Characterization of high-speed electro-optic phase modulators based on heterodyne carrier mapping at a fixed low-frequency

Ying Xu; Shangjian Zhang; Xinhai Zou; Zhongtao Ruan; Yutong He; Heping Li; Yali Zhang; Zhiyao Zhang; Yong Liu; Ninghua Zhu; Ninghua Zhu

doi:10.1364/OE.477679

1. Introduction

Electro-optic phase modulators (EOPMs) feature the inherent characteristics of bias-free and linear modulation due to the single-waveguide structure, which supports the applications in optical fiber communication and microwave photonics [1,2], such as microwave signal generation, transmission, processing [3–5], photonic microwave up- or down-conversion [6,7] and optical frequency comb generation [8,9]. The key metrics for EOPMs mainly include modulation depth, half-wave voltage (V_π), and response bandwidth (BW). For example, the half-wave voltage refers to the voltage required to achieve a π-phase shift. It not only reflects the modulation efficiency of EOPMs at different frequencies but also represents the response bandwidth and the efficiency degradation within the interesting frequency range. Hence, the precise characterization of modulation depth and half-wave voltage is critical for device optimization and system application [10,11].

As the phase modulation inherently keeps a constant envelope of the optical carrier, the direct measurement can only be performed in the optical domain, where the modulation depth and the half-wave voltage of EOPMs are obtained by analyzing the amplitude ratio of the sideband and carrier of the modulated spectrum [12,13]. However, the measurement of this method is limited by the resolution of the commercially available optical spectrum analyzers (OSAs), resulting in difficult measurements in the low-frequency regime and rough measurements in the high-frequency range. For high-resolution measurement, the phase modulated signal is eventually converted into the intensity modulated signal through a Mach-Zehnder [14], Sagnac interferometer [15], optical discriminator [16] or filter [17,18]. Nevertheless, the phase modulation to intensity modulation (PM-to-IM) conversion should be operated at small-signal driving conditions in order to reduce the nonlinearity of PM-to-IM conversion. Moreover, the major disadvantage of these methods is that the assistant photodetector (PD) should be qualified for broadband photodetection and the roll-off response of the PD must be decoupled in the measurement link through extra calibration especially when the high-frequency operation is involved.

In order to relieve the bandwidth requirement, we proposed a low-speed photonic sampling scheme for measuring electro-optic Mach-Zehnder modulators, in which the optical modulated sidebands are down-converted to the duplicate components in the first Nyquist frequency range by a low-speed photodetector and measured via low-frequency detection [19]. Low-speed photonic sampling has been also implemented for measuring the frequency response of EOPMs with the help of PM-to-IM conversion in a Sagnac loop [20]. However, the phase modulator is assumed to feature phase modulation for the CW-propagating optical carrier while modulation-free for the anti-CW-propagating one, which is not always valid. Furthermore, this approach cannot measure frequency responses at frequencies equal to multiples of the repetition frequency of the optical pulse sampling source.

For self-calibration measurement, we also proposed heterodyne spectrum mapping (HSM) to obtain the modulation depth and the half-wave voltage of EOPMs without extra calibrating the PD’s roll-off response [21,22]. In this method, the optical carrier and its phase-modulated sidebands are mapped from the optical domain to the electrical domain through a heterodyne Mach-Zehnder interferometer, and we can equivalently observe the desired optical spectrum lines through closely spaced half-frequency components [23]. In this case, the uneven frequency response of the PD is eliminated by configuring the half-frequency relationship between the modulation frequency and the heterodyne frequency. Nevertheless, the HSM method is implemented by a half-frequency broadband photodetector and two microwave sources swept with a half-frequency relationship. Therefore, the methods, which are capable of characterizing EOPMs with fixed low-frequency detection and without the calibration of PD, and at the same time keeping the simplest single-tone microwave driving, are attractive and of great interest.

As is known to all, the phase modulation will generate a serial of upper and lower sidebands at both sides of the optical carrier. The optical carrier and its sidebands are equally spaced by the modulation frequency with the relative amplitude of Bessel function J₀(m) and J_k(m). The heterodyne spectrum mapping provides an equivalent electrical measurement of the optical carrier and its phase-modulated sidebands, where the heterodyning ratio in the electrical spectrum corresponds to the amplitude ratio of the first-order sideband with respect to the optical carrier in the optical spectrum.

In this paper, we propose a self-referenced scheme for characterizing EOPMs based on heterodyne carrier mapping (HCM). Inspired by the HSM method, this method also employs a heterodyne interferometer to perform the conversion of the phase modulated signal to heterodyne signal. Different from the previous one, our method only maps the optical carrier to a fixed low-frequency replica in the electrical domain by means of the heterodyne interferometer. The modulation depths and the half-wave voltages are determined by referencing the electrical replica of the optical carrier when the EOPM is under on and off single-tone modulation. Outperforming the HSM scheme, our method only requires a single-tone signal to drive the EOPM under test and completely eliminates the roll-off responsivity of the PD with fixed low-frequency detection. Theoretical description and experimental demonstration are presented to elaborate our method. The frequency-dependent modulation depths and half-wave voltages of a commercial phase modulator are experimentally measured, and the results are compared to those obtained using conventional methods to check the consistency and accuracy.

2. Operation principle

The schematic of the proposed method is shown in Fig. 1. A continuous light wave with a frequency of f₀ from a laser diode is injected into a heterodyne Mach-Zehnder interferometer (HMZI). In the upper arm of the HMZI, the optical carrier is sent into the EOPM under test and modulated by the microwave signal v(t) = V₁sin(2πf₁t+θ). The output optical field of EOPM is expressed as

(1)$${E_{u\_on}}(t )= {E_0}\exp [{j2\pi {f_0}t + jm\sin ({2\pi {f_1}t + \theta } )} ]\textrm{ = }{E_0}\sum\limits_{n ={-} \infty }^\infty {{J_n}(m )} \exp [{j2\pi ({{f_0} + n{f_1}} )t + jn\theta } ]$$

where J_n(·) is the nth Bessel function of the first kind, m is the modulation depth of the phase modulator and is defined by

(2)$$m = {{\pi {V_1}} / {{V_\pi }}},{V_1} = \sqrt {2P{Z_L}}$$

with the half-wave voltage V_π of the EOPM at the frequency of f₁, the peak amplitude V₁ and the average power P of the microwave signal and the characteristic impedance Z_L of 50 ohms. According to Eq. (1), the amplitude of the optical carrier of the EOPM is proportional to J₀(·). In the case of off-modulation, that is, when there is a null signal applied on the EOPM, Eq. (1) will be reduced to be

(3)$${E_{u\_off}}(t )= {E_0}\exp ({j2\pi {f_0}t} )$$

In the lower arm of the HMZI, the frequency of the optical carrier is shifted by f_s with an acousto-optic frequency shifter (AOFS) and is written as

(4)$${E_l}(t )= \gamma {E_0}\exp ({j2\pi {f_0}t + j2\pi {f_s}t + j\varphi } )$$

where γ represents the relative amplitude ratio and φ represents the phase difference between the two interferometer arms.

Fig. 1. Schematic scheme of the proposed heterodyne carrier mapping. LD: laser diode; PM: phase modulator; AOFS: acousto-optic frequency shifter; LSPD: low-speed photodetector; MNA: microwave network analyzer; OSA: optical spectrum analyzer.

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From the output of the HMZI, the phase-modulated optical signal and the frequency-shifted optical carrier are combined by an optical coupler and collected by a low-speed photodetector (PD) with the responsivity of R to convert to a photocurrent signal. With the help of Jacobi-Anger expansion, the generated photocurrent signal can be expressed as

(5)$$i(t )= R{|{{E_{u\_on}}(t )+ {E_l}(t )} |^2} = RE_0^2\left[ {1 + {\gamma^2} + 2\gamma \sum\limits_{n ={-} \infty }^{ + \infty } {{J_n}(m )\cos ({2\pi n{f_1}t - 2\pi {f_s}t + n\theta - \varphi } )} } \right]$$

From Eq. (5), it is easy to quantify the amplitude of the frequency component at f_s as

(6)$${A_{on}}({{f_s};m} )= 2\gamma RE_0^2{J_0}(m )$$

It can be seen that the optical carrier with the parameter of modulation depth m is mapped to an electrical replica at a fixed low-frequency f_s by frequency-shifted heterodyning. The modulation depth m and the half-wave voltage V_π can be extracted with the following equations

(7a)$$H(m )= \frac{{{A_{on}}({{f_s};m} )}}{{{A_{off}}({{f_s};0} )}} = {J_0}(m )$$

(7b)$$H({{V_\pi };{V_1}} )= \frac{{{A_{on}}({{f_s};{{\pi {V_1}} / {{V_\pi }}}} )}}{{{A_{off}}({{f_s};0} )}}\textrm{ = }{J_0}\left( {\pi \frac{{{V_1}}}{{{V_\pi }}}} \right)$$

where A_on and A_off represent the amplitudes of f_s that the EOPM is under on-modulation and off-modulation by switching on and off the microwave source.

From Eqs. (7a) and (7b), the modulation depth m and the half-wave voltage V_π can be extracted by referencing the amplitude of the fixed frequency electrical replica f_s of the optical carrier in the case that the EOPM is under on- and off- modulation, respectively. It is worth mentioning that the self-reference measurement could be operated with fixed low-frequency detection, which eliminates the roll-off responsivity of PD and reduces the bandwidth requirement of the PD to tens of MHz level or even lower. Besides, our measurement is insensitive to the power unbalance and phase difference of the interferometer, since the heterodyning ratio is independent of the relative amplitude γ and phase difference φ. It should be pointed out that the Mach-Zehnder interferometer is operated at heterodyning mode instead of interference mode and it does not introduce any nonlinear transfer function, which is totally different from the traditional interferometer method. Eventually, with the aid of the heterodyne carrier mapping, our method enables the extraction of the modulation depth and the half-wave voltage of wideband EOPMs under on-modulation and off-modulation at fixed low-frequency.

3. Experimental demonstration

An experiment is set up as shown in Fig. 1 to verify the proposed method of an EOPM based on heterodyne carrier mapping. A continuous light wave at the wavelength of 1550.12 nm is injected into a heterodyne Mach-Zehnder interferometer (HMZI) through an optical coupler. In the upper arm of the HMZI, the optical carrier is sent into the EOPM under test and modulated by the microwave signal f_m. The optical carrier in the lower arm is frequency shifted by f_s (80 MHz) with an acousto-optic frequency shifter. The phase-modulated optical signal and the frequency-shifted optical carrier are combined by another optical coupler and detected by a low-speed PD (Thorlabs DET01CFC). The HMZI is driven at the electrical port of the EOPM under test and collected at the output port of the PD by an MNA (Agilent E8361), which allows independently changing the swept frequency of the source (port 1) and remaining the fixed frequency of receiver (port 2).

Figure 2 shows the typical output optical spectra of the EOPM under test and the electrical spectra of the electrical replica f_s (80 MHz), where the EOPM is driven by a microwave signal with a frequency of 10 GHz and the driving level V₁ is set to be 0 V, 0.70 V, 1.00 V, 1.40 V and 2.00 V respectively. The corresponding electrical power of f_s is measured to be 26.78 µW, 24.66 µW, 22.75 µW, 19.11 µW and 12.89 µW respectively, from which the heterodyne ratio H(m) is determined to be 0.921, 0.850, 0.714 and 0.482. Therefore, the modulation depths can be calculated to be 0.403 (rad), 0.564 (rad), 0.803 (rad) and 1.154 (rad) based on Eq. (7a) under the driving settings. The power ratio of the phase-modulated optical carrier is also measured to be 0.928, 0.846, 0.710 and 0.479 at the same driving conditions and proves the feasibility of the heterodyne carrier mapping. Besides, the ratios of the first-order sideband with respect to the optical carrier are also extracted to be 0.207, 0.230, 0.445 and 0.706, which corresponds to modulation depths of 0.405 (rad), 0.574 (rad), 0.813 (rad) and 1.160 (rad) respectively. The result coincides with that of the proposed heterodyne carrier mapping. It is worth noting that the measurement is achieved by only detecting and analyzing a fixed-frequency component f_s at 80 MHz.

Fig. 2. Measured (a) optical spectra of the EOPM under test and (b) electrical spectra of electrical replica f_s (80 MHz) under on- and off- modulation at microwave frequency of 10 GHz with driving amplitudes V₁ of 0.70 V, 1.00 V, 1.40 V and 2.00 V respectively.

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The modulation depth of the EOPM at other frequencies can be also measured by changing the modulation frequency f₁ in the cases of on- and off-modulation. Figure 3 illustrates the measured amplitudes of the electrical replica f_s under the driving amplitudes of 0.70 V, 1.00 V, 1.40 V and 2.00 V, and the electrical amplitude A(f_s; 0) (the solid black line) of f_s that EOPM is under off-modulation is shown as a benchmark for normalization. As can be seen, the amplitude variation of f_s becomes stronger and easier to observe as the driving amplitude increases. The modulation depths under different driving amplitudes are calculated by solving Eq. (7a), as shown by the solid lines in Fig. 4, and the microwave signal driving amplitudes are obtained with the average power measured by a microwave power meter and displayed by the short dotted lines for reference. Meanwhile, the OSA-based method is executed for comparison, as exhibited by the open circles in Fig. 4. It is easy to find out that the results of the proposed method have the same trend at different frequencies as those of the OSA-based method.

Fig. 3. Measured electrical power of electrical replica f_s under different frequencies when the EOPM is under off-modulation and on-modulation with driving amplitudes V₁ of 0.70 V, 1.00 V, 1.40 V and 2.00 V respectively.

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Fig. 4. Meausred modulation depths of the EOPM with this scheme (the solid lines) and OSA-based method (the open circles) with driving amplitudes V₁ of 0.70 V, 1.00 V, 1.40 V and 2.00 V respectively.

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Before resolving the half-wave voltage, the error transfer is firstly investigated by the total differential of Eqs. (7) and (2), and the relative error transfer equation of modulation depth and half-wave voltage is derived and is written as

(8a)$$\frac{{\delta m}}{m} = F(m )\cdot \frac{{\delta H(m )}}{{H(m )}}$$

(8b)$$\frac{{\delta {V_\pi }}}{{{V_\pi }}} = \frac{{\delta {V_1}}}{{{V_1}}} - \frac{{\delta m}}{m}$$

where F is the error transfer factor and expressed as

(9)$$F = \frac{{{J_0}(m )}}{{m{J_1}(m )}}$$

Figure 5 represents the error transfer factor F as a function of the modulation depth m and the driving amplitude V₁. With the increase of the modulation depth from 0 to 2.404 (the corresponding V₁ is 0.765V_π), F shows a dramatic decline, which means that the measurement accuracy can be improved with a high power driving signal within this range. According to the technical specification of Agilent E8361, the accuracy is typically less than 0.58% when the electrical power ranges from 100 nW to 100 µW. The maximum measurement error of the driving amplitude is 0.27%. Therefore, the uncertainty of half-wave voltage can be calculated by the equation of 0.27%+F*2*0.58%. To achieve a measurement uncertainty of less than 5%, the error transfer factor F is desirable to be less than 4.078, and the reasonable minimum modulation depth is 0.679 (rad). From Fig. 4, when the driving amplitude is 2.00 V, the minimum modulation depth is 0.681 (rad) and the error transfer factor is 4.052, which corresponds to the maximum measurement uncertainty of half-wave voltage of 4.97%. Hence, the driving amplitude of 2.00 V is selected to calculate the half-wave voltages over the entire measurement frequency range, and the results with the error bar are plotted in Fig. 6.

Fig. 5. Error transfer factor F as a function of modulation depth and driving amplitude.

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Fig. 6. Half-wave voltage with error bar at the driving amplitude of 2.00 V.

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Fig. 7. Measured frequency-dependent half-wave voltage V_π and relative frequency response S₂₁ of the EOPM with our method and heterodyne spectrum mapping method.

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The calculated frequency-dependent half-wave voltage is compared with the result of the HSM method which shares the same setup as our method and is driven by the microwave amplitude of 2.00 V as shown by the blue line with open squares in Fig. 7. The excellent consistency between these two methods confirms that the proposed method achieves the self-referenced frequency response measurement of the electro-optic phase modulator. The relative frequency response can be obtained by the equation of -20log₁₀(V_π/V_π(1 GHz)), which is referenced to the frequency of 1 GHz and is compared with the data from the manufacturer. It should be pointed out that the absolute frequency response parameter V_π over the whole frequency range cannot be derived by the relative frequency response unless the half-wave voltage at a certain frequency is known. Compared with the conventional heterodyne spectrum mapping method, our method eliminates the use of multiple high-frequency microwave sources and enables wideband measurements of the modulation depth and half-wave voltage of EOPM with a low-speed PD.

4. Discussion and conclusion

This method employs a frequency-shifted heterodyne interferometer to convert the phase modulated signal to the heterodyne signal and transforms the electro-optic response measurement from the optical domain to the electrical domain, which improves tremendously the frequency resolution and is also feasible in the extremely low-frequency region. Compared with the conventional PM-to-IM conversion method, our method enables the measurement of the electro-optic frequency response parameters such as modulation depth and half-wave voltage without the need for extra calibration by standard broadband PD. In contrast to the previous heterodyne spectrum mapping method, it only maps the optical carrier to a fixed low-frequency electrical replica, replacing the half-frequency mapping of the first-order sidebands. Moreover, this scheme requires only a high-frequency microwave source to drive the frequency-shifted heterodyne interferometer and demands only a low-speed PD for photoelectric conversion.

In summary, the proposed method enables self-referenced frequency response parameter measurement of high-speed EOPMs based on heterodyne carrier mapping. In this method, the output optical carrier of the EOPM is mapped to a fixed-frequency electrical replica through a frequency-shifted heterodyne interferometer. The modulation depth and the half-wave voltage of the EOPM are extracted by monitoring the amplitude variation of the electrical replica under the case that the EOPM is on- and off- modulation. The bandwidth requirement of the PD is reduced to the level equal to the frequency of the detected signal and the uneven frequency response of the PD is fully eliminated. Additionally, the measurement range can be greater than 40 GHz, as long as the stimulus source supports the operation.

Funding

National Natural Science Foundation of China (61927821); National Key Research and Development Program of China (2019YFB22003500); Fundamental Research Funds for the Central Universities (ZYGX2019Z011).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Characterization of high-speed electro-optic phase modulators based on heterodyne carrier mapping at a fixed low-frequency

Abstract

1. Introduction

2. Operation principle

3. Experimental demonstration

4. Discussion and conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (7)

Equations (11)

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