Theory of the ultrafast mode-locked GaN lasers in a large-signal regime

Igor V. Smetanin; Peter P. Vasil’ev; Dmitri L. Boiko

doi:10.1364/OE.19.017114

1. Introduction

Compact laser sources generating sub-picosecond pulses in the blue/violet wavelength range [1] can meet the requirements for various applications in different fields of science and technology. They can be used for high-density optical data storage systems, biomedical diagnostic and fluorescence imaging, optical combs etc. For such an application, understanding the dynamics of GaN-based semiconductor lasers is particularly essential. Mode locking is often considered as the most promising technique to generate picosecond and femtosecond optical pulses with high repetition rates and low timing jitter [2,3].

One of the commonly used theoretical approaches to treat the mode-locking pheno-menon in lasers with a slow gain and absorber has been elaborated by Haus and New [4,5]. The approach utilizes a solitary time-domain master equation and considers localized gain and absorber. Several attempts have been made to adapt this efficient method to monolithic passively mode-locked laser diodes (LDs) [6,7]. However, all previous developments assume low pulse energy as compared with the saturation energy of absorber. This assumption is not valid for the blue-violet InGaN/GaN lasers: Using typical parameters from Table 1 , one obtains the saturation energy of absorber E_sA = ħωv_gdh/Γs(∂g*/∂n) of 2.5 pJ. The expected energy W_p of the pulse travelling in the cavity is in the range 1-10pJ, which is comparable or higher then E_sA. This renders previous time-domain developments not suitable for GaN LDs.

Table 1. Dynamic model parameters for monolithic mode-locked InGaN/GaN laser diodes [10,12].

View Table

In this paper, we develop a new analytical approach for passively mode-locked diode lasers with slow absorber and amplifier. Our approach utilizes pulse-energy domain analysis and is suitable at high pulse energies, much exceeding the saturation energy of absorber section. We find that, two analytic solutions are possible, which are referenced herewith as a high- and low-energy branches. Ultrafast pulse operation in mode-locking regime and its stability are discussed in application to InGaN/GaN monolithic multi-section laser diode.

As a model system, we utilize monolithic multisection cavity, in which absorber section is located in the middle of the laser cavity (see Fig. 1 .(a)). The gain and absorption section are treated in localized dynamic model approximation and a single pulse traveling in the cavity is assumed.

Fig. 1 Schematic of illustration of InGaN/GaN ridge-waveguide multi-section laser diode (a). Sellmeier model predictions for the group velocity dispersion $D^{*} = ω_{L}^{2} \partial^{2} k / \partial ω^{2}$ (b) and group velocity (c).

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2. Haus-New’s theoretical model in time-domain representation

As a starting point, we use the steady-state master equation of the time-domain Haus-New’s theory [4,5] which in the case of semiconductor laser diode can be written as [6,7]

\begin{array}{l} {1 + (1 - i α_{A}) q_{i} \exp (- E (t) / E_{s A}) - (1 - i α_{L}) g_{i} \exp (- E (t) / E_{s L}) + ξ^{2} + i (ψ + ξ) + \\ + (1 + δ - 2 i ξ) \frac{1}{ω_{L}} \frac{d}{d t} - (1 + i D) \frac{1}{ω_{L}^{2}} \frac{d^{2}}{d t^{2}}} a (t) = 0. \end{array}

Here

a (t)

is the complex slow varying amplitude of the electric field component of the laser pulse,

E (t) = (d h / Γ) (e^{2} / 8 π c) \int_{- \infty}^{t} | a (t^{'}) | d t^{'}

and

W_{p} = (d h / Γ) (e^{2} / 8 π c) \int_{- \infty}^{\infty} | a (t^{'}) | d t^{'}

are the instantaneous (cumulative) and total pulse energies,

ξ = Δ ω / ω_{L}

is the frequency detuning of the carrier from the center of the gain line reported to the gain spectral half-width,

E_{s A}

and

E_{s L} = s E_{s A}

are the saturation energies of absorber and amplifier sections,

l_{A}

and

l_{L}

are their lengths. All material and geometrical parameters of the structure (Table 1) are normalized with respect to the cold cavity loss

α_{C} l_{C} = α_{i} l_{C} - (1 / 2) \ln (R_{1} R_{2})

[see the first term in Eq. (1)].

The carrier relaxation time T_L in the gain-section quantum wells (QWs) is significantly longer than generated picosecond pulses so as the relaxation during the pulse can be neglected (see Table 1). Even though the recovery time T_A of absorber is only few times longer the pulse, the absorber relaxation during the pulse is neglected as well. (The effect of partial relaxation within the pulse has been accounted for in the numerical model of Ref. [8] and appears to be insignificant.) As such, large-signal absorption (second term in (1)) and gain (third term) are represented by instantaneous solutions of the corresponding rate equations during the build-up of an ultrafast pulse [6,7]. In particular, q_i and g_i are the saturated absorption and gain in the cavity at the beginning of a pulse, q_iexp(-W_p/E_sA) and g_iexp(-W_p/E_sA) represent the same large-signal parameters at the end of the pulse. Solving the rate equations during the cavity roundtrip T_rep time, when these relax back to the initial absorption q_i. and gain g_i, one obtains the relationships between the steady-state saturated and unsaturated small-signal values of the absorption and gain in the cavity [6,7]:

q_{i} = q_{0} \frac{1 - \exp (- T_{r e p} / T_{A})}{1 - \exp (- W_{p} / E_{s A} - T_{r e p} / T_{A})}, g_{i} = g_{0} \frac{1 - \exp (- T_{r e p} / T_{L})}{1 - \exp (- W_{p} / E_{s L} - T_{r e p} / T_{L})}

These expressions account for the carrier relaxation between the subsequent pulses. For the model system in Fig. 1, the normalized small-signal roundtrip absorption $q_{0}$ and gain $g_{0}$ in the cavity are controlled by the external bias V_a and pump current density J:

q_{0} = \frac{l_{A} Γ s}{v_{g} α_{C} l_{C}} (\frac{\partial g}{\partial n}) n_{t} (1 - V_{a}), g_{0} = \frac{l_{L} Γ}{v_{g} α_{C} l_{C}} (\frac{\partial g}{\partial n}) [\frac{J T_{L}}{e h} - n_{t}],

where n_t is the transparency carrier density, and

V_{a}

is the normalized bias parameter. The last four terms in Eq. (1) accounts for the dispersion coefficient

D^{*} = ω_{L}^{2} \partial^{2} k / \partial ω^{2}

(Fig. 1(b)), the round trip phase shift

ψ_{0}

and time delay

Δ T

of the pulse. In Eq. (1), these are represented by normalized parameters

D = 2 D^{*} / α_{C}

,

ψ = 2 ψ_{0} / α_{C} l_{C}

,

δ = 2 ω_{L} Δ T / α_{C} l_{C}

[6,7].

3. Analytic solution in the pulse-energy domain

So far the analytic solutions of the master Eq. (1) have been utilizing an assumption of small pulse energy E(t),W_p<<E_sA,_sL, yielding a series expansion of the gain and absorption terms in the time-domain [6,7]. We find an original approach to solve Eq. (1) for high pulse energies,developing our analysis in the pulse energy domain.

We introduce new variable $x = E (t) / W_{p}$ , measuring the progress of instantaneous (cumulative) pulse energy towards the total energy in the pulse, so as 0≤x≤1. It allows us to apply the following anzatz for the slowly varying pulse amplitude

a = {[A G^{1 / 2} (x)]}^{1 + i β},

where A is the peak pulse amplitude,

A = {(4 π c W_{p} / e^{2} τ_{p})}^{1 / 2}

, βis the pulse chirp parameter, and the function G(x) is the pulse intensity envelope. It reaches maximum value

\max (G) = 1

at the peak of the pulse, and vanishes at the beginning and the end of the pulse, G(0) = G(1) = 0. Substituting (4) in (1) and separating the real and imaginary parts, we transform it to the master equation for steady-state mode-locking regime in the energy domain representation:

\begin{array}{l} 2 (1 - β D) B^{2} G G'' + (1 - β^{2} - 2 β D) B^{2} {(G')}^{2} - (1 + δ + 2 ξ β) B G' - ξ^{2} = \\ = 1 + q_{i} \exp (- μ x) - g_{i} \exp (- μ x / s), \\ 2 (β + D) B^{2} G G'' + (2 β + D - β^{2} D) B^{2} {(G')}^{2} - (β + δ β - 2 ξ) B G' = \\ = ψ + ξ - α_{A} q_{i} \exp (- μ x) + α_{L} g_{i} \exp (- μ x / s) \end{array}

where

μ = W_{p} / E_{s A}

and

B = 1 / 4 ω_{L} τ_{p}

are the normalized pulse energy and the inverse pulse width.

We obtain the analytic solution of Eq. (5) in the steady-state mode-locking regime by introducing a series expansion for the pulse envelop $G (x) \approx 1 - 4 {(x - 1 / 2)}^{2}$ in the vicinity of the peak at x = 1/2. Note that its inverse transform $t / τ_{p} = \int_{1 / 2}^{x} d x^{'} / G (x^{'})$ to the time-domain representation yields the hyperbolic secant pulse shape ${| a (t) |}^{2} = A^{2} \cosh^{- 2} (t / τ_{p})$ . The term in the right-hand side of the first Eq. (5) is the net cavity gain $f (x) = - 1 - q_{i} \exp (- μ x) + g_{i} \exp (- μ x / s)$ . It reaches maximum at the pulse energy $x_{\max} = s \ln [s q_{i} / g_{i}] / μ (s - 1)$ . Therefore, we substitute in Eq. (5) the series expansion $f (x) \approx s Δ_{1} - 1 - \frac{1}{2} μ^{2} Δ_{1} {(x - x_{\max})}^{2}$ . In a similar way we introduce a series expansion in the right-hand side of the second Eq. (5). Finally, considering terms at each power of x, we obtain the following steady-state solution [Eqs. (6)-(10)], which is valid at any energy of the pulse, including the special case of our interest $μ \equiv W_{p} / E_{s A} \geq 1$ :

The chirp β of the pulse (4) is

β = - \frac{2}{3} \frac{γ + D}{1 + D^{2}} Y .

where

γ = {[\frac{α_{L}^{s}}{α_{A}}]}^{1 / (s - 1)}, Y = \frac{3}{2} \frac{(1 + D^{2})}{(γ + D^{2})} {\sqrt{2 + \frac{9}{4} \frac{{(1 - γ D)}^{2}}{{(γ + D)}^{2}}} - \frac{3}{2} \frac{(1 - γ D)}{(γ + D)}} .

Using the notation

\begin{array}{l} T_{1} = Y^{- 1} + 4 Y {(\frac{β + γ}{1 + β^{2}})}^{2} + \frac{2}{3} \frac{4 (γ + D) - 3 (γ D - 1)}{1 + D^{2}}, T_{2} = 4 b_{1} + 8 Y {(\frac{β + γ}{1 + β^{2}})}^{2} \frac{β b_{1} + γ b_{2}}{β + γ} \\ T_{3} = 16 (s - \frac{1}{Δ_{1}}) - 8 b_{1}^{2} - 16 Y {(\frac{(β + γ) (β b_{1} + γ b_{2})}{(1 + β^{2}) (β + γ)})}^{2},, \\ Δ_{1} = \frac{s - 1}{s} {[\frac{{(g_{i} / s)}^{s}}{q_{i}}]}^{1 / (s - 1)}, b_{1} = \frac{s}{s - 1} \ln [\frac{s q_{i}}{g_{i}}], b_{2} = \frac{s}{s - 1} \ln [\frac{α_{A}}{α_{L}} \frac{s q_{i}}{g_{i}}] \end{array}

we can represent the normalized energy µ and duration τ _p of the mode-locked pulse in a simple form

μ_{\pm} = \frac{T_{2}}{T_{1}} \pm \sqrt{\frac{T_{2}^{2}}{T_{1}^{2}} + \frac{T_{3}}{T_{1}}}, τ_{p, \pm} = \frac{4}{ω_{L}} \frac{Y^{1 / 2}}{μ_{\pm} Δ_{1}^{1 / 2}},

where the indexes “+” and “-”distinguish two solutions referenced here as the high-energy and low-energy branches, respectively; the FWHM of the hyperbolic secant pulse is 1.76τ _p. Finally, the normalized frequency detuning from the line center and the pulse delay read

ξ = \frac{1}{2} \frac{β + γ}{1 + β^{2}} Y^{1 / 2} Δ_{1}^{1 / 2} [μ_{\pm} - 2 \frac{β b_{1} + γ b_{2}}{β + γ}], 1 + δ = \frac{1}{2} \frac{1 - β γ}{1 + β^{2}} Y^{1 / 2} Δ_{1}^{1 / 2} [μ_{\pm} - 2 \frac{b_{1} - b_{2} β γ}{1 - β γ}] .

The Lyapunov’ analysis of dynamic stability of the two solutions (9) is a complicated problem and will be reported elsewhere. Instead of it, we utilize below a set of conditions, which define domains of stable mode-locking regime for each solution.

4. Results and discussion

We apply our approach to analyze the stability and performances of a monolithic mode-locked InGaN/GaN diode laser. All major laser parameters are indicated in Table 1 and the ratio of the absorber length to the overall cavity length is $l_{A} / l_{C} = 0.1$ . In Table 1, the short recovery time of absorber is settled by the tunneling of carriers from the QWs to the tilted QW barriers and by the carrier drift time through the intrinsic region of the multiple QW heterostructure barriers of about 30-40 nm thick. Thus, due to a large reverse bias, the absorber recovers much faster than the carrier relaxation takes place in the gain-section QWs. The group refractive index in InGaN/GaN lasers is close to 3.5 [9,10]. To estimate the group velocity dispersion we use Sellmaier model for the refraction index dispersion in GaN [9]. The predicted group velocity value (Fig. 1(c)) is in good agreement with the value from Table 1, therefore we are confident in the estimated group velocity dispersion of $D^{*} \approx 60$ cm⁻¹ ( $D \approx 3$ ) in Fig. 1(b).

The model predictions for the output peak power $P_{\pm}$ and FWHM pulse width (1.76τ_{p, ±} ) are plotted in the Fig. 2(a) as a function of the pump current. The output peak power per cavity facet is calculated as a fraction $\propto - \ln (R) / 2 (α L - \ln R)$ of the pulse peak power in the cavity $W_{p} / 2 τ_{p}$ . In Fig. 2(b), the corresponding normalized pulse energy µ _± = W_{p, ±} /E_sA and absorber bias parameter are displayed.

Fig. 2 Solution Eq. (9) of the mode-locking master Eq. (1): (a) FWHM pulse width 1.76τ_{p, ±} (left axis) and output peak power per facet $P_{\pm}$ (righ axis) v.s. pumping current for high-energy (black curves) and low energy (red curves) solution branches; (b) Bias voltage at which the self-starting modelocking regime is stable (left axis) and the relative pulse energy µ _± = W_{p, ±} /E_sA (righ axis)

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As expected, the higher the pulse energy, the shorter the pulse width is. This applies equally to the comparison of the pulse parameters between the high- (index “+”) and low-energy (index “-”) branches of Eq. (9). The relative energy µ _± of the mode-locked pulses traveling in the cavity is comparable to or greater than 1. Therefore our approach to solution of the Haus-New’s master equation suitable for mode-locked pulses of arbitrary energy W_p is not just an interesting feature of the model. As indicated in Fig. 2(b), in InGaN/GaN monolithic mode-locked lasers, the pulse energy systematically exceeds the absorber saturation energy E_sA, and the absorber operates at high saturation level. Previously known solutions [6,7] are not valid in this operation regime of absorber. In the mode-locking regime corresponding to the high–energy branch of Eq. (9) the pulses of ~500fs width can be produced at the output peak pulse power of ~1.5W per laser facet.

The necessary conditions for stable regime of mode-locking has been formulated in Refs [4–7]: The roundtrip net gain in the cavity must be negative all the time but the lasing pulse so as all spurious field fluctuations are dumped. In terms of the energy-domain solution (6)-(10), this requirement leads to the boundary conditions for f(x):

1 + q_{i} - g_{i} > 0 (x = 0), 1 + q_{i} \exp (- μ_{\pm}) - g_{i} \exp (- μ_{\pm} / s) > 0 (x = 1) .

However during the pulse, the roundtrip net gain should become positive somewhere on the interval 0<x<1 in the energy domain representation. Therefore the maximum of the roundtrip net gain f(x) located at x = x _max [see the discussion after Eq. (5)] assumes that $s Δ_{1} - 1 > 0$ ,yielding us the third condition for the steady-state mode-locking regime:

g_{i} \geq s q_{i}^{1 / s} / {(s - 1)}^{(s - 1) / s}

It follows that there exists the minimum value of the absorption coefficient $q_{i \min} = 1 / (s - 1)$ so as at $q_{i} < q_{i \min}$ both the high-energy and low pulse energy mode-locking regimes are unstable. Finally we have to add also the necessary start-up condition

1 + q_{0} < g_{0},

which means the net gain should be positive to launch the amplification from zero amplitude level.

In the case of GaN laser diode, the domain (11)-(13) of self-starting stable mode-locking regime for each solution (9) is very narrow, as indicated in the absorber bias map in Fig. 2(b). The domains of stability of the two solutions do not overlap, indicating that the two solutions may not coexist simultaneously. Interestingly, in the limit of very small pulse energies, the difference between two solutions vanishes, in agreement with results of Ref. [11].

5. Conclusion

In conclusion, we have developed a new analytical approach to the theory of passively mode-locked diode lasers at moderate pulse energies $μ \geq 1$ , which is based on the energy-domain representation of the pulse envelope evolution. In this approach, we calculate the parameters of the mode-locked pulse in the vicinity of the peak intensity at the arbitrary pulse energies, while usually [6,7] the small-energy approximation $μ < < 1$ is used which drops at pulse energies exceeding the absorber’s saturation energy.

Our approach, in fact applies well to any cavity arrangement, e.g. with absorber section located at the facet or in the external cavity configuration. Furthermore, it can be easily tailored to the case of colliding pulse mode locking (CPM) regime, which may occur in a cavity with absorber situated in the middle (as in Fig. 1 (a)). The overlap of two CPM pulses at the absorber is taken in to account by replacing the roundtrip time by T_rep/2 in all equations so as to account for the reduced time elapsed between propagation of pulses. The second modification consists in substituting the double pulse energy 2W_p in the equation for steady state saturated absorption q_i in Eq. (2) so as to account for the saturation effect of the two pulses overlapping in the absorber.

The developed approach allows us to determine the mode-locked operation parameters for the high-power sub-picosecond blue-violet GaN diode laser.

This research is supported by the EC Seventh Framework Programme under grant agreement #238556 (FEMTOBLUE).

References and links

1. S. Nakamura, M. Senoh, S.- Nagahama, N. Iwasa, T. Matsushita, and T. Mukai, “Blue InGaN-based laser diodes with an emission wavelength of 450 nm,” Appl. Phys. Lett. 76(1), 22–24 (2000). [CrossRef]

2. K. Saito, H. Watanabe, T. Miyajima, M. Ikeda, and H. Yokoyama, “Mode locking of an external-cavity bisection GaInN blue-violet laser diode producing 3 ps duration optical pulses,” Appl. Phys. Lett. 96(3), 031112 (2010). [CrossRef]

3. P. P. Vasil'ev, Ultrafast diode lasers: fundamentals and applications (Artech House, Boston, 1995).

4. G. H. C. New, “Pulse evolution in mode-locked quasi-continuous lasers,” IEEE J. Quantum Electron. 10(2), 115–124 (1974). [CrossRef]

5. H. A. Haus, “Theory of mode-locking with a slow saturable absorber,” IEEE J. Quantum Electron. 11, 736-746 (1975); “Mode-locking of lasers,” IEEE J. Sel,” Topics in QE 6(6), 1173–1185 (2000). [CrossRef] [CrossRef]

6. R. G. M. P. Koumans and R. van Roijen, “Theory for passive Mode-Locking in Semiconductor Laser Structures Including the Effects of Self-phase Modulation, Dispersion, and Pulse Collisions,” IEEE J. QE. 32(3), 478–492 (1996). [CrossRef]

7. J. Leegwater, “Theory of Mode-Locked Semiconductor Lasers,” IEEE J. Quantum Electron. 32(10), 1782–1790 (1996). [CrossRef]

8. J. L. A. Dubbeldam, J. A. Leegwater, and D. Lenstra, “Theory of mode-locked semiconductor lasers with finite absorber relaxation times,” Appl. Phys. Lett. 70(15), 1938 (1997). [CrossRef]

9. J. Piprek, Semiconductor optoelectronic devices (Academic, 2002)

10. W. G. Scheibenzuber, U. T. Schwarz, L. Sulmoni, J. Dorsaz, J.-F. Carlin, and N. Grandjean, “Recombination coefficients of GaN-based laser diodes,” J. Appl. Phys. 109(9), 093106 (2011). [CrossRef]

11. A. G. Vladimirov and D. Turaev, “Model for passive mode locking in semiconductor lasers,” Phys. Rev. A 72(3), 033808 (2005). [CrossRef]

12. J. Dorsaz, D. L. Boïko, L. Sulmoni, J.-F. Carlin, W. G. Scheibenzuber, U. T. Schwarz, and N. Grandjean, “Optical bistability in InGaN-based multi-section laser diodes,” Appl. Phys. Lett. 98(19), 191115 (2011). [CrossRef]

Parameter	Value
ħω, photon energy	2.9 eV
∂g*/∂n, differential gain	2.2 × 10⁻⁶ cm³/s
s, differential absorption to gain ratio	3
n_t transparency carrier density	1.6 × 10¹⁹ cm⁻³
h active layer thickness (QW width × QW number product)	4 nm
c/v _g, group velocity index	3.5
Γ mode confinement factor	0.02
d waveguide width	2 µm
α_i intrinsic material loss	20 cm⁻¹
R ₁ = R ₂ cavity facet reflection coefficients	18.5%
2ħω_L gain linewidth	70 meV
α_A,L absorber (A) and amplifier (L) linewidth enhancement factors	3
D* dispersion coefficient	60 cm⁻¹
T_L carrier lifetime in the gain section	910 ps
T_A absorber recovery time	5 ps
l_C cavity length	0.8mm

Theory of the ultrafast mode-locked GaN lasers in a large-signal regime

Abstract

1. Introduction

2. Haus-New’s theoretical model in time-domain representation

3. Analytic solution in the pulse-energy domain

4. Results and discussion

5. Conclusion

References and links

Cited By

Figures (2)

Tables (1)

Equations (13)

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