Signal processing scheme for broadband heterodyne gigahertz interferometry with a broadband and a second low-noise photodetector with limited bandwidth

There is a need for highly accurate vibration measurements in the gigahertz range. To measure these vibrations with heterodyne interferometers, methods in the state of the art require both high photodetector bandwidths and high carrier frequencies. However, conventional methods such as acousto-optic modulators rarely achieve frequency shifts above 500 MHz and are inefficient at higher frequencies. Additionally, detector bandwidths are limited, or the noise level of high bandwidth detectors is insufficient. In this paper, we propose a solution to these limitations by using a setup with two phase-locked lasers to create a beat frequency in combination with a signal processing scheme that utilizes a broadband and a second low-noise photodetector with a much smaller bandwidth and low noise. Our method could enable gigahertz heterodyne vibration measurements with high resolution. The novelty of our concept is that we only detect the lower sidebands and are still insensitive to AM. This is achieved by two consecutive measurements with frequency shifting of the lasers, effectively swapping the upper and lower sidebands.


Introduction
Due to the expansion of global networks of technical devices in the "Internet of Things" (IoT), there is an increased need for communication in high-frequency radio systems.This need is illustrated by the 5G mobile radio standard and is anticipated to be further advanced in 6G.Microacoustic filters are pivotal in enabling the utilization of gigahertz frequency channels by means of channel selection and noise suppression [1][2][3][4] Furthermore, they are increasingly used in quantum computing [5][6][7][8].For technical applications in communication and quantum technology, it is essential to detect the resulting microacoustic vibrations, or their harmonics, in the GHz range and with vibration-amplitude resolutions well below one picometer [9].A vibration measurement with high resolution enables the determination of material properties as well as the optimization of the performance, for example, by analyzing dissipation mechanisms and occurring destructive interferences [9].
Current state-of-the-art methods are categorized into homodyne and heterodyne techniques.The best results of heterodyne interferometers are around 10 GHz at a resolution of 360 fm/√Hz [10] and the current best commercial device Polytec MSA-600 allows the detection of vibrations up to 2.5 GHz at 12.5 fm/√Hz.The system can measure at frequencies higher than the carrier frequency by measuring just one sideband and reconstructing the other one [11].The device is suitable up to 6 GHz, but no resolution is specified for this [12].The main limitation with heterodyne interferometers is the generation of a high enough carrier frequency and the lack of detectors with low noise at a high bandwidth.
Another approach is to use a homodyne interferometer with a pulse laser.In this setup, a frequency comb is generated up to the gigahertz range [9,13,14].This enables the measurement of narrowband vibrations up to a frequency of 12 GHz [9].The resolutions achieved with the pulsed laser at about 55 fm/√Hz are approximately a factor of 10 below the theoretically achievable resolutions defined by photon shot noise.
This paper aims to further optimize heterodyne methods for the detection of gigahertz vibrations because the transfer characteristic of this interferometer can be calibrated which makes accurate measurements possible [15].We discuss the essential aspects of phase demodulation of the photodetector signal in heterodyne interferometers, which is required to extract the signal from the detector signal.
Traditionally, phase demodulation uses both sidebands of the signal to ensure resistance to amplitude modulation.However, we propose a new approach that uses a special setup with phase-locked lasers and a novel signal processing scheme.Our approach demonstrates the feasibility of utilizing just a single sideband while maintaining immunity to amplitude modulation effects.
In combination with a setup that uses two photodetectors, one with high bandwidth and relatively high noise and a second with low noise and small bandwidth, out method could enable high gigahertz measurements at high resolutions.
For a better understanding of the signal processing techniques, we will introduce sidebands and phasors for amplitude and phase modulation.

Heterodyne Interferometer
The fundamental concept of heterodyne interferometry is well-known and described in detail in various literature [15][16][17][18][19].A schematic representation is shown in Figure 1(a).A laser source is split into two beams: the measurement (red) and reference (orange) beam.The measurement beam impinges on the measurement object that oscillates with () and its phase is shifted by   in accordance with the doppler shift.With the wavelength of the laser  it is [18]   () = 4  ⋅ (). ( One of the beams is additionally shifted by a carrier frequency  c .Conventionally,  c is generated by an acousto-optic-modulator (AOM), as shown in Figure 1(a).
The primary limitations for heterodyne interferometers at GHz frequency measurements are AOMs and detectors.Commercial AOMs are only available up to a few hundred MHz and their efficiency decreases sharply at higher frequencies.
To circumvent the limitation by AOMs, we use a setup that employs frequency-shifted lasers to achieve higher and freely selectable carrier frequencies, as shown in Figure 1(b).The carrier frequency is generated by the difference in the frequencies of the master  M and slave laser  S .The photodetector (PD) detects the frequency difference, which is mixed with a signal from a local oscillator.The local oscillator sets the desired carrier frequency.The frequency of the slave laser is controlled via a phaselocked loop (PLL) to achieve the desired carrier frequency.This setup was used to achieve GHz frequency measurements, and the setup is described in more detail in [20,21].Note that in the previous setup, two high-bandwidth (H-BW) detectors are used.In a new iteration we will use the illustrated low bandwidth (L-BW), detector whose function we explain below.With this approach, the limitations of AOM can be overcome and heterodyne measurements in the GHz range become feasible.However, the problem persists that detectors with an acceptably high bandwidth usually exhibit a high level of noise.This is where we aim to apply our new signal processing scheme.
In order to explain it, we will go into more detail on sidebands and modulation as well as phasors.

Sideband detection
The information about the vibration is contained in the phase of the signal from the detector.Due to the phase modulation by the single-frequency vibration of the object, there is an upper and a lower sideband around the carrier frequency.With the carrier frequency, the simplified signal from the detector () is () ∝ cos(2   −   ()) = cos (2   − 4  ⋅ cos(2 vib )). ( To demodulate the signal from a heterodyne interferometer, typically both sidebands of the vibration frequency  vib are required and therefore a high bandwidth, as shown in the qualitative frequency spectrum of () in Figure 2(a).However, we will explain how demodulation can be implemented with only one frequency-adjusted sideband that can be detected by the L-BW detector with low noise.To achieve this, the second detector measures the carrier frequency with a high bandwidth and more noise, as shown in Figure 2(b).With the setup shown in Figure 1(b), the carrier frequency can be selected so that the lower sideband always lies within the bandwidth of the L-BW detector.

Phasors
Understanding sidebands and phasors, along with their visual representation, is essential for understanding the concepts presented in this paper, we will discuss them in this subsection.

H-BW L-BW
Beginning with Figure 3(a), we examine the time signal of both the carrier and an amplitudemodulated carrier signal (AM) across a single period, accompanied by their corresponding complex value representation.In the time domain, the phase of the amplitude-modulated signal matches the unmodulated carrier signal, with differing amplitudes.This is also evident in the complex plot.
Figure 3(b) illustrates the origin of the amplitude-modulated signal based on the carrier (red) and the two sidebands (blue).Phasors representing the sidebands are plotted in relation to the carrier in 45° steps from 0 to 270°.This visual approach underscores the phase relationship of the sidebands with the carrier, complementing earlier depictions of the carrier frequency and sidebands in Figure 2.
The resulting amplitude-modulated phasor, derived from the sum of the red carrier and the blue sideband phasors, is depicted in green.It can also be seen that the phase of the amplitude modulated phasor is identical to the carrier.The green phasor in Figure 3(b) is an equivalent representation to the signal shown in the time domain in Figure 3(a) at the top.Hence, the amplitudes of the AM modulated phasors in Figure 3 (b) correspond to the time signal shown in Figure 3(a).For example, the amplitude is at the maximum at 0° and at the minimum at 180°.Figures 4(a) and 4(b) extend these concepts to phase modulation.Figure 4(b) illustrates that the green phasor of the phase-modulated signal initially leads the unmodulated signal and subsequently falls behind at approximately 180 degrees.This effect is consistent in Figure 4(a) as well.In the complex display, no difference can be seen in the static representation, whereas in a moving representation, the additional oscillation around the carrier frequency, as shown to some extent with the green phasor in Figure 4(b), would be recognizable.

Amplitude and Phase modulation
Detecting only the lower sideband and reconstructing the upper sideband from it, introduces the problem that it is no longer possible to distinguish between AM and PM.As described in Eq. 3, in heterodyne interferometers, the information about the displacement s(t) is in the phase of the signal.Therefore, AM is considered as noise since it does not contain information of the out-of-plane vibration.
To address this, although our method uses only the lower sideband it still remains resistant to AM.We accomplish this by performing two separate measurements with the setup shown in Figure 1(b).One measurement with  S >  M and the second with  S <  M .In both measurements, the absolute value of the carrier frequency is | S −  M | =  c .
By changing the frequencies of the lasers in such a way, we are effectively switching the position of the upper and lower sideband, allowing us to reconstruct the vibration signal with limited bandwidth and eliminating the influence of AM.

Results
We demonstrate this concept by considering a raw signal ()with displacement information contained in the PM and disturbance by AM where  AM is the amplitude of the AM and  PM the amplitude of the PM. 1 and  2 are phase offsets.
To demonstrate the impact of the above-mentioned switching of the frequencies, the complex phasors of the raw signal are shown in Figure 5. .
As can be seen in Figure 5(a) with  S >  M and 5(b) with  S <  M , the amplitudes for the carrier frequency and the sidebands are identical.However, the phases are different.The most important thing is that the values are conjugate complex mirrored around the carrier frequency respectively.For example, the complex value of the lower sideband at   −  vib with  S >  M is  1 ⋅ e  1 (Figure 5(a)), whereas the complex value of the upper sideband at   +  vib with  S <  M is the complex conjugate of the former  1 ⋅  − 1 (Figure 5(b)).
When both sidebands are known, the measurement is insensitive to AM, and the correct signal can be obtained through phase demodulation.
If it is only possible to detect the lower sideband because the bandwidth is too small, the upper sideband can be reconstructed from it mathematically.Typically, in this case, it would be assumed that only PM is present, since a possible AM component cannot be quantified.If no AM in the signal is present, it's possible to reconstruct the signal correctly.However, if both AM and PM are present, the demodulated signal is still vulnerable to AM, as illustrated in the demodulated signal in Figure 6(a).This is due to the falsely reconstructed phase information from the original signal, as shown in Figure 6(c).
However, performing two measurements using the method described above allows for accurate reconstruction of the vibration signal, rendering the demodulated signal immune to AM, as demonstrated in Figure 6(b).Here, the reconstructed phase information is identical to the original, as shown in Figure 6(d).Note that the amplitude of the sidebands in Figure 6(c) and 6(d) are identical, but their phases are different.
To reiterate, the phases in Figure 6(d) are derived from the two measurements shown in Figure 5.In both measurements, only the lower sidebands are detected.By the described complex conjugate mirroring, it is thus possible to effectively obtain the lower and upper sidebands of one measurement.

Figure 2 :
Figure 2: Qualitative frequency spectrum of () with required bandwidth for (a) demodulation with both sidebands, =  c +  vib (b) demodulation with just the lower sideband.

A
c  c +  vib  c −  vib H-BW  A  c  c +  vib  c −  vib

Figure 3 :
Figure 3: Visual representation of amplitude modulation in the time domain (a).Representation of how the amplitude-modulated signal is formed from the combination of the carrier signal at  c with the upper  c +  vib and lower sideband  c −  vib (b).Carrier and carrier phasors are depicted in red, AM modulated signal and AM phasors in green and sideband phasors in blue.

Figure 4 :
Figure 4: Visual representation of phase modulation in the time domain (a).Representation of how the phase modulated signal is formed from the combination of the carrier signal at  c with the upper  c +  vib and lower sideband  c −  vib (b).Carrier and carrier phasors are depicted in red, PM modulated signal and PM phasors in green and sideband phasors in blue.

Figure 5 :
Figure 5: Complex phasors of the sidebands and the carrier of the raw signal () with (a)  S >  M and (b)  S <  M .Note that the sequence of the illustrated sidebands has been adjusted to make comparison easier. AM = 0.25,  PM = 0.1,  1 =  2 =  4