Silicon unique modulation mechanism based on free-carrier dispersion (FCD) effect determines that there is operation and performance difference from LiNbO3 modulator when achieving various optical modulation formats. In this paper, the influence of nonlinear FCD and free carrier absorption (FCA) effect on the return-to-zero (RZ)-DPSK generation scheme is numerically analyzed. Silicon waveguide with p-n diode is adopted and the reverse bias is the key factor which should be chosen carefully. Performance analysis includes two parts: the property of the generated optical signal and the dispersion penalty which is related to chirp. The simulation results show that the output phase of the optical RZ-DPSK signal has undesirable distortion and the power has considerable loss. Furthermore, the simulation of modulator with 20 dB extinction ratio is also performed for relative analysis. The poor extinction ratio will further impact the characteristic. Even the push-pull operation is utilized, there is a residual chirp resulting from FCA and nonlinear FCD effect. This kind of chirp is characterized by the dispersion penalty.
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Advanced optical modulation formats have become a key ingredient to the design of modern wavelength-division multiplexed (WDM) optically routed networks . Choosing the right modulation format is of great importance for high performance fiber communication . It is expected that, in the future optical network nodes, the added signal and transmission signal can be modulated and converted to arbitrary optical format respectively to meet the requirement of more flexible and multiplex optical communication networks. If the combination of the discrete devices is employed to achieve this, the system will become bulky and expensive.
Recently, silicon photonics has emerged as the premier candidate for electronic-photonic integration due to its small footprints and compatibility with electronics while keeping the fabrication cost relatively low . This integration fundamentally impacts a wide range of applications, including computing, communications and signal processing . One of the major applications is high-speed modulators for multi-format modulation. The integration of multi-format modulators on a single silicon chip will have a profound effect on future optical communication networks. Among the advanced optical modulation formats, differential binary phase shift keying (DBPSK, or simply DPSK) has the advantage of 3-dB receiver sensitivity improvement  and can provide the solution to the long haul and metropolitan networks being upgraded from 10Gb/s to 40Gb/s . Silicon micro-ring has been theoretically and experimentally demonstrated for DPSK modulators, yet it will generate frequency chirping [6, 7].
DPSK generation based on Mach-Zehnder modulator (MZM) with push-pull operation can eliminate the frequency chirping in a broad spectral range . It is also the basic scheme for QPSK. The modulator has been fabricated using optical waveguides made of polar crystals such as LiNbO3 and InP . However, unlike LiNbO3 or InP material’s linear electro-optical effect, silicon has remarkable free-carrier dispersion (FCD) effect which is nonlinear and will introduce considerable loss resulting from free-carrier absorption (FCA) effect . There is bad impact on the generated optical DPSK signal such as distortion in output phase and considerable loss in power. Moreover, some modulation characteristics like chirp will also be influenced by these factors . To the best of our knowledge, the impact of FCA and nonlinear FCD on the performance of Silicon DPSK generation has never before discussed.
In this paper, we utilize the MZM with reversed p-n diodes which has high modulation efficiency for the need of push-pull operation. According to the simulation, optimal DC reverse bias is selected. Then the influence of FCA and nonlinear FCD on the performance of silicon DPSK modulation is numerically analyzed. The performance analysis includes two parts: the property of the generated optical DPSK signal and the dispersion penalty which is related to chirp. The paper is aimed to provide the theoretical basis for constructing the silicon DPSK modulator.
We will limit our scope to RZ-DPSK format because RZ format is generally more robust to intersymbol interference (ISI) and many nonlinear propagation distortions . Besides, sinusoidally driving an MZM makes the output power of RZ-DPSK signal having a gentle change during the whole bit period. This gentle change can help better explain the influence of FCA and nonlinear FCD.
2. Theory and device structure
RZ-DPSK generation scheme usually needs an external NRZ-DPSK modulator and an MZM-based RZ pulse carver which may suffer from relatively high cost. There has been the research about using a single-stage dual-electrode MZM to generate RZ-DPSK . This makes RZ-DPSK more attractive. In my paper, we use the structure as can be seen in Fig. 1(a) to analyze the influence of nonlinear FCD and FCA. The original data is a pseudorandom bit sequence and the block of NRZ to RZ converter  produces electronic RZ format data V1(t) and V2(t). According to FCD effect in silicon, the electronic drive signal V1(t) and V2(t) will introduce the effective index change (ΔNeff) and absorption coefficient variation (Δα) through the process of carrier depletion. Thus, the free carrier induced phase shift in the waveguide is Δφ = ΔNeff2πL/λ and the electric field transmission coefficient is t = exp (−ΔαL/2) (t ≈1 for LiNbO3), where L is the MZI arm length and λ is the optical wavelength of 1.55 μm. Push-pull operation is crucial for chirp-free DPSK modulation. Generally, the drive RF signal should have opposite polarity (V1 = − V2). Reverse bias is applied to the RF phase shifters for high-speed modulation in carrier depletion mode. Thermo optic phase shift (φbias) introduced by DC phase shifter is used to set the operation point at null transmission.
The MZM power transfer function TP(V1,V2) and the phase change Φ of output signal can be expressed as:
α1, α2 and φ(V1), φ(V2) are the absorption coefficient change and phase shift induced by V1 and V2. In addition, α0 is the absorption coefficient introduced by reverse bias. The parameter γ represents that the loss in one arm is different from the other and is related to the extinction ratio of the modulator.
Figure 1(b) shows the difference of output power and phase between LiNbO3 MZM and silicon MZM. For chirp-free based on push-pull operation, P1 = P2 and φV1 = − φV2 should be satisfied just like in LiNbO3 MZM. When applying voltage between the two phase shifters of LiNbO3 MZM, the optical phase in arm 1 rotates clock-wise, while that in arm 2 rotates counterclockwise. At last, the phase of the combined optical signal only has two values and can change abruptly from 180° to 0°. However, linear electro-optic effects known as Pockels and Stark effects do not exist in silicon. Nonlinear FCD and FCA are existent together and inevitable. Nonlinear FCD means that the relationship between the voltage of driving RF signal and the corresponding phase shift Δφ is nonlinear. When push-pull is operated (V1 = − V2), the phase shift doesn’t have the condition φV1 = − φV2. On the other hand, according to the FCA effect, the voltages applied to two phase shifters must have opposite signs to achieve push-pull, that means there will be two kinds of physical process between phase shifter1 and 2, carrier injection and carrier depletion. Carrier injection can result in the loss increase and carrier depletion can cause the loss decrease. That is the power imbalance. P1 is shorter than P2 as can be seen in Fig. 1(b). Therefore, under the influence of nonlinear FCD and FCA, the generated optical signal P after interference in silicon is shorter than that in LiNbO3 and is deviating from the real axis. The phase between the signal P and real axis is varying with time. The phase deviation is defined here as phase (T/2), where T is the bit period. From above analysis, we can see that the property of the generated optical signal based on silicon has some distortions like power loss and phase deviation. Furthermore, it can’t be chirp-free and the simulation of dispersion penalty also should be performed.
2.2 Device structure
The cross section of the p-n diode of the MZM we applied to achieve DPSK modulation is shown in Fig. 2 . Carrier depletion in a reverse biased p-n diode has an advantage on the operational speed which can meet the requirement of high-speed optical networks. This structure with high modulation efficiency is first introduced by . The rib width and height are 500 and 220 nm respectively, the etching depth is 70 nm. Two moderately doped slits with equal concentration of 1018/cm3 and opposite polarities form a p-n junction inside the rib waveguide. Two heavily doped regions (1020/cm3) to form ohmic contacts are situated 1μm apart from the rib and the regions in the middle are lightly doped to 2 × 1017/cm3. The simulation model of p-n diode with incomplete ionization, concentration dependent mobility, Shockley Read Hall recombination and ohmic contact is built by using the software ATLAS . Then the 2-D carrier distributions are calculated and imported into a finite-difference program. At last we obtain the free-carrier-induced effective refractive index change (ΔNeff) and the absorption coefficient variation (Δα) as shown in Fig. 2(b). For ideal p-n junction, its effective mode index shift at a reverse bias of 9.9-V is ΔNeff = 3.98 × 10−4, which leads to a static Vπ∙L figure of merit of 1.93-V∙cm.
3. Performance analysis
To investigate the influence of nonlinear FCD and FCA on the performance of DPSK signal, numerical simulations are performed.
3.1 Reverse bias
From Fig. 2(b) we can see that, for various values of the reverse bias, the slope of ΔNeff curve around that is different which can result in the different levels of nonlinear FCD. Finding a good reverse bias is essential to the design of DPSK modulator based on p-n diode. Firstly, we set the length of phase shifter at 4.5 mm because this length can meet the requirement of π phase shift. The operation point at null transmission is achieved by φbias = π. Secondly, we choose one single value of reverse bias such as 3.5 V, then RF signal with different swing Vpp are applied to phase shifters. Lastly, we get the maximal output power, the corresponding phase deviation and swing Vpp of the driving signal. Figure 3(a) shows the different reverse bias voltages lead to the variation of maximal output power (a.u.) and phase deviation. We can also get the corresponding swing Vpp of the driving signal as the function of reverse bias (Fig. 3(b)). The maximal output power has an approximate monotone increasing relationship with the reverse bias because the bigger voltage of reverse bias will result in stronger carrier depletion, then the power loss caused by FCA will reduce. On the other hand, the phase deviation is also influenced owing to the nonlinear FCD. A good reverse bias can improve the characteristic of the generated optical signal. We hope that phase deviation and swing Vpp is small and the output power is large. In addition, when reverse bias is above 5 V and RF signal with Vpp is applied, p-n diode has the danger of breakdown. Taking all these factors into account, we choose 5 V as the optimal reverse bias value.
3.2 Numerical simulations
The reverse bias is 5 V and the length of phase shifters are 4.5 mm. NRZ drive signal as the original data is a 5 Gb/s pseudorandom bit sequence and after the circuit of NRZ-to-RZ converter, it becomes RZ format with 50% duty cycle. We first investigate the performance of the generated optical RZ-DPSK signal because the imperfection of the generated optical signal reflects the influence of nonlinear FCD and FCA directly. Figure 4(a) shows that the phase shift introduced by the electrical drive signal has a small distortion compared with the ideal sinusoidal shape. This is because the curve of ∆Neff in Fig. 2(b) is not linear which determines that the increase of ∆Neff between 5 and 7.4 V is smaller than that between 2.6 and 5 V. Although V1 = –V2 = 2.4 V, the phase shift φV1≠ –φV2 which is caused by nonlinear FCD.
We select a differentially encoded data sequence of (11001), the generated optical signal has a phase pattern of (ππ00π). Figure 4(b) and 4(c) shows the power and phase behavior. It can be seen that there is still considerable 1.97 dB loss that the modulator will encounter from the FCA and nonlinear FCD even the optimal reverse bias is selected. Furthermore, there is an abrupt phase change when data is transformed from 1-bit to 0-bit and vice versa. However, whether 0-bit or 1-bit is transmitted, the phase deviation is about 7.38°. This can be regarded as transmitter phase noise and be optimized using phase-sensitive amplifier [15, 16].
For the actual silicon MZM, the input optical field can’t be equally split into the two arms. We assume that dc extinction ratio is 20 dB and the corresponding parameter γ is 0.82. From Fig. 4(e) we can see that there is no exact π phase jumps and the speed of phase transitions is about 32 ps which is limited by the finite extinction ratio. If the silicon modulator has the worse extinction ratio, the time of phase transitions will be longer. The phase changes in a wide range which is about 9.6° and 2.3° during 1-bit and 0-bit respectively. On the other hand, the finite extinction ratio determines that the output optical intensity (a.u.) has a slight difference between 1-bit and 0-bit. It is 0.62 for 0-bit and 0.64 for 1-bit. This intensity difference will become bigger when extinction ration is worse as can be seen in Fig. 4(f). In conclusion, the quality of the phase and the power loss can be improved when modulator has higher extinction ratio and more equal optical power splitting.
From above simulation about the generated optical signal we can conclude that the signal is not chirp-free. The chirp parameter αchirp is defined in 
We use the instantaneous frequency (dΦ/ dt) to characterize the chirp of the optical signal as shown in Fig. 5 . The instantaneous frequency change is negative during the rising edge of the signal and positive during the falling edge. According to Eq. (3), this will result in the negative αchirp. This negative chirp characteristic induced by silicon material is believed to weaken the pulse broadening introduced by unavoidable optical chromatic dispersion and account for the difference in the dispersion penalty [17, 18]. If the modulator has the finite 20 dB extinction ratio, the chirp parameter αchirp is also negative and larger (11.7 GHz) in Fig. 5(b), this is because the optical phase change in Fig. 4(e) is faster than that in Fig. 4(c).
In order to investigate the evolution of transmitted silicon-MZM-generated BPSK signal, the simulation about optical spectrum and dispersion penalty which is related to the frequency chirp is performed. The dispersion penalty is the increase in energy per bit required to maintain the BER value (10−9) when dispersion is present. The conventional single-mode fiber (17ps/km/nm) is used as a dispersive medium. Figure 6(a) shows the simulated optical spectrum. We note that the silicon-based modulation produces a spectrum very similar to that given by the LiNbO3 MZM, although it is a litter broader in the low-power regions which mainly results from the chirp in the generated optical signal. The power-penalties, referenced to the 0-km propagation of the LiNbO3-modulated RZ-DPSK, are summarized in Fig. 6(b). The fiber-propagated behavior between LiNbO3 MZM, silicon MZM and silicon MZM with γ = 0.82 are similar, exhibiting gradually decreasing dispersion-penalties. As indicated by analysis, the silicon MZM with 20 dB extinction ratio suffers from the most chirp than the others. More importantly, the dispersion penalty of the silicon MZM with γ = 1 relative to the LiNbO3 is less than 0.63 dB throughout the 100-km propagation. This demonstration shows that the chirp resulting from FCA and nonlinear FCD in silicon improve the dispersion penalty as the propagation distance increasing.
The influence of the FCA and the nonlinear FCD on RZ-DPSK generation scheme based on silicon MZM was numerically analyzed. The impacts are inevitable. According to the push-pull operation and the working principle of p-n diode, DC reverse bias should be introduced and there is an optimal value for the specific device structure. For the structure in Fig. 2(a), 5 V DC reverse bias is selected. In addition, the 5 GHz pseudorandom bit sequence signals after NRZ-to-RZ converter with Vpp 4.8 V are applied. The performance analysis includes two parts: the property of the generated optical signal and the dispersion penalty which is related to chirp. The simulation results show that nonlinear FCD and FCA can lead to 7.38° output phase deviation and considerable 1.97 dB power loss. The modulator with 20 dB extinction ratio is also simulated for relative analysis. There is no exact π phase jump and the speed of phase transitions rises to 32 ps. If the silicon modulator has the worse extinction ratio, this transition time will be longer. Besides, output optical intensity (a.u.) has a slight difference between 1-bit and 0-bit and this intensity difference will become bigger when extinction ration is worse. Therefore, the higher extinction ratio can improve the quality of the generated optical signal.
Even the push-pull operation is adopted, the nonlinearity of FCD and the power imbalance between two phase shifters will introduce residual chirp. To some extent, this kind of chirp induced by silicon material will weaken the pulse broadening and make a difference in the optical transmission systems. From the simulation of the dispersion penalty we can see that the chirp resulting from FCA and nonlinear FCD in silicon can improve the dispersion penalty as the propagation distance increasing.
Although the DPSK signal suffers from the inevasible distortion and 4.5 mm is too large for integration, the silicon devices will still play an important role in the advanced format modulation. The length and the performance such as phase deviation, power loss and dispersion penalty can be further improved by adopting more effective p-n junction which is more excellent in phase modulation.
This work is supported by the Natural Science Foundation of China (No. 6177055) and the 863 project under Grant 2012AA012203.
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