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Laser Beam Profile
Developed by Ernest Behringer  Published July 16, 2016
This set of exercises guides the student to model the results of an experiment to determine the profile of a laser beam using a knifeedge technique. It requires the development of the model of the knifeedge profile, and fitting of the model profile to experimental data. Here, the computational tasks are handled by builtin functions of the computational tool being used to complete these exercises.
Subject Area  Waves & Optics 

Level  Beyond the First Year 
Available Implementations  Python and Easy Java Simulations 
Learning Objectives 
Students who complete this set of exercises will be able to
* express an equation predicting the profile of a laser oscillating in the TEM00 mode in terms of dimensionless ("scaled") variables suitable for coding (**Exercise 1**);
* produce both line plots and contour plots of the (scaled) irradiance of the beam versus (scaled) position(s) (**Exercises 1 and 2**);
* develop a model of, and plot, the knifeedge profile of the laser beam (**Exercise 3**); and
* fit the model, i.e., of the irradiance versus knifeedge position, to experimental data (**Exercise 4**)

Time to Complete  120 min 
These exercises are not tied to a specific programming language. Example implementations are provided under the Code tab, but the Exercises can be implemented in whatever platform you wish to use (e.g., Excel, Python, MATLAB, etc.).
### Exercise 1: The TEM$_{00}$ Mode: "Line Cuts"
The irradiance $I$ (power per unit area) of a laser oscillating in the TEM$_{00}$ mode can be expressed as
$$
I(x,y) = I_0\exp\Bigg[{{2(x^2+y^2)}\over{w_0^2}}
\Biggr]
$$
where $I_0$ is the maximum irradiance of the beam and $w_0$ is a parameter that describes the width (i.e., the spatial extent) of the beam. You can show that when $y=0$, the locations $x_{1/2}$ at which $I = I_0/2$ are $x_{1/2} = \pm(0.5\ln{2})^{1/2}w_0 = \pm 0.59w_0$. Show that when $y=0$ and $x = w_0$, the irradiance is $I = I_0\exp(2) = 0.135$, and show that the above expression for the irradance is equivalent to
$$I(x,y) = I_0\exp\bigg[2\bigl(\tilde{x}^2+\tilde{y}^2\bigr)
\biggr]
$$
where the "scaled variables" $\tilde{x}$ and $\tilde{y}$ are defined as $\tilde{x}\equiv x/w_0$ and $\tilde{y}\equiv y/w_0$. We can relate this expression to the total power $P_0$ transmitted by the beam:
$$
I(x,y) = {{2P_0}\over{\pi w_0^2}}\exp\bigg[2\bigl(\tilde{x}^2+\tilde{y}^2\bigr)
\biggr]
$$
(a) Assume $P_0 = 1.0$ mW and $w_0 = 1.0$ mm to generate a line plot of the irradiance $I$ on the vertical axis versus $\tilde{x}$ on the horizontal axis for $2\leq \tilde{x} \leq 2$ for different values of $\tilde{y}=0.0, 0.2, 0.4, 0.6, 0.8,$ and $1.0$.
(b) Repeat (a) but now plot the irradiance divided by the maximum irradiance for that particular value of $\tilde{y}$ (in other words, plot $I/I_{max,{\tilde{y}}}$) versus $\tilde{x}$ on the horizontal axis. Comment on the resulting plot.
### Exercise 2: The TEM$_{00}$ Mode: A Contour Plot
(a) A filled contour plot is a plot that indicates value by color as a function of two variables. Make a plot of the irradiance versus the variables $x$ and $y$ for $2w_0 \leq x,y \leq 2w_0$, with $w_0 = 0.5$ mm and $P_0 = 1.0$ mW.
(b) Using the same values for $w_0$ and $P_0$, make a filled contour plot of the scaled irradiance $I(x,y)/(2P_0/\pi w_0^2)$ versus the scaled variables $\tilde{x}$ and $\tilde{y}$ for $2 \leq \tilde{x},\tilde{y} \leq 2$.
### Exercise 3: The KnifeEdge Profile
You can experimentally determine the value of the parameter $w_0$ that describes the width of the beam by measuring the profile of the laser beam using a knifeedge mounted on a linear translation stage together with an optical detector connected to a power meter. The idea is that you allow the laser beam to enter the detector while you translate the knifeedge across the beam and monitor the power meter reading. Initially, when the beam is not blocked, the reading gives the total power of the beam; finally, the beam is completely blocked from entering the detector by the knifeedge and the reading would be zero (in the absence of background light). A schematic is shown below; we assume the beam is centered on the origin.
![Alt Figure](images/laser_beam_profile/Beam_Profile_by_knifeedge.png "")
Imagine that the knifeedge travels in the $+x$direction, and that the position of the knifeedge is $x$. Then the power received by the detector is
$$P(x) = {{2P_0}\over{\pi w_0^2}}\int_{\infty}^{+\infty}dy\int_{x}^{+\infty}\exp\bigg[2\bigl(\tilde{x}^2+\tilde{y}^2\bigr)\biggr]dx
$$
where $P_0$ is the total power of the laser beam. Show that this expression is equivalent to
$$
P(x) = {{P_0}\over{2}}\Biggl[1  {\rm erf}\Bigl({{2x}\over{w_0}}\Bigr)\Biggr]
$$
where ${\rm erf}(u) = \int_u^{+\infty}\exp(t^2)dt$ is the error function. Calculate $P(x)/P_0$ and plot this quantity versus $x$, the position of the knife edge, when $w_0 = 0.5$ mm and for $3w_0 \leq x \leq 3w_0$.
### Exercise 4: Fit Experimental Data to the Model KnifeEdge Profile
Imagine that you have gone to the lab, set up the experiment to determine the beam profile, and obtained the data shown in the table below. Uncertainties in $x$ are $\Delta x = \pm 0.0001$ in., and the uncertainties in $P(x)/P_0$ are $\pm 0.001$.
$x$ [in.]  $P(x)/P_0$  $x$ [in.]  $P(x)/P_0$
::  ::  ::  ::
0.0366  1.000  0.0014  0.420
0.0346  1.000  0.0034  0.315
0.0326  1.000  0.0054  0.229
0.0306  0.999  0.0074  0.156
0.0286  0.999  0.0094  0.105
0.0266  0.998  0.0114  0.066
0.0246  0.998  0.0134  0.040
0.0226  0.998  0.0154  0.025
0.0206  0.995  0.0174  0.016
0.0186  0.992  0.0194  0.012
0.0166  0.983  0.0214  0.010
0.0146  0.968  0.0234  0.008
0.0126  0.947  0.0254  0.007
0.0106  0.915  0.0274  0.007
0.0086  0.870  0.0294  0.007
0.0066  0.814  0.0314  0.006
0.0046  0.738  0.0334  0.006
0.0026  0.638  0.0354  0.005
0.0006  0.532  
On the same graph, plot: the experimental data with error bars: the "guess" function $P(x)/P_0$ with $w_0 = 0.02$ mm; and the fit function.
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Credits and Licensing
Ernest Behringer, "Laser Beam Profile," Published in the PICUP Collection, July 2016.
The instructor materials are ©2016 Ernest Behringer.
The exercises are released under a Creative Commons AttributionNonCommercialShareAlike 4.0 license