ABSTRACT
Ψ = ∫ ∆ (19.4) This basic equation of the Fresnel theory of diffraction was derived by making use of interference concepts only. However, a complete picture of diffraction phenomena requires that ( ),P r tΨ includes effects such as a phase shift at the aperture or an angular dependence on the inclination to the surface normal. These fundamental properties of wave propagation, which are lacking in the Fresnel treatment, are specified by the Kirchhoff formulation of diffraction phenomena. Consider two scalar wave functions 'and ΨΨ which are solutions to the scalar wave equation (15.14), so that, using Green’s theorem derived in Problem 15.5, one obtains
( )' ' S
dS 0Ψ ∇Ψ − Ψ∇Ψ ⋅ =∫ G (19.5) on any closed surface S around a given point P. Let Ψ be an unspecified scalar wave and be a spherical wave (16.30) of the form'Ψ
( ) ( )tkrie r
tr ω−=Ψ A,' (19.6) where r is measured from P, that is Eq.(19.6) represents spherical waves converging to the point P. There is a singularity at P, where 0=r , so that this point must be excluded from the region enclosed by S, and this can be achieved by subtracting an integral over a small sphere σ of radius centred at P. 0r
σ P y
x
z
S r0
Figure 19.2. Integration of spherical waves converging to a point P Equation (19.5) takes the form
e e e edS d r r r rσ
σ =
⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞∇ − Ψ∇ ⋅ + ∇Ψ − Ψ∇ ⋅ =⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦∫ ∫ G G (19.7)
where the common factor has been cancelled out. The unit normal t-ie ωA σe G to the small
sphere σ points toward the origin at P, and the gradient is directed radially outward, so that ( ./ re )∂∂−=∇ σG In terms of the solid angle Ωd measured at P, the element of area on the sphere is Ω==⋅ drdde 20σσσ GG and Eq.(19.7) becomes
e e e edS r d r r r r r rσ =
⎡ ⎤ ⎡⎛ ⎞ ⎛ ⎞∂Ψ ∂ ⎤∇Ψ − Ψ∇ ⋅ = − Ψ Ω⎢ ⎥ ⎢⎜ ⎟ ⎜ ⎟∂ ∂⎢ ⎥ ⎢⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣∫ ∫ ⎥⎥⎦ G
r r ik e r d e d
rσ σ= ∂Ψ⎡ ⎤= − Ψ Ω + Ψ Ω⎢ ⎥∂⎣ ⎦∫ ∫
In the limit as approaches zero, the first integral on the right-hand side vanishes and
so that the second integral becomes 0r
,10 →ikre 4 .Pπ Ψ We then obtain for an arbitrary scalar wave function a relation between its value PΨ at any point P inside a closed
1 4
e e dS r rπ
⎡ ⎤⎛ ⎞Ψ = ∇Ψ − Ψ∇ ⋅⎢ ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦∫ ⎥ G
which is called the Kirchhoff integral theorem. The diffraction produced by an aperture
of arbitrary shape can now be described by including Σ Σ as part of the closed surface of integration and by assuming that the contributions to the integral from from elements of S which lie outside the aperture, may be neglected in the geometric configuration illustrated in Figure 19.3.
