PICUP Laser Beam Profile EXERCISE 4: FIT EXPERIMENTAL DATA TO THE MODEL KNIFE-EDGE PROFILE

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Laser Beam Profile

Developed by E. Behringer

This set of exercises guides the student to model the results of an experiment to determine the profile of a laser beam using a knife-edge technique. It requires the development of the model of the knife-edge profile, and fitting of the model profile to experimental data. Here, the computational tasks are handled by built-in functions of the computational tool being used to complete these exercises.

Subject Area	Waves & Optics
Level	Beyond the First Year
Available Implementation	Python
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 knife-edge profile of the laser beam (Exercise 3); and fit the model, i.e., of the irradiance versus knife-edge position, to experimental data (Exercise 4)
Time to Complete	120 min

Exericse 4

EXERCISE 4: FIT EXPERIMENTAL DATA TO THE MODEL KNIFE-EDGE 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 are in., and the uncertainties in are .

[in.]		[in.]
-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 with mm; and the fit function.

Beam_Profile_Exercise_4.py

# Beam_Profile_Exercise_4.py

# This file will generate a plot of

# beam profile data, guess function, and fit function

# for the TEM00 laser mode

# The fit is generated using the curve_fit function of Python

# Written by:

# Ernest R. Behringer

# Department of Physics and Astronomy

# Eastern Michigan University

# Ypsilanti, MI 48197

# (734) 487-8799

# This email address is being protected from spambots. You need JavaScript enabled to view it.

# Selected and normalized power data from graduate student:

# Najwa Sulaiman

# Department of Physics and Astronomy

# Eastern Michigan University

# Ypsilanti, MI 48197

# This email address is being protected from spambots. You need JavaScript enabled to view it.

# 20160627 by ERB

# import the commands needed to make the plot and fit the data

from __future__ import print_function

from pylab import xlim,xlabel,ylim,ylabel,grid,show,plot,legend,title

from numpy import linspace,ones,sqrt

from scipy.special import erf

from scipy.optimize import curve_fit

from matplotlib.pyplot import errorbar

# Inputs

npts_guess = 200 # number of intervals in the guess function

# Initialize the fit parameter(s)

w0 = 0.020 # The initial guess for w0 [in.]

print('The initial guess for w0 is ',w0,' in.')

# The knife-edge position data (corrected from raw data) [in.]

Xcorr_points = -linspace(0.0366, -0.0354, 37)

# The power data (corrected and normalized from raw data)

Powercorr_points = [1.000, 1.000, 1.000, 0.999, 0.999, 0.998, 0.998, 0.998, 0.995, 0.992, 0.983, 0.968, 0.947, 0.915, 0.870, 0.814, 0.738, 0.638, 0.532, 0.420, 0.315, 0.229, 0.156, 0.105, 0.066, 0.040, 0.025, 0.016, 0.012, 0.010, 0.008, 0.007, 0.007, 0.007, 0.006, 0.006, 0.005]

# Error bar data here

Xcorr_points_err = 0.0001*ones(len(Xcorr_points)) # position error [in.]

Powercorr_points_err = 0.01*ones(len(Powercorr_points)) # scaled power error

# Define the fit function: 0.5*(1.0 - erf(sqrt(2.0)*x/w0))

def func(x, w0):

return 0.5*(1.0 - erf(sqrt(2.0)*x/w0))

# Generate the guess function

# First, the knife-edge position values

Xcorr_inch_guess = linspace(min(Xcorr_points),max(Xcorr_points),npts_guess)

# Second, the scaled power guess

Powercorr_guess = func(Xcorr_inch_guess, w0)

# Use the curve_fit function to fit the model to the experimental data

# Note that the output consists of two parts:

# The first part is an array of the values of the fit parameters

# as determined by a least-squares fitting algorithm;

# the second part is the covariance array (a square, symmetric matrix)

# that has the uncertainties in the parameters on the diagonal of the array.

curve_fit_output = curve_fit(func, Xcorr_points, Powercorr_points, w0)

print(curve_fit_output)

w0_fit = curve_fit_output[0]

# Generate the array needed to make a smooth plot of the fit function

Powercorr_fit = func(Xcorr_inch_guess,w0_fit)

# Define the limits of the horizontal axis

xlim(min(Xcorr_points),max(Xcorr_points))

# Label the horizontal axis, with units

xlabel("Knife-edge position \(x\) [inch]", size = 16)

# Define the limits of the vertical axis

ylim(min(Powercorr_points),max(Powercorr_points))

# Label the vertical axis, with units

ylabel("Scaled transmitted power \(P(x)/P_0\)", size = 16)

# Make a grid on the plot

grid(True)

# Plot the data as symbols with error bars: magenta (m) circles (o).

errorbar(Xcorr_points,Powercorr_points,xerr=Xcorr_points_err,yerr=Powercorr_points_err,fmt="mo",label="Data")

# Plot the guess as a line: black (k) dashed (--)

plot(Xcorr_inch_guess,Powercorr_guess,"k--",label="Guess")

# Plot the fit as a line: blue (b)

plot(Xcorr_inch_guess,Powercorr_fit,"b",label="Fit")

# Make the legend

legend(loc=1)

# Make title

title('Guess value of \(w_0 = \)%.2e mm, fit value of \(w_0 = \)%.2e mm'%(w0,w0_fit))

show()

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Credits

Fremont Teng; Loo Kang Wee

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Details: Written by Loo Kang Wee; Parent Category: 06 Modern Physics; Category: 04 Laser and Semi-conductors; Published: 30 April 2018; Created: 30 April 2018; Last Updated: 20 July 2018; Hits: 6943