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Optical AmpUfler or

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Amplifier Coupl.~r

FIBER OPTIC SENSORS

Extrinsic:

Signal In

Signal Out

Environmental

Changes

Intrinsic:

.

EnvironmentalFactors: Temp, Press,Chem

.

(

~ 1iQ

.c ... (D - I N > '0 'E

. n. ~ 0" ~ "' ~ ~ Q.

t")

:r ~ ... ~ t")

'(; .., (;;.

- n. "' 0 ...,

:!) 0- ~ ... .::;,

'0 ~ ~

n O VI

~ r- ", V1

V1 m X " m Z V1

< ~ ~ O "' --4 m x "'0 m z \I\ < ~ r- m > VI

-1 m ><

"'0 m Z VI

< '"

~ ~ ~ r- n J8.

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m -4 ~ m m z n m z -4 ~ ~ r- O on on n m ~

Ut

c ~ ~ ~ ~ z m n ~ CJJ

r- m Ut

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c c rn -4 O "' ~ )80

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:1:

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:1:

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C')

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< rn ~ -< < rn ~ -< r- > ~ Q rn < m ~ -< r- > ~ G)

m r- > ~ Q m

~ m O c ~ m \ft r- > "' m ~VI

O C ~ n "' r- > "' m ~ O ... r- m O r- > V\

m ~ O ... r- m O

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n 0 .. .

n D 0. ~ ~ ~

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ANATOMY ~fOPTICAL FIBER-S'

Graded-index fiber

r ~ r --+-

Core- radius (a); cladding thickness (h-a).

Step index profile.

Graded index profile.

. There are two approaches to analyzing the propagation of light throughfibersa.) Field propagation using exact solutions to Maxwell's equations.

b.) Ray tracing

When the core radius is much larger than the wavelength A (ieee a ~ lOA),can use geometrical optics description with high accuracye

.

Geometrical optics is very insightful for many situations..

2

Geometrica~' Descr Beam Pro a ation in O tical,-Fibers:

The geometrical description of light propagation in fibers is based on thephenomena

.

When a beam is incident to an interface from a medium with higher refractiveindex a critical angle occurs at which light is refracted at 90° to the surfacenormal between to dielectrics.

.

Beyond this angle light is no longer transmitted into the second medium

(appreciably).

.

3

Snell's Law:

(}1 = sin-~Sin(}2nl

Since nl > n2 there is a limit to O2 (i.e. O2 = 900) and this limit is the critical

angle Oc. It occurs when

.

n2

nl

= ()c

Surrounding a medium with refractive index nl with a medium of lower indexn2 will form a trap for light that is incident at angles > 8co

.

This is the basis for an optical waveguide..

Optical Waveguide n2

n1

TE Polarized Fields:

4

TE Intensity Coefficients:

Ti =

TM Polarized Fields:

TM Intensity Coefficients:

~ I = 1'1112

12111 =It II I

Can also consider that for angles of incidence that are equal and greater than the

critical angle ec :

JI2)2nl

n2,

.2 nSIll (71-

write:

5

B = [ ( ;; ) sin2 el -1 ]112

coefficients as

cos '¥ -j sin '¥

= exp(2j'¥)

exp(-j'¥)

tan'¥ = ! = n2 [ ( nl I n2 )2 sin2 01 _lJI/2

A nl cos 01

= [ sin201-(n2/nI)2J/2

cos 01

Similarly for TM reflection:

Therefore for each case there is a

of the incident field.

6

and n2 = 1.0 is shown below:

$ 1-

2.. .

~

II

..L

O 40~--1---io---L-LO

Incident angleFIGURE 3.13. The phase change upon total reflection for each of the tWo polarizations. Forthis calculation. we assumed that the index of the dense medium was 1.5 and the index of the

less dense medium was 1.0.

r2 (.2 '11'01 "" -!q I'-C-. "c , ~"'r .,-;c "?

.

f /I1.f#J"' .:.., (, ~ :.\,-.1.

Fiber Numerical Aperture:

The acceptance angle for a fiber defines its numerical aperture (NA).

It is an important parameter for determining the beam propagation and couplingcharacteristics of optical waveguides.

.

The NA is defined as:.

NA.

n.sma.1 1

where ai is the largest acceptance angle that is coupled into the waveguide.

This can be determined from the TIR condition at the interface between nl and

n2.

.

~

Jn1

sin-1mill( ~ )

90° -~

5

This leads to the simple result:.

Example:

III = 1.50; ll2 = .49; NA = 0.1729 and a.I = 9.96°,

An important parameter based on the numerical aperture is thefrequency or v#.

.

This parameter is used to determine mode characteristics of the fiber structure..

The V# is defined as:.

a= 2J'[- NA

A

Notice that a fiber with a particular radius and NA can change simply by using

a different wavelength.

.

6

Approximatg.../!f ethod _for M odes:

. Consider the optical waveguide shown in the figure below with nl > n2.

02 A

~

e

,..01

! i...

...~..i r"

k = 27tDl /A2a

B

A ray propagates from A to B to C reflecting at interfaces between n 1 and n2 at

locations A and B.

.

In order jror a wavefront to be stable within the waveguide the wavefront mustbe continuous at all locations within the guide.

.

The total phase delay between the wave front at points A and c:.

<PI is thelryhase change upon reflection (there are two reflections), and <P2 is thephase change due path ABC.

.

7

The geometrical path ARC:.

L ABC =

-

Converting to an.

(j)2 =

Therefore a exist when:.

t;?!~~ J~ ~ ~~oc.{A~ tN\~ A...

J , (::! f=.,..;.,.;t \I1A /,.; ~ 6' M

A. , ~ I A.J fIo'1,

2 8JZ'"n1rPtot = rPl + a cas () = 2JZ'"m

~

{.A)~

~p,~~with m = 0, 1, 2, ...

8

Example:

nl = 1.50

n2 = 1.40

A. = 1.0 Jlma= d/2 = 25 J.lm;

Assume that <p 1 ~ 00

In this case ec = 68.96°; aI = 32.583°

~c~~ lA.Jt.X:il1~~

~

w~ >-and

rounded to the next lowest integer .

There will also be one mode for M = O making the total estimated modes equal to

55.

It shoulrll be noted that this is only an approximate expression to providean intuit jive concept of modes. The actual number of modes is determinedfrom solution to the field equations and boundary conditions for the

specific waveguide.

.

If the ~r radius and NA is known it can quickly be deteffilined if the fiber isoperating in a single mode condition since all modes except the fundamentalwill be cut off when

.

v ~ 2.405.

9

EXAMPLE:For a SMF the NA ~ 0.1.

Therefore if the core index nl = 1.50, n2 = 1.496662.

With NA = 0.10 a fiber with

A. = 0.85 J.Lm ~ a= 3.254 J.Lm

A. = 1.30 J.Lm ~ a= 4.976 J.Lm

A. = 1.55 J.Lm ~ a= 5.933 J.Lm

.

V2

2 .# Modes ~

. The z-component of the propagation constant k = 27tnl/A is referred to as the

eigenvalue of the waveguide and is usually assigned the

10

The field propagating in a waveguide (axis along the z) can be expressed as.

where ko = 27t/A.

Notice that the argument of the radical can become < 0..

K IS and signifies an oscillating field. It is defined as:.

This applies when ~ < kon I. Oscillatory (propagating) form for the electric

field.

.

Similarly when J3 > kon2:.

y specifies an attenuation coefficient. The field decays -does not propagate.

The values for 13 are deteffilined from solutions to the field equations andboundary conditions.

.

A characteristic equation can be formed that specifies the allowed modes forthe waveguide.

.

14

F or a symmetric waveguide the

modes are:

for TE even and odd.

y

Ktan KQ =

K

y

-

and for the even and odd

2

n1

n2

y

Ktan KQ =

2

nl

n2

K

y

-

The characteristic equations are solved either numerically or graphically bysetting the left-hand side of the equation equal to the right hand side.

.

The intersection of the two curves gives the.

Once K is found, the coefficients /3 and yare determined to specify the allowedfiber modes.

.

12

ALTERNATIVE REPRESENTATION:

Symmetric waveguide propagation constants:.

Let a equal the of the waveguiding ( core) region..

Multiply both sides by a2 and combine expressions:.

2-

Let X = Ka and y = ya.

This allows the relation to be written in the form:.

22lZ"a

A

R2 = X2 + y2 =

The characteristic equations become:.

ay == aK tan

-ay == ay tan

or

Y=XtanX

y=-XcotX

13

TEaYad t:-I R

TE 1 /

TE2

-0

> II

>-

><

!1

TEJ><

-

:1

~h~I;I.1, rz.t.,r /t,.J~Iy'.!1f -1ttf"~.

tL ~~~L /JI.'-C~ L

,~ r..~ r-4?~e.. ~ ~

~:1\J- ..Qf r~~~~~ ~~

A 1'..1 4--"+~

gJ-

~J

)(-o j1.1

)(I j

TE~~

c-)o

-

~i

~

~ X b ~ .f :-/ ~~v* ,

~ ~ ; d.('L-"'(~-127'- I-ry,,-

-.~v'" If!-4 v J I e F :. r..1~ La;)t.-J )

21T

x = 1=( d

I-.i~ure ~-2 Char~lL"l<.'risliL" <.'4U;.ltiIJn Ji;.lgr;.lm T[: m\.\J j l)i Ji l..:l.'trlL" ~1;.1h \I. ~iv<.,~l1iJt:.

J r (1\-C- vV~.~~ r..~ .~..v v' ~ ,..1'-L = /?""l.f(.-

L

ftd .,LY(}! I

$11I/51l I...V~{'

t~~- xy:::.x

j (.YY"-t" ~~c~ ("<~v-x) ~--P L-)(C,yt,.i ~ ~ ~ c..1

~-.//O1r..f) ~ ~ ~-"-£-k tE.1(~ ,.,.,.R.-- fr ~ ;>("

~ 1~i..." /L<C.. J.;l.l. C7'-"'-+'I'-'V ~ t:...~ 1L Wort()~.

\/ ..; lrc.

This gives a more intuitive representation for the characteristic equation.

ModesA

Ya

Ka

SM,

Cut-off

. The allowed values for K and yare given by the intersection of the two curves.

All fonns of characteristic equations basically work in this manner ..

Notice that a single mode operating condition occurs when the V# is less than avalue corresponding to the second to lowest allowed tan or cotangent function.

.

14