Photonic microwave waveform generation based on phase modulation and tunable dispersion

Yongsheng Gao; Aijun Wen; Hanxiao Zheng; Dong Liang; Lin Lin

doi:10.1364/OE.24.012524

1. Introduction

Versatile microwave waveforms, such as sinusoidal, triangular, square and sawtooth waveforms, are widely used in signal processing [1,2], modern radar [3,4], and optical communication systems [5,6]. Usually, electric frequency synthesizers or digital-analog converters (DAC) are used to generate arbitrary waveforms. However, conventional waveform generator based on electronic techniques is bandwidth limited (usually below 20 GHz) and cannot meet high-frequency and large-bandwidth requirements of future electric systems.

Photonic-assisted microwave waveform generation technique draws lots of attention due to its advantages of large bandwidth, light weight, immunity to electromagnetic interference, and compatible with other optical systems [7,8]. Photonic generation of sinusoidal waveforms with high and tunable frequency has been extensively studied, especially using the optoelectronic oscillator (OEO) [9,10] and external-modulation based frequency multiplication techniques [11,12]. Other versatile waveforms, including triangular, square and sawtooth waveforms, contain more abundant frequency components, and thus are relatively difficult to generate. In the past ten years, photonic techniques including direct space-to-time pulse shaping [13,14], spectral-shaping [7,15,16], photonic DAC [17,18], and external modulation of a continuous-wave light [19,20] have been utilized to generate microwave waveforms. In the direct space-to-time pulse shaping methods, an ultra-short optical pulse is sent to an optical pulse shaper to generate a pulse burst with increased pulse space, where the optical pulse shaper may be a space optical modulator [13] or an arrayed waveguide grating (AWG) [14], and the desired pulses are generated after photodetection of the pulse burst. In the spectral-shaping methods [7,15,16], the spectrum of an optical pulse is manipulated and a dispersive device is followed to realize wavelength-to-time mapping. Microwave waveforms can also be generated by photonic DAC techniques. In principle, arbitrary waveforms can be obtained using this method by defining the digital bit steam to be processed. However, the resolution is seriously limited by current photonic DAC capabilities [17,18].

In a typical external modulation scheme, a continuous-wave (CW) light is often modulated by a sinusoidal local oscillator (LO) signal through an external modulator, such as a phase modulator (PM), single or nested intensity modulator, or polarization modulator. After optical processing and photodetection by a photodiode (PD), harmonics of the LO signal are generated and thus forming the desired microwave waveforms. External modulation is one of the most promising microwave waveform generation methods since the LO signals with high and tunable frequency are easily available, and the external modulation system is simple in structure and expected to be integrated in the future. In recent years, various optical processing techniques have been applied with external modulation to generate microwave waveforms. Triangular waveforms are generated by optical phase manipulation in a dispersive fiber [19,20] or nonlinear polarization rotation in a highly nonlinear fiber [21]. Nested external modulator, dual-parallel Mach-Zehnder modulator (DPMZM) for example, is used for simultaneously electro-optic modulation and optical processing by properly biasing the sub-modulators to generate triangular waveforms [22]. In [23], an OEO based microwave photonic filtering technique is utilized to generate fundamental or frequency doubling triangular waveforms. The demerit of the above schemes is that the repetition rate is difficult to tune since inflexible fiber, electrical phase shifter and band pass filter are used. In [24–27], optical carrier manipulation technique is applied to generate microwave waveforms, where the intensity and phase of the optical carrier is processed by stimulated Brillouin scattering (SBS) [24], DPMZM based nonlinear modulation [25], or bidirectional use of a polarization modulator in a Sagnac loop [26]. Except for triangular waveforms, other waveforms with square or sawtooth shapes can also be generated in [20,25,26]. Previously, we have demonstrated the generation of triangular waveforms by phase modulation and fiber Bragg grating (FBG)-based spectrum manipulation [27]. However, the frequency tunability and system stability is challenged due to the use of inflexible and environment-sensitive FBG.

In this paper, a photonic triangular and square waveform generator based on phase modulation in a Sagnac loop and tunable dispersion is proposed. Similar to our previous scheme in [27], the PM is located in a Sagnac loop to realize carrier manipulation. Instead of optical filtering, however, a compact electrically-controlled tunable dispersion compensation module (TDCM) is utilized in this paper to introduce phase shifts on each order sidebands. The PM-based structure is simple and free from bias drift. In addition, optical filters and electrical devices are avoided, which makes the photonic microwave waveform generator flexible in wave shape and repetition rate. In the demonstration experiment, both triangular and square waveforms with the repetition rates of 5 and 10 GHz (bandwidths of 15 and 30 GHz) are generated. The bandwidth could be improved to above 120 GHz if larger-bandwidth measurement instruments are used.

2. Principle

2.1 Optical processing for phase modulated signal using Sagnac loop and TDCM

The schematic diagram of the microwave waveform generator based on phase modulation and tunable dispersion is illustrated in Fig. 1. A CW light wave from a laser diode (LD) is sent to a polarization beam splitter (PBS) through an optical circular (OC). By adjusting the polarization controller (PC1) after the LD, the two light waves at the output of PBS is set to be equal in power. The phase modulator is placed along the clockwise (CK) direction, with its input and output pigtails connected to the two output ports of the PBS. In the Sagnac loop, the light wave along the CK direction is phase modulated by an LO signal, while the light wave along the counter-clockwise (CCK) direction is negligible due to velocity mismatch [28]. The two light waves then arrive at the PBS with orthogonal polarizations and travel out the Sagnac loop through the OC. The polarization multiplexed light waves are sent to a polarizer to coupling the two light waves and then a PD is used to detect the waveforms.

Fig. 1 Schematic diagram of the microwave waveform generator based on phase modulation and tunable dispersion. LD: laser diode; PC: polarization controller; OC: optical circular; PBS: polarization beam splitter; PM: phase modulator; LO: local oscillator; Pol: polarizer; TDCM: tunable dispersion compensation module; PD: photodiode. Spectra of the optical signals (a) after PM (b) after polarizer and (c) after TDCM.

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Assuming the light wave output from the LD is E_in(t), the light wave output the PM along the CK direction can be expressed as

\begin{matrix} E_{C K} (t) = \frac{1}{\sqrt{2}} E_{i n} (t) \exp [j m \sin (Ω t)] \\ = \frac{1}{\sqrt{2}} E_{i n} (t) \cdot \sum_{n = - \infty}^{\infty} J_{n} \exp (j n Ω t) \end{matrix}

where Ω is the angular frequency of the LO signal, m is the modulation index, and J_n is the n-th order Bessel function of the first kind at m. The phase modulated light wave (CK) contains the optical carrier and each order sidebands, shown in Fig. 1(a). Higher order sidebands are ignored considering their small power.

The light wave output from the polarizer can be expressed as [29]

\begin{matrix} E_{P o l} (t) = \overset{C K}{\overset{︷}{\frac{1}{\sqrt{2}} E_{i n} (t) \exp [j m \sin (Ω t)]}} \cos α + \overset{C C K}{\overset{︷}{\frac{1}{\sqrt{2}} E_{i n} (t)}} \sin α \exp (j θ) \\ = \frac{\cos α}{\sqrt{2}} E_{i n} (t) \cdot [\sum_{n = - \infty}^{\infty} J_{n} \exp (j n Ω t) + \tan α \exp (j θ)] \end{matrix}

where α is the angle between the principal axes of the PBS and the polarizer, and θ is the phase difference between the two orthogonal light waves, which can be both adjusted by PC2. After the polarizer, the intensity and phase of the phase modulated signal is changed due to its combination with the carrier along the CCK direction, as shown in Fig. 1(b).

The TDCM is used to introduce different phases to each order sidebands, and the light wave after the TDCM is expressed as

\begin{matrix} E_{T D C M} (t) = E_{P o l} (t) \cdot \exp (j n^{2} φ) \\ \approx \frac{\cos α}{\sqrt{2}} E_{i n} (t) \cdot {\begin{cases} J_{0} + \tan α \exp (j θ) \\ + J_{1} \exp [j (Ω t + φ)] - J_{1} \exp [j (- Ω t + φ)] \\ + J_{2} \exp [j (2 Ω t + 4 φ)] + J_{2} \exp [j (- 2 Ω t + 4 φ)] \\ + J_{3} \exp [j (3 Ω t + 9 φ)] - J_{3} \exp [j (- 3 Ω t + 9 φ)] \end{cases}} \end{matrix}

where

φ = - D λ^{2} Ω^{2} / (4 π c)

is the dispersion-induced phase shift on the ± 1st order sidebands, c is the light velocity, λ is the wavelength of the optical carrier and D is the dispersion value [30]. For each order sideband, the dispersion-induced phase shift after dispersion is proportional to the square of its frequency relative to the optical carrier, as shown in Fig. 1(c).

After optical processing by the Sagnac loop and the TDCM, the phase modulated signal is converted to intensity modulation and harmonics of the LO signal can be detected by the PD. The current of the electrical signal output from the PD is written as

i (t) = η E_{D} (t) \times E_{D}^{*} (t) \propto Γ_{1} \sin (Ω t) + Γ_{2} \cos (2 Ω t) + Γ_{3} \sin (3 Ω t)

where η is the responsivity of the PD. According to (3), the coefficients of the fundamental, second and third harmonics of the LO signal are approximately calculated as

{\begin{matrix} Γ_{1} \approx - 2 J_{0} J_{1} \sin φ - 2 J_{1} J_{2} \sin 3 φ - 2 J_{2} J_{3} \sin 5 φ - 2 J_{1} \tan α \sin (φ - θ) \\ Γ_{2} \approx - J_{1}^{2} + 2 J_{0} J_{2} \cos 4 φ + 2 J_{1} J_{3} \cos 8 φ + 2 J_{2} \tan α \cos (4 φ - θ) \\ Γ_{3} \approx 2 J_{1} J_{2} \sin 3 φ - 2 J_{0} J_{3} \sin 9 φ - 2 J_{3} \tan α \sin (9 φ - θ) \end{matrix}

2.2 Generation of triangular waveforms

As we all know, the Fourier expansion of a triangular waveform with a full duty cycle can be expressed as

T_{t r i} (t) \propto \sin (Ω t) - \frac{1}{3^{2}} \sin (3 Ω t) + \frac{1}{5^{2}} \sin (5 Ω t) + \dots

The ideal triangular waveform contains infinite harmonics, with the even harmonics suppressed and the odd harmonics with specific relation in intensity and phase. Due to the limitation in bandwidth, the ideal waveform cannot be realized in practice, and an approximation using finite harmonics is often considered in the generation of microwave waveforms. As described in [26], the first three harmonics can be used as an approximation of a triangular waveform since the intensity of higher order harmonics are much smaller. According to (6), in order to generate a triangular waveform, the coefficients of the fundamental term, second and third harmonics should satisfy the following condition

Γ_{2} = 0, Γ_{1} / Γ_{3} = - 9

which indicates that the second harmonic should be suppressed and the power ratio between the fundamental term and the third harmonic should be 19.1 dB. In addition, the initial phase difference between the fundamental term and the third harmonic should be 180°. The above equation has multiple solutions and can be easily satisfied by adjusting the modulation index, the polarization state before the polarizer, and the dispersion value. For example, the following values can be used as a generation condition of a triangular waveform

m = 0.91, φ = - {17.1}^{0}, α = 45^{0}, θ = - 15^{0}

2.3 Generation of square waveforms

The Fourier expansion of a square waveform with a full duty cycle can be expressed as

T_{s q u} (t) \propto \sin (Ω t) + \frac{1}{3} \sin (3 Ω t) + \frac{1}{5} \sin (5 Ω t) + \dots

Similarly, the first three harmonics can be used as an approximation of a square waveform when the coefficients of the fundamental term, second and third harmonics satisfy the following condition

Γ_{2} = 0, Γ_{1} / Γ_{3} = 3

The second harmonic should also be suppressed and the power ratio between the fundamental term and third harmonic should be 9.5 dB. Additionally, the fundamental term and third harmonic have the same initial phase, unlike that in the triangular waveforms. The following set of values is an example of the multiple solutions that can be used as the generation condition of a square waveform

m = 2.2, φ = - {7.2}^{0}, α = 60^{0}, θ = - 111^{0}

The dispersion-induced phase shift φ is dependent on the LO frequency and dispersion value. If the waveform generation condition expressed in (8) and (11) is used, the dispersion values required by each LO frequency when generating triangular and square waveforms are drawn in Fig. 2, respectively. The wavelength of the optical carrier is set as 1550.12 nm in the calculation.

Fig. 2 Required dispersion values as a function of LO frequency when generating triangular and square waveforms.

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3. Experimental results

A proof-of-concept experiment is set up according to Fig. 1. A CW light wave with an average power of 10 dBm and a wavelength of 1550.12 nm is generated from a laser source (Yokogawa AQ2200-136) and sent to the Sagnac loop. The phase modulator (Photline, MPZ-LN-40) in the loop has a half-wave voltage of 6–7 V and a bandwidth of above 30 GHz, and is driven by an LO signal from a microwave source (Agilent, N5183A MXG). The TDCM after the polarizer has a tunable dispersion range from −1200 to 1200 ps/nm. A wideband PD (U2T DPRV2022A) is applied to realize the photodetection and the detected electrical signal is measured by a 50-GHz electrical spectrum analyzer (R&S, FSW-50) and a 30-GHz real-time oscilloscope (LeCroy, SDA830Zi-A).

The frequency of the LO signal is first set as 5 GHz. In order to generate triangular waveforms, the modulation index, polarization state, and dispersion-induced phase shift are set according to the example given by (8). The power of LO signal is set as 15 dBm to achieve a modulation index of about 0.9, and the measured spectrum of phase modulated signal is drawn in Fig. 3 using a black line. The dispersion value is set as 480 ps/nm, and the dispersion-induced phase shift φ is calculated as −17.3°. The polarization of the light wave injected into the polarizer is manually controlled through a PC to change α and θ. After properly tuning the PC, the measured spectrum of optical signal before the PD is drawn in Fig. 3 using a red line, and the spectrum of the detected electrical signal is shown in Fig. 4(a). As can be seen from Fig. 4(a), harmonics of the drive signal are generated. The power ratio between the fundamental term (5 GHz) and third harmonic (15 GHz) is 19.6 dB, which is close to the ideal value (19.1 dB). The second harmonic is well suppressed, with a power 43.8 dB lower than the fundamental term. As a result, a full-duty-cycle triangular waveform is observed in the oscilloscope. The eye diagram of the generated triangular waveform is drawn in Fig. 4(b) with blue dots. The period of the generated waveform is 200 ps, indicating a repetition rate of 5 GHz, which is equal to that of the drive signal. For comparison, the normalized ideal triangular waveform is also drawn in Fig. 4(b) with a red line. The measured waveform is close to the ideal one and the calculated root-mean-square error (RMSE) is 0.0705.

Fig. 3 Measured spectra of the optical signals after the PM and before the PD when generating triangular waveforms using a 5-GHz LO signal.

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Fig. 4 Measured spectra (a, c) and waveforms (b, d) of the generated triangular waveforms with the repetition rate of 5 GHz (a-b) and 10 GHz (c-d).

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To demonstrate the tunability in repetition rate, the frequency of the LO signal is changed to 10 GHz. The polarization state of the light wave before the polarizer remains unchanged. The power of the LO signal is increased by about 1 dB because of the increase of the half-wave voltage at high frequency. The dispersion value is set to 120 ps/nm to obtain the same dispersion-induced phase shift (φ = −17.3°). The spectrum of the detected electrical signal is shown in Fig. 4(c), where the fundamental term is 35.4 and 19.1 dB larger than the second and third harmonics. The measured eye diagram of the generated triangular waveform is drawn in Fig. 4(d). The period is 100 ps and the repetition rate is 10 GHz, which is consistent with the expectation. Similarly, the measured waveform is close to the ideal one and the RMSE is 0.0751.

In experiment demonstrating the generation of square waveforms, the modulation index, polarization state of the light before the polarizer, and the dispersion value is set according to (11). An LO signal with the frequency of 5 GHz and the power of 23 dBm is applied to drive the phase modulator. The modulation index is calculated as about 2.2 and the spectrum of the phase modulated signal is drawn in Fig. 5 using a black line. The dispersion-induced phase shift is set to −7.2° by setting the dispersion value of the TDCM at 200 ps/nm. After properly tuning the PC before the polarizer, the spectrum of the optical signal before the PD is drawn in Fig. 5 using a red line, and the measured electrical spectrum is shown in Fig. 6(a). It can be seen that the power of the fundamental term is 9.6 dB larger than that of the third harmonic. The second harmonic at 10 GHz is well suppressed and is 46.1 dB lower than the fundamental term in power. Then, a full-duty-cycle square waveform with a period of 200 ps is observed in the oscilloscope, and its eye diagram is drawn in Fig. 6(b) with blue dots. The repetition rate is 5 GHz, the same as that of the drive signal. The normalized ideal square waveform is also drawn in Fig. 6(b) with a red line for comparison and the calculated RMSE is 0.3323.

Fig. 5 Measured spectra of the optical signals after the PM and before the PD when generating square waveforms using a 5-GHz drive signal.

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Fig. 6 Measured spectra (a, c) and waveforms (b, d) of the generated square waveforms with the repetition rate of 5 GHz (a-b) and 10 GHz (c-d).

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In order to generate a square waveform with larger bandwidth, the frequency of the LO drive signal is then changed to 10 GHz. Similarly, the polarization state of the light wave before the polarizer remains unchanged, and the power of the LO signal is increased a little to get the same modulation index m = 2.2. The dispersion value of the TDCM is changed to 50 ps/nm to achieve the same dispersion-induced phase shift at −7.2°. The spectrum of the detected electrical signal are shown in Fig. 6(c). As expected, the power of the fundamental term is 9.5 dB larger than that of the third harmonic, and the second harmonic is suppressed. The eye diagram of the generated square waveform is drawn in Fig. 6(d). A period of 200 ps or a repetition rate of 10 GHz is observed. Compared with the normalized ideal square waveform, the calculated RMSE is 0.3415.

4. Discussion

This paper aims to providing a simple, realizable and reconfigurable waveform generator, and a photonic microwave waveform generator based on phase modulation and tunable dispersion is proposed. The presented method is based on external phase modulation of a CW light wave using a sinusoidal LO signal. By properly optical processing through a Sagnac loop and a TDCM, the photonic system can convert the sinusoidal LO signal to both triangular and square waveforms, as shown in Fig. 4 and Fig. 6.

The repetition rate of the generated waveform can be changed by tuning the frequency of the LO signal, which is experimentally verified using a 5- and 10-GHz LO signal, respectively. The fine tunability of the system results from the use of Sagnac loop and TDCM. The Sagnac loop based carrier manipulation is independent of the light wavelength and the LO frequency. Although the phase shift introduced by the TDCM is proportional to the light wavelength and the square of the LO frequency, the dispersion value of a commercially available TDCM is tunable in a large range. Considering that the bandwidth of the commercially available modulator and photodiode is 40 GHz and 120 GHz, respectively, and the sinusoidal LO signal as high as 40 GHz is easily available, the repetition rate of the triangular or square waveforms generated using this scheme can exceed 40 GHz (or 120-GHz bandwidth). In practical applications, the polarization state of the light before the polarizer can be adjusted by an electrically-controlled PC or a polarization modulator, which is predicted to significantly improve the tuning speed and accuracy, since the current polarization modulator can support an ultrahigh-speed of above 50 Gbps [31].

The performance of the generated waveforms is evaluated by comparing with the ideal waveforms. The generated triangular waveforms in the experiment are close to the ideal waveforms and the calculated RMSE is about 0.07. When generating square waveforms, however, relatively larger RMSEs (above 0.3) are observed. This issue results from the fact that the relative power of the high-order harmonics of the square waveforms is larger than that of the triangular waveforms. As only the first three harmonics is considered in the proposed method, the approximate square waveforms lost more Fourier components.

From the eye diagrams in Fig. 4 and Fig. 6, we can also observe that the waveforms in different periods are not exactly the same, indicating the instability of the generated waveforms. The main challenge for the system stability may result from the polarization instability and light reflection in the Sagnac loop. In order to alleviate the environment sensitivity of the polarization state, the pigtails of the PBS and the modulator in the loop are preferred to be polarization maintaining fiber. Improved polarization stability is also predicted if the manually-controlled PC is replaced by an electrically-controlled PC or a polarization modulator. The light reflection is mainly caused by the return loss of fiber connectors, and can be reduced by using less connectors or choosing the connectors with large return loss.

5. Conclusion

A photonic microwave waveform generator based on phase modulation in a Sagnac loop and tunable dispersion is proposed. In the demonstration experiment, full-duty-cycle triangular and square waveforms with repetition rates of 5 and 10 GHz are successfully generated by simply adjusting the power and frequency of the LO signal, the polarization state before the polarizer and the dispersion value. Thanks to the frequency and wavelength independent of the Sagnac loop and the large tuning range of compact TDCM, the generated waveforms exhibits fine tunability in wave shape and repetition rate. The repetition rate of the generated waveforms can be improved to above 40 GHz (120-GHz bandwidth) if larger-bandwidth measurement instruments are used. The phase modulation based structure also features the advantages of easy implementation and free from bias drift.

Acknowledgments

This work was supported in part by National Natural Science Foundation of China (NSFC) under Grant 61306061, in part by China 111 Project under Grant B08038, and in part by National Key Laboratory Foundation of China under Grant 9140C530202140C53011.

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Photonic microwave waveform generation based on phase modulation and tunable dispersion

Abstract

1. Introduction

2. Principle

2.1 Optical processing for phase modulated signal using Sagnac loop and TDCM

2.2 Generation of triangular waveforms

2.3 Generation of square waveforms

3. Experimental results

4. Discussion

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (6)

Equations (11)

Optics Express