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Transcript of Waveguides p 2
8/9/2019 Waveguides p 2
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WaveguidesPart 2
• Rectangular Waveguides – Dielectric Waveguide – Optical Fiber
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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
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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%
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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
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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
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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
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Dielectric Waveguide
$ 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
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Dielectric Waveguide
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$r ε =
2 1 (air)n =
)he ma+imum freBuenc' at which thisslab will support onl' one mode is
( )( ) ( ) ( )2 2
1 2
%
ma& 2 23
1 1
2 2
3 1#1.2 '
1## 1# 1. 1.#
c
n n f
a −− −
×= = =
×
5+ample
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Dielectric WaveguideField 5Buations% )he field eBuations can be obtained b' solving ?a+well"s eBuationswith the appropriate boundar' conditions
( )( ) ( )
( )
( )( ) ( )
2 1
1
2 1
2 sin1
sin1
2 sin1
cos cos
cos cos (m * #+ 2+ , ...)
For & - a 2 cos cos
For & / a 2 : 2
: 2
i
i
i
x a j z y o i
j z y o i
x a j z y o i
E E e e
E E x e
E E e e
a
a
α β θ
β θ
α β θ
β θ
β θ
β θ
− − −
−
+ + −
==
=
( )( ) ( )
( )
( )( ) ( )
2 1
1
2 1
2 sin1
sin1
2 sin
1
sin cos
sin cos (m * 1+ 3+ ...)
For & - a 2: sin cos
For & / a 2: 2
2
i
i
i
x a j z y o i
j z y o i
x a j z
y o i
E E e e
E E x e
E E e e
a
a
α β θ
β θ
α β θ
β θ
β θ
β θ
− − −
−
+ −
==
= −
5ven ?odes%
Odd ?odes%
( ) 22
2 1 2 1sin
i n nα β θ = −
1 sine iβ β θ =)he phase constant in medium 3 is
)he attenuation in medium 2 is
1sin sinu oe
i in
λ λ λ θ θ = =)he effective wavelength in the guide is
1 sin p
e i
cu n
ω β θ = =
3
2
.
3
2
.
3
2
.
3
2
.
)he propagation velocit' is
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D, @% Findλe and u p at 8 - 9:; for the )5 / mode in a -/ mm thic( n 3 7 2 /
dielectric in air 1$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 °
Dielectric Waveguide
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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/-
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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
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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
−= =
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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$& can bewritten as
21 sin f c
o
n NA
n
θ −=
Fiber Laser !ource
( )sin ci crit
f
n
nθ =
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5+ample , 8% Let"s find the critical angle within the fiber described in 5+ample
, . )hen we"ll find the acceptance angle and the numerical aperture
1 1., #sin sin %1.% .
1.,$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
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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
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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