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Photon field in the presence of a nanofiber

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2015 Jpn. J. Appl. Phys. 54 072001

(http://iopscience.iop.org/1347-4065/54/7/072001)

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Photon field in the presence of a nanofiber

Makoto Morinaga1*, Ramachandrarao Yalla2, and Kohzo Hakuta2

1Institute for Laser Science, University of Electro-Communications, Chofu, Tokyo 182-8585, Japan2Center for Photonic Innovations, University of Electro-Communications, Chofu, Tokyo 182-8585, JapanE-mail: morinaga@ils.uec.ac.jp

Received November 19, 2014; accepted April 13, 2015; published online June 10, 2015

The presence of a nanofiber modifies the photon field near the nanofiber. The emission of photons into a certain direction can be enhanced byplacing a nanofiber near the photon emitter. We discuss the modification of the photon field by the presence of a nanofiber and evaluate the effectof emission enhancement for a realistic experimental scheme. © 2015 The Japan Society of Applied Physics

1. Introduction

Manipulation of single photons stored in a single modewaveguide is one of the ultimate controls of the photon fieldwhich are required to use photons as carriers of quantuminformations in the field of quantum information processingand communication.1) In this context, the use of a nanofiberfor the efficient coupling of photons emitted by a singleemitter into a single mode optical fiber has been extensivelystudied recently.2–19) For that purpose, an emitter is placed inthe vicinity of the nanofiber surface, and since the photonfield is modified by the presence of the nanofiber, one has touse this modified field instead of the free space photon fieldto discuss the interaction between the emitter and theelectromagnetic field. The guided modes through the nano-fiber have already been discussed from various aspects,3,20–27)

however, to determine the coupling efficiency into the guidedmodes experimentally, for example, the knowledge on theradiation modes is also inevitable. The coupling efficiencyinto the nanofiber-guided modes is given by

�c ¼ Pg

Pg þ Pr¼ Ig

Ig þ Ir; ð1Þ

where Pg (Ig) and Pr (Ir) are the photon emission probabilities(intensities) into the guided and radiation modes, respec-tively. To determine the coupling efficiency ηc, one has toknow the value of Ir along with Ig. Ir is the total intensityof photons emitted towards outside the nanofiber and isexpressed as

Ir ¼Z�

Ik̂ dk̂: ð2Þ

Here, Ik̂ is the intensity of the photon field emitted into the k̂direction and the integration is carried out over the wholesolid angle Ω = 4π. Experimentally however, due to technicaldifficulties, photons are usually collected over a finite solidangle Ωex only:

Iex ¼Z�ex

Ik̂ dk̂: ð3Þ

In free space, assuming an isotropic emitter, Ir is deducedfrom Iex simply as Ir = αIex with α = Ω=Ωex. However, whenthe emitter is placed near the nanofiber surface, the presenceof the nanofiber induces anisotropy in the emission patterneven if the emitter itself is isotropic, and introduces an addi-tional factor � into α as � ¼ �=ð��exÞ. In this paper, we firstcalculate how the presence of a nanofiber modifies the photon

emission probability as a function of emission direction, andthen estimate the correction factor � for a realistic experi-mental scheme used in Ref. 12.

2. Emission probability

First, we discuss in general the emission probability of aphoton into a specific mode from an emitter via dipoletransition.

2.1 Mode expansion of the photon fieldThe quantized electric field of angular frequency ω can beexpanded using a set of mode functions in the following form(in the interaction picture):

Eðr; tÞ ¼ Eð�Þðr; tÞ þ EðþÞðr; tÞ¼XM

fEð�ÞM ðrÞayMei!t þ EðþÞ

M ðrÞaMe�i!tg: ð4Þ

Here, (+) and (−) denote the positive and negative frequencyparts, respectively, and Eð�Þ

M ðrÞ are the mode functions ofmode M. ayM and aM are creation and annihilation operatorsof mode M photons, respectively. We are interested in onlythe photons of single frequency, so the normalization factors(which are functions of frequency) are omitted in Eq. (4).

2.2 Emission probability of a photon into a specifiedmodeThe probability that an emitter at a position r emits aspontaneous photon into mode M through a transition froman upper energy state ∣u⟩ to a lower energy state ∣l⟩ via dipoletransition can be written as

emission probability / jh1MjhljEðr; tÞ � pjuij0ij2¼ jEðþÞ

M ðrÞ � hljpjuij2; ð5Þwhere ∣0⟩ is the vacuum state (i.e., no photons), ∣1M⟩ isthe single photon state (1 photon in mode M), and p is thedipole moment of the emitter. Thus, if we are interested in thetransition between specific states of the emitter, the electricfield along the dipole moment should be evaluated. In thecase of quantum dots, the transition dipole moment of theemitter is randomly oriented and Eq. (5) becomes

emission probability / jEðþÞM ðrÞj2; ð6Þ

i.e., the emission probability is proportional to the square ofmodulus of the mode function of the electric field at theposition of the emitter. Hereafter, we assume the randomorientation of the dipole moment. In the following, we willconcentrate on the positive frequency part of the electric field

Japanese Journal of Applied Physics 54, 072001 (2015)

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and write E(+) simply as E. The (real) electric field can beobtained from E as 1

2ðE þ E�Þ [or 1

2ðE þ EyÞ].

3. Emission probability of a photon into a specifieddirection

In this section, we calculate the emission probability of aphoton into a certain direction k̂ (k̂ is a unit vector thatspecifies the direction) in the presence of a nanofiber. FromEq. (6), we need to calculate the corresponding modefunction Ek̂ðrÞ.

3.1 Mode expansion in a system of axial symmetryWe expand the electric field in terms of cylindrical waves.In the region where matter is absent, such expansion can bedivided into two parts: the incoming component (expressedby the Hankel functions of the second kind: Hð2Þ

n ) and theoutgoing component (expressed by the Hankel functions ofthe first kind: Hð1Þ

n ):

EðrÞ ¼ EðinÞðrÞ þ EðoutÞðrÞ:In free space, the field that propagates out into the directionk̂ is, of course, a plane wave: Ek̂0ðrÞ ¼ E0 expðik � rÞ, wherek ¼ kk̂ with k = ω=c being the photon wavenumber and E0 aconstant vector perpendicular to k. We write Ek̂0ðrÞ as a sumof incoming and outgoing components:

Ek̂0ðrÞ ¼ EðinÞk̂0

ðrÞ þ EðoutÞk̂0

ðrÞ:In the presence of a nanofiber on the z-axis (the axis of axialsymmetry), the field that propagates out into the direction of awavevector k should have the form (outside the nanofiber)

Ek̂ðrÞ ¼ EðinÞðrÞ þ EðoutÞk̂0

ðrÞ; ð7Þwhere E(in)(r) is to be determined from the boundarycondition on the nanofiber surface with respect to EðoutÞ

k̂0ðrÞ.

Details of the derivation of the mode function [Eq. (7)]are given in the appendix.

3.2 Numerical examplesWe consider photon emission from a single small emitterwith a randomly oriented dipole moment deposited on thenanofiber surface. We assume that the size of the emitter ismuch smaller than the wavelength of the emitting light andalso much smaller than the radius of the nanofiber so that thepresence of the emitter itself does not alter the modes of thelight field. According to Eq. (6), jEk̂ðrÞj2 gives the prob-ability of emitting a photon into the k̂ direction (r is the posi-tion of the emitter). The geometry considered here is shownin Fig. 1. The direction of k̂ is specified by the polar angle θand azimuthal angle ϕ. We call the mode TM (TE) for whichthe magnetic field (electric field) is perpendicular to thenanofiber axis (z-axis). In Fig. 2, the emission probabilityrelative to the free space value (i.e., [emission probability inthe presence of a nanofiber]=[emission probability in freespace]) is plotted as a function of ϕ for θ = 90, 75, and 60°.The size parameter (wavenumber × radius of the nanofiber)ka = 1.47 is used for this figure, which corresponds to thefiber radius a = 200 nm assuming the cesium D2 line wave-length λ = 852 nm. The refractive index n = 1.453 of thenanofiber corresponding to quartz glass is used throughoutthis paper. In Fig. 3, we plot the relative emission probabilityinto the θ = π=2, ϕ = 0 direction for various size parameter

ka values. We note that the emission probability is enhancedin this direction by the presence of the nanofiber withincreasing size parameter up to ka ∼ 8, especially for the TMmode. An intuitive explanation for this enhancing effect isthat the nanofiber acts as a cylindrical lens and tends tocollect the emitted light into the direction behind the nano-fiber. Such a focusing effect increases with the nanofiberradius a until it saturates when the size of the nanofiber

photon

x

y

z

o

emitter

φ

θnanofiber

2a

Fig. 1. A nanofiber of radius a is placed along the z-axis and an emitter isput on the surface at ð�a; 0; 0Þxyz ¼ ða; �; 0Þr�z.

0

0.5

1

1.5

2

2.5

3

0 20 40 60 80 100 120 140 160 180

rela

tive

emis

sion

pro

babi

lity

φ [degree]

TM: 90°TM: 75°TM: 60°TE: 90°TE: 75°TE: 60°

Fig. 2. Relative emission probability versus the azimuthal direction of theemitted photon for polar angle θ = 90, 75, and 60°. The size parameterka = 1.47 and refractive index n = 1.453 are assumed for the nanofiber.

0

2

4

6

8

10

12

14

0 2 4 6 8 10

rela

tive

emis

sion

pro

babi

lity

size parameter ka

TMTE

Fig. 3. Relative emission probability of photons into the directionθ = π=2, ϕ = 0 as a function of size parameter ka.

Jpn. J. Appl. Phys. 54, 072001 (2015) M. Morinaga et al.

072001-2 © 2015 The Japan Society of Applied Physics

reaches the geometrical optics region (a ≳ �, i.e., ka ≳ 2�).Note that in the small fiber radius limit ka → 0 (i.e., inthe static field limit), the relative emission probability ofthe TM mode converges to 1, whereas that of the TE modeapproaches ½2=ðm2 þ 1Þ�2 < 1. For the general azimuthalangle ϕ and θ = π=2, this limit of the TE mode is shown to be4(m4 sin2ϕ + cos2ϕ)=(m2 + 1)2, whereas that of the TMmode is always 1.

4. Enhancement factor of detection probability

We consider the experimental scheme used in Ref. 12 andestimate how much detection by an external detector isenhanced (or reduced) by the presence of a nanofiber. Thegeometry of the experimental setup to measure the emissioninto the radiation modes is shown in Fig. 4. An objective lenscentered at the x-axis collects emitted photons from anemitter into a photodetector. All emitted photons directedwithin a solid angle determined by the numerical aperture NAof the objective lens (NA = 0.6) move into the detector, i.e.,photons in the direction ð�; �Þ satisfying sin � cos� �ð1 � NA2Þ1=2 are detected. Thus, the enhancement factor(ratio of the detection rates of photons in the presence=absence of a nanofiber) for this system is obtained byaveraging the relative emission probabilities calculated in theprevious section over the solid angle of the objective lens.The azimuthal position of the emitter has also some distribu-tion around ϕ = π (ϕe = 0). Figure 5 shows the estimatedenhancement factors for an emitter at ða; �; 0Þr�z for the TMand TE modes, and their average, whereas in Fig. 6 we showthe total enhancement factors (averages of the TE and TMmodes) for different azimuthal positions of the emitter ϕe = 0,15, 30, 45, 90, 135, and 180°. From Fig. 6, we see that theenhancing effect decreases rapidly with increasing ϕe forka ≳ 4. This can be qualitatively explained by noting thatbecause of the strong focusing effect in that parameter region,the photon emission pattern is highly directional and theenhancement occurs only within the angle of this directivity.Under the experimental conditions in Ref. 12, emitters aresupposed to distribute over ��e0 < �e < �e0 with �e0 ≲90°. In Fig. 7, the enhancement factors 90� and 45� areplotted assuming a uniform distribution of azimuthal positionϕe of the emitter over −90 < ϕe < 90° and −45 < ϕe < 45°,respectively. Also, the enhancement factor ζtotal for theradiation into the whole solid angle Ω = 4π is plotted. � ¼

90�=total or 45�=total gives the correction factor introducedat the end of Sect. 1 that is used to estimate the couplingefficiency from the experimental data. In the experimentin Ref. 12, the correction factor � ¼ 90�=total was used todetermine the coupling efficiency.

x

y

z

o

emitterφe

nanofiber

objective lensphoto-detector

φ

θ

Fig. 4. External detection scheme of Ref. 12. An emitter (quantum dot) isput on the nanofiber surface at ða; � � �e; 0Þr�z, and an objective lens placedon the + side of the x-axis collects emitted photons into a photodetector.

0.5

1

1.5

2

2.5

3

3.5

4

0 2 4 6 8 10

enha

ncem

ent f

acto

r

size parameter ka

TMTE

average

Fig. 5. Enhancement factor for the external detection rate assuming thatthe emitter is positioned at ða; �; 0Þr�z (i.e., ϕe = 0).

0.5

1

1.5

2

2.5

3

3.5

4

0 2 4 6 8 10

enha

ncem

ent f

acto

r

size parameter ka

0°15°30°45°90°

135°180°

Fig. 6. Enhancement factors for various azimuthal emitter positions.

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

0 2 4 6 8 10

enha

ncem

ent f

acto

r

size parameter ka

-45°..45°-90°..90°

total

Fig. 7. Enhancement factors 45� and 90� assuming a uniform distributionof azimuthal position ϕe of the emitter over the ranges −45 to 45° and −90 to90°, respectively. The enhancement factor ζtotal for the radiation into thewhole solid angle is also plotted.

Jpn. J. Appl. Phys. 54, 072001 (2015) M. Morinaga et al.

072001-3 © 2015 The Japan Society of Applied Physics

5. Conclusions

The photon field modified by the presence of a nanofiber ismode-expanded by a set of mode functions classified by theoutgoing direction. Using this expansion, the relative emis-sion probability into a specific direction in the presence ofa nanofiber is calculated. Also, the enhancing effect on thedetection rate of photons by a detector placed outside thenanofiber is estimated for a realistic experimental scheme. Bychanging the incoming wave (Hð2Þ

n ) with the outgoing wave(Hð1Þ

n ), this method can also be applied to evaluate the photonfield when an external radiation is incident on the nanofiber.Such examples include the fabrication of a photonic crystalstructure on an optical nanofiber using a femtosecond laserthat has been realized recently.28)

Acknowledgements

We thank Fam Le Kien and Yoko Miyamoto for helpfulcomments and suggestions. This work is partly supported bya Grant-in-Aid for Scientific Research 21540407 and theStrategic Innovation Project by Japan Science and Technol-ogy Agency.

Appendix: Mode function Ek̂ðr Þ

In this appendix, we derive the explicit form of the modefunction Ek̂ðrÞ introduced in Eq. (7) by solving the boundarycondition on the nanofiber surface. The derivation is self-contained assuming only a small amount of knowledge ofbasic properties of Bessel functions. The symbol m is usedfor the refractive index of the material (nanofiber) in this

appendix, which we adopt from the notation in Ref. 29. Thegeneral description of the light scattering by a dielectric rightcircular cylinder is found in Refs. 30 and 31. We choosea coordinate such that the z-axis is the axis of the nanofiberand the wavevector k lies in the xz-plane, so that k ¼kðsin �0; 0; cos �0Þxyz with θ0 being the angle between k andthe z-axis. For a plane wave (i.e., in free space),

Ek̂0ðrÞ ¼ E0eik�r ¼ E0 e

ikr sin �0 cos�eikz cos �0 ðA:1Þwith E0 ⊥ k. From r E ¼ �ð@H=@tÞ,

Hk̂0ðrÞ ¼ H0 eikr sin �0 cos�eikz cos �0 ðA:2Þ

with H0 ¼ i k! k̂ E0. In the following, we use the formula

eikr sin �0 cos� ¼X1n¼�1

inJnðkr sin �0Þe�in�:

A.1 TM modeFor the TM mode, E0 and H0 in (A.1) and (A.2) are given by

E0 ¼ ETM0 E0ð�cos �0; 0; sin �0Þxyz

¼ E0ð�cos �0 cos�;�cos �0 sin�; sin �0Þr�zH0 ¼ HTM

0 H0ð0;�1; 0Þxyz¼ H0ðsin�;�cos�; 0Þr�z; ðA:3Þ

where H0 ¼ ikE0=ð!Þ, and we use in this appendix thenotation ( )rϕz to express a vector by its components tangentialto the cylindrical coordinates [i.e., v ¼ ðvr; v�; vzÞr�z withvr ¼ vx cos� � vy sin� and v� ¼ vx sin� þ vy cos�]. Eachcomponent of the electric field ETM

k̂0ðrÞ Ek̂0ðrÞ with E0

defined in Eq. (A·3) is transformed as

ETMk̂0r

ðrÞ ¼ �E0 cos �0 cos�X1n¼�1

inJnðkr sin �0Þe�in�eihz

¼ � 1

2E0 cos �0

X1n¼�1

inþ1fJnþ1ðkr sin �0Þ � Jn�1ðkr sin �0Þge�in�eihz

¼ E0 cos �0X1n¼�1

inþ1J0nðkr sin �0Þe�in�eihz;

ETMk̂0�

ðrÞ ¼ �E0 cos �0 sin�X1n¼�1

inJnðkr sin �0Þe�in�eihz

¼ � 1

2E0 cos �0

X1n¼�1

infJnþ1ðkr sin �0Þ þ Jn�1ðkr sin �0Þge�in�eihz

¼ �E0 cos �0X1n¼�1

innJnðkr sin �0Þ

kr sin �0e�in�eihz;

ETMk̂0z

ðrÞ ¼ E0 sin �0X1n¼�1

inJnðkr sin �0Þe�in�eihz;

with h k cos �0. Defining � kr sin �0, finally

ETMk̂0

ðrÞ ¼ E0

X1n¼�1

JLn ð�; �0Þe�in�eihz;

where JLn are defined as

JLn ð�; �Þ ¼ in

i cos �dJnð�Þd�

� cos �nJnð�Þ

�

sin �Jnð�Þ

0BBBB@

1CCCCA

r�z

:

In the case of magnetic field HTMk̂0

ðrÞ Hk̂0ðrÞ with H0 defined in Eq. (A·3), similarly,

Jpn. J. Appl. Phys. 54, 072001 (2015) M. Morinaga et al.

072001-4 © 2015 The Japan Society of Applied Physics

HTMk̂0

ðrÞ ¼ H0

X1n¼�1

JTn ð�Þe�in�eihz; ðA:4Þ

where JTn are defined as

JTn ð�Þ ¼ in

nJnð�Þ�

idJnð�Þd�

0

0BBBB@

1CCCCA

r�z

:

In the presence of a nanofiber, outside the nanofiber, the mode function ETMk̂

can be written as

ETMk̂

ðrÞ ¼ 1

2E0

X1n¼�1

fb0nHLð2Þn ð�Þ þ a0nH

Tð2Þn ð�Þ þHLð1Þ

n ð�Þge�in�eihz

¼ E0

X1n¼�1

fbnHLð2Þn ð�Þ þ anH

Tð2Þn ð�Þ þ JLn ð�; �0Þge�in�eihz; ðA:5Þ

where bn ¼ 12ðb0n � 1Þ, an ¼ 1

2a0n,

HLðqÞn ð�Þ ¼ in

i cos �0dHðqÞ

n ð�Þd�

� cos �0nHðqÞ

n ð�Þ�

sin �0HðqÞn ð�Þ

0BBBBB@

1CCCCCA

r�z

with q = 1, 2, and

HTð2Þn ð�Þ ¼ in

nHð2Þn ð�Þ�

idHð2Þ

n ð�Þd�

0

0BBBBB@

1CCCCCA

r�z

:

The coefficients bn and an [and dn and cn in Eq. (A·7)] are to bedetermined from the boundary conditions on the nanofibersurface. Similarly, for the magnetic field,

HTMk̂

ðrÞ ¼ H0

X1n¼�1

fbnHTð2Þn ð�Þ � anH

Lð2Þn ð�Þ þ JTn ð�; �0Þge�in�eihz: ðA:6Þ

Inside the nanofiber,

ETMk̂

ðrÞ ¼ E0

X1n¼�1

fdnJLn ð�0; �Þ þ cnJTn ð�0Þge�in�eihz

HTMk̂

ðrÞ ¼ mH0

X1n¼�1

fdnJTn ð�0; �Þ � cnJLn ð�0Þge�in�eihz;

ðA:7Þ

where θ satisfies mk cos θ = h and �0 ¼ ðm2k2 � h2Þ1=2r ¼ mkr sin �. The boundary conditions under which the tangentialcomponents of the electric and magnetic fields are continuous inside and outside the nanofiber surface can be written in the form

Au ¼ v; ðA:8Þwhere

A ¼

n

�scos �0H

ð2Þn ð�sÞ �iHð2Þ0

n ð�sÞ � n

�0scos �Jnð�0sÞ iJ0nð�0sÞ

� sin �0Hð2Þn ð�sÞ 0 sin �Jnð�0sÞ 0

�iHð2Þ0n ð�sÞ � n

�scos �0H

ð2Þn ð�sÞ imJ0nð�0sÞ

mn

�0scos �Jnð�0sÞ

0 sin �0Hð2Þn ð�sÞ 0 �m sin �Jnð�0sÞ

0BBBBBBB@

1CCCCCCCA

with �s ¼ ka sin �0 ¼ ðk2 � h2Þ1=2a, �0s ¼ ðm2k2 � h2Þ1=2a ¼mka sin � (a is the radius of the nanofiber), and

u ¼

bn

an

dn

cn

0BBB@

1CCCA; v ¼

� n

�scos �0Jnð�sÞ

sin �0Jnð�sÞiJ0nð�sÞ

0

0BBBBB@

1CCCCCA: ðA:9Þ

The boundary conditions for the normal components of theelectric and magnetic fields are automatically satisfied if

Eq. (A·8) is fulfilled. (A.8) can be solved as u = A−1v, or

ui ¼ detAi

detA; ðA:10Þ

where Ai is A with the ith column replaced by v. In the case ofθ0 = θ = π=2, TM and TE modes are decoupled and theboundary conditions are reduced to an = cn = 0 and

bnH0ð2Þn ð�sÞ þ J0nð�sÞ ¼ mdnJ

0nðm�sÞ

bnHð2Þn ð�sÞ þ Jnð�sÞ ¼ dnJnðm�sÞ; ðA:11Þ

Jpn. J. Appl. Phys. 54, 072001 (2015) M. Morinaga et al.

072001-5 © 2015 The Japan Society of Applied Physics

leading to

bn ¼ � mJ0nðm�sÞJnð�sÞ � Jnðm�sÞJ0nð�sÞmJ0nðm�sÞHð2Þ

n ð�sÞ � Jnðm�sÞHð2Þ0n ð�sÞ

: ðA:12Þ

A.2 TE modeIn the case of the TE mode, similarly, E0 in Eq. (A·1) isgiven by

E0 ¼ ETE0 E0ð0;�1; 0Þxyz ¼ E0ðsin�;�cos�; 0Þr�z:

ðA:13ÞComparing this with HTM

0 in Eqs. (A·3) and (A·4), we obtain

ETEk̂0ðrÞ ¼ E0

X1n¼�1

JTn ð�Þe�in�eihz:

In the presence of a nanofiber, outside the nanofiber,

ETEk̂ðrÞ ¼ E0

X1n¼�1

fbnHLð2Þn ð�Þ þ anH

Tð2Þn ð�Þ þ JTn ð�Þge�in�eihz

HTEk̂ðrÞ ¼ H0

X1n¼�1

fbnHTð2Þn ð�Þ � anH

Lð2Þn ð�Þ � JLn ð�Þge�in�eihz

instead of Eqs. (A·5) and (A·6). Inside the nanofiber, the fields are again the sum of JLn and JTn as in the case of the TM mode[Eq. (A·7)]:

ETEk̂ðrÞ ¼ E0

X1n¼�1

fdnJLn ð�0; �Þ þ cnJTn ð�0Þge�in�eihz

HTEk̂ðrÞ ¼ mH0

X1n¼�1

fdnJTn ð�0; �Þ � cnJLn ð�0Þge�in�eihz:

ðA:14Þ

Note that we use the same symbols bn, an, dn, and cn for thefield coefficients, but their values are different for the TM andTE modes. The boundary conditions and their solution havethe same form as Eqs. (A·8) and (A·10), but v is nowreplaced with

v ¼

iJ0nð�sÞ0

n

�scos �0Jnð�sÞ

� sin �0Jnð�sÞ

0BBBB@

1CCCCA: ðA:15Þ

In the case of θ0 = π=2, boundary conditions reduce to

anHð2Þn ð�sÞ þ Jnð�sÞ ¼ mcnJnðm�sÞ

anH0ð2Þn ð�sÞ þ J0nð�sÞ ¼ cnJnðm�sÞ

ðA:16Þ

and bn = dn = 0, which lead to

an ¼ � J0nðm�sÞJnð�sÞ � mJnðm�sÞJ0nð�sÞJ0nðm�sÞHð2Þ

n ð�sÞ � mJnðm�sÞHð2Þ0n ð�sÞ

:

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