Exploring the HF2LI-MOD: Measuring Bessel Functions

Today, I will explore the MOD option, which allows direct sideband demodulation with the HF2LI. First of all, I will explain the difference between direct sideband demodulation and tandem demodulation. Then, I will show you loopback examples, where I generate a frequency modulation (FM) signal on the output channel and feed this signal back to the input for direct sideband demodulation. These loopback measurements demonstrate the unique capabilities of the HF2LI-MOD. Advanced application examples are mentioned at the end of this post.

The MOD option is not included in the basic HF2LI and additionally requires the multifrequency option MF. If you already have a HF2LI and are interested in the MOD option, the upgrade is possible on the field without shipping the HF2LI back to ZI.

Tandem Sideband Demodulation with the HF2LI

Suppose I want to demodulate an input signal consisting of a carrier at a frequency ωc and two sidebands at frequencies +-ωm from the carrier. Tandem demodulation is the standard approach that requires two lock-in amplifiers in a tandem configuration: firstly, the input signal is demodulated to baseband at the carrier frequency with a bandwidth larger ωm; then the demodulated signal (either R or Theta according to whether the signal is an amplitude of frequency modulated signal) is fed to the second lock-in, where it is demodulated at the modulation frequency ωm. Additional instrumentation or measurement effort is paid, because the demodulation oscillator in the first demodulation step needs to be in phase with the carrier at ωc of the incoming signal.

As one single HF2LI contains two lock-in amplifiers, tandem demodulation is feasible. The first channel demodulates the incoming signal at the carrier frequency. From the three available demodulators of channel 1, I would use one demodulator with a narrow bandwidth to analyze the carrier frequency and another demodulator with sufficient bandwidth for the baseband signal. For further demodulation with the modulation frequency, the baseband signal is fed to the input of the second lock-in channel via the analog auxiliary output. In doing so, up to three sidebands can be analyzed in the second channel. The core limitation of this technique is set by the highest possible 200 kHz demodulation bandwidth and another 200 kHz filter on the auxiliary output channel.

Direct Sideband Demodulation with the HF2LI-MOD

Instead of tandem demodulation, the sidebands can also be analyzed directly by demodulating at frequencies (ωc – ωm) and (ωc + ωm). The HF2LI-MOD implements this scheme by using local oscillators at the corresponding sum and difference frequency of ωc and ωm. The assignment of the 6 internal demodulators to these local oscillators is shown in the figure below: Demodulators (1,2,3) and (4,5,6) are grouped into two MOD channels, MOD1 and MOD2, respectively. The 2 carrier frequencies can be equal, harmonics of each other (e.g. ωc1 = n * ωc2) or set arbitrarily. The same holds for the 4 different sideband spacings ωm1, .., ωm4. In the example of the figure below, each of these frequencies is set arbitrarily.  In total, you can therefore detect 4 sidebands (one upper and one lower sideband for each MOD channel). With such many degrees of freedom, you have plenty of possibilities for direct sideband demodulation with a frequency range up to 50 MHz in just one single box. This direct scheme has several advantages: it reduces lab complexity (only one lock-in used) and minimizes the SNR, because only one analog to digital conversion is required, which is not the case for tandem demodulation schemes.

HF2LI-MOD Output Signals

As stimuli for your measurements, you can choose between sinusoidal excitation, amplitude modulated (AM) excitation or frequency modulated (FM) excitation for either the MOD1 or MOD2 channel. Note that the FM generation requires two oscillators and one demodulator, thus leaving the other MOD available for demodulating 1 carrier and 2 sidebands. For sinusoidal excitation and AM excitation, you have all the 6 demodulators available to measure 4 independent sidebands.

The FM Spectrum

Only few people are excited about the mathematical description of an FM signal s(t), because

s(t) \phantom{_{-n}} = \sum\limits_{n=-\infty}^\infty J_n(h) \cos[(\omega_c + n\omega_m)t + \varphi_c + n\varphi_m]

with the modulation index

h\phantom{J_{-n}()} = \frac{\Delta\omega}{\omega_m}
containing the peak frequency derivation Δω. s(t) therefore describes a signal, whose instantaneous frequency sinusoidally changes between (ωc – Δω) and (ωc + Δω) at a rate given by the modulation frequency ωm. φc and φm account for the constant phase shift of the corresponding oscillator. The spectrum of s(t) is an infinite number of sidebands around ωc. Each of sideband pair is weighted with the associated Bessel function Jn (of the first kind) evaluated at the modulation index. Due to this weighting, you observe only a few low order harmonics in practice, whereas high order harmonics vanish. For odd negative n, the sign of the Bessel function is inverted, because
J_{-n}(h) = (-1)^n J_n(h)
The Bessel functions of the first kind are depicted below for n = 0,1,2. For a more intuitive approach on the Bessel functions, I recommend you Michele’s post on amplitude and frequency/phase modulation.

Measuring the Bessel Functions

To demonstrate direct sideband demodulation with the HF2LI-MOD, I will sweep the modulation index h and directly measure the Bessel functions shown above with the demodulated sidebands. In a first step, I will generate an FM signal and demodulate 2 sidebands by using one HF2LI-MOD. In a second step, I use two HF2LI-MOD, one to generate the FM signal and another one to demodulate 4 sidebands.

Single Instrument FM Loop-back

To generate the FM signal, I use MOD1. I set the carrier frequency ωc to 1 MHz, the modulation frequency ωm to 1 kHz and the output amplitude to 1 V. The frequency deviation Δω is the sweep parameter. This FM signal is fed back to the input and routed to MOD2, where all sidebands are demodulated with a bandwidth of 6.8 Hz. I set demodulator 4 to the carrier frequency ωc, demodulator 5 to  ωc + ωm and demodulator 6 to ωc – 2ωm, which is done in the MF panel by selecting the harmonic 2 in demodulator 6.
I implemented the modulation index sweep in a LabVIEW routine, where I step the frequency derivation Δω in 100 steps from 0 to 10 kHz and record a single measurement value per demodulator for each step. The modulation index is thus swept from 0 to 10 with a step size of 0.1. Before the measurement, my routine provides an auto-zero function to eliminate the phase shifts φc and φm. I therefore directly obtain the Bessel functions on the (in-phase) X channels of each demodulator. One detail is a factor of √2 due to the RMS reading of X, which I accord for before plotting the Bessel functions. The resulting demodulator signals plotted against the modulation index h are shown in the figure below. The carrier corresponds to J0, the first sideband to J1 and the second sideband to J2. While the second sideband actually is a lower sideband (n=-2), I directly get J2 due to the even n.

In a next step, I then set φm to -90 degrees, which leads to a phase shift of -90 degrees between each created sideband. The first sideband is therefore shifted by -90 degrees and needs to be detected on the (quadrature) Y channel. The demodulated signal will correspond to -J1. The second sideband is shifted by 180 degrees and is measured on X. In this case, also the second sideband will have a different sign and therefore correspond to -J2. The figure shown below exactly reproduces the expectations.

In case you are interested in the LabVIEW implementation, you can download the vi [drain file 3 url here]. Provided you have a HF2LI-MOD, you simply need to connect output 1 to input 1 and run the vi with the same settings as shown in the former figure. If you also want to reproduce the latter figure, you need to shift the phase of oscillator 2 by -90 degrees.

Two Instrument FM Loop-back

To measure 4 sidebands at a time, I use a second HF2LI device with the MOD extension to generate an FM signal. Think of the FM signal generated by the second lock-in I use as the measurement signal resulting from an experiment. Consider for instance a laser, intensity modulated at 1 MHz and phase modulated at 1 kHz. You could use such a setup to measure the phase modulation sidebands at higher frequencies, where the laser noise is lower than at kHz frequencies. Hereto, you need a 1 MHz and a 1 kHz sinusoidal carrier to excite the modulation and then have exactly the same situation as I create here with a second lock-in amplifier as FM generator.

I use exactly the same settings for the generated FM signal as in the example with one device. On the HF2LI used for sideband demodulation, I use the second demodulator for J1, the third for J2, the fifth for J3 and the sixth for J4. The third and sixth demodulator both measure a lower sideband, but are here equal to the upper due to the properties of the Bessel functions Jn for even n. The resulting plots are shown in the figure below. On the left hand side, I have set the phases φc and φm to zero and thus get all the signals on the X channels. On the right hand side, I used a phase shift of -90 degrees for φm. I therefore detect the odd sidebands on the Y channels and the even sidebands on the X channels. According to the expectations, J1 and J2 are negative, whereas J3 and J4 are again positive.

Application Examples
You are welcome to browse the publications section of the Zurich Instruments website for the HF2LI-MOD. You find there examples on frequency modulation Kelvin probe force microscopy (FM-KPFM) and on advanced gyroscopes for inertial navigation. These examples combine the MOD option with the PLL option to measure sidebands of mechanical resonances, which is a very powerful tool for AFM and MEMS applications.