Experiment 12

Exp 12
Diffraction

Purpose: To use diffraction of light to determine the length of CD grooves (width [a] of the openings) by measuring the length between the reflected rays using diffraction grating equations.

Materials: 
Diode laser
CD
Reflecting surface (white board)
Meter stick(s)
Stands (wooden blocks)
Diffraction film

Procedure:
Since λ of diode laser was not known, it has to be calculated first.
To calculate the wavelength of the laser, a diffraction film of known d (distance between slits) was used. In this case, it was 500 slits / mm. Reciprocating it,
d = 2*10^-3 mm.
The formula used was

d sin(θ) = m λ,

solving for λ,

λ = d sin (θ)/ m

The angle could be found experimentally. First, the laser was aimed perpendicular to the diffraction film; the ray would diffract and at least three rays would be shown on the reflecting surface. The angle was the arctangent of the distance between the diffraction film to the reflecting surface over the distance between the first light (1st maxima) to third light (3rd maxima).
Using formula above, θ was 40.32 degree. λ then, is  647.6 nm.

With λ known, the grooves can be calculated. The formula used is

a sin (θ) = m λ

a = m λ/ sin (θ

The CD was positioned so that when laser was aimed towards it, its ray would reflect to the surface orthogonal to it. The distance (d) from the CD to the surface was 68.5 + 0.1 cm The distance between the two lights observed was 28.8 + 0.1 cm. Using phytagorean principles, the angle was calculated to be 22.8 degrees.

Using the equation previously mentioned and letting m = 1 (first maxima), a was calculated to be 1671 nm.

Uncertainty and Error Analysis:

*The slits was assumed to be a constant number without error (it was not measured in the lab). 

ud = 0

*The angle, however, had uncertainty. The triangle was found using two length measurements. 

tan (θ) = (Length / Width);

θ = arctan (Length / Width);

uθ = ((L ln (W) 1/(1+L/W) uW)^2 + (1/W* (1/(1+L/W)) uL)^2)^(0.5)

uθ = 0.45 degree

λ(d, θ) = d sin (θ)/ m

dλ/dd = sin (θ) / m 

dλ/dθ = d cos(θ) / m

uλ = ((sin (θ) / m  * ud)^2 + (d cos(θ) / m * uθ)^2)^(0.5)

uλ = 3.43*10^-7 m = 343 nm

λ then, is 647.6 nm + 343 nm

The second calculation has angle with different uncertainty. Using the same formula:

uθ = ((L ln (W) 1/(1+L/W) uW)^2 + (1/W* (1/(1+L/W)) uL)^2)^(0.5)

uθ = 8.57 degrees

a(θ,λ)  =  λ / sin (θ)

da/ dλ = 1 / sin(θ)

da/ dθ = - λ cot(θ)/ sin(θ)

ua = ((1/ sin(θ) * uλ)^2 + (- λ cot(θ)/ sin(θ) * uθ)^2)^(0.5)

ua = 34070 nm

a = 1671 + 34070 nm

Normal CD grooves usually range between around 1500 nm. The calculated values, including the uncertainties, definitely falls within the range. 

Errors (comment on the large uncertainty value):

The large uncertainty value could be caused by several reasons, one of them being having to do the experiment twice. The wavelength of the laser was unknown initially and had to be determined by performing  an additional experiment. When performing this, some errors were recorded and accumulated throughout the lab. Those error later further were being accumulated and added up along with the second experiment's uncertainties, causing the values to substantially increased to what was calculated above. 

Further error analysis:

Although the calculated value of 1671 nm falls within about 10% of the expected value of 1500 nm, it was not exact due to several factors. One of them was the experimented CD had been used previously; it was a burned CD; a burned CD probably had a different slit density than a new CD. Secondly, the CD was from at least 2002, 10 years prior. CD-ROM technology had change the past decade. CDs today had an improved memory capacity than a decade ago; an increase in memory capacity implied that it had different slit density. 

Another room for error was when experiment was performed, it was not truly observed whether the ray was perfectly orthogonal to the surface or not. If it was not, then the calculated angle of 22.8 degrees would not be perfectly accurate, hence the calculated a value would be different. 

The inconsistency of the distance could not be found using the obtained data, because it would require two reflections on two different locations on the CD, which was not performed in this experiment, unfortunately.