End week 4

 For the first day this week, I went in to the office in Palaiseau. I feel like I didn't accomplish much this week, although the Laser 50 conference was enjoyable.

I read two papers today, and scanned the samples I'll be ablating (hopefully...) soon. I'll post 

the pics in a separate post as blogger seems to throw a fit when I try to put too many images into a post.

Measurement of Gaussian Laser Beam Radius Using the Knife-Edge Technique: Improvement on Data Analysis

 de Araújo, Silva, de Lime, Pereira, de Oliveira; Applied Optics Vol 48, No 2, 10 January 2009

 The authors begin by assuming the incident beam has a Gaussian spatial profile, as follows:

Where w is the beam radius at the point where the intensity declines as 1/e, and xo and yo are the coordinates of the center of the transverse profile of the beam. If one were to drag an edge (usually a knife via a precision micrometer) in a direction, say x, such that the knife slices out lines of constant x, the power which reaches passes by the knife edge is given by:

Where x' is simply a variable of integration. The denominator is for normalization and the top involves an error function in x. The general solution to the above equation is given by:

The difficulty with this equation is that the error function is impractical for use in fitting experimental data. Alternative methods include using the derivative of P with respect to x, but such differentiation results in amplification of fluctuations and consequently increases error.  None of this is new. The authors reference a previous paper by Khosrofian and Garetz, who suggest using a purely analytic which approximately represents P(x) to fit the data. The function they suggest is:

Where the a's are simple coefficients. To extend the results to negative s, K&G claim that f(-s)=1-f(s), that is to say, the function is odd with respect to P(x)=.5 as seen in the graph below.

The addition of the authors of this paper is that claim that p(s) may only have odd order terms, the original authors having included both odd and even order terms. Frankly, I'm not sure I buy this yet, and I want to give it some more thought over the weekend. In any case, the authors find:

a1=-1.5954086

a3=-7.3638857e-2

a5=6.4121343e-4

Which they then use to determine w of a HeNe beam experimentally to an accuracy of +/- .06 um.

Measuring a Narrow Bessel Beam Spot by Scanning a CCD Pixel

Tiwari, Ram, Jayabalan, Mishra, Measurement Scence and Technology 21 (2010)

Although the authors specify their measuring technique as useful for Bessel beams, the same precision makes the process useful for Gaussian beam profiles as well. I will summarize this procedure very generally, since it seems a little computation heavy/involves some software munching as compared to what we're doing.

 The idea is as follows. First, assume that the number of "counts" per pixel on a CCD is linear with applied intensity. Assume the beam is propagating in the z direction. If one wants to find the beam profile at some z', tag a single pixel on a CCD array. Arrange the pixel at the center of the beam in the y direction by scanning until the maximum number of counts on that pixel is found, applying filters as needed.  The, scan that particular pixel in the y direction to map out an intensity profile of the beam in the z' plane. One may then plot the (normalized) pixel counts as a function of y and compare with the expected profile while varying a parameter. In our case, one would expect a Gaussian profile and would vary the beam width.

 The authors intended to use said method for Bessel beams because the knife method was not available due to the significant amount of beam power contained in the spot rings.