- Email: [email protected]
CW lidar system using a dual parallel Mach–Zehnder modulator
Optics Communications 403 (2017) 266–270
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
A stepped FM/CW lidar system using a dual parallel Mach–Zehnder modulator Cheng-hua Yang, Yong Zhang *, Zhi-qiang Shen, Lu Xu, Xu Yang, Yue-hao Liu, Chen-fei Jin, Yuan Zhao * Department of physics, Harbin Institute of Technology, Harbin, Heilongjiang province 150001, China
a r t i c l e
i n f o
Keywords: Frequency modulation Heterodyne Lidar
a b s t r a c t A stepped frequency-modulated continuous-wave lidar system using a dual parallel Mach–Zehnder modulator with heterodyne detection is demonstrated. The lidar transmitter utilizes a modulation loop circuit composed of an electro-optic dual parallel Mach–Zehnder modulator and a fiber amplifier, as well as a tunable Fabry–Perot filter to generate a bandwidth-enhanced stepped frequency-modulated signal. In addition, a centroid algorithm is used in the receiver to process the signal with both high precision and high accuracy. The simulation results demonstrated that the ranging precision of the proposed lidar was 2.09 cm and the ranging accuracy was 0.16 cm. A validation experiment verified that the data obtained in the simulation had a modulation bandwidth of 4 GHz using a 200-MHz signal source. The improvements resulted in a frequency-modulated continuous-wave lidar system with high precision and accuracy by operating a modulated signal with a wider modulation bandwidth using signal sources with low bandwidth in a low-cost and compact structure. © 2017 Published by Elsevier B.V.
1. Introduction Lidar systems have been used extensively for estimating atmospheric parameters [1,2], measuring velocity and vibrations [3,4], and imaging [5,6]. The frequency-modulated continuous-wave (FM/CW) lidar with heterodyne detection has become a common detection mechanism because of the fine temporal resolution and ranging precision provided by the bandwidth of the chirp with low bandwidth detectors and analog-to-digital converters (ADCs). For an FM/CW lidar system, the ranging precision 𝜎𝑅 is generally determined by the speed of light in a vacuum 𝑐, the modulated bandwidth 𝐵 and the detection signal to noise ratio (SNR) [7]. The most pressing technical bottleneck in the system is the bandwidth, which requires improving because a higher precision is required for practical lidar applications such as metrology and tomography, while there are technical problems in producing a high-frequency signal source to drive the electro-optic modulator; this results in an increased volume and additional costs. As a solution, tunable lasers have been adapted to generate a wideband frequency-modulated (FM) signal with a bandwidth of more than 1 THz; however, the stability, frequency sweep rate, and frequency stability of these laser sources are insufficient in high dynamic applications [8]. As a result, there is an intense need for high-speed wideband modulation methods in FM/CW lidar systems. * Corresponding authors.
E-mail addresses: [email protected] (Y. Zhang), [email protected] (Y. Zhao). http://dx.doi.org/10.1016/j.optcom.2017.06.038 Received 24 April 2017; Received in revised form 8 June 2017; Accepted 10 June 2017 0030-4018/© 2017 Published by Elsevier B.V.
In recent years, research into the use of dual parallel Mach–Zehnder modulators (DPMZM) in orthogonal frequency-division multiplexing has increased because of the improved performance of the DPMZM with regard to carrier-suppressed single side band modulation [9,10]. It has been demonstrated that a loop circuit consisting of an electrooptic DPMZM and a fiber amplifier can be used in the generation of a flat frequency comb with an ultra-wide bandwidth as large as tens of GHz [11]. Compared to tunable lasers, this modulation method can provide a much higher modulation rate as well as good stability and the method demonstrates great potential for transferring photonics techniques for improving FM/CW lidar systems. In this paper, we propose and experimentally demonstrate a stepped FM/CW lidar system based on a DPMZM. A modulation loop circuit consisting of a DPMZM and an Erbium Doped Fiber Amplifier (EDFA) was used to generate a wideband FM signal using a low bandwidth signal source; an optical heterodyne module was adopted in the receiver. In addition, a centroid algorithm is used in the receiver to process the signal with high accuracy. The proposed lidar system is flexible and suitable for practical applications due to the high ranging precision and accuracy when employing a wider modulation bandwidth with low bandwidth signal sources in an economical approach.
C.-h. Yang et al.
Optics Communications 403 (2017) 266–270
Fig. 2. Heterodyne IF signal of the proposed FM/CW lidar system, (a) frequency-to-time waveform; (b) frequency spectrum distribution. Fig. 1. Block diagram of the proposed FM/CW lidar system with coherent heterodyne detection.
filter changes half a wavelength. According to the frequency selective characteristic of the tunable F–P filter [14], 2. System description and operating principle Δ𝑓 = Δ𝑉 ∕𝑉π ⋅ 𝐹 𝑆𝑅, Fig. 1 shows the block diagram of the proposed bandwidth-enhanced stepped FM/CW lidar system with coherent heterodyne detection. As shown in the solid box in Fig. 1, a DPMZM consists of two Mach–Zehnder (M–Z) intensity modulators, named I and Q. Both of the modulators are biased at the minimum work point with a bias voltage of 𝑉𝜋 , and the phase shifter between the two modulators is biased to generate a phase shift of 𝛷 = 90◦ for carrier-suppressed single sideband (SSB) modulation. The form of the two voltage signals used to drive the arms of the modulator can be expressed as, 𝑉𝐼 (𝑡) = 𝑉0 cos (2𝜋Δ𝑓 ) ,
(1)
𝑉𝑄 (𝑡) = 𝑉0 sin (2𝜋Δ𝑓 ) ,
(2)
the light field of the transmission light can be derived as, ( ) 𝑁−1 ∑ [ ( ) ] 𝑡 − 𝑚𝑡0 𝐸𝑡 (𝑡) = 𝐸𝑆 rect cos 2𝜋 𝑓0 + 𝑚Δ𝑓 𝑡 . 𝑡0 𝑚=0
(3)
and the optical field of the local oscillator (LO) can be expressed as, ) ( 𝑁−1 ∑ [ ( ) ] 𝑡 − 𝑚𝑡0 cos 2𝜋 𝑓0 + 𝑚Δ𝑓 𝑡 , (9) 𝐸LO (𝑡) = 𝐸𝑙 rect 𝑡 0 𝑚=0
where 𝑓0 is the optical frequency of the laser and 𝐸𝑆 is the amplitude of the input optical field. Eq. (3) represents a carrier-suppressed SSB optical signal with the frequency shift Δ𝑓 from the central frequency 𝑓0 . The output optical field of the modulation loop circuit after a plurality of modulation cycles can be expressed as, 𝐸comb =
𝑁−1 ∑
{ [ ] } 𝐸𝑆,𝑚 cos 2𝜋 𝑓0 + (𝑚 + 1) Δ𝑓 𝑡 ,
(7)
Eq. (7) indicates that the output of the tunable F–P filter is a stepped FM signal with a step time duration of 𝑡0 and a step frequency of Δ𝑓 , which can be used in the coherent heterodyne detection. At the receiver, the optical field of the echo signal scattered by the target can be expressed as, ] { [( ) 𝑁−1 ∑ 𝑡 − 𝜏𝑅 − 𝑚𝑡0 𝐸Echo (𝑡) = 𝐸𝑟 rect 𝑡0 𝑚=0 (8) } [ ( )( )] ⋅ cos 2𝜋 𝑓0 + 𝑚Δ𝑓 𝑡 − 𝜏𝑅 ,
where 𝑉0 is the amplitude of the driving voltage waveform. The optical field at the modulator output can be derived as [12], [ ( ) ] 𝐸0 = 𝐸𝑆 cos 2𝜋 𝑓0 + Δ𝑓 𝑡 ,
(6)
where 𝜏𝑅 = 2𝑅∕𝑐 is the time delay of the echo, R is the distance between the target and the lidar system, and 𝐸𝑟 and 𝐸𝑙 are the constant optical fields of the echo and the LO. When 𝑓𝑅 ∈ (0, 𝐵∕2), the heterodyne IF signal can be expressed as, { [ ( ) ] ( ) cos 2𝜋 𝑚𝑅 − 1 Δ𝑓 𝑡 , 𝑡 ∈ 𝑚𝑅 𝑡0 , 𝜏𝑅 𝐼𝐼𝐹 = (10) ( ) ( ( ) ), cos 2𝜋𝑚𝑅 Δ𝑓 𝑡 , 𝑡 ∈ 𝜏𝑅 , 𝑚𝑅 + 1 𝑡0
(4)
𝑚=0
where 𝐸𝑆,𝑚 is the amplitude of output optical field of frequency 𝑓0 + (𝑚 + 1) ⋅ Δ𝑓 , and N is the amount of the frequency components. Eq. (4) indicates that the output of the modulation loop circuit is equivalent to a series of frequency comb signals with a constant frequency shift of Δ𝑓 . In an ideal case, the energy loss in the modulation loop circuit can be entirely compensated by the EDFA during each loop, the constant output optical field of each frequency would be equivalent to 𝐸𝑆 , and N is equal to infinity. Under practical conditions, the compensation function of the EDFA cannot offset the loss; therefore 𝐸𝑆,𝑚 decreases gradually along with m and N is equal to infinity. It has been reported that using a value of 100 for N is achievable for this type of modulation loop circuit where 𝐸𝑆,𝑚 is used as a constant [13]. A tunable Fabry–Perot (F–P) filter with a pass bandwidth of Δ𝑓 was used in this study. When the free spectral range (FSR) of the F–P filter is equal to the modulation bandwidth corresponding to the finite length of the frequency comb, expressed as 𝐹 𝑆𝑅 = 𝑁 ⋅ Δ𝑓 , the stepped voltage waveform used to drive the filter can be expressed as, ( ) ( ) 𝑁−1 ∑ 𝑉𝜋 𝑡 − 𝑚𝑡0 𝑉F–P (𝑡) = 𝑡 ∗ rect , (5) 𝑇 𝑡0 𝑚=0
where 𝑚𝑅 = 0,1,2,. . . is the integer part of 𝜏𝑅 ∕𝑡0 . Eq. (10) shows that the heterodyne IF signal consists of two frequency components spaced at Δ𝑓 , which are relative to 𝑚𝑅 . A more intuitive view of 𝐼𝐼𝐹 is shown in Fig. 2(a). Subsequently, the IF signal is filtered using a bandpass filter and processed using a Fast Fourier Transform (FFT) function. The frequency spectrum is shown in Fig. 2(b), which indicates a dual-peak phenomenon in the frequency spectrum corresponding to the distance of the target. When the time delay 𝜏𝑅 is less than 𝑡0 , the peak with the lower frequency of the heterodyne signal is covered by the zero frequency noise. In this case, the signal laser is delayed by 𝑡0 before being transmitted to the telescope as shown in Fig. 1. Thus Eqs. (8) and (10) should be adapted as, [ ( ) ] 𝐸Echo (𝑡) = 𝐸𝑟 cos 2𝜋𝑓𝑠 𝑡 − 𝜏𝑅 − 𝑡0 𝑡 , and 𝐼𝐼𝐹 =
where 𝑇 = 𝑁 ⋅ 𝑡0 is the modulation period of the FM signal, 𝑉𝜋 is the driving voltage of the filter when the cavity length of the F–P
{
[ ( ) ] ( ) cos 2𝜋 𝑚𝑅 + 1 Δ𝑓 𝑡 , 𝑡 ∈ 𝑚𝑅 𝑡0 , 𝜏𝑅 [ ( ) ] ( ( ) ), cos 2𝜋 𝑚𝑅 + 2 Δ𝑓 , 𝑡 ∈ 𝜏𝑅 , 𝑚𝑅 + 1 𝑡0
𝑚𝑅 = 0, 1, 2, … . 267
(11)
(12)
C.-h. Yang et al.
Optics Communications 403 (2017) 266–270
Table 1 Simulation parameters. Parameter
Value
Wavelength of laser Frequency shift Δ𝑓 Step time duration 𝑡0 Number of frequency steps N Bandwidth of FM signal B Sampling frequency Modulation period of FM signal T
1550 nm 200 MHz 100 ns 21 4 GHz 1 GHz 2.1 μs
It can be drawn from Eq. (12) that the proportion of each frequency component of the dual-peak changes along with the fractional part of 𝜏𝑅 ∕𝑡0 . In other words, when the fractional part of 𝜏𝑅 ∕𝑡0 increases, the proportion of the higher frequency component increases as well, while the proportion of the lower frequency component decreases, and vice versa. Thus, the detection range can be determined with a high degree of ranging accuracy by both the frequencies of the dual-peak and the proportion of each frequency component. Therefore, a centroid algorithm was used to calculate the distance of the target from the lidar system as, [( ] ) ( ) 𝑐𝑡0 𝑚𝑅 + 1 ⋅ 𝐴𝑓− + 𝑚𝑅 + 2 ⋅ 𝐴𝑓+ 𝑐𝑡0 𝑅= , (13) − ( ) 2𝑛 𝑟𝑒𝑓 2𝑛 𝐴 +𝐴 𝑟𝑒𝑓
𝑓−
Fig. 3. Simulation results of the ranging precision and accuracy as functions of 𝑁𝑟 .
𝑓+
where 𝐴𝑓 − and 𝐴𝑓 + are the amplitudes of 𝑓− = (𝑚𝑅 + 1)Δ𝑓 and 𝑓+ = (𝑚𝑅 + 2)Δ𝑓 in the frequency spectrum and 𝑛𝑟𝑒𝑓 is the refractive index of the transmission media. 3. Simulation experiment A simulation experiment of the proposed FM/CW lidar system was demonstrated using MATLAB under laboratory conditions. The simulation parameters are shown in Table 1. In the simulation experiment, a 1550 nm continuous laser was used as the laser source, and a stepped FM signal with 21 frequency steps was generated in the transmitter. The step time duration of the signal was 100 ns and the frequency shift was 200 MHz. The noise conditions are simulated as 𝑆𝑁𝑅 = 10 dB, which is given by [6], 𝑆𝑁𝑅 = 10 lg
𝐼𝑠 𝐼𝑛
Fig. 4. Ranging precision changing with the frequency step and the number of frequency steps.
the simulation ranging precision and accuracy. The simulation results indicate that the proposed lidar system has the advantages of both low variability and low range error. In order to evaluate the influence of the modulation bandwidth on the ranging precision, an additional simulation of the ranging precision was conducted with different values of N and Δ𝑓 ; the interpolation curve is shown in Fig. 4. It can be seen from the figure that the ranging precision improves along with N and Δ𝑓 . The improvement in precision is considerable when N is less than 10; beyond 10, the curve gradually flattens. On the other hand, the ranging precision gradually improves as Δ𝑓 increases from 100 MHz to 1 GHz. The simulation result shows that the ranging precision is determined by the modulation bandwidth of the stepped FM signal, which can be enhanced with the proposed lidar system. The simulation was conducted with 𝑡0 = 100 ns.
2 2
,
(14)
where 𝐼𝑠 and 𝐼𝑛 are the heterodyne IF signal current and the noise current respectively. The lidar system was used to detect a simulated plane target at a distance of 10 m. The frequency components of the heterodyne IF signal can be computed as 200 and 400 MHz, which indicates that the simulated sampling frequency of 1 GHz satisfies the Nyquist condition. The ranging precision and accuracy are used to quantify the performance of the proposed lidar system. The range standard deviation of the detection results is defined as the ranging precision, which is calculated as, √ √ 𝑁𝑟 ( )2 √ 1 ∑ 𝑅𝑖 − 𝑅 , (15) 𝜎𝑅 = √ 𝑁𝑟 𝑖=1
4. Validation experiment Finally, a validation experiment was conducted in the laboratory based on the simulation results. In the validation experiment, a stepped FM signal with 21 frequency steps was generated in the transmitter. The step time duration of the signal was 100 ns and the frequency shift was 200 MHz, which was limited by the signal generator. A 2-channel waveform generator with a 240-MHz modulation bandwidth (AFG3252C) was used to drive both the DPMZM (LN86S-FC) and a customized tunable F–P filter. In the receiver, the heterodyne IF signal was collected by a high-speed detector (DET08CFC, 8 GHz bandwidth) and recorded by a mixed domain oscilloscope with an analog bandwidth of 1 GHz (DPO5104B), which satisfied the Nyquist condition for the beat frequencies of 200 MHz and 400 MHz. A target with a cross-sectional area of 20 cm × 20 cm was
and the mean range error of the detection results is defined as the ranging accuracy, which is calculated as, 𝛿𝑅 = 𝑅 − 𝑅real =
𝑁𝑟 1 ∑ 𝑅 − 𝑅real , 𝑁𝑟 𝑖=1 𝑖
(16)
where 𝑁𝑟 is the number of repeated measurements. Fig. 3 shows the simulation results of the ranging precision and accuracy as functions of 𝑁𝑟 . According to the convergence of the curves, the performance limits of the proposed lidar system are 𝜎𝑅 = 2.09 cm and 𝛿𝑅 = 0.16 cm respectively, which can be considered as 268
C.-h. Yang et al.
Optics Communications 403 (2017) 266–270
Fig. 5. Result of validation experiment with the proposed FM/CW lidar system (𝑁 = 21): (a) 2-D plot of the detected distance to the target; (b) histograms of distance distribution of the target.
Fig. 6. Result of contrasting experiment with the proposed FM/CW lidar system (𝑁 = 2): (a) 2-D plot of the detected distance to the target; (b) histograms of the distance distribution of the target.
composed of four paperboards with an equal size of 20 cm × 5 cm, which were placed at 10 m (A), 10.02 m (B), 10.05 m (C), and 10.10 m (D) from the lidar system. The validation experiment was then conducted using a manuallycontrolled scanning detection approach. The length of the scanning step was 5 mm in light of the diameter of the laser spot illuminating the target, which was 4 mm. The total number of scanning points was 40 × 40, and the experimental result is shown in Fig. 5. The ranging precision and accuracy of the validation experiment were computed using Eqs. (15) and (16) (Table 2). The average results were 𝜎𝑅 = 2.38 cm and 𝛿𝑅 = 0.66 cm, respectively. In order to study the influence of the modulation bandwidth on the performance of the proposed system, another experiment was conducted with a stepped FM signal with only 2 frequency steps under the same conditions, which is referred to as the contrasting experiment. The corresponding results are shown in Fig. 6. Similarly, the ranging precision and accuracy of the four targets were computed according to Eqs. (15) and (16), and the precision and accuracy of the contrasting ′ = 8.56 cm and 𝛿 ′ = 1.72 cm, respectively (Table 2). experiments were 𝜎𝑅 𝑅 A comparison of the results of the two experiments indicates that the ranging precision and accuracy of the validation experiment, which adopted a stepped modulation signal with 𝑁 = 21, are superior to that of the contrasting experiment using a stepped modulation signal with 𝑁 = 2. Thus, the advantage of the proposed stepped modulation system
is evident, which proves the feasibility of the proposed stepped FM/CW lidar system by generating a stepped FM signal with an enhanced modulation bandwidth. It is noteworthy that there is a difference between the experimental ranging precision (2.38 cm) and the simulation value shown in Fig. 3 (2.09 cm). This difference is caused by the difference of averaging times between the two conditions. The simulation precision is reduced by averaging more data, which follows the central limit theorem. On the other hand, there is a difference between the experimental ranging accuracy (0.66 cm) and the simulation value (0.16 cm). The difference is mainly caused by the system vibration and the environmental temperature changes in the experiment. These factors will affect the stability of the adopted DPMZM and F–P filter, and result in the differences between the simulation parameters and the experimental parameters [13,15]. 5. Conclusion We propose and experimentally demonstrate a stepped FM/CW lidar system using DPMZM with heterodyne detection. The lidar transmitter utilizes a modulation loop circuit composed of a DPMZM, a fiber amplifier, and a tunable F–P filter to generate a bandwidth-enhanced stepped FM signal. The simulation experiment indicates that the simulation ranging precision and accuracy of the proposed lidar system are 2.09 cm and 0.16 cm with a modulation bandwidth of 4 GHz. The validation 269
C.-h. Yang et al.
Optics Communications 403 (2017) 266–270 Table 2 Experimental results (in cm). Frequency step number
Results
A
B
C
D
Average
𝑁 = 21
Precision 𝜎 Accuracy 𝛿
2.47 0.72
2.45 0.57
2.28 0.70
2.33 0.65
2.38 0.66
𝑁 =2
Precision 𝜎 ′ Accuracy 𝛿 ′
8.72 1.81
8.14 1.54
8.33 1.66
9.05 1.87
8.56 1.72
experiment resulted in a ranging precision of 2.38 cm and a ranging accuracy of 0.66 cm with a modulation bandwidth of 4 GHz by using a 200 MHz signal generator. The results indicate the feasibility of the proposed lidar system with high ranging precision and accuracy. Available high-frequency electric signal sources usually adopt time interleaving (TI) and other methods to generate signal with a modulation bandwidth of 4 GHz [16]. However, these methods lead to increased power consumption, volume, and even incur degradation in the effective number of bits (ENOB) of the generated signal [17]. This degradation cannot be completely limited despite additional equalization and calibration, which causes severe distortion in the generated optical signal and the LO, and eventually results in a decrease in the ranging precision and accuracy of the proposed lidar system. By contrast, the proposed system adopts a modulation circuit loop based on a DPMZM and a tunable F–P filter to broaden the modulation bandwidth. This is a novel approach that uses optical method to partially replace conventional electric modulation. It allows the utilization of high-precision low-frequency electric signal sources to generate wideband modulated signals without degradation in ENOB. The proposed lidar system is also characterized with tunability, compact structure, low power consumption and low-cost. For an electric modulation bandwidth of 4 GHz, the corresponding optical modulation bandwidth (the spectral bandwidth) is only 0.032 nm at a wavelength of 1550 nm, and the modulation bandwidths of the adopted devices are in nanometer magnitude. Hence, it is convenient to generate a 4 GHz signal as a narrow-band signal, and the proposed lidar system has the potential for further expansion of the modulation bandwidth up to 100 GHz. The potential for practical applications of the proposed system can be expected since the key devices used in the proposed system are easily obtainable and the parameters of the devices were improved. Specifically, the bandwidth of the DPMZM in our experiment was 40 GS/s and a frequency comb with more than 100 steps has been reported, from which a step-modulated signal with the same number of
frequency steps can be expected [18]. In addition, the use of advanced tunable F–P filters has been reported with both a high Q-factor and a quick response of less than one nanosecond [19]. By combining these advanced technologies, the proposed FM/CW lidar system is expected to achieve satisfactory ranging precision and accuracy when operating at a much wider modulation bandwidth with low-frequency signal sources and the system has a compact structure. References [1] J.T. Dobler, F.W. Harrison, E.V. Browell, B. Lin, D. McGregor, S. Kooi, Y. Choi, S. Ismail, Appl. Opt. 52 (2013) 2874–2892. [2] Y. Zhang, F. Yi, W. Kong, Y. Yi, Appl. Opt. 53 (2014) 7312–7320. [3] W.C. Swann, N.R. Newbury, Opt. Lett. 31 (2006) 826–828. [4] S. Gao, M. O’Sullivan, R. Hui, Opt. Express 20 (2012) 25867–25875. [5] W. He, B. Sima, Y. Chen, H. Dai, Q. Chen, G. Gu, Opt. Commun. 308 (2013) 211–217. [6] P.F McManamon, Opt. Eng. 51 (2012) 060901-1–060901-13. [7] F. Yang, Y. He, J. Shang, W. Chen, Appl. Opt. 48 (2009) 6575–6582. [8] Z.W. Barber, J.R. Dahl, A.B. Mateo, S. Crouch, R. Reibel, Imaging and Applied Optics, OSA Technical Digest, 2015, LM4F.2,. [9] Y. Zhao, X. Pang, L. Deng, X. Yu, X. Zheng, B. Zhou, Idelfonso Tafur Monroy, Opt. Express 19 (2011) B681–B686. [10] X. Li, Y. Xu, J. Yu, Opt. Lett. 41 (2016) 4162–4165. [11] W. Li, L.X. Wang, J.Y. Zheng, M. Li, N.H. Zhu, IEEE Photonics Technol. Lett. 25 (2013) 1875–1878. [12] T. Kobayashi, A. Sano, E. Yamada, E. Yoshida, M. Miyamoto, J. Lightwave Technol. 27 (2009) 3714–3720. [13] M. Iodice, G. Cocorullo, F.G. Della Corte, Rendina, Opt. Commun. 183 (2000) 415– 418. [14] A.D. Kersey, T.A. Berkoff, W.W. Morie, Opt. Lett. 18 (1993) 1370–1372. [15] S.J. Fabbri, C. O’Riordan, S. Sygletos, A.D. Ellis, Electron. Lett. 49 (2013) 135–136. [16] E. Olieman, A.J. Annema, B. Nauta, IEEE J. Solid-State Circuits 50 (2015) 704–713. [17] C. Laperle, M. O’Sullivan, Signal Processing in Photonic Communications. Optical Society of America, 2014. SM3E. 1. [18] T. Kawanishi, T. Sakamoto, S. Shinada, M. Izutsu, IEICE Electron. Express 1 (2004) 217–221. [19] W. Zhang, N. Ehteshami, W. Liu, J. Yao, Opt. Lett. 40 (2015) 3153–3156.
270
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
A stepped FM/CW lidar system using a dual parallel Mach–Zehnder modulator Cheng-hua Yang, Yong Zhang *, Zhi-qiang Shen, Lu Xu, Xu Yang, Yue-hao Liu, Chen-fei Jin, Yuan Zhao * Department of physics, Harbin Institute of Technology, Harbin, Heilongjiang province 150001, China
a r t i c l e
i n f o
Keywords: Frequency modulation Heterodyne Lidar
a b s t r a c t A stepped frequency-modulated continuous-wave lidar system using a dual parallel Mach–Zehnder modulator with heterodyne detection is demonstrated. The lidar transmitter utilizes a modulation loop circuit composed of an electro-optic dual parallel Mach–Zehnder modulator and a fiber amplifier, as well as a tunable Fabry–Perot filter to generate a bandwidth-enhanced stepped frequency-modulated signal. In addition, a centroid algorithm is used in the receiver to process the signal with both high precision and high accuracy. The simulation results demonstrated that the ranging precision of the proposed lidar was 2.09 cm and the ranging accuracy was 0.16 cm. A validation experiment verified that the data obtained in the simulation had a modulation bandwidth of 4 GHz using a 200-MHz signal source. The improvements resulted in a frequency-modulated continuous-wave lidar system with high precision and accuracy by operating a modulated signal with a wider modulation bandwidth using signal sources with low bandwidth in a low-cost and compact structure. © 2017 Published by Elsevier B.V.
1. Introduction Lidar systems have been used extensively for estimating atmospheric parameters [1,2], measuring velocity and vibrations [3,4], and imaging [5,6]. The frequency-modulated continuous-wave (FM/CW) lidar with heterodyne detection has become a common detection mechanism because of the fine temporal resolution and ranging precision provided by the bandwidth of the chirp with low bandwidth detectors and analog-to-digital converters (ADCs). For an FM/CW lidar system, the ranging precision 𝜎𝑅 is generally determined by the speed of light in a vacuum 𝑐, the modulated bandwidth 𝐵 and the detection signal to noise ratio (SNR) [7]. The most pressing technical bottleneck in the system is the bandwidth, which requires improving because a higher precision is required for practical lidar applications such as metrology and tomography, while there are technical problems in producing a high-frequency signal source to drive the electro-optic modulator; this results in an increased volume and additional costs. As a solution, tunable lasers have been adapted to generate a wideband frequency-modulated (FM) signal with a bandwidth of more than 1 THz; however, the stability, frequency sweep rate, and frequency stability of these laser sources are insufficient in high dynamic applications [8]. As a result, there is an intense need for high-speed wideband modulation methods in FM/CW lidar systems. * Corresponding authors.
E-mail addresses: [email protected] (Y. Zhang), [email protected] (Y. Zhao). http://dx.doi.org/10.1016/j.optcom.2017.06.038 Received 24 April 2017; Received in revised form 8 June 2017; Accepted 10 June 2017 0030-4018/© 2017 Published by Elsevier B.V.
In recent years, research into the use of dual parallel Mach–Zehnder modulators (DPMZM) in orthogonal frequency-division multiplexing has increased because of the improved performance of the DPMZM with regard to carrier-suppressed single side band modulation [9,10]. It has been demonstrated that a loop circuit consisting of an electrooptic DPMZM and a fiber amplifier can be used in the generation of a flat frequency comb with an ultra-wide bandwidth as large as tens of GHz [11]. Compared to tunable lasers, this modulation method can provide a much higher modulation rate as well as good stability and the method demonstrates great potential for transferring photonics techniques for improving FM/CW lidar systems. In this paper, we propose and experimentally demonstrate a stepped FM/CW lidar system based on a DPMZM. A modulation loop circuit consisting of a DPMZM and an Erbium Doped Fiber Amplifier (EDFA) was used to generate a wideband FM signal using a low bandwidth signal source; an optical heterodyne module was adopted in the receiver. In addition, a centroid algorithm is used in the receiver to process the signal with high accuracy. The proposed lidar system is flexible and suitable for practical applications due to the high ranging precision and accuracy when employing a wider modulation bandwidth with low bandwidth signal sources in an economical approach.
C.-h. Yang et al.
Optics Communications 403 (2017) 266–270
Fig. 2. Heterodyne IF signal of the proposed FM/CW lidar system, (a) frequency-to-time waveform; (b) frequency spectrum distribution. Fig. 1. Block diagram of the proposed FM/CW lidar system with coherent heterodyne detection.
filter changes half a wavelength. According to the frequency selective characteristic of the tunable F–P filter [14], 2. System description and operating principle Δ𝑓 = Δ𝑉 ∕𝑉π ⋅ 𝐹 𝑆𝑅, Fig. 1 shows the block diagram of the proposed bandwidth-enhanced stepped FM/CW lidar system with coherent heterodyne detection. As shown in the solid box in Fig. 1, a DPMZM consists of two Mach–Zehnder (M–Z) intensity modulators, named I and Q. Both of the modulators are biased at the minimum work point with a bias voltage of 𝑉𝜋 , and the phase shifter between the two modulators is biased to generate a phase shift of 𝛷 = 90◦ for carrier-suppressed single sideband (SSB) modulation. The form of the two voltage signals used to drive the arms of the modulator can be expressed as, 𝑉𝐼 (𝑡) = 𝑉0 cos (2𝜋Δ𝑓 ) ,
(1)
𝑉𝑄 (𝑡) = 𝑉0 sin (2𝜋Δ𝑓 ) ,
(2)
the light field of the transmission light can be derived as, ( ) 𝑁−1 ∑ [ ( ) ] 𝑡 − 𝑚𝑡0 𝐸𝑡 (𝑡) = 𝐸𝑆 rect cos 2𝜋 𝑓0 + 𝑚Δ𝑓 𝑡 . 𝑡0 𝑚=0
(3)
and the optical field of the local oscillator (LO) can be expressed as, ) ( 𝑁−1 ∑ [ ( ) ] 𝑡 − 𝑚𝑡0 cos 2𝜋 𝑓0 + 𝑚Δ𝑓 𝑡 , (9) 𝐸LO (𝑡) = 𝐸𝑙 rect 𝑡 0 𝑚=0
where 𝑓0 is the optical frequency of the laser and 𝐸𝑆 is the amplitude of the input optical field. Eq. (3) represents a carrier-suppressed SSB optical signal with the frequency shift Δ𝑓 from the central frequency 𝑓0 . The output optical field of the modulation loop circuit after a plurality of modulation cycles can be expressed as, 𝐸comb =
𝑁−1 ∑
{ [ ] } 𝐸𝑆,𝑚 cos 2𝜋 𝑓0 + (𝑚 + 1) Δ𝑓 𝑡 ,
(7)
Eq. (7) indicates that the output of the tunable F–P filter is a stepped FM signal with a step time duration of 𝑡0 and a step frequency of Δ𝑓 , which can be used in the coherent heterodyne detection. At the receiver, the optical field of the echo signal scattered by the target can be expressed as, ] { [( ) 𝑁−1 ∑ 𝑡 − 𝜏𝑅 − 𝑚𝑡0 𝐸Echo (𝑡) = 𝐸𝑟 rect 𝑡0 𝑚=0 (8) } [ ( )( )] ⋅ cos 2𝜋 𝑓0 + 𝑚Δ𝑓 𝑡 − 𝜏𝑅 ,
where 𝑉0 is the amplitude of the driving voltage waveform. The optical field at the modulator output can be derived as [12], [ ( ) ] 𝐸0 = 𝐸𝑆 cos 2𝜋 𝑓0 + Δ𝑓 𝑡 ,
(6)
where 𝜏𝑅 = 2𝑅∕𝑐 is the time delay of the echo, R is the distance between the target and the lidar system, and 𝐸𝑟 and 𝐸𝑙 are the constant optical fields of the echo and the LO. When 𝑓𝑅 ∈ (0, 𝐵∕2), the heterodyne IF signal can be expressed as, { [ ( ) ] ( ) cos 2𝜋 𝑚𝑅 − 1 Δ𝑓 𝑡 , 𝑡 ∈ 𝑚𝑅 𝑡0 , 𝜏𝑅 𝐼𝐼𝐹 = (10) ( ) ( ( ) ), cos 2𝜋𝑚𝑅 Δ𝑓 𝑡 , 𝑡 ∈ 𝜏𝑅 , 𝑚𝑅 + 1 𝑡0
(4)
𝑚=0
where 𝐸𝑆,𝑚 is the amplitude of output optical field of frequency 𝑓0 + (𝑚 + 1) ⋅ Δ𝑓 , and N is the amount of the frequency components. Eq. (4) indicates that the output of the modulation loop circuit is equivalent to a series of frequency comb signals with a constant frequency shift of Δ𝑓 . In an ideal case, the energy loss in the modulation loop circuit can be entirely compensated by the EDFA during each loop, the constant output optical field of each frequency would be equivalent to 𝐸𝑆 , and N is equal to infinity. Under practical conditions, the compensation function of the EDFA cannot offset the loss; therefore 𝐸𝑆,𝑚 decreases gradually along with m and N is equal to infinity. It has been reported that using a value of 100 for N is achievable for this type of modulation loop circuit where 𝐸𝑆,𝑚 is used as a constant [13]. A tunable Fabry–Perot (F–P) filter with a pass bandwidth of Δ𝑓 was used in this study. When the free spectral range (FSR) of the F–P filter is equal to the modulation bandwidth corresponding to the finite length of the frequency comb, expressed as 𝐹 𝑆𝑅 = 𝑁 ⋅ Δ𝑓 , the stepped voltage waveform used to drive the filter can be expressed as, ( ) ( ) 𝑁−1 ∑ 𝑉𝜋 𝑡 − 𝑚𝑡0 𝑉F–P (𝑡) = 𝑡 ∗ rect , (5) 𝑇 𝑡0 𝑚=0
where 𝑚𝑅 = 0,1,2,. . . is the integer part of 𝜏𝑅 ∕𝑡0 . Eq. (10) shows that the heterodyne IF signal consists of two frequency components spaced at Δ𝑓 , which are relative to 𝑚𝑅 . A more intuitive view of 𝐼𝐼𝐹 is shown in Fig. 2(a). Subsequently, the IF signal is filtered using a bandpass filter and processed using a Fast Fourier Transform (FFT) function. The frequency spectrum is shown in Fig. 2(b), which indicates a dual-peak phenomenon in the frequency spectrum corresponding to the distance of the target. When the time delay 𝜏𝑅 is less than 𝑡0 , the peak with the lower frequency of the heterodyne signal is covered by the zero frequency noise. In this case, the signal laser is delayed by 𝑡0 before being transmitted to the telescope as shown in Fig. 1. Thus Eqs. (8) and (10) should be adapted as, [ ( ) ] 𝐸Echo (𝑡) = 𝐸𝑟 cos 2𝜋𝑓𝑠 𝑡 − 𝜏𝑅 − 𝑡0 𝑡 , and 𝐼𝐼𝐹 =
where 𝑇 = 𝑁 ⋅ 𝑡0 is the modulation period of the FM signal, 𝑉𝜋 is the driving voltage of the filter when the cavity length of the F–P
{
[ ( ) ] ( ) cos 2𝜋 𝑚𝑅 + 1 Δ𝑓 𝑡 , 𝑡 ∈ 𝑚𝑅 𝑡0 , 𝜏𝑅 [ ( ) ] ( ( ) ), cos 2𝜋 𝑚𝑅 + 2 Δ𝑓 , 𝑡 ∈ 𝜏𝑅 , 𝑚𝑅 + 1 𝑡0
𝑚𝑅 = 0, 1, 2, … . 267
(11)
(12)
C.-h. Yang et al.
Optics Communications 403 (2017) 266–270
Table 1 Simulation parameters. Parameter
Value
Wavelength of laser Frequency shift Δ𝑓 Step time duration 𝑡0 Number of frequency steps N Bandwidth of FM signal B Sampling frequency Modulation period of FM signal T
1550 nm 200 MHz 100 ns 21 4 GHz 1 GHz 2.1 μs
It can be drawn from Eq. (12) that the proportion of each frequency component of the dual-peak changes along with the fractional part of 𝜏𝑅 ∕𝑡0 . In other words, when the fractional part of 𝜏𝑅 ∕𝑡0 increases, the proportion of the higher frequency component increases as well, while the proportion of the lower frequency component decreases, and vice versa. Thus, the detection range can be determined with a high degree of ranging accuracy by both the frequencies of the dual-peak and the proportion of each frequency component. Therefore, a centroid algorithm was used to calculate the distance of the target from the lidar system as, [( ] ) ( ) 𝑐𝑡0 𝑚𝑅 + 1 ⋅ 𝐴𝑓− + 𝑚𝑅 + 2 ⋅ 𝐴𝑓+ 𝑐𝑡0 𝑅= , (13) − ( ) 2𝑛 𝑟𝑒𝑓 2𝑛 𝐴 +𝐴 𝑟𝑒𝑓
𝑓−
Fig. 3. Simulation results of the ranging precision and accuracy as functions of 𝑁𝑟 .
𝑓+
where 𝐴𝑓 − and 𝐴𝑓 + are the amplitudes of 𝑓− = (𝑚𝑅 + 1)Δ𝑓 and 𝑓+ = (𝑚𝑅 + 2)Δ𝑓 in the frequency spectrum and 𝑛𝑟𝑒𝑓 is the refractive index of the transmission media. 3. Simulation experiment A simulation experiment of the proposed FM/CW lidar system was demonstrated using MATLAB under laboratory conditions. The simulation parameters are shown in Table 1. In the simulation experiment, a 1550 nm continuous laser was used as the laser source, and a stepped FM signal with 21 frequency steps was generated in the transmitter. The step time duration of the signal was 100 ns and the frequency shift was 200 MHz. The noise conditions are simulated as 𝑆𝑁𝑅 = 10 dB, which is given by [6], 𝑆𝑁𝑅 = 10 lg
𝐼𝑠 𝐼𝑛
Fig. 4. Ranging precision changing with the frequency step and the number of frequency steps.
the simulation ranging precision and accuracy. The simulation results indicate that the proposed lidar system has the advantages of both low variability and low range error. In order to evaluate the influence of the modulation bandwidth on the ranging precision, an additional simulation of the ranging precision was conducted with different values of N and Δ𝑓 ; the interpolation curve is shown in Fig. 4. It can be seen from the figure that the ranging precision improves along with N and Δ𝑓 . The improvement in precision is considerable when N is less than 10; beyond 10, the curve gradually flattens. On the other hand, the ranging precision gradually improves as Δ𝑓 increases from 100 MHz to 1 GHz. The simulation result shows that the ranging precision is determined by the modulation bandwidth of the stepped FM signal, which can be enhanced with the proposed lidar system. The simulation was conducted with 𝑡0 = 100 ns.
2 2
,
(14)
where 𝐼𝑠 and 𝐼𝑛 are the heterodyne IF signal current and the noise current respectively. The lidar system was used to detect a simulated plane target at a distance of 10 m. The frequency components of the heterodyne IF signal can be computed as 200 and 400 MHz, which indicates that the simulated sampling frequency of 1 GHz satisfies the Nyquist condition. The ranging precision and accuracy are used to quantify the performance of the proposed lidar system. The range standard deviation of the detection results is defined as the ranging precision, which is calculated as, √ √ 𝑁𝑟 ( )2 √ 1 ∑ 𝑅𝑖 − 𝑅 , (15) 𝜎𝑅 = √ 𝑁𝑟 𝑖=1
4. Validation experiment Finally, a validation experiment was conducted in the laboratory based on the simulation results. In the validation experiment, a stepped FM signal with 21 frequency steps was generated in the transmitter. The step time duration of the signal was 100 ns and the frequency shift was 200 MHz, which was limited by the signal generator. A 2-channel waveform generator with a 240-MHz modulation bandwidth (AFG3252C) was used to drive both the DPMZM (LN86S-FC) and a customized tunable F–P filter. In the receiver, the heterodyne IF signal was collected by a high-speed detector (DET08CFC, 8 GHz bandwidth) and recorded by a mixed domain oscilloscope with an analog bandwidth of 1 GHz (DPO5104B), which satisfied the Nyquist condition for the beat frequencies of 200 MHz and 400 MHz. A target with a cross-sectional area of 20 cm × 20 cm was
and the mean range error of the detection results is defined as the ranging accuracy, which is calculated as, 𝛿𝑅 = 𝑅 − 𝑅real =
𝑁𝑟 1 ∑ 𝑅 − 𝑅real , 𝑁𝑟 𝑖=1 𝑖
(16)
where 𝑁𝑟 is the number of repeated measurements. Fig. 3 shows the simulation results of the ranging precision and accuracy as functions of 𝑁𝑟 . According to the convergence of the curves, the performance limits of the proposed lidar system are 𝜎𝑅 = 2.09 cm and 𝛿𝑅 = 0.16 cm respectively, which can be considered as 268
C.-h. Yang et al.
Optics Communications 403 (2017) 266–270
Fig. 5. Result of validation experiment with the proposed FM/CW lidar system (𝑁 = 21): (a) 2-D plot of the detected distance to the target; (b) histograms of distance distribution of the target.
Fig. 6. Result of contrasting experiment with the proposed FM/CW lidar system (𝑁 = 2): (a) 2-D plot of the detected distance to the target; (b) histograms of the distance distribution of the target.
composed of four paperboards with an equal size of 20 cm × 5 cm, which were placed at 10 m (A), 10.02 m (B), 10.05 m (C), and 10.10 m (D) from the lidar system. The validation experiment was then conducted using a manuallycontrolled scanning detection approach. The length of the scanning step was 5 mm in light of the diameter of the laser spot illuminating the target, which was 4 mm. The total number of scanning points was 40 × 40, and the experimental result is shown in Fig. 5. The ranging precision and accuracy of the validation experiment were computed using Eqs. (15) and (16) (Table 2). The average results were 𝜎𝑅 = 2.38 cm and 𝛿𝑅 = 0.66 cm, respectively. In order to study the influence of the modulation bandwidth on the performance of the proposed system, another experiment was conducted with a stepped FM signal with only 2 frequency steps under the same conditions, which is referred to as the contrasting experiment. The corresponding results are shown in Fig. 6. Similarly, the ranging precision and accuracy of the four targets were computed according to Eqs. (15) and (16), and the precision and accuracy of the contrasting ′ = 8.56 cm and 𝛿 ′ = 1.72 cm, respectively (Table 2). experiments were 𝜎𝑅 𝑅 A comparison of the results of the two experiments indicates that the ranging precision and accuracy of the validation experiment, which adopted a stepped modulation signal with 𝑁 = 21, are superior to that of the contrasting experiment using a stepped modulation signal with 𝑁 = 2. Thus, the advantage of the proposed stepped modulation system
is evident, which proves the feasibility of the proposed stepped FM/CW lidar system by generating a stepped FM signal with an enhanced modulation bandwidth. It is noteworthy that there is a difference between the experimental ranging precision (2.38 cm) and the simulation value shown in Fig. 3 (2.09 cm). This difference is caused by the difference of averaging times between the two conditions. The simulation precision is reduced by averaging more data, which follows the central limit theorem. On the other hand, there is a difference between the experimental ranging accuracy (0.66 cm) and the simulation value (0.16 cm). The difference is mainly caused by the system vibration and the environmental temperature changes in the experiment. These factors will affect the stability of the adopted DPMZM and F–P filter, and result in the differences between the simulation parameters and the experimental parameters [13,15]. 5. Conclusion We propose and experimentally demonstrate a stepped FM/CW lidar system using DPMZM with heterodyne detection. The lidar transmitter utilizes a modulation loop circuit composed of a DPMZM, a fiber amplifier, and a tunable F–P filter to generate a bandwidth-enhanced stepped FM signal. The simulation experiment indicates that the simulation ranging precision and accuracy of the proposed lidar system are 2.09 cm and 0.16 cm with a modulation bandwidth of 4 GHz. The validation 269
C.-h. Yang et al.
Optics Communications 403 (2017) 266–270 Table 2 Experimental results (in cm). Frequency step number
Results
A
B
C
D
Average
𝑁 = 21
Precision 𝜎 Accuracy 𝛿
2.47 0.72
2.45 0.57
2.28 0.70
2.33 0.65
2.38 0.66
𝑁 =2
Precision 𝜎 ′ Accuracy 𝛿 ′
8.72 1.81
8.14 1.54
8.33 1.66
9.05 1.87
8.56 1.72
experiment resulted in a ranging precision of 2.38 cm and a ranging accuracy of 0.66 cm with a modulation bandwidth of 4 GHz by using a 200 MHz signal generator. The results indicate the feasibility of the proposed lidar system with high ranging precision and accuracy. Available high-frequency electric signal sources usually adopt time interleaving (TI) and other methods to generate signal with a modulation bandwidth of 4 GHz [16]. However, these methods lead to increased power consumption, volume, and even incur degradation in the effective number of bits (ENOB) of the generated signal [17]. This degradation cannot be completely limited despite additional equalization and calibration, which causes severe distortion in the generated optical signal and the LO, and eventually results in a decrease in the ranging precision and accuracy of the proposed lidar system. By contrast, the proposed system adopts a modulation circuit loop based on a DPMZM and a tunable F–P filter to broaden the modulation bandwidth. This is a novel approach that uses optical method to partially replace conventional electric modulation. It allows the utilization of high-precision low-frequency electric signal sources to generate wideband modulated signals without degradation in ENOB. The proposed lidar system is also characterized with tunability, compact structure, low power consumption and low-cost. For an electric modulation bandwidth of 4 GHz, the corresponding optical modulation bandwidth (the spectral bandwidth) is only 0.032 nm at a wavelength of 1550 nm, and the modulation bandwidths of the adopted devices are in nanometer magnitude. Hence, it is convenient to generate a 4 GHz signal as a narrow-band signal, and the proposed lidar system has the potential for further expansion of the modulation bandwidth up to 100 GHz. The potential for practical applications of the proposed system can be expected since the key devices used in the proposed system are easily obtainable and the parameters of the devices were improved. Specifically, the bandwidth of the DPMZM in our experiment was 40 GS/s and a frequency comb with more than 100 steps has been reported, from which a step-modulated signal with the same number of
frequency steps can be expected [18]. In addition, the use of advanced tunable F–P filters has been reported with both a high Q-factor and a quick response of less than one nanosecond [19]. By combining these advanced technologies, the proposed FM/CW lidar system is expected to achieve satisfactory ranging precision and accuracy when operating at a much wider modulation bandwidth with low-frequency signal sources and the system has a compact structure. References [1] J.T. Dobler, F.W. Harrison, E.V. Browell, B. Lin, D. McGregor, S. Kooi, Y. Choi, S. Ismail, Appl. Opt. 52 (2013) 2874–2892. [2] Y. Zhang, F. Yi, W. Kong, Y. Yi, Appl. Opt. 53 (2014) 7312–7320. [3] W.C. Swann, N.R. Newbury, Opt. Lett. 31 (2006) 826–828. [4] S. Gao, M. O’Sullivan, R. Hui, Opt. Express 20 (2012) 25867–25875. [5] W. He, B. Sima, Y. Chen, H. Dai, Q. Chen, G. Gu, Opt. Commun. 308 (2013) 211–217. [6] P.F McManamon, Opt. Eng. 51 (2012) 060901-1–060901-13. [7] F. Yang, Y. He, J. Shang, W. Chen, Appl. Opt. 48 (2009) 6575–6582. [8] Z.W. Barber, J.R. Dahl, A.B. Mateo, S. Crouch, R. Reibel, Imaging and Applied Optics, OSA Technical Digest, 2015, LM4F.2,. [9] Y. Zhao, X. Pang, L. Deng, X. Yu, X. Zheng, B. Zhou, Idelfonso Tafur Monroy, Opt. Express 19 (2011) B681–B686. [10] X. Li, Y. Xu, J. Yu, Opt. Lett. 41 (2016) 4162–4165. [11] W. Li, L.X. Wang, J.Y. Zheng, M. Li, N.H. Zhu, IEEE Photonics Technol. Lett. 25 (2013) 1875–1878. [12] T. Kobayashi, A. Sano, E. Yamada, E. Yoshida, M. Miyamoto, J. Lightwave Technol. 27 (2009) 3714–3720. [13] M. Iodice, G. Cocorullo, F.G. Della Corte, Rendina, Opt. Commun. 183 (2000) 415– 418. [14] A.D. Kersey, T.A. Berkoff, W.W. Morie, Opt. Lett. 18 (1993) 1370–1372. [15] S.J. Fabbri, C. O’Riordan, S. Sygletos, A.D. Ellis, Electron. Lett. 49 (2013) 135–136. [16] E. Olieman, A.J. Annema, B. Nauta, IEEE J. Solid-State Circuits 50 (2015) 704–713. [17] C. Laperle, M. O’Sullivan, Signal Processing in Photonic Communications. Optical Society of America, 2014. SM3E. 1. [18] T. Kawanishi, T. Sakamoto, S. Shinada, M. Izutsu, IEICE Electron. Express 1 (2004) 217–221. [19] W. Zhang, N. Ehteshami, W. Liu, J. Yao, Opt. Lett. 40 (2015) 3153–3156.
270