Download Unit 1.5 Propagation in Fibre

How many light rays??
yRecall that only light rays which enter the core with an angle less than the acceptance angle
will propagate
yThere are an infinite number of possible ray angles, all less than acceptance angle
Optical Communications Systems
yIn theory then there are an infinite number of light rays?
Range of angles
over which light
will not be
transmitted
Propagation in Fibre
Range of angles
over which light
will be transmitted
Electromagnetic Modes in Fibre
Observing Modes Experimentally
yVisible light is used as a source, typically HeNe laser (Red, 670 nm)
yOutput from a fibre is projected onto a reflective surface, such as a white card
in a darkened room
yTo obtain an improved model for propagation in a fibre, EM wave theory must be used.
yRay diagram or Geometric Optics approach remains useful as a way to visualise
propagation in a fibre.
yBasis of EM analysis is a solution to Maxwells equations for a fibre.
yFor ease of analysis a fibre is frequently replaced by a planar optical waveguide, that is
a slab of dielectric with a refractive index n1, sandwiched between two regions of
lower refractive index n2.
Output from singlemode fibre, HE11 mode
n2
n1
Output from a fibre supporting two
modes
Output from a multimode fibre, a socalled speckle pattern
n2
Planar waveguide
Formation of Modes in
a Fibre (I)
Formation of Modes in
a Fibre (II)
B
F
B
θ
E
F
A
d
θ
E
A
d
C
D
yPropagation of an individual ray takes place in a zigzag pattern as shown
yIn practice there is at the fibre input an infinite number of such rays, called more properly
plane rays.
yEach ray is in reality a line drawn normal to a wavefront, for example the wavefront shown by
the dotted line FC above.
yFor plane waves all points along the same wavefront must have identical phase.
yThe wavefront intersects two of the upwardly travelling portions of the same ray at A and C.
D
C
yUnless the phase at point C differs from that at point A by a multiple of 2π then destructive
interference takes place and the ray does not propagate.
yMoving along the ray path between A and C involves a phase change caused by the
distance AB and BC and a phase change caused by reflection
yCombining these two phase changes and setting the result equal to a multiple of 2π we
get a condition for propagation of a "ray", more properly now called a mode.
Types of Optical Fibre
yThree distinct types of optical fibre have developed
yThe reasons behind the development of different fibres are explored later
yConcern here is to examine propagation in the different fibres
Step Index Fibre
The three fibre types are:
• Step index fibre
• Graded index fibre
Multimode fibres
• Singlemode fibre (also called monomode fibre)
Step Index Fibre
0
N2 N1
Normalised Frequency for a Fibre
yFor an optical fibre we can define the so-called normalised frequency "V"
Cladding
Diameter
yConvenient dimensionless parameter that combines some key fibre variables
Cladding
yIt is defined thus:
Core
Diameter
Core
V=
2π
πa
λ
2
2
. n1 - n2
where a is the fibre radius and λ is the operating wavelength
Refractive index profile for a
step index optical fibre
V is also very commonly defined using the
numerical aperture NA thus:
ySimplest and earliest form of fibre
yThe larger the core diameter the more modes propagate
Relative Refractive Index
yNormally the symbol ∆ is used
y∆
∆ is defined thus:
2
∆=
2
n1 - n2
2
2n1
if ∆ is << 1 then ∆ is given by:
The normalised frequency V can be
written in terms of ∆ :
∆=
V=
2π
π
λ
a.NA
We will use this definition
yWith a large core diameter many thousands of modes can exist
yIt is also possible to define a so called relative refractive index for a fibre
V=
Modes in a MM Step Index Fibre
yIn a multimode step index fibre, a finite number of guided modes propagate. Number
of modes is dependent on:
ƒ
Wavelength λ, Core refractive index n1
ƒ
Relative refractive index difference ∆, Core radius a
yNumber of propagating modes (M) is normally expressed in terms of the normalised
frequency V for the fibre:
n1 - n2
n1
2π
π
λ
a.n1 2∆
∆
M=
V
2
2
Problem: A step index fibre with a core diameter of 80 µm has a relative refractive index
difference of 1.5%, a core refractive index of 1.48 and operates at 850 nm.
Show (a) that the normalised frequency for the fibre is 75.8 and (b) that the number of modes
is 2873
Influence of Core Size and
Wavelength
yAs the core diameter increases and with it the normalised frequency, the number of
modes increases with a square law dependency on core size
yAs the wavelength increases the number of modes decreases
850 nm
1320 nm
Graded Index Fibre
Cladding
Diameter
0
Graded Index
Fibre
Propagation in a Graded Index
Fibre
N2 N1
Cladding
Core
Diameter
Core
Parabolic variation in
refractive index
yAn expanded ray diagram for a graded index fibre, showing a discrete number
yTypical core diameter for this fibre type: 50 to 120 µm
of refractive index changes n1 to n6 for the fibre axis to the cladding.
yDifferent refractive index profiles have developed
yResult is a gradual change in the direction of the ray, rather than the sharp
change which occurs in a step index fibre
Propagation in a Graded Index
Fibre
Graded Index Fibre Profiles
The index variation n(r) in a graded
index fibre may be expressed as
a function of the distance (r)
Cladding
from the fibre axis
Fibre Axis
b
a
Core
n(r) =
∆ (r/a))
n1 (1-2∆
α
for r < a (core)
Light ray (a) and (b) are refracted progressively within the
fibre. Notice that light ray (a) follows a longer path
within the fibre than light ray (b)
n(r) =
n1 (1-2∆
∆)
= n2
for r > a (cladding)
yMeridional (axial) rays follow curved paths in the fibre as shown
yBenefits of using graded index design are considered later
Most common value of the profile
parameter α is 2, a so called
parabolic profile. An infinite profile
Refractive index profiles for Graded Index
fibres
parameter implies a step index fibre
Modes in a Graded Index Fibre
Calculating the number of modes in a graded index fibre is very involved
As an approximation it can be shown that the number of modes is dependent on the
normalised frequency V and on the profile parameter α.
That is
M=
where ∆, is again given by:
α
α+2
∆=
V
2
2
n1 - n2
n1
if ∆ is << 1
Exercise
For the most common value of α show that for fibres with similar relative
refractive indices, core radii and operating wavelengths, the number of
modes propagating in a step index fibre is twice that in a graded index fibre
Singlemode
Fibre
Refractive Index Profiles
for SM Fibres
Singlemode Optical Fibre
Multimode graded index
Multimode step index
Cladding
Small Core
0 N2
N1
Cladding
Diameter
Cladding
Small Core
Diameter
Refractive index
Core
Conventional singlemode fibre
(so called matched cladding)
Depressed cladding singlemode fibre
(less susceptible to bend loss)
Up-and-down profile singlemode fibre
(used in dispersion flattened fibre)
also called multicladding fibre
Triangular profile singlemode fibre
(used in dispersion shifted fibre)
profile
Energy Distribution in a
Singlemode Fibre
Normalised Frequency for SM
Fibres
ySinglemode fibre exhibits a very large bandwidth and has thus become the fibre of choice in
most high speed communications systems.
yThe amplitude distribution of the optical energy in a singlemode fibre mode is not uniform, nor is
it confined only to the core
ySinglemode operation is best considered with the aid of the fibre normalised frequency V:
yIn multimode fibres if we assume a mode model instead of ray diagram approach then some
small percentage of the energy is contained within the cladding close to the core, but typically <
1% so the ray model is still a valid view
V=
2π
π
a . NA
λ
yRay diagram model does not work for singlemode fibre
7-9 µm
> 50 µm
ySingle mode operation takes place where V is less than the so-called cutoff value of Vc = 2.405.
yThe single mode is the lowest order mode that the waveguide will support, referred to as the
HE11 mode. This mode cuts off at V=0.
Core
Core
Cladding
Cladding
yAs will be explained practical V values are normally between about 2 to 2.4
ySinglemode operation is achieved by altering the fibre radius, NA or the wavelength in use so
that V lies in the range above.
Multimode energy
distribution is confined
to the core
Singlemode energy distribution
peaks in the centre of the core
(Darker shading = higher energy)
Mode Field Diameter and Spot Size
(II)
Mode Field Diameter and Spot Size (I)
yMode field diameter (MFD) is an important property of SM fibres.
yThe amplitude distribution of the HE11 mode in the transverse plane is not uniform, but is
approximately gaussian in shape, as shown below
Fibre centre
The MFD is defined as the width of the
amplitude distribution at a level 1/e (37%)
from the peak or for power 13.5%from the
peak
- 3/2
= 0.65 + 1.619V
yFor V < 2 the spot size is significantly
larger than the core size.
yFor V < 2 the beam is partially contained
within the cladding and loss increases
yFor this reason V should be between about
2 and 2.4
The spot size is the mode field radius w. Its
value relative to core radius is given by the
expression:
w
yAs the V value approaches 2.4 the spot
size approaches the fibre radius.
yMFD or spot size is frequently specified as
well as core radius or diameter for the fibre
Normalised spot size as a
function of the fibre V value
-6
+ 2.879V
a
Cutoff Wavelength
Singlemode operation only takes place above a theoretical cutoff wavelength λc
where V < Vc = 2.405
λc =
2π
π
Vc
a NA
In practice the theoretical cutoff wavelength is difficult to measure. An alternative is EIA
(Electronics Industry Association of America) cutoff wavelength, which states that the
cutoff wavelength is:
The wavelength at which the power in the HE21
mode is 0.1 dB of the power in the HE11
(fundamental mode)
SM Fibre Summary and Problem
yCore size is a useful parameter for multimode fibres, but is not so useful for SM
fibres.
yTelecommunications systems are normally designed to work close to the cutoff
wavelength for good power confinement (small spot size), but not close enough to
cutoff so that significant power is carried in higher modes.
Exercise
A singlemode fibre has a core refractive index of 1.465 and a cladding refractive index of 1.46. What
is the maximum core size if the fibre is to support only one mode at 1300 nm?
Answer: core radius 4.11 microns, 8.23 microns core diameter.
If the wavelength is increased to 1550 nm what is the new fibre V value, the spot size and the MFD?
Answer: V = 2.02, Spot size 5.18 microns,
The EIA cutoff wavelength can be 100 nm less than the theoretical cutoff
wavelength
MFD 10.4 microns