Physics equations used in real world photonics problem?

[flux!]

We typically solve idealized problems in physics coursework. How about when we deal with real world Photonics Industry Problems, what Physics equations we will be talking of here?

 

[jfizzix]

The equations would be particular to the problems being solved, but it's not any different than conventional physics; it's just more specialized.

 

[flux!]

At last! I though I could not apply those equation if not only in coursework. Could you post down one equation in photonics you typically use?

 

[jfizzix]

A lot of the equations I use come from the same starting points as you might see in course work.
One neat example is the paraxial Helmholtz equation, which described the evolution/propagation of laser beams.

For example, from Maxwell's equations, we can get the wave equation for light.
$\frac{\partial^{2}\vec{E}}{\partial x^{2}}+\frac{\partial^{2}\vec{E}}{\partial y^{2}}+\frac{\partial^{2}\vec{E}}{\partial z^{2}}=\frac{1}{c^{2}}\frac{\partial^{2}\vec{E}}{\partial t^{2}}$
If we only consider the amplitude of the electric field $E$ as important, and assume the light is monochromatic enough that the electric field can be factored into a time-dependent function $f(t)$, and a space-dependent function $A(x,y,z)$, the space-dependent part is described by the Helmholtz equation
$\frac{\partial^{2}A}{\partial x^{2}}+\frac{\partial^{2}A}{\partial y^{2}}+\frac{\partial^{2}A}{\partial z^{2}}=-k^{2} A$
Now, if we also say that the light is predominantly moving along the $z$ direction, or that the $z$-component of the momentum of the field is much larger than the $x$ or $y$ components, we can further approximate the helmholtz equation, by basically taking a small angle approximation.
When we do this, we get the paraxial Helmholtz equation, named because it describes light predominantly moving along one axis.
$-\frac{\partial^{2} A}{\partial x^{2}}-\frac{\partial^{2} A}{\partial y^{2}}=2 i k \frac{\partial A}{\partial z}$
Solutions to this equation show how laser beams change as the propagate through free space.

One of the more popular solutions to this equation is the Gaussian laser beam. What makes this equation nice to use is that it is often easier to solve explicitly than the more complicated, but more fundamental equations seen in coursework.

 

[flux!]

That was very informative, Thank you! Could I let this topic open so others could add their frequently encountered equation?

 

[jfizzix]

sure thing!