Download Fiber Waveguide

ECE 182
FIBER WAVEGUIDE
I.
OBJECTIVES
To study coupling efficiency as a function of numerical aperture for a fiber waveguide, and
determine the linear attenuation for a length of the fiber.
II.
GENERAL BACKGROUND
In its simplest form optical fiber consists of a cylindrical core of silica glass surrounded
by a cladding whose refractive index is lower than that of the core. Because of an abrupt index
change at the core–cladding interface, such fibers are called step-index fibers. In a different type
of fiber, known as graded-index fiber, the refractive index decreases gradually inside the core.
Considerable insight in the guiding properties of optical fibers can be gained by using a ray
picture based on geometrical optics. The geometrical-optics description, although approximate,
is valid when the core radius is much larger than the light wavelength. When the two become
comparable, it is necessary to use the wave-propagation theory.
Numerical Aperture of a Fiber
Since transmission of light inside the fiber waveguide relies on total internal reflection,
there exists a maximum acceptance angle. Light rays whose angles of incidence exceed this limit
do not undergo total internal reflection. Referring to Fig.1, the critical angle at the core–cladding
interface is
(1)
where n1 and n2 are the refractive indices of the core and cladding. The corresponding cutoff
incident angle jc is, from Snell’s law
(2)
n0 - the refractive index of air.
n0
jc
θc
Guided ray
Core index n1
Cladding index n2
Fig. 1. Light confinement through total internal reflection in step-index fiber.
Rays for which j < jc are guided rays.
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Combining Eqs. (1) and (2) gives
(3)
Equation (3) specifies the maximum incident angle to the end face of a fiber. The quantity
nosinjc is called the numerical aperture (NA), and
(4)
for n1 º n2 the NA can be approximated by
(5)
where D is the fractional index change at the core-cladding interface.
Graded-Index Fibers
The refractive index of the core in graded-index fibers is not constant but decreases
gradually from its maximum value n1 at the core center to its minimum value n2 at the core–
cladding interface.
(6)
where
(7)
a is the core radius and p, called the grade profile parameter, determines the steepness of the
profile. Most graded-index fibers are designed to have a nearly parabolic-index profile (p = 2).
The step-index fiber is the special case of graded-index fiber with p = ¶. The advantage of
graded-index fiber is the much smaller modal dispersion than that of the multimode step-index
fiber.
Modal Dispersion
Modal dispersion can be understood by referring to Fig.1. Using the boundary conditions
at the fiber core and cladding interface, one can show that within the acceptance angle of the
fiber only those rays of certain discrete angles can propagate down a long fiber. Rays at each of
these discrete angles are related to a spatial mode which can be excited. Mode dispersion is
caused by the difference in propagation time for rays of different angles.
A short pulse (called an impulse) would broaden considerably as a result of different path
lengths. Pulse broadening can be estimated by considering the shortest and longest ray paths.
The shortest path occurs for θi = 0 and is just equal to the fiber length L. The longest path occurs
for jc given by Eq. (3) and has a length L/sinθc. By taking the velocity of propagation v = c/n1,
the time delay in step-index fiber is given by
2
(8)
Modal dispersion is much smaller in graded-index fibers than in step-index fibers since
the group velocities are equalized and the differences between the delay times of the modes are
reduced. It is easy to understand qualitatively. Similar to the case of step-index fibers, the path is
longer for more oblique rays. However, the ray velocity changes along the path because of
variations in the refractive index. More specifically, the ray propagating along the fiber axis
takes the shortest path but travels most slowly as the index is largest along this path. Oblique
rays have a large part of their path in a medium of lower refractive index, where they travel
faster. It is therefore possible for all rays to arrive together at the fiber output by a suitable choice
of the refractive-index profile.
Obviously there is no mode dispersion if only one mode can propagate in a fiber. Such
fiber is called a single-mode fiber. But dispersion of an optical fiber is not limited to mode
dispersion. Even a single-mode fiber suffers from additional dispersions, such as material
dispersion, waveguide dispersion, and nonlinear dispersion.
Coupling of Light into a Fiber by Use of a Lens
Coupling of light into a fiber is not a simple task, especially for single-mode fiber having
a core diameter of a few microns. A simplified coupling setup is shown in Fig.2. The most
important parameter of any coupling setup is a coupling efficiency. In the case where the light
source is a collimated laser beam of uniform intensity, the derivation for coupling efficiency is
given below.
Fig. 2. Coupling of a collimated light beam into a fiber by use of a lens.
F is lens focal distance and r is lens aperture radius.
The numerical aperture of the lens is defined by
NAlens = sinθL
(9)
For small θL and n0 = 1, the above becomes
NAlens @ θL
(10)
The dependence of the maximum acceptance angle θF of the fiber is related to its critical angle
φc and
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n 0 sin θ F = n12 − n 22
(11)
NAfiber @ θF
(12)
For small acceptance angle θF,
The coupling efficiency is defined as the ratio of the light power accepted by the fiber
core to light power incident on fiber. The light power gathered by the core can be described in
terms of the fiber acceptance angle by
Pf = Iiπx2 = Iiπ (FÿtanθF)2
(13)
where Ii is the incident light intensity. The total incident power is
P0 = Iiπr2 = Iiπ(FÿtanθL)2
(14)
And the coupling efficiency is
2
P  tanθ F   NA (fiber) 
η= f =
=
Po  tanθ L   NA (lens) 
2
(15)
When the light is coupled into the fiber using a lens, not only the angle of incidence of
the converging beam but also the diameter of the incident light beam is restricted. The diameter
of the incident beam has to be equal to or smaller than the core diameter for efficient coupling of
light to the fiber. Multimode fibers have a relatively large core, resulting in a high coupling
efficiency.
Light Attenuation in Fiber
Light propagating inside an optical fiber are governed by Beer’s law:
dP/dz = −αP
(16)
where α is the attenuation coefficient. Although denoted by the same symbol as the absorption
coefficient, α includes not only material absorption but also other sources of attenuation. If Pin is
the power launched at the input end of a fiber of length L, the output power Pout from Eq. (14) is
given by
Pout = Pin exp{-αL}
(17)
Here attenuation coefficient α is measured in km-1. In telecommunication systems the attenuation
coefficient usually defined in units of dB/km:
γ =−
10
P
log10 out
L
Pin
(18)
The attenuation coefficient of fused silica glass (material used to make fibers) is strongly
dependent on wavelength.
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Material absorption can be divided into two categories. Intrinsic absorption losses
correspond to absorption by fused silica whereas extrinsic absorption is related to losses caused
by impurities within silica.
Another source of attenuation is a light scattering arising from local microscopic
fluctuations in density. Density fluctuations lead to random fluctuations of the refractive index.
Light scattering in such a medium is known as Rayleigh scattering.
In practice, imperfections at the core-cladding interface (e.g., random core-radius
variations) can also lead to additional losses which contribute to the net fiber loss.
III.
SYSTEM DESCRIPTION
The optical system starts with a HeNe laser. Its output beam is expanded with a spatial filter,
and collimated with a lens. An iris is used to control the size of the collimated beam. Following the
iris is an adjustable microscope objective mount. Objectives can be threaded into and removed from
this mount. There are transverse adjustments for centering the objective with respect to the input beam.
This mount will be used to measure optical efficiencies of various objectives.
The next mount is a compound unit. The front part has a fixed-position objective mount for
insertion and removal of objectives. The rear part has a fiber positioner that contains a fiber chuck, and
a short length of fiber is housed within the chuck. The positioner has x, y, and z movements to allow
optimal positioning of the fiber tip with respect to the focal spot of the particular objective used.
The final group of components consists of a compound mount, a spool of fiber, and a fiber
positioner. The compound unit, as before, is used for coupling light into the input end of the fiber spool.
The fiber positioner is used to hold the fiber chuck at the output end so that output power can be
measured. The fiber is a graded-index type with a core diameter of 90µm. The fiber spool contains a
1.1km length of the fiber.
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IV.
EXPERIMENTS
The fiber tip must never come in contact with anything. In addition, when performing coupling of light
into the fiber using microscope objectives, monitor the distance between the fiber tip and the objective
lens surface, to prevent contact between the lens and the fiber, as this will damage both units.
A. Optical Efficiency of Microscope Objectives
Set of five microscope objectives with different numerical apertures will be used.
(a) Check that the iris is open to pass the full width of the collimated beam.
(b) Thread the NA=0.1 objective into the adjustable objective mount. Position the power meter
detector after the objective so that the expanded beam fills approximately 50% of the detector area.
(c) Adjust the transverse controls to center the objective with respect to the input beam in order to
maximize detected power.
(d) Slowly close the iris down until the output power just begins to drop. Record the output power P0.
(e) Move the detector in front of the objective, and measure the input power Pi.
(f) Calculate the optical efficiency E0 = P0/ Pi for the objective.
(g) Repeat (a) - (f) for objectives with NA=0.25, 0.3, 0.35, and 0.5. Note that Pi will vary depending on
the iris size set for each objective.
B. Fiber Coupling Efficiency
The compound mount will be used for this part.
(a) Thread the NA=0.1 objective into the front part of the mount.
(b) Slowly slide the objective until the fiber tip is located at the approximate focal distance away from
the objective lens. Gently lock the objective in place with its lock-screw.
(c) Using a white card to observe the output from the fiber, adjust the x, y, and z controls to
visually maximize the output.
(d) Place the detector close to the output side of the fiber chuck, but do not allow the detector to
come in contact with the chuck.
(e) Optimize the detected output power by fine tuning the x, y, and z controls. Record this as the fiber
output power Pf.
(f) Determine the fiber coupling efficiency as Ef = Pf / P0.
(g) Loosen the objective lock-screw and slide the objective off the mount.
(h) Repeat (a) - (g) for the objectives with NA=0.25, 0.3, 0.35, and 0.5.
(i) Plot coupling efficiency Ef vs. NA.
C. Linear Attenuation of the Fiber
(a) Using the optimum objective found in part B, couple light into the input end of the fiber spool.
(b) Using a white card to observe the output from the fiber spool, adjust the x, y, and z controls to
visually maximize the output. As before, be careful of the fiber attached to the fiber chuck when
adjusting the z control.
(c) Place the detector close to the output fiber chuck. Optimize the detected output power by fine
tuning the x, y, and z controls on the input mount. Record this as the fiber output power I0.
(d) Define the power coupled into the fiber as Ii = Pf where Pf is the coupling power measurement
for the particular objective used.
(e) Calculate the fiber attenuation γ = -10log10( I0 / Ii)/L in dB/km. Fiber length is 1.1km.
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