Backscattering Efficiency Validation

Scott Prahl

9 Feb 2019, Version 1



In [1]:

    
%matplotlib inline

import numpy as np
import matplotlib.pyplot as plt
import miepython as mp

Wiscombe tests

Since the backscattering efficiency is $|2S_1(-180^\circ)/x|^2$, it is easy to see that that backscattering should be the best comparison. For example, the asymmetry factor for this test case only has three significant digits and the scattering efficiency only has two!

A typical test result looks like this:

MIEV0 Test Case 12:  Refractive index:  real    1.500  imag  -1.000E+00,  Mie size parameter =     0.055
                    NUMANG =   7 angles symmetric about 90 degrees

  Angle    Cosine           S-sub-1                     S-sub-2               Intensity  Deg of Polzn
   0.00  1.000000   7.67526E-05   8.34388E-05   7.67526E-05   8.34388E-05   1.28530E-08        0.0000
                   (  1.000000)  (  1.000000)  (  1.000000)  (  1.000000)  (  1.000000)
  30.00  0.866025   7.67433E-05   8.34349E-05   6.64695E-05   7.22517E-05   1.12447E-08       -0.1428
                   (  1.000000)  (  1.000000)  (  1.000000)  (  1.000000)  (  1.000000)
  60.00  0.500000   7.67179E-05   8.34245E-05   3.83825E-05   4.16969E-05   8.02857E-09       -0.5999
                   (  1.000000)  (  1.000000)  (  1.000000)  (  1.000000)  (  1.000000)
  90.00  0.000000   7.66833E-05   8.34101E-05   3.13207E-08  -2.03740E-08   6.41879E-09       -1.0000
                   (  1.000000)  (  1.000000)  (  1.000000)  (  1.000000)  (  1.000000)
 120.00 -0.500000   7.66486E-05   8.33958E-05  -3.83008E-05  -4.17132E-05   8.01841E-09       -0.6001
                   (  1.000000)  (  1.000000)  (  1.000000)  (  1.000000)  (  1.000000)
 150.00 -0.866025   7.66233E-05   8.33853E-05  -6.63499E-05  -7.22189E-05   1.12210E-08       -0.1429
                   (  1.000000)  (  1.000000)  (  1.000000)  (  1.000000)  (  1.000000)
 180.00 -1.000000   7.66140E-05   8.33814E-05  -7.66140E-05  -8.33814E-05   1.28222E-08        0.0000
                   (  1.000000)  (  1.000000)  (  1.000000)  (  1.000000)  (  1.000000)

  Angle          S-sub-1                     T-sub-1                     T-sub-2
   0.00   7.67526E-05   8.34388E-05   3.13207E-08  -2.03740E-08   7.67213E-05   8.34592E-05
         (  1.000000)  (  1.000000)  (  1.000000)  (  1.000000)  (  1.000000)  (  1.000000)
 180.00   7.66140E-05   8.33814E-05   3.13207E-08  -2.03740E-08   7.66453E-05   8.33611E-05
         (  1.000000)  (  1.000000)  (  1.000000)  (  1.000000)  (  1.000000)  (  1.000000)

            Efficiency factors for             Asymmetry
    Extinction    Scattering    Absorption        Factor
      0.101491      0.000011      0.101480      0.000491
  (  1.000000)  (  1.000000)  (  1.000000)  (  1.000000)

Perfectly conducting spheres



In [2]:

    
print("                                 miepython      Wiscombe")
print("     X       m.real   m.imag        Qback         Qback      ratio")

m=complex(1.55, 0.0)
x   = 2*3.1415926535*0.525/0.6328
ref = 2.92534
qext, qsca, qback, g = mp.mie(m,x)
print("%9.3f   % 8.4f % 8.4f   % 8e % 8e %8.5f" % (x,m.real,m.imag,qback,ref,qback/ref))

m=complex(0.0, -1000.0)
x=0.099
ref = (4.77373E-07*4.77373E-07 +  1.45416E-03*1.45416E-03)/x/x*4
qext, qsca, qback, g = mp.mie(m,x)
print("%9.3f   % 8.4f % 8.2f   % 8e % 8e %8.5f" % (x,m.real,m.imag,qback,ref,qback/ref))
x=0.101
ref = (5.37209E-07*5.37209E-07 +  1.54399E-03*1.54399E-03)/x/x*4
qext, qsca, qback, g = mp.mie(m,x)
print("%9.3f   % 8.4f % 8.2f   % 8e % 8e %8.5f" % (x,m.real,m.imag,qback,ref,qback/ref))
x=100
ref = (4.35251E+01*4.35251E+01 +  2.45587E+01*2.45587E+01)/x/x*4
qext, qsca, qback, g = mp.mie(m,x)
print("%9.3f   % 8.4f % 8.2f   % 8e % 8e %8.5f" % (x,m.real,m.imag,qback,ref,qback/ref))
x=10000
ref = abs(2.91013E+03-4.06585E+03*1j)**2/x/x*4
qext, qsca, qback, g = mp.mie(m,x)
print("%9.3f   % 8.4f % 8.2f   % 8e % 8e %8.5f" % (x,m.real,m.imag,qback,ref,qback/ref))
print()









    



                                 miepython      Wiscombe
     X       m.real   m.imag        Qback         Qback      ratio
    5.213     1.5500   0.0000    2.925341e+00  2.925340e+00  1.00000
    0.099     0.0000 -1000.00    8.630007e-04  8.630064e-04  0.99999
    0.101     0.0000 -1000.00    9.347779e-04  9.347732e-04  1.00001
  100.000     0.0000 -1000.00    9.990254e-01  9.990256e-01  1.00000
10000.000     0.0000 -1000.00    9.999998e-01  9.999997e-01  1.00000

Spheres with a smaller refractive index than their environment



In [3]:

    
print("                                 miepython      Wiscombe")
print("     X       m.real   m.imag        Qback         Qback      ratio")
m=complex(0.75, 0.0)
x=0.099
ref = (1.81756E-08*1.81756E-08 + 1.64810E-04*1.64810E-04)/x/x*4
qext, qsca, qback, g = mp.mie(m,x)
print("%9.3f   % 8.4f % 8.4f   % 8e % 8e %8.5f" % (x,m.real,m.imag,qback,ref,qback/ref))
x=0.101
ref = (2.04875E-08*2.04875E-08 + 1.74965E-04*1.74965E-04)/x/x*4
qext, qsca, qback, g = mp.mie(m,x)
print("%9.3f   % 8.4f % 8.4f   % 8e % 8e %8.5f" % (x,m.real,m.imag,qback,ref,qback/ref))
x=10.0
ref = (1.07857E+00*1.07857E+00 + 3.60881E-02*3.60881E-02)/x/x*4
qext, qsca, qback, g = mp.mie(m,x)
print("%9.3f   % 8.4f % 8.4f   % 8e % 8e %8.5f" % (x,m.real,m.imag,qback,ref,qback/ref))
x=1000.0
ref = (1.70578E+01*1.70578E+01 +  4.84251E+02* 4.84251E+02)/x/x*4
qext, qsca, qback, g = mp.mie(m,x)
print("%9.3f   % 8.4f % 8.4f   % 8e % 8e %8.5f" % (x,m.real,m.imag,qback,ref,qback/ref))
print()









    



                                 miepython      Wiscombe
     X       m.real   m.imag        Qback         Qback      ratio
    0.099     0.7500   0.0000    1.108554e-05  1.108554e-05  1.00000
    0.101     0.7500   0.0000    1.200381e-05  1.200382e-05  1.00000
   10.000     0.7500   0.0000    4.658441e-02  4.658462e-02  1.00000
 1000.000     0.7500   0.0000    9.391602e-01  9.391600e-01  1.00000

Non-absorbing spheres



In [4]:

    
print("                                 miepython      Wiscombe")
print("     X       m.real   m.imag        Qback         Qback      ratio")

m=complex(1.5, 0)

x=10
ref = abs(4.322E+00 + 4.868E+00*1j)**2/x/x*4
qext, qsca, qback, g = mp.mie(m,x)
print("%9.3f   % 8.4f % 8.5f   % 8e % 8e %8.5f" % (x,m.real,m.imag,qback,ref,qback/ref))

x=100
ref = abs(4.077E+01 + 5.175E+01*1j)**2/x/x*4
qext, qsca, qback, g = mp.mie(m,x)
print("%9.3f   % 8.4f % 8.5f   % 8e % 8e %8.5f" % (x,m.real,m.imag,qback,ref,qback/ref))

x=1000
ref = abs(5.652E+02 + 1.502E+03*1j)**2/x/x*4
qext, qsca, qback, g = mp.mie(m,x)
print("%9.3f   % 8.4f % 8.5f   % 8e % 8e %8.5f" % (x,m.real,m.imag,qback,ref,qback/ref))
print()









    



                                 miepython      Wiscombe
     X       m.real   m.imag        Qback         Qback      ratio
   10.000     1.5000  0.00000    1.695064e+00  1.695084e+00  0.99999
  100.000     1.5000  0.00000    1.736193e+00  1.736102e+00  1.00005
 1000.000     1.5000  0.00000    1.030309e+01  1.030182e+01  1.00012

Water droplets



In [5]:

    
print("                                                  old")
print("                                 miepython      Wiscombe")
print("     X       m.real   m.imag        Qback         Qback      ratio")

m=complex(1.33, -0.00001)

x=1
ref = (2.24362E-02*2.24362E-02 +  1.43711E-01*1.43711E-01)/x/x*4
qext, qsca, qback, g = mp.mie(m,x)
print("%9.3f   % 8.4f % 8.5f   % 8e % 8e %8.5f" % (x,m.real,m.imag,qback,ref,qback/ref))
x=100
ref = (5.65921E+01*5.65921E+01 +  4.65097E+01*4.65097E+01)/x/x*4
qext, qsca, qback, g = mp.mie(m,x)
print("%9.3f   % 8.4f % 8.5f   % 8e % 8e %8.5f" % (x,m.real,m.imag,qback,ref,qback/ref))
x=10000
ref = abs(-1.82119E+02  -9.51912E+02*1j)**2/x/x*4
qext, qsca, qback, g = mp.mie(m,x)
print("%9.3f   % 8.4f % 8.5f   % 8e % 8e %8.5f" % (x,m.real,m.imag,qback,ref,qback/ref))
print()









    



                                                  old
                                 miepython      Wiscombe
     X       m.real   m.imag        Qback         Qback      ratio
    1.000     1.3300 -0.00001    8.462445e-02  8.462494e-02  0.99999
  100.000     1.3300 -0.00001    2.146326e+00  2.146327e+00  1.00000
10000.000     1.3300 -0.00001    3.757191e-02  3.757215e-02  0.99999

Moderately absorbing spheres



In [6]:

    
print("                                 miepython      Wiscombe")
print("     X       m.real   m.imag        Qback         Qback      ratio")

m=complex(1.5, -1.0)
x=0.055
ref = abs(7.66140E-05 + 8.33814E-05*1j)**2/x/x*4
qext, qsca, qback, g = mp.mie(m,x)
print("%9.3f   % 8.4f % 8.4f   % 8e % 8e %8.5f" % (x,m.real,m.imag,qback,ref,qback/ref))
x=0.056
ref = (8.08721E-05*8.08721E-05 +  8.80098E-05*8.80098E-05)/x/x*4
qext, qsca, qback, g = mp.mie(m,x)
print("%9.3f   % 8.4f % 8.4f   % 8e % 8e %8.5f" % (x,m.real,m.imag,qback,ref,qback/ref))
x=1.0
ref = (3.48844E-01*3.48844E-01 +  1.46829E-01*1.46829E-01)/x/x*4
qext, qsca, qback, g = mp.mie(m,x)
print("%9.3f   % 8.4f % 8.4f   % 8e % 8e %8.5f" % (x,m.real,m.imag,qback,ref,qback/ref))
x=100.0
ref = (2.02936E+01*2.02936E+01 +  4.38444E+00*4.38444E+00)/x/x*4
qext, qsca, qback, g = mp.mie(m,x)
print("%9.3f   % 8.4f % 8.4f   % 8e % 8e %8.5f" % (x,m.real,m.imag,qback,ref,qback/ref))
x=10000
ref = abs(-2.18472E+02  -2.06461E+03*1j)**2/x/x*4
qext, qsca, qback, g = mp.mie(m,x)
print("%9.3f   % 8.4f % 8.4f   % 8e % 8e %8.5f" % (x,m.real,m.imag,qback,ref,qback/ref))
print()









    



                                 miepython      Wiscombe
     X       m.real   m.imag        Qback         Qback      ratio
    0.055     1.5000  -1.0000    1.695493e-05  1.695493e-05  1.00000
    0.056     1.5000  -1.0000    1.822196e-05  1.822197e-05  1.00000
    1.000     1.5000  -1.0000    5.730026e-01  5.730036e-01  1.00000
  100.000     1.5000  -1.0000    1.724214e-01  1.724214e-01  1.00000
10000.000     1.5000  -1.0000    1.724138e-01  1.724138e-01  1.00000

Spheres with really big index of refraction



In [7]:

    
print("                                 miepython      Wiscombe")
print("     X       m.real   m.imag        Qback         Qback      ratio")

m=complex(10, -10.0)
x=1
ref = abs(4.48546E-01 + 7.91237E-01*1j)**2/x/x*4
qext, qsca, qback, g = mp.mie(m,x)
print("%9.3f   % 8.4f % 8.4f   % 8e % 8e %8.5f" % (x,m.real,m.imag,qback,ref,qback/ref))
x=100
ref = abs(-4.14538E+01  -1.82181E+01*1j)**2/x/x*4
qext, qsca, qback, g = mp.mie(m,x)
print("%9.3f   % 8.4f % 8.4f   % 8e % 8e %8.5f" % (x,m.real,m.imag,qback,ref,qback/ref))
x=10000
ref = abs(2.25248E+03  -3.92447E+03*1j)**2/x/x*4
qext, qsca, qback, g = mp.mie(m,x)
print("%9.3f   % 8.4f % 8.4f   % 8e % 8e %8.5f" % (x,m.real,m.imag,qback,ref,qback/ref))









    



                                 miepython      Wiscombe
     X       m.real   m.imag        Qback         Qback      ratio
    1.000    10.0000 -10.0000    3.308997e+00  3.308998e+00  1.00000
  100.000    10.0000 -10.0000    8.201273e-01  8.201267e-01  1.00000
10000.000    10.0000 -10.0000    8.190044e-01  8.190052e-01  1.00000

Backscattering Efficiency for Large Absorbing Spheres

For large spheres with absorption, backscattering efficiency should just be equal to the reflection for perpendicular light on a planar surface.



In [8]:

    
x = np.logspace(1, 5, 20)  # also in microns

kappa=1
m = 1.5 - kappa*1j
R = abs(m-1)**2/abs(m+1)**2
Qbig = R * np.ones_like(x)

qext, qsca, qback, g = mp.mie(m,x)
plt.semilogx(x, qback, '+')
plt.semilogx(x, Qbig, ':')
plt.text(x[-1],Qbig[-1],"$\kappa$=%.3f" % kappa,va="bottom",ha='right')

kappa=0.001
m = 1.5 - kappa*1j
R = abs(m-1)**2/abs(m+1)**2
Qbig = R * np.ones_like(x)

qext, qsca, qback, g = mp.mie(m,x)
plt.semilogx(x, qback, '+')
plt.semilogx(x, Qbig, ':')
plt.text(x[-1],Qbig[-1],"$\kappa$=%.3f" % kappa,va="bottom",ha='right')

plt.ylim(0,0.2)
plt.title("Backscattering Efficiency for m=1.5 - i $\kappa$")
plt.xlabel("Size Parameter")
plt.ylabel("$Q_{back}$")
plt.grid()