Given a [electromagnetic wave](Electromagnetic%20wave.md) described by a [light-field](Light-field.md) of the form $E(x,y,z)=u(x,y,z)e^{ikz},$ its amplitude $u(x,y,z)$ under the [paraxial Fresnel approximation](Fresnel%20approximation.md#Paraxial%20propagation) is a solution to the _paraxial wave equation_, which is given as
$\nabla_T^2u+2ik\frac{\partial u}{\partial z}=0$
Thus this equation is incredibly useful for modeling [coherent light](Coherent%20light.md) which has an intensity concentrated along the z-axis.
It is also implied by the paraxial approximation that $\Delta z=\frac{\lambda}{2\pi}$ and $|2k \frac{\partial u}{\partial z}|>>|\frac{\partial^2 u}{\partial z^2}|.$ That is the the amplitude of $u$ varies slowly in $z.$ For example, in the case where we consider [gaussian beam](Gaussian%20beam.md) solution, this would correspond with a slowly varying beam envelope.
# Solutions
## Huygens-Fresnel diffraction integral
The [diffraction integral](Huygens-Fresnel%20Principle.md#Diffraction%20Integral) following the [Huygens-Fresnel Principle](Huygens-Fresnel%20Principle.md) is a solution to the differential equation
### Proof
# Derivation
Consider the [scalar Helmholtz equation](Helmholtz%20equation%20(optics).md#Scalar%20diffraction%20theory), which is given as .md#%5E27a30d), and the following ansatz, $E(x,y,z)=u(x,y,z)e^{ikz},$ which we then plug into the Helmholtz equation to obtain
$\Bigg[\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}+\frac{\partial}{\partial z}\Bigg(\frac{\partial u(x,y,z)}{\partial z}+iku\bigg)\Bigg]e^{ikz}=0$
which simplifies to
$\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}+2ik\frac{\partial u}{\partial z}+\frac{\partial^2 u}{\partial z^2}=0$
where we generally condense the equation to write:
$\nabla_T^2u+2ik\frac{\partial u}{\partial z}+\frac{\partial^2 u}{\partial z^2}=0$
Where the subscript $T$ refers to the transverse components. Finally we apply the [paraxial approximation](Paraxial%20approximation.md) by assuming the $z$ component of $u(x,y,z)$ is approximately constant (corresponding to the small-angle approximation). Thus we may write $\frac{\partial^2 u}{\partial z^2}=0,$ obtaining the [paraxial wave equation](Paraxial%20Wave%20Equation.md) given as
$\nabla_T^2u+2ik\frac{\partial u}{\partial z}=0.$
Or alternatively it may be written as
$\nabla_T^2u-2ik\frac{\partial u}{\partial z}=0$
if we follow the ansatz $E(x,y,z)=u(x,y,z)e^{-ikz}.$
# In cylindrical coordinates
In [cylindrical coordinates](Nabla.md#In%20different%20coordinate%20systems) the equation takes the form
$\frac{1}{r}\frac{\partial}{\partial r}\Bigg(r\frac{\partial u}{\partial r}\Bigg)-2ik\frac{\partial u}{\partial z}=0,$ ^054e1d
## Solution
A common cylindrically symmetric ansatz motivated by the form of the [[Gaussian beam]] beam amplitude is given as
$u(z,r)=e^{-i(P(z)+\frac{kr^2}{2q(z)})}.$ ^c56b00
where $P(z)$ is referred to as the _phase shift_
# Limitations of the Paraxial wave equation
In general the paraxial wave equation is accurate in modeling systems with [laser light](Coherent%20light.md) as long as the wavefronts are tilted by more than $~30°.$ We can see this based on the following argument...
## Corrections
### Higher order corrections
#Electromagnetism/Optics/waveOptics