Single Photon Collection Efficiency

Figure 4.2.: Confocal microscope. In (a) the working principle of a confocal microscope is shown. A light source is focused on a single point of the sample and the detected light coming back from this point is spatially filtered by a pinhole. As (b) and (c) show, in this way background light from adjacent points gets suppressed.

the increased resolution is the increased signal to noise ratio achieved in confocal microscopy. Squaring the original PSF heavily reduces unwanted side-lobes.

The confocal microscope described so far only gets information from one point.

In order to form an image, the sample has to scanned, either by moving the sample or by moving the excitation and detection spot. Often, confocal microscopy is implemented as fluorescence confocal microscopy, where a laser is used to excite fluorescing molecules (or other emitters) and only the fluorescence is detected. With the possibility to selectively attach dyes to objects of interest, this method has great success in biology, where it is routinely used. More information on this can be found in References [152] and [154]. In this thesis, confocal microscopy is not used to image tissue, but to have a good suppression of unwanted background light when addressing single photon emitters.

4.2. Single Photon Collection Efficiency

After the introduction of optical microscopy in Section 4.1, in this section different approaches to enhance the collection efficiency of single photons beyond what is possible with standard microscopy schemes are introduced. This is very important, since fundamental experiments and applications in nanophotonics deal with small amounts of light, down to the limit of single photons emitted from single emitters (see Section 2). Moreover, it is crucial not to lose the photons due to a low collec-tion efficiency, since an ideal single photon source, which is crucial for linear optics quantum computing (see Section 2.3) [34], has to emit exactly one photon per ex-citation – never two and never zero. For such a photon gun [46], two requirements have to be met: (1) The quantum efficiency of the emitter has to approach unity, and (2) the photon collection efficiency of the optical system has to be near 100 %.

The first requirement can be met by choosing an efficient emitter and if necessary further enhancing its emission resonantly via Purcell enhancement in resonant op-tical structures (see Section 2.2) [158]. The second requirement needs either the

θc

a b c

Figure 4.3.: Single photon collection schemes. (a) Due to total internal reflection, only photons propagating inside a cone with opening angle Θc can be collected.

(b) Using a geometrical approach the number of photons being collected can be increased. (c) In the resonant approach a resonant structure enhanced the emitter and directs the light into one desired mode.

collection optics to cover a solid angle of 4π – which is impracticable for almost all applications – or the photons to be directed exclusively into a small solid angle.

Directing the photons can again be achieved via the Purcell effect by resonantly enhancing emission in one desired spatial mode, or by employing geometrical optics to non-resonantly direct the photons [159].

The challenges in working with single photon sources lie not only in identifying a suitable single photon emitter, but also in collecting the emitted single photons efficiently. For solid state quantum emitters like molecules, quantum dots, and defect centres, the photons need not only to be directed to the collection apparatus, they also need to be extracted from their host material. Diamond for example has an index of refraction of 2.41 at 638 nm [160]. This leads to a critical angle Θc

of less than 25 degrees at a diamond-air interface. So only photons inside a cone with that opening angle can pass through the interface, the rest is lost due to total internal reflection (see Figure 4.3 (a)). There are mainly two approaches to address this problem, a geometrical one as shown in Figure 4.3 (b) and a resonant one shown in Figure 4.3 (c) [159].

4.2.1. Geometrical Approach

In this approach, the geometry of the material surrounding the single photon emit-ter is changed in a way that more of the photons are directed towards the collection optics. A simple example of this is the hemispherical solid immersion lens (SIL) in Figure 4.3 (b) [161]. The problem with total internal reflection is avoided by placing the emitter in the centre of a hemisphere. By doing so, the light will hit the sur-face only under normal incidence and can therefore exit the host material without being hindered by total internal reflection. Since SILs act as an immersion medium analogous to the immersion oil in oil immersion microscopy, due to their elevated

4.2. Single Photon Collection Efficiency

refractive index, they increase the effective numerical aperture of a light collecting system [161, 162]. Commercial SILs made from transparent high-index materials such as zirconium dioxide [144] are commonly only available as half spheres. Using micro-structuring processes like focussed ion beam milling [163] or a combination of mechanical and laser techniques [164], SILs can be fabricated directly from the material hosting an emitter. Since in this approach the emitter is embedded in a homogeneous medium, the maximum amount of collected photons is 50 %. This limit can be overcome by introducing a dielectric interface, because light will be emitted preferably in the direction of the optically thicker material, i.e., the direc-tion of the SIL, if the refractive index of the SIL is the higher one [165]. However, with a dielectric interface present, hemispherical SILs tend to direct most of the light from an emitter on its surface to very shallow angles, which makes collection via subsequent optics very difficult [144]. One way to improve on this disadvantage is to use the so called Weierstrass or super-hemisphere geometry (a sphere cut at a height of h = 1 +_n¹ with n being the SIL’s index of refraction). In this geometry, photons are diffracted into a smaller solid angle [162]. In Section 4.3 another – more efficient – variant of a SIL geometry is investigated.

When not relying on SILs, there are other possibilities to enhance the collection efficiency geometrically: With a carefully designed dielectric slab consisting of a sapphire cover glass and polyvinyl alcohol (PVA), it was shown that nearly every photon can be directed to the collection optics [166]. To achieve this, molecules of known orientation were placed in the middle the PVA layer and imaged through the sapphire substrate with a high numerical aperture objective. However, this large value comes at a cost. Very high numerical aperture collection optics are needed in order to collect all the photons. By employing an additional mirror layer, these large angels can be reduced [167], but still, the resulting mode-profile has little overlap with a Gaussian beam, making fibre coupling extremely challenging. Both aspects are disadvantageous, for example when it is difficult to use large numerical aperture optics, a situation often present when extracting light from emitters in cryostats.

Furthermore, the emitter has to be fully embedded in a medium, restricting this approach to to very simple single photon emitters. Application to more complex nano-engineered structures is not possible [91, 108, 144, 168–172].

Another possibility of exploiting the effects arising from the sample geometry is coupling of the emitters directly to a waveguiding structure. This can be done by coupling directly to the evanescent field from tapered fibres [113, 173, 174], coupling to the mode directly by putting the emitter on the waveguides facet [175] (see also Section 6.2), or embedding the emitter into the waveguide’s material [176, 177] (see also Section 7.4).

In Section 7.5.2, an approach to enhance the collection efficiency using a reflecting parabolic antenna is introduced and experimentally verified.

4.2.2. Resonant Approach

In this approach, the local density of optical states (LDOS) is engineered [159]. Via Fermis’s golden rule [178]:

Wi→f = 2π

¯h |V_{f i}|²ρ(E_f), (4.6) a direct dependence of the transition probability on the LDOS is found. Here, Wi→f denotes the transition probability from an initial state i to a final state f, V_{f i} is the transition matrix element, andρ(E_f) is the density of states at the final state’s energyE_f (see also Chapter 9, where the LDOS is measured). This provides the possibility to enhance certain transitions by increasing the LDOS and also to suppress others by reducing the LDOS [158, 179]. In this way, the spatial emission pattern as well as the spectral properties of the emitted light can be influenced, so that the collection efficiency can be improved. On resonance with optical cavities, where the LDOS is high, the enhancement is called Purcell effect (see Section 2.2 for details) [30]. The Purcell enhancement factor P reads:

P = 3 4π²

λ n

3 Q

V , (4.7)

where λ is the vacuum wavelength, n the refractive index, Q the cavity’s quality factor andV the mode volume [158]. An emitter’s emission falling into the spectral line of the mode will be enhanced by this factor. From Equation 4.7 it is clear, that for a high enhancement a high quality factor and, at the same time, a small mode volume is needed. An example of enhancing the emission of the zero phonon line of a nitrogen vacancy centre can be found in Section 6.1.