What is the difference between your results and the "exact" results?  Are you getting a Bragg like response, but just at a shifted frequency?  Is there not stop band at all in  your results?  If you increase grid resolution, does it eventually converge to the correct results?  

If you are getting at Bragg looking results, I think your FDTD algorithm is working.  You are probably not representing the period of the grating correctly.  The most critical dimension to get correct in this simulation is the period of the grating.  Is your period what you are calling S?  If so divide by your grid resolution (maybe dz?) and see what king of number you get.  If it is something like 20.6, that means you are probably not representing the grating period accurately.  Adjust your grid resolution so that the grating period is an exact integer number of grid cells.

Another common mistake is to set the permittivity values in the algorithm to the refractive index.  Permittivity is actually refractive index squared.

Another thing you can try is to set the refractive indices of your grating to both 1.0.  This will force your FDTD to simulate transmision through free space.  You should get 100% transmission and 0% reflection.  That is a good test of your code and boundary conditions.

Take a look at Lecture 1 here:

http://emlab.utep.edu/ee5390fdtd.htm

It outlines how to build your FDTD code in stages to help troubleshoot your problem.  

Hope something here helps.  Seeing your results and the exact results would help me identify what is wrong.

Thank you for your answer. I obtained Bragg looking results at the right frequency but the band of reflection is wider and I have bigger peaks at the sides of the band of reflection.

The quarter-wave Bragg mirror consists of five periods of layers with Bragg wavelength at 800 nm. The refractive indices are nh=2.5 and nl=1.5 and the thickness of the layers is dh=80nm and dl=133.3 nm, so the optical thickness per layer (nh*dh=nl*dl=200nm) is a quarter of 800 nm.

I chose the space increment, dx=13.33334 nm so that dh=6*dx and dl=10*dx, the time increment is dt=S*(dx/c0) where c0 is the speed of light in vacuum and like I mentioned S=0.5. As you mentioned, I set permittivity as the refractive indices squared.

To compute reflection coefficient, I used a reference simulation of free space to compare with the mirror simulation and I obtained the results mentioned above. Is this the proper way to compute the reflection of the structure?

I tested this, by setting both refractive indices to 1.0, and got 0 % reflection.

I attach the results obtain by FDTD "Reflectivity.png" and the exact results "Exact results" shown by the solid line. The results are in function of the wavenumber normalized to the Bragg wavenumber.

For a discussion on calculating reflection and transmission in 1D-FDTD, check out Lecture 7 here:

http://emlab.utep.edu/ee5390fdtd.htm

Looking at your plots, I see two potential problems.

1. The grating period is not represented perfectly.  is your period in integer multiple of dx exactly?  If not, that will throw off your results because your dimensions are not represented accurately.  Adjust dx until it the period is exactly an integer multiple of dx.  For a discussion on this, see Lecture 6 at the above link.

2. Grid dispersion.  Essentially a wave in FDTD propagates slightly slower than a physical wave would.  It is sort of like the permittivity is too high so it has the effect of pushing the response of devices to slightly lower frequencies.  You can compensate for the grid dispersion pretty easily.  See Lecture 10 at the same link above for a discussion.

A simple test to confirm either of the above is to reduce your value of dx while keeping your grid the same physical size (i.e. will need to add points).  Does the spectra get better after doing this?  if not, you have another problem.  If so, it is probably one or both of the above.

Hope this helps!