Electric field in coating layer is calculated. This will be used in loss calculation in AlGaAs coatings.

• In each coating layer, there are two E waves, transmitted and reflected  waves. The two interfere and become an effective field.
• The averaged electric field will depend only on the transmitted beam inside each layer, see the calculation.
• The effective transmissivity can be calculated, for coatings with N layers between air and substrate, there will be an N+1 vector representing the effective transmission, called tbar in the code. This tbar(n) is the transmissivity in the nth layer, similar to rbar in Evans etal calculation.
• The ratio of E field/ E input in nth layer will be tbar(1)*tbar(2)*...tbar(n)
•  |E field/ E input |^2 of the final transmitted beam is the transmission of the coatings.  The numbers from this calculation agrees to the calculation from before.

==supplementary information==

1) average E field in layer is the transmitted E field in the layer.

 I attached a short matlab file for a simulation of the combined field. Ein in each layer will be the transmitted beam through the layers. For a value of r close to 1, we get a standing wave. Try changing the value of r in test_refl.m to see the effect

2) Calculation for the transmitted field in each layer

I borrow the notation from Evns etal paper (rbar), the calculation code multidiel_rt.m is attached below. Note: the final transmission calculated in the code is the transmission from the coating to the substrate. To calculate the transmission to the air, multiply the last transmission by 2*n_sub/(n_sub + n_air) which is the transmission from sub to air. Since the thickness of the substrate is not known with the exact number, it will not be exact to the transmision calculated in GWINC or Matt A's code (which do not take the sub-air surface into account), but they will be close, because the reflected beam in the last interface will be small compare to those in the coatings.

==result==

The penetration of E field for QWL and different optimized coatings are shown here. The transmissions in the legend are calculated from MattA./GWINC and the values in the parenthesis are calculated from multidiel_rt.m which include the effect from the substrate-air surface. This makes the values in the parenthesis smaller (as more is reflected back and less is transmitted).