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Analysis of optical arbitrary waveform generation based on time-domain synthesis in a dual-parallel Mach–Zehnder modulator
Optik - International Journal for Light and Electron Optics 176 (2019) 549–558
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Original research article
Analysis of optical arbitrary waveform generation based on timedomain synthesis in a dual-parallel Mach–Zehnder modulator
T
⁎
Jin Yuan, Tigang Ning, Jing Li, Li Pei , Jingjing Zheng, Yueqin Li Key Lab. of All Optical Network and Advanced Telecommunication Network of EMC, Beijing Jiaotong University, Beijing, 100044, China
A R T IC LE I N F O
ABS TRA CT
Keywords: Microwave photonics Arbitrary waveform Time-domain synthesis Dual-parallel Mach–Zehnder modulator
A photonic-assisted microwave arbitrary waveform generator is theoretically analyzed and verified by simulation. The basic principle of the generator is synthesis of optical field envelops. By using a sinusoidal signal to modulate a CW via a dual-parallel Mach–Zehnder (DP-MZM), squareshaped waveforms are directly generated firstly, whose envelops or harmonic power ratios are able to be modified by changing the RF driving voltage. When two identical square-shaped pulses with harmonics power ratio (9:1) between the 1st- and the 3rd-order components suffer a differential envelope phase shift (π/2), the superposition of these envelops contributes to a triangular-shaped waveform. Similarly, we generate flat-top waveform and Gaussian waveforms by properly setting the bias drifts of DP-MZM, time delay of the tunable time delay line, and modulation index. As DP-MZM is a key component in our proposal, we discuss the influence of the bias drifts on the generated waveforms, which make the scheme more practical.
1. Introduction Recently versatile microwave waveforms, such as sinusoidal, triangular, square, trapezoid, and sawtooth waveforms have attracted much attention because of their applications in optical communication links [1], all-optical microwave signal processing and manipulation [2,3], wire and wireless communications [4,5]. Conventionally, electric frequency synthesizers or digital-analog converters are used to generate arbitrary waveforms. However, waveform generator based on electronic techniques is bandwidth limited and cannot meet high-frequency and large-bandwidth requirements of future electric systems [4]. In order to break these limitations, photonic approaches are highly expected for generating arbitrary waveforms [6–16]. One basic photonic generation of microwave waveform is Fourier synthesis [6]. An optical comb acts as a coherent broadband source to generate the tailored optical signal by using an optical spectral shaper and the desired waveform can be obtained by detecting the tailored optical signal in a photodetector (PD). Furthermore, microwave waveform can also be generated based on optical spectral shaping incorporating frequency-to-time mapping (FTTM) [7,8]. The desired spectral envelope can be obtained by using a dispersive element, which can be mapped to the temporal waveform via FTTM. However, the flexibility of this method is restricted and the mode-locked laser (MLL) which used as the optical source is high-cost. Besides the approaches mentioned above, microwave waveform can also be generated using external modulation of a continuous wave (CW) optical signal [9,7–16]. Due to the nonlinearity of the external modulator, a series of optical sidebands are generated. By controlling the phase and amplitudes of the optical sidebands, the desired microwave waveform can be obtained. For example, approaches based on stimulated Brillouin scattering (SBS) in optical fiber are proposed [9,10]. However, the SBS effect in sensitive to the environment, which may make the system unstable. Li et al. reported proposals to
⁎
Corresponding author. E-mail address: [email protected] (L. Pei).
https://doi.org/10.1016/j.ijleo.2018.09.120 Received 19 July 2018; Received in revised form 16 September 2018; Accepted 20 September 2018 0030-4026/ © 2018 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 176 (2019) 549–558
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generate triangular-shaped waveform by using a Mach–Zehnder modulator (MZM) in conjunction with an optical interleaver [11] or a dispersive element [12]. It is noted that the schemes mentioned above can only generate triangular-shaped waveform and are limited to generate other waveforms. Recently, approaches reported are extended to generate other types of microwave waveform. For example, arbitrary waveform generation using a polarization modulator (PolM) in a Sagnac loop is reported, in which the primary problem is the stability of the system [13]. Dai et al. carried out a versatile waveform generator based on frequency comb generation [14,15]. A photonic approach of microwave waveform generator based on time-domain synthesis is also proposed [16]. Despite of two Mach–Zehnder modulators which decrease the system integration, other optical devices are needed, which increase the complexity of the system. In this paper, we propose photonic approaches to generate optical arbitrary waveforms based on optical waveform synthesis in time-domain processing. In our scheme, a DP-MZM formed by two sub-MZMs (MZ-a and MZ-b) lying on each arm of a parent MZM (MZ-c) acts as the key component in our scheme. A tunable time delay line (TDL) is applied between the driving signals of two subMZMs. Firstly, setting three bias points of DP-MZM at quadrature points and properly tuning the modulation index β of DP-MZM, two square-shaped waveforms can be generated at the output of MZ-a and MZ-b. When the time delay τ of the TDL is 0, square-shaped waveform is obtained at the output of DP-MZM. When τ is tuned to 1/(4fRF), waveform with triangular-shaped envelopes can be generated by superimposition of two square-shaped waveforms. According to this principle, by properly controlling the bias points of DP-MZM, β, and τ, flat-top and Gaussian waveforms are generated. We also analyzed the impact of the extinction ratio and bias voltage drifts of the DP-MZM on the characteristics of the generated triangular-shaped waveform, which makes the scheme more practical. Moreover, the key significance of our approach is that we use a compact structure of DP-MZM which is expected to be more stable. 2. Model and theory Fig. 1 shows the schematic diagram of the proposed arbitrary waveform generator. In the setup, a CW light is applied to a DPMZM which is formed by two sub-MZMs (MZ-a and MZ-b) and a parent MZM (MZ-c). Because of x-cut design, MZ-a and MZ-b are configured for push–pull operation and the two sub-MZMs and the parent MZM have independent DC biases. A radio frequency (RF) signal VRF(t) with a frequency of ωRF and amplitude of VRF is divided into two paths by a RF splitter to drive MZ-a and MZ-b. Vbias1, Vbias2, and Vbias3 denote the three bias voltages applied to MZ-a, MZ-b and MZ-c. A TDL is used to set the phase between driving signals of MZ-a and MZ-b. In the system, by properly setting the bias points of the two sub-MZMs and the parent MZM, controlling the powers of the microwave driving signals and tuning the time-delay τ of the TDL, different waveforms can be obtained by overlapping the optical field envelopes at point A and point B. In the following, we will perform the generation of arbitrary waveforms. 2.1. Square-shaped waveform generation In this case, three bias points of DP-MZM are all biased at quadrature point (Vbias1, 2, 3 = Vπ/2, where Vπ is the half-wave switching voltage of DP-MZM). The optical field at the output of MZ-a and MZ-b can be expressed as [17]:
EA (t ) =
Ein (t ) π V cos ⎡ (V (t ) + Vbias1) ⎤ exp ⎛j bias1 π ⎞ ⎥ ⎢ 2Vπ RF 2 ⎦ ⎝ 2Vπ ⎠ ⎣
EB (t ) =
Ein (t ) π V cos ⎡ (VRF (t + τ ) + Vbias2) ⎤ exp ⎛j bias2 π ⎞ ⎥ ⎢ 2 ⎦ ⎝ 2Vπ ⎠ ⎣ 2Vπ
⎜
⎟
⎜
(1)
⎟
(2)
Ein(t) = E0exp(jω0t) is the optical field at the input of the DP-MZM. E0 and ω0 are the amplitude and angular frequency respectively. The optical field at the output of the DP-MZM can be written as
Fig. 1. Schematic setup of the proposed arbitrary waveform generator; (CW, continuous-wave laser; DP-MZM, dual-parallel Mach–Zehnder modulator; TDL, tunable time delay line; PD, photodetector; ESA, electrical spectrum analyzer). 550
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V EC (t ) = EA (t ) + EB (t )exp ⎛j bias3 π ⎞ ⎝ Vπ ⎠ ⎜
⎟
(3)
When the optical signal at the output of the DP-MZM is sent to a photodetector (PD) for intensity detection, we acquire an electrical signal at the output of the PD as below
⎧ cos2 ⎡ π (VRF (t ) + Vbias1) ⎤ + cos2 ⎡ π (VRF (t −τ ) + Vbias2) ⎤+⎫ ⎦ ⎪ ⎣ 2Vπ ⎦ ⎣ 2Vπ ⎪ ⎪ R |Ein |2 ⎪ π π ∗ Iout (t ) = R⋅EC (t )⋅EC (t ) = 2 cos ⎡ 2V (VRF (t ) + Vbias1) ⎤ × cos ⎡ 2V (VRF (t −τ ) + Vbias2) ⎤ ⎦ ⎬ ⎣ π ⎦ ⎣ π ⎨ 2 ⎪ ⎪ Vbias2 − Vbias1 + 2Vbias3 π ⎪ ⎪ × cos 2Vπ ⎭ ⎩
(
)
(4)
When τ is set to τ = 0, Eq. (4) can be simplified as π
Iout (t ) =
π
⎧ 2 cos2 ⎡ (VRF (t ) + Vbias1) ⎤ + 2 cos ⎡ (VRF (t ) + Vbias1) ⎤⎫ R |Ein |2 ⎪ ⎦⎪ ⎣ 2Vπ ⎦ ⎣ 2Vπ ⎬ ⎨ × cos π (V (t ) + V ) × cos Vbias2 − Vbias1 + 2Vbias3 π 2 ⎡ 2Vπ RF bias 2 ⎤ ⎪ ⎪ 2Vπ ⎦ ⎣ ⎭ ⎩
(
)
(5)
Assuming that parameter β = πVRF/Vπ is defined as the modulation index. Substituting Vbias1 = Vbias2 = Vbias3 = Vπ/2 into Eq. (5), the optical intensity can be further simplified to
Iout (t ) =
R | Ein |2 2
{cos ⎡⎣
πVRF (t ) Vπ
π
}
+ 2⎤ + 1 ⎦
∝ IDC + sin [β sin (ωRF t )] ∝ IDC + J1 (β )sin (ωRF t ) + J3 (β )sin (3ωRF t ) + J5 (β )sin (5ωRF t ) + …
(6)
J2k-1(·) is the Bessel function of the first kind of order 2k-1. The Fourier series expansion of a square-shaped waveform Tsq(t) is given by ∞
Tsq (t ) = DC +
∑ n = 1,3,5...
1 1 1 sin (nωm t ) = DC + sin (ωm t ) + sin (3ωm t ) + sin (5ωm t ) + ... n 3 5
(7)
Comparing Eqs. (6) and (7), we can conclude that in order to obtain a square-shaped waveform, J3(β)/J1(β) = 1/3 should be satisfied. This indicates that the electrical power difference between the first-order and third-order harmonic should be roughly 9.5 dB. Fig. 2(a) shows the relationship between J2k-1(·) and modulation index, β. It is obvious that when β is within a range from 1 to 3, J5(β) is far smaller than J1(β) and J3(β). This proves that the impact of harmonics higher than 5th-order is negligibly small and only the 1storder 3rd-order sidebands are considered to form the desired waveform. Fig. 2(b) shows the power ratio of J1(β) and J3(β). It can be seen that when β is tuned to 2.3, the power ratio is 9.5 dB, which satisfies with the property of square-shaped waveform. In this case, the optical intensity of the generated square-shaped waveform can be expressed as
Iout , sq (t ) ∝ IDC + sin (ωRF t ) +
J3 (β ) 1 sin (3ωRF t ) = IDC + sin (ωRF t ) + sin (3ωRF t ) 3 J1 (β )
(8)
2.2. Triangular waveform generation It has been reported that triangular waveform can be generated by synthesizing two properly modulated square-like envelopes
Fig. 2. Relationship between (a) Bessel function J2k-1(·) and (b) J3(β)/J1(β) versus modulation index, β. 551
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Fig. 3. Triangular-shaped waveform generation by the superimposition of two square-shaped waveforms with π/2 phase shift.
[15]. As shown in Fig. 3, when two square-shaped waveforms suffer an envelope phase shift of π/2, the superposition of these signals contributes to a triangular waveform in a PD. In our scheme, when τ is set to τ = 1/(4fRF), the two identical square-shaped waveforms at the output of MZ-a and MZ-b suffer an envelope phase shift of π/2. According to Eq. (8), after synthesizing of two square-shaped envelopes, the output current of PD can be written as
Iout , tr (t ) ∝ IDC + sin (ωRF t ) +
(
2 2
sin (3ωRF t ) + sin [ωRF (t + τ )] +
) + cos (ω t ) ⎤⎦ + cos (3ω ⎡cos (ω t + ) + ⎣
= IDC + ⎡cos ωRF t + ⎣ = IDC +
J3 (β ) J1 (β )
RF
3π 2
RF
π 4
J3 (β ) J1 (β )
J3 (β ) ⎡cos J1 (β ) ⎣
RF t +
3π 4
)
(3ω
RF t
+
3π 2
J3 (β ) J1 (β )
sin [3ωRF (t + τ )]
) − cos (3ω
RF t ) ⎤
⎦
⎤ ⎦
(9)
The Fourier series expansion of a triangular-shaped waveform Ttr(t) is given by ∞
Ttr (t ) = DC +
∑ n = 1,3,5...
1 1 1 cos (nωm t ) = DC + cos (ωm t ) + cos (3ωm t ) + cos (5ωm t ) + ... n2 9 25
(10)
According to Eqs. (9) and (10), in order to achieve triangular-shaped pulses, the amplitude ratio between the +1-order sideband and +3-order sideband should be 1/9. In spectrum, the relationship is displayed by power ratio. As J2 3(β)/J2 1(β) = 1/81, 10lg[J2 3(β)/J2 1(β)] is equal to19. So the power difference between the +1-order sideband and +3-order sideband is around 19.08 dB. As shown in Fig. 2(b), when β is tuned to 1.51, the power ratio between the first-order and third-order sidebands is 19.08 dB, which satisfies the property of triangular-shaped waveform. 3. Simulation and discussion In order to verify the mechanism of the proposed scheme, simulations are carried out via OptiSystem 10.0. As shown in Fig. 1, a light wave from a CW laser with central wavelength of 1550 nm and line-width of 0.8 MHz is sent to the DP-MZM. The DP-MZM has insertion loss of 5.5 dB, half-wave switching voltage of 4 V and extinction ratio of 30 dB. The optical spectrum and waveform of Iout (t) are captured by an OSA and an oscilloscope. An electrical spectrum analyzer is used to measure the electrical spectrum. 3.1. Square-shaped waveform generation When properly controlling the magnitude of the electrical signal, β can be tuned to 2.3. The driving frequency fRF is set to be 10 GHz. By tuning time delay of the TDL to τ = 0, simulation results of square-shaped waveform are shown in Fig. 4(a–b). The simulated electrical spectrum of the generated waveform is shown in Fig. 4(a). As can be seen, the third-order harmonic at 30 GHz is 9.6 dB (close to the theoretical value of 9.5 dB) lower than the fundamental tone at 10 GHz. The second-order harmonic is well suppressed. Also note that the fifth-order harmonic is around 30 dB lower than the first –order tone. This means that fifth-order and higher order harmonics can be neglected. Then, a half-duty-cycle square waveform with a period of 100ps is observed in the oscilloscope as shown in Fig. 4(b). The normalized ideal square waveform is also drawn in Fig. 4(b) with a red line for comparison. It is obvious that the simulated square waveform fits well with the idea one. In order to show the frequency tunability of the proposed waveform generator, the frequency of the driving signal applied to the DPMZM is changed to 20 GHz. As shown in Fig. 4(c–d), a square waveform with a period of 50 ps is obtained. When β is tuned within a range from 1.5 to 3, square-like envelopes can also be obtained. Fig. 5(a)–(f) illustrate the simulated temporal waveforms and the corresponding electrical spectrum when β is tuned to 1.51, 2.3 and 2.5. Fig. 5(a) and (d) show the generated square-shaped waveform and its corresponding electrical spectrum when β = 1.51. The power ratio between the first-order and third-order harmonics is 19.1 dB, which presents a good approximation with the Fourier series expansion of a triangular-shaped waveform, but the phase relationship determines a square-shaped waveform with rising and falling time of 45 ps. When the modulation index increases, i.e., 2.3, the rising and falling time are 24 ps. A square-shaped waveform with flat top and bottom is obtained 552
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Fig. 4. Simulated electrical spectra and waveforms of the generated square-shaped waveforms with repetition rates of (a, b)10 GHz; and (c, d) 20 GHz. (For interpretation of the references to colour in the text, the reader is referred to the web version of this article.)
Fig. 5. Simulated square-shaped waveforms and the corresponding electrical spectra: (a, d) β = 1.51; (b, e) β = 2.3 and (c, f) β = 2.5.
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Fig. 6. Simulated waveforms at the output of (a) MZ-a; (b) MZ-b; (c) MZ-c; and (d) corresponding electrical spectrum of triangular-shaped waveform.
and the power ratio between the first two harmonics in closed to the Fourier series expansion of an ideal square waveform (9.5 dB). In these cases, all the waveforms have duty cycles of 50%. 3.2. Triangular-shaped waveform generation Through tuning τ to 25 ps, modulation index β to 1.51, triangular-shaped waveform can be generated. The frequency of driving signal is set to 10 GHz. Fig. 6(a) and (b) show the simulated temporal waveforms at the output of MZ-a and MZ-b with a time delay of τ = 25 ps. These two signals superpose in a PD and triangular-shaped waveform can be obtained in Fig. 6(c). Fig. 6(d) shows that the power ratio between the 10 GHz and 30 GHz components is 19.08 dB roughly, which goes well with the property of the triangularshaped waveform. To investigate the tunability of the scheme, we simulate the generation of triangular-shaped waveform at different repetition rate. By adjusting the driving frequency of the DP-MZM, the following three cases are simulated: (a) fRF = 2.5 GHz; (b) fRF = 5 GHz; (c) fRF = 7.5 GHz. Pictures in Fig. 7 plot the triangular-shaped temporal waveforms of Iout(t) at with different period of 400 ps, 200 ps, and 133.33 ps according to the above cases. In this way, we can implement the repetition rate’s continuous tenability of the repetition rate simply by changing the driving frequency of the local oscillator and the time delay of the TDL. 3.3. Arbitrary waveform generation Next, we will show that the proposed waveform generator is also capable of generating other waveforms, e.g. flap-top waveform and Gaussian waveform. Again, a 10 GHz sinusoidal RF signal was applied to the DPMZM. When MZ-a and MZ-b are biased at MATP, MITP or MITP, MATP, τ and β are adjusted to 0 and 1.6, a flat top waveform with a duty ratio of 50% is obtained. Fig. 8(a)–(c) show simulation results for the flat top waveform. This flat top waveform with a duty ratio of 50% will be used for return-to-zero signals in optical fiber communication, especially differential phase-shift-keyed transmission [18]. Similarly, when τ and β are adjusted to 25 ps and 2.35 respectively, Gaussian waveform with a repetition rate of 10 GHz is generated as shown in Fig. 9(a). When β is tuned to 3.14, half-wave cosine waveform is generated as shown in Fig. 9(c). The corresponding electrical spectra of the generated waveforms are shown in Fig. 9(b) and (d).
Fig. 7. Simulated temporal waveforms with repetition rates of: (a) 2.5 GHz, (b) 5 GHz, (c) 7.5 GHz. 554
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Fig. 8. (a) Simulated flat top waveform with a duty cycle of 50%; (b) corresponding optical spectrum; and (c) corresponding electrical spectrum.
Fig. 9. Simulated Gaussian and half-wave cosine waveforms with a repetition rate of 10 GHz (a, c) and corresponding electrical spectra (b, d).
In the simulations below, MZ-a and MZ-b are biased at MATP, MATP or MITP, MITP, fRF is tuned at 10 GHz and τ is tuned to 0. When the two sub-MZMs are biased at MATP and MATP and β is set to 3.14, a Gaussian waveform with a repetition rate of 20 GHz is generated as shown in Fig. 10(a). When the two sub-MZMs are biased at MITP and MITP and β is set to 3.14, a flat-top waveform, with a repetition rate of 20 GHz is obtained as shown in Fig. 10(c). Fig. 10(b) and (d) show the corresponding spectra of the generated waveforms. Different from the previous results, the generated waveforms in this case is frequency-doubled. According to the previous discussion, by properly controlling bias points of the DP-MZM, time-delay τ of the TDL, and the 555
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Fig. 10. Simulated Gaussian and flat-top waveforms with a repetition rate of 20 GHz (a, c) and corresponding electrical spectra (b, d). Table 1 Satisfying conditions and corresponding waveforms. Bias modes
π/2, π/2, π/2 0, π, π/2; or π, 0, π/2
0, 0, π/2; or π, π, π/2
Conditions Time delay τ (ps)
Modulation index, β
Generated waveforms
0 25 0 25
2.3 1.51 1.6 2.35 3.14 3.14
Square-shaped Waveform Triangular-shaped Waveform Flat top Waveform Gaussian Waveform Half-wave Cosine Waveform Flat-top Waveform; Gaussian Waveform (frequency doubled)
0
modulation index β, square, triangular, Gaussian and flat top waveforms can be generated. Table 1 below makes a summary about these cases.
3.4. Bias point drifts of DP-MZM As DP-MZM is a key component in our approach, it is necessary to discuss the bias drifts of the DP-MZM. In order to verify the influence of this parameter on the output waveforms, we take triangular-shaped waveform for example. In the previous generation of triangular-shaped, DP-MZM is biased at three independent quadrature bias points. However, the bias drifts will degrade the stability of the generated triangular-shaped waveform to some extent. As the second harmonic is the main distortion harmonic, we focus on P1Ω/P2Ω and P1Ω/P3Ω to calculate the affordable range of the bias drift. The parameter ΔVbias = (ΔV/Vπ) × 100% is defined as the bias drift. Assume that when P1Ω/P2Ωis higher than 29 dB and P1Ω/P3Ω is within a range of 18 dB to 20 dB, the undesired distortion is acceptable. Fig. 11 shows the numerical simulations about P1Ω/P3Ω and P1Ω/P2Ω versus the three bias drifts varying from −10% to +10%. As shown in Fig. 11, when ΔVbias1, ΔVbias2 and ΔVbias3 change within a range of ± 10%, the tolerable range of the bias drifts of MZ-a, MZ-b and MZ-c prove to be −4% ≤ ΔVbias1 ≤ 4.5%, −4.48% ≤ ΔVbias2 ≤ 5.25%, and −2% ≤ ΔVbias3 ≤ 3.3%. Fig. 12 shows the temporal waveform of Iout(t) and the corresponding electrical spectrum in three different bias drift cases. Fig. 12(a) shows the temporal waveform and the corresponding spectrum with no bias drift, which agree well with the characteristics of the triangular-shaped waveform. In Fig. 12(b), when ΔVbias1 and ΔVbias2 are 2%, and ΔVbias3 is 3%, within the acceptable range mentioned in Fig. 11, the temporal waveform kept triangular-shape with linearly upping-and-falling edges roughly. However, as in Fig. 12(c), when the bias drifts turn to ΔVbias1 = 6%, ΔVbias2 = 6% and ΔVbias3 = 6%, the undesired harmonics are obvious and the ratio between 1-order harmonic and 3-order harmonic is higher 25 dB. At this time, the upping-and-falling edges of the pulse are not linearly any more. In practical applications, by using a modulator with larger half-wave voltage or bias-controlled circuit to stabilize the mentioned three bias voltages within an acceptable range.
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Fig. 11. Simulated P1Ω/P2Ω and P1Ω/P3Ωcurves versus MZM bias voltage: (a) MZ-a bias drift, (b) MZ-b bias drift, (c) MZ-c bias drift.
Fig. 12. Temporal waveforms of Iout(t) and the corresponding electrical spectra in different bias drift cases: (a) ΔVbias1 = 0, ΔVbias2 = 0 and ΔVbias3 = 0; (b) ΔVbias1 = 2%, ΔVbias2 = 2% and ΔVbias3 = 3%; (c) ΔVbias1 = 6%, ΔVbias2 = 6% and ΔVbias3 = 6%.
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4. Conclusion In conclusion, we reported arbitrary waveform generation based on time-domain synthesis in a DP-MZM. The key components in the system are a DP-MZM and a TDL. By properly adjusting three bias points of DP-MZM, time delay τ of TDL, and modulation index β, square-shaped, triangular-shaped, flat top, and Gaussian waveform are generated. This proves that our scheme has potential to generate arbitrary waveforms. Furthermore, simulations are taken to evaluate the impact of bias point drift of the DP-MZM on the generated waveforms. These discussions make the scheme more practical. Waveform generation in our proposal can be realized by overlapping and controlling optical envelopes, which is more stable and avoids the complex manipulation of spectral lines. Our scheme needs no optical filtering or dispersive element, which make the scheme simpler. Also, waveforms with tunable rate can be obtained only by tuning the driving frequency and the time delay of the TDL. Acknowledgements This work is jointly supported by the National Natural Science Foundation of China (NSFC) (61525501, 61471033), and the Fundamental Research Funds for the Central Universities (2018JBM006). References [1] S. Myunghun, P. Kumar, Optical microwave frequency up-conversion via a frequency-doubling optoelectronic oscillator, IEEE Photonics Technol. Lett. 19 (2007) 1726–1728. [2] A.I. Latkin, S. Boscolo, R.S. Bhamber, et al., Doubling of optical signals using triangular pulses, Opt. Soc. Am. B 26 (2009) 1492–1496. [3] Francesca Parmigiani, et al., Efficient optical wavelength conversion using triangular pulses generated using a superstructured fiber Bragg Grating, Optical Fiber Communication Conference Optical Society of America, (2008). [4] J. Yao, Photonic generation of microwave arbitrary waveforms, Opt. Commun. 284 (2011) 3723–3736. [5] R.S. Bhamber, A.I. Latkin, S. Boscolo, et al., All-optical TDM to WDM signal conversion and partial regeneration using XPM with triangular pulses, Presented at the 34th European Conference on Optical Communication (ECOC 2008) (2008). [6] Z. Jiang, C.B. Huang, D.E. Leaird, A.M. Weiner, Optical arbitrary waveform processing of more than 100 spectral comb lines, Nat. Photonics 1 (2007) 463–467. [7] J. Ye, L. Yan, W. Pan, B. Luo, X. Zou, A. Yi, S. Yao, Photonic generation of triangular-shaped pulses based on frequency-to-time conversion, Opt. Lett. 36 (2011) 1458–1460. [8] H. Jiang, L. Yan, Y. Sun, J. Ye, W. Pan, B. Luo, X. Zou, Photonic arbitrary waveform generation based on crossed frequency to time mapping, Opt. Express 21 (2013) 6488–6496. [9] W. Sun, W. Li, W. Wang, W. Wang, J. Liu, N. Zhu, Triangular microwave waveform generation based on stimulated brillouin scattering, Photonics J. 6 (2014) 1–7. [10] X. Liu, W. Pan, X. Zou, D. Zhang, L. Yan, B. Luo, B. Lu, Photonic generation of triangular-shaped microwave pulses using SBS-based optical carrier processing, J. Lightwave Technol. 32 (2014) 3797–3802. [11] J. Li, J. Sun, W. Xu, T. Ning, L. Pei, J. Yuan, Y. Li, Frequency-doubled triangular-shaped waveform generation based on spectrum manipulation, Opt. Lett. 41 (2016) 199–202. [12] J. Li, X. Zhang, B. Hraimel, T. Ning, L. Pei, K. Wu, Performance analysis of a photonic-assisted periodic triangular-shaped pulse generator, J. Lightwave Technol. 30 (2012) 1617–1624. [13] W. Liu, J. Yao, Photonic generation of microwave waveforms based on a polarization modulator in a Sagnac loop, J. Lightwave Technol. 32 (2014) 3637–3644. [14] B. Dai, Z. Gao, X. Wang, Versatile waveform generation using single-stage dual-drive Mach–Zehnder modulator, Electron. Lett. 47 (2011) 336–338. [15] B. Dai, Z. Gao, X. Wang, Generation of versatile waveforms from CW light using a dual-drive Mach–Zehnder modulator and employing chromatic dispersion, J. Lightwave Technol. 31 (2013) 145–151. [16] Y. Jiang, C. Ma, G. Bai, X. Qi, Y. Tang, Z. Jia, T. Wu, Photonic microwave waveforms generation based on time-domain processing, Opt. Express 23 (2015) 19442–19452. [17] F. Zhang, X. Ge, S. Pan, Triangular pulse generation using a dual-parallel Mach–Zehnder modulator driven by a single-frequency radio frequency signal, Opt. Lett. 38 (2013) 4491–4493. [18] A.H. Gnauck, P.J. Winzer, S. Chandrasekhar, C. Dorrer, Spectrally efficient (0.8 b/s/Hz) 1-Tb/s (25$ times $42.7 Gb/s) RZ-DQPSK transmission over 28,100-km SSMF spans with 7 optical add/drops, Proc. ECOC, (2004).
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Original research article
Analysis of optical arbitrary waveform generation based on timedomain synthesis in a dual-parallel Mach–Zehnder modulator
T
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Jin Yuan, Tigang Ning, Jing Li, Li Pei , Jingjing Zheng, Yueqin Li Key Lab. of All Optical Network and Advanced Telecommunication Network of EMC, Beijing Jiaotong University, Beijing, 100044, China
A R T IC LE I N F O
ABS TRA CT
Keywords: Microwave photonics Arbitrary waveform Time-domain synthesis Dual-parallel Mach–Zehnder modulator
A photonic-assisted microwave arbitrary waveform generator is theoretically analyzed and verified by simulation. The basic principle of the generator is synthesis of optical field envelops. By using a sinusoidal signal to modulate a CW via a dual-parallel Mach–Zehnder (DP-MZM), squareshaped waveforms are directly generated firstly, whose envelops or harmonic power ratios are able to be modified by changing the RF driving voltage. When two identical square-shaped pulses with harmonics power ratio (9:1) between the 1st- and the 3rd-order components suffer a differential envelope phase shift (π/2), the superposition of these envelops contributes to a triangular-shaped waveform. Similarly, we generate flat-top waveform and Gaussian waveforms by properly setting the bias drifts of DP-MZM, time delay of the tunable time delay line, and modulation index. As DP-MZM is a key component in our proposal, we discuss the influence of the bias drifts on the generated waveforms, which make the scheme more practical.
1. Introduction Recently versatile microwave waveforms, such as sinusoidal, triangular, square, trapezoid, and sawtooth waveforms have attracted much attention because of their applications in optical communication links [1], all-optical microwave signal processing and manipulation [2,3], wire and wireless communications [4,5]. Conventionally, electric frequency synthesizers or digital-analog converters are used to generate arbitrary waveforms. However, waveform generator based on electronic techniques is bandwidth limited and cannot meet high-frequency and large-bandwidth requirements of future electric systems [4]. In order to break these limitations, photonic approaches are highly expected for generating arbitrary waveforms [6–16]. One basic photonic generation of microwave waveform is Fourier synthesis [6]. An optical comb acts as a coherent broadband source to generate the tailored optical signal by using an optical spectral shaper and the desired waveform can be obtained by detecting the tailored optical signal in a photodetector (PD). Furthermore, microwave waveform can also be generated based on optical spectral shaping incorporating frequency-to-time mapping (FTTM) [7,8]. The desired spectral envelope can be obtained by using a dispersive element, which can be mapped to the temporal waveform via FTTM. However, the flexibility of this method is restricted and the mode-locked laser (MLL) which used as the optical source is high-cost. Besides the approaches mentioned above, microwave waveform can also be generated using external modulation of a continuous wave (CW) optical signal [9,7–16]. Due to the nonlinearity of the external modulator, a series of optical sidebands are generated. By controlling the phase and amplitudes of the optical sidebands, the desired microwave waveform can be obtained. For example, approaches based on stimulated Brillouin scattering (SBS) in optical fiber are proposed [9,10]. However, the SBS effect in sensitive to the environment, which may make the system unstable. Li et al. reported proposals to
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Corresponding author. E-mail address: [email protected] (L. Pei).
https://doi.org/10.1016/j.ijleo.2018.09.120 Received 19 July 2018; Received in revised form 16 September 2018; Accepted 20 September 2018 0030-4026/ © 2018 Elsevier GmbH. All rights reserved.
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generate triangular-shaped waveform by using a Mach–Zehnder modulator (MZM) in conjunction with an optical interleaver [11] or a dispersive element [12]. It is noted that the schemes mentioned above can only generate triangular-shaped waveform and are limited to generate other waveforms. Recently, approaches reported are extended to generate other types of microwave waveform. For example, arbitrary waveform generation using a polarization modulator (PolM) in a Sagnac loop is reported, in which the primary problem is the stability of the system [13]. Dai et al. carried out a versatile waveform generator based on frequency comb generation [14,15]. A photonic approach of microwave waveform generator based on time-domain synthesis is also proposed [16]. Despite of two Mach–Zehnder modulators which decrease the system integration, other optical devices are needed, which increase the complexity of the system. In this paper, we propose photonic approaches to generate optical arbitrary waveforms based on optical waveform synthesis in time-domain processing. In our scheme, a DP-MZM formed by two sub-MZMs (MZ-a and MZ-b) lying on each arm of a parent MZM (MZ-c) acts as the key component in our scheme. A tunable time delay line (TDL) is applied between the driving signals of two subMZMs. Firstly, setting three bias points of DP-MZM at quadrature points and properly tuning the modulation index β of DP-MZM, two square-shaped waveforms can be generated at the output of MZ-a and MZ-b. When the time delay τ of the TDL is 0, square-shaped waveform is obtained at the output of DP-MZM. When τ is tuned to 1/(4fRF), waveform with triangular-shaped envelopes can be generated by superimposition of two square-shaped waveforms. According to this principle, by properly controlling the bias points of DP-MZM, β, and τ, flat-top and Gaussian waveforms are generated. We also analyzed the impact of the extinction ratio and bias voltage drifts of the DP-MZM on the characteristics of the generated triangular-shaped waveform, which makes the scheme more practical. Moreover, the key significance of our approach is that we use a compact structure of DP-MZM which is expected to be more stable. 2. Model and theory Fig. 1 shows the schematic diagram of the proposed arbitrary waveform generator. In the setup, a CW light is applied to a DPMZM which is formed by two sub-MZMs (MZ-a and MZ-b) and a parent MZM (MZ-c). Because of x-cut design, MZ-a and MZ-b are configured for push–pull operation and the two sub-MZMs and the parent MZM have independent DC biases. A radio frequency (RF) signal VRF(t) with a frequency of ωRF and amplitude of VRF is divided into two paths by a RF splitter to drive MZ-a and MZ-b. Vbias1, Vbias2, and Vbias3 denote the three bias voltages applied to MZ-a, MZ-b and MZ-c. A TDL is used to set the phase between driving signals of MZ-a and MZ-b. In the system, by properly setting the bias points of the two sub-MZMs and the parent MZM, controlling the powers of the microwave driving signals and tuning the time-delay τ of the TDL, different waveforms can be obtained by overlapping the optical field envelopes at point A and point B. In the following, we will perform the generation of arbitrary waveforms. 2.1. Square-shaped waveform generation In this case, three bias points of DP-MZM are all biased at quadrature point (Vbias1, 2, 3 = Vπ/2, where Vπ is the half-wave switching voltage of DP-MZM). The optical field at the output of MZ-a and MZ-b can be expressed as [17]:
EA (t ) =
Ein (t ) π V cos ⎡ (V (t ) + Vbias1) ⎤ exp ⎛j bias1 π ⎞ ⎥ ⎢ 2Vπ RF 2 ⎦ ⎝ 2Vπ ⎠ ⎣
EB (t ) =
Ein (t ) π V cos ⎡ (VRF (t + τ ) + Vbias2) ⎤ exp ⎛j bias2 π ⎞ ⎥ ⎢ 2 ⎦ ⎝ 2Vπ ⎠ ⎣ 2Vπ
⎜
⎟
⎜
(1)
⎟
(2)
Ein(t) = E0exp(jω0t) is the optical field at the input of the DP-MZM. E0 and ω0 are the amplitude and angular frequency respectively. The optical field at the output of the DP-MZM can be written as
Fig. 1. Schematic setup of the proposed arbitrary waveform generator; (CW, continuous-wave laser; DP-MZM, dual-parallel Mach–Zehnder modulator; TDL, tunable time delay line; PD, photodetector; ESA, electrical spectrum analyzer). 550
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V EC (t ) = EA (t ) + EB (t )exp ⎛j bias3 π ⎞ ⎝ Vπ ⎠ ⎜
⎟
(3)
When the optical signal at the output of the DP-MZM is sent to a photodetector (PD) for intensity detection, we acquire an electrical signal at the output of the PD as below
⎧ cos2 ⎡ π (VRF (t ) + Vbias1) ⎤ + cos2 ⎡ π (VRF (t −τ ) + Vbias2) ⎤+⎫ ⎦ ⎪ ⎣ 2Vπ ⎦ ⎣ 2Vπ ⎪ ⎪ R |Ein |2 ⎪ π π ∗ Iout (t ) = R⋅EC (t )⋅EC (t ) = 2 cos ⎡ 2V (VRF (t ) + Vbias1) ⎤ × cos ⎡ 2V (VRF (t −τ ) + Vbias2) ⎤ ⎦ ⎬ ⎣ π ⎦ ⎣ π ⎨ 2 ⎪ ⎪ Vbias2 − Vbias1 + 2Vbias3 π ⎪ ⎪ × cos 2Vπ ⎭ ⎩
(
)
(4)
When τ is set to τ = 0, Eq. (4) can be simplified as π
Iout (t ) =
π
⎧ 2 cos2 ⎡ (VRF (t ) + Vbias1) ⎤ + 2 cos ⎡ (VRF (t ) + Vbias1) ⎤⎫ R |Ein |2 ⎪ ⎦⎪ ⎣ 2Vπ ⎦ ⎣ 2Vπ ⎬ ⎨ × cos π (V (t ) + V ) × cos Vbias2 − Vbias1 + 2Vbias3 π 2 ⎡ 2Vπ RF bias 2 ⎤ ⎪ ⎪ 2Vπ ⎦ ⎣ ⎭ ⎩
(
)
(5)
Assuming that parameter β = πVRF/Vπ is defined as the modulation index. Substituting Vbias1 = Vbias2 = Vbias3 = Vπ/2 into Eq. (5), the optical intensity can be further simplified to
Iout (t ) =
R | Ein |2 2
{cos ⎡⎣
πVRF (t ) Vπ
π
}
+ 2⎤ + 1 ⎦
∝ IDC + sin [β sin (ωRF t )] ∝ IDC + J1 (β )sin (ωRF t ) + J3 (β )sin (3ωRF t ) + J5 (β )sin (5ωRF t ) + …
(6)
J2k-1(·) is the Bessel function of the first kind of order 2k-1. The Fourier series expansion of a square-shaped waveform Tsq(t) is given by ∞
Tsq (t ) = DC +
∑ n = 1,3,5...
1 1 1 sin (nωm t ) = DC + sin (ωm t ) + sin (3ωm t ) + sin (5ωm t ) + ... n 3 5
(7)
Comparing Eqs. (6) and (7), we can conclude that in order to obtain a square-shaped waveform, J3(β)/J1(β) = 1/3 should be satisfied. This indicates that the electrical power difference between the first-order and third-order harmonic should be roughly 9.5 dB. Fig. 2(a) shows the relationship between J2k-1(·) and modulation index, β. It is obvious that when β is within a range from 1 to 3, J5(β) is far smaller than J1(β) and J3(β). This proves that the impact of harmonics higher than 5th-order is negligibly small and only the 1storder 3rd-order sidebands are considered to form the desired waveform. Fig. 2(b) shows the power ratio of J1(β) and J3(β). It can be seen that when β is tuned to 2.3, the power ratio is 9.5 dB, which satisfies with the property of square-shaped waveform. In this case, the optical intensity of the generated square-shaped waveform can be expressed as
Iout , sq (t ) ∝ IDC + sin (ωRF t ) +
J3 (β ) 1 sin (3ωRF t ) = IDC + sin (ωRF t ) + sin (3ωRF t ) 3 J1 (β )
(8)
2.2. Triangular waveform generation It has been reported that triangular waveform can be generated by synthesizing two properly modulated square-like envelopes
Fig. 2. Relationship between (a) Bessel function J2k-1(·) and (b) J3(β)/J1(β) versus modulation index, β. 551
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Fig. 3. Triangular-shaped waveform generation by the superimposition of two square-shaped waveforms with π/2 phase shift.
[15]. As shown in Fig. 3, when two square-shaped waveforms suffer an envelope phase shift of π/2, the superposition of these signals contributes to a triangular waveform in a PD. In our scheme, when τ is set to τ = 1/(4fRF), the two identical square-shaped waveforms at the output of MZ-a and MZ-b suffer an envelope phase shift of π/2. According to Eq. (8), after synthesizing of two square-shaped envelopes, the output current of PD can be written as
Iout , tr (t ) ∝ IDC + sin (ωRF t ) +
(
2 2
sin (3ωRF t ) + sin [ωRF (t + τ )] +
) + cos (ω t ) ⎤⎦ + cos (3ω ⎡cos (ω t + ) + ⎣
= IDC + ⎡cos ωRF t + ⎣ = IDC +
J3 (β ) J1 (β )
RF
3π 2
RF
π 4
J3 (β ) J1 (β )
J3 (β ) ⎡cos J1 (β ) ⎣
RF t +
3π 4
)
(3ω
RF t
+
3π 2
J3 (β ) J1 (β )
sin [3ωRF (t + τ )]
) − cos (3ω
RF t ) ⎤
⎦
⎤ ⎦
(9)
The Fourier series expansion of a triangular-shaped waveform Ttr(t) is given by ∞
Ttr (t ) = DC +
∑ n = 1,3,5...
1 1 1 cos (nωm t ) = DC + cos (ωm t ) + cos (3ωm t ) + cos (5ωm t ) + ... n2 9 25
(10)
According to Eqs. (9) and (10), in order to achieve triangular-shaped pulses, the amplitude ratio between the +1-order sideband and +3-order sideband should be 1/9. In spectrum, the relationship is displayed by power ratio. As J2 3(β)/J2 1(β) = 1/81, 10lg[J2 3(β)/J2 1(β)] is equal to19. So the power difference between the +1-order sideband and +3-order sideband is around 19.08 dB. As shown in Fig. 2(b), when β is tuned to 1.51, the power ratio between the first-order and third-order sidebands is 19.08 dB, which satisfies the property of triangular-shaped waveform. 3. Simulation and discussion In order to verify the mechanism of the proposed scheme, simulations are carried out via OptiSystem 10.0. As shown in Fig. 1, a light wave from a CW laser with central wavelength of 1550 nm and line-width of 0.8 MHz is sent to the DP-MZM. The DP-MZM has insertion loss of 5.5 dB, half-wave switching voltage of 4 V and extinction ratio of 30 dB. The optical spectrum and waveform of Iout (t) are captured by an OSA and an oscilloscope. An electrical spectrum analyzer is used to measure the electrical spectrum. 3.1. Square-shaped waveform generation When properly controlling the magnitude of the electrical signal, β can be tuned to 2.3. The driving frequency fRF is set to be 10 GHz. By tuning time delay of the TDL to τ = 0, simulation results of square-shaped waveform are shown in Fig. 4(a–b). The simulated electrical spectrum of the generated waveform is shown in Fig. 4(a). As can be seen, the third-order harmonic at 30 GHz is 9.6 dB (close to the theoretical value of 9.5 dB) lower than the fundamental tone at 10 GHz. The second-order harmonic is well suppressed. Also note that the fifth-order harmonic is around 30 dB lower than the first –order tone. This means that fifth-order and higher order harmonics can be neglected. Then, a half-duty-cycle square waveform with a period of 100ps is observed in the oscilloscope as shown in Fig. 4(b). The normalized ideal square waveform is also drawn in Fig. 4(b) with a red line for comparison. It is obvious that the simulated square waveform fits well with the idea one. In order to show the frequency tunability of the proposed waveform generator, the frequency of the driving signal applied to the DPMZM is changed to 20 GHz. As shown in Fig. 4(c–d), a square waveform with a period of 50 ps is obtained. When β is tuned within a range from 1.5 to 3, square-like envelopes can also be obtained. Fig. 5(a)–(f) illustrate the simulated temporal waveforms and the corresponding electrical spectrum when β is tuned to 1.51, 2.3 and 2.5. Fig. 5(a) and (d) show the generated square-shaped waveform and its corresponding electrical spectrum when β = 1.51. The power ratio between the first-order and third-order harmonics is 19.1 dB, which presents a good approximation with the Fourier series expansion of a triangular-shaped waveform, but the phase relationship determines a square-shaped waveform with rising and falling time of 45 ps. When the modulation index increases, i.e., 2.3, the rising and falling time are 24 ps. A square-shaped waveform with flat top and bottom is obtained 552
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Fig. 4. Simulated electrical spectra and waveforms of the generated square-shaped waveforms with repetition rates of (a, b)10 GHz; and (c, d) 20 GHz. (For interpretation of the references to colour in the text, the reader is referred to the web version of this article.)
Fig. 5. Simulated square-shaped waveforms and the corresponding electrical spectra: (a, d) β = 1.51; (b, e) β = 2.3 and (c, f) β = 2.5.
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Fig. 6. Simulated waveforms at the output of (a) MZ-a; (b) MZ-b; (c) MZ-c; and (d) corresponding electrical spectrum of triangular-shaped waveform.
and the power ratio between the first two harmonics in closed to the Fourier series expansion of an ideal square waveform (9.5 dB). In these cases, all the waveforms have duty cycles of 50%. 3.2. Triangular-shaped waveform generation Through tuning τ to 25 ps, modulation index β to 1.51, triangular-shaped waveform can be generated. The frequency of driving signal is set to 10 GHz. Fig. 6(a) and (b) show the simulated temporal waveforms at the output of MZ-a and MZ-b with a time delay of τ = 25 ps. These two signals superpose in a PD and triangular-shaped waveform can be obtained in Fig. 6(c). Fig. 6(d) shows that the power ratio between the 10 GHz and 30 GHz components is 19.08 dB roughly, which goes well with the property of the triangularshaped waveform. To investigate the tunability of the scheme, we simulate the generation of triangular-shaped waveform at different repetition rate. By adjusting the driving frequency of the DP-MZM, the following three cases are simulated: (a) fRF = 2.5 GHz; (b) fRF = 5 GHz; (c) fRF = 7.5 GHz. Pictures in Fig. 7 plot the triangular-shaped temporal waveforms of Iout(t) at with different period of 400 ps, 200 ps, and 133.33 ps according to the above cases. In this way, we can implement the repetition rate’s continuous tenability of the repetition rate simply by changing the driving frequency of the local oscillator and the time delay of the TDL. 3.3. Arbitrary waveform generation Next, we will show that the proposed waveform generator is also capable of generating other waveforms, e.g. flap-top waveform and Gaussian waveform. Again, a 10 GHz sinusoidal RF signal was applied to the DPMZM. When MZ-a and MZ-b are biased at MATP, MITP or MITP, MATP, τ and β are adjusted to 0 and 1.6, a flat top waveform with a duty ratio of 50% is obtained. Fig. 8(a)–(c) show simulation results for the flat top waveform. This flat top waveform with a duty ratio of 50% will be used for return-to-zero signals in optical fiber communication, especially differential phase-shift-keyed transmission [18]. Similarly, when τ and β are adjusted to 25 ps and 2.35 respectively, Gaussian waveform with a repetition rate of 10 GHz is generated as shown in Fig. 9(a). When β is tuned to 3.14, half-wave cosine waveform is generated as shown in Fig. 9(c). The corresponding electrical spectra of the generated waveforms are shown in Fig. 9(b) and (d).
Fig. 7. Simulated temporal waveforms with repetition rates of: (a) 2.5 GHz, (b) 5 GHz, (c) 7.5 GHz. 554
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Fig. 8. (a) Simulated flat top waveform with a duty cycle of 50%; (b) corresponding optical spectrum; and (c) corresponding electrical spectrum.
Fig. 9. Simulated Gaussian and half-wave cosine waveforms with a repetition rate of 10 GHz (a, c) and corresponding electrical spectra (b, d).
In the simulations below, MZ-a and MZ-b are biased at MATP, MATP or MITP, MITP, fRF is tuned at 10 GHz and τ is tuned to 0. When the two sub-MZMs are biased at MATP and MATP and β is set to 3.14, a Gaussian waveform with a repetition rate of 20 GHz is generated as shown in Fig. 10(a). When the two sub-MZMs are biased at MITP and MITP and β is set to 3.14, a flat-top waveform, with a repetition rate of 20 GHz is obtained as shown in Fig. 10(c). Fig. 10(b) and (d) show the corresponding spectra of the generated waveforms. Different from the previous results, the generated waveforms in this case is frequency-doubled. According to the previous discussion, by properly controlling bias points of the DP-MZM, time-delay τ of the TDL, and the 555
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Fig. 10. Simulated Gaussian and flat-top waveforms with a repetition rate of 20 GHz (a, c) and corresponding electrical spectra (b, d). Table 1 Satisfying conditions and corresponding waveforms. Bias modes
π/2, π/2, π/2 0, π, π/2; or π, 0, π/2
0, 0, π/2; or π, π, π/2
Conditions Time delay τ (ps)
Modulation index, β
Generated waveforms
0 25 0 25
2.3 1.51 1.6 2.35 3.14 3.14
Square-shaped Waveform Triangular-shaped Waveform Flat top Waveform Gaussian Waveform Half-wave Cosine Waveform Flat-top Waveform; Gaussian Waveform (frequency doubled)
0
modulation index β, square, triangular, Gaussian and flat top waveforms can be generated. Table 1 below makes a summary about these cases.
3.4. Bias point drifts of DP-MZM As DP-MZM is a key component in our approach, it is necessary to discuss the bias drifts of the DP-MZM. In order to verify the influence of this parameter on the output waveforms, we take triangular-shaped waveform for example. In the previous generation of triangular-shaped, DP-MZM is biased at three independent quadrature bias points. However, the bias drifts will degrade the stability of the generated triangular-shaped waveform to some extent. As the second harmonic is the main distortion harmonic, we focus on P1Ω/P2Ω and P1Ω/P3Ω to calculate the affordable range of the bias drift. The parameter ΔVbias = (ΔV/Vπ) × 100% is defined as the bias drift. Assume that when P1Ω/P2Ωis higher than 29 dB and P1Ω/P3Ω is within a range of 18 dB to 20 dB, the undesired distortion is acceptable. Fig. 11 shows the numerical simulations about P1Ω/P3Ω and P1Ω/P2Ω versus the three bias drifts varying from −10% to +10%. As shown in Fig. 11, when ΔVbias1, ΔVbias2 and ΔVbias3 change within a range of ± 10%, the tolerable range of the bias drifts of MZ-a, MZ-b and MZ-c prove to be −4% ≤ ΔVbias1 ≤ 4.5%, −4.48% ≤ ΔVbias2 ≤ 5.25%, and −2% ≤ ΔVbias3 ≤ 3.3%. Fig. 12 shows the temporal waveform of Iout(t) and the corresponding electrical spectrum in three different bias drift cases. Fig. 12(a) shows the temporal waveform and the corresponding spectrum with no bias drift, which agree well with the characteristics of the triangular-shaped waveform. In Fig. 12(b), when ΔVbias1 and ΔVbias2 are 2%, and ΔVbias3 is 3%, within the acceptable range mentioned in Fig. 11, the temporal waveform kept triangular-shape with linearly upping-and-falling edges roughly. However, as in Fig. 12(c), when the bias drifts turn to ΔVbias1 = 6%, ΔVbias2 = 6% and ΔVbias3 = 6%, the undesired harmonics are obvious and the ratio between 1-order harmonic and 3-order harmonic is higher 25 dB. At this time, the upping-and-falling edges of the pulse are not linearly any more. In practical applications, by using a modulator with larger half-wave voltage or bias-controlled circuit to stabilize the mentioned three bias voltages within an acceptable range.
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Fig. 11. Simulated P1Ω/P2Ω and P1Ω/P3Ωcurves versus MZM bias voltage: (a) MZ-a bias drift, (b) MZ-b bias drift, (c) MZ-c bias drift.
Fig. 12. Temporal waveforms of Iout(t) and the corresponding electrical spectra in different bias drift cases: (a) ΔVbias1 = 0, ΔVbias2 = 0 and ΔVbias3 = 0; (b) ΔVbias1 = 2%, ΔVbias2 = 2% and ΔVbias3 = 3%; (c) ΔVbias1 = 6%, ΔVbias2 = 6% and ΔVbias3 = 6%.
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4. Conclusion In conclusion, we reported arbitrary waveform generation based on time-domain synthesis in a DP-MZM. The key components in the system are a DP-MZM and a TDL. By properly adjusting three bias points of DP-MZM, time delay τ of TDL, and modulation index β, square-shaped, triangular-shaped, flat top, and Gaussian waveform are generated. This proves that our scheme has potential to generate arbitrary waveforms. Furthermore, simulations are taken to evaluate the impact of bias point drift of the DP-MZM on the generated waveforms. These discussions make the scheme more practical. Waveform generation in our proposal can be realized by overlapping and controlling optical envelopes, which is more stable and avoids the complex manipulation of spectral lines. Our scheme needs no optical filtering or dispersive element, which make the scheme simpler. Also, waveforms with tunable rate can be obtained only by tuning the driving frequency and the time delay of the TDL. Acknowledgements This work is jointly supported by the National Natural Science Foundation of China (NSFC) (61525501, 61471033), and the Fundamental Research Funds for the Central Universities (2018JBM006). References [1] S. Myunghun, P. Kumar, Optical microwave frequency up-conversion via a frequency-doubling optoelectronic oscillator, IEEE Photonics Technol. Lett. 19 (2007) 1726–1728. [2] A.I. Latkin, S. Boscolo, R.S. Bhamber, et al., Doubling of optical signals using triangular pulses, Opt. Soc. Am. B 26 (2009) 1492–1496. [3] Francesca Parmigiani, et al., Efficient optical wavelength conversion using triangular pulses generated using a superstructured fiber Bragg Grating, Optical Fiber Communication Conference Optical Society of America, (2008). [4] J. Yao, Photonic generation of microwave arbitrary waveforms, Opt. Commun. 284 (2011) 3723–3736. [5] R.S. Bhamber, A.I. Latkin, S. Boscolo, et al., All-optical TDM to WDM signal conversion and partial regeneration using XPM with triangular pulses, Presented at the 34th European Conference on Optical Communication (ECOC 2008) (2008). [6] Z. Jiang, C.B. Huang, D.E. Leaird, A.M. Weiner, Optical arbitrary waveform processing of more than 100 spectral comb lines, Nat. Photonics 1 (2007) 463–467. [7] J. Ye, L. Yan, W. Pan, B. Luo, X. Zou, A. Yi, S. Yao, Photonic generation of triangular-shaped pulses based on frequency-to-time conversion, Opt. Lett. 36 (2011) 1458–1460. [8] H. Jiang, L. Yan, Y. Sun, J. Ye, W. Pan, B. Luo, X. Zou, Photonic arbitrary waveform generation based on crossed frequency to time mapping, Opt. Express 21 (2013) 6488–6496. [9] W. Sun, W. Li, W. Wang, W. Wang, J. Liu, N. Zhu, Triangular microwave waveform generation based on stimulated brillouin scattering, Photonics J. 6 (2014) 1–7. [10] X. Liu, W. Pan, X. Zou, D. Zhang, L. Yan, B. Luo, B. Lu, Photonic generation of triangular-shaped microwave pulses using SBS-based optical carrier processing, J. Lightwave Technol. 32 (2014) 3797–3802. [11] J. Li, J. Sun, W. Xu, T. Ning, L. Pei, J. Yuan, Y. Li, Frequency-doubled triangular-shaped waveform generation based on spectrum manipulation, Opt. Lett. 41 (2016) 199–202. [12] J. Li, X. Zhang, B. Hraimel, T. Ning, L. Pei, K. Wu, Performance analysis of a photonic-assisted periodic triangular-shaped pulse generator, J. Lightwave Technol. 30 (2012) 1617–1624. [13] W. Liu, J. Yao, Photonic generation of microwave waveforms based on a polarization modulator in a Sagnac loop, J. Lightwave Technol. 32 (2014) 3637–3644. [14] B. Dai, Z. Gao, X. Wang, Versatile waveform generation using single-stage dual-drive Mach–Zehnder modulator, Electron. Lett. 47 (2011) 336–338. [15] B. Dai, Z. Gao, X. Wang, Generation of versatile waveforms from CW light using a dual-drive Mach–Zehnder modulator and employing chromatic dispersion, J. Lightwave Technol. 31 (2013) 145–151. [16] Y. Jiang, C. Ma, G. Bai, X. Qi, Y. Tang, Z. Jia, T. Wu, Photonic microwave waveforms generation based on time-domain processing, Opt. Express 23 (2015) 19442–19452. [17] F. Zhang, X. Ge, S. Pan, Triangular pulse generation using a dual-parallel Mach–Zehnder modulator driven by a single-frequency radio frequency signal, Opt. Lett. 38 (2013) 4491–4493. [18] A.H. Gnauck, P.J. Winzer, S. Chandrasekhar, C. Dorrer, Spectrally efficient (0.8 b/s/Hz) 1-Tb/s (25$ times $42.7 Gb/s) RZ-DQPSK transmission over 28,100-km SSMF spans with 7 optical add/drops, Proc. ECOC, (2004).
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