the source not to have a DC component. In this case the derivative of the Gaussian pulse is used:

dg(t) =twg(t) 0 = −(t−t0) tw e −(t−_t0)2 t2w _(5.23)

Anyway, in most of physical situation, it is required to simulate the propagation of a time-limited signal with a finite bandwidth, centered on a given frequency. As instance, it can be the case of a laser which is turned on and emits light in a very small bandwidth around the telecom frequency λ = 1.55µm. In this case, the source is simulated as a modulated Gaussian pulse:

m(t) =g(t)sin(2π fmt) (5.24) where fm is the signal frequency, or modulation frequency, and g(t)the Gaussian pulse (eq. 5.21 ) with bandwidth f_bw. The upper frequency used for the spatial dis-cretization is: fhigh = fm+2 fbw.

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Figure 5.4: Modulated Gaussian pulse at the telecom wavelength (1.55µm), shown in both frequency and time domain.

As instance, 5.4 illustrates a modulated Gaussian source at telecom wavelength (1.55µm) generated in Lumerical’s FDTD simulation software.

5.1.7 FDTD Boundary Conditions

Typically we need to model an unbounded physical situation, where light can propa-gate indefinitely in all directions. For example, we can think of modelling an electro-magnetic pulse that, while propagates in a waveguide, encounters a defect and scatters in free space. As computational resources are limited, it is not possible to simulate a too large volume of space, respecting at the same time the required meshing condi-tion∆ ≤ λ

20. If the radiation that propagates away from the center of the simulation region is not supposed to be reflected back, a possibility is to truncate the simulation space with a special boundary region that absorbs all incoming light and does not reflect it back. Nowadays, the most versatile solution to absorb light at the simulation boundary is offered by the Perfectly Matched Layer (PML). The PML, invented by Jeanne-Pierre Bérenger in 1990 [11], owes its name to its reflection-less nature: any arbitrarily polarized and broad band waves, normal to PML layers, would penetrate into it without any reflection. Furthermore, PML medium parameters are chosen so that the penetrating electromagnetic waves are rapidly absorbed, as their amplitudes rapidly decay. The outer PML boundary consists of a perfectly electric conductor (PEC), which

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reflects back the survived electromagnetic field. Hovewer, it is molsty absorbed dur-ing the second round-trip in the PML layer: the PML thickness d is chosen so that the electromagnetic radiation not absorbed after a propagation of 2d in the PML layer is negligible.

As instance, it is possible to show that a plane wave (k, Einc, Hinc)impinging on a PML layer (fig. 5.5) is not reflected only if the PML material constants perfectly match the inner FDTD simulation region material constants. An isotropic PML layer presents the following permittivity and permeability tensors:

¯¯e2=e2   a 0 0 0 b 0 0 0 b   (5.25) ¯¯e2=e2   a 0 0 0 b 0 0 0 b   (5.26)

For an isotropic material described only by e1and µ1(fig. 5.5), it can be shown that the perfectly matching condition requires the PML paramters (eq., ) to satisfy e1=e2,

µ1 = µ2, b = d = a−1 = c−1. Thanks to this choice, the incident oblique plane wave enters in the PML layer Ω2 without any reflection. In the PML layer the plane wave propagates with the new wavevector k2 = bk1xx+k1yy, independently on simulated

isotropic material and plane wave frequency and polarization.

Figure 5.5: A plane wave(k, Einc, Hinc)incident on the PML layerΩ2. [27] Therefore the transmitted plane wave amplitude evolves as a complex exponential ∝ e−jbk1xx _{while the wave propagates along the x direction. The signal is attenuated as}

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it travels in the PML layer if the PML parameter b is set as a complex number with an imaginary part:

b=kx+

σx jωe0

(5.27) In particular, the attenuation rate is given by:

e−jbk1xx ₌_e−jkxk1xx_e−Ccos(Φinc)x _(5.28)

whereΦincis the wave angle of incidence and C=√erµrνincludes the relative

per-mittivity, permeability and the free space impedance. We notice that the attenuation rate, given by e−Ccos(Φinc)x , is indipendent on the polarization and frequency, so that the overall attenuation is only due to the signal propagation length in the PML.

Finaly, we mention that PML layers perfectly match anisotropic media as well. This is important also in the case of simulating isotropic materials: as the PML layer en-closes the simulation region, at the boundary corners two normal PML layers overlap. Even if they are both anisotropic, they match each other.

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6 Fabrication equipment

6.1 Introduction: fabrication process

Once the device has been designed, we can proceed with its fabrication. The fabrica-tion process consists of several sequential steps, as fig. 6.1 illustrates.

An AlGaAs wafer is grown by Molecular Beam Epitaxy (MBE). The wafer is then cleaved into smaller samples, whose size is approximately 1×1cm2. The sample is cleaned to remove eventual organic contaminants from its surface. Then, an electro-sensible resist is deposited on its surface. The sample is thus ready to be exposed to the Electron Beam Lithography (EBL) electronic beam. During EBL, the electronic beam draws the desired pattern on the resist. In the exposed area, the molecules of resist change their chemical properties, becoming more or less soluble than the other molecules not hit by the electrons. As the sample is immersed in a solvent in the following development step, a resist pattern generates on the sample surface.

Such pattern is then transfered to the sample through an etching process, as Induc-tive Coupled Plasma etching (ICP). During ICP, the resist pattern works as a mask, avoiding the etching of the underlying surface while the rest of the AlGaAs material is being etched. The etched depth is set by the time duration of the ICP etching, under specific condition of plasma density, gas chemical composition and acceleration volt-age. As the etching is finished, the remaining pattern of resist is removed from the sample. Eventually, the input and output facet are cleaved.

The fabrication process is not a trivial process due to the high number of required steps. In each step the fabrication tools (e.g. MBE growth chamber, resist spin-coater, EBL and ICP equipments) and human processing (e.g. cleaning, development) are subjected to systematic non-idealities which limit the fabrication accuracy. Only by taking care of all the processing steps it is possible to fabricate a device under the desired tolerances. As instance, the MBE growth is responsible for the refractive index values, since it control the thicknesses and composition of each layer. The EBL expo-sure and development determine the device patterns width, while the ICP fixes the device height. Also the device quality is important: the absence of dirt and contami-nants, thank to the cleaning process, and an acceptable sidewall roughness, resulting from both EBL and ICP, are fundamental to minimize the propagation losses due to the scattering of light.

In the following paragraphs we will discuss all the fabrication steps. In particular, we will present the fabrication processing from a general point of view, with focus on

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6 Fabrication equipment

Figure 6.1: Steps to fabricate the device. The AlGaAs wafer is grown by MBE, then cleaved into several smaller pieces. A small sample of AlGaAs is cleaned and covered by resist. The EBL draws the device pattern on the resist, which is developed afterwards. As last steps, an ICP etching transfers the device pattern from the resist mask to the AlGaAs sample. The resist is finally removed and the device input and output facet are cleaved.

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physical and chemical principles.

The implementation of the fabrication steps for our device is presented in the Device Fabrication chapter.

6.2 Growth: Molecular Beam Epitaxy

6.2.1 Introduction

Molecular Beam Epitaxy (MBE) is an epitaxial growth technique commonly used in material science. The epitaxial growth is based on a flux of atoms sent onto a heated substrate.

Figure 6.2: Molecular Beam Epitaxy chamber at LPN laboratory, CNRS, Paris. MBE equipment (fig. 6.2) is based on crucibles, which are heated under vacuum to evaporate atoms fluxes. These atoms fluxes are directed onto the substrate via small shutters on the crucibles and their intensity is controlled by the crucibles temperature. As the atoms reach the heated substrate, a percentage of them stick on it and diffuse until they enter into a lattice site or join a growing island.

The main advantage of MBE is its growth sensitivity: a change in the composition of the growing material within the monolayer range is achievable. Such performance is possible thanks to a relatively small growth rate (≈ 1ML/s) and the presence of mechanical shutters, which can stop abruptly the molecular source beams. Since this flexibility is not offered by bulk crystal growth, MBE is used for the growth of semi-conducting heretostructures and engineered band gap structures, such as quantum wells. The diffusion of dopants and atoms across the interfaces is reduced in compar-ison to other techniques, thanks to the lower temperatures of the growth. As instance, in GaAs T = 600◦C in MBE, lower than the Vapor-Phase Epitaxy (VPE) T = 700◦C. MBE is also used to realize active devices if impurity dopants are introduced during

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the epitaxial growth (eg. Si or Ge in GaAs) . Among different classes of materials, MBE is particularly suitable for the preparation of III-V active compounds, such as GaAs and AlGaAs [52]. The ultra-high vacuum growth environment (P≤10−9mbar) allows the growth of highly pure semiconductors, whose electron mobilities reach

µ=105cm2V−1s−1at T=77K and carrier concentration N=1014cm3.

Finally, the high accuracy in the molecular growth comes with reliable realtime diagnostic techniques. The growth process is continuously monitored by optical re-flectance or RHEED (Reflection of High-Energy Electrons from Diffraction).

6.2.2 MBE Principle

MBE is based on the adsorption of species, coming from molecular beams, on a heated crystalline substrate, in ultra-high vacuum (UHV) conditions (P ≈ 10−10torr). Thus the three main ingredients of MB are UHV, molecular regime and heated crystalline substrate.

UVH is necessary to minimize the presence of contaminants on the growing surface. Indeed, according to the kinetic theory of gases, the number of molecules which im-pinge on the growing surface, in unit of area and time, depends mainly on the pres-sure:

Φ= P

(2πmKBT)1/2

(6.1) where T is the average temperature of the gas molecules, m their mass and KB the Boltzmann’s constant. In the case of a GaAs (100) surface, which presents about 6.3 · 1014atoms/cm2, under atmospheric conditions and an ideal sticking coefficient of one, the surface would be covered by a monolayer (ML) of residual contaminant molecules in just a few seconds.

Concerning the molecular regime, the molecular beams of the growing species are generated by effusion cells, in which pure materials are evaporated or sublimated. It is possible to do an estimation of the generated molecular flux by assuming a condition of thermal equilibrium in the cell between the solid (or liquid) and the evaporated molecules [64]. The relationship between the flux of atoms impinging on the growing surface and the effusion cell, placed at a perpendicular distance d from the substrate, depends on the cell aperture area A and pressure P, according to:

Φ= (AP/πd2)(NA/2πMKBT)1/2 (6.2) where M is the molecular weight of the molecules and T their absolute tempera-ture T. However, in practice the flux generated depends also on the geometry of the

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effusion cell and the thermal equilibrium is not always reached [73]. It is hard to pre-dict the exact flux, so a common practice is to experimentally measure the flux as a function of the effusion cell temperature, prior to the growth.

We should notice that the flux depends also on the relative position and angle be-tween the growing surface and the effusion cell, so that there are different fluxes on different substrate areas. If an uniform layer is desired, the substrate should be con-tinuously rotated azimuthally during the growth. On the other hand, by stopping the rotation of the substrate, it is possible to take advantage of this property in order to grow materials with thicknesses or composition of doping concentration radially graded on the substrate.

Finally, we remind that the flux is not the only parameter to determine the growing rate of an atomic specie: the sticking coefficient must be kept into account as well. For example, As is more easily desorbed than Ga. To compensate, As flux is higher than the Ga flux. Moreover, As flux is maintained during growing interruptions, so that the GaAs lattice does not decompose.

The last important aspect for MBE of MBE is the substrate, the template for the growth. To have a high quality crystalline growth, the substrate usually has the same crystal structure and lattice constant of the growing layers. As instance, a GaAs sub-strate is used for the growth of AlGaAs. Usually, for economical reason the quality of the substrate is lower than the epitaxial layers. This is not an issue as the migration of crystal defects from the substrate to the overhead epitaxial layer becomes negligi-ble as the growth proceeds. Thus a buffer layer is placed on the substrate to filters out the defects, especially in the growth of active devices. For a high quality growth, the condition of the substrate surface is critical. Due to the presence of dangling bonds (bonds missing an atom at the exposed extremity), superficial surfaces reorga-nize with a different disposition of atoms compared to the underlying bulk crystal. To satisfy the dangling bonds and minimize the surface energy, both reconstruction and incorporation of atmospheric atoms, like oxygen and carbon, occur. To remove these impurities, which otherwise would form the first monolayers of the substrate surface, an annealing at high temperature is performed. Concerning the reconstruction, in or-der to minimize the number of dangling bonds, the substrate is cut with an angle so that the crystal surface is not a major crystal plane, which has the highest number of interplane bonds. For example, in GaAs an angle of 2◦ from the [100] plane leads to a less reacting surface. Concerning the heating of the substrate, a high temperature improves the mobility of sticked atoms, so the probability per unit of time that they encounter a lattice site or growing island increases. However the temperature cannot be too high, as it increases the desorption coefficient as well.

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Figure 6.3: RHEED measurement: a high-energy collimated electron beam impinges at a grazing angle θ on the MBE growing sample. A CCD records the diffracted beam intensity and pattern, which depends on the surface of the growing crystalline material.

6.2.3 RHEED

The MBE ultra-high vacuum (UHV) environment is compatible with various tech-niques to characterize the growing crystal surface, such as the RHEED (Reflection of High-Energy Electrons from Diffraction). In RHEED (scheme in fig. 6.3), a high-energy collimated electron beam (5−50keV) impinges on the surface at a grazing angle (≈1◦), penetrating the growing material only by a few atomic layers. The electronic beam is diffracted by the superficial crystal lattice. The diffracted electrons satisfy the condi-tion of constructive interference, known as Laue law [37] in the reciprocal wavevectors space.

Figure 6.4: RHEED principle in reciprocal space: the diffracting electrons have a wavevector K given by the impinging electrons K0and the lattice

AlGaAs on-chip device emitting a light beam carrying spin and orbital angular momentum

POLITECNICO DI MILANO

Master Degree in Physics Engineering

AlGaAs on-chip device emitting a

light beam carrying Spin and

Orbital Angular Momentum

Supervisor:

Prof. Roberto Osellame

Cosupervisor: Prof. Sara Ducci

Master degree thesis of:

Giorgio Maltese

Matr. 819242

Acknowledgments

1 Abstract

2 Estratto

Contents

3 Introduction

3.1 AlGaAs photon pairs sources

3.2 AlGaAs device to generate a light beam carrying OAM

4 Spin and Orbital Angular Momentum of

light

4.1 Angular momentum of light

4.2 Spin Angular Momentum

4.3 Orbital Angular Momentum

∑

5 Numerical Simulation: Finite-difference

Time-domain

5.1 The Finite-difference Time-domain simulation method

6 Fabrication equipment

6.1 Introduction: fabrication process

6.2 Growth: Molecular Beam Epitaxy