Impact of Grating Duty-Cycle Randomness on DFB Laser Performance

Abstract

The duty-cycle randomness (DCR) of the Bragg grating of the distributed feedback (DFB) lasers introduced by the fabrication process is inevitable, even with state-of-the-art technologies such as electron beam lithography and dry or wet etching.This work investigates the impact of grating DCR on DFB laser performance through numerical simulations. The result reveals that such randomness causes a reduction in the side mode suppression ratio (SMSR), and deteriorates the noise characteristics, i.e., broadens the linewidth and increases the relative intensity noise (RIN). With the grating DCR, the effective grating coupling coefficient decreases as evidenced by the reduced Bragg stopband width. However, the longitudinal spatial hole burning (LSHB) effect in the DFB lasers can somewhat be diminished by the grating DCR. The seriousness of these effects depends on different grating structures and their coupling strengths. Our simulation shows that a degradation of 17 dB can be brought to the SMSR of the uniform grating DFB lasers with their duty cycles taking a deviation of ±25% in a uniformly distributed random fashion. It also broadens the linewidth of the quarter-wavelength phase-shifted DFB lasers by more than 2.5 folds. The impact of this effect on the RIN is moderate—less than 2%. All the performance deteriorations can partially be attributed to the effective reduction in the grating coupling coefficient of around 20% by such a DCR.

1. Introduction

Distributed feedback (DFB) lasers are widely used in fiber-optic communication systems and networks due to their stable single-mode operation characteristics [1]. However, the randomness in their grating duty cycles introduced by imperfect fabrication processes, including the grating pattern formation by electron beam lithography with insufficient electron beam resolution, and subsequent processes such as development, etching, and embedding, can potentially lead to detrimental effects on the device’s performance. This is because the disruption to the periodicity of the grating jeopardizes the phase condition required to constructively establish the lasing mode coherence. Despite the possible significant impact of the grating duty-cycle randomness (DCR) on DFB laser performance, there seems to be a lack of systematic study on this topic so far.

Numerical simulation is an efficient and reliable way to tackle this problem, as reproducing the grating DCR through an experimental approach is costly, and the simulation tools for the DFB lasers are mature and accurate [2]. Many models and associated numerical solution techniques have been developed for the DFB laser’s static and dynamic performance simulation. To capture the grating effect in the DFB lasers, those one-dimensional (1D) models along the wave propagation direction (i.e., along the laser cavity) are sufficient. In those 1D approaches, there still exist the traveling-wave method (TWM) [3,4,5,6,7,8] and the standing-wave method (SWM) [9,10,11]. In principle, either method can be exploited to solve our problem. However, the complex root-searching algorithm employed by the SWM often misses the true root or finds the false root in dealing with complicated grating structures. The TWM is therefore preferred in our case, for it only involves a marching algorithm. Although the coupled-mode equation-based TWM [5,12,13] is more popular in dealing with the DFB lasers with “ideal” gratings without randomness, we need to start with the transfer matrix model (TMM) that deals with the grating in a pitch-by-pitch fashion [14,15,16], rather than in a section-by-section fashion, for the grating DCR with variations from pitch to pitch must be incorporated in our model.

Similar to those reported methods in the literature [14,15,16] that divide the grating down to pitches, in this work, we firstly developed a modified TMM that breaks the grating all the way down to the sub-pitch to capture the DCR from pitch to pitch. After being validated through comparison with the existing results, this model was exploited to simulate the DFB laser performance with the grating DCR incorporated.

This paper is organized as follows: the modified TMM and the extended solution technique are described in Section 2, in which the approach is also validated. The simulated device characteristics on various aspects for DFB lasers with the grating DCR are shown and discussed in Section 3. This work is finally summarized in Section 4.

2. Theoretical Model for DFB Lasers with Grating DCR

2.1. The Modified TMM and Numerical Solution Technique

Figure 1 shows a side view of a DFB laser. It is clearly visible that the duty cycle of the grating is not uniform. The wave equation governing light propagation in a weakly guided structure of semiconductor lasers can be expressed as [2]:

$${\nabla }^{2}E=\frac{1}{c^2}\frac{{\partial }^2}{\partial t^2}E+\frac{1}{c^2{\epsilon }_0}\frac{{\partial }^2}{\partial t^2}P+{\mu }_0\frac{\partial }{\partial t}{J}_{sp},$$

(1)

where:

• E—electric field in [V/m];
• P—induced polarization of the host medium in [C/m²];
• $J_{sp}$—spontaneous emission source in [A/m²];
• ${\epsilon }_0$—permittivity in a vacuum in [F/m];
• ${\mu }_0$—permeability in a vacuum in [H/m];
• c—speed of light in vacuum in [m/s].

With a built-in Bragg grating, the solution of of Equation (1) can be considered to be a pair of contra-propagating traveling waves modulated by slowly varying envelopes, where the slowly varying envelopes can further be decomposed into a stationary transverse mode and slowly varying amplitude along the propagation direction (z) and in time (t) [2]:

$$E\left(x,y,z,t\right)=\left[e^f{\left(z,t\right)e}^{j\left({\omega }_{0}t-\beta z\right)}+e^b\left(z,t\right)e^{j\left({\omega }_{0}t+\beta z\right)}\right]\varphi \left(x,y\right),$$

(2)

with $e^{f,b}$ denoting the slowly varying amplitudes in [V] of the forward and backward traveling waves, respectively, $\varphi$ the transverse mode in [1/m], ${\omega }_0=2\pi c/{\lambda }_0$ the reference angular frequency, and $\beta =2\pi n_{eff}/{\lambda }_0$ the propagation constant in [1/m], $n_{eff}$ the effective index of the transverse mode, and ${\lambda }_0$ the reference wavelength (chosen close to the lasing wavelength).

Figure 2 shows partial DFB grating with DCR, comprising two periods labeled as m and $m+1$. To treat wave propagation in DFB lasers as described by Equation (1) with its solution taking the form of Equation (2), by following the TMM [14,15,16], we further divide each grating period into four subsections: an interface between two sections with effective indices $n_a$ and $n_b$, respectively, a smooth waveguide section with effective index $n_b$ and a length of $l_m$, another interface between $n_b$ and $n_a$, and a smooth waveguide section with $n_a$ and a length of $l_m^{\prime }$ as illustrated in Figure 2. Due to the grating DCR, we must let the lengths of the smooth waveguides with effective indices $n_b$ and $n_a$ vary randomly but keep their summation to be fixed as $\Lambda$. Since the multiplication of the four subsections is different from period to period, no simplification approach for the identical matrix multiplication by exploiting matrix diagonalization can be applied, which leaves us with a substantially increased burden on the numerical computation. Fortunately, such a problem can still be handled by a personal computer.

By letting $e_m^f$ and $e_m^b$ be the forward and backward propagating wave amplitudes at the left-hand side of the m^th period, $e_{m+1}^f$ and $e_{m+1}^b$ be the forward and backward propagating wave amplitudes at the right-hand side of the $m^{th}$ period, (which are also the forward and backward propagating wave amplitudes at the left-hand side of the ${(m+1)}^{th}$ period), we can express these amplitudes as [17]:

$$\left[\begin{array}{c}{e}_{m+1}^f\\ e_{m+1}^b\end{array}\right]=\left[\begin{array}{cc}{e}^{-j{\beta }_a{l}_m^{\prime }}& 0\\ 0& e^{j{\beta }_a{l}_m^{\prime }}\end{array}\right]\frac{1}{t_{ab}}\left[\begin{array}{cc}1& -r\\ -r& 1\end{array}\right]\left[\begin{array}{cc}{e}^{-j{\beta }_b{l}_m}& 0\\ 0& e^{j{\beta }_b{l}_m}\end{array}\right]\frac{1}{t_{ba}}\left[\begin{array}{cc}1& r\\ r& 1\end{array}\right]\left[\begin{array}{c}{e}_m^f\\ e_m^b\end{array}\right],$$

(3)

$$wherer=\frac{n_b-n_a}{n_b+n_a},t_{ab}=\frac{2n_a}{n_b+n_a},t_{ba}=\frac{2n_b}{n_b+n_a},$$

(4)

$$and{\beta }_{a,b}=2\pi n_{a,b}/{\lambda }_0+\left(j-{\alpha }_m\right)\Gamma g/2-j\alpha /2,$$

(5)

with $l_m^{\prime }$, $l_m$, and ${\beta }_{a,b}$ indicating the lengths and the complex propagation constants of the local guided mode within sections of effective indices $n_a$ and $n_b$ in the $m^{th}$ period, respectively.

Defining X as a random number ranging from 0 to 1, and R ranging from 0 to 100% as a given fixed parameter describing the extent of the randomness, we have:

$$l_m=\Lambda \left[0.5+\left(X-0.5\right)R\right]and{l}_m^{\prime }=\Lambda -l_m.$$

(6)

Finally, the optical amplitudes at the laser facets can be connected by sequentially multiplying the amplitude at the very left end with all the matrices in order from left to right:

$$\left[\begin{array}{c}{e}_L^f\\ e_L^b\end{array}\right]=A_L{A}_M{{\cdots A}_m{A}_{m-1}\cdots A_{2}A}_1{A}_0\left[\begin{array}{c}{e}_0^f\\ e_0^b\end{array}\right],$$

(7)

$$where{A}_m=\left[\begin{array}{cc}{e}^{-j{\beta }_a{l}_m^{\prime }}& 0\\ 0& e^{j{\beta }_a{l}_m^{\prime }}\end{array}\right]\frac{1}{t_{ab}}\left[\begin{array}{cc}1& -r\\ -r& 1\end{array}\right]\left[\begin{array}{cc}{e}^{-j{\beta }_b{l}_m}& 0\\ 0& e^{j{\beta }_b{l}_m}\end{array}\right]\frac{1}{t_{ba}}\left[\begin{array}{cc}1& r\\ r& 1\end{array}\right],$$

(8)

$$and{A}_0=\frac{1}{\sqrt{1-r_0^2}}\left[\begin{array}{cc}1& r_0\\ r_0& 1\end{array}\right],A_L=\frac{1}{\sqrt{1-r_L^2}}\left[\begin{array}{cc}1& {-r}_L\\ {-r}_L& 1\end{array}\right],$$

(9)

with $r_0$ and $r_L$ representing the reflectivity of the left and right facets, respectively, $e_0^f$ and $e_0^b$ the optical amplitudes at the left facet, and $e_L^f$ and $e_L^b$ the optical amplitudes at the right facet.

The carrier rate equation that links the injection current density and the carrier density, and the gain model that links the carrier density to the material gain are given as [2]:

$$\frac{\partial N\left(z,t\right)}{\partial t}=\eta \frac{J\left(z,t\right)}{ed}-\left[AN\left(z,t\right)+B{N}^2\left(z,t\right)+C{N}^3\left(z,t\right)\right]-v_{g}g\left(z,t\right)S\left(z,t\right),$$

(10)

$$g\left(z,t\right)=\frac{g_N\left[N\left(z,t\right)-N_T\right]}{2\left[1+\epsilon S\left(z,t\right)\right]},$$

(11)

with the photon density in Equations (10) and (11) defined as [18]:

$$S\left(z,t\right)=\frac{n_{eff}\sqrt{\frac{{\epsilon }_0}{{\mu }_0}}\left[{\left|e^f\left(z,t\right)\right|}^2+{\left|e^b\left(z,t\right)\right|}^2\right]}{{2v}_g{\Sigma }_{ar}\hslash {\omega }_0}.$$

(12)

Finally, the output optical power can be found as [2]:

$$Power\left(L,t\right)=n_{eff}\left(1-{\left|r_L\right|}^2\right){\left|e^f\left(L,t\right)\right|}^2/2\sqrt{{\mu }_0/{\epsilon }_0},$$

(13)

$$Power\left(0,t\right)=n_{eff}\left(1-{\left|r_0\right|}^2\right){\left|e^b\left(0,t\right)\right|}^2/2\sqrt{{\mu }_0/{\epsilon }_0},$$

(14)

with $Power\left(L,t\right)$ and $Power\left(0,t\right)$ representing the output optical power from the right and left facets, respectively.

In the above equations and expressions, we have the following definitions:

g—material gain in [1/m];

N—carrier density inside the active region in [1/m³];

J—injection current density in [A/m²];

S—photon density in [1/m³];

$\eta$—injection efficiency, dimensionless;

A—nonradiative carrier recombination rate through SRH process in [1/s];

B—carrier recombination coefficient through spontaneous emission and bimolecular processes in [m³/s];

C—Auger recombination coefficient in [m⁶/s];

d—active-region thickness in [m];

e—elementary charge in [C];

$v_g=c/n_g$—group velocity in [m/s], with $n_g$ indicating the group index;

$g_N$—differential gain [m²];

$N_T$—transparency carrier density in [1/m³];

$\epsilon$—nonlinear gain saturation factor in [m³];

$\Gamma$—optical confinement factor, dimensionless;

$\alpha$—internal optical loss in [1/m³];

${\Sigma }_{ar}$—active-region cross-sectional area in [1/m²];

$\hslash {\omega }_0=hc/{\lambda }_0$—single photon energy in [J];

${\alpha }_m$—linewidth enhancement factor, dimensionless.

By following the TWM, we need to solve the equation through a time-domain marching algorithm so that the complicated root-searching can be avoided. To address the initial values provided on both facets, we must reformulate Equation (3) according to [18,19,20] by letting the wave follow a contra-propagation scheme in accordance with the time sequence. In conjunction with Equation (3), the carrier rate equation Equation (10) posed as an ordinary differential equation (ODE) can readily be solved through the Runge–Kutta method [21].

Our tractable numerical solution technique follows the following procedure: (1) set the injection current; (2) start with an initial traveling wave amplitude distribution at zero and an initial carrier density distribution at transparency; (3) update the carrier density distribution along the cavity by solving Equation (10) through the Runge–Kutta method; (4) find the gain distribution from expression Equation (11); (5) calculate the traveling wave amplitude distribution through the transfer matrix equation Equation (3) in the contra-propagation scheme; (6) find the photon density distribution from expression Equation (12), which completes one time step; and (7) update the traveling wave amplitude distribution with what has been obtained from the previous time step, and repeat (3)∼(7) until both the traveling wave amplitude distribution and carrier density distribution converge, which gives the steady-state laser performance under the given injection current at step (1). By merging steps (1) and (2) to update the injection current at any given time step, we can readily obtain the dynamic laser performance following the above procedure.

2.2. Model Validation

We validate our model through comparisons made on the device static and dynamic properties with those obtained from the well-established SWM [9] for the same DFB laser structure with the same set of parameters. The devices being investigated are typical DFB lasers with uniform gratings without DCR. The modeling parameters are listed in

Table 1. DFB laser parameters.

Parameters	Values
Grating period $\Lambda$ [nm]	244.5
Active-region thickness d [μm]	0.15
Active-region cross-sectional area ${\Sigma }_{ar}$ [μm2]	0.3
Facet reflectivity $r_1$, $r_2$	0
Laser cavity length L [μm]	300
Optical confinement factor $\Gamma$	0.3
Effective index under zero injection $n_{eff}$	3.2
Group index $n_g$	3.6
Optical modal loss $\alpha$ [cm−1]	50
Differential gain $g_N$ [10−16 cm2]	2.5
Transparent carrier density $N_T$ [${10}^{18}$ cm−3]	1.0
Nonlinear gain saturation factor $\epsilon$ [${10}^{-17}$ cm3]	6.0
Linewidth enhancement factor ${\alpha }_m$	4.0
Nonradiative carrier recombination rate through SRH process A [${10}^9$ s−1]	0.1
Carrier recombination coefficient through spontaneous emission and bimolecular processes B [10−10 cm3s−1]	1.0
Auger recombination coefficient C [10−29 cm6s−1]	7.5
Grating coupling coefficient $\kappa$ [cm−1]	50 1

¹ $\kappa =\frac{\pi }{{\lambda }_0}\frac{n_2-n_1}{2}$ [12].

. The solid and dash lines in Figure 3 and Figure 4 represent the result calculated by the TMM and SWM, respectively.

Figure 4a,b show the spatial distribution of the carrier and photon densities along the laser cavity, respectively. Again, the results calculated by the two different models agree well.

Figure 3 shows the optical power–current curve, from which we find that the two models produce almost exactly the same result.

3. Simulation Result and Discussion

In this section, all laser parameters in simulation are quoted from Table 1, with the exception that $\kappa$ is set to different values. $\kappa$ is introduced in the coupled mode equation model as a measure of the grating coupling strength. In the TMM, the grating coupling strength is altered by the effective index difference between the two sections in each period (i.e., $n_b-n_a$); there is no need to involve $\kappa$. However, since $\kappa$ is a popular design parameter in DFB lasers, we converted the change on the index contrast ($n_b-n_a$) into the normalized gating coupling coefficient $\kappa L$ with $\kappa$ given by the formula shown in the footer of Table 1. As such, instead of the index contrast ($n_b-n_a$), $\kappa L$ is used to indicate the varying of the grating coupling strength. All the DFB laser performance is calculated under a bias current of $1.5I_{th}$ to $2.0I_{th}$. In the simulation, we assume that the duty cycle of the grating is completely random and follows a uniform distribution. The grating exhibits a variation in its duty cycle by ±25% to match real-world conditions.

3.1. Effect of Grating DCR on SMSR

Figure 5 shows the impact of the grating DCR on the SMSR of uniform grating DFB lasers with different $\kappa L$. The result for DFB lasers with a quarter-wavelength phase-shifted grating at $\kappa L=2.5$ is also shown for comparison.

The simulation result clearly shows that the SMSR deteriorates significantly. Measured by the median value, the deterioration on SMSR increases from ∼5 dB at $\kappa L$ = 3.0 to ∼17 dB at $\kappa L$ = 2.0 for uniform grating DFB lasers. The quarter-wavelength phase-shifted DFB laser is more immune to the grating DCR as evidenced by only a ∼3 dB drop in its SMSR at $\kappa L$ = 2.5 as compared to a ∼12 dB drop in the uniform grating DFB laser’s SMSR.

It is well known that the quarter-wavelength phase-shifted DFB laser provides a stable single mode operation with high SMSR, as it does not have the inherent dual mode degeneracy problem in uniform grating DFB lasers [22]. However, it suffers from the severe longitudinal spatial hole burning (LSHB) effect especially for high $\kappa L$ [23]. To mitigate the LSHB effect in quarter-wavelength phase-shifted DFB lasers, the corrugation pitch modulation (CPM) with the grating phase-shift distributed within a phase-arranging region (PAR) was proposed [24,25] to replace the conventional grating with an abrupt phase change. As shown in Figure 6, in ideal case (black circular spots), the quarter-wavelength phase-shifted DFB laser with the CPM grating indeed has a much higher SMSR. With the grating DCR, however, its SMSR drops drastically. For DFB lasers with the CPM grating, their median values of SMSR have no significant advantage as compared to the DFB laser with an abrupt phase-shifted grating, regardless of the CPM grating PAR length ($L_p$) ratio over the total cavity length (L). Moreover, the fluctuation of the SMSR for DFB lasers with the CPM grating is much higher than that for the DFB laser with the abrupt phase-shifted grating. This result reveals that the DCR has a stronger impact on the SMSR for CPM gratings. Namely, the CPM grating is advantageous to be adopted by the quarter-wavelength phase-shifted DFB lasers only if the grating DCR can be reduced to below a certain extent. Otherwise, the effort to make CPM gratings for DFB lasers cannot be justified.

3.2. Effect of Grating DCR on Linewidth

Figure 7 shows the impact of the grating DCR on the linewidth of DFB lasers. As compared to the ideal grating without the DCR, while the linewidth is moderately broadened by ∼40% for uniform grating DFB lasers with $\kappa L$ = 2∼2.5, it drastically increases by 1.6∼2.5 folds for quarter-wavelength phase-shifted DFB lasers with $\kappa L$ falling in the same range. The fluctuation range of the linewidth for the quarter-wavelength phase-shifted DFB lasers is also substantially broader than that for the uniform grating DFB lasers.

The grating DCR brings in a significant linewidth broadening effect, with a greater impact on the quarter-wavelength phase-shifted DFB lasers as compared to the uniform grating DFB lasers with the same $\kappa L$. There is also a trend that the linewidth broadening effect is more pronounced as $\kappa L$ increases, especially for the quarter-wavelength phase-shifted DFB lasers.

3.3. Effect of Grating DCR on RIN

Figure 8 shows the effect of the grating DCR on the relative intensity noise (RIN) for DFB lasers with $\kappa L=2.0$. As compared to the ideal grating without the DCR, the median RIN of the uniform grating and the quarter-wavelength phase-shifted DFB lasers with the grating DCR increase by 1.10 dB/Hz and 1.56 dB/Hz, respectively, with a fluctuation of RIN at 0.19 dB/Hz and 1.91 dB/Hz, respectively. This result indicates that the grating DCR only has a moderate impact on the RIN of either uniform grating or quarter-wavelength phase-shifted DFB lasers.

3.4. Effect of Grating DCR on Coupling Strength

Since the lasing spectrum can directly be obtained from the TMM approach [26,27], from which the Bragg stopband width can readily be found, we can therefore extract the effective grating coupling coefficient ($\overline{\kappa }$) [28] when the DCR exists.

Figure 9a,b present the calculated Bragg stopband widths of the uniform grating and the quarter-wavelength phase-shifted DFB lasers, respectively, with $\kappa L$ varying from 1.57 to 2.5. Shown on the left and right vertical axes are the Bragg stopband width and the corresponding effective grating coupling coefficient, respectively. With the grating DCR, an average reduction of ∼20% on the grating coupling coefficient is found relative to the ideal grating coupling coefficient ($\overline{\kappa }$/$\kappa$), for either uniform grating or quarter-wavelength phase-shifted DFB lasers. Although the relative weakening to the grating coupling strength is the same for DFB lasers with different grating structures, the fluctuation of the effective grating coupling coefficient of the quarter-wavelength phase-shifted DFB lasers is more than doubled as compared to that of the uniform grating DFB lasers.

The grating DCR leads to an effective reduction in its coupling strength regardless of the grating structure and its coupling strength in the ideal case (i.e., without the grating DCR) as evidenced by the shrinkage of its Bragg stopband width. This effect is attributed to the fact that the grating DCR jeopardizes the phase-matching condition on the reflected and transmitted waves from pitch to pitch, thereby diminishing the coherence of the coupling between the contra-propagating waves.

The effective reduction in the grating coupling strength due to the DCR should consequently reduce the LSHB effect. This is true as evidenced by the calculated photon and carrier density distributions in DFB lasers with different grating structures as shown in Figure 10 and Figure 11.

4. Conclusions

Based on a modified TMM approach that can handle the sub-period variation in Bragg grating, we studied the impact of the grating DCR on DFB laser performance. The grating DCR effectively reduces the grating coupling strength, and consequently reduces the SMSR and broadens the linewidth. However, its effect on the RIN is moderate. The grating DCR brings a greater SMSR degradation to the uniform grating DFB lasers, whereas it gives a wider linewidth broadening to the quarter-wavelength phase-shifted DFB lasers. In general, the impact of the DCR is more pronounced as the grating coupling strength increases. Lastly, we find that the CPM grating can hardly be justified if the grating DCR cannot be diminished to some extent.