Statement
Equation statement
The Helmholtz equation is a time-independent wave equation that governs the behavior of wave-like phenomena in steady states (when the system is oscillating at a particular frequency).
For the electric field:
$${{(\nabla^2+k^2)\tilde E(\vec r,\omega)=0}}$$
Where:- \(k={{n(\omega)\frac{\omega}{c} }}\)
- \(\nabla^2\) : the laplacian operator
Paraxial approximation
Helmotz equation in the paraxial approximation
Under the paraxial approximation, we assume that the wavevector is predominantly unidirectional. This corresponds to considering a small angle between the wavevector and one axis, such that \(k_z = k_0 \cos(\alpha) \approx k_0\) and \(\frac{k_x}{k_z}, \frac{k_y}{k_z}\lt \lt 1\). Consequently, the spectrum is narrowly centered around \(||\vec{k}|| = k_0.\)
Within this approximation, the Helmholtz equation can be rewritten as:
$${{\frac{\partial A}{\partial z}=i\frac{1}{2k_0}\nabla_{\perp}^2A}}$$
With:
- \(\nabla ^2_\perp= \frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\) and \(z\) is the direction of propagation
- \(||\vec k||=k_0\)
- \(A\) : the wave envelope
Solutions
Fresnel diffraction
Appendix
How to find the Helmotz equation
Starting from the
Equations de Maxwell for a linear dielectric medium (\(\vec j=\vec 0,\rho=0,\vec H=\frac{\vec B}{\mu 0}\)):
$$\vec{rot}(\vec E)=-\frac{\partial\vec B}{\partial t}\quad,\quad \vec{rot}(\vec B)=\mu_0\frac{\vec D}{\partial t}\quad ,\quad div(\vec E)=0 $$
We have the following relation between the differential operators: \(\vec{rot}(\vec{rot}(\vec E))=\vec{grad}(div(\vec E))-\Delta \vec E\)
Thus using the equations bellow : $$\vec{rot}(\vec{rot}(\vec E))=-\frac{\partial \vec{rot}(\vec B)}{\partial t}=\mu_0\frac{\partial^2\vec D}{\partial t^2}=-\Delta \vec E$$
We know that \(\vec D=\epsilon_0(1+\chi)\vec E\) (
Susceptibilité électrique) for a linear and isotropic medium.
So :
$$\Delta \vec E+\frac{\partial ^2\vec E}{\partial t^2}\left(1+\chi\right)\mu_0\epsilon_0=0$$
There is the following relation between the fundamental constants: \(\epsilon_0\mu_0c^2=1\)
$$\Delta \vec E+\frac{\partial ^2\vec E}{\partial t^2}\left(1+\chi\right)\frac{1}{c^2}=0$$
If we look at the time Fourier transform (\(t\to\omega\)):
$$\Delta\tilde E+\frac{\omega^2}{c^2}\left(1+\chi(\omega)\right)\tilde E=0$$
And the Fourier transform in space domain (\(\vec r\to\vec k\)):
$$k^2\tilde{\tilde E}=\frac{\omega^2}{c^2}\left(1+\chi(\omega)\right)\tilde{\tilde E}$$
Finally, we find the Helmotz equation:
$$\Delta\tilde E+k^2\tilde E=0\implies (\nabla^2+k^2)\tilde E(\omega,\vec r)=0$$
Paraxial approximation
We are looking at a monochromatic plane wave \(E(\vec r,t)=A(\vec r)e^{i(k_0z-\omega_0 t)}+c.c.\)
In the time Fourier space: \(E(\vec r,\omega)=A(\vec r)e^{ik_0z}\delta(\omega-\omega_0)+c.c.\)
We put this wave into the Helmotz equation: $$\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right)A(\vec r)+\frac{\partial ^2 A(\vec r)}{\partial z^2}+2ik_0\frac{\partial A(\vec r)}{\partial z}-\left(k_0^2-\underbrace{k^2(\omega_0)}_{k_0^2}\right)A(\vec r)=0$$
By looking at the Fourier transform on the space variables:
$$-\left(k_x^2+k_y^2\right)\tilde A-k_z^2\tilde A+2ik_0k_z\tilde A=0$$
However, we can easly see that \(k_z-k_0=k_0\cos(\alpha)\) and \(k_y=k_x=k_0\sin(\alpha)\) where \(\alpha\) is the angle between \(\vec k\) and the \(z-\)axis.
If we consider the paraxial approximation (\(\alpha\lt \lt 1\)): \(k_z\approx \frac 12 \alpha^2k_0\) and \(k_x,k_y\approx\alpha k_0\)
By replacing these expression in the equation below:
$$-\alpha^2k_0^2\tilde A-\underbrace{k_0\alpha^4\tilde A}_{\alpha\lt \lt 1,\text{ neglect} }-2\alpha k_0^2 \tilde A=0$$
Therfore, when we go back in the direct space:
$$\frac{\partial A}{\partial z}=\frac{i}{2k_0}\nabla_\perp^2 A\quad\text{Helmotz equation in paraxial approximation}$$
With \(\nabla^2_\perp=\frac{\partial ^2}{\partial x^2}+\frac{\partial ^2}{\partial y^2}\)
Nota bene: The paraxial approximation we describe just below is the same that consider the slowly varying envelope \(\frac{\partial^2 A}{\partial z^2}\lt \lt k_0\frac{\partial A}{\partial z}\) .
