Experiment 3: Interference

The wave-particle duality of light is one of nature’s most baffling realities – and the confusion is rightly so. The true nature of light had, in fact, perplexed scientists for decades before the strange concept of wave-particle duality came to light. The likes of Isaac Newton vehemently backed the idea that light was a particle, yet on the other corner, existing and forthcoming theory from prominent names such as Christopher Huygens, seemed to indicate that light was wave. It wasn’t until the year 1803, that English physicist Thomas Young finally confirmed the apparent behaviour of light as a wave, in his famous yet surprisingly elegant demonstration, the double slit experiment. Yet the solidification of the wave aspect of light did not mean the end of its understanding as a particle. As Albert Einstein eloquently put it, "It seems as though we must use sometimes the one theory and sometimes the other, while at times we may use either. We are faced with a new kind of difficulty. We have two contradictory pictures of reality; separately neither of them fully explains the phenomena of light, but together they do". It then became apparent that these two seemingly irreconcilable ideas, when put together, revealed the true nature of light.

Here’s a thought experiment. Imagine aiming a loaded shotgun at two adjacent walls, standing a few feet apart from each other. Aiming straight at a slit in the first wall, fire a stream of bullets using a shotgun (these guns spread their bullets out). Looking at the other side of the wall, what kind of impact mark would these bullets leave on the second wall? Well, the answer is quite intuitive. The bullets that made it through would obviously leave a line of tiny craters on the wall’s surface. Next, cut a second slit into the wall, just beside the first one. Firing the shotgun once again, what would you expect to happen? Well, a duplicate of the first turnout should happen. Now you have two lines of bullet marks on the second wall. Now let’s look at how waves would interact with this setup. This time, imagine the two walls half submerged in a pool of water. Let’s take it a step further and say the back wall lights up where it is hit with the most intensity. A single ball bobbing up and down in the water will serve as a wave source. As these waves travel through the single slit and hit the back wall, the area of brightness left on the wall would be at a maximum directly in front of the slit, and decrease as it goes further to the sides. Repeating the process with the double slit setup, what pattern would you expect to emerge on the back wall? Intuitively, it should be once again a duplicate of the single slit setup, which would be two bands of light side-by-side with decreasing intensity from the center. However, intuition can only get us so far. What actually happens is this. 

Figure 1: Double slit diffraction

Suddenly, it is not just two bands that appear. As you can see, the ratio of slit to band is not one-to-one. So if that’s not what’s going on, then what is? In order to get a grasp of what is happening, we must first understand the concept of interference.

Interference is one of the things that make waves and particles fundamentally different in behaviour. When two particles collide, you could get a change in direction, a transfer of momentum, or even obliteration. However, when two wave pulses with the same amplitude meet, depending on their amplitude’s signs (this can be interpreted as facing up or down), two possible things could happen. In the case that the two wave pulses exhibit the same sign in amplitude, what would happen is the two waves would add to each other to form a single wave of with an amplitude equal to the sum of the two waves’ initial amplitudes. This is called constructive interference. On the other hand, if two wave pulses with opposite signs in amplitude meet, they would simply cancel each other out, leaving their location of collision flat and undisturbed. This is called destructive interference. It is these two concepts that explain the behaviour of a wave when passed through two adjacent slits. As the initial wave hits the slits, it splits into two separate source points, generating two different waves beyond the first wall. These two waves then interact with one another as they radiate forward. The alternating light and dark bands on the wall are representative of the parts of the wave where instructive or destructive interference occurred respectively.  The resulting array is called a diffraction pattern.

Now let’s take that shotgun we were talking about a while back, and shrink it down to a quantum level, where instead of having bullets for ammunition, we fire photons. Starting with the single slit setup, we fire photons into the wall. What results is a splay of photon marks plastered on the second wall in the shape of a slit. Typical particle behaviour. Moving on to the double slit setup, we fire the photon gun once more. But this time, instead of seeing two slit-like patterns on the second wall, we observe a diffraction pattern splayed across the wall. Could it be the photons colliding with each other to cause more than two bands to appear on the wall? We then fire photons one at a time to remove the possibility of the photons colliding with each other. Surely a particle can’t collide with itself. Upon firing, however, we see the same result – another diffraction pattern. This is in fact, very similar in essence to Thomas Young’s double-slit experiment. And it was this experiment that led to the inescapable conclusion that light is indeed simultaneously both a particle and a wave.

The experiment we conducted (we’re talking about the experiment done in this very study), however, takes the slit diffraction concept to a bit of a more complex level. All the previous examples worked on the assumption that the slits were just wide enough to fit a single particle. In reality, mechanical constraints obviously prevent us from creating slits of such infinitesimal widths; so what we end up working with are slits of considerably small widths, but at the same time are very large when compared to the photons we shoot through the slits. According to Huygen’s principle, each point on a wave front can be thought of as point source of an individual wave. So in essence, what it means is that a single wave front is in fact an infinite number of wave fronts that, when combined, result in a single and coherent wave. It may be hard to grasp, but it can be understood better when we recognize that all of these individual waves interfere with one another. The reason why we see only a single wave front is that everything else has destructively interfered, and what remains is the only result of constructive interference. When a laser beam is fired through an actual and realistic slit, the light that makes it through the slit diffracts into its individual source-points, causing an enormous number of waves to occur and interfere simultaneously, in turn causing a diffraction pattern to appear on the second wall. And in fact, this is how the first part of the experiment went.
A monochromatic red laser was fired through a single slit, producing a diffraction pattern on the wall. Although pictures were not able to be taken, here is a graphical representation of what turned out.

Figure 2: Intensity plot of diffraction pattern as a result of monochromatic light passing through a slit

A series of fringes were splayed across the wall, with the brightest and widest of them all, called the central maxima, at the center. The goal of this experiment was to determine the width of the slit, using the other measureable factors of the experiment. Namely, these other measureable factors were, the distance L of the pattern from the slit, the wavelength of the light source λ, and the angle θ indicated on the figure. By setting y=0 to be the center of the central maximum (the brightest intensity maximum), the locations of the intensity minimums (dark fringes) could also be measured. To solve for the width of the slit a, a simple and elegant equation was used

Before calculating for the slit width, the wavelength of the light source was first calculated for. After manually measuring all variables (using rulers and meter sticks) except and using the given specs of the stup, the measured quantities were tabulated and the wavelength was calculated for in two different situations. The first using a slit with a known width of 0.02 mm, and the second using a slit with a given width of 0.04 mm.

Table 1: Wavelength of the Laser diode

	m = 1, slit width = 0.02mm	m = 1, slit width = 0.04mm
distance between side orders	10.3 cm	2.7 cm
center to side	5.15 cm	1.35 cm
Calculated wavelength	624 nm	327 nm
% difference	4%	49.7%
slit to screen distance	1.65 m	1.65m
Average wavelength	475.5 nm

The first calculated wavelength turned out pretty well, with just a 4% deviation from its known value. The second setup, however, deviated by 49.7%. Next, the slit width was calculated for using the given wavelength of 650 nm.

Table 2: Data and results for the 0.04 mm single slit

	m=1	m=2
Distance between side orders	2.7 cm	5 cm
Distance from center to side	1.35 cm	2.5 cm
Calculated slit width	0.08 mm	0.08 mm
Percent difference	114.5%	114.5%

The calculated slit difference turned out to be consistent for both values of m. On a side note, the precision of the results makes me doubt the integrity of our measurements, and this is furthered by the fact that if we use the distance between side orders instead of the distance from center to side in our calculations, we end up with a calculated slit width if 0.04 mm. Curious. However, with what we have right now, 114.5% is the percent devation of our results.
The following portion of the experiment, this time, dealt with a double-slit setup. The result was akin to a combination of what went on during the single slit setup. However, with two sources of light, one from each slit, a second source of diffraction occurred. Just like the pattern produced in the single slit setup, what is called a diffraction envelope was displayed on the wall. Here are some borrowed pictures of the two setups (Figure 3).

Figure 3. Difference between single slit and double slit diffraction patterns.

Figure 4. Diffraction envelope and interference fringes of double slit setup

however, a noticeable difference between the two setups is the presence of many small, equally spaced, and alternating light and dark fringes occurring inside each diffraction envelope. These are called interference fringes. This is, as mentioned above, the result of the two light sources (coming from each slit) interfering with each other. It is also worth mentioning that like the single slit diffraction pattern, the diffraction envelope loses brightness as you move to its sides. Thus, the tiny fringes inside the envelope follow suit (Figure 4).

Since the diffraction envelope is highly similar to the diffraction pattern produced by the single-slit setup, the same equation (1) was used to calculate the slit width using data gathered from the double slit setup.

Table 3: Data and results for the a = 0.04 mm, d = 0.25 mm double slit

	m=1	m=2
distance between side orders	5.1 cm	6.85 cm
center to side	2.55 cm	3.425 cm
Calculated slit width	.0421 mm	.063 mm
% difference	5.30%	57.50%

Once again, the first order’s results turned out relatively fine at 5.3% deviation, but the second order’s calculations yielded a 57.5% error.
Finally, the last part of the experiment involved the simple counting and measuring of the interference fringes in the diffraction envelopes.

Table 4: Data and results for double-slit interfence

	Slit width: 0.04 mm Slit separation: .25 mm	Slit width: 0.04 mm Slit separation: .50 mm	Slit width: 0.08 mm Slit separation: .25 mm	Slit width: 0.08 mm Slit separation: .50 mm
number of fringes	12	25	14	12
width of central maximum	5.1 cm	5.35 cm	5 cm	2.1 cm
Fringe Width	.425 cm	.214 cm	.357 cm	.175 cm

At the end of the experiment, here were some notable observations. As the slit width decreases, the diffraction envelopes increase sin length. Also, the fringe width increases as the wavelength increases, and likewise, as the distance of slit to surface increases, so does the fringe width. These are all in line with the experimental observations, yet can also be inferred from equation (1). Similarly, since the patterns in the double slit setup go by the equation 

The mentioned relationships above are practically the same. It was also predicted that when the slit width was changed, the diffraction envelope and the interference patterns would remain the same in width; only changing in focus and clarity. However, due to faulty data (most likely due to error in the measuring process), the data was not in line with these predictions (refer to Table 4). Lastly, it was seen that as the slit separation was increased, the number of interference fringes within the diffraction envelope increase. At the same time, the width of the fringes decreased. These were consistent with observations (refer to Table 4).

In totality, the experiment demonstrated some key characteristics of light's nature as a wave, while also showing us that not everything can be derived from intuition alone.

Sources:

[1] Dr Quantum: Wave Particle Duality and the Observer! Youtube. 12 Mar. 2011. Television.

[2] "Experiment 8 Interference and Diffraction." Physics 103.1 Experiment Manual. Quezon City:               National Institute of Physics, 2011. N. pag. Print.
[3] Double slit diffraction. Digital image. Youtube. N.p., n.d. Web. 24 Feb. 2016.                                       <https://i.ytimg.com/vi/yaEym48DGkw/maxresdefault.jpg>.

Optical Disk - Reflection and Refraction

Optical Disk - Reflection and Refraction

I. Brief Background

Light, as it was discovered in the 1800's, exists as electromagnetic radiation. This groundbreaking realization hit home, when the concepts of electric and magnetic fields were finally summarized in the four famous Maxwell Equations.

 In this experiment, the behavior and propagation of light was studied using a method known as geometric optics, wherein the wavelength of light is assumed to be smaller than any object or obstacle in its path.

II. What Went On

The experiment was divided into six distinct parts. Alignment and calibration of the optics was first performed, to ensure the quality of collected data and measurements. The experiment kicked off with the observation of a light ray's behavior when reflected from a plane mirror. The mirror, mounted on a rotating disk was rotated in angle increments of ten degrees, and each corresponding angle of reflection was measured. The same procedure was then done on concave and convex mirrors, with each mirror set undergoing three trials. Here are the results:

Table 1: Part 1 of the experiment

Angle of Incidence°	Angle of Reflection°
	Plane mirror	Convex Mirror	Concave Mirror
10	10	12	10
20	20	22.5	21
30	30	32.5	31

The next step practically involved the same procedure, except this time, five parallel light rays were allowed to hit the mirrors. For each of the three different mirrors, the resulting directions of the reflected lines were recorded. The mirrors were not rotated this time around. 

Fig. 1: Sketches of the reflected light paths 

The fourth part the the experiment involved the beaming of light through a semicircular prism. Using a prism allowed for the splitting of the light ray into reflected and refracted light beams. The prism was set on the rotating disk with the flat side facing the light source, and made to rotate in angled increments of 10° , starting at 10°  and ending at 50°. At each stop, the angles of the reflected and refracted light rays relative to the central axis were measured. The index of refraction was then calculated using Snell's Law. 

Table 2: Part 4 of the experiment

Angle of Incidence°	Angle of Reflection°	Angle of Refraction°	Index of Refraction
10	10	6.5	1.533
20	21	13.5	1.465
30	31.5	20	1.462
40	41.5	25.5	1.493
50	51.5	31	1.487

Next, the same thing was done for the prism, but this time with its curved side receiving the light beam. 

Table 3: Part 4.1 of the experiment

Angle of Incidence°	Angle of Reflection°	Angle of Refraction°	Index of Refraction
10	11	16	1.587
20	21	31.5	1.527
30	31	49	1.509
40	41	75	1.503
50	50

Additionally, the angle at which the refraction ray disappeared (critical angle) was also recorded. From the critical angle, the index of refraction could also be calculated for. using Snell's Law, but with the assumption that refracted light would no longer be visible (where its angle would be 90°). The speed of light inside the glass was also calculated using the gathered data and the equation v = nc where v is the speed of light in the given medium, n the refractive index, and c being the speed of light in a vacuum.

Table 4: Part 5 of the experiment

Critical Angle	43°
Index of Refraction of glass n	1.466
Speed of light inside the semi-circular glass	4.398x10^8 m/s

For the sixth and final segment of the experiment, five more prisms of varying shapes were placed one at a time on the rotating disk. Multiple parallel rays of light were then beamed at each prism. The corresponding light patterns for each prism were then recorded. 

Fig. 2: Sketches of the refracted light rays

One remarkable thing about light is that it obeys the law of reflection regardless of whether it is met by a plane or spherical mirror. This is because when a light particle (which is very, very small) hits even a curved surface, the fact that it is just so tiny makes the surface it hits practically flat; in the same sense that a curve can be seen as a connected series of infinitely small straight lines. So, to put things simply, for a travelling light particle about to reflect off a surface, that surface will always be flat regardless of its overall shape. There was, however, one aspect of the experiment that one could initially think of as disregarded. It is easy to forget that even without the glass, light around us travels through the medium of air we breathe. Being a medium, its possible effects on the passing light as well as its index of refraction should be taken into account. Well, in fact it is taken into account, and the apparent "disregard" for it is rightfully so. This is because air has a refractive index of 1, so in essence it doesn't do much at all when it comes to the bending of light passing through it. 

When passing through the planar face of the semi-circular glass, both an angle of incidence and an angle of refraction were observed. Interestingly, after recording different angles from different degrees of rotation, sufficient data was able to be gathered in order to theoretically deduce that if the contact surface of the glass were to be switched to its curved side, and the angle of incidence were to be set at the same values as the angles of refraction that were previously measured, the resulting angle of refraction would be the same as the experimentally measured angles of incidence. All it took to deduce this was to plug in the switched values into Snell's Law. But in an intuitive sense, switching the receiving surface would only rotate the glass by 180° which in effect would only change to initial direction to its opposite, but keep the initial angles as they were. Thus, resulting in the predicted angles of incidence and refraction. 

In the experiment, the obtained critical angle was 43°. Beyond this angle, what is known as total internal reflection would occur, where the supposedly refracted light is directed back into the glass and is left to continuously reflect making the refracted ray essentially disappear. 

To conclude, the long ago established theories on optics and its laws were once again verified, thus proving to stand the tests of time. Before we left, we all took pictures of the light patterns that came out of the experiment.

References:

Lab Manual Authors. Experiment 6: Optical Disk - Reflection and Refraction. N.p.: n.p., 2013. Print.