Travelling wave analysis on high-speed performance of Q-modulated distributed feedback laser

Jiankun Zhi; Hongli Zhu; Dekun Liu; Lei Wang; Jian-Jun He

doi:10.1364/OE.20.002277

1. Introduction

High-speed, low-chirp, low power consumption semiconductor lasers and modulators are important components for next generation optical networks. A commonly used directly modulated laser (DML) has fundamental speed limits, and has severe wavelength chirp that limits the transmission distance [1–3]. An external Mach-Zehnder modulator based on electro-optic effect in LiNbO₃ has excellent performance in high speed modulation with low chirp [4,5]. However, such an external modulator is very expensive and cannot be integrated with III-V semiconductor lasers. An electro-absorption modulated laser (EML) can monolithically integrate a modulator with the laser [6–8]. However, the typical length of an electro-absorption modulator (EAM) is more than 100 micrometers in order to achieve high extinction ratio, which leads to a high capacitance that compromises the speed. Although a travelling wave electrode design can eliminate the capacitance limitation, the structure of the device is very complicated [9]. Besides, an EAM introduces an inherent energy loss by absorbing the light in the OFF state and is therefore not power efficient.

While the advanced modulation formats such as the differential quadrature phase shift keying (DQPSK) are being developed for ultra-long-haul optical transmission, the traditional on-off keying (OOK) will remain the predominant solution for shorter distance communications due to the simplicity, small size and low cost of the transceivers. Recently we proposed a semiconductor laser structure incorporating a monolithically integrated electro-absorptive Q-modulation mechanism that has potential advantages of high-speed, low-chirp, high extinction ratio and high power efficiency [10]. Rate equation analysis has shown that the Q-modulated laser (QML) can achieve 40Gb/s return-to-zero (RZ) modulation with excellent signal quality [11]. While its bandwidth is much higher than that of directly modulated laser (DML), an additional advantage is that its wavelength chirp due to carrier density perturbation decreases with increasing bit rate, in opposite to the case of DML and EML. Besides, the QML does not emit light when the signal is at the OFF state, as opposed to the case of EML where the light is emitted and then absorbed. This increases significantly the power efficiency of QML as compared to EML since in the latter case the power is absorbed wastefully 50% of the time for non-return-to-zero (NRZ) signal or 75% of the time for RZ signal, even in the case of ideal waveforms.

The rate equation model does not consider any particular structure of the laser cavity. It only uses a simple parameter, the photon lifetime, to characterize the laser cavity. It can be considered as a “zero-dimension” (in spatial domain) model of the laser. As a result, it can only show the general behavior and mechanism of Q-modulation, but cannot provide any insight on the effect of structural parameters such as the cavity length and phase. Nor can it be used as a simulation tool for structural design because it only shows the modulation behavior when the photon life time is modulated, but does not concern how the photon lifetime modulation is achieved. In particular, the rate equation analysis does not take into account the wave propagation in the laser structure. It provides a good approximation of the laser behavior when the cavity round-trip propagation time is much smaller than the bit duration. However, for 40 Gb/s and beyond, the round-trip propagation time is no longer negligible for typical cavity lengths. In this paper, we analyze the high-speed performance of Q-modulated distributed feedback (DFB) laser using the travelling wave method that can take into account the wave propagation in the cavity. The effect of the cavity length on the modulation performance of the laser is revealed for the first time.

2. Theoretical model

The structure of the Q-modulated DFB laser is schematically shown in Fig. 1 . It consists of a gain section, a phase section, and a Q-modulator section at the rear end of the laser. The modulator section is placed between two deep etched trenches which constitute an anti-resonant Fabry–Pérot cavity acting as a rear reflector. The gain section incorporates a DFB grating for providing feedback to the laser cavity. The phase section is adjusted by current injection so that the reflection by the rear Q-modulator interferes constructively with the distributed feedback mechanism. Although the grating is not necessary in the modulator and phase sections, it is incorporated throughout the structure for fabrication simplicity so that a uniform grating can be patterned using holographic method. Contrary to the direct modulated laser, the QML keeps the pump current constant in the gain section. By varying the absorption coefficient of the modulator waveguide through a current injection or reverse-biased electro-absorption effect, the reflectivity of the rear reflector is modulated, resulting in the modulation of the Q-factor (or the photon lifetime) of the laser cavity, and consequently the lasing threshold and output power.

Fig. 1 Schematic of the Q-modulated laser

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The modulator section is typically very short, in the order of 10-20μm, which is over an order of magnitude smaller than the gain section. Therefore, in this paper, for simplicity, we neglect the wave propagation in the modulator section and use the transfer matrix method (TMM) [12] to simulate the reflectivity of the rear reflector. When a modulation signal is applied on the modulator section, the effective reflectivity r_eff of the rear reflector can be considered to change instantaneously. Also, the effect of the phase adjustment by the phase section can be considered by adding a phase in the effective reflectivity r_eff of the rear reflector, while the length of the phase section can be included in the overall DFB cavity length for the dynamic simulation using the traveling wave method [13].

The optical field in the DFB laser cavity can be written in the following form

E (z, t) = [F (z, t) e^{- i β_{0} z} + R (z, t) e^{i β_{0} z}] e^{i w_{0} t}

where

F (z, t)

and

R (z, t)

represent, respectively, the forward and backward waves propagating along the longitudinal direction z with an effective propagation constant β₀ and angular frequency ω₀. The time-dependent coupled wave equations in the waveguide can be written as

\frac{1}{v_{g}} \frac{\partial F (z, t)}{\partial t} + \frac{\partial F (z, t)}{\partial z} = (\frac{1}{2} g - \frac{1}{2} α_{0} - i δ_{b} - i \frac{ω_{0}}{c} Δ n) F (z, t) + i κ R (z, t) + s_{f}

\frac{1}{v_{g}} \frac{\partial R (z, t)}{\partial t} - \frac{\partial R (z, t)}{\partial z} = (\frac{1}{2} g - \frac{1}{2} α_{0} - i δ_{b} - i \frac{ω_{0}}{c} Δ n) R (z, t) + i κ F (z, t) + s_{r}

where

v_{g}

is the group velocity,

κ

is the coupling coefficient,

α_{0}

is the waveguide loss, and

g

is the gain expressed by

g (z, t) = \frac{Γ α_{g} [N (z, t) - N_{t}]}{1 + ε P (z, t)}

Here $Γ$ is the confinement factor of the mode field in the gain medium, $α_{g} = \frac{d g}{d N}$ is the differential gain, $N (z, t)$ is the carrier density at position z and time t, $N_{t}$ is carrier density at transparency, $ε$ is the gain compression coefficient, and $P (z, t)$ is the photon density. $δ_{b}$ is the deviation from the Bragg condition given by

δ_{b} = \frac{ω_{0}}{c} n_{e f f}^{} - \frac{π}{Λ}

with

Λ

the pitch of the grating, and n_eff the effective refractive index. The effective refractive index change

Δ n

due to the carrier density variation is given by

Δ n = - \frac{λ_{0}}{4 π} Γ α α_{g} [N (z, t) - N_{t}]

where

α

is the linewidth enhancement factor. The variation of the carrier density

N (z, t)

is governed by the following time-dependent carrier rate equation

\frac{d N (z, t)}{d t} = \frac{J}{e d} - A N (z, t) - B N {(z, t)}^{2} - C N {(z, t)}^{3} - \frac{α_{g} [N (z, t) - N_{t}]}{1 + ε P (z, t)} v_{g} P (z, t)

where

J

is the injected current density,

d

is the thickness of the active layer, and A, B, C are the nonradiative recombination coefficient, bimolecular recombination coefficient and auger recombination coefficient, respectively.

s_{f}

and

s_{r}

are the spontaneous noises coupled into the forward and backward waves, respectively, which are expressed by

〈 s_{f, r} (z, t) s_{f, r}^{*} (z^{'}, t) 〉 = β K R_{s p} δ (t - t^{'}) δ (z - z^{'}) / v_{g}

〈 s_{f, r} (z, t) s_{f, r} (z^{'}, t) 〉 = 0

where

R_{s p} = B N^{2} / L

is the spontaneous emission per unit length,

β

is the coupling coefficient of spontaneous emission into the lasing mode; and

K

is the transverse Petermann’s factor [14].

The coupled wave Eqs. (2) and (3) can be rewritten in the following matrix format

(\frac{1}{v_{g}} \frac{\partial}{\partial t} \pm \frac{\partial}{\partial z}) [\begin{matrix} F (z, t) \\ R (z, t) \end{matrix}] = [\begin{matrix} A^{11} (z, t) & A^{12} (z, t) \\ A^{21} (z, t) & A^{22} (z, t) \end{matrix}] [\begin{matrix} F (z, t) \\ R (z, t) \end{matrix}] + [\begin{matrix} S^{f} (z, t) \\ S^{r} (z, t) \end{matrix}]

where

A^{11} (z, t) = A^{22} (z, t) = \frac{1}{2} g - \frac{1}{2} α_{0} - i δ_{b} - i \frac{ω_{0}}{c} Δ n

, and

A^{12} (z, t) = A^{21} (z, t) = i κ

_.

We now follow the box scheme to search for the numerical solution of this equation. First, a mesh in the propagation direction z and time t is set up with intervals of Δz and Δt, respectively.

\begin{array}{l} t = k Δ t, k = 0, 1, 2, .... \\ z = n Δ z, n = 1, 2, 3, ... \end{array}

Then, the left hand side of Eq. (10) is calculated by

\frac{\partial F (z, t)}{\partial t} = \frac{1}{2} (\frac{F_{n + 1, k + 1} - F_{n + 1, k}}{Δ t} + \frac{F_{n, k + 1} - F_{n, k}}{Δ t})

\frac{\partial F (z, t)}{\partial z} = \frac{1}{2} (\frac{F_{n + 1, k + 1} - F_{n, k + 1}}{Δ z} + \frac{F_{n + 1, k} - F_{n, k}}{Δ z})

\frac{\partial R (z, t)}{\partial t} = \frac{1}{2} (\frac{R_{n + 1, k + 1} - R_{n + 1, k}}{Δ t} + \frac{R_{n, k + 1} - R_{n, k}}{Δ t})

\frac{\partial R (z, t)}{\partial z} = \frac{1}{2} (\frac{R_{n + 1, k + 1} - R_{n, k + 1}}{Δ z} + \frac{R_{n + 1, k} - R_{n, k}}{Δ z})

where

F_{n, k}

and

R_{n, k}

denote the values of

F (n Δ z, k Δ t)

and

R (n Δ z, k Δ t)

, respectively. Similarly, the right hand side of Eq. (10) is given by

\begin{array}{l} [\begin{matrix} A^{11} (z, t) & A^{12} (z, t) \\ A^{21} (z, t) & A^{22} (z, t) \end{matrix}] [\begin{matrix} F (z, t) \\ R (z, t) \end{matrix}] + [\begin{matrix} C^{f} (z, t) \\ C^{r} (z, t) \end{matrix}] = \frac{1}{4} [\begin{matrix} A_{n + 1, k + 1}^{11} & A_{n + 1, k + 1}^{12} \\ A_{n + 1, k + 1}^{21} & A_{n + 1, k + 1}^{22} \end{matrix}] [\begin{matrix} F_{n + 1, k + 1} \\ R_{n + 1, k + 1} \end{matrix}] + \\ \frac{1}{4} [\begin{matrix} A_{n + 1, k}^{11} & A_{n + 1, k}^{12} \\ A_{n + 1, k}^{21} & A_{n + 1, k}^{22} \end{matrix}] [\begin{matrix} F_{n + 1, k} \\ R_{n + 1, k} \end{matrix}] + \frac{1}{4} [\begin{matrix} A_{n, k + 1}^{11} & A_{n, k + 1}^{12} \\ A_{n, k + 1}^{21} & A_{n, k + 1}^{22} \end{matrix}] [\begin{matrix} F_{n, k + 1} \\ R_{n, k + 1} \end{matrix}] + \frac{1}{4} [\begin{matrix} A_{n, k}^{11} & A_{n, k}^{12} \\ A_{n, k}^{21} & A_{n, k}^{22} \end{matrix}] [\begin{matrix} F_{n, k} \\ R_{n, k} \end{matrix}] + \\ \frac{1}{4} [\begin{matrix} C_{n + 1, k + 1}^{f} \\ C_{n + 1, k + 1}^{r} \end{matrix}] + \frac{1}{4} [\begin{matrix} C_{n + 1, k}^{f} \\ C_{n + 1, k}^{r} \end{matrix}] + \frac{1}{4} [\begin{matrix} C_{n, k + 1}^{f} \\ C_{n, k + 1}^{r} \end{matrix}] + \frac{1}{4} [\begin{matrix} C_{n, k}^{f} \\ C_{n, k}^{r} \end{matrix}] \end{array}

with the following boundary condition at

z = 0

and

N Δ z

R (0, t) = F (0, t) \cdot r_{e f f}

F (N Δ z, t) = R (N Δ z, t) \cdot r'

where r’ is the reflectivity of the cleaved facet at the front end and

r_{e f f}

is the effective reflectivity of the rear reflector which can be calculated by the TMM method. When a signal is applied on the modulator section, it is assumed that

r_{e f f}

would change instantaneously along with the signal.

By plugging Eqs. (12)-(16) into Eq. (10) and using the boundary conditions in Eqs. (17)-(18), we can calculate the values of $F (z, t)$ and $R (z, t)$ for all positions $z = n Δ z$ and time $t = k Δ t$ .

The output power from the front end of the laser can be calculated by

P_{o u t}^{F} (t) = \frac{d w}{Γ} v_{g} \frac{h c}{λ} ({| F (L, t) |}^{2} - {| R (L, t) |}^{2})

Since $F (z, t)$ and $R (z, t)$ contain not only the magnitude but also the phase, the frequency chirp Δf of the output signal can be easily obtained by calculating the time derivation of the phase $Φ_{F} (z, t)$ of $F (z, t)$ as follows

Δ f = \frac{1}{2 π} \frac{\partial Φ_{F} (L, t)}{\partial t}

3. Results and discussions

We consider a Q-modulated DFB laser operating at 1.55μm. The grating pitch is $Λ = 241.2 n m$ , with the effective index varying from $n_{1} = 3.215$ to $n_{2} = 3.21$ . The length of the active region is $L_{a} = 241.2 μ m$ (1000 pitches), with the corresponding normalized coupling coefficient $κ L_{a} = \frac{2 (n_{1} - n_{2}) L_{a}}{(n_{1} + n_{2}) Λ} = 1. 56$ .

The two etched trenches are designed with a width equal to an odd-integer multiple of quarter wavelength. The modulator section is placed in an anti-resonant cavity which satisfies

\frac{4 π n}{λ} L_{m} + Φ_{1} + Φ_{2} = (2 m + 1) π

where n and

L_{m}

are the effective refractive index and length of the modulator section, respectively,

Φ_{1}

and

Φ_{2}

are the reflection phases of the two deep trenches.

In order to achieve a high reflectivity modulation, the trench between the modulator and the phase section (Trench 1) needs to have a relatively low reflectivity while the trench at the other end of the modulator (Trench 2) needs to have a high reflectivity. This can be achieved by using trenches with widths equal to an odd integer multiple of quarter-wavelength and filling Trench 1 with a high index dielectric material while Trench 2 can be simply an air trench. As an example, Trench 1 is filled with silicon nitride with a refractive index $n^{'} = 2.0$ and the trench width is $w_{1} = 5 λ_{0} / (4 n^{'})$ , while Trench 2 is filled with air with a width $w_{2} = 5 / 4 λ_{0}$ . In this case, the reflection phases of the two deep trenches is $Φ_{1} = Φ_{2} = 0$ . The length of the modulator section between the two trenches in our numerical example is $L_{m} = 25.25 λ_{0} / n = 12.2 μ m$ .

Figure 2 shows the reflectivity and phase of the rear reflector. We assume that when a reversed biased voltage signal is applied on the modulator section, its absorption coefficient changes from 0 to 2000 cm⁻¹, corresponding to the ON (1) and OFF (0) states, respectively. We can see that for the center wavelength $λ_{0} = 1.55 μ m$ which corresponds to the anti-resonant condition, when the absorption coefficient changes from 0 to 2000 cm⁻¹, the effective reflectivity of the rear reflector changes by the largest amount from 0.85 to 0.2, while the phase change is minimal. This leads to a large threshold modulation (as we will see later, while producing a minimal adiabatic wavelength chirp.

Fig. 2 Magnitude (a) and phase (b) of the effective reflectivity spectrum of the rear reflector when the absorption coefficient of the modulator section is 0 or 2000 cm⁻¹.

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The lower level reflectivity can be further reduced by using a filler with a larger refractive index for Trench 1. Figure 3 shows the variation of the effective reflectivity of the rear reflector as a function of the absorption coefficient. We can see that when there is no gap filler, corresponding to the refractive index of gap filler = 1, the effective reflectivity changes from 0.95 to 0.68 when the absorption coefficient of the modulator waveguide changes from 0 to 2000cm⁻¹. When a gap filler with a refractive index of 2 is used, the effective reflectivity changes from 0.85 to 0.2. And when the refractive index of gap filler is 2.5, the variation range is further enlarged, becoming from 0.78 to 0.07. Note that the lower level reflectivity can also be reduced by using a trench width (for Trench 1) that deviates away from an odd integer multiple of quarter wavelength or by using a shallowly etched trench, but the reflectivity is more sensitive to the trench width or etching depth, resulting in a more stringent fabrication tolerance. Besides, the lower the reflectivity of Trench 1, the larger the phase variation of the effective reflectivity of the rear reflector due to the refractive index variation associated with the absorption modulation in the Q-modulator, which leads a larger wavelength chirp [10,11]. Therefore, it is important to choose suitable gap filler so that Trench 1 has a high reflectivity while producing sufficient reflectivity modulation of the rear reflector. For example, as we will see later, when the length of active region is 1000~3000 pitches, good performance is obtained when the ON state reflectivity is 0.85 and the OFF state reflectivity is 0.2~0.3, which means that a gap filler with refractive index of 2 is appropriate.

Fig. 3 Effective reflectivity of the rear reflector as a function of the absorption coefficient of the modulator waveguide, with the refractive index of the gap filler for Trench 1 as a parameter.

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In the QML structure, both the DFB grating and the rear reflector contributed to the feedback mechanism of the laser. A phase section is necessary to ensure the constructive interference of the two feedback mechanisms. It plays the role of adjusting the phase of the effective reflectivity of the rear reflector. Figure 4 shows the variation of the threshold gain and the resonance wavelength of the laser when the phase changes. As we can see in Fig. 4(a), there is a large threshold gain difference between the ON and OFF states of the modulator (i.e. when its absorption coefficient changes from 0 to 2000 cm⁻¹). When the phase is at the optimal value, which is about 7π/4 in our numerical example, the threshold gain is the lowest for the ON state while the threshold difference between the ON and OFF states is very large (from 20 cm⁻¹ to 95 cm⁻¹). Also, from Fig. 4(b), we can see that a mode-hop can occur at a certain phase. However, by adjusting the phase section, the lasing wavelength at the ON state can be tuned for over 1 nm without significant increase in threshold or mode-hop. While the mode hop can also occur for the OFF state, it occurs at a very high threshold value which is normally above the operating point.

Fig. 4 Threshold gain (a) and center wavelength (b) versus the phase.

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Figure 5 shows the transmission gain spectra for the ON and OFF states when the phase is set to 7π/4 and the injected current is near the threshold of the ON state. As we can see, for the ON state ( $α = 0$ ), the transmission gain for the lasing mode is more than 45dB, which is a clear sign of start-oscillation. Yet for the OFF state ( $α = 2000 c m^{- 1}$ ), the transmission gain is negative, which indicates failure to oscillate. Therefore, the mechanism of the Q-modulation works effectively from the static simulation.

Fig. 5 Transmission gain spectra of the ON and OFF states

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For the dynamic analysis, we use the traveling wave model as described in Section 2 with parameters given in Table 1 . Figure 6 shows the output waveform (left figure), and the frequency chirp and the average carrier density (right figure) when a 40Gb/s return-to-zero (RZ) signal is applied to the Q-modulator. The corresponding duration of the ON state signal in the “1” bit is 12.5ps. Figure 6(a) shows the case where the injected current into the gain section is 60mA and the reflectivity of the rear reflector changes from 0.3 to 0.85 when the signal changes from the OFF to the ON state. For Fig. 6(b)-6(c), the injected current is 80mA and the reflectivity of the rear reflector changes from 0.3 to 0.5 (b), from 0.3 to 0.85 (c) or from 0.5 to 0.85 (d). The length of the active-region is 241.2μm (1000 pitches), corresponding to a round trip propagation time of 5.8ps. The phase is optimized at about 1.8π. For all the above cases, we can first observe that the carrier density variation is relatively slow. It does not follow the input signal or output waveform because the carrier density change cannot respond to high speed modulation due to limited carrier life time. Unlike the case of directly modulated laser where the carrier density variation is a prerequisite for output power modulation, the carrier density does not need to be modulated for generating output power modulation in QML. In fact, the carrier density variation is only a consequence of photon density modulation, and it decreases with increasing frequency. This is the reason why the QML can produce a lower adiabatic chirp compared to DML [10], especially at high modulation frequency. Note that the frequency chirp calculated by Eq. (20) includes both adiabatic chirp (related to the carrier density variation) and transient chirp (inherently associated with the output signal modulation).

Table 1. Parameters used in the travelling wave model

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Fig. 6 Output waveform (left) and frequency chirp and carrier density (right) of the QML under 40Gb/s RZ signal modulation when the injected current into the gain section is 60mA and the reflectivity of the rear reflector changes from 0.3 to 0.85 (a); and when the injected current is 80mA and the reflectivity of the rear reflector varies from 0.3 to 0.5 (b); from 0.3 to 0.85 (c); and from 0.5 to 0.85 (d).

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On the detailed modulation characteristics, let us first look at the case where the gain section is injected with a constant current of 80mA.When the reflectivity of the rear reflector is modulated from 0.3 to 0.5 (Fig. 6(b)), the lasing threshold changes from 60 mA to 28 mA. The output power is low and the extinction ratio of the modulation is also low. The slowly varying adiabatic chirp follows the carrier density variation (in opposite direction) and is relatively small. The frequency chirp is dominated by transient chirp which is about 15 GHz. By increasing the ON state reflectivity of the rear reflector to 0.85 (Fig. 6(c)), which corresponds to a threshold current of 7.5 mA, not only the peak power of the ON state is increased, the OFF state power is reduced due to limited carrier recovery time after carrier depletion by stimulated emission at the ON state, as shown in Fig. 6(b). This results in an excellent extinction ratio, even though the 80mA injected current into the active region is higher than the threshold (60mA) of the OFF state. The transient frequency chirp is increased to about 35GHz. Note that if we reduce the injected current to the threshold or below, as shown in Fig. 6(a), although the power level at the OFF state can be further reduced, the peak power at the ON state is much more pattern dependent, which is therefore not desirable. When the OFF state reflectivity is increased to 0.5, the output power at the OFF state increases as expected (Fig. 6(d)), and consequently the extinction ratio is reduced. At the same time, the variation of the peak power at the ON state decreases, resulting in less pattern dependency. The frequency chirp is also reduced.

The above results are very similar to those obtained by using rate equations [11], since the length of the DFB structure is relatively short and the round trip propagation time is much shorter than the bit duration. To investigate the effect of the cavity length on the modulation performance, we calculated the output waveforms under 40Gb/s RZ modulation for cavity length varying from 1000 to 4000 pitches (i.e. 241.2μm to 964.8μm), as shown in Fig. 7 . The normalized coupling coefficient of the DFB is kept constant at $κ L_{a} = 1. 56$ and the reflectivity of the rear reflector at the ON state is fixed at 0.85 (or 0.75 in the case of 4000 pitches), while the OFF state reflectivity, the injected current on the active region, and the phase are individually optimized for achieving the best extinction ratio. The values of those parameters are listed in Table 2 for the four different cases. We can see that when the cavity length increases, the OFF state power level increases and the ON state peak power become more pattern dependent, which leads to the reduction of the extinction ratio. While the power level at the OFF state can in principle be reduced by increasing the threshold or by reducing the injected current in the gain region, the pattern dependency increases at the same time which reduces the lowest peak power for the “1”bits. Therefore, the compromise between the two considerations determines the best achievable extinction ratio. Besides, a secondary peak following the bit “1” peak becomes more visible with increasing cavity length, due to reflections of Bloch waves at the boundaries of the DFB grating [15]. The interval between the main peak and the secondary peak corresponds to the round trip propagation time. For the case of cavity length equal to 4000 pitches (i.e. 964.8μm), the round trip time is about 23.2ps, resulting in the secondary peak almost overlapping with the next bit. This degrades the signal quality, especially if the following bit is “0”. Figure 8 plots the extinction ratio versus the cavity length. Here we take into account the pattern dependency in the extinction ratio by calculating the ratio between the lowest bit “1” peak power and the highest bit “0” level over the calculation time window. We can see that an extinction ratio of about 7dB can be achieved with a cavity length of 500μm, and the limit for the cavity length is about 300μm for achieving an extinction ratio above 10dB.

Fig. 7 Output waveform under 40Gb/s RZ signal modulation when the cavity length is (a) 1000 pitches; (b) 2000 pitches; (c) 3000 pitches; and (d) 4000 pitches.

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Table 2. Parameters used for the four cases of Fig. 7

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Fig. 8 Extinction ratio versus cavity length. The solid line is obtained by fitting the calculated data (circles) with a third-order polynomial.

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It should be pointed out that the effect of the cavity length on the achievable extinction ratio cannot be revealed by the rate equation model. Some of the results in this paper from the travelling wave method (e.g. the output waveform) can only be qualitatively compared with those from the rate equation model since there is no simple relationship between the photon lifetime and the complex cavity structure considered in this paper. The photon lifetime is related to many parameters including DFB coupling constant, mirror reflectivity and phase, and cavity length, and therefore it cannot uniquely describe or determine all characteristics of the laser such as the relationship between the cavity length and the modulation performance.

4. Conclusions

In conclusion, we have designed and simulated the structure of a Q-modulated distributed feedback laser. A large reflectivity modulation of the rear reflector is achieved by using an anti-resonant cavity formed by two deep trenches with the one between the modulator and phase section filled by a high index dielectric material. The high speed performance of the laser is investigated using the travelling wave model. Due to the cavity dynamics related to the wave propagation in the structure, the modulation extinction ratio decreases when the cavity length increases. It is shown that 40Gb/s RZ signal modulation can be achieved with an extinction ratio of 7dB and 10dB with a cavity length of 500μm and 300μm, respectively. Due to its simplicity, compactness, and power efficiency with respect to external modulation schemes and superior performance over directly modulated laser, the Q-modulated laser has great potential for applications in high speed communications and optical interconnects.

Acknowledgments

The authors thank Profs. Xun Li and Wei-Ping Huang of McMaster University, Hamilton, Canada for help and discussions on the travelling wave modeling. This work was supported by the National High-Tech R&D Program of China (grant No. 2011AA010305), the Natural Science Foundation of Zhejiang Province (grant No. Z1110276) and the Fundamental Research Funds for the Central Universities of China.

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8. M. Suzuki, Y. Noda, H. Tanaka, S. Akiba, Y. Kushiro, and H. Isshiki, “Monolithic Integration of InGaAsP/InP Distributed Feedback Laser and Electroabsorption Modulator by Vapor Phase Epitaxy,” J. Lightwave Technol. 5(9), 1277–1285 (1987). [CrossRef]

9. U. Westergren, M. Chaciński, and L. Thylén, “Compact and efficient modulators for 100 Gb/s ETDM for telecom and interconnect applications,” Appl. Phys., A Mater. Sci. Process. 95(4), 1039–1044 (2009). [CrossRef]

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$Γ$	Optical confinement factor	0.1
A	Nonradiative recombination coefficient	1.0 × 10⁸ s⁻¹
B	Bimolecular recombination coefficient	1.0 × 10⁻¹⁰ cm³s⁻¹
C	Auger recombination coefficient	7.0 × 10⁻²⁹ cm⁶s⁻¹
$v_{g}$	Group velocity	8.33 × 10⁹ cm/s
$a_{g}$	Differential gain coefficient	1.65 × 10⁻¹⁶ cm²
$ε$	Nonlinear gain coefficient	9 × 10⁻¹⁷ cm⁻³
$α$	Linewidth enhancement factor	3.0
$N_{t}$	Transparent carrier density	1 × 10⁻¹⁸ cm⁻³
w	Width of active waveguide	3 μm
d	Thickness of active region	44 nm
r’	Front facet reflectivity	0.001
n_eff	Effective refractive index	3.2125
Λ	Grating pitch	0.2412 μm
β	Coupling coefficient of spontaneous emission	2 × 10⁻⁵
K	Transverse Petermann’s factor	1

Cavity Length	Phase of the Phase Section	Injected Current	ON state Reflectivity	OFF State Reflectivity
241.2μm (1000 pitches)	1.8π.	80 mA	0.85	0.3
482.4μm (2000 pitches)	0.	60 mA	0.85	0.25
723.6μm (3000 pitches)	0.2π	60 mA	0.85	0.2
964.8μm (4000 pitches)	0.3π	55 mA	0.75	0.1

$Γ$	Optical confinement factor	0.1
A	Nonradiative recombination coefficient	1.0 × 10⁸ s⁻¹
B	Bimolecular recombination coefficient	1.0 × 10⁻¹⁰ cm³s⁻¹
C	Auger recombination coefficient	7.0 × 10⁻²⁹ cm⁶s⁻¹
$v_{g}$	Group velocity	8.33 × 10⁹ cm/s
$a_{g}$	Differential gain coefficient	1.65 × 10⁻¹⁶ cm²
$ε$	Nonlinear gain coefficient	9 × 10⁻¹⁷ cm⁻³
$α$	Linewidth enhancement factor	3.0
$N_{t}$	Transparent carrier density	1 × 10⁻¹⁸ cm⁻³
w	Width of active waveguide	3 μm
d	Thickness of active region	44 nm
r’	Front facet reflectivity	0.001
n_eff	Effective refractive index	3.2125
Λ	Grating pitch	0.2412 μm
β	Coupling coefficient of spontaneous emission	2 × 10⁻⁵
K	Transverse Petermann’s factor	1

Cavity Length	Phase of the Phase Section	Injected Current	ON state Reflectivity	OFF State Reflectivity
241.2μm (1000 pitches)	1.8π.	80 mA	0.85	0.3
482.4μm (2000 pitches)	0.	60 mA	0.85	0.25
723.6μm (3000 pitches)	0.2π	60 mA	0.85	0.2
964.8μm (4000 pitches)	0.3π	55 mA	0.75	0.1

Travelling wave analysis on high-speed performance of Q-modulated distributed feedback laser

Abstract

1. Introduction

2. Theoretical model

3. Results and discussions

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (8)

Tables (2)

Equations (21)

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