# Gaussian Beam Propagation ## Import files ```python import BeamProp_Script as bs # This is the script that handles the propagation import sympy as sym # For Symbolic examples import numpy as np # Handling of lists and for plotting import matplotlib.pyplot as plt # Plotting ``` ### Let's show what BeamProp_Script has ```python help(bs) ``` Help on module BeamProp_Script: NAME BeamProp_Script - Created on Wed Feb 19 15:51:54 2020 DESCRIPTION @author: wrighta FUNCTIONS W0(zr, lam) Parameters ---------- zr : float, integer, symbol Rayleigh range in meters lam : float, integer, symbol Wavelength of light in meters Returns ------- w0 : float, integer, symbol Beam waist radius in meters Zr(wo, lam) Parameters ---------- wo : float, integer, or symbol Beam waist radius in meters. lam : float, integer, or symbol Wavelength of light in meters. Returns ------- zr : float, int, symbols Rayleigh range for given beam waist and wavelength. lens(f) Parameters ---------- f : float or integer or sympy symbol in meters Thin lens focal length in meters Returns ------- mat : 2x2 matrix [ [ 1, 0], [-1/f, 1] ] mult(mat1, *argv) Parameters ---------- mat1 : 2x2 ABCD matrix Last matrix light interacts with. *argv : 2x2 ABCD matrices From left to right, the matrices should be entered such that the leftmost matrix interacts with light temporally after the rightmost matrix. Returns ------- Mat : 2x2 matrix The ABCd matrix describing the whole optical system. plot(func, var, rang=array([0. , 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, 0.1 , 0.11, 0.12, 0.13, 0.14, 0.15, 0.16, 0.17, 0.18, 0.19, 0.2 , 0.21, 0.22, 0.23, 0.24, 0.25, 0.26, 0.27, 0.28, 0.29, 0.3 , 0.31, 0.32, 0.33, 0.34, 0.35, 0.36, 0.37, 0.38, 0.39, 0.4 , 0.41, 0.42, 0.43, 0.44, 0.45, 0.46, 0.47, 0.48, 0.49, 0.5 , 0.51, 0.52, 0.53, 0.54, 0.55, 0.56, 0.57, 0.58, 0.59, 0.6 , 0.61, 0.62, 0.63, 0.64, 0.65, 0.66, 0.67, 0.68, 0.69, 0.7 , 0.71, 0.72, 0.73, 0.74, 0.75, 0.76, 0.77, 0.78, 0.79, 0.8 , 0.81, 0.82, 0.83, 0.84, 0.85, 0.86, 0.87, 0.88, 0.89, 0.9 , 0.91, 0.92, 0.93, 0.94, 0.95, 0.96, 0.97, 0.98, 0.99, 1. , 1.01, 1.02, 1.03, 1.04, 1.05, 1.06, 1.07, 1.08, 1.09, 1.1 , 1.11, 1.12, 1.13, 1.14, 1.15, 1.16, 1.17, 1.18, 1.19, 1.2 , 1.21, 1.22, 1.23, 1.24, 1.25, 1.26, 1.27, 1.28, 1.29, 1.3 , 1.31, 1.32, 1.33, 1.34, 1.35, 1.36, 1.37, 1.38, 1.39, 1.4 , 1.41, 1.42, 1.43, 1.44, 1.45, 1.46, 1.47, 1.48, 1.49, 1.5 , 1.51, 1.52, 1.53, 1.54, 1.55, 1.56, 1.57, 1.58, 1.59, 1.6 , 1.61, 1.62, 1.63, 1.64, 1.65, 1.66, 1.67, 1.68, 1.69, 1.7 , 1.71, 1.72, 1.73, 1.74, 1.75, 1.76, 1.77, 1.78, 1.79, 1.8 , 1.81, 1.82, 1.83, 1.84, 1.85, 1.86, 1.87, 1.88, 1.89, 1.9 , 1.91, 1.92, 1.93, 1.94, 1.95, 1.96, 1.97, 1.98, 1.99, 2. , 2.01, 2.02, 2.03, 2.04, 2.05, 2.06, 2.07, 2.08, 2.09, 2.1 , 2.11, 2.12, 2.13, 2.14, 2.15, 2.16, 2.17, 2.18, 2.19, 2.2 , 2.21, 2.22, 2.23, 2.24, 2.25, 2.26, 2.27, 2.28, 2.29, 2.3 , 2.31, 2.32, 2.33, 2.34, 2.35, 2.36, 2.37, 2.38, 2.39, 2.4 , 2.41, 2.42, 2.43, 2.44, 2.45, 2.46, 2.47, 2.48, 2.49, 2.5 , 2.51, 2.52, 2.53, 2.54, 2.55, 2.56, 2.57, 2.58, 2.59, 2.6 , 2.61, 2.62, 2.63, 2.64, 2.65, 2.66, 2.67, 2.68, 2.69, 2.7 , 2.71, 2.72, 2.73, 2.74, 2.75, 2.76, 2.77, 2.78, 2.79, 2.8 , 2.81, 2.82, 2.83, 2.84, 2.85, 2.86, 2.87, 2.88, 2.89, 2.9 , 2.91, 2.92, 2.93, 2.94, 2.95, 2.96, 2.97, 2.98, 2.99])) Parameters ---------- func : Sympy function of one variable Sympy function defining the beam width after the last optical element. var : sympy variable Variable in func that will be plotted. rang : numpy array Array of the values along the optical axis to be plotted Returns ------- plot : matplotlib graph Graph of the beam width of var prop(d) Parameters ---------- d : float or integer or sympy symbol Distance light is propagating along the z-axis. Returns ------- mat: 2x2 matrix [ [1, d], [0, 1] ] q1_func(z, w0, lam, mat) Parameters ---------- z : float, int, symbol Position of the beam waist in meters. w0 : float, int, symbol Radial waist size in meters (of the embedded Gaussian, i.e. W0/M). lam : float, int, symbol Wavelength of light in meters. mat : float, int, symbol The ABCD 2x2 matrix describing the optical system. Returns ------- z: float, int, symbol Position of the beam waist after the optical system zr: float, int, symbol Rayleigh range of the beam after the optical system q1_inv_func(z, w0, lam, mat) Parameters ---------- z : float, int, symbol Position of the beam waist in meters. w0 : float, int, symbol Radial waist size in meters (of the embedded Gaussian, i.e. W0/M). lam : float, int, symbol Wavelength of light in meters. mat : float, int, symbol The ABCD 2x2 matrix describing the optical system. Returns ------- R : float, int, symbol Radius of curvature of the wavefront in meters. w : float, int, symbol Radius of the beam in meters. ray(y, theta) Parameters ---------- y : float or integer or sympy symbol in meters The vertical height of a ray. theta : float or integer in radians The angle of divergence of the ray. Returns ------- mat : 2x1 matrix [ [y], [teta] ] DATA oo = oo FILE c:\users\wrighta\documents\beamprop\beamprop_script.py ## Let's first see how we define a beam and how we can visualize it propagating. ### A Gaussian beam can be defined by it's (radial) waist, $w_0$, it's Rayleigh range, $z_R = \frac{\pi * w_0^2}{\lambda}$, and the location of its waist, $z_0$. ```python w0 = 1E-3 # 1mm beam waist lam = 355E-9 # wavelength of 355 nm (UV) zR = bs.Zr(w0, lam) # Rayleigh range in m z0 = 0 # location of waist in m ``` ### We now want to define our "optical system" using matrices. For this first example, we will just use a free space propagation matrix, and let the beam propagate a distance $d$ which we will define using a symbol. ```python d = sym.symbols('d') M = bs.prop(d) ``` ### We now use the bs script to do all the ABCD and q-parameter math, and return the waist and radius of curvature functions ```python R, w = bs.q1_inv_func(0, w0, lam, M) ``` ```python print('w = {}'.format(w)) ``` w = 0.001*(0.0127690021685256*d**2 + 1)**0.5 ### And as simple as that, we have a function for our waist. Let's plot it and see what it looks like ```python bs.plot(w, d, rang = np.arange(0,10)) ``` ![png](output_14_0.png) ### Let's show what happens when a beam travels through a lens. We use the "mult" function to multiply multiple ABCD matrices together. ```python w0 = 1E-3 # 1mm beam waist lam = 355E-9 # wavelength of 355 nm (UV) zR = bs.Zr(w0, lam) # Rayleigh range in m z0 = 0 # location of waist in m d = sym.symbols('d') M = bs.mult(bs.prop(d), bs.lens(.5), bs.prop(1)) R, w = bs.q1_inv_func(0, w0, lam, M) bs.plot(w, d, rang = np.arange(0,1,.01)) ``` ![png](output_16_0.png) ### Lets look at how to expand and collimate a beam with a two lens system ```python w0 = 1E-3 # 1mm beam waist lam = 355E-9 # wavelength of 355 nm (UV) zR = bs.Zr(w0, lam) # Rayleigh range in m z0 = 0 # location of waist in m d1, d2, d3, f1, f2 = sym.symbols('d1 d2 d3 f1 f2') M = bs.mult(bs.prop(d3),bs.lens(f2),bs.prop(d2), bs.lens(f1), bs.prop(d1)) R, w = bs.q1_inv_func(0, w0, lam, M) ``` ### For example, lets say the beam travels 1 m before hitting the first lens, and we want the beam to be 5x w0 after coming out of the second lens. We substitute d1 for 1 meter, since the beam propagates 1 meter, and we substitute d3 for 0, since we only care about the beam size right at the second lens. This gives us a relation between f1 and d2 (the separation between the lenses). ```python w = w.subs(d1,1).subs(d3,0) f1_eq = sym.solve(w - 5*w0, f1)[0] print('f = {}'.format(f1_eq)) ``` f = 1.0084642216545e+15*d2*(1.12051580183833e+27*d2 - 4.41556446152598e+29*sqrt(1 - 0.000504320418227052*d2**2) + 8.88733242867719e+28)/(1.13000009595246e+42*d2**2 + 2.26000019190491e+42*d2 - 2.12276362486616e+45) #### Suppose we wanted the distance between the lenses to be 1 meter, we could find what f1 we need. ```python print('f1 = {:.2f} m, for a lens separation of 1 meter'.format(f1_eq.subs(d2, 1))) ``` f1 = 0.17 m, for a lens separation of 1 meter ### Now we need to collimate the beam. Lets still assume the beam propagates 1 m, and f1 = .17 m. There are a couple different ways to think about collimation. One is that the beam size doesn't change over a long distance. The other is that the radius of curvature is infinite (i.e. a plane wave). Lets us the latter interpretation. Thus, we want to find the focal length f2 that makes R infinite, or that makes 1/R =0. ```python R_coll = R.subs(d1,1).subs(d2,1).subs(f1,.17).subs(d3,0) f2_coll = sym.solve(1/R_coll,f2)[0] print('f2 = {:.2f}, for a collimated beam, 5x the original waist, after propagating 1m to the first lens of f1 = .17m, and propagating another 1m to the second lens'.format(f2_coll)) ``` f2 = 0.83, for a collimated beam, 5x the original waist, after propagating 1m to the first lens of f1 = .17m, and propagating another 1m to the second lens ### Lets plot the beam profile after the second lens, and see if it is collimated. ```python M = bs.mult(bs.prop(d3),bs.lens(.83),bs.prop(1), bs.lens(.17), bs.prop(1)) R, w = bs.q1_inv_func(0, w0, lam, M) bs.plot(w,d3) ``` ![png](output_27_0.png) ### Looks very collimated. Lets check the beam size (to make sure its 5* w0) and check the collimation ```python expansion_factor = w.subs(d3,0)/ w0 print('beam is w = {:.2f} x w0'.format(expansion_factor)) ``` beam is w = 4.90 x w0 ```python beam_size_change = (w.subs(d3,10) - w.subs(d3,0)) / w.subs(d3,0) * 100 print('Over 10 m after second lens, beam changes by {:.0f}%'.format(beam_size_change)) ``` Over 10 m after second lens, beam changes by 1% ```python ```