Propagation in Optical Fibres - Çankaya...

ECE 474 - HTE Ocak 2013 Sayfa 1

Propagation in Optical FibresHTE - 29.01.2013

1. Basics

For sinusoidal variations, the basic relation between speed of light, denoted by c frequency (of light)denoted by f and wavelength (of light) denoted by is

(1.1)c f

where 8 82.997925 10 m / s 3 10 m / sc in vacuum (free space), i.e. in atmosphere freeenvironment. Depending on the refractive index of the medium, n , the speed of light and therelated quantities will change as

: speed of light in free space , : speed of light in medium with refractive index

: wavelength of light in free space , : wavelength of light i

2 2 , ,

n

n

n n nn

c c n

c nc k knn n

n medium with refractive index

: wave number in free space , : wave number in medium with refractive index (1.2)n

n

k k n

The implication in (1.2) is that if the refractive index of the medium is 1n , then the speed of light,wavelength of light and the associated wave number will remain as that of vacuum, that is theparameters without any index. But if 1n , then nc c , n and nk k , since in vacuum,

1n and in other mediums 1n , for instance, 1.33n in water, 1.5n in glass, 2.42n indiamond. Another implication of (1.2) is that when light changes the medium of propagation, itsspeed, its wavelength and the wave number change but its frequency (i.e. the number of cycles itmakes per unit time) remains the same such that c f is always satisfied. The refractive indexproperty is related to the dielectric characteristics of the medium and is also measured by thedielectric constant, relative permittivity, as we shall see later.

It is worth noting that refractive index may change according to spatial coordinates, even with time(temporal coordinate). Being aware of such characteristics allows us to guide light and a medium,thus achieve communication using light frequencies as carriers. It is important to realize that opticalfrequencies are quite high, for instance at a wavelength of 1 m , the frequency becomes

8143 10 m / s 3 10 Hz 300 THz (1.3)

1 mcf

Hence it is reasonable to assume that we can place, multiplex (in FDM sense) a lot of message signalsonto such high frequency carriers. It is customary to speak in terms of wavelengths rather thanfrequencies for the optical range.

1. 1 Simple Laws of Reflection, Refraction, Principle of Total Reflection


By taking a simple two dimensional geometry, we illustrate in Fig. 1.1, an optical ray incident on aboundary where there occurs a change in the refractive index and how it is reflected and refracted.

Fig. 1.1 Illustration of simple two dimensional geometry for incident, reflected and refracted rays.

As seen from Fig. 1.1, when a ray is incident upon a boundary where there is a refractive indexchange from 1n to 2n , there is refraction as well as reflection. The amount of reflection andrefraction is related to the individual refractive indices of the two mediums. Since we have arrangedthat 2 1n n , then 1 2 , and according to Snell’s law, these angles are connected as follows

1 1 2 2sin sin (1.4)n n

Furthermore, the angle of incidence and the angle of reflection are the same, i.e. they are both 1 asshown in Fig. 1.1.

From (1.4), we deduce that there will be an angle of 1 such that 2 / 2 , then

1 1 2 2sin sin (1.5)2n n n

At this point, we let 1 c and call c the critical angle, hence

12 2 21 1 1

sin or sin asin (1.6)c cn n nn n n

Under the circumstances that 1 reaches c , the refracted component will vanish and there will be

total reflection as shown in Fig. 1.2. Note that since 1 1 / 2 , the condition in (1.6) can also beexpressed as

))

) ))

Plane of normal

Refracted ray

Medium with n1

n2 < n1

Medium with n2

Boundary

2

1

1

2

1

Reflected rayIncident ray


12 2 21 1 1

cos or cos acos (1.7)c cn n nn n n

Fig. 1.2 Illustration of reaching the critical angle and total reflection stage.

Assuming that propagation of light can be represented in terms of ray tracing as given in Figs. 1.1and 1.2, the above total reflection phenomena is exactly the principle of light guidance in opticalfibres.

Exercise 1. 1 : Consider a boundary between air (vacuum) and water. Find the critical angles c and

c for total reflection to occur. Discuss if total reflection is possible both ways, i.e. when the incidentray is on the side of air medium and when the incident ray is on the side of water medium. Make thenecessary plots and indicate the necessary angles.

Comments on the total reflection phenomena shown in Fig. 1.2

a) It is obvious then when 1 2n n , then it is impossible to reach a stage of critical angle and totalreflection. In this case mostly refraction will take place.

b) Light propagation can be studied both in terms or rays and waves. When we do wave analysis,then we find that even after the critical angle stage, refraction (or energy loss due to refraction) ispossible due to electromagnetic tunnelling, depending on how much the field extends into theother medium beyond the boundary.

c) In real life, we have a three dimensional situation and the boundary is not a flat surface. Despitethis however, Fig. 1.1 provides the basic principle of reflection and refraction mechanisms.

2. Optical Fibres

2.1 Mechanical Construction of Optical Fibres

))

) )) )) ) ) ))

n1

n2

1

2

Refracted ray

Incident ray

2

2 = / 2

Refracted ray

1 =

c1

1Reflected rayIncident ray

Reflected ray

Normal incidence Critical angle view Total reflection stage

1 =

cIncident ray

1 >

c

1 =

c

1 <

c

1 <

c

Totallyreflected ray


An optical fibre cable is a cylindrical shaped closed medium that guides or carries light in the form ofelectromagnetic waves. The mechanical construction of the optical fibre is shown in Fig. 2.1. Due tothe existence of boundary conditions set by the physical dimensions, only those waves, called modes,are allowed to propagate. This propagation is mostly confined to the core region. The refractiveindex difference between the core and the cladding provides the necessary guidance of theelectromagnetic waves or the total internal reflection mechanism along the fibre propagation axis.

Fig. 2.1 Mechanical construction of a fibre cable.

We classify three types of fibre, depending on core radius (or diameter), i.e., a , refractive indexprofile in the core, i.e. 1.n These are step index multimode, graded index multimode and step indexsingle mode fibres, as illustrated in Fig. 2.2.

Fig. 2.2 Types of fibres.

As the names given in Fig. 2.2 imply, many modes propagate in multimode fibres, since many modessatisfy the propagation conditions, this causes dispersion, because the time it takes for each mode totraverse the fibre length is different. This is balanced to a certain extent in graded index fibres. Onthe other hand, single mode fibres are constructed so that they transmit only the fundamental mode,

Protective coating

Protective coating

b

Cladding

Cladding

n2

n2

n1

n1 Core

Core

aFibre axis

Cla

ddin

g

Cla

ddin

g

Cla

ddin

g

Cla

ddin

g

Cla

ddin

g

Cla

ddin

g

n1n2 n2n2

a

CoreCore

Core Core

Multimodestep index fibre

Multimodegraded index fibre Singlemode

step index fibre

n1 n1

n2

ba

a

n2n2

Core

Core


called 11HE , thus these fibres do not suffer from multimode dispersion. Almost 100 % of the fibresused in public communication facilities are single mode fibres (single mode fibres have always steprefractive index profile).

The physical core and cladding dimensions (radius or diameter) are more or less standardized asfollows

a) The cladding diameter is always, 2 125 mb .b) The core diameter in multimode fibres is 2 50 ma .c) The core diameter in single mode fibres may vary 2 2 m 10 ma . As will be seen later the

single mode property is achieved by arranging the parameters of core radius, wavelength ofoperation and the refractive index difference between core and cladding, i.e. between 1n and 2n .

2.2 Numerical Aperture

Now based on the simple rules of reflection and refraction depicted in Fig. 1.1 and 1.2, we try todetermine, the ray collecting cone of the fibre front face, i.e. fibre entrance. So this is finding themaximum incidence angles of rays, which will propagate in fibre. Fig. 2.3 shows the associated twodimensional view of this situation for a simple multimode step index fibre.

Fig. 2.3 View of fibre end face for calculation of numerical aperture.

Looking at Fig. 2.3, we note that the reflection and the refraction phenomena encountered on thecore cladding boundary (interface) is exactly the same as the one in Figs. 1.1 and 1.2. Hence from(1.6), we write the for the condition of an incident ray from the light source to be totally internallyreflected from the core cladding boundary as

21

sin (2.1)cnn

where c is the critical value of the angle which means that for total internal reflection to takeplace, the minimum value that can be taken up by is c . So total internal reflection will take place

)

)

)

Core n1

Core n1

b

ra

Fibre axis

Incident ray fromlight source thatwill be refracted atcore cladding bondary

Incident ray fromlight source thatwill be internallyand totally reflected

0

n2Refracted ray

Internally reflected ray

n2Cladding

Cladding

Air with n0 = 1

Core claddingboundary (interface)

Fibre entranceFibre front faceAir fibre boundary


provided that c . Rearranging (2.1) and relating the result to the angle by keeping in mindthat / 2 , we get

2 2 22 2 22 1 2

2 21 1

-12 2 2

1 1 1

sin , 1 sin sin

cos , cos cos (2.2)

c c c

c c

n n nn n

n n nan n n

Thus similar to discussion for , we can say that, total internal reflection will take place if c .Using the Snell’s law, we can also relate everything to the angle of fibre entrance, 0 as

0 0 1 0 0

0 1

sin sin sin , 1

sin sin (2.3)c c

n n n

n

Using (2.2), we define the acceptance angle called the numerical aperture (NA) as sine of the

maximum angle subtended by a ray at fibre entrance, i.e. 0sin c or alternatively as the product ofthe core refractive index, 1n and the sine of maximum angle of a refracted ray at fibre entrance,hence NA will be given by

0.52 20 1 1 2NA sin sin (2.4)c cn n n

It is important to realize that NA is solely determined by the total internal reflection conditions at thecore cladding boundary. Otherwise rays with any value of 0 will enter (refract into) the fibre, since

1 0 1n n .

As can be seen (2.4) is in the form of refractive index difference between core and cladding. In otherwords, the principle of light guidance in fibres is based on the (minute) refractive index differencebetween core and cladding, rather than the absolute values of 1n and 2n . For this reason we definethe following (normalized) refractive index difference term

2 21 2 1 2

1 1 2 1 1 221 1

1 , , 2 (2.5)2n n n n n n n n n nn n

In terms of , NA will become

0.5

1 2 1 2 1NA 2 (2.6)n n n n n

It is important to summarize the conditions of total internal reflection and refraction in a step fibre asfollows

0 0

0 0

If or or total internal reflection ito the core or propagation along fibre axis

If or or refr

c c c

c c c

action out to cladding (2.7)


Example 2.1 : A step index optical fibre has a core diameter is large enough so that the ray theory isconsidered applicable. In this fibre, 1 1.5n and 2 1.47n . Determine the followings

a) The critical angle at core cladding boundary for total internal reflection to occur, i.e. c .b) NA of the fibre.c) The acceptance angle, 0 c at the air fibre boundary.

Solution :

a) From (2.1), the angle at core cladding boundary for total internal reflection to occur is

1 1 02

1

1.47sin sin 78.5 (2.8)1.5c

nn

b) From (2.4), NA is

0.5 0.52 1 2 21 2NA 1.5 1.47 0.3 (2.9)n n

c) From (2.4), the acceptance angle, 0 c at the air fibre boundary is

1 00 sin NA 17.4 (2.10)c

As seen in this numeric example, the refractive index difference between the core and the cladding isindeed small (minute).

Exercise 2.1 : Examine the dependence of c , NA and 0 c on 1n and 2n in a step index fibre.Comment whether the larger or smaller values of c , NA and 0 c would be useful.

The above analysis is quite useful to given an insight into propagation of rays in a fibre. But weshould not forget that ray theory is applicable if the dimensions of the physical objects are muchlarger than the wavelength. For instance in the above case, where the (multimode) step index fibrewas considered, the fibre core diameter is 50 μm , hence compared with a wavelength of 1

μm ,

we can safely say that ray theory is applicable. But in other cases, we certainly have to use wave ormode theory which is applicable independent of dimensions of the physical objects present on theway of propagation. In particular, we need to see how we can approach the single mode limit. We dothis in the next section.

3. Mode Theory (Modal Analysis)

For this analysis, convenient form of Maxwell’s equations is required. Because of the cylindricalgeometry of the fibre, we use cylindrical coordinates, where the respective coordinates will be

, , zr as shown in Fig. 3.1.


Fig. 3.1 Coordinate axes in a fibre.

In (3.1), we find the relevant Maxwell’s equations.

Curl of :

Curl of :

Div of : 0 , Div of : 0 (3.1)

t

t

HE E

EH H

E E H H

E and H are electric and magnetic field vectors, t indicates time. and stand for the permeabilityand permittivity of the medium so that they represent the magnetic and electrical properties of themedium. and are further expanded as given below in (3.2)

0

0

0

0

: Relative permeability of medium : Permeability of free space (vacuum)

: Relative permitivity of medium :

r r

r r

Permitivity of free space (vacuum) (3.2)

The speed of light in vacuum and in other mediums, the wave number in free space and in othermediums are further related to the quantities of (3.2) as follows

r

zFibre axis

C ore

C ore


0 0

1Free space : : speed of light in free space2

: wavelength of light in free space

c f ck

: frequency of light in free space : angular frequency of light in free

f

1 1 1

1

1

1 1

1 0 1 0 0

1 1 1 11 11 1 0 1 0 0 1 0 0

1

1

spaceIn a medium with , , 1

1 1 1 1 ,2

: speed of light in a medium with arbitry permitivity and 1

: wav

r r r

r

r

r r

w w cc f nk n n

c

1 1

1 11

elength of light in a medium with arbitry permitivity and 1

: wavenumber of light in a medium with arbitry permitivity and 1 (3.3)r r

r rk

From (3.3), we understand that the refractive index is equal to the square root of the relativepermittivity of the medium. Since the fibre material has no magnetic properties, then 1r in fibre.This means, the electromagnetic propagation or the guidance of light in fibre will entirely rely onrefractive index changes.

Now returning to (3.1), we take the curl of the equation on the first line and benefit from theequation on the second line and the vector identities to write

2 2 2 2 22 2

2 2 2 2 2

2 2 2 2 22 2

2 2 2 2 2

, 0

, 0 (3.4)

n nt c t c t

n nt c t c t

E E EE E

H H HH H

The equations in (3.4) are known as Helmhotz’s equations. Our time dependence will be sinusoidal,

i.e., exp exp 2j t j ft and the propagation axis, i.e. z dependence will be exp j z ,where is known as the propagation constant (the wave number) of the modes propagating infibre. Using these and the equivalence of 2 in (3.4), we obtain the following differential equationfor the z component of the electric and magnetic fields

2 22 2

2 2 2

2 22 2

2 2 2

1 1 0

1 1 0 (3.5)

z z zz

z z zz

E E E k Er r r rH H H k Hr r r r

The other components rE and E , rH and H can be expressed in terms of zE , and zHcomponents as shown below.


2 2 2 2

2 2 2 2

1 1 ,

1 1 , (3.6)

z z z zr

z z z zr

E H E HE E wr r r rj k j k

H E H EH Hr r r rj k j k

zE will have functionally separable dependence on , , andr z t which can be written as

(3.7)z r z tE AF r F F z F t

where A is a constant to be determined by taking into account the boundary conditions (at corecladding interface). We have already postulated the dependence on andz t in the form of

exp exp 2 (3.8)z tF z F t j z j t

We further envisage that the dependence on is also sinusoidal, such that

exp (3.9)F jv

where for completeness v has to be an integer. Thus (3.5) will turn into a differential equation interms of rF as stated below

2 22 2

2 2

1 0 (3.10)r r rF F vk Fr r r r

(3.10) is the well known Bessel equation. The solution to (3.10) can be any of the four types of Bessel

functions, symbolically denoted as , , ,v v v vJ x Y x I x K x , where v is called the order of theBessel function, while x is name the argument. We note we have not applied the boundaryconditions as yet, so (3.10) covers the propagation both in the core and the cladding. Keeping in mindthat propagating modes (physically corresponding to those rays that satisfy the total internalreflection properties) should have finite amplitudes and some oscillatory behaviour, when inside thecore and exponential decay from the core cladding interface radially outwards.


Fig. 3.2 The behaviours of , , ,v v v vJ Y I K .

Fig. 3.2 illustrates the behaviours of all possible Bessel functions against the argument, x . In our casethis argument will be the radial coordinate r , coupled with some multiplicative factor. In line withthe physical interpretation made above for the propagating mode, a suitable choice (from theinspection of the Bessel curves in Fig. 3.2) is the representation of the mode field by vJ inside thecore and by vK inside the cladding. Considering the magnetic field component as well, we can

express zE and zH fields inside and outside the core as follows

exp exp exp 2 inside core

exp exp exp 2 inside core

exp exp exp 2 inside cladding

exp exp exp 2 inside

z v

z v

z v

z v

E r a AJ ur jv j z j t

H r a BJ ur jv j z j t

E r a CK wr jv j z j t

H r a DK wr jv j z j t

cladding (3.11)

where B , C , D are new constants and the other newly introduced terms are defined as

2 2 2 11 1 1

2 2 2 22 2 2

2 ,

2 , (3.12)

nu k k n k

nw k k n k

Note that the propagation constant is limited such that

2 2 1 1 (3.13)k n k k n k

In this manner, 2u and 2w are always positive or u and w are always real. This is essential forpropagating modes. 2u and 2w becoming negative corresponds to the case of refracting modes.

Now we come to the application of boundary conditions at r a , i.e. the core cladding boundary. Atthis boundary, for continuity, we have to match the following fields

0 2 4 6 8 10 12 14 16 18 20-1.5

-1

-0.5

0

0.5

1

1.5

x

Bess

el fu

nctio

nsJ v

(x

) ,Y v

(x

) , ,I v

(x

) , ,K v

(x

)

Jv ( x )Y

v ( x )

Iv ( x )

Kv ( x )


core at cladding at

core at cladding at

core at cladding at

core at cladding at (3.14)

z z

z z

E r a E r a

E r a E r a

H r a H r a

H r a H r a

From the matching conditions of (3.14) and the equalities from (3.6), we get only four constants tobe determined from these boundary conditions. These are the constants or the coefficients given in(3.11) and named as A , B , C , D . Here our ait is to find, what is called the characteristic equation,that will yield the propagation constants, i.e. values of those modes that satisfy the conditions offorward propagation, i.e. propagation to the positive side of the z axis. For this we seek a nontrivialsolution to (3.14), which exists if the determinant of the coefficients A , B , C , D are zero. Thisrequirement will deliver the following characteristic equation (CE).

2 21 2 2 2

' ' ' ' 1 1 (3.15)v v v vv v v v

J ua K wa J ua K wa vk kuJ ua wK wa uJ ua wK wa a u w

where prime, ‘ indicates the derivative of the Bessel function with respect to its argument. By usingthe following relations

1 1 1 1

1 1 1 1

'

'

0.5 , 0.5

0.5 , 0.5 (3.16)

v v v v v v

v v v v v v

xJ x J x J x J x J x J xv

xK x K x K x K x K x K xv

It is possible to replace all Bessel function derivatives in (3.15), an act we shall carry out when wewrite the different versions of CE. Firstly, we examine the case of 0v . Here the right hand side of(3.15) becomes zero. Thus we have two CEs separated by the square brackets on the left hand side of(3.15). After using the derivative equivalents given in (3.16), these two CEs will become

1 1

0 0

2 21 1 2 1

0 0

0 CE for TE modes

0 CE for TM modes (3.17)

n n

n n n n

n n

n n n n

J u K wu J u w K w

k J u k K wu J u w K w

where we have used the normalized definitions of, ,n nu ua w wa . TE mode means that 0zE , while 0zH , i.e. there is no electric field component along the propagation axis. TM mode is

defined on the other hand as, 0zE , 0zH . From (3.12), bearing in mind that

1 21 2

2 2 , (3.18)n nk k

The CE given in (3.17) for TM can be converted into


2 21 1 2 1

0 0

0 CE for TM modes (3.19)n nn n n n

n J u n K wu J u w K w

Finding the zeros (the roots) of the CEs for TE and TM modes will provide us with values of thosemodes that will propagate in the fibre. To solve the CEs in (3.17) in this manner, first we remove the

singularity in the denominator due to 0 nJ u by multiplying both sides of the equation by 0 nJ u ,hence (3.17) and (3.19) will turn into

2 2 21 1 2 0 1 2 12

1 1 00 0

1 0 1 11 0

0 0

0 , 0 CE for TM modes

0 , 0 CE for TE modes (3.20)

n n n nn n n

n n n n n

n n n nn n n

n n n n n

n J u n J u K w n K wn J u u J u

u w K w w K w

J u J u K w K wJ u u J u

u w K w w K w

To generalize the process of finding s and make this process applicable for any fibre, we define afibre parameter called normalized frequency, V as follows

2 2 2 2 2 2 2 2 21 1

2 2 2 2 2 2 2 2 22 2

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 2 1 2

, ,

NA (3.21)

n n n

n n

n n

n n

u au w aw a

u a u a k a k

w a w a k a k

V u w a u w a k k a k n n a k

When finding the roots of (3.20), we have to plot of the left hand side of the CEs against twovariables, i.e. nu and nw . For TM modes, we also need to know (the numeric values of) 1n and 2n .So in the search for roots of CEs, three dimensional plots will emerge. One sample plot of the CE formTM modes is shown in Fig. 3.3. As seen from this figure, there are several the zero crossings, i.e. theroots, but they are hardly visible. In this sense a contour plot provides a much clearer picture. Wecombine this contour plot with the first equation on the last line of (3.21), thus obtain what is calledV circles. Such combined plot can be found in Fig. 3.4 for the CE of TM modes. As seen from Fig. 3.4,the indices of modes are designated as TMvm where v is the circumfrential mode number(remember that for TE and TM modes, 0v ) and m is the order of root. For instance, the first zerocrossing (root) is given the index of 1m , while the second root is given the index of 2m and soon. Thus we see on the graph of Fig. 3.4, the mode labels of 01TM and 02TM .

Another point of interest is that, since in a fibre, the refractive index difference is much smaller thanunity, or 1 2n n , there s hardly any difference between the zero crossings of TE and TM modes. Thismeans CE of TE modes and CE of TM modes placed on the first and second lines of (3.20) areessentially one single equation. This case is graphically illustrated in Fig. 3.5. It is worth mentioningthat the status of 1 2n n is named as weak guidance in literature.


Fig. 3.3 The three dimensional view of the CE for TM modes.

Fig. 3.4 Contour plot of the CE for TM modes and the associated V circles.

Using (3.21), the cross points in Fig. 3.4 can give us the (normalized) propagation constant, i.e. n or

of that particular mode at that particular V value. A sample marking is made on Fig. 3.4 for 01TMat 8.V Of course to arrive at the actual numeric value of n or , according to (3.21) we need toknow the core radius, core refractive index, cladding refractive index or the refractive indexdifference and the wavelength of the source i.e. 1 2, , or ,a n n . In practice, such details can befound on manufacturer's sheet (of course depends on our choice of light source). But with thedefinitions given in (3.12), we are able to treat the propagation in fibres by a single parameter. Thismeans, although there are three separate parameters defining the propagation behaviour of fibre,these are core radius, refractive index difference and the wavelength of operation, they can becombined into a single parameter called normalized frequency parameter. And the propagationcharacteristics of a fibre can be analysed purely in terms of that normalized frequency parameter.

02

46

810

02

46

810

-0.5

0

0.5

1

un

Zero crossings (roots) of TM02

Three dimensional plot of characteristic equation (CE) for TM modes


wn0

00

0

00

00

0

2 4

4

6

6

6

8

8

8

8

10

10

10

10

1012

12

Contour plot of CE for TM modes and different V circles

un

wn

V = 6 circle

V = 12 circle



1 2 3 4 5 6 7 8 9 10

1

2

3

4

5

6

7

8

9

10

Cross point for TM01

at V = 8


Fig. 3.5 Contour plots of the CE for TE and TM modes and the associated V circles.

Now we return to the general CE given in (3.15) and analyse the case of 0v . With this setting, byusing (3.16) we obtain

1 2

2 21 1 1 11 2

2 21 1 1 11 2

3 4

v n v n v n v n

n v n n v n n v n n v n

v n v n v n v n

n v n n v n n v n n v n

T T

T T

J u K w J u K wk k

u J u w K w u J u w K w

J u K w J u K wk k

u J u w K w u J u w K w

0 (3.22)

Again plotting (3.22) will give us new zero crossings, new roots and new modes for which thepropagation constants can be calculated as explained above. Further simplification can be achievedfor the CE of (3.22). That is under the principle of weak guidance, i.e. 1 2n n or 1 2k k . The termsof (3.22) can be approximated as

2 2 2 24 1 1 2 1 2 1 3 2 3

2 2 2 21 2 3 4 1 1 3 3 1 1 1 2 3 3 2 1

2 21 2 3 4 1 1 3 2 1 3 1 3

,0

2 2 0 0 (3.23)

T k T k T T k T k TTT TT Tk T T k T Tk T T k TTT TT k TT k TT TT

The last (approximate) equality means (after converting the approximation into exactness)

1 1 1 1 0 (3.24)v n v n v n v nn v n n v n n v n n v n

J u K w J u K wu J u w K w u J u w K w

Setting the first and the second square bracketed terms individually to zero, we get

0

00

0

00

00

0

2 4

4

6

6

6

8

8

8

8

10

10

10

10

10 12

12

Contour plots of CEs for TE and TM modes and different V circles

un

wn

Zero crossings (roots) of

TM01

and TE01

modes

Zero crossings (roots) of

TM02

and TE02

modes

1 2 3 4 5 6 7 8 9 10

1

2

3

4

5

6

7

8

9

10


1 1

1 1

0 CE for EH modes

0 CE for HE modes (3.26)

v n v n

n v n n v n

v n v n

n v n n v n

J u K wu J u w K w

J u K wu J u w K w

In order to arrive at the limit of single mode fibre, it is instructive to plot the CEs for HE and HEmodes as well. Such a plot can be found in Fig 3.6. We see from Fig. 3.6 that only one mode does nothave a cut-off, which means it can theoretically propagate what the V value of the fibre is. Thatmode is 11HE which is the mode chosen for propagate in single mode fibres. Although this mode willcontinue to exist regardless of fibre V value, for physical reasons, we wish to make the core radius aslarge as possible to the limit of prevention other modes from propagation. Such a limit can be set bythe joint examination of Figs. 3.5 and 3.6. From there and from the horizontal axis of Fig. 3.5, we seethat the closest modes to 11HE are 01TE and 01TM which will start to propagate if 2.4V . The

more exact value is 2.45V , which corresponds to the first zero crossing of Bessel function 0J x .Thus we can say that if we keep the fibre V parameter below the value of 2.45, it will be a singlemode fibre.

Fig. 3.6 Contour plots of the CE for HE and EH modes and the associated V circles.

As can be noted from (3.26) and Fig. 3.6, it is possible to obtain many modes of EH and HE, since weare at liberty to vary from 1v upwards. But those modes, whose zero crossings (roots) fall outsidethe V circle of the fibre will not of course propagate, hence we can terminate the process of findingroots from such v values onwards. Below we do an example to illustrate this point.

Example 3.1 : Assume that we have a fibre with 4V and we wish to find which TE and TM modespropagate in this fibre at what propagation constants.

Solution : From Fig. 3.5, we find that 4V circle only encloses the zero crossings (roots) of

01 01TE , TM . Hence from the point of view of TE and TM modes, only 01 01TE , TM will propagate in

1 2 3 4 5 6 7 8 9 10

1

2

3

4

5

6

7

8

9

10

un

wn

Contour plots of HE and EH modes for = 1 and different V circles

0

00

0

00

0

00 0

0

0

00

2 4

4

6

6

6

8

8

8

8

10

10

10

10

10 12

12

= 1

HE11

EH11

HE12

EH12

HE13


this fibre. As explained above, the exact numeric value of propagation constants i.e., n or will bedetermined by the knowledge of other fibre parameters. Assuming, we are dealing with a fibre of

1 2, 1.48, =1.47a n n and we use a light source of 1.55 m , then from the last line of (3.21),the core radius to result in a fibre frequency parameter of 4V is

0.52 2 0.52 2

1 2 1 2

5.745 m (3.27)2V Va

k n n n n

From Fig. 3.5, we read at the cross point of zero crossings (roots) of 01 01TE , TM and the 4V

circle, 3, 2.6n nu w . Benefiting from (3.21) and (3.27), we can calculate n and as follows

0.5 0.52 2 2 2 2 21 26

34.3368 or 34.3334

5.976 10 (3.28)

n n n n

n

a k u a k w

a

As a final point, using (3.13), we check that

2 2 1 1

6 6 62 15.9589 10 , 5.976 10 , 5.9994 10 (3.29)k n k k n kk k

Exercise 3.1 : Repeat the calculations in Example 3.1, by switching to 6V and 10V and byincluding 1 1HE and EHm m from Fig. 3.6. Assume that the larger values of V is obtained by enlargingthe core radius.

As an alternative to the displays given in Figs. 3.4 to 3.6 is to plot the variation of the propagationconstant or its wavelength and refractive index normalized version against V . Such a graph is givenin Fig. 3.7 (copied directly from Fig. 2.5 of Ref. [2], pay attention that in this figure normalizedpropagation constant is shown by b , below we use the notation nb , since b is used in these notes toindicate cladding radius).

Here the normalized propagation constant, denoted by nb is related to the actual propagationconstant as

2

1 2

/ (3.30)nk nbn n

Since in this definition wavelength and the refractive indices are taken into account, we talk aboutthe variation of nb against V , it effectively means the variations against the core radius, a . Asstated in (3.29), for propagating modes is limited as 2 1n k n k , this way nb will range between0 to 1 as shown on the left hand side vertical axis of Fig. 3.7. The vertical axis on the right hand sideof Fig. 3.7 is on the other hand marked in the scaling of / k , named as n .


Fig. 3. 7. The variation of normalized propagation constant against the normalized frequency of thefibre for some of the lowest order modes.

Example 3.2 : By adopting the refractive index values and the wavelength of operation, i.e.

1 21.48, =1.47, 1.55 mn n find nb using the formulation given in (3.20) and the plots in Figs.3.5 and 3.6 at 4V , 6V and 10V for 01 01TE , TM , 11 21HE and HE . Compare thesecalculations with those readings form Fig. 3.7.

Solution : We start with 4V and 01 01TE , TM , because is calculated in Example 3.1 readily.

6 621 2

1 2

201 01

1 2

/ 2 inserting 5.976 10 , 1.48 , =1.47 , 4.0537 10

/ 0.4209 for TE , TM (3.30)

n

n

k nb n n kn nk nbn n

Drawing a vertical line in Fig. 3.7, starting at the point of 4V on the horizontal axis, we find thatthis vertical line will approximately intersect with 01 01TE , TM curves around 0.4209nb , that is,the value given on the second line of (3.30). Such action is marked in Fig. 3.7 with coloured lines.

Now we turn to nb of 11HE at 4V . From Fig. 3.6 that the intersection of 4V circle with 11HE

curve is at 1.914, 3.5n nu w . Again benefiting from (3.21), (3.27), (3.29) and (3.30), we can

calculate and nb as follows


0.52 2 21 61

6

6 621 2

1 2

inserting 5.745

μm , 5.9994 10 , 1.

914

5.9507 10/ 2 inserting 5.9507 10 , 1.48 , =1.47 , 4.0537 10

0.8257

nn

n

n

a k ua k u

a

k nb n n kn n

b

(3.31)

Again by following the drawn vertical line in Fig. 3.7, starting at the point of 4V on the horizontalaxis, we find that this vertical line will approximately intersect with 11HE curve around 0.8257nb ,that is, the value given on the final line of (3.31).

Now we handle, 6V . For this we recalculate the core radius using (3.27)

0.5 0.52 2 6 2 21 26 7.5325 m (3.32)

4.0537 10 1.48 1.47Va

k n n

From Fig. 3.6, we the intersection of 6V with 11HE curve at 2.061, 5.6n nu w , then as in

(3.31) and nb are calculated as

0.52 2 21 61

6

6 621 2

1 2

inserting 7.5325

μm , 5.9994 10 , 2

.061

5.9537 10/ 2 inserting 5.9537 10 , 1.48 , =1.47 , 4.0537 10

0.8825

nn

n

n

a k ua k u

a

k nb n n kn n

b

(3.33)

We again see that, the value of 0.8825nb agrees well with the value read from the left side of thevertical axis in Fig. 3.7. The case of other modes at 6V is left as lab exercise.

It is interesting to find a simple relationship between nb and V for 11HE curve. This is given by

21.1428 0.996 / (3.34)nb V V

which is known to be accurate within 0.2 % for V in the range 1.5-2.5, the range where mostpractical single mode fibres lie in.

4. Single Mode Fibres

As mentioned above and as seen from Fig. 3.6, only one mode, namely 11HE is free from cut off, i.e.

it propagates regardless of the V value. This means that 11HE will continue to propagate even when,the V approaches zero, which corresponds to having core radius, assuming finite wavelength andrefractive index difference. Of course have a fibre of nearly zero radius would create practicalproblems, such as alignment, handling etc. Therefore to obtain single mode fibre, we seek the largest


value of V , yet eliminate the other modes. Looking at Fig. 3.5, we see that, modes closest 11HE are

01 01TE and TM . Hence it will be sufficient to set V just below the cut off values of 01 01TE and TM ,

that is 2.45V . From (3.17) and (3.20), this is the first root (the zero crossing) of 0 2.45nJ u ,when 0nw .

After suppressing the time variations, by using (3.6), (3.7) and (3.11), the radial component of thesingle mode can be written as

0 0

0 0

0 0

0 0

exp exp , / inside core (4.1)

exp exp inside cladding

n nn

n n

r

n n

n n

J ur J u rj z j z r r a r a

J u J uE

K wr K w rj z j z r a

K w K w

It is possible to plot the radial component of the electric field, using formulation given in (4.1). This isshown in Fig. 4.1.

Fig. 4.1 The radial electric field distribution of 11HE in a single mode fibre at different V values.

Quite a number of observations are possible regarding Fig. 4.1. They are listed below.

The field in a single fibre distributes itself more or less evenly between the core and the cladding.This is in contrast to the propagating modes of the multimode fibre where the field decaysconsiderably before the core cladding boundary. But in a single mode fibre, as seen from Fig. 4.1,an important portion of the field propagates in the cladding region and this portion increases ininverse proportionality with V values. For this reason illustrated in Fig. 4.1 and to minimizealignment problems, we tend to operate at V values (of course we must bear in mind that in allcases, we cannot exceed 2.45V ).

In a single mode fibre, cladding material has to be high quality (in addition to the core), due to theexistence of a sizable portion of the field in the cladding. This means that a constant refractive

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

V = 1.5V = 1.8

V = 2.1

V = 2.4

Radial electric field distribution of a single mode fibre at different V values

Normalized radial distance - rn

Fiel

d - |

E r |

J0 ( un rn )

K0 ( un rn )

Core Cladding

rn = 1r = a


index value, 2n must be maintained uniformly along the radial, circumferential and axialdirections, at least in the regions of close proximity to the core cladding boundary.

Inside the core, field distribution is governed by the Bessel function, 0 n nJ u r , while this is

handed over to the Bessel function 0 n nK w r inside the cladding. Thanks to the boundarymatching conditions applied in (3.14), there is perfect continuity between the two fields at thecore cladding boundary, that is 1 ornr r a .

Note that in Fig. 4.1 variation of V (or the fibre variation) is actually achieved by changing the coreradius, yet the horizontal axis is normalized with respect to the core radius of each fibre separatelyand is common to all curves. In this manner, if the plot of the radial field for each fibre was along itsown absolute radial distance axis, the part of the field in the core and the cladding would changesomewhat. This fact does not alter the essence of above observations.

When a cross section of the single mode fibre is viewed, the illumination created by 11HE mode willlook like the image given Fig. 4.2. As we see from this figure, the central part of the core area is highlyilluminated, but as we move to the cladding the, light intensity reduces.

Fig. 4.2 Light illumination of 11HE mode in a single mode fibre of 2V .

It is possible to gather and approximate the core and the cladding parts of the radial field of (4.1)under a single Gaussian exponential as expressed below

2

2exp exp (4.2)nr

s

rE A j zw

where sw is known as the spot size determining basically how much the field is confined to the core

or how much it extends into the cladding. An approximate relation between sw and V exists and itvalid in the range 1.2 2.4V and is given by

Normalized x coordinate - rnx

Nor

mal

ized

yco

ordi

nate

-r n

y

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3


1.5 60.65 1.619 2.879 (4.3)sw V V

By choosing several V values, in Figs. 4.3 and 4.4, we try to illustrate how close (4.2) models the twopart field expression of (4.1).

Fig. 4.3 Bessel function representation and the Gaussian exponential approximation of the radial fieldin a single mode fibre at 2.4V .

Fig. 4.4 Bessel function representation and the Gaussian exponential approximation of the radial fieldin a single mode fibre at 1.5V .

Here the separate figures for 2.4V and 1.5V are chosen because of the common axis problemmentioned above. Consequently, the normalized horizontal axis of Figs. 4.3 and 4.4 are according tothe core radius values given in these figures. Additionally in both figures, the amplitude of the field(i.e. the vertical axis) for Bessel function representation is also normalized with respect to its peak inorder to get a meaningful comparison between Bessel function representation of (4.1) and the

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 V = 2.4

Comparison of Bessel function and Gaussian exponential approximated representations of radial field distributions at V = 2.4


Nor

mal

ized

fiel

d - |

E r |

Gaussian exponential approximation

Bessel function representation

Core Cladding

rn = 1r = a = 3.45 m

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Comparison of Bessel function and Gaussian exponential approximated representations of radial field distributions at V = 1.5


Nor

mal

ized

fiel

d - |

E r |

Core Cladding

V = 1.5

Gaussian exponential approximation

rn = 1r = a = 2.154 m

Bessel function representation


Gaussian exponential approximation of (4.3). Comparing Figs. 4.3 and 4.4, we see that at larger Vvalues, the Gaussian exponential approximation of (4.3) is indeed a close match to the Besselfunction representation of (4.1). But as V is reduced, the difference between the two curves seemsto increase as displayed in Fig. 4.4. But this is considered to be major problem, since most practicalfibres are operated closer to 2.4V .

Finally on single mode fibres, we calculate the percentage of mode power contained inside the core.For this we take (4.2) and define a confinement factor, c as follows

2112

200

222

20 0

21.5 6

2exp21 exp

2exp

2 1 exp0.65 1.619 2.879

nnr n

scorec

total snr n n

s

r drE dr wPP wrE dr dr

w

V V

(4.4)

where on the last line of (4.4), we have used the V equivalence of sw given in (4.3).

In Fig. 4.5, c is plotted against V .

Fig. 4.5 Plot of confinement factor showing the percentage of power contained in the core duringpropagation in single mode fibre.

According to Fig. 4.5, just above 80 % of power of the 11HE mode will be contained in the core when2.4V , but this percentage drops as the value of V is lowered.

These notes are based on

1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.40.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9Plot of percentage of power contained in the core, confinement factor against V

Normalized frequency - V

Con

finem

ent f

acto

r -

c


1) Gerd Keiser, “Optical Fiber Communications” 3nd Ed. 2000, McGraw Hill, ISBN : 0-07-116468-5.2) Govind P. Agrawal, “Fiber-Optic Communication Systems” 2002, Jon Wiley and Sons, ISBN : 0-471-

21571-6.3) B. E. A. Saleh, M.C. Teich “Fundamentals of Photonics” , 2007, Wiley, ISBN No : 978-0-471-35832-

9.4) My own ECE 474 Lecture Notes.

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