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Young’s Double-Slit Interference Pattern
1. Two slits interference
When a beam of light of wavelength 𝜆𝜆 is incident at two slits separated by a distance a , the light arriving at an observation screen forms an intensity pattern illustrated in the Figure below.
As a function of angle 𝜃𝜃 the light intensity is given by:
𝐼𝐼(𝜃𝜃) = 𝐼𝐼𝑜𝑜 �𝑐𝑐𝑐𝑐𝑐𝑐 �𝜋𝜋^ 𝑎𝑎^ 𝑠𝑠𝑠𝑠𝑠𝑠 𝜆𝜆 (𝜃𝜃)��
2 � 𝑠𝑠𝑠𝑠𝑠𝑠�^
𝜋𝜋 𝑏𝑏 𝑠𝑠𝑠𝑠𝑠𝑠 𝜆𝜆 (𝜃𝜃)� 𝜋𝜋 𝑏𝑏 𝑠𝑠𝑠𝑠𝑠𝑠 𝜆𝜆(𝜃𝜃) �
2 (1)
where a is the center-to-center separation between the two slits and b is the width of each slit. The intensity pattern is a product of two
terms. The first term on the right hand side, �𝑐𝑐𝑐𝑐𝑐𝑐 � 𝜋𝜋^ 𝑎𝑎^ 𝑠𝑠𝑠𝑠𝑠𝑠 𝜆𝜆 (𝜃𝜃)��
2 , corresponds
to the interference pattern of the two slits, while the second
term, (^) � 𝑠𝑠𝑠𝑠𝑠𝑠�^
𝜋𝜋 𝑏𝑏 𝑠𝑠𝑠𝑠𝑠𝑠 𝜆𝜆 (𝜃𝜃)� 𝜋𝜋 𝑏𝑏 𝑠𝑠𝑠𝑠𝑠𝑠 𝜆𝜆(𝜃𝜃) �
2 , corresponds to the diffraction pattern due to the
finite size of the width of the each slit. Because 𝑎𝑎 > 𝑏𝑏 then the
interference pattern shows a rapid change with the angle, 𝜃𝜃, while the diffraction pattern shows a slower modulation with the angle, 𝜃𝜃. In the experiments described below, you will explore patterns formed by two slits of fixed width, b , but with different separations, a.
Under the small angle approximation 𝑐𝑐𝑠𝑠𝑠𝑠(𝜃𝜃)^ can be calculated from:
𝑐𝑐𝑠𝑠𝑠𝑠(𝜃𝜃)^ ≅ 𝑡𝑡𝑎𝑎𝑠𝑠(𝜃𝜃) = (𝑦𝑦^ − 𝐷𝐷^ 𝑦𝑦 0 ) (2)
where 𝑦𝑦 is the transverse distance for a pattern centered at 𝑦𝑦𝑜𝑜, and 𝐷𝐷 is the distance between the plate with the double slits and the plane of observation.
Accordingly, the interference pattern �𝑐𝑐𝑐𝑐𝑐𝑐 �𝜋𝜋^ 𝑎𝑎^ 𝑠𝑠𝑠𝑠𝑠𝑠 𝜆𝜆 (𝜃𝜃)��^2 will be zero
where:
𝑐𝑐𝑠𝑠𝑠𝑠(𝜃𝜃) = (𝑦𝑦𝑟𝑟^ − 𝐷𝐷^ 𝑦𝑦^0 )= (2𝑟𝑟 + 1) 2𝑎𝑎𝜆𝜆 (3)
By combining Equations (2) and (3), we obtain that the separation between two darks bands is given:
𝑦𝑦𝑟𝑟+1 − 𝑦𝑦𝑟𝑟 = ∆𝑦𝑦 = 𝜆𝜆𝑎𝑎^ 𝐷𝐷
On the other hand, the slower modulation from the diffraction pattern will be zero at:
Measuring the width of each slit
From the slow variation on the intensity pattern, measure the
position of the first zero in the diffraction pattern: 𝑦𝑦𝑠𝑠=1 − 𝑦𝑦 0 = 𝜆𝜆𝑏𝑏^ 𝐷𝐷 and
determine the width of each slit.
Using different the separation for the double slits
Repeat the measurements for another double slit pattern. Choose a pattern with the same slit width but with a different separation between the slits. What happen when the separation between the slits increase or decrease?
3. Computing a the interference pattern
Use a personal computer and your preferred software to plot the
theoretical results for intensity (^) 𝐼𝐼𝐼𝐼𝑜𝑜 versus (𝑦𝑦 − 𝑦𝑦𝑜𝑜)^ , as described by
Equations (1)-(3). Make sure to enter the values that you measured for the separation between the slits, 𝑎𝑎, the slit width, 𝑏𝑏, and the light wavelength 𝜆𝜆. Compare this calculation with your experiment. Include a screen dump of this graph in your lab report.
