The quality factor, Q, of a cavity is proportional to the ratio of the energy stored to the energy lost per second [84]:
Q= 2πνW
P (7.4)
whereW is the energy stored in the cavity,P is the power dissipated, andνis the frequency of the radiation. The higher theQfactor of the cavity, the more effective it will be as a tool to increase the sensitivity of the spectroscopic setup. The “loadedQ” (including coupling
losses) can be calculated from the cavity spectrum using [85]:
QL=
ν
∆ν (7.5)
whereν is the frequency of the cavity mode, and ∆ν is the width of the cavity mode. The ring down time of the cavity can furthermore be calculated using [85]:
τ = QL
2πν (7.6)
From the spectrum in Figure 4, we find that the cavity modes have a width, ∆ν, of 2.0 MHz in the the 90◦ off-axis configuration. This gives QL= 1.5 ×105, andτ = 0.8µs.
The Q factor can be used to calculate further characteristics of the cavity, and it is also a useful measure to assess the fundamental limits of cavities and what can be done to optimize their performance. Since theQ factor is dependent on the angular frequency of the radiation, this somewhat obscures the comparison of cavities between different frequency ranges. Another useful measure therefore is the effective path length of the radiation in the cavity, which is given by [82]:
Lef f =
λQL
2π (7.7)
From equation (7.7), we see that the effective path length is directly proportional to the wavelength of the photons. Since THz photons (λ∼0.1 - 1 mm) have wavelengths approxi- mately two orders of magnitude shorter than those at MW frequencies (λ∼1 - 10 cm), the implication is that an increase inQ of approximately two orders of magnitude is needed to achieve similar path lengths for the two regimes. Indeed, if we calculate the path length in the experiment above, where λ= 1 mm, and QL = 1.5 × 105, we find that Lef f = 24 m.
This is only a factor of a few higher than what is regularly achieved in our flow cell system that is several meters in length, and which can be efficiently double-passed.
To determine how theQfactor, and thus the path length, can be improved compared to the system tested here, we next investigate the aspects of the cavity that limit theQ factor. As mentioned previously, if the optics inside the cavity are chosen with a diameter large compared to that of the beam, diffraction losses can practically be eliminated. In this case, two factors, in principle, limit the quality of the cavity: reflection (and diffraction) losses off the wire-grid polarizers that serve as the input and output coupling mirrors, and power losses due to the finite conductivity of the metal in the mirrors. To examine the effects of each, we will consider the limiting cases where the losses are dominated by only one of the effects. First, the limit where reflection losses are negligible compared to ohmic losses in the metal is considered. In this case the Q factor is given by [85]:
Q= L
2δ (7.8)
whereδ is the skin depth of the metal, which is a measure of the power loss due to conduc- tance in the surface of the mirror. The skin depth is given by [85]:
δ = √ 1
πνµσ (7.9)
whereµis the permeability of the metal, andσ is the conductivity. Gold has a permeability of 4π·10−7 N/A2 and a conductivity of 4.4·107 (Ωm)−1 (at 300 K) [86], which gives a skin depth at 300 GHz of 1.39 × 10−5 cm. The cavity length of our off-axis configuration is 30.5 cm, giving a theoretical Q = 1.1 ×106. In reality this number will be slightly lower in our case, since the polarizer wires are made of tungsten, which has a somewhat lower
conductivity than gold; but this idealized calculations is nonetheless useful for comparative purposes.
Next, we consider the case where the ohmic losses are negligible and the Q factor is dominated by reflection loss of the wire-grid polarizers. In this case the ring down time is give by [87]:
τ = L
c(1−R) (7.10)
where R is the reflectivity of the mirror. This equation can be combined with equation (7.6) to give:
Q= 2πνL
c(1−R) (7.11)
With a reflectivity of 0.999 at 300 GHz, this gives a theoretical limit of Q = 1.9 × 106. Again, this is an idealized calculation because we have not included any diffraction caused by the wires themselves.
From these two limiting cases we can calculate the expected total Q factor by:
1 Qt = 1 QC + 1 QR (7.12)
where the QC is that due solely to conductance losses, and the QR to reflection losses.
Substituting in the numbers we found above gives a theoretical limit on the total quality factor ofQt= 7.0×105. As mentioned previously this number is lowered somewhat because
the wire-grid polarizers are made from tungsten rather than gold, which brings the total maximum Q into the range of 5-6 × 105. We determined a QL = 1.5 × 105 from the
spectrum in Figure 4, andQ’s as high as 3× 105 have been measured in our system with very careful alignment of the cavity and coupling optics.
These results show that in the current setup it is the conductance losses in the mirror surface that outweigh the reflection losses at the polarizers, although they are similar in magnitude. It should be noted that an increase in frequency causes a rapid decrease in the reflectivity of the polarizer, which corresponds to a decrease in the Q factor. For example, the reflectivity of the G45×10 polarizer in our setup decreases from ∼0.999 at 300 GHz to
∼0.99 at 1 THz, which givesQR= 6.3×105with this increase in frequency the only change
in the system. On the other hand, as we see from equations (7.8) and (7.9), an increase in frequency causes a decrease in the ohmic loss in the metal and a concomitant increase in
QC,QC ∼
√
ν. At 1 THz, the skin depth of gold decreases to 7.6 ×10−6 cm, and QC =
2.0× 106.
Put more generally, for mixed mirror/polarizer cavity designs, at low frequencies the dominant loss term is conductance loss in the metal surfaces of the optics and at high frequencies the dominant loss term is reflection loss at the wire-grid polarizers – with a transition point at several hundred GHz. Increasing the reflectivity of the polarizers will increase the frequency at which this transition point occurs. In principle, it should be possible to increase the reflectivity of the polarizer by decreasing the width and spacing of the wires to the point where ohmic losses always dominate over reflection loss. Because of the relationship in equation (7.12), the reflection loss should be at about an order of magnitude less than the conductance loss for ohmic loss to dominate. Especially at higher THz frequencies this is difficult to achieve with free-standing wire-grid polarizers due to the mechanical challenges of winding very fine wires onto an open aperture under tension. A more promising method for significant increases in reflectivity is therefore the deposition of a wire grid array onto a substrate transparent at THz frequencies, such as high resistivity silicon. This method should allow for the manufacture of a much finer and more closely
spaced array of wires, ensuring that the system will always operate in the regime where conductance loss is the dominant term. The main advantage of operating in this regime is that discussed above, namely that the conductance loss decreases with frequency (QC
∼ √ν). This partially compensates for the decrease in path length due to the shorter wavelengths at THz frequencies, and would maximize the performance of the cavity.
After decreasing the reflection losses at the wire grid polarizers, the second route to improving the quality factor of the cavity is by decreasing the ohmic loss in the metal surface. Gold wires on high resistivity silicon (a combination that has long been used in THz photomixer designs) would thus lead to an additional gain in the Q factor over free- standing wire grid polarizers as gold is a better conductor than tungsten. In the system we tested here, high reflectivity deposited gold wire grid arrays should increase the Q closer to values near 1.1 × 106 at room temperature. Further gains can be made by cooling the mirrors as the conductivity of metals increases at lower temperatures. For example, at 77 K the conductivity of gold increases to 2.1 × 108 (Ωm)−1, or for aluminum at this temperature the conductivity is 4.2 × 108 (Ωm)−1 [86], resulting in QC values of 2.4 ×
106 and 3.4 × 106, respectively. Cooling to liquid Helium temperatures (4.2 K) or using
superconducting metal wires should allow further improvements, at the cost of considerable experimental complexity of course. Finally, equations (7.8) and (7.11) both show thatQ is linearly proportional to the length of the cavity, so larger cavities would result in a further increase inQ.