Chapter VIII: Future Directions and Perspectives
A.4 High Precision Measurements and Noise Abatement
High precision measurements to 0.001 were required for the measurements of near perfect transmission and near perfect reflection samples in Chapter 5 and Chapter 6 in addition to collaborative work with other students, such as the work with trans- mission confocal microscopy. These measurements required consideration of every component in the system and its contribution to the overall noise. While developing this procedure, sources of noise were found in unexpected components, components
Figure A.4
that were designed to have lower noise, and were remedied. Leveraging the capa- bilities of lock-in amplifiers is key to this end, and a deeper understanding of the origins of noise is required. A great resource is the Stanford Research Systems 830 Lock-in Amplifier product manual or similar. This discussion can serve as a topic primer. The actual electronics setup of the lock-in circuit is described in Reference [83] Appendix B.
Often, a superstition or myth was spread about picking a chopper reference fre- quency: Use a prime number. The reasoning behind this is that there would be no lower harmonics as a source of noise else where in the system. While true, the wisdom of this advice stops there. Noise can come from anywhere at any frequency, and thus the best strategy is to know the system being measured.
Start with knowing the sample. A first question is "What is the time constant of the response?" The time constant is defined as the time the signal takes to reach 63.2% of its final value in response to a step function stimulus/excitation. So, a signal that takes 0.1s to reach 63.2% its final value, will need roughlytτ=0.5s to be within 1% its final value assuming naturally exponential rates. As a first heuristic, chopping rates should be at a frequency lower than the required multiples of τ to reach the desired level of precision. Fortunately, photodiode time constants are in the nanoseconds range, but if an a signal is the product of heat transfer or a chemical reaction, this will likely not be the case. The lock-in also has a time constant, and that time constant setting should be greater than the time the sample requires to reach its final desired value, or ideally more than it. As an exampleτLI A > 5τsample should give greater than 99% of the final value and a roughly 1% precision.
This time constant determines the required filter bandwidth you will need to get an
arbitrarily precise signal from a lock-in amplifier. Lock-ins equivalent function in frequency space is to act as a notch-pass filter, passing only the signal at the reference frequency through to processing, and ideally would be perfect. However, due to the discrete nature of the measurement, there is some band-width to this filtering.
The time constant is related to the minimum modulation frequency according to f = 1/(2πτ). This gives the maximum frequency of chopping to get a desired level of precision from the lock-in amplifier. Likely, this will be in the kHz range, but it is good to be aware that chopping at higher frequencies is an option, because often strong noise sources can come from low frequencies. This is especially true of "1/f" noise, which is the name for a naturally occurring spectrum of noise that decays from 0Hz with a slow slope of inverse cut-off frequency. Each system has its own 1/f noise spectrum and its own cut-off frequency for this type of noise. So, this paradigm of time constant gives an upper limit, and it is more likely that longer time constants and slower frequencies will be needed.
Picking a Reference Frequency
1. Start with connecting the signal amplifier output, in this instrument the tran- simpedence amplifiers, to a spectrum analyzer or oscilloscope.
2. Set the amplifier to intended or the maximum amplification anticipate to encounter, this should ideally create a high level of white noise on the analyzer read out. Taking the FFT in the scope should show an even power distribution across the frequencies near the frequency given by the previously determined lock-in time constant; there should not be a 1/f slope in that portion of the noise spectrum, or else another amplifier should probably be used.
3. Connect the response device to the scope, which is usually a photodiode for reflection/transmission measurements, but it could also be a custom fabricated photo-responsive device or a potentiostat. Look for the same white noise spectrum from this circuit, in the dark. Next look for the same noise spectrum in the steady state, such as constant illumination. There should be an equal magnitude spectrum of white noise.
4. Component by component, build the measurement circuit and examine its noise spectra in this fashion, with and without a steady-state stimulus. If any component introduces 1/f noise close to where you need to modulate, replace it.
onto sample) conditions, but priority can be given to the illuminated state.
Find a part of the spectrum with only white noise in a range of±bandwidth (the bandwidth is the f = 1/(2πτ)determined by your precision level). The lock- in amplifier should only pass through signal at that band to the signal processor.
As long as it is flat, and the sample signal is higher than the background, the lock-in should be able to reliably read-out the signal magnitude.
6. Start the response modulation (the chopper) with the scope and FFT read out connect still by using a BNC Tee connector. The signal should be the largest peak at that signal. There will be harmonics from the square wave signal, but they should be much smaller or outside the determined bandwidth.
If there are two signals with comparable magnitude in the desired band- width, another reference frequency should be tried. Alternatively, increasing the sample signal by increasing illumination power or otherwise can be a solution.
A common source of equal magnitude noise near the reference frequency in mechanically chopped signals such as this is phase jitter. The mechanical nature of the chopper wheel means that it can sometimes have trouble main- taining a steady frequency due to defects, bends, wear and tear from years of lab use and other non-idealities that create instabilities. A change in frequency can sometimes help mitigate these effects.
7. At this point, a measurement can be taken to characterize the overall level of noise. The time constant of the lock-in, in practice, should be long enough to experience at least a few if not tens of signal cycles. The lock-in becomes more precise with the number of samples input. An important nuance: the final value reported by the lock-in is not the steady state value of the sample, it is the magnitude of the input frequency. So, a more precise magnitude will come from multiple samplings of the photodiodes "stead-state" value. A helpful thought is to consider the discrete nature of the signal seen by the lock-in and how accurately one can determine a frequency without an infinite time sampling – the longer the time sample, the better resolved the frequency becomes.