Boost Your Optical Signal Detection with Lock-in Amplifiers and Boxcar Averagers – Q&A

This blog post answers the many appreciated questions asked by the audience during the webinar "Boost Your Optical Signal Detection with Lock-in Amplifiers and Boxcar Averagers”. The event was recorded, and the video is available here. 

The webinar covers the following topics: 

• Lock-in amplifier working principles 
• Lock-in amplifier measurement optimization 
• Boxcar averager working principles

We also focus on the following application examples: 

• Fluorescence analysis at multiple wavelengths  
• Pump-probe spectroscopy and microscopy 
• Terahertz time-domain spectroscopy 
• Stimulated Raman scattering (SRS) microscopy 

Below you can read a selection of questions asked during the webinar, divided into four different topics. 

Working principles of lock-in amplifiers 

For a detailed discussion and rigorous mathematical treatment of the lock-in amplifier working principle, see our lock-in detection white paper or contact us directly. 

What is your approach to finding the ideal bandwidth? 

The answer to this question is not unique and depends on your signal's properties. For example, if you have a large signal to begin with, you would start with a relatively large bandwidth and then narrow it down until the SNR reaches an acceptable level. On the other hand, if the starting signal is particularly small, one might not be able even to see it if the filter bandwidth is too large.  Therefore, in this case, one would start with a narrower bandwidth and increase it to the point where the signal is not smeared out. 

A more detailed discussion can be found in this video and our blog posts Frequency-Domain Response of Lock-in Filters and Time-Domain Response of Lock-in Filters. 

By locking the frequency often we see that the signal is shaking and it's kind of difficult to lock the exact desired frequency. So how can we improve our measurements? 

I am assuming you mean that the internal locking to an external reference provided to the instruments is not stable. Generally, one should make sure that the external reference signal is clean. If your reference signal is noisy, then it should not be used as a reference in the first place. As the external reference is locked with a phase-locked loop (PLL for short, for more information on the topic, take a look at our white paper), such reference should be free from frequency noise or jitter, as they could negatively affect the PLL locking. In contrast, the amplitude noise is typically not relevant for correct frequency locking. 

What is the sampling rate of the lock-in amplifier? Kindly explain the mechanism. 

For a lock-in Amplifier, the sampling rate indicates the frequency at which the analog input signal is sampled by the analog-to-digital (ADC) converter; that is, how many samples per second are recorded before the signal is digitized. According to the Nyquist theorem, to correctly resolve a certain frequency f in a signal, one must sample such a signal at a rate that is at least double that frequency, i.e., a sampling rate of 2f or larger.
The UHFLI Lock-in Amplifier, for example, has a sampling rate of 1.8 GSa/s, 3 times larger than the 600 MHz maximum input frequency. The reason for this oversampling (theoretically a sampling rate of 1.2 GSa/s would be enough for a 600 MHz signal) is to improve anti-aliasing performance, increase resolution, and reduce noise. 

The filters in the presentation are analog filters or digital filters? 

The filters that were shown in the presentation are digital filters since all the operations performed within Zurich Instruments lock-in amplifiers are carried out in the digital domain. The analog input signal is digitized as soon as it enters the instrument and receives a low-noise pre-amplification; therefore, all the subsequent operations (mixing with reference, filtering, etc.) are performed in the digital domain. 

To achieve a faster measurement, is it possible to have a larger bandwidth during signal transitions and then lower it to have better precision? 

The optimization of the bandwidth always depends on the actual application and therefore a general automatic procedure is not easily achieved. Still, the APIs provide all the tools needed to allow a quick implementation by the user for their application. 

We normally get X and Y outputs from the lock-in but there is no easy way to get R and Theta measurements directly from the lock-in. Is there a way to measure R and Theta directly rather than X and X? 

Our lock-in amplifiers can measure and calculate the amplitude and phase of your signal in both polar (R and Theta) and cartesian (X and Y) coordinates. Then one can easily route these signals to the Auxiliary Outputs of the instruments in case an interface to other instruments is needed. 

In some of your slides, you talk about demodulation at multiple frequencies…how many signals can I demodulate simultaneously with your lock-in amplifiers? 

The number of simultaneous demodulations one can perform depends on the number of demodulators, which in turn depends on the specific instruments.
When equipped with the Multi-Frequency option, our instruments have the following properties:  

• MFLI: 4 oscillators and 4 demodulators 
• HF2LI: 6 oscillators and 6 demodulators  
• UHFLI, GHFLI, and SHFLI: 8 oscillators and 8 demodulators 

An important aspect to mention is that since our lock-in amplifiers operate with digital signal processing techniques, multiple demodulation channels do not come at the cost of signal losses, in contrast to analog lock-in amplifiers.

How do you determine the frequency and amplitude of the reference signal in the lock-in amplifier?  

Lock-in amplifiers lock their internal oscillator to the reference signal with a phase-locked-loop (PLL), which defines the Frequency and Phase of the signal. The amplitude is not relevant. The multiplication of the signal with the references is performed with a proper coefficient so the measured amplitude, after the low pass filter, is the root-mean-square amplitude of the signal at the Signal Input connector. 

Could you elaborate a bit on the optimum selection of filter order with lock-in amplifiers? 

The typical scenario where one would need to choose a high-order filter is when there are undesired frequency components close to the measurement frequency which can leak through the filter into your signal (e.g., powerline peak). Otherwise, it is generally advisable to keep the filter order low to minimize the settling time of the signal. This is especially relevant if the demodulated signal is part of a feedback loop, as high filter orders might introduce propagation delays that could affect the stability of the loop.  

Boxcar Averager working principle 

For a detailed discussion and rigorous mathematical treatment of the Boxcar Averaging technique, please refer to our Boxcar Averager white paper. 

Considering your SHFLI, how is this increase in bandwidth up to 8 GHz related to the boxcar performance? Can you choose smaller time windows, what about retriggering and deadtime? For example, is it possible to use it for common 80 MHz laser systems? 

An instrument with a higher sampling rate will indeed allow for shorter window widths and higher zero dead time repetition rates. Yet, for common 80 MHz laser systems, the Boxcar Averager unit already on the UHFLI would work perfectly, ensuring zero dead time for repetition rates below 450 MHz. 

How to correctly set the width of the boxcar window to have the best SNR? Is there an optimum width? 

For a Gaussian-shaped pulse, it can be shown that one obtains the maximum SNR by choosing a window that covers about 90% of the pulse power.  In a real measurement, however, the signal might not be Gaussian, or its power might not be evenly distributed across the pulse. In that case, one can optimize the SNR by starting with a large boxcar window and then reducing its width until the SNR peaks. This type of optimization can be done directly with the Sweeper module embedded in our LabOne software. One can easily sweep the starting point and/or the window width to find the best set of parameters to maximize the SNR. More information about the Sweeper module can be found in this blog post. 

Do your instruments have multiple boxcar averagers? and do you also provide a baseline subtraction option? 

Each UHFLI equipped with the Boxcar Averager option comes with 2 independent Boxcar Averager units. 
Furthermore, each of the units comes with an embedded baseline suppression feature. The LabOne software allows the user to arbitrarily set the position of a signal and reference window. The difference between these two windows is then automatically calculated. 
As an example, the baseline suppression can be used to remove slowly varying components from the signal (slower than the shot-to-shot fluctuation) or to perform differential measurements by taking the difference of two consecutive pulses. The latter is often done in pump-probe spectroscopy to directly obtain the pump-induced differential intensity change ΔA/A.
Moreover, the instruments have also an Arithmetic Unit that allows the user to perform mathematical operations on various internal signals, e.g. the ratio between the results of the two independent boxcar units. 

General questions 

In slide 12, there are green and red dots in the mechanical chopper, what are they? 

Those dots simply depict laser beams of different sizes/shapes that could be placed at different positions along the chopper radius. The idea is to show that depending on the beam size and profile, one can achieve different modulation functions (pure sinusoidal, square wave, low-duty cycle). 

What is the limitation on the internal manual freq. modulation on HF2LI and UHFLI? 

The limitation of the internal frequency of the instruments is the same as the one on the input signal, meaning that all the frequencies that can be generated by the numerical oscillators of the instruments follow the same rules given by the Nyquist limit. So for the HF2LI and UHFLI the maximum internal/external frequency is 50 MHz and 600 MHz, respectively. 

Is the reference frequency the same as the frequency of the signal of interest? If not, what is the frequency deviation? 

Yes, the frequency of the reference provided to the Lock-in Amplifier is the same as the signal that one wants to measure.  For example, if you have a train of optical pulses modulated by a chopper and measured by a photodetector, the reference signal coming from the chopper and provided to the Lock-in Amplifier defines the frequency at which the signal of interest appears.

Use cases 

Hi, when we do a pump-probe experiment, the amplitude always is a positive number, how do you offset/normalize the signal to get the positive and negative number, which means ΔA/A? 

In a typical pump-probe experiment, the Lock-in amplifier output is indeed proportional to ΔA. To normalize the signal to the intensity of the probe pulses, one normally needs to calibrate the system and measure the intensity of such pulses separately and subsequently compute ΔA/A. 
On the other hand, by using the Boxcar Averager of the UHFLI, you could automatically compute ΔA/A. Since the UHFLI has two independent units, you can measure delta A with one unit and capture the pulse without any signal using the other unit to provide a baseline. Using the Arithmetic Unit (AU) of the instrument, one then computes the ratio between the 1st and 2nd unit to directly obtain ΔA/A.  

If I would like to measure various frequencies simultaneously during a time-resolved Raman measurement, I assume that I would need as many detectors plus Lock-in amplifiers as the number of frequencies I want to measure. Or could I just recover the whole signal by a Fourier Transform from one detector? 

If with “various frequencies” you refer to the possibility of demodulating an individual signal modulated at multiple frequencies (e.g. like the fluorescence analysis example on slide 15 in the presentation), then one detector coupled with a Lock-in amplifier with multiple demodulators would do the job. 
If you are referring to the frequency content of your time-resolved signal instead, then taking the Fourier Transform of that signal will give you the frequencies related to the molecular response of your sample. This bandwidth is generally limited by the bandwidth of your laser pulses. 

How could one use lock-in amplification to boost the signal from a camera in an imaging system?  

To perform lock-in amplification with a camera in an imaging system, one would in principle need a Lock-in Amplifier for each pixel/detector in the camera, which is not a feasible solution. In the imaging examples I showed in the presentations, the signal was always detected by a single photodetector and the sample was spatially scanned to measure the signal at different positions. 

Is the LIA capable to recognize a fluctuation in the number of ultra-fast digital pulses over a certain duration of time and demodulate the fluctuation in pulse count according to the modulation frequency? 

If you are asking whether our instruments can count a certain number of pulses in a certain duration of time, then the UHFLI can indeed do it. When equipped with the pulse counter option (UHF-CNT), the UHFLI can analyze up to 4 pulse trains in parallel and operate in several different modes (free-running, gated, etc.) depending on the specific use case.