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Abstract
An all-photonic approach of microwave waveforms generation and transformation is proposed and experimentally demonstrated. From the perspective of envelope function operation in time domain, an initial triangular waveform is transformed into square waveform and sawtooth (or reversed-sawtooth) waveform via two types of differentiators, respectively. In addition, by using a SOA as a multiplier, both bright and dark parabolic pulses are achieved, which are further transformed into sawtooth (or reversed-sawtooth) waveform by taking the first derivative operation. The feasibility of the system is verified by theoretical analysis and simulation. In experiment, all of the expected results are successfully demonstrated and agree with the theoretical analysis well. This scheme provides a novel access to implement all-optical microwave waveforms generation, transformation, signal processing and computing.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Microwave waveforms generation and transformation is one of the fundamental demands for signal processing, radar, sensors and wireless communication [1–4]. Conventionally, such tasks are performed by electronic systems, which have limited bandwidth and difficulties for high speed signal processing in an optical network. To solve the problem, photonic approach is an attractive way because photonic systems exhibit wide bandwidth and can directly process optical signals without optical-to-electrical (OE) and electrical-to-optical (EO) conversions [5,6]. Therefore, numerous photonic solutions for waveforms generation and transformation have been proposed in recent decades [7–12].
Fourier synthesis is the typical method to implement waveforms generation and transformation. Since an optical signal can be expressed as the summation of Fourier components with suitable weights and phases, waveforms generation or transformation is able to be performed by manipulating spectral lines from optical frequency combs. Although this method has been successfully demonstrated by many previous works [7,8], the environmental sensitivity and complex spatial operation make it inconvenient for practical applications. External modulation technique provides a relatively simple way to generate required Fourier components [13–18]. By controlling modulation parameters and manipulating modulation sidebands, the wanted waveforms are synthesized. However, modulation process has limited bandwidth as well as the number of modulation sidebands, which affects the accuracy of high-frequency signal generation. In addition, waveform transformation from an original waveform could not be flexibly carried out by modulation technique based Fourier synthesis method.
Different from the principle of Fourier synthesis, time-domain synthesis is another effective method to perform waveform generation and transformation. When the waveforms are treated as geometrical patterns, some desired signals can be synthesized by means of carving, splitting joint and overlapping the available signal envelopes [9–10,19–20]. This method avoids inconvenient spectral lines manipulation and provides a visualized operation for waveform generation, but this technology presents weaker adaptivity, because nonlinear switching manipulations are usually difficult to be realized.
In fact, the basic character of a signal is a temporal intensity distribution, which can be mathematically expressed as a function of time. It means that flexible photonic waveform generation or transformation can be achieved by performing mathematical operation of envelope functions [11]. This principle has been widely adopted in electrical signal generation or processing through arithmetic circuit, but it is less applied in optical domain due to the relatively immature development of photonic signal processors. Nevertheless, there are still some photonic processors have been reported, such as integrators [21] and differentiators [22,23], which are utilized for digital signal processing, ultrawideband (UWB) signal generation or optical pulse shaping and coding [24,25]. However, as far as we know, the potential and feasibility of microwave waveform generation or transformation based on photonic arithmetic circuit has not been sufficiently exploited yet.
In this paper, several fundamental waveforms generation and transformation based on all-optical temporal multiplier and differentiators are proposed and demonstrated. In our system, two types of differentiators with different rules of operation are employed, which are called differentiator type 1 [22] and differentiator type 2 [23] respectively. Starting from triangular waveform, square waveform is directly obtained through differentiator type 1. If the triangular pulse passes through differentiator type 2, sawtooth (or reversed-sawtooth) waveform with half duty cycle are generated, which can be further multiplexed to synthesize frequency doubling sawtooth (or reversed-sawtooth) waveform with full duty cycle. By using a semiconductor optical amplifier (SOA) as a multiplier, parabolic pulse is produced by the product operation of two triangular envelope functions. In addition, sawtooth (or reversed-sawtooth) waveform can be further obtained by carrying out the first derivation of parabolic envelope function. All of the mentioned results are verified by simulation and experimental demonstration. This work not only exhibits a way to realize microwave waveforms generation or transformation, but also provides an access to all-optical signal processing or computing.
2. Principle
The conceptual diagram of the proposed microwave waveform generation and transformation is illustrated by Fig. 1. In this scheme, the initial triangular waveform with the second-order approximation is generated by a waveform generator, whose envelope function is denoted as ${I_{tr}}(t)$. In order to approach the edges of triangular waveform with better linearity, the Fourier expansion of triangular waveform is optimized as ${I_{tr}}(t) = \cos ({\omega _m}t) + (1/13)\cos (3{\omega _m}t)$, where ${\omega _m}$ is the angular frequency of triangular pulses [11]. For the following waveform transformation, the initial triangular signal is processed by three processing units, respectively. In unit 1, the first derivation of ${I_{tr}}(t)$ is performed by differentiator type 1, which gives a result of square pulse written as ${I_{tr}}^{\prime}(t)$. In unit 2, the operation result of ${I_{tr}}^{\prime}(t) \times {I_{tr}}(t)$ is carried out by differentiator type 2, which transforms the triangular waveform into sawtooth waveform with 50% duty cycle. After a time division multiplexer (TDM), frequency doubling sawtooth waveform with full duty cycle is achieved. In unit 3, when the triangular pulse is launched into the multiplier, the operation results of ${I_{tr}}^2(t)$ and $- {I_{tr}}^2(t) + \textrm{c}$ (c is a constant) can be obtained at the same time, which means bright and dark parabolic pulses are generated. By taking the first derivation again, parabolic pulse is transformed into sawtooth waveform with full duty cycle.
Fig. 1. Conceptual diagram of the waveform generation and transformation. TDM: time division multiplexer, Itr: triangular envelope function, Isq: square envelope function, Isa: sawtooth envelope function, IBpa: bright parabolic envelope function, IDpa: dark parabolic envelope function.
2.1 Differentiator type 1
The schematic structure of the differentiator type 1 is shown by Fig. 2, which is firstly reported in [22]. The operating principle is based on the mathematical definition of differentiation, and the mathematical express can be written as
In our case, the initial triangular waveform, ${I_{tr}}(t)$, has the characteristic of time symmetry within one period. Therefore, the corresponding envelope function $- {I_{tr}}(t) + c$ is equivalent of the original signal with half period shift. As shown by Fig. 3, the expected square waveform can be derived by overlapping two identical triangular waveforms with relative time shift. In order to evaluate the quality of the generated signal under different $\triangle t$, we introduce the concept of goodness of fit from mathematical statistics to analyze the fitting degree between the differential results and the standard waveform. This concept is defined as the fitting degree between the regression line and the observed value, which is expressed by
Fig. 3. The simulation result of square waveform generation based on the temporal differentiation of triangular waveform with second-order approximation.
Figure 4 shows the fitting degree of the transformed square waveform with different $\triangle t$. Here, the triangular waveform with the second-order approximation is taken into account. From the figure, the higher fitting degree is approaching along with the decrease of $\triangle t$, but the direct current (DC) is increased due to the term $c/\triangle t$ in Eq. (1). To balance the DC and fitting degree of transformed waveform, the value of $\triangle t = 0.2T$ is chosen, which corresponds fitting degree of 0.821.
2.2 Differentiator type 2
The schematic structure of differentiator type 2 is shown by Fig. 5, which is based on phase modulation cascading an optical filter with linear edge [23]. For an optical signal with intensity of $I(t)$, the light field can be expressed as
where ${\omega _C}$ and ${\Phi _0}$ are the angular frequency and the initial phase of the optical field. When the light field is phase modulated by a signal $\varphi (t)$, the optical field becomes where $\beta$ is the phase-modulation index. In this case, the light field has instantaneous angular frequency of $\omega = {\omega _c} + \beta \varphi ^{\prime}(t)$. Now, let’s consider an optical bandpass filter with linear edge, whose frequency response is shown by Fig. 6, and the spectral transfer function is written asAssume that ${\omega _C}$ is located at the linear slope region of the filter and the bandwidth of the chirped optical field can be covered by the edge, the filtered optical field in frequency domain can be described by
where $E(\omega )$ is the Fourier transform of $E(t )$. By taking the inverse Fourier transform to Eq. (6), the optical field in time domain can be expressed asIn Eq. (8), the first term on the right side is a constant, which presents the DC on the output signal. Under small phase-modulation index, the second term can be neglected because it is a quadratic term. Thus, the optical intensity can be simplified as
In this differentiator, the optical filter acts as a frequency discriminator to detect the chirp variation of the phase modulated signal, which may give the first temporal derivative of phase variation. Theoretically, accurate results can be obtained if the filter window has ideal linear edge, as shown in Fig. 6. Usually, such filter is not easy to get, but Gaussian filter, delay interferometer and specially designed FBG can be employed to implement the task, because those filters have the windows with quasilinear edges. These replacements can also perform well and the corresponding results have been demonstrated and verified in [23,28–29].
In our case, the initial signal ${I_{tr}}(t)$ is an optical triangular pulse. Through self-phase modulation (SPM) effect in high nonlinear fiber (HNLF), the phase variation on the light field can be given by
where $\gamma$ represents the nonlinearity coefficient and ${L_{eff}}$ is the effective length of the HNLF. According to Eq. (9), the outputs in time domain areThe operation process can also be diagrammatically understood by Fig. 7. In Fig. 7(a), the operation result of $[C - D{I_{tr}}^{\prime}(t)]$ is a square pulse with half duty cycle, and the $[C - D{I_{tr}}^{\prime}(t)]{I_{tr}}(t)$ gives the reversed-sawtooth waveform with 50% duty cycle. By taking multiplexing technique, the frequency doubling reversed-sawtooth waveform with full duty cycle can be obtained. Similarly, the sawtooth waveform with 50% duty cycle and the frequency doubling sawtooth waveform with full duty cycle are generated when ω has the condition of ${\omega _0} < \omega < {\omega _2}$, as illustrated by Fig. 7(b).
2.2 Multiplier
Essentially, the temporal multiply operation of optical intensity is a process of intensity modulation, which can be implemented by electro-optical modulator or other modulation devices. In our scheme, an optical-optical modulation is carried out by a SOA. As shown in Fig. 8, when the initial triangular pulses on the wavelength of λ1 is power split and injected into the SOA with opposite directions, optical-optical modulation is occurred due to carrier dynamics [30–31]. At the same time, a CW probe light with wavelength of λ2 is also launched into the SOA in the same direction as ${I_{tr1}}(t)$. Here, the CW probe light plays two important roles in this process. At first, the carrier dynamics of the SOA can be adjusted by controlling the injection strength of the CW, which is able to suppress pattern effect and approach linear optical-optical modulation [30]. Secondly, the CW probe light may suffer cross-gain modulation (XGM) effect within the SOA and form the negative pattern of the product [31]. To simply understand the process of the multiplier, the qualitative description is given by Fig. 9.
Fig. 8. Schematic structure of unit 3 including the multiplier and the first type of differentiator.
Assuming that ${I_{tr2}}(t)$ and CW with proper strengths are sent into the SOA, the carriers will be adequately consumed. Therefore, the normalized gain function in the SOA is negative to ${I_{tr2}}(t)$, which is expressed as
Clearly, Eq. (13) presents a bright parabolic function. Meanwhile, due to the XGM effect, the intensity of injected CW will be converted into
Fig. 10. Simulation results of the generated parabolic pulses. (a): Bright parabolic pulse. (b): Dark parabolic pulse.
One interesting thing is that, the generated bright and dark parabolic pulses are just satisfied with the requirement of the differentiator type 1. By properly controlling the relative time delay $\triangle t$, we can realize the differential operation as shown in Fig. 11, which are sawtooth or reversed-sawtooth pulses with full duty cycle.
Fig. 11. The simulation results of sawtooth waveforms generation based on the temporal differentiation of parabolic pulses. (a): Differentiation of bright parabolic pulse. (b): Differentiation of dark parabolic pulse.
Similarly, Fig. 12 shows the fitting degree of the achieved reversed-sawtooth waveform with different $\triangle t$. When the value of $\triangle t = 0.2T$ is chosen, the fitting degree is 0.801.
3. Experimental results and discussion
To investigate the flexibility of the proposed scheme, the experiment demonstration is carried out based on the configuration illustrated by Fig. 13. In this scheme, the waveforms generation or transformation is started from an initial triangular waveform, but the generation of triangular waveform is not the focal point in this work. To exhibit the universality, here, the method of triangular waveform generation reported in [32] is directly employed, because it is a purely optical spectrum manipulation method from a sinusoidal signal without assistant envelope detection. Briefly, a laser diode (LD1) emits a continuous wave with wavelength of 1547.18 nm, which is modulated by a MZM with 5-GHz sinusoidal drive signal, where the MZM is biased at minimum transmission point and the modulation index is 1.29. Because the MZM is a polarization dependent device, if the polarization direction of the incident optical field has an angle with the principal axis (the X axis in Fig. 13) of the MZM, only the optical component on the X polarization state will be modulated and the optical carrier on the Y polarization state keeps unchanged. Therefore, the modulated optical field has odd-order sidebands and orthogonal optical carrier. After a band-pass filter with square filter window, the negative odd-order sidebands will be filtered out. The phase relationship among optical carrier and remained sidebands may satisfy the requirement of Fourier expansion of triangular signal. Once the optical carrier and the modulation sidebands are projected on the same polarization direction through a polarizer (POL) as shown in Fig. 14(a), the beating signal of those components contribute the desired triangular waveform. The detailed operation procedure is also illustrated in Fig. 13. In our case, the generated triangular waveform has the coefficient ratio of 1/13 between the first-order and the third-order harmonics, which provides a triangular waveform with better linearity. The measured optical spectrum, waveform and electrical spectrum of the 5-GHz triangular pulses are shown by Fig. 14, which are monitored by optical spectrum analyzer (OSA, YOKOGAWA AQ6370C), oscilloscope(OSC, Agilent 86100D Infiniium DCA-X) and electrical spectrum analyzer (ESA, Agilent N9010A).
Fig. 13. Experiment setup. LD: laser diode, PC: polarization controller, MZM: Mach-Zehnder modulator, EDFA: erbium-doped fiber amplifier, TBPF: tunable band-pass filter, POL: polarizer, ODL: optical delay line, PBC: polarization beam combiner, HNLF: high nonlinear fiber, HBF: high-birefringence fiber, SOA: semiconductor optical amplifier, Cir: circulator, WDM: wavelength division multiplexer.
Fig. 14. Measured results of the generated triangular waveform. (a) Optical spectrum, (b) Corresponding electrical spectrum and waveform.
From the initial 5-GHz triangular waveform, the first type differential operation is performed by unit 1. As shown by Fig. 13, when the initial triangular pulse is power split by a 3-dB coupler, the signal in the lower branch has envelope function of ${I_{tr}}(t)$. In the upper branch, the envelope function of $- {I_{tr}}(t - \varDelta t) + c$ can be achieved by properly tuning ODL1. Once these two pulses are recombined by PBC1, the first type differential operation is implemented, where the PC3 and PC4 are used to balance the signal powers on two branches, and PBC1 prevents optical interference. As described in the theoretical analysis, the relative time delay may affect the result of the differential operation. Here, a relative time delay of 40 ps (corresponding to one-fifth period) is taken. Figure 15 shows the measured results. Obviously, it presents a square envelope function yielding from the differential operation of triangular envelope function.
To implement the second type differential operation, the power of the initial triangular pulses are amplified to be 20 dBm by EDFA 2 and sent into differentiator type 2, as shown by the unit 2 in Fig. 13. Within this differentiator, there is a 700-m HNLF with nonlinear coefficient of 11 W−1 km−1. When the amplified triangular pulses pass through the HNLF, SPM effect occurs. Due to the effect of phase modulation, the light field shows more modulation sidebands. However, the characteristics of phase modulation do not change the profile of the envelope, but only introduce a chirp on the light field. The corresponding optical spectrum is shown by Fig. 16(a) (the red dash line). As we know, a high-birefringence fiber loop mirror (HB-FLM) can be employed as an optical filter [33], which has periodical filter windows with sine-squared profile and presents quasilinear transmission characteristic at the middle region of window edges. In our experiment, a 10-m high-birefringence fiber (HBF) with beat length of 3.81 mm is inserted in the Sagnac loop, which determines a spectral period of 0.59 nm for the HB-FLM (seen the black dot line in Fig. 16(a)). By properly tuning the wavelength of the triangular pulses and cascading this HB-FLM, the chirped triangular pulses are filtered by the rising edge of the optical filter. After the filter, the differential operation of $[C + D{I_{tr}}^{\prime}(t)]{I_{tr}}(t)$ is performed, and the 5-GHz sawtooth pulses with half duty cycle is consequently achieved. The measured optical spectrum and waveform are shown by Fig. 16(a) (the blue solid line) and Fig. 16(b), respectively. Furthermore, 10-GHz sawtooth pulse with full duty cycle can be obtained after a multiplexer, as shown by Fig. 17. One interesting thing is that, if the chirped triangular pulse is filtered by the falling edge of the optical filter, reversed-sawtooth pulses are achieved. The measured results are shown in Fig. 18(a) and Fig. 18(b). Again, Fig. 19 shows the 10-GHz reversed-sawtooth pulses with full duty cycle after the TDM.
Fig. 16. Measured results of differentiator type 2. (a): The spectrum after SPM (red dash line), filter window (black dot line), optical spectrum after the rising edge of the optical filter (blue solid line). (b): The corresponding waveform.
Fig. 18. Measured results of differentiator type 2. (a): The spectrum after SPM (red dash line), filter window (black dot line), optical spectrum after the falling edge of the optical filter (blue solid line). (b): The corresponding waveform.
When the initial triangular pulses are launched into unit 3, the multiply operation is performed at first. As shown by Fig. 13, the initial 5-GHz triangular pulse with wavelength of 1547.18 nm is divided into two paths by a 3-dB coupler and simultaneously injected into the SOA in opposite directions. The SOA is biased at 420 mA and the injection optical powers of two signals are both 0.5 dBm. At the same time, a CW probe light with wavelength of 1552.48 nm is emitted from LD 2 and also launched into the SOA alone path 1. The injection power of the CW probe light is set at −4 dBm. Under these injection power conditions, linear optical-optical modulation in the SOA is occurred. By properly tuning ODL3, the time relationship between gain variation and the propagated signal on path 1 can be controlled. Now, the optical-optical modulation, which is also a multiply operation, transforms the triangular pulse on path 1 into bright parabolic pulse and converts the CW probe light into dark parabolic pulse through XGM effect in SOA. The outputs from the multiplier can be independently measured after a WDM. In this case, the wavelengths of the bright and dark parabolic pulses are 1547.18 nm (the wavelength of initial triangular pulse) and 1552.48 nm (the wavelength of CW probe light) respectively. The optical spectra and corresponding waveforms are given by Fig. 20. Compared with the optical spectrum of triangular pulse, the optical spectrum of bright parabolic pulse has more sidebands due to the optical-optical modulation in SOA as shown in Fig. 20(a). Similarly, in Fig. 20(c), the optical spectrum of dark parabolic pulse also shows similar result to the bright parabolic pulse.
Fig. 20. Measured optical spectra and waveforms after multiply operation. (a), (b) The optical spectrum and waveform of the bright parabolic pulse. (c), (d) The optical spectrum and waveform of the dark parabolic pulse.
Furthermore, the generated bright and dark parabolic pulses provide the required envelope functions for implementing the first type differential operation. Through a WDM, bright and dark parabolic pulses are separated into the upper and lower wavelength branches, which are then recombined with 40-ps (one-fifth period) relative time delay by tuning ODL4 to contribute the results of differential operation. Figure 21 gives the results, which are 5-GHz reversed-sawtooth pulse and sawtooth pulse with full duty cycle.
Fig. 21. Measured waveforms. (a) The first order differential result of bright parabolic pulse. (b) The first order differential result of dark parabolic pulse.
According to the experimental demonstration above, the feasibility of the proposed scheme has been confirmed, which agrees with the theoretical expectations well. Limited by the bandwidth of our test instrumentation (the bandwidth of the used OSC and ESA are 38 GHz and 26.5 GHz), the signals with repetition frequency of only 5 GHz are demonstrated. However, the scheme has great potential to process signals with much higher frequency.
4. Conclusion
In conclusion, an all-optical method for microwave waveform generation and transformation based on photonic temporal operators is proposed and verified. By employing two types of photonic temporal differentiator and one multiplier, the square pulse, frequency doubling sawtooth (or reversed-sawtooth) pulse, parabolic pulses and sawtooth (or reversed-sawtooth) pulse are achieved from an initial triangular waveform. All of the experimental results are consistent with theoretical analysis. Different from the ideas of Fourier synthesis and time-domain synthesis, this method takes the view of functional operation to perform waveform transformation, which makes the implementation procedure mathematically intuitionistic and provides a new access for flexible microwave waveform generation or transformation.
Funding
National Natural Science Foundation of China (61835003, 62105076); High Level Innovation Talent Program of Guizhou Province, China (2015-4010); Platform and Talent Program of Guizhou Province, China (2018-5781-1); Guizhou Provincial Science and Technology Projects (ZK-2021-327).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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