Download the measurement of the refractive index of glass beads used for

G. SMITH
THE MEASUREMENT OF
THE REFRACTIVE INDEX OF
GLASS BEADS USED FOR
TRAFFIC MARKINGS
ABSTRACT
The present Australian Standard E42-1967, Glass Beads
for Traffic Markings, requires the refractive index of the
beads to be measured by the Becke line method, which
is of the liquid immersion type. Though it is sufficiently
accurate for practical purposes, it is time consuming
and requires a wide range of liquids of different refractive indices to be available. An alternative, non-immersion method is presented which is much quicker to perform and for typical beads used for pavement markings
has an accuracy of ± 0.001. The method is based on the
same principle as the formation of the well known rainbow. This alternative method, if used with a white light
source also can be used to give a rapid, accurate determination of the dispersion of the glass.
INTRODUCTION
Spherical glass beads or 'ballotini' are increasingly being
used to improve the reflecting properties of painted pavement markings and signs, where increased brightness is required to improve visibility under poor weather conditions
and at night. In the production and testing of these beads it
is often necessary to have some knowledge of their refractive index.
Most methods for the precise measurement of the
refractive index of glass rely upon a large enough piece
being available, one surface of which can be ground flat. If
this cannot be done due to smallness of size or because the
sample cannot be destroyed, alternative methods have to be
used and these generally rely upon the immersion of the
sample in a mixture of two liquids. The mixture refractive index is varied until the sample optically disappears using
some sensitive technique such as the transmission of a narrow spectral line through the sample and liquid. The index of
the mixture can be measured precisely and simply with an
Abbe refractometer. In general, the liquid immersion
methods are time consuming, tedious and perhaps most importantly require the availability of two miscible liquids, one
whose index is greater than and one whose index is less
than that of the sample. This requirement becomes increasingly more difficult to satisfy as the index of the glass
increases.
The measurement of the refractive index of glass beads
used in traffic markings poses special problems. Firstly, they
are too small to be measured by the first of the above
methods. Secondly, depending upon the source of the
beads, the refractive index may vary from bead to bead in
the same sample. This variation, even though it may be
26
small, can cause sufficient light scatter to prevent the optical
disappearance of a bulk sample for any liquid index, including that which is the same as the mean of the beads. The
only alternative to bulk immersion methods is to measure the
index of individual beads. Depending upon the variation of
index from bead to bead in the sample, an accurate
measurement of mean index may require the individual
measurement of a large number of beads.
In the Australian Standard (1967) for glass beads, the
prescribed method is one of the immersion type and examines the index of individual beads. For traffic marking
beads, the index accuracy required is not great, being approximately 0.01 to 0.05. This is not high enough to warrant
the use of the two-liquid mixture method already discussed.
Instead, the bead is immersed in a single liquid of known index, viewed under a microscope and a decision made of
relative index. As the microscope tube is raised or lowered
the movement of the Becke line is observed. The Becke line
is a thin band of bright light formed near the boundary of two
media of different indices. As the tube is raised relative to
the sample stage the Becke line moves into or towards the
medium of higher refractive index.
For any liquid, since the index determination is only
relative, i.e. the bead index is only either greater or less than
that of the surrounding liquid, more accurate determination
involves using successive liquids whose indices are chosen
closer and closer to that of the sample. Two liquids are found
- one whose index (n H ) is higher than that of the bead and
the other whose index (n L ) is lower than that of the bead.
The bead index is then specified as lying somewhere between the corresponding upper and lower bounds n H and
n L respectively. For an accuracy of 0.01 this implies the
availability of 10 stable liquids of known index for each 0.1
in index range. Since beads are produced in the range from
1.5 to over 2.0, to cover this range with an accuracy of 0.01
would require at least 50 stable liquids. As the index increases it becomes increasingly difficult to find stable liquids. One further disadvantage of the Becke line method is
that if for a particular liquid, the indices do not match, it is
very difficult to quantitatively assess the magnitude of the
difference.
The source of the Becke line has been described by
Wahlstrom (1969) and as originally defined is associated
with a vertical contact of two substances of different indices
as shown in Fig. 1. The bright line is formed where the
refracted rays such as ray 1 and the total internally reflected
rays such as ray 2 intersect the remaining rays such as ray 3.
Australian Road Research, Vol. 7, No. 1, March 1977
SMITH - REFRACTIVE INDEX OF GLASS BEADS
However for the size of beads used in traffic markings (0.1 to
1.0 mm diameter) the geometrical theory should be adequate.
Fig. 3 shows a cross-section of a sphere. A ray refracted
into the sphere is partly reflected off the rear surface. This
reflected ray is refracted out of the sphere, finally making an
angle 0 with the incident ray. For a collimated beam incident on the sphere the angle 0 will vary with the point of incidence P on the surface. The rainbow is formed because of:
(1) the variation of index with wavelength; and
(2) as P moves across the surface of the sphere from A to B
the angle 0 increases positively, passes through a maximum
value 6 max then decreases.
n
It is this maximum which gives rise to a localised high
ray density and a corresponding localised maximum intensity of light.
Fig. 1 — The formation of the Becke line for refractive indices n and N when N
>n
The Becke line is used mainly in geology to determine
the index of small crystals. However, there are some
problems in applying the principle to spherical glass beads
which unlike crystals do not have flat surfaces and have instead strong refracting power. The deviation of all the rays
passing through the sphere not only makes the Becke line
less bright and more diffuse but also produces a dark band at
the edge of the bead. When the index of the bead is greater
than that of the liquid the dark band is formed above the
bead and, when the bead index is less than that of the liquid, below the bead. The source of the band is shown in
Fig. 2 for the case where the bead index is higher than that
of the liquid. In this case, collimated light is used. The bead
acts as a positive lens and deviates the light or rays towards
the axis of the bead as shown. As a result, no light or rays
pass through or into the dotted region and this dotted region
corresponds to a dark band at the edge of the bead. From
the author's experience, this dark band is sometimes easier
to see than the Becke line and thus may be a more useful
guide to relative index.
An alternative method of measuring the refractive index
of glass beads is presented here which makes use of the
fact that the beads are spherical or nearly so. The method,
which in practice is much simpler and quicker to carry out
than the immersion methods, is based on the observation of
the 'rainbow'. It is well known that when the sun (a collimated beam of white light) is incident upon a number of
transparent spheres (raindrops) part of the light after entering the drops is reflected backwards in the form of a rainbow. The different spectral colours are seen because the
refractive index of water varies with wavelength and increases with decrease in wavelength. The angular diameter
of the rainbow, or more specifically the angle subtended by
the arcs of different colours (wavelengths), depends only on
the refractive index of water for that wavelength and not on
the size of the raindrop.
THEORY
The geometric optical theory describing the formation of the
rainbow has been given by Longhurst (1967). Of course, an
exact treatment must include the theory of diffraction.
Australian Road Research, Vol. 7, No. 1, March 1977
Fig. 2 — Refraction and deviation of rays through a sphere of index N in a
medium of index n, for collimated light and N > n. The deviation gives rise to a
dark band (dotted region) at the edge of the sphere
27
SMITH - REFRACTIVE INDEX OF GLASS BEADS
Fig. 3 — Refraction and reflection of rays in a sphere which lead to the formation of the rainbow
For a given wavelength and hence a given refractive index n, Longhurst shows that 0 max is related to n by the
equations
° max = 4r — 2i
cos i
2
1
(1)
3
where r and i are as shown in Fig. 3.
The angle a subtended by the rainbow at the observer is
then given by
a
= 20 max
(2)
Eqns (1) and (2) were solved most conveniently by choosing
values of n and then finding the corresponding values of a.
The results are plotted in Fig. 4. Two aspects of these results
are important and worth further discussion.
(1) The mathematical solution for the angular diameter
a of the rainbow gives a range of a from 0° to 360°. The solutions in the range 180° to 360° require a special physical interpretation. Firstly, when a is less than 180°, as it is for the
meteorological rainbow, the bow is formed by rays travelling
in a backwards direction, hence it can only be viewed by an
observer looking away from the light source. However, when
a is greater than 180° the bow is formed by rays travelling in
a forward direction and hence now it can only be viewed by
an observer looking in the direction of the source. In this
case, physically, the bow has an angular diameter of (360 a) degrees.
(2) In eqn (1) when n is greater than 2, cos i is greater
than 1 and therefore there is no solution to the above equations. When n is equal to 2, it can be seen from Fig. 4. that
the rainbow has zero angular diameter. The physical interpretation of these results is that when the sphere refractive index is greater than or equal to 2, no rainbow is formed.
This is because for n greater or equal to 2, as P (Fig. 3) goes
from A to B, the angle 8 does not go initially positive as
28
before, but is immediately negative and progressively increases negatively without passing through any stationary
value. Taking the refractive index of water to be 1.333 (at
20°C, sodium light) the angular diameter of the meteorological rainbow is 84.16°.
Because the refractive index of glass and water increases with decrease in wavelength it can be seen on
recourse to Fig. 4 that for the (primary) rainbow, red must be
on the outside and purple on the inside. Higher order rainbows are also produced, due to multiple reflections inside
the sphere. These are of greater angular subtense and have
progressively weaker intensities than the primary rainbow.
One can easily distinguish the primary from the secondary,
apart from their relative intensities, by the fact that for the
secondary bow the colour order is reversed with red on the
inside and blue on the outside.
In applying the above theory to the measurement of the
refractive index of glass beads the assumption has to be
made that the beads are spheres. In practice, within a sample, a significant proportion may be non-spherical depending upon the type of manufacturing process and the degree
of subsequent sorting of spheres from non-spheres. Because
of this, and the fact that the efficiency of the beads as retroreflectors depends upon them being spheres or nearly so the
Australian Standard E42-1967 sets limits to the allowed
departure from sphericity of a sample as a whole. According
to the Standard, the degree of sphericity of the beads in the
sample is defined in terms of the percentage of 'rounds' in
the sample. The percentage of 'rounds' in turn is determined
by the proportion which roll down a given slope, the conditions being set out in the Standard. The Standard sets a
minimum of 70 per cent 'rounds' per sample.
Since refractive index varies with wavelength, precise
determination of index must be carried out for individual
wavelengths. However, the Australian Standard specifies a
source of natural light or that from a microscope illuminant.
The use of a broad spectrum band source is another factor
limiting the accuracy of the method laid down by the
Australian Standard. However, using a white light source the
'rainbow' method described here can be used to assess the
index at any wavelength; in particular it can be used to
measure the dispersion of the glass.
METHOD
The beads were placed one layer thick on matt black paper
at the centre of a circular graduated spectrometer table. A
collimated beam of light from a quartz iodine lamp was used
to illuminate the beads. The beam, initially horizontal as
shown in Fig. 5, was reflected down onto the beads by a 45
deg. mirror. By looking horizontally at the image of the beads
formed in the mirror, the 'rainbow' could be clearly seen.
The angular subtense (a) of any spectral colour of the
rainbow could be measured by aligning the crosshairs of the
telescope pivoted about the table centre, focused to infinity,
in turn, onto the diagonally opposite sides on the bow. The
angular subtense (a) of the bow was then the difference of
these two angular readings. In this study, the spectral colour
chosen to define the mean index was that at the
yellow/green boundary. The wavelength corresponding to
this colour was estimated to be 550 nm.
For the most common types of beads used in pavement
markings the refractive indices are in the range 1.5 to 1.6,
but usually closer to 1.5 with a minimum value set at 1.5 by
the Australian Standard. Once the 'rainbow' angular subtense (a) is known, the corresponding index n can be read
Australian Road Research, Vol. 7, No. 1, March 1977
REFRA C TIVE
SMITH - REFRACTIVE INDEX OF GLASS BEADS
90
180
270
360
ANGULAR DIAMETER OF 'RAIN BOW" (degrees)
Fig. 4 — Relation between the angular subtense (a) of the 'rainbow' and the refractive index (n) of the sphere. For water, n = 1.333 (at 20°C in Sodium light)
and a= 84.16°.
off a graph of the type shown in Fig. 4. To gain maximum
precision for graph reading various sections of the n versus
a graph were expanded and replotted as shown in Fig. 6.
RESULTS
SAMPLE 1
For the first sample of beads tested, ten separate readings of
a the angular subtense at the yellow/green boundary, were
made. The angle a was found to be 42.44 ± 0.15 deg. From
the graph in Fig. 6, the corresponding index was found to be
1.518 ± 0.001. To compare this value with the Becke line
method, six beads were drawn at random from the sample.
Using sodium light (wavelength = 589 nm), all the beads
had indices less than 1.518, three had indices greater and
three less than 1.514, and all had indices greater than 1.512.
The mean value is thus approximately 1.514. The difference
in the two means may be partly due to the small sample size
used for the Becke line method, but more likely is due to the
fact that the index at 550 nm ('rainbow') is greater than that
at 589 nm (Becke line). Nevertheless, average values for the
two methods differ by only 0.004.
Australian Road Research, Vol. 7, No. 1, March 1977
SAMPLE 2
As explained previously, the 'rainbow' method assumes the
beads are spherical. In practice, there is some departure
from sphericity in many beads in the samples depending to
a greater or a lesser extent from bead to bead. In order to
empirically examine the effects of non-sphericity on the
estimated index, different sub-samples of beads were obtained, each containing a different proportion of rounds as
defined by the Australian Standard E42-1967. Six sub-samples were used containing from 47.3 per cent to 90.4 per
cent rounds, as given in Table I. In all cases the rainbow was
easily seen and well defined. Five measurements of a were
obtained for each sub-sample, the means and standard
deviations are shown in Table 1. From the results it can be
seen that there is some scatter of standard deviations but
the mean indices are the same within ± 0.001 which is
the same magnitude as the standard deviation obtained
from the ten readings in Sample 1.
The dispersion of this sample using the 90.4 per cent
round sub-sample was determined by measuring the index
for the extreme edges of the 'rainbow'. Three measurements
were made of the angular subtense of the red and purple
edges of the 'rainbow'. The mean index at the red edge was
29
SMITH - REFRACTIVE INDEX OF GLASS BEADS
Mirror
0000000000
HORIZONTAL VIEW
VERTICAL VIEW
Telescope
Fig. 5 — Apparatus and layout for viewing and measuring the angular subtense
of the 'rainbow'.
1.512 and that for the purple edge was 1.535. Thus the
range of refractive index over the visible region for this sample was 0.023. Typical values for high quality optical glass
are 0.017 (Schott BK7) and 0.019 (Schott K3).
SAMPLE 3
Although the index of beads used in traffic markings is in the
region of 1.5, it was decided to apply the method to higher
index beads. A high index sample was obtained whose precise index was not known but thought to be about 1.9.
From the graph in Fig. 4 the predicted angular subtense
was about 3°. Such a small angle could not be measured by
the set-up used previously for the low index beads. This was
because the viewing telescope inclined at an angle of 1.5°
to the incident beam obscured the beam. To measure such
small angles, the sample and mirror assembly were set
further back on the table and a beam splitting mirror inserted
in the beam at the table centre. The sample and 'rainbow'
could then be viewed in the beam splitting mirror.
With this sample the full spectrum of the 'rainbow' could
not be seen. Only the red, orange and yellow colours could
be distinguished, the centre of the bow being white. In this
case the angular subtense of the red boundary of the bow
was measured. The average of three readings was 3.4 ± 0.1
degrees: the corresponding refractive index was found to be
1.890 ± 0.002.
The index for other colours or wavelengths must be
higher than this value. Dispersion generally increases with
30
index and a typical index range for high index, high quality
optical glass is 1.899 — 1.999 (Schott SF 58). If the beads
tested have had a similar dispersion this would explain the
non-appearance of the green and blue colours in the 'rainbow'.
TABLE I
per cent Rounds
a
Mean Index
47.3
52.6
61.5
73.9
81.2
90.4
42.32 ± 0.19
42.42 ± 0.12
42.58±0.07
42.50 ± 0.17
42.46±0.05
42.44±0.08
1.519
1.518
1.517
1.518
1.518
1.518
The angular diameters and corresponding mean indices of samples containing
different proportion of rounds. An error of 0.20 in angle corresponds to an error of 0.001
in the refractive index.
CONCLUSIONS
The 'rainbow' method of the measurement of refractive index of glass beads used in traffic marking gives results
which agree well with the Becke line method and has an accuracy of 0.001 assessed by the level of reproducibility of
successive readings.
Although the theory requires the beads to be spherical,
departures from sphericity well beyond the maximum
Australian Road Research, Vol. 7, No. 1, March 1977
SMITH - REFRACTIVE INDEX OF GLASS BEADS
allowed by the Australian Standard have no significant effect
on measured mean index.
The 'rainbow' method is much quicker to carry out than
the Becke line procedure and not only gives the mean index
but can also be used to find the dispersion of the glass.
On the negative side, the method cannot be used to
measure indices greater than or equal to 2.
1.54
1.53w
z
APPENDIX
1.51
1.50
38 39 40 41 42 413 414 45 46
ANGULAR DIAMETER oC
(degrees)
Fig. 6 — Relation between Le and n in the index range of the most convnon
traffic marking beads
REFERENCES
G. SMITH, B.Sc., Ph.D.
The major components of the equipment required for the
'rainbow' method are as follows: a light source, preferably a
quartz iodine lamp because of its high intensity and small
compact filament; a condenser lens; an opaque screen containing a small circular hole approximately 2 mm in
diameter; an achromatic collimating lens; a front surface mirror; a graduated circular table and a telescope. A divided circle spectrometer ideally combines the graduated table and
the telescope. However, it also usually contains a collimating system but this could be used in place of the achromatic
collimating lens above. The total cost of this equipment is
about the same as a medium priced microscope that would
be used for the Becke line method.
No special techniques are required in setting the equipment up apart from ensuring that the beam incident on the
beads is collimated and the telescope is focused on infinity.
The ideal method for collimation is the 'auto-collimation'
method described in various optical text books.
AUSTRALIAN STANDARD E42-1967. Glass Beads for Traffic Markings. Standards
Assoc. of Aust.
LONGHURST, R.S.(1966). Geometrical and Physical Optics. (Longman: London)
WAHLSTROM, E.E. (1969). Optical Crystallography. 4th Edn. (J. Wiley: New York.)
Dr George Smith graduated from the University
of Melbourne with a B.Sc. degree in 1964, majoring in Physics. In 1972 he took out a Ph.D.
degree from the University of Reading (U.K.) in
the field of Applied and Modern Optics. In
1974 he spent a year at the Australian Road
Research Centre working in the visual science
aspect of Human Factors. Since then he has
held a position as lecturer in Applied Optics at
the Department of Optometry, University of
Melbourne. His major fields of interest are applied optics and visual science, particularly as
applied to road research.
ACKNOWLEDGEMENTS
The author wishes to thank Mr G. Hind of Potters Industries who supplied the various samples of beads and
Dr A. Cundari of the Geology Department, University of Melbourne, for help and advice on the Becke line
tests.
Australian Road Research, Vol. 7, No. 1, March 1977
31