Determining the ROC for ZM4 and ZM5

[Sheila, Camilla, Ryan, Eric]

We would like to verify that our recent mode measurements after ZM5 ( 90783) and before ZM4 (90815 ) make sense by connecting the two.  We decided to use the q value from the measurement at the nominal ZM2 strain in 90815 (ZM2 strain = 3.15V) and propagate that mode through the path containing ZM4 and ZM5 and calculate the overlap with the q values from 90783 measured at different strain settings for ZM4/ZM5.  The goals here are as follows:

1. Improve our model of the ZM4 - to - SEC path and verify that we're interpreting our mode measurements (with M^2 > 1) correctly.  
2. Determine the ROC for ZM4 and ZM5 as a function of the strain gauge voltage.  This is particularly important since we are going to replace ZM5 and Camille would like to know what ROC to shoot for with the pre-stressing procedure.

 

First, I address item 1.

Mode Measurements with M² > 1:

Our system seems to be adding some higher order abberations to the beam.  As a result, our mode measurements indicate that we have an M^2 number significantly above 1 (between 1.2 - 1.5 depending on the PSAM settings).  When M^2 is > 1, the presence of HOM content in the beam prevents one from focusing down to as tight of a waist, for the same divergence angle, the beam radius at the waist will be larger by a factor of M.  The thorlabs beam profiler accounts for this by fitting the data to the following formula (which we confirmed by doing our own independent fit):

w(z)² = w_M²[1 +(z - z₀)² (pi*w_M²/(M²*lambda))²]

Where w_M² = M²*w₀²  Is the waist for a beam with M²>1, and w₀ is the waist for the TEM00 component of the beam (ie for M² = 1). 

The q parameter ends up the same as before:

q(z) = (z-z₀) + i*z_R

where z_R = pi*w₀²/lambda = pi*w_M²/(lambda* M²)

Knowing that M² > 1 tells us that our beam is a mixture of TEM00 and some higher order mode content.  However, from the M² value alone we don't know which higher order modes are excited (in principle one might be able to make some rough projections using the surface abberation measurements of the PSAMs from Caltech, but that sounds tricky and is beyond the scope of today's post).  If we want to do mode matching calculations, the only thing we can do at the moment is back propagate the TEM00 component and do all mode calculations for TEM00.  

We use the same beam propagation matricies as always to back propagate the TEM00 component to determine what the TEM00 mode looks like in HAM 7.  

Determination of the ZM4 and ZM5 ROCs

I then took the q value (for the nominal ZM2 = 3.15V) from the measurement before ZM4, back propagated it to ZM4 using our length measurements.  I then propagated the q through ZM4 and ZM5 and calculated the overlap with the q values measured after ZM5 for various values of the ZM4/ZM5 strain gauge settings in ( 90783)

Then, the ROCs for ZM4 and ZM5 were chosen for each strain gauge settings to maximize the overlap.  The overlap is => 98% over the entire 2D grid of ZM4/ZM5 strain gauge values, which gives us some confidence that the ROC values are accurate.  One thing that gives us pause is that the change in ROC for ZM5 doesn't appear to change linearly in diopters with the strain gauge reading.  ZM4, on the other hand is roughly consistant with a 5 mD/V change though because the beam spot is quite small on ZM4, we are relatively insensitive to its ROC value.

ZM4 Strain (V)	ZM4 ROC (m)
2.0	-12
4.0	-11
6.0	-10
8.0	-9

 

ZM5 Strain(V)	ZM5 ROC (m)
-4.5	3.8
-2.0	4.05
0.0	4.4
2.0	4.55

 

These values give the following overlaps for the x and y direction (our mode measurements indicate we have non-negligible asitgmatism on this path) for propagating the nominal q value from (90815 where ZM2 strain = 3.15) to the q vales from ( 90783) .

ZM4 \ ZM5	-4.5	-2.0	0.0	2.0
2.0	x = .994, y = .995	x = .998, y = .997	x = .990, y = .995	x = .9874, y = .993
4.0	x =.996, y = .997	x = .994, y = .995	x = .986, y = .992	x = .983, y = .991
6.0	x =.995, y = .997	x =.992, y = .993	x =.983, y = .989	x =.980, y = .984
8.0	x =.993, y = .995	x =.990, y = .992	x =.980, y = .984	x =.977, y = .980

The fact that this set of ROC values gives good overlap over the entire 2D grid suggests that these ROCs are a resonable model for ZM4 and ZM5 at these strain gauge settings.

 

Attached is an a la mode file for doing the beam propagation.  One could do some more intellegent fitting of the data to extract the best ROC estimates; I'm just sorta hand fitting it at the moment.