27 Fraunhofer Diffraction

Transcription

1 27 Fraunhofer Diffraction Contents 27. Fraunhofer approximation 27.2 Rectangular aperture Keywords: Fraunhofer diffraction, Obliquity factor. Ref: M. Born and E. Wolf: Principles of Optics; R.S. Longhurst: Geometrical nd Physical Optics;. Sommerfeld: Optics. 27. Fraunhofer approximation In this chapter we start with Kirchoff s result and apply Fraunhofer approximation to it. In this the observation point P is very far from the source and the aperture dimensions are very small compared to the distance between the aperture and he observation point. In Fraunhofer diffraction one is concerned with only plane waves. The fig. below shows a small aperture and a very distant observation point P. We begin with (26.32) of last chapter. ψ(p) = i 2λ e ik(r+s) rs (cosβ cosδ)ds (27.) Now if P is very far from the aperture and the aperture is very small, then

2 2 27 Fraunhofer Diffraction O (x,y,z ) r r Q R (ζ,η,) (,,) s δ s P (x,y,z) Fig. 27.: perture and observation variables r r, s s and β = πδ. So all these variables, r, s and δ can be treated as constant and they come out of the integral in (27.). The exponential factor remains inside the integral since it can change drastically even with a slight change in r or s due to the smallness of λ compared to r and s. ψ(p) = i cosδ r s λ e ik(r+s) ds. (27.2) Now, r 2 = (x ζ) 2 +(y η) 2 +z 2 = r 2 2(x ζ +y η)+(ζ 2 +η 2 ), s 2 = (xζ) 2 +(y η) 2 +z 2 = s 2 2(xζ +yη)+(ζ 2 +η 2 ); [ r = r 2(x ζ +y η) r 2 + (ζ2 +η 2 /2 ) r 2, [ 2(xζ +yη) s = s s 2 + (ζ2 +η 2 /2 ) s 2. (27.3)

3 27. Fraunhofer approximation 3 Keeping the terms upto second order in ζ and eta in (27.3), we have, r r (x ζ +y η) r 2r (x ζ +y η) 2 s s (xζ +yη) s 2s 2s 3 r+s r +s (x ζ +y η) (xζ +yη) r s (x ζ +y η) 2 2r 2s 2r 3 2r 3 +O(ζ 3,η 3 ), (xζ +yη)2 +O(ζ 3,η 3 ), (xζ +yη)2 2s 3. (27.4) We substitute (27.4) in (27.2), ψ(p) = i e ik(r +s ) cosδ r s λ e iφ(ζ,η) dζdη. (27.5) where, φ(ζ,η) = (x ζ +y η) (xζ +yη) r s 2r 2s (x ζ +y η) 2 (xζ +yη)2 2r 3 2s 3. (27.6) If we denote first two direction cosines for the points O and P by (l,m ) and (l, m) respectively, l = x /r, m = y /r ; l = x/s, m = y/s. (27.7) then (27.6) becomes, φ(ζ,η) = (l l)ζ +(m m)η + [( + )(ζ 2 +η 2 ) (l ζ +m η) 2 2 r s r (lζ +mη)2. (27.8) s

4 4 27 Fraunhofer Diffraction One has to now perform the integration with respect to ζ and η in (27.5). If for a situation the quadratic terms could be neglected in the function, φ(ζ, η), we call that a Fraunhofer approximation and the resulting disturbance at P is due to a Fraunhofer diffraction. In Fraunhofer diffraction only linear terms in ζ and η participate, so it is a case where only the plane waves take part in the diffraction phenomenon. On the other hand where the quadratic terms in φ(ζ,η) cannot be neglected one would have a Fresnel diffraction. It is obvious that if r and s the quadratic terms vanish. So in this case one has a Fraunhofer diffraction. This is a situation where both, the source and the screen, are effectively placed at infinite distance from the aperture. In this case one has to assume tending to infinity. In fact there are other situations where the contributions from the quadratic terms are not appreciable. This would happen if, ( k 2 + )(ζ 2 +η 2 ) (l ζ +m η) 2 r s r (lζ +mη)2 s << 2π. (27.9) The condition can be satisfied for the following two situations, a) r >> (ζ2 +η 2 ) max λ using the inequalities of the type and s >> (ζ2 +η 2 ) max. (27.) λ (l 2 +m 2 )(ζ 2 +η 2 ) (l ζ +m η) 2 (27.)

5 27. Fraunhofer approximation 5 and b) r + s = and r λ (ζ 2 +η 2 ) max >> l 2, m 2, l 2, m 2. (27.2) Problem : Show that (27.) and (27.2) are indeed the conditions for which the condition (27.9) is satisfied. The conditions a) set lower limits to the distances r and s for given aperture size and wavelength for observing a Fraunhofer diffraction. The conditions b) tell that it can also occur when the point of observation is situated in a plane parallel to that of the aperture and if both point of observation and source are very close to the z-axis. If one puts lenses before or/and after the aperture Fraunhofer diffraction occurs on a plane conjugate to the source. So, Fraunhofer diffraction patterns are actually the images formed by optical instruments. So, in th Fraunhofer limit (27.5) becomes, ψ(p) = i e ik(r +s ) cosδ r s λ e i[(l l)ζ+(m m)η dζdη. (27.3) Factors in front of the integral in (27.3) contribute a constant, say C, ψ(p) = C e i(pζ+qη) dζdη, (27.4) where p = l l and q = m m.

6 27 Fraunhofer Diffraction Rectangular aperture We can now evaluate the integral in (27.4) for a rectangular aperture of width 2a and height 2b and have the Fraunhofer diffraction. ψ(p ) = C Z a a Z bb ei(pζ+qη) dζdη, (27.5) 4C sin(kpa) sin(kqb) k 2 pq sin(kpa) sin(kqb) = 4abC kpa kqb = (27.6) (27.7) = p Csinc(kpa)sinc(kqb). (27.8) In (27.8) p is aperture area. The intensity at p will be the modulus square of the amplitude, If the plane waves are incident on the aperture from one side, l and m and hence I(p) = ψ(p ) 2 = I sinc2 ( 2bπy 2aπx )sinc2 ( ) s λ s λ 3 2 2b 2a Fig. 27.2: Rectangular aperture and its Fraunhofer diffraction pattern Problem 2: How would this pattern look when one makes b? (27.9)