Viale.pdf

8/9/2019 Viale.pdf

http://slidepdf.com/reader/full/vialepdf 1/5

Abstract

To modelize infinite photonic crystal fibre(PCF) with 2D-finite-geometry mode solver, itis necessary to use a perfectly matched layer(PML). We have performed a new type of PMLdesign to simulate propagation in PCFs. Theresults obtained with index-guiding PCFs are invery good agreement with previous theoretical

published results. Our PML is quicklyoptimize. In the case of bandgap-guiding PCFs,the transmission bandgap of the fibre isidentical in both cases, however variations ofconfinement loss are due to the high number ofdegrees of freedom necessary to solve theproblems. Morever, the link betweenMATLAB and FEMLAB offers the possibilityto modelise realized profile to estimate lossbefore experimental results.

Keywords FEMLAB 3.1 – Photonic crystal fibre– index-guiding PCF – hollow-core PCF – Perfectlymatched layer – Confinement loss

1 Introduction

Software based on the finite-element method, likeFEMLAB, solves two-dimensional electromagneticproblems. One drawback of this technique for the

application to optical fibre is that Maxwell’sequations are solved with a finite geometry. It isnecessary to define an equivalent geometry as infiniteto determine characteristics of propagation. Forcompensating this difficulty, we use a perfectlymatched layer (PML) [1]. PML is an absorbing layerspecially studied to absorb without reflection theelectromagnetic waves. Using this layer, we can

estimate the confinement loss of any optical fibre. Inthis report, we have performed calculations on newgeneration of optical fibres called Photonic CrystalFibres (PCFs).Some calculations about optical fibres have alreadybeen performed using a PML [2-8]. In previous work,the PML was defined as a square surrounding thefibre structure as shown in Figure 1 [3]:

Figure 1: Square PML applied for optical fibre

computations

In this case, the square PML is decomposed in 8

areas. Each region is defined by a parameter “si”. Thisterm is introduced in the Maxwell’s equations [9] anddepends on directions x or y. Instead of simulation in(x,y)-dimension, we have the possibility to work in(,θ)-dimension. We have to introduce a one-regionPML invariant following the radius . So thisgeometry is circular.A circular design for PML was defined to improvecalculations modes propagation in optical fibres [10].This work puts in evidence the impact of the PML onthe different propagating modes in optical fibres.However, definition and use of the PML is difficult,due to the high number of parameters. In this report,

we discuss the application of an uniform parameter“s” in a circular PML.

2 Definition of the circular PML

In 2-dimensional case, a circular PML is an absorbingregion surrounding the fibre structure. The thicknessis noted e and the internal radius rin. The PML isschematically shown in Fig. 2:

Structure of optical fibre

1

635

2

4 87Pure silica

PML

x

y

Pierre Viale, Sébastien Février, Frédéric Gérôme, Hervé Vilard

Confinement Loss Computations in Photonic Crystal Fibres using a

Novel Perfectly Matched Layer Design

P. VialeIRCOM, CNRS UMR 6615, 123 avenue Albert Thomas87060 Limoges, France,Tel. : (33) 5 55 45 72 68Fax : (33) 5 55 45 72 53E-Mail : [email protected]

Excerpt from the Proceedings of the COMSOL Multiphysics User's Conference 2005 Paris

8/9/2019 Viale.pdf


Figure 2: Cross section of optical fibre surrounded by

circular PML

In PML region, Maxwell’s equations can be written[2,11]:

Esn jH 20 ⋅=∧∇

(1)

Hs jE 0 ⋅−=∧∇ (2)

with0

m

20

e

j1

n

j1s ⋅−=⋅−=

(3)

where:E : Electric field;H : Magnetic field; : Angular frequency;0 and µ 0 : Permittivity and permeability ofvacuum, respectively;n : Refractive index;e and m : Electric and magneticconductivities of PML, respectively.

The absorption of the PML will be optimal whenreflection coefficient R at the interface between PMLand pure silica region is small. This small coefficient

corresponds to an impedance adaptation whetherconductivities satisfy relation (3). So:

0

m

20

e

n

= (4)

To optimize R, electric conductivity in the PMLpresents an m-power profile as shown in Fig. 3 and inequation (5):

Figure 3: ’s continuity between pure silica and PML

regionm

inmaxe

e

r-

⋅= (5)

We can estimate parameter “s” as [2]:

⋅⋅−

=

regionsothersin,1

regionPMLin,e

r- j1

s

2

in

(6)where κ is defined for λ (wavelength) by

⋅=

R1ln

ne43

(7)

By specifying refractive index of PML using theparameter “s”, the imaginary part of propagationconstant of the propagating mode represents theconfinement loss in Neper/m. So, calculations aboutphotonic crystal fibres can be performed. However,using imaginary parameters, the calculations arememory consuming. The geometry has to be optimizefor computations.

3 Definition of photonic crystal fibres

A new generation of optical fibres, called PhotonicCrystal Fibre (PCF), was proposed for the first time in1996 [12]. The core is surrounded by a crystalcladding composed of air holes embedded in puresilica.Two different principles of light propagation exist inPCFs:

- Total Internal Reflection (TIR), if the corerefractive index is higher than the average refractiveindex of the cladding. There are called index-guidingPCFs,

- Photonic Band-Gap (PBG), if the othercase. There are called bandgap-guiding PCFs.

The variation of radial electric field is different inboth cases as shown in Fig.4:

Figure 4: Variation of electric field E in TIR propagation fibre (a) and in PBG fibre (b)

For index-guiding PCFs, the electric field isevanescent in the cladding, whereas for bandgap-guiding PCFs, E oscillates to 0. When using the PML,it is necessary positioned where E is highlyattenuated. For (a) case, there are no evident problemswhereas in (b) case, the position of PML should becrucial due to high oscillations in the end of thecladding.

Circular PML

Structure of optical fibre

Pure silica region

rin

e

rine

e

max

’s continuity

P P u u r r e e s s i i l l i i c c a a r r e e g g i i o o n n P P M M LL r r e e g g i i o o n n


8/9/2019 Viale.pdf


4 Optimization of the PML

By definition, the PML should be placed far from thestructure. It should be large in order to get variationof “s” is not abrupt. For a 6-µm radius structure, wehave performed calculations by varying position rin

and varying thickness e. The results of confinementloss are shown in Fig. 5:

0.00

0.05

0.10

0.15

0.20

0.25

10 15 20 25 30 35 40 45 50

rin (µm)

c o n f i n e m e n t l o s s ( d B / k m )

e=1µm

e=5µm

e=10µm

Figure 5: Variation of confinement loss in 6-µm radius

structure

For a thickness e=10µm, variation of confinementloss is almost zero. In this case, the position of thePML is not a crucial parameter. For thicknessesinferiors at 10 µm, the term rin is important. Theconfinement loss varies around the value of0.1dB/km.It is important to note that there is no ideal PML. Foreach structure of fibre, there is a different optimizedPML.

5 Calculations with index-guiding PCFs

For these calculations, we have compared ourcalculations with those of Saitoh using squared PML[3] and in first step (classical PCFs) with othertotally-different method: multipole method [13,14].This method is faster, but can only be used withclassical PCFs. The core is made of silica, and thecladding is composed of 1.61-µm diameter air holes(d) separated by a pitch of 2.3µm (Λ).In the in-set ofFig. 6, the structure of PCF is shown.

2.00E-03

2.00E-02

2.00E-01

1.00E+00

6.00E+00

3.47E-03

3.04E-02

2.26E-01

1.41E+00

7.74E+00

3.06E-03

2.57E-02

2.06E-01

1.40E+00

7.73E+00

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000

Wavelength (nm)

C o n f i n e m e n t l o s s ( d B / m )

Calculation with squared PML

Calculation with circular PML

Multipole method

e=10 µm

rin=15 µm

d =1.61 µm

Λ=2.3 µm

Figure 6a: Comparison of calculations with squared and

circular PML surrounding index-guiding PCF, andmultipole method with 2 layers of air holes

6.00E-08

1.00E-03

9.95E-07

3.06E-05

2.50E-06

1.00E-02

6.00E-05

7.94E-08

2.45E-06

1.39E-02

1.04E-03

5.95E-05

1.21E-02

7.97E-04

1.E-08

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000

Wavelength (nm)

C o n f i n e m e n t l o s s ( d B / m )



Multipole method

e=10 µmrin=20 µm

d =1.61 µm

Λ=2.3 µm

Figure 6b: Comparison of calculations with squared and

circular PML surrounding index-guiding PCF and

multipole method with 3 layers of air holes

Whatever the methods used, values of confinementloss () are of the same order of magnitude. Relativeerror between both FEM values of (PML) andmultipole method values (MM) is shown in Fig. 7.

0

20

40

60

80

100

120

140

160

1000 1250 1500 1750 2000

wavelength (nm)

R e l a t i v e E r r o r E r ( % )

Squared PML

Circular PML

Squared PML

Circular PML

PCF 3 layers

PCF 2 layers

Figure 7: Variation of relative error Er of confinement loss

between FEM and MM method for both cases of PCFs

For high confinement loss (), Er is minor than 40%.For low , this error is higher. It is due to the limit ofutilisation of multipole method. For values of minorthan 10-5dB/km, multipole method is inefficient. Sofor classical PCF, the circular PML is efficient tostudy the confinement loss.Assuming this result, we have performed calculationson complex index-guiding PCF called “dual-concentric PCF” [7]. These fibres allow obtaining avery high negative chromatic dispersion coefficient.The structure of this fibre is given in the inset of Fig.

8. The result of the comparison between two modelsof PML is plotted in Fig. 8.

1.00E-06

3.00E-04

8.00E-05

5.00E-06

2.00E-05

2.65E-04

7.13E-05

4.25E-06

1.80E-05

1.E-06

1.E-05

1.E-04

1.E-03

1450 1470 1490 1510 1530 1550 1570 1590 1610 1630 1650

wavelength (nm)

c o n f i n e m e n t l o s s ( d B / m )

Calcuation with squared PML

calculation with circular PML

e = 10 µmrin = 20 µm

Figure 8: Comparison of calculations with squared andcircular PML surrounding concentric dual-core PCF

(a)

(b)

MM

MMPML

Er

−=


8/9/2019 Viale.pdf


As previously, variations of confinement loss in bothcases are small. So for index-guiding PCFs, thecircular PML is efficient and quickly optimized.

6 Calculations with bandgap-guiding PCFs

The core of the structure is generally in air,surrounded by an air/silica cladding, so these fibresare called hollow-core PCFs. The light is guided by aphotonic bandgap effect. The thickness of silica websis very small (50nm) as is shown in Fig. 9:

Figure 9: Structure of Hollow-core PBG fibre

For computation, it is needed to highly mesh thinweb. In consequence, the number of degrees of

freedom (dof) is higher. The computer used is a2.66Ghz-Pentium IV with 512MB RAM. Thethreshold of dof is 210,000 in this case withUMFPACK solver and with complex propagationconstant. For computation, it is needed to use thesymmetry of the structure. To compare our methodand that developed by Saitoh et al, calculations areperformed with a 4.7-µm pitch structure of a hollow-core PBG fibre [8]. The results are plotted in Fig.10a:

0.001

0.01

0.1

1

10

100

1000

1.4 1.45 1.5 1.55 1.6 1.65 1.7 1.75wavelength (µm)

C o n f i n e m e n t l o s s ( d B / k m )



Figure 10a: Comparison of calculations with squared and

circular PML surrounding hollow-core PBGF

Figure 10b: Propagation modes in bandgap-guided PCFs.

1. Fundamental mode2. Coupling between surface and fundamental

modes

3. Surface mode

We can observe variations between confinementlosses with the two different PMLs. Although limit ofdof is reached in this case, we can observe thattransmission bandgap is identical in the two cases[1.4µm-1.75µm]. Moreover, we can observe inFig.10b, fundamental and surface modes typical insuch type of fibre.So, the circular PML is efficient for bandgap-guiding

PCFs. For better computation, we have to use anothersolver, or more performing computer. However,results obtain with this PML are in good agreementwith published results. Also, the validity of circularPML using bandgap-guiding PCFs is proved.

7 PML used on with realized fibre profile

In this last part, we have just introduced thepossibility offered by MATLAB to performcalculations with realized fibre profile. Indeed,computed program in MATLAB permits to load

profile from Scanning Electron Micrograph (SEM)photograph (Fig. 11a).

Figure 11: Modelisation with a realized profile

a. SEM photograph

b. Simulated profile

c. Mesh of the structure

d. Solution computed with Femlab

We obtain a simulated profile (Fig 11b) from thepicture (Fig. 11a). A good mesh (Fig. 11c) is then

ΛΛΛΛ=4.7µm

d/ ΛΛΛΛ=0.98

Λ

d


8/9/2019 Viale.pdf


applied to solve the propagation using this profile(Fig. 11d). In our case, with this fibre, confinementloss is in order to 100dB/m. We have performed atour knowledge the first computation of confinementloss in realized profile HC-PBG fibre.

8 Conclusions

We have evidenced the capabilities of a new PMLable to perform calculations about confinement lossin photonic crystal fibres with FEMLAB. The circularPML studied is efficient and easier to use that currentsolution to perform calculations about index-guidingPCFs. The circular PML is quickly optimized. For thebandgap-guiding PCFs, the use of PML is moredifficult. The optimization of position and thicknessof PML is highly important. Results are depended onmesh of the structure. High mesh needs use of

Random Access Memory (RAM). Using of a newsolver [15], less intensive memory for large problemscan be the solution to resolve high-number-dofproblem. Calculations of confinement loss withrealized profile have been performed. For the firsttime to our knowledge that type of calculations havebeen done using realized profile.

References1. J.P. BERENGER “A perfectly matched

layer for the absorption of electromagneticwaves”, J. Comput. Phys., 114, pp 185-200(1994).

2. Y. TSUJI et al “Guided-mode and leaky-mode analysis by imaginary distance beampropagation method based on finite elementscheme”, Journal of Lightwave Technology,18, pp 618-623 (2000).

3. K. SAITOH et al “Full-vectorial imaginary-distance beam propagation method based ona finite element scheme: Application tophotonic crystal fibers”, Journal of Quantum

Electronics, 38, pp 927-933 (2002).4. K. SAITOH et al “Confinement losses in

air-guiding photonic bandgap fibers”,Photonics Technology Letters, 15, pp 236-

238 (2003).5. K. SAITOH et al “Chromatic dispersion

control in photonic crystal fibers: applicationto ultra-flattened dispersion”, Optics

Express, 11, pp 843-852 (2004).6. C.-P. YU et al “Yee-mesh-based finite

difference eigenmode solver with PMLabsorbing boundary conditions for opticalwaveguides and photonic crystal fibers”,Optics Express, 12, pp 6165-6177 (2004).

7. K. SAITOH et al “Highly nonlineardispersion-flattened photonic crystal fibersfor supercontinuum generation in a

telecommunication window”, Optics Express, 12, pp 2027-2032 (2004).

8. K. SAITOH et al “Air-core photonic band-gap fibers: the impact of surface modes”,Optics Express, 12, pp 394-400 (2004).

9. K. SAITOH et al “Full-vectorial finiteelement beam propagation method withperfectly matched layers for anisotropic

optical waveguides”, Journal of LightwaveTechnology, 19, pp 405-413 (2001).

10. H. ROGIER et al “Berenger and leaky modein optical fibers terminated with a perfectlymatched layer” Journal of Lightwave

Technology, 20, pp 1141-1148 (2002).11. Ü. PEKEL et al “A finite-element-method

frequency-domain application of theperfectly matched layer (PML) concept”,

Microwave and Optical Technology Letters,9, pp 117-122 (1995).

12. J.C. KNIGHT et al “All-silica single-modeoptical fiber with photonic crystal cladding”,

Optics Letters, 21, pp 1546-1549 (1996).13. T.P. WHITE et al “Multipole method for

microstructured optical fibers: I.Formulation”, JOSA B, 19, pp 2322-2330(2002).

14. T.P. WHITE et al “Multipole method formicrostructured optical fibers: II.Implementation and results”, JOSA B, 19, pp2331-2340 (2002).

15. J.YSTROM “Multigrid methods”,http://www.comsol.se/multigrid/ .


Documents

Viale.pdf