## Webinar Summary – Optimize signal acquisition for optics and photonics measurements

This blog post accompanies the two webinars “Optimize the Signal Acquisition for Optical Measurements” and “Optimize the Signal Acquisition for Optics and Photonics measurements”. Thank you to everybody who joined the live event and participated actively by asking so many questions. The answers for both events are provided in the blog post Q&A – Optimize signal acquisition for optical measurements. In this blog I want to summarize the content of the webinar and elaborate on the following aspects:

• Lock-in amplifier working principle
• Lock-in amplifier measurement optimization
• Boxcar averager working principle
• Differences between lock-in amplifier and Boxcar averager

In the webinar, we used the Zurich Instruments MFLI Lock-in Amplifier for the demonstration of the tunable diode laser absorption spectroscopy (TDLAS) case study and the UHFLI Lock-in Amplifier and Boxcar Averager for the case study on pump-probe measurements. The instruments are controlled using our LabOne software. The measurements were performed on signals coming from one of our Arbitrary Waveform Generators according to typical signals in actual experiments; I did not have access to an optical setup.

## Lock-in working principle

The working principle of a lock-in amplifier is described in detail in the white paper Principles of Lock-in Detection.

Figure 1: Lock-in amplifier scheme: The mixer multiplies the input signal with the reference signal. The result is low-pass-filtered and represents the X component of the measurement. The same operation is applied to the Y quadrature.

As depicted in Figure 1, the input signal is multiplied in a mixer with the reference signal, which is generated by the lock-in amplifier or provided externally. The result is a band shift of the input signal by the reference frequency, as shown in Figure 2d – the signal of interest is located at DC now and can be implemented using a low pass filter. These two signal processing steps result in a precise and narrow band-pass filter that follows the reference frequency as the center frequency and is superior in signal quality to direct implementation of a passive band-pass filter, for example.

Figure 2: Signal processing steps in a lock-in amplifier. a) Input signal in the time domain – one clear oscillation frequency is visible. b) Representation in the frequency domain: the single frequency shows as a distinct peak. After mixing with the reference signal a fast oscillation is visible in c with a finite DC offset. The low-pass filter applied by the lock-in amplifier smoothens the fast oscillation resulting in the red graph. d) Band-shifted spectrum after the mixing and low-pass filter depicted as the red dashed line.

The schematic included here represents analog signal processing components. In modern lock-in amplifiers such as the ones from Zurich Instruments, this functionality is implemented digitally to provide a better dynamic range, lower noise, and a wider filter range.

## Lock-in measurement optimization

To optimize your signal acquisition with a lock-in amplifier, you have three direct parameters to optimize: reference frequency, filter bandwidth, and filter order. How to choose them depends on the specific noise floor, which should be characterized first. Figure 3 shows an example of measurement. Settings of the signal input parameters are essential to keep the noise floor as low as possible and avoid pick-up of spurious signals. Input range, impedance, and AC coupling need to match the incoming signal. How to choose them best, is elaborated in the video “6 tips to improve your lock-in measurements” [1]. Still, three noise components are always present: 1/f noise, white noise, and some noise peaks. If the measurement setup allows for this, select the reference frequency from the regime where only white noise is present and stay away from noise peaks.

Figure 3: Amplitude spectral density. Noise floor measurement with illustrated lock-in amplifier filter response with filter bandwidth f_co centered around reference frequency f_{mod}.

The next step is the choice of the best filter parameters – this is crucial to achieving the highest signal-to-noise-ratio. To optimize it, we need to capture all signals and suppress the noise. The filter has two  parameters: filter bandwidth and filter order. The frequency and step response for different filter bandwidths are depicted in Figure 4.

Figure 4: Frequency and step response for different filter bandwidths.

The smaller the cut-off bandwidth (f_co), the more the step response is smeared out in time. A more detailed description can be found in this two blogs [2] [3]  The filter bandwidth should be chosen so that it is as narrow as possible without affecting the shape of a signal feature. Figure 5 depicts the effect on two absorption peaks. A measurement with too large a filter bandwidth leads to a noisy signal (see panel a). Panel b, by contrast, illustrates the ideal situation: no noise and narrow peaks. The signal shown in panel c is distorted because the filter bandwidth is too small – still, all noise is gone.

Figure 5: Effect of different filter bandwidths. Measurement of two absorption lines a) with too large a filter bandwidth, b) with a good filter bandwidth, and c) with too narrow a filter bandwidth.

There are two ways to find the best settings. One is to calculate the response from known parameters of the experiment and from the details provided on the filter in the Signal Processing Basics chapter in the User Manual. Alternatively, the experimental approach consists in starting with a narrow filter bandwidth and increasing it as long as the features in the measurement are not smeared out. The innovative approach is very challenging if the signal is buried deep in noise. Then a first dark measurement helps to have a reference with all spurious features in it to discriminate against the signal in the actual measurements.

The filter order defines the slope of the filter edge, as depicted in Figure 6. The higher the filter order, the steeper the slope of the filter edge and the more smeared out and delayed the response on the step function.

Figure 6: Frequency and step response for a low-pass filter with different filter orders.

A higher filter order can suppress a noise peak close to the signal in the frequency domain. Figure 7 shows the absorption peak with a lower filter order filter and pick-up leaking into the measurement. This manifests in the ringing on top of the signal of interest. Panel b shows the same with identical bandwidth but higher order. The ringing does not influence the measurement anymore. A Summary of this part is given in the video “Low-pass filter settings done right” [4].

Figure 7: Measurement of the same signal with a) a 3rd-order filter, and b) an 8th-order filter with identical cut-off frequency.

## Boxcar averager working principle

While lock-in amplifiers are ideally suited to analyze sinusoidal signals, periodic non-sinusoidal signals can be analyzed with a better scheme – the boxcar averager.

Let’s explain the working principle using a pulsed signal such as the one generated by a pulsed laser impinging on a photodetector. For pump-probe measurements, THz-time-domain spectroscopy or various non-linear microscopy techniques, this kind of signal needs to be analyzed. As depicted in Figure 8, the information in the signal (orange line) is contained in a short time interval and is therefore spread out in frequency.

Figure 8: Boxcar averager working principle. The Boxcar averager integrates the input signal (orange line) in the time domain over a Boxcar window of width T_DC. The shorter this interval, the further the frequency response extends to high frequencies.

To capture all information, the boxcar averager integrates the signal over a window of duration T_DC (blue lines). All noise components outside of this window are entirely suppressed, and only noise synchronized in time compromises the measurement. As a consequence, higher frequency noise is not averaged out but captured as shown in the frequency domain – the shorter the Boxcar window f_BOX ~1/T_DC, the broader the frequency response. The Boxcar window is phase-locked to the periodic signal. As a consequence, the Boxcar window will match the signal even if the repetition rate of the source varies. By matching the Boxcar window width to the signal, the Boxcar averager response inherently matches the bandwidth in which the signal is above the noise floor. The Boxcar averager exploits the fact that it captures signal components far beyond the fundamental frequency where the noise floor is typically lower.

The best SNR is achieved by setting the Boxcar window so that it contains ~95% of the signal power, as depicted in Figure 9. Nevertheless, it is always better to choose a Boxcar window broader than the pulse. This reduces the SNR only slightly; making it too short can easily have a dramatic impact on the SNR.

Figure 9: Boxcar window setting. Two Boxcar averager units are displayed on separate panels. Both operate on the same signal and the Boxcar integration window is shown as the grey shaded areas. Each unit provides one boxcar window and one boxcar baseline window.

Typically, in pump-probe measurements there is one probe pulse carrying the signal and one or multiple in the same period which are unaffected. Using a second Boxcar baseline window on such a pulse helps to reduce noise and cancel any DC component of the entire signal – hence the name ‘baseline suppression’. The window is always of width identical to the associated Boxcar window to balance the signal averaging. In Figure 9, the same signal is displayed in the two Boxcar units of the UHFLI with a Boxcar window and a Boxcar baseline window synchronized to 4 individual pulses.

## Difference between lock-in amplifier and boxcar averager

A pulse train with modulation at the Nyquist frequency is an excellent example (see orange line in Figure 10) to elaborate on the difference between a lock-in amplifier and a Boxcar averager. Here one pulse has information imprinted on it, whereas the subsequent one doesn’t. The fastest possible modulation allows for detection with the highest sensitivity with a lock-in amplifier. The reference signal of the lock-in amplifier is depicted in dark blue.

Figure 10: Lock-in amplifier and Boxcar averager response function. The input signal is depicted in orange. The response function of the lock-in amplifier (dark blue) is a sine wave in the time domain and an individual peak in frequency domain. The Boxcar window coincides with the sub-pulse train carrying the signal. The Boxcar baseline window is centered on every other pulse.

Its positive peak coincides with the information-carrying sub-pulse, the negative peak with the reference sub-pulse train. In the frequency domain, this corresponds to an individual line at the fundamental frequency. The Boxcar window (light blue) is placed on the information-carrying pulse – the associated boxcar baseline window on the unaffected pulse train. The response in the frequency domain detects the signal at all odd harmonics.

For the situation described above, we perform the measurement with a lock-in amplifier and a Boxcar averager simultaneously with identical measurement bandwidths. The result is plotted over time in Figure 11. Both techniques recover the sinusoidal signal, but the difference in SNR is apparent. The Boxcar averager signal (blue) has an SNR about 4 times better than the signal from the lock-in amplifier (orange).

Figure 11: Lock-in amplifier and Boxcar averager measurement. An input signal is analyzed with a lock-in amplifier and Boxcar averager simultaneously with identical measurement bandwidth. The recoverd signal – a sinusoidal oscillation – is depicted over time for the lock-in amplifier (orange line) and the Boxcar averager (blue line). The SNR on the latter is significantly better.

I hope I could answer all your questions, and help identify the best signal recovery technique for your experiment or save time when you search for the best measurement settings. If you have further questions, do not hesitate to call me or write to me – I will be happy to continue the conversation.

### References

[2] Blogpost Mehdi Alem, Time-Domain Response of Lock-in Filters

[3] Blogpost Mehdi Alem, Frequency-Domain Response of Lock-in Filter

[4] YouTube, Low-pass filter settings done right