1.
Introduction

Semiconductor lasers have compact structure, high photoelectric transformation efficiency and wide range of lasing wavelengths, so they have great application prospects in optical communication, laser processing, laser ranging, biomedical equipment and so on^[1-3]. Single chip integrated tapered semiconductor lasers can amplify the power of wave mode from the seed light source, obtaining higher output power with little effect on beam quality. Their fabrication is simple and the fabrication cost is low. Consequently, they have become one of the research hot-spots of laser technology. Paschke and Sumpf et al.^[4] have found that using different electrodes to separately drive the master area and amplify area can obtain high power and nearly diffraction limited output laser. Since a DBR laser with first order gratings emits two degenerate modes, there are lobes in the far field. However, the second order gratings produce two order diffraction, corresponding to the grating coupling and the surface emission loss. Surface emission loss can cause a large threshold gain difference between the main mode and the lowest order mode, which eliminates pattern degeneracy. Consequently, second order gratings are used to improve the mode characteristics^[5].



In this paper, we use traveling wave coupling theory to investigate the time domain characteristics of tapered semiconductor lasers with second order DBR gratings^[6], which it is separately driven by two different electrodes. The tapered laser structure, as shown in Fig. 1, is divided into four sections, as follows: the master-oscillator (MO) region, with a length of L₂ and a width of 4 μm; the rear grating, with a length of L₁; the front grating region, with a length of L₃; and the second order grating^[7], whose the coupling coefficient is $kappa $ and surface emission loss is $varepsilon $. The front and back gratings play significant roles in determining the stability and operating characteristics of the device. Typically, the longer front grating provides additional isolation from the residual facet reflections and lowers the facet anti-reflection coating requirements. The duty cycle is 0.3 and the tapered amplify region with a length of L₄ and a flare angle of about 6°^[8]. R and S represent forward wave and backward wave, respectively.






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Figure1.
Schematic top view of the laser.




In the following section, the traveling wave theory and coupled mode equation are introduced and the finite difference method is used to solve the equation. In the third section, the simulation results are given, the variation of output optical power and charge carriers with time of tapered laser with DBR grating is analyzed, and the influence of different length front grating on longitudinal mode characteristics is compared. In the last section, the work of this paper is summarized.

2.
Method

In semiconductor lasers with second order gratings, the coupling of the front and rear traveling waves must be considered. In this paper, R(t, z) stands for the forward wave, S(t, z) represents the backward wave, and the time domain coupling wave equation that they satisfy is as follows^[9-11]:

$$left( {frac{1}{{{{{v}}_{ m{g}}}}} frac{partial }{{partial t}} + frac{partial }{{partial z}}} ight)R = [g - a - varepsilon (z)]R + [i kappa (z) - varepsilon (z)]S + {Phi _{ m{R}}}tag{1a},$$

$$left( {frac{1}{{{{{v}}_{ m{g}}}}} frac{partial }{{partial t}} - frac{partial }{{partial z}}} ight)S =[g - a - varepsilon (z)]S + [i kappa (z) - varepsilon (z)]R + {Phi _{ m{S}}}tag{1b},$$

where v_g represents the group velocity of the light wave, g represents the amplitude gain and phase change of the active region, $a$ represents the energy loss caused by free carrier absorption and scattering, the value in MO area is a constant: the value of ${a_0}$ in this paper is 2 cm^–1, in the tapered amplify area, $a = {a_0} + dfrac{1}{{2r}}$ (r is the curvature radius of the current light field), $varepsilon $ represents radiation loss caused by gratings, $kappa $ is the grating coupling coefficient, $varepsilon $ and $kappa $ are zero away from the grating area. The output power is the product of the square of output light intensity |R|² and the lateral width of the field W. Gain g is a function of time t and position z, and the expression is as follows:

$${{g}}(t,z) = frac{{G(z,t)}}{2} (1 - i alpha ),$$

(2)

where $alpha $ is the line-width enhancement factor, the expression of G(z,t)^[12] is as follows:

$$G(z,t) = A (N(z,t) - {N_0}),$$

(3)

where A is the differential gain coefficient, N(z,t) is the carriers sheet density, N₀ is the transparency sheet density. The last term of the coupled wave equation ${Phi _{
m{R}}}$ and ${Phi _{
m{S}}}$ represent the contribution of spontaneous emission to the light field^[13,14], and the expression is as follows:

$$begin{split}& leftlangle {{Phi _i}(z,t){Phi _i}(z',t')} ight angle = delta (z - z') frac{{{text{π}} Delta upsilon }}{2}{{ m{e}}^{ - {text{π}} Delta upsilon left| {t - t'} ight|}} & qquad!! qquadqquadqquad times (G {n_{ m{sp}}}(z) hv frac{{{w_{ m{act}}}}}{{{W^2}}}),end{split}$$

(4)

where i represent R and S. The product of the upper formula and the lateral width W of the light field is the ratio of spontaneous emission to the forward wave or the backward wave. $Delta upsilon $ corresponds to the full-width at half-maximum of the Lorentzian line-shape function, which is associated with the direct electron and hole radiative transition. Hence, $dfrac{{pi Delta upsilon }}{2} {e^{ - pi Delta upsilon left| {t - t'}
ight|}}$ in the Eq. (4) can be reduced to a delta function, $delta (t - t')$, w_act is the width of active layer, ${n_{
m{sp}}}$ is spontaneous emission factor, whose expression is:

$${n_{ m{sp}}}(z) = frac{1}{{1 - exp left{ {[{{hv + {E_{ m{FV}}}(z) - {E_{ m{FC}}}(z)]} / {{k_{ m{B}}}T}}} ight}}}.$$

(5)

As can be seen from this formula, ${n_{
m{sp}}}$ varies with the change of position z. For many applications, the incomplete inversion factor is adequately treated as a constant.



The time and spatial distribution of the carrier sheet N(t,z) is as follows:

$$frac{{partial N(z,t)}}{{partial t}} = frac{{{eta _i}}}{e}J(z,t) - Re (N) - frac{G}{{hv}}({left| R ight|^2} + {left| S ight|^2}),$$

(6)

where ${eta _i}$ is the injection efficiency, $J(z,t)$ is the density of current, $Re (N) = dfrac{{N(z,t)}}{{{tau _{
m{e}}}}}$ represents non-radiation loss, ${tau _{
m e}}$ is equal to the recombination lifetime.



Next, the finite difference method is used to solve the Eqs. (1a), (1b) and (6). Before that, the difference recurrence relation must be derived. Let $t$$ = k Delta t$, $z = m Delta z$, $R(t,z) =$$ R(k,m),S(t,z) = S(k,m)$, $N(t,z) = N(k,m)$, chooses approximate $Delta t$, makes it satisfy the correlation: $Delta t = Delta z/{v_{
m{g}}}$$R$ satisfies the recurrence relations^[15]:

$$begin{array}{l}R(t + Delta t,z + Delta z) - R(t,z) ;;= R(t + Delta t,z + Delta z) - R(t,z + Delta z) + R(t,z + Delta z) - R(t,z);; = dfrac{{partial R}}{{partial t}}left| {_{z + Delta z} } ight.Delta t + dfrac{{partial R}}{{partial z}}left| {_t} ight. Delta z ;; = dfrac{{partial R}}{{partial t}}left| {_{z + Delta z}} ight. Delta t - dfrac{{partial R}}{{partial t}}left| {_z} ight. Delta t + dfrac{{partial R}}{{partial t}}left| {_z} ight. Delta t + dfrac{{partial R}}{{partial z}}left| {_{Delta t} Delta z} ight. ;; = dfrac{{{partial ^2}R}}{{partial {z^2}}} {v_{ m{g}}} Delta t Delta z + left( {dfrac{1}{{{v_{ m{g}}}}} dfrac{{partial R}}{{partial t}} + dfrac{{partial R}}{{partial z}}} ight) Delta z.end{array}$$

According to the slowly varying envelope approximation, the second order reciprocal of R to z can be ignored, and the following recurrence relation can be obtained:

$$R(k + 1,m + 1) = R(k,m) + Delta z {Psi _R},tag{7a}$$

where ${Psi _R}$ represents all terms on the right side of Eq. (1a). Similarly, we can deduce the recurrence relationship about S:

$$S(k + 1,m - 1) = S(k,m) + Delta z {Psi _S},tag{7b}$$

where ${Psi _S}$ represents all terms on the right side of Eq. (1b). However, this method is a truly daunting and time-consuming calculation, it requires that $Delta z$ is equal to 1–5 μm. Simulating $1;mu {
m s}$ of device operation would require ${10^7}$ updates in time for each grid point on the axis of the laser. Consequently, we employ an accurate procedure for transferring the iterations step into a larger distance. Without affecting the accuracy, the iterative step size can be increased by 5 to 10 times. Considering an instantaneous moment, $R$ and $S$ have the following correlation^[16]:

$$frac{{ m{d}}}{{{ m{d}}z}}left[ {begin{array}{*{20}{c}}{bar R}{bar S}end{array}} ight] = left[ {begin{array}{*{20}{c}}{g - a - varepsilon }&{i kappa - varepsilon }{ - i kappa + varepsilon }&{ - g + a + varepsilon }end{array}} ight]left[ {begin{array}{*{20}{c}}{bar R}{bar S}end{array}} ight].$$

(8)

According to the algorithm of differential matrix, the following results can be obtained:

$$left[ {begin{array}{*{20}{c}} {bar R(z + Delta z)} {bar S(z + Delta z)} end{array}} ight] = exp (tilde M Delta z)left[ {begin{array}{*{20}{c}} {bar R(z)} {bar S(z)} end{array}} ight],$$

(9)

where $tilde M$ represents the matrix in Eq. (8). Similarly, let $Delta z' = - Delta z$, we can obtain the following relationships:

$$left[ {begin{array}{*{20}{c}} {bar R(z - Delta z)} {bar S(z - Delta z)} end{array}} ight] = exp (tilde M ( - Delta z))left[ {begin{array}{*{20}{c}} {bar R(z)} {bar S(z)} end{array}} ight].$$

(10)

Let matrix $F !=! exp (tilde M Delta z)$, matrix $B !=! exp (tilde M ( - Delta z))$, so:

$$ {{bar R}_{{{m}} + 1}} = {F_{11}} {{bar R}_{{m}}} + {F_{12}} {{bar S}_{{m}}},tag{11a}$$

$${{bar S}_{{{m}} - 1}} = {B_{21}} {{bar R}_{{m}}} + {B_{22}} {{bar S}_{{m}}}.tag{11b}$$

Considering the effects of time and spontaneous emission, the final expression is obtained:

$$R_{{{m}} + 1}^{k + 1} = {F_{11}}R_{{m}}^k + {F_{12}}S_{{m}}^k + {Phi _R} Delta z,tag{12a}$$

$$S_{{{m}} - 1}^{k + 1} = {B_{21}}R_{{m}}^k + {B_{22}}S_{{m}}^k + {Phi _S} Delta z.tag{12b}$$

Boundary conditions:

$${R_{{{m}} = 1}} = {r_{ m b}} {S_{{{m}} = 1}},tag{13a}$$

$${S_{{{m}} = {{n}}}} = {r_{ m{f}}} {R_{{{m}} = {{n}}}},tag{13b}$$

where $n,{r_{
m{f}}},{r_{
m{b}}}$ are the numbers of the last step size, the reflectivity of the front cavity surface and the reflectivity of the rear cavity surface, respectively.



The differential recurrence relationship of carriers is as follows:

$$N_{{m}}^{k + 1} = N_{{m}}^k + Delta t cdot Im ,$$

(14)

where $Im $ represents the right terms in Eq. (6), and $Delta t$$ = dfrac{{Delta z}}{{{v_{
m{g}}}}}$. Simultaneous Eqs. (12), (13) and (14) can be used to solve the carrier change and light field change in the device. It is suitable for grating-free Fabry-Perot tapered lasers, tapered lasers with DBR gratings and DFB lasers.

3.
Simulation results and analysis

In an ideal case, the reflectivity of the front cavity facet of the amplifier is zero, which would only display system dynamics originating in the master oscillator. The tapered amplify region amplifies the mode produced by the master oscillator region. In practice, the reflectivity of the front cavity surface is not zero, which will introduce additional models and aggravate the mode competition. For the noise term^{[17, 18]}, each cyclic iteration produces the random number of Gaussian distribution with mean value of 0 and variance of 1, but must satisfy Eq. (4). The simulation parameters are shown in Table 1.

Parameter	Typical value
Back grating length (L1)	100 μm
MO area length (L2)	600 μm
Front grating length (L3)	50, 75, 100, 200 μm
PA area length (L4)	1200 μm
MO area width	4 μm
Front cavity width	134 μm
Current injection efficiency (?i)	0.95
Spontaneous emission factor (nsp)	2.6
Differential gain coefficient (A)	1.0 × 10 –11 cm
Transparency carrier density (N0)	1.0 × 1012 cm–2
Total recombination time (τe)	1.3 × 10–9 s
Grating cross coupling coefficient (?)	30 cm–1
Grating radiation loss (ε)	10 cm–1
Approximately emission wavelength (λ)	980 nm
Front cavity facet power reflective (Rf)	0.002
Back cavity facet power reflective (Rb)	0.3
Group refractive index (ng)	4.6
MO area injection current (IMO)	120 mA
PA area injection current (IPA)	3.0 A

Table1.
Parameters of simulation.



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Parameter	Typical value
Back grating length (L1)	100 μm
MO area length (L2)	600 μm
Front grating length (L3)	50, 75, 100, 200 μm
PA area length (L4)	1200 μm
MO area width	4 μm
Front cavity width	134 μm
Current injection efficiency (?i)	0.95
Spontaneous emission factor (nsp)	2.6
Differential gain coefficient (A)	1.0 × 10 –11 cm
Transparency carrier density (N0)	1.0 × 1012 cm–2
Total recombination time (τe)	1.3 × 10–9 s
Grating cross coupling coefficient (?)	30 cm–1
Grating radiation loss (ε)	10 cm–1
Approximately emission wavelength (λ)	980 nm
Front cavity facet power reflective (Rf)	0.002
Back cavity facet power reflective (Rb)	0.3
Group refractive index (ng)	4.6
MO area injection current (IMO)	120 mA
PA area injection current (IPA)	3.0 A

When the power reflectivity of the front-end face is 0.002 and the length of the front grating is 50 μm, as shown in Fig. 2, the carrier concentration of the device varies with time. In the first few nanoseconds, the carrier concentration rises sharply. The reason for this phenomenon is that the photons in the device have not yet been produced, and after the photons are excited, the carrier concentration decreases rapidly, and finally tends to be stable, and the generation of carriers and the transformation of photons reach a dynamic equilibrium. The following is the total number of photons in the device, which tends to be stable after the initial rapid change, slightly lagging behind the change of carrier concentration. Fig. 3 shows the spectrum after Fourier transform of the output light field power. As can be seen, there are multiple longitudinal mode patterns whose spacings are about 50 GHz when the device works stably, displaying serious mode competition between longitudinal modes.






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Figure2.
(Color online) The top panel shows the change of total charge carriers and the bottom panel shows the change of total photons with the change of time with a 50 μm front grating.






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Figure3.
(Color online) The frequency of output light with a 50 μm front grating.




Lengthening the front grating L₃ can improve longitudinal mode characteristics. Increasing the front grating to 75 μm and the other conditions remain unchanged. It can be seen from the following figure that the number of longitudinal modes is obviously reduced, and the single longitudinal mode characteristics are more obvious with only three modes left, as shown in Fig. 4.






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Figure4.
(Color online) The frequency of output light when front grating is 75 μm.




By extending the length of the grating L₃ to 100 μm, we observe that the laser has high power and excellent longitudinal mode characteristics. In the simulation, mode competition occurs only in the first few nanoseconds. During this time, the master oscillator settles into a single longitudinal mode. As shown in Fig. 5, the above is the spectrum of the initial unstable state, and the below is the stable output spectrum.






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Figure5.
(Color online) The top panel shows the transient frequency of output light during 0–5 ns and the bottom panel shows the stable state frequency of output light with a 100 μm front grating.




Fig. 6 shows the variation of carriers and photons under the condition that the length of the front grating L₃ is 100 μm. Fig. 7 shows the output power of the end facet, the output power above is the full time domain output power, and the bottom is the stable output power. As can be seen, under the stable condition, the output power tends to be stable. When the front grating is continued to increase to 200 μm, there is almost one longitudinal mode in the device (as shown in Fig. 8) but the power of the output light field is slightly lower than that of the front grating is 100 μm (as shown in Fig. 9).






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Figure6.
(Color online) The top panel shows the change of total charge carriers and the bottom panel shows the total photons with a 100 μm front grating.






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Figure7.
(Color online) The top panel shows the change of output power with the change of time during all simulation time and the bottom panel shows the output power in stable state with a 100 μm front grating.






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Figure8.
(Color online) The stable state frequency of output light with a 200 μm front grating.






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Figure9.
(Color online) The top panel shows the change of output power with the change of time during all simulation time and the bottom panel shows the output power in stable state with a 200 μm front grating.




From these frequency figures it can be found that along with the increase of the length of the front grating, there are fewer and fewer stable modes in the device. When the front grating is extended to 100 μm, almost one longitudinal mode exists in the compound-cavity, for which the front grating with suitable length can eliminate the mode degeneracy and ensure the stable operation of the single mode in the device.

4.
Conclusion

We use coupled-wave equation to accurately analyze model the single longitudinal mode and the multiple longitudinal mode in a compound cavity. The integration algorithms in Eq. (12) can help us to take larger time steps while maintaining computational efficiency and accuracy. To get nearly ideal single longitudinal mode and high-power output, we optimize the length of front gratings. When the front grating is extended to no less than 100 μm, mode competition occurs over only the first few nanoseconds, at which time the master oscillator settles into a single longitudinal mode. Meanwhile, considering that the structure scale of the laser must be controlled in a certain range and maintain its high photoelectric transformation efficiency, the grating length cannot be too long to avoid affecting its practicability. Therefore, the ideal light output can be obtained by increasing the length of front grating to about 100 μm, when MO and PA area are driven separately using different electrodes^[19].



The calculated results in this paper have significance for the actual fabrication of tapered lasers with DBR gratings. The model that we have developed in this article can be used to analyze 3D simulations when lateral and transverse optical fields are considered^[20] and can then be used to simulate various types of semiconductor lasers, including DBR gratings, DFB gratings and so on^{[21, 22]}.