Wave Scattering, Inhomogeneities, and the Born Approximation

Wave Scattering, Inhomogeneities, and the Born Approximation

The earlier wave articles treated propagation in uniform media, reflection at sharp interfaces, normal modes in bounded regions, and forced response.

A new effect appears when the medium varies through space.

A wave entering an inhomogeneous region can generate a new outgoing field.

That process is wave scattering.

We begin with a scalar wave equation whose local wave speed depends on position:

(1)

For harmonic time dependence,

(2)

the spatial field satisfies

(3)

where

(4)

Choose a uniform background medium with wave speed

(5)

and wave number

(6)

Define the scattering potential

(7)

Then

(8)

The inhomogeneity therefore acts like a distributed source whose strength depends on the field itself.

1 Incident, scattered, and total fields

Write

(9)

The incident field is a solution of the background equation:

(10)

For a plane incident wave,

(11)

with

(12)

The scattered field is the additional field produced by the inhomogeneity.

For an isolated scatterer in open space, the physical scattered solution must be outgoing.

2 Outgoing free-space Green function

The background Helmholtz Green function is defined by

(13)

In three dimensions, the outgoing solution is

(14)

The outgoing exponential selects waves propagating away from the source point

(15)

3 The Lippmann-Schwinger equation

Starting from

(16)

use the Green function to write

(17)

This is the Lippmann-Schwinger equation for the scalar scattering problem.

It has an important physical interpretation.

The total field inside the inhomogeneity drives secondary outgoing waves.

Those secondary waves add to the incident field to produce the total field.

4 Why the scattering problem is implicit

The integral contains

(18)

the unknown total field.

Therefore the equation does not merely evaluate the scattered wave from a known source.

The source itself depends on the unknown field.

Strong scattering can therefore include repeated interactions:

• the incident field excites the inhomogeneity,
• the generated field propagates,
• that field interacts with the inhomogeneity again,
• further scattered contributions are produced.

This repeated interaction is multiple scattering.

5 Born series

Insert the Lippmann-Schwinger equation back into itself.

Symbolically,

(19)

One iteration gives

(20)

Continuing formally,

Φ	= Φinc + G0UΦinc	(21)
	+ G0UG0UΦinc + .	(22)

This is the Born series.

The terms represent:

• zeroth order: incident wave,
• first order: one scattering interaction,
• second order: two scattering interactions,
• higher orders: repeated scattering.

6 First Born approximation

If the inhomogeneity is sufficiently weak, the field inside the scattering region remains close to the incident field:

(23)

Then

(24)

This is the first Born approximation.

It is a single-scattering approximation.

It neglects the modification of the internal field by repeated scattering.

7 Far-field approximation

Suppose the scatterer is localized near the origin and the observation point is far away:

(25)

Then

(26)

and

(27)

Therefore

G0(r,r′)

≈e−ik0r⋅r′.

(28)

Define the scattered wave vector

(29)

8 Scattering amplitude

For an incident plane wave of unit amplitude,

(30)

the far-field Born scattered wave becomes

(31)

The first Born scattering amplitude is

(32)

Define the momentum-transfer or scattering vector

(33)

Then

(34)

The Born scattering amplitude is therefore the spatial Fourier transform of the scattering potential, apart from the factor

(35)

9 Magnitude of the scattering vector

For elastic scattering,

(36)

If

(37)

is the angle between

(38)

and

(39)

then

q2	= |k s − ki|2	(40)
	= k02 + k 02 − 2k 02 cos 𝜃	(41)
	= 2k02(1 − cos 𝜃).	(42)

Using

(43)

we obtain

(44)

Forward scattering has

(45)

Backscattering has

(46)

10 Scattering as spatial-frequency measurement

The first Born amplitude has the form

(47)

Therefore changing the scattering angle changes the spatial Fourier component of the inhomogeneity that is sampled.

Small angles correspond to small

(48)

and therefore large-scale structure.

Large angles correspond to larger

(49)

and therefore finer spatial structure.

This Fourier-transform interpretation is fundamental in diffraction, imaging, radar, acoustics, and many inverse problems.

11 Differential scattering cross section

If the incident plane wave has unit amplitude and the scattered far field is

(50)

the differential scattering cross section is

(51)

The dimensions are:

(52)

and

(53)

Since

(54)

the Born integral indeed gives

(55)

12 Spherically symmetric scatterer

Suppose

(56)

Choose the

(57)

axis along

(58)

Then

(59)

The angular integral gives

(60)

Therefore

(61)

The three-dimensional Fourier transform has reduced to a one-dimensional radial integral.

13 Example 1: Gaussian inhomogeneity

Consider

(62)

The three-dimensional Fourier transform is

(63)

Therefore

(64)

The differential cross section is

(65)

The scattering becomes increasingly forward-peaked when

(66)

is large because larger scattering angles correspond to larger

(67)

14 Example 2: uniform spherical inhomogeneity

Let

(68)

Then

fB(q)

= U0 ∫ 0ar2dr.

(69)

Evaluating the integral gives

(70)

Equivalently, with

(71)

(72)

15 Forward limit of the uniform sphere

As

(73)

use

(74)

and

(75)

Then

(76)

Therefore

(77)

This also follows directly from

(78)

For a uniform sphere,

(79)

16 Example 3: numerical sphere amplitude

Take

(80)

(81)

and

(82)

At

(83)

q	= 2k0 sin 30∘	(84)
	= 10.0 m−1.	(85)

Therefore

(86)

The Born amplitude is

fB	= U0a3	(87)
	≈ 8.13 × 10−4 m.	(88)

The differential cross section is approximately

(89)

17 Why scattering minima appear

Different parts of an extended scatterer radiate secondary waves with different phases.

At some observation angles, these contributions reinforce.

At others, they cancel.

The oscillatory factor

(90)

is therefore an interference effect across the finite sphere.

Scattering patterns contain information about scatterer size and shape because of this internal phase structure.

18 Born validity

The first Born approximation assumes

(91)

inside the scattering region.

It is best when:

• the contrast is weak,
• the scatterer is not too large in accumulated phase shift,
• multiple scattering is small,
• the system is not near a strong internal Resonance.

A useful heuristic is that the phase perturbation through the object should remain small.

If

(92)

is a characteristic change in wave number and

(93)

is a characteristic traversal length, one often wants

(94)

for a comfortably weak phase perturbation.

This is a heuristic, not a universal theorem.

Different scattering geometries can require more careful validity criteria.

19 A dimensionless weak-scattering estimate

Since

(95)

if

(96)

with

(97)

then

U	= (k0 + Δk)2 − k 02	(98)
	≈ 2k0Δk.	(99)

Therefore a rough small-phase condition

(100)

can also be written approximately as

(101)

Again, this is an order-of-magnitude guide.

20 When the first Born approximation fails

Important failure mechanisms include:

• strong wave-speed contrast,
• large accumulated phase through the scatterer,
• repeated internal reflection,
• multiple scattering among separated objects,
• resonant trapping or cavity-like behavior,
• very strong forward phase distortion.

Then higher Born terms or a different solution method may be required.

Possible approaches include:

• retaining additional Born-series terms,
• direct numerical solution of the Helmholtz equation,
• boundary-element methods,
• finite-element methods,
• exact partial-wave methods for symmetric scatterers.

21 Relation to sharp-interface reflection

The earlier interface articles treated an abrupt material boundary.

Scattering from a distributed inhomogeneity generalizes that idea.

A sharp interface can be viewed as an extreme spatial variation.

A weak smooth inhomogeneity instead produces many small secondary contributions distributed throughout its volume.

The Born approximation sums those weak contributions coherently.

22 Relation to diffraction

Diffraction and scattering are closely related wave phenomena.

Both involve:

• phase differences,
• superposition,
• Fourier transforms,
• far-field angular patterns.

In the first Born approximation, the connection becomes explicit:

(102)

This is the same mathematical structure that appears in far-field diffraction from apertures and objects.

23 Relation to quantum scattering

The time-independent Schrödinger equation can also be rearranged into a Helmholtz-like equation containing a potential term.

That produces a mathematically similar Lippmann-Schwinger equation and Born approximation.

The interpretation is different:

• here Φ is a classical scalar wave amplitude,
• in quantum mechanics the wavefunction has a probabilistic interpretation.

The mathematical analogy is nevertheless extremely useful.

24 Scattering from several weak objects

Suppose

(103)

In the first Born approximation,

fB(q)	= ∫ ∑ jUj(r)e−iq⋅rd3r	(104)
	= ∑ jfB,j(q).	(105)

Thus the amplitudes add before taking the squared magnitude.

The resulting cross section contains interference terms between different scatterers.

This is why arrays, lattices, and distributed targets can produce sharp angular structure.

25 Translated identical scatterers

Let one scatterer have potential

(106)

If a copy is translated to

(107)

its potential is

(108)

Its Fourier transform gains a phase:

(109)

For many identical scatterers,

(110)

The first factor describes one object.

The second factor describes the arrangement of objects.

This is the basic separation between a form factor and a structure factor.

26 Common mistakes

1. Forgetting that the total field is incident plus scattered field.
2. Defining the scattering potential with one sign and then using formulas derived for the opposite sign.
3. Forgetting the outgoing boundary condition on the Green function.
4. Treating the Lippmann-Schwinger equation as explicit even though the unknown field appears inside the integral.
5. Calling the first Born approximation an exact solution.
6. Replacing the internal field by zero rather than by the incident field.
7. Forgetting the far-field approximation when defining a scattering amplitude.
8. Using q = k_i − k_s while keeping an exponential convention derived for q = k_s − k_i.
9. Forgetting that q = 2k₀ sin(𝜃∕2) for elastic scattering.
10. Confusing amplitude with differential cross section.
11. Forgetting that dσ∕dΩ = |f|² has units of area.
12. Treating large-angle scattering as sampling only large spatial scales.
13. Forgetting the factor sin(qr)∕(qr) for a spherical Fourier transform.
14. Using the sphere formula directly at q = 0 without taking its finite limit.
15. Applying the Born approximation to strong or resonant scatterers without checking validity.

27 Practice exercises

1. Starting from a spatially varying wave speed, derive the Helmholtz scattering equation in terms of U(r) = k²(r) − k ₀².
2. Verify that the outgoing three-dimensional Green function satisfies the required dimensions.
3. Derive the Lippmann-Schwinger equation using the background Green function.
4. Generate the first three terms of the Born series by repeated substitution.
5. Derive the first Born approximation.
6. Derive the far-field approximation to G₀(r,r′).
7. Derive the Born scattering amplitude for a plane incident wave.
8. Show that q = 2k₀ sin(𝜃∕2).
9. Verify dimensionally that the scattering amplitude has units of length.
10. Derive the radial Born integral for a spherically symmetric potential.
11. Derive the Gaussian Born amplitude.
12. Derive the uniform-sphere Born amplitude and its forward limit.
13. Reproduce the numerical uniform-sphere example.
14. Explain physically why a finite sphere produces angular minima.
15. Show that translating a scatterer multiplies its Born amplitude by a phase factor.

28 Summary

For a spatially varying medium,

(111)

where

(112)

The outgoing Green function is

(113)

The exact integral equation is

(114)

The first Born approximation replaces the internal total field by the incident field:

(115)

In the far field,

(116)

with

(117)

For elastic scattering,

(118)

The differential scattering cross section is

(119)

The first Born approximation therefore turns weak wave scattering into a Fourier-transform problem.

References

References