Laser Coherence Length Calculation

Rev: 0.1

Last Update: December 10, 2013

Setup Environment



In [1]:

    
#Import sympy
from sympy import *

# Setup iPython for formatted output
%load_ext sympy.interactive.ipythonprinting









    



/Users/Michael/Library/Enthought/Canopy_64bit/User/lib/python2.7/site-packages/sympy/utilities/exceptions.py:149: SymPyDeprecationWarning: 

using %load_ext sympy.interactive.ipythonprinting has been deprecated
since SymPy 0.7.3. Use from sympy import init_printing ;
init_printing() instead. See
http://code.google.com/p/sympy/issues/detail?id=3914 for more info.

  warning(see_above, SymPyDeprecationWarning)

Theory

In theory the laser coherence will drop to zero when the extremes of the laser linewidth wavelengths are out of phase by $\pi$ radians. Due to the slight difference in wavelength between the high and low edges of the laser line width they will see a slightly different phase change as the light propagates. The location where the difference in the phase change at the high an low edges of the linewidth is $\pi$ radians is loosely called the coherence length. Typically one does not want the coherence dropping to zero in the application and the coherence length might be specified at a different reduction in coherence. For simplicity, the derivation below will be for the case where the coherence drops to zero.

Derivation of Coherence Length

Define symbols:



In [2]:

    
n, d, L, L1, L2, dL = symbols("n, d, lambda, lambda_1, lambda_2, Delta_lambda")

n is the index of refraction, d is the distance the light travels and $\lambda$ is the center wavelength of the laser. $\lambda_1$ and $\lambda_2$ are the upper and lower wavelength cutoffs for the laser linewidth, the extremes in wavelength. Finally, $\lambda_\delta$ is the difference in wavelength between the upper and lower cutoffs of the linewidth. The phase of the light at $\lambda_1$ and $\lambda_2$ after propagation are given by:



In [3]:

    
phi1 = 2*pi*n*d / L1
phi2 = 2*pi*n*d / L2

$\phi_1$ is the phase change for $\lambda_1$ and $\phi_2$ is the phase change for $\lambda_2$.

Setting the difference between the phase changes equal to $\pi$ and solving for d results in the coherence length.



In [4]:

    
cl = solve((phi2-phi1)-pi,d)
cl[0]









    Out[4]:





$$\frac{\lambda_{1} \lambda_{2}}{2 n \left(\lambda_{1} - \lambda_{2}\right)}$$

Assuming that the linewidth is symmetrical one can substitute $\lambda^2$ for $\lambda_1 \lambda_2$. In addition, $\lambda_1-\lambda_2$ can be replaced by $\lambda_\delta$, where $\Delta_\lambda$ is the linewidth of the laser.



In [5]:

    
cl = cl[0].subs(L1*L2,L**2).subs(L1-L2,dL)
cl









    Out[5]:





$$\frac{\lambda^{2}}{2 \Delta_{\lambda} n}$$

Linewidth in Frequency Space

More often than not the linewidth of the laser is defined in frequency space, $\Delta_\nu$. Before substituting, the relationship between $\Delta_\lambda$ and $\Delta_\nu$ needs to be developed.

Define new symbols:



In [6]:

    
c, nu, dnu = symbols("c, nu, Delta_nu")

The governing relationship between wavelenght and frequency is:

$$\nu = \dfrac{c}{\lambda}$$

Taking the derivative provides the following relationship:



In [7]:

    
deltaNu = diff(c/L,L)*dL
deltaNu









    Out[7]:





$$- \frac{\Delta_{\lambda} c}{\lambda^{2}}$$

Solveing for $\Delta_\lambda$ facilitates a simple substitution.



In [8]:

    
deltaLambda=solve(deltaNu-dnu,dL)
deltaLambda[0]









    Out[8]:





$$- \frac{\Delta_{\nu} \lambda^{2}}{c}$$

Substituting for $\Delta_\lambda$ in the coherence length formula above casts the expression in terms of a linwidth defined in frequency space.



In [9]:

    
cl.subs(dL,deltaLambda[0])









    Out[9]:





$$- \frac{c}{2 \Delta_{\nu} n}$$

Results

Linewidth in wavelength space $$l_c = \dfrac{\lambda^2}{2n\Delta_\lambda}$$

Linewidth in frequency space $$l_c = \dfrac{c}{2n\Delta_\nu}$$