A photonic microcavity is an optical resonator with dimensions that are similar to the wave-length of the light they couple to. A cavity introduces cavity modes (CMs) which are res-onances of distinct linewidth and spacing. Confinement of light modes in microcavities can come from a high reflectivity on the cavity borders, which is achievable with mirrors, like the Fabry-Pérot resonator, or a large contrast of the refractive index of the cavity material compared to the ambient, like for the CBR. Also a periodic lattice, modulating the refractive index, like in photonic crystal cavities, can restrict the propagation of light and, therefore, form a cavity. [49] If a cavity photon is absorbed faster than decaying due to strong coupling, it forms new modes called polaritons that are half-light and half-matter. Weak coupling cavities, however, feature a preferential emission of photons, making light collection out of solid-state sources more efficient and consequently, is of higher interest within this thesis.
2.3.1 Quality Factor
The quality factor is a useful quantity, describing the amount of optical energy dissipation from the cavity, and is dependent on the linewidth at the full width at half maximum (FWHM) δωc of a CM at ωc:
Q= ωc
δωc, (2.42)
2.3.2 Light-Matter Interaction
In order to understand optical resonators better, it is necessary to briefly discuss basic princi-ples and equations of light-matter interaction. In a semi-classical approach, light couprinci-ples to quantized matter within three processes, spontaneous emission with the probability density p(t) of decayp(t) = exp(−t/τ) and the emitter’s characteristic mean lifetimeτ, stimulated emission, where the presence of an external photon promotes the emission of a clone and absorption which is the inverse, yet symmetric process of stimulated emission.
Fermi’s Golden Rule
In a full quantum-mechanical interaction picture, one can make use of Fermi’s golden rule which is the result of a weak time-dependent perturbation H′ on a system with a known spectrum of the unperturbed Hamiltonian and reads
1 τ = 2π
~2⟨f|H′|i⟩2ρf, (2.43) with the transition matrix element of initial to final state and the density of final states ρf. In the case of an excitation in a single atom in vacuum, the atom’s dipole d couples to the (empty) photon states of the vacuum field, Ekn with wavevectors kn, and emits with a probability related to the transition dipole matrix element, i.e., the off-diagonal elements of the perturbation matrix Mkn ∝ |d·Ekn|. Using the electric dipole approximation, i.e., the field wavelength is much larger than the size of the atom, one can find the decay rate τ0−1 of an excited state of an atom in vacuum with the transition frequency ω0 and the dipole magnituded, representing the coupling strength:
1
τ0 = ω03
3π~c3ϵ0d2, (2.44)
2.3.3 Purcell Effect
When studying the physics of weak coupling cavities, the Purcell effect is of great interest, since it is a figure of merit of a photonic cavity, stating that a change of the photon density of states that couple to the emitter’s dipole can enhance or inhibit its emission rate. A weak coupling justifies the use of a perturbation theory and, therefore, of Fermi’s golden rule. Changing equation 2.44 from vacuum to a medium with refractive index nand effective volume Veff confining the electric field, including the Lorentzian mode density within the cavity and taking the ratio of τ0 and the lifetime τc inside the cavity, one finds
τ0
τc =FP δλ2c δλ2c+ 4(λc−λe)2
|E(r)|2
|Emax|2
d·E dE
2
, with FP= 3Λ3Q
4π2Veff, (2.45) where Λ = λnc. This effect was reported by Purcell in 1946 [50] and its factor FP, called Purcell factor, only features cavity properties, making it a handy number for designing optical resonators. Equation 2.45 shows also limiting factors of a potential lifetime reduction, namely the spectral matching of emitterλeand cavity modeλcwithin the Lorentzian, as well as the spatial matching of the emitter to regions with high electric field density. The last term shows that the Purcell enhancement also appears polarization-selective, as seen in Wang et al. [48].
Furthermore, the lifetime shortening effect can also be limited by non-radiative recombination channels and leaking modes.
2.3.4 Circular Bragg Resonator and Highly-Efficient Broadband Reflector The circular Bragg resonator (CBR) is a broadband microcavity, showing an optimized ex-traction efficiency of embedded light emitters, while simultaneously enhancing the radiative rate through the Purcell effect [29][51]. The nanostructure operates by including three mech-anisms. A circular second-order dielectric Bragg grating with a period Λ = λ/neff enables, contrary to a first-order grating, not only lateral reflection of emitted light back to the center but also surface-normal radiation [52], greatly boosting the signal yield [53]. The center of the rings is basically a grating defect that acts as a cavity confining the electromagnetic field in a disc with diameter 2Λ and effective volume Veff to exploit Purcell enhancement. Un-derneath the structure, a highly-efficient broadband reflector (HBR), consisting of an oxide layer and an Ag or Au film acting as a mirror, makes light emitted towards the substrate also available for collection by an objective or an optical fiber [30]. In Figure 2.8 (a) and (b), one can see a schematic and a micrograph of the microcavity nanostructure, while (c) provides characteristic cavity properties.
To design the cavity [28], a simulation of the 1D radial section of the grating on an oxide layer is carried out, altering the grating period such that perpendicular reflection of a lateral applied electric field yields a broadband surface-normal radiation, as in Figure 2.9 (a). Then, the radius of the central cylinder is tuned by a 3D finite difference time domain (FDTD) simulation to obtain a Purcell factor matching the typical emitter wavelength, cf.
Figure 2.9 (b). Finally an Au reflector is added underneath and, through optimization of the oxide spacer, leaking modes are reflected back to the cavity, being made accessible for light collection. Figure 2.9 (c) shows the cross-sectional electric field distribution and a highly directed beam. The inset shows the Gaussian far-field profile of the cavity emission.
To gain flexibility with the material system, Dr. Santanu Manna carried out further simulations to change the material of the HBR dielectric from SiO2 to Al2O3, using FDTD within the softwareLumerical. The results are plotted in Figure 2.10, showing that a change of the oxide material is possible with similar results. The parameters used in the experiment for the structures for both material systems are given in Table 2.1. Simulations also show that
Λ
740 760 780 800 820
0.6 0.7 0.8 0.9
1.0 Collection efficiency Purcell factor
Wavelength (nm)
Collection efficiency
0 5 10 15 20 25
Purcell factor
Y 1 mµ
Λ
Figure 2.8– (a)Cross-sectional and(b) SEM top view (taken from Ref. [31]) of the CBR-HBR cavity. (c)Simulation results (taken from Ref. [28]) of the collection efficiency using a numerical aperture (NA) objective with NA=0.65, where NA= sinθ(half angle) in air, and
the Purcell factor of a CBR-HBR cavity designed for 785nm.
-2 -1 0 1 2 -0.2
0.
0.2 0.4 0.6
SiO2 Au CBR
Air
Band-edge=820nm
R nm R nm R nm
Figure 2.9– (a)The stop-band of the 1D grating. (b)Tuning of the central cavity radius to match the Purcell enhancement. (c) An XZ-cross-section of the electric field distribution
in the cavity. The inset shows the far-field beam profile. Taken from Ref. [28].
Wavelength (nm)
760 765 770 775 780 785 790 795 800 805 810
Extraction efficiency (%)
40 50 60 70 80 90 100
Al2O3: 140nm Al2O3: 160nm Al2O3: 180nm Al2O3: 200nm Al2O3: 220nm Al2O3: 240nm Al2O3: 260nm
Wavelength (nm)
770 775 780 785 790 795
Purcell factor
0 5 10 15 20 25
Al2O3: 140nm Al2O3: 160nm Al2O3: 180nm Al2O3: 200nm Al2O3: 220nm Al2O3: 240nm Al2O3: 260nm
Half angle (°)
0 10 20 30 40 50 60 70 80 90
Extraction efficiency (%)
0 10 20 30 40 50 60 70 80 90 100
NA=0.65 87.8%
NA=0.85 95.2%
Al2O3: 140nm Al2O3: 160nm Al2O3: 180nm Al2O3: 200nm Al2O3: 220nm Al2O3: 240nm Al2O3: 260nm
Al2O3: 140nm Al2O3: 160nm Al2O3: 180nm Al2O3: 200nm Al2O3: 220nm Al2O3: 240nm Al2O3: 260nm
Al2O3: 140nm Al2O3: 160nm Al2O3: 180nm Al2O3: 200nm Al2O3: 220nm Al2O3: 240nm Al2O3: 260nm
Al2O3: 140nm Al2O3: 160nm Al2O3: 180nm Al2O3: 200nm Al2O3: 220nm Al2O3: 240nm Al2O3: 260nm
Figure 2.10– (a) The extraction efficiency and (b) Purcell factor for various oxide thick-nesses and (c) different NA objectives, using the material constants nAl0.4Ga0.6As = 3.3 and
nAl2O3 = 1.61. Simulations from Dr. Santanu Manna.
two polarized cavity modes with a peak separation P emerge for elliptic structures, whereP shows a linear behavior for small eccentricities E= (1−ab22)12 with a slope 69(3) nmand an offset -8(1) nm for the elliptic principal axes a andb, assuming b > a. As an example, this yields P = 7nm for a rather small eccentricity e = 0.21 and is in the range of δωc of the cavity mode.
Grating Groove Center Oxide material Reflector Membrane period width diameter and thickness thickness thickness
330 nm 90 nm 660 nm 220 nm SiO2 150 nm 140 nm
330 nm 90 nm 650 nm 160 nm Al2O3 150 nm 140 nm Table 2.1 – Geometrical CBR-HBR parameters obtained in simulations by Yao et al. [30]
(SiO2) and Dr. Santanu Manna (Al2O3).