Quantitative Discussion of the role of the AR coating

In this section we will compare performance of the cases with and without an AR coating and and give an analysis of the lasing modes and linewidth on the example of the 11.1 µm laser. Without an AR coating, the intra-cavity facet has an as-cleaved refectivity of 28%. If used in the EC setup, the useful power in the grating-selected modes is severely limited by the dynamic range in which the laser can be driven without oscillation on the chip’s own Fabry- Perot modes. For a 5-mm QCL from the same wafer as the QCL used in the previous section, in the center of the gain region, the threshold current for lasing on the grating-selected mode was 1.757 A, while the threshold for the FP modes was 1.802 A. This resulted in a maximum power in the grating-selected modes of no more than 380 µW. Owing to the limitation on the drive current, the tuning range is also limited, in this case to about 36 cm−1_{, which is 4%}

of the center wavelength.

The 5-mm QCL used in the previous section (which had a slightly higher waveguide loss than the 6-mm QCL), before coating had a threshold current for the FP modes of 1.96 A. When coated with an AR coating of 9.0% reflectivity, the threshold rose to 2.28 A, while the threshold for the EC-selected modes was lowered to 1.86 A. While the uncoated QCL could be driven 2.6% above its lasing threshold in the EC, the coated one can now be driven up to 17.2% above its threshold, which leads to a higher output power and a broader tuning range. In the center of the gain region with a pulse duration of 100 ns and a repetition rate of 200 kHz, 2 mW average power on the grating-selected modes was reached and the tuning

range almost doubled to 67 cm−1_{, which equals 7.5% of the center wavelength.}

As discussed in Section1.5.6, another drawback of an imperfect AR coating is associated with coupled-cavity effects that lead to intensity fluctuations depending on whether or not the center of the grating-selected bandwidth (the grating or Littrow wavenumber) coincides with an FP mode of the QCL chip. For the extreme case of not using an AR coating at all, the intensity periodically vanishes with a period of 1/2nl ≈ 0.308 cm−1 _{(for the 5 mm}

QCL), i.e. the period of the chip’s FP modes. Since the Littrow reflection of the grating has a bandwidth of δν ≈ ν/N = 0.166 cm−1_{, where ν = 900 cm}−1 _{and N ≈ 6800 is the number}

of illuminated grooves on the grating, it selects as many as 2-3 external cavity modes, spaced at ≈ 0.046 cm−1_{. Closer analysis of the setup with the model given in [22] reveals that all}

of these modes get strongly suppressed when the grating bandwidth center is detuned from the FP modes. Figures 3.14 a) and b) show the threshold gain envelope function (wavy line) as a function of wavenumber for two different settings of grating angle. Figure 3.14a) shows the threshold gain for the case where the center of the grating feedback bandwidth is tuned onto an FP mode (drawn as thick gray vertical lines) at 900 cm−1 _{and b) where it is centered}

between two modes at 900 cm−1 _{and 899.846 cm}−1_{. The black vertical lines indicate allowed}

modes of the external cavity, i.e. the phase condition is satisfied on a round trip including the free space path toward the grating. The horizontal line shows the gain corresponding to the pump current the laser is driven at. The intersecting points of the black vertical lines with the threshold gain envelope indicate the thresholds for those particular modes. All modes that have thresholds below the actual gain will lase (in pulsed mode, i.e. without full mode competition or gain saturation). The lower these points lie, i.e. the greater the difference between gain and threshold gain, the stronger the mode will lase. In Fig. 3.14b) it is apparent that the modes (black lines) are sufficiently strongly suppressed for the laser to shut off. The greater the distance between FP modes, i.e. the shorter the QCL stripe, the wider the tuning gap, i.e. the range in which there is no intensity.

Preliminary results show, that on a 1-cm laser the power does not completely vanish between FP modes due to the high density of the FP modes whose spacing now is then exceeded by the wavelength chirp during the pulse.

For the AR coated stripe, Fig. 3.15 shows the output power as a function of the grating- selected wavenumber (center of bandwidth). Between FP modes, the laser does not shut off, but the power decreases to about 26% of its next maximum value. The oscillations have a period of 0.308 cm−1 _{which is equal to 1/2nl with n =3.25 and l = 0.5 cm. Also, the}

maximum power very slightly oscillates with a period that is equal to about 4 periods of the FP modes, as is to be seen in Fig. 3.16, which is a magnification of some of the oscillations in Fig. 3.15.

The first of these results is explained in Fig. 3.17 using the same model used for Fig. 3.14, the second is explained in more detail in the next section. Figures 3.17a) and c) show cases where the grating is tuned to FP modes of the QCL at 900 cm−1 _{and at 899.692 cm}−1

yielding power maxima, and b) where it is tuned halfway between these two modes, resulting in a power minimum. The EC modes are marked with black lines and the FP modes with

899.5 900. 900.5 6. 8. 10. 12. 899.5 900. 900.5 6. 8. 10. 12. wavenumber @cm-1D threshold gain @cm - 1 D gain @cm - 1 D

(a) Grating is tuned to FP modes of the QCL at

900 cm−1 899.5 900. 900.5 6. 8. 10. 12. 899.5 900. 900.5 6. 8. 10. 12. wavenumber @cm-1D threshold gain @cm - 1 D gain @cm - 1 D

(b) Grating is tuned midway between FP modes to

899.846 cm−1.

Figure 3.14 Analysis of the lasing longitudinal modes of the uncoated laser in the external cavity.

The wavy line indicates the threshold gain as a function of wavenumber when the grating is tuned to a) 900 cm−1 _{and b) 899.846 cm}−1_{. Fabry-Perot modes of the chip are drawn as gray lines at}

899.692 cm−1_{, 900.00 cm}−1_{, and 900.308 cm}−1_{. Black vertical lines indicate allowed longitudinal}

modes of the entire cavity, where the phase condition is satisfied not just within the chip. The horizontal line indicates the gain at which the laser is pumped, since it is driven with short pulses, gain clamping can be ignored to first approximation. The intersecting point of the wavy line with a black vertical line (marked with dots) is the threshold gain for that particular mode, if the crossing point is below the horizontal line, it will lase. The lower the threshold gain, the stronger this mode will lase. In a) there are two lasing modes, in b) the laser has shut off.The parameters for the QCL used in this model were: waveguide loss αw= 4 cm−1; gain in the limit of no amplifier saturation γ0(I) = 3.33 × I[W]; drive

current, I = 1.9 A, free space length l = 9 cm, coupling constant between chip and free-space cavity η= 0.9, grating efficiency RG= 0.93.

899.5 900. 900.5 6. 8. 10. 12. 899.5 900. 900.5 6. 8. 10. 12. wavenumber @cm-1D threshold gain @cm - 1 D gain @cm - 1 D

(a) Grating is tuned to FP mode of the QCL at 900 cm−1

899.5 900. 900.5 6. 8. 10. 12. 899.5 900. 900.5 6. 8. 10. 12. wavenumber @cm-1D threshold gain @cm - 1 D gain @cm - 1 D

(b) Grating is tuned midway between FP modes.

899.5 900. 900.5 6. 8. 10. 12. 899.5 900. 900.5 6. 8. 10. 12. wavenumber @cm-1D threshold gain @cm - 1 D gain @cm - 1 D

(c) Grating is tuned to FP mode of the QCL at

899.692 cm−1

Figure 3.17 Threshold gain envelope (wavy line), Pump gain (horizontal line), FP-modes of chip

(black vertical lines), EC modes (thick gray lines) of the coated QCL. a) and c) result in power maxima, b) results in a minimum. The parameters for the QCL used in this model were: waveguide loss αw= 4 cm−1; gain in the limit of no amplifier saturation γ0(I) = 3.33×I[W]; drive current, I = 2.2 A,

free space length l = 9 cm, coupling constant between chip and free-space cavity η = 0.9, grating efficiency RG= 0.93.

thicker grey lines. In these plots, the intersections of the allowed modes and the loss curve are also marked by dots for clarity. The higher the loss of a mode, the lower the power in this given mode for a certain Littrow wavenumber νg given by the grating angle.

The difference between cases a) and c) in this simulation is that in a), the effective free- space cavity length between the intra-cavity facet and the grating does not allow for a standing wave at exactly 900 cm−1 _{in a) (because of the phase jump of π upon reflection from the}

grating), but in c) at 899.692 cm−1 _{it does. Thus in a) the effective EC reflectivity decreases}

at exactly 900 cm−1_{, decreasing laser action there. The next EC modes are very close though,}

and laser action is strong on both of these. The output power of each mode is proportional to the difference between gain and threshold gain (for short pulses, i.e. disregarding mode competition). EC Loss [cm -1] 5.5 6.0 6.5 7.0 7.5 8.0 νg [cm-1] 900.0 900.5 901.0 901.5 902.0 902.5 903.0

Figure 3.18 Loss values of all the allowed EC modes as a function of grating wavenumber. Grating

pivot point 2.5mm from optical axis.

EC Loss [cm -1] 5.80 5.85 5.90 5.95 6.00 νg [cm-1] 900.0 900.5 901.0 901.5 902.0 902.5 903.0

Figure 3.19Zoom-in on loss values of all the allowed EC modes as a function of grating wavenumber.

Grating pivot point 2.5mm from optical axis.

3.4.3 Quantitative Discussion of mode-hopping for the AR-coated