1) Without plotting a Cornu spiral, find the hypothetical value of Dv for such a spiral plotted for the diffraction pattern of a single slit of width 0.80mm, assuming Fresnel zone parameters a = 40.0 cm and b = 50.0 cm with red light of 6400 Å. From this result obtain the relative intensity I.
Dv = S Ö{ 2 (a + b)/ abl } =
0.08 cm Ö{ 2 (40 cm + 50 cm )/ (40cm) (50 cm) 6.4 x 10 -5 cm }
Dv = 3.0
And from table of
Fresnel integrals:
(Dx) = 0.6058 – 0.0000 = 0.6058
(Dy) = 0.4963 – 0.0000 = 0.4963
A2 = (Dx) 2 + (D y) 2
I » A2 = (0.6058) 2 + (0.4963) 2
2) A student is given the top section of a Cornu spiral to analyze for an exam:
Using this, obtain the relative amplitude for the particular diffraction pattern. From this find the relative intensity.
Soln.
From the diagram, on inspection of v along the spiral, we see:
(D v) = (1.4 - 0.9) = 0.5
Obtain the (Dx, Dy) limits from the table of Fresnel integrals for these values, e.g.
(Dx) = 0.5431 - (0.7648) = (-0.222)
(Dy) = 0.7135 - 0.3398 = 0.374
A2 = (Dx) 2 + (D y) 2
The relative amplitude is:
A = Ö (Dx) 2 + (D y) 2
= Ö (-0.222) 2 + (0.374) 2
= 0.435
And the relative intensity is:
I » A2 = (0.435) 2 = 0.189
3) Plot the graph for a Cornu spiral for a single slit diffraction pattern at intervals of Dv = -0.10 to 3.0 and Dx from 0 to 0.90. On the graph draw a chord from x = 0.40 to x = 0.75 and from this estimate Dv along the Cornu arc. Hence or otherwise obtain the relative amplitude and the intensity.
This is extracted from the full spiral graphed below:
From the given information and the graph segment (upper right quadrant of Cornu spiral) we obtain:
A = Ö (Dx) 2 + (D y) 2
A =
Ö (0.35) 2 + (0.24) 2
A = 0.424
I » A2 = (0.424) 2 = 0.18
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