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The step azimuthal optical fiberI.M. Fabbri a & A. Lucianetti ba Centro di Ricerca di Fisica Matematica , Sesto San Giovanni 20099, Italyb Physics Department , University of Florida , Gainesville, FL 832611, USAPublished online: 20 Apr 2010.

To cite this article: I.M. Fabbri & A. Lucianetti (2010) The step azimuthal optical fiber, Journal of Modern Optics, 57:7,528-535, DOI: 10.1080/09500341003675323

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Journal of Modern OpticsVol. 57, No. 7, 10 April 2010, 528–535

The step azimuthal optical fiber

I.M. Fabbria and A. Lucianettib*y

aCentro di Ricerca di Fisica Matematica, Sesto San Giovanni 20099, Italy;bPhysics Department, University of Florida, Gainesville FL 832611, USA

(Received 18 November 2009; final version received 1 February 2010)

We present a new kind of optical fiber by introducing a perturbation of the classical core–cladding fiber design.The azimuthal symmetry of the refractive index is modified by a narrow parabolic perturbation. The perturbationacts like an azimuthal potential well against the propagating pulse. Theoretical calculations lead to a strongselection of the azimuthal modes. We show that by conveniently choosing the parameters, the fundamental modeis not reflected as a result of its radial field distribution.

Keywords: optical fibers; propagation optics; fiber laser; beam propagation; specialty fiber; transmission controls

1. Introduction

Mode selectivity is one of the most important issues inthe study of propagation of pulses in optical fibers. Oneapproach to obtain highmode selectivity is to externallycoil the wire or to use the bend-loss suppression ofhigher order modes [1–3] and the selection tapers [4].These methods have some drawbacks such as Ramanscattering and Brillouin scattering, which limit theavailable output powers of fiber lasers [5]. Using bendloss to select high-order modes allows the core diameterto be increased well beyond the single mode limit.Unfortunately, coiling the optical fiber leads tounwanted high losses and limits the achievable peakpower. Another approach is to break the azimuthalconstant distribution of the inner refractive index, as inthe case of chirally coupled, D-shaped, rectangular core,or PANDA fibers [6].

Recently, a new design based on spiral indexdistribution has been proposed. In this kind of fiber,starting from the transverse electric field of a particularpropagating mode, two eigenvalue equations werefound [7], where the azimuthal one was an HermiteSturm–Liouville eigenvalue differential problem.

In this paper the eigenvalue Fourier equation willbe used to study a step azimuthal perturbation of therefractive index which leads to higher modal selectivity.By choosing the ‘step-window knife’ parameters, onlyLP0,n modes are allowed to propagate. The windowtechnique provides a comparable effect by wrappingthe fiber around a cylindrical mandrel, i.e. low loss for

the fundamental mode LP01 and high loss for LP11 andthe other high-order modes. Wrapping the fiber causesthe optical path length to be modified producing aperturbation of the effective refractive index distribu-tion [8,9]. The window produces interference betweenthe reflected and transmitted pulses at its interfacewhich is destructive for higher order modes.

Finally we analyze the higher order mode selectionin the spiral optical fiber previously studied [7]. Thespiral fiber avoids non-linear effects like four-wave-mixing and reduces the elastic scattering effects [5].

2. Field equations and boundary analysis

Let us consider the parabolic refractive profilesn2effð�, rÞ given by

n2eFð�, rÞ ¼

n2clad ��Fð�Þ

r2r 2 ½a, b�,

n2core ��Fð�Þ

r2r 2 ½�u, a�,

n2core r 2 ½0, �uÞ, � 2 ½�p, p�:

8>>>><>>>>:

�Fð�Þ ¼

G1 � 2 ½�p,��� [ ½�, p�,

G2 � 2 ½��, ��,

G1 4G2,

8><>:

where g, jG1j, jG2j� 1 are constants which representthe very small perturbation functions �F(�) of the

*Corresponding author. Email: antonio.lucianetti@polytechnique.eduyPresent address: Laboratoire LULI, Ecole Polytechnique, 91128 Palaiseau cedex, France

ISSN 0950–0340 print/ISSN 1362–3044 online

� 2010 Taylor & Francis

DOI: 10.1080/09500341003675323

http://www.informaworld.com

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classic step index refractive profile ncore/clad. Here a is

the core radius of the fiber and �u the radius of a small

unperturbed zone (see Figure 1).The transverse electric field of a particular mode in

the fiber satisfies the scalar wave equation, which can

be written in polar coordinates r and � as

@2�

@r2þ1

r

@�

@rþ

1

r2@2�

@�2þ n2eðr, �Þk

2 � �2� �

� ¼ 0,

0 � r � 1, �p5 � � p,

k ¼!

c,

where �(r, �) is the field, k is the free-space wave-

number, and ne(r, �) is the effective refractive index

distribution.The boundary conditions associated with the field

equation due to the electric charge distribution and

finite energy are:

�ðr, �Þ 2 C1½½�p,p� � ð0,1Þ� \ L2½½�p, p� � ð0,1Þ�,

limr!1

�ðr, �Þ ¼ 0,

@�

@r

���r¼0¼ 0:

The field �(r, �) can be conveniently expressed as

�ðr, �Þ ¼ �ðrÞ f ð�Þ:

By substituting the equation of ne(r, �) into the field

equation, we obtain two eigenvalue equations

1

f ð�Þ

@2f ð�Þ

@�2� k2�Fð�Þ ¼ �W, �p5 � � p,

r2

�ðrÞ

@2�ðrÞ

@r2þ

r

�ðrÞ

@�ðrÞ

@r�Wþ n2core=cladk

2 � �2h i

r2 ¼ 0,

0 � r51:

These are the Fourier and Bessel equations,

respectively.Let’s analyze the azimuthal Fourier equation in the

following linear second-order differential form

Lf ð�Þ :¼ �f 00ð�Þ þ qð�Þ f ð�Þ ¼Wf ð�Þ:

The L-operator acts in the Hilbert space L2[�,]\C1[�, ], with q(�) given by

qð�Þ ¼

�1

� 2 ½�p,��� [ ½�, p�,

�2

� 2 ½��, ��,

�1 4 �2 05 �5p,

8>><>>:

where �1, �2, � and � are constants (see Figure 2), with

the following periodic boundary value conditions

f ðpÞ ¼ f ð�pÞ; f 0ðpÞ ¼ f 0ð�pÞ:

The Fourier boundary values equation with D� (seeFigure 2) is used to introduce an azimuthal step

perturbation DG (see Figure 1) of the refractive index.

This enables us to control the light-modes. This is

the classical rectangular-well operator of Quantum

Mechanics [10].Let’s analyze the solution of the eigenvalue equa-

tion LfW(�)¼WfW(�) subdividing the solution into two

parts

fWð�Þ ¼a1 e

�h� þ a2 e� �h�, � 2 ½��, ��,

a3 eh� þ a4 e

�h�, � 2 ½�p,��� [ ½�,p�,

(

h ¼ ð�1 �W Þ1=2,

�h ¼ ð�2 �W Þ1=2,

h2 � �h 2 ¼ �1 � �2 ¼ D�:

The derivative of the solution fW (�) is given by

f 0Wð�Þ ¼a1 �h e

�h� � a2 �h e��h�, � 2 ½��, ��,

a3h eh� � a4h e

�h�, � 2 ½�p,��� [ ½�, p�:

(Figure 1. One-azimuthal-step refractive index fiber.

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The identities

a3 e�hp þ a4 e

hp ¼ a3 ehp þ a4 e

�hp,

a3 e�h� þ a4 e

h� ¼ a1 e� �h� þ a2 e

�h�,

a3 eh� þ a4 e

�h� ¼ a1 e�h� þ a2 e

� �h�,

a3h e�hp � a4h e

hp ¼ a3h ehp � a4h e

�hp,

a1 �h e��h� � a2 �h e

�h� ¼ a3h e�h� � a4h e

h�,

a1 �h e�h� � a2 �h e�

�h� ¼ a3h eh� � a4h e

�h�,

8>>>>>>>><>>>>>>>>:

ensure that the boundary conditions are satisfied by

the eigenfunction fW (�).The first and the fourth identities are satisfied if

and only if sin(ih)¼ 0, i.e. h¼ im, m2Z and�hm ¼ iðm2 þ D�Þ1=2, the system reduces to

�a1 e� �hm� � a2 e

�hm� þ a3 e�im� þ a4 e

im� ¼ 0,

�a1 e�hm� � a2 e

� �hm� þ a3 eim� þ a4 e

�im� ¼ 0,

a1 �hm e��hm� � a2 �hm e

�hm� � a3im e�im� þ a4im eim� ¼ 0,

a1 �hm e�hm� � a2 �hm e�

�hm� � a3im eim� þ a4im e�im� ¼ 0,

Wm ¼ m2 þ �1:

8>>>>>>><>>>>>>>:

A non-trivial solution is obtained by imposing the

determinant D of the system to vanish, i.e.

D ¼ ð2m2 þ D�Þ sin ð2m�Þ sin 2ðm2 þ D�Þ1=2�� �

þ 2mðm2 þ D�Þ1=2

�

�cos ð2m�Þ cos 2ðm2 þ D�Þ1=2�

� �� 1

�¼ 0:

Because D�¼ k2(G1�G2)¼ (42/2)DG40, the deter-

minant D is always a real number. The fundamental

solution m¼ 0 satisfies the equation D¼ 0 and thesystem reduces to

�a1 exp½�iðD�Þ1=2�� � a2 exp½iðD�Þ

1=2�� þ a3 þ a4 ¼ 0,

�a1 exp½iðD�Þ1=2�� � a2 exp½�iðD�Þ

1=2�� þ a3 þ a4 ¼ 0,

a1iðD�Þ1=2 exp½�iðD�Þ1=2�� � a2iðD�Þ

1=2

� exp½iðD�Þ1=2�� ¼ 0,

a1iðD�Þ1=2 exp½iðD�Þ1=2�� � a2iðD�Þ

1=2

� exp½�iðD�Þ1=2�� ¼ 0:

8>>>>>>>>><>>>>>>>>>:

If we choose �, D� so that �(D�)1/2¼ (2�/)�(DG)1/2� 1 or �(DG)1/2� /2 then we can approxi-mate exp[i(D�)1/2�]’ exp[�i(D�)1/2�]’ 1, and thesystem reduces to

�a1 � a2 þ a3 þ a4 ’ 0,

a1 � a2 ’ 0:

The fundamental solution becomes f�1=21ð�Þ ’ 2a1

� 2 ½�p, p�.The same approximation for higher order modes

m� 1 leads to the system

�a1 e�im� � a2 e

im� þ a3 e�im� þ a4 e

im� ¼ 0,�a1 e

im� � a2 e�im� þ a3 e

im� þ a4 e�im� ¼ 0,

a1ðm2 þ D�Þ1=2 e�im� � a2ðm

2 þ D�Þ1=2 eim�

�a3m e�im� þ a4m eim� ¼ 0,a1ðm

2 þ D�Þ1=2 eim� � a2ðm2 þ D�Þ1=2 e�im�

�a3m eim� þ a4im e�im� ¼ 0:

8>>>>>><>>>>>>:

Then for m 6¼ 0 the factor D� can’t be completelyeliminated because it is not multiplied by � and thedeterminant becomes

D ¼ ð2m2 þ D�Þ � 2mðm2 þ D�Þ1=2 6¼ 0:

Figure 2. The function q(�).

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For example, if we want the first modes to sense the

perturbation we have to choose D�’ 0.1, �� 1, which

means (DG)1/2’ /2.It is worth pointing out that the perturbation of the

refractive index G1,2/r2 is undetectable in the cladding

region where r is large (r410 mm), then the e.l.m.

well could be considered a core effect. A possible

choice DG’ 0.25� 10�14 corresponds to a perturba-tion ðjDn2jÞ1=2 ¼ ðjn2eF � n2corejÞ

1=2¼ G1=2

1 =r ’ 5� 10�2

at r¼ 1 mm and (jDn2j)1/2’ 5� 10�3 at r¼ 10 mm, an

unperturbed radius �u of a fraction of a mm and

�’ 1/36 rad.Special cases are described by the following

theorems.

Theorem 2.1: If 2m�¼s1, s12N, �52, the condi-tion D¼ 0 reduces to

cos ps1 1þD�m2

�1=2 !

¼ ð�Þs1 , 05 s1 5 4m:

The constants �1, �2 must satisfy the relation

D� ¼ m2 ‘1s1

�2

�1

" #� 1,

‘1 2 N, ‘1 � s1 even,

15‘1s1

��������� 1þ

1

m2

�1=2

5 21=2:

Proof: It is enough to solve the system

2m� ¼ ps1,

2ðm2 þ D�Þ1=2� ¼ p‘1:

œ

We observe that

052m�

p5 ‘1 5

2ð21=2Þm�

p5 4ð21=2Þm:

Theorem 2.2: If 2(m2þD�)1/2�¼s2, s22N, �52,

the condition D¼ 0 reduces to

cosmps2

ðm2 þ D�Þ1=2

�¼ ð�Þ

s2 :

The constants �1, �2 must satisfy the relation

D� ¼ m2 s2‘2

�2

�1

" #,

‘2 2 N, ‘2 � s2 even,

15��� s2‘2

���� 1þ1

m2

�1=2

5 21=2,

05 ‘2 5 4ð21=2Þm:

Proof: It is enough to solve the system

2ðm2 þ D�Þ1=2� ¼ ps2,

2m� ¼ p‘2:

(

œ

The step azimuthal window periodically introducesfurther internal refractions into the optical fiber. Assketched in Figure 3, the rays that enter the fiberintersect the core–cladding interface and strike the stepazimuthal window. Each time that the ray of a highazimuthal order mode intersects the azimuthalwindow, it undergoes a small reflection and after arelatively short optical path becomes an unboundedray, unless the geometry of the window preciselymatches the conditions described above.

Because of their radial symmetry, the modes withm¼ 0 are less sensitive to the azimuthal windowperturbation. In a standard step-index fiber with coreradius a and core and cladding indices ncore, nclad, thenumber of guided modes is determined by theV value [11]

V ¼2paðn2core � n2cladÞ

1=2:

The V number must be less than 2.405 for the classicalfiber to be single mode, while in the case of the insertionof our azimuthal window this range is larger and thenumber must be less than 3.832 which corresponds tothe mode labeled LP02. For a photonic crystal fiber theV number transforms into an effective Veff value

Veff ¼2pLðn20 � n2effÞ

1=2,

where n0 is the index of silica (the core material) andneff is determined by the propagation constant of thefundamental space-filling mode [12], the pitch of the

Figure 3. Unbound rays: angled view.

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holes L theoretically ensures the fiber to be endlesslysingle mode. The limiting value of Veff depends on therelative size of the holes, smaller holes make single-mode guidance more likely but makes the fiber moresusceptible to bend loss.

From the point of view of the wave equation (seeFigure 4), the azimuthal part of the pulse is partiallytransmitted and partially reflected by the step window.Both the reflected and the transmitted pulses move inthe same direction and interfere because of thecontinuity due to the circular shape of the fiber.

The interference is destructive unless the geometri-cal phase conditions are satisfied or the azimuthalcomponent of the pulse is near zero as in the case ofLP0,� modes which are insensitive to the azimuthalperturbation. Pulses propagating along the fiber peri-odically encounter the window well, and the producedmultiple destructive interferences cause the highazimuthal mode to become evanescent.

3. Solutions of the Maxwell equations for the

step-azimuthal optical fiber with G1^ 0

We propose in this section a particular case (seeFigure 5) of the window for G1¼ 0 and then �1¼ 0,G250, D�40, Wm¼0¼ 0.

This kind of fiber can be built by modifying the

fundamental core–cladding design with a radial para-

bolic concentration of dopants. Several groups have

obtained various levels of suppression of high-order

modes by adjusting fiber index and dopant distribu-

tions [13,14].In this section we discuss the case m¼ 0 by

calculating the solutions of Maxwell’s equations and

all the field components of the modes.As in [15–17], we define

Dn ¼ 1��clad�core¼ 1�

ncladncore

�2

,

k� ¼ kancore 2pa

,

v2 ¼ u2 þ w2 ¼ ða!Þ2��coreDn,

u ¼ aðk2n2core � �2Þ

1=2,

w ¼ að�2 � k2n2cladÞ1=2,

where �clad, �core and, nclad, ncore are the dielectric

constants and the refractive indices of the cladding and

the core, respectively, k* is the normalized propagation

constant, k¼!/c, a is the core radius, and v is the

normalized frequency.

Figure 4. Interference of the azimuthal components of the incident pulse.

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Assuming Dn� 1 we have a simplification in

matching the field components at the core–cladding

interface [15] and we use the Cartesian coordinates

in order to express the components of the field

vectors.We may express E� in term of Ex and Ey as

E� ¼ �Ex sin � þ Ey cos �:

Thus, the continuity of E� is equivalent to the conti-

nuity of Ex or Ey.The y-polarized solutions are expressed in the

following form: Core (�u5r5a)

Exðr, �, z, tÞ ¼ 0,

Eyðr, �, z, tÞ ¼ AJ0u

ar

� �exp½ið!t� �zÞ�,

Ezðr, �, z, tÞ ¼ �Aiu

a�sin �J 00

u

ar

� �exp½ið!t� �zÞ�,

Hxðr, �, z, tÞ ¼ ��

!�AJ0

u

ar

� �exp½ið!t� �zÞ�,

Hyðr, �, z, tÞ ’ 0,

Hzðr, �, z, tÞ ¼ Aiu

a!�cos �J 00

u

ar

� �exp½ið!t� �zÞ�:

Cladding (r4a)

Exðr, �, z, tÞ ¼ 0,

Eyðr, �, z, tÞ ¼ BK0w

ar

� �exp½ið!t� �zÞ�,

Ezðr, �, z, tÞ ¼ �Biw

a�sin �K 00

w

ar

� �exp½ið!t� �zÞ�,

Hxðr, �, z, tÞ ¼ ��

!�BK0

w

ar

� �exp½ið!t� �zÞ�,

Hyðr, �, z, tÞ ’ 0,

Hzðr, �, z, tÞ ¼ Biw

a!�cos �K 00

w

ar

� �exp½ið!t� �zÞ�:

Unperturbed zone (0� r5�u)

Exðr, �, z, tÞ ¼ 0,

Eyðr, �, z, tÞ ¼ AJ0u

ar

� �exp½ið!t� �zÞ�,

Ezðr, �, z, tÞ ¼ �Aiu

a�sin �J 00

u

ar

� �exp½ið!t� �zÞ�,

Hxðr, �, z, tÞ ¼ ��

!�AJ0

u

ar

� �exp½ið!t� �zÞ�,

Hyðr, �, z, tÞ ’ 0,

Hzðr, �, z, tÞ ¼ Aiu

a!�cos �J 00

u

ar

� �exp½ið!t� �zÞ�:

Figure 5. A special case with "1¼ 0 and "250.

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The field of the perturbed fiber does not change

with respect to the unperturbed one. The x-polarized

solution is analogous. The continuity condition for Ey

at the core boundary gives the constant B:

B ¼ AJ0ðuÞ

K0ðwÞ:

The constant A is then determined by the normaliza-

tion condition. By considering the continuity of Ez at

r¼ a, we obtain the following mode condition

uJ 00ðuÞ

J0ðuÞ¼ w

K 00ðwÞ

K0ðwÞ:

The mode field is continuous with its derivative at the

interface with the small unperturbed zone.The time averaged Poynting vector along the

waveguide is

Sz ¼1

2<ðExH

�y � EyH

�x Þ:

Substituting the field components we obtain

Sz ¼

�

!�jAj2J20

u

ar

� �, �u 5 r5 a,

�

!�jBj2K2

0

w

ar

� �, r4 a:

8>><>>:

The amount of power that is contained in the core/

cladding and in the unperturbed region is given by

Pcore ¼

ð2p0

ða�u

Szr dr d�,

Pclad ¼

ð2p0

ð1a

Szr dr d�,

Pun ¼

ð2p0

ð�ua

Szr dr d�:

Using some classical integrals of Bessel functions

[18], the powers Pcore, Pclad, and Pun can be written as:

Pcore ¼�

2!�pa2jAj2½J20ðuÞ þ J21ðuÞ� � Pun,

Pclad ¼�

2!�pa2jBj2½�K2

0ðwÞ þ K21ðwÞ�,

Pun ¼�

2!�p�2ujAj

2 J20u

a�u

� �þ J21

u

a�u

� �h i:

The total power flow is thus given by

P ¼�

2!�pa2jAj2

"J21ðuÞ þ

J20ðuÞK21ðwÞ

K20ðwÞ

#� Pun:

It is worth pointing out that

lim�u!0

Pun ¼ 0:

Note that the expression of the power for the funda-

mental mode is identical to the classical step azimuthal

fiber case [19].

4. High-order modes for the spiral optical fiber

In a previous paper [7] the spiral optical fiber was

initially studied in fundamental mode operation, and

the solution was obtained in terms of the Hermite func-

tions in � and Bessel’s in r. In Cartesian coordinates

these kinds of perturbations are also known as ‘a square

law medium for a lens waveguide’ [19].Because of the particular polynomial dependence of

the fields in the angle �, the matching conditions are

possible only for pure TE or pure TM mode.In order to propagate the fundamental mode, the

g value of the spiral perturbation must be

315 g5 2:2:

In order to propagate the higher order modes, the

azimuthal continuity of the e.l.m. field must be

satisfied. According to the solution of the azimuthal

Hermite eigenvalue equation [20], the gp, j perturbation

values must satisfy the relation

gp, j ¼

2

tp, j

p2, p even:

Hence, the solution in the core region is

Ez

Hz

� ¼

A

B

� JðWp, jÞ

1=2

u

ar

� �Hp �

!gp, jc

� �1=2�

� exp �!gp, j2c

�2� �

, r5 a � p5 � � p:

and in the cladding region

Ez

Hz

� ¼

C

D

� KðWp, jÞ

1=2

w

ar

� �Hp �

!gp, jc

� �1=2�

� exp �!gp, j2c

�2� �

, r4 a � p5 � � p:

The eigenvalues are

Wp, j ¼ 2vp, j pþ1

2

�:

The v values must satisfy the following equation

vp, j ¼ðtp, jÞ

2

p2:

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where the points �¼ tp, j are in the range [20],

��p, j 5 tp, j 5 �þp, j,

�p, j ¼ ð2pþ 1Þ1=2 cos ð ~Fp, jÞ Oð1=pÞ,

~Fp, j ¼~Tpj2þX1p¼1

Jpð pÞ

psin ð p ~Tpj Þ,

~Tpj ¼�2þ 3þ 4j

2pþ 1p, j � 1:

The complete solution of the wave equation [7] canthen be expressed in terms of the longitudinal fields.

5. Conclusion

In this paper a new type of perturbation of therefractive index for optical fiber has been proposed.The classical Fourier equation with special boundaryvalues was used in order to solve the Maxwellequations. The purpose of the work is to theoreticallyintroduce an azimuthal step index optical window inthe classical core/cladding fiber. The window acts as anextremely high selector of azimuthal modes allowingone to achieve high peak powers in single modeoperation.

We have found that for a thin window �� 1, thelower azimuthal mode for m¼ 0 is a solution of theazimuthal Fourier equation, while for m40 a strictrule of selection between the parameters D�, � must besatisfied. Moreover, for m� 1 the propagating modesencounter a small well and destructive interferencephenomena limit their propagation along the fiber.

References

[1] Limpert, J.; Schreiber, T.; Nolte, S.; Zellmer, H.;

Tunnermann, T.; Iliew, R.; Lederer, F.; Broeng, J.;

Vienne, G.; Petersson, A.; Jakobsen, C. Opt. Express

2003, 11, 818–823.

[2] Koplow, J.P.; Kliner, D.A.V.; Goldberg, L. Opt. Lett.2000, 25, 442.

[3] Clarkson, W.A.; Cooper, L.J.; Wang, P.; Williams, R.B.;Sahu, J.K. Trends Opt. Photon. 2003, 83, 261–267.

[4] Alvarez-Chavez, J.A.; Grudinin, A.B.; Nilsson, J.;Turner, P.W.; Clarkson, W.A. Mode Selection in High

Power Cladding Pumped Fiber Lasers with TaperedSection. In Conference on Lasers and Electro-Optics;OSA Technical Digest; Optical Society of America:

Washington, DC, 1999; pp 247–248.[5] Agrawal, G.P. Nonlinear Fiber Optics, 2nd ed.;

Academic: New York, 1995.

[6] Tunnermann, A.; Schreiber, T.; Roser, F.; Liem, A.;Hofer, S.; Zellmer, H.; Nolte, S.; Limpert, J. J. Phys. B:At. Mol. Opt. Phys. 2005, 38, S681–S693.

[7] Fabbri, I.M.; Lauto, A.; Lucianetti, A. J. Opt. A: Pure

Appl. Opt. 2007, 9, 963–971.[8] Marcuse, D. Appl. Opt. 1982, 21, 4208–4213.[9] Marcuse, D. J. Opt. Soc. Am. 1976, 66, 311–320.

[10] Cohen-Tannoudji, C.; Diu, B.; Laloe, F. QuantumMechanics; Wiley: New York, 1977.

[11] Snyder, A.W.; Love, J.D. Optical Waveguide Theory;

Chapman Hall: New York, 1983.[12] Birks, T.A.; Knight, J.C.; Russell, P.St.J. Opt. Lett.

1997, 22, 961–963.

[13] Offerhaus, H.L.; Broderick, N.G.; Richardson, D.J.;Sammut, R.; Caplen, J.; Dong, L. Opt. Lett. 1998, 23,1683–1685.

[14] Sousa, J.M.; Okhotnikov, O.G. Appl. Phys. Lett. 1999,

74, 1528–1530.[15] D. Gloge, D. Appl. Opt. 1971, 10, 2252–2258.[16] Snyder, A.W. IEEE T. Microw. Theory 1969, 17,

1130–1138.[17] Snitzer, E. J. Opt. Soc. Am. 1961, 51, 491–498.[18] Watson, G.N. A Treatise on the Theory of Bessel

Functions, 2nd ed.; Cambridge University Press:Cambridge, UK, 1966.

[19] Marcuse, D. Light Transmission Optics, 2nd ed.;

Van Nostrand Reinhold: New York, 1982.[20] Fabbri, I.M.; Lucianetti, A.; Krasikov, I. On a Sturm–

Liouville Periodic Boundary Values Problem. IntegralTransforms and Special Functions 2009, 20, 353–364.

[21] Szego, G. Orthogonal Polynomials; AMS ColloquiumSeries, 4th ed.; American Mathematical Society:Providence, RI, 1975; Vol. 23.

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