Microwave signal is widely used in modern radar, wireless communication, signal processing and medical imaging, and their quality has a decisive impact on the performance of the corresponding system [1–4]. Unfortunately, the microwave generation methods in the electric domain are not so satisfactory. For example, traditional electronic microwave generation is faced with great challenges in terms of cost and power consumption [5]. Therefore, some photonic microwave signals with the advantages of high bandwidth, high frequency and anti-electromagnetic interference have attracted widespread attention. As one of the basic photonic microwave signal waveforms, triangular waveform generation methods have been proposed by different research groups. The triangular waveform signal could be generated by using spectrum shaping and frequency-to-time mapping (FTTM) [6–8]. Based on FTTM, the external modulation method was proposed and it provided a simple way to generate the desired Fourier components [9–13]. Unfortunately, the symmetry of the triangular waveform is invariant. A triangular wave signal with symmetry adjustable is required in many application fields such as all-optical time division multiplexing (TDM) to wavelength division multiplexing (WDM) signal conversion [14] and doubling of optical signals [15].

The scheme based on cascaded SD-MZM and BPD could generate triangular wave signals with an adjustable symmetry index [16]. However, the two identical sub-photoelectric detectors with high frequency added cost to the system. Another triangular waveform generator with variable symmetry based on a dual-polarization modulation was proposed [17]. In this system, a continuous optical wave was divided into two beams, and they were modulated by a radio frequency signal through two Mach-Zehnder modulators, respectively. Then, the polarization state of one beam was kept the same and that of the other beam was rotated 90° by a polarization rotator. By setting the modulation index, the optical delay and the phase shift of the phase shifter appropriately, the superposition of the two beams could generate an optical triangular waveform with different symmetry factors. Unfortunately, using two modulators would complicate the system. Although an I/Q (I, in-phase; Q, quadrature phase) modulator can be substituted for these two modulators in this system [18], which added the cost of the system. Recently, a cascaded SD-MZM and polarization-maintaining fiber architecture was proposed to generate triangular waveforms with adjustable symmetry through simulation [19]. However, according to the theoretical derivation [19], the symmetry index could not be varied continuously under the conditions of a constant length polarization-maintaining fiber (PMF). Besides, changing the length of the PMF in real-time is inconvenient, which means it is not applicable.

In this paper, a triangular wave signal generation scheme is proposed, of which the key component is a cascaded SD-MZM and optical polarization beam splitter-combiner architecture. In the triangular microwave signal generator, parameters including the modulation index and bias-induced phase shift of the SD-MZM and the controllable delay can be changed to generate a triangular waveform with variable symmetry. To get the target waveform, a neighborhood-region-search method is presented to find the optimum values of the parameters. By changing these parameters, the symmetry index can be continuously changed from 20% to 80%. Compared with the previous schemes, this generator just contains one SD-MZM and the BPD is not needed, which means the costs and energy consumption are significantly reduced.

The scheme of the optical triangular wave generator with adjustable symmetry is shown in Fig. 1. An optical continuous wave with an angular frequency of ω₀ is generated by a laser diode (LD) and then injected into a SD-MZM after passing through a polarization controller (PC1). A radio frequency signal is used to modulate the input optical continuous wave. The modulated optical field is

Figure 1. Schematic diagram of the optical triangular wave generator with adjustable symmetry. LD, laser diode; EDFA, erbium-doped optical fiber amplifier; RF, radio frequency source; SD-MZM, the single-drive Mach-Zehnder modulator; PD, photoelectric detector; PC, polarization controller; PBC, polarization beam combiner; OC, optical coupler; TODL, tunable optical delay line; 90° MPS, 90° microwave phase shifter; ESA, electrical spectrum analyzer; OSC, oscilloscope.

where E_in (t) = E₀ exp( jω₀t) is the optical field of the optical continuous wave. E₀ and ω₀ is the amplitude and the angular frequency of the optical continuous wave, respectively. m = _πV_RF /V_π is the modulation index. V_π is a half-wave voltage of the modulator and V_RF is an amplitude of the radio frequency signal, respectively. Ω = 2_πf_RF is an angular frequency of the radio source. φ = _πV_bias/V_π is a bias-induced phase shift and V_bias is the bias voltage of the SD-MZM.

The modulated optical field is amplified by an erbium-doped optical fiber amplifier (EDFA) to compensate for the loss of the SD-MZM and then divided into two branches by a 50:50 optical coupler (OC). The upper branch contains a tunable optical delay line (TODL), while the lower branch just contains a polarization controller (PC2). The optical fields of the two branches can be coupled into the principal optic axis of the PBC through a polarization beam combiner (PBC). The PC2 is to ensure the orthogonal components of the PBC are almost equal. The phase shift induced by TODL is Ωτ. Obviously, the phase change of a 90-degree introduced by the 50:50 OC can be compensated by the TODL. Therefore, the optical fields of the PBC output can be expressed by two orthogonal components as follows:

where the E_∥ and the E_⊥ are parallel component and vertical component, respectively. ℜ is the amplification factor of the EDFA and the τ is the controllable delay difference.

After photoelectric conversion by using a photoelectric detector (PD), a corresponding alternating current (AC) signal can be obtained:

where γ is the PD response factor.

According to Jacobi-Anger identify, i(t) can be rewritten as,

and

In the experiment, the angular frequency (Ω) of a given radio frequency (RF) source is constant. Therefore, the phase shift introduced by the TODL is only related to the controllable delay difference. From Eqs. (4) and (5), the odd (a_2k−1) and even harmonic (a_2k) coefficients can be controlled by three variables (m, φ and τ) of the system. Therefore, Eqs. (4) and (5) can be further simplified as,

In the proposed system, the phase of π/2 can be compensated by a 90° microwave phase shifter (MPS). Therefore, the generated AC signal can also be expressed as,

Theoretically, a triangular waveform can be shown as follows:

where T is the entire period. The δ is the symmetry index, which is defined as the ratio of the rise time to the T. The corresponding Fourier series can be expressed as

and

where _ω is the angular frequency. From the Eqs. (7) and (9), one can find that symmetry index of the generated triangular waveform varies with the three variables (m, φ and τ) of the system. Obviously, any triangular waveform can be generated when harmonics of correspondence in Eqs. (7) and (9) are equal except for a certain time delay. In practice, the main orders of harmonics in Eq. (9) may turn out to be a very good approximation. For example, if the power of those harmonics higher than the third is lower than that of the first three harmonics, a favorite precision of the generated triangular waveform can be realized under the condition of retaining the first three harmonics [20].

To evaluate the quality of the generated waveforms, one can use similarity η to compare the similarity between generated waveform and the theoretical waveform. The similarity η is defined as,

where Y_k is the k-th sampling value of the simulated or experimental triangular wave and X_k is the k-th sampling value of the desired triangular wave. n represent the total sampling points of the desired triangular wave in a period.

If the components above the third order can be neglected, Eq. (7) can be further simplified as:

From Eqs. (9) and (12), one can find that a target triangular waveform can be achieved at a₁ = b₁, a₂ = b₂, a₃ = b₃. For a theoretical triangular wave with a certain symmetry index, the power of the first three harmonics (b₁, b₂, b₃) can be calculated by Eq. (9), which are shown in Table 1. In the proposed triangular generator, one can get these corresponding index (a₁, a₂, a₃) with the same values (i.e. a₁ = b₁, a₂ = b₂, a₃ = b₃) by setting different parameters (m, φ and τ). Based on the Eqs. (4) and (5), these parameters can be calculated as,

TABLE 1. The parameter values of the triangular wave generator with different symmetry indexes, and the characteristic parameters of the theoretical and the simulated triangular wave with different symmetry indexes.

Case	A	B	C	D	E	F	G
δ (%)	20	30	40	50	60	70	80
b1	1	1	1	1	1	1	1
b2	0.405	0.294	0.155	0	−0.155	−0.294	−0.405
b3	0.180	0.042	−0.069	−0.112	−0.069	0.042	0.180
m	1	0.5	1.3	1.5	1.3	0.5	1
φ	−2.466	−2.610	1.208	1.571	−1.934	0.532	0.676
τ (ps)	80.739	82.685	32.120	31.035	32.120	82.685	80.739
a1	1	1	1	1	1	1	1
a2	0.410	0.290	0.153	0	−0.153	−0.290	−0.410
a3	0.181	0.042	−0.065	−0.110	−0.065	0.042	0.181
η (1)	0.996	0.997	0.997	0.998	0.997	0.997	0.997

and

Equations (13) and (14) indicate that the φ and τ are both the function of m. For a triangular waveform with a certain symmetry index, many values of the m can be selected. Correspondingly, the φ and τ will be changed. In other words, there are many sets of parameter (m, φ and τ) values that satisfy the Eqs. (13) and (14) for a specific symmetry index. For example, for a triangular waveform with a symmetry index of 20%, these parameters collection [(m = 1.1, φ = 0.423 and Ωτ = 1.829), (m = 1.3, φ = 0.002 and Ωτ = 1.572), (m = 1.5, φ = 0.332 and Ωτ = 1.412)] can be obtained, which is shown in Fig. 2. Certainly, there are also many other parameters collection for the same symmetry index. But it should be pointed out that not all these parameters collection can generate the target triangular waveform. It can be explained by the simulated results as shown in Fig. 3. The simulated triangular wave is based on the parameters (m, φ and τ) values of (1.4, 0.174 and 59.036). Although these parameters satisfy the Eqs. (13)–(14), the fourth-order component of the generated signal is higher than that of the third-order component, which reduces the similarity between the generated waveform and the theoretical waveform.

Figure 2. The parameter values of the theoretical waveform with different symmetry coefficients when the components above the third order are neglected. (a) The φ values under different symmetry indexes, (b) the Ωτ values under different symmetry indexes.

Figure 3. The theoretical and simulation waveforms with a symmetry index of 20% and the corresponding spectrum of the simulated waveform. (a) The theoretical waveform (solid blue line) and simulation (dotted orange line) with a symmetry index of 20%; (b) The corresponding spectrum of the simulated waveform with a symmetry index of 20%.

To improve the similarity, a neighbor-region-search method was adopted. We firstly propose a target triangular waveform with a certain symmetry index. Based on Eq. (12), the optimum triangular waveform is searched within the searching range. The searched values of the variables will be recorded if the η of the searched waveform is greater than 0.9 and the fourth-order component of the generated signal is lower than that of the third-order component. When the search is complete, the similarity index close to 1 mostly represents the best parameter to generate the target triangular and the corresponding optimum variables are reserved. If there is no record, it means that the triangular wave generator can not generate the target waveform. The optimal variables of the target triangular waveform with different symmetry indexes are shown in Table 1.

The corresponding simulated triangular waveform signals with the symmetry indexes range of 20–80% are shown in Fig. 4. The fourth-order harmonic component can be ignored. It can be seen from Figs. 4(a)–4(g) that the simulated waveform based on the search parameters is consistent with the target waveform.

Figure 4. The simulated triangular waveforms and the spectrum of the simulated triangular waveform with different symmetry coefficients. (a)–(g) The normalized simulation (solid blue line) and target (dotted orange line) triangular waveforms with symmetry index of 20–80%; (h)–(n) The spectrum of the simulated triangular waveform with the symmetry indexes range of 20–80%.

For example, for a triangular waveform with a symmetry index of 20%, the power ratio of a₁/a₂ is 7.743 dB and the power ratio of a₁/a₃ is 14.867 dB in simulation as shown in Table 1. They are very close to the spectral relationship of the theoretical waveform, of which the power ratio of b₁/b₂ is 7.691 dB and the power ratio of b₁/b₃ is 14.367 dB. The similarity between the simulated and theory triangular with different symmetry indexes were calculated and they are shown in Table 1 as η (1). It proves that the simulated waveform is very similar to the theoretical waveform.

4.1. Experimental Setup

A proof-of-concept experiment is carried out based on the setup of Fig. 1 to prove the feasibility of the proposed scheme. Since the oscilloscope in our laboratory has a maximum observation limit of 22 GHz, the frequency of the RF is 4 GHz in the experiment. A LD (8164B; Keysight Technologies, CA, USA) is used to generate the continuous light wave. The continuous wave is modulated by a SD-MZM (KG-AMBOX; Conquer Photonics Co Beijing, China) with a cosine drive signal of 4 GHz. The half-wave voltage of the modulator is 4.4 V. The optical polarization beam splitter-combiner consists of an upper branch, a lower branch and a PBC. The upper branch contains a TODL, and the lower branch contains a PC2. The operation bandwidth of 90° MPS (SHWHB-02002650-90K) is 2–26.5 GHz. By setting the parameters including the modulation index and bias-induced phase shift of the SD-MZM and the controllable delay based on the simulated results, a desired triangular waveform could be generated. An oscilloscope (OSC) (Agilent 86100D; Keysight Tecnologies) and an electrical spectrum analyzer (ESA) (Agilent N9010A; Keysight Tecnologies) are used to measure the generated signal.

4.2. Experimental Results and Discussion

Based on the simulation results, the generation of triangular waveforms with different symmetry indexes is verified in the experiment. Figures 5(a)–5(g) show the experimental results of the normalized triangular waveforms with the symmetry indexes range of 20–80% and the corresponding spectral are shown in Figs. 5(h)–5(n). The first three orders of harmonics components are consistent with the theoretical values in the experiment. For example, when the symmetry index is 20%, the power ratio of $a_1^{\text{'}}$/$a_2^{\text{'}}$ is 8.397 dB and the power ratio of $a_1^{\text{'}}$/$a_3^{\text{'}}$ is 14.514 dB in the experiment as shown in Table 2. Their values are very close to the spectral relationship of the target waveform, in which the power ratio of b₁/b₂ is 7.691 dB and the power ratio of b₁/b₃ is 14.367 dB. The similarities in the experiment are shown in Table 2 as η (2). From the similarity and spectral relationship, the experimental results agree well with the theoretical and simulated results.

Figure 5. The experimental triangular waveforms and the corresponding spectrum with different symmetry coefficients. (a)–(g) The experimental triangular microwave signals waveforms with the symmetry indexes range of 20–80%, (h)–(n) the corresponding spectrum.

TABLE 2. The first three harmonics of the experimental triangular waveform, and the similarity between the theoretical and the experimental triangular waveform.

Case	A	B	C	D	E	F	G
δ (%)	20	30	40	50	60	70	80
a1'	1	1	1	1	1	1	1
a2'	0.380	0.305	0.175	0.053	0.157	−0.276	−0.407
a3'	0.188	0.063	−0.082	−0.133	−0.076	0.061	0.209
η (2)	0.981	0.991	0.990	0.999	0.980	0.933	0.984

In conclusion, a triangular waveform signals generator with adjustable symmetry index was theoretically analyzed and experimentally demonstrated. In the triangular microwave signal generator, a triangular waveform with variable symmetry can be generated by appropriately adjusting the parameters which include the modulation index and the bias-induced phase shift of the SD-MZM and the controllable delay difference. By changing these parameters, the symmetry index can be continuously changed from 20% to 80%. Compared with the previous schemes, this generator just contains one SD-MZM and the BPD is not needed, which means the costs and energy consumption are significantly reduced.

The National Natural Science Foundation of China (Grant No. 61751102, 61965004); National Key Research and Development Program of China (Grant No. 2021YFB2206302); Introduction talent research start-up fund of Guizhou University [Guida Ren Ji He Zi (2018-14)].

The authors declare no conflict of interest.

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