Eindhoven University of Technology MASTER Optical control of the quality factor of photonic crystal nanocavities for cavity quantum electrodynamics Swinkels, M.Y.

Eindhoven University of Technology

MASTER

Optical control of the quality factor of photonic crystal nanocavities for cavity quantum electrodynamics

Swinkels, M.Y.

Award date:

2011

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Department of Applied Physics Photonics and Semiconductor Nanophysics group

Optical control of the quality factor of photonic crystal nanocavities for cavity

quantum electrodynamics

M.Y. Swinkels

November 2011

Research group : Photonics and Semiconductor Nanophysics (PSN) Technische Universiteit Eindhoven

Supervisors : Prof. Dr. A (Andrea) Fiore Dr. C. (Chaoyuan) Jin

Abstract

Photonic crystals are artificial materials with a photonic bandgap, where light of certain frequencies cannot propagate in the crystal. If a defect is introduced inside this crystal a localized state is created, which can have ultra-high confinement of light in a volume of the order of . Due to these unique properties these nanocavities have been used for several cavity quantum electrodynamics investigations. These could have applications in quantum information processing and quantum computing.

However, in order for these processes to be effective, it is necessary to read-out the information after the manipulations have been performed. Here the strong confinement of the optical field limits the read-out efficiency and speed. To overcome this problem it would be desirable to dynamically change the confinement inside a photonic nanocavity, to get high confinement and thus high interaction when the photon should be manipulated, but then decrease the confinement when the photon needs to be extracted. If an emitter (e.g. a quantum dot) is also introduced in the cavity, which couples to the cavity mode, this could lead to the control of Rabi oscillations between the emitter and the cavity.

In this research photonic crystals with embedded quantum dots have been fabricated. Inside the photonic crystal a double heterostructure (DHS) cavity was created. This was done by fabricating a photonic crystal waveguide where the lattice constant was changed locally. This formed a cavity with an experimental Quality-factor of up to 13,000. Close to this cavity a larger cavity was placed by again changing the lattice constant. In this larger cavity Fabry-Perot (FP) modes arise which have a much lower Q than the DHS cavity. The modes of the two cavities can now be spectrally aligned by wavelength tuning to introduce a coupling between the two modes. When the modes are coupled the confinement of the DHS cavity is lowered.

This system has been studied using coupled mode theory, Finite Difference Time Domain (FDTD) and Finite Element Method (FEM) simulations and micro- and time-resolved photoluminescence. The coupled mode theory showed that the highest change in Q-factor was obtained in the transition between weak and strong coupling between the cavities. Apart from that, the structure was simulated where the FP-modes were spectrally aligned with the DHS cavity mode, using thermo- optic tuning to observe the change in Q-factor. This way the possibility of getting high Q changes was confirmed and was again found in the transition between weak and strong coupling.

The tuning of the DHS cavity mode over several FP modes has been shown and the corresponding Q change has also been found. This effect has been found in both the weak and strong coupling regime between the cavities with a maximum dQ/Q of more than 50%.

Tuning the FP modes was much harder because of the larger size of the cavity. However also here it was found that the Q of the DHS cavity dropped 20% when the FP mode was spectrally aligned with the DHS cavity mode.

Finally time-resolved measurements have been performed to investigate the change in dynamic behaviour of the emitters when the two cavities couple. A decrease in lifetime of 66% was found when the cavities were brought out of resonance.

Table of Contents

1 INTRODUCTION 1

1.1 GENERAL INTRODUCTION 1

1.2 QUANTUM INFORMATION PROCESSING 1

1.3 CAVITY QUANTUM ELECTRODYNAMICS 2

1.3.1 STRONG COUPLING 3

1.3.2 WEAK COUPLING 4

1.3.3 REALISATION 5

1.4 PHOTONIC CRYSTALS 6

1.4.1 BASIC PRINCIPLE 6

1.4.2 MATHEMATICAL DESCRIPTION 7

1.4.3 MEMBRANE TYPE CRYSTALS 9

1.4.4 CAVITIES 9

1.4.5 WAVELENGTH TUNING 10

1.4.6 Q-TUNING 11

1.4.7 OUR STRUCTURE 12

1.5 GOAL 13

2 MODELLING 14

2.1 COUPLED MODE THEORY 14

2.1.1 MATRIX DEFINITION 14

2.1.2 RESULTS 15

2.1.3 MASTER EQUATION 17

2.1.4 RESULTS 19

2.2 COMSOL SIMULATIONS 21

2.2.1 FINITE ELEMENT CALCULATIONS USING COMSOL 21

2.2.2 HEAT TRANSFER SIMULATIONS 21

2.2.3 ELECTROMAGNETIC SIMULATIONS 22

2.3 MEEP AND MPB CALCULATIONS 24

3 EXPERIMENTAL METHODS 26

3.1 MICRO-PL SETUP 26

3.1.1 LASER CALIBRATION 28

3.2 SNOM SETUP 29

3.3 SAMPLE INFORMATION 30

3.3.1 SAMPLE GROWTH 30

3.3.2 DEVICE FABRICATION 30

3.3.3 SAMPLE CHARACTERISATION 31

3.3.4 CAVITY CHARACTERIZATION 32

4 EXPERIMENTAL RESULTS 35

4.1 THERMAL TUNING 35

4.1.1 GAAS 35

4.1.2 INGAASP 36

4.2 ONE BEAM EXPERIMENTS 39

4.2.1 GAAS 39

4.2.2 INGAASP 41

4.3 TWO BEAM EXPERIMENTS 43

4.4 TIME-RESOLVED MEASUREMENTS 44

5 CONCLUSION AND RECOMMENDATIONS 51

5.1 RECOMMENDATIONS 52

APPENDIX A 53

APPENDIX B 55

REFERENCES 57

1

1 Introduction

This chapter gives an introduction to the work performed for this thesis. The chapter starts out with a general introduction and motivation for the research. This is emphasized by explaining one of the more important applications of photon manipulation in general, namely quantum information processing. Afterwards, a brief description of Cavity Quantum ElectroDynamics (CQED), which is one of the ways to describe matter-photon interactions, will also be given. Finally this chapter will explain photonic crystals and how they have been used in the experiments performed for this thesis.

1.1 General introduction

The field of photonics emerged halfway the twentieth century with the invention of the laser. Since then light has been used in various applications and the typical size of these devices have been reducing constantly. Optical signals are now being used to for example send information across the ocean through optical fibers, also for optical interconnects between computers. All the new applications also place ever increasing demands on photonic devices.

One of the problems with optical information is the difficulty to manipulate the photons. All the optical signals have to be converted to an electronic signal, processed and then switched back to an optical signal. This optics-electronics-optics interface severely limits the processing speed and efficiency and is causing a high power demand on modern data centres. To improve this, all-optical processes could be a way towards faster and more efficient data processing.

Because of the bosonic character of photons, they do not interact with each other. This has advantages for the transportation of information using photons, but is a problem when non-linear interaction between photons is required. For this purpose, light-matter interaction can be used.^[1]

Switching to the quantum regime for communication opens up new possibilities. This is because the quantum nature of photons enables quantum information processing with significant differences to classical information. For example, due to its unique nature, quantum communication can ensure a perfectly safe communication network.

This thesis describes the work performed to control matter-photon interactions, manipulating the photon at will. This is done by trapping and releasing a photon inside a photonic cavity with an embedded two-level system, by optical control of the cavity Q-factor. This research could be used as a first step towards a fully controllable single photon sources and also provides the basis for more complex quantum information processes in the solid state.

1.2 Quantum information processing

The main difference between quantum information and classical information is that where in classical information, one bit can take the value of either “0” or “1”, in quantum information one bit can take any superposition of these values.^[1] Only a measurement on a certain axis will force the bit to take either of the values. This means that a measurement on a quantum bit inherently destroys its original state.

One way to achieve this superposition of states is by control of the polarization state of a photon.^[3]

By choosing a random basis the polarization of the photon can represent a superposition of the two polarizations in that basis. This basis can consist of two linear or two circular polarization directions.

2 One of the main applications of quantum information processing lies in quantum communication.

This is a new way of preventing eavesdropping on signals. By using quantum information, the information will inherently be impossible to eavesdrop without the receiver noticing a change in information.^[3] This is assuming that only one photon codes the bit, so the eavesdropper cannot take part of the signal and perform measurement on that part without disturbing the other part.

Therefore single and/or entangled photon sources and detectors are the crucial parts for optical quantum information processes.

At the moment commercial quantum communication devices are already available. However, these devices function with strongly attenuated laser sources.^[4] These laser sources have a Poissonian distribution so there will always be a chance of emitting two or more photons, which would still enable eavesdropping. Apart from that, the stronger the laser is attenuated, the bigger the chance that no photon at all is send, thus severely limiting the efficiency of the device. Therefore inherent single photon sources could be a huge step towards optical quantum information processes.

Additionally, optical quantum computers have also been proposed.^[5] For these applications an efficient single photon source is crucial as well, but apart from that CQED is desirable to provide the required interactions.

1.3 Cavity Quantum ElectroDynamics

CQED describes the interaction between a two-level system and a photon inside a photonic cavity. In a system like this the photon and the two-level system become a coupled system. The system is described by the coupling rate between the two-level system and the photon g, the loss rate for the emitter γ and the loss rate for the cavity κ. A schematic diagram of a typical CQED system is drawn in Figure 1.

Figure 1: Schematic drawing of a typical CQED system, with γ the losses of the emitter, g the coupling between the emitter and the cavity and κ the losses of the cavity. Image adapted from: [6]

Depending on the ratio between the coupling and the losses, the system can display very different behaviour. If g is much larger than κ (κ << g) the system is said to be in the strong coupling regime. In the other case, the system is said to be in the weak coupling regime. The different behaviour in the different regimes is displayed in Figure 2, where the probability of either a photon inside the cavity or the emitter being in the excited state is plotted for different losses.

3

It is possible to change the coupling strength between a photon and an emitter by changing the spectral overlap using for example quantum Stark tuning of the emitter.^[7] However in this project the aim is to change the confinement to go from the strong coupling between to weak coupling emitter and photon. In this way the photon and emitter can remain spectrally aligned.

0,5 1,0

Quantum Dot Optical cavity

0,5 1,0

0,5 1,0

Occupation probability (-)

Time

Figure 2: Occupation probability of the excited state of an emitter and a one-photon state for different losses. (a) The case for a perfect cavity and emitter, without any losses (κ=γ=0). (b) A more realistic scenario with some losses in the optical cavity. In (c) the losses in the optical cavity have become so large that there are no more Rabi oscillations (κ>g).

This means the system has reached the weak coupling regime. The lossless scenario clearly shows the Rabi oscillations but with increasing losses these dampen out more.

1.3.1 Strong coupling

In the strong coupling regime the energy will be exchanged between the emitter and the photon state periodically. The periodic change of occupation probability is called Rabi oscillations. These are the oscillations where the maximum probability periodically switches between the photon and the excited state of the emitter. In the ideal case, without losses, these oscillations would go on forever.

However if the loss rate becomes comparable to the Rabi frequency, the oscillation will dampen out.

This is also displayed in Figure 2, where it can be seen that Rabi oscillations are only obtained if the losses are sufficiently small.

From these graphs it can be seen that light-matter interactions are most effective in the strong coupling regime, so this is the regime where photon manipulation is most efficient. However, when photon transportation is needed, interaction is an undesired effect and so it would be preferred for the system to be in weak coupling mode, with high losses or low coupling strength.

(a)

(b)

(c)

4 1.3.2 Weak coupling

The coupling strength between light and matter is proportional to the lifetime of the photon and inversely proportional to the effective volume of the light field, where the effective volume is defined as^[9]:

(1)

In the weak coupling regime, the alteration of photonic density of states still causes a change in electrodynamic behaviour for the system. This effect is called the Purcell effect.

The Purcell effect is a modification of the spontaneous emission rate inside a microcavity in the weak coupling regime. The basic requirement for the Purcell effect to take place is that light needs to be confined into a sufficiently small volume, thus locally altering the photonic density of states locally.

According to Fermi’s Golden Rule the transition probability for an emitter is proportional to the density of photonic states. The transition rate for an electric dipole (the two-level system) interacting with a light field , as derived from Fermi’s Golden Rule is^[10]:

(2)

Where ρ is the optical density of states. This shows that the emission rate of an emitter is proportional to the optical density of states. Using standard methods from solid state physics^[8] the optical density of states in free spaces can be derived to be:

(3)

However inside a microcavity discrete modes will arise. These are displayed as discrete peaks in the density of states with spectral width , which is proportional to the Q-factor of that cavity:

(4)

The Purcell factor is now the ratio in local density of states for a microcavity and in free space

(5)

When plugging this ratio into equation (2) the ratio in emission rate is obtained, which is the definition of the Purcell factor:

(6)

A more detailed quantum-mechanical derivation of the Purcell factor yields different pre-factors, but the Purcell factor remains proportional to .^[10] Therefore it is important for all processes in weak coupling regime where light-matter interaction is still important to have the as high as possible.

Several approaches for this have been proposed and devised, one of which is the photonic crystal nanocavity, which was used in this project and will be described in more detail in the next section.

5

Note that for enhancement of emission due to the Purcell effect two conditions must be satisfied.

First, the emission spectrum must match one of the cavity mode frequencies. If not, there will not be a peak in the density of states and emission will be suppressed instead of being enhanced. Secondly, the spontaneous emission spectrum needs to be narrower than the cavity resonance peak to be able to use Fermi’s golden rule.^[12]

1.3.3 Realisation

Both quantum emitters and photonic nanocavities can be created in several ways. For the emitter one could for example use trapped ions, colour centres in diamond or cold atoms. In this project a solid-state system based on self-assembled InAs quantum dots is used. These quantum dots are constituted of a small volume of semiconductor material with a small bandgap, surrounded by semiconductor material with a higher bandgap. Due to this difference in bandgap electrons inside a quantum dot are confined in three dimensions. If the size of the quantum dot is sufficiently small, a quantum dot will display quantized energy states, much like an atom. This becomes an important effect once the energy separation between the modes is smaller than the thermal energy. Therefore quantum dots are often called “artificial atoms” which makes them very suitable as our emitter.

In solid-state systems a large variety of photonic cavities have been designed and created. And overview of some of these types of cavities is given in Figure 3. In this project the photonic crystal is used because it is a solid state system, it is more easily integrated inside a larger photonic chip and it has the highest Purcell factor, due to the smallest mode volume.

Figure 3: Comparison of systems exhibiting Rabi oscillations, consisting of an optical cavity and an emitter. Image adapted from: [11]

6

1.4 Photonic crystals

1.4.1 Basic principle

Photonic crystals are structures with a periodic modulation of the refractive index.^[13] This modulation can be done in 1 dimension, when the crystal is also called a distributed Bragg reflector, or in 2 or 3 dimensions. A schematic drawing of all these types of photonic crystals is shown in Figure 4. In the experiments described in this report quasi 2-D photonic crystals were used, so the rest of this thesis will focus on this kind of photonic crystals.

At the edges where the refractive index changes, Bragg reflection will take place. Due to this standing waves will arise, which construct discrete optical modes. Because of the symmetry in this system, one of these modes is maximized in the high-index region while the other is maximized in the low-index region. The wave in the high-index region will have an increased energy due to the larger portion of the mode in the high refractive index region, while the reverse happens for the other mode. This results in a splitting of the two standing waves which is dependent on the difference in refractive index as shown schematically in Figure 5. If this difference in refractive index is chosen correctly, a complete photonic bandgap can arise. For this region of frequencies no light can propagate through the photonic crystal. This process is completely analogous to the arising of an electronic band gap in a solid state material. ^[8]

Figure 4: Schematic drawing of a 1-D, 2-D and 3-D photonic crystal. The different colours represent materials with a different refractive index. Image adapted from: [13]

7

Figure 5: The photonic band structures for on-axis propagation, as computed for three different multilayer films. In all three cases each layer had a width of 0.5a. Left: Every layer has the same dielectric constant, Center: Layers alternate between ε=13 and ε=12. Right: Layers alternate between an ε of 13 and 1. Image adapted from: [13]

1.4.2 Mathematical description

The modes in a photonic crystal obey Maxwell’s equations, which without any free currents are^[14]:

(7)

To simplify these equations a bit, a nonmagnetic medium is assumed, so and a sinusoidal dependence of the electric and magnetic field is assumed. This doesn’t limit the solution a lot since all solutions can be written as a superposition of sinusoidal solutions. Inserting this all into equation (7) the master equation for photonic crystals is obtained^[13]:

(8) Where is an hermitian operator, of which the eigenvectors represent the spatial field distributions and the eigenvalues represent the corresponding frequencies of those modes. It is also a linear operator, which means that any superposition of two solutions is also a solution.

Up to this point, no assumptions have been made regarding the refractive index. If we now assume the refractive index to be periodic we can plug in Bloch’s theorem.^[8] This means that the mode distributions are given by a fast-varying function with a similar periodicity to the lattice constant, multiplied by an envelope function. This periodicity in the modes ensures that the wavevectors components must obey , where p is an integer and a is the lattice constant for the photonic crystal. Because of this periodicity the whole investigation can be limited to one unit cell, which in 1-D is called the Brillouin zone.

8 Another important feature of this operator is its scaling invariance. If we would scale the periodicity of the dielectric materials by setting thus making a change of variables from and and put this into equation (8) we get:

(9)

Thus the frequencies have changed with the same factor as the periodicity. Therefore the only demand for a photonic crystal is that the periodicity in refractive index has a length scale similar to the wavelength investigated.

Because of this scale invariance, equation (8) can be rewritten with dimensionless parameters and can then be used to calculate normalized band diagrams by solving the eigenvalues of equation(8) . These give the allowed frequencies for certain k-points. An example of such a band diagram is shown in Figure 6 where the band diagram for a photonic crystal consisting of a 2-D hexagonal lattice of air holes in a semiconductor slab is plotted.

Figure 6: Band diagram of a photonic crystal consisting of a hexagonal lattice of air holes in a semiconductor slab. Image adapted from: [13]

9

1.4.3 Membrane type crystals

Although 3-D photonic crystals can and have been fabricated ^[15], for a lot of processes a membrane type quasi 2-D photonic crystal already suffices. Total internal reflection (TIR) ensures the light confinement in the third direction.^[16] TIR is the process where light with an angle larger than the critical angle is reflected from an interface between a material with a high refractive index and a material with a low refractive index.

In this project a membrane type photonic crystal was used. A schematic drawing of this type of crystal with a cavity is shown in Figure 7. The crystal consists of a semiconductor slab, with air on both sides and air holes inside the slab. The air on both sides creates the index contrast required for TIR while the air holes constitute the photonic crystal.

Figure 7: Schematic drawing of a membrane type photonic crystal nanocavity. The holes constitute the photonic crystal while the air layer on top and below the crystal ensure vertical confinement through total internal reflection. Image adapted from: [16]

Because TIR only occurs when the light has certain angles with the surface, smaller than the critical angle, certain k-values will not be reflected at the surface of the crystal. These modes will therefore not be confined inside the crystal and are dubbed “leaky modes”. The k-values are usually depicted by a light line in 2-D or a light cone in 3-D, inside which the vertical k-value is too large for TIR.

1.4.4 Cavities

If the properties of the crystals are changed locally, this will create a region with a different photonic bandgap than the surrounding crystal. This will confine the light at the defect, much like a defect in a solid-state electronic crystal.^[17] The local properties can be changed in several ways. A few examples are, removing air holes, changing the lattice constant or changing the radius of some air holes. By combining all of these modifications many types of nanocavities can be created. An example of these is drawn schematically in Figure 8 (a) where a schematic drawing of a heterostructure cavity^[26] is given, which is created by locally increasing the lattice constant of a photonic crystal waveguide.

Each of these cavities allows for certain modes to arise. All these modes have their own particular mode profile. An example of one of these mode profiles is shown in Figure 8(b) where the electric field distribution of one of the modes of the Noda cavity is shown.

10

Figure 8: Schematic drawing of a Noda cavity (a) and the spatial pattern of the electric field of one of the modes that would could arise in such cavity (b). The lattice mismatch in (a) has been exaggerated to make the effect more visible.

The electric field distribution shown in (b) is of one of the modes of the Noda cavity

Because the in-plane confinement is now basically assured by the surrounding photonic crystal, the most important element for the losses in the cavity and thus the Q-factor are the out of-plane losses. To prevent out of plane losses cavities can be designed in such a way that they have small Fourier components inside the light line, thus preventing out-of-plane losses. This is usually achieved by creating smooth edges for the mode, since most of the scattering takes place at the edges of the cavity. This because a sharp edge creates a wide distribution in k-space.

1.4.5 Wavelength tuning

For a lot of applications and also for the project described in this thesis, it is necessary to change the resonant wavelength of a photonic crystal cavity. This can be realised during fabrication or afterwards during operation.

During the fabrication process one can change the design of the crystal to cater the specific needs, this is called lithographic tuning. An increasing lattice spacing or decreased hole size will cause a redshift of the modes.^[19]

Another way is to change the crystal after fabrication. The general way to do this is by changing the refractive index of the material. An increase in refractive index will cause a redshift of the cavity mode. There are several ways to change the refractive index but the most important ways for this project are thermal tuning or by carrier injection.

Thermal tuning of the refractive index takes place because heating a material will change its bandgap due to changes in the crystalline structure.^[20] These changes in the bandgap mediate a change in refractive index through the Kramers-Kronig (KK) Relations. Since heating the material usually causes the bandgap to decrease, the refractive index will increase and the mode will thus redshift. This effect is much larger than the thermal expansion of the crystal, which would also cause a redshift due to the increased lattice spacing of the crystal.^[21] Thermal tuning takes place on a time- scale of μs and is thus one of the slow tuning mechanisms available.^[21]

There are two effects playing a role in tuning by carrier injection. One effect is the decreased interband absorption due to the free carriers, which changes the refractive index again according to the KK relations.^[22] This process is strongly wavelength dependent. Another effect is the inducing of a polarization which changes the refractive index according to the Drude model.^[23] Both processes

(a) (b)

11

can be very fast, the refractive index change will happen almost instanteously, the wavelength shift could take place in a time proportional to the lifetime of the cavity. ^[21] However, removing the carriers from the system is usually a much slower process, therefore it can be useful to remove the carriers from the system again by applying an electric field.

For both tuning methods the amount of wavelength shift induced by the change in refractive index can roughly be estimated by:

(10)

1.4.6 Q-tuning

In general a high-Q cavity is accompanied by a waveguide for in- and out-coupling of photons into the cavity.^[24] The total losses of the cavity can now be divided into vertical losses into free space (given by Qv) and in-plane losses to the waveguide (given by Qinplane). The total Q-factor is given by equation (11). To change the Q-factor of the cavity either of these losses needs to be changed significantly.

(11)

The vertical losses are usually caused by scattering processes and are thus governed by the cavity design and fabrication imperfections. After creation of the cavity it is hard to change any of these parameters. The in-plane losses are usually negligible compared to the vertical losses and can therefore usually be ignored when determining the Q-factor. However when another cavity or waveguide with lower confinement is sufficiently close to get spatial overlap the photon could leak from the high-Q cavity to the low-Q cavity thus creating an in-plane loss channel.

In the dynamic Q-tuning first introduced by Tanaka et al.^[24] the in-plane losses are mainly to a waveguide. The losses in this waveguide are controlled by placing a mirror on one side and creating interference effects. This interference effect can be either constructive or destructive, thus enhancing or decreasing in-plane losses.

In the experiments described in this thesis a high-Q cavity is coupled to a different cavity with a lower Q-factor. A mode inside a high-Q cavity can only couple to a second cavity when one of the cavity modes is spectrally aligned with the high-Q mode. In this case photons can escape through the second cavity and thus in-plane losses are significantly increased. If the cavity mode is out of resonance, the photon cannot leak to the second cavity anymore and the in-plane losses are reduced. This is the basis for the dynamic Q-tuning used in the project described in this thesis. In this way the Q-factor of a cavity can be changed after the cavity has been created, on the timescale needed to spectrally align the two cavities.

The high-Q cavity is brought into resonance with a nearby low-Q cavity by spectrally aligning the two modes using any of the methods described in previous section. When the modes have spectral and spatial overlap, they will couple, thus creating a superposition of the two modes. This will cause the Q-factor to change to a value between the two extremes. In this way the light-matter interactions inside the high-Q cavity can be altered depending on the resonance frequency of the low-Q cavity nearby.

12 1.4.7 Our structure

In this project two kinds of cavities have been used, both based on the photonic crystal waveguide.

The photonic crystal waveguide is a long line defect inside a crystal. The band diagram of such a waveguide is shown in Figure 9. This band diagram has been calculated using MPB which is described in more detail in section 2.3.

0,0 0,1 0,2 0,3 0,4 0,5

frequency (c/2a)

k (2/a)

Photonic bandgap Even mode

0,18 0,20 0,22 0,24 0,26 0,28 0,30 0,32

Odd mode

Figure 9: Photonic bands of a waveguide with the lattice constant shifted in one direction. The band diagram clearly shows in yellow the photonic bandgap induced by the crystal and the two modes in the bandgap. The green circles indicate the slow-light region.

The photonic crystal waveguide has certain characteristics which are different from a regular dielectric waveguide. This is mainly caused by the different dispersion curve in a photonic crystal which could cause light to slow down considerably more than in a normal waveguide.^[25] This slow- light behaviour is displayed for the modes in the flat region of the band diagram, at the edge and the centre of the Brillioun zone.

To create a defect inside the photonic crystal waveguide the lattice constant was changed in the direction of the waveguide in two regions, thus forming two defects. One of these defects was only 2 periods wide. This kind of double-heterostructure cavity has first been proposed in the group of Professor Noda^[26] as a suitable design for a cavity with an extremely high Q. Therefore this cavity will be referred to as Noda cavity for the remainder of this thesis. The other defect was 80 periods long and thus displayed Fabry-Pérot (FP) like resonances. Therefore this cavity will be referred to as FP cavity in the remainder of this thesis.

Between the Noda cavity and the FP cavity, a small barrier was placed with of 4-10 periods, with the original smaller lattice constant. Due to this smaller lattice constant the modes allowed in this region

13

had higher energies than the modes in the Noda and FP cavity and therefore this barrier part effectively confined the light inside the cavities. However due to the finite length of the barrier, a spatial overlap between the Noda mode and the FP mode was still expected, thus making it possible to couple them by bringing them in spectral resonance. A schematic drawing of the structure, along with a photonic potential is given in Figure 10.

Figure 10: Schematic drawing of the structure used for the project described in this thesis. The coloured rectangles show the different barriers for this system. The red barriers are used to control the losses, while the blue one is changed to control the coupling constant. Beneath the drawing is the band diagram. The lattice mismatch of the FP cavity has been altered over the experiments and thus also the energy difference has been altered during the experiments, as well as the position of the excitation area.

The lattice mismatch between the FP cavity and the Noda cavity was changed during the experiment to change the spectral distance between the modes of the two cavities. Apart from that the barrier between the Noda and the FP cavity was changed to change the coupling strength between two modes of these cavities. Finally the barrier at the far end of the FP cavity was changed to control the losses and thus the Q-factor of the FP cavity.

The ultra-high Q-factor of the Noda cavity can be understood from the small difference in lattice constant that is used to create it, 1.01 to 1.03 times the original lattice constant. Due to this, the energy step at the edge of the barrier is rather small, producing a small influence in k-space.

Therefore the Fourier-distribution does not widen up much and thus vertical scattering at the edges is diminished significantly. Because of this, the in-plane losses start to play a significant role in the Q- factor, thus making it possible to significantly change the total Q by changing the in-plane losses.

1.5 Goal

The goal of this project is to show a proof of principle of the Q-tuning of the Noda cavity by bringing an FP mode in spectral overlap with the Noda mode. Besides that a study to find the ideal Q- switching conditions is performed.

Collection spot

Noda cavity FP cavity

Excitation spot

14

2 Modelling

In order to study its behaviour, the system it was first simulated using coupled mode theory and Finite Element (FE) and Finite Difference Time Domain (FDTD) simulations using COMSOL, MEEP and MPB. These simulations and their results will be described in this chapter. This chapter will start with a description of coupled mode theory, which gives a general description of two coupled modes and will end with the FDTD and FE simulations, which are more specific for our system.

2.1 Coupled mode theory

To study the effect of two coupled cavities, a simple coupled mode theory has been derived and used. In this chapter this theory will first be explained and the results of it will be described in this section. The derivation of the theory will be given in Appendix A. This section will end with a description of the master equation, which has been used to study the dynamic behaviour of the system and with the results obtained using it.

2.1.1 Matrix definition

In general two (or more) coupled modes can be described by an eigenvalue problem, .^[28]

In the case of uncoupled modes the matrix describing the Hamiltonian would not have any off- diagonal terms and the on-diagonal terms would describe the Hamiltonian for that particular mode when it is not coupled to any other mode. In coupled mode theory these off-diagonal terms are no longer zero. The resulting perturbation matrix than looks like (depending on the basis)^[27]:

(12)

Where γi the loss rate for mode i, g is the coupling constant between both of the modes and Δω is the detuning between the systems. Note that this matrix only describes the detuning from the unperturbated system, since coupled mode theory is a perturbation theory to the uncoupled case.

To get a more physical interpretation of the coupling constant, the perturbation matrix has been derived for our system of two coupled photonic cavities. The derivation is described in Appendix A but the perturbation matrix turns out to be:

(13)

Where the ωj is are the frequencies of each of the unperturbated systems and αj is the spatial overlap between both modes as defined in .

This matrix equation can be solved for different detuning and different overlaps. The real part of the resulting eigenvalues will give the detuning from ω1, which is chosen as reference, while the imaginary part gives the losses and can thus be related to the Q-factor.

15

When two modes couple there are two regimes of coupling, namely weak and strong coupling, depending on the coupling strength relative to the losses.^[29] These two regimes display very different coupling behaviour and will thus be treated separately in the remainder of this chapter.

This is similar to the two coupling regimes described in section 1.3, however since the main thing changed in the experiments is the coupling between the cavities, strong and weak coupling will now only be used to describe the coupling between the cavities, independent of the coupling between the cavity and the emitter.

When the combined losses of the system are larger than the coupling, it is called the weak coupling regime. In this case the losses are still stronger than the coupling so even though the two modes still influence each other, they can still be considered separately and therefore can cross normally.

If the coupling is stronger than the combined losses of the system, it is called the strong coupling regime. In this regime the two modes cannot be considered separately anymore. This is most apparent in the anti-crossing of the energies which shows that the two modes never cross, since one physical system with two solutions cannot have degeneracy in those solutions, since it is represented by a Hermitian matrix.^[30]

2.1.2 Results

The perturbation matrix equation (13) has been solved numerically using Matlab. The values of the coupling strength and the losses of the second mode were changed, while the frequency of the second mode was increased linearly to investigate the effect of the detuning. In this way the behaviour of the system in both strong and weak coupling regime could be studied. The calculation results were plotted in detuning-frequency and detuning-Q plots. The detuning is the relative detuning of the central uncoupled Fabry-Pérot cavity from the uncoupled Node cavity resonance frequency. The frequencies of the uncoupled modes were also plotted as a reference. Each colour in this graph represents a different mode.

Figure 11(a)+(b) shows the behaviour of the coupled system in the weak coupling regime. The crossing of the modes can be recognized. The fact that the colours still do not cross is the result of the numerical method, which sorts the results and thus does not allow for crossing. This is also apparent from the fact that the mode with the highest Q-factor remains on the resonant frequency of the Noda cavity, even though the colours change. The Q-tuning range in the weak coupling regime is small and broad, due to the small influence of the FP modes on the Noda mode.

Figure 11(c)+(d) show typical results for the coupled system in the strong coupling regime. The anti- crossing of the modes is now clearly visible in the detuning-frequency diagram, especially from the zoom in inset, where the red and the blue line are bending away from each other at the crossing point. This is a clear signature of the strong coupling regime. It can also be seen that the dip in Q- factor is much sharper than in the weak coupling case. This is caused by the crossing of the highest Q mode from one mode to the other. The highest Q mode however remains closest to the frequency of the uncoupled Noda cavity. The Q-factor of this mode shows oscillatory behaviour similar to the weak coupling case.

From the comparison of Figure 11 (b) and (d) it can also be seen that even though the losses for the uncoupled Noda cavity were the same in both cases, the maximum Q-factor for the coupled Noda is much lower in the strong coupling regime. This is caused by the fact that in this case the coupling

16 range is much larger than the spacing between the FP modes. Therefore, the coupling between the Noda cavity and the FP cavity is never completely destroyed and thus the Q-factor will always be lowered effectively.

Figure 11: Results of the coupled mode theory calculations in weak (a)+(b) (α=1/9000) and strong (c)+(d) (α=1/4000) regime.

Graph (a) and graph (c) give the detuning-frequency dependence. Graph (b) shows resulting Q-factor of the highest Q mode, while graph (d) shows the resulting Q-factor of all modes. The different colours represent different eigenvalues of the matrix, or different modes. The dashed lines represent the frequencies of the modes if they wouldn’t be coupled, while the solid lines represent the frequencies resulting from the coupling. The latter modes are also included in the Q-factor graphs. The colours in all these graphs represent the same mode and can thus be compared. The inset in graph (c) shows the anti-crossing behaviour for the strongly coupled modes. The Q-factor of both the Noda mode and the FP modes was kept the same in both cases, at 5000 and 2000 respectively

(d) (b)

(c)

(a)

17

To find the highest possible Q-tuning range, the system has been simulated for several relative coupling strengths, ranging from the strong coupling regime to a regime in the weak coupling. This was done by changing the loss rate in the FP cavity and the coupling constant between the cavities, while keeping the other parameters constant. From this the ΔQ/Q was investigated systematically and also the wavelength shift of the highest Q mode is studied. This is done because for several quantum information processes, a wavelength shift would be undesirable.^[31] The result of this investigation is shown in Figure 12.

Figure 12: ΔQ/Q (a) and wavelength shift (b) of two modes brought into resonance as a function of the losses and the coupling constant as determined from coupled mode theory. A clear transition between the weak and the strong coupling regime is observed and the ideal coupling regime is found to be at this transition.

As can be seen from Figure 12 the ΔQ/Q is largest and the wavelength shift is the smallest for the regime between the weak and the strong coupling regime. This can also be understood intuitively. In the weak coupling regime the coupling between the two modes is so small that the Noda mode is barely influenced when brought into resonance with the FP mode. This results in a small Q-tuning range. The small wavelength shift is within the linewidth of the FP mode and is caused by the superposition of the modes when the two modes are coupled. In the strong coupling regime the Q- factor of the FP increases strongly as well. Since the Q-factor of the Noda mode can only drop to minimal the Q-factor of the FP mode, a higher Q-factor for the FP mode results in a smaller Q-tuning range. The ΔQ/Q depends on the Q-factor of the uncoupled Noda cavity.

2.1.3 Master equation

To study the final dynamic behaviour this system would display in the single-photon regime, the master equation for the system has been derived and solved numerically. With the master equation the density matrix elements , which give the population of state m if m=n, or the coherence of state m and n if m≠n, can be calculated.^[32]These equations are the quantum mechanical equivalents to semi-classical ensemble bloch wave equations. Both equations give similar results in this case, but the master equation can be extended to include decoherence, which cannot be simulated using a classical model.

Strong coupling

Weak coupling

Strong Coupling

Weak Coupling

2 4 6

2 4 6

_FP/_Noda(-)

Coupling constant ( Noda/) 0,0

0,036 0,072 0,110,14 0,160,18 0,200,22 0,230,25 0,270,29

Strong Coupling Q/Q

Weak Coupling

(a)

2 4 6

2 4 6

_FP/_Noda(-)

Coupling constant (/ Noda) _0,0

0,064 0,13 0,19 0,26 0,32 0,38 0,45 0,51

(nm)

Strong Coupling

Weak Coupling

(b)

18 The master equation has been derived for the excited state of one two-state emitter (the quantum dot) and two photon modes Apart from that a ground state was included to have an input channel for the excited state of the quantum dot. The system is drawn schematically in Figure 13. The two- state emitter could not couple directly to the FP cavity but apart from that all states could couple.

The derivation of the equations and the used Hamiltonian are given in Appendix B. The basis for the system is:

(14)

The resulting set of differential equations is:

(15)

Figure 13: Schematic drawing of the coupled system consisting of two coupled cavities (coupling constant η, losses κ) and an emitter (coupling constant g, losses γ) coupled to one of the cavities.

19

Where g is the coupling between the quantum dot and the Noda cavity, γ is the population decay rate in the quantum dot, ωi is the respective resonance frequency of state i, κi is the intrinsic loss rate of cavity i and η is the coupling between the cavities.

The three equations for state 4 represent the ground state dynamics and are completely independent from the other equations, as can be seen from the fact that these transitions are depending only on the ground state probabilities. These states were only included for physical and mathematical completeness, but the interest lies with the cavities and the excited state of the quantum dot so these 3 equations have been omitted for the simulations.

These equations were solved simultaneously, setting the losses in the quantum dot and the Noda cavity to zero. The Noda and the FP cavities were first out of resonance, but the FP was brought into resonance with the Noda cavity at different times by changing the resonance wavelength of the FP mode. This introduced an exit channel to the system through which the photon could tunnel out.

The quantum dot and the Noda were kept in strong coupling but the Noda and the FP were again investigated in both the weak and the strong coupling regime. Some typical results of these calculations are shown in Figure 14.

2.1.4 Results

As can be seen from Figure 14, where the occupation of each state is plotted versus time, the coupling of the two cavities also has severe influences on the dynamics of the system. Also the timing of introducing the coupling is important in this sense. The importance of the timing can be seen by comparing the graphs on the left with the graphs on the right. The graphs on the left represent the case where the FP cavity is brought into resonance with the Noda cavity at the time where the maximum probability is for the excited state of the quantum dot. This ensures the FP cavity can remove the energy from the Noda cavity without energy going back to the quantum dot.

On the other hand, the graphs on the right represent the case where the FP cavity is brought into resonance when the maximum probability is in the Noda cavity, this causes a highly asymmetric lineshape for the Noda cavity. Apart from that, the quantum dot will start to get excited immediately by the Noda cavity, which delays extraction to the FP cavity, since the quantum dot and the FP cavity don’t directly couple.

The master equation simulations also showed the most efficient way to extract the photon was in the transition regime between strong and weak coupling between the cavities. This because strong coupling would change the resonant wavelength of the Noda and thus destroy the coupling between the Noda cavity and the quantum dot, displayed by the fact that the quantum dot remains occupied after the coupling has been turned on. On the other hand, too weak coupling would result in too small an exit channel, resulting in back-coupling from the Noda to the quantum dot and thus again not resulting in a symmetric photon.

20

Figure 14: Master equation investigations of a coupled system of a two-state emitter, an infinite Q photonic cavity and a low Q cavity which is brought into resonance at the time indicated by the position of the dashed line. The top two graphs represent the typical behaviour for the strong coupling between the cavities, the middle two for the transition regime between strong and weak coupling and the bottom two for the weak coupling between the cavities. For the graphs on the left the low Q cavity was brought into resonance at the point of maximum probability for the quantum dot while with the graphs on the right, the low Q cavity was brought into resonance at the time where the quantum dot had zero occupation probability.

21

2.2 Comsol simulations

For this project the commercial software package COMSOL has been used to study both the electromagnetic field and the heat transfer in our system. COMSOL relies on finite element calculations to simulate a system. Finite element calculations, our COMSOL structure and the results of these simulations will be described in this section.

2.2.1 Finite element calculations using COMSOL

Because analytical solutions are generally hard to find and often hard to solve with numerical methods, most numerical methods rely on finite element calculations to find an approximate value close to the exact solution. The numerical calculation will converge onto the exact solution until a certain tolerance condition has been met.

In finite element calculation both the time and the spatial coordinates are discretized. In practice this means space is divided into points, which should obey certain conditions.^[33] Time differentials are also discretized after which they can be numerically integrated using standard integration methods as Runge-Kutta.

The heat transfer calculations were done using the heat transfer module. The laser was simulated by a rectangle source input, as is shown in Figure 15. The boundaries at the edge of the system were chosen to act as a heat sink and were chosen at approximately the position where the undercut would end, while the air holes were chosen to be thermally insulating. These simulations were done in 2-D due to computational limits.

In COMSOL the electromagnetic solutions for the photonic crystal are found by solving the Maxwell’s equations using a finite element calculation. For this purpose the RF module has been used, which is described in more detail in [34]. The boundaries are chosen to be absorbing, to ensure a finite system without reflections from the edges. The field was excited using a point source. All these simulations have also been done in 2-D because of computational limits.

2.2.2 Heat transfer simulations

The heat transfer simulations were performed to investigate the effects of a laser spot as a local heating source on the sample temperature and more specifically if it would be possible to heat and thus shift the Noda and the FP cavity independently. For the calculations the following heat transfer coefficients were used^[35]:

The optical absorption is considered to be the same for both GaAs and InP. Although this is not completely true, taking into account the uncertainties in the reflectance due to the large numerical aperture of our objective, the difference in absorption can be neglected.

The results of the heat transfer simulations are shown in Figure 15.

22

Figure 15: Calculated temperature distribution using the heat transfer simulations performed for our structure for GaAs (a) and InP (b). The laser spot was placed at the rectangle and the absorbed laser power was the same (100μW) in both cases. The Noda cavity is on the left side of the structure.

It can be seen from Figure 15 that both materials have a similar temperature distribution but the maximum temperature reached is severely different. The higher maximum temperature for InGaAsP is caused by the much lower thermal conduction, which requires a higher temperature difference to reach thermal equilibrium. The temperature distribution is mainly dictated by the boundaries, which are, apart from a scaling factor, similar for both materials. This is because the boundaries have been set as a heat sink so the position of the maximum and minimum temperature are forced upon the system

It can also be seen that although the Noda cavity is still heated somewhat the FP cavity is heated much more and it should thus be possible to tune the two cavities nearly independently. In the actual experiments, due to practical limitations, the laser spot will be much closer to the Noda cavity, causing a larger portion of the heat shifting the Noda cavity. This shows that the position of the second spot is crucial for performing the right thermo-optic experiments.

2.2.3 Electromagnetic simulations

The electromagnetic simulations served several purposes; testing the structure and its ideal parameters, comparing spectra found in experiments with simulations and to get the mode distributions. For the electromagnetic simulations a wavelength dependent effective refractive index library has been used for the electrodynamic material properties.

Figure 16 shows the wavelength spectra found in 2-D simulations for the Noda cavity and the FP cavity and corresponding mode distributions in the combined structure. In these simulations both the excitation and collection spot are a point and thus the spectrum from only that particular cavity can be studied. In practice however, both spot sizes are larger than the Noda cavity. This causes for the modes from the barrier to become visible on the spectra. Therefore these two spectra have been combined for the spectra of the Noda cavity.

0 10 20 30 40 50 60 70 80 90 100 0 1 2 3 4 5 6 7 8 9 10

(a)

(b)

ΔT:

ΔT:

23

1304 1308 1312 1316 1320

Noda mode Barrier modes

|E y|2 (a.u.)

|E y|2 (a.u.)

Wavelength (nm)

1366 1368 1370 1372 1374 1376

|E y|2 (a.u)

Wavelength(nm)

From Figure 16 it can be seen that the Noda mode, combined with the barrier mode, shows one distinct peak, which is attributed to the Noda cavity and a series of Fabry-Pérot like peaks from the barrier. This barrier acts as an FP cavity due to reflections at the edge of the crystal.

The FP mode has two different resonances, the even and the odd modes, at higher energies. This is also displayed in Figure 9 in the previous chapter. Both these resonances show clear slow-light behaviour where the modes are squeezed together and get an increased Q-factor. This is in agreement with the band calculations of next section. In Figure 16 only the even modes have been plotted.

The resonant wavelength of the FP cavity was now changed by decreasing the refractive index in the most distant half of the cavity. In the meantime the Q-factor of the Noda was determined. The Q- factor was much higher than the expected value due to the fact that these were 2-D simulations and thus there were no losses in the third dimension, which in practice would be the main loss channel.

However for the purpose of this project 2-D simulations would suffice since the goal is to change the in-plane losses. The ΔQ/Q will only be lower in practice because of the lower Q-factor for the Noda cavity.

Figure 17 shows the resulting Q-factor which was acquired from these simulations. Figure 17(a) shows the Q change in the weak coupling regime. The weak coupling regime in this case was achieved by reducing the barrier at the end of the FP cavity to be the same size as the barrier between Noda and FP. Since the main loss channel in this case is the far end of the FP cavity, as apparent by the high Q of the uncoupled Noda, this ensured that the losses and the coupling were comparable. This should be the ideal Q-tuning region, which is also apparent in the high dQ/Q displayed. Figure 17(b) shows the Q change in case of strong coupling, where the barrier at the other end of the FP cavity was increased and the losses thus decreased.

Figure 16: Spectra of the Noda mode and Barrier mode (a) and the Fabry-Pérot mode (b) as acquired from 2-D COMSOL simulations. The FP spectrum shows the FP modes (green arrows) close to the band edge. The high Q of the modes can be explained by the fact that the simulations were done in 2-D and there were thus no vertical losses.

(a) (b)

24 A resemblance between Figure 17 and Figure 11 can also be seen. Especially the smaller Q-tuning region in the strong coupling region is visible in the COMSOL simulations as well. At the lowest point, 2 peaks could be observed which shows the mode anti-crossing and crossing of the Q-factor of the highest Q mode from one mode to the other. This indicates that the effect observed in the COMSOL simulations is really a coupling effect and strengthens the validity of both theoretical descriptions.

2.3 Meep and MPB calculations

Meep and MPB are both free Finite Difference Time Domain (FDTD) simulation packages developed by MIT.^[36] Both these programs have been designed specifically for photonic crystals. Unlike COMSOL, Meep and MPB work in Fourier space and therefore implicitly assume periodic boundaries.

Meep was used to calculate the quality factor and mode distribution in 3D. This was used to see if the results obtained from the COMSOL simulations were still valid in 3D. The quality factor was of course significantly lower because of the out-of-plane losses, but the overall conclusions for the Q- tuning were still valid. Apart from that the mode profiles were also similar as can be seen from Figure 17 where the mode profile of the Noda from the 2D comsol simulations and the 3D Meep simulations has been compared.

MPB uses a similar method as Meep to calculate the photonic bands in the waveguide. This was mainly done to check the modes and the relative spacing of the modes in the FP cavity. These calculations were performed in 2D as well since differences with fabricated samples were expected to be mainly due to fabrication imperfections and not due to the difference between 2D and 3D. This due to the fact that, for the solution, the values of k within the x,y-plane would be used.

0.980 0.985 0.990 0.995 1.000 5.0x10⁵

1.0x10⁶ 1.5x10⁶ 2.0x10⁶

Q ual it y fact or ( -)

n/n

0,0 5,0x10⁴ 1,0x10⁵ 1,5x10⁵ 2,0x10⁵

Quality factor (-)

n / n₀

Figure 17: Results of 2D COMSOL simulations for our system showing the change in Q-factor in the Noda cavity as the refractive index of the FP cavity is changed. (a) the result in the weak coupling regime, where the barrier at the far end of the FP cavity is the same as the barrier between the Noda and the FP cavity (5 periods). (b) the result in the strong coupling regime, where the barrier at the far end of the FP was much higher (20 periods) than the barrier between the Noda and the FP cavity, which was still 5 periods.

(a) (b)