La dignidad dei hombre
XIV. Voluntad y Fortuna
The gain and the loss can both be plotted over the continuum of wavenumbers (Figs. 1.13 and 1.15). However, in a laser, according to Eq. (1.6), only a discrete set of wavenumbers fulfill the resonance condition required for the positive feedback loop that enables laser action. These are the modes of the resonator. For a Fabry-Perot laser, the reflection coefficients r1
and r2 are normally constant over the wavenumber range of interest. In case of the Fresnel
reflectivities of (nearly) transparent materials, they are also (nearly) real, thus the argument is either 0 or ±π, and the modes are equidistant and can be numbered such as in Eq. (1.9). For an EC, Eq. (1.6) translates into
2lβ(ν) + arg(r1) + arg(ˆr(ν, νg)) = 2mπ, (1.47)
where arg(ˆr(ν, νg)) is now a highly oscillatory function (within the grating’s reflective band)
because of the large distance traveled to the grating and back, which enters the argument through the terms form e2iLβair = e4iπLλ.
These modes can be marked in the loss curve of the EC laser, e.g. Fig. 1.13, to give Fig. 1.17. The Fabry-Perot modes – i.e. the modes far from the grating wavenumber, where the grating reflectivity is negligible and thus the free-space contribution is as well – are spaced at 1/(2nl), where l is the chip length. The EC modes, i.e. the modes close to the grating wavenumber, are spaced at 1/(2(nl + L)) since the resonator length is now the chip length plus the free-space path. The loss curve, however, oscillates with a period 1/(2L), since its oscillatory part is only determined by the distance between facet and grating. Therefore, the loss curve is intersected by the allowed modes in a complicated manner, that depends on the exact values νg, l, and L, with no straight-forward rule about when it is the dips or the peaks
2266 2268 2270 2272 2274 4 6 8 10 12 wavenumber @cm-1D alpha @cm - 1 D
Figure 1.17 Wavenumber-dependent loss of the external cavity with an AR coating reflectivity of
1.5%, with a grating tuned to νg = 2270 cm−1). The allowed modes (only plotted between 2268 cm−1 and 2272 cm−1) are marked with vertical red lines.
that are intersected. This rule can be given for the limiting case that the free-space length is much larger than the chip length. In this case, when the grating is tuned to coincide with an FP mode, the EC modes coincide with dips of the loss curve. When the grating is tuned half way between FP modes, the the EC modes coincide with peaks of the loss curve.
2266 2268 2270 2272 2274 4 6 8 10 12 wavenumber @cm-1D alpha @cm - 1 D
Figure 1.18 Wavenumber-dependent loss of the external cavity with an AR coating reflectivity of
10−4, with a grating tuned to ν
g = 2270 cm−1). The allowed modes (only plotted between 2269 cm−1 and 2271 cm−1) are marked with vertical red lines.
Since the loss curve is identical to the threshold gain curve, its value at a particular mode is the threshold for that mode. The mode with the lowest threshold will be the dominant mode for laser action, and the others can be disregarded after a very brief transient phase after the onset of the pump current. For instance, it can be seen in Fig. 1.17 that the mode at exactly 2270.0 cm−1is strongly suppressed, while the third mode toward the left and
the right (counting from the central mode), at 2269.7 cm−1 and 2270.3 cm−1, are the ones
that start to oscillate first. The higher the lowest intersection lies for a given grating angle, the lower the overall threshold of the laser and the stronger its emission at a given pump current. Thus, as the grating is tuned and the EC modes intersect the loss curve at varying
loss values, the overall intensity oscillates. The phenomenon that the selected mode does not tune continuously, but “hops” to wherever the threshold is lowest, is called mode-hopping. As is clear from Fig. 1.12, the oscillations of the effective EC reflectivity are damped with better AR coatings, since the coupled-cavity effect is suppressed. The extreme case for an AR coating with a residual reflectivity of 10−4 is plotted in Fig. 1.18. Since the loss curve is
much smoother now, it is clear that if the grating is tuned, the lowest threshold stays close to the minimum which is close to the grating-selected wavenumber.
To avoid mode hopping different approaches can be taken. One uses a long external cavity in conjunction with a long chip driven in pulsed mode. Here, the oscillatory heating of the active region chirps the wavelength across the densely-spaced EC modes through variation of the chip’s refractive index. The drawback of this approach is that the emission is multimode and the linewidth is relatively large, since it is the envelop of the modes. However, the great advantage is simplicity, and the overall linewidth can be limited by using a very large beam diameter to make the grating-envelope narrow. This approach is taken in Chapter3.4.
The second approach combines a good AR coating with a QCL driven in continuous-wave (CW) mode and a grating with a precisely positioned pivot point for the grating. This approach is given in [127]. The pivot point is chosen so that as the grating is rotated, the cavity length varies precisely in such a way as to leave the same mode in the minimum of the grating-selected band. Thus, the angle of the grating and the overall length of the cavity have to tune in synchron, so that a single grating-selected mode “surfs” along on the loss- minimum as it is tuned. The advantage of this approach is that it allows mode-hop-free broadband tuning with a chirp-less CW linewidth. The drawback is that the setup has to be immensely stable, since if the pivot point moves by as much as a micron, mode-hops reappear during tuning.
Two other approaches to avoid mode-hopping during CW operation take the very obvious route of adjusting the cavity length at each tuning step by maximizing the output with a closed feedback loop. One adjusts the effective chip length by thermally altering the refractive index. For this, the heat sink temperature of the QCL needs to be adjusted at each tuning step, which is very slow and not practical for actual continuous tuning, only for stepwise continuous tuning. The other adjusts the grating’s pivot point by mounting the entire grating on a piezo actuator. Although a piezo crystal can actuated very quickly, if the grating is rocked too abruptly, it starts to vibrate at its mechanical resonance. Since a grating is a very bulky load, the resonance frequency is very low and in the worst case, its mechanical resonance is close to that of the entire setup, which can lead to serious damage. Thus this approach is also neither fast, nor stable. Quite oppositely, it is very technology intensive, and therefore even less fail-safe. Despite all of this, it is quite commonly used in commercial External Cavity QCLs as discussed in Section 1.3.
2
The Anti-Reflection coating of
the QCL facet
2.1 Introduction
The facet of the QCL facing the external reflector of the External Cavity – this is normally called the intra-cavity facet – needs to receive an optical coating that eliminates its inherent Fresnel reflectivity. Without an anti-reflection (AR) coating, the two distinct spatial regions of the External Cavity resonator, the QCL chip and the free-space region, behave as two separate cavities that are coupled through the intra-cavity facet’s finite reflectivity, see Section 1.5.2. This coupling results in unwanted interaction that leads to strongly fluctuating intensities of the output beam as the laser is tuned, or to the extreme case of discrete tuning with gaps between the allowed modes. This is called mode hopping and is discussed in detail in Section 1.5.6.
We fabricate these coatings in our labs using reactive magnetron sputtering with quasi in- situ measurement of laser output to determine the quality of the coating during the deposition process. Due to the key role the AR coating plays for EC operation, this thesis dedicates this chapter to its design and fabrication as well as its non-trivial characterization.