where ∇2 is the Laplace operator, k2 is the eigenvalue, and f is the (eigen)function. When the equation is applied to waves, k is known as the
wave number. The Helmholtz equation has a variety of applications in physics and other sciences, including the
wave equation, the
diffusion equation, and the
Schrödinger equation for a free particle.
The Helmholtz equation often arises in the study of physical problems involving
partial differential equations (PDEs) in both space and time. The Helmholtz equation, which represents a time-independent form of the
wave equation, results from applying the technique of
separation of variables to reduce the complexity of the analysis.
For example, consider the wave equation
Separation of variables begins by assuming that the wave function u(r, t) is in fact separable:
Substituting this form into the wave equation and then simplifying, we obtain the following equation:
Notice that the expression on the left side depends only on r, whereas the right expression depends only on t. As a result, this equation is valid in the general case if and only if both sides of the equation are equal to the same constant value. This argument is key in the technique of solving linear partial differential equations by separation of variables. From this observation, we obtain two equations, one for A(r), the other for T(t):
where we have chosen, without loss of generality, the expression −k2 for the value of the constant. (It is equally valid to use any constant k as the separation constant; −k2 is chosen only for convenience in the resulting solutions.)
Rearranging the first equation, we obtain the (homogeneous) Helmholtz equation:
Likewise, after making the substitution ω = kc, where k is the
wave number, and ω is the
angular frequency (assuming a monochromatic field), the second equation becomes
The two-dimensional analogue of the vibrating string is the vibrating membrane, with the edges clamped to be motionless. The Helmholtz equation was solved for many basic shapes in the 19th century: the rectangular membrane by
Siméon Denis Poisson in 1829, the equilateral triangle by
Gabriel Lamé in 1852, and the circular membrane by
Alfred Clebsch in 1862. The elliptical drumhead was studied by
Émile Mathieu, leading to
Mathieu's differential equation.
If the edges of a shape are straight line segments, then a solution is integrable or knowable in closed-form only if it is expressible as a finite linear combination of plane waves that satisfy the boundary conditions (zero at the boundary, i.e., membrane clamped).
If the domain is a circle of radius a, then it is appropriate to introduce polar coordinates r and θ. The Helmholtz equation takes the form
We may impose the boundary condition that A vanishes if r = a; thus
the method of separation of variables leads to trial solutions of the form
where Θ must be periodic of period 2π. This leads to
It follows from the periodicity condition that
and that n must be an integer. The radial component R has the form
and ρ = kr. The radial function Jn has infinitely many roots for each value of n, denoted by ρm,n. The boundary condition that A vanishes where r = a will be satisfied if the corresponding wavenumbers are given by
Writing r0 = (x, y, z) function A(r0) has asymptotics
where function f is called scattering amplitude and u0(r0) is the value of A at each boundary point r0.
Three-dimensional solutions given the function on a 2-dimensional plane
Given a 2-dimensional plane where A is known, the solution to the Helmholtz equation is given by:[3]
where
is the solution at the 2-dimensional plane,
As z approaches zero, all contributions from the integral vanish except for r=0. Thus up to a numerical factor, which can be verified to be 1 by transforming the integral to polar coordinates .
This solution is important in diffraction theory, e.g. in deriving
Fresnel diffraction.
where u represents the complex-valued amplitude which modulates the sinusoidal plane wave represented by the exponential factor. Then under a suitable assumption, u approximately solves
This equation has important applications in the science of
optics, where it provides solutions that describe the propagation of
electromagnetic waves (light) in the form of either
paraboloidal waves or
Gaussian beams. Most
lasers emit beams that take this form.
The assumption under which the paraxial approximation is valid is that the z derivative of the amplitude function u is a slowly varying function of z:
This condition is equivalent to saying that the angle θ between the
wave vectork and the optical axis z is small: θ ≪ 1.
The paraxial form of the Helmholtz equation is found by substituting the above-stated expression for the complex amplitude into the general form of the Helmholtz equation as follows:
Expansion and cancellation yields the following:
Because of the paraxial inequality stated above, the ∂2u/∂z2 term is neglected in comparison with the k·∂u/∂z term. This yields the paraxial Helmholtz equation. Substituting u(r) = A(r) e−ikz then gives the paraxial equation for the original complex amplitude A:
Two sources of radiation in the plane, given mathematically by a function f, which is zero in the blue region
The
real part of the resulting field A, A is the solution to the inhomogeneous Helmholtz equation (∇2 + k2) A = −f.
The inhomogeneous Helmholtz equation is the equation
where ƒ : Rn → C is a function with
compact support, and n = 1, 2, 3. This equation is very similar to the
screened Poisson equation, and would be identical if the plus sign (in front of the k term) were switched to a minus sign.
in spatial dimensions, for all angles (i.e. any value of ). Here where are the coordinates of the vector .
With this condition, the solution to the inhomogeneous Helmholtz equation is
(notice this integral is actually over a finite region, since f has compact support). Here, G is the
Green's function of this equation, that is, the solution to the inhomogeneous Helmholtz equation with f equaling the
Dirac delta function, so G satisfies
The expression for the Green's function depends on the dimension n of the space. One has
^Mehrabkhani, S., & Schneider, T. (2017). Is the Rayleigh-Sommerfeld diffraction always an exact reference for high speed diffraction algorithms?. Optics express, 25(24), 30229-30240.
^J. W. Goodman. Introduction to Fourier Optics (2nd ed.). pp. 61–62.
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Saleh, Bahaa E. A.; Teich, Malvin Carl (1991). "Chapter 3". Fundamentals of Photonics. Wiley Series in Pure and Applied Optics. New York: John Wiley & Sons. pp. 80–107.
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ISBN978-0126546569.
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