THE LEAKY MODE EFFECT ON MONOMODE ANISOTROPIC FIBRES

In document Studies of anisotropic optical fibres (Page 43-47)

yx x birefringence.

THE LEAKY MODE EFFECT ON MONOMODE ANISOTROPIC FIBRES

3.1 Introduction

In the previous Chapter we have considered fibres constructed

from anisotropic material. We have seen that on weakly guiding,

weakly anisotropic fibres the refractive index profiles for x-

and y-polarised light generally couple very little through the wave

equation, so that they behave essentially independently. On the

basis of this we have shown how to analyse the fundamental modes of these fibres. This led us to propose a design for an ideal single­ polarisation fibre.

However, no matter how small the coupling between the two polarisation states of light on the fibre, it never completely

vanishes and in fact is present even on isotropic fibres. As a

consequence the fundamental modes of optical fibres are plane

polarised only as a first approximation, but possess components (even though generally smaller in amplitude) of the orthogonal

coupling is in fact quite small, the orthogonal component being of order A^ ~ 6 << 1 (see Table 2.2 on page 22 ) and thus may generally be neglected as was assumed in the previous Chapter. However there is an important exception and that is when this component leads to leakage. In this Chapter we will show that the minor orthogonally polarised field component can indeed lead to a substantial loss if the relative birefringence 6../A. is

ij i

sufficiently high. To our knowledge apart from our work [1,2] this kind of loss mechanism has not been previously reported. We will also show how it can be used to design another type of single­ polarisation fibre.

Throughout this Chapter we will again confine ourselves to weakly guiding, weakly anisotropic fibres, ie. A 6.. « 1 . All

il

parameters are defined in Table 2.2 on page 22 . For convenience we will also assume that the fibre axes are aligned with the principal axes of refractive index as shown in Fig. 2.4 on page 21.

We will start by giving a physical description of the loss effect. Understanding the physics of the effect will lead us to a simple derivation of the condition, ie. minimum amount of birefringence, required for this mechanism to work. We then give an indication of the magnitude of the loss involved and present a design for a new type of single-polarisation fibre. Even though we were able to derive the "cutoff-condition" for the loss effect just by intuitive ideas, we have to solve the wave equation to derive the actual loss. In this Chapter we present a derivation of this loss by the Green's function method. We will first introduce and formulate the method for the example of stress-induced birefringence. For this case of practical importance the method becomes particularly simple. Then we go on and present the results for arbitrary, but weak, anisotropy. As a byproduct of this method we will also obtain the full fields of the fundamental modes to first order in We will continue by investigating the

i

ij

properties of the fundamental modes. Then we investigate the fibre loss and present a formula for the loss of fibres with arbitrary, but weak, anisotropy. We conclude the chapter by giving an overview of alternative derivations for the loss which can be found in our published work [1]. All the mathematical details have been

33

relegated to the appendices to streamline the presentation.

The material presented in this Chapter is the result of

research undertaken in collaboration with Prof. A.W.Snyder [1-3].

3.2 Leakage Losses in Monomode Anisotropic Fibres

3.2.1 Physical Description of the Loss Mechanism

Our description of the fundamental modes of anisotropic fibres given in Chapter 2 treats each modal polarisation state as if it were in isolation of the other. This is valid only when the modal fields are plane-polarised, which is never exactly the case for non- planar waveguides. Even on the isotropic fibre, the "x"-polarised modal field has a very small y-component of order A and analogously the "y”-polarised modal field a very small x component [4,5]. Thus, a fraction of power propagating in the "x"-polarised mode "sees" something of the ny profile and analogously the "y"-polarised mode "sees" something of the nx profile. In other words, there is a small amount of coupling between the two polarisation states.

On isotropic fibres, the phase velocity w/ß of a mode can not exceed the speed c/n ^ of a z- directed plane wave in the cladding

without it losing power to radiation, ie. bound modes obey

ß > kn , , with k = 2tt/ A

cl Generalizing to anisotropic fibres,

x . ., . x x y

bound modes must satisfy ß > k n ^ assuming that n ^ > n ^ ,

otherwise the x-polarised portion of the modal field will radiate because it travels faster than an x-polarised plane wave in the cladding medium. Quite obviously the "y"-polarised mode of the fibre of Fig. 2.5 (on page 24 ) is leaky. Furthermore, the "y"- polarised mode of the fibre in Fig. 1 will also "leak" if the

birefringence is sufficiently high to satisfy ß < kn , where

y cl

ßy is defined by eqs. (2.7) and (2.8).

From an alternative viewpoint, leakage occurs when some of the "y"-polarised mode of the fibre in Fig.l "sees" a cladding index n*^ which is higher than the effective index ß /k "seen" by the

y-polarised mode. Then, total internal reflection is frustrated for

3 4

Fig.l. Design for the ’Leaky-Mode' Single-Polarisation

Fibre. In this example = A . Then the y-polarised

mode is leaky, provided eq.(l) is satisfied. All parameters

3 5

w a v e g u i d e w i t h n < n . On the o t h e r hand, the e f f e c t i v e i ndex

co cl

y

of the x - p o l a r i s e d mode is h i g h e r than n ^ , so that this mode suffers no loss.

F r o m ei t h e r v i e w p o i n t d i s c u s s e d above, r a d i a t i o n losses occur

X X

w h e n e v e r $ < k n ^ . U s i n g the l i m i t i n g c o n d i t i o n 3^ = k n ^ > we s how d i r e c t l y ( A p p e n d i x A.3) that the m i n i m u m r e lative b i r e f r i n g e n c e n e c e s s a r y for lea k a g e loss is

6 yx s

A ( 1.14

- i

V )2J

(

1

)

a ssumes the step p r o f i l e fibre of Fig. 1 , w i t h A^ = A^

V - V - V of T a b l e 2.2 w h e r e 6 and A are d e p i c t e d in

x y yx

W h e n 6 /A = 1, then the core index of the y - p r o f i l e to w i t h i n an a c c u r a c y of 1 per c e n t w h e n 1.4 < V < 2.5 . This = A so

~ x y

that Fig. 1 .

eq u a l s the clad d i n g index of the x - p r o f i l e , as in Fig. 2a . W h i l e w h e n 6 /A = V2 » we h ave the s i t u a t i o n in Fig. 2b.

A p a r t f rom our w o r k [1,2], this lea k a g e loss has not been p r e v i o u s l y reported. It occurs on n o n - p l a n a r w a v e g u i d e s only as it r e q u i r e s the modal e l e c t r i c field to h ave both x and y components.

3 . 2 . 2 Magnitude of the Fibre Loss

The m i n i m u m b i r e f r i n g e n c e n e c e s s a r y for l eakage loss is d e t e r m i n e d us i n g only e l e m e n t a r y phys i c a l arguments, whe r e a s we must solve the w a v e e q u a t i o n to o b t a i n the m a g n i t u d e of this loss.

W e shall derive the loss in Sections 3 .3 and 3 .4 . For the mo m e n t we w ill d r a w u p o n one r esult f rom S e c t i o n 3 . 6 w h e r e we d i scuss the fibre loss in detail. T h e p ower f low e of the y - p o l a r i s e d mode of Fig. 1 a t t e n u a t e s at a rate (see eq.(22) in S e c t i o n 3 . 6 ) .

Y “

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