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Message ID: 58     Entry time: Fri Jun 25 00:11:13 2010
Author: James K 
Type: Misc 
Category: Hartmann sensor 
Subject: SURF Log -- Day 7, SLED Beam Characterization 

BACKGROUND:


In order to conduct future optical experiments with the SLED and to be able to predict the behavior of the beam as it propagates across the table and through various optics, it is necessary to know the properties of the beam. The spot size, divergence angle, and radius of curvature are all of interest if we wish to be able to predict the pattern which should appear on the Hartmann sensor given a certain optical layout.

It was therefore necessary to conduct an experiment to measure these properties. The wavefront emanating from the SLED is assumed to be approximately Gaussian, and thus has an intensity of the form:

 

where A is some amplitude, w is the spot size, x and y are the coordinates transverse to the optical axis, and x0 is the displacement of the optical axis in the x-direction from the optical axis. The displacement of the optical axis in the y-direction is assumed to be zero (that is, y0=0). A and w are both functions of z, which is the coordinate of displacement parallel to the optical axis.

 

Notice that the total intensity read by a photodetector reading the entire beam would be the double integral from negative infinity to infinity for both x and y. If a opaque plate was placed such that the the beam was blocked from some x=xm to x=inf (where xm is the location of the edge of the plate), then the intensity read by a photodetector reading the entire non-blocked portion of the beam would be:

 

Mathematica was used to simplify this integral, and it showed it to be equivalent to:

where Erfc() is the complementary error function. Note that for fixed z, this intensity is a function only of xm. If an experiment was carried out to measure the intensity of the beam blocked by a plate from x=-inf to x=xm for multiple values of xm, it would therefore be possible via regression analysis to compute the best-fit values of A, w, and x0 for the measured values of Ipd and xm. This would give us A, w and x0 for that z-value. By repeating this process for multiple values of z, we could therefore find the behavior of these parameters as a function of z.

Furthermore, we know that at z-values well beyond the Rayleigh range, w should be linear with respect to z. Assuming that our measurements are done in the far-field (which, for the SLED, they almost certainly would be) we could therefore find the divergence angle by knowing the slope of the linear relation between w and z. Knowing this, we could further calculate such quantities as the Rayleigh range, the minimum spot size, and the radius of curvature of the SLED output (see p.490 of "Lasers" by Milonni and Eberly for the relevant functional relationships for Gaussian beams).


EXPERIMENT:

An experiment was therefore carried out to measure the intensity of of beam blocked from x~=-inf to x=xm, for multiple values of xm, for multiple values of z. A diagram of the optical layout of the experiment is below:

 

(top view)


The razor blade was mounted on a New Focus 9091 Translational Stage, the relative displacement of which in the x-direction was measured with the Vernier micrometer mounted on the base. Tape was placed on the front of the razor so as to block light from passing through any of its holes. The portion of the beam not blocked by the razor then passed through a lens which was used to focus the beam back onto a PDA1001A Large Area Silicon Photodiode, the voltage output of which was monitored using a Fluke digital multimeter. The ruler stayed securely clamped onto the optical table (except when it was translated in the x-direction once during the experiment, as described later).

The following is a picture of this layout, as constructed:

 

 
The procedure of the experiment was as follows: first, the translational stage was clamped securely with the left-most edge of its base lined up with the desired z-value as measured on the ruler. The z-value as measured on the ruler was recorded. Then, the translational stage was moved in the negative x-direction until there was no change in the voltage measured on the DMM (which is directly proportional to the measured intensity of the beam). When no further DMM readout change was yielded from -x translation, it was assumed that the the razor was no longer blocking the beam. Then, the stage was moved in the +x direction until the voltage output on the DMM just began to change. The micrometer and DMM values were both recorded. The stage was then moved inward until the DMM read a voltage which was close to the nearest multiple of 0.5V, and this DMM voltage and micrometer reading were recorded. The stage was then translated until the DMM voltage dropped by approximately 0.5V, the micrometer and DMM readings were recorded, and this process was repeated until the voltage reached ~0.5V. The beam output was then covered by a card so as to completely block it, and the voltage output from the DMM was recorded as the intensity from the ambient light from that measurement. The stage was then unclamped and moved to the next z-value, and this process was repeated for 26 different values of z, starting at z=36.5mm and then incrementing z upwards by ~4mm for the first ten measurements, then by increments of ~6mm for the remaining measurements.
 
The data from these measurements can be found on the attached spreadsheet.
 
A few notes on the experiment:
 
The vernier micrometer has a measurement limit of 13.5mm. After the tenth measurement, the measured xm values began to exceed this limit. It was therefore necessary to translate the ruler in the negative x-direction without translating it in the z-direction. Plates were clamped snugly to either side of the ruler such that the ruler could not be translated in the z-direction, but could be moved in the x-direction when the ruler was unclamped. After securing these plates, the ruler was moved in the negative x-direction by approximately 5mm. The ruler was then clamped securely in place at its new x location. In order to better estimate the actual x-translation of the ruler, I took the following series of measurements: I moved the stage to z-values at which sets of measurements were previously taken. Then, I moved the razor out of the beam path and carefully moved it back inwards until the output on the DMM matched exactly the DMM output of the first measurement taken previously at that z-value. The xm value corresponding to this voltage was then read. The translation of the stage should be approximately equal to the difference of the measured xm values for that DMM voltage output at that z-value. This was done for 8 z-values, and the average difference was found to be 4.57+-0.03mm, which should also be the distance of stage translation (this data and calculation is included in the "x translation" sheet of the attached excel workbook).
 
At this same point, I started using two clamps to attach the translational stage to the table for each measurement set, as I was unhappy with the level of secureness which one clamp provided. I do not, however, believe that the use of one clamp compromised the quality of previous sets of measurements.

 

RESULTS:


A MATLAB function 'gsbeam.m' was written to replicate the function:

and then another function 'beamdata.m' was written to input each dataset, fit the data to a curve of the functional form of the previous function for each set of data automatically, and then output PDF files plotting all of the fit curves against each other, each individual fit curve against the data from that measurement, and a plot showing the widths w as a function of z. Linear regression was done on w against z to find the slope of the w(z) (which, for these measurements, is clearly shown by the plot that the beam was measured in the far-field and thus w is approximately a linear function of z). An array of the z-location of the ruler, the fit parameters A, x0, x, and the 2-norm of the residual of the fit is also outputted, and is shown below for the experimental data:

 

z(ruler) A x0 w 2normres
36.5 7.5915 11.089 0.8741 0.1042
39.9 5.2604 11.1246 1.048 0.1013
44 3.8075 11.1561 1.2332 0.1164
48 2.777 11.1628 1.4479 0.0964
52 2.1457 11.1363 1.6482 0.1008
56 1.6872 11.4206 1.858 0.1029
60 1.3831 11.2469 2.0523 0.1021
64 1.1564 11.1997 2.2432 0.1059
68 0.972 11.1851 2.4483 0.0976
72 0.8356 11.1728 2.6392 0.1046
78 0.67 6.8821 2.9463 0.0991
84 0.5559 6.7548 3.2375 0.1036
90 0.4647 6.715 3.5402 0.0958
96 0.3993 6.7003 3.8158 0.1179
112 0.2719 6.8372 4.6292 0.0924
118 0.2398 6.7641 4.925 0.1029
124 0.2117 6.7674 5.2435 0.1002
130 0.189 6.8305 5.5513 0.0965
136 0.1709 6.8551 5.8383 0.1028
142 0.1544 6.8243 6.1412 0.0981
148 0.1408 6.7993 6.4313 0.099
154 0.1286 6.8062 6.7322 0.0948
160 0.1178 6.9059 7.0362 0.1009
166 0.1089 6.904 7.3178 0.0981
172 0.1001 6.8817 7.6333 0.1025
178 0.0998 6.711 7.6333 0

 

All outputted PDF's are included in the .zip file attached. The MATLAB functions themselves are also attached.The plots of the fit curves and the plot of the widths vs. the ruler location are also included below:

 

(note that I could probably improve on the colormap that I chose for this. note also that the 'gap' is because I temporarily forgot how to add integers while taking the measurements, and thus went from 96mm on the ruler to 112mm on the ruler despite going by a 6mm increment otherwise in that range. Also, note that all of these fit curves were automatically centered at x=0 for the plot, so they wouldn't necessarily intersect so neatly if I tried to include the difference in the estimated 'beam centers')

(note that the width calculated from the 26th measurement is not included in the regression calculation or included on this plot. The width parameter was calculated as being exactly the same as it was for the 25th measurement, despite the other parameters varying between the measurements. I suspect that the beam size was starting to exceed the dimensions blocked by the razor and that this caused this problem, and that would be easy to check, but I have yet to do it. Regardless, the fit looks good from just the other 25 measurements)

These results are as expected: that the beam spot-size should increase as a function of z and that it should do so linearly in the far-field. My next step will be to use the results of this experiment to calculate the properties of the SLED beam, characterizing the beam and thusly enabling me to predict its behavior within further optical systems.

 

Attachment 1: BeamData.xlsx  21 kB
Attachment 2: beam_pdfs.zip  136 kB  Uploaded Mon Jun 28 01:42:21 2010
Attachment 3: beamdata.m  3 kB  Uploaded Mon Jun 28 08:57:35 2010  | Hide | Hide all
function D=beamdata(M,guess)
%Imports array of beam characterization measurements. Structure of M is 
% [z, x, I, a] where z is the displacement of the beam blockage along
% the optical axis, x is the coordinate of razor edge, I is the measured
% output of the photodetector and a is the ambient light level
%and guess is an estimate of the parameters [Amplitude x0 width] for the
%first measurement
%Output Structure [z A x0 w residual_2norm]
thisfile=mfilename('fullpath');
thisdir=strrep(thisfile,mfilename(),'');
... 105 more lines ...
Attachment 4: gsbeam.m  80 Bytes  Uploaded Mon Jun 28 08:57:58 2010  | Hide | Hide all
function I=gsbeam(x,xdat)
I=pi/4*x(1)*x(3)^2*erfc(sqrt(2)*(x(2)-xdat)/x(3));
end
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