A Tutorial on the Classical Theories of Electromagnetic Scattering and Diffraction

1 Introduction

The classical theories of electromagnetic (EM) scattering and diffraction were developed throughout the nineteenth century by the likes of Augustine Jean Fresnel (1788-1827), Gustav Kirchhoff (1824-1887), John William Strutt (Lord Rayleigh, 1842-1919), and Arnold Sommerfeld (1868-1951).¹, ², ³ A thorough appreciation of these theories requires an understanding of the Maxwell-Lorentz electrodynamics⁴, ⁵, ⁶, ⁷, ⁸, ⁹, ¹⁰, ¹¹ and a working knowledge of vector calculus, differential equations, Fourier transformation, and complex-plane integration techniques.¹² The relevant physical and mathematical arguments have been covered (to varying degrees of clarity and completeness) in numerous textbooks, monographs, and research papers.¹³, ¹⁴, ¹⁵, ¹⁶, ¹⁷, ¹⁸, ¹⁹, ²⁰, ²¹, ²², ²³ The goal of this tutorial is to present the core concepts of the classical theories of scattering and diffraction by starting with Maxwell’s equations and deriving the fundamental results using mathematical arguments that should be accessible to students of optical sciences as well as practitioners of modern optical engineering and photonics technologies. A consistent notation and uniform terminology is used throughout the paper. To maintain the focus on the main results and reduce the potential for distraction, some of the longer derivations and secondary arguments have been relegated to the appendices.

The organization of the paper is as follows. After a brief review of Maxwell’s equations in Sec.2, we provide a detailed analysis of an all-important Green function in Sec.3. The Huygens-Fresnel-Kirchhoff scalar theory of diffraction is the subject of Sec.4, followed by the Rayleigh-Sommerfeld modification and enhancement of that theory in Sec.5. These scalar theories are subsequently generalized in Sec.6 to arrive at a number of formulas for vector scattering and vector diffraction of EM waves under various settings and circumstances. Section 6 also contains a few examples that demonstrate the application of vector diffraction formulas in situations of practical interest. The famous Sommerfeld solution to the problem of diffraction from a perfectly electrically conducting half-plane is presented in some detail in Sec.7.

Applying the vector formulas of Sec.6 to far-field scattering, we show in Sec.8 how the forward scattering amplitude for a plane-wave that illuminates an arbitrary object relates to the scattering cross-section of that object. This important result in the classical theory of scattering is formally known as the optical cross-section theorem (or the optical theorem).¹, ⁹, ²⁴

An alternative approach to the problem of EM scattering when the host medium contains a region of weak inhomogeneities is described in Sec.9. Here, we use Maxwell’s macroscopic equations in conjunction with the Green function of Sec.3 to derive a fairly simple formula for the far-field scattering amplitude in the first Born approximation. The related problem of slow neutron scattering from ferromagnetic media is treated in Sec.10. The paper closes with a few conclusions and final remarks in Sec.11.

2 Maxwell’s equations

The standard equations of the classical Maxwell-Lorenz theory of electrodynamics relate four material sources to four EM fields in the Minkowski spacetime (r, t).⁴, ⁵, ⁶, ⁷, ⁸, ⁹, ¹⁰, ¹¹ The sources are the free charge density ρ_free, free current density J_free, polarization P, and magnetization M, while the fields are the electric field E, magnetic field H, displacement D, and magnetic induction B. In the SI system of units, where the free space (or vacuum) has permittivity ε₀ and permeability μ₀, the displacement is defined as D =ε₀E + P, and the magnetic induction as B =μ₀H + M. Let the total charge-density ρ(r, t) and total current-density J(r, t) be defined as

The charge-current continuity equation, ∇ ⋅ J + ∂_tρ = 0, is generally satisfied by the above densities, irrespective of whether their corresponding sources are free (i.e., ρ_free and J_free), or bound electric charges within electric dipoles (i.e., −∇ ⋅ P and ∂_tP), or bound electric currents within magnetic dipoles (i.e., μ0−1 ∇×M). Invoking Eqs.(1) and (2), Maxwell’s macroscopic equations are written as follows:

Taking the curl of both sides of the third equation, using the vector identity ∇ × ∇ × V = ∇(∇ ⋅ V) − ∇²V, and substituting from the first and second equations, we find

Here, c=(μ0 ε0)−1/2 is the speed of light in vacuum. Similarly, taking the curl of both sides of Eq.(4) and substituting from Eqs.(5) and (6), we find

The scalar and vector potentials, ψ(r, t) and A(r, t), are defined such that B = ∇ × A and E = −∇ψ − ∂_tA. With these definitions, Maxwell’s third and fourth equations are automatically satisfied. In the Lorenz gauge, where ∇ ⋅ A + c⁻²∂_tψ = 0, Maxwell’s second equation yields

Similarly, substitution into Maxwell’s first equation for the E-field in terms of the potentials yields

Since monochromatic fields oscillate at a single frequency ω, their time-dependence factor is generally written as exp(−iωt). Consequently, the spatiotemporal dependence of all the fields and all the sources can be separated into a space part and a time part. For example, the E-field may now be written as E(r)e^−iωt, the total electric charge-density as ρ(r)e^−iωt, and so on. Defining the free-space wavenumber k₀ = ω/c, the Helmholtz equations (7)-(10) now become

In regions of free space, where ρ(r) = 0 and J(r) = 0, the right-hand sides of Eqs.(11)-(14) vanish, thus allowing one to replace k02 E(r) with −∇²E(r), and similarly for B(r), A(r), and ψ(r), whenever the need arises. These substitutions will be used in the following sections.

3 The Green function

In the spherical coordinate system (r, θ, φ), the Laplacian of the spherically symmetric function G(r)=eik0r/r equals ∂2(rG)/r∂r2=−k02 G(r) everywhere except at the origin r = 0, where the function has a singularity. Thus, (∇2+k02)G(r)=0 at all points r except at the origin. A good way to handle the singularity at r = 0 is to treat G(r) as the limiting form of another function that has no such singularity, namely,

The Laplacian of our well-behaved, non-singular function is readily found to be

The first two functions appearing inside the square brackets on the right-hand side of Eq.(16) are confined to the vicinity of the origin at r = 0; they are tall, narrow, symmetric, and have the following volume integrals (see Appendix A for details):

Thus, in the limit of sufficiently small ε, the first two functions appearing on the right-hand side of Eq.(16) can be represented by δ-functions,^[†] and the entire equation may be written as

This is the sense in which we can now state that (∇2+k02)G(r)=−4πδ(r) in the limit when ε → 0. Shifting the center of the function to an arbitrary point r₀, we will have

The gradient of the Green function G(r,  r₀), which plays an important role in our discussions of the following sections, is now found to be

Appendix B provides an analysis of Eq.(20), an inhomogeneous Helmholtz equation, via Fourier transformation.

4 The Huygens-Fresnel-Kirchhoff theory of diffraction.¹, ², ³, ⁹

Consider a scalar function ψ(r) that satisfies the homogeneous Helmholtz equation (∇2+k02)ψ(r)=0 everywhere within a volume V of free space enclosed by a surface S. Two examples of the geometry under consideration are depicted in Fig.1.⁹ In general, ψ(r) can represent the scalar potential or any Cartesian component of the monochromatic E-field, B-field, or A-field associated with an EM wave propagating in free space with frequency ω and wave-number k₀ = ω/c. In Fig.1(a), the EM wave arrives at the surface S₁ from sources located on the left-hand side of S₁, and enters the volume V contained within the closed surface S = S₁ + S₂. In Fig.1(b), the EM waves emanate from inside the closed surface S₁ and permeate the volume V enclosed by S₁ on one side and by a second closed surface S₂ that defines the outer boundary of V. In both figures, the point r₀, where the field is being observed, is located inside the volume V, and the surface normals n^ everywhere on the closed surface S are unit-vectors that point inward (i.e., into the volume V).

Fig.1.

Two surfaces S₁ and S₂ bound the scattering region, which is assumed to be free of sources and material bodies. All the radiation within the scattering region comes from the outside. In (a) the sources of radiation are on the left-hand side of S₁, while in (b) the radiation emanates from the sources inside the closed surface S₁. The observation point r₀ is an arbitrary point within the scattering region. All the points located on S₂ are assumed to be far away from r₀, so that the fields that reach S₂ do not contribute to the fields observed at r₀. At each and every point on S₁ and S₂, the surface normals n^ point into the scattering region.

The essence of the theory developed by G. Kirchhoff in 1882 (building upon the original ideas of Huygens and Fresnel)¹ is an exact mathematical relation between the observed field ψ(r₀) and the field ψ(r) that exists everywhere on the closed surface S. This relation is derived below using the sifting property of Dirac’s δ-function, the relation between the δ-function and the Green function given in Eq.(20), some well-known identities in standard vector calculus, and Gauss’s famous theorem of vector calculus, according to which ∫V∇ ⋅ Vdv=∮SV ⋅ ds. We have

Example.

In the extreme situation where S is a small sphere of radius ε centered at r₀, we will have ψ(r)≅ψ(r0), G(r,r0)=eik0 ε/ε, and ∂n G(r,r0)=−(ik0−ε−1)eik0 ε/ε. Given that the surface area of the sphere is 4πε², the second term in Eq.(22) makes a negligible contribution to the overall integral when ε → 0. The first term, however, contains eik0 ε/ε2, which integrates to 4π in the limit of ε → 0, yielding ψ(r₀) as the final result.

Taking the spherical (or hemi-spherical) surface S₂ in Fig.1 to be far away from the region of interest, the field ψ(r) everywhere on S₂ should have the general form of f(θ,φ)eik0 r/r and, consequently, ∂_nψ(r)∼(ik₀ − r⁻¹)ψ(r). Similarly, across the surface S₂, the Green function has the asymptotic form G(r,r0)∼eik0 r/r and, therefore, ∂_nG(r, r₀)∼(ik₀ − r⁻¹)G(r, r₀). Thus, on the faraway surface S₂, the integrand in Eq.(22) must decline faster than 1/r², which means that the contribution of S₂ to ψ(r₀) as given by Eq.(22) should be negligible. Kirchhoff’s diffraction integral now relates the field at the observation point r₀ to the field distribution on S₁, as follows:

To make contact with the Huygens-Fresnel theory of diffraction, Kirchhoff suggested that both ψ(r) and ∂_nψ(r) vanish on the opaque areas of the screen S₁, whereas in the open (or transparent, or unobstructed) regions, they retain the profiles they would have had in the absence of the screen. These suggestions, while reasonable from a practical standpoint and resulting in good agreement with experimental observations under many circumstances, are subject to criticism for their mathematical inconsistency, as will be elaborated in the next section.

5 The Rayleigh-Sommerfeld theory.

In the important special case where the surface S₁ coincides with the xy-plane at z = 0, one can adjust the Green function in such a way as to eliminate either the first or the second term in the integrand of Eq.(23). These situations would then correspond, respectively, to the so-called Neumann and Dirichlet boundary conditions.¹, ⁹ Let the observation point, which we assume to reside on the right-hand side of the planar surface S₁, be denoted by r0+=(x0,y0,z0), while its mirror image in S₁ (located on the left-hand side of S₁) is denoted by r0− = (x₀, y₀, − z₀). If we use G(r,r0+) in Eq.(23), we obtain ψ(r₀) on the left-hand side of the equation, but if we use G(r,r0−) instead, the integral will yield zero—simply because the peak of the corresponding δ-function now resides outside the integration volume. Thus, we can replace G(r, r₀) in Eq.(23) with either of the two functions G(r,r0+)±G(r,r0−). On the S₁ plane, where z = 0, we have G(r,r0−)=G(r,r0+) and ∂n G(r,r0−)=−∂n G(r,r0+). The resulting diffraction integrals, respectively satisfying the Neumann and Dirichlet boundary conditions, will then be

Note that, while Eq.(23) applies to any arbitrary surface S₁, the Rayleigh-Sommerfeld equations (24) and (25) are restricted to distributions that are specified on a flat plane. Given the scalar field profile ψ(r) and/or its gradient on a flat plane, all three equations are exact consequences of Maxwell’s equations. To apply these equations in practice, one must resort to some form of approximation to estimate the field distribution on S₁. The conventional approximation is that, in the opaque regions of the screen S₁, either ψ(r) or ∂_nψ(r) or both are vanishingly small and, therefore, negligible, whereas in the transparent (or unobstructed) regions of S₁, the field ψ(r) and/or its gradient ∂_nψ(r) (along the surface-normal) retain the profile they would have had in the absence of the screen. In this way, one can proceed to evaluate the integral on the open (or transparent, or unobstructed) apertures of S₁ in order to arrive at a reasonable estimate of ψ(r₀) at the desired observation location.

As a formula for computing diffraction patterns from one or more apertures in an otherwise opaque screen, the problem with Eq.(23) is that, when combined with Kirchhoff’s assumption that both ψ and ∂_nψ vanish on the opaque regions of the physical screen at S₁, it becomes mathematically inconsistent. This is because an analytic function such as ψ(r) vanishes everywhere if both ψ and ∂_nψ happen to be zero on any patch of the surface S₁. In contrast, Eq.(24), when applied to a physical screen, requires only the assumption that ∂_nψ be zero on the opaque regions of the screen. While still an approximation, this is a much more mathematically palatable condition than the Kirchhoff requirement.⁹ Similarly, Eq.(25) requires only the approximation that ψ be zero on the opaque regions. Thus, on the grounds of mathematical consistency, there is a preference for either Eq.(24) or Eq.(25) over Eq.(23). However, given the aforementioned approximate nature of the values chosen for ψ and ∂_nψ across the screen at S₁, it turns out that scalar diffraction calculations based on these three formulas yield nearly identical results, rendering them equally useful in practical applications.¹, ⁹

An auxiliary consequence of Eqs.(24) and (25) is that, upon subtracting one from the other, the integral of ∂_n(Gψ) over the entire flat plane S₁ is found to vanish; that is,

We will have occasion to use this important identity in the following section.

6 Vector diffraction.

Applying the Kirchhoff formula in Eq.(22), where the integral is over the closed surface S = S₁ + S₂, and the function ψ(r) is any scalar field that satisfies the Helmholtz equation, to a Cartesian component of the E-field, say, E_x, we write

Given that Eq.(27) is similarly satisfied by the remaining components E_y, E_z of the E-field, the vectorial version of Kirchhoff’s formula may be written down straightforwardly. Algebraic manipulations (using standard vector calculus identities described in Appendix C) simplify the final result, yielding the following expression for the E-field at the observation point:⁹

Once again, it is easy to show that the contribution of the spherical (or hemi-spherical) surface S₂ to the overall integral in Eq.(28) is negligible. This is because, in the far field, n^ ⋅ E→0 and n^×B→E/c, while |E|∼eik0 r/r, G∼eik0 r/r and ∇G∼−(ik0−r−1)eik0 rn^/r. Consequently,

A similar argument can be used to arrive at the vector Kirchhoff formula for the B-field, namely,

Needless to say, Eq.(30) could also be derived directly from Eq.(29), or vice versa, although the algebra becomes tedious at times; see Appendix D for one such derivation.

We mention in passing that the arguments that led to Eqs.(29) and (30) could not be repeated for the vector potential A(r), since in arriving at Eq.(28), in the step where ∇ ⋅ E or ∇ ⋅ B are set to zero (see Appendix C), we now have ∇ ⋅ A = iωψ/c².^[‡]

In parallel with the arguments advanced previously in conjunction with the Rayleigh-Sommerfeld formulation for scalar fields, one may also modify Eqs.(29) and (30) by setting G(r, r₀) = 0 and ∇G=2∂n G(r,r0)n^, provided that S₁ is a planar surface. This is equivalent to applying Eq.(25) directly to the x, y, z components of the E-field (or the B-field). It is not permissible, however, to retain G and remove ∇G (again, in the case of a planar S₁), because the gradient of G(r,r0+)+G(r,r0−) has a nonzero projection onto the xy-plane.

In those special (yet important) cases where S₁ coincides with the xy-plane at z = 0, one could begin by applying either Eq.(24) or Eq.(25) to the x, y, z components of the field under consideration. Manipulating the resulting equation with the aid of vector-algebraic identities in conjunction with the fact that ∇G = −∇₀G, leads to vector diffraction formulas that could be useful under special circumstances. For instance, the vectorial equivalent of Eq.(25) yields

It is easy to see that the first two terms of the second integral on the right-hand side of Eq.(31) vanish since both GE_x and GE_y go to zero in the far out regions of the xy-plane. The vanishing of the third term, however, requires invoking Eq.(26) as applied to the z-component of the E-field. Note, as a matter of consistency, that on the left-hand side of Eq.(31), ∇0 ⋅ E(r0)=0 and that, on the right-hand side, the divergence of the curl is always zero.

A similar argument can be advanced for the B-field and, therefore, the following vector diffraction equations are generally valid for a planar surface S₁:

In similar fashion, the vectorial equivalent of Eq.(24) yields

On the penultimate line of Eq.(34), the first two terms in the integral are seen to vanish when Eq.(26) is applied to the x and y components of the E-field; the 3^rd and 4^th terms go to zero due to the vanishing of GE_x and GE_y in the far out regions of the xy-plane. Equation (34) thus yields the same expression for E(r₀) as the one reached via Eq.(31).

Example 1.

A plane-wave E(r,t)=(Ex0 x^+Ey0 y^+Ez0 z^)ei(kx x+ky y+kz z−ωt) arriving from the region z < 0 is reflected from a perfectly conducting plane mirror located in the xy-plane at z = 0. The exact cancellation of the tangential components of the E-field at the mirror surface means that the scattered E-field in the xy-plane at z = 0⁺ is given by^[§]

Beyond the mirror in the region z > 0, the scattered E-field is obtained from Eq.(32), as follows:

The 2D Fourier transform appearing in Eq.(36) is readily found to be (see Appendix E):

This is a valid equation whether the incident beam is of the propagating type (i.e., homogeneous plane-wave, with real-valued k_z), or of the evanescent type (i.e., inhomogeneous, with imaginary k_z). Substitution into Eq.(36) now yields

As expected, in the half-space z > 0, the scattered field precisely cancels out the incident field. This result is quite general and applies to any profile for the incident beam, not just plane-waves, for the simple reason that any incident beam can be expressed as a superposition of plane-waves. It should also be clear that, in the absence of the perfectly conducting reflector in the xy-plane, the distribution of the tangential E-field (or the tangential B-field) throughout the xy-plane at z = 0 can be used to reconstruct the entire distribution in the z > 0 half-space via either Eq.(32) or Eq.(33).

Example 2.

Consider a screen in the xy-plane at z = 0 consisting of obstructing segment(s) in the form of thin sheet(s) of perfect conductors and open regions otherwise. A monochromatic incident beam creates surface charges and surface currents on the metallic segment(s) of this planar screen. The scattered fields produced by the induced surface charges and currents can be described in terms of the scattered scalar and vector potentials ψ_s(r, t) and A_s(r, t). Given that the induced surface current has no component along the z-axis, the vector potential will likewise have only x and y components. Application of Eq.(24) to A_sx(r) and A_sy(r) yields

Now, Bs(r)=∇×As(r)=−∂z Asy x^+∂z Asx y^+(∂x Asy−∂y Asx)z^. Consequently,

The symmetry of the scattered field ensures that B_sx and B_sy are zero in the open areas of the screen, so the integral in Eq.(40) need be evaluated only on the metallic surfaces of the screen. We will have

In situations where a thin, flat metallic object acts as a scatterer, Eq.(41) provides a simple way to compute the scattered field provided, of course, that an estimate of the magnetic field at the metal surface (or, equivalently, an estimate of the induced surface current-density) can be obtained. Needless to say, considering that, in the absence of the scatterer, the continued propagation of the incident beam into the z > 0 half-space can be reconstructed from the tangential component of the incident B-field in the z = 0 plane (see Example 1), one may add the incident B-field to the scattered field of Eq.(41) and obtain, unsurprisingly, the general vector diffraction formula of Eq.(33).

Example 3.

Figure 2 shows a perfectly conducting thin sheet residing in the upper half of the xy-plane at z = 0. The incident beam is a monochromatic plane-wave of frequency ω, wavelength λ₀ = 2πc/ω, linear-polarization aligned with y^, and propagation direction along the unit-vector σinc=(sin⁡θinc)x^+(cos⁡θinc)z^, as follows:

Fig.2.

A perfectly conducting thin sheet sits in the upper half of the xy-plane at z = 0. The incident plane-wave, which is linearly polarized along the y-axis, has amplitude E_inc, frequency ω, wave-number k₀ = ω/c, and propagation direction σinc=sin θinc x^+cos θinc z^. The observation point r₀ is in the xz-plane.

Using Eq.(32), we derive the diffracted E-field at the observation point r0=x0 x^+z0 z^. (The symmetry of the problem ensures that the field profile is independent of the y-coordinate of the observation point.) In what follows, we use the 0^th and 1^st order Hankel functions of type 1, H0(1)(ζ) and H1(1)(ζ), as well as the (complex) Fresnel integral F(ζ)=∫ζ∞exp(ix2)dx.²⁶, ²⁷ For a graphical representation of the Fresnel integral via the so-called Cornu spiral,²⁸ see Appendix F. The following identities will be needed further below:

The E-field at the observation point r₀ is found to be

Here, we have used the large-argument approximate form of H1(1)(ζ) given by Eq.(45). Assuming that z₀ ≫ |x − x₀|, we proceed by invoking the following approximation:

Note that this approximation is not accurate when |x−x0| acquires large values; however, the rapid phase variations of the integrand in Eq.(47) ensure that the contributions to the integral at points x that are far from x₀ are insignificant. Substitution from Eq.(48) into Eq.(47) yields

At the edge of the geometric shadow, i.e., the straight line where x₀ = z₀sin θ_inc, we have F(0)=1/2πeiπ/4. On this line, the E-field amplitude is 1/2E_inc. Above the shadow’s edge, the field amplitude steadily declines, whereas below the edge, there occur a large number of oscillations before the field settles into what is essentially the incident plane-wave.¹Figure 3 shows a typical plot of the far-field intensity (i.e., the square of the E-field amplitude) versus the distance (along the x-axis at a fixed value of z₀) from the edge of the geometric shadow.

Fig.3.

Normalized intensity in the far field of a sharp edge as a function of distance along the x-axis (at fixed z₀) from the edge of the geometric shadow, where x₀ = z₀sin θ_inc.

Example 4.

Shown in Fig.4 is a circular aperture of radius R within an otherwise opaque screen located in the xy-plane at z = 0. A plane-wave E(r, t) = E_incexp[i(k₀  σ_inc ⋅ r − ωt)], where the unit-vector σ_inc specifies the direction of incidence, arrives at the aperture from the left-hand side. Maxwell’s 3^rd equation identifies the incident B-field as B = σ_inc × E/c. The observation point r₀ = r₀σ₀ is sufficiently far from the aperture for the following approximation to apply:

Fig.4.

A circular aperture of radius R inside an otherwise opaque screen located at z = 0 is illuminated by a plane-wave whose E-field amplitude and propagation direction are specified as E_inc and σ_inc, respectively. The observation point r₀= r₀σ₀ is in the far field; that is, r₀ ≫ R. The unit-vectors σ_inc and σ₀ have respective polar coordinates (θ_inc, φ_inc) and (θ₀, φ₀). The projection of the vector σ_inc− σ₀ onto the xy-plane of the aperture is a vector of length ζ that makes an angle φ with the position vector r=xx^+yy^.

Let us assume that the screen is a thin sheet of a perfect conductor on whose surface the tangential E-field necessarily vanishes. The appropriate diffraction equation will then be Eq.(32), with the tangential E-field outside the aperture allowed to vanish (i.e., n^×E=0 on the opaque parts of the screen). Approximating the E-field within the aperture with that of the incident plane-wave, we will have

Here, ζ is the magnitude of the projection of σ_inc− σ₀ onto the xy-plane of the aperture, while φ is the angle between that projection and the position vector r=xx^+yy^; that is,

Consequently,

The E-field at the observation point r₀ = r₀σ₀ (which is in the far field of the aperture, i.e., r₀ ≫ R) is now found by computing the curl of the expression on the right-hand side of Eq.(54). Considering that z^×Einc is independent of r₀, we use the vector identity ∇ × (ψV₀) = ∇ψ × V₀, then ignore the small (far field) contributions to ∇ψ due to the dependence of ζ on (θ₀, φ₀), to arrive at

The approximate nature of these calculations should be borne in mind when comparing the various estimates of an observed field obtained via different routes.⁹ For instance, had we started with Eq.(33) and proceeded by setting n^×B=0 on the opaque areas of the screen, we would have arrived at the following estimate of the observed E-field:

The differences between Eqs.(55) and (56), which, in general, are not insignificant, can be traced to the assumptions regarding the nature of the opaque screen and the approximations involved in equating the E and B fields within the aperture to those of the incident plane-wave.

7 Sommerfeld’s analysis of diffraction from a perfectly conducting half-plane.

A rare example of an exact solution of Maxwell’s equations as applied to EM diffraction was published by A. Sommerfeld in 1896.²⁵ The simplified version of Sommerfeld’s original analysis presented in this section closely parallels that of Ref.[1], Chapter 11. Consider an EM plane-wave propagating in free space and illuminating a thin, perfectly conducting screen that sits in the upper half of the xy-plane at z = 0. The geometry of the system is shown in Fig.5, where the incident k-vector is denoted by k_inc, the oscillation frequency of the monochromatic wave is ω, the speed of light in vacuum is c, the wave-number is k₀ = ω/c, and the unit-vector along the direction of incidence is σ_inc. The plane-wave is linearly polarized with its E-field amplitude E₀ along the y-axis and H-field amplitude H₀ = E₀/Z₀ in the xz-plane of incidence. (Z0=μ0/ε0 is the impedance of free space.) The electric current density induced in the semi-infinite screen is denoted by J(r, t). In the chosen geometry, all the fields (incident as well as scattered) are uniform along the y-axis; consequently, it suffices to specify the position of an arbitrary point in the Cartesian xyz space by its x and z coordinates only; that is,

Fig.5.

A plane, monochromatic EM wave propagating along the unit-vector σinc=cos⁡θ0 x^+sin⁡θ0 z^ arrives at a thin, semi-infinite, perfectly electrically conducting screen located in the upper half of the xy-plane. The plane-wave is linearly polarized, with its E-field aligned with the y-axis, while its H-field has components along the x and z directions. The induced current sheet, denoted by J(r, t), oscillates parallel to the incident E-field along the direction of the y-axis. (The system depicted here is essentially the same as that in Fig.2, with the exception of the angle of incidence θ₀ being the complement of θ_inc of Fig.2.)

Here, r=x2+z2, and the angle θ is measured clockwise from the positive x-axis. The range of θ is (0, π) for points r on the right-hand side of the screen, and (−π, 0) for the points on the left-hand side. Figure 5 shows that k_inc makes an angle θ₀ ∈ (0, π) with the positive x-axis. The incident plane-wave is, thus, fully specified by the following equations:

With the aid of the step-function step(x) and Dirac’s delta-function δ(z), we now express the electric current density J(r, t) excited on the thin-sheet conductor as a one-dimensional Fourier transform, namely,

In the above equation, σ_x represents spatial frequency along the x-axis, and J(σx) is the (complex) amplitude of the induced surface-current-density along y^, having spatial and temporal frequencies σ_x and ω, respectively. The first goal of our analysis is to find the function J(σx) such that its Fourier transform vanishes along the negative x-axis—as demanded by the step-function in Eq.(61)—while its radiated E-field cancels the incident plane-wave’s E-field at the surface of the screen along the positive x-axis. Now, a current sheet J(σx)ei(k0 σx x−ωt)δ(z)y^ that fills the entire xy-plane at z = 0 radiates EM fields into the surrounding free space that can easily be shown to have the following structure:

Here, k₀ = ω/c is the wave-number in free space, while k_x = k₀σ_x, and kz=k02−kx2 are the k-vector components along the x and z axes. In general, k_z must be real and positive when |k_x|≤k₀, and imaginary and positive otherwise. The ± signs associated with H_x indicate that the plus sign must be used for the half-space on the right-hand side of the sheet, where z > 0, while the minus sign is reserved for the left-hand side, where z < 0. The discontinuity of H_x at z = 0 thus equals the surface-current-density J(σx) along the y-direction, in compliance with Maxwell’s requisite boundary conditions. Needless to say, Eqs.(62) and (63) represent a single EM plane-wave on either side of the screen’s xy-plane, which satisfy the symmetry requirement of radiation from the current sheet, and with the tangential component H_x of the radiated H-field chosen to satisfy the requisite boundary condition at the plane of the surface current. The plane-waves emanating to the right and left of the xy-plane of the current sheet are homogeneous when k_z is real, and inhomogeneous (or evanescent) when k_z is imaginary.

When working in the complex σ_x-plane, we must ensure that k_z has the correct sign for all values of the real parameter k_x = k₀σ_x from −∞ to ∞. Considering that k_z/k₀ is the square root of the product of (1 − σ_x) and (1 + σ_x), we choose for both of these complex numbers the range of phase angles (−π, π], as depicted in Fig.6(a). The corresponding branch-cuts thus appear as the semi-infinite line segments (−∞, −1) and (1, ∞), and the integration path along the real axis within the σ_x-plane, shown in Fig.6(b), will consist of semi-infinite line segments slightly above and slightly below the real axis, as well as a short segment of the real-axis connecting −1 to 1. This choice of the integration path ensures that kz=k01−σx2 acquires the correct sign for all values of σ_x.

Fig.6.

(a) In the complex σ_x-plane, where the phase of the complex numbers 1 ± σ_x is measured counterclock-wise from the positive σx′ axis, the range of both angles is confined to the (−π,π] interval. (b) The integration path along the σx′ axis is adjusted by shifting the part from −∞ to −1 slightly upward, and the part from 1 to ∞ slightly downward, so that kz/k0=(1−σx)(1+σx) has the correct sign everywhere. The pole at σx=cos⁡θ0 is handled by locally deforming the contour into a semi-circular path below the real axis.

For reasons that will become clear as we proceed, Sommerfeld suggested the following mathematical form for the surface current-density J as a function of σ_x:

The domain of this function is the slightly deformed real axis of the σ_x-plane depicted in Fig.6(b). The proposed function has a simple pole at σx=cos⁡θ0, where θ₀ is the orientation angle of k_inc shown in Fig.5, and a (complex) constant coefficient J0 that will be determined shortly. In addition, J(σx) contains the term 1−σx, whose branch-cut in the system of Fig.6(a) is the semi-infinite line-segment (1, ∞) along the real axis of the σ_x-plane. It is now possible to demonstrate that the proposed J(σx) satisfies its first required property, namely,

For x < 0, the integration contour of Fig.6(b) can be closed with a large semi-circular path in the lower half of the σ_x-plane. The only part of the integrand in Eq.(65) that requires a branch-cut is 1−σx, whose branch-cut is the line segment (1, ∞). The integration contour of Fig.6(b), when closed in the lower-half of the σ_x-plane, does not contain this branch-cut. Moreover, the pole at σx=cos⁡θ0 is outside the closed loop of integration, and the contributions to the integral by the singular points σ_x = ±1 are zero. Consequently, the current-density J(r, t) in the lower half of the xy-plane turns out to be zero, exactly as required.

The radiated E-field, a superposition of contributions from the various J(σx) in accordance with Eqs.(62) and (64), must cancel out the incident E-field at the surface of the screen; that is,

For x > 0, the contour of integration in Eq.(66) can be closed with a large semi-circle in the upper-half of the σ_x-plane. The branch-cut for 1+σx in the denominator of the integrand is the line-segment (−∞, −1), which is below the integration path and, therefore, irrelevant for a contour that closes in the upper-half-plane. For the singularities at σ_x = ±1, the residues are zero, whereas for the pole at σx=cos⁡θ0, the residue is eik0 cos⁡θ0 x/1+cos⁡θ0. The requisite boundary condition at the screen’s surface is thus seen to be satisfied if J0 is specified as

Having found the functional form of J(σx), we now turn to the problem of computing the scattered E-field E_s(r) at the arbitrary observation point r=xx^+zz^=r(cos⁡θx^+sin⁡θz^)in accordance with Eqs.(62), (64), and (67); that is,

The integral in Eq.(68) must be evaluated at positive as well as negative values of x, and also for values of z on both sides of the screen. The presence of k_z in the exponent of the integrand requires that the branch-cuts on both line-segments (−∞, −1) and (1, ∞) be taken into account. For these reasons, the integration path of Fig.6(b) ceases to be convenient and we need to change the variable σ_x to something that avoids the need for branch-cuts. We now switch the variable from σ_x to φ, where σx=cos⁡φ, with the integration path in the complex φ-plane shown in Fig.7(a). Considering that

the depicted integration path represents the continuous variation of σ_x from −∞ to ∞. Similarly,

is positive real on the horizontal branch, and positive imaginary on both vertical branches of the depicted contour. It is also easy to verify that 1−σx=2sin(φ/2) has identical values at corresponding points on the contours of Figs.(6b) and (7a), as does 1+σx=2cos(φ/2). Recalling that dσx=−sin⁡φ dφ=−1−cos2⁡φ dφ, we can rewrite Eq.(68) as

Fig.7.

(a) Contour of integration in the complex φ-plane corresponding to the integration path in the σ_x-plane shown in Fig.6(b). By definition, σ_x = cosφ, which results in kz=k0 σz=k0⁡sin⁡φ, a well-defined function everywhere in the φ-plane that does not need branch-cuts. The small bump in the integration path around φ = θ₀ corresponds to the small semi-circular part of the contour in Fig.6(b). The negative values of σx″ on the semi-circle translate, in accordance with Eq.(69), into positive value of φ″ on the corresponding bump. (b) Along the steepest-descent contour S(θ) that passes through the saddle-point φ = θ, Re[cos(φ − θ)] is constant. At the saddle-point, S(θ) makes a 45° angle with the horizontal and vertical lines, which represent contours along which Im[cos(φ − θ)] is constant. S(θ) also has the property that Im[cos(φ − θ)], which is zero at the saddle-point, rises toward ∞ (continuously and symmetrically on both sides of the saddle) as φ moves away from the saddle-point along the contour.

The above equation yields the scattered E-field on both sides of the screen, with the minus sign in the exponent corresponding to z > 0, and the plus sign to z < 0. In what follows, noting the natural symmetry of the scattered field on the opposite sides of the xy-plane, we confine our attention to Eq.(71) with only the minus sign in the exponent; the scattered E-field on the left hand side of the screen is subsequently obtained by a simple change of the sign of z.

The integration path of Fig.7(a) is now replaced with the steepest-descent contour S(θ) depicted in Fig.7(b). Passing through the saddle-point of exp[ikr⁡cos(φ−θ)] at φ = θ, this contour has the property that Re[cos(φ − θ)] everywhere on the contour equals 1. In contrast, Im[cos(φ − θ)] starts at zero at the saddle, then rises toward infinity (continuously and symmetrically on opposite sides of the saddle) as φ moves away from the saddle-point at φ = θ. It is easy to show that the original integration path of Fig.7(a) can be joined to S(θ) to form a closed loop with negligible contributions to the overall loop integral by the segments that connect the two contours at infinity.^[**]

The only time when the single pole of the integrand at φ = θ₀ needs to be accounted for is when θ < θ₀, at which point the pole is inside the closed contour, as Fig.7(b) clearly indicates. In the vicinity of the pole, the denominator of the integrand in Eq.(71) can be approximated by the first two terms of the Taylor series expansion of cosφ, as follows:

The residue at the pole is thus seen to be −exp[ik0 r⁡cos(θ0−θ)], with a corresponding contribution of −E0 y^exp[ik0 r⁡cos(θ−θ0)] to the scattered E-field. It must be emphasized that the scattered field contributed by the pole at φ = θ₀ is relevant only when θ < θ₀, in which case it cancels the contribution of the incident plane-wave of Eq.(59) to the overall EM field on the right-hand side of the screen, where the screen casts its geometric shadow. Outside this geometric shadow, where θ₀ < θ ≤ π, the incident beam spills into the z ≥ 0 region—without the counter-balancing effect of the scattered field produced by the pole at φ = θ₀.

Returning to the scattered E-field produced by the integral in Eq.(71) over the steepest-descent contour S(θ), the first term in the integrand can be further streamlined, as follows:

The scattered E-field on the right-hand side of the screen may thus be written as

Note that we have factored out the imaginary part of the exponent and taken it outside the integral as eik0 r; what remains of the exponent, namely, ik0 r[cos(φ−θ)−1], is purely real on S(θ). We proceed to compute the integral in Eq.(74) only for the first term of the integrand and refer to it as E_s,1; the contribution of the second term, E_s,2, will then be found by switching the sign of θ₀.

A change of the variable from φ to φ − θ would cause the steepest-descent contour to go through the origin of the φ-plane; this shifted contour will now be denoted by S₀. Taking advantage of the symmetry of S₀ with respect to the origin, we express the final result of integration in terms of the integral on the upper half of S₀, which is denoted by S0+. We will have

Another change of variable, this time from φ to the real-valued ζ=−exp(iπ/4)sin(φ/2), where ζ ranges from 0 to ∞ along the steepest-descent contour S0+, now yields^[††]

The integral appearing in the above equation is evaluated in Appendix G, where it is shown that

Here, F(α)=∫α∞exp(ix2)dx is the complex Fresnel integral defined in Eq.(46). We thus find

The expression for E_s,2(r) is derived from Eq.(78) by switching the sign of θ₀. One has to be careful in this case, since sin[(θ − θ₀)/2] may be negative. Given that both θ and θ₀ are in the (0, π) interval, the sign of sin[(θ − θ₀)/2] will be positive if θ > θ₀, and negative if θ < θ₀. The scattered E-field of Eq.(74) is thus given by

To this result we must add the contribution of the pole, namely, −E0 y^exp[ik0 r⁡cos(θ−θ0)] when θ < θ₀, and the incident beam E0 y^exp[ik0 r⁡cos(θ−θ0)]for the entire range 0 ≤ θ ≤ π. Recalling that

the total E-field on the right-hand side of the xy-plane, where 0 ≤θ ≤ π, becomes

On the left-hand side of the screen, where z < 0 and, therefore, −π ≤ θ ≤ 0, the scattered E-field is obtained by replacing θ with |θ| in Eq.(79) as well as in the contribution by the pole at φ = θ₀. (Appendix H shows that the scattered E-field in the z < 0 region can also be evaluated by direct integration over a modified contour in the complex φ-plane.) Once again, adding the incident E-field and invoking Eq.(80), we find

One obtains Eq.(82) by replacing θ in Eq.(81) with 2π + θ, the latter θ being in the (−π, 0) interval. Thus, Eq.(81) with 0 ≤ θ ≤ 2π covers the entire range of observation points r. A clear understanding of Eq.(81) requires familiarity with the general behavior of the Fresnel integral F(α); Appendix F contains a detailed explanation in terms of the Cornu spiral representation of F(α).^[‡‡] The general expression for the total (i.e., incident plus scattered) E-field given in Eq.(81) is somewhat simplified in terms of the new function Φ(α)=e−iα2F(α), as follows:

It is worth mentioning that, the contribution to the scattered E-field by the pole at φ = θ₀, namely, Es,3(r)=−E0 y^exp[ik0 r⁡cos(θ−θ0)], which exists only in the intervals 0 ≤ θ < θ₀ and 2π − θ₀ < θ ≤ 2π, cancels the incident E-field in the shadow region behind the screen, while acting as the reflected field in front of the perfectly conducting half-mirror.

7.1 The magnetic field.

The total H-field is computed from the E-field of Eq.(83) with the aid of Maxwell’s equation ik0 Z0 H(r)=∇×E(r)=−(r−1 ∂θ Ey)r^+(∂r Ey)θ^. The identity Φ′(α)=−1−2iαΦ(α)will be used in this calculation. To simplify the notation, we introduce the new variables ξ1=−2k0 r sin[(θ−θ0)/2] and ξ2=2k0 rsin[(θ+θ0)/2]. We find

The Cartesian components H_x = H_rcos θ − H_θ sin θ and H_z = H_r sin θ + H_θ cos θ of the magnetic field may now be obtained from the polar components H_r and H_θ, as follows:

It is readily verified that H_z = 0 at the surface of the conductor, where θ = 0, and that H_x equals the x component of the incident H-field in the open half of the xy-plane, where θ = π.

8 Far field scattering and the optical theorem.

In the system of Fig.1(b), let the object inside the closed surface S₁ be illuminated from the outside, and let the observation point r₀ = r₀σ₀ be far away from the object, so that the approximate form of G(r, r₀) given in Eq.(50) along with its corresponding gradient ∇G≅−ik0 σ0 G(r,r0) would be applicable. Denoting by E_s(r) and B_s(r) the scattered fields appearing on S₁, Eq.(28) yields the E-field at the observation point as

Considering that the local field in the vicinity of r₀ has the character of a plane-wave, the last term in the above integrand, which represents a contribution to E_s(r₀) that is aligned with the local k-vector, is expected to be cancelled out by an equal but opposite contribution from the first term.⁹ We thus arrive at the following simplified version of Eq.(88):

With reference to Fig.8, suppose now that the object is illuminated (and thus excited) by a plane-wave arriving along the unit-vector σ_inc, whose E and B fields are

Fig.8.

A monochromatic plane-wave propagating along the unit-vector σ_inc is scattered from a small object in the vicinity of the origin of the coordinate system. The scattered electric and magnetic fields on the closed surface S₁ surrounding the object are denoted by E_s(r) and B_s(r). The surface normals n^ at every point on S₁ are outward directed. The scattered light reaching the far away observation point r₀ = r₀σ₀ has the k-vector k₀σ₀ and the EM fields E_s(r₀) and B_s(r₀).

The time-averaged total Poynting vector on the surface S₁ is readily evaluated, as follows:

If we dot-multiply both sides of Eq.(92) by −n^, then integrate over the closed surface S₁, we obtain, on the left-hand side, the total rate of the inward flow of EM energy, which is the absorbed EM power by the object. On the right-hand side, the first term integrates to zero, because (Ei⋅Ei∗)σinc is a constant and ∮S1n^ds=0. The integral of the second term will be the negative time-rate of the energy departure from the object via scattering, which can be moved to the left-hand side of the equation. The combination of the two terms on the left-hand side now yields the total EM power that is taken away from the incident beam—either by absorption or due to scattering. We will have

Comparison with Eq.(88) reveals that the integral in Eq.(93) is proportional to the scattered E-field in the direction of σ₀ = σ_inc as observed in the far field. (Recall that the term (n^⋅ES)σ0 in the integrand of Eq.(88) has been deemed inconsequential.) If Eq.(93) is normalized by the incident EM power per unit area, namely, Pinc=1/2 Z0−1 Re(Ei ⋅ Ei*), the left-hand side will become the scattering cross-section of the object, while the right-hand side, aside from the coefficient ik0 eik0 r0/(4πr0), will be the projection of the forward-scattered E-field (i.e., σ₀ = σ_inc) on the incident E-field. This important result in the theory of scattering has come to be known as the optical theorem (or the optical cross-section theorem).¹, ⁹, ²⁴

9 Scattering from weak inhomogeneities.⁹

Figure 9 shows a monochromatic plane-wave of frequency ω passing through a transparent, linear, isotropic medium that has a region of weak inhomogeneities in the vicinity of the origin of coordinates. The host medium is described by its relative permittivity ε(r, ω) and permeability μ(r, ω), which consist of a spatially homogeneous background plus slight variations (localized in the vicinity of r = 0) on this background; that is,

Fig.9.

A monochromatic plane-wave of frequency ω and E-field amplitude E_inc propagates along the unit-vector σ_inc within a mildly inhomogeneous host medium of refractive index nh(ω)=(μh εh)1/2. The inhomogeneous region of the host, a small patch in the vicinity of the origin of coordinates, is specified by its relative permittivity ε_h(ω) + δε(r, ω) and relative permeability μ_h(ω) + δμ(r, ω). The scattered field is observed at the faraway point r₀ = r₀σ₀.

The displacement field is thus written as D(r, ω) = ε₀ε(r, ω)E(r, ω) and the magnetic induction is given by B(r, ω) = μ₀μ(r, ω) H(r, ω). In the absence of free charges and currents, i.e., when ρ_free(r, ω) = 0 and J_free (r, ω) = 0, Maxwell’s macroscopic equations will be⁴, ⁵, ⁶, ⁷, ⁸, ⁹, ¹⁰, ¹¹

Noting that the D and B fields depart only slightly from the respective values ε₀ε_hE and μ₀μ_hH that they would have had in the absence of the δε and δμ perturbations, we write

Recalling that the refractive index of the homogeneous (background) material is defined as nh=(μh εh)1/2, and that the free-space wave-number is k₀ = ω/c, Eq.(97) may be rewritten as

This Helmholtz equation has a homogeneous solution, which we denote by D_h(r, ω), and an inhomogeneous solution, which arises from the local deviations δε and δμ of the host medium. Recalling that the Green function G(r, r₀) = exp (ik₀n_h|r − r₀|)/|r − r₀| is a solution of the Helmholtz equation ∇²G + (k₀n_h)²G = −4πδ(r − r₀), an integral relation for the scattered field solution D_s(r, ω) of Eq.(98) in terms of the total fields E(r, ω) and H(r, ω) will be

Using a far-field approximation to G(r, r₀) similar to that in Eq.(50), we will have

The vector identity (∇×V)e−ik  ⋅  r=ik×Ve−ik ⋅  r+∇×(Ve−ik  ⋅  r) can be used to replace the first term in the integrand of Eq.(100) with −(k02 εh nh/c)e−ik0 nh σ0 ⋅  rσ0×δμH. The volume integral of ∇×(Ve−ik  ⋅  r) becomes the surface integral of e−ik ⋅ r V×ds, which subsequently vanishes because, for away from the inhomogeneous region, δμ → 0. Similarly, the second term of the integrand is replaced by iε0k0nhσ0×[∇×(δεE)]e−ik0nhσ0 ⋅ r. Another application of the vector identity then replaces the remaining term with −ε0 k02 nh2 σ0×(σ0×δεE)e−ik0 nh σ0 ⋅  r. We thus arrive at

In the first Born approximation, the E(r) and H(r) fields in the integrand of Eq.(101) are replaced with the solutions E_h(r) and H_h(r) of the homogeneous Helmholtz equation. When the homogeneous background wave is a plane-wave, we will have

A final substitution from Eqs.(102) and (103) into Eq.(101) yields

Thus, in the first Born approximation, the scattered field D_s(r₀, ω) = ε₀ε_hE_s(r₀, ω) is directly related to the host medium perturbations δε(r, ω) and δμ(r, ω) via the volume integral in Eq.(104). Here, E_inc embodies not only the strength but also the polarization state of the incident plane-wave, the unit-vector σ_inc is the direction of incidence, σ₀ = r₀/r₀ is a unit-vector pointing from the origin of coordinates to the observation point r₀, and q = k₀n_h(σ_inc − σ₀) is the difference between the incident and scattered k-vectors.

10 Neutron scattering from magnetic electrons in Born’s first approximation.

The scattering of slow neutrons from ferromagnetic materials can be treated in ways that are similar to our analysis of EM scattering from mild inhomogeneities discussed in the preceding section. The wave function ψ(r, t) of a particle of mass m in the scalar potential field V(r, t) satisfies the following Schrödinger equation:⁶, ⁷, ⁸, ²⁰

When the potential is time-independent and the particle is in an eigenstate of energy ℰ_n, we will have the time-independent Schrödinger equation, namely,

The Green function for Eq.(106) is G(r,r′)=eik|r – r′|/|r−r′|, where ℏk=(2 m ℰn)1/2. If, in the absence of the potential V(r), the solution of the homogeneous equation is found to be ψ₀(r), then, when the potential is introduced, we will have

Note that Eq.(107) is not an actual solution of Eq.(106); rather, considering that the desired wave-function ψ(r) appears in the integrand on the right hand-side, Eq.(107) is an integral form of the differential equation (106). In the case of Born’s first approximation, one assumes that V(r) is a fairly weak potential, in which case ψ0(r′) can be substituted for ψ(r′), yielding

In a typical scattering problem, an incoming particle of mass m and well-defined momentum p=ℏk has the initial wave-function ψ0(r)=eik  ⋅  r. Upon interacting with a weak scattering potential V(r), the wave-function will change in accordance with Eq.(108). Let V(r) have significant values only in the vicinity of the origin, r = 0, and assume that the scattering process is elastic, so that the momentum p˜ of the particle after scattering will have the same magnitude ℏk as before, but the direction of propagation changes from that of k to that of k˜. At an observation point r far from the origin, that is, |r|≫|r′|, the scattered particle’s momentum is expected to be p˜=ℏk˜=ℏkr^, and the Green function may be approximated as

Substitution into Eq.(108) now yields

Denoting the change in the direction of the particle’s momentum by q=p−p˜, and noting that e^ikr/r is simply a spherical wave emanating from the origin, the scattering amplitude f(q) is readily seen to be

Here, f(q) has the dimensions of length (meter in SI). Note that the presence of e^ikr/r in Eq.(110) makes the wave-function ψ(r) dimensionless. Let dΩ=sin⁡θ dθ dφ be the differential element of the solid angle viewed from the origin of the coordinates. The differential scattering cross-section will then be dσ/dΩ=|f(q)|2, with the total cross-section being σ=∫|f(q)|2dΩ.

Example 1

A particle of mass m is scattered by the spherically symmetric potential V(r) corresponding to a fixed particle located at r = 0. The scattering amplitude, computed from Eq.(111), will be

Considering that q=p−p˜, and that p and p˜ have the same magnitude p, we denote by θ the angle between p and p˜, and proceed to write q = 2p sin (θ/2). The scattering amplitude thus has circular symmetry around the direction of the incident momentum p. The ambiguity of Eq.(112) with regard to forward scattering at θ = 0 is resolved if the forward amplitude f(0) is directly computed from Eq.(110)—with the destructive interference between the incident and scattered amplitudes properly taken into account.

For the Yukawa potential V(r) = v₀e^−αr/r, with α > 0 being the range parameter, the scattering amplitude is obtained upon integrating Eq.(112), as follows:

In the limit when α → 0, the Yukawa potential approaches the Coulomb potential V(r) = v₀/r. When a particle having electric charge ±e and energy ℰ=p2/2m is scattered from another particle of charge ±e, the scattering cross-section will be

This is the famous Rutherford scattering cross-section.⁹

Example 2

In low-energy scattering, k≅0 and the scattering amplitude in all directions becomes f(θ,φ)≅−(m/2πℏ2)∫−∞∞V(r)dr. In the case of low-energy, soft-sphere scattering, where V(r) = V₀ when r ≤ r₀ and zero otherwise, we find

Let us now consider the case of a polarized neutron entering a ferromagnetic medium and getting scattered from the host’s magnetic electrons.²⁹ To obtain an estimate of the corresponding scattering potential V(r), we begin by noting that the magnetic field surrounding a point-dipole mδ(r)z^ in free space is

A second magnetic point-dipole m^’, located at r≠0, will have the energy ℰ=−m′ ⋅ H(r), which may be written as follows:

These results are consistent with the Einstein-Laub formula F = (m ⋅ ∇)H for the force as well as T = m × H for the torque experienced by a point-dipole in a magnetic field.³⁰, ³¹ Recall that, in contrast to the standard formula for the dipole moment, our definition of B as μ₀H + M maintains that the magnitude of m equals μ₀ times the electrical current times the area of a small current loop. Consequently, the aforementioned expression for ℰ coincides with the well-known expression ℰ = −m ⋅ B(r) of the energy of the dipole m in the external field B(r) = μ₀H(r).⁶, ⁹

Equation (117) must be augmented by the contact term −2m ⋅ m^’δ(r)/3μ₀ to account for the energy of the dipole pair when m and m^’ overlap in space.²⁹ We will have

Suppose the electron has wave-function ψ(r_e) and magnetic dipole moment μ_e, while the incoming neutron has wave-function exp(ik ⋅ rn), magnetic dipole moment μ_n, and mass m_n. We assume the scattering process does not involve a spin flip, so that both μ_e and μ_n retain their orientations after the collision. Moreover, we assume the electron—being bound to its host lattice—does not get dislocated or otherwise distorted, so that ψ(r_e) is the same before and after the collision. Thus, the potential energy distribution across the landscape of the incoming neutron is the integral over r_e of the product of the electron’s probability-density function |ψ(r_e)|² and the dipole-dipole interaction energy ℰ(r_e − r_n) between the neutron and the electron. Consequently, in the first Born approximation, the scattering amplitude from an initial neutron momentum p=ℏk to a final momentum p′=ℏk′, is given by

Defining the electronic magnetization (i.e., magnetic moment density of the electron) by M(re)=|ψ(re)|2μe and its Fourier transform by M˜(k)=∫−∞∞M(re)exp(ik ⋅ re)dre, upon substitution from Eq.(118) into Eq.(119) and setting r = r_n – r_e, we find

Defining q=p−p′ and changing the variables from (r_e, r_n) to (r_e, r = r_n − r_e) —whose transformation Jacobian is 1.0 —substantially simplifies the above integral, yielding

Appendix I shows that the exact evaluation of the integral in Eq.(121) leads to

This is the same result as given in Ref.[29], Eq.(23), in the case of λ = 1. The coefficient 4πμ₀ appears here because we have worked in the SI system of units with B = μ₀H + M.

11 Concluding remarks.

In this paper, we described some of the fundamental theories of EM scattering and diffraction using the electrodynamics of Maxwell and Lorentz in conjunction with standard mathematical methods of the vector calculus, complex analysis, differential equations, and Fourier transform theory. The scalar Huygens-Fresnel-Kirchhoff and Rayleigh-Sommerfeld theories were presented at first, followed by their extensions that cover the case of vector diffraction of EM waves. Examples were provided to showcase the application of these vector diffraction and scattering formulas to certain problems of practical interest. We did not discuss the alternate method of diffraction calculations by means of Fourier transformation, which involves an expansion of the initial field profile in the xy-plane at z = 0 into its plane-wave constituents. In fact, with the aid of the two-dimensional Fourier transform of G(r, 0) given in Eq.(37), it is rather easy to establish the equivalence of the Fourier expansion method with the Rayleigh-Sommerfeld formula in Eq.(25), and also with the related vector formulas in Eqs.(32) and (33). Appendix J outlines the mathematical steps needed to establish these equivalencies.

The Sommerfeld solution to the problem of diffraction from a thin, perfectly conducting half-plane described in Sec.7 is one of the few problems in the EM theory of diffraction for which an exact analytical solution has been found; for a discussion of related problems of this type, see Ref.[1], Chapter 11. Scattering of plane-waves from spherical particles of known relative permittivity ε(ω) and permeability μ(ω), the so-called Mie scattering, is another problem for which an exact solution (albeit in the form of an infinite series) exists; for a discussion of this and related problems the reader is referred to the vast literature of Mie scattering.¹, ⁴, ⁹, ¹⁰, ¹¹²²

In our analysis of neutron scattering from ferromagnets in Sec.10, we used the contact term −2m ⋅ m′ δ(r)/3μ0 to account for the interaction energy of the dipole pair m,m′ when they happen to overlap at the same location in space. This is tantamount to assuming that the dipolar magnetic moments are produced by circulating electrical currents. The contact term would have been m ⋅ m′δ(r)/3μ0 had each magnetic moment been produced by a pair of equal and opposite magnetic monopoles residing within the corresponding particle. Since the Amperian current loop model has been found to agree most closely with experimental findings, we used the former expression for the contact term in Eq.(118); see Ref.[9], Sec.5.7, and Ref.[29] for a pedagogical discussion of the experimental evidence—from neutron scattering as well as the existence of the famous 21 cm astrophysical spectral line of atomic hydrogen—in favor of the Amperian current loop model of the intrinsic magnetic dipole moments of subatomic particles.

Although we did not discuss the Babinet principle of complementary screens that is well known in classical optics, it is worth mentioning here that a rigorous version of this principle has been proven in Maxwellian electrodynamics.¹, ⁹ The original version of Babinet’s principle is based on the Kirchhoff diffraction integral of Eq.(23), and the notion that, if S₁ consists of apertures in an opaque screen, then the complement of S₁ would be opaque where S₁ is transmissive, and transmissive where S₁ is opaque. Considering that, in Kirchhoff’s approximation, ψ(r) and ∂_nψ(r) in Eq.(23) retain the values of the incident beam in the open aperture(s) but vanish in the opaque regions, it is a reasonable conjecture that the observed field in the presence of S₁ and that in the presence of S₁’s complement would add up to the observed field when all screens are removed—i.e., when the unobstructed beam reaches the observation point. Similar arguments can be based on either of the Rayleigh-Sommerfeld diffraction integrals in Eqs.(24) and (25), provided, of course, that the Kirchhoff approximation remains applicable. Appendix K describes the rigorous version of the Babinet principle and provides a simple proof that relies on symmetry arguments similar to those used in Example 2 of Sec.6.

Finally, to keep the size and scope of this tutorial within reasonable boundaries, we did not broach the important problem of EM scattering from small dielectric spheres, nor that of EM scattering from small perfectly conducting spheres. The interested reader can find a detailed discussion of these problems in Appendices L and M, respectively.