Diffraction. Monday Nov. 18, 2002. Diffraction theory (10.4 Hecht). We will first develop a formalism that will describe the propagation of a wave – that is develop a mathematical description of Huygen’s principle. Wavefront U. What is U here?. We know this. Diffraction theory.
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Monday Nov. 18, 2002
What is U here?
We know this
where r is measured from point P in V and U1 = “Green’s function”
Thus (4) can be written,
Now as 0 U2(r) UP (its value at P)
Now designate the disturbance U as an electric field E
This gives the value of disturbance at P in terms of values on surface enclosing P.
It represents the basic equation of scalar diffraction theory
Have infinite screen with aperture A
Let the hemisphere (radius R) and screen with aperture comprise the surface () enclosing P.
Radiation from source, S, arrives at aperture with amplitude
E=0 on .
Also, E = 0 on side of screen facing V.
Fresnel Kirchoff diffraction formula
In Fraunhofer diffraction, both incident and diffracted waves may be considered to be plane (i.e. both S and P are a large distance away)
If either S or P are close enough that wavefront curvature is not negligible, then we have Fresnel diffractionFraunhofer vs. Fresnel diffraction
Now calculate variation in (r+r’) in going from one side of aperture to the other. Call it
sin ≈ h/d
sin’ + sin = ( h’/d + h/d )
Second term = measure of curvature of wavefront
Fraunhofer or far field limit
Assume S on axis, so
Assume small ( < 30o), so
r’ >> so is constant over the aperture
r will not vary much in denominator for calculation of amplitude at any point P
consider r = constant in denominator
y = b
r = ro - ysin
dA = L dy
where L ( very long slit)
Fraunhofer single slit diffraction pattern