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C: Plasmonics; Optical, Magnetic, and Hybrid Materials

Lasing Conditions of Transverse Electromagnetic Modes in Metallic-Coated Micro- and Nanotubes Nicolás Passarelli, Raúl Alberto Bustos-Marún, and Ricardo A Depine J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b01808 • Publication Date (Web): 23 Apr 2019 Downloaded from http://pubs.acs.org on April 23, 2019

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The Journal of Physical Chemistry

Lasing Conditions of Transverse Electromagnetic Modes in Metallic-Coated Micro- and Nanotubes †

Nicolás Passarelli,

Raúl Bustos-Marún,

∗,†

‡

and Ricardo Depine

†Instituto de Física Enrique Gaviola (IFEG-CONICET) and Facultad de Ciencias

Químicas, Universidad Nacional de Córdoba, Ciudad Universitaria, Córdoba 5000, Argentina ‡Grupo de Electromagnetismo Aplicado, Departamento de Física, FCEN, Universidad de Buenos Aires and IFIBA, Consejo Nacional de Investigaciones Cientícas y Técnicas (CONICET), Ciudad Universitaria, Pabellón I, C1428EHA, Buenos Aires, Argentina E-mail: [email protected]

Abstract In this work, we study the lasing conditions of the transverse electric modes (TE) of micro- and nanotubes coated internally with a thin metallic layer.

This geometry may tackle some of the

problems of nanolasers and spasers as it allows the recycling of the active medium while providing a tunable plasmonic cavity. The system presents two types of TE modes: cavity modes (CMs) and whispering-gallery modes (WGMs). On the one hand, we show that the lasing of WGM is only possible in nanoscale tubes.

On the other hand, for tubes of some micrometers of diameter, we

found that the system presents a large number of CMs with lasing frequencies within the visible and near-infrared spectrum and very low gain thresholds. Moreover, the lasing frequencies of CMs can be accurately described by a simple one-parameter model. Our results may be useful in the design of micro- and nanolasers for lab-on-chip devices, ultra-dense data storage, nanolithography, or sensing.

INTRODUCTION

troscopies, as a way of counteracting the effect of optical losses in dierent plasmonic de-

Micro-

and

Nanolasers

nanoscale dimensions)

13

(lasers

of

micro-

or

vices, or directly as a source of radiation for

and spasers (a type

lab-on-chip

devices,

ultra-dense data stor-

711

of laser which connes light at subwavelength

age,

scales

polari-

of nanolasers and spasers have been exten-

can be used to provide a controllable

sively discussed in the literature over the last

tons)

4,5

by

exciting

surface

plasmon

source of on-demand electromagnetic elds at very small size scales.

6

or

nanolithography.

Dierent

decades, both from the theoretical

These devices can

the experimental

nd numerous applications for enhanced spec-

13,11,1825

forms

5,1117

and

point of view. There

has been studied a wide variety of structures

ACS Paragon Plus Environment 1

The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 2 of 22

medium (thanks to microuidics

Active Media Noble Metal

23

), provides a

tunable resonant cavity (by modifying the di-

Polymer

electric constant of the solution of the core,

Incident wave

physically separates the active medium from the external medium (where analytes may be placed for instance), presents large volumes of hot-spots, low gain threshold, and a large number of resonances within the visible and near-

Figure 1: Scheme of the system treated. An in-

infrared spectrum.

nite multiwall cylinder where the core allows

The work is organized as follow.

the circulation of a solution with the dye (ac-

General remarks, we discuss some generalities

tive medium), the intermediate wall provides

of the studied system.

the plasmonic cavity, and the outer wall gives

In section Scattering

Problem, we explain how to solve the scat-

the physical support to the system. The inci-

tering problem in the linear-response regime.

dent wave is just used to numerically nd the poles of the scattering cross-section.

In section

There, we give the equations of the electromag-

See text

netic elds in terms of cylindrical harmonics,

for more details.

the boundary conditions, the expressions for the cross-sections, and the relation between the supporting

scattering cross-sections and the near elds at

the spaser/nanolaser as well as dierent com-

the lasing condition. In section Modeling the

pounds to act as the optical-gain medium. This

optical active medium, we discuss the dier-

includes cavities in thin lms, dierent forms

ent approximations used to describe the active

of waveguides made of lithographic or nano-

medium and how we obtained the parameters

printed surfaces, graphene sheets, carbon nan-

of the dyes used. In section Lasing conditions,

otubes, nanoparticle or quantum dot arrange-

we discuss the lasing conditions in terms of las-

ments, etc.

ing frequencies and gain thresholds, providing

and

materials/metamaterials

for

4,6

There are, in general, two main problems in-

also some approximations for their estimation.

volved in the design of spasers and nanolasers.

In section Results, we show the main results

One is the need for a good matching between

of the work and summarize them in the con-

the plasmonic resonances (for spasers) or the

clusions section.

cavity resonaces (for nanolasers) and the emission cross-section of the active medium.

The

THEORY

other main problem is photobleaching, or the irreversible photo-degradation of the molecules or quantum dots

21

26

General remarks.

that form the active media.

Other features are also desirable, such as large volumes of hot-spots,

27

The rst one is an extension of well known ana-

to allow the study of

lytical solution of an innite two-wall concentric

large molecules, or the presence of several reso-

cylinder

nances that can be used for lasing with dierent active media.

25

to three-wall concentric cylinders,

extinction cross-sections. The second axis is the modeling of the active medium.

bleaching problem but should also oer some

We use two

dierent models: a wideband approach, which

exibility to nely tune the frequencies of res-

24,25,28

2932

including the expressions for the scattering and

The ideal design of a nanolaser

or a spaser should not only solve the photo-

onances

The theoretical treatment

of the system is sustained in two major axes.

neglects the frequency dependence of the active

while providing large volumes of

medium, and a Lorentz-like model for the fre-

hot-spots and low gain thresholds.

quency dependence of the active medium.

In this work, we study the lasing conditions

Although we performed several calculations

of a geometry that seems promising to meet all

with dierent geometries, most of the work is

the above requirements. The proposed geome-

focused on one representative example.

try, see Fig. 1, allows the recycling of the active


This

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The Journal of Physical Chemistry 51

geometry consists of three concentric cylinders,

can be thought as Fabry-Pérot modes

see Fig.

nite systems, or they can be treated as disper-

1.

The internal core of 2000nm of

radius consists of a dielectric solvent and a dis-

sion relations for innite cylinders.

for -

52

solved dye. The inner wall of the tube is a layer

Regarding the feasibility of the proposed ge-

of 25 nm of a plasmonic material (gold), and the

ometry, modern techniques of fabrication show

outer one, of 475 nm, is a dielectric (polymer)

that tubes of some few microns, suitable for

that provides mechanical support to the cavity

microuidics, can be produced

and separates it from the external medium.

even possible to make them of dierent materi-

Due to its sizes, the system lays between the microlaser and nanolaser categories.

als.

33,34,5860

5357

and it is

On the other hand, coating of sur-

Further-

faces with very thin layers of noble metals has

more, due to the metallic coating, the lasing of

been employed in several experimental works

some resonances turn the system into a spaser.

using dierent techniques.

61,62

For simplicity, throughout this paper we will

The incident wave, shown in Fig. 1, is used in

use the generic term nanolaser to refer to the

the calculations to nd the poles of the scatter-

system, but keeping in mind these considera-

ing cross-section of the system, which leads us

tions.

to the lasing conditions. We will only consider

The geometry was inspired by recent works

the case of electric elds perpendicular to the

that present polymer microtubes suitable for

z

uidics

magnetic materials.

21,3336

The fact that the system allows

axis of the cylinders, as we are studying nonFor this polarization, the

the circulation of a uid through it, can be

resonances will be found when the perpendic-

used for recycling the active media as well as

ular component of the incident wave matches

to deposit the metal layer by chemical meth-

the resonant condition, and this is why we need

ods.

only to consider the perpendicular incidence.

3739

The recycling of the active media may

help to counteract the photobleaching of the

The dielectric constant of gold as a function

63

dye molecules acting as the optically active

of the wavelength was taken from Ref.

medium.

dielectric constant of the polymer was consid-

The intermediate layer provides the

ε1 = 2.22. 64

The

plasmonic cavity that will control the resonance

ered constant with

frequencies and the outer layer gives the phys-

treatment of the dielectric constant of the ac-

ical support to the system. In this way, if for

tive medium is discussed in details in section

example the nanolaser is used for sensing pur-

Modeling the optical active medium.

Scattering problem.

poses, the analyte is physically separated from

The theoretical

The analytical solu-

the active medium. The latter not only avoids

tion for the scattering of electromagnetic plane

undesired chemical interferences but can also be

waves incident perpendicularly on a homoge-

used to control the resonances of the nanolaser

neous circular cylinder of innite length is well

by tunning the dielectric constant of the inner

known.

cylinder.

tional invariance of the geometry, the solution

31

In this case, and due to the transla-

Similar geometries have been studied before,

of the rigorous electromagnetic vectorial prob-

including wires, tubes, pores, as well as bers

lem can be reduced to the treatment of two

or capillaries.

21,36,4043

The sizes of these sys-

independent scalar problems, corresponding to

tems ranged from tens of nanometers up to hun-

the following two basic polarization modes in

dreds of micrometers of radius. The geometries

which the polarization of the interior and scat-

within the nanometers scale have been stud-

tered elds maintain the polarization of the in-

ied theoretically for perpendicular illumination,

cident wave: i)

which excites the TE modes of lasing, but gen-

the incident electric eld is directed along the

erally within quasi-static approximations. The

cylinder axis, and ii)

larger geometries have been studied mainly in

when the incident magnetic eld is directed

the context of low loss communication, where

along the cylinder axis In the rst case, it is

the longitudinal modes of lasing are the impor-

clear that the electric eld can only induce elec-

tant ones.

tric currents directed along the cylinder axis but

4450

The resonances in such a case


s

(or TM) polarization, when

p

(or TE) polarization,


Page 4 of 22

j.

not along the azimuthal direction, thus prevent-

fractive index of medium

ing the existence of localized surface plasmons

separation of variables approach,

in metallic cylinders for For

p

s (or TM) polarization.

Following the usual

Fj (r, θ) can be

represented with the following multipole expansions

polarization, on the other hand, the elec-

tric eld, contained in the main section of the cylinder, can induce azimuthal electric currents

∞ X

Fj (r, θ) =

which, under appropriate conditions, give place

[ajn Jn + Ajn Hn ] einθ

(2)

n=−∞

to localized surface plasmon resonances. number of layers,

29,30,65

ajn , Ajn (j = 0, . . . 3), are complex amJn (kj r) is the n-th Bessel function of rst kind and Hn (kj r) is the n-th Hankel

where

For multilayered cylinders with an arbitrary

plitudes,

similar conclusions can

be obtained: the solution of the rigorous elec-

the

tromagnetic vectorial problem can be separated

function of the second kind, with asymptotic

into two independent (s and

behavior for large argument (|x|

p)

scalar polar-

ization modes and, when metallic layers are

r

involved, localized surface plasmon resonances occur for

Hn (x) −→

p, but not for s, incident polarizations.

Taking into account that we are interested in

The amplitudes

the TE modes of nanolasers, in what follows we restrict our attention just to the case of

a0n

→ ∞)

2 −i(x−nπ/2−π/4) e . πx

(3)

in the medium of incidence

are determined by the incident wave. Assum-

p

ing, without loss of generality, a unit-amplitude

polarization.

incident plane wave propagating along the x-

To model the linear response of our system

axis, and using the Jacobi-Anger expansion.

we consider an innitely long circular cylinder (the core) covered with two concentric layers of

e

homogeneous materials and illuminated from a

−ik0 r cos θ

∞ X

=

(−i)n Jn (k0 r) einθ

66

(4)

n=−∞

semi-innite ambient medium. We assume that all the materials involved are isotropic and lin-

the amplitudes

a0n

are given by

εj and a magnetic permeability µj ,(j = 0, 1, 2, 3), where j = 0 corresponds to the ambient and j = 3 corresponds to the core (see gure 1).

For physical reasons, Hankel functions of the

The Gaussian system of units is used and an

second kind, singular at the origin, are not al-

exp(+iωt) time-dependence is implicit throughout the paper, with ω the angular frequency, c the speed of light in vacuum, t the time, and √ i = −1.

lowed in the eld representation of the core re-

ear and characterized by a dielectric constant

Fields representation.

nates (r, θ ), where

θ

a0n = (−i)n .

gion. Thus

A3n = 0.

is the angle respect to the

axis of a vector contained in the

tions,

everywhere with expansions

(2),

a

innity

sextuple

be found.

of

∇ × (Fj zˆ) = i

Fj (r, θ) must satisfy Helmholtz equations

Boundary conditions.

kj = ωmj /c

and

ampli-

A0n , a1n , A1n , a2n , A2n , a3n , must ~ j in p poFinally, the electric eld E in terms of Fj (r, θ) can be obtained

According to Maxwell's equa-

(∇2 + kj2 )Fj = 0 ,

unknown

from Ampère's Law

in each region

with

~ j = Fj zˆ H

larization

eld along the axis of the cylinder and evalu-

j.

eld

tudes, namely,

the non-zero component of the total magnetic ated in region

(6)

Therefore, in order to evaluate the magnetic

Using polar coordi-

x − y plane of modulus r , all the elds in the p polarization case can be written in terms of Fj (r, θ),

x

(5)

r = Rj (j = 1, . . . 3)

(1)

ω ~ εj Ej . c

(7)

At the boundary

between two homoge-

neous regions, the continuity of the tangential

mj = (εj µj )

1/2

components of the electric and magnetic eld

the re-


Page 5 of 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


must be fullled and these conditions are equiv1 ∂Fj . These alent to the continuity of Fj and εj ∂r boundary conditions provide a system of six lin-

A0n , a1n , A1n , a2n , A2n

ear equations for the six unknown amplitudes

ent multipole order

and

a3n .

Due to the ro-

tational symmetry of the system, the boundary conditions do not couple amplitudes with dier-

n.

The continuity of the

z

component of the magnetic eld gives equations

a3n Jn (k3 R3 ) = a2n Jn (k2 R3 ) + A2n Hn (k2 R3 ) a2n Jn (k2 R2 ) + A2n Hn (k2 R2 ) = a1n Jn (k1 R2 ) + A1n Hn (k1 R2 ) a1n Jn (k1 R1 ) + A1n Hn (k1 R1 ) = (−i)n Jn (k0 R1 ) + A0n Hn (k0 R1 )

(8) (9) (10)

whereas the continuity of the tangential component of the electric eld gives the equations

k3 k2 a3n Jn0 (k3 R3 ) = [ a2n Jn0 (k2 R3 ) + A2n Hn0 (k2 R3 ) ] ε3 ε2 k2 k1 [ a2n Jn0 (k2 R2 ) + A2n Hn0 (k2 R2 ) ] = [ a1n Jn0 (k1 R2 ) + A1n Hn0 (k1 R2 ) ] ε2 ε1 k1 k 0 [ a1n Jn0 (k1 R1 ) + A1n Hn0 (k1 R1 ) ] = [ (−i)n Jn0 (k0 R1 ) + A0n Hn0 (k0 R1 ) ] ε1 ε0

(11)

(12)

(13)

where primes denote derivatives of the Bessel and Hankel functions with respect to their arguments. The above equations can be written in matrix form

~ =B ~ MX with

(14)

h iT ~ = [A0n , a1n , A1n , a2n , A2n , a3n ]T , B ~ = (−i)n Jn (k0 R1 ), k0 J 0 (k0 R1 ), 0, 0, 0, 0 X ε0 n

−Hn (k0 R1 ) − kε0 Hn0 (k0 R1 )  0  0 M=  0   0 0 

Jn (k1 R1 ) k1 0 J (k R ) ε1 n 1 1 Jn (k1 R2 ) k1 0 J (k R ) ε1 n 1 2 0 0

and

 Hn (k1 R1 ) 0 0 0 k1  H (k R ) 0 0 0 ε1 n 1 1   Hn (k1 R2 ) −Jn (k2 R2 ) −Hn (k2 R2 ) 0  k1 k2 0 k2 0 0  0 H (k R ) − ε2 Jn (k2 R2 ) − ε2 Hn (k2 R2 ) ε1 n 1 2  0 Jn (k2 R3 ) Hn (k2 R3 ) −Jn (k3 R3 )  k2 0 k2 0 J (k R ) H 0 (k R ) − kε33 Jn (k3 R3 ) ε2 n 2 3 ε2 n 2 3

Far elds, near elds, and the lasing thresholds. The coecients A0n determines

where

Csca

and

Cext

are the scattering and ex-

tinction cross-sections respectively.

the amplitude of the scattered wave and can

The poles of

Csca

and

Cext

for each order

n,

be used to calculate the cross-sections per unit

can be found as the points in the parameter's

length of the cylinder:

space where the determinant of the matrix

Csca =

Cext

in Eq.

∞ 4ε20 X |A0n |2 k0 n=−∞

M

There, the elds associ-

ated with a given order

(15)

∞ 4 X = 2 Re{A0n } ε0 k0 n=−∞

14 cancels.

n

diverge.

This con-

dition corresponds to resonances with compensated optical losses and they are identied with the lasing threshold.

(16)

As we have already reported for spheres,

67

there is a proportionality relation between the



Page 6 of 22

near and far eld at the lasing condition. This

or divergences of the electric/magnetic elds in

can readily be understood by considering the

an arbitrary position. In principle, the last two

following reasonings. When the determinant of

strategies could be cumbersome because of dark

M approaches zero, all the coecients of modes ~ = M−1 B ~ . Let n and −n goes to innity as X n us recall that as j−n = (−1) jn and H−n = (−1)n Hn , the coecients aj,n and aj,−n should

modes or the presence of nodes in the harmonics

be equal.

ally small, which can safely be marked as dark

n = 0). The same can be Csca and Cext , and

sections

A0n

modes as

done to the cross-

gences of

the electric elds

ample, in the upper panel of Fig.

Csca

easily found numerically. For ex-

~ 2 E0n ,

Then,

Eθ = −

F

n

A0n almost zero, just

merically as divergences of

Csca .

Modeling the optical active medium The

solution of the scattering problem discussed in

∂F . ∂r

the previous sections requires the value of the 0 00 complex dielectric constant ε = ε + iε of each

(17)

component for the wavelengths of interest. The

the square modulus of the scat-

imaginary part of it reects the amount of ab-

tered electric eld, at the lasing condition of modes

9 one can

over the y-axis of the gure, can be found nu-

function as

−inF r

re-

than the average value per mode, show diver-

can be written in term of the

Er =

A0n

is orders of magnitude smaller

check that even poles with

E

strictly zero.

Fn and F−n (or F0

According to Eq. 7 the component of the electric eld

A0n

In our case, even modes with values of

2 can be replaced by the

summation of only two terms

~j. E

lematics only for modes with

Then, at the lasing condition the

summation in Eq. for

respectively. However, dark modes can be prob-

−n, results in: 2 ~ 2 2 ~ E0n = |A0n | Mn

sorption or emission of light. In our treatment 00 and due to the temporary dependence used, ε

and

has a positive sign for an active medium and a (18)

negative sign for an absorbing medium.

The

Clausius-Mossotti relation determines the di-

where

electric constant of a mixture of molecules in terms of the number of molecules of each type

2 ~ 2 2 Mn = k02 |Hn−1 | + |Hn+1 |  2  2 cos (nθ) n even × 2 sin2 (nθ) n odd  1 n=0 2

per unit of volume larizability

Ni

Ni αi (ω) ε−1 X = 4π ε+2 3 i

dye) dissolved in a non-absorbing host medium (subscript h), the rst order Taylor expansion in Ndye /Nh leads to:

2∂x Zn (x) = Zn+1 + Zn−1 and (2n/x)Zn (x) = Zn−1 − Zn+1 , where Zn is Jn or Hn .

dye (subscript

the Bessel's and Hankel's functions,

Comparing Eqs. 15 and 18 gives the relation

E0

Note that,

Csca . 2 k ~ 2 ~ E0n = 2 Csca M n 4ε and

ε ≈ εh + (19)

4πNdye αdye (ω)

(21)

68

which,

assuming a complete population inversion of the

goes to innity

the same should happen in every region. Then, the lasing thresholds can be found as zeros of the determinant of Eq. 14, divergences of

2

modeling the polarizability of the dye,

are connected due to the continuity condition,

0

2 + εh 3

One can take a simple two-level system for

as the elds in dierent regions

when the elds in the region

(20)

Considering a diluted solution of an excited

We have used above the recurrent relations of

between

and with molecular po-

αi :

Csca ,


Page 7 of 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


dye and the sign convention for

α (ω) =

|µ12 |2 ~

ε00 ,

One additional approximation that is nor-

results in

mally used to describe an optical active medium

i γ2

ω0 − ω + 2 2 γ 2 1 + 2 ωγR (ω0 − ω) + 2

is the wideband approximation, which comes from taking

|ω − ω0 | γ/2.

In this case Eq.

ε = εh + iεl .

(25)

23 turns into

(22) where

µ ~ 12

is the transition dipole moment be-

tween the ground and excited states with an en-

It should be mentioned that the wideband

∆E = ~ω0 , γ is a phenomeno ~ /~ is logical damping term, and ωR = µ ~ e→g · E

ergy dierence of

approximation

does

not

Kramers-Kronig relation

comply

31,32,68

with

the

nor does it take

the Rabi frequency. As can be noticed, Eq. 22

into account the spectral line of any particular

takes into account saturation eects. However,

dye. However, at a maximum of absorption or

in the present paper we will not take them into

emission, that is to say at the resonant wave-

account for the following reasons: Given that

length of the dissolved dye, it happens that the

the electromagnetic elds inside the structures

contribution to the imaginary part of the refrac-

analyzed (with a large radius) is not homoge-

tive index presents a maximum while the real

neous, the inclusion of saturation eects will

part goes to zero. Under this condition Eq. 25

introduce spatial inhomogeneities into the di-

becomes exact.

electric constant of the active medium.

This

is useful for a rst exploration of the system as

will break the symmetry of the problem, mak-

it allows one to calculate the optical response

ing the expansion of the elds in Eq. 2 not very

of a system independently of the characteristic

useful, as now additional boundary conditions

of the active medium. Then, after choosing the

are required, see section Boundary conditions.

poles of

In such a case, the present approach does not of-

selected and a better approach can be used.

references,

67,69

to be used, a specic dye can be

We used water as the host medium,

fer advantages over other methods such as nite elements for example.

Csca

The wideband approximation

67

εh = 1.77,

Lasing conditionsin most of the calculations

As discussed in several

except in Fig.

not taking into account satura-

6 where we also use ethanol,

tion eects has the consequence that elds go

εh = 1.85.

to innity once optical losses are compensated.

ent dyes were adjusted from their emission spec-

However, this divergences can indeed be used

tra using the Füchtbauer-Ladenburg method as

to nd the lasing conditions, which is the main

described in the next section.

21 and Eq.

γ

and

Parameters of the dyes.

point of the present work. Combining Eq.

The values of

εl

for the dier-

While the spec-

tral shape of uorescence light is relatively easy

22 (neglecting

to measure, it is much more challenging to mea-

saturation eects) results in

(ω0 − ω + i γ2 ) γ2 , ε = εh + εl (ω0 − ω)2 + ( γ2 )2

sure its absolute values due to the diculty of measuring accurately some quantities such as

(23)

doping concentration, the degree of electronic excitation, collection and detection eciencies,

εl is the maximum value of the imaginary 00 of the dielectric constant ε ! 2 2 + εh 8πNdye |µ12 |2 εl = (24) 3 ~γ

70

where

etc.

part

used to obtain the absolute scaling of the emis-

The Füchtbauer-Ladenburg method

sion cross-section spectrum.

is

In this method,

one exploits the fact that the excited-state lifetime is close to the radiative lifetime

τrad ,

which itself is determined by the emission crossNote that, as we are assuming a complete popu-

section

σ(ω)

for transitions to any lower-lying

lation inversion of the dye, every molecule con-

energy level.

This is quantitatively described

tributes to the stimulated emission of radiation, thus

Ndye

is directly its concentration.



to a Lorentzian function

by the equation

1 τrad

Page 8 of 22

2εh ≈ 2 πc

Z

ω 2 σ(ω)dω.

I(ω) =

(γ/2)2 . (ω0 − ω)2 + (γ/2)2

Knowing the spectral intensity of uorescence

Since typically the width of the emission spec-

I(ω), proportional to σ(ω), and using the above

trum

the integrand of Eq. 26. This yields

I(ω) πc2 R σ(ω) = 2 2εh τrad ω I(ω)dω Typically, the spectra of the dyes

(26)

σ(ω) =

I(ω) are tted

Table 1: Parameters of dyes used in Fig. 3. state required to obtain

εl = 0.05.

Ni

and

The values of

71

γ,

c2 εh τrad ω02 γ

I(ω).

(27)

is the concentration of molecules in the excited

τrad

and the spectra of the dyes, from which we

were taken from Ref. The values of µ12 are calculated from Eq. 30. Values in 14 the third and fourth columns are in 10 Hz, or in nm between parentheses. obtained

ν0

γ

is much smaller than the position of the 2 resonance ω0 , one can take the ω factor out of

equation one nds

τrad

Alexa-uor-680 Oregon-green-488 Atto-rho3b Atto-725 cy3b tCO

|µ12 |

ν0

γ/2π

1.200

4.27 (702)

0.226 (13265)

6.847

6.781

4.100

6.03 (497)

0.331 (9057)

2.203

0.480

1.500

5.03 (596)

0.336 (8923)

4.786

2.231

0.500

3.96 (757)

0.266 (11270)

11.843

17.262

2.800

5.37 (558)

0.375 (7995)

3.171

0.878

3.000

6.40 (468)

1.248 (2402)

2.358

0.146

(ns)

On the other hand, the relation between the

the value of

µ12

68

4πεh ω Im[α(ω)]. c

σaligned (ω) =

|µ12 | = (28)

value

orientation-averaged,

of

the

be obtained by inserting Eq.

σmax ,

.

(30)

The dyes used in this work, see Fig.

3, were

reported by the manufacturer, the values

22 into 28 with

and

γ

71

τrad of ν0

obtained from the tting of the reported

spectra, the values of

σaligned (ω0 ) 8π |µ12 |2 εh ω0 = . 3 3~γc

1/2

Table 1 show the dyes used, the values of

can

ω = ω0 , σmax =

3~ 8πε2h τrad

and radiative lifetimes are easily accessible.

macroscopic,

cross-section

3/2

quantum yields, and their uorescence spectra

between the electric eld and the molecule. maximum

c ω0

chosen because they are soluble, present high

where one is assuming a perfect alignment The

Ni (µM )

that ts the experiments,

emission cross-section and the polarizability (in cgs) can be written as

(D)

µ12

obtained from Eq. 30,

and the concentration required to reach (29)

0.05.

The value of

εl

εl =

used ensures the lasing

condition over all the frequency range studied. Note that the calculation of the concentrations

Comparing Eqs. 27 and 29 allow us to obtain

of the dyes shown in the table, of the order of

µM ,

assumes a complete population inversion,

which may require a very intense pumping laser.


However, even if only 1 out 1000 molecules is

3

3

in the excited states the concentrations of the

0.8

dyes require to ensure the lasing conditions are

mM ,

which does not seem too

high considering that all dyes are highly soluble

0.8 1

y(µm) |E| (a.u.)

of the order of

1 y(µm)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.4

−1

0.4

−1

|E| (a.u.)

Page 9 of 22

in water.

Lasing conditions Within the interval of pa-

−3

rameters explored, see section Results, the analyzed system presents two types of modes, the

1

0.0 −3

3

x(µm)

−3

−1

0.0 1

3

x(µm)

Left panel - Near eld of a typical

cavity mode close to its gain threshold, ν = 4.29 × 1014 Hz ε003 = 0.0124. Near eld of a typical whispering-gallery mode 14 close to its gain threshold, ν = 4.25 × 10 Hz ε003 = 0.1532. The geometry is that of Fig. 1

Right panel -

The former corre-

sponds to modes where the elds are mostly concentrated within the region of the active media and there is a wave pattern along the radius.

−1

Figure 2:

cavity and the whispering-gallery modes (CMs and WGMs respectively).

−3

The WGMs are modes where the elds

R3 = 2000nm, R2 − R3 = 25nm, and R1 − R2 = 475nm. The electric elds are normalized with

are concentrated close to the metal layer and there is a wave pattern along the circumfer-

to their respective maximum values.

ence of the cylinder. Typical near elds of both types of modes are shown in Fig. 2. Note that

boundary between regions 2 and 3,

the lasing of WGMs turns the system into a spaser, where the light is highly conned close face plasmon polaritons.

On the other hand,

un,i

and due to the elds distribution, the lasing of

where

CMs turns the system into what can be considered a metallic-coated nanolaser (o microlaser

Jn Bessel's (mh is the

depending on the sizes).

active medium),

11

c un,i mh R3

ωn,i =

to the metal layer due to the excitation of sur-

Therefore, the pro-

is the

r = R3 ,

i-th

i.e.

(31)

non trivial zero of the

function, we are taking

m3 ≈ mh

refractive index of the host of the

R3

is dened in Fig.

1, and

posed system has the peculiarity of being, at

ωn,i

the same time, a spaser and a metallic-coated

call this approximation Bessel's zero approxi-

nanolaser, depending on which mode is lasing.

mation (BZA). When the radius of the cylinder

is the resonant angular frequency. We will

As we will see in section Results, the cavity

is large with respect to the wavelength, one can

modes present far lower gain thresholds, which,

use the asymptotic forms of the Bessel's func-

due to the large number of poles, makes the

tions to describe the radial component of the

whispering-gallery modes unreachable within

eld of each mode.

the visible spectrum, for the main geometry

Jn

studied. For that reason, we will focus on the

ω π π mh r ∝ cos mh r − n − c c 2 4

ω

lasing conditions of cavity modes.

Lasing frequencies of cavity modes.

Under this approximation, the resonant fre-

To

quencies result simply

nd the resonant frequencies of the system we will assume a cavity-like model where the stand-

ωn,p =

ing wave pattern arises from that of waves conned in a cylindrical cavity. This is justied by the gold layer surrounding the active medium.

where

Under this model and neglecting the imaginary part of

k3 ,

Jn

p

cπ 4mh R3

(2p + 2n + 1)

is an odd integer.

(33)

We will call this

Fabry-Pérot approximation as it coincide with

the resonant condition occurs when

the argument of the Bessel's function

(32)

the resonances of a Fabry-Pérot interferometer.

corre-

In Eqs. 31 and 33 we assumed the elds at the

sponding to region 3 (k3 r ) becomes zero at the

boundary of regions 2 and 3 were strictly zero and neglected the imaginary part of


k3 ,

which


Page 10 of 22

is not the case for realistic systems. To correct

the same cavity-like model as before. We start

the formulas we propose a minimum model that

by the gain-loss compensation condition,

only adds a constant shift

δνcorr

to the lasing

Wabs = −Wsca

frequencies,

ν=

cun,i + δνcorr . 2πmh R3

where

(34)

Wabs

and

Wsca

(35)

are the energy absorbed

and scattered per unit time. The former can be calculated from

Note, that, because of consistency, the same correction should be applied to both Eq. 31 and Eq 33.

As we will see, introducing

δνcorr

Wabs

im-

proves considerably the agreement with numerical results but, in exchange, now one should

a priori

know

at least one lasing frequency. De-

Z 2 c 00 ~ = − ε3 k E dV 8π V∈ 3 Z 2 c 00 ~ − ε2 k E dV 8π V∈ 2

(36)

spite this, one is characterizing the whole spectrum of resonant frequencies by a single parameter.

where indexes

Withing the Fabry-Pérot approximation, the proportionality between

ω

and

in Fig. 1. The energy scattered per unite time

p causes the dif-

Wsca

can be calculated from

ferent modes to be equispaced in frequency. Considering the geometry studied,

1.33

m3 ≈ mh =

Wsca

R3 = 2.0µm, the gap between two contiguous zeros (p and p + 2) belonging to the same mode n is and

c ' 0.56 × 1014 Hz. νp+2 − νp = 4 × (8mh R3 )

trum, for

n

n

dominate,

small. This large amount of poles

2

emission cross-section.

ε003 =

Given the sizes of the tubes used in the calculations, where the largest one has a radius of where

400 and 800 nm, deviations respect to the

En

R

|E~ n |

V ∈2 l|A0n |2

We

factor

8

dV

2

|E~ n |

,

(38)

dV |2

is the electric eld produces by the

excitations of modes

will analyze this in more details in section Re-

n

and

−n.

should be replaced by a

n=0 factor 4. For

the

The integrals over the electric eld in Eq. 38

sults and discussion.

Gain thresholds of cavity modes.

8ε0 − R

ε002

V ∈3 l|A0n

nm, and the wavelengths considered, be-

Fabry-Pérot approximation are expected.

n and −n.

Using Eqs. 36 and 37, one 00 can write the required condition for ε3 to reach the gain threshold. For n 6= 0 it results in

mode within the visible spec-

some resonance close to the maximum of its

tween

(37)

ing condition, two coecients of the summation

shows that, given a dye, there will always be

2000

∞ c 4lk02 X = |A0n |2 8π k n=−∞ ( 1 n 6= 0 c ≈ lε0 k |A0n |2 × 1 π n=0 2

where we are using the fact that, at the las-

This means that there are approximately 7 poles for each

2 and 3 refers to regions 2 an 3

can be written in terms of Bessel's and Hankel's

To nd

functions. The result for

an expression for the gain threshold we will use


n 6= 0

is

Page 11 of 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


2 ~ En

Z V ∈2

l |A0n |2

ZR2 " dV

= 2π R3

|a2n |2 2 2 |J (k r)| + |J (k r)| n−1 2 n+1 2 |A0n |2

A2n a∗2n ∗ ∗ +Re Hn−1 (k2 r) Jn−1 (k2 r) + Hn+1 (k2 r) Jn+1 (k2 r) |A0n |2 # |A2n |2 2 2 + |k2 |2 rdr 2 |Hn−1 (k2 r)| + |Hn+1 (k2 r)| |A0n |

and

2 ~ En

Z V ∈3

|a3n |2 dV = 2π l |A0n |2 |A0n |2

ZR3

|Jn−1 (k3 r)|2 + |Jn+1 (k3 r)|2 |k3 |2 rdr

and

Hn .

For

n=0

a

1/2

(40)

0

where we have used the recurrent relations of

Jn

(39)

dierent systems.

factor should be

Eq.

added to the right-hand side of both equations.

For examples, according to

41, concentrating the elds on the active |a3n |2 ) should reduce

medium region (increasing

As we are dealing with cavity modes where

the gain thresholds.

the electric eld is concentrated mostly within the cavity, one can assume that the integral of |E|2 over region 2 in Eq. 38 is negligible. Then,

RESULTS

ε003 ≈ R R3 0

4ε0 |A0n | π|a3n |2

2

Using the methodology described in the previ-

. |Jn−1 |2 + |Jn+1 |2 |k3 |2 rdr

(41)

ous sections, we study dierent geometries of multiwall micro and nanotubes. In all the gures, except in Figs.

Note that the right hand side of Eqs. 38 and 00 41 depends on ε3 , as the imaginary part of k3 does. Therefore, the equation have to be solved

Fig.

Given a value of

k3 ,

The upper panel of Fig. 3 shows the scattering cross-section

34 to estimate and thus

k0 ,

ajn

and

Ajn

Csca

as a function of the imagi-

nary part of the dielectric constant of the gain 00 medium ε3 and the frequency of the incident light ν . As discussed previously, the poles of

one

can solve the set of linear equations given in Eqs. 8-13 to obtain the

We rst used the wideband approxi-

mation, Eq. 25, to nd the poles of the system.

that, is to use the BZA discussed in the previous

k3 ∈ Re.

1 and discussed in section General re-

marks.

One approximation that can be made to avoid one can use Eq.

8 and 9, we used, as a

representative example, the geometry shown in

either iteratively or by an optimization method.

section, i.e.

DISCUS-

SION

Eq. 38 turns into

AND

coecients

Csca

as well as the integrals involving the Bessel's

correspond to the full loss compensation

condition, or the lasing condition. These con-

and Hankel's functions in Eq. 38 (or 41).

ditions are dicult to visualize in this gure

Despite the dierent approximations that one 00 can make to estimate the value of ε3 , which as we will see do not work that well, Eqs.

due to the large number of poles that the sys-

a priori

tem present. In the lower panel of Fig. 3, we show only the position of the poles, to better

38 and 41 can be useful for other reasons, e.g. to

visualize them.

nd the lasing conditions by directly applying

To nd the position of the poles, we used a

an optimization method on Eqs. 38. Further-

simplex

more, the equations can also help to rational-

72 00

algorithm to maximize the function

Csca (ν, ε ),

ize the observed trends in the gain threshold of


starting from the maxima of

Csca


750

5

600

λ(nm)

λ(nm) 450

740

5

700

660

100

3 10−2

1

Csca (cm)

10−2

ε′′3 (102 )

3

Csca (cm)

ε′′3 (102 )

100

1 10−4

4

5 6 ν(1014 Hz)

750

600

λ(nm)

10−4

7

4

4.2 4.4 ν(1014 Hz)

4.6

λ(nm) 450

740

5

700

660

100

n=0 4

n=2 n=3

2

n=4

4.0

5.0

6.0

10−2

1

n=5 0

3

10−4

7.0

4

ν(1014 Hz) Figure 3: section

Upper panel -

Csca

as function of the imaginary part

imation, Eq. 25.

Lower panel -

the upper panel but Eq.

Csca

The same as

was calculated using

The 00 Gray lines in both gures are the values of ε3 obtained using Alexa-uor-680, with εl = 0.05.

up to 5 only, obtained within the

wideband approximation. Gray lines shows the 00 value of ε3 resulting from using dierent dyes, with εl = 0.05. From left to right, "Alexa-

23 with Alexa-uor-680 as the dye.

quantity of poles that can be achieved with rel-

uor-680", "Oregon-green-488", "Atto-rho3b", "Atto-725", "cy3b", and "tCO".

4.6

Upper panel -

Lower panel n (−n)

4.2 4.4 ν(1014 Hz)

Figure 4: Csca as function of 00 ε3 and ν obtained using the wideband approx-

Scattering cross-

of the dielectric constant of the gain medium ε003 and the frequency of the incident light ν . Position of the poles of Csca , for

Csca (cm)

n=1

ε′′3 (102 )

ε′′3 (102 )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 22

atively low gains, lower than

See section

0.05.

In the lower

panel of the gure, we only show the poles up to

Parameters of the dyes.

order

5 for clarity, but in total there are around

200 poles within the visible spectrum. Superobtained from a systematic variation of

ν

and

imposed on the gure, we also show the imag-

To distinguish true divergences from simple

inary part of the dielectric constant of the ac-

maxima, we use the fact that in a pole the itera-

tive media resulting from using dierent dyes,

tive process will change drastically the value of

see section Parameters of the dyes. If we con-

Csca

sider that the dyes only contribute to the imag-

ε003 .

meanwhile in a maximum the value should

Csca .

We

inary part of the dielectric constant of the gain

consider a maximum as a true pole only if

Csca

medium, all the poles below a gray curve will

times its initial value.

be reached with the dye and concentration be-

converge without much changes in changes more than

100

We checked the poles found with this procedure

ing studied.

and conrm they are in perfect agreement with

obtain TE modes of lasing all along the visible

Eq. 38.

spectrum, and using moderate concentrations

The upper panel of Fig. 3 shows the enormous

As can be seen, it is possible to

of the dyes, see Table 1 in section Parameters


Page 13 of 22 Numeric Bessel’s zeros approx. Fabry-P´erot approx.

served in such cases.

tion of the dyes, the real part of the dielectric

450

constant is basically that of the host medium

p = 15 p = 13

5

600

p = 11

Re (ε) ≈ εh , while around 3% of |ε|.

λ(nm)

ν(1014 Hz)

p = 17

6

This is reasonable once

one realizes that, due to the low concentra-

p = 19

7

its imaginary part is small,

p=9

4 1

2

3

4

λ(nm)

750

p=7

0

750

600

750

450

600

450

10−1

5

Figure 5: Comparison of the values of

νpole

ε′′3

n ob-

10−2

tained numerically, with the BZA, Eq. 31, and the Fabry-Pérot approximation, Eq.

33.

For

Figure 7:

just above the corresponding blue lines.

panel -

4

λ(nm)

5

6

4

7

5

6

7

Left panel - Position of the poles of Right

Csca for all n (−n) obtained numerically.

of the dyes.

600

4

ν(1014 )Hz

in the Fabry-Pérot approximation are indicated

750

Bessel’s zero approx.

Numeric

10−3

both approximations we used a constant shift 14 of δνcorr = 0.23 × 10 Hz. The values of p used

The same, but poles are obtained us-

ing the Bessel's zeros approximation, i.e. ν is 14 calculated from Eq. 34 (δνcorr = 0.23 × 10 Hz) 00 and ε3 is calculated from Eq. 41 with k3 ∈ Re estimated from Eq. 34. The color of the dots

450

H2On=0 EtOHn=0

stands for the order and high

ε′′3 (102 )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


n

n, low n are plotted in blue

in red.

Fig.

5 compares the position of the poles 00 obtained numerically, maximizing Csca (ν, ε ),

2

with the positions predicted by the BZA, Eq.

0

31, and the Fabry-Pérot approximation, Eq. 33.

4

5 6 ν(1014 Hz)

7

As can be seen, the agreement with the BZA is almost perfect. On the other hand, the agreement with the

Figure 6: Example of the variation of the position of the poles induced by a change in

εh .

Fabry-Pérot approximation is not bad at low

n. Csca

n,

the agreement wors-

ens and modes with the same value of

It is interesting to compare the position of the poles of

However, at higher

p

signi-

cantly deviate from the predicted linear depen-

obtained with the wideband

n.

approximation with that obtained with the ex-

dence with respect to

plicit consideration of a particular dye, using

ple approximation estimate, surprisingly well,

Eq. 23. This is done in Fig. 4. As expected,

the frequency separation between modes with

when the resonant frequency of a pole

νpole

the same

is

of

close to the maximum of the emission cross-

n

but dierent

Despite this, this sim-

p,

even at high values

n.

section of the dye, the results for both approx-

Note that in Fig. 5, only one parameter has

νpole

been used to adjust both approximations, the

is quite far from the maximum, the wideband

global correction to the frequency of the poles

imations coincide.

However, even when

δνcorr .

approximation seems to work pretty well espe00 cially at predicting the values of εpole . Only small deviations in the values of νpole were ob-

Besides its obvious application for enhanced spectroscopies, the type of system proposed can



Page 14 of 22 1

7 6 600

4

b) 2000/500

a) 2000/25/475

0.8

0.6

750

ε′′3

5

λ(nm)

ν(1014 Hz)

450

7 450 6 600

5 4

d) 400/100

c) 400/5/95 0

10

20

n

30

40

0

10

20

n

30

40

0.4 λ(nm)

ν(1014 Hz)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

0.2

750 0

50

ν and ε003 , found numerically for dierent geometries and modes n. The crosses are the values of ν obtained using the BZA, Eq. 31. Geometries are indicated above each gure, e.g. panel (a) corresponds to R3 = 2000nm, R2 − R3 = 25nm, and R1 − R2 = 475nm. The values of δνcorr used are 0.23, 0.13, 0.92, and 0.90 14 (10 Hz) for panels (a), (b), (c), and (d) respectively. In Fig. (a) and (c), lled dots without crosses nearby (at large n) correspond to poles of whispering-gallery modes. Figure 8: Filled dots corresponds to the position of the poles,

be used for sensing, through changes in the di-

global shift used to correct Eqs.

electric constants.

36,73,74

31 and 33,

In this case, the fact

δνcorr , depends not only on the geometry of the

that the width of the peaks in the spectrum

system but also on its dielectric constants. For

of micro- and nanolasers are very narrow, is a

example, according to Eqs.

great advantage for the sensitivity of any tech-

the position of the peak at the higher frequency 14 should be 0.23 × 10 Hz, but the actual value is 0.162 × 1014 Hz.

niques based on the shift of peaks. Fig. 6 shows the large shift of the poles with

n=0

caused

by a slight change in the dielectric constant of the host medium

εh ,

from

1.77

(water) to

Fig.

1.85

BZA for to the calculation of both, ν (Eq. 34 δνcorr = 0.23 × 1014 Hz) and ε003 (Eq. 41 with k3 given by Eq. 34). As can be seen, the 00 predicted values of ε3 deviate signicantly for the lowest frequency modes where there is an 00 unrealistic wide spread of ε3 values. For inter-

From the values of Fig. 6, it is possible to es-

with

timate the refractive index sensitivity (RIS), widely used for sensor characterization and dened as the ratio between the resonance shift

∆λ

and the refractive index variation of the

∆n. 73,74

7 compares the lasing conditions ob-

tained numerically with those found using the

(ethanol).

analyte,

31 the change in

RIS is usually measured in

mediate and high frequencies (larger than

5Hz

nm/RIU, where RIU is the abbreviation of re-

approximately) the dierences are smaller and

fractive index unit. In Fig. 6, the RIS ranges a relatively large value according to the bibli-

at least the BZA estimates well the order of 00 magnitude of ε3 . The reason for the large errors is that both, the coecients ajn and Ajn ,

ography.

In principle Eq. 33 could have been

and the integrals over the Bessel and Hankel

used to predict the variation of the position of

functions, are very sensitives to the exact com-

the peaks, and the values of RIS. However, this

plex value of

from

305

to

73

648

nm/RIU for

n = 0,

which is

does not lead to good estimations, since the

k3 .

To compare the eect of the geometry of the


Page 15 of 22

system on the lasing conditions, in Fig.

8 we

indicates that the numerator of Eq.

41 varies

show the position of the poles corresponding

signicantly with the mode and the geometry of

to four dierent geometries.

The geometry

the system, or that the power dissipated by the

2000/500 is the same as 2000/25/475(the

gold layer is not completely negligible. By in-

geometry studied so far) but the gold layer

spection of our results, we concluded that both

has been replaced by a thicker polymer layer.

contributions are responsible for the deviations.

The

For WGM, it does not seem to be a correlation 2 2 00 between |A0n | /|a3n | and ε3 , see upper panel

geometry

400/5/95

is

the

same

as

2000/25/475 but rescaled by a factor 5. The same is true for the last geometry, 400/100, with respect to 2000/500.

In Fig.

crosses indicated the values of

ν

8, the

power dissipated by the gold layer plays a cen-

predicted by

tral role in determining the gain thresholds in

the BZA while the squares are the values of found numerically.

of Fig. 9, which is natural considering that the

ν

this case.

In panels (a) and (c) the

0.6

squares without crosses nearby correspond to WGMs. As the gure shown, the BZA is able to predict the resonant frequency

ν

of all the

0.4

cavity modes, but only when there is a metal For systems without a metal layer, see

ε′′3

layer.

panels (b) and (d), the BZA can also work but

0.2

only in limited spectral regions. As mentioned

CM 2000/25/475 WGM 2000/25/475 CM 400/5/95 WGM 400/5/95

at the beginning of section Lasing conditions, the WGMs have much higher gain thresholds

0

than those of cavity modes for the main geom-

0

0.2

0.4

0.6

etry studied, see panel (a) of Fig 8. However, for smaller systems, the increase of the gain

1

1.2

1.4

1.2 CM 2000/25/475 2000/500 CM 400/5/95 400/100

thresholds of cavity modes makes them comparable. This is better appreciated in the upper

0.9

panel of Fig. 9 where one can check that the 00 values of ε3 of cavity modes and WGMs are comparable for the geometry 400/5/95. Figs.

0.8

|A0n /a3n |2

ε′′3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


8 can also be used to analyze the ef-

0.6 0.3

fects of dierent geometries on the gain threshold of cavity modes. Broadly speaking, the g-

0

ure shows that the gold layer reduces the gain

0.1

thresholds, while reducing the size of the system

1

10

100

|A0n /a3n |2

increases the gain thresholds. This is better ap00 preciated in Fig. 9 where we plot ε3 as function 2 2 of |A0n | /|a3n | , which can be taken as an indi-

Figure 9: The imaginary part of the dielectric 00 constant of the active medium, ε3 , as a function 2 2 of the connement, calculated as |A0n | /|a03 | ,

cator of the connement of the elds. Although

a rigorous comparison of gain thresholds is di-

for dierent geometries and type of modes, in-

cult for dierent geometries, since the resonant

dicated in the insets.

frequency of the same modes changes with the geometry, the general trend for cavity modes (CM in the gure) is the more conned the

CONCLUSIONS

eld, the lower the gain threshold or, to be 2 2 more precise, smaller values of |A0n | /|a3n | are 00 correlated with smaller values of ε3 . This is in

The studied geometry presents two types of TE modes:

agreement with the expected trend of Eq. 41,

cavity modes and whispering-gallery-

modes. Within the visible spectrum, there are

although the lack of a proportionality relation



Page 16 of 22

a large number of both. Cavity modes present 00 −2 very low gain thresholds (ε3 around 10 ) while the WGMs exhibit gain thresholds an order of

found that increasing the size of the systems,

magnitude higher.

The small separation be-

as increases the number of lasing frequencies

tween consecutive cavity modes and their much

within a given frequency interval. We have also

smaller gain thresholds makes WGMs hard to

shown that adding even a very small layer of a

use for lasing. The reason for that is that the

plasmonic material such a gold (a

nite width of the emission cross-section of the

in a microtube whose total

active medium will make the population inver-

dramatically reduces the gain threshold of the

sion of the active medium will be depleted by

nanolaser (up to one order of magnitude). The

some cavity mode way before the lasing condi-

eect of the geometry on the gain thresholds

tions of a WGM is reached.

Therefore, for the

can be rationalized, at least qualitatively, in

type of systems studied here, only in tubes with

terms of the connement of electromagnetic

radii in the nanoscale, the WGMs may be used

elds into the region of the active media.

67

Comparing

dierent

geometries,

we

have

considerably reduces the gain threshold as well

for lasing.

25nm layer radius is 2.5µm)

Contrary to the longitudinal modes of las-

The very low gain thresholds of cavity modes

ing, which only emits plane waves, the elec-

imply that the required concentration of the −6 dyes can be as low as 10 M . On the other

tromagnetic elds emitted by the TE modes

hand, the geometry studied allow one the re-

nanostructures. Those hot spots are relatively

cycling of the active medium and thus can be

large and can easily accommodate large macro-

used to counteract the eects of photobleach-

molecules or even greater analytes such as en-

ing, an eect that may limit the practical ap-

tire cells. Moreover, the electromagnetic elds

plications of other proposals such as active core-

can be further enhanced with the aid of ad-

shell-nanoparticles.

sorbed nanoparticles, if required. In micro- and

present hotspots without requiring additional

In the geometry studied, the frequencies of

nanolasers, longitudinal modes can in principle

lasing can always be tunned by changing the

interfere with the TE modes since they com-

geometry of the system but also by changing

pete for the excited states to produce stimu-

the dielectric constant of the gain medium, us-

lated emission of radiation. However, their gain

ing dierent solvents for example.

The latter

threshold can be independently changed by con-

can also be useful for sensing purposes. On the

trolling their connement, e.g. by leaving or not

other hand, giving that the system posses so

the tubes opened at the ends or using absorb-

many tunable resonances, it should not be dif-

ing materials there. A possible future research

cult, in principle, to tune a system where the

direction is to compare, for the same system,

maximum of the absorption cross-section of the

the lasing conditions of longitudinal and trans-

dye and the frequency of the pump laser coin-

verse modes, addressing explicitly the problem

cide with one resonance, while the maximum of

of competition for the excited states.

the emission cross-section of the dye coincides

In this work, we used the divergences of the

with another one. This should greatly enhance

scattering cross-section to nd the spasing con-

the performance of the devices.

ditions, but it would be computationally useful

We have proposed a simple model that can

to compare this strategy with other methods,

accurately predict the wideband limit of the

e.g. using point-dipole emitters to look for di-

frequencies of lasing for the geometry studied.

vergences of near elds, or directly solving itera-

This may result especially useful in the design

tively Eq. 38, which just express compensation

of the devices.

of gains and losses.

On the other hand, we have

Moreover, for theoretical

shown that the deviations from the wideband

completeness, it is important to study the scat-

approximation are not so important even for

tering of multiwall nanotubes for oblique illu-

modes whose frequencies do not coincide with

mination.

the maximum of the emission cross-section of

Although the system studied seems promising

the dyes.

for several applications such as sensing or as


Page 17 of 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


part of a lab-on-chip device, more studies are

W, T. Structural engineering in plasmon

required to evaluate, for example, the intensity

nanolasers.

of the electromagnetic elds attainable for the

2881.

proposed geometry, or the true potential of the

Chem. Rev. 2017, 118,

2865

(7) Berini, P.; De Leon, I. Surface plasmon

system for sensing purposes.

polariton ampliers and lasers.

Ultimately it would be important to experi-

ton. 2012, 6, 16.

mentally test the studied geometry to contrast the theoretical results and, more importantly,

Nat. Pho-

(8) Hill, M. T.; Gather, M. C. Advances in

the performance of the microuidic-based recy-

small lasers.

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Nat. Photon. 2014, 8, 908.

(9) Törmä, P.; Barnes, W. L. Strong coupling between surface plasmon polaritons and

ACKNOWLEDGEMENTS

emitters: a review.

78, 013901.

This work was supported by Consejo Nacional de Investigaciones Cientícas y Técnicas (CON-

(10) Amendola, V.;

Rep. Prog. Phys. 2015,

Pilot, R.;

Frasconi, M.;

ICET), Argentina; Secretaría de Ciencia y Tec-

Marago, O. M.; Iati, M. A. Surface plas-

nología de la Universidad Nacional de Cór-

mon resonance in gold nanoparticles: a re-

doba (SECYT-UNC), Córdoba, Argentina; and

view.

Universidad de Buenos Aires,

203002.

Project UBA

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20020100100327, Buenos Aires, Argentina. (11) Ma, R.-M.; Oulton, R. F. Applications of nanolasers.

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