DBS C
7.6 Scalar diffraction theory
S
P O
FIGURE 7.17: The schematics of Fresnel diffraction.
Combining Eqs. (7.104) and (7.105), except for a constant we have dEP =
Es
rr′
eik(r+r′)da. (7.106)
Thus the field at P due to the entire aperture can be written as EP =Es
Z Z 1 rr′
eik(r+r′)da. (7.107) Eq. (7.107) is incomplete on two counts: (i) it does not incorporate the obliq- uity factorF(θ), and (ii) it does not have theπ/2 phase change of the diffracted wave with respect to the incoming wave. After incorporating these changes the formula now reads
EP = −ikEs
2π Z Z
F(θ) eik(r+r′) rr′
!
da, (7.108)
where−i= exp(−iπ/2) accounts for the phase shift andF(θ) is given by F(θ) =1 + cos(θ)
2 . (7.109)
Eq. (7.109) holds still under the approximationλ < b < r, r′. The integration is to be performed over a closed surface, including the aperture. Kirchhoff’s approximations amount to the fact that the wave function and the derivative vanish right behind the opaque part of the screen. The vector fieldE is ap- proximated by a scalar having the same value at the aperture as in the case of its absence.
Interference and diffraction 151 medium, the vector fieldsEorHsatisfy the vector Helmholtz equations. Fur- ther, for a homogeneous isotropic medium, any of the Cartesian components of the electric or the magnetic field satisfy the same Helmholtz equation, albeit for the scalar component, However, for light propagating through a step-index medium (interface between two dielectric media) or any localized or distributed inhomogeneity (as in diffraction problems), the situation is not so simple. The assumptionsisotropic andhomogeneous break down and there is mixing of the various components ofE andH, and even coupling between them via the boundary conditions. In such cases, if the mixing and couplings are strong, the rigorous theory must incorporate the inherent vector character of the fields. In fact, a comparison of the rigorous vector theory with the scalar counterpart reveals ripples (in the step-index example) in both amplitude and phase in the rigorous treatment, while they are absent in the scalar theory [57].
The differences are noticeable only in the immediate vicinity of the interface.
In the case of apertures, in typical diffraction problems the differences show up near the edge of the apertures. After several wavelengths away from the in- homogeneity both the rigorous and the scalar approximation produce similar results and the mixing effects can be ignored. Diffraction from sub-wavelength structures thus may need full vectorial treatment (see Chapter 14). Except for such cases, a scalar theory is a widely accepted tool for diffraction problems.
The scalar theory starts with Green’s second identity and leads to Kirchhoff’s integral theorem. Applied to a specific problem of diffraction from an open aperture in an otherwise opaque screen, this leads to the Fresnel-Kirchhoff diffraction integral. As we go along we will briefly mention the limitations of Kirchhoff’s approximation in the boundary conditions, which led to Bethe’s theory [58] and the recent developments on extraordinary transmission (see Chapter 14).
7.6.1 Helmholtz-Kirchhoff integral theorem
Consider a volumeV enclosed by a surfaceS(seeFig. 7.18). Let the scalar fieldU satisfy the Helmholtz equation inV. the Kirchhoff integral relates the field at a point P to the value of the field and its first derivative on the boundary. In order to arrive at the integral theorem we invoke the second Green’s identity, applicable to two functionsU andG, which are continuous along with their first and second derivatives in V as well as on S. Green’s second identity can be written as
Z Z Z
V
(U∇2G−G∇2U)dV =− Z Z
S
U∂G
∂n −G∂U
∂n
da, (7.110) where ∂F∂n = ∇F ·n is the directional derivative along unit inward normal n. In the context of diffraction problems it is more convenient to use the in- ward normal, though a standard form of Green’s identity involves the outward normal [31]. Let G also satisfy the Helmholtz equation so that for both the functions similar relations hold:∇2G=−k2G and ∇2U =−k2U. Thus the
n
n (x,y,z)
V
P
S
S’ ε
FIGURE 7.18: Schematics of the domain of integration for the derivation of the Helmholtz-Kirchhoff integral theorem.
integrand on the left-hand side of Eq. (7.110) reduces to zero and we have Z Z
S
U∂G
∂n −G∂U
∂n
da= 0. (7.111)
Let the auxiliary Green’s functionG(sometimes referred to as the point func- tion) be given by
G(x, y, z) =eiks
s , (7.112)
wheresis the distance from pointPto (x, y, z). The field given by Eq. (7.112) represents a spherical wave emanating from a point source atP and thus has a singularity at P (s = 0). The closed surface in Eq. (7.111) must exclude any singularity. To this end we exclude a spherical region with surfaceS′ with center atP and with radius ǫ(see Fig. 7.18). The integration in Eq. (7.111) can now be performed on S∪S′, where the enclosed volume does not have any singularity. We get
Z Z
S
U∂G
∂n −G∂U
∂n
da=− Z Z
S′
U ∂
∂n eiks
s
−eiks s
∂U
∂n
da′. (7.113) Since nis the outward normal to S′ and s is along the same direction, the directional derivative∂n∂ can be replaced by ∂s∂, and for the directional deriva- tive of the auxiliary Green’s function we have
∂G
∂n = ∂
∂s eiks
s
= eiks s
ik−1
s
. (7.114)
Interference and diffraction 153 In order to evaluate the integral on the right-hand side of Eq. (7.113), we replace the surface element da′ by the element of the solid angle dΩ and rewrite it in the form
− Z Z
Ω
U
ik−1
ǫ
eikǫ ǫ
−eikǫ ǫ
∂U
∂s
ǫ2dΩ. (7.115) Finally, taking the limitǫ→0 on the right-hand side of Eq. (7.115), we have finite contribution only from the second term, reducing the integral overS′to 4πU(P). Thus Eq. (7.113) reduces to
U(P) = 1 4π
Z Z
S
U ∂
∂n eiks
s
−eiks s
∂U
∂n
da. (7.116)
It is clear from Eq. (7.116) that a knowledge of the field and its first derivative on the surface is adequate for its evaluation at an interior point.
7.6.2 Fresnel-Kirchhoff diffraction integral
Consider now a typical diffraction scenario as depicted in Fig. 7.19. A point source is placed atP0 and let the observation point be atP. The source and the observation points are separated by an opaque screenB with an opening (aperture)A. Let the distance between an arbitrary pointPAon the aperture to the observer (source) be denoted bys(r). We assume the linear dimension of the opening to be larger than the wavelengthλ, though much smaller than bothrands(see Chapter 14 for near and far-field definitions). Let the closed surface be formed by the part of a spherical surfaceSR with center atP with
SB SA P0
SR R s P
r
PA n
θnr θns
FIGURE 7.19: Schematics of the domain of integration for the derivation of the Fresnel-Kirchhoff diffraction integral.
a large radiusR, opaque screen SB and the opening SA. Thus the Kirchhoff integral given by Eq. (7.116) can be broken up as follows:
Z Z
S
= Z Z
SA
+ Z Z
SB
+ Z Z
SR
. (7.117)
It can be shown that for sufficiently largeR, the contribution fromSR is neg- ligible [31]. In order to evaluate the contributions fromSA andSB, Kirchhoff made the following approximations.
• On the opaque screenSB, both the function and its derivatives vanish:
U = 0, ∂U
∂n = 0. (7.118)
• On the opening SA, both the function and its derivatives are the same as created by the source, as if no screen were there:
U =Ui, ∂U
∂n = ∂Ui
∂n, (7.119)
whereUiis the field produced by the source atPAin absence of any obstacles.
There are several mathematical and physical inconsistencies in Kirchhoff approximations in the context of realistic systems. The mathematical incon- sistency stems from the fact that in standard boundary value problems, we use the Dirichlet or the Neumann boundary conditions but not both in general.
Kirchhoff’s approximations given by Eq. (7.118) lead to the mathematical conclusion that the field must be zero everywhere in space. Rayleigh showed that eitherU = 0 or ∂U∂n = 0 is enough to derive another integral, namely, the Rayleigh-Sommerfield diffraction integral [31]. Moreover, in reality most of the opaque screens in diffraction optics are made of metals with finite thickness and conductivity. Such screens, along with the holes, can support surface plas- mons and localized plasmons. Also as discussed earlier, scalar theory can break down in the near vicinity of the aperture. A deeper understanding of these limitations led to the recently discovered area of extraordinary transmission (see Chapter 14).
Referring back toFig. 7.19and assuming an amplitudeAfor the spherical wave, we have
Ui=Aeikr
r , ∂Ui
∂n =Aeikr r
ik−1
r
cosθnr. (7.120) For the auxiliary Green’s function we have similar expressions:
G= eiks
s , ∂G
∂n =Aeiks s
ik−1
s
cosθns. (7.121) In Eqs. (7.120) and (7.121),θnr (θns) is the angle between the inward normal and r(s). In view of the boundary conditions given by Eq. (7.118), there is
Interference and diffraction 155 no contribution fromSB. Further, in the limitkr≫1, ks≫1, the field atP can be written as
U(P) =−iA 2λ
Z Z
SA
eik(r+s)
rs (cosθnr−cosθns))da. (7.122) Eq. (7.122) is known as the Fresnel-Kirchhoff diffraction integral.
For symmetric (with respect to the aperture) illumination when cosθnr= 0, Eq. (7.122) simplifies to
U(P) =−iA 2λ
eikr0 r0
Z Z
SA
eiks
s (1 + cosχ))da, (7.123) whereχ=π−θnsandr0is the radius of the spherical wavefront reaching the aperture. In Eq. (7.123) we can easily see the manifestation of the Huygens- Fresnel principle (see Eq. (7.108)) by recognizing the obliquity/inclination factor F(χ) given by Eq. (7.109). We can further distinguish between the Fraunhofer and Fresnel regimes of diffraction, which are covered extensively in the standard optics literature [31]. We covered the scalar theory mainly in order to understand the inconsistencies in the Kirchhoff diffraction theory.