M. RUDAN — KIRCHHOFF’S THEORY OF DIFFRACTION 1 Kirchhoff’s Theory of Diffraction M. Rudan I. KIRCHHOFF’S THEORY OF DIFFRACTION where the derivatives are calculated with respect to the com- q At a point p internal to V , the solution of the Helmholtz ponents of : equation by means of the Green function, using a unit vector rn = (ξ − xn)i1 + (η − yn)i2 + (ζ − zn)i3 , ν that lies along the outward normal to S, reads ∂r ∂ 1/2 ξ − x n = (ξ − x )2 + (η − y )2 + . = n , ∂f ∂G ∂ξ ∂ξ n n r 4πf(p)= G − f dS − Gb dV . n ZS ∂ν ∂ν ZV whence grad rn = rn/rn. As a consequence, The above renders f(p) as a function of b, of the boundary values of and , and of the auxiliary function . exp(jkrn) exp(jkrn) 1 f ∂f/∂ν G grad = jk − grad rn . Although both f and ∂f/∂ν appear in the integral above, it rn rn rn can be demonstrated that the values of either f or ∂f/∂ν are ′ If ρn is the angle between ν and rn, then sufficient to specify f at every point within V . ′ Here, f indicates the Fourier transform with respect to time cos ρn = ν • grad rn = cos(ρn + π)= − cos ρn , of the scalar potential ϕ or of any component of the vector where ρn is the angle between the inward normal −ν and rn. potential A, and b indicates the transform of −ρ/ε0 or −µ0J , i It follows i =1, 2, 3. As a consequence, f and b depend on the angular frequency ω besides the spatial coordinates, with k2 = ω2/c2 ∂f 1 exp(jkrn) = cos ρ − jk f 0 . ∂ν n r n r in vacuum. Xn n n If a material is present within V , the constants ε0, µ0 must be The above takes a simpler form if the sources are far enough replaced with the material’s parameters ε, µ. Similarly, the from A to fulfill the relation r ≫ λ, so that k ≫ 1/r . Also, expression for the angular frequency takes the form k2 = n n it will be assumed that the positions of the points p are such ω2/u2, with u the phase velocity. n as to make the angles ρ small. This yields In the following it will be assumed that the charge density n within V is zero, so that b =0. ∂f exp(jkrn) ≃−jk f 0 = −jkf(q) . Now, consider the case in which the closed surface S is made ∂ν n r Xn n of three parts, namely, an opaque screen B, an aperture A within B, and a portion C of a large sphere of radius R In a similar manner one finds centered at p ≡ (x,y,z). ∂G exp(−jks) = ν • grad . Let q ≡ (ξ,η,ζ) be any point of A, and p0, p1, p2,..., pn . ∂ν s a set of points belonging to the half space on the side opposite where the derivatives are calculated with respect to the com- to p with respect to the surface A∪B, with pn ≡ (xn,yn,zn). ponents of p: Also, let rn = q − pn and s = p − q. Each point pn is assumed to be a point source from which a spherical wave of s = (x1 − ξ1)i1 + (x2 − ξ2)i2 + (x3 − ξ3)i3 , the form fn0 exp(jkγn)/γn is emitted, where fn0 is a complex ∂s ∂ 2 2 1/2 x1 − ξ1 constant and γn the distance between pn and an arbitrary point = (x1 − ξ1) + (x2 − ξ2) + . = , ∂x1 ∂x1 s belonging to the same half space as pn. In turn, it is k =2π/λ, with λ the wavelength. whence grad s = s/s. As a consequence, Adding up all the spherical waves at q yields the boundary exp(−jks) exp(−jks) 1 condition grad = − jk + grad s . s s s exp(jkrn) f(q)= fn0 . ′ rn If σ is the angle between ν and s, then Xn ′ Note that ν points from q towards the half space of the pn. cos σ = ν • grad s = cos(σ + π)= − cos σ , As a consequence, ν • rn < 0. The other boundary condition is found from where σ is the angle between the inward normal −ν and s. It follows ∂f exp(jkrn) ∂G 1 = ν • grad f = ν • fn0 grad , = cos σ + jk G . ∂ν r Xn n ∂ν s Also in this case one assumes that p is far enough from A to E. De Castro Advanced Research Center on Electronic Systems, and Dipar- timento di Elettronica, Informatica e Sistemistica, Universit`adi Bologna, Viale fulfill the relation s ≫ λ, so that k ≫ 1/s, to yield ∂G/∂ν ≃ Risorgimento 2, I-40136 Bologna, Italy. March 27, 2008 jk cos σ G. M. RUDAN — KIRCHHOFF’S THEORY OF DIFFRACTION 2 In order to apply the solution of the Helmholtz equation to the simplicity. The distance between the two mirrors is typically calculation of f(p) it is necessary to perform the integration much larger than the mirrors’ diameters. As a consequence, over the closed surface S made of the union of A, B, and C. one may replace 1+cos σ with 2. The field fp is given by In many cases one is interested in calculating f(p) at any point belonging to the half space opposite to that of the sources (still j fp(p)= − G(|p − q|) fq(q) dA , keeping the case where p is far enough from A to maintain λ ZA the validity of the simplifications used above). For this, one may take the limit R → ∞ and assume that the Sommerfeld where the integration is carried out over the domain A of q. asymptotic conditions hold. This makes the contribution of C However, for self-consistency fp must be the same function to the integral to vanish. as fq apart from a multiplicative complex constant. As a Next, the Kirchhoff boundary conditions are imposed. They consequence the suffixes may be dropped. Moreover, the factor consist in assuming that the contribution of B to the integral −j/λ may be incorporated into the multiplicative constant be- vanishes, and that the contribution of A is the same as if cause the wavelength here is a fixed parameter, yet unknown. B were not present. This is equivalent to assuming that In conclusion the above becomes the perturbation due to the edges of B is negligible. The theory based on these boundary condition is called Kirchhoff’s f(p)= η G(|p − q|) f(q) dA . diffraction theory. ZA In conclusion, remembering that k =2π/λ one finds The latter is an integral equation, specifically, a homogeneous j Fredholm equation of the second kind. The parameter η is the f(p)= − G(|p − q|) f(q) (1+cos σ) dA , 2λ ZA eigenvalue, while f is the corresponding eigenfunction. The solutions of the above provide the modes of the resonator. which is also called the Fresnel-Kirchhoff diffraction formula Once the solutions are found, the corresponding eigenfunctions and is the mathematical expression of the Huygens-Fresnel may be used as boundary conditions to determine the field at Principle. The Huygens coefficient is (1 + cos σ) /(2jλ). all internal points of the resonator. If the position of p is such as to make the angle σ small, the above becomes The theory depicted above is due to Fox and Li. j f(p) ≃− G(|p − q|) f(q) dA . λ Z A IV. RECIPROCITY THEOREM OF HELMHOLTZ II. BABINET’S PRINCIPLE Consider the special case where there is only one source at, Consider the dual situation in which A is the opaque screen say, p0. Letting r = |q − p0|, cos ρ = −ν • grad r, it follows and B the aperture. The dual situation will be indicated with a prime. As exp(jkr) f(q)= f(|q − p |)= f00 , + = 0 r ZA ZB′ Z∞ one finds fA(p) + fB′ (p) = f∞(p), where fA(p) is the ∂f 1 exp(jkr) p ′ p = cos ρ − jk f00 ≃−jkf(q)cos ρ . solution at in the original case, fB ( ) is the solution in ∂ν r r the dual case, and f∞(p) is the solution in the case where no obstacle is present. This result is called Babinet’s principle. In the above it has been assumed that k ≫ 1/r, whereas no An immediate consequence of the above is that, if at some assumption has been made about the position of p0. As a con- point p it is fA =0, at the same point it is fB′ (p)= f∞(p). sequence, the angle ρ is not necessarily small. Remembering Namely, where the intensity is zero in presence of one of the that s = |p − q|, cos σ = −ν • grad s one finds screens, the intensity in presence of the dual screen is the same as if no screen were present. j exp[jk(r − s)] f(p)= f00 ν • grad(r + s) dA . 2λ ZA rs III. APPLICATION TO THE FABRY-PEROT RESONATOR The Fresnel-Kirchhoff diffraction formula Consider now the case where the source is at p and the solution j is sought at p0. The same procedure used so far yields f(p)= − G(|p − q|) f(q) (1+cos σ) dA , 2λ ZA j exp[jk(s − r)] may also be exploited to improve the description of the optical f(p0)= − f00 ν • grad(r + s) dA , 2λ ZA rs resonators with respect to that given by the geometrical optics. In this case one must assume that p belongs to the first mirror where the minus sign comes from the replacement ν ←−ν.