Modelo IDEAL 51

There are two ways of reflecting light which are relevant in this thesis, either by metallic mirrors or by distributed Bragg reflectors (DBRs).

Below the plasma frequency, which lies for most metals well in the ultra-violet spectral region, electrons in metals can be approximately described as free electrons. As such, the absorption (∝k) is a lot higher than the polarisability (∝n) of the medium and the light penetrates very little into the mirror. We can express the reflectivityR of a surface via the reflection coefficient r from Fresnel equations (as will be discussed later, see Equation 3.7). At normal incidence,Ronly depends on the complex refractive indices, ˜n1, ˜n2and for ˜n1=1 (i.e., vacuum or air) and ˜n2=n−ik(i.e., an arbitrary, absorbing material) it will become85

Rmetal=|r|2= (n−ik)−1 (n−ik) +1 2 = (n−1)2+k2 (n+1)2_+k2. (2.9)

It follows that ifkn, as is the case for metals,R→1. In practice, the reflectance for metals of good optical quality (e.g. silver, gold, Al) is limited toRmetal = 0.9· · ·0.98. There, even thoughk is large, not much energy is dissipated because the light only penetrates for small

distancesdpen into the metal.85The effective cavity lengthd is thus increased with respect to the distance between the two mirrors, d_c: d = d_c+2d_pen, whered_c d_pen. If the thickness of the metal mirror is on the order of the penetration depth, some light will be transmitted through the mirror (i.e., it will be semitransparent) and the reflectance will be reduced.

The reflectance of a DBR, by contrast, relies on interference. A DBR consists of alternating layers of two different, transparent materials with refractive indices n1 and n2, see Figure 2.4b. Both layers have optical thicknessdopt=λc4, which corresponds to a phase shift of π2 for light with wavelengthλ_c propagating through the layer. According to the Fresnel equations (Equation 3.7), light reflected on the surface from n1 to n2 will experience a phase shift of π if n1 < n2, i.e, only on every other interface. Transmission is never accompanied by a phase shift. As a consequence, all reflected light will interfere constructively, while a double- reflection into forward direction leads to cancellation of the electric field (see two bottommost layers in Figure 2.4b).

The concept of interference makes the DBR a highly tunable system, where the region and amount of reflection can be freely chosen by changing the thickness of the layers or the number of mirror pairs,N. The larger the number of mirror pairs in the DBR stack, the more effective the interference cancellation becomes and hence, the higher the reflectivity atλ_c. In quantative terms, this can be stated as90

RDBR=1−4 _n 1 n2 ‹2N , n1<n2. (2.10)

Equation 2.10 gives the maximum reflectance at normal incidence at λ_c. By increasing the number of layers in a DBR, one can thus increase the reflectance of the mirror to well above 99% and is only limited by the losses (scattering or absorption) in the mirror.91 A full quan- titative description with an explicit wavelength dependence including polarisation and angle dependence goes beyond the requirements for understanding the rest of this thesis and can be found for example in Panzariniet al.[92]. The overall shape of the DBR reflection spectrum will be described qualitatively, though, and is schematically illustrated in Figure 2.5a. From the considerations regarding the interference above, it is clear that both a change in angle and a change in wavelength will lead to a reduction of the maximum reflectance. Around the central wavelengthλ_c, however, this change is small, so that a so-called stop band forms. Its width depends on the refractive index contrast_|n1−n2|. Outside this stop band, the reflectivity

Figure 2.5: Properties of DBRs and DBR-based cavities. (a) Exemplary reflection spectrum of a DBR with indication of the parameters affecting the spectrum. DL stands for double layer, #DL for number of double layers. (b) Transfer matrix (TM) calculation of the mode shift originating from the phase matching when keeping the cavity thickness and energy (Ec) constant and only changing the stopband position (EDBR) via the thickness of the DBR layers.

drops drastically and fringes are observed due to partially constructive or entirely destructive interference at different wavelengths. The relatively high transparency outside the stop band allows for coupling light efficiently into the cavity.

The confinement of the optical mode by interference leads to two particularities of the spatial mode profile inside a microcavity. First, the field does not necessarily vanish on the edges of the microcavity. If the refractive index adjacent to the cavity is smaller than that of the second material, the standing wave will have antinodes at the edges of the cavity (and a node in the centre of a λ₂-cavity).90In the opposite case, the field distribution resembles that of a metal cavity, with nodes on the edges. Second, the interference principle leads to a large penetration depth into the mirrors,70

d_penDBR₌ λc 2n_c ·

n1n2 |n1−n2|

, (2.11)

which also depends on the refractive index inside the cavity,nc. For a typical cavity used for the experiments presented in this thesis, d_penDBR ≈ 700 nm ≈ 3d_cn_c. This large value results in the photonic mode volume being significant larger than in metal cavities, which will be of importance for the light–matter interaction discussed further below. A second effect of the large penetration depth into the DBR is a possible frequency shift of the mode due to phase matching. When the energy of maximum reflectance of the DBR,EDBR= hc_λc, differs from the cavity modeE_c(Equation 2.7), the energyE_mof the observed mode depends on bothE_MCand

EDBR:92 Em= d_c·EMC(Θ) +dpenDBR·EDBR(Θ) d_c+dDBR pen ≈ 1+sin2Θeff dc·EMC(Θ=0 ◦_{) +d}DBR pen ·EDBR(Θ=0◦) dc+dpenDBR (2.12)

One thus observes a mode pulling towards the centre of the stop band upon detuning Ecfrom EDBR. This is illustrated in the calculation of Figure 2.5b, where the mode (low reflectance in the stop band) is shifted upon changingEDBR, even thoughEcis kept constant. The second line in Equation 2.12 used the fact that EDBR also depends on its internal angle ΘDBR as EDBR∝

1

cosΘDBR. On the assumption thatn1,n2,nc≈neff, it becomes clear that Emis also expected to

follow a parabola as function of sinΘ0.