CAPÍTULO 2: MARCO TEÓRICO
2.1.1 COMPRAS Y ALMACÉN (BODEGAS)
2.1.1.1 Área de Compras
2.1.1.1.3 Selección de Proveedores
In the experiment we derive the measured signalνmeas(t) from the correction voltage,
which compensates for both, frequency drift and fluctuations of the laser δνlaser[t]
as well as changes of the cavity’s resonance frequency. Here we separate the fre- quency detuning of the cavity into a contribution νsignal(t) that represents the in-
tended frequency response to refractive index changes from the analyte (e.g., single molecules), and a noise contributionδνcavity[t], which is due to involuntary thermal
drift of the resonance or thermo-refractive frequency noise. Consequently we can write the recorded signal as the sum of these contributions.
νmeas(t) = νsignal(t) +δνcavity[t] +δνlaser[t]
In this relationδνcavity[t] and δνlaser[t] set the noise background of the measurement
that determines the smallest resolvable frequency shift and thus the sensitivity of the device. The cavity noise contribution will be discussed in the course of general sensitivity considerations in chapter 5. In general δνlaser dominates overδνcavity for
the diode laser.
Here, we address the issue of laser noise δνlaser[t], which can be eliminated from
the recorded signal, by means of a stable frequency reference. To this end, we set up an ultra stable reference cavity, which is based on a Fabry-P´erot (FP) cavity machined from Zerodur ultra low expansion (ULE) glass. Figure 2.9 (a) shows a photograph of the FP spacer with highly reflective, low loss, mirrors attached to the end facets. The mirrors are designed for the Nd:YAG’s fundamental wavelength at 1064 nm and they are clamped to the spacer with leaf springs. The assembly forms a plano-concave FP cavity and the spacer length of 115 mm corresponds to a free spectral range of 1.3 GHz. A second high finesse FP cavity at 635 nm wavelength was likewise assembled and used for the calibration of the diode laser (cf. Figure
2.5).
To achieve maximum stability against temperature changes, the cavity is placed in- side a temperature controlled housing at high vacuum (2·10−7mbar). Photographs of the vacuum chamber and a copper cylinder with resistive wire for temperature
2.2 The optical setup 39
b
a
c
1.0 0.0 0 5 -5 0.2 0.4 0.6 0.8 20 30 -30 -20 -10 0 10 nor maliz ed r eflec tion frequency detuning [MHz] er ror sig nal [mV ]d
Figure 2.9.: Photographs of the ultra-stable reference cavity assembly. (a) The Fabry-P´erot cavity consists of a ultra low expansion (ULE) glass spacer and two high reflectivity low loss mirrors that are mounted with three leaf springs. (b) The cavity is placed in a temperature controlled copper cylinder, which is surrounded by two additional aluminum shieldings (not shown in the picture). (c) The full assem- bly with the Fabry-P´erot cavity inside the vacuum housing. A connected ion pump
(Gamma Vacuum TiTan 10S) maintains a high vacuum of ∼2·10−7mbar. (d) The
reflected intensity from the cavity (lower graph) and the PDH error signal that is used to stabilize the reference laser to the cavity (upper curve). The line width cor- responds to a finesse of>10,000 and is currently limited by the integration time of the photodiode.
actuation are shown in Figure 2.9 (b) and (c). Such isolated optical cavity expe- riences little exposure to temperature fluctuations of the environment provides a frequency reference which is at least two orders of magnitude more stable than the free running Nd:YAG and the microtoroid.
A second Nd:YAG laser at 1064 nm wavelength is aligned on the cavity (cf. Figure
2.5), such that intensity dip in the reflected signal is minimized for the TEM00 res- onance. Figure 2.9 (d) shows the sharp reflection minimum (linewidth in the order of 100 kHz) that is observed when the laser is scanned over the resonance.
Next, we use a PDH scheme to lock the second Nd:YAG laser to the stabilized cav- ity, while the frequency doubled Nd:YAG laser at 532 nm wavelength is locked to a WGM of the toroid resonator. The measurement laser also features a port for the fundamental wavelength at 1064 nm, which is perfectly correlated with the measure- ment frequency. We overlap the two laser beams on a fast (1.8 GHz) photodiode and record the beat note. In the case, where the first Nd:YAG laser is locked to the WGM cavity, in the absence of an analyte (i.e. νsignal = 0), the noise characteristics
of the beat note reflect purely the noise of the WGM cavity δνcavity[t]. In section
5.1 we will show that the cavity noise is best characterized by the Allan deviation (or variance) as a measure of frequency stability.
40 2. The experimental setup 2 4 6 8 10 12 14 16 18 0 5 10 15 20 25 time [s] frequency [MHz] voltage data counter data voltage data counter data time [s] frequency [MHz] 15.2 15.4 15.6 15.8 16 16.2 16.4 16.6 16.8 17 10 9 8 7 6 5
zoom on trace below
A
Figure 2.10.: Comparing the correction voltage to the absolute laser de- tuning. We measure the frequency detuning using two different schemes. The blue curve shows the calibrated signal obtained from the correction signal that is fed back on the laser piezo (before×20 amplification). The orange curve shows the beat note between the laser locked to the resonator and the reference laser, which is stabilized to an ultra-stable Fabry-P´erot cavity. The beat signal does not require any calibra- tion, as the counter directly issues a frequency. Moreover the beat signal does not contain laser noise, within the locking bandwidth of 30 kHz. This is well illustrated in the lower panel, where an external frequency detuning of amplitude ‘A’ is applied to the laser (change of crystal temperature), which triggers a compensation via the laser piezo (blue curve), but does not show in the absolute frequency (orange curve).
Indeed, it turns out, that cavity noiseδνcavity is the limiting factor for the sensitivity
on short time scale below 1 s, where it dominates over the laser noise of the Nd:YAG. On longer time scales laser drift slightly dominates over the WGM resonator drift. This is, however, only true if no analyte is added. Under realistic measurement con- ditions, additional temperature variations lead to an increased drift of the WGM resonator. In any case Nd:YAG laser noise does not impose a limit on the sensitivity for fast measurements, e.g. of SUV.
Recording the beat note not only permit us to measure the toroid-resonator noise, but it also provides a mean of calibration. To test the signal scaling deriving from the frequency dependence on the piezo voltage, we record the beat note in parallel to the correction signal in an SUV measurement. Counting the beat note directly provides a frequency and does not need additional calibration and scaling (besides a factor ×2 to account for frequency doubling). In Figure 2.10 we overlap the converted correction signal with the absolute beat note signal and confirm that the correction
2.2 The optical setup 41
signal indeed reflects the frequency detuning of the laser. For practical reasons it is required that the correction voltage does not exceed a certain limit. Therefore we implemented an additional slow feedback loop that acts on the laser crystal temperature and induces an opposite frequency drift when the piezo voltage becomes too large. Such a temperature correction and its effect on the laser frequency shows in the correction signal in Figure2.10(lower panel). The beat note is not affected by the temperature correction, as it reflects the detuning between the reference cavity and the resonator and does not contain any laser frequency drift.