Theory of Passively Mode-Locked Photonic Crystal Semiconductor Lasers

: We report the ﬁrst theoretical investigation of passive mode-locking in photonic crystal mode-locked lasers. Related work has investigated coupled-resonator-optical-waveguide structures in the regime of active mode-locking [Opt. Express 13, 4539-4553 (2005)]. An extensive numerical investigation of the inﬂuence of key parameters of the active sections and the photonic crystal cavity on the laser performance is presented. The results show the possibility of generating stable and high quality pulses in a large parameter region. For optimized dispersion properties of the photonic crystal waveguide cavity, the pulses have sub picosecond widths and are nearly transform limited.


Introduction
A mode-locked (ML) laser based on photonic crystal (PC) waveguides (PCW) gives unique possibilities for controlling the pulse properties.Thus, by operating in the slow-light regime, very compact devices can be realized while maintaining a pulse repetition rate compatible with the requirements of optical communication systems.Pulse properties such as the width and chirp can be tuned via the PC dispersion.Also, engineering of the waveguide geometry provides control over the optical density of states, which changes the spontaneous emission and thereby affects the dynamics and noise properties of the laser [1,2].High quality monolithic passive mode-locked lasers suitable for optical communication systems have already been demonstrated [3], but for dense integration on photonic chips, the PC structures offer the possibility for the necessary size reduction [4].Fig. 1 schematically shows the type of structure considered here, where a cavity is created inside the PC structure by removing a line of holes.Mode-locked operation is facilitated by introducing a gain and absorber material in the line defect [5,6].Changing the shape and position of holes near the cavity provides vast possibilities for engineering the dispersion.However, it is necessary to perform a detailed investigation of the interplay between the dispersion of the slow-light waveguide and the dynamics of the gain and absorber sections.It is well-known that there is a trade-off between the achievable light slowdown with which the size reduction scales and the bandwidth over which the higherorder dispersion is small [7,8].In order to obtain a better understanding of these effects, it is necessary to carefully account for them in a model describing the laser output.
The model that we have developed is described in Section 2, while Section 3 presents the results obtained from numerical solutions of the model equations.In Section 4 the conclusions are summarized.

Model
Our model is a modification of that of Vladimirov et al. [9], which assumes a ring cavity containing a gain section, an absorber, and a bandwidth limiting filter, see Fig. 2. The dynamics of the carrier density, N(z,t), of the active materials is described by a set of rate equations for the gain material (r = g) and the absorber (r = q), respectively.They are coupled to a traveling wave equation for the complex electric field envelope, E(z,t), The parameters v b g , g r , Γ r , and α r are the group velocity in bulk material, differential gain, transverse confinement factor, and line width enhancement factor, respectively.Furthermore, N tr r is the transparency carrier density, γ r is the carrier density relaxation rate, J g is the injected particle current density in the gain material, and for the absorber J q = 0.The use of a reverse biased absorber further means that γ q γ g .As seen from Eq. ( 1), the model assumes a unidirectionally propagating field, which is unable to describe effects such as colliding-pulse mode-locking [10].This is an essential assumption of our model, since the system under consideration here is a linear cavity, where forwards and backwards propagating waves are expected to play a similar role.However, support for this model is found in [11], where qualitative agreement between bifurcation diagrams showing different solution regimes have been observed for a unidirectional ring cavity model and a model with two counter propagating waves.
Our model is a lumped element model, where the dispersion of the laser cavity and bandwidth limitation of the active sections are treated via a separate component.This approximation is allowed, as long as the effect of the dispersion and bandwidth limitation on the pulse reshaping mechanism is limited [12].Hence, with the use of a strongly dispersive PCW, this approximation is important to keep in mind.The pulse widths found in this paper are on the order of 1 ps, where gain nonlinearities such as spectral hole burning and carrier heating start to play a role [13].Following the model in [9], these effects are, however, not included here.Random spontaneous emission noise [1,2] is also not considered in this work.The model does not assume small round trip gain or loss as in [12], which makes it suitable for semiconductor active materials.
The transmission of the field through the filter is given by where FPCW (ω −ω 0 ) is the complex transfer function of the filter.By integrating over the spatial coordinate and using a Lorentzian transfer function, a set of nonlinear coupled delay differential equations can be obtained [9].In the case considered here, a more general form of the transfer function is used, which provides the necessary freedom to describe the influence of the strong dispersion of the PCW on the laser output The expression in Eq. ( 3) is obtained by considering the passage of the electric field through a homogeneous material with susceptibility χ(ω), which is related to the wavenumber by k 2 (ω) = ω 2 /c 2 (1 + Re{χ(ω)}).In Eq. ( 3) this expression is replaced by the first 4 terms of its Taylor expansion.The dispersion relation, k(ω), of the PCW is found from a band structure calculation and the parameter values used in this work are chosen using the results of Settle et al. [14] as a guideline.The real function f H (ω) accounts for the spectral amplitude filtering.All physical transfer functions are subject to a causality condition, which imposes the restriction that F PCW (t ≤ 0) = 0. This, in turn, gives rise to the following relation between the real and imaginary parts of FPCW (ω) [15] +∞ Since φ (ω) is completely determined by the dispersion relation, f H (ω) is determined by Eq. ( 5).However, the Taylor expansion used in Eq. ( 3) is only a good approximation within a limited bandwidth, so solving Eq. ( 5) without any restrictions on f H (ω) is not a good approach.For this reason, we have chosen the definition in Eq. ( 4) with the adjustable parameters {a p , b p , c p }, where the values of b p are of course restricted.An iteration approach is used to determine these parameters using Eq. ( 5).This is done by inserting an initial guess on the LHS and fitting the result of the integral to the form of the RHS.The new parameter values {a p , b p , c p } are then reinserted on the LHS and so on.For small values of k (3) ω 0 and large filter bandwidths, the transfer function has a significant value for t ≤ 0 if a single Gaussian with fixed values of {a 1 , b 1 , c 1 } is used.By using Eq. ( 4) and the iteration procedure described above, it is, however, possible to make F PCW fulfill the causality condition to a far better extent.For the specific parameter values used here, however, it makes no significant difference if a single Gaussian is used for f H without the iteration procedure.In general, it is important to check the result of the iteration method as there is no guarantee of convergence.The causality condition is not addressed in the related work of Heck et al. [16], where a transfer function approach is also used to describe a passive dispersive element.
With FPCW (ω) given by Eq. ( 2), the resulting model equations are with The time domain transfer function in Eq. ( 8) is calculated from Eq. ( 2) using a numerical Fourier transformation.The normalizations used in Eqs. ( 6)-( 8) are The round trip non-resonant loss, √ κ, accounts for propagation and mirror losses and ) is the normalized cold cavity round trip time.The length of the gain section (absorber) is 2L g (2L q ).The delay term, u − T 0 , in Eq. ( 8) arises from the periodic boundary condition of the ring cavity: E(z+2L tot ,t) = E(z,t), where 2L tot = 2L PCW +2L g +2L q is the total round trip length of the cavity.

Simulation Results
We have solved Eqs. ( 6)-( 8) numerically to investigate how the laser output depends on the parameters.Using the results in [9] as a starting point, we use parameter values that are typical of standard ridge waveguide in-plane quantum well lasers [9,17].Unless otherwise stated, we use the parameter values given in Table 1.The output power of the laser is where A σ is the cross sectional area of the waveguide perpendicular to the z-axis and t out is the output coupling factor.Designs like the one shown in Fig. 1 can have large quality factors resulting in a value of κ much closer to 1 than the one given in Table 1.This, in turn, means that the threshold gain, g th 0 = Γ[q 0 − ln(κ)] = 0.014, becomes smaller and the laser can be driven by smaller pumping currents.The values of the parameters in Eq. ( 4) are a 1 = 0.33, a 2 = 0.37, With these parameters f H (ω − ω 0 ) looks like a Gaussian with a FWHM width of 1.25 THz.In Figs.3-9 the linewidth enhancement factors are zero, and in Fig. 10 an investigation of the effects of changing the α-parameters is presented.

Phase Diagram
Eqs. ( 6)-( 8) have qualitatively different types of solutions depending on the values of the unsaturated gain, g 0 , and loss, q 0 .The left plot in Fig. 3 shows a phase diagram of the different solution types.The large size of the green domain representing ML solutions shows that mode locked behavior is not critically dependent on the exact values of g 0 and q 0 , which shows promise for an experimental realization of the laser design.The smaller green regions at larger Fig.
Phase diagram of possible solution types as a function of the unsaturated gain, g 0 , and loss, q 0 .The green domain represents ML solutions, red is intensity modulated pulses, blue is irregular solutions with no periodicity, yellow is higher harmonic solutions and orange is CW solutions.Pulse properties of solutions restricted to the black line are shown in Fig. 4. Right: Plots of the different solution types depending on the value of (q 0 , g 0 ).I: (q 0 , g 0 ) = (5, 1.5), II: (q 0 , g 0 ) = (5, 2.5), III: (q 0 , g 0 ) = (5, 4.2), IV: (q 0 , g 0 ) = (5, 5).ratios of g 0 /q 0 correspond to higher harmonic ML regimes with several pulses per round trip time.Solutions in the narrow green region around (q 0 , g 0 ) = (2.5, 2.5) have 2 pulses in each round trip time whereas solutions in the uppermost green region have 3.The relative positions of the different regions in the left plot of Fig. 3 qualitatively agree with FIG. 5 in [9].Note that the values of g 0 and q 0 in Fig. 3 are very large compared to the transparency value, meaning that the laser is pumped very hard in this regime.
The graphs in the right plot of Fig. 3 show examples of the different solution types for values of g 0 and q 0 indicated by the location of the corresponding roman numerals in the left plot.Fig. 4. Variation of the pulse width (black) and duty cycle (red) for (q 0 , g 0 ) restricted to the line g 0 = 0.65q 0 − 0.2 in the left plot of Fig. 3.
The variation of the full width at half maximum (FWHM) pulse width, Δt FWHM , and duty cycle, T /Δt FWHM , for (q 0 , g 0 ) restricted to the black line in the left plot of Fig. 3 is shown in Fig. 4. T is the period of the output pulse train.The pulse width is seen to be large and vary significantly for small values of q 0 .The duty cycle expresses how closely the pulses are spaced, and from Fig. 4 it is clear that q 0 must be sufficiently large in order for the pulses to become well separated.

Influence of PC Dispersion
To obtain short pulses it is important to have a large frequency interval over which the dispersion is nearly linear so that broadening of the pulses due to group velocity dispersion (GVD) is minimized.Designs for "semi-slow" light waveguides are therefore of interest [14,18].Fig. 5(a) shows a plot of the Taylor expansion of k(ω) in Eq. (3).The gray region marks the frequency interval, where the group index is approximately constant.For a small filter bandwidth the transform limit sets a lower bound on the achievable pulse width and for a large bandwidth the pulse width is limited by either GVD or the dynamics of the interaction of the field with the active sections.In Fig. 5(b) the dependence of the pulse width and pulse bandwidth, Δ f FWHM , on the filter bandwidth, Δ f PCW , is depicted.It clearly shows that the pulse width increases significantly as the filter bandwidth becomes small.The ratio Δ f FWHM /Δ f PCW also becomes large, which indicates that the small filter bandwidth is the cause of the pulse broadening.The flattening of the black curve along with the decrease of Δ f FWHM /Δ f PCW for large filter bandwidths indicate that in this parameter region, the pulse width is determined by the gain and loss dynamics rather than GVD.The largest filter bandwidth in Fig. 5(b) is smaller than the PCW transmission window and typical gain bandwidths, so it does not correspond to any of the components of the system described in Fig. 1.This bandwidth limitation is however necessary in our model in order for the Taylor expansion in Eq. ( 3) to be a good approximation.With a filter bandwidth of 1.25 THz, we operate close to the domain where the pulse width is predominantly determined by gain and loss dynamics, and thus it has a limited physical impact that we use a filter bandwidth, which is smaller than the gain bandwidth and the PCW transmission window.1.The group velocity is constant along the dashed black line.
The frequencies on the abscissa are f − ω 0 /2π, were ω 0 is the fast oscillating part of the total field, E (z,t) = E(z,t)e ik 0 z−iω 0 t .(b) Variation of the pulse width in the time (black) and frequency domain (red) with the filter bandwidth, Δ f PCW .

Compact Laser Design
By designing the structure shown in Fig. 1 to have a large group index, it is possible to reduce the overall size of the laser.For this reason it is important to understand the influence on the pulse properties of changing the PCW length and the group index.Fig. 6(a) shows the results of simulations with different waveguide lengths, 2L PCW .The pulse width is observed to increase linearly for large values of 2L PCW , but it gradually starts deviating from this behavior as 2L PCW decreases to 40 μm, where a transition to an irregular solution type occurs.Notice that Δt FWHM varies only 20% when 2L PCW is increased by 250%.The repetition rate, f r , obtained under cold cavity conditions is Here, S = v b g /v g is the slowdown factor and v g is the group velocity in the PCW.The value of f r obtained from Eq. ( 9) is plotted in Fig. 6(a) and it is seen to agree well with the repetition rate found from the solution of Eqs. ( 6)- (8).The small deviation is attributed to pulse reshaping in the active sections, which depends on the pulse energy.
To investigate the effect of reducing the device size by increasing the slowdown factor, we have varied 2L PCW and S simultaneously while keeping their product constant to keep f r approximately constant.The result is depicted in Fig. 6(b).The repetition rate is 66 GHz and decreases about 5% in the considered interval.The length reduction factor, ρ R = [(S − 1)2L PCW + 2L tot ] /2L tot , expresses how much smaller the laser can be made compared to a conventional design without the PCW structure if their cold cavity repetition rates are equal.The reason for using S − 1 in this definition is to ensure that ρ R = 1 for S = 1.There is a difference between this expression and Eq. ( 9), where the contribution 2L PCW is counted twice.T 0 = (γ q /v b g )2L tot has to be the total normalized cold cavity round trip time due to the periodic boundary condition, but the time it takes to pass the PCW is added to T 0 in the model, which results in the term 2L PCW /v b g being counted twice in Eq. ( 9).This does not affect any of the conclusions drawn here, and the issue could be circumvented by replacing k (1) g in the first order term in Eq. ( 3).This would subtract the term, which is counted twice in Eq. ( 9) while maintaining the correct value of k  issue is that ρ R = 20 when S = 60 in Fig. 6(b), but from the design shown in Fig. 1, the length reduction factor is expected to be approximately equal to the slowdown factor since the PCW fills up the entire cavity.By setting L g = L q = 0 this would indeed be the case, but we keep the Fig. 7. Variation of the pulse width (black) and repetition rate (red) as a function of the slowdown factor.The dashed blue line shows f r from Eq. ( 9). more general formulation with nonzero lengths and merely note that ρ R is expected to be larger than the values shown in Fig. 6(b).From Fig. 6(b) it is also seen that the length reduction comes at the price of an increasing pulse width, which varies significantly for small values of 2L PCW .In Fig. 7 the influence of varying only the slowdown factor is investigated.The pulse width is seen to have a minimum, but the overall variation of Δt FWHM is only about 13% indicating that the pulse width is not strongly dependent on S for the parameter values shown here.Fig. 7 also shows that the repetition rate deviates more from Eq. ( 9) than when varying the cavity length in Fig. 6(a).
When the slow light effect of a PCW is used to reduce the size of a laser structure, the scattering losses due to fabrication imperfections increase with decreasing group velocity [19].This is an important effect to keep in mind, and it could be implemented in our model by letting κ be a function of the group velocity.red dashed line shows that the shape is Gaussian to a good approximation.The time-bandwidth product (TBP), Δt FWHM Δ f FWHM , is very close to the transform limit for Gaussian pulses in the region of small pulse width in Fig. 9(b).It increases by a factor of 2 for the widest pulses, due to pulse broadening induced by the PCW dispersion described by the phase part of Eq. ( 2).In Fig. 9 we varied the higher order dispersion coefficients freely, but it is an open question whether all points in this phase space are obtainable by changing the PCW geometry.Investigations of shifting the neighboring rows along [7] and perpendicular to [8] the propagation direction do, however, suggest that the size of the phase space used in Fig. 9 is well within the physically obtainable limits.

Linewidth Enhancement Factors
So far, all the simulations have been performed with the linewidth enhancement factors, α g and α q , equal to zero.Since they cause self phase modulation (SPM) of the pulses, it is, however, important to investigate their influence on the laser output.Fig. 10(a) shows a phase diagram of solution types depending on α q and α g .The colors refer to the same solution types as in Fig. 3.
It is observed that the range of α-parameters, for which stable ML behavior occurs, is rather limited.This is in line with the results of Leegwater [21], who also found a narrow stability region.However, in [21] this region was observed to become larger when the frequency dependence of the gain and absorption was taken into account.Based on these considerations, our simulations are expected to be conservative estimates of the laser properties.The results in [22] also indicate that it could be possible to tune the α-parameters by changing the bias voltage without affecting the dynamics for certain values of the voltage.The results of simulations performed with α g = α q for two different values of the second order term, [k (1) ω 0 , in Eq. ( 3) are shown in Fig. 10(b).The behavior of the pulse width and the TBP is observed to be highly dependent on the sign of [k (1) ω 0 .For negative values, the pulse width increases with α g = α q , while it decreases for positive values.This behavior agrees with the results in [16].Fig. 10(b) also shows that when the second order term is positive, the time-bandwidth product stays relatively close to the Gaussian transform limit of 0.44 as the pulse width decreases.The black solid line in Fig. 10(b) shows that sub-picosecond pulses are achievable even with nonzero α-parameters.However, mode-locked solutions are found only for α g = α q < 3.6, which is why the solid lines terminate at this value.This trend becomes more pronounced as k (2) ω 0 gets closer to the optimum value of −2.1 × 10 6 ps 2 /km found in Fig. 9(b), where mode-locked solutions occur for α g = α q < 1.6.Thus, the linewidth enhancement factors limit the attainable pulse width.However, this also shows that dispersion engineering can be used to achieve mode-locking for certain values of the α-parameters, although at the price of larger pulse widths.

Conclusion
In conclusion, we have presented a model for a new type of photonic crystal passively modelocked laser.The laser model predicts that a PCW laser should be capable of emitting stable, sub picosecond, and nearly transform limited pulses of high quality.Extensive numerical investigations of the interplay between the gain dynamics and the strong dispersion of the PCW show that the laser is predicted to exhibit mode locking behavior for a large range of parameter values.This fact shows promise for a realization of the type of design shown in Fig. 1.The repetition rate of the laser is shown to be tunable over a wide range by adjusting the group velocity of the pulses in the PCW inside the cavity.The proposed laser design is to the best of our knowledge the first to combine a passively mode-locked scheme with a PC slow-light structure in order to obtain very compact and highly versatile ML lasers suitable for integration on photonic chips.The conclusions have been reached with the simplifying assumption of the ring cavity model, and future work should address linear cavities featuring bidirectionally propagating fields.

Fig. 1 .
Fig. 1.Illustration of the PC ML laser design.The gain and absorber sections are placed in a line defect cavity, which is created by removing holes of the perfectly periodic PC structure.By altering the shape and position of holes close to the cavity, it is possible to engineer the dispersion properties of the cavity.

Fig. 2 .
Fig. 2. Illustration of the ring cavity model.The pulse circulates in the cavity, passing through each element.

Fig. 5 .
Fig. 5. (a) A plot of the Taylor expansion (solid black) of the dispersion relation with parameter values from Table1.The group velocity is constant along the dashed black line.The frequencies on the abscissa are f − ω 0 /2π, were ω 0 is the fast oscillating part of the total field, E (z,t) = E(z,t)e ik 0 z−iω 0 t .(b) Variation of the pulse width in the time (black) and frequency domain (red) with the filter bandwidth, Δ f PCW .

Fig. 6 .
Fig. 6.(a) Variation of the pulse width (black) and the repetition rate, f r , (red) with the length of the PCW, 2L PCW .The dashed blue line shows f r from Eq. (9).(b) Variation of the pulse width (black) and length reduction factor, ρ R , (red) as the length of the PCW and the slowdown factor are varied simultaneously, while keeping their product S × 2L PCW = 1200 μm constant.

Fig. 9 .
Fig. 9. (a) The dependence of the ratio between the trailing and main pulse peak powers on the second and third order expansion coefficients in Eq. (3).(b) Pulse width as a function of the second and third order expansion coefficients in Eq. (3).

Fig. 10 .
Fig. 10.(a) A phase diagram of solution types found by varying the linewidth enhancement factors.The color code is the same as in Fig. 3. (b) Variation of the pulse width (black) and time-bandwidth product, Δt FWHM Δ f FWHM , (red) as a function of the linewidth enhancement factors.The values of [k (1)

Table 1 .
Parameter values used unless otherwise stated in the figure captions.