Cavity efficiency including the phonon wings : theoretical approach 

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Free-space to cavity switching

Before giving more details about the additional parts of the setup, outside the cryostat, it is necessary to describe step by step how the switch from free-space to cavity configuration is done :
• First, a raster scan of the sample is done, and a carbon nanotube is selected (the criterion being its intensity of emission, its spectrum, etc…).
• The steering mirror is set to its medium position and the lens is moved so that the emitter is at its focal point.
• The excitation is switched to the backside. This step enables to keep the very same excitation in free-space and cavity. It ensures that the nanotube excited remains the same and that the effects observed, such as increased emission rate are due to the presence of the cavity and not to a change in the excitation. See section 5.2.1 for further details.
• The location of the nanotube is recorded by a camera which images the plane of the sample, as shown in figure 5.4.
• The fiber-lens ensemble is brought towards the sample. Light is injected in the fiber, so that the location of its tip can be monitored on the above mentioned camera. With this monitoring the position of the fiber can be corrected compared to the location of the nanotube even if the fiber travel path deviates from the lens optical axis.
• The collection path is switched to the cavity configuration. A telescope can be added and adjusted to ensure an optimal collection (it compensates for any misalignment of the lens in the cryostat which cannot be moved once the cryostat is closed).
• Finally, the cavity length is adjusted to fit the emission wavelength of the nanotube, as shown in figure 5.5.
This process is non-destructive so it is possible to switch back and forth from free-space to cavity configuration as many times as needed.
Figure 5.4 – Schema of the complete setup (the steering mirror and the confocal pinhole were not drawn for the sake of simplicity). The excitations paths are depicted in green, the dashed lines corresponds to a usual confocal microscope setup, the solid lines correspond to an excitation through the sample mirror. The collection paths are shown in red lines. The dashed ones corresponds to confocal microscopy while the solid lines correspond to cavity experiments. The switching is done with a removable mirror.
Figure 5.5 – Intensity of the photoluminescence (in colors) of a carbon nanotube embedded in cavity as a function of the emission energy and the cavity length. As the cavity length increases, the cavity modes shifts towards low energies. Here the cavity mode is resonant with the CNT around 1 407 eV. Details : (6, 4) nanotube. Excitation : 760 nm, linear polarization (along CNT axis), P = 0 7 • Psat. This nanotube is referred to as Aleph in further figures.


Lens-Fiber mounting principle

One of the goals of the setup described here was the ability to switch from free-space experiments to cavity experiments, reproducibly, with as few re-alignment as possible. One possibility to couple the objective and the fiber was to stick the later on the side of the former. A disadvantage of this method comes from the fact that the lens width (including the mount) is about 6 mm. Then, the lateral shift to switch from cavity to free-space would have been 3 mm, a large distance which would have implied inconvenient drifts.
Thus, a second method was implemented : it consists in drilling the lens in its center to stick the fiber inside, as shown in figure 5.1. This way, a switch from free-space to cavity requires only a longitudinal move and very small lateral corrections. In the following, the drilling process is described and its consequences for confocal microscopy are investigated.

Lens drilling principle

The free-space experiments, as introduced in section 4.1, rely on a lens of numerical aperture 0.68, focal length 3 1 mm, and working distance 1 76 mm which collects the emission of the nano-object.
The hole diameter is about 400 µm, sufficiently large to let the fiber and its coating (diameter 250 µm) pass easily and sufficiently small to be able to stick the fiber once aligned. The first generation of lenses was processed in the mechanical workshop of the department by Nabil Garroum. For that, he used a drill of radius 0 2 mm to remove material by concentric circles. In order to avoid any crack in the lens, or any scratch at its surface, a layer of polymer (PMMA) was deposited before the drilling and removed afterwards. Currently, the drilling is performed by a company3, by means of pressurized air.

Collection efficiency

The greatest care was taken to measure how the drilling of the objective changed its properties for confocal microscopy. In appendix B, the ability of the objective to focus is experimentally assessed. The focalized spot waist is increased by less than 10%, and the resolving power change is estimated to be negligible. This is important important to keep the zone observed in confocal microscopy as low as possible. However, the most important characteristic for the confocal measurement is the collection efficiency, as it directly affects the estimation of the quantum efficiency of the emitter.
Let’s consider that the emission of a nanotube is dipolar, so that the intensity I in a direction with angle χ with respect to the nanotube axis writes :
This leads to the following expression for the intensity integrated over a cone of angle θmax :
Iintegrated ∝ 2π θmax I (θ, φ) sin θdθdφ (5.4) Here θmax corresponds to the numerical aperture of the lens : θmax = asin(N A). One can take into account the drilling by defining a minimum angle θmin (for a hole radius of 200 µm and a focal length of 3 mm, θmin = 4°) and perform the integration between θmin and θmax. The loss expected for the typical lenses used is this work, compared to the case without drilling, is expected to be below 2% (in good agreement with the experimental data of appendix B). The impact of the lens drilling on confocal microscopy is thus considered to be negligible.


Fiber sticking and angle control

The final goal of the fiber sticking is to enable the creation of a high finesse microcavity with a low volume. The challenges are twofold : first one wants to center the fiber on the optical axis of the lens, so that a simple translation of the ensemble is sufficient to switch between cavity and free-space configuration. Second, the angle made by the fiber with the optical axis of the lens has to be small so that the cavity axis is parallel to it.
After the fiber is shot and a mirror is deposited at its apex, the unprocessed end of the fiber is inserted into the lens. The fiber is then pulled until less than a centimeter protrudes. The fiber is held (by the part below the lens) by a holder mounted on a three axis micrometer stage. It enables to center the fiber precisely and to tune finely how much it protrudes from the lens. See the picture in figure 5.7.
The lens itself is held by a mirror mount which enables to control precisely its orientation, and thereby the angle between the optical axis and the fiber axis.
In order to control the centering and the angle, the position of the fiber is monitored with a long working distance objective either from the top or horizontally in two perpendicular directions. When the fiber position is considered correct, a drop of UV glue4 is deposited on the fiber just above the lens, as shown in figure 5.8. Then by lowering and raising repeatedly the fiber, the glue flows down into the drilled hole. The process is repeated three times to ensure that a sufficient amount of glue is present.

Table of contents :

1 Spontaneous Emission Control with a Cavity 
1.1 Light – Matter Coupling
1.1.1 Atom in free-space
1.1.2 Cavity coupling: the Jaynes-Cummings model
1.2 Purcell effect
1.2.1 Purcell factor
1.2.2 High quality factor cavities
1.3 Purcell effect in condensed matter
1.3.1 Purcell factor for a condensed matter emitter
1.3.2 Light confinement in semi-conductor structures
1.3.3 The case of carbon nanotubes
2 Single Carbon Nanotube Properties 
2.1 Structural and electronic properties
2.1.1 Structural properties
2.1.2 Electronic properties
2.2 Optical properties
2.2.1 Single electron model
2.2.2 Excitonic properties
2.3 Samples
2.3.1 Synthesis of Carbon Nanotubes
2.3.2 Micellar suspension
2.3.3 Spin coating
3 A Fiber Fabry-Perot Microcavity 
3.1 Cavity Planar-Concave modes
3.1.1 Resonant condition
3.1.2 Spectrum of an empty cavity
3.2 Cavity mode volume
3.2.1 Manufacturing of fibered mirrors
3.2.2 Effective mode volume
3.3 Fiber – Cavity coupling
3.3.1 Single mode fibers
3.3.2 Beyond single mode fibers
3.4 Finesse and storage time
3.4.1 Losses
3.4.2 Measuring the finesse
4 Scanning confocal microscopy of carbon nanotubes 
4.1 A scanning Confocal Microscope
4.1.1 Principle
4.1.2 Excitation
4.2 Low temperature single nanotube photoluminescence
4.2.1 Single carbon nanotube photoluminescence
4.2.2 Coupling to acoustic phonons
4.3 Spectral diffusion in Carbon Nanotubes
4.3.1 Spectral diffusion of the ZPL
4.3.2 Spectral diffusion of the Phonon Wings
5 A Purcell enhanced single-photon source 
5.1 Coupling a tunable cavity to the confocal microscope
5.1.1 Cavity Setup
5.1.2 Lens-Fiber mounting principle
5.2 Measuring the Purcell Factor
5.2.1 Photon counts method
5.2.2 Time domain method
5.2.3 Experimental Results
5.3 Single-photon source
5.3.1 Statistics of a light source
5.3.2 Second order correlation function
5.3.3 Hanbury-Brown and Twiss setup
6 Efficiency of a cavity coupled nanotube 
6.1 Cavity efficiency including the phonon wings : theoretical approach
6.1.1 Evolution of the populations
6.1.2 Cavity efficiency
6.2 Experimental derivation of the cavity efficiency
6.2.1 Experimental “reconstructed” spectra
6.2.2 Fit of the experimental data
6.2.3 Evolution with cavity volume


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