The basic Eikonal equation of geometric optics

1.15.1 General theory

The Eikonal equation is an approximation to the wave equation but is useful when we are only interested in the propagation of wave fronts, ie, constant phase surfaces not on the amplitude of the wave. It is a first order equation in the spatial arguments and is of quadratic algebraic degree. Although we have derived it from the standard three dimensional wave equation or equivalently from the standard three dimensional Helmhohltz equation that describe propagation at a definite frequency, it is applicable to even more general situations like Helmholtz equations of the form

a ,0=1

+fc² (x)tp(x) = 0

which describe wave propagation in inhomogeneous and anisotropic media. The Eikonal equation involves what is called the geometric optics approximation in which the wave number is very large or equivalently, the wave length is much smaller than the dimensions of the medium in which it propagates. One can improve upon this approximation by using higher order perturbation theory but then the problem of investigating the convergence of the perturbation series would remain. In what follows, we present first the Eikonal equation as an approximate nonlinear partial differential equation for the phase of the wave and then present the nonlinear Schrodinger wave equation obtained by assuming that in a given direction the dimensions of the medium are much larger than the corresponding wavelength of the wave. This is some sort of restricted geometric optics approximation. The second order partial derivative terms appear only for the transverse variables x, y while along the transverse directions x, y, first order partial derivatives with Eikonal like quadratic terms appear along with linear terms. Thus one gets a Schrodinger equation with a nonlinearity if the z axis is interpreted as time. Alternately, we may have a geometric optics approximation in the transverse directions but not in the longitudinal directions. Then, we get another version of the nonlinear Schrodinger equation with quadratic terms in the first order partial derivative along the longitudinal direction and along the transverse direction, we have only linear terms in the second order transverse partial derivatives. It is an interesting exercise to generalize such an idea to n > 3 dimensions where geometric optics approximation holds along the first p co-ordinate directions and along the remaining n—p coordinate directions linear terms in the second order partial differentials of the wavefront appear.

The Helmholtz equation for a wave field is obtained by Fourier transforming the wave equation w.r.t the time argument. It is given by

(V² + k²)^r) = 0---(1)

where = ip(x, y, z) = k) is defined by

I ip(r. t)exp(—iwt)dt, k = o/c

Jr

and V>(r, t) satisfies

(c²V² — c)²)V>(r, t) = 0

We now substitute

V’(r) = exp(iS(r))

into (1). Evaluating the partial derivatives,

V-0 = iS7 S.exp(iS),

V²0 = (-| VS|² + iV²S).exp(iS)

so our Helmholtz equation transforms to a nonlinear pde for the phase field S.

| VS|² - k² - iV²S = 0---(1)

Now, if we assume that propagation takes place only along the x direction, then —S" (x) / S'(x)² = d/ dx()./S') and S' is the local wave number which is proportional to the reciprocal of the wavelength. Its rate of change with x is small. Equivalently, writing S(r) = k.r + 8S(r), we find the exact equation

k + VJS(r)|² - k² - iV²8S(r) = 0

or equivalently,

|V5S(r)|² + 2A-.Vd'S’(r) = iV²6S(r)

so we have approximately upto first order terms,

iV²6S(r) = 2kS78S(r)

and hence, approximately,

|Jb + Vd'S(r)|² - k² - 2k.V6S(r) = 0

which can he expressed as

|VS(r)|² = k²+ 2k.V6S(r)

and if we neglect the last term being of the first order of smallness compared to the other terms, we get the Eikonal equation

|VS(r)|² = k²

Note that

|fc.V2 « |Vd-S(r)|/|fc|

which is very small for small values of the wavelength compared with the dimensions of the region where the wave propagates, ie in geometrical optics. There is another approximation to the wave equation known as the nonlinear Schrodinger equation. Suppose we decompose the wave motion into propagation in the xy plane and along the z direction. Then, Helmholtz equation can be expressed as

^i + d²_z+k²)^x,y,z)=0

where

Vl=d² ₊ d²

Now writing

ip(x,y, z) = exp(iS(r)) = exp(i(So(x,y,z) + k_zz))

ie,

S(r) = S(x, y, z) = S₀(x, y, z) -I- k_zz = S₀(r) + k_zz

and substituting this into (1) gives us

|VS₀ + M² - ^k'² ~ *V²S₀ = 0

This equation is exact and expands to

I VSoI² + k² + 2k_zd_zS₀ - k² = iV²S₀

In the special case when k_z = k and | ¿9*S’₍₎ | /A-1 *S’_O | is very small and so also are |c)²>So|/|c)xSo|² the above equation simplifies to the non-linear Schrodinger equation (with z being interpreted as the time variable and x, y as the spatial variables):

id_zS₀ = (-l/2fc)V15₀ - (i/2k)|V_±S₀|²

This is the celebrated non-linear Schrodinger equation. The approximation involved here is that the dimensions of the region in a particular direction are much larger than the wavelength.

1.15.2 Points to remember

[1] When we substitute a pure phase factor exp)iS(r)) into the three dimensional Helmholtz equation (which is a time Fourier transformed version of the wave equation, or equivalently the wave equation at a fixed frequency) and make the geometric optics approximation, namely that the dimensions of the region in which the wave propagates are much larger than the wavelength then second order partial derivatives of the phase S can be neglected and we get the Eikonal equation |VS|² = k². where w = fee is the frequency. The surfaces of constant phase are S(r) — art = constt from which, we deduce that the local wave velocity, ie, the velocity of the wavefronts is given by v = dr/dt, where

S(r + dr) — omegaft + dt) = S(r) — art

or equivalently,

dr.VS(r) — a>dt = 0

ie,

t'.VS(r) = w = fee

Since the velocity of the wave is directed along the normal to the wavefronts, we can deduce that

|v| = c, |VS(r)| = fe

which is precisely the Eikonal equation. Thus, the Eikonal equation has a natural interpretation in terms of the velocity of wavefronts or equivalently, of constant phase surfaces.

[2] The nonlinear Schrodinger equation is derived from the wave equation by assuming that in certain spatial directions, though not all, the geometric optics approximation is valid.

[3] The approximate refractive index of the medium can be expressed as n(r) = c/|v| = |V5(r)|/fc in a medium in which the waves travel at lesser than the speed c of light. In that case, the Eikonal equations has to be modified accordingly.

[4] The Eikonal equation can also be arrived at from Schrodinger’s wave equation. Specifically, writing the wave function at energy E as ifr(r) = exp(iS(r)/h), we get on substituting into the Schrodinger equation

(—h²/2m)V²^(r) + V(r)V>(r) = Eip{r)

that

(-h²/2m)(iV²S/h - |VS|²/h²) = (E - V(r))

and neglecting the O(l/h) term in comparison with the O(l//»²) term gives us the Eikonal equation for matter waves:

|VS(r)|² = 2m(E-V(r))=_P(r)²

where p(r) is the momentum of a nonrelativistic particle having energy E and moving in the potential V(r):

p(r)²/2m + V(r) = E

[5] The Eikonal approximation involving neglecting partial derivatives of second order can be obtained for more general wave equations, for example, for wave equations in curved space-time having the form

N N

57 afcm(a;)ÖA.ömV’(a’) + 57 bk(x)dk^(x) + k2(x)t/>(x) = 0

k,m=l k=l