MODELING OF SPONTANEOUS EMISSION IN PRESENCE OF CYLINDRICAL NANOOBJECTS: THE SCATTERING MATRIX APPROACH Текст научной статьи по специальности «Физика»

Modeling of spontaneous emission in presence of cylindrical nanoobjects:

the scattering matrix approach

V.V. Nikolaev1, E.I. Girshova1, M.A. Kaliteevski2 1 Submicron Heterostructures for Microelectronics Research and Engineering Center of the Russian Academy of Science, 194021 St Petersburg, Russia, Politekhnicheskaya, 26;

2ITMO University, 197101, St. Petersburg, Russia, Kronverksky Pr., 49

Abstract

We propose a method of analysis of spontaneous emission of a quantum emitter (an atom, a luminescence center, a quantum dot) inside or in vicinity of a cylinder. At the focus of our method are analytical expressions for the scattering matrix of the cylindrical nanoobject. We propose the approach to electromagnetic field quantization based of eigenvalues and eigenvectors of the scattering matrix. The method is applicable for calculation and analysis of spontaneous emission rates and angular dependences of radiation for a set of different systems: semiconductor nanowires with quantum dots, plasmonic nanowires, cylindrical hollows in dielectrics and metals. Relative simplicity of the method allows obtaining analytical and semi-analytical expressions for both cases of radiation into external medium and into guided modes.

Keywords: spontaneous emission, scattering matrix, cylindrical symmetry.

Citation: Nikolaev VV, Girshova EI, Kaliteevski MA. Modeling of spontaneous emission in presence of cylindrical nanoobjects: the scattering matrix approach. Computer Optics 2023; 47(1): 16-26. DOI: 10.18287/2412-6179-CO-1143.

Introduction

Modification and control of spontaneous radiation of quantum emitters is a hot topic of recent decades [1 - 12]. The aim of spontaneous emission manipulation can be achieved with the structures with cylindrical geometry which became technologically accessible and widely researched. Such nanoobjects include semiconductor nanowires [5 - 9, 11 - 17], metallic wires with nanoscale diameters [10], cylindrically perforated metals and dielectrics. Potential applications lay in the arear of singlephoton sources [1 - 3, 5, 6], plasmonic metamaterials and nanolasers. Theoretical understanding of spontaneous emission process in such structures is of crucial importance.

Initially the theoretical consideration of spontaneous emission in cylindrical structures was limited to the problem of mode excitation in cylindrical waveguides with approaches based on classical electrodynamics [18, 19]. There were several theoretical methods published which account for radiation from inside a cylinder into external medium [9, 20 - 24, 26 - 28]. Chu and Ho approximated the cylindrical waveguide with a rectangular Fabry-Perot resonator [20]; Nha and Jhe used electrodynamic Green's function technique to calculate radiative-dumping constant of a quantum emitter [21]; Zakowicz and Janowicz used decomposition into cylindrical modes and Fermi's Golden rule with complicated mode quantization approach to resolve a problem of luminescence from dielectric cylinder [22]; Sondergaard and Tromborg [23] used formalism of Green's tensor and quantum-mechanical current operator to study spontaneous emission in active waveguides; Klimov and Ducloy [24] calculated radiation

of a classical electric dipole near dielectric nanofiber; Fam Le Kien et al. [25] concentrated attention on emission into the fundamental HE11 mode of cylindrical dielectric waveguide. In the paper of A.V. Maslov, M.I. Bakunov, and C.Z. Ning [26] a method based on Fourier integrals [18, 19] is generalized to allow calculation of a dipole radiation intensity from inside a waveguide or a finite-length dielectric cylinder; Henderson et al. [27] made another approach to the problem of classical dipole radiation in vicinity of a waveguide. We recently proposed a relatively simple method for calculation of spontaneous emission from dielectric nanowires which is limited to the case of radial light extraction (with no account for polarization mixing) [28].

Relatively recent advance in this field was made in the paper of Paniagua-Dominguez et al. [9] where external radiation from a dielectric cylinder was treated using so called leaky modes, i.e. continuation of guided modes in the region where these modes acquire imaginary part of propagation wavenumber and become unconfined. The method of Ref. [9] exploits semi-phenomenological approach of a one-dimensional (1D) optical resonator and the induced 1D current to account for the nanowire finate length. In the framework of this approach the researchers examine analogy between Mie resonances and leaky modes [29] and apply their theory to model angular distribution of quantum-dot luminescence from nanowires [30]. In Refs. [31, 32] the emission from nanowire systems with metallic parts are considered.

Despite the wealth of theoretical results on this subject and significant advances made, many researchers use purely numerical methods designed for calculation of a classical dipole radiation in arbitrary dielectric surround-

ings. Such situation might be due to complexity of existing quantum-mechanical methods and leaves limited space to analytical interpretation of numerical and experimental results. The aim of our work is to develop a relatively simple, rigorous and universal theoretical method and present analytical and semi-analytical formulas for spontaneous emission in guided modes and external space for arbitrary position of a quantum emitter in a system containing a cylinder. In the center of our approach is an analytical expression of the optical scattering matrix of a cylinder which could be metallic or dielectric with arbitrary refractive index. We propose a relatively straightforward method of quantization of the modes radiating into external space and solve the problem analytically using eigenvalues and eigenvectors of the scattering matrix of the cylinder. The treatment of the guided modes is consistent with that for radiating modes and is based on the same principles.

1. Optical scattering matrix of homogeneous cylinder

In this work we consider a quantum emitter placed in an arbitrary point inside or outside of an infinitely-long cylinder with the radius po. The external embedding medium is characterized by the values of dielectric permittivity ei and magnetic permeability |i at the emitter wavelength; and the cylinder is described by parameters e2 and |2. As a starting point we consider the mode structure of classical electromagnetic fields in a system containing an infinite cylinder. Monochromatic electromagnetic fields can be expressed in a form Re{E (r) exp (-irat)} where the real part is taken from the product of a coordinate-dependent complex- vector field E (r) and a time-dependent factor. If a structure possesses cylindrical symmetry in respect to the z axis, one may consider an electric field written in a cylindrical coordinate system (p, 9, z) as E(r)=Em (p) exp(i [kz + m<]) where kz is the wavenumber of propagation along the symmetry direction, m is an integer number standing for a photon angular momentum. In this work we consider modes which freely propagate along z axis, hence kz is a real number.

For the subsequent analysis the following dimention-less parameters are important:

q — Po ko, z — Po kz, x —-y/eiiiq2 - z2, X2

— Ve2|2q2 - z2,

(1)

where ko = ra / c is the wavevector of emitted light in vacuum, q is the size parameter, z is dimensionless propagation number and xi(2) are external (internal) transverse wavenumbers kpi — ^ ik2 - k2 multiplied by the radius po of the cylinder. Although in this work we consider nonmagnetic materials, we keep magnetic permeabilities in following expressions for the sake of completeness presuming that their values are real and positive. The dielectric permittivities are presumed to be real and can have an arbitrary sign; a negative value of permittivity is reserved for description of a metallic medium below plas-

mon resonance [io]. Thus, the dimensionless parameters xi, X2 can be either real or pure imaginary depending on the wavelength, the values of the propagating constant kz and the dielectric permitivities. In case the expression e,- |,q2- Z2 is negative one should choose the value of the square root from the upper half of complex plain.

The method which we develop in this paper allows to analyze several distinct types of structures and their operating regimes. The type I structure is characterized by the permittivity of the cylinder which is larger than the permittivity of the surrounding medium e2 > ei, both per-mitivities are positive. This type of structure corresponds to semiconductor nanowires and nanopillars which can be considered dielectric at the wavelength of a quantum emitter. In this case parameter X2 is real and positive whereas Xi is imaginary for guided modes and real for the modes radiating in the external medium.

We designate the opposite case e2 < ei as Type II structure; this model structure can be applied to cylindrical hollows (e2 = i for air permittivity) etched in a dielectric. In this case Xi is always real corresponding to the modes radiating into a surrounding dielectric whereas the parameter X2, which describes the internal field, can be ether real or imaginary.

Our method will also allow to model metallic structures below plasmon resonance where either e2 or ei are negative. The negative e2 and positive ei (Type III structure) corresponds to metallic nanowires [io] whereas the opposite case (Type IV) can be used to model cylindrical hollows in a metallic film.

If Xi is real, the field with fixed m and kz in the external medium can be written as a superposition of converging and diverging propagating waves of TM and TE polarizations:

EExt _ m —

Ext++A-E Exm-

(2)

where A+M[[TE are constant amplitude coefficients; the plus (minus) superscript stands for diverging (converging) waves. The axial component of magnetic (electric) field in TM (TE) polarized cylindrical wave is zero; the explicit form of the dimensionless field profiles EETMiTE,m can be found in Appendix A, Eqs. (Ai-2).

If parameter xi is imaginary the transverse wave-number kpi is imaginary as well and the fields outside the structure are changing exponentially with the distance from the axis. For the general description of the external fields we can use Eq. (2) with the convention that for imaginary xi plus (minus) superscripts stand for the fields exponentially increasing (decreasing) with the growth of the distance p from the structure axis; the field profiles for exponentially-changing waves are presented in Appendix A, Eqs. (A4-5). For physically meaningful guided modes exponentially increasing fields are absent.

Due to the requirement of continuity of tangential components of electric and magnetic fields on the cylin-

drical interface the four coefficients ATM-ite are connected to each other. One can write the relation between the plus and minus coefficients of two different polarizations in the matrix form:

ATt

\ AT

(3)

In this expression quantities r can be considered as analogues of reflection coefficients and the 2*2 matrix s composed of these coefficients is a form of the scattering matrix written in a particular basis. Using continuity conditions and the requirement of the finiteness of the fields on the structure axis we derived the explicit expression of the scattering matrix in a general form:

T(2)T(1) - m21 - | V2

Rm (1 )©

rTE^TE — "

rTE^TM — i

T(1T;(2) - m2 f - J V2

Rm(x)© : m^ s1;1 z V

(4)

Rm (X1 )© qQm (x )'

rTM^TE — rTE^TM ,

with the following designations introduced:

©— TS(1)T;(1) - m2 fzJ V2, V — — -—, \ q ) X22 X12

Ts(') — eF^ (X1 )-62Gm (X2 ),

T(i ) — ^F« (X1 ) - ;2Gm (X2 ).

(5)

These expressions contain functions which are defined differently for real and imaginary arguments. For real arguments these functions are:

H1 (X ) w

Rm (X)— Hf)(X) , Qm (X)— i 4 X2 H™ (X)HS) (X ),

G () 1 •/m(x) () 1 m» (x) °m (x )——Ti\, Fm (x)—■

(6)

Jm (X )

H« (x)

where H(m2> are Hankel functions. For a pure imaginary argument x=i | x | these functions are defined as follows:

Rm (X) = "TUT, 6m (X)=|xf Im (\x|)Km (\x|),

Km (|X|)'

Gm (x) — F® (X) — -i

/m(x| )

Im (XI)'

(7)

Fm2 (x )—- p

K,(X| )

where Im and Km are modified Bessel functions. One can see that for zero photon angular momentum (m = 0) or for direction perpendicular to the structure axis (kz = 0, z = 0) the scattering matrix is diagonal, which manifests absence of polarization mixing for such conditions [28].

In further analysis we would need to know a relation between the fields outside and inside the cylinder. The field inside the structure can be written as a superposition of field profiles of TM and TE polarizations:

EInt _ D Û Int _1 D Û Int

m — UTM^TM m^^TE^TE ,m-

(8)

Explicit expressions of field profiles ETMTTEm can be found in Appendix B, Eqs. (B1,2,5,6). We have found a matrix which connects internal-field coefficients Btm te to the external coefficients A-mite of converging or exponentially decreasing waves:

— B

ATM ate

BInt — —

1 ; 1

1

x zm2) (x1 )

2 V ;1 © Qm (X1 ) X2 Zl1 (X2 )

I—z ^ ; m £2;^ V

q

(9)

T;(1)

-imy] £1 ;2 — V yl;; T1 q

Here we introduced designations for the functions Z(m2) (x). For real arguments they coincide with the following Bessel functions:

Zm* (x ) — Jm (x ), Zm2 (x ) — Hm ) (x )

(10)

and for imaginary x these designations stand for the modified Bessel functions:

Z« (x) — Im (x|), Z™ (x) — Km (x|).

(11)

Similarly to the scattering matrix, the matrix BInt is diagonal for m = 0 or kz = 0, z = 0.

2. Spontaneous emission of a quantum emitter

Let us consider a quantum emitter placed at the point r0 with an optical transition characterized by the dipole matrix element df = |d/,|ed (ed is a complex vector of optical-transition polarization). Spontaneous emission rate can be calculated using Fermi's Golden Rule:

W — ^ ld À S Ie p (r0 ) •ed |2 8( i - hap ), (12)

where hrafi is the energy of the optical transition and Ep (r0) is the complex amplitude of the electric field of a quantized mode at the location of the emitter. The subscript p stands for the mode quantization numbers; the real electric field of the mode is calculated as Re{Epexp (-irapt)}. The optical mode should be quantized and nor-

X

malized so that the total electromagnetic energy inside a quantization volume is equal to the single photon energy hrap. Summation in Eq. (i2) is taken over all possible quantized optical modes.

It is natural for the considered problem to take quantization figure in a form of a cylinder which z axis coincides with the symmetry axis of the structure; the radius R and the height Lz of the quantization cylinder are put to infinity at the final stage.

On the top and bottom facets of a quantization cylinder the standard Born-von Karman periodic boundary condition can be used [2o] which discretizes the propagation wavenumber kz or the parameter z. The differential of the mode number connected with the light propagation along the axis is

dNz = —dkz. 2n

(13)

In the expression (i2) one can substitute summation over the discrete quantum number Nz with integration over the propagation wavenumber differential dkz.

2.1. Radiation into external medium

Emission of a quantum emitter into external medium can be analyzed as follows. If the conditions are such that the parameter xi and the transverse wavenumber kpi are real (see Eq. (i)), the field outside the cylinder is a superposition of converging and divergent cylindrical waves of TM and TE polarizations, Eq. (2). For spontaneous-emission calculation a quantization procedure for such modes should be devised.

We propose a simple and straightforward quantization condition: the electric field components tangential to the cylindrical surface of the quantization cylinder should be equated to zero. Such procedure has a distant analogy to quantization of electron-hole pairs in quasi-continuous part of spectrum (the Elliott formula).

Let us introduce an auxiliary matrix:

M = exp

2 R n

2 x1--mn--

Po 2

rTM ^TM rTE^TM

rTM

rti

(14)

where s can be obtained from the cylinder scattering matrix s Eqs. (3, 4) by multiplication of the lower row by -i. Using the relation between coefficients of convergent and divergent waves Eq. (3) and asymptotics of Hankel functions at large arguments one can show that the requirement of zero transverse electric field components at the quantization cylindrical surface in the limit R ^ ® is equivalent to equality of the matrix M eigenvalue to minus one:

exp

2 R n

2 x1--mn--

Po 2

: P ± = -1,

(15)

where p± are the eigenvalues of the matrix s:

P ± =

rTM^TM rTE^TE , rTM^TM + rTE^TE

D = 1-4-

'TE^TM'TM ^TE

(16)

0

"TM ^TM + rTE^TE

o2

In addition, for the quantized mode the pair of coefficients {A-M, A-e } is an eigenvector of matrix M and consequently is an eigenvector of the modified scattering matrix of the cylinder s. This conclusion allows to obtain relations between polarizations in the quantized radiating modes:

A- =- rTM ^te

rt-TE — _ " ± ATM?

rTE^TE + P~ A- =- Vte^tm A-

^TM — _ " ± ^TE-

rTM^TM - p±

(17a) (176)

Coefficients for the divergent waves can be expressed through the coefficients of the convergent waves using the scattering matrix Eq. (3). Thus we show that the proposed quantization procedure produces analytical expressions for the polarization mode structure which depends on the cylinder reflection properties (the analytical expressions of r coefficients for different types of cylinders were presented in the previous chapter, see Eq. (4, 8)).

Let us analyze the obtained results. If the conditions are such that mixing of TM and TE polarizations are weak or absent (when m = o or the radiation propagates perpendicularly to the axis, i.e. kz « o, z « o, 8 « n /2) then cross-coefficients go to zero Ftm^te ^ o, Fte^tm ^ o and one obtains

D ^ 1,

P + ^ rTM ^TM , P ^ -rTE^TE.

(18)

Thus at weak mixing conditions the quantized mode corresponding to the eigenvalue p+ transforms into a pure TM mode, whereas p- modes approach TE modes. Such transformation is typical for a cylindrical waveguide mode: EH-modes approach pure TM modes and HE-modes go to TE modes with the decrease of the waveguide dielectric contrast. Therefore we can label radiating modes at perpendicular propagation angle corresponding to p+ as EH (or TM-like) modes, and the quantized modes corresponding to pcan be considered as HE (or TE-like) modes. At an arbitrary propagation angle we choose the sign in Eq. (18) in such manner that the angular dependencies of p+(9) are everywhere continuous. Considering Eq. (17) one can see that the expression (17a) can be more conveniently used for EH modes and expression (176) suits HE modes since in this case correct polarizations are obtained in the vicinity of the perpendicular propagation angle. Therefore we can state that another quantum number designating external radiative modes labels polarization and can be either EH or

HE corresponding to the different eigenvalues p+ of the modified scattering matrix.

To be able to use these modes for spontaneous-emission calculation Eq. (12) one needs to normalize them:

f up (r)dV = ha , (19a)

JVc ^ '

Up (r)= I^(e(r)|Ep (r) + M(r)|Hp (r)), (19b)

where integration of electromagnetic energy density Up(r) over the volume Vc of the quantization cylinder is performed (in this work the Gaussian unit system is used). In case of modes radiating into external space this integral goes to infinity with the increase of the size of quantization cylinder. Therefore, for normalization procedure we can consider the asymptotic form of cylindrical waves at large distances which transform into local plain waves (see Appendix A, Eq. (A3)). Hence we obtain the normalization condition

| ATM I + IAT

|2 4ft x1

LZR P0S1

ha.

(20)

In (20) it is presumed that any light absorption or generation in the cylindrical structure can be neglected.

The quantization condition Eq. (15) can be analyzed as follows. In the absence of absorption or generation the absolute values of the modified scattering matrix eigenvalues are always equal to unity |p±|=1, which is the consequence of the energy-flux conservation. Thus Eq. (15) imposes restriction on the phase of the complex number on the right hand side. When the quantization-cylinder diameter R is large the main contribution into phase variation stems from the exponential factor whereas variation of the phase of the eigenvalue p± can be neglected. This gives us a quasi-discrete spectrum of the radiating modes, and the differential of the mode quantum number related to the transverse field distribution is:

dNp=

ft Po

(21)

As a result, the summation over photon modes in Fermi's Golden Rule can be expressed as

X ^X X XX ^

p m J = EH,HE Nz NP

X X 2fti

(22)

m J=EH ,HE

ft" Po

A number of radiating modes with fixed m and polarization J = EH / HE in an interval of photon energies [hap, hap + d(hap)\ can be written as:

dNm-J = dN pdNz = s1^1 RLr ^Oj d8d (ha p ), (23)

2ft'

(hc )2

where 8 is the angle between the symmetry axis and the propagation direction of cylindrical waves at large dis-

tances (tan 8 = xi/z = , see Appendix A, Eq. (A3)). To obtain the analogue of the photonic density of states Eq. (25) from Eq. (24) we used the dispersion relation between the light wavevector and frequency in the external medium k = Jk21~+kf = ^/s^a/c with the help of which an area element in the {kP1, kz} half-plane can be transformed

dX

dkz — = dkzdkpi ^ Po

^ kjdBdkj = Sj^.!— d Qd a>.

(24)

Rewriting Eq. (12) using Eqs. (20, 22, 23) we obtain the expression for the spontaneous emission rate Wmd into external radiating modes:

Wmd =- W.: 4

' Ej,m IP0) •

xX X jJ-, C sin 8d 8, (25)

m=-oo J=EH,HE 0 1 + Cm

w1=4 m-^»^!^! aft ld f l2,

where the coefficients Cm were introduced (see Eq. (17))

C = ate 2 ATM 2

ATM ß=ß+ ate ß=ß-

\rTM^TE | I'TE^TM I

(26)

Kte^te + ß+ wtm ^TM ß

which stem from the mode normalization Eq. (19). One can see that Cm = 0 when the polarization mixing is absent.

In Eq. (25) the quantity E J=EH/HE,m is a dimensionless electric-field profile of the quantized radiating mode of the structure. This complex vector field is formed as follows. For EH (HE) polarization one should put A-M = 1 ( A-e = 1) and the equation (17a) (equation (17b)) should be used for calculating another converging-wave coefficient. The coefficients of the outgoing cylindrical waves (AM , A}e ) and of the field inside the cylindrical structure (Btm, Bte) are obtained using the matrix expressions Eq. (4) and Eq. (10) respectively. Since the elements of the matrices s and BInt were obtained analytically, one can find an analytic expressing for E J m in any point inside and outside the cylindrical structure and use it in Eq. (25) to calculate spontaneous emission of a quantum emitter placed at that point.

One can see that parameters of quantization cylinder R and Lz do not enter the final expression Eq. (25). If the quantum emitter is placed on the cylinder axis, one should restrict angular momentum numbers to m = - 1, 0, 1 because the modes with |m| >1 have zero electric field at the axis (see Appendix B).

2.2. Spontaneous emission into guided modes

Let us consider the case when the parameter X1 is imaginary (or the transverse wavenumber kP1 is imaginary, see Eq. (1)). This situation may occur for the nanowire (type I structure) in a waveguide regime or for the cylindrical hollow in a metal film (type IV structure). In this case the plus coefficients ATM/TE in Eq. (2) at exponentially increasing fields should be equal to zero. In the matrix form, Eq. (3), this condition is written as

= 5 >

(27)

and similarly for the modified scattering matrix s. This leads to the requirement of degeneracy of the scattering matrices:

det [5 (q, z )] = det [5 (q, z ) = 0,

'TE—TE'TM—TM — 'TM—TE'TE—TM ■

(28)

For the type I structure this condition reduces to the standard dispersion equation of the cylindrical waveguide. The equation (28) produces the dispersion dependence between the mode frequency and the propagation wavenumber for a particular guided mode v with an angular number m:

q = q™,v (z ),

«=«m,v (kz ).

(29)

From such dispersion relations group velocities of guided modes can be found

d«m,v (kz ) dkz

(30)

The relations between external exponentially-decreasing fields of different polarizations are defined by Eq. (27):

A- = rTM —TE A-^TE — _ ^TM?

rTE—TE A- = rTE—TM A-

(31)

The fields inside the cylinder can be found from Eq. (10).

The summation over guided modes in Eq. (12) takes the form:

S —ZEE —

p m v Nz

z z t í dkz—à Z Z 2ï d (Äm>

m v m v «m v m,v

(32)

where Eqs. (13) and (30) were used; ram.v is the cut-off frequency of the guided mode. Normalization of the electromagnetic energy of a guided mode in a cylinder of

height Lz leads to the following expression for the spontaneous emission rate into a particular guided mode with quantum numbers m and v:

Wm,v =

"WG —

1 «fi Ih P --Hfi —

2 hvgv S,

1

x |E^bv(Pü, ra fl )• erf |2 H (ra fl -ram,v),

where H (ra) is the Heaviside function, E; electric field and

(p)PdP

(33)

is the mode

(34)

is the normalization integral of the mode energy density Eq. (19b) over the plain perpendicular to the symmetry axis. In Eq. (34) the field profile E^g can be scaled arbitrary provided that that the same field is used in calculation of the energy density Eq. (19b), Eq. (34). For calculation of field profiles in an arbitrary point of the structure one can employ Eq. (2) for external fields and Eq. (9) for internal fields putting AMTE = 0 ; the relation between coefficients Atm/te is fixed by Eq. (31) and the internal-field coefficients Btm/ te are given by the general matrix relation Eq. (10).

3. Results and discussion

We apply obtained theoretical results to calculate Pur-cell factors for quantum emitters placed in GaN nan-owires. The Purcell factor is defined as a ratio of the spontaneous emission rate of an emitter into a particular channel to the emission rate of that emitter placed into homogeneous bulk GaN. The refractive index of GaN is taken to be equal to 2.4 which approximately corresponds to the yellow-green region of the spectrum [33]. The emitter emission wavelength is taken to be 550 nm to simulate radiation of InGaN quantum dots [34]. The radiating quantum dot is placed at the center of the nanowire; the length of the nanowire is considered to be large enough to apply an infinite waveguide approximation.

2.5

¡2 2 0

o

^ * c

,2 1.5

"Ö

y 1.0

CL

0.5 0.0

Axial dipole

Radial dipole

0

Diameter (nm) Diameter(nm)

Fig. 1. Purcell factors of an InGaN quantum dot (emission wavelength 550 nm) placed in the center of a GaN nanowire as functions of nanowire diameter. In the left panel (a) the optical-transition matrix element is directed along the nanowire axis (axial orientation); in panel (b) the matrix element is perpendicular to the axis (radial orientation)

In Fig. 1 calculation of the Purcell factors for a InGaN quantum dot in a GaN nanowire (NW) is shown. In the

m,v

600

left panel the radiating optical transition is described by a linear dipole directed along the NW axis; in the right panel the direction of the dipole is perpendicular to the axis. The latter case can be considered as corresponding to the circularly-polarized transition or as a radiation of a linear dipole averaged by its direction perpendicular to the axis. One may see that both external and guided channels of radiation significantly influence the total spontaneous emission.

Maxima in the external emission seen in Fig. 1 occur due to microcavity effect [35] or the Purcell enhancement induced by the light confinement in the lateral directions. One may see in Fig. 1 that distance between diameters at which maxima appear approximately coincide with the emission wavelength in the material of the cylinder (230 nm in the considered example). In the simple case when an emitter is placed in the center of a one-dimensional (1D) Fabry-Perot cavity resonant conditions are satisfied when the length of the resonator is exactly equal to the odd number of half-wavelengths in the cavity material. The considered case of cylindrical geometry and the emitter which can radiate light at any angle is more complicated than the 1D case. This can be seen in Fig. 1 where Purcell enhancement peaks are broad and the positions of their maxima do not follow the simple Fabry-Perot rule, but the tendency to have wavelength distance between maxima is recovered at larger diameters.

As the diameter of the nanowire increases new guided modes appear and start to participate in the radiation process. When the diameter exceeds the cut-off value the emission in that particular mode quickly reaches maximum but with further diameter increase the electric field of the guided mode at the cylinder axis decreases. Thus the decrease of the guided-mode confinement factor at the emitter position stipulates the decline of the mode interaction with the emitter.

In Fig. 1 one may see that emergence of new guided modes (TM-polarized for axial-dipole case and EH-polarized for radial case) coincide with the decrease of the luminescence in radiative modes. In the axialtransition case the diameter at which radiation in external space is maximal approximately coincides with that for guided-mode luminescence.

Thus an important finding evident in our calculation results (Fig. 1) is a direct correlation between emission into external space and the waveguide emission. We would like to stress attention that our formalism of spontaneous emission into radiative modes summarized in the expression (25) does not contain any artificial provision for guided modes and their cut-off frequencies. Therefore, the attained correlation between external and waveguide radiation is naturally embedded in our formalism and shows that the two parts of the developed theory are self-consistent.

From Fig. 1b one may see that dependencies of the HE Purcell factors on diameter are qualitatively different from that for EH modes. Indeed, the radiation into HE-

polarized modes commences evenly starting from zero values at cut-off diameters whereas luminescence into EH modes shows threshold-like character at cut-off conditions with an abrupt jump of the Purcell factor. This effect is due to different behavior of the group velocities of HE and EH modes at cut-off.

Our results for GaN nanowires presented in Fig. 1b are qualitatively similar to purely-numerical calculations obtained by the J.M. Gerard scientific group for GaAs nanowires of finite length and radial dipole orientation, see Refs. [3, 6 - 8]. In the paper of Claudon et al. Ref. [6] in Fig. 1 numerical calculations of external and guided mode Purcell factors are shown and one can see sharp drop in the external emission at the cut-off diameter of a new guided mode. This behavior closely matches our semi-analytical calculations in Fig. 1b, although the authors of Ref. [6] attribute the guided mode as HE12 whereas we conclude that it is EH11 mode.

In the limiting case of small diameters, the spontaneous emission is suppressed by dielectric screening of a dipole inside a nanowire [6-8, 18, 26]. In thin nanowires the analytical expressions for Purcell factors of axial (Pax) and radial (Prad) dipoles are known [26]:

Pax Prod =M i—f. (35)

Vs2 VS2 +S2 J

For a GaN nanowire (the refractive-index value is 2.4) in air these Purcell factors are Pax=0.417, Prad =0.0365 and strong suppression of the radial-transition radiation is evident.

r 1 1 1 /' —

/ _____ -

/ / ■ External axial

. /^T » .

: >/ i External radial;

.----- ■ HE11'' . I . . (a) ' i ■

0 100 200 300 0 200 400 600 800

Diameter (nm) Diameter (nm)

Fig. 2. (a) External-radiation Purcell factors from a GaN nanowire for smaller diameters shown in logarithmic scale; dashed horizontal lines correspond to Eq. (35) . The Purcell factor for a radial dipole radiation into the fundamental HE11 mode is also shown. (b) External radiation factors as functions of the nanowire diameterfor axial and radial optical-transition orientation

In Fig. 2a the Purcell factors in smaller-diameter region are shown in logarithmic scale. One can see that our calculations exactly match the analytical values Eq. (35) when the diameter goes to zero (dashed lines in Fig. 2a). It can be seen in Fig. 2a that despite the existence of the fundamental HE11 mode in the small-diameter region the emission in that mode is negligible due to the weak light localization inside NW. The onset of spontaneous emission into the fundamental HE11 mode happens when the nanowire diameter exceeds the value of approximately 130 nm.

In Fig. 2b external radiation factors are shown which are defined as ratios of a luminescence intensity in exter-

nal space to the total luminescence intensity including radiation in guided modes. One can see that the fraction of radiation that goes directly into surrounding space is always significant for axial optical transitions. This suggests that one cannot limit consideration of the spontaneous emission process to guided modes in such material system. At small diameters there are no guided modes interacting with the axial transition, thus all luminescence is transmitted by external radiating modes. At the same time the radial transition directs almost all its luminescence in external space despite the formal existence of the fundamental HEn mode at small diameters (see Fig. 2a). It is evident that in the radial case the variation of the external radiation factor is strong and diameter regions exist where relative external luminescence is either suppressed or enhanced; but the external radiation factor newer falls to zero value showing that it is not possible to direct total spontaneous emission in guided modes in the considered system.

.(a) A ■

ii \

/r v11 h V

J I

(i)

200 400

Diameter D (nm)

0.15 0.30 0.4 D=285 nm

(V) .„ D=325 nm

Fig. 3. Angular dependence of spontaneous emission from GaN nanowires of different diameters D: 70 nm (I), 165 nm (II), 210 nm (III), 285 nm (IV) and 325 nm (V). Black (gray) curves show normalized spontaneous emission rate inside a solid angle for axial (radial) dipole direction; dashed (dash-dotted) curves depict angular distribution of emission for an axial (averaged radial) dipole transition in bulk GaN. The panel (a) shows external-emission Purcell factors; arrows point the NW diameters under investigation

In Fig. 3 we examine angular dependence of spontaneous emission of a quantum emitter placed in the center of a nanowire; the emission rates in particular solid angles are normalized to the total emission rate of a dipole in air. It is evident that directionality of emission strongly depends on the nanowire diameter for both axial and radial dipole-transition orientations. One may list the following findings: a) at NW diameters when the total external emission of the axial dipole reaches local maxima (panels (I) and (IV) in Fig. 3) the radiation of that type of optical transition has preferable direction perpendicular to the NW axis resembling the case of a dipole emission in homogeneous medium; b) when the NW diameter approaches the cut-off condition from below (panel (II) for TM mode interacting with the axial dipole and panel (IV) for EHn mode interacting with the radial dipole) the spontaneous radiation of the corresponding optical transition becomes highly directional and the light tends to propagate at sharp angles to the NW axis; c) when the diameter passes the cut-off (panels III and V) the angular distributions of emission intensity change abruptly

switching from a highly directional regime to more homogeneous one.

Conclusions

We have developed a theoretical approach which allows calculation and analysis of spontaneous radiation of a quantum emitter in a presence of a cylinder. The position of the emitter can be arbitrary (inside or outside the cylinder) and the refractive index of the cylinder can exceed or be less than the refractive index of the surrounding medium. The dielectric permittivity of the cylinder or the medium in which it is placed can be negative as in the case of metals below plasmonic resonance. Therefore, our method allows modeling of spontaneous emission in a variety of structures such as semiconductor nanowires, cylindrical hollows in dielectric substrates, plasmon waveguides and metals with cylindrical perforation. Our method accounts for emission into external space as well as for radiation channeling into guided modes.

We applied our method to investigation of spontaneous emission from GaN nanowires. We had researched dependence of Purcell factors on the nanowire diameter; comparison of the emission intensities into external medium and into different guided modes was performed. We have found strong correlation between radiation in external space and into guided modes. In particular, we have shown that when the values of parameters of the structure cross cut-off conditions for guided modes qualitative and abrupt quantitative changes in spontaneous emission occur. It is shown that directionality of spontaneous emission strongly depends on the diameter of the cylindrical structure. We demonstrate that the angular dependence of radiation in external space for diameters just below guided-mode cut-off is highly directional whereas above cut-off emission is channeled into a newly formed guided mode.

Acknowledgments

The work has been supported by the Russian Science Foundation 21-12-00304.

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Supplementary A: Cylindrical waves outside a cylinder

If the transverse wavenumber kPi and the parameter xi are real (see Eq. (1)) the cylindrical waves in external medium are propagating and the components of TM polarized waves are proportional to the following dimensionless field profiles:

(Ei-)z = * (xip), (ES+M =- f^ ^^M

2Vei^i q v ^ ^ei^i q xip (A1)

(Ai)

(e %£) = i z h;(i'2) (xip),

V /p 2 Je^ q

where p = p /po is the distance from the symmetry axis reduced to the radius of the cylindrical structure. Hankel functions of the first (second) kind enter expressions designated by plus (minus) superscript and stand for divergent (convergent) waves.

TE polarized waves have two nonzero electric components:

¡~ \ i / x ¡~ \ i Hmi/2)(xip)

(E") = - iHm(i/2) (xip), (eam~) =-1m m . (A2)

v 't 2 v 'p 2 xip

The asymptotic form of the cylindrical waves at large distances from the symmetry axis is:

emtexp I +/1 m — + —

EEMt (r) = EMm (p)x exp(i(z + mt]) * -ee--— exp([kpip + kzz] )

J2-xp

(( - ^ (A3)

eimtexpI +iI m — + — I I

Egm- (r) = EEtJ- (p)x exp(i^z + mt]) * e*-\ ^ 2 4 ^ exp([kpip + kzz]).

^/2-xip

where ee is a unit vector corresponding to the spherical coordinate 8 . Thus at large distances from the axis and in local regions of space cylindrical waves transform into orthogonal plain waves. For imaginary kpi and xi the TM-polarized field profile has a form:

(eS-)z = i-1 *zr)(xip), (-)=-i m z Zr)(xip)

2 VEiMi q v ' 7t 2 VEii^i q xip (A

(A4)

11 z

(E em +m-)p = 2-^=qz;(i/2) (xip).

Eiii q

where expressions with plus superscripts contain functions Z® (x) = Im (x|) which exponentially increase with the distance from the symmetry axis and waves with minus superscript contain exponentially decreasing functions zm2) (x ) = Km (|x|) (see Eq. (12)).

TE-polarized exponentially-changing electric fields are

1 1 Z(1/2) (x p)

(eEEtm-) =-1 zm(i/2)(xip), (m-) =-1 m m \x1}. (A5)

v 7t 2 v 7p 2 x1p

At large distance from the axis E+/-« exp(+|xi |p) for both polarizations.

Supplementary B: Cylindrical fields inside a cylinder

If the value of parameter x2 is real (i.e. the internal transverse wavenumber kp2 is real, see Eq. (1)) TM-polarized electric-field profile inside the cylinder can be written as:

(E '"> ) = 1 x2 J (x p) (E M ) =- m z Jm (x2p)

\*^TM,m I ~ I Jm\X2YU I^TMI _ I Z ;

v /z VE2I2 q v 7t VE212 q x2p _

1 _ (B1)

(ETM,m ) = i I Jm (x2p),

v /p VE2I2 q

where Jm is the Bessel function of the first order. The nonzero components of TE field are:

(E Tim) = -iJm(x2p), ) = -mJm ^ ). (B2)

v 't v 'p x2p

Let us consider electric field at the symmetry axis. Only TM polarized mode demonstrate nonzero axial electric field at the axis:

ETM,m=o (r = 0)=-pL=—ez , (B3)

vs2m*2 q

The nonzero radial electric field at the axis is present only for TM/TE modes with m = +1:

E TM ,m=+i (r = o) = ETM ,m=+i (p = 0)exp (+it) = ±i-^-=L= - x^ ( + iey),

V2 V62^2 q V2 (B4)

^H,m=±i (r = o) = ETt,m=±i (p = o)exp(+it) = --j= x (ex + iey) .

The modes with |m| >1 have zero electric field at the symmetry axis. The expression of TM-polarized field for imaginary X2 is

1 1 X2 /i i_\ t~ Int \ 1 m z 1

(ETMm ) = —Im (klp), (ETMm ) =- Im (klp)

V 'z 2 q 1 ' V ■ !♦ q X2P ^ 1 >

(B5)

(ETMm ) = iZI»flx21p),

v /p ^62^2 q

where only the modified Bessel function Im is preserved due to its finiteness at zero argument.

The TE-polarized field inside the cylinder for imaginary X2 can be written as:

(m ) = -iim (x21 p), (E Ttm )p = - fz Im ((2 |p). (B6)

The electric fields at the axis for modes with imaginary X2 are similar to the aforementioned case of real X2.

Authors' information

Valentin Nikolaev, (b. 1978) graduated from Peter the Great St. Petersburg Polytechnic University in 2001. He received his PhD in Physics degree from the University of Exeter, UK, in 2002 and his Candidate in Physics & Maths degree from Ioffe Institute, St. Petersburg, Russia in 2005. He worked as a researcher in University of York, Ioffe Institute and Submicron Heterostructures for Microelectronics Research and Engineering Center of the Russian Academy of Science. His current research interests include nanophotonics and optics of semiconductor nanostructures. E-mail: valia.nikolaev@gmail.com .

Elizaveta Ilinichna Girshova (b. 1991) graduated from Saint-Petersburg State University in 2016. Currently she works as an engineer in Research Center for Advanced Functional Materials and Laser Communication Systems. Research interests are nanophotonics, optoacoustics, light-matter interaction. E-mail: ilinishna@gmail.com .

Mikhail Alekseevich Kaliteevski, (b. 1969), Dr, chief reseacher at ITMO University. Research interests: light-matter interaction, plasmonics, mesoresonators, nanophotonics, numerical simulation. E-mail: m.kaliteevski@gmail.com.

Received April 6, 2022. The final version - June 4, 2022.