Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It's 100% free.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am interested in the finite-difference beam propagation method and its applications. I try to solve the Helmholtz equation. At first, i would like to solve numerically it for the easiest case, without nonlinearities. Just to make sure I'm on the right way. But i really don't understand how to wright the boundary condition. I chose the transparent boundary condition and i need to write it properly to solve numerically the equation.

So, for a linear, homogeneous and instantaneous medium the Helmholtz equation is writen (in 3D case, z is the propagation direction) $$ \frac{\partial^{2} E(x,y,z)}{\partial x^{2}} + \frac{\partial^{2} E(x,y,z)}{\partial y^{2}} + \frac{\partial^{2} E(x,y,z)}{\partial z^{2}} = - (k_{0} n )^{2} E(x,y,z) $$

It can be solved if the initial condition is known, $E(x,y,0)$.

Introducing operator $\hat{S}$ $$ \hat{S} = \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + (k_{0} n )^{2} $$ The equation can be written in the following form $$ \frac{\partial^{2} E(x,y,z)}{\partial z^{2}} = -\hat{S} \ E(x,y,z) $$

The solution of this equation is $$ E(x,y,z) = \exp \left [ - i \sqrt{\hat{S}} z \right ] E^{+}(x,y,0) + \exp \left [ i \sqrt{\hat{S}} z \right ] E^{-}(x,y,0) $$

Considering only the forward propagating component and introducing the propagation operator $\hat{P}^{+}$ the electric field at $z=\Delta z$ can be written through the value of the field at $z=0$ (initial condition written earlier) and so on. $$ E(x,y,\Delta z) = \hat{P}^{+}(\Delta z) \ E(x,y,0) $$ where $$ \hat{P}^{+}(\Delta z) = \sum \limits_{n=0}^{\infty} \frac{1}{n!}\left[- i \sqrt{\hat{S}} \right]^{n} \Delta z^{n} $$

Obtained expression can be adopted to the Crank-Nicholson scheme. But it is also necessary to write the boundary condition. How to write the boundary condition if the medium is confined in the transparent walls ?

share|cite|improve this question

closed as off-topic by Chris White, ja72, Waffle's Crazy Peanut, user1504, Manishearth Jul 12 '13 at 5:59

  • This question does not appear to be about physics within the scope defined in the help center.
If this question can be reworded to fit the rules in the help center, please edit the question.

I think you'd get some good answers if you ask this on – David Ketcheson Jan 13 '12 at 20:01
This question appears to be about implementation of a numerical method, rather than about the physics of the problem. – Chris White Jul 10 '13 at 14:28

Since you seem to be using finite differences, you should look at the paper of Hadley, titled 'Transparent Boundary Condition for the Beam Propagation Method' - without any treatment of the boundary values you're automatically assuming a Dirichlet boundary condition.

You can incorporate the boundary conditions into your square-root differential operator $\sqrt{\hat{S}}$ by approximating it via a Padé approximant, in short: quotients of polynomials in $\hat{S}$. With this you can arrive at equations containing $\hat{S}$ and $E$, and satisfy your boundary conditions for your $E$ field by choosing the right 'boundary entries' for $\hat{S}$'s matrix representation. That's the only way I managed to find and implement, as it is similar to doing the same for the paraxial approximation that reduces the Helmholtz-equation to a form that would only contain $S$ instead of $\sqrt{\hat{S}}$ (I'm not a clever man).

As you'll see from the paper of Hadley, the simplest form of the TBC (transparent boundary condition) involves treating the field at the boundary as a lateral plane wave and making sure that no reflected plane wave gets back into the computation window by modifying the fields at the boundary.

When I started to look into solving the Helmholtz-Equation I came around a dissertation by Filippo Pigozzo that was valuable for getting a very good overview of the matter.

share|cite|improve this answer
Thanks for your answer, mike. I asked this question exactly after i had read Pigozzo's dissertation. I'll look through the article of Hadley for further information. thanks. – jacksonslsmg4 Nov 29 '11 at 10:57
@mike the link to the dissertation seems to be broken; could you please include the dissertation's title in your answer? – WetSavannaAnimal aka Rod Vance Jul 10 '13 at 6:53

Not the answer you're looking for? Browse other questions tagged or ask your own question.