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4*10 Gbps WDM communication system based on a tunable V-cavity semiconductor laser

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Abstract

This paper is about the V-cavity tunable semiconductor laser with a 1550 nm band used as a transmitter to build a wavelength division multiplexing (WDM) optical fiber communication link. In the experiment, a 20 km optical fiber communication link with a reasonable eye diagram and low bit error rate (BER) transmitted by 40 Gbps can be established. The experimental results show that a single laser can achieve a wavelength tuning range of 25 nm, reach 32 channels at a 100 GHz frequency interval, and the average side mode suppression ratio (SMSR) is above 39 dB. The advantages and application potential of V-cavity tunable semiconductor laser (VCL) for wavelength routing in optical communication networking are verified by experiments.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

With the development of communication technology and various intelligent devices, the smart city based on All Optical Network (AON) is expected to become a new platform in the future, and the realization of the desire is based on the design and planning of optical network [13]. The progress of WDM technology provides solutions to meet the demand of large-capacity and high-rate communication brought by the construction of smart cities [46]. WDM technology realizes multi-signal transmission by multiplexing multiple wavelengths of light on a single fiber, and can multiply the capacity of the communication system without changing the existing network infrastructure [7,8]. WDM fiber communication system has superior compatibility, high network flexibility and reliability, which is one of the key technologies to realize AON. However, the development of access network technology based on WDM is facing the challenge of limited capacity expansion and low resource utilization efficiency. Each channel requires a laser of the corresponding wavelength as an optical transmitter, which also greatly increases the cost of WDM system. The rapid development of Photonics Integrated Circuits (PICs) technology has reduced the manufacturing cost of semiconductor lasers. Therefore, low-cost tunable lasers have great application potential in WDM systems [9,10].

The wavelength tunable lasers that have been extensively studied in recent years are mainly as follows. Distributed feedback (DFB) lasers have been widely used in optical communication networks because of their high intensity, high SMSR, and decent direct modulation performance. DFB semiconductor laser realizes single longitudinal mode laser through the periodic change of refractive index or gain. Furthermore, it realizes the wavelength tuning through temperature tuning, but the wavelength tuning range is relatively narrow [11,12]. In 2018, Wang Y et al. proposed a low-cost tunable small shape factor pluggable (SFP) module based on a DFB laser array, which can realize the tunable wavelength of 16 channels in a dense wavelength division multiplexing (DWDM) system with a channel interval of 100 GHz [13]. The distributed bragg reflector (DBR) lasers have much smaller device sizes compared to DFB lasers. Zhu Y et al. proposed a direct modulated DBR laser with 12 operating channels in 2019 and explored its application in 5 G mobile fronthaul (MFH) [14]. The simulation results show that the modulation rate of 25 Gbps and extinction ratio (ER) of 6.2 dB can be obtained by direct modulation. However, due to the complexity of grating structure processing, achieving higher modulation rates and smaller volumes at low cost is challenging. Professor Jian-Jun He and his team, first proposed a tunable semiconductor laser based on V-cavity in 2008 [15]. Designing a specific coupler structure makes it unnecessary to create a complex grating structure and secondary epitaxial growth. The parameters of various prevalent tunable semiconductor lasers are extensively compared in Table 1. The external cavity laser exhibits a substantial tuning range and a narrow line width; however, its miniaturization is hindered by a multitude of discrete components and a voluminous structure. The commercially available array DFB laser suffers from multi-mode interference (MMI) coupling output loss, low yield, and high cost. The fabrication of DBR lasers entails the production of gratings, resulting in a bulky and expensive design. Notably, the digital supermode-distributed bragg reflector (DS-DBR) variant necessitates a significant number of tuning electrodes and presents challenges in terms of packaging. The three-stage DBR laser, while cost-effective, compromises its tuning range by eliminating the forward grating. Conversely, the VCL laser possesses several advantageous attributes, including the absence of gratings, elimination of multiple epitaxial growth, compact dimensions (approximately 0.3*0.5 mm2), extensive adjustment range, and low cost. These features align exceptionally well with the growing demands for network bandwidth and the cost-effective requisites of tunable lasers in WDM system [16].

Tables Icon

Table 1. Comparison of major tunable semiconductor lasers

In this paper, a tunable semiconductor laser based on V-cavity laser is proposed. A single laser can realize 32 channels adjustable in 100 GHz frequency spacing, and the SMSR is stable between 35 dB and 40 dB. The BER is less than 10−9 in 20 km optical fiber transmission. In addition, a 20 km WDM optical fiber communication link with four VCLs tunable semiconductor lasers as transmitter is constructed and its transmission characteristics are demonstrated. The same laser can achieve tunable multi-wavelength in different systems under the different wavelength requirements; only one tunable laser can be realized. Future optical networks necessitate enhanced transmission rates, heightened dynamic flexibility, and reduced costs. They not only serve as backup for fixed-wavelength lasers in dense wavelength division multiplexing systems but also enable automated wavelength configuration, wavelength conversion, and wavelength routing. These capabilities facilitate dynamic allocation and optimization of network resources, bolster bandwidth utilization, curtail network construction expenses, and ensure the overall reliability of the network system [17]. Consequently, the monolithic wavelength tunable laser and its corresponding WDM communication system hold significant significance and exhibit vast prospects in the realm of future optical communication.

2. Theoretical basis of VCL design

The basic structure of VCL is shown in Fig. 1. The main body of the laser consists of a half-wave coupler with a coupling phase of $\mathrm{180^\circ }$, two waveguide resonators and three electrodes covered above. The shorter cavity is used to provide the fixed gain of the laser, called the fixed gain cavity, and adjust the interval between each channel of the laser. The longer resonator is used to change the refractive index by changing the injection current and temperature in the process of wavelength tuning, and then the output mode is switched in different channels, which is called the channel selector cavity. The length difference between the two resonators enables V-cavity lasers to amplify tunable wavelength ranges using Vernier effects [18,19]. The half-wave coupler is used to select the laser mode to achieve a single longitudinal mode output with high SMSR. At the front and rear ends of the two resonators, there are three deep etched regions, which are used to precisely control the length of the two resonators and the half-wave coupler, forming the etched facet as the reflection surface of the laser. The deep etching facet of the fixed gain cavity and the half-wave coupler is coated with a high reflection film of titanium and gold to improve the reflectivity. The deep etching facet of the cavity is selected as the output port of the laser. The yellow areas above the waveguide are the coupler electrode, the fixed-gain cavity electrode and the channel-selective cavity electrode. The coupler electrode acts as the central gain electrode and the direct modulation electrode. Two isolation gaps separate the three electrodes to power the two resonators and the half-wave coupler. The common ground electrode is located on the backside of the laser chip.

 figure: Fig. 1.

Fig. 1. Schematic diagram of V-cavity tunable semiconductor laser.

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In the VCL, the variation of amplification frequency is due to Vernier effects, but there is a lack of mode suppression mechanism, which leads to low SMSR of some lasers. Therefore, in the design of VCL, a half-wave coupler is designed to connect two waveguide resonators to provide a mode selection function and achieve a single longitudinal mode output with a SMSR of more than 39 dB.

For N*N multimode interference (MMI) coupler, according to the coupling mode theory [20], the coupling phase from the ith input to the jth output satisfies the following relation:

$${\varphi _{ij}}\textrm{ = }\left\{ {\begin{array}{{c}} {{\varphi_0} + \pi + \frac{\pi }{{4N}} \times ({j - i} )({2N - j + i} )}\\ {{\varphi_0} + \frac{\pi }{{4N}} \times ({j + i - 1} )({2N - j - i + 1} )} \end{array}} \right.\textrm{ }\begin{array}{*{20}{c}} {i + j\textrm{ is even number}}\\ {i + j\textrm{ is odd number}} \end{array}$$
where ${\varphi _0}$ represents the phase introduced by the propagation constant, and when $N = 2$, the bar-coupling phase is ${\varphi _{11}} = {\varphi _{22}} = {\varphi _0} + \pi$ and the cross-coupling phase is ${\varphi _{12}} = {\varphi _{21}} = {\varphi _0} + \pi /2$. The phase difference between them is $\mathrm{90^\circ }$. The V-cavity laser is expanded with the reflection surface of the half-wave coupler as the symmetry axis, and the incidence and reflection of photons in the V-cavity can be equivalent to a 2*2 four-port system, as shown in Fig. 2.

 figure: Fig. 2.

Fig. 2. Schematic diagram of the V-cavity laser after expansion.

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Assuming that the input phase difference is $\theta$ at the two input terminals and the total energy equals to 1. The phase difference between the bar-coupling coefficient and the cross-coupling coefficient is$\mathrm{90^\circ }$. Then the optical field at the output of the coupler is as follows:

$${P_1} = \frac{1}{2}[{{{|{{C_{11}}} |}^2} + {{|{{C_{21}}} |}^2} + 2|{{C_{11}}} ||{{C_{21}}} |\cos ({\varphi + \theta } )} ]$$
$${P_2} = \frac{1}{2}[{{{|{{C_{12}}} |}^2} + {{|{{C_{22}}} |}^2} + 2|{{C_{12}}} ||{{C_{22}}} |\cos ({\varphi - \theta } )} ]$$
${C_{11}}$, ${C_{12}}$, ${C_{21}}$ and ${C_{22}}$ are the coupling coefficients, bar-coupling coefficient ${C_{11}} = {C_{22}}$ and cross-coupling coefficient ${C_{12}} = {C_{21}}$, according to Eqs. (2) and (3), the normalized power of the two output terminals changing with the phase difference of the optical input field can be obtained, as shown in Fig. 3(a). Similarly, when a half-wave coupler is used, the phase difference $\varphi = 180\mathrm{^\circ }$ and the optical field normalized power changes at the optical output field are shown in Fig. 3(b). Where${C_{11}} = {C_{22}} = 0.755$, ${C_{12}} = {C_{21}} ={-} 0.245$.

 figure: Fig. 3.

Fig. 3. The normalized power curve of input phase difference. (a) $\varphi = 90\mathrm{^\circ }$; (b) $\varphi = 180\mathrm{^\circ }$.

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Figure 3(a) shows that the energy distribution at the output of a coupler with a coupling phase of $\mathrm{90^\circ }$ is complementary. This $\mathrm{90^\circ }$ phase coupler is commonly used for optical splitting, and as the two arms of the interferometer. If the VCL is expected to achieve good mode selection, then the output energy of both ports needs to be maximized at a specific resonant wavelength and minimized at other non-resonant wavelengths. The transmission function in the same direction can be realized by using couplers with cross-coupled phase $\varphi = m\pi ({m = 0, \pm 1, \pm 2 \ldots } )$. The energy of the two outputs of a half-wave coupler with a coupling phase of $\mathrm{180^\circ }$ always has the same variation trend, that is, it reaches the maximum value when the phase difference at the input end is $\mathrm{180^\circ }$ and the minimum value when the phase difference at the input end is 0, as shown in Fig. 3(b). The selected wavelength in the optical field at both ends of the coupler has a phase difference of $\mathrm{180^\circ }$. Therefore, the selected wavelength has the maximum energy output returned to the V-cavity after passing through the half-wave coupler. However, for other wavelengths, the light fields at both ends of the coupler have the same phase. After passing through the half-wave coupler, the light interferes and the energy is reduced as radiation escaping from the substrate, so that only a small amount of energy is returned to the V-cavity.

Ideally, when both ends of the coupler input light of the same amplitude and opposite phase, there should be no transmission loss at the output of the coupler relative to the input. However, according to Eqs. (2) and (3), in reality, for a coupler whose coupling phase is $\pi$, the additional loss is defined by the following equation:

$${\varepsilon _1} = 10{\log _{10}}({{{|{{C_{11}}} |}^2} + {{|{{C_{12}}} |}^2} + 2|{{C_{11}}} ||{{C_{12}}} ||{\cos (\varphi )} |} )$$
$${\varepsilon _2} = 10{\log _{10}}({{{|{{C_{22}}} |}^2} + {{|{{C_{21}}} |}^2} + 2|{{C_{21}}} ||{{C_{22}}} ||{\cos (\varphi )} |} )$$
In addition, the half-wave coupler can select the mode matching the V-cavity structure and suppress other modes to improve the SMSR of the selected working mode. This method enables the single-mode performance of VCL to approach that of a tunable laser with grating.

3. Experimental characterization

3.1 Experimental setup

In this paper, a WDM communication system based on V-cavity tunable semiconductor laser was designed. The structure diagram of the experimental system is shown in Fig. 4, and the BtB test experiment of the system is shown in Fig. 5. The system consisted of four VCL lasers, a multiplexer, a dispersion compensator module (DCM, YOSC AD-020), a demultiplexer, a bit error rate tester (BERT, Textronix BSA125C), an optical wide-range oscilloscope (OWO, Agilent 86100C), an optical spectrum analyzer (OSA, YOKOGAWA AQ6370D), an arbitrary waveform generator (AWG, Keysight, M9502A), an optical power meter (OPM, Agilent 8163B), and 20 km single-mode fiber (SMF). The VCL laser was encapsulated in the transmit optical sub-assembly (TOSA) test board in the form of a SFP [21,22]. The VCL-TOSA was equipped with 3 direct current (DC) laser diode drivers (LDD) and a radio frequency (RF) LDD. Three DC LDDs supplied the power to the three electrodes respectively. RF LDD was used to amplify the signal of the external differential input and control the modulation depth by programming the modulation current (Imod value). In addition, RF LDD could also shape the input signal's waveform to provide a threshold chirp to improve the transmission effect and lower noise. The temperature control module was also integrated into the VCL-TOSA and is equipped with an aluminum heat sink, which required a 5 V constant voltage drive when in use.

 figure: Fig. 4.

Fig. 4. (a) The structure diagram of the experimental system. (b) The BtB test experiment of the system.

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 figure: Fig. 5.

Fig. 5. Full-channel spectral diagram.

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The control program on personal computer (PC) set the temperature of VCL-TOSA and the current value of three electrodes to make the laser work in the selected channel. BERT provided pseudo-random bit sequence (PRBS23) code type and signals at the required rate. The signals were shaped and amplified by RF LDD on VCL-TOSA and then loaded onto the coupler electrodes of VCL. Four VCL-TOSA output optical signals are multiplexed by a multiplexer, transmitted by the single-mode fiber, demultiplexed by a demultiplexer, and then received by VCL- receiver optical sub-assembly (VCL-ROSA) as a receiver. The length of the transmission link was 20 km. An OPM detected the received power. In the experiment, OSA was used to observe the received spectrum, and OWO was used to observe the electric eye image of the data transmission system.

3.2 Transmission characteristic analysis

The purpose of the preliminary experiment was to test the direct modulation capability of VCL-TOSA. All 4 VCLs of InGaAlAs material used in this experiment could realize the spectral output of 32 channels, and the full-channel spectral diagram after superposition is shown in Fig. 5. Tunable range was 1543-1568 nm, greater than 25 nm. The average output power of continuous operation at room temperature was 7.029 dBm, and the average SMSR could reach over 39 dB.

The channel with a wavelength of 1550.95 nm was taken as an example. The specific parameters of the channel are shown in Table 2. In the table, ILD1 represents the injection current of the electrode of the coupler, ILD2 represents the injection current of the channel selection cavity, and ILD3 represents the injection current of the fixed-gain cavity. In the experiment, the output signal of VCL-TOSA was received directly by OSA and OWO under back-to-back (BtB) conditions. The program gradually increased the modulation current from 0 mA and then increased the modulation depth. The changes of the eye diagram and spectrum were also observed. The parameters to focus on in the eye diagram are extinction ratio (ER) and signal-to-noise ratio (SNR). Too low ER would reduce the sensitivity of the receiver; too high ER or jitter and chirp would result in a high transmission power penalty.

Tables Icon

Table 2. Typical channel parameters

As shown in Fig. 6, at the modulation rate of 10 Gbps, the eye diagram and spectrum changes were observed when the modulating electrode's current was 7.5 mA, 14.5 mA, and 21.5 mA, respectively. The test results of the eye diagram and spectrum were presented in pairs. When the Imod value was 7.5 mA, the results are shown in Fig. 6(a) and (b), with ER was 4.16 dB, SNR was 4.32 dB, and SMSR was 38.48 dB. When the Imod value was 14.5 mA, the result is shown in Fig. 6(c) and (d), ER was 6.38 dB, SNR was 4.05 dB, and SMSR was 36.26 dB. When the Imod value was 21.5 mA, the result is shown in Fig. 6(e) and (f), ER was 8.69 dB, SMSR rapidly drops to 7.52 dB, and SNR was as low as 2.91 dB. According to the data analysis, with the increase of Imod value, ER gradually increased, while SMSR gradually decreased. When the modulation depth was increased to a certain extent, the SMSR would decrease significantly, and the eye diagram's noise points would obviously increase. The reason should be that the distribution of photons in the cavity changed in the dynamic process (refers to the state with the modulated signal), which will lead to the reduction of the difference of the threshold value of the main side mode. The working state of the half-wave coupler could deviate with the addition of the modulating signal, which would also lead to the reduction of the difference of the main side mode threshold, futher result in the reduce of the SMSR under dynamic conditions. When the carrier injection of the modulating signal is too large, the refractive index of the waveguide will deviate significantly, and the phase difference between the cross-coupling coefficient and the through-coupling coefficient would deviate 180 degrees so that the half-wave coupler will separate from the operating condition and lose the extra mode selection characteristic. Improper setting of the amplitude of the modulated signal would lead to low SMSR and poor transmission quality, so selecting the appropriate working point could be indispensable. For access to a 10 Gbps modulated signal, PC controlled the modulation electrode current (Imod value) and the spectrum changes were observed in OSA. When the left and right-side peaks were consistent, fine-tuned the coupling electrode current size to achieve SMSR maximization. Then the state could be considered as the best working state, and the noise generated by the mode distribution could be suppressed. Mode distribution noise was generated by the energy distribution of the jitter in different modes when the laser works, which was an inherent attribute of each mode. When the side mode power increased, mode distribution noise was significantly enhanced, while with the increase of the modulation current, the ER would further increase the mode distribution noise and affect the transmission effect, so the inappropriate high ER should not be adopted. According to the experimental result, it was considered that the modulation current should be controlled to ensure proper SMSR, and the ER of 4-6 dB after 20 km fiber transmission was the best. All the signals of four VCL-TOSA were adjusted above to select the best operating point for the construction of the WDM system.

 figure: Fig. 6.

Fig. 6. Changes of eye diagram and edge mode rejection ratio with increasing modulation depth.

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After the laser was adjusted to the optimal operating point of each channel, the transmission characteristics of the four-channel WDM communication system were tested. The parameters of each channel are shown in Table 3. In the experiment, AWG (Keysight, M9502A) was used to generate modulated signals and load them onto four VCL-TOSAs. The multiplexed four VCL-TOSAs output optical signals were transmitted by 20 km single-mode fiber and then demultiplexed by a demultiplexer. The output signals characteristic is detected by OSA and OWO. The optical carrier center wavelength of the four channels was 1549.37 nm, 1550.89 nm, 1552.56 nm and 1554.00 nm, respectively.

Tables Icon

Table 3. Optimal operating parameters of four typical VCL channels

In the experiment, 10 Gbps signal is generated by an AWG and divided into four signals by a power divider to load on four laser modulation electrodes, the whole system would transmit at the rate of 4*10 Gbps. Since the four lasers share the same structure and exhibit similar performance, one of the lasers has been selected for analysis. The eye diagrams are detected under BtB and 20 km optical fiber transmission, respectively; the test results are shown in Fig. 7. (a) and (b). After demultiplexing, ER was 4.69 dB, and SNR was 7.42 dB under BtB conditions with a central wavelength of 1550.89 nm; ER was 4.16 dB and SNR was 6.75 dB after fiber transmission of 20 km. The modulation rate of each channel was set at 12.5 Gbps so that the whole system was transmitted at a rate of 4*12.5 Gbps. The eye diagram of BtB and 20 km optical fiber transmission in the same channel is shown in Fig. 7. (c) and (d). In the BtB condition, ER was 3.06 dB and SNR was 4.46 dB. After 20 km of optical fiber transmission, ER was 2.81 dB, and SNR was 3.11 dB. According to the results, it could be analyzed that the eye diagram were able to open typically and transmitted effectively at the modulation rate of 4*10 Gbps. In contrast, the quality of the eye diagram decreased obviously at the modulation rate of 4*12.5 Gbps, and the jitter and noise increased significantly. After the optical fiber transmission of 20 km, the eye diagram was barely opened, and the laser carrier transmission with high SNR cannot be realized. In summary, a single laser working at the modulation rate of 10 Gbps and below could achieve a high transmission effect; that is, the WDM communication system composed of four VCL would achieve fiber communication with high SNR at the modulation rate of 4*10 Gbps.

 figure: Fig. 7.

Fig. 7. Eye diagrams of two transmission modes at different modulation rates.

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In addition, the BERT and digital receiver were added to test the BER of the communication system. After the program controlled the modulation amplitude, the RF signal sent by the error meter was transmitted to the variable optical attenuator (VOA) and the 3 dB coupler through VCL-TOSA under the condition of BtB. One of the two optical fibers of the 3 dB coupler output to the optical power meter and the other to OWO. Since the receiver of the BER used the transmitter's clock, clock synchronization was required to calibrate the delay before testing the BERT data.

The four-channel parameters are the same as those in Table 3 when the whole system was transmitted at a 4*10 Gpbs modulation rate and tested under the PRBS23 code type. The BER after BtB and 20 km optical fiber transmission were tested, respectively. The same reference receiver was adopted for testing, and the receiver's power was adjusted to −7 dBm. The BER began to decrease significantly, and a BER value was recorded for each 0.5 dB attenuation.

The experimental result of BER under different conditions is described in Fig. 8, when the received power was increased to more than −8.25 dBm, the BER of each channel after 20 km fiber transmission was less than 10−9, which could be considered to achieve error-free transmission. After transmitting 20 km of optical fiber, the power penalty was only 0.5 dB. The power penalty was generated because of the frequency noise, such as chirp generated in the direct modulation process, was affected by dispersion in the transmission process, which further worsened the SNR relative to the case of BtB.

 figure: Fig. 8.

Fig. 8. The results of BER.

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4. Conclusion

This paper analyzes the high-speed modulation performance of VCL-TOSA and explores the transmission characteristics of the WDM communication system constructed by VCL-TOSA. The proposed WDM communication system can realize the direct modulation rate of 4*10 Gbps fiber communication under the PRBS23 code. After 20 km of optical fiber transmission, the BER is below 10−9, and the ER is between 4-6 dB. The designed single laser can realize 32 tunable channels with 100 GHz channel spacing, and the output power of each channel is more than 6 dBm. In practical application, low cost, high speed, and stable optical fiber communication transmission can be achieved. In this paper, the VCL laser is first applied to the WDM communication system, which proves that it has great potential in wavelength routing applications. Because of its simple tuning characteristics, it is expected to become the core device of the optical network unit in the next generation optical network terminal, which provides more possibilities for the development of the optical communication industry.

Funding

National Natural Science Foundation of China (62027825); Natural Science Foundation of Jilin Province (20220101130JC, YDZJ202301ZYTS495); Natural Science Foundation of Shandong Province (ZR2019MC069); 111 Project (D21009).

Acknowledgments

I thank my advisor for providing me with some academic suggestions and assisting me in reviewing the paper.

Disclosures

The authors declare no conflicts of interest.

Data availability

No data were generated or analyzed in the presented research.

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Data availability

No data were generated or analyzed in the presented research.

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Figures (8)

Fig. 1.
Fig. 1. Schematic diagram of V-cavity tunable semiconductor laser.
Fig. 2.
Fig. 2. Schematic diagram of the V-cavity laser after expansion.
Fig. 3.
Fig. 3. The normalized power curve of input phase difference. (a) $\varphi = 90\mathrm{^\circ }$; (b) $\varphi = 180\mathrm{^\circ }$.
Fig. 4.
Fig. 4. (a) The structure diagram of the experimental system. (b) The BtB test experiment of the system.
Fig. 5.
Fig. 5. Full-channel spectral diagram.
Fig. 6.
Fig. 6. Changes of eye diagram and edge mode rejection ratio with increasing modulation depth.
Fig. 7.
Fig. 7. Eye diagrams of two transmission modes at different modulation rates.
Fig. 8.
Fig. 8. The results of BER.

Tables (3)

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Table 1. Comparison of major tunable semiconductor lasers

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Table 2. Typical channel parameters

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Table 3. Optimal operating parameters of four typical VCL channels

Equations (5)

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φ i j  =  { φ 0 + π + π 4 N × ( j i ) ( 2 N j + i ) φ 0 + π 4 N × ( j + i 1 ) ( 2 N j i + 1 )   i + j  is even number i + j  is odd number
P 1 = 1 2 [ | C 11 | 2 + | C 21 | 2 + 2 | C 11 | | C 21 | cos ( φ + θ ) ]
P 2 = 1 2 [ | C 12 | 2 + | C 22 | 2 + 2 | C 12 | | C 22 | cos ( φ θ ) ]
ε 1 = 10 log 10 ( | C 11 | 2 + | C 12 | 2 + 2 | C 11 | | C 12 | | cos ( φ ) | )
ε 2 = 10 log 10 ( | C 22 | 2 + | C 21 | 2 + 2 | C 21 | | C 22 | | cos ( φ ) | )
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