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Abstract
We propose a novel method for the frequency response measurement of electro-optic phase modulators by time-frequency analyzing the instantaneous frequency of a sinusoidal frequency-modulated (SFM) signal. The SFM signal is generated by a frequency-shifting loop (FSL) in which the electro-optic phase modulator under test generates a dual-sideband frequency shift per roundtrip. And the bandwidth of the generated SFM signal is dozens of times of the input radio-frequency (RF), which makes it a precise method to measure the frequency response of electro-optic phase modulators at lower cutoff frequencies. Based on the previously proposed time-delayed interference model [H. Yang et al., J. Lightwave Technol. 39, 3112 (2021) [CrossRef] ], we establish the relationship between the “peak-to-peak” frequency of the SFM waveforms and the modulation index of the phase modulators. And a short-time Fourier transformation (STFT) is adopted to characterize the time-frequency properties of output waveforms. Experimental results are well supported by the simple theoretical model. Finally, the accuracy of the half-wave voltage measurement is discussed.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Electro-optic phase modulators (EOPMs) have attracted much attentions in the applications of microwave photonics [1–2], optical communication [3] and fiber-optic sensing [4–6] because of their tunability and bias-free. Frequency response of phase modulators is one of the fundamental parameters that characterize the bandwidth and efficiency of the electro-optical conversion. Several methods have been demonstrated for measuring the frequency response of EOPMs by analyzing either optical or electrical intensity spectrum. The simplest method is measuring the output optical spectrum of phase modulators with an optical spectrum analyzer [7,8]. However, the spectral resolution is on the order of $10\; MHz$, limited by the resolution of optical spectrum analyzer (OSA). In order to achieve a higher frequency resolution characterization, the optical spectrum should be transferred to the electrical domain and be measured with a vector network analyzer (VNA) or an electrical spectrum analyzer (ESA) [9–12]. The disadvantage of this method is that an extra calibration should be implemented to remove the frequency response of photodetectors [9]. Several self-calibrating methods via two-tone modulation or photonic down-conversion sampling are also proposed to measure the frequency response of phase modulators [10–12].
The frequency response of phase modulators is usually investigated and measured in high frequency range. While high-speed electro-optic phase modulators are widely used, the measurement of the low frequency response is also crucial in applications such as fiber-optic gyroscope [13–15] and electro-optic switch [16–18]. However, in the measurement of the low frequency response, such as in the kHz range, the above methods are more susceptible to the environmental factors, such as the temperature variation and the vibration. In this paper, we propose a novel method for measuring the half-wave voltage of the phase modulators in the low frequency range based on the time-frequency analysis method. Our method consists of a single-frequency laser and a fiber loop in which the EOPM under test induces a dual-sideband frequency shift per roundtrip. By slightly detuning the modulation frequency of the EOPM off an integer multiple of the fundamental loop frequency, a sinusoidal frequency-modulated pulse doublet was observed [19]. Furthermore, the influence of the half-wave voltage on the sinusoidal frequency-modulated pulse doublet should be theoretically and experimentally investigated.
This paper is organized as follows. In section 2, the relationship between the half-wave voltage, the RF driving voltage and the “peak-to-peak” frequency of the SFM pulse-doublets is theoretically investigated. The third section presents the experimental results obtained with standard components at $1.064\mu m$, focusing on the half-wave voltage measurement and the accuracy analysis. Finally, we discuss possible extensions of this work.
2. Method
2.1 Set-up principle
We consider a fiber loop as depicted in Fig. 1. A single-frequency fiber laser is connected to the input port of a $1 \times 2$ fiber coupler. One of the outputs is injected into a fiber loop, which contains an EOPM, an optical amplifier (OA) and a tunable bandpass filter (TBPF). The optical amplifier is used to compensate for the loop losses and enhance the number of relevant round-trips inside the loop. The EOPM generates a dual-sideband frequency shift per roundtrip. Besides, the tunable bandpass filter limits the output bandwidth and efficiently reduces parasitic loop oscillations. The round-trip time is $\tau = nL/c$, where n is the group index of the loop fiber, L is the loop length and c is the velocity of light in the vacuum. A 2 × 2 optical coupler enables to seed the loop and to extract a fraction of the circulating laser power. Then, the FSL output and a portion of the single-frequency laser output are combined to increase the intensity of RF-modulated pulses and limit the intra-comb beatings. The EOPM under test is driven by a synthesizer (SYN).
Fig. 1. Schematic of the proposed photonic architecture for the frequency response measurement of EOPM. SFL: single-frequency laser; OC: optical coupler; TBPF: tunable bandpass filter; OA: optical amplifier; EOPM: electro-optical phase modulator; SYN: synthesizer; PD: photodiode.
2.2 Model
In order to measure the frequency response of phase modulators, we need to investigate the relationship between the time-frequency characteristic of the output waveforms and the modulation index $\delta = \pi V/{V_\pi }$ (${V_\pi }\; is$ the half-wave voltage and V is the driving voltage). Here we focus on a small detuning case in which the modulation frequency is slightly detuning off an integer multiple of the fundamental loop frequency. Previous studies have investigated the output power of the phase-modulated FSL in the small detuning case [19], which is expressed as
It’s worthwhile to note that ${f_{tm}}$ is independent of the optical frequency of the seed laser. When the modulation frequency is much less than ${f_c}$, namely $q = 0$, Eq. (4) could be simplified to
And the “peak-to-peak” frequency ${f_{tm}}$ will be in the vicinity of the fundamental frequency ${f_c}$. In other words, a RF signal with the kHz range through FSL is transformed to the MHz range. And the modulation index $\delta $ is proportional to ${f_{tm}}$. Finally, the relationship between the “peak-to-peak” frequency ${f_{tm}}$ and the modulation index $\delta $ is established. This property will significantly contribute to precisely measuring the low frequency response of the EOPMs. Besides, using the equation $\delta = \pi V/{V_\pi }$, Eq. (5) could be expressed as
To theoretically demonstrate the relationship between $\delta $ and ${f_{tm}}$, we perform simulations based on Eq. (1) with $p = 0.97$, ${\omega _0}\tau = 0.4\pi $, ${f_c} = 10MHz$. Limited by the loop loss, the largest integer number of roundtrip N is about ∼ 300 [20,21]. To fully characterize the time-frequency characteristics of the generated signal, N is set to 500. Moreover, a short-time Fourier transformation is performed to obtain the instantaneous frequency. Figure 2 depicts the simulations results with $\delta $ increasing from $3\pi /4$ to $5\pi /4$ and ${f_m} = 40kHz\; ({q = 0} )$, showing that the “peak-to-peak” frequency ${f_{tm}}$ linearly increases with $\delta $. The horizontal axis of the upper graph in Fig. 2 is the same as the horizontal axis of the graph below. And when $\delta = \pi $, ${f_t}$ is equal to ${f_c}$. The simulations results agree well with the analysis.
Fig. 2. Temporal and short-time Fourier transformation simulations results with ${f_m} = 40kHz$: the influence of modulation index on the “peak-to-peak” frequency.
3. Frequency response measurement experiments
3.1 Experimental parameters
To demonstrate the ability of the proposed scheme to measure frequency response of electro-optical phase modulators, we experimentally investigate the time and frequency response of the phase-modulated FSL as depicted in Fig. 1. The seed is a single-frequency fiber laser that delivers $5\; mW$ at $1064\; nm$, with a linewidth of approximately $2\; kHz$. In order to avoid parasitic oscillations when the gain is raised, we take a bandwidth of 0.35 nm optical tunable bandpass filter inside the loop. The center frequency of the TBPF is fixed to match the frequency of the seed laser. To improve the stability, the loop is designed with all-polarization fiber components and the experiments are carried out in a relatively closed box to isolate the system from the vibrations. The loop round-trip frequency presents high stability that is measured to be ${f_c} = 14.605131MHz$ in the mode-locking regime. The operating frequency range of the EOPM is 5 kHz to 10 GHz. The half-wave voltage of the EOPM is highly dependent on the natural refractive index of the crystal and the electro-optic coefficient. And the temperature changes will simultaneously affect the refractive index and electro-coefficient of the crystal. In order to precisely measure the half-wave voltage of the electro-optic phase modulator, the temperature of the EOPM is controlled and adjusted to different values. The maximum RF voltage applied to the EOPM is 10 V with 50$\mathrm{\Omega }$ impedance. In the experiment, the pump current of the YDFA is adjusted to increase the gain coefficient in the loop while avoiding parasitic oscillation. The detection setup consists of a 200 MHz bandwidth photodiode and a 1 GHz bandwidth oscilloscope.
3.2 Half-wave voltage
To experimentally demonstrate the ability of precisely measuring the half-wave voltage of the EOPM, a thermistor and a thermo electric cooler (TEC) are attached to the surface of the EOPM and constitute a temperature control loop. Then we keep $T = 17\circ{C}$ and ${f_m} = 30\; kHz\; ({q = 0} )$; Figs. 3(a)–3(h) show that by increasing the RF voltage ${V_{RF}}$ from 4 V to 10 V, which corresponds to increasing the modulation index in the theoretical model, we find that (i) the output waveforms are the SFM pulse-doublets and (ii) the “peak-to-peak” frequency, namely the sum of two frequency peaks in one modulation period, increases from $7.05\; MHz\; ({3.89\; MHz + 3.16\; MHz} )$ to $17.39\; MHz({10.21MHz + 7.18\; MHz} )$.
Fig. 3. Temporal and the corresponding STFT experimental results with ${f_m} = 30\; kHz$: (a)-(b) ${V_{RF}} = 4\; V$; (c)–(d) ${V_{RF}} = 6\; V$; (e)–(f) ${V_{RF}} = 8\; V$; (g)–(h) ${V_{RF}} = 10\textrm{V}$.
In order to further demonstrate the ability to measure the frequency response of the EOPM, we measure the “peak-to-peak” frequency ${f_{tm}}$ and investigate ${f_{tm}}$ with respect to the RF voltages ${V_{RF}}$ at different modulation frequencies and temperatures. The scattered points in Fig. 4 are the measurement values and the solid lines are the fitting of the scattered points. Figure 4 (a) reports the experimental results with $T = 17\circ{C}$. The relationship between the “peak-to-peak” frequency ${f_{tm}}$ and the RF voltage ${V_{RF}}$ is shown to be linear in Fig. 4(a), in good agreement with Eq. (6). As expected, all solid lines are nearly passing through point (0,0). By comparing different traces, we find the slope $k = {f_c}/{V_\pi }$ could be used to denote the half-wave voltage ${V_\pi }$. Specifically, when ${f_{tm}}$ on the vertical coordinate is equal to ${f_c}$, the corresponding value on the horizontal axis is the half-wave voltage of the specific frequency ${f_m}$. As shown in Fig. 4(a), the half-wave voltage reduces from more than 11 V to less than 8 V as the modulation frequency increase from 5 kHz to 100 kHz. Besides, temperature is another important factor influencing the half-wave voltage of the EOPM. We first keep ${f_m} = 40\; kHz$; Fig. 4 (b) shows that by increasing temperature from $17\circ{C}$ to $23\circ{C}$, we find that (i) ${f_{tm}}$ is still proportional to ${V_{RF}}$ and (ii) the slope become smooth. The first feature (i) further demonstrates the feasibility of the proposed scheme. The second feature (ii) indicates that the half-wave voltage increases as the temperature increases.
Fig. 4. The “peak-to-peak” frequency ${f_{tm}}$ with respect to the RF voltages ${V_{RF}} $ (a) different modulation frequencies; (b) different temperatures
Following the preceding conclusions, we further reduce the modulation frequency of the EOPM and investigate its frequency response. The temperature in the experiment is set to $17\circ{C}$ and RF voltage keeps 10 V. Figures 5(a)–5(d) report the experimental results for ${f_m} = 2\; kHz$ and ${f_m} = 1\; kHz$, respectively. At variance with the experimental results in Fig. 3, in which the instantaneous frequency varies sinusoidally with time, the curves in Fig. 5(b) and (d) present more complex waveforms. In analyzing the waveforms, we recall that it is assume that the modulation function of the EOPM is $\Upsilon (t )= {e^{j\delta sin({2\pi {f_m}t} )}}$. Theoretically, if the modulation function changes, the instantaneous frequency of the frequency-modulated waveform will also change. For example, if the modulation function changes to ${e^{j\omega t}}$ (frequency modulation), we will observe the linear frequency modulated waveforms or the chirp waveforms [22]. Besides, the operating frequency of the EOPM is 5 kHz to 10 GHz, which indicates the modulation function can’t be simply modeled by the traditional format at frequency below 5 kHz. Thus, the difference occurs because the modulation function at such a low frequency has changed.
Fig. 5. The temporal and the corresponding STFT waveforms with (a)-(b) ${f_m} = 2\; kHz$ and (c)-(d) ${f_m} = 1\; kHz$
3.3 Accuracy analysis
For completeness, we investigate the error dependence of the measured half-wave voltage on the uncertainty of the RF power ${V_{RF}}$, the fundamental frequency ${f_c}$ and the measured “peak-to-peak” frequency ${f_{tm}}$ by the error transfer function as the following:
The RF power is generated by a synthesizer and the relative error of RF power $d{V_{RF}}/{V_{RF}}$ could be precisely measured. The fundamental frequency of the loop ${f_c}$ is determined by the loop length and is experimentally measured in the mode-lock regime. To reduce the jitter of the fundamental frequency, the loop is designed with all-polarization fiber components and the experiments are carried out in a relatively closed box to isolate the system from the vibrations. Thus, the frequency jitter can be completely constrained to an extremely low range. The measured “peak-to-peak” frequency ${f_{tm}}$ is the primary factor that determines the error of the half-wave voltage.
As discussed previously, the “peak-to-peak” frequency ${f_{tm}}$ is measured by taking STFT. In practice, the STFT is computed as a succession of FFTs of windowed data frames, where the window slides forward through time. Therefore, there is a contradiction between the time resolution and the frequency resolution. The narrower the temporary window function is, the higher the time resolution is; but meanwhile the passband of the frequency become wide, which leads to a reduction in the frequency resolution. In order to characterize the frequency accuracy, we simulate the SFM waveforms $S(t )= sin({ksin({2\pi {f_m}t} )} )$ with $k = 200$ and ${f_m} = 10kHz$, respectively. The desirable peak-frequency is $2MHz$. Table 1 is the simulation results with different temporary window lengths. The relative frequency is calculated for the minimum value of 0.05%, ranging from $2\mu s$ to $10\mu s$. Theoretically as long as we take an appropriate time window, the relative measurement accuracy can be kept in a low range, regardless of the modulation frequency.
Table 1. The simulation peak-frequency and relative error of STFT
4. Conclusion
We have proposed and experimentally demonstrated a fiber loop seeded by a single-frequency laser for measuring the frequency response of the EOPM in the low frequency range. This experiment shows an alternative approach to other optical and electrical intensity spectrum analysis methods. In contrast to the intensity measurement of the modulation sidebands, our scheme transfers the intensity observations to the “peak-to-peak” frequency of the SFM pulse doublets, possessing the advantage of fast, high-accuracy and low frequency response measurement. And the bandwidth of the generated SFM signal is dozens of times of the input RF, which makes it a precise method to measure the frequency response of the EOPMs at lower cutoff frequencies. And if a high-frequency signal applied to the EOPM, such as GHz, a SFM waveform with a bandwidth of THz will be generated [19,22]. By employing a faster photodetector, the proposed method could also measure the high frequency response. Besides limited by the STFT algorithm, the measurement accuracy is about 0.05%. By adopting the state-of-art time and frequency analysis method, such as wavelet transform or Hilbert-Huang transform, the measurement accuracy can be further increased.
Funding
National Natural Science Foundation of China (62105361); Foundation for Distinguished Young Scholars of China Academy of Space Technology.
Acknowledgments
The authors thank Dr. Haiyang Zhang for assistance on the experiment.
Disclosures
The authors declare no conflicts of interest.
Data availability
All the data and calculations that support the findings of this study are available from the corresponding author upon reasonable request.
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