waveguides p 2

8/9/2019 Waveguides p 2

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WaveguidesPart 2

• Rectangular Waveguides – Dielectric Waveguide – Optical Fiber



Dielectric WaveguideLet us consider the simpler case of a rectangularslab of waveguide

r ε

i r θ θ =

1 1 1 2 2 2andβ ω µ ε β ω µ ε = =

!nell"s Law of Reflection

1

2

sin

sint

i

θ β β θ

=!nell"s Law of Refraction

( ) 21

1

sin r i critical

r

ε θ

ε −=

÷ ÷

#ritical $ngle%

( )Case(1): i critical i θ θ < ( )Case(ii): i critical i θ θ >

When the incident angle is greater than the critical angle& the wave is totall'reflected bac( and this phenomenon is (nown as Total internal reflection

)otal internalreflection

*ncidentwave

Reflectedwave

Refractedwave

*ncidentwave



Velocity of light in Free Space

Velocity of light in the medium u

r r c

nu

µ ε = = =

Dielectric Waveguide

)he index of refraction &n& is the ratio of the speed of light in a

vacuum to the speed of light in the unbounded medium& or

*n nonmagnetic material

r n ε =

( ) 1 2

1

sini critical

n

nθ −=

÷ 1

2

sin

sint

i

n

n

θ

θ =

1 1 1 1u

o r o r o o r r r r

cu

µε µ µ ε ε µ ε µ ε µ ε = = = =

1

o o

c µ ε

=

1r µ =

Where

#ritical $ngle%

!nell"s Law of Refraction%

!nell"s Law of Refraction can be e+pressed in terms of refractive inde+%

*nde+ of refraction%



D, -% $ slab of dielectric with inde+ of refraction . // sits in air What is the

relative permittivit' of the dielectric0 $t what angle from a normal to theboundar' will light be totall' reflected within the dielectric0 1$ns% & 3 - °4

Dielectric Waveguide5+ample

What is the relative permittivit' of the dielectric0

1 3n =

2 1 (air)n =

( )Criticaliθ

11 r n ε = 2

1 1r nε =2

1 3 9

r ε = = $t what angle from a normal to the boundar' willlight be totall' reflected within the dielectric0

( )1 12

1

1sin sin 3 19.i critical

n

nθ − −

=

= = ÷ ÷ o



Dielectric Waveguide)5 wave

1 2

1 2

cos cos

cos cosi t

i t

n n

n n

θ θ θ θ

−Γ =+

( )222 11 sin

2tancos

i

TE

i

n nθ φ

θ

− −=

÷ ÷

E x

Hy

Hz

( )( )

222 1

222 1

cos sin

cos sin

i i

TE

i i

j n n

j n n

θ θ

θ θ

+ −Γ =− −

TE Γ

6sing !nell"s Law of refraction

)he reflection coefficient of a )5 plane wave

1!ee #hapter -4 is given b'

( ) 22

2 11sincos

tan2 2 cos

ii

i

n na m θ β θ π

θ

−− = ÷

)5 modes 1-/ mm thic( dielectric of εr 7 8 orn72 operating at 8 - 9:;4

)5 wave

L:! R:!

R:!

L:!

For this e+ample onl' three )5modes are possible<

$4 )5 / at θi 7 ,8 8 °&=4 )5 3 at θi 7 -, °& and#4 )5

2 at θ

i 7 . > °

1$41=4 1#4

Possible modes can be obtained b' evaluatingthe phase e+pression for various values of m



Dielectric Waveguide)? wave

E x

Hy

E z

( )

( )

222 1

222 1

cos sin

cos sin

i i

TE

i i

j n n

j n n

θ θ

θ θ

+ −Γ =− −

TE Γ

6sing !nell"s Law of refraction

)he reflection coefficient of a )? plane wave

1!ee #hapter -4 is given b'

)? modes 1-/ mm thic( dielectric of εr 7 8 or n72 operating at 8 - 9:;4

)? wave1 2

1 2

cos cos

cos cost i

TM

t i

n n

n n

θ θ

θ θ −Γ =+

( )

( )

222 11

2

2 1

sincostan

2 2 cos

ii

i

n na m

n n

θ β θ π

θ

−− = ÷

For this e+ample onl' three )?modes are possible<

$4 )? / at θi 7 ,3 @°&=4 )? 3 at θi 7 -2 °& and#4 )? 2 at θi 7 .. °

R:!

L:!

1$4

1=41#4

L:! R:!

Possible modes can be obtained b' evaluatingthe phase e+pression for various values of m




$ larger ratio of n 3An2 results ina4 a lower critical angle and thereforeb4 more propagating modes

( )( )

222 11

2

2 1

sincos

tan 2 2 cos

ii

i

n na m

n n

θ β θ π

θ

−− =

÷

L:!R:! for various mR:!

For single mode operation%

2 21 2

1 1

2o

a

n nλ

<−

2 21 2

1

2

o

n n

a λ <

−1or4

:Sla! thic"nessa

2 21 2

1

2

c

n n f

a<

−o

c

f λ =6sing




D, @% !uppose a pol'eth'lene dielectric slab of thic(ness 3// mm e+ists inair What is the ma+imum freBuenc' at which this slab will support onl' onemode0

1## mma =

2 21 2

1

2

c

n n f

a<

−

1 1.n =

2 1 (air)n =

From )able 5 2& for pol'eth'lene

1 2.2$ 1.n = =1 2.2$r ε =

2 1 (air)n =

)he ma+imum freBuenc' at which thisslab will support onl' one mode is

( )( ) ( ) ( )2 2

1 2

%

ma& 2 23

1 1

2 2

3 1#1.2 '

1## 1# 1. 1.#

c

n n f

a −− −

×= = =

×

5+ample



Dielectric WaveguideField 5Buations% )he field eBuations can be obtained b' solving ?a+well"s eBuationswith the appropriate boundar' conditions

( )( ) ( )

( )

( )( ) ( )

2 1

1

2 1

2 sin1

sin1

2 sin1

cos cos

cos cos (m * #+ 2+ , ...)

For & - a 2 cos cos

For & / a 2 : 2

: 2

i

i

i

x a j z y o i

j z y o i

x a j z y o i

E E e e

E E x e

E E e e

a

a

α β θ

β θ

α β θ

β θ

β θ

β θ

− − −

−

+ + −

==

=

( )( ) ( )

( )

( )( ) ( )

2 1

1

2 1

2 sin1

sin1

2 sin

1

sin cos

sin cos (m * 1+ 3+ ...)

For & - a 2: sin cos

For & / a 2: 2

2

i

i

i

x a j z y o i

j z y o i

x a j z

y o i

E E e e

E E x e

E E e e

a

a

α β θ

β θ

α β θ

β θ

β θ

β θ

− − −

−

+ −

==

= −

5ven ?odes%

Odd ?odes%

( ) 22

2 1 2 1sin

i n nα β θ = −

1 sine iβ β θ =)he phase constant in medium 3 is

)he attenuation in medium 2 is

1sin sinu oe

i in

λ λ λ θ θ = =)he effective wavelength in the guide is

1 sin p

e i

cu n

ω β θ = =

3

2

.

3

2

.

3

2

.

3

2

.

)he propagation velocit' is



D, @% Findλe and u p at 8 - 9:; for the )5 / mode in a -/ mm thic( n 3 7 2 /

dielectric in air 1$ns% .- mm and 3 @ + 3/>

mAs4

5+ample

( )( ) ( ) ( )1 1

%

9sin sin sin sin

3 1#3 mm

,. 1# 2 0,.,u o

e

i i i

c

n fn

λ λ λ

θ θ θ = =

×= = =

× o

)he effective wavelength in the guide is

( )( ) ( )1

%

%

sin sin

3 1#1.$ 1# m s

2 0,., p

e i

cu

n

ω β θ

= =×

= = ×o

)he propagation velocit' is

# mma =1 2.#n =

2 1 (air)n =

2 1 (air)n =From Fig , 3@& the critical incident angle forthe )5 / mode

)5 / at θi 7 ,8 8 °




Optical Fiber $ t'pical optical fiber is shown in Figure )he fiber core is completel' encased in a fiber cladding that has a

slightl' lesser value of refractive inde+ !ignalspropagate along the core b' total internal reflection atthe coreCcladding boundar'

f cn n>

$ cross section of the fiber with ra's traced for twodifferent incident angles is shown *f the phase

matching condition is met& these ra's eachrepresent propagating modes)he abrupt change in n is characteristic of a stepCinde+ fiber Optical fiber designed to support onl'one propagating mode is termed single-mode fiber ?ore than one mode propagates in multi-mode

fiber

2 2

#1

2 f ca n n

k

π λ

−>

*n stepCinde+ optical fiber & a single mode will propagate so long as the wavelength isbig enough such that

where ( /3 is the first root of the ;eroth order=essel function& eBual to 2 8/-



Optical Fiber For stepCinde+ multiCmode fiber& the total number ofpropagating modes is appro+imatel'

( )2

2 22 f c

a N n n

π λ

= − ÷

5+ample , .% !uppose we have an optical fiber core of inde+ 3 8@- sheathed incladding of inde+ 3 8-/

#1

2 22 f c

k a

n n

λ

π <

−2 2

#1

2 f ca n n

k

π λ

−>

What is the ma+imum core radius allowed if onl' one mode is to be supported at awavelength of 3--/ nm0

:ow man' modes are supported at this ma+imum radius for a source wavelength of>-/ nm0

( ) ( )9

2 2

2.,# 1 # 1#or 2.%,

2 (1.,$ ) (1., #)

x ma a m µ

π

−

< <−

( )2$

2 2

9

(2.%, 1# )2 (1.,$ ) (1., #) 9.$

% # 1#

x m N

x m

π −

−= − = ÷

)he fiber supports modes



Optical Fiber Eumerical $perture

Light must be fed into the end of the fiber to initiate

mode propagation $s Figure shows& upon incidencefrom air 1n o4 to the fiber core 1n f 4 the light is refractedb' !nell"s Law%

Fiber Laser !ource

sin sino a f bn nθ θ =

( )cos cos 9# sinb c cθ θ θ −= =o

2 2sin cos 1b b

θ θ + =

2sin 1 coso a f bn nθ θ = −

2

sin 1 sino a f cn nθ θ = −

9# 1%#c bθ θ + + =o o

9#b cθ θ = −o

)he sum of the internal angles in atriangle is 3>/ deg

9# o

)he numerical aperture & E$& is defined as

21 sinsin f c

a

o

n NA

n

θ θ

−= =



Optical Fiber Eumerical $perture)he incident light ma(e an angle θc with a normal tothe core–cladding boundar' $ necessar' conditionfor propagation is that θc e+ceed the critical angle1θi4critical & where

( )sin ci crit

f

n

nθ =

2 2 f c

o

n n NA

n

−=

)herefore& the numerical aperture & E$& can bewritten as

21 sin f c

o

n NA

n

θ −=

Fiber Laser !ource

( )sin ci crit

f

n

nθ =



5+ample , 8% Let"s find the critical angle within the fiber described in 5+ample

, . )hen we"ll find the acceptance angle and the numerical aperture

1 1., #sin sin %1.% .

1.,$c

c

f

n

nθ −= = =

÷ ÷ o

2 21 (1.,$ ) (1., #)

sin 12.1 .1

aθ − −= = ÷ ÷

o

)he critical angle is

)he acceptance angle

Finall'& the numerical aperture is

sin #.2#9.a NA θ = =

Optical Fiber Eumerical $perture



Optical Fiber !ignal Degradation

*ntermodal Dispersion % Let us consider the case when a singleCfreBuenc' source 1called amonochromatic source4 is used to e+cite different modes in a multiCmode fiber 5ach modewill travel at a different angle and therefore each mode will travel at a different propagationvelocit' )he pulse will be spread out at the receiving end and this effect is termed as theintermodal dispersion

Waveguide Dispersion% )he propagation velocit' is a function of freBuenc' )he spreadingout of a finite bandwidth pulse due to the freBuenc' dependence of the velocit' is termed as

the waveguide dispersion

?aterial Dispersion % )he inde+ of refraction for optical materials is generall' a function offreBuenc' )he spreading out of a pulse due to the freBuenc' dependence of the refractiveinde+ is termed as the material dispersion

$ttenuation

5lectronic $bsorption % )he photonic energ' at short wavelengths ma' have the right amountof energ' to e+cite cr'stal electrons to higher energ' states )hese electrons subseBuentl'release energ' b' phonon emission 1i e & heating of the cr'stal lattice due to vibration4

ibrational $bsorption% *f the photonic energ' matches the vibration energ' 1at longerwavelengths4& energ' is lost to vibrational absorption



Optical Fiber 9radedC*nde+ Fiber

One approach to minimi;e dispersion in a

multimode fiber is to use a graded inde+ fiber 1or9R*E& for short4

)he inde+ of refraction in the core has anengineered profile li(e the one shown in Figure:ere& higher order modes have a longer path totravel& but spend most of their time in lower inde+of refraction material that has a faster propagationvelocit'

Lower order modes have a shorter path& but travelmostl' in the slower inde+ material near the centerof the fiber

)he result is the different modes all propagatealong the fiber at close to the same speed )he9R*E therefore has less of a dispersion problemthan a multimode step inde+ fiber

waveguides p 2

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