Reconfigurable microwave signal processor with a phase shift of &#x03C0;

Hao Sun; Xinyi Zhu; Wei Li; Ninghua Zhu; Ming Li

doi:10.1364/OE.26.010358

1. Introduction

Signal processors, such as a Hilbert transformer (HT) and a differentiator, are critical blocks for radar systems, telecommunication systems, optical computing, signal generation and ultrafast signal detection [1–3]. To overcome the bottleneck of traditional electrical circuits, microwave photonics signal processors have been broadly investigated in the last decades due to the broader operating bandwidth and lower power consumptions. For instance, a HT or a fractional Hilbert transformer (FHT) could be realized by using a Bragg grating [4–10], a multi-tap microwave photonics filter [11–13] or ring resonators [14, 15]. Moreover, a tunable fractional-order photonic differentiator has been proved by using Bragg gratings [17–19] or a distributed feedback semiconductor optical amplifier (DFB-SOA) [16]. What is more, due to the inherent convolution process between the input optical energy spectrum and the intensity profile of the modulation signal [25–28], a conventional incoherent photonic signal processor is ideal to achieve the temporal intensity integration [20, 21].

In recent years, many on-chip photonic signal processors have shown the fully reconfigurable capability for optical signal processing. For example, a fully reconfigurable on-chip photonic signal processor has been proposed based on an InP-InGaAsP material system [22]. Various signal processing functions could be performed by tuning the injection currents into active components of the signal processor, such as temporal integration, temporal differentiation and Hilbert transform. Similarly, an DFB-SOA is capable of performing several different functions including ordinary differential equation solving and temporal intensity differentiation by controlling the injection currents into the DFB-SOA [23], which is a simple and effective solution for all-optical signal processing and computing. Very recently, it was reported that a multipurpose silicon photonics signal processor core could implement over twenty different functions by using a two-dimensional photonic waveguide mesh [24], which shows a high potential of an all-optical field-programmable gate array in the future.

However, the reconfigurability of the photonic-assisted microwave signal processor is still a big challenge for its application in microwave band. Many reported microwave signal processors could only realize one certain signal processing function, such as the FHT [11, 12, 14]. Therefore, considering its inherent advantage of the reconfigurability and the simple structure, the time-spectrum (TSC) convolution system [25] has a high potential to achieve a reconfigurable microwave signal processor. Nevertheless, the random phase of the incoherent optical source would cause a temporal intensity convolution process. As a result, the signal processing functions with phase shifts, such as temporal Hilbert transformation or differentiation, could hardly be implemented by directly using the TSC system [25, 26].

In this paper, we propose and demonstrate a reconfigurable microwave signal processor based on a TSC system [25], where a balanced photodetector (BPD) is introduced to provide a phase shift of $π$ for the impulse response of our technique. Theoretically, the proposed signal processor is capable of performing any microwave signal processing function with a phase shift of $π$ . In our experiment, two kinds of signal processing functions including temporal intensity Hilbert transformations and temporal intensity differentiations have been achieved by programming the user-defined optical filter. Note that amplitude variations of the radio frequency (RF) intensity response of the proposed HT are less than 3dB over a bandwidth around 10GHz. To test the performance of the system, Gaussian pulses with pulse widths (full-width-at-half-maximum (FWHM)) around 125ps, 85ps and 68ps are selected as the input microwave signals. Then, temporal intensity Hilbert transformations and temporal intensity differentiations of those input microwave signals are achieved, with root mean square errors (RMSEs) fluctuating between 4.68% and 7.69%.

2. Principle

2.1. Principle of TSC system and our proposal

The basic principle of the conventional TSC system is shown in Fig. 1. The broadband incoherent optical source is firstly shaped by programming a user-defined optical filter. Then the shaped incoherent light is modulated by a microwave signal through an intensity modulator. After the optical dispersive delay line, an intensity convolution process between the modulation signal and a time-scaled version of the shaped optical energy spectrum could be implemented [25, 26]. Mathematically, the intensity profile of the output electrical signal could be written as,

I_{o u t p u t} (t) \propto S (f_{o}) \otimes I_{m w} {(t)}_{2 π f_{o} = t / ψ_{2}},

where

ψ_{2}

,

f_{o}

and

I_{m w} (t)

represent the group delay dispersion, optical frequency and the intensity of the modulation signal, respectively; in particular,

ψ_{2} = 2 π \partial^{2} β / \partial ω^{2} L

, where

β

is the propagation constant of the optical signal and

L

is the length of the dispersive element [29]. Note that the following inequality should be satisfied,

| Δ ω_{m}^{2} ψ_{2} / 8 | ≪ π

[25], so that the induced distortion on the modulation signal could be ignored. The symbol

Δ ω_{m}

is the full-width frequency bandwidth of the microwave signal.

Fig. 1 Principle of the conventional TSC system. PD: Photodetector. The symbol $\otimes$ represents the convolution process. The intensity profile of an output signal is a convolution between the time-mapped optical energy spectrum profile and the temporal intensity modulation waveform. Note that the orange line denotes the optical path and the green line represents the electrical path.

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Considering the time-mapped optical energy spectrum profile as the impulse response of the TSC system, signal processing functions such as Fourier transformation of the input microwave signal [30] could be performed by suitably reshaping the input optical energy spectrum profile. However, due to the random phase induced by the incoherent optical source [25, 26], it can hardly achieve the temporal Hilbert transformation or the temporal differentiation of the input microwave signals. Therefore, we replace the photodetector (PD) with a BPD to induce a phase shift of $π$ . The principle of our proposal is shown in Fig. 2. By programming the user-defined optical filter, the time-mapped optical energy spectrum profile is reshaped and divided into two groups (see Figs. 2(b)-2(d)). One group is connected with the positive branch of the BPD while the other one links up with the negative one. Based on our concept, not only the HT and differentiator, but also the impulse response with more than one phase jump could be implemented.

Fig. 2 Basic principle of the reconfigurable microwave signal processor. (a) Schematic diagram of our proposal; (b) Principle of the proposed HT; (c) Principle of the proposed differentiator. (d) Principle of a signal processor with two phase jumps. The orange and green lines represent the optical and electrical paths, respectively.

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2.2. Hilbert transformer

The Hilbert transform is given by,

h {(t)}_{H i l b e r t} \otimes I (t) = 1 / (π t) \otimes I (t),

where

I (t)

is the input signal and

h {(t)}_{H i l b e r t}

is the impulse response of the HT. Its frequency response could be written as,

H {(f)}_{H i l b e r t} = {\begin{cases} - j f \geq 0 \\ j f < 0 \end{cases} .

To emulate the flat frequency response of the ideal HT, the impulse response should be a sinc-like function. Mathematically, the designed impulse response could be written as,

h {(f)}_{H i l b e r t} \propto S {(f_{o})}_{2 π f_{o} = t / ψ_{2}} = {\begin{cases} | \sin c (k / ψ_{2} \cdot (t - t_{c})) | Δ t / 2 > t - t_{c} > 0 \\ 0 t - t_{c} = 0 \\ - | \sin c (k / ψ_{2} \cdot (t - t_{c})) | - Δ t / 2 < t - t_{c} < 0 \end{cases},

where

k

is a variable and

t_{c}

represents the time duration that the central frequency component travels in the dispersive element. More generally, the RF bandwidth of the frequency response

Δ f_{r}

is inversely related to the temporal resolution of the corresponding impulse response

δ t

[29]. In the designed impulse response, the temporal resolution depends on two factors: 1) the coefficient

k / ψ_{2}

, which is related to the number of the side lobes in the time aperture,

Δ t = 2 π \cdot Δ f \cdot | ψ_{2} |

; and 2) the optical spectral resolution of the programmable optical filter

δ f

, which provides the theoretical minimum temporal resolution when the coefficient

k / ψ_{2}

is a constant.

Firstly, a larger coefficient $k / ψ_{2}$ brings more side lobes in the time aperture so that temporal resolution of the impulse response would be higher. Secondly, according to the frequency-to-time mapping process, the minimum temporal resolution of the impulse response $δ t_{\min}$ is inherently determined by the optical spectral resolution of the optical filter and the group delay dispersion; mathematically $δ t_{\min} = ψ_{2} \cdot 2 π \cdot δ f$ . Therefore, when the dispersion is a constant, the optical spectral resolution of the optical filter provides a theoretical maximum operating bandwidth for our technique.

According to the simulated results (see in Fig. 3), a larger $k$ and a smaller dispersion could lead to a broader processing bandwidth of our technique. The maximum RF bandwidth of the simulated HT is around 23GHz and the phase shift is $π$ . Particularly, experimental results show that the incoherent optical spectrum profile is relatively flat in the range from 191.2THz to 193.7THz. As a result, the optical bandwidth in our simulation is set as 2.5 THz.

Fig. 3 Simulated intensity responses and phase responses of our proposal. (a) and (b) represent the intensity response and the phase response of the proposed Hilbert transform, respectively; (c) and (d) represent the intensity response and the phase response of the proposed differentiator. The blue, red and yellow lines are simulation results while the violet dash line denotes frequency responses of ideal signal processing functions. Note that $k_{1} = 4 π p s$ , $k_{2} = 1.4 π p s$ , $ψ_{2}^{1} = - (544 / 2 π) p s^{2} / r a d$ , $ψ_{2}^{2} = - (3888 / 2 π) p s^{2} / r a d$ and the bandwidth of the optical spectrum is around 2.5THz.

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2.3. Differentiator

The frequency response of a differentiator is given by,

H (f) = {\begin{cases} - 2 π j f f \geq 0 \\ 2 π j f f < 0 \end{cases} .

According to Eq. (3) and Eq. (5), there is a phase difference of π between the phase responses of the HT and the differentiator. Therefore, the two branches of the BPD should be switched compared to the proposed HT (see Figs. 2(b) and 2(c)). In order to emulate the frequency response of an ideal differentiator, we introduce the sinc² function. It is obvious that the Fourier transform of sinc² waveform is a triangular wave and one of its sides is a linear function. Thus, the impulse response could be written as,

h {(t)}_{d i f f} \propto S {(f_{o})}_{f_{o} = t / ψ_{2}} = {\begin{cases} - {| \sin c (k / ψ_{2} \cdot (t - t_{c})) |}^{2} Δ t / 2 > t - t_{c} > 0 \\ 0 t - t_{c} = 0 \\ {| \sin c (k / ψ_{2} \cdot (t - t_{c})) |}^{2} - Δ t / 2 < t - t_{c} < 0 \end{cases} .

According to the simulation in Fig. 3(c), the maximum bandwidth of the simulated differentiator is around 10GHz and the phase shift is

π

. It can be seen that the operating bandwidth varies along with the coefficient

k / ψ_{2}

.

2.4. Impulse response with more than one phase jump

By defining multiple non-overlapping bands in the user-defined optical filter, the proposed microwave signal processor also can realize an impulse response with more than one phase jump of $π$ . One example is shown in Fig. 2(d), where the center frequency component is chosen as the negative group and others are connected with the positive branch. To make a quantitative analysis, we design a sinc-like impulse response with two phase jumps. The simulated results are shown in Fig. 4. In the simulation, the designed impulse response is similar to a sinc-like function but part of the main lobe is negative over 0.7ns (see in Fig. 4(a)). The specially designed impulse response brings a unique frequency response, as shown in Fig. 4(b). There are several notches with the phase shift of $π$ and the power of the intensity response is higher in the range from 0.08GHz to 1.08GHz. According to Fig. 4(c), the input microwave signal is converted to a waveform with three wave crests. Compared with the simulation in Section 2.2, these two phase jumps in the impulse response lead to a frequency response with more than one phase shift. Meanwhile, the output signal is a convolution between the impulse response and the input microwave signal. Therefore, it is proved that our technique has the capacity for implementing any signal processing function with the phase shift of $π$ .

Fig. 4 Simulated frequency response of a signal processor with two phase jumps. (a) represents the designed impulse response. The impulse response is a sinc-like function while the part of the main lobe is negative. Note that $k = 1.4 π p s$ , $ψ_{2}^{2} = - (3888 / 2 π) p s^{2} / r a d$ and the temporal duration of the negative part is around 0.7ns. (b) shows the intensity response of the designed impulse response. (c) is the comparison between profiles of the input microwave signal and the output microwave signal. The RF bandwidth of the input Gaussian pulse is around 1GHz. (d) The phase response of the designed impulse response.

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3. Experimental results and discussion

3.1. Experimental setup

The experimental setup is shown in Fig. 5. A broadband incoherent light emitted from a super luminescent diode (SLD) is firstly modulated by using a 20 GHz-bandwidth Mach-Zehnder modulator (MZM) (Oclaro). The modulation waveform is provided by an arbitrary waveform generator (AWG) (Tektronix AWG70001A). After going through a single mode fiber (SMF) with a total GVD of around −544ps², the output of the intensity modulator is stretched. Then a programmable optical filter, WaveShaper (WS) (Finisar WaveShaper 4000s), is used to shape and divide the input optical energy spectrum into two groups which connect with different branches of a BPD (BPRFV2125AM). The bandwidth of the BPD is about 43GHz. Before the BPD, two tunable delay lines (TDLs) are applied to realize a fine tuning of the time delays. Finally, the output signals are monitored by implementing a real-time oscilloscope (Tektronix DPO73304D). In addition, we also measured the RF intensity response and phase response by using a Vector network analyzer (VNA) (Agilent 8722ET). To verify our concept, we hope to achieve an HT and a differentiator by specially programming the user-defined optical filter. Note that in our experiment, the variable k1 and y2 should be set as $k_{1} = 4 π p s$ and $ψ_{2} = - (544 / 2 π) p s^{2} / r a d$ .

Fig. 5 Experimental setup. Link 1: measurement of the frequency response. Link 2: measurement of output signals. Link 3: measurement of the shaped optical spectrum. SLD: Super luminescent diode, IM: Intensity modulator, AWG: Arbitrary waveform generator, SMF: Single mode fiber, VNA: Vector network analyzer, EDFA: Erbium-doped optical fiber amplifier, OSA: Optical spectrum analyzer, WS: WaveShaper, BPD: Balanced photodetector, OSC: Oscilloscope, TDL: Tunable delay line. The orange line represents the optical path while the green line shows the electrical path.

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3.2. Experimental results of the Hilbert transformer

Measured results are shown in Figs. 6(a)-6(c). Based on Eq. (4), we realized a sinc-like optical energy spectrum profile by programming the user-defined optical filter (see Fig. 6(a)). The shaped incoherent light is divided into two groups (Port1 and Port2) and the bandwidth of the optical spectrum is around 2.5THz. As the optical signal with higher frequency components travels faster in the SMF, the time-mapped optical energy spectrum profile in Port2 (higher frequency components) should enter the negative branch of the BPD and Port1 (lower frequency components) should be connected with the positive one. Then the designed impulse response of the proposed HT can be realized(see Eq. (4)).

Fig. 6 Measured frequency responses of our proposal and shaped optical energy spectrums. (a-c) shaped optical spectrums, RF intensity responses and phase responses of the HT; (d-f) Corresponding results of the differentiator. Comparisons between the simulation and measured results are also shown in these figures. Note that Port2 connects with the negative branch of the BPD while Port1 links the positive branch. The bandwidth of the optical spectrum is around 2.5THz and these experimental results are measured without averaging.

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Simulated results show that the operating bandwidth of the HT is around 23GHz when $k_{1} = 4 π p s$ and $ψ_{2}^{1} = - (544 / 2 π) p s^{2} / r a d$ . However, limited by the bandwidth of the MZM, the maximum bandwidth of the proposed HT should be around 20GHz. According to the experimental results, amplitude fluctuations of the measured RF intensity response are less than 3dB between 50MHz (the start frequency of the VNA) and 10GHz. It can be seen that there is a significant attenuation of the power between 11GHz and 15GHz (see Fig. 6(b)). This is mainly caused by uneven frequency responses of the intensity modulator (IM), the electrical amplifier (EA) and the BPD. Detailed analyses and measured frequency responses of these devices are shown in Section 3.4. Even though there is a power attenuation, the measured intensity responses and phase responses of our proposal fit well with the simulation. The phase shift of the HT is around −90° over the bandwidth of 20GHz.

Temporal intensity Hilbert transformations of Gaussian-like pulses with FWHM around 125ps, 85ps and 68ps have also been achieved (see in Figs. 7(a)-7(c)). Note that the inequality, $| Δ ω_{m}^{2} ψ_{2} / 8 | ≪ π$ , is satisfied. Compared with ideal outputs, the normalized peaks of the measured signals in Figs. 7(a)-7(c) are around 0.68, 0.63 and 0.65, respectively. In order to make a quantitative analysis, the RMSEs of the output signals in Figs. 7(a)-7(c) are also calculated over a time window around 36ns, which are 5.54%, 5.14% and 4.68%.

Fig. 7 Measured outputs of the proposed HT and differentiator. The blue, red dash and yellow lines represent the measured input signals, outputs of the ideal signal processing functions and the measured output signals, respectively. (a), (b) and (c) Experimental and simulated results of temporal intensity Hilbert transformation. (d), (e) and (f) Experimental and simulated results of temporal intensity differentiation. The input signals in each row are the same and these experimental results are measured with averaging.

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The RMSE variation is around 0.86%. Experimental results show an outstanding capacity for high-speed microwave signal processing with a bandwidth up to tens of gigahertz.

3.3. Experimental results of the differentiator

Based on Eq. (6), the designed impulse response of the proposed differentiator could be realized by programming the user-defined optical filter. However, as the difference between the phase response of an HT and that of a differentiator is $π$ , the optical signal with higher frequency components should connect with Port1 of the BPD and the lower counterparts should link Port2. The shaped optical spectrum is shown in Fig. 6(d). Note that the bandwidth of the optical spectrum is also around 2.5THz.

Measured RF intensity responses and phase responses are shown in Figs. 6(e)-6(f), where experimental results are in a great agreement with the simulation. There is also a significant attenuation of the power between 11GHz and 15GHz which is caused by the optoelectronic devices (see Section 3.4). The phase shift of the differentiator is around 90º within 20GHz, which is opposite to the HT.

We also test the performance of our proposal by choosing the same Gaussian-like pulses mentioned in Section 3.2 as input signals. The measured output signals are shown in Figs. 7(d)-7(f). The normalized peaks of the measured signals in Figs. 7(d)-7(f) are around 0.79, 0.75 and 0.67. Also, RMSEs of output signals in Figs. 7(d)-7(f) are around 7.69%, 6.25% and 4.98% over 8ns, respectively. The RMSE variation is around 2.71%. Thus, it is proved that the proposed signal processor is able to realize a high-speed temporal intensity differentiation and the bandwidth can be up to tens of gigahertz.

3.4. Discussion

To measure frequency responses of the optoelectronic devices, another experiment has been conducted based on the schematic diagram shown in Fig. 8. The whole system can be considered as a negative microwave photonics filter (NMWF) [31]. The measured results of the NMWF and the EA are shown in Fig. 9, where the passband of the NMWF could be adjusted by tuning the difference of the time delay between those two taps. Experimental results show that a power attenuation always exists in the measured RF intensity responses between 11GHz and 15GHz, such as 3.45dB for the EA and 7.7dB for the NMWF. Thus, the power attenuation in measured frequency responses could be mainly ascribed to the EA, MZM and BPD. To improve the performance of our system, photoelectrical devices with flat frequency responses and better designed impulse responses of different signal processing functions are needed.

Fig. 8 Experimental setup for measuring frequency responses of the EA, IM and BPD. TLS: Tunable laser source; IM: Intensity modulator; EA: Electrical amplifier; VNA: Vector network analyzer; BPD: Balanced photodetector; TDL: Tunable delay line. An optical coupler (50/50) is used before the TDLs.

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Fig. 9 Comparison between RF intensity responses of the proposed HT, the negative microwave photonics filter and the EA. Note that the negative microwave photonics filter is measured with three different time delays, Delay1, Delay2 and Delay3.

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Furthermore, the difference of the time delay between Port1 and Port2 could also have a significant impact on the RF intensity response of the proposed microwave signal processor. More generally, the impulse response could be written as,

h (t) = h_{1} (t) \otimes δ (t + Δ t /2) - h_{2} (t) \otimes δ {(t - Δ t /2)}_{t = 2 π f_{o} \cdot ψ_{2}},

where

h_{i} {(t)}_{i = 1, 2}

and

Δ t

are the designed impulse responses and the difference of the time delay between Port1 and Port2, respectively. Therefore, its RF intensity response is given by,

H (ω) = H_{1}^{2} (ω) + H_{2}^{2} (ω) - 2 H_{1} (ω) H_{2} (ω) \cos (ω Δ t),

where

H_{i} {(ω)}_{i = 1, 2}

represents the Fourier transform of the

h_{i} {(t)}_{i = 1, 2}

. Based on Eq. (8), the frequency response is decided by

h_{i} {(t)}_{i = 1, 2}

and the difference

Δ t

. According to the simulation, the spaces between two notches in frequency domain,

Δ f_{s p a c e}

, are around 7GHz and 9GHz as shown in Fig. 10(c).

Fig. 10 Simulation of the proposed HT with different time-delayed difference. (a), (b) and (c) represent frequency responses of the proposed HTs with different time-delayed differences (the red line) and the HT with no time-delayed difference (the blue line). (d), (e) and (f) represent the corresponding output signals (the red line), which are compared with the ideal outputs (the yellow line). The RF bandwidth of the input Gaussian pulse (the blue line) is around 20GHz. Note that $k =4 π p s$ , $ψ_{2} = - (544 / 2 π) p s^{2} / r a d$ and the bandwidth of the optical signal is 2.5THz. Based on our calculation, $Δ t =0 .004 n s$ is the maximum value to make the frequency response invariant.

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Due to the strong disturbance from the designed function $h_{i} {(t)}_{i = 1, 2}$ in lower frequency part, $Δ f_{s p a c e}$ is not inversely related to $Δ t$ . However, the $Δ f_{s p a c e}$ is a reciprocal of $Δ t$ in higher frequency range since the influence of $h_{i} {(t)}_{i = 1, 2}$ is weaker, as shown in Fig. 11. In addition, deviations between the measured output signals and the ideal outputs increase when the $Δ t$ is larger (see in Fig. 10). The calculated RMSEs of output signals in Figs. 10(d)-10(f) are 5.7%, 6.2% and 7.2% over a time window around 2ns, respectively. Based on the simulation, the difference between the time delays should be smaller than 0.004ns so that the frequency response of the proposed HT could remain unchanged, as shown in Fig. 10(a). In this case, the RMSE between the output signal and the ideal output should be less than 5.7% (in simulation). What is more, the optical spectral resolution $δ f$ of the user-defined optical filter leads to a maximum operating bandwidth of the proposed microwave signal processor, $1/(2 π \cdot ψ_{2} \cdot δ f)$ , as mentioned in Section 2.2. Therefore, for a certain input signal, the lowest optical spectral resolution of the user-defined optical filter should satisfy the condition that the RMSE of the output signals is less than 5.7% over 2ns (in simulation).

Fig. 11 Details about the intensity responses of the proposed HTs with different time-delayed differences. The blue lines represent the frequency response of the HT with no time-delayed difference while the red lines give the results with different time-delayed differences. Note that all the parameters are same with the simulation in Fig. 10.

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In order to implement a broader operating bandwidth, a user-defined optical filter with a higher optical spectral resolution and a tunable delay line with fine adjustment are needed. Also, the optoelectronic devices with flat frequency responses could lead to a higher quality and a lower RMSE of the output signal. Nevertheless, the devices with higher performances will be costly and the cooperation between them will be extremely complex.

4. Conclusion

We proposed and experimentally demonstrated a reconfigurable microwave signal processor which is capable of performing any signal processing function with a phase shift of π. To validate its reconfigurable capacity, two signal processing functions, including temporal intensity Hilbert transformation and temporal intensity differentiation, have been achieved by programming the user-defined optical filter. Theoretically, the bandwidth of our proposal could be adjusted by tuning the dispersion or reshaping the input optical intensity spectrum. In our experiment, the operating bandwidth could be up to tens of gigahertz even thought a significant power attenuation exists in the measured frequency responses between 11GHz and 15GHz. Based on the analyses, this power attenuation is mainly caused by the uneven frequency responses of the optoelectronic devices in our system. In addition, the difference of time delay between the two photodetected signals and the optical spectral resolution of the optical filter also have a significant impact on the output signals. Therefore, to increase the operating bandwidth and reduce the RMSE, optoelectronic devices with flat frequency responses, broader bandwidth and higher optical spectral resolution are needed.

Funding

National Natural Science Foundation of China (61522509, 61377002 and 61090391), Beijing Natural Science Foundation (4152052), National High-Tech Research and Development Program of China (2015AA017102). M. L. was supported partly by the Thousand Young Talent Program.

Acknowledgments

We acknowledge Shuqian Sun, Nuannuan Shi and Tengfei Hao for comments and discussion.

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Reconfigurable microwave signal processor with a phase shift of $π$

Abstract

1. Introduction

2. Principle

2.1. Principle of TSC system and our proposal

2.2. Hilbert transformer

2.3. Differentiator

2.4. Impulse response with more than one phase jump

3. Experimental results and discussion

3.1. Experimental setup

3.2. Experimental results of the Hilbert transformer

3.3. Experimental results of the differentiator

3.4. Discussion

4. Conclusion

Funding

Acknowledgments

References and links

Cited By

Figures (11)

Equations (8)

Optics Express