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Measuring variable refractive indices using digital photos

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2010 Phys. Educ. 45 83
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www.iop.org/journals/physed

Measuring variable refractive
indices using digital photos

S Lombardi, G Monroy, I Testa1 and E Sassi

Department of Physical Sciences, ‘Federico II’ University of Napoli,
Complesso Monte S Angelo, Via Cintia, 80126 Napoli, Italy

E-mail: [email protected]

Abstract
A new procedure for performing quantitative measurements in teaching
optics is presented. Application of the procedure to accurately measure the
rate of change of the variable refractive index of a water–denatured alcohol
mixture is described. The procedure can also be usefully exploited for
measuring the constant refractive index of distilled water.

Introduction as is the case in the Thomson-like apparatus,
or of a water wave in a ripple tank, or of
In a recent issue of Physics Education [1], a laser beam propagating in different media.
some demonstration experiments to illustrate The basic idea is to model the real trajectory
the physical concepts involved in mirages were by means of fundamental geometrical entities
suggested. One of these is a well-known (line, segment, angle) or figures (circumference,
demonstration [2] which shows the bending of a square) and then to measure distances, angular
laser light beam in an aquarium tank filled with amplitudes and figure parameters using the well-
water and a solution of sugar or salt. As in other known educational software Cabr`ı Ge´ometre´®2.
cases dealing with optics contents, the suggested An important aspect of this software is that point
demonstrations deal only qualitatively with the coordinates of the trajectories can also be obtained
gradient of refractive index of the solution; and exported in common spreadsheet software,
hence students do not perform quantitative allowing one to easily perform linear or parabolic
measurements, which could be valuable from the fits.
educational viewpoint.
With this procedure, accurate measurements
In this article, we briefly present a measure- (about ±0.05%) of the wavelength of straight
ment procedure aimed at including also quantita- and circular water waves in a ripple tank and of
tive aspects in demonstration experiments dealing electron charge to mass ratios have been obtained
with optics. (more details are presented in [3, 4]).

Basically, the procedure allows one to The procedure allows quantitative measure-
make quantitative and accurate measurements ments also in the case of optics. In the first in-
on digital pictures of phenomena in which a stance, it can be used in place of traditional labo-
visible trajectory, photographed via a digital ratory activities to measure constant refractive in-
camera, can be approximated by a geometrical
entity or figure. For instance, the trajectory 2 Cabr`ı Ge´ometre´® is available at www.cabri.com. Costs vary
can be that of an accelerated electron beam, according to the type of software and licence purchased. For
the purpose of this study, Cabr`ı II Plus has been used. For this
1 Author to whom any correspondence should be addressed. product, a school site licence costs about 700e.

0031-9120/10/010083+10$30.00 © 2010 IOP Publishing Ltd P H Y S I C S E D U C A T I O N 45 (1) 83

S Lombardi et al

Figure 1. A laser beam trajectory in a water–ethyl alcohol mixture. The laser beams enters in the tank from the
left. The points where the laser beam undergoes total internal reflection are circled in yellow and white.

dices (e.g. that of distilled water). In the following, medium a
we will use the procedure to perform quantitative φi na
measurements in situations where light propagates
in media characterized by a variable refractive
index.

Experimental setting separation
surface
A (150 × 800 × 50) mm3 tank was used to observe
the trajectory of the light beam in a mixture of medium b φr
water and denatured ethyl alcohol. The length of nb
the tank is crucial to observing a clear bending in
the trajectory. A 20 mW red (632.8 nm) DPSS Figure 2. Schema of Snell’s law in the case na > nb.
laser has been used.
y x
The experiments discussed here have been 0
performed using a mixture of about 200 ml of φ0 y0 n0
denatured ethyl alcohol in about 3 l of water. The
height of the water from the base of the water tank Figure 3. Schema of the experimental situation for
was about half of the height of the tank (∼7.5 cm). the measurement of the rate of change of the variable
refractive index of a water and ethyl alcohol mixture.
200 ml of denatured ethyl alcohol were slowly
added to the water using a dropper to allow the where ϕi and ϕr are the angles (called respectively
alcohol to gradually mix with the water. The time the angle of incidence and the angle of refraction)
needed for the quantities of water and of denatured formed respectively in media a and b by the
ethyl alcohol involved to become a homogeneous direction of propagation of light and the normal
compound is long enough (from 6 to 10 h) for to the (ideal) separation surface between them
performing the experiment safely and observing (figure 2).
the phenomenon for a reasonable period (about
1 h). A typical observed laser beam trajectory in When na > nb the refraction angle is greater
the mixture obtained is shown in figure 1, where than the incidence angle; if the incidence angle
the inversions of the trajectory are also indicated. is such that the angle of refraction becomes 90◦,
all the light will be reflected back in the medium
Theory a. The lowest incident angle for which there is
no more refraction (or, the largest incident angle
In terms of geometrical optics, when light for which refraction can still occur) is the critical
travels across two media, a and b, characterized
respectively by refraction indices na and nb, its
direction of propagation changes according to
Snell’s law:

na sin(ϕi) = nb sin(ϕr) (1)

84 P H Y S I C S E D U C A T I O N January 2010

Measuring constant and variable indices of refraction

y x

x1 x2 x3 xi xi +1 n0

y0 n1
y1
n2
y2
y3 n3
yi
yi + 1 ni
ym
ym + 1 ni + 1

xm total internal reflection
nm +1

angle. The existence of the critical angle is related fact that the light beam propagates from mixture
to the fact that, from Snell’s law, it is possible to layers of higher index to mixture layers of lower
find values of ϕi such that index of refraction until it undergoes total internal
reflection, resulting in a first inversion of the
na sin(ϕi) 1. (2) trajectory. The second inversion of the light beam
nb trajectory (the white circled area) corresponds to
another total internal reflection due to the fact that
The lowest value ϕi min that satisfies inequal- the light beam, after the first inversion, propagates
from mixture layers of lower index to mixture
ity (2) is layers of higher index of refraction until it hits the
separation surface between the mixture and the air
ϕi min = arcsin nb . (3) at an angle which is greater than the critical angle
na between the two substances (the critical angle
between ethyl alcohol and air is about 47◦, having
Equations (1) and (3) can be used to describe assumed the refractive index of ethyl alcohol to be
not only simple cases of light propagation in about 1.36).
materials, but also the rather more complex
propagation of the light beam in the water–ethyl To derive an analytic expression for the light
alcohol mixture when slowly added. The layer of path trajectory in figure 1, for didactical purposes,
alcohol ‘floats’ on the water and spreads slowly we will limit our attention to the propagation of the
towards the bottom of the tank, with the result laser beam in the x y plane (figure 3). Moreover,
that the mixture can be considered a medium with the simple case in which the refraction index n of
variable index of refraction for a time period of the mixture varies continuously only along the y
about 1 h. In this case, light does not propagate axis, decreasing with depth, will be considered.
rectilinearly but follows a curved trajectory, since
its direction of propagation changes continuously. For the derivation of a model of the
Moreover, since the denatured ethyl alcohol has a phenomenon studied (see, for instance, [5]),
greater index of refraction than water (about 1.36 we will assume that for depths |y| > |y0|
versus 1.33), the mixture layers on top have a the light beam travels across liquid layers each
greater index of refraction than the bottom ones. characterized by a constant index of refraction.
This produces the two minima visible in figure 1. Thus, in each of these layers, light propagates
The minimum circled in yellow corresponds to the

January 2010 PHYSICS EDUCATION 85

S Lombardi et al

in a straight line and due to multiple subsequent depth y and abscissa x; as a consequence, the
refractions, its direction changes when passing refraction index will be a function of the generic
from one layer to the following one (figure 4). depth y:

Consider two of these consecutive layers, the y 2 1
i th and the i + 1th, and let ϕi and ϕi+1 be the x sin2(ϕ0)(n0
angles of refraction in these layers, respectively. 1+ = )2 n2 ( y ). (12)
Due to the assumption that the light beam travels
in straight lines in each of the layers, ϕi is also the Considering the limit of the infinitesimal layers,
angle of incidence at which the light beam hits the equation (12) can be written in derivative form:
separation surface between the i th and the i + 1th
layers. Hence, Snell’s law for these two layers can dy 2 1
be written as dx (ϕ0)(n0
1+ = sin2 )2 n 2( y). (13)

ni sin(ϕi) = ni+1 sin(ϕi+1). (4) Differentiating equation (13) once with respect to
variable x we get

Let s1, s2, . . . , si , si+1, . . . be the dis- dy d dy = 1 d n2(y)
tances travelled in the subsequent layers of liquid. 2 dx dx sin2(ϕ0)(n0)2 dx
For the i + 1th layer, in particular, one has
dx

(14)

( si+1)2 = (xi+1 − xi )2 + (yi+1 − yi )2. (5) which can also be written as

Defining xi+1 ≡ (xi+1 − xi ) and yi+1 ≡ 2 dy d2 y = 1 dy dn2(y) .
(yi+1 − yi ), equation (5) can be written as dx dx 2 sin2(ϕ0)(n0)2 dx dy

2 2 (15)

si +1 =1+ yi +1 . (6) Simplifying and rearranging both sides of (15),

x i +1 x i +1 we finally obtain a differential equation for the

Recalling that trajectory of the light beam in the water–ethyl

alcohol:

xi+1 = si+1 sin ϕi+1, (7) d2 y 1 dn2(y) .
dx 2 2 sin2(ϕ0)(n0)2 dy
= (16)

equation (6) becomes

1 2 yi+1 2 (8) As expected, if n is constant w.r.t. y, the trajectory
x i +1 of the light beam is linear in y. In all other cases,
sin ϕi+1 =1+ to solve equation (16), a form for n2(y) must be
hypothesized. In general, n would depend on the
which, taking into account (4), can be written as concentration of the water–ethyl alcohol mixture;
however, in our case, such a concentration varies
yi +1 2 1 ni+1 2 with depth since the experiment is performed
x i +1 sin2 (ϕi) during the diffusion of the ethyl alcohol in
1+ = ni . (9) the water. Since this diffusion is very slow,
the concentration also varies very slowly with
Since light travels across layers of constant depth; hence it can be assumed that there is
refractive index the following equalities hold: a linear dependence between the concentration
and the depth. More problematic is finding a
n0 sin ϕ0 = n1 sin ϕ1 = n2 sin ϕ2 = · · · (10) unique relationship between the refractive index
= ni sin ϕi = · · · and the concentration of the ethyl alcohol–
water mixture. Usually, such a relationship is
as it is possible to easily verify from figure 3. empirically determined. The values reported in
Taking into account (10), equation (9) can be the literature (see, for instance, [6]) mostly suggest
written as a quadratic dependence of n on the concentration
of the ethyl alcohol. However, due to the small
yi +1 2 1 difference between the refractive indices of ethyl
x i +1 sin2(ϕ0)(n0
1+ = )2 (ni+1 )2. (11)

The subscript i + 1 can now be omitted in
equation (11), which can be written for a generic

86 P H Y S I C S E D U C A T I O N January 2010

Measuring constant and variable indices of refraction

alcohol and water, such quadratic relationships Removing subscript i + 1 and taking the limit for
can be inferred from the experimental data also infinitesimal layers, one finally has
for n2. Therefore, since the propagation of the
light beam occurs in a region of the mixture dy = cot arcsin n0 sin ϕ0 . (24)
where the percentage concentration of alcohol dx n
reasonably assumes values very near to the lower
extreme of the interval [0, 100] (with the quantities For x → 0, y → y0 and n → n0 for continuity
used in this experiment the final concentration reasons. Hence the condition to be imposed on the
of ethyl alcohol in the water would be 7%), the first derivative of y can be written as follows:
relationship between n2 and the concentration can
be well approximated as a linear one. Such an dy = limn→n0 cot arcsin n0 sin ϕ0
assumption will be later verified by investigating n
the agreement with experimental data. For the dx x=0
particular case under study, assume
= cot ϕ0. (25)

Integrating (19) twice, with conditions (20)
and (25), one obtains

n2(y) = h − k|y| (17) y = − 4n20 k x2 + cot ϕ0x + y0. (26)
sin2
where h and k are positive constants3 and ϕ0

n2(y0) ≡ n02. (18) To verify the linear dependence of n2 on the depth
Substituting (17) in (16) one has
and hence determine the unknown parameters

k, n0, y0, φ0 in (26) a parabolic fit procedure [7]

applied to the digital image obtained in the

dy2 = k (19) experiment can be performed. The complete
dx 2 − 2n20 sin2 ϕ0 .
procedure in Cabr`ı is described in the following

section.

This second-order differential equation can be Using conditions (17) and (18) it is possible
solved by imposing conditions on the first
derivative of y and on y at a given abscissa. We to obtain an expression for the percentage gradient
seek solutions such that for x = 0 y = y0. Hence
we can impose that n2 (y) of the square of the index of refraction of

n 2
0
the mixture between the value at the ordinate y0

and that at the ordinate y in terms of measurable

y|x=0 = y0. (20) parameters. Defining

To derive a condition for the first derivative of y, n2(y) ≡ n2(y) − n 2 (27)
let us return to the finite liquid layers in figure 4. 0
Note that for the i + 1th layer it holds that
and taking into account (17) one obtains

n2 = (h − k|y|) − (h − k|y0|)
n02 (y)
yi+1 = xi+1 cot(ϕi+1) (21) n 2
0

which from Snell’s law can be written as k (28)
= − n20 (|y| − |y0|).
ni
yi+1 = xi+1 cot arcsin ni+1 sin ϕi . For any given |y| > |y0|, i.e. for increasing values
(22)
of the negative ordinate, n2 ( y ) is a negative

n 2
0
quantity according to the fact that, in our case,
Recalling equalities (10), equation (22) becomes

n0 the index of refraction decreases with depth. The
ni+1
yi+1 = xi+1 cot arcsin sin ϕ0 . positive ratio k , expressed as [length]−1, is the
(23)
n 2
0
rate of decrease of the square of the index of

refraction with depth in the water–ethyl alcohol

3 The parameter h represents the value of n2 when y = 0 and mixture.
can be determined according to the physical situation studied.
Given the chosen form of n2, this parameter is inessential for To obtain a more compact and straight form

the determination of the trajectory of the light path. For the of equation (28), the ratio n2 at y = ymin,

meaning of the parameter k, see later in the article. n 2
0
where ymin is the depth at which the trajectory

January 2010 PHYSICS EDUCATION 87

S Lombardi et al

Figure 5. Digital photo of the red (632.8 nm) laser beam trajectory in a mixture of 3 l of water and 210 ml of
denatured alcohol. The laser beam enters the tank at the left.

reaches a minimum, can be calculated. In the Table 1. Values of the parabolic best-fit parameters for
proposed mathematical model, such a depth can be the trajectory of the laser beam propagating in the
determined from (26), setting the first derivative water–denatured alcohol mixture (figure 5).
of y with respect to x equal to zero. After few
calculations, one obtains a (c.u.−1) = 0.0052 ± 0.0003
b = −0.075 ± 0.004
n20 c (c.u.) = −0.62 ± 0.01
k
| ymin | = |y0| + (cos ϕ0)2 . (29)

Results

But, from (28), one has The couples (xi , yi ) have been fitted using a
parabola of the form y = ax 2 + bx + c. In
n2 k (30)
n20 y=ymin = − n02 (|ymin| − |y0|) figure 8, the plots of the experimental data (with

and hence, taking into account (29), we obtain their uncertainties) and of the best fit curve are

shown.

n2 = −(cos ϕ0)2, (31) The parameters of the best fit are shown
n02
ymin in table 1.

i.e., the percentage gradient of the mixture square The best fit curve approximates the experi-

refraction index at the maximum depth reached mental data well. This evidence gives empirical

(negative) depends only on the incident angle of support to the initial hypothesis of the linear de-

the light beam. pendence of n2 on y. The measurement procedure

also provides good accuracy in the determination

Measurements of the parabolic fit parameters. The physical pa-

Initially, a digital image of the laser beam rameters k, n0, y0, φ0 in (26) can be determined

trajectory obtained from the experiment (figure 5) from these parameters, according the following re-
has to be imported into the Cabr`ı environment.
An x–y reference system for calculating the lationships:
coordinates of any point in the x–y plane is
k = − 1 4a (32)
required in Cabr`ı. In such a reference system n20 + b2
the x-axis is parallel to the upper front edge
of the mixture surface and the y-axis is parallel ϕ0 = arctan 1 (33)
to the short edge of the water tank (figure 6)4. b

Then, the laser beam trajectory is pinpointed using y0 = c. (34)
several points (figure 7). With the couples (xi , yi ),
expressed in c.u.5, statistical calculations can be Their estimated values are reported in table 2,
performed. together with the estimates of the percentage

4 In this way, the same reference system in which the depends on the resolution of the screen; as a consequence, to
theoretical calculations have been developed is reproduced in get an estimation of real distances, a reference object has to be
Cabr`ı. included in the photo. In the experiment described, the length
5 We indicate as c.u. the unit used by the Cabr`ı software of the water tank has been adopted as the reference unit. The
when providing to the user results involving distances. ‘c.u.’ size of a pixel in Cabr`ı units (0.03 c.u.) has been adopted as the
uncertainty on the measured coordinates.

88 P H Y S I C S E D U C A T I O N January 2010

Measuring constant and variable indices of refraction

Figure 6. Reference system used to measure the laser beam trajectory in figure 5.

Figure 7. Tracing of the laser beam trajectory in figure 5 in Cabr`ı.

gradient of the mixture square refraction index at Table 2. Values of the physical parameters describing
the propagation of the laser beam in the
the maximum depth n|n22 ymin (equation (31)), of water–denatured alcohol mixture (figure 5).
0
the maximum depth reached by the light beam

ymin (cm) (equation (29)) and of the difference y0 (cm) = −2.13 ± 0.04
between the initial and final depths |ymin| − |y0|.
ϕ0(deg) = −85.7 ± 0.2

The results show that the square of the index k (cm−1) = 0.0061 ± 0.0003
n20
|n2
of refraction decreases at a rate of about 0.6% = −0.0056 ± 0.0006
n02 ymin
ymin (cm) = −3.1 ± 0.1
for every centimetre of depth in the mixture. At
|ymin| − |y0| (cm) = 0.9 ± 0.1
the minimum of the trajectory, the gradient of the

square of the index of refraction of the medium

with respect to the value at the entry point of the

laser beam in the tank is 0.56%. different digital photo (figure 9) of the same light
beam (figure 5). The measurements carried out
To prove the reliability of the procedure, in Cabr`ı are shown in figure 10. The plot of this

the measurements have been repeated on a

January 2010 PHYSICS EDUCATION 89

c.u.S Lombardi et al

c.u.
0 2 4 6 8 10 12 14 16
0

–0.1
–0.2
–0.3
–0.4 y = 0.0052x2 – 0.075x – 0.62
–0.5 R2 = 0.988
–0.6
–0.7
–0.8
–0.9
–1.0

Figure 8. Data obtained from the tracing of the laser beam trajectory shown in figure 5. The full curve is a
parabolic fit to the experimental data.

Figure 9. Second digital photo of the red (632.8 nm) laser beam trajectory in a mixture of 3 l of water and 210 ml
of denatured alcohol. The laser beam enters the tank at the left.

Figure 10. Tracing of the red (632.8 nm) laser beam trajectory of figure 9 in Cabr`ı.

second experimental data set obtained in Cabr`ı is Conclusions
shown in figure 11.
The procedure illustrated in this article may
The values of best fit curve and those of the plausibly help teachers in performing quantitative
physical parameters of the trajectory are reported measurements and in introducing modelling
in tables 3 and 4. of fairly complex phenomena related to light
propagation via a simple software tool suitable for
Since the results obtained in the two data sets 16–18 year old students.
are comparable, the measurement technique can be
considered reliable.

90 P H Y S I C S E D U C A T I O N January 2010

Measuring constant and variable indices of refraction

c.u.c.u.
0 2 4 6 8 10 12 14 16
0

–0.1
–0.2

–0.3

–0.4 y = 0.0051x2 – 0.073x – 0.63
–0.5 R2 = 0.9873
–0.6

–0.7
–0.8

–0.9
–1.0

Figure 11. Plot of the data set obtained from the tracing of the laser beam trajectory in figure 9. The full curve
is a parabolic fit to the experimental data.

Table 3. Values of the parabolic best fit parameters for treating a 3D phenomenon in a 2D approximation.
the trajectory of the laser beam propagating in the In this case, many software packages (Photoshop,
water–denatured alcohol mixture (figure 9). Photopaint) are nowadays available for correcting
geometric distortions in digital photos. However,
a (c.u.−1) = 0.0051 ± 0.0003 software filters may not be enough. In these cases,
b = −0.073 ± 0.004 a qualitative treatment may be more effective; for
c (c.u.) = −0.63 ± 0.01 instance, in the case of measurements involving
point coordinates, the associated uncertainties can
Table 4. Values of the physical parameters describing be suitably modified so as to also take into account
the propagation of the laser beam in the distortion effects.
water–denatured alcohol mixture (figure 9).
In conclusion, the above results show that
y0 (cm) = −2.18 ± 0.05 the accuracy of the proposed procedure can be
considered as quite satisfactory for a secondary
ϕ0 (deg) = −85.8 ± 0.2 school laboratory.

nkn02n202(|cymmin−1=) = 0.0058 ± 0.0003 Acknowledgments
−0.0053 ± 0.0006
Special thanks go to Luca Murrone and Maria
ymin (cm) = −3.1 ± 0.1 Rosaria Colombo for help with carrying out the
proposed experiments.
|ymin| − |y0| (cm) = 0.9 ± 0.1
Received 7 July 2009, in final form 2 November 2009
Some limitations of the procedure have to be doi:10.1088/0031-9120/45/1/010
pointed out here. Firstly, the need for a good
quality digital photograph of the phenomenon References
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later on, if a good digital camera is not available, nature and in the laboratory Phys. Educ.
the measurements on digital photos of the same 44 165–74
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geometric distortions necessarily introduced when [2] Cornwall M G 1992 Light travels in straight
lines?—a physical simulation of light
6 Digital photos of laser beams propagating in substances
with constant and variable refractive indices are available on
demand.

January 2010 PHYSICS EDUCATION 91

S Lombardi et al

propagation in a graded index optical fibre Phys. learning at different levels and the use of models in physics
Educ. 27 273–9 education.
[3] Monroy G, Testa I and Lombardi S 2006 Teaching
wave physics through modeling images. Use of Gabriella Monroy is associate professor at the University of
Cabr´ı® to address water wave geometrical Naples ‘Federico II’, Italy. In collaboration with the Italian
models and basic laws Modelling in Physics and Ministry of Education, she has developed teaching materials
Physics Education (Proc. GREP vol 20) ed for secondary education. Her research interests are issues in
E van den Berg, T Ellermeijer and teacher training as pedagogical content knowledge, teacher
O Slooten (Amsterdam: University of competences and the impact of teaching innovation.
Amsterdam) pp 299–307
[4] 2007 Esperimenti didattici e immagini: misure Italo Testa has a physics degree from the University of
quantitative con Cabr`ı Ge´ometre` (in Italian) G. Naples, and a PhD in physics education from the University of
Fis. 48 151–69 Udine. He is now full-time researcher at the Department of
[5] Rossi B 1957 Optics (Reading, MA: Physical Sciences at the University of Naples. His main
Addison-Wesley) research interests focus on teachers’ and students’
[6] Yeh Y-L and Lin Y-P 2008 Opt. Commun. understanding and use of models in physics, and on the
281 744–9 development of innovative teaching strategies in secondary
[7] Taylor J R 1982 An Introduction to Error Analysis school education.
(Mill Valley, CA: University Science Book)
Elena Sassi is full professor of physics and physics education
Sara Lombardi teaches in secondary schools. She holds a at the Faculty of Science, University of Naples ‘Federico II’,
maths degree from the University of Naples ‘Federico II’, Italy. She is a member of the European Physical Society and
Italy. Her research interests include the design and study of the International Commission for Physics Education. She has
ICT-based environments for improving physics teaching and also coordinated a teacher education program in Uganda. Her
main research interests are lab work and ICT-based modelling
in physics teaching.

92 P H Y S I C S E D U C A T I O N January 2010