## Abstract

In this paper, we investigated the optical Kerr lensing effect in quantum-cascade lasers with multiple resonance levels. The Kerr refractive index *n*
_{2} is obtained through the third-order susceptibility at the fundamental frequency *χ*
^{(3)}(*ω*; *ω*, *ω*,-*ω*). Resonant two-photon processes are found to have almost equal contributions to *χ*
^{(3)}(*ω*; *ω*, *ω*,-*ω*) as the single-photon processes, which result in the predicted enhancement of the positive nonlinear (Kerr) refractive index, and thus may enhance mode-locking of quantum-cascade lasers. Moreover, we also demonstrate an isospectral optimization strategy for further improving *n*
_{2} through the band-structure design, in order to boost the multimode performance of quantum-cascade lasers. Simulation results show that the optimized stepwise multiple-quantum-well structure has *n*
_{2}≈10^{-8} cm^{2}/W, a twofold enhancement over the original flat quantum-well structure. This leads to a refractive-index change Δ*n* of about 0.01, which is at the upper bound of those reported for typical Kerr medium. This stronger Kerr refractive index may be important for quantum-cascade lasers ultimately to demonstrate self-mode-locking.

© 2008 Optical Society of America

## 1. Introduction

Quantum-cascade lasers (QCLs) designed with harmonic levels are attractive for multiple-color light generation, due to the potentially large optical nonlinearities associated with intersubband transitions in these structures. Until now, most studies of the nonlinearities of QCLs focus on second- and third-harmonic generation through multi-harmonic levels as determined by the nonlinear optical susceptibilities *χ*
^{(2)} and *χ*
^{(3)} [1–5]. However, in addition to harmonic generation, the multi-resonance subband structure also affects other nonlinear performance resulting from multi-photon transitions. In this work, we investigate the Kerr nonlinearity, i.e., the intensity-dependent refractive index, in a QCL structure with multi-resonant levels.

The Kerr nonlinearity is of interest in QCLs with regard to the ultrafast properties of the gain medium [6] as well as the nature of the lasing mode itself. The laser-radiation pulse, propagating through the Kerr nonlinear medium with positive *n*
_{2}, experiences greater refractive index in central part of the beam, where the intensity is highest, in both transverse and longitudinal directions. In the transverse direction, the higher refractive index increases the beam confinement in the central part and thus decreases the waveguide loss. Combined with the ultrafast carrier relaxations in QCLs, the net result is a fast saturable-absorber mechanism [7], which may contribute ultimately to the emission of self-mode-locked ultra-short pulses [8]. When the intracavity intensity is well below saturation value, the fast saturable absorber can open and close the net gain window generated by the pulse immediately before and after the pulse. The strength of the fast absorber is proportional to *n*
_{2} [8, 9]. Along the pulse-propagation direction, the pulse undergoes self-phase-modulation (SPM) due to the additional phase shift induced by nonlinear refractive index. It is known in other types of lasers that SPM can broaden the frequency spectrum and concurrently shorten the pulse duration [10]. The pulse duration is inversely proportional to *n*
_{2}. [10]. While the saturable absorber can stabilize the mode-locked pulses, the SPM can shape the pulse as chirped and compress the pulse even more. Thus, analysis of both transverse and longitudinal Kerr effects suggest that pulse shortening through the Kerr medium will be enhanced by increasing *n*
_{2}. Self-pulsation of mid-infrared QCLs had been reported in [11]. This behavior is attributed [12] to the coherent Risken-Nummedal-Graham-Haken (RNGH) [13, 14] like instability rather than the Kerr effect due to QCLs shorter upper state lifetime compared to photon lifetime [15]. More recently [16], study on multimode operation regimes in QCLs shows that the fast gain recovery of QCLs promotes two multimode regimes, i.e., the spatial hole burning (SHB) and the RNGH instability. However, the Kerr effect may still play an important role in enhancing short-pulse generation in QCLs. There are several reasons to think that this may be so: one is that the saturable absorber affects the single-mode instability as to lower the RNGH instability threshold [12] as well as to exhibit a more pronounced Rabi splitting [16]; second, the photon lifetime can be tailored by the laser cavity length; and third, the Kerr-lensing effect can effectively sustain the short-pulse generation after it is initiated [7]; and also, in case that the laser dephasing rate is much faster than the Rabi oscillation, coherent emission in the lasing cavity will be destroyed and the self-pulsating behavior might be dominated by conventional saturable-absorber mechanism.

We take the active region of the GaInAs/AlInAs mid-infrared QCL structure in [1] as an example to illustrate our analysis. We find that in a suitably designed structure, multi-photon processes may lead to a twofold enhancement of the Kerr nonlinearity, with a predicted total refractive index change Δ*n* up to 0.005. To further explore the mechanisms of that may underlie short-pulse generation and perhaps self-mode-locking (SML) in QCLs, it is desirable to design a QCL structure with a higher Kerr nonlinearity, an issue that has not been addressed in the literature. In the later part of this paper, we demonstrate a systematic optimization method to further increase *n*
_{2} through tailoring the quantum-well structure composing the active region of the QCL. Simulation results show that the optimization brings additional twofold increase on the Kerr refractive index, which makes the nonlinear refractive index change higher than other QCLs studied for SML. The optimized structure is a candidate for future study on resolving the interaction between the two kinds of single mode instabilities resulting in short-pulse generation in QCLs.

## 2. Enhancement of the Kerr nonlinearity due to multi-resonance

The current study is based on the structure presented in [1], which is a lattice-matched GaInAs-AlInAs QCL designed for enhanced intracavity SHG. Our analysis of the Kerr nonlinearity starts with a model of this structure developed in our previous work [5]. In the model, carrier transport and power output of the structure are analyzed by self-consistently solving rate equations for carriers and photons. The model accounts for all active portions of the QCL containing an active region, an injector and a collector, which is equal to 1.5 periods of the full cascade. Boundary conditions are applied based on the periodicity of the QCL structure. Various levels relevant to lasing, nonlinear optical processes, as well as reservoir and sink levels for the carriers are included. Carrier transports both within and between cascades are accounted for. The lasing output power and nonlinear responses are evaluated through steady-state electron and photon populations. The coherent response is left out in the model due to the weak coherence resulted from much faster dephasing rate than the Rabi frequency.

Specifically, the band structure, as shown in Fig. 1, retains 15 energy levels, with 5 energy levels within each region. The energy states in the active region, where linear and nonlinear radiative transitions take place, are identified as *E*
_{1}, *E*
_{2}, *E*
_{3}, *E*
_{4}, and *E*
_{5}. The lasing transition takes place between levels *E _{3}*-

*E*

_{2}. The band-structure is originally designed for enhanced resonant SHG, thus the energy separations of

*E*

_{3}-

*E*

_{4}and

*E*

_{4}-

*E*

_{5}are in resonance with the lasing energy. This band structure is solved by the finite difference method with band nonparabolicity included. The subband energies and dipole matrix elements (DMEs) obtained in our model agree well with those listed in [1].

The Kerr nonlinearity describes a nonlinear contribution to the refractive index (Kerr effect), i.e., the refractive index is *n*=*n _{o}*+

*n*[17], with

_{2}I*n*the linear refractive index,

_{o}*n*

_{2}the nonlinear refractive index,

and *I*=*n _{o}*

*cε*

_{o}E^{2}/2 the light intensity.

Following Eq. (1), *n*
_{2} can be obtained from the third-order optical susceptibility *χ*
^{(3)}(*ω*; *ω*, *ω*,-*ω*), which itself can be evaluated from the steady-state rate equations for the carrier populations in the various subbands by the model demonstrated in [5]. According to the formulations in [17], two kinds of resonant terms contribute to *χ*
^{(3)}(*ω*; *ω*, *ω*,-*ω*), i.e., terms related to single-photon processes as shown in Fig. 2(b) and those related to two-photon process as in Fig. 2(c),

Single-photon transitions exist between any two states whose separation is resonant with the fundamental frequency *ω*, i.e. states *E*
_{2}-*E*
_{3}, *E*
_{3}-*E*
_{4}, and *E*
_{4}-*E*
_{5}. In any lasing structure there are sequential single-photon processes contributing to *χ*
^{(3)}(*ω*; *ω*, *ω*,-*ω*), which lead to absorption saturation. The explicit expression of *χ*
^{(3)}
_{1p} (*ω*; *ω*, *ω*, -*ω*) obtained by summing up all the contributions from the single-photon processes is

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}+\left({N}_{4}-{N}_{3}\right)\frac{{M}_{43}^{4}}{{\left({E}_{43}-\u045b\omega -i{\gamma}_{43}\right)}^{2}\left({E}_{43}-\u045b\omega +i{\gamma}_{43}\right)}$$

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}+\left({N}_{5}-{N}_{4}\right)\frac{{M}_{54}^{4}}{{\left({E}_{54}-\u045b\omega -i{\gamma}_{54}\right)}^{2}\left({E}_{54}-\u045b\omega +i{\gamma}_{54}\right)}],$$

from which we can see that *χ*
^{(3)}
_{1p} (*ω*; *ω*, *ω*, -*ω*) depends on the population inversion (*N _{i}*-

*N*) between resonant levels, the dipole matrix elements (DMEs)

_{j}*M*, and energy broadening terms |

_{ij}*E*-

_{ij}*ħω*-

*iγ*|. Due to the multiple-resonance nature of the subbands, additional contributions to

_{ij}*χ*

^{(3)}(

*ω*;

*ω*,

*ω*,-

*ω*) from two-photon processes also exist for resonant cascades

*E*

_{2}-

*E*

_{3}-

*E*

_{4}and

*E*

_{3}-

*E*

_{4}-

*E*

_{5}, where two photons with frequency

*ω*are absorbed simultaneously or sequentially and stimulate the upward electronic transitions across two consecutive resonant levels

*E*

_{2}-

*E*

_{4}or

*E*

_{3}-

*E*

_{5}. The expression for resonant contributions due to these two-photon processes is [17]

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}+{N}_{3}\frac{{M}_{34}^{2}{M}_{45}^{2}}{{\left({E}_{43}-\u045b\omega -i{\gamma}_{43}\right)}^{2}\left({E}_{53}-2\u045b\omega +i{\gamma}_{53}\right)}].$$

In Eqs. (3) and (4), the steady-state populations *N _{i}* (

*i*=1,…, 5) at the subbands in the active region are solved from the rate-equation model [5]. Our calculations show that

*n*

_{2}from the sequential single-photon processes is ~2.6×10

^{-9}cm

^{2}/W and that from two-photon processes is ~2.4×10

^{-9}cm

^{2}/W. The two kinds of contributions are both positive and of comparable magnitude, which means that the additional harmonic resonant levels in the lasing active region significantly enhance (actually double)

*n*

_{2}. The maximum optical intensity in the fundamental frequency is about 1 MW/cm

^{2}. This will result in a total refractive index change Δ

*n*up to 0.005, which is within the range of 1×10

^{-2}to 1×10

^{-4}for a typical Kerr medium [18].

As mentioned above, it may be expected that QCLs with enhanced *n*
_{2} may exhibit some tendency toward conventional SML due to the Kerr lensing mechanism and SPM. Therefore, a QCL structure with a significantly enhanced Kerr nonlinearity merits further study of the interactions between the RNGH instability and Kerr nonlinearities. In the next section, we demonstrate an optimization strategy to further increase the nonlinear refractive index *n*
_{2} of the structure shown in Fig. 1.

## 3. Supersymmetric optimization of the Kerr refractive index

The optimization strategy for *n*
_{2} can be determined by first observing the expressions of *χ*
^{(3)}(*ω*; *ω*, *ω*,-*ω*). As seen from Eqs. (3) and (4), both the single- and two-photon contributions to *χ*
^{(3)} (*ω*; *ω*, *ω*,-*ω*) are proportional to the population distribution *N _{i}* at each energy state and the DMEs

*M*, while

_{ij}*N*depends on both the band-structure and on the doping, but

_{i}*M*relates only to the band-structure. Via band-gap engineering, the band diagram of the QCLs can be flexibly tailored by a judicious choice of the geometries and compositions of the constituent quantum wells. Overall, optimization of

_{ij}*χ*

^{(3)}(

*ω*;

*ω*,

*ω*,-

*ω*) through optimizing the DMEs is preferred. To carry out an effective optimization procedure, the DME product chosen for optimization should be associated to the most dominant terms in Eqs. (3) and (4), respectively. In Eq. (3), the broadening between subbands 2 and 3 has the smallest magnitude |

*E*

_{23}-

*ħω*-

*iγ*

_{23}| and there is

*N*

_{3}-

*N*

_{2}≈

*N*

_{3}-

*N*

_{4}≈

*N*

_{3}-

*N*

_{5}for steady-state populations, so the first term associated with

*M*

^{4}

_{23}is dominant. In Eq. (4), the much larger population

*N*

_{3}(

*i.e.*,

*N*

_{3}≫

*N*

_{2},

*N*

_{4},

*N*

_{5}) in the second term suppresses the smaller broadening magnitude |

*E*

_{23}-

*ħω*-

*iγ*

_{23}| in the first term, so the second term associated with

*M*

^{2}

_{34}

*M*

^{2}

_{45}dominates. In order to increase these DME products and at the same time to maintain the energy-level positions to preserve the important resonances and, further, to facilitate carrier relaxation and period-to-period tunneling, the optimization method we adopted here is the supersymmetric quantum mechanics (SUSYQM) technique. In this technique, a family of isospectral potentials is generated depending on a single parameter

*λ*; the

*λ*value corresponds to the maximum DME which then determines the optimized potential shape in the active region. We used this technique in the optimization of second-harmonic generation in mid-infrared QCLs [4].

Details of the SUSYQM optimization procedure are described in [4]. The enhancement of DME products *M*
^{2}
_{23} and *M*
_{34}
*M*
_{45} with dependence on *λ* are shown in Fig. 3. In Fig. 3(a), the optimization base-function is the wavefunction of the second subband level, i.e., *θ*(*z*)=*ψ*
^{(2)}
_{0}(*z*), while *θ*(*z*)=*ψ*
^{(4)}
_{0}(*z*) in Fig. 3(b). As seen in Fig. 3(a)
*M*
^{2}
_{23} can be enhanced by up to 1.7 times while *M*
_{34}
*M*
_{45} remains unchanged; in Fig. 3(b), *M*
_{34}
*M*
_{45} increased 1.35 times and *M*
^{2}
_{23} is unchanged. Due to the fact that *N*
_{3}≫*N*
_{2} at steady-state for lasing QCLs, the enhancement in Fig. 3(a) is stronger than that shown in Fig. 3(b). With *λ _{opt}*=0.22, the optimized potential can then be evaluated through

${V}_{\mathrm{opt}}\left(z\right)={V}_{o}\left(z\right)-\frac{{\u045b}^{2}}{\sqrt{m\left(z\right)}}\frac{d}{dz}\left[\frac{1}{\sqrt{m\left(z\right)}}\frac{d}{dz}\left\{\mathrm{ln}\left[{\lambda}_{\mathrm{opt}}+{\int}_{0}^{z}dt{\theta}^{2}\left(t\right)\right]\right\}\right],$

where *V*
_{o}(*z*) is the original potential shape of the active region.

The nonlinear transitions exist only in the active region, so the band structure modification by SUSYQM mainly lies in the active region. The optimized potential shape yielded by SUSYQM in comparison with the original potential shape in the active region is shown in Fig. 4. The curved potential shape can be obtained by modifying the mole fractions of the ternary alloys, i.e., *E _{g}*=324+700

*x*+400

*x*

^{2}(meV) for Ga

_{x}In

_{1-x}As and

*Eg*=357+2290

*x*(meV) for Al

_{x}In

_{1-x}As [19]. The ideal smoothness of the potential shape can be obtained by continuous grading, which will make fabrication difficult. Digital alloy growth may be a favorable alternative. The digitalization period (DP) is the thickness of each growth layer with the same composition. Figure 4 shows the digitalized potential shape of the optimized structure with one monolayer DP. Based on the spectrum-preserving property of the SUSYQM procedure, the subband energies of the ideal-optimized structure is the same as that of the original structure; however, the digitalization will cause some undesirable small shifts of the subbands. Our simulation results show that with monolayer DP, the energy levels in the digitalized structure only have maximum 5 meV deviations from the original structure, which have only minor influence on the detuning factors in the evaluation of

*n*

_{2}due to the short dephasing times. In the original structure, the Ga

_{0.47}In

_{0.53}As/Al

_{0.48}In

_{0.52}As material system is lattice-matched to the InP substrate. In the optimized structure, due to the change of mole-fractions, there will be strain generated between layers, while our simulation results show that the total strain for one module of the QCL structure is compensated. The strain-induced deformation potential is also included in the model of the optimized structure. The enhancement of

*z*

^{2}

_{23}in the digitalized optimal structure is about 1.74, while that for the ideally smooth structure is 1.67. According to Eq. (3)

*χ*

^{(3)}

_{1p}(

*ω*;

*ω*,

*ω*, -

*ω*) can increase 3.3 times. Since

*χ*

^{(3)}

_{1p}(

*ω*;

*ω*,

*ω*, -

*ω*) and

*χ*

^{(3)}

_{2p}(

*ω*;

*ω*,

*ω*-

*ω*) have almost equal contributions to

*n*

_{2}in the original structure, the overall enhancement to

*n*

_{2}due to the SUSYQM optimization is a factor of ~2, with a value of 10

^{-8}cm

^{2}/W. This will result in a refractive index change of 0.01, which is much higher than those QCLs studied for spectrum broadening or multimode dynamics [11, 12].

## 4. Conclusion

In conclusion, we analyzed the Kerr nonlinearity in QCLs with multiple harmonic resonance in the active region. The study is based on a self-consistent rate-equation model for QCLs, in which all the carrier scattering mechanisms are accounted. Coherence response is neglected due to fast carrier dephasing rate.

Simulation results show that additional contribution to the Kerr nonlinearity from resonant two-photon processes is comparable to that from the single-photon processes. However, in order to investigate SML mechanism of QCLs with high Kerr nonlinearity, further increase of *n*
_{2} is necessary. We therefore illustrate an optimization strategy based on SUSYQM. The resultant stepwise QW structure is expected to have two-fold enhancement of *n*
_{2} compared with the original flat quantum-well structure, which will bring a refractive index change of 0.01. This is much higher than that of the typical QCL structures available to date. The optimized structure may be used for continuing study on the relationship between Kerr nonlinearity and RNGH instability, while both lead to multimode performance of QCLs. Higher values of *n*
_{2} may be possible by removing some constraints inherent in SUSYQM by using other optimization procedures. The optimized structure, while incorporating strain, is strain compensated over an entire period of the QCL. Finally, the optimized structure can be fabricated by the digital alloy technique with one-monolayer DP while substantially preserving the predicted enhancement. It has to be pointed out that the optimized structure has the drawback of fabrication complexity. The SUYQM optimization will work better for longer wavelength QCLs due to the easier digital alloy growth for wider QW layers. However, the optimized structure provides a lasing medium with unique combination of ultrafast carrier dynamics and high Kerr nonlinearity, which is desired for analyzing the interplay between different SML mechanisms in QCLs as well as their dominance in pulse initiation and stabilization. Experimental demonstration of SML in some lasers has been disappointing due to the fact that one mechanism is strong enough to sustain mode-locking but is typically unable to initiate it. Based on the optimized structure presented in the current work, we will further investigate the pulse formation and generation through a QCL medium with high Kerr nonlinearity by including the SPM and group velocity dispersion (GVD) effects.

## Acknowledgment

This work was supported by the start-up research fund from the University of Minnesota Duluth and the National Science Foundation by grant ECCS 0523923. David S. Citrin would also like to acknowledge the support of the CNRS, France.

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