Understanding Light

Light exhibits many properties, one of which is that it can behave either as a particle or a wave. Furthermore, light manifests even more properties when one considers it either as a particle or a wave. It is because of this that scientists were baffled and argued among themselves whether to consider light exclusively as a particle or a wave.

In this experiment, we consider light to be wave as we investigate two of its more known properties, Reflection and refraction. As light also exhibits diffraction and interference as a particle, that will be tackled in separate blog.

Reflection, to put it simply, is the bouncing off of light on a surface or object. When you shine light onto any object, light will inevitably bounce off on it. This is pretty intuitive as we see this phenomena everyday when our eyes our glared by light due to it reflecting off of a surface going in our direction.

Refraction, on the other hand, is the bending of light through a surface or object. Contrary to reflection, light refracts only when it goes through different media (or objects). This is because light slows down when it travels through different media.

Figure 1: Probable Trajectory of the President
through the Room

The bending and slowing down of light can be explained using a scenario. Let's say that the president of a company has an important meeting, however to go to this meeting, he/she has to go through a room where the company's staff members are having a party. As refined as he is, the president walks with a constant velocity. Before entering the room, the president follows a straight path walking at normal pace. Upon entering however, the his/her path is immediately blocked by the staff members and he/she is forced to slow down and go in different directions before finally going in his normal pacing and path upon exiting the room. Figure 1 shows this visually, showing the probable path the president would take upon entering the room.

In this scenario, the room is the medium,  the staff members are its molecules, and the president the light particle. Note that the president had to slow down and change path due to staff members in the room. Similar to the him/her, light particles slow down and 'bend' because of its interactions between the media's particles.

Figure 2: Illustration of Reflection and Refraction

While Reflection and Refraction are both phenomena we encounter everyday there are laws upon which we can measure the extent to which they occur. Such laws are the Law of Reflection and Snell's Law.

Using Figure 2, we can illustrate these laws and explain what they and embody.

The Law of Reflection simply states that if you have an incident ray of light travelling towards a flat, smooth surface, as in the figure, the angle that ray of light forms with the normal of the surface (incident angle) will be equal to the angle the ray of light forms with normal when it bounces off the surface (reflected angle). That is, given that $\theta_\text{1}$ is the angle of incidence and $\theta_\text{1}^\prime$ the reflected angle,
\begin{equation}
\theta_1=\theta_1^\prime
\end{equation}
 In the figure, the normal line is the dashed line going through Medium 2. One should note that the normal line of a surface is the line perpendicular to the plane of that surface.

Recall that when light passes through some media, it slows down. Say that Medium 1 of Figure 2 is air and Medium 2 is some other material. When light passes through Medium 2, it slows down and continues to travel at some speed $v$. We can take the ratio of the speed of light in a vacuum and its speed in that medium, and define it to be media's the index of refraction $n$. Taking $c$ to be the speed of light in a vacuum with a value of $3\times10^8$ m/s, then we can define $n$ to be:
\begin{equation}
n\equiv\dfrac{c}{v}
\end{equation}

With the concept of index of refraction, we can finally express Snell's Law, which states that the ratio of the indices of refraction of two media is equal to the ratio of the sines of the refracted angle and incident angle. Using the given variables in Figure 2, this is expressed as:
\begin{eqnarray}
\nonumber \dfrac{n_1}{n_2}&=&\dfrac{\sin\theta_2}{\sin\theta_1} \\
n_1\,\sin\theta_1&=&n_2\,\sin\theta_2
\end{eqnarray}

Figure 3: Illustration for Critical Angle
(source: Physics for Scientists and Engineers)

Note that if $n_2>n_1$, as in the figure, then light travels faster in Medium 1 and the ray of light tends towards the normal ($\theta_1>\theta_2$), and vice versa. Furthermore, since $v<c$ for all cases light travels through different media, then all indices of refraction $n>1$.

An interesting phenomena occurs when light passes through an object of greater index of refraction to one with lesser index. Such effect is known as total internal reflection. Recall that light is refracted away from the normal when going from high to low index of refraction. Given that, there exists a critical angle $\theta_c$ such that the refracted ray moves parallel to the plane of the surface ($\theta_2=90^\circ$ as in the figure). Using Snell's Law to calculate for this critical angle:
\begin{eqnarray}
\nonumber n_1\,\sin\theta_c&=&n_2\,\sin(90^\circ) \\
\sin\theta_c&=&\dfrac{n_2}{n_1}
\end{eqnarray}
One should remark that the critical angle only exists when $n_2>n_1$.

The experiment essentially tackles all of the discussed concepts above, and tries to verify it given experimental data.

Some sort of 'photonics' kit was used in the experiment. Firstly, the optics were lined up. An optic disk was placed next to the light source, with a slit plate in between the two. Multiple parallel rays were made with the central one coinciding the $0^\circ$-$0^\circ$ mark on the disk. Then a slit mask was put between the disk and slit plate to focus the central ray of light. This was the main set up for most of the experiment.

Figure 4: Placement of Materials on the Optical Bench

For the first part of the experiment, the Law of Reflection was verified. A plane, convex, and concave mirrors were put on the optical disk such that the mirrors coincided with the plate's $90^\circ$-$90^\circ$ mark. Then the disk was rotated such that incident ray strikes the mirror's center at an angle, and the resulting angle of reflection was recorded.

Table 1: Resulting Angles of Reflection for All Mirrors

Angle of Incidence ($^\circ$)	Angle of Reflection ($^\circ$)
	Plane Mirror	Convex Mirror	Concave Mirror
$30$	$30$	$30$	$30$
$45$	$45$	$45$	$45$
$60$	$60$	$60$	$60$

As expected, the angle of incidence will be equal to the angle of reflection, hence verifying the Law of Reflection to be true. Interestingly I have only expected this to be true for a plane mirror, and thought that it would work otherwise for the convex and concave mirrors due to it being curved. Anyway, one observation made during this part of the experiment was that when the convex and concave mirrors were used, light rays behaved differently. Light rays seemed to become wider when reflected off of convex mirrors and narrower for concave mirrors, as shown below in Figure 5.

Figure 5: 'Widening' of Light Rays due to Reflection on a Convex Mirror

Allowing three rays of light to pass through and letting the angle of incidence of the middle ray to be $0^\circ$ one can see that light reflects off of the mirrors in a different manner. The plane mirror simply reflects the light rays back, the concave mirror makes light rays converge to some point and convex mirrors makes light rays diverge.

Figure 6: Behavior of Light when Reflecting off of a Convex Mirror

The following diagram simplifies the behavior of the light rays as it bounces off of the mirrors.

Figure 7: Ray Tracing for (a) Plane, (b) Concave, and (c) Convex Mirrors

In the next part of the experiment, Snell's Law was verified. A semicircular lens was placed on the disk instead of the mirror. First with the flat side facing the incident ray. Then the disk was rotated at different angles to determine the resulting angle of refraction. Then, we linearize using Snell's Law in equation (3) with $\theta_1$ and $n_1$ referring to air, and $\theta_2$ and $n_2$ referring to the lens.

Manipulating equation (3) such that $\sin\theta_2$ is on one side:

\begin{equation}
\sin\theta_2=\dfrac{n_1}{n_2}\,\sin\theta_1
\end{equation}

The first media light traveled in is air, which has an index of refraction $n_1=1.00$. Hence the index of refraction of the glass $n_2$ is the reciprocal of the slope of the plot of $\sin\theta_2$ vs. $\sin\theta_1$.

Figure 8: The Linearized Plot for when the Flat Side of the Lens is Facing the Incident Ray.
$\sin\theta_2$ is $y$, and $\sin\theta_1$ is $x$. From the graph one sees that the reciprocal
of the index of refraction of the glass is $0.6772$ with a root mean square value of $1$. Furthermore
implying that the index of refraction of the glass $n_2=1.4767$. This shows that Snell's Law holds
true for this case.

Hence for the case that the incident ray hits the flat side of the semicircular lens, $n=1.4767$. Since $1.4767>1$, then one can say that when light passes through the flat side of the semicircular lens, then the refracted ray would tend toward the normal.

In the other part of the experiment, this time, the curved surface is facing the incident ray and the same process repeats. However, by the last angle ($50^\circ$) an amazing thing happened -- the refracted ray of light disappeared! Interestingly, instead of getting five data points, only four were gathered here. This means that the critical angle has been reached. Looking for that critical angle, and applying the usual linearization as the previous part, one gets the following plot, with the reciprocal of the slope $n=0.6738$. A value not greater than one.

Figure 9: The Linearized Plot for when the Curved Side of the Lens is Facing the Incident Ray.
From here, one sees that the reciprocal of the index of refraction of the glass is
$n=0.6738$, a value not greater than 1. Hence this does not follow Snell's Law.

Furthermore, when looking at the rays of light as it goes through the lens, one sees that compared to when the flat side of the lens was facing the incident ray, the refracted ray from this case tends away from the normal.

Figure 10: Ray Diagrams for the Behavior of Light when it passes through
the (a) Flat and (b) Curved Side of the Lens. One notes that the Refracted Ray
for (a) tends towards the normal and for (b) away from the normal.

Upon thinking hard, an idea lit up. When the incident ray hits the flat side of the lens, the ray makes an angle with respect to it, and that gets refracted. When it hits the curved side of the lens however, one can imagine it hitting the normal of the curved side, because a circle can be described as a polygon of infinite number of sides. So any line directed to it from the center, must be perpendicular to the tangent at that point. Hence instead of considering air as the first media and lens as second (as in the previous part), in this case, we consider the glass to be the first media, and air as the second. Which makes sense because in the previous part we said that the index of refraction of the glass is $1.4767$, which is greater than $1$. Hence when light goes to air from the glass, it tends away from the normal.

We plot the data similarly as before, with $\sin\theta_2$ as the $x$ values and $\sin\theta_1$ as the $y$ values, giving us:

Figure 11: The Corrected Linearized Plot for when the Curved Side of the Lens is
Facing the Incident Ray. $\theta_2$ is $y$ and $\sin\theta_1$ is $x$. From the graph,
one sees that the reciprocal of the index of refraction of the glass is $0.6738$ with a
root mean square value of $1$. Furthermore implying that the index of refraction of the
glass $n_2=1.4841$. This shows that Snell's Law holds true for this case.

Hence giving us an index of refraction of $n_2=1.4841$, relatively close to the $1.4767$ obtained in the previous part.

Given that, one can finally gather the general properties of the glass lens as follows:

Table 2: General Properties of the Semicircular Glass Lens

Critical Angle $\theta_c$	$42^\circ$
Index of Refraction of Glass $n$	$1.476$
Speed of Light Inside the Semicircular Glass	$2.032\times10^8$ m/s

The critical angle was measured by turning the disk to some point that the refracted ray disappears. The index of refraction is the average of the indices obtained in the experiment. Note that the index of refraction is defined as the ratio of the speed of light in the vacuum and that of light in a medium. So to calculate for the speed of light in some medium $v$:
\begin{equation}
v=c\,n
\end{equation}
In the last part of the experiment, light was shown through different refracting media, similar to what's shown below.

Figure 12: One of the Refracting Media Used in the Experiment
Light Rays seem to reflect in the media itself before going out of it.

There were five of them, each of which ray tracing was done, and the following figure summarize the path in which light took when passing through each of them.

Figure 13: Ray Tracing in Different Media as (a) (b) Right Triangle Lens
(c) Trapezoidal Lens, (d) Double Convex Lens, and (e) Double Concave Lens

Note that in some of them, such as figure 13-a and figure 13-b, light bounces off of the inside of the lens before going out of it. This is light exhibiting total internal reflection, some great stuff.

Generally, I had lots of fun during the experiment. Especially after the experiment proper as me and my group mates played around with the lenses. Such as this:

Figure 14: Fun with Optics!

Hopefully other experiments will be like this. Cheers!