ENE 423
Lecture IV
Integrated Optic Components
We may distinguish the integrated optic devices into two kinds in passive and
active components.
Passive devices: directional couplers, beam splitters, isolators, lenses, and
prisms. An example of passive integrated optic directional coupler is shown in the
figure below. Ideally, relative output powers are given by
 L 
P2
 cos 2 

P1
 2 Lc 
 L 
P3
 sin 2 

P1
 2 Lc 
Where Lc is called the coupling length. It is the length which there is complete
transfer from the upper to the lower waveguide.
Directional coupler
Beam splitter
Active devices: modulators, switches, light sources, and light detectors.
An integrated-optic modulator called Mach-Zehnder interferometer consists of
parallel Ti:LiNbO3 indiffused waveguide. An external modulator like this is very
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important for high speed network. Modulation could be externally produced at higher
frequencies than that of direct modulation. Also, direct modulation of a light source
might cause a change in output wavelength and spectral width while external
modulation does not.
Mach-Zehnder Interferometer
Optical Fiber Waveguides
cladding, n2
2b
2a
core, n1
cladding, n2
If we consider index profile in fibers, we may categorize them into 2 types:
step-index fiber and graded-index fiber.
Step-Index Fiber (SI fiber)
n
n 1
n2
for r  a (core)
for a  r  b (cladding )
The critical angle of SI fiber is the same as in slab waveguide that is
 n2 

 n1 
c  sin 1 
3
Graded-Index Fiber (GRIN fiber)
 n( r )
n
 n2
for r  a (core)
for a  r  b (cladding )


r
n1 1  2    ; r  a
n( r )  
a

;r  a
 n1 1  2
where  = parameter describing the refractive index profile variation

n12  n22 n1  n2

2n12
n1
n(r)
=
n1
=2
=1
ra
4
Butted coupling from the light source to a GRIN fiber is more efficient near
the axis than further out. Unlike the SI fiber for which NA remains the same
regardless the entry point. Therefore, coupling efficiency is generally higher for SI
fibers than for GRIN fibers.
Advantages of GRIN fiber is that several modes can be lumped together and
cause the effective number of modes to decrease. NA of GRIN fiber may be written as

n 2(1  (r / a)
NA(r )   1
0


; for r  a
; for r  a
As seen above, NA of GRIN decreases from n1 2 to zero as r moves from
the fiber axis to the core-cladding boundary.
Attenuation
The length of fiber is limited by dispersion and attenuation. Attenuation or
loss in fiber may be classified as absorption, scattering, geometric effects, connectors,
or splicing.
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Bending loss
When fiber is bent, fields break away and radiate into cladding. This bending
loss can be reduced by increasing radius of curvature R.
Attenuation can be measured by a cut-back method and OTDR (optical time
domain reflectometer).
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Ex. Consider a fiber whose core index is 1.5 and whose cladding index is 1.485. The
core radius is 100 micron. At what bending radius does a ray traveling along the fiber
axis strike the cladding at the critical angle in the bend?
Soln
Cut back method
1
2
Pin
P02
P01
l
 (dB / km) 
10log10 ( P02 / P01 )
l
OTDR
This method requires only one end of the fiber to be measured. This OTDR
transmits an optical pulse down the fiber and measures the reflections. Reflections
occur owing to discontinuities due to splices, connectors, and fiber breaks and to
scattering. The Rayleigh scattering gives a continuous return signal. The time delay of
reflections is a measure of their location along the fiber.
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Ex. A 50 km fiber link between transmitter and receiver consists of 3 km segments
that are spliced together. Losses are fiber attenuation at 0.5 dB/km, 0.3 dB/splice loss,
1 dB/connector. Calculate
(a) Total loss in the link
(b) Calculate output power when Pin = 1 mW
Soln
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Ex. A fiber has n1 = 1.5 and n2 = 1.49 and core diameter 50 micron. Consider the
guided ray traveling at the steepest angle with respect to the fiber axis. How many
reflections are there per meter for this ray?
Soln
Modes in SI fibers
The geometry of fiber causes modes like the case of slab waveguide. The fiber
mode chart is normalized by plotting neff vs. normalized frequency V (famously
known as V-number).
V
2 a

n12  n22
where a = core radius
Conventional fibers do not preserve the polarization of the wave due to some
external forces to the fibers (such as bending, twisting, or splicing). The polarization
of light in fiber is random. To preserve polarization, impurities are added to the core
and core geometry is adjusted. This can produce “polarization maintaining”(PM)
fiber. Another special fiber called “polarizing”(PZ) fiber is done by designing the
asymmetry in the fiber such that the undesired polarization state has a higher
attenuation than that of the desired state.
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Some light propagation in the fiber is similar to that in the slab. We have
c   

2
and   k0 neff . At a fixed value of V, several modes may propagate, each
having a different neff. This V-number can determine the number of modes (N) for
V>10 by
N
V2
2
For single mode fibers, it needs all modes expect HE11 to be cut off. This
occurs at V  2.405 and this yields
a


2.405
2 n  n
2
1
2
2

2.405
2 .NA
10
Because of 2-D con confinements, modes in fibers are designated by 2
subscripts, e.g. TE01 or TM01. There are still some modes called “hybrid” that are HE
and EH. The hybrid mode is the mode that contains components of both electric and
magnetic field.
Modes in GRIN fiber
Generally, the expression for neff is presented instead of producing a mode
chart in case of GRIN fibers. The effective refractive index of GRIN is expressed as
neff 
 pq
k0
 n1  ( p  q  1)
2
k0 a
where p, q are numbers to describe a mode. The lowest mode can be
obtained with p = q = 0.
Again, the allowed modes of light propagation have the range of neff as
n2  neff  n1
In this case, cutoff occurs at neff = n2. If we want only single mode to guide in
GRIN fiber, we substitute p = 1, q = 0 along with neff = n2 into the equation
neff  n1  ( p  q  1)
2
, it yields
k0 a
a


1.2
 n1 (n1  n2 )
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If we use V-number and it is large, the number of modes in GRIN fiber can be
approximated by N 
V2
. Some predict the number of modes using -profile as
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N

 2
a 2 k 2 n12 
Ex. Consider an SI fiber with n1 = 1.5 and n2 = 1.485 at 0.82 μm. If the core radius is
50 μm, how many modes can propagate?
Soln
Distortion in fibers
Fiber links are limited in path length by attenuation and pulse distortion.
Distortion in signal due to fiber includes
1. Material dispersion
2. Waveguide dispersion
3. Multimode dispersion
Waveguide dispersion in a silica fiber
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Total dispersion for various kinds of fibers
Distortion in SI fibers
Total pulse spreading  could be expressed as
  ( m   g )2  ( mm ) 2
where m, g, and mm are the same expression as in slab
waveguide.
 m   M m ..l
 g   M g ..l
n (n  n ) n 2 
 
   1 1 2  1
cn2
cn2
 l mm
If n1  n2
n1 ( NA)2
 
  

c
2cn1
 l mm
m and g are much less than mm in SI fiber. Especially, g can be
neglected for the short wavelength such as  < 1.2 μm.
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Distortion in GRIN fibers
GRIN fiber have effective number of modes less than in SI fiber since rays
travel in shorter routes and faster. This minimizes multimode pulse spreading. The
multimode distortion can be expressed as
n 2
 
   1
2c
 l mm
By using the same expression of m and g as in slab waveguide for the
GRIN fiber, the total pulse spreading can be written as
  total 
2
( m   g )2  ( mm
)
Length dependence of the pulse spread in multimode fibers
Pulse spreading increases linearly with fiber length for short distance of the
link. For longer link, pulse broadening is proportional to the square root of the length.
This
l is caused by the mode mixing in multimode fiber. In short path, this
mixing is still incomplete. After traveling further, an equilibrium modal power
distribution is reached. The length in which equilibrium is reached called “equilibrium
length (le)” or some call it “critical length (lc). Therefore, we can write the multimode
distortion as

 
; l  le
 l.  l 

 
  
 l.l .    ; l  l
e
 e  l 
where (/l) is the spread per unit length in linear region.
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Ex. The equilibrium length of a multimode fiber is 2 km. The modal spread is 25
ns/km. The light source emits at 800 nm and has a spectral width of 50 nm. Compute
the optical 3-dB bandwidth of a 5-km length of this fiber.
Soln
 
      m   g     M m  M g  .
 l  dis
ps
From figure at  = 800 nm, M m  115
nm.km
ps
and M g  1.8
nm.km
 
     115  1.8   50  5840 ps/km
 L  dis
  dis  5840  5  29200  29.2 ns
Ex. Calculate the multimode dispersion for the fiber with n1 = 1.48 and n2 = 1.46 if
(a) the fiber is a SI fiber.
(b) the fiber is a GRIN fiber.
Soln