Does Hyugens principle apply in three dimensions? Does Hyugens principle apply in three dimensions ?
If a surface wave (for simplicity an ocean wave) is propagating along the x axis we know that this wave ray is a point source for wavelets on the y axis but what about the z axis.
If this diagram was 3 d would we see a spherical wave front expanding from each point 

http://physics.ucdavis.edu/Classes/Physics9B_Animations/ReflRefr.html
 A: Yes, absolutely, in general. The Huygens' principle is an intuitive picture of the solution of the Helmholtz equation through superposition of Green's functions. The basic solution is $E(\mathbf{r})=\frac{\exp(i\,k\,|\mathbf{r}|)}{|\mathbf{r}|}$ and you're simply building solutions out of sums of this one ("sums" in the broad sense of "linear combination" that includes integrals).
The building of solutions to quantum field theory problems using basic solutions called "propagators" is also often referred to as "Huygens' principle". It's the same basic idea. 
The exact Green's function depends on the dimensionality of your problem and also the boundary conditions. However, Huygens's principle is the approximation that for most boundary conditions an approximate solution can be built by assuming sources of waves of the form $\frac{\exp(i\,k\,|(\mathbf{r}-\mathbf{r}_0|)}{|\mathbf{r}-\mathbf{r}_0|}$ with centers $\mathbf{r}_0$ on the "primary" wavefront. Moreover, the Green function changes for two dimensional problems. If we have a two dimensional problem wherein there is only variation in $x$ and $y$, the Green's function is no longer $E(\mathbf{r})=\frac{\exp(i\,k\,|\mathbf{r}|)}{|\mathbf{r}|}$ but rather one expressed through the Hankel function:
$$E(\mathbf{r})\propto H_0^\pm(k\,|\mathbf{r}|)\sim \sqrt{\frac{2}{\pi\,|\mathbf{r}|}}\,\exp(\pm i\,k\,|\mathbf{r}|)\;\text{as}\;k\,|\mathbf{r}|\to\infty$$
the asymptotic expression becoming pretty accurate for $k\,|\mathbf{r}|$ greater than about 10.
