Analysis of the intensity variation in a diffraction pattern from a single slit of width a shows that the intensity is given by sin (Ta sin 0/A) I I, (37.2) Ta sin 0/A max
How much diffraction spreading does a light beam undergo? One quantitative answer is the full width at half maximum of the central maximum of the single-slit Fraunhofer diffraction pattern. You can evaluate this angle of spreading in this problem. (a) as shown, define φ = πa sin φ/λ and show that at the point where I = 0.5Imax we must have φ = √2 sin φ. (b) Let y1 = sin φ and y2 = φ = /√2. Plot y1 and y2 on the same set of axes over a range from φ = 1 rad to φ = π/2 rad. Determine φ from the point of intersection of the two curves. (c) Then show that if the fraction λ/a is not
large, the angular full width at half maximum of the central diffraction maximum is θ = 0.885λ/a. (d) What If? Another method to solve the transcendental equation φ = √2 sin φ in part (a) is to guess a first value of φ, use a computer or calculator to see how nearly it fits, and continue to update your estimate until the equation balances. How many steps
(iterations) does this process take?
![Analysis of the intensity variation in a diffraction pattern from a single slit of width
a shows that the intensity is given by
sin (Ta sin 0/A)
I I,
(37.2)
Ta sin 0/A
max](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff9f03da3-4e26-4892-899c-52e584df7b6d%2Fdeadccdd-28a2-4d84-a014-c0863869f39d%2Ff4g93z.png&w=3840&q=75)
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