ARTICLE IN PRESS

Physica E 41 (2009) 843–847

Contents lists available at ScienceDirect

Physica E

1386-94

doi:10.1

� Tel.:

E-m

journal homepage: www.elsevier.com/locate/physe

A generalized modeling strategy of quantum-cascade structures withresonance-enhanced harmonic oscillations

Jing Bai �

Department of Electrical and Computer Engineering, University of Minnesota, 1023 University Drive, Duluth, MN 55812, USA

a r t i c l e i n f o

Article history:

Received 14 November 2008

Accepted 9 January 2009Available online 19 January 2009

PACS:

42.50.Ct

42.55.Px

42.65.Ky

Keywords:

Quantum-cascade lasers

Nonlinear effects

Second-harmonic generation

Third-harmonic generation

77/$ - see front matter & 2009 Elsevier B.V. A

016/j.physe.2009.01.001

+1218 726 8606; fax: +1218 726 7267.

ail address: jingbai@d.umn.edu

a b s t r a c t

In this work, the modeling strategy of quantum-cascade lasers is generalized to including the harmonic-

resonance-enhanced nonlinear oscillations up to the third order. In addition to the lasing power at

fundamental frequency, the second-harmonic and third-harmonic output power can all be evaluated

through the model. The model is built on the rate-equations for subband electrons and photons, in

connection with Maxwell wave equations for propagation coupling between different emission modes.

Various radiative and nonradiative carrier scattering mechanisms are accounted in the model. In

addition to single-photon processes, multi-photon processes in the nonlinear cascades are also

accounted. The model is based on a full cascade structure containing the injector, active region, and

collector in order to account for the nonideal injection efficiency between periodic stages of quantum-

cascade lasers. The simulation results on two fabricated quantum-cascade structures capable of

nonlinearly harmonic generations agree well with the experiment measurements. The model can be

used to evaluate and further optimize the nonlinear performances of quantum-cascade lasers.

& 2009 Elsevier B.V. All rights reserved.

1. Introduction

The considerable optical nonlinearities associated with inter-subband transitions in quantum-cascade lasers (QCLs) can beutilized to design QCLs as strongly nonlinear oscillators withsecond-harmonic generation (SHG) or third-harmonic generation(THG) in the mid-infrared (MIR) or far-infrared (FIR) regions of thespectrum using the monolithic integration approach [1–5],providing for the unlike re-absorption due to that the emittedSHG or THG photons are well below the bandgap of theconstituent materials. These aspects make higher order harmonicgenerations in QCLs a very attractive way to access the MIR(3–5mm) where both conventional laser diodes and QCL’s havehad limited success [6]. This also opens the possibility of compactmulticolor MIR sources, which are of intense interest for chemicalsensing. Rapid experimental development has stimulated thenecessity of an explicitly theoretical description which canconnect all contributing factors to the nonlinear power outputof QCLs in order to further improve SHG or THG capabilities ofQCL’s. In addition, various effects brought by the featured designof nonlinear region should also be accounted in the modelingscheme for QCLs. Here a generalized simulation approach for QCLsis developed from the previous rate-equation model for QCLslasing without nonlinear oscillations [7–9].

ll rights reserved.

The self-consistent rate-equation approach is a semi-classicalor Boltzmann-like treatment, in which the entire multiplequantum well (QW) is treated as a single quantum mechanicalsystem with a well-defined Hamiltonian. This approach has beensuccessfully applied to both THz and MIR QCLs without SHG orTHG [7–9]. All the subband energy levels are eigenstates (whichare stationary by definition) of this Hamiltonian. The transportprocess is the collective effect of intersubband scattering betweenthe various subbands (eigenstates) involved, and can be calculatedusing Fermi’s golden rule. The electrons in each subband maintaina Fermi-Dirac distribution due to rapid scattering [10]. However,there are some limitations in the existing modeling strategy: (1)only the linear response, i.e., single-photon processes, is included;(2) the rate-equations are only built for subband electronpopulations, thus the power output cannot be evaluated fromthe model; (3) the wave mode propagation coupling and phaseinformation are not reflected in the model. In the current work, animproved model for the QCL structure is developed with allabove-mentioned limitations removed. The model can be used toevaluate both linear and nonlinear performances of QCLs. Inaddition to linear optical transitions, nonlinear optical transitions,such as multi-photon processes brought by the resonant SHG orTHG cascades are built into the model.

The paper is organized as follows. In Section 2, the rate-equation formulation with built-in nonlinear optical transitions ispresented. Section 3 describes the derivation of linear andnonlinear powers based on the self-consistent solution of rateequations and Maxwell wave equations, while Section 4 presents

ARTICLE IN PRESS

J. Bai / Physica E 41 (2009) 843–847844

some simulation results in comparison with the experimentalmeasurements. The conclusions are given in Section 5.

2. Formulation of rate-equations with nonlinear opticaltransitions

A fabricated QCL structure capable of THG [1] is taken as anexample to illustrate the generalized rate-equation approach. Theband structure for 1.5 periods is shown in Fig. 1, which includesthe 13 subband levels accounting for the full injector/active-region/collector cascade under a bias of 42 kV/cm. Both bandnonparabolicity and position-dependent effective electron massare accounted in the band structure calculation. The states in theactive region, i.e., states 2, 5, 7, 11, and 13, are indicated in blue inFig. 1. The injector region includes states 1, 3, 4, and 6 and activeregion includes states 8, 9, 10, and 12. States 5, 7, 11, and 13 areequally spaced with energy intervals resonant with the lasingenergy E75, which means a triply harmonic-resonance cascade. Inaddition to single-photon absorption and emission, this resonancecascade enables two/three-photon transitions both sequentiallyand simultaneously. The two-photon transitions take place in SHGcascades 5-7-11 and 7-11-13, while three-photon transitions incascade 5-7-11-13. Due to the reduced population inversionbetween lasing levels 5-7, the lasing performances might bedegraded by some of these multi-photon processes, such asabsorption between states 7-13, emission between 11-5, andemission between 13-5. The intriguing single/two/three-photonprocesses, as well as their interplay with the time-varying photondensity are all incorporated into the rate-equation model in orderto investigate the laser output. The coherent response is left out ofthe model due to the weak coherence resulted from much fasterdephasing than the Rabi oscillations [11]. The single-photonstimulated emission rate is expressed as [12]

Wpnm ¼

e2z2nmo

2�gnm

ðEnm � _oÞ2 þ ðgnm=2Þ2mo, (1)

where mo is the photon density, znm is the dipole matrix element(DME) between levels n and m, gnm is the FWHM of the n-m

transition, Enm is the energy difference between levels n and m, e isthe permittivity of the lasing medium, and o is the incidentphoton frequency. Unlike the single-photon transition rate, whichis linear with the intracavity photon density (light intensity), thetwo-photon stimulated emission/absorption rate is proportional

900

800

700

600

500

400

300

200

100

00 10 20 30

Quantum well gr

Ene

rgy

leve

ls (

meV

)

F = 42 kV/cm

Fig. 1. Band structure for the 1.5 perio

to the squared photon density. With the two-photon transitioncascade g-m-n, the two-photon stimulated emission/absorptionrate is given by [12]

W2pngðoÞ ¼

e4z2nmz2

mg

4_�2

_oEmg � _o

� �2 gng

ðEng � 2_oÞ2 þ ðgng=2Þ2m2

o. (2)

Similarly, the three-photon absorption and stimulated emissionprocesses also exist for triply harmonic levels g-m-n-o and thescattering rate is proportional to the cubic photon density as

W3pog ðoÞ ¼

q6z2ogz2

nmz2mg

4_�3r �

30

ð_oÞ3

ðEmg � _oÞ2ðEng � 2_oÞ2

�ðgog=2Þ

ðEog � 3_oÞ2 þ ðgog=2Þ2m3

o. (3)

Eqs. (1)–(3) describe the stimulated emission/absorption rates.In the model, it is assumed that there is no photon initially in thecavity and first photon is generated by single-photon spontaneousemission with the rate expressed as

Wspnm ¼

e2z2nmo�V

gnm=2

ðEnm � _oÞ2 þ ðgnm=2Þ2, (4)

where V is the volume for one period of the QCL structure. Thetwo-photon/three-photon spontaneous emission is ignored due tothat the transition rate is much smaller than that of the single-photon. In addition to the radiative transitions in the active region,the carrier transport within each stage and between adjacentstages are dominated by the electron–electron scattering andelectron–longitudinal (LO) phonon scattering, which are studiedin detail in Ref. [13].

The periodic boundary conditions are introduced using theapproach presented in Ref. [8]. The subband populations in theinjector region are equivalent to those in the collector. Fortransition between any two levels of the nonlinear cascade 5-7-11-13 in the active region, the total scattering rate includes notonly the nonradiative ones, but also the radiative single/two/three-photon transitions, which are linearly/quadratically/cubi-cally dependent on the incident photon intensity

Wij ¼WLOij þWe�e

ij þWp;2p;3pij þWsp

ij . (5)

For transitions between any other subbands, the transition rateis the sum of the electron-electron and electron-LO phononscattering rates and thus independent of the photon density in thecavity.

40 50 60 70

owth axis z (nm)

E13

E11

E7

E5E2

ds of the QCL structure in Ref. [1].

ARTICLE IN PRESS

J. Bai / Physica E 41 (2009) 843–847 845

As an example for the subband populations in the injector/collector region, the rate equation for subband 1, which isequivalent to subband 8 in the collector, is written as

dn1

dt¼X

j

ðWj1nj þWj8nj �W1jn1 �W8jn8Þ þX

k

ðWk1nk �W1kn1Þ,

(6)

where Wpq represents the total scattering rate between subbandsp, and q. In the above expression, indices j ¼ 2,5,7,11,13 andk ¼ 3,4,6. The rate equation is similar for any other injector orcollector state. The rate equation for the subband population inthe active region is given by

dnj

dt¼X13

i¼1;iaj

ðWijni �WjinjÞ, (7)

where j ¼ 2,5,7,11,13.In addition to the rate equation for the subband population,

there is one another rate equation required for the density ofphotons resulting from both the single- and multi-photonprocesses, viz.

dmo

dt¼ G½Wsp

75n7 þWp75ðn7 � n5Þ þWsp

ð11Þ7n11 þWpð11Þ7ðnð11Þ � n7Þ

þWspð13Þð11Þn13 þWp

ð13Þð11Þðnð13Þ � nð11ÞÞ�

þ 2G½W2pð13Þ7ðn13 � n7Þ þW2p

ð11Þ5ðn11 � n5Þ�

þ 3G½W3pð13Þ5ðn13 � n5Þ� �

mo

top, (8)

where G is the mode confinement factor, and tpo is the photon

lifetime related to the total loss as tp ¼ (vgao)�1 [14].The set of 14 rate equations involves electron populations in 13

subbands together with the photon density. As shown in Eq. (8),the photon-density variation depends on the transient scatteringrates and subband electron populations in the active region, whilethe radiative rates are also linearly or nonlinearly dependent onthe photon density in the cavity. Apart from this, in each subband,there is also the interplay between the electron population andscattering rates, which are connected by the quasi-Fermidistribution functions. The steady-state electron populations andphoton density are obtained by iteratively solving the entire set of14 rate equations. The current density can be evaluated from thesteady-state subband electron populations and scattering rates asillustrated in Ref. [8].

The nonlinear susceptibilities can be evaluated based on thesteady-state electron populations, i.e.,

wð2Þð2oÞ � 2pe3

�0

z75zð11Þ7zð13Þð11Þ

Eð11Þ5 � 2_o� igð11Þ5

n7 � n11

Eð11Þ7 � _o� igð11Þ7

"

þn7 � n5

E75 � _o� ig75

�þ

zð11Þ7zð13Þð11Þzð13Þ7

Eð13Þ7 � 2_o� igð13Þ7

�n11 � n13

Eð13Þð11Þ � _o� igð13Þð11Þ

þn11 � n7

Eð11Þ7 � _o� igð11Þ7

!#, (9)

and

wð3Þ2p ðo;o;o;�oÞ ¼4pe4

�0

z75zð11Þ7zð13Þð11Þzð13Þ5

Eð13Þ5 � 3_o� igð13Þ5

1

Eð11Þ5 � 2_o� jgð11Þ5

�

�n7 � n11

Eð11Þ7 � _o� jgð11Þ7þ

n7 � n5

E75 � _o� jg75

� �

�1

Eð13Þ7 � 2_o� jgð13Þ7

n11 � n13

Eð13Þð11Þ � _o� jgð13Þð11Þ

"

�n11 � n7

Eð11Þ7 � _o� jgð11Þ7

�. (10)

The multi-resonance band structure leads to the minimumfrequency detuning from the resonance and substantially in-creases those nonlinear susceptibilities. In the next section, thelinear and nonlinear power evaluation based on the steady-statesolution of rate-equations is introduced.

3. Evaluation of linear and nonlinear output power

The linear output power can be calculated from the photondensity mo in the cavity as

Po ¼ ANmodð_oÞmoc=no, (11)

where A is the cross-section area transverse to the lightpropagation direction, Nmod is the number of periods in the lasingcavity, and the factor c/no is the speed of light in the lasing cavity.The SHG and THG powers can be derived from the propagationcoupling with the fundamental wave (FW). Only TM modes areexcited efficiently since the polarization associated with electro-nic intersubband transitions contains only the x component. Thewave propagates along the waveguide in the z direction. In orderto simplify the solution procedure, it is assumed that thetransverse wave profile is only x dependent. Here only thederivation procedure for THG power is presented, and that forSHG can be obtained in a similar manner. The magnetic fieldsHy,o(3o) of the FW and THG modes can be represented asHy;oð3oÞ ðx; zÞ ¼ Aoð3oÞðzÞFoð3oÞ ðxÞe

�ikoð3oÞz. Ao,3o(z) is the magnetic-field amplitude varying along the waveguide direction, with theassumption that Ao(z) varies slowly with the coordinate z andA3o(0) ¼ 0. Fo,3o(x) represents the mode profile in the transversedirection and satisfies the Helmholtz equation

q2Foð3oÞðxÞ

qx2þ

o2�oð3oÞðxÞc2

� k2oð3oÞ

� �Foð3oÞðxÞ ¼ 0, (12)

where eo(3o) is the frequency- and position-dependent dielectricconstant at the fundamental and third-harmonic frequencies. Theequations are solved by the finite-difference approach, in whichdifferent TM modes at both frequencies are obtained with phaseconstants ko(3o) obtained as eigenvalues and the transverse modeprofiles Fo(3o)(x) as eigenvectors. The linear and nonlinear poweroutputs relate to the magnetic-field amplitude as

Poð3oÞðzÞ ¼noð3oÞW

2�0cjAoð3oÞðzÞj

2

ZF2

y;oð3oÞðxÞ

�oð3oÞðxÞdx, (13)

where W is the width of the waveguide. The transversecomponent of electric field Ex is related to Hy as

qHyðx; z;oÞqz

¼ io�ðx;oÞExðx; z;oÞ. (14)

Combining Eqs. (13) and (14), the magnitude of the third-harmonic magnetic field at output can be derived as

A3oðLÞ ¼

3on3oA3

o4c3�2

0

eiðDkTH�ia3oÞDgL � 1

DkTH � ia3o

� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� R3

p1� R1

R 1

�3oðxÞ

wð3ÞðxÞF3y;oðxÞFy;3oðxÞdx

RF2

y;3oðxÞdx,

(15)

where no ¼ koc/o and n3o ¼ k3oc/(3o) are refractive indices ofthe FW and THG modes, a3o is the total loss including thewaveguide loss a3o

w and mirror loss a3om for the THG mode, and

DkTH ¼ 3ko�k3o is the phase constant mismatch. The propagationloss ao in the fundamental mode wave has been included in therate Eq. (8) through the photon lifetime. R1 and R3 are reflectioncoefficients at the fundamental and second-harmonic frequencies,respectively.

ARTICLE IN PRESS

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

00 2 4 6 8 10 12

QW growth direction x (µm)

Mag

netic

fie

ld H

y (a

.u.)

Mag

netic

fie

ld H

y (a

.u.)

Active region

TH TM00, n = 3.28

SH TM00, n = 3.34

FM TM00, n = 3.20

0.8

0.6

0.4

0.2

0

-0.2

-0.4

-0.6

Active region

TH TM00, n = 3.20

SH TM00, n = 3.23

FW TM00, n = 3.20

J. Bai / Physica E 41 (2009) 843–847846

From Eqs. (13) and (15), the nonlinear THG output power canbe obtained from the linear power as

P3o ¼9p2jwð3Þj2½e�2a3oL � 2e�a3oL cosðDk3HLÞ þ 1�ð1� R3Þ

I23n3

on3ol2c�2

0ðDk2þ a2

3oÞð1� R1Þ3

P3o, (16)

where l is the wavelength of the FW and I3 is the effectiveinteraction cross-section decided by the overlap between thefundamental and THG modes,

I3 ¼1

n3on3o

RF2

3oðxÞdxh i R 1

�oðxÞF2oðxÞdx

� �3=2

W

R 1

�3oðxÞðxÞF3

oðxÞF3oðxÞdx

�������� R 1

�3oðxÞF2

3oðxÞdx

� �1=2. (17)

Similarly, the SHG power can be derived as

P2o ¼2p2jwð2Þj2½e�2a2oL � 2e�a2oL cosðDkLÞ þ 1�ð1� R2Þ

I2n2on2ol

2c�0ðDk2þ a2

2oÞð1� R1Þ2

P2o, (18)

where I2 is the effective interaction area determined by theoverlap between the linear and third-harmonic waves

I2 ¼1

n4on2

2o

RF2

2oðxÞdx 2 R 1

�oðxÞF2oðxÞdx

� �2

W

R 1

�2oðxÞ

F2oðxÞF2oðxÞdx

��������

� �2 R 1

�2oðxÞF2

2oðxÞdx

. (19)

From Eqs. (16)–(19), it is seen that the THG and SHG powers areproportional to |w(3)(3o)|2 and |w(2)(2o)|2, respectively. In addi-tion, the mode-profile overlap and phase mismatch also playimportant roles.

-0.80 2 4 6 8 10 12

QW growth direction x (µm)

Fig. 3. Mode profiles for the structure in Ref. [1]: (a) the lowest order of

fundamental, second-, and third-harmonic modes; (b) second- and third-harmonic

modes with the best phase-matching to the linear mode.

4. Results

By applying the simulation procedure described in theprevious section, several straightforward calculations are per-formed regarding the nonlinear response of the existed QCLstructures with THG [1] and SHG [2]. Fig. 2 shows the simulationresult for the linear power output under the applied electric bias42 kV/cm for the structure in Ref. [1]. At steady-state, the linearoutput power at 11.1mm is about 0.2 W. For this structure, themaximum mode-profile overlap between the SHG/THG and theFW modes occurs for the TM00 modes at these three frequencies,whose profiles are shown in Fig. 3(a). From Eqs. (16) and (18), itcan be seen that the mode overlap is inversely proportional to theeffective interaction area. To achieve higher SHG/THG outputpower, the interaction area should be as small as possible. The

0.2

0.18

0.16

0.14

0.12

0.1

0.08

0.06

0.04

0.02

00 1 2 3 4 5 6 7 8 9 10 11 12

Time (ps)

Lin

ear

outp

ut p

ower

(W

)

Fig. 2. Simulation results for linear output power of the structure in Ref. [1].

effective interaction area for the FW and THG TM00 modes is114mm2, with a linear to THG (at 3.7mm) conversion efficiency,i.e., Z3o ¼ P3o/Po

3, estimated to be 40mW/W3. The linear to THGconversion efficiency obtained is close to the experimental resultreported in Ref. [1]. The effective area for the FW and SHG TM00

modes is 445mm2 and the linear to SHG (at 5.5mm) conversionefficiency Z2o ¼ P2o/Po

2 is about 100mW/W2. The TM00 modes atthe linear, second-, and third-harmonic frequencies can achievesubstantial mode overlap, but with large phase mismatch, asshown in Fig. 3(a). At the second/third-harmonic frequency, thephase mismatch DksH(TH) is about 100 times larger than the lossa2o(3o), the nonlinear power will be decreased to about 10�4 ofthe power under phase-matched conditions with the same as themode overlap. It is found that higher order SHG and THG modeshave better phase matching with the TM00 FW mode. As shown inFig. 3(b), the best phase-matching could be achieved betweenTM02 SHG /TM05 THG mode and the TM00 FW mode, but withsome sacrifice of mode profile overlap. The effective interactioncross-section between TM02 SHG and TM00 FW increases to0.15 mm2, and the Z2o drops to 6mW/W2; the effective cross-section between TM05 THG and TM00 FW increases to 154mm2,and the Z3o increases to 0.4 W/W2. It can be seen that under thebest phase-matching condition, for the SHG, the poor modeprofile overlap suppresses the reduced phase-mismatch factor,and finally leads to the one-order of decrease of the linear to SHGconversion efficiency Z2o; for the THG, the modal overlap onlydecreased slightly, while the much better phase-matching condi-tion is achieved, which results in 104 times improvement on theconversion efficiency Z3o. The model also applied to another

ARTICLE IN PRESS

J. Bai / Physica E 41 (2009) 843–847 847

experimental QCL structure capable of SHG [2]. For the TM00

modes at the three frequencies, the linear to SHG conversionefficiency Z2o is estimated to be 400mW/W2, while the linear toTHG conversion efficiency Z3o is 2.6mW/W3. The linear to SHGconversion efficiency obtained is close to the experimental resultreported in Ref. [2]. The deviation of results from this model andthe experimental results is likely in part due to the assumptionthat the nonlinear susceptibilities are constant within the activeregion. Comparing the nonlinear conversion efficiencies for theTM00 modes at the fundamental, SHG, and THG frequencies forthe two structures in Refs. [1] and [2], respectively, the structurein Ref. [1] has high conversion efficiencies for both SHG and THG,while the one in Ref. [2] has high SHG efficiency but low THGefficiency. The main reason for this is that the first structure hasmuch higher (almost double) DMEs z(13)5 and z(13)(11) in the activeregion than the corresponding DMEs in the second structure,which leads to higher value DME product for the third-ordernonlinear susceptibility |w(3)(3o)|.

5. Summary

A modeling strategy for the MIR QCL’s with SHG or THGcapability is presented. The model is based on rate-equations forsubband electron populations and the photon density in the lasingcavity. The model is built on a full cascade structure containinginjector, active region and collector. Nonunity pumping from oneperiod to next is taken into account by including all relevantelectron–electron and electron–LO phonon scatterings betweeninjector/collector and active regions. In addition to the single-photon radiative transitions, multi-photon transitions brought bythe nonlinear cascades are incorporated in the rate-equations.With the linear power evaluated from the steady-state solution ofphoton density in the cavity, the nonlinear SHG/THG power can beevaluated from its wave propagation coupling and model profile

overlap with the linear wave. The model is applied on twofabricated QCL structures designed with SHG and THG capabil-ities, respectively. Linear and nonlinear outputs evaluated fromthe model agree well with the measurement results reported. Themodel is also used to identify the higher order SHG/THG TMmodes with the best phase match to the lowest TM00 fundamentalwave. Even though the quasi-phase matching can greatly enhancethe nonlinear output, the poor mode profile overlap maysubstantially weaken such enhancement. The model presentedcan be used to estimate and further optimize the performance ofQCLs, especially those with enhanced nonlinearly optical interac-tions.

References

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