# Gain competition in dual wavelength quantum cascade lasers.

**ABSTRACT** We investigated dual wavelength mid-infrared quantum cascade lasers based on heterogeneous cascades. We found that due to gain competition laser action tends to start in higher order lateral modes. The mid-infrared mode with the lower threshold current reduces population inversion for the second laser with the higher threshold current due to stimulated emission. We developed a rate equation model to quantitatively describe mode interactions due to mutual gain depletion.

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**ABSTRACT:**We demonstrate high power, room temperature, single-mode THz emissions based on intracavity difference frequency generation from mid-infrared quantum cascade lasers. Dual active regions both featuring giant nonlinear susceptibilities are used to enhance the THz power and conversion efficiency. The THz frequency is lithographically tuned by integrated dual-period distributed feedback gratings with different grating periods. Single mode emissions from 3.3 to 4.6 THz with side-mode suppression ratio and output power up to 40 dB and 65 µW are obtained, with a narrow linewidth of 5 GHz.Optics Express 01/2013; 21(1):968-73. · 3.55 Impact Factor - SourceAvailable from: Augustinas Vizbaras
##### Article: Terahertz sources based on Čerenkov difference-frequency generation in quantum cascade lasers

Karun Vijayraghavan, Robert W. Adams, Augustinas Vizbaras, Min Jang, Christian Grasse, Gerhard Boehm, Markus C. Amann, Mikhail A. Belkin[Show abstract] [Hide abstract]

**ABSTRACT:**We report room-temperature terahertz sources based on Čerenkov difference-frequency generation in dual-wavelength mid-infrared quantum cascade lasers with giant resonant optical nonlinearities originating from intersubband transitions. A Čerenkov difference-frequency generation scheme allows for extraction of THz radiation along the whole length of the laser waveguide and provides directional terahertz emission. Experimentally, our sources demonstrate a conversion efficiency of up to 70 μW/W2 approximately an order of magnitude improvement over the previous reports.Applied Physics Letters 06/2012; 100(25). · 3.52 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Performance of a slot metal clad THz QCL waveguide to suppress higher order lateral modes is elucidated here. Results demonstrated here show that the proposed structure provides as high as 4-fold rise in gain threshold for the higher order modes.IEEE Journal of Selected Topics in Quantum Electronics 01/2013; 19(1). · 4.08 Impact Factor

Page 1

Gain competition in dual wavelength quantum

cascade lasers

Markus Geiser,1, 4 Christian Pflügl,1,* Alexey Belyanin,2 Qi Jie Wang,1 Nanfang Yu,1

Tadanaka Edamura,3 Masamichi Yamanishi,3 Hirofumi Kan,3 Milan Fischer,4 Andreas

Wittmann,4 Jérôme Faist,4 and Federico Capasso1,5

1School of Engineering and Applied Sciences, Harvard University, 9 Oxford Street, Cambridge, Massachusetts

02138, USA

2Department of Physics, Texas A&M University, College Station, Texas 77843, USA

3Central Research Laboratories, Hamamatsu Photonics K.K., Hirakuchi 5000, Hamakita-ku, Hamamatsu 434-8601,

Japan

4 Institute of Quantum Electronics, ETH Zürich, CH-8093 Zürich, Switzerland

5capasso@seas.harvard.edu

*pflugl@seas.harvard.edu

Abstract: We investigated dual wavelength mid-infrared quantum cascade

lasers based on heterogeneous cascades. We found that due to gain

competition laser action tends to start in higher order lateral modes. The

mid-infrared mode with the lower threshold current reduces population

inversion for the second laser with the higher threshold current due to

stimulated emission. We developed a rate equation model to quantitatively

describe mode interactions due to mutual gain depletion.

©2010 Optical Society of America

OCIS codes: (140.3070) Infrared and far-infrared lasers; (140.5965) Semiconductor lasers;

(230.5590) Quantum well devices.

References and links

1. J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science

264(5158), 553–556 (1994).

2. S. Y. Zhang, D. G. Revin, J. W. Cockburn, K. Kennedy, A. B. Krysa, and M. Hopkinson, “λ–3.1µm room

temperature InGaAs/AlAsSb/InP quantum cascade lasers,” Appl. Phys. Lett. 94(3), 031106 (2009).

3. M. Rochat, D. Hofstetter, M. Beck, and J. Faist, “Long-wavelength (λ~16 µm), room-temperature, single-

frequency quantum-cascade lasers based on a bound-to-continuum transition,” Appl. Phys. Lett. 79(26), 4271

(2001).

4. A. Lyakh, C. Pflügl, L. Diehl, Q. J. Wang, F. Capasso, X. J. Wang, J. Y. Fan, T. Tanbun-Ek, R. Maulini, A.

Tsekoun, R. Go, C. Kumar N. Patel, “1.6 W high wall plug efficiency, continuous-wave room temperature

quantum cascade laser emitting at 4.6 µm,” Appl. Phys. Lett. 92, 111110 (2008).

5. A. Lyakh, R. Maulini, A. Tsekoun, R. Go, C. Pflügl, L. Diehl, Q. J. Wang, F. Capasso, and C. K. N. Patel, “3

Watt continuous-wave room temperature single-facet emission from quantum cascade lasers based on non-

resonant extraction design approach,” Appl. Phys. Lett. 95(14), 141113 (2009).

6. R. Maulini, A. Lyakh, A. Tsekoun, R. Go, C. Pflügl, L. Diehl, F. Capasso, and C. K. N. Patel, “High power

thermoelectrically-cooled and uncooled quantum cascade lasers with optimized reflectivity facet coatings,” Appl.

Phys. Lett. 95(15), 151112 (2009).

7. Y. Bai, S. Slivken, S. R. Darvish, and M. Razeghi, “Room temperature continuous wave operation of quantum

cascade lasers with 12.5% wall plug efficiency,” Appl. Phys. Lett. 93(2), 021103 (2008).

8. C. Gmachl, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, F. Capasso, and A. Y. Cho, “Quantum cascade

lasers with a heterogeneous cascade: two-wavelength operation,” Appl. Phys. Lett. 79(5), 572 (2001).

9. R. Maulini, A. Mohan, M. Giovannini, J. Faist, and E. Gini, “External cavity quantum-cascade laser tunable from

8.2 to 10.4 µm using a gain element with a heterogeneous cascade,” Appl. Phys. Lett. 88(20), 201113 (2006).

10. A. Hugi, R. Terazzi, Y. Bonetti, A. Wittmann, M. Fischer, M. Beck, J. Faist, and E. Gini, “External cavity

quantum cascade laser tunable from 7.6 to 11.4 µm,” Appl. Phys. Lett. 95(6), 061103 (2009).

11. E. Rosencher, A. Fiore, B. Vinter, V. Berger, Ph. Bois, and J. Nagle, “Quantum engineering of optical

nonlinearities,” Science 271(5246), 168–173 (1996).

12. C. Sirtori, F. Capasso, J. Faist, L. N. Pfeiffer, and K. W. West, “Far-infrared generation by doubly resonant

difference frequency mixing in a coupled quantum well two-dimensional electron gas system,” Appl. Phys. Lett.

65(4), 445 (1994).

13. N. Owschimikow, C. Gmachl, A. Belyanin, V. Kocharovsky, D. L. Sivco, R. Colombelli, F. Capasso, and A. Y.

Cho, “Resonant second-order nonlinear optical processes in quantum cascade lasers,” Phys. Rev. Lett. 90(4),

043902 (2003).

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Received 19 Jan 2010; revised 25 Mar 2010; accepted 1 Apr 2010; published 27 Apr 2010

10 May 2010 / Vol. 18, No. 10 / OPTICS EXPRESS 9900

Page 2

14. M. Austerer, C. Pflügl, S. Golka, W. Schrenk, A. M. Andrews, T. Roch, and G. Strasser, “Coherent 5.35 µm

surface emission from a GaAs-based distributed feedback quantum-cascade laser,” Appl. Phys. Lett. 88(12),

121104 (2006).

15. M. A. Belkin, F. Capasso, A. Belyanin, D. L. Sivco, A. Y. Cho, D. C. Oakley, C. J. Vineis, and G. W. Turner,

“Terahertz quantum-cascade-laser source based on intracavity difference-frequency generation,” Nat. Photonics

1(5), 288–292 (2007).

16. M. A. Belkin, F. Capasso, F. Xie, A. Belyanin, M. Fischer, A. Wittmann, and J. Faist, “Room temperature

terahertz quantum cascade laser source based on intracavity difference-frequency generation,” Appl. Phys. Lett.

92(20), 201101 (2008).

17. C. Pflügl, M. A. Belkin, Q. J. Wang, M. Geiser, A. Belyanin, M. Fischer, A. Wittmann, J. Faist, and F. Capasso,

“Surface-emitting terahertz quantum cascade laser source based on intracavity difference-frequency generation,”

Appl. Phys. Lett. 93(16), 161110 (2008).

18. J. Faist, D. Hofstetter, M. Beck, T. Aellen, M. Rochat, and S. Blaser, “Bound-to-continuum and two-phonon

resonance quantum-cascade lasers for high duty cycle, high-temperature operation,” IEEE J. Quantum Electron.

38(6), 533–546 (2002).

19. A. Belyanin, M. Troccoli, and F. Capasso, “Raman Injection and Inversionless Intersubband Lasers” in

Intersubband Transitions in Quantum Structures, R. Paiella (McGraw-Hill Companies, 2006), chapter 6.3.5.

1. Introduction

Quantum cascade lasers [1] (QCLs) are semiconductor lasers based on intersubband

transitions between electronic states in coupled quantum wells (QWs). The transition

wavelength is a function of the QW and barrier widths in the active region and can be

designed over a broad spectral range for a given material. Mid-infrared (mid-IR) QCLs

working at room temperature (RT) have been demonstrated in the spectral range from about

3µm [2] to 15µm [3] and Watt-level output power in continuous wave operation was reported

[4–7]. A unique feature of these devices is the cascading scheme, in which electrons traverse a

stack of many, typically 30-50 active regions. Due to the flexibility in design, QCLs featuring

heterogeneous cascades can be realized, which lase in several wavelength bands

simultaneously [8]. Heterogeneous active regions are also employed in broadband tunable

lasers for spectroscopic applications [9,10]. Besides the advantage of delivering multiple

wavelengths simultaneously, these devices are also very interesting for nonlinear light

generation in QCLs. Giant nonlinearities in semiconductor QW structures have been predicted

[11] and experimentally demonstrated [12]. Integration of coupled QWs with a large nonlinear

susceptibility in the active region of QCLs has led to the demonstration of intracavity second

harmonic [13,14] and sum-frequency generation [13] in QCLs. Recently, difference frequency

generation (DFG) has been demonstrated in dual wavelength mid-IR QCLs [15,16]. Two mid-

IR pump beams sharing the same waveguide generate their difference-frequency in the THz

regime by means of intersubband nonlinearities integrated in the active regions of these

devices. For efficient DFG a large overlap of the two pump beams is desired, as DFG power is

proportional to the product of their intensities. In recent work, we found that the two pump

lasers tend to operate in different lateral modes, leading to reduced efficiency in nonlinear

light generation [17]. In this paper we study in detail the multimode behavior of dual

wavelength QCLs and describe the experimental effects by a rate equation model which

includes the interaction of the two lasers.

2. Active region and waveguide design

We investigated two slightly different dual wavelength mid-IR QCL designs. The active

region of the first structure (S1) consists of two stacks of stages with their band structures

shown in Fig. 1. The first stack is based on a double phonon resonance (DPR) design [18]

lasing at about 10.5µm (Fig. 1(a), main laser transition: green-red) and the second stack

features a bound-to-continuum (BTC) design [18] lasing at about 8.9µm (Fig. 1(b), main laser

transition: green-blue). As described in detail in [16], the BTC design also includes a large

nonlinear susceptibility χ(2) for efficient generation of the difference-frequency of the two

mid-IR lasers. To obtain a large value for χ(2), the lasing transition of the DPR laser is close to

resonance (difference is ~5 meV) with the transition 4 → 3 (see Fig. 1 (b) and (d) transition

green-red) in the BTC design.

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Figure 1(c) shows a schematic of the waveguide. 30 stages of each active region including

thin low-doped InGaAs layers surrounding the active regions were grown on a low doped

(~1*1017cm−3) InP wafer consecutively, followed by a 3.5µm thick cladding layer

(~5*1016cm−3) and a 200nm thick highly doped plasmon layer (~5*1018cm−3). This dielectric

waveguide confines both mid-IR modes in the vertical direction [16].

A second structure (S2) was grown with the same waveguide and a very similar active

region design. The only difference was that the layers in the active region were about 10%

thicker than the ones in the original wafer and the active region doping was lower. This led to

a different performance as discussed in detail in section 4.

Fig. 1. Band diagram for (a) the double phonon resonance and (b) the bound to continuum

active region of structure S1 calculated for an applied field of 34KV/cm and 41kV/cm,

respectively which are the injection resonance conditions for each active region. The lasing

transitions ((a) green-red, (b) green-blue) are shown with wavy arrows. The layer sequence in

nm, starting from the injection barrier, for the BTC structure is 4.0/ 2.4/ 0.7/ 6.5/ 0.8/ 6.4/ 0.8/

5.8/ 2.2/ 4.0/ 1.3/ 3.8/ 1.4/ 3.7/ 1.5/ 3.6/ 1.9/3.6 / 2.5/ 3.6/ 2.5/ 3.5. The layer sequence for one

period of a DPR section is 4.0/2.0/ 0.7/ 6.0/ 0.9/ 5.9/ 1.0/ 5.2/ 1.4/ 3.8/ 1.2/ 3.2/ 1.2/ 3.2/ 1.6/

3.1/ 1.9/ 3.1/2.2/ 3.0/ 2.2/ 2.9. The barriers are underlined. (c) Cross sectional view of the ridge

showing schematically the waveguide design. (d) Schematic energy diagram of the BTC active

region with states 4, 3, and 2. State 1 is the injector ground state. States 4, 3 and 2 are marked

in bold green, red and blue in (b), respectively. Radiation ω1 (green to red) comes from the

DPR laser (Fig. 1a).

3. Device processing and measurement techniques

The devices were processed into ridge waveguide lasers. The processing started with reactive

ion etching of 20-28µm wide ridges. We also processed narrower devices (12-18µm) which

were wet etched. A 300nm thick Silicon Nitride insulation layer was deposited and opened on

top of the ridge. After that, extended Ti/Au contacts were evaporated. The wafers were

thinned to approximately 250µm and cleaved into laser bars, which were Indium mounted on

Copper heat sinks and wire bonded. Some of the devices had a second-order grating for THz

surface emission etched into the top of the ridges since they were used for the work discussed

in [17], which also includes a detailed description of their fabrication. This grating is designed

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to couple efficiently to the generated THz radiation and is expected to have only little

influence on the mid-IR modes. For electroluminescence measurements, circular mesa

structures were processed with wet etched sidewalls to eliminate cavity feedback.

For the laser measurements, our devices were operated in pulsed mode with 60ns pulses at

a 20kHz repetition rate. The lasers were mounted in Nitrogen flow cryostats. Mid-IR spectra

were taken with a Fourier transform IR spectrometer (FTIR). 1D far fields were collected with

a Mercury-Cadmium-Telluride (MCT) detector, which is mounted on a motorized rotation

stage and rotated around the sample. 2D far fields are composed of multiple 1D far fields. A

9.5µm short pass filter was placed in front of the detector to measure the far field of the short

wavelength pump beam separately. The far field of the long wavelength pump beam was

obtained by subtracting the short wavelength far field from the far field measured without

filter.

Electroluminescence (EL) data was collected with a liquid Nitrogen cooled MCT detector,

using lock-in technique and a FTIR spectrometer operated in step scan mode.

4. Device performance and far field behavior

The devices fabricated from the first wafer S1, showed lasing at two wavelengths up to RT.

We observed sets of longitudinal modes around 10.5µm and 8.9µm corresponding to the peaks

of the gain spectrum of the DPR and BTC active regions, respectively. In the following, we

refer to the set of modes lasing at ~10.5µm and to the set of modes lasing at ~8.9µm as the

DPR laser and the BTC laser, respectively since the main contribution to lasing at these

wavelengths is made by the respective active regions. The DPR laser has a lower threshold

current density than the BTC laser and typically also a significantly larger slope efficiency. A

typical set of data – mid-IR spectra, light vs. current and voltage vs. current characteristics of

a 28µm wide and 1.5mm long device are shown in Fig. 2. EL measurements show comparable

peak intensities for both wavelengths (see Fig. 3) over a large range of applied voltages,

which indicates that the significant difference in performance of the two lasers cannot be

explained by large differences in gain for the two lasers.

Fig. 2. IR-characterization of a typical device of structure S1, taken at a temperature of 150K.

(a) shows the spectra at different current densities. The spectra are shifted vertically for clarity.

Light output vs. current density and voltage vs. current density characteristics of the same

device can be seen in (b).

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Fig. 3. EL spectra of structure S1 at different voltages, taken from mesa structures at 78K. The

two laser transitions of the two different active regions can be distinguished.

We measured the far field profile of several devices at different temperatures. Figure 4

shows a 2D far field of the mid-IR emission. All of the lasers with a ridge width equal to or

larger than 20µm showed lasing in TM00 mode for the DPR laser and higher order lateral

mode operation (typically TM02 or TM03) for the BTC laser, except for one device, which

showed TM00 operation for both wavelengths. Only for very narrow devices (~12µm wide),

both pump beams lased consistently in TM00. Observed differences in performance (e. g.

appearance of different higher order modes) between nominally identical devices can be

explained by losses due to sidewall roughness scattering which can vary from device to

device. In narrower ridges, where only TM00 modes are present, higher order modes overlap

significantly with the lossy SiN insulation layer at the sidewall of the ridges, which suppresses

the appearance of these modes.

Fig. 4. 2D far field profile for a typical device from wafer S1 with the DPR laser operating in

TM00 and the BTC laser in TM03, taken at a temperature of 78K.

The devices fabricated from the second wafer (S2) showed a slightly different behavior.

These devices emitted around 10.2µm (DPR laser) and 8.8µm (BTC laser). The threshold

current density for the DPR laser was lower at low temperatures, but the difference between

the thresholds of the two lasers was smaller than in the devices processed from the first wafer.

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Experimental data shows, that threshold current densities rise slightly slower with temperature

for the BTC laser than for the DPR laser. This leads to threshold inversion in a particular

24µm wide sample at approx. 120K, resulting in the BTC laser having a lower Jth above this

temperature.

Below 120K the far field characteristics of this device are similar to the farfield

characteristics of devices from wafer S1: the device operates in TM00 mode for the DPR laser

and TM02 for the BTC laser (see Fig. 5a). Above 120K, however, when the BTC laser has a

lower threshold current density than the DPR laser, both lasers operate in TM00 mode (Fig.

5b).

Fig. 5. Far field profile for a device of structure S2, taken at (a) 80K and (b) 180K. While the

10.2µm DPR laser operates at both temperatures in the TM00 mode, the 8.8µm BTC laser

changes from TM02 to TM00 with increasing temperature. The asymmetry of the BTC mode in

(a) is attributed to a not perfectly horizontally mounted sample.

This result indicates that the high intracavity photon density around 10.5µm influences the

performance and the properties of the BTC active region. If the threshold current for the DPR

laser is lower than the one of the BTC laser, the high intracavity intensity at around 10.5µm

leads to the reduction of the electron population in the upper laser state of the BTC active

region due to stimulated emission, because this design has transitions that are close to

resonance with the DPR laser mode. This reduces population inversion in the BTC stack and

degrades the performance of the 8.9µm pump beam, while the BTC active region contributes

to lasing around 10.5µm. This effect depends on the intracavity photon density and is

therefore strongest in the center of the ridge device, where the 10.5µm TM00 mode has its

intensity maximum. This leads to a stronger reduction of the gain in the center of the ridge

compared to the edges of the ridge, favoring lasing in higher order modes of the BTC laser.

In the next section we present a rate equation model which can numerically explain this

behavior.

5. Rate equation model

We use steady state rate Eqs. (1) – (4) to model the position dependent electron population of

the different states in the BTC active region. The rate equations describe four energy levels

shown in Fig. 1(d): an upper laser state (4), a lower laser state (2), a state almost in resonance

with the 10.5µm radiation (3) and state (1), which includes the additional active region and

injector states. As we are interested in the electron population of these states below threshold

of the BTC laser, we neglect the effect of the radiation at 8.9µm on the electron populations

and include only the stimulated transitions due to the presence of the 10.5µm DPR laser mode

in the cavity. This intensity is deduced from experimental data. The rate of stimulated

transitions is dependent on the 10.5µm photon density, and therefore on the position in the

BTC active region. The most significant contribution to the gain reduction is caused by the

transition between the upper laser state 4 and state 3. Additionally, stimulated transitions to

other levels have to be included. For the present active region, the only other considerable

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contribution is caused by transitions from state 4 to the lower laser level 2. The rate equations

are:

44

143242

4.

( , )( , )

n x y

τ

( , )(

S x y n x y

( , )( , ))

n x y

( , )(

S x y n x y

( , )( , ))

n x y

0,

inject

j

nonrad

dn x y

dtq

σσ

=−−−−−=

(1)

334

143

433.

( , )( , )

n x y

τ

( , )

n x y

τ

( , )(

S x y n x y

( , )( , ))

n x y

0,

nonrad

dn x y

dt

σ

=+−−=

(2)

3242

242

423221

( , )

n x y

τ

( , )( , )

n x y

τ

( , )

n x y

τ

( , )(

S x y n x y

( , )( , ))

n x y

0,

dn x y

dt

σ

=+−+−=

(3)

3

142

413121

( , )

n x y

τ

( , )( , )

n x y

τ

( , )

n x y

τ

0,

inject

j

dn x y

dtq

=++−=

(4)

where jinject is the injection current density, τij are the transition lifetimes from state i to state j,

τi nonrad is the total nonradiative lifetime of state i and ni is the local sheet carrier density in

state i. x and y describe the spatial position within the device cross-section as explained

below. S(x, y) is the photon flux density with a spatial distribution which is determined by the

TM00 mode profile of the cold cavity at 10.5µm wavelength. σ1 and σ2 are the cross sections

for stimulated transitions from the upper state 4 to the states 3 and 2 respectively. They are

defined in Eq. (5):

22

43

22

42 43

+

42

+

12

2222

00 43 4342 42

2

ε

2

2

ε

2

,

() (2)()(2)

pp

e z

λ

e z

λ

n Ln LEEEE

πγ

πγ

σσ

γγ

==

−−

(5)

Here, e is the elementary charge, zij is the dipole matrix element for the transition i → j, n

is the effective refractive index of the 10.5µm mode, λ is the vacuum wavelength, Lp is the

thickness of one stage in the active region (60nm), γij is the half width at half maximum

(HWHM) of the respective transition i → j (in our model we assumed γij = 7.5meV for the

mid-IR transitions 4-3 and 4-2), E is the energy of the 10.5µm photons and Eij is the energy

difference between states i and j.

We calculated the following nonradiative lifetimes at RT: τ4 nonrad = 1ps, τ43 = 12.5ps,

τ42 = 2.3ps, τ41 = 2.1ps, τ3 nonrad. = 0.13ps, τ32 = 3.0ps, τ31 = 0.15ps and τ21 = 0.18ps. These

calculated lifetimes are based on LO-phonon scattering at an applied field of 33kV/cm

(corresponds to the threshold voltage of the BTC laser). Matrix elements are: z43 = 1.57nm

and z42 = 1.80nm. The photon flux density of the 10.5µm mode is deduced from the measured

edge emission power of a typical device slightly below the threshold current density of the

BTC laser.

The equations are solved for each of the 30 stages of the BTC active region (y-direction)

and for 100 lateral positions across the ridge (x-direction) for each stage. This divides the

BTC active region into 3000 pixels. The injection current density of 6kA/cm2 used in the

model is higher than the 10.5µm DPR laser threshold current density at RT and lower than the

threshold current density for the 8.9µm BTC laser. Figure 6 shows the calculated spatially

dependent population inversion in the BTC active region in the presence of the 10.5µm TM00

mode of intracavity power corresponding to 30mW facet emission in a 24µm wide ridge.

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Fig. 6. Position dependent population inversion in the BTC active region. Depopulation via

stimulated emission of the 10.5 µm photons is strongest in the middle of the ridge, where their

density is the highest. (a) is a schematic, while (b) quantifies the effect.

The gain for the different lateral modes TM0a, a = 0, 1, 2,… of the 8.9µm laser is given by

the sum of the contributions of each pixel gTM0a(x,y), x = 1…100, y = 1…30. The gain for one

pixel is shown in Eq. (6):

2

22

42

42

Ω

32

8.9

00

2

0

34

42

3243

1

|( , )| /

x y

1

( , )

x y

( , )

x y

2

| ( , )| ( ( , )

x y

( , ))

n x y

(( , )

n x y

( , ))

n x y

m

cn L

ℏ

TM aTM a

p

G

G

e z

g

n x y

G G

µ

ω

ε

×

+ Ω

=Γ× ℜ

−

+−

(6)

The gain term includes the usual intersubband gain, as well as a contribution from an Anti-

Stokes Raman scattering of the 10.5µm laser mode into the 8.9µm laser mode [19]. Note that

the scattering involves emission or absorption of the intersubband polariton resonant to the

transition 2-3 (see Fig. 1b and 1d); it does not involve phonons. The absolute value of the

Raman contribution is approximately one third of the total value of the gain but has a negative

sign because population of the upper laser state 4 is higher than that of state 3. Here, ω8.9µm is

the angular frequency of the 8.9µm mode, e is the elementary charge, z42 is the dipole

transition matrix element from 4 to 2, c is the speed of light in vacuum, n is the effective

refractive index, Lp is the thickness of one stage in the active region (60nm) and ΓTM0a(x,y) is

the overlap of the transversal mode TM0a with the current pixel. Furthermore, Eq. (7)

(

ijij

Gi

γ

=+∆

)

/

ijij

E

ω

−

ℏ

(7)

defines the complex detunings, with γij being the HWHM of the transition between states i and

j (7.5meV for the mid-IR transitions 4→2 and 4→3 and 0.75meV for the 3→2 transition), ∆Eij

is the energy difference between levels i and j. Ω(x,y) = ez34 Eω1(x,y)/ħ is the Rabi frequency,

where Eω1 (x,y) is the electric field in the 10.5µm laser mode.

The total modal gain for each mode is now determined by summing over all pixels. Table

1 shows the resulting gain values in cm−1 for different lateral modes for a 24µm wide ridge as

shown in Fig. 6. The total modal gain for the 8.9µm mode was calculated in the presence and

without a 10.5µm mode.

If no 10.5µm radiation is present, higher order modes have slightly higher modal gain. The

small differences are due to different effective refractive indices for different modes. Higher

order modes also have higher waveguide losses as they overlap more with the lossy SiN

insulation layer at the sidewall of the ridges. The difference in waveguide loss for the TM00

and TM03 mode for example was calculated to be ~1.25cm−1 with the TM00 mode having

lower losses. The difference in waveguide losses is larger than the difference in gain, leading

to a larger net gain (modal gain minus waveguide losses) for the TM00 mode compared to

higher order modes.

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Received 19 Jan 2010; revised 25 Mar 2010; accepted 1 Apr 2010; published 27 Apr 2010

10 May 2010 / Vol. 18, No. 10 / OPTICS EXPRESS 9907

Page 9

In the presence of 10.5µm radiation, however, the modal gain for the 8.9µm mode is

reduced due to stimulated emission as discussed above. This reduction depends on the

intracavity photon intensity of the 10.5µm TM00 mode and is smaller for higher order modes

than for the TM00 mode. This effect increases the difference in modal gain for different

modes. Table 1 shows that already for small photon intensities –corresponding to an output

power of 30mW at 10.5µm - the difference between the TM00 and TM03 mode is about ~2cm-

1. This difference overcompensates for the larger waveguide losses for the higher order mode,

leading to a higher net gain for the TM03 mode compared to the TM00 mode. This is in

accordance with our experimental results which show higher order mode operation for the

8.9µm mode in broad ridges (>20µm).in the presence of 10.5µm radiation.

Table 1. Modal Gain for Different Lateral Modes With and Without 10.5µm Radiation a)

Mode TM00 TM01 TM02 TM03

No radiation at 10.5 µm

20.04 20.08 20.25 20.49

30mW radiation at 10.5 µm

15.07 16.61 16.83 17.03

a) Calculated gain in cm−1 for the different lateral modes without and with the presence of a 10.5µm mode.

The simulated device is 24µm wide and 2mm long.

6. Summary and conclusions

We performed a detailed experimental study of the lateral mode behavior of dual wavelength

QCLs, comprising heterogeneous active regions. Unexpected and disadvantageous appearance

of higher order lateral modes was observed. To describe and understand the underlying

physics we suggest a rate equation model that includes the mutual interactions of the two laser

modes. Based on these assumptions about electronic population of the active region obtained

by the rate equation model, the gain for different lateral modes can be calculated. In this

model, the mid-IR mode with the lower lasing threshold current would reduce population

inversion in the laser with the higher threshold current due to stimulated emission. This effect

is more pronounced in the center of the ridge where the lower threshold mode, which typically

is TM00, has its intensity maximum. This leads to a stronger reduction of the gain in the center

of the ridge compared to the outer parts of the ridge, favoring lasing in higher lateral modes of

the second wavelength. The predictions of this model match the experimental findings. The

model developed in this work can also be expanded to any type of multi-wavelength and

broadband QCL based on heterogeneous cascades which are interesting for many applications

such as sensing.

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Received 19 Jan 2010; revised 25 Mar 2010; accepted 1 Apr 2010; published 27 Apr 2010

10 May 2010 / Vol. 18, No. 10 / OPTICS EXPRESS 9908