General properties of Nd3+:Gd2O3 nanocrystals

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FEM simulations of silica microtoroids

Microtoroids are another interesting microcavities [16], since WGMs in these cavities have the advantage to achieve smaller mode volumes, compared to spherical microcavities of the same outer diameter. However, unlike microsphere cavity, the Helmholtz’s equations in this cavity is not separable, thus there is still no analytical theory for its WGMs structure and positions. In general, the WGMs of toroids can be characterized by their polarizations and three integer orders (n, m, q), where the azimuthal mode number m and q (the latter being defined as the number of polar antinodes minus one), are similar to the spherical ones, while the order n denotes a radial-like mode number.
Recently, several methods have been developed to numerically solve Maxwell’s equations based on techniques like Finite Difference Time Domain (FDTD) [52] or Finite element method (FEM) [53]. Among these techniques, the FEM method has been applied for the simulations in WGM toroidal microcavities [54, 55]. In fact, the three dimensional problem in axis symmetric WGMs cavities is reduced to a two dimension problem and can be easily simulated using a commercial software (Comsol Multiphysics)3. Here, this software is used to study the WGMs in toroidal microcavities.

Higher order modes

For a better understanding of the WGMs of a silica microtoroid, the higher order modes are also studied using FEM simulations. Figure 1.10 gives the electric energy density distributions of different WGMs with TE polarization and m = 160. The silica toroid is set to possess an outer diameter D = 30 m and a minor diameter d = 6 μm, in order to match the parameters of the toroids experimentally studied in Chapter 2. In this figure, one clearly observe that a WGM labeled with integers q and n possesses q + 1 antinodes in the polar direction and n antinodes in the radial direction. In the figure, the corresponding resonance positions are also provided.
Table 1.3 provides the comparison of mode spacings for this toroid (FEM) and a silica microsphere (approximation from its analytical solution). From this table, one observes large spacing of different q order modes for microtoroid compared to microsphere. This is because the large curvature of a toroid induced by small d value has strong effect on the q order modes. It should be mentioned that all these spacing values are dependent on both D and d values.

Fabrication of silica microspheres

Over the past decades, several techniques have been devoted to fabricate ultrahigh Q dielectric microspheres. Features like an extremely smooth surface and axi symmetric shape are basic requirements for the desired high-Q factor, as already discussed. To achieve this purpose, melting is the favorite technique, because it can easily produce dielectric microspheres of both good sphericity and surface smoothness thanks to surface tension. Generally speaking, silica glass has melt point above 800C. For example, it is about 1650C for pure silica. To achieve the melting point for glass materials, several heating methods have been successfully applied:
• Gas flame Using a microtorch with propane or hydrogen is the most ancient and still rather common technique to melt glass and the early work on solid optical microspheres was based on it [13].
• Carbon dioxide laser First introduced in our group[14], it has become the most common technique, because Carbon dioxide (CO2) laser can be well controlled to fabricate several kinds of WGMs microcavities.
• Electric arc Electric arc is another way to achieve the melting process of glass. This technique is generally used with fiber splicing equipment[56].
• Plasma torch A microwave plasma torch can also be used to fabricate various active microspheres from the corresponding powders, and produces extremely good sphericity[31].

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Fabrication of on-chip microtoroids

In recent years, the development of silicon microfabrication techniques has permitted the invention of novel ultra-high Q factor microcavities of toroidal shape and directly integrated on a silicon chip [16]. These microtoroids also own the advantage of precisely controllable size. Toroidal microcavities, as we already mentioned in section 1.1.3, feature at the same time a smaller number of WGM (due to a reduced size in the polar direction) and smaller mode volumes (thanks to a better confinement).
In this section, the fabrication procedure of on-chip toroidal microcavities is described. It is separated into two parts. First, silicon technology is used to produce silica microdisks sitting on a silicon pedestal. This step is performed in Sinaps laboratory (Service de Physique des Matéiaux et Microstructures (SP2M) at CEAGrenoble) with which we collaborate. Subsequently, the microdisks are melted using a CO2 laser. This melting step, also termed as “laser reflow”, leads to microtoroids having a reduced outer diameter. By this way, the typical Q factor of about 104 for microdisks is upgraded to be in excess of 100 million for microtoroids.

Table of contents :

Introduction
1 Whispering gallery mode microcavities 
1.1 General properties of WGMs
1.1.1 A simple approach for WGMs
1.1.2 WGMs theory in microspheres
1.1.2.1 Solution of electromagnetic field
1.1.2.2 Resonance positions and spacing
1.1.2.3 Mode splitting due to small ellipticity
1.1.2.4 Optical field distribution
1.1.2.5 Quality factor
1.1.2.6 Mode volume
1.1.3 FEM simulations of silica microtoroids
1.1.3.1 Mode volume
1.1.3.2 Higher order modes
1.2 Fabrication of silica microspheres
1.2.1 CO2 laser source
1.2.2 Experimental setup
1.2.3 Results
1.3 Fabrication of on-chip microtoroids
1.3.1 Fabrication of microdisks
1.3.2 Fabrication of microtoroids
2 WGM excitation with tapers 
2.1 Tapered fiber couplers
2.1.1 Introduction
2.1.2 Taper fabrication
2.1.3 Results and discussion
2.2 Modeling the Coupling
2.2.1 Description of the model
2.2.1.1 Equations of the fields
2.2.1.2 Effects of the coupling gap g adjustment
(a) The critical coupling region: C = I or g = gc
(b) The undercoupled region: C I or g > gc
(c) The overcoupled region: C I or g < gc
2.2.2 WGM Doublets
2.3 Excitation of WGMs in microspheres
2.3.1 Experimental setup
2.3.2 Excitation mapping of WGMs in a microsphere
2.4 Excitation of WGMs in microtoroids
2.4.1 Experimental setup
2.4.2 Typical WGM resonance spectra
2.4.3 The impact of the gap
2.4.4 Excitation mapping of toroid WGMs
3 Microlaser characterization 
3.1 Thermal bistability
3.1.1 Theoretical model
3.1.2 Numerical and experimental results
3.2 Experimental setup and method
3.2.1 Experimental setup
3.2.2 Step-by-step recording method
3.3 Results
3.3.1 Evidence of lasing
3.3.2 Real-time laser characteristic measurement
4 Nd3+:Gd2O3 based lasers 
4.1 Photoluminescence of a doped sphere
4.1.1 General properties of Nd3+:Gd2O3 nanocrystals
4.1.2 Photoluminescence in the WGM
4.2 Lowest threshold recording
4.2.1 Q factors
4.2.2 Power calibration
4.2.3 Evidence of lasing
4.2.4 Threshold and slope efficiency
4.3 Sub-μW threshold single-mode microlaser
4.3.1 Fundamental polar mode q = 0 for pumping
4.3.2 Emission spectra and threshold
4.3.3 Laser performance vs coupling conditions
4.3.4 Microlaser characterization using scanning Fabry-Perot interferometer
5 Other results in microlasers 
5.1 Microsphere lasers using Yb3+:Gd2O3 nanocrystals
5.1.1 General properties of Yb3+ ions
5.1.2 Q factors of the active microsphere
5.1.3 Laser results
5.2 Neodymium implanted silica microtoroid lasers
5.2.1 Fabrication of a rolled-down microtoroid
5.2.2 Q factors
5.2.3 Emission spectra
5.2.4 Single mode lasing threshold
Conclusion 151
Bibliography 

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