Interference and Diffraction

Terms to take note:

Central maximum : central bright fringe,  widest and brightest among the bright fringes (maxima)

mth intensity minimum : For single-slit. Ranges from ±1, ±2, ±3, …, ±n. Positive when above or right of the center of pattern, negative when below or left of the center of pattern.

mth intensity minimum : For double-slit. Ranges from ±1, ±2, ±3, …, ±n. Positive when above or right of the center of pattern, negative when below or left of the center of pattern.

diffraction envelope : the outline of the single-slit diffraction pattern


We are all familiar that light has a dual nature; it can behave as a particle or a wave. Diffraction occurs when a wave encounters an obstacle or a slit that is comparable in size to its wavelength, whereas Interference is the phenomenon where waves meet each other and combine constructively or destructively to form composite waves. The main difference between the two is that diffraction involves a wave and some obstacle which deflects the wave or bends it and interference involves a wave which combine with other waves.[1] 

However, fundamentally, these two phenomena are the same, as they both arise from wave interference that follows the principle of superposition which states that when waves encounter each other, they can either add up (constructive interference) or cancel each other out (destructive interference), depending on the properties of each wave.[2]

A0tY5
Figure 1: Diffraction Pattern [2]
QKoqU
Figure 2: Interference Pattern [2]

The objective of this experiment is to quantitatively relate the single-slit and double-slit diffraction pattern obtained to the slit width. Also, to differentiate the patterns produced by single-slit and double-slit diffraction and determine the quantitative relationship between a double-slit diffraction pattern and the corresponding slit separation.

124.png
Figure 3. Experimental Set-Up. (L) laser,  (S) slit, (OB) optical bench, (SC) screen [3]

The experiment started by assembling the optical materials like the one in Figure 3. Slit and pattern were of the same level vertically .The first part of the experiment was the single-slit diffraction. Here, 0.004 mm, 0.002 mm, and 0.008 mm width single slit were taken into  account. Boundaries of dark fringes on white sheet were marked and important data were recorded and calculated in Table W1 and W2. The second part was the double-slit interference I: calculating the slit width. Here, the double slit with 0.04-mm slit width and 0.25-mm slit separation were selected. Boundaries of dark fringes in the diffraction envelope were marked on the white sheet. Important data were recorded and calculated in Table W3. The last part pf the experiment was double-slit interference II: changing the slit width and slit separation. Here, we considered 4 double-slits combination which can be found in Table W4. The number of interference fringes located inside the central maximum were counted and the width of the central maximum were recorded. Important data were also calculated in Table W4.

Table W1. Wavelength of the laser diode

a = 0.02 mm, m = 1

a = 0.04 mm, m = 1

Distance between side orders, Δy1

143 mm

61.6 mm

Distance from center to side, y1

71.5 mm

30.8 mm

Calculated wavelength 7.7089 x 10-4mm

6.6415 x 10-4mm

Percent difference

18.60%

2.18%

Average wavelength (λ)

7.1752 x 10-4mm

Slit-to-screen distance (L)

1855 mm

In Table 1, the wavelength was calculated using the equation:

ym,diff = mλLa-1                                                                        (1)

Rearranging Equation 1,

λ = a ym,diff m-1L-1                                                                   (2)

where λ corresponds to the wavelength, a is the single-slit width,   ym,diff  is the distance from center to side, m is the intensity minimum, in this case, ±1, L is the distance from slit to screen. Distance between side orders , Δy1, was measured between the first order  (m = ±1) minima. The measurement started at the midpoint of the dark fringes.  Dividing it by two, you’ll get the distance from center to side, ym,diff.

Theoretical wavelength is 650 nm or 6.5 x 10-4mm. It can be observed that in the second column (a = 0.002 mm, m =1), the percent difference is relatively high but in the third column (a = 0.004 mm, m =1), percent difference is relatively low. Percent difference for the average wavelength is 10.5 %. Possible source of error may come from slightly different vertical level of slit and pattern in which the small angle of inclination gave rise to a slightly different lengths of fringes.

Increasing the wavelength of the laser would increase the width of the fringes as suggested by Equation 2. Also, placing the screen farther from the slit would have a larger value of L in which case would also give rise to an increase in width of the fringes.

Table W2. Data and results for the 0.04 mm single slit

m = 1 m = 2
Distance between side orders, 61.6 mm 130
Distance from center to side 30.8 mm 65
Calculated wavelength

6.6415 x 10-4 mm

7.0081 x 10-4mm

Percent difference 2.18% 7.82%

The procedure and equations used in Table W1 were the same in Table W2 only that the second-order minima (m = ±2 ) was taken into account. The percent differences in Table W2 are relatively low making the data acceptable.

graph 0.02
Figure 1: Diffraction Pattern
graph 0.04
(b) a = 0.04mm
graph 0.08
(c) a = 0.08 mm

Figure W1. Sketches of diffraction pattern for various slit widths and fixed slit-to-screen distance

In Figure W1, you can see the obtained measurements of the diffraction patterns perfectly labeled. The widest is the central maximum, it is the brightest fringe. As you move away from the central maximum, the brightness and the width decreases. In between the bright fringes are the dark fringes in which destructive interference occurred. To understand more, take a look at Figure 4 below. You can also observe that the first-order minima have the same values for both the left and the right side, the same goes with second-order minima. Also, in Figure W1, as the slit width is decreased, diffraction envelope became smaller.

plot intensity
Figure 4. Intensity plot of a diffraction pattern through a single slit [3]

In Figure 4, the highest crest gives rise to the brightest fringe which is the central maximum. The crests in Figure 4 are the bright fringes while the troughs are the dark fringes. You can see that the crest’s height goes shorter and shorter as you move away from the center, this corresponds to the decreasing brightness of the fringe as you move away from the center.

Table W3. Data and results for the a = 0.04 mm, d = 0.25 mm double slit

m = 1 m = 2
Distance between side orders, 70 mm 136 mm
Distance from center to side 35 mm 68 mm
Calculated slit width 0.03803 mm 0.03914 mm
Percent difference 4.925% 2.132%
Slit-to-screen distance (L) 1855 mm

In Table W3, slit width was calculated using the equation:

ym,diff = mλLd-1                                                                                        (3)

Rearranging Equation 3,

= mλL ym,diff-1                                                                                     (4)

Equation 3 is similar in form with Equation 1 with the slit width, a, replaced by the slit separation, d. However, Equation 1 gives the position of intensity minima in a single-slit setup while Equation 3 gives the position of intensity maxima in a double-slit setup.

Taking the average wavelength obtained in Table W1, the percent differences are relatively low. Thus, the values obtained are valid.

In this part of the experiment – double-slit interference I, diffraction envelope was observed. Inside the diffraction envelope are equally spaced dark and bright fringes but of different brightness.

In double-slit interference, the width of the interference fringes are controlled by the slit separation, d. On the other hand, the diffraction envelope is controlled by the slit separation, a.

Similar to the one stated in Table W1, nncreasing the wavelength of the laser would increase the width of the fringes as suggested by Equation 4. Also, placing the screen farther from the slit would have a larger value of L in which case would also give rise to an increase in width of the fringes.

Table W4. Data and results for double-slit interference II

a = 0.04 mm

d = 0.25 mm

a = 0.04 mm

d = 0.50 mm

a = 0.08 mm

d = 0.25 mm

a = 0.08 mm

d = 0.50 mm

Number of fringes 13 25 5 11
Width of central maximum 61.5 61.5 28.2 28.2
Fringe width 4.73 2.46 5.64 2.56

In Table W4, it can be seen that the larger the slit separation d in the same slit width, a, the greater the number of fringes and the width of central maximum remained unchanged. Theoretically, the diffraction envelope is independent of the slit separation. However, the interference fringes are dependent of the slit separation. The larger the slit separation, the greater the number of the interference fridges, the smaller the fringe width. And our observation agreed with this.

a
(a) a= 0.04 mm, d = 0.25 mm
b
(b) a =0.08 mm, d = 0.25 mm
c
(c) a = 0.04 mm, d = 0.50 mm
d
(d) a = 0.08 mm, d = 0.50 mm

Figure W2. Sketches  of interference pattern for various slit widths and slit separation

In Figure W2, you can see the obtained measurements of the diffraction patterns in a diffraction envelope perfectly labeled. The small fringes are assumed to be of equal widths. The intensity of light in these fringes varies. The end ones are the dullest while the center is the brightness. To see how this works, take a look at the Figure 5 below.

plot intensity
Figure 5. Intensity plot of a double-slit interference pattern [3]

Figure 5 shows how the dashed line or the interference envelope groups the fringes. This is similar to the single-slit pattern (left).  You can see that inside the envelope are smaller single-slit patterns, this is the reason why the brightness of the fringes varies (right).

If light were to behave as a particle for the single slit and double slit, the pattern would look like the one in Figure 6, not that far when light behaves a a wave. The concentrated regions give rise to your constructive interference or the bright fringes while the barren or not concentrated regions are your destructive interference or the dark fringes. The central maximum have the most concentrated region which is the middle part horizontally from Figure 6.

particle light
Figure 6: Light as a particle [4]
REFERENCES:

[1] Nikos M. (http://physics.stackexchange.com/users/44176/nikos-m), What is the difference between diffraction and interference of light?, URL (version: 2014-09-29): http://physics.stackexchange.com/q/137863

[2] Subodh Ghulaxe (http://physics.stackexchange.com/users/22316/subodh-ghulaxe), What is the difference between diffraction and interference of light?, URL (version: 2014-09-29): http://physics.stackexchange.com/q/137865

[3] Tipler, P., Physics for Scientists and Engineers, 4th ed., W.H. Freeman & Co., USA (1999)

[4] http://www.perceptions.couk.com/uef/imgs/2slit3.gif

SAMPLE CALCULATIONS:

For Table W1:

λ = a ym,diff m-1L-1  = (0.02 mm)(71.5 mm)/(1)(1855mm) = 7.70819 x 10-4 mm = 771 nm

For Table W3:

= mλL ym,diff-1 = (2)(7.1752 x 10-4mm)(1855mm)/(68mm) = 0.03914 mm

PICTURES

 

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