Light travels in straight lines?a physical simulation of light propagation
in a graded index optical fibre
~
M G Cornwall
Largescale demonstrations of the propagation of
a HeNe laser beam in graded Index opHcal medla
have been developed, using interditfusing liquid
isyarn. The sinuous paths Of a ray in a GRIN
optical fibre and lenses have been simulated.
of physical demonstrations of light propagation in
graded index media. The physical simulations are
achieved by using various liquids to create different refractive index profiles within a rectangular
tank.
A concept once treated as an interesting but
peripheral topic in introductory physics-total
internal reflection-has become the basis of the
fibre optic communications revolution. Whilst
modern A-level and College physics textbooks
now mention this technological application of an
old principle, the simple idea of a light ray ‘bouncing’ along inside a light pipe is usually the limit of
the explanation given. In fact, there are relatively
few actual fibre optic applications for which this
simple model of light propagation is correct.
Furthermore few, if any, simple demonstrations of
the way in which light actually travels in fibres
seem to have been developed.
The simulations described here are the result of
a series of undergraduate projects carried out by
second- and final-year students in the Department
of Mathematical Sciences at Brighton Polytechnic
over several years. Originally, computer programs
using simple ray-tracing techniques were developed
to demonstrate the properties of stepped and
graded index fibres. However, whilst computer
simulations are excellent ways in which to model
the real world in order to illustrate principles and
to try out ‘what if‘ experiments, they lack the
credibility of a real demonstration or experiment.
We therefore developed a complementary package
Beam propagation in a graded index medium
Malcolm Cornwall is Principal Lecturer in charge of
Physics at the University of Brighton (formerly Brighton
Polytechnic) and a member of the Editorial Board for
this journal. He often addresses schools and colleges
on optics education. holography, lasers and related
subjects.
M31~912019LX15m73t07
104.5001992 IOP Publtshmg Lld
The first and simplest arrangement was intended
to demonstrate the bending of a ray in a medium
with a continuously varying index. A layer of
sugar is placed at the bottom of a large rectangular
glass or plastic tank, covered with water and
allowed to diffuse for several days. If sufficient
sugar is used, a solid layer will still be present at
the bottom of the tank, with a gradual merger
between solid and solution, creating a ‘graded
index’ medium. Sugar is used because, besides
being a cheap and completely benign ‘chemical’, it
has an extremely large solubility in water and
produces large changes in refractive index, n.
Moreover, values of n as a function of sugar
concentration are listed in standard physical data
handbooks (e.g. The Handbook of Chemistry and
Physics). When a 0.5-1.0 mW HeNe laser beam
is directed into the tank so as to travel through
a relatively strongly graded index region it is
observed to bend in a very obvious way. Figure 1
is a multiple exposure photograph showing the
laser beam paths for several incident angles on the
left-hand side of the tank.
We have found that even for experienced physicists this bending is quite startling at first sight, so
familiar are we with the ‘fact’ that, on a macroscopic scale, light rays always travel in straight
lines. The equation for the path of a light ray in an
optically inhomogeneous medium with a refractive
index gradient in they direction given by n(y) and
a constant index in the x direction is
d2y/dx2=[l/(2n: sinzO0)]dn2(y)/dx
273
Flgure 1. Propagation of a H e N e laser beam in a graded index medium. The photograph shows an approximately
50 cm long 'sugar tank' (seetext) with a laser beam launched from t h e left into t h e tank at different heights. (The
bright area at the bottom of t h e t a n k is the undissolved sugar layer.)
where n, is the refractive index of the medium at
the point of entry of the ray, and 00 the corresponding angle of incidence relative to the vertical.
Note that the curvature of the ray is proportional
to the gradient of the square of the refractive index
function (Ghatak and Thyagarajan 1978).
The bending, being related to the gradient ofthe
refractive index, is very much more P " m c e d
near to the interface between the undissolved solid
sugar and the solution. Indeed, significant bending
can be observedclose to this interface just a day or
so after the water layer is added.
An initial problem with the sugar solution was
that the region with the highest index gradient was
also optically diffuse and caused unacceptably
large scattering of the beam and hence loss of
beam definition. The use of chemically pure sugar
in the form of high purity sucrose, and distilled
water, avoidd this problem, H
~the absence
~
of impurities in the solution then produ& the
opposite problem of insufficient scattering centres
to make the beam visible except in virtual darkness. A 'beam visualizing' additive is a small drop
n4
of white emulsion paint, which contains Ludoxt,
a major component of some types of paint.
Very small samples of the solution were taken
with a fine capillary tube at various depths in the
tank, and the refractive index measured with an
Abbe refractometer. The index profile was found
to be exponential within the experimental error.
(The solid-liquid diffusion in the tank would,
theoretically, be expected to give a 'complementary error function' (erfc) dependence of index on
distance from the interface (Jost 1960); this function differs little from an exponential). This simulation incidentally illustrates the principle of the
mirage, as observed in desert conditions (or, more
mundanely, above a road on a hot day).
manufactured
by DuPont, consists of
~t This material,
~
,
a suspension of extremely small 1-12 nm) particle of
amorphous silica, each having a negative charge, such
that they are maintained in suspension in a suitably
buffered me,jium,with an approximately unifom den.
sity within the suspension.
The 'sugar tank' is simple to set up but requires a
large tank and considerable waiting time (although
this can be reduced by using warm water initially).
we have fOund that an equally effective graded
index medium can be produced by carefully floatink? 2-3 Cm Of isopropyl alcohol (IPA) on water.
Beam visibility is greatly enhanced if Ludox-doped
water is used (Ludox flocculates in alcohol).
Interdiffusion of the liquids is quite fast, and
within a few minutes a beam launched into the
region of the alcohol-water interface is strongly
refracted upwards even in a relatively short tank
about 10 cm in length (figure 2(d. As the refractive index is 1.38 and is greater than that of water
(1.33) the index gradient has the opposite sense
from that in the sugar tank; the beam is therefore
bent upwards as in a mirage. (In general. rays bend
towards the layer of higher refractive index.)
The dependence of the beam Cw-"Ure on the
position at which the beam enten the tank. is
illustrated in figure 2@). This shows the appearance of a 'fan' of rays incident on the tank such
that *e top ofthe fan travels through a region Of
high index gradient near the liquid interface and
has a high curvature, whilst the lower part of the
fan travels through almost pure water and is
fan was p r o d u d by
virtually undeflected. pe
reflec.ing the laser beam from a small mirror
attached to a loudspeaker driven at a few tens of
hertz ( ~ ~ and~williams
~ ~lggl).)
1 1
If the laser beam is launched into the IPA layer at
a suitable angle immediately after it is poured Onto
the water, the beam can be made to 'bounce'
&tween the water-air and water-rpAinterfaces.
The beam is simply totally internally reflected at
the upper surface, but is much mOre gradually
curved at the liquid interface, due to the initial
index gradient that exists as the two liquids begin
to interdiffuse. me gradual d i r e c t i o n of the
km
observed here is the basic mechanism by
which a graded index optical fibre operateS.
Several other liquids can be used instead of IPA,
but the only other relatively benign solvent we
have tried that produces interesting results is ethanol, which we use in the form of industrial methylated spirit (IMS).This also floats on water and
gradually diffuses into it, but it has a refractive
index of 1.325 which is slightly lower than that of
water. As a result a laser beam bends downward
when it is launched into the tank. Because of the
small difference in the refractive indices and the
smaller solubility of IMS in water, the bending is
less dramatic and can be seen only by careful
manipulation of the incident beam height and
angle.
Graded Index fibres
Flgura 2. (a) Path of a HeNe laser beam launchedfrom
the right in a 10 cm long tank in which a 3 cm layer of
iSOprOpYl alcohol (IPA) has been floated onto water.
( b )A 'fan' of rays, producedas described in the text.
enteringan IPA tank from the left. The uppermostpart of
the fan is strongly refracted:the lowest part of the fan
traVeiS through almost pure water in a straight line.
The success of the sugar and IPA tanks suggested
the desirability of producing a physical simulation
of a complete graded fibre, by using three layers of
suitable refractive index and of suitable density SO
that they would float on one another and gradually
interdiffuse. Within such a structure the sinuous
path of a light ray in a graded index fibre could
be shown.
A literature search produced a reference in
R W Wood's classic book Physical Optics (Wood
1935) to a much earlier paper by Wood (1899), in
which he describes a simulation of a mirage, and
of a light beam in a layered medium. We found,
however, that the solutions used by Wood were
not as effective as the following sequence of
solutions:
Bottom layer: concentrated potassium alum
solution.
Middle layer: a mixture of 30% glycerol in IPA.
Top layer:
a mixture of 30% water in IPA.
275
The relative thicknesses of the layers are not
important, but the middle one should be about
3-4 cm thick for a tank of about 50 cm in length.
The layers need to be poured very carefully on top
of one another and then left for some time.
The three layers diffuse slowly into one another
and we have found that at room temperature
the optimum conditions are produced after about
18 hours. The layers need to be poured very carefully on top of one another, to avoid premature
mixing. We have used a simple ‘pourer’ in the form
of a piece of paper bent into ‘U’ shape, and held so
that the bottom of the ‘U’ just touches the surface
of the liquid layers as they are added. Launching a
laser beam horizontally into the tank at various
heights produced the beam paths shown in figure 3.
The beam paths are apparently sinusoidal in
shape and all have about the same periodic length.
This is the path produced by a so-called ‘parabolic’
index gradient, which is approximately the gradient created in graded index fibres. (In such a fibre,
the propagation time for any ray path along the
fibre is (very nearly) the same, so that there is little
‘intermodal‘ dispersion due to the different times
taken by rays travelling along different paths,
which occurs in a step index fibre. So in a graded
index fibre, digital signal pulses are transmitted
with little distortion over long distances.)
If a beam is launched into the tank immediately
after it is set up, the beam ‘bounces’ between the
interfaces, simulating the propagation of a ray in
a stepped index fibre. However, due to optical
distortions near the interface between the liquids
the beam looks sinusoidal. In addition, the beam
visually appears to be trapped in the interface layer
for a significant distance before being totally internally reflected. To minimize the visual distortions
the beam should be launched very close and
parallel to the side wall of the tank through which
it is observed.
Using the technique mentioned earlier, we have
measured the value of n as a function of depth
after 18 h, and have found a variation from 1.342
near the top of the layers to 1.359 near the bottom
ofthe tank, through a smooth maximum of 1.384
in the middle of the central layer. It is perhaps
surprising that such small variations in n are
nevertheless capable of producing such very
obvious undulations in the beam path. (In fact for
actual fibres the variation can be well under I%.)
Unfortunately, it is not possible to use Ludox to
enhance the beam visibility, because the colloid
comes out of suspension in the alum and produces
a floccular precipitate. We have had some success
with other combinations of liquids within which
Ludox is stable, and also using other visibilityenhancing techniques.
276
An improved fibre simulation
It would obviously be desirable to develop a
‘three-layer’ simulation, which has a much larger
change in refractive index across the layers, and
hence produces larger and more obvious undulations of the laser beam and, ideally, with a much
shorter ‘waiting time’. The criteria for the choice
of liquids is that, if their densities are d,, d2 and d3
and their indices n , , n2 and n3, then we require that
d , > d 2 > d 3 and n l < n 2 > n , , with the difference
between n2 and the other two indices being as large
as possible. In addition the liquids should be
mutually miscible, should be able to be handled
without serious hazards and should ideally be
inexpensive. These criteria proved to be remarkably difficult to satisfy. An extensive literature
search and experimentation resulted in the following set of liquids, which most nearly match the
requirements:
Bottom layer: dichloromethane.
Middle layer: cinnamaldehydet .
ethanol (IMSis suitable).
Top layer:
A 5 cm layer of IMS, a 1-2 cm layer of cinnamaldehyde and a 5 cm layer of dichloromethane were
successively poured into a tank about 15 cm in
length using the ‘U’ shaped pourer. The width of
the tank should be small-we used one about
2 cm wide-to reduce the required volumes of the
liquids. To make the beam clearly visible a few
crystals of rhodamine 6G, a fluorescent dye, were
dissolved in the cinnamaldehyde before it was
added to the tank. The laser beam is then visible as
a bright pinkish red fluorescence.
The three layers in this tank need to be allowed
to interdiffuse for a few minutes for optimum
performance. Measurement of the refractive index
as a function of depth immediately after setting up
the tank gave a value of n varying smoothly from
1.384 near the surface, through a maximum of
1.598 in the middle layer to 1.470 just above the
bottom of the tank. The waiting time, the most
effectivethickness of the middle layer, and the best
angle and position of incidence of the laser beam
are interdependent. Typical beam paths are shown
in figure 4(a) and are easy to produce, but even
more ‘wavelengths’ can be produced by optimization of the variables. Although fewer ‘waves’ are
t Cinnamaldehyde has an extremely strong and linger-
ing smell (of cinnamon) and is classifiedas an ‘irritant’.
A few people may be allergic to the vapour, and it is
desirable therefore to handle the liquid with care and in
a fume cupboard. It therefore does not satisfy one of our
criteria. However, all other high index candidates for the
middle layer ofwhich we are aware are more hazardous.
propagation of a laser beam in t h e three-iayer 35 cm long tank simulating a two-dimensional
graded index fibre (seetext). This iS a multiple exposure photograph showing the paths for a horizontal beam
incident from the leftat five different heights.
ngure 3. The wave-like
visible here than in figure 3, it should be noted that
the tank used for figure 4 is very much shorter and
the curvature of the laser beam much greater, due
to the very large index gradients produced in the
‘cinnamaldehyde’tank.
By scanning the incident beam as described
earlier, a fan of rays can be launched into the
simulated ‘fibre’ as shown in figure 4(b). This is a
more realistic simulation of the actual state of
affairsin a fibre.
If the laser beam is directed into the top layer it
behaves like a ‘cladding ray’ (i.e. a ray that initially
travels in the outer, lower index layer in a fibre)
and is not properly propagated in the ‘core’ layer.
Figure 4(c) shows that such a ray in fact initially
diverges as it enters the ‘core’ and then reconverges
as it enters the bottom ‘cladding layer’.
Deficiemies In the simulations
The most obvious deficiency in the physical simulations is that they are two-dimensional representations of a three-dimensional process, and cannot
represent the more complex ‘skew’ modes that can
occur in real fibres; all the ray paths simulated are
in effect meridional, i.e. they are restricted to a
plane passing through the axis of the ‘fibre’. The
simulations are based on a ‘ray’ model of light
propagation. This means that they cannot represent mode selection, mode conversions and evanescent modes-phenomena that are important in
fibres, and which are essentially a result of the
interference properties of waves. In particular the
model cannot simulate the characteristics of a
single-mode fibre.
A~ interesting theoretical COntraStexists between
a computer simulation and the physical simulations. The former is rigidly a ‘ray’-based model
and illustrates directly the inadequacy of that
model; a ray that is incident exactly parallel to the
fibre axis will propagate along a path of constant
index and will therefore not suffer any refraction.
This is the caSe not only for a ray originally
incident in an axial direction but alsp for any ray
that is eventually refracted into an axial direction;
such a ray will not experience any further ‘bending’ k a u s e it is then travelling in a constant index
layer. This obviously fallacious conclusion from
the ray model is demonstrated by the physical
simulation, in which we are dealing with a real
light beam with a finite width, which is of course
refracted even when it is incident exactly normal
to the index gradient. In a computer model the
inadequacy of the ray model is disguised by the
fact that the fibre is usually simulated by a set of
finite layers of constant index; only if by chance a
ray is incident at exactly the critical angle at a
layer interface will it be refracted parallel to the
layers. With the precision to which a computer
operates, this is extremely unlikely and the ray is
either completely reflected or refracted. The effect
of the finite width of the laser beam is also obvious
when the beam propagates through a region of
sufficiently rapidly varying index to smear out the
beam by bending different parts of it to varying
extents (see figure 4(c)).
Fu*rhvelopmmh
We have investigated other combinations of
liquids with larger index differences, but none has
m
Figure4. HeNe beam paths in the 'cinnamaldehyde'
tank (seetext): the beam is visible as a resuit of t h e
fluorescenceof Rhodamine 6G dissolvec in t h e
cinnamaldehyde layer. ( a )Beam launched horizontally
from t h e iefl into t h e middle layer. The tank length is
onlyabout 15cm.comparedwithabout50cm in figure3
( b )An approximately parallel fan of rays is incident
from t h e left. The initially 'collimated' beam is 'focused'
and 'defocused as it propagates. (c) The path of a
beam launched into the upper 'cladding' layer. Note that
it first diverges and then reconverges a s it enters and
leaves the 'core' layer.
been as effective as the system described above.
We are also looking a t various possibilities for
producing 'solid versions of the tanks, which
would have the virtue of permanence and threedimensionality. A long solid Perspex (Plexiglass)
tube ahout 2 cm in diameter can be used to show
skew modes, but the beam is not easily visible.
Much more effective is a long hollow glass Or
Perspex tube, filled with Ludox-doped water.
A suitably launched laser beam clearly shows the
characteristic skew paths of non-meridional rays
in a stepped fibre' In tbis caSe total internal
reflection takes place a t the plastic (or dass)-air
interface, and the weaker reflection a t the water278
solid interface is strong enough to cause a smeared
appearance of the beam a s it propagates along the
tube. A relatively short glass tube filled with
cinnamaldehyde doped with Rhodamine 6G shows
skewed modes much more dramatically. Because
the liquid has a much higher refractive index than
glass, not only does TIR takes place at the liquidsolid interface, resulting in a much 'cleaner' beam,
but the beam is 'trapped' even for high incident
angles (i.e. the simulated fibre has a very large
numerical aperture).
Wood (1935) described the in-diffusion of a
sugar solution into gelatin a s a way of producing
what we would now call a graded index (GRIN)
lens. GRIN lenses-also
called sewocs-usually
consist of a short length of graded index rod,
having a length of either 0.25 or 0.27 times the
'period' of the sinusoidal path (a few mm) within
the fibre. These two lengths produce, respectively,
a 'lens' that will collimate a diverging beam, or a
'lens' that focuses an initially parallel beam. In two
dimensions we have simulated the operation of a
GRIN lens by using a small rectangular glass
'cell' (as used for holding liquid samples in spectrometry) standing inside a longer tank. By filling
the cell with Ludox-doped water and moving it to
different positions along the beam path, the focusing or collimating action of a GRIN lens is graphically demonstrated (figure 5).
The growing importance of fibre-optic-based
communications and sensor technology, and the
emergence of integrated optics and optical computing as the next innovative wave, suggest that
our students should have more than a peripheral
understanding of the optical principles involved.
These simulations provide one very graphic means
of demonstrating how light can be propagated in
Figure 5. A demonstration of t h e operation of a
collimating GRIN lens. The fan of rays is incident from
t h e right. and a Small cell (filled with Ludox-doped
water) is placed at a n appropriate position to show that
a parallel beam can be produced from a point source,
for a GRIN lens of length equal to0.25 'wavelengths'.
The main tankcontainsa thre&ayer 'cinnamaldehyde'
fibre simulation.
real fibres and some of the characteristics of these
fibres.
There is undoubtedly considerable potential for
further development of the ideas described here.
In particular, it would be very desirable to find
more pleasant and less expensive alternatives to
cinnamaldehyde. Development of solid threedimensional simulations, perhaps along the lines
suggested by Wood, would provide a more permanent and convenient teaching aid.
Acknowledgments
I am indebted to my students, notably Mark
Greenway Stanley and Yanik Le ROY, and to the
Technical Support team of my department for
their invaluable practical contributions.
Ghatak A K and Thyagarajan K 1978 Conremporury
Oprics (New York: Plenum)
Handbook of Chemisrry und Physics 54th edn (Boca
Raton, FL: CRC)
Jost W 1960 Diffusion 3rd edn (New York: Academic)
Wood R W 1935 PhysicdOptics (London: Macmillan)
-1899 Phil. Mug.
Further reading
Halley P 1987 Fibre Optic Sysfems(NewYork: Wiley)
lizuka K 1986 Engineering Oprics 2nd edn (Berlin:
Springer)
Jones K A I987 lnrroducrion IO Opricol E/ecrronics (New
York: Wiley)
ZangerHandZangerC 1991FibreOplic
CommunicarionsundOrher Applicurions (NEW York
Merril)
References
Cornwall M G and Williams G T 199I Laser
I
Applicuriom in Science Educarion: U Teacher's
Hondbook (Brighton Polytechnic: LASE Publications)
279