Since we are trying to optimize a layer structure for AlGaAs coatings. It is a good idea to summarize some notes about all the coatings details. Thanks Koji for the discussion about the coaitngs.

==some background about SiO2/Ta2O5 QWL with 1/2 wave cap coatings==

 For quarter wave layer stack (QWL) SiO2/Ta2O5 coatings, SiO2 and Ta2O5 are the material with low (nl) and high refractive indices (nh), respectively. Due to the stronger structure of SiO2, we usually have a cap of SiO2  as a protective layer on top. This cap has thickness of 1/2 wave length. The reason is that the reflected beam from the interface between the cap and the next layer will be in phase with the first reflected beam at the air-coating surface, see the figure below (top).

If the SiO2 cap is 1/4 thick, the reflected beam from the interface between the cap and the next layer will destructively interfere, causing the reflectivity to go down (see the picture below, middle). 

However, if the cap is Ta2O5 (nH) material, it can be QWL thickness, and the phase from every reflected beams still interferes constructively (picture below, bottom).

Note: As we can see, the incoming beam and the reflected beam are 180 degree out of phase. It means that the E field at the coatings surface will always be zero. This will prevent the burning on the surface of the coating. With this, the standing wave in the cavity will always have zero E field at the coating surface, see below picture.

This is not AR coat, since all the reflected beams interfere constructively. The reflected beams from AR coating will destructively interfere among each layer.

To sum up for the SiO2/Ta2O5 coatings:

• SiO2 is stronger than Ta2O5, so we use it for the end cap.
• Because SiO2 has lower n than that of Ta2O5, the cap thickness has to be 1/2 wave thick so that all the reflected beams interfere constructively.
• We want the reflected phase to be 180 degree away from the incident beam so that the surface won't get burnt from the building up E field. (If the E field is non zero, it will be amplified by a factor of Finesse/pi).  My previous optimization for AlGaAs that used 1/8 cap was wrong because the reflection phase was not 180. This means that by adjusting the cap thickness to optimize the TO noise is not a good method, since the reflection phase is not close to 180 anymore. The optimization has to take the phase into account.

==AlGaAs coatings==

 For GaAs/Al_0.92Ga_0.08As (AlGaAs) coatings, the situation is a bit different from SiO2/Ta2O5. The cap has to be GaAs (nH) because Al_0.92Ga_0.08As will oxidize and change its material properties. Now that the cap will be nH, the thickness has to be 1/4 wavelength.  The last layer next to the substrate has to be GaAs (nH) too (I think because of both the better reflectivity and the fabrication process).

==optimization code==

 There is an assumption about the layer structure used in the optimization code that the cap is nL(SiO2), 1/2 layer. The coatings layers are even number ( doublets of SiO2/Ta2O5). I'm making sure all the assumptions in the code are fixed. Here is a preliminary result.

above: Layer structure, the first layer (cap) is GaAs (nH). In the optimization, I keep the cap thickness to be 1/4, and vary the rest.

above: Noise budget of the optimized layer. TO noise is below BR noise from DC up to 1kHz.

The reflectivity of the coatings is -0.9997 + 0.0209i  (reflection phase = 180 - 1.2 degree). I'm not sure if this is good enough, maybe better optimization can be done.

Note: My layer structure is really different from what rana did in T1200003. For my structure, the layers near the cap vary a lot before getting close to 0.25 when the layers are close to the substrate. The result from 1200003 is the opposite. The layers near the cap are about 0.25, and start to diverge when the layers are close to the substrate.

above:  Optimized coatings result from T1200003. The optimization probably assume the cap of low index material, but the following layers evolution are opposite of what I got. That's why I'm not sure about my optimization.

I'll upload my codes soon so that people can check my optimization.