Improving accuracy through double acousto-optic modulators in photonics-assisted mmWave heterodyne interferometric sensing

Yiwen Lu; Tong Yang; Qizhuang Cen; Xinpeng Wang; Chong Liu; Feifei Yin; Kun Xu; Ming Li; Yitang Dai

doi:10.3788/COL202523.073901

1. Introduction

Fiber-optic interferometric sensing has flourished over the past few decades and could find numerous applications in many fields such as hydrophones^[1], optical gyroscopes^[2,3], and gravitational wave detection^[4,5]. By detecting the phase change introduced by the delay measurand applied on the sensing fiber, it can provide excellent sensitivity corresponding to the optical frequency. Note that the detection delay deviation caused by the fractional frequency stability of the optical source can be expressed as $Δ τ = \frac{Δ f}{f_{c}} τ_{d}$ , where $τ_{d}$ is the delay imbalance between two interference arms, and $Δ f$ and $f_{c}$ represent the frequency drift and central frequency of the optical source. Once the lengths of the two interference arms are not equal, the frequency drift of the laser in traditional laser interferometers will introduce a large phase error, which is destructive for precision measurements^[6]. A highly stable laser source is expected to fundamentally reduce the noise impact, and one of the most common laser frequency stabilization methods is to lock the laser to a molecular or atomic transition^[7] or an ultra-stable optical cavity^[8]. Unfortunately, such methods might be quite expensive, bulky, and dependent on complex lock-acquisition schemes. In contrast, utilizing the photonics-assisted mmWave as the optical source, which consists of two comb lines from an optical frequency comb (OFC), is a low-cost alternative solution. Two optical frequencies are used in a single interferometer to obtain different phase changes, and the sensing information can be extracted by difference. Thanks to the wide coverage bandwidth and the phase coherence between comb lines, mmWave-based sensing offers greater flexibility and accuracy than laser interferometers, albeit at the expense of a little sensitivity.

To determine the phase change, heterodyne interrogation is attractive because it effectively avoids the confusion of low-frequency noise. The key part is an optical frequency shifter, typically represented by the acousto-optic modulator (AOM), to generate the reference frequency-shifted light, which then interferes with the measured light to form a difference frequency carrier signal. In the past, the noises of most optical and electrical components (such as thermal noise, shot noise, and laser noise)^[9,10] have been focused on, but the effect of the AOM on heterodyne interferometric sensing has been minimally studied. To our knowledge, in addition to the phase noise of AOM’s driving source^[11], Liu et al. considered the intensity noise caused by the relative diffraction efficiency fluctuation of the AOM^[12]. Recently, there have been studies on frequency-shifted feedback lasers that indicate that the residual zero-order diffraction may interfere with the frequency-shifted first-order beam at the output port^[13,14]. The AOM cannot therefore be considered simply as an ideal frequency shifter, especially in the field of precision interferometric measurement.

Here, we analyze the accuracy degradation caused by residual zero-order diffraction in photonics-assisted-mmWave heterodyne interferometric sensing and compare it with double AOM system configurations. Theory well demonstrates the effectiveness of using double AOMs to overcome the effect of zero-order perturbation, especially when the two AOMs are in parallel. In our experiment, a dither up to 0.15 ps with a single AOM is successfully eliminated through double AOMs, and the system performance is limited to the noise floor, which is $- 110 dBc / Hz$ , excelling among similar works.

2. Theory

2.1. AOM’s modeling

It is known that the acousto-optic effect involves multiple physical processes where the electrically driven acoustic wave in the medium can change the localized refractive index of the waveguide, thereby manipulating photons^[15]. When the light is incident at the Bragg angle, the zero-order diffracted light is transmitted along the original optical path, while the first-order diffracted light will produce a deflection angle and a Doppler frequency shift. For industrial and scientific applications, the AOM is usually designed to output only the first-order diffracted light, but in fact, the outgoing light may be mixed with other weak diffraction orders. Generally, we assume that the outgoing light is expressed as $E_{out} (t) = E_{in} (t) [\sqrt{η_{1}} \exp (- i w_{m} t) + \sqrt{ε_{0}}]$ , where $E_{in} (t)$ is the photoelectric field of incident light, $w_{m}$ is the angle frequency of AOM’s driving source, and $η_{1}$ and $ε_{0}$ denote the diffraction efficiency of the first order and residual zero order, respectively.

2.2. System configuration

Figure 1 illustrates the schematic diagram of photonics-assisted mmWave heterodyne interferometric sensing. The two continuous waves (CWs) operating at optical frequencies $f_{1} and f_{2}$ are injected into a Mach–Zehnder interferometer (MZI) with a sensing fiber placed on one of the interference arms. Heterodyne interrogation is implemented through the AOM with single, cascading, and parallel configurations, at heterodyne frequencies of $f_{m 1}$ , $f_{m 1} + f_{m 2}$ , and $f_{m 1} - f_{m 2}$ , respectively, where $f_{m 1}$ and $f_{m 2}$ are the electrical driving frequencies of the AOM, as shown in Figs. 1(a)–1(c). The polarization controller (PC) is used to match the polarization of light propagating through the two arms. The two optical spectral regions are separated at the output of the interferometer and then photoelectrically detected. As with traditional laser interferometers, the two electrical signals will carry the phase changes of $2 π f_{1} τ (t)$ and $2 π f_{2} τ (t)$ , respectively, where $τ (t)$ represents the delay measurand. Consequently, the phase difference of the two branches can be calculated as $2 π (f_{1} - f_{2}) τ (t)$ , which indicates that the delay measurand is scaled up by the difference of the two optical frequencies. When an electro-optical frequency comb is employed, the sensitivity can be adjusted by selecting two comb lines with different frequency intervals. The photonics-assisted mmWave can be expressed as $k f_{RF}$ , where $k$ indicates the number interval between two comb lines, and $f_{RF}$ is the frequency of the RF reference driving the electro-optic modulator. The frequency stability of the RF reference can easily reach 5 × 10^-14 at 1 s averaging time^[16], but correspondingly, the laser frequency drift controlled within 10 Hz is challenging. Successfully, the mmWave-based sensing can bypass the narrow linewidth requirement of the laser.

Figure 1.Schematic diagram of photonics-assisted mmWave heterodyne interferometric sensing in different AOM configurations. (a) Single AOM, (b) double AOM cascading, and (c) double AOM parallel. DAQ, data acquisition. The corresponding optical sidebands on each interference arm are also displayed.

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As shown in Fig. 1(a), the photonics-assisted-mmWave sent to the MZI is divided into two arms by an optical coupler, one of which, through the sensing fiber, can be represented by $E_{1 s} (t) \propto E_{in} (t) \exp [i w_{1, 2} τ (t)]$ . On the other arm, the photoelectric field through a single AOM is represented as $E_{1 r} (t) \propto E_{in} (t) [\sqrt{η_{1}} \exp (- i w_{m 1} t) + \sqrt{ε_{0}}]$ . The two arms are coupled, and then the detected signals around $w_{m 1}$ are extracted, which can be written as $V_{1 a, 1 b} = A_{1 a, 1 b} \cos (w_{1} t + ψ_{1 a, 1 b}) \propto 2 \sqrt{η_{1}} R_{PD} P_{in} \cdot {\cos [w_{m 1} t + w_{1, 2} τ (t)] + \sqrt{ε_{0}} \cos (w_{m 1} t)},$ (1a)where $w_{1} = w_{m 1}$ , $A_{1 a, 1 b} \propto 2 \sqrt{η_{1}} R_{PD} P_{in}$ when $ε_{0} ≪ 1$ , $R_{PD}$ is the responsivity of the PD, $P_{in}$ is the input optical power, and the phases are calculated as $ψ_{1 a, 1 b} = w_{1, 2} τ (t) + \tan^{- 1} {\frac{- \sqrt{ε_{0}} \sin [w_{1, 2} τ (t)]}{1 + \sqrt{ε_{0}} \cos [w_{1, 2} τ (t)]}} \approx w_{1, 2} τ (t) - \sqrt{ε_{0}} \sin [w_{1, 2} τ (t)] .$ (1b)

The approximation stands when $ε_{0} ≪ 1$ . As can be seen, in addition to the phase change introduced by the measurand weighted by the corresponding optical frequency, there is additional phase perturbation due to the same frequency crosstalk caused by residual zero-order diffraction. Thus, the detection delay obtained from phase difference and sensitivity mapping is calculated as $τ_{1} (t) = \frac{ψ_{1 a} - ψ_{1 b}}{w_{1} - w_{2}} = τ (t) - \frac{\sqrt{ε_{0}}}{w_{1} - w_{2}} {\sin [w_{1} τ (t)] - \sin [w_{2} τ (t)]} .$ (1c)

The inability to distinguish the phase perturbation from the valuable phase difference information causes the delay measurand to be disturbed, leading to a reduction in sensing accuracy.

In Fig. 1(b), a second AOM is added to form a cascading configuration. In this case, the output of the reference arm is represented as $E_{2 r} (t) \propto E_{in} (t) [\sqrt{η_{1}} \exp (- i w_{m 1} t) + \sqrt{ε_{0}}] [\sqrt{η_{1}} \exp (- i w_{m 2} t) + \sqrt{ε_{0}}]$ , where the diffraction efficiency is simplified to be the same. After being detected and filtered, the two electrical signals with the same frequency, equal to the sum of the AOM’s driving frequencies, can be expressed as $V_{2 a, 2 b} = A_{2 a, 2 b} \cos (w_{2} t + ψ_{2 a, 2 b}) \propto 2 η_{1} R_{PD} P_{in} \cdot {\cos [(w_{m 1} + w_{m 2}) t + w_{1, 2} τ (t)] + ε_{0} \cos [(w_{m 1} + w_{m 2}) t]},$ (2a)where $w_{2} = w_{m 1} + w_{m 2}$ , $A_{2 a, 2 b} \propto 2 η_{1} R_{PD} P_{in}$ when $ε_{0} ≪ 1$ , the phases are approximated as $ψ_{2 a, 2 b} \approx w_{1, 2} τ (t) - ε_{0} \sin [w_{1, 2} τ (t)],$ (2b)and the actual recovered delay becomes $τ_{2} (t) = τ (t) - \frac{ε_{0}}{w_{1} - w_{2}} {\sin [w_{1} τ (t)] - \sin [w_{2} τ (t)]} .$ (2c)

Comparing Eq. (2c) with Eq. (1c), we find that cascading AOMs can square the coefficient of zero-order perturbation from $\sqrt{ε_{0}}$ to $ε_{0}$ , thereby suppressing the disturbance.

For the double AOM parallel configuration shown in Fig. 1(c), the photoelectric field of the sensing arm is written as $E_{3 s} (t) \propto E_{in} (t) \exp [i w_{1, 2} τ (t)] [\sqrt{η_{1}} \exp (- i w_{m 2} t) + \sqrt{ε_{0}}]$ . Since both arms involve frequency shifts, the beat of residual zero-order diffracted light and any sideband will not constitute the same frequency crosstalk, so the two electrical signals are given by $V_{3 a, 3 b} = A_{3 a, 3 b} \cos (w_{3} t + ψ_{3 a, 3 b}) \propto 2 η_{1} R_{PD} P_{in} \cdot \cos [(w_{m 1} - w_{m 2}) t + w_{1, 2} τ (t)],$ (3a)where $w_{3} = w_{m 1} - w_{m 2}$ , $A_{3 a, 3 b} \propto 2 η_{1} R_{PD} P_{in}$ , and the phases contain only the amplified sensing information, as shown in the following equation: $ψ_{3 a, 3 b} = w_{1, 2} τ (t) .$ (3b)

Also, by phase difference and sensitivity mapping, we then recover the measurand without perturbation: $τ_{3} (t) = τ (t) .$ (3c)

As a result, the degradation of sensing accuracy due to residual zero-order diffraction can be mitigated by cascading AOMs, while the effect can even be eliminated by parallel AOMs.

Figure 2 plots the theoretical simulation of zero-order perturbation in different AOM configurations. The mmWave frequency is set to 100 GHz, and then the sensing sensitivity is $0.2 π$ rad/ps. In the simulation, $ε_{0}$ is assumed to be 5.6 × 10^-2, and a delay measurand with a linear change of 256 ps/s over time is preset. As shown, during the observation time of 0.1 s, the detection delay changes by 25.6 ps, but due to the common $π$ -phase constraint^[17], the actual detection range is 10 ps. For the single AOM configuration, we clearly find that the change is periodic and the largest zero-order perturbation contribution occurs at the half-period. The zoomed-in view shows that the residual zero-order diffraction results in a dither with a peak-to-peak value of as high as 1.5 ps. This is because a small delay jitter corresponds only to a slight phase change in the sensing mmWave domain, but will lead to a large variation in the optical phase. Fortunately, it can be effectively suppressed by cascading AOMs, and the peak-to-peak value is compressed to 0.3 ps. When the diffraction efficiency of the residual zero-order is smaller, the suppression effect will be more significant. As a contrast, the simulation result of the AOM parallel configuration is also presented, which is not disturbed by residual zero-order diffraction and is consistent with the preset delay measurand.

Figure 2.Theoretical simulation of zero-order perturbation in different AOM configurations.

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2.3. Noise floor discussion

Thanks to I/Q digital demodulation technology, we can obtain the phase of each branch and then calculate the phase difference, which can be expressed as $ψ_{a} - ψ_{b} = (w_{1} - w_{2}) τ (t) + ψ_{ε 0} + n_{ψ},$ (4)where $ψ_{ε 0}$ refers to the zero-order perturbation and $n_{ψ}$ denotes the noise floor. When $ψ_{ε 0}$ is larger than $n_{ψ}$ , we can observe the effect of residual zero-order diffraction on the sensing accuracy; otherwise, it will be masked by the higher noise floor. Following the noise model described in Ref. [18], we give the expression of the power spectral density (PSD) of the noise floor as $N_{ψ} (f) = 2 \frac{{| H (f) |}^{2}}{A^{2}} [N_{a} (f + f_{0}) + N_{a} (f - f_{0})],$ (5)where $H (f)$ is the frequency response in digital demodulation and $A$ is the signal amplitude. $N_{a} (f)$ represents the PSD of input additive noise for each branch, which is usually expressed as the sum of the thermal noise $N_{th} = 4 k T$ , the shot noise $N_{sh} = 2 e i_{D} Z$ , and the laser’s relative intensity noise (RIN) $N_{rin} = RIN i_{D}^{2} Z$ . The photocurrents in different AOM configurations are expressed as $i_{D 1} \propto R_{PD} P_{in} (1 + η_{1}), i_{D 2} \propto R_{PD} P_{in} (1 + η_{1}^{2}), i_{D 3} \propto R_{PD} P_{in} (2 η_{1}) .$ (6)

Note that the noise floor is determined not only by the dominant noise source type but also by the first-order diffraction efficiency. Figure 3 depicts the noise floor deterioration using double AOMs. When the first-order diffraction efficiency is greater than 0.8, there is little noise floor deterioration, despite the system being in different AOM configurations or limited to different noise source types. Only when the first-order diffraction efficiency is very small will an obvious difference be shown, but this is rarely seen in commercial AOMs. In addition, the phase noise of the AOM’s driving source contributes equally to each branch and will be perfectly offset when calculating the phase difference. We then conclude that it is reasonable to use double AOMs to address the accuracy degradation problem caused by residual zero-order diffraction without worsening the noise floor.

Figure 3.Noise floor deterioration using double AOMs.

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Finally, we turn to study the effect of the optical source on the sensing results. The additional phase noise introduced by photonics-assisted mmWave phase noise can be derived as $N_{Θ} (f) = 4 \sin^{2} (π f τ_{d}) {| Θ_{1} (f) - Θ_{2} (f) |}^{2} \approx {(2 π f τ_{d})}^{2} {| k Θ_{RF} (f) |}^{2},$ (7)where $f$ is the frequency offset and $τ_{d}$ is the delay imbalance between the two interference arms. $Θ_{1} (f), Θ_{2} (f)$ denote the phase noise of the optical frequencies $f_{1}, f_{2}$ , and $Θ_{RF} (f)$ denotes the phase noise of the RF reference. The approximation stands when $π f τ_{d} ≪ 1$ . This shows that in mmWave-based sensing, the additional phase noise depends on the relative phase relation between the two optical frequencies rather than the single frequency itself. Therefore, the strict balance of the arm delay and high stability of the RF reference are conducive to improving the sensing accuracy.

3. Experiment

Figure 4 shows the experimental setup of photonics-assisted mmWave heterodyne interferometric sensing in different AOM configurations. The light emitted from the CW laser is power-amplified by an erbium-doped fiber amplifier (EDFA) and then injected into a phase modulator (PM), where an OFC is generated, with two adjacent comb lines spaced at the frequency of the RF reference. The direct amplification of the CW before OFC generation is to prevent the possibly uneven gain during the optical amplification from causing the coherence between comb lines to become poor. At the MZI output, a waveshaper (WS) is utilized to filter out the two required optical spectral regions at separate ports, and then the respective optical/electrical (O/E) conversion and processing are performed. The electrical amplifier (EA) on the branch is used to enhance the electrical signal from the photodetector (PD), and the BPF1 removes the higher harmonics arising from the electrical amplification. To keep the frequency of the test signal injected into the DAQ (PCIe-9852) consistent, we down-convert the two electrical signals to $f_{0} = 24 MHz$ with the help of a tunable frequency source $f_{x}$ . The collected ADC signal is mixed with an in-phase component and an orthogonal component, respectively, so that the I/Q orthogonal terms are obtained by low-pass filtering. The phase can be extracted by the arctangent.

Figure 4.Experimental setup of photonics-assisted mmWave heterodyne interferometric sensing in different AOM configurations. (a) Single AOM, (b) double AOM cascading, and (c) double AOM parallel. EDFA, erbium-doped fiber amplifier; PC, polarization controller; PM, phase modulator; WS, waveshaper; PD, photodetector; EA, electrical amplifier; BPF, band-pass filter; LPF, low-pass filter.

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First, the noise floor measurement is carried out to assess the minimum detection delay. The RF frequency is 25 GHz, and the number interval is set to 4, so the mmWave frequency is 100 GHz. When the MZI does not load the measurand, we collect the two test signals simultaneously and perform digital demodulation to get the phases, and then subtract. The phase difference is used to calculate the single sideband (SSB) phase noise spectral density as the noise floor, and the result is shown in Fig. 5. In addition, the mmWave contribution can be plotted by the measured phase noise of the RF reference according to Eq. (7). The introduction increases at a slope of 20 dB/decade with an increasing frequency offset, resulting in an upward trend after the frequency offset of 1 kHz. However, thanks to the small delay difference between the two arms of the MZI, the additional phase noise is significantly below the noise floor and can be negligible. Regardless of the burrs that may be generated during digital processing, the noise floor is approximately $- 110 dBc / Hz$ over an analysis bandwidth of 1 MHz. After the average phase noise power is obtained by integrating, the minimum detection delay can be calculated by taking the square root and converting it by sensitivity, and the result is 3.6 fs. In Table 1, we compare this work with recent similar works. Although these sensors are targeted at different measurands, they are essentially converted into phase measurements, and the noise floor of this work is outstanding. Furthermore, selecting any two comb lines with a tunable frequency interval to form the photonics-assisted mmWave as the optical source can provide considerable flexibility in sensitivity.

Table 1. Comparison With Recent Similar Works

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Table 1. Comparison With Recent Similar Works

Measurand	Optical source	Sensitivity (depend on)	Noise floor (dBc/Hz)
Displacement^[19]	Two close frequencies from two laser sources	Central frequency	−87
Vibration^[20]	Two close frequencies from broadband light	Central frequency	−88
Strain rate^[21]	Broadband light	Central frequency	−105
Acoustic^[17]	Two narrow-linewidth tunable lasers	Frequency interval	−105
Delay (This work)	Photonics-assisted mmWave	Frequency interval	−110

Figure 5.The noise floor and mmWave contribution.

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In the next experiment, we demonstrate the effect and suppression of zero-order perturbation by simulating rapidly changing delay jitter. A motorized delay line (MDL) with a speed set to 256 ps/s acting as a delay measurand is loaded on the MZI. Relying on the devices available in the lab, we construct the following three AOM configurations for comparison: 1) the single AOM with a frequency downshift of 80 MHz; 2) the AOM with a frequency downshift of 80 MHz cascaded with the other AOM with a frequency upshift of 120 MHz; and 3) the AOM with a frequency upshift of 120 MHz paralleled with another AOM with a frequency upshift of 200 MHz, respectively. The driving power of the AOM is tuned close to the saturation point^[12], about 30 dBm.

As shown in Fig. 6(a), the detection delay changes with time, where the approximate rising slope is consistent with the setting, but the curve is not ideally linear. We believe this is due to the extra delay jitter caused by the external environment when the MDL is operating, such as mechanical vibration generated by the motor. There is a 5.12 ps delay change in the acquisition time, which spans over half a detection range. From the local zoomed-in display, we find that the detection delay has a peak-to-peak dither of 0.15 ps in the single AOM configuration, while the dither is not observed in the cascading or parallel configuration. According to the peak-to-peak value, the diffraction efficiency of the residual zero-order is calculated to be 5.6 × 10^-4. Theoretically, cascading AOMs can suppress the dither to 3 fs, which is lower than the current noise floor. Further reduction of the noise floor may be achieved by optimizing the power configuration in the link, and the MZI should be placed in a box isolating thermal and vibration. The normalized power spectrum of the detection results is shown in Fig. 6(b). The $\pm 0.15 MHz$ spurious band that occurs when using a single AOM is effectively suppressed, which also confirms the effectiveness of using double AOMs to improve the sensing accuracy.

Figure 6.Experimental results of zero-order perturbation in different AOM configurations. (a) Waveform in time domain. (b) Normalized power spectrum.

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4. Conclusion

In summary, we propose to use double AOMs to improve the measurement accuracy in photonics-assisted mmWave heterodyne interferometric sensing. In this system, the delay measurand loaded on the MZI is synchronously sensed by utilizing two comb lines from an electro-optical frequency comb and successfully recovered by phase difference and sensitivity mapping. It is noted that, in the interference process, the beat tone generated by the first-order and residual zero-order diffracted light of the AOM is the same frequency as the actual sensing signal, so the crosstalk will affect the detection results. Both theory and experiment show that the effect of zero-order perturbation can be effectively suppressed or even removed using double AOMs, which provides optimization possibilities for sensing applications where the noise floor has been strictly compressed.

Category: Microwave Photonics

Received: Dec. 17, 2024

Accepted: Feb. 11, 2025

Published Online: Jun. 4, 2025

The Author Email: Yitang Dai (ytdai@bupt.edu.cn)

DOI:10.3788/COL202523.073901

CSTR:32184.14.COL202523.073901