Multimode Emission in GaN Microdisk Lasers [original]

RESEARCH ARTICLE

www.lpr-journal.org

Multimode Emission in GaN Microdisk Lasers

Monty L. Drechsler,* Luca Sung-Min Choi, Farsane Tabataba-Vakili, Felix Nippert,

Aris Koulas-Simos, Michael Lorke, Stephan Reitzenstein, Blandine Alloing,

Philippe Boucaud, Markus R. Wagner, and Frank Jahnke

Quantum well nanolasers usually show single-mode lasing, as gain saturation

suppresses emissions in other modes. In contrast, for whispering gallery

mode microdisk lasers with GaN quantum wells as active material, above

threshold multimode laser emission is observed. This intriguing emission

feature is manifested in the fact that several modes simultaneously show the

characteristic kink in the input–output curve at the onset of lasing. A quantum

theory for nanolasers is used to support the experimental ﬁnding and to

analyze this behavior in the presence of gain saturation. Coupling eﬀects

between neighboring modes are identiﬁed as the origin of multimode lasing,

which initiate photon exchange between modes via population pulsations

similar to classical wave-mixing eﬀects. A reduction of this type of mode

coupling with increasing mode spacing is demonstrated. The results can pave

the way for multimode application of nanolasers in integrated

photonic circuits.

M. L. Drechsler, M. Lorke, F. Jahnke

Institute for Theoretical Physics

University of Bremen

Otto-Hahn-Allee 1, 28359 Bremen, Germany

E-mail: [email protected]

L.S.-M.Choi,F. Nippert, A. Koulas-Simos,S.Reitzenstein,M.R.Wagner

InstituteofSolidStatePhysics

Technische Universität Berlin

Hardenbergstr.36, 10623 Berlin, Germany

F. Tabataba-Vakili

Fakultät fürPhysik,MunichQuantumCenter, and CenterforNanoScience

(CeNS)

Ludwig-Maximilians-UniversitätMünchen

Geschwister-Scholl-Platz1,80539München,Germany

F. Tabataba-Vakili

MunichCenterforQuantumScienceandTechnology(MCQST)

Schellingtraße4,80799München,Germany

B.Alloing,P. Boucaud

UniversitéCôted’Azur, CNRS

CRHEA,rueBernardGrégory,Sophia-Antipolis06905,France

M.R.Wagner

Paul-Drude-InstitutfürFestkörperelektronik

Leibniz-InstitutimForschungsverbundBerline.V.

Hausvogteiplatz5-7,10117Berlin,Germany

The ORCID identiﬁcation number(s) for the author(s) of this article

can be found under https://doi.org/10.1002/lpor.202400221

GmbH. This is an open access article under the terms of the Creative

Commons Attribution License, which permits use, distribution and

reproduction in any medium, provided the original work is properly cited.

DOI: 10.1002/lpor.202400221

1. Introduction

Microdisk resonators are important

building blocks for integrated photonic

circuits.[1–3]Due to their large Q-factors

they can serve as a platform for coherent

light sources.[4,5]Multimode operation is

of particular interest, as it opens up the

prospect for communication over mul-

tiple channels in a single waveguide,[6,7]

as well as sensing applications.[8]The-

oretical understanding of multimode

operation provides the fundamentals

for on-chip mode division multiplexing

without a spatial multiplexer. Nanolasers

typically have a large free spectral range

(FSR) due to their small resonator di-

mensions, and therefore single-mode

lasing usually prevails.[9–11]This is in

particular the case for quantum well

(QW) lasers, where emission into the modes is provided by a

shared gain medium. Under steady-state conditions, stimulated

and spontaneous emission need to balance the cavity losses.

Above the laser threshold, this leads to gain saturation, which

favors stimulated emission for one mode with the highest modal

gain. The side-mode suppression through gain saturation of

the joint active material acts as a mode-coupling eﬀect. Never-

theless, we observe multimode-lasing in GaN-based whispering

gallery mode (WGM) microdisk lasers with a large FSR of several

nanometers. Emergence of multimode lasing is well understood

in the framework of semiclassical theories. Population pulsations

with the beat frequencies between the modes lead to additional

mode coupling,[12]which results in light scattering between the

modes, analogous to non-degenerate four-wave mixing eﬀects.

Laser emission from microdisk resonators has been success-

fully demonstrated in the visible or ultra-violet spectral range un-

der pulsed optical excitations using III-nitride heterostructures

and quantum wells as active regions.[13–17]Various microdisks

were obtained using techniques like photo-electro-chemical etch-

ing, sacriﬁcial layers, underetching, electron-beam lithography,

nanosphere lithography in combination with diﬀerent types

of substrates like sapphire or silicon. Single mode[13,14,16,18]or

multimode[15,17,19–23]lasing was reported in diﬀerent types of mi-

crodisk lasers with various diameters. However, the nature and

mechanisms behind multimode lasing and the appropriate the-

ory and modeling to support experimental observations have re-

mained elusive.

In the WGM microdisk lasers studied in this article, sev-

eral resonator modes appear within the gain bandwidth of a

www.advancedsciencenews.com www.lpr-journal.org

Figure 1. a) Scanning electron micrograph of a III-nitride microdisk resonator. The III-nitride layers highlighted in false colors are epitaxially grown on

a silicon substrate. The layer stacking is indicated on the image. The 220 nm thick AlN layer appears in violet color at the bottom. b) μ-PL spectra of a

microdisk resonator with a diameter of 4 μm at diﬀerent excitation powers. The zoomed inset shows a mode with a visible secondary feature.

multi-quantum well active medium. Light-scattering eﬀects be-

tween the modes as a result of population pulsations compete

with gain saturation and lead to multimode stimulated emis-

sion. In our samples, a reduced mode volume (estimated as

15 ×(𝜆∕n)3following ref. [24]) together with the elevated Q-factor

of about 7000 alter the spontaneous emission into the laser mode

in comparison to the total spontaneous emission. In order to ad-

dress the inﬂuence of spontaneous emission and the mode cou-

pling due to wave-mixing eﬀects, we use a quantum optical mul-

timode laser theory. In this article, we show that wave-mixing ef-

fects explain the occurrence of multimode lasing also in a quan-

tum optical laser theory. The strength of the wave-mixing eﬀect

is discussed in terms of the FSR.

2. Results and Discussion

2.1. Fabrication of III-Nitride Microdisk Lasers

The investigated microlasers consist of mushroom-type mi-

crodisk resonators, which were fabricated from a sample grown

by metal organic chemical vapor deposition on a Si(111) sub-

strate. The structure is grown on the wurtzite c-axis and consists

of a 220 nm AlN buﬀer layer followed by a 520 nm thick GaN

layer and the optically active region. The active region consists

of ﬁve 3 nm thick In0.1Ga0.9N quantum wells separated by 7 nm

thick GaN barriers. The whole structure is terminated by a 20 nm

GaN cap layer. Figure 1ashows a false color scanning electron

microscope image of the processed microdisk. The fabrication

of the microdisks was achieved using standard cleanroom pro-

cessing. A SiO2hard mask was deposited on the wafer by plasma

enhanced chemical vapor deposition. A UV5 positive resist was

used to deﬁne the pattern of the microdisk resonators by electron-

beam lithography and the oxide layer was etched with reactive ion

etching using CH2F2and CF4gases. The III-nitride layers were

subsequently etched by inductively-coupled plasma dry etching

using Cl2and BCl3gases and the remaining SiO2was removed.

To achieve optical conﬁnement and to avoid leaking of the modes

into the silicon, the substrate was partially underetched using

XeF2gas leading to a mushroom-type structure with a narrow

pedestal as shown in Figure 1a. Microdisks with diameters from

3to6μm were obtained, in which the WGMs are conﬁned to the

periphery of the disks with slightly inclined sidewalls. Additional

information on the III-nitride material characterization can be

found in the Supporting Information of ref. [23].

2.2. Experimental Optical Spectroscopy of the Microdisks

The microdisk lasers of diﬀerent diameter were examined us-

ing a micro-photoluminescence (μ-PL) setup. For excitation a

frequency-doubled Yb ﬁber laser, emitting pulses with a fre-

quency of 76 MHz, a wavelength of 515 nm and a pulse du-

ration of approx. 1.5 ps was used. The pulses were frequency

doubled again to reach 257 nm and focused onto the sample in

backscattering geometry. The collected photoluminescence was

guided into a 80 cm focal length Czerny-Turner monochromator

with a 1200mm−1grating providing a 0.03 nm spectral resolu-

tion and detected by a liquid-nitrogen cooled CCD. All measure-

ments were performed in a nitrogen atmosphere at room tem-

perature and the pressure was selected to minimize degradation

of the sample, presumably caused by laser-induced oxidation or

reduced thermal transport.[25–27]

Figure 1b depicts the power-dependent emission spectra for

a microdisk laser with a diameter of 4 μm, where the excita-

tion power ranges between 5 and 45 mW. Four distinct peaks

with a FSR splitting of 28.0±0.4meV clearly arise at 33±5mW

(≈100𝜇Jcm

−2per pulse threshold energy) due to the GaN-

quantum well emission coupling to the WGMs of the microdisk

resonator. The emission of all four modes is present even at

the highest excitation power of 45 mW, well above the thresh-

old pump power, thus exhibiting multimode laser operation. Ad-

ditionally, secondary features are evident next to the resonator’s

modes at high excitation powers, shown in the zoomed in-

set in Figure 1b and also in the measurements of ref. [23].

These originate from the lifting of the degeneracy between clock-

wise and counter-clockwise modes due to imperfections of the

resonator.[5,28,29]Lasing was independently veriﬁed by second-

order photon-autocorrection measurements at 5 K. At room tem-

perature, these measurements are not possible due to low signal

to noise ratio.

18638899, 2024, 10, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/lpor.202400221 by Technische Universitaet Berlin, Wiley Online Library on [17/10/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License

www.advancedsciencenews.com www.lpr-journal.org

Figure 2. Input-output characteristic of a In0.1Ga0.9N QW nanodisk laser with a diameter of 4 μm. Experimental results for the intensity (open circles)

are compared to calculations (solid lines) with (a) and without (b) wave mixing. The experimental intensities and pump rates are scaled with constant

factors for direct comparison with the theory. The gray line in (a) shows the background intensity. The insets in (a) and (b) show the calculated emission

spectrum of the laser evaluated at the pump rates marked with crosses.

The experimental results obtained from the 4 μm structure

were used as a basis for the development of a theoretical model

explaining the underlying mechanism leading to multimode

laser operation. From Figure 1b Q-factors up to approx. 7000

are deduced.

2.3. Theory-Experiment Comparison

For the WGM microdisk lasers discussed in this paper, Figure 2a

shows the intensities of the individual modes, as well as the total

output intensity as a function of the excitation power. The results

for the individual mode intensities (solid lines: theory, circles: ex-

periment) reveal a clear kink, which indicates the transition to

lasing for all four observed modes. Band ﬁlling leads to a small

shift of the gain maximum. As a result, the weighting of the in-

dividual modes changes with increasing excitation power toward

higher energies.

In the following, we investigate the inﬂuence of population

pulsations and the associated wave-mixing eﬀects on the emis-

sions. For a calculation without wave mixing, Figure 2b, the total

intensity (black solid line) behaves similarly to the previous case.

The lasing threshold occurs at smaller pump rates because the

main mode (mode with the largest intensity, mode 3) has more

photons available and thus higher stimulated emission. For ex-

ample, a calculated total photon number of ten is reached for a

pump rate of 5.2×10−4ps−1with wave mixing and for a pump

rate of 4.9×10−4ps−1without wave mixing. In the absence of

wave mixing, the other modes do not show a transition to lasing,

as the corresponding kink is missing.

The behavior of the individual modes in the input-output char-

acteristics is also reﬂected in the calculated emission spectra,

which are provided as insets to Figure 2a,b. When the redistri-

bution of photons due to wave-mixing eﬀect is included, a multi-

mode spectrum is obtained below and above the laser threshold.

If the wave-mixing eﬀect is omitted, below threshold pumping

leads to an emission spectrum showing several modes. For above

threshold pumping, however, the mode with the highest modal

gain wins the mode competition and saturates the gain, thereby

suppressing emissions to the other modes.

In the wave-mixing process, two modes interact with the quan-

tum wells and generate pulsations in the electron population

with the diﬀerence frequency.[12]As a result, photons can be ex-

changed between the modes. The mode with the highest gain

scatters photons into other modes, thereby allowing them to pass

the laser threshold. This leads to an additional loss channel for

the main mode and the gain may exceed the resonator losses

for this mode. This increase in gain allows the side modes to

pass the laser threshold. Figure 3illustrates this phenomenon.

Below threshold, a multimode spectrum occurs as spontaneous

emission dominates, gain saturation is absent, and a weak wave-

mixing process is still present.

Figure 3. Schematic illustration of the relationship between gain satura-

tion and photon scattering due to wave mixing. The vertical dotted lines

mark the spectral positions of the resonator modes. The orange line shows

the gain due to stimulated emission and without wave-mixing contribu-

tions, which is saturated in such a way that stimulated and spontaneous

emission compensate the resonator losses. Due to redistribution of pho-

tons by wave mixing, the gain can exceed the resonator losses (green

dashed line) as it exhibits additional losses in the total balance.

www.advancedsciencenews.com www.lpr-journal.org

Figure 4. Intensities of the individual modes relative to the intensity of the main mode (mode with the largest intensity) as a function of the FSR, above

(a) and below (b) the laser threshold. The red dashed lines are exponential ﬁts.

2.4. Mode Coupling Eﬃciency

The described eﬀects of mode coupling, i.e., the gain satura-

tion of the common active material by the strongest mode and

the scattering of photons between the modes, are in them-

selves generic eﬀects. In addition to the nitride-QW-based WGM

microdisk lasers studied here, multimode lasing has been re-

ported in microdisk resonators incorporating arsenide-QWs,[30]

phosphide-QWs,[31]and even perovskites.[32]Multimode lasing

was also observed in microdisk resonators with quantum dots

as the active medium.[28,33,34]In this case, however, multimode

operation occurs due to the inhomogeneous broadening of the

discrete emitter ensemble, such that each mode only saturates

the respective quantum dots in close spectral vicinity. For a large

FSR, each laser mode is supported by a separate set of quantum

dots and the gain saturation of one mode has no inﬂuence on the

other modes. We exclude QD-like states due to disorder-induced

localization in the used sample. Local ﬂuctuations induced by in-

dium, with a potential in the tens of meV range, are screened

under high carrier injection, resulting in delocalized charge car-

riers at room temperature.[35]

For the mode coupling due to photon scattering between the

modes, the FSR in relation to the gain bandwidth plays a cen-

tral role, indirectly also the strength of scattering and dephasing

processes. This ultimately makes the processes dependent on the

material system and resonator type and lastly dependent on the

strength of gain saturation versus population pulsations.

To investigate the eﬃciency of mode coupling, we examine the

dependence of the emission spectrum on the FSR. Figure 4pro-

vides the intensities of the individual modes relative to the main

mode. The relative intensities decrease exponentially with the

FSR (red dashed ﬁts in Figure 4). Thus, multimode lasing is less

apparent for a larger FSR and photon scattering is less eﬃcient.

Below the laser threshold, emission into several modes is sup-

ported even with a larger FSR as there is no gain saturation, while

the photon scattering remains present albeit at weaker eﬃciency.

The photon scattering between the modes is represented by

processes in which a photon is added into one mode qwhile

being removed from another mode q′. This can be expressed

in terms of photon creation and annihilation operators, b†

qand

bq′, respectively. The corresponding expectation values ⟨b†

qbq′⟩

with q≠q′are the oﬀ-diagonal photon density matrix elements.

These transition amplitudes quantify the eﬃciency of photon

scattering between the modes and are a central part of the used

quantum optical multi-mode laser theory. Their calculation is

outlined in Section 4. The relative coupling strength (normalized

to the square root of the respective photon numbers)

|⟨b†

qbq′⟩|

√⟨b†

qbq⟩⟨b†

q′bq′⟩

(1)

between modes two and three and the largest pump rate in

Figure 2a has a value of 15%, which emphasizes the magnitude

of the process.

3. Conclusion

High Q-factor WGM microdisk resonators with InGaN quantum

wells as active material have been used to demonstrate multi-

mode laser operation. A direct analysis of the experimental re-

sults with a semiconductor laser theory attributes the simultane-

ous occurrence of a threshold transition for several laser modes

to strong mode coupling eﬀects. Quantum optical semiconductor

laser models using a quantization of the optical ﬁeld play an im-

portant role in the analysis of emission properties when the tran-

sition from conventional lasers with low Q-factor and low sponta-

neous emission coupling to nanolasers is performed. While mul-

timode lasing was observed in our present and some previous ex-

periments, quantum optical semiconductor models have focused

on treatments with a single laser mode. In this paper we provide

the necessary extension to a multi-mode semiconductor theory

by including oﬀ-diagonal photon density matrix elements. The

resulting mode coupling eﬀects turn out to be of strong magni-

tude and enable multi-mode operation despite the fact that all

modes use a common quantum well gain medium and gain sat-

uration is present.

www.advancedsciencenews.com www.lpr-journal.org

4. Quantum Optical Multimode Laser Theory

Starting from the Lindblad-von-Neumann equation, the equa-

tions of motion were derived for relevant expectation val-

ues like the mean photon number and ﬁrst-order pho-

ton correlations deﬁning the emission spectrum. The light-

matter interaction gives rise to a hierarchy of coupled equa-

tions, which was truncated at doublet level of the cluster-

expansion approximation.[36–38]Interaction of the light ﬁeld

within a two-band model in eﬀective-mass approximation

was treated for the In0.1Ga0.9N quantum well (QW) active

medium.

For the light-matter-coupling, a Jaynes-Cummings-type inter-

action was considered with several quantized resonator modes.

The system is described by the Hamiltonian



H=∑

k(𝜀e

ke†

kek+𝜀h

kh†

khk)+∑

ℏ𝜔qb†

qbq

+∑

k′,q′(gk′q′b†

q′hk′ek′+g∗

k′q′bq′e†

k′h†

k′)(2)

where ek(hk)and e†

k(h†

k)are annihilation and creation opera-

tors for conduction band electrons (valence band holes), bqand

b†

qare annihilation and creation operators of photons in the q-th

mode, respectively. krepresents the state described by wave vec-

tor 

kand spin s. The energy of the electrons (holes) in state

kis denoted by 𝜀e

k(𝜀h

k),𝜔qis the frequency of the q-th cavity

mode, and gkq is the light-matter coupling strength between the

q-th mode and electrons (holes) in state k. The equations of mo-

tion for the photon numbers nq=⟨b†

qbq⟩, the electron occupa-

tions fe

k=⟨e†

kek⟩and the hole occupations fh

k=⟨h†

khk⟩are given

ℏd

dtnq=2∑

Re{−igkq ⟨b†

qhkek⟩}−2𝜅nq(3)

ℏd

dtfe∕h

k=−2∑

Re {−igkq ⟨b†

qhkek⟩}

−𝛾rel (fe∕h

k−Fe∕h

k(T))−𝛾r⟨e†

kh†

khkek⟩

+pk(1−fe

k−fh

k+⟨e†

kh†

khkek⟩)(4)

Resonator losses were taken into account with the photon decay

rate 𝜅, which was related to the Q-factor of the q-th resonater

mode Qq=ℏ𝜔q

2𝜅. The electrons were optically excited high into

the conduction band with the pump rate pk. A relaxation time ap-

proximation was used for the scattering of electrons (holes) with

rate 𝛾rel toward a Fermi-Dirac distribution Fe

k(T)(Fh

k(T))at lat-

tice temperature T. Emission in non-lasing modes was described

by loss rate 𝛾r. The corresponding modes were either strongly

detuned or have low Q-factors. The occupations nqand fe∕h

kare

driven by the photon-assisted polarization ⟨b†

qhkek⟩,whichobeys

the diﬀerential equation

ℏd

dt⟨b†

qhkek⟩=i∑

k′

g∗

k′q⟨e†

k′h†

k′hkek⟩+i(fe

k+fh

k−1)∑

q′

g∗

kq′⟨b†

qbq′⟩

−[iΔkq +𝜅+𝛾rel +Γ

El +Γ

Phot +pk+𝛾r

⋅⟨b†

qhkek⟩(5)

where Δkq =𝜀e

k+𝜀h

k−ℏ𝜔qis the detuning of the mode qto

the electronic transition at k. Following the cluster-expansion

scheme, many-body correlations up to second order were ex-

plicitly included in the set of equations used here. On subse-

quent orders, electron–electron, electron–phonon, and higher-

order electron–photon correlations appear, which lead to de-

phasing contributions in the photon-assisted polarization, de-

scribed by Equation (5). As higher-order correlations were not

treated explicitly, their contributions were added as phenomeno-

logical dephasing constants. Here, ΓEl represents dephasing due

to electron-electron and electron–phonon contributions and ΓPhot

contains dephasing due to higher-order electron–photon correla-

tions. The k=k′contribution to ⟨e†

k′h†

k′hkek⟩in Equation (5)de-

scribes the spontaneous emission contribution due to recombi-

nation of an electron-hole pair with momentum k.Termswith

k≠k′were polarization-like quantities that account for spon-

taneous emission processes due to the coupling of diﬀerent k-

states. The dynamics of these processes is given by the equation

ℏd

dt⟨e†

k′h†

k′hkek⟩=

⎧

⎪

⎨

⎪

⎩

−i∑

q[gk′q(fe

k′+fh

k′−1)⟨b†

qhkek⟩−g∗

kq (fe

k+fh

k−1)⟨bqe†

k′h†

k′⟩]

−[i(𝜀e

k+𝜀h

k−𝜀e

k′−𝜀h

k′)+2𝛾rel +𝛾r+pk+pk′

2+2ΓEl]⋅⟨e†

k′h†

k′hkek⟩

for k≠k′,

−2∑

Re{−igkq ⟨b†

qhkek⟩}−𝛾r⟨e†

kh†

khkek⟩

−2𝛾rel ⋅(⟨e†

kh†

khkek⟩−Fe

k(T)Fh

k(T))+pk(1−fe

k−fh

k+⟨e†

kh†

khkek⟩)for k=k′.

(6)

This expectation value was in turn driven by the photon-assisted

polarization ⟨b†

qhkek⟩. The second term in Equation (5) describes

the stimulated emission for q=q′and photon scattering for q≠

q′. To account for wave mixing, a separate equation is formulated:

ℏd

dt⟨b†

qbq′⟩=−i∑

k(gkq′⟨b†

qhkek⟩−g∗

kq ⟨bq′e†

kh†

k⟩)

−[i(ℏ𝜔q′−ℏ𝜔q)+2𝜅+2ΓPhot (1−𝛿q,q′)]⟨b†

qbq′⟩

(7)

Note that for q=q′the equation for the photon number, Equa-

tion (3), was recovered. For this case, the dephasing contribution

ΓPhot was omitted.

The Equations (3)–(7) form a closed set of self-consistent laser

equations, which could be used to determine the input–output

characteristics of the nanolasers. The description goes beyond

www.advancedsciencenews.com www.lpr-journal.org

Table 1. Set of parameters for modeling a In0.1Ga0.9N QW nanodisk laser.

m0denotes the mass of an electron. The eﬀective masses are determined

from.[41]

𝜀g2967 meV

me0.17 ⋅m0

mh0.74 ⋅m0

ℏ𝜔12929 meV

ℏ𝜔22957 meV

ℏ𝜔32985 meV

ℏ𝜔43013 meV

g00.03 1

A107nm2

𝛾rel 10 1

T50 K

𝛾r0.055 1

𝜅0.373 1

ΓPhot 71

ΓEl 10 1

rate equations, since the polarizations, photon scattering and

couplings between diﬀerent k-states were calculated in a phase

sensitive manner (all quantities oscillate with the corresponding

transition energy) according to the microscopic interaction pro-

cesses. An advantage of the approach was that it allowed to study

the inﬂuence of speciﬁc correlation between modes or diﬀerent

k-states.

Furthermore, the theory provides access to the emission spec-

trum S(𝜔) via the ﬁrst-order photon correlation function G(1)(t, 𝜏)

together with the Wiener-Khinchin theorem,[39]

S(𝜔)={G(1)(t, 𝜏)}with G(1)(t, 𝜏)

=ℏ

2𝜖0∑

q,q′√𝜔q𝜔q′u∗

q(

rD)uq′(

rD)⟨b†

q(t)bq′(t+𝜏)⟩(8)

where uq(

r) are the mode functions of the q-th mode and 

rDde-

notes the position of the detector. Using the quantum regression

theorem,[40]equations of motion for the contributing correlation

functions are obtained

ℏd

d𝜏⟨b†

q(t)bq′(t+𝜏)⟩=−i∑

gkq′⟨b†

q(t)hkek(t+𝜏)⟩

−(iℏ𝜔q′+𝜅+Γ

Phot)⟨b†

q(t)bq′(t+𝜏)⟩(9)

ℏd

d𝜏⟨b†

q(t)hkek(t+𝜏)⟩=i(fe

k+fh

k−1)∑

q′

g∗

kq′⟨b†

q(t)bq′(t+𝜏)⟩

−[i(𝜀e

k+𝜀h

k)+Γ

El +𝛾rel +pk+𝛾r

×⟨b†

q(t)hkek(t+𝜏)⟩(10)

The coupled set of Equations (3)–(7) together with Equations (8)–

(10) allowed to calculate the emission spectrum of nanolasers in

the presence of photon redistribution between diﬀerent modes.

Table 1shows the parameters used for the calculation.

Acknowledgements

M.L.D., L.S.-M.C., and F.T.-V. contributed equally to this work. The Bremen

group acknowledges ﬁnancial support from the Deutsche Forschungsge-

meinschaft (DFG) via project JA619/18-1 and a grant for CPU time at the

HLRN (Berlin/Göttingen). The microdisk fabrication was partly supported

by the French RENATECH network as well as by the French Agence Na-

tionale de la Recherche (ANR) under MILAGAN convention (No. ANR-

17-CE08-0043-02) and Labex GANEX (Grant no. 314 ANR-11-LABX-0014).

Support from the Comb-on-GaN convention of Labex GANEX is also ac-

knowledged. A.K-S. and S.R. acknowledge ﬁnancial support from the DFG

via project Re2974/21-1.

Open access funding enabled and organized by Projekt DEAL.

Conﬂict of Interest

The authors declare no conﬂict of interest.

Data Availability Statement

The data that support the ﬁndings of this study are available from the cor-

responding author upon reasonable request.

Keywords

gallium-nitride, microdisk, microlasers, multimode, quantum-optical, the-

ory, whispering-gallery

Received: February 14, 2024

Revised: April 12, 2024

Published online: May 3, 2024

[1] G. Roelkens, L. Liu, D. Liang, R. Jones, A. Fang, B. Koch, J. Bowers,

Laser Photonics Rev. 2010,4, 751.

[2] N. Kryzhanovskaya, A. Zhukov, E. Moiseev, M. Maximov, J. Phys. D:

Appl. Phys. 2021,54, 453001.

[3] S. Rodt, S. Reitzenstein, APL Photonics 2021,6, 010901.

[4] X. Jiang, C. Zou, L. Wang, Q. Gong, Y. Xiao, Laser Photonics Rev. 2015,

10, 40.

[5] L. He, ¸S. K. Özdemir, L. Yang, Laser Photonics Rev. 2012,7, 60.

[6] S. J. Choi, K. Djordjev, S. J. Choi, P. Dapkus, IEEE Photonics Technol.

Lett. 2003,15, 1330.

[7] D. Gostimirovic, W. N. Ye, Opt. Lett. 2021,46, 388.

[8] T. Reynolds, N. Riesen, A. Meldrum, X. Fan, J. M. M. Hall, T. M.

Monro, A. François, Laser Photonics Rev. 2017,11,2.

[9] W. W. Chow, S. Reitzenstein, Appl. Phys. Rev. 2018,5,4.

[10] S. Kreinberg, K. Laiho, F. Lohof, W. E. Hayenga, P. Holewa, C. Gies, M.

Khajavikhan, S. Reitzenstein, Laser Photonics Rev. 2020,14, 2000065.

[11] A.Koulas-Simos,J.Buchgeister,M.L.Drechsler,T.Zhang,K.Laiho,

G. Sinatkas, J. Xu, F. Lohof, Q. Kan, R. K. Zhang, F. Jahnke, C. Gies, W.

W. Chow, C.-Z. Ning, S. Reitzenstein, Laser Photonics Rev. 2022,16,9.

[12] M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics, Addison-Wesley

Publishing Company, Boston 1974.

[13] E. D. Haberer, R. Sharma, C. Meier, A. R. Stonas, S. Nakamura, S. P.

DenBaars, E. L. Hu, Appl. Phys. Lett. 2004,85, 5179.

www.advancedsciencenews.com www.lpr-journal.org

[14] A. C. Tamboli, E. D. Haberer, R. Sharma, K. H. Lee, S. Nakamura, E.

L. Hu, Nat. Photonics 2006,1, 61.

[15] H. W. Choi, K. N. Hui, P. T. Lai, P. Chen, X. H. Zhang, S. Tripathy, J.

H. Teng, S. J. Chua, Appl. Phys. Lett. 2006,89, 21.

[16] D. Simeonov, E. Feltin, H.-J. Bühlmann, T. Zhu, A. Castiglia, M.

Mosca, J.-F. Carlin, R. Butté, N. Grandjean, Appl. Phys. Lett. 2007,90,

061106.

[17] D. Simeonov, E. Feltin, A. Altoukhov, A. Castiglia, J.-F. Carlin, R. Butté,

N. Grandjean, Appl. Phys. Lett. 2008,92, 171102.

[18] I. Aharonovich, A. Woolf, K. J. Russell, T. Zhu, N. Niu, M. J. Kappers,

R. A. Oliver, E. L. Hu, Appl. Phys. Lett. 2013,103, 021112.

[19] X. Zhang, Y. F. Cheung, Y. Zhang, H. W. Choi, Opt. Lett. 2014,39,

5614.

[20] J. Sellés, C. Brimont, G. Cassabois, P. Valvin, T. Guillet, I. Roland, Y.

Zeng, X. Checoury, P. Boucaud, M. Mexis, F. Semond, B. Gayral, Sci.

Rep. 2016,6, 21650.

[21] J. Sellés, V. Crepel, I. Roland, M. El Kurdi, X. Checoury, P. Boucaud, M.

Mexis, M. Leroux, B. Damilano, S. Rennesson, F. Semond, B. Gayral,

C. Brimont, T. Guillet, Appl. Phys. Lett. 2016,109, 231101.

[22] Y. Li, L. Feng, F. Li, P. Hu, M. Du, X. Su, D. Sun, H. Tang, Q. Li, F. Yun,

ACS Photonics 2018,5, 4259.

[23] F. Tabataba-Vakili, C. Brimont, B. Alloing, B. Damilano, L. Doyennette,

T. Guillet, M. El Kurdi, S. Chenot, V. Brändli, E. Frayssinet, J.-Y.

Duboz, F. Semond, B. Gayral, P. Boucaud, Appl. Phys. Lett. 2020,117,

121103.

[24] K. Srinivasan, M. Borselli, O. Painter, A. Stintz, S. Krishna, Opt. Ex-

press 2006,14, 1094.

[25] I. Aharonovich, N. Niu, F. Rol, K. J. Russell, A. Woolf, H. A. R. El-Ella,

M. J. Kappers, R. A. Oliver, E. L. Hu, Appl. Phys. Lett. 2011,99, 11.

[26] I. Rousseau, G. Callsen, G. Jacopin, J.-F. Carlin, R. Butté, N.

Grandjean, J. Appl. Phys. 2018,123, 113103.

[27] J. Chrétien, Q. M. Thai, M. Frauenrath, L. Casiez, A. Chelnokov, V.

Reboud, J. M. Hartmann, M. El Kurdi, N. Pauc, V. Calvo, Appl. Phys.

Lett. 2022,120,5.

[28] P. Jaﬀrennou, J. Claudon, M. Bazin, N. S. Malik, S. Reitzenstein, L.

Worschech, M. Kamp, A. Forchel, J.-M. Gérard, Appl. Phys. Lett. 2010,

96, 071103.

[29] R. R. Galiev, N. M. Kondratiev, V. E. Lobanov, A. B. Matsko, I. A.

Bilenko, Phys. Rev. Appl. 2021,16,6.

[30] M. J. Ries, E. I. Chen, N. Holonyak, G. M. Iovino, A. D. Minervini,

Appl. Phys. Lett. 1996,68, 1540.

[31] T. Zhou, K. W. Ng, X. Sun, Z. Zhang, Nanophotonics 2020,9, 2997.

[32] Q. Zhang, S. T. Ha, X. Liu, T. C. Sum, Q. Xiong, Nano Lett. 2014,14,

5995.

[33] P. Michler, A. Kiraz, L. Zhang, C. Becher, E. Hu, A. Imamoglu, Appl.

Phys. Lett. 2000,77, 184.

[34] F. Albert, C. Hopfmann, A. Eberspächer, F. Arnold, M. Emmerling,

C. Schneider, S. Höﬂing, A. Forchel, M. Kamp, J. Wiersig, S.

Reitzenstein, Appl. Phys. Lett. 2012,101,2.

[35] A. David, N. G. Young, M. D. Craven, Phys.Rev.Appl.2019,12,

044059.

[36] C. Gies, J. Wiersig, M. Lorke, F. Jahnke, Phys. Rev. A 2007,75, 013803.

[37] S. Ates, C. Gies, S. M. Ulrich, J. Wiersig, S. Reitzenstein, A. Löﬄer, A.

Forchel, F. Jahnke, P. Michler, Phys. Rev. B 2008,78, 155319.

[38] M. Kira, S. W. Koch, Semiconductor Quantum Optics, Cambridge Uni-

versity Press, Cambridge 2011.

[39] L. Mandel, E. Wolf, Optical Coherence and Quantum Optics,Cam-

bridge University Press, Cambridge 1995.

[40] H.-P. Breuer, F. Petruccione, The Theory of Open Quantum Systems,

Oxford University Press, Oxford 2007.

[41] M. Goano, E. Bellotti, E. Ghillino, G. Ghione, K. F. Brennan, J. Appl.

Phys. 2000,88, 6467.