About

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

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 x$x$ are Δx=±0.0001$\mathrm{\Delta }x=\pm 0.0001$ in., and the uncertainties in P(x)/P0$P(x)/P_0$are ±0.001$\pm 0.001$.

x$x$ [in.]	P(x)/P0$P(x)/P_0$	x$x$ [in.]	P(x)/P0$P(x)/P_0$
-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)/P0 with w0=0.02mm; and the fit function.

#

# 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()

 

Translations

Code	Language		Translator	Run

Software Requirements

	Android	iOS	Windows	MacOS
with best with	Chrome	Chrome	Chrome	Chrome
support full-screen?	Yes. Chrome/Opera No. Firefox/ Samsung Internet	Not yet	Yes	Yes
cannot work on	some mobile browser that don't understand JavaScript such as.....		cannot work on Internet Explorer 9 and below

 

Credits

Fremont Teng; Loo Kang Wee

end faq