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Anomalous diffraction theory

Scott Prahl

9 Feb 2019, Version 2





In [1]:



    


%matplotlib inline

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

First thing is to define some reasonably accurate approximations for the efficiencies for large spheres. This way we can ensure that the limiting cases behaive as they should.

These formulas used below are from Moosmüller and Sorensen Single scattering albedo of homogeneous, spherical particles in the transition regime





In [2]:



    


def Qabs_adt(m,x):
    """
    Anomalous diffraction theory approximation for absorption efficiency
    """
    n = m.real
    kappa = abs(m.imag)

    if kappa == 0:
        return np.zeros_like(x)
    return 1+2*np.exp(-4*kappa*x)/(4*kappa*x)+2*(np.exp(-4*kappa*x)-1)/(4*kappa*x)**2

def Qext_adt(m,x):
    """
    Anomalous diffraction theory approximation for extinction efficiency
    """
    n = m.real
    kappa = abs(m.imag)
    rho = 2*x*np.abs(m-1)
    beta = np.arctan2(kappa,n-1)
    ex = np.exp(-rho * np.tan(beta))

    qext_adt = 2
    qext_adt += -4*ex*np.cos(beta)/rho*np.sin(rho-beta)
    qext_adt += -4*ex*np.cos(beta)**2/rho**2*np.cos(rho-2*beta)
    qext_adt += 4*np.cos(beta)**2/rho**2*np.cos(2*beta)
    return qext_adt


def Qabs_madt(m,x):
    """
    Modified anomalous diffraction theory approximation for absorption efficiency
    """
    n = m.real
    kappa = abs(m.imag)

    if kappa == 0:
        return np.zeros_like(x)
    
    qabs_adt = Qabs_adt(m,x)
    epsilon = 0.25 + 0.61*(1-np.exp(-8*np.pi/3*kappa))**2
    c1 = 0.25*(1+np.exp(-1167*kappa))*(1-qabs_adt)
    c2 = np.sqrt(2*epsilon*x/np.pi)*np.exp(0.5-epsilon*x/np.pi)*(0.7393*n-0.6069)
    return (1+c1+c2)*qabs_adt


def Qext_madt(m,x):
    """
    Modified anomalous diffraction theory approximation for extinction efficiency
    """
    n = m.real
    kappa = -np.imag(m)
    
    qext_adt = Qext_adt(m,x)
    epsilon = 0.25 + 0.61*(1-np.exp(-8*np.pi/3*kappa))**2
    c2 = np.sqrt(2*epsilon*x/np.pi)*np.exp(0.5-epsilon*x/np.pi)*(0.7393*n-0.6069)
    Qedge = (1-np.exp(-0.06*x))*x**(-2/3)

    return (1+0.5*c2)*qext_adt+Qedge

No Absorption Case m=1.5





In [3]:



    


m = 1.5
x = np.logspace(-1, 5, 50)  # also in microns
qext, qsca, qback, g = mp.mie(m,x)
qabs = qext-qsca



    











In [4]:



    


plt.semilogx(x, qabs, 'b-+', label="miepython")
plt.semilogx(x, Qabs_adt(m,x), 'r', label="ADT")
plt.semilogx(x, Qabs_madt(m,x), 'g+:', label="MADT")

plt.ylabel("$Q_{abs}$")

plt.xlabel("Size Parameter")
plt.title("Absorption Efficiency for m=%.3f-%.3fi" % (m.real,abs(m.imag)))
plt.legend()
plt.grid()
plt.show()



    


















    
























In [5]:



    


plt.semilogx(x, qext, 'b-+', label="miepython")
plt.semilogx(x, Qext_adt(m,x), 'r', label="ADT")
plt.semilogx(x, Qext_madt(m,x), 'g+:', label="MADT")

plt.ylabel("$Q_{ext}$")

plt.xlabel("Size Parameter")
plt.title("Extinction Efficiency for m=%.3f-%.3fi" % (m.real,abs(m.imag)))
plt.legend()
plt.grid()
plt.show()



    


















    
























In [6]:



    


Qsca_adt = Qext_adt(m,x)-Qabs_adt(m,x)
Qsca_madt = Qext_madt(m,x)-Qabs_madt(m,x)

plt.semilogx(x, qsca, 'b', label="miepython")
plt.semilogx(x, Qsca_adt, 'r:', label="ADT")
plt.semilogx(x, Qsca_madt, 'g--', label="MADT")

plt.xlabel("Size Parameter")
plt.ylabel("$Q_{sca}$")
plt.title("Scattering Efficiency for m=%.3f-%.3fi" % (m.real,abs(m.imag)))
plt.legend()
plt.grid()
plt.show()



    


















    
























In [7]:



    


Qpr_adt = Qext_adt(m,x)-g*Qsca_adt
Qpr_madt = Qext_madt(m,x)-g*Qsca_madt

plt.semilogx(x, qext - g * qsca, 'b', label="miepython")
plt.semilogx(x, Qpr_adt, 'r:', label="ADT")
plt.semilogx(x, Qpr_madt, 'g--', label="MADT")


plt.xlabel("Size Parameter")
plt.ylabel("$Q_{pr}$")
plt.title("Radiation Pressure for m=%.3f-%.3fi" % (m.real,abs(m.imag)))
plt.legend()
plt.grid()
plt.show()



    


















    
























In [8]:



    


plt.semilogx(x, g, 'b', label="miepython")

plt.xlabel("Size Parameter")
plt.ylabel("g")
plt.title("Anisotropy for m=%.3f-%.3fi" % (m.real,abs(m.imag)))
plt.legend()
plt.grid()
plt.show()



    


















    
























In [9]:



    


## No absorption means that the argument that the backscatter
## efficiency goes as the surface reflection fails.  See 09_backscatter.ipynb
## for tests that show that miepython correctly calculates qback

plt.semilogx(x, qback, '+', label="miepython")
plt.semilogx(x, x**0.5, ':', label="limit")

plt.xlabel("Size Parameter")
plt.ylabel("g")
plt.title("Backscattering Efficiency for m=%.3f-%.3fi" % (m.real,abs(m.imag)))
plt.legend()
plt.grid()
plt.show()

A little absorption m=1.5-0.001j





In [10]:



    


m = 1.5-0.001j
x = np.logspace(-1, 5, 50)  # also in microns
qext, qsca, qback, g = mp.mie(m,x)
qabs = qext-qsca



    











In [11]:



    


plt.semilogx(x, qabs, 'b-+', label="miepython")
plt.semilogx(x, Qabs_adt(m,x), 'r', label="ADT")
plt.semilogx(x, Qabs_madt(m,x), 'g+:', label="MADT")

plt.ylabel("$Q_{abs}$")

plt.xlabel("Size Parameter")
plt.title("Absorption Efficiency for m=%.3f-%.3fi" % (m.real,abs(m.imag)))
plt.legend()
plt.grid()
plt.show()



    


















    
























In [12]:



    


plt.semilogx(x, qext, 'b-+', label="miepython")
plt.semilogx(x, Qext_adt(m,x), 'r', label="ADT")
plt.semilogx(x, Qext_madt(m,x), 'g+:', label="MADT")

plt.ylabel("$Q_{ext}$")

plt.xlabel("Size Parameter")
plt.title("Extinction Efficiency for m=%.3f-%.3fi" % (m.real,abs(m.imag)))
plt.legend()
plt.grid()
plt.show()



    


















    
























In [13]:



    


Qsca_adt = Qext_adt(m,x)-Qabs_adt(m,x)
Qsca_madt = Qext_madt(m,x)-Qabs_madt(m,x)

plt.semilogx(x, qsca, 'b', label="miepython")
plt.semilogx(x, Qsca_adt, 'r:', label="ADT")
plt.semilogx(x, Qsca_madt, 'g--', label="MADT")

plt.xlabel("Size Parameter")
plt.ylabel("$Q_{sca}$")
plt.title("Scattering Efficiency for m=%.3f-%.3fi" % (m.real,abs(m.imag)))
plt.legend()
plt.grid()
plt.show()



    


















    
























In [14]:



    


Qpr_adt = Qext_adt(m,x)-g*Qsca_adt
Qpr_madt = Qext_madt(m,x)-g*Qsca_madt

plt.semilogx(x, qext - g * qsca, 'b', label="miepython")
plt.semilogx(x, Qpr_adt, 'r:', label="ADT")
plt.semilogx(x, Qpr_madt, 'g--', label="MADT")


plt.xlabel("Size Parameter")
plt.ylabel("$Q_{pr}$")
plt.title("Radiation Pressure for m=%.3f-%.3fi" % (m.real,abs(m.imag)))
plt.legend()
plt.grid()
plt.show()



    


















    
























In [15]:



    


plt.semilogx(x, g, 'b', label="miepython")

plt.xlabel("Size Parameter")
plt.ylabel("g")
plt.title("Anisotropy for m=%.3f-%.3fi" % (m.real,abs(m.imag)))
plt.legend()
plt.grid()
plt.show()



    


















    
























In [16]:



    


Qbacks = abs(m-1)**2/abs(m+1)**2
Qback = Qbacks * np.ones_like(x)

plt.semilogx(x, qback, '+', label="miepython")
plt.semilogx(x, Qback, ':', label="limit")

plt.xlabel("Size Parameter")
plt.ylabel("g")
plt.title("Backscattering Efficiency for m=%.3f-%.3fi" % (m.real,abs(m.imag)))
plt.legend()
plt.grid()
plt.show()

Some Absorption m=1.5-0.1j





In [17]:



    


m = 1.5-0.1j
x = np.logspace(-1, 5, 50)  # also in microns
qext, qsca, qback, g = mp.mie(m,x)
qabs = qext-qsca



    











In [18]:



    


plt.semilogx(x, qabs, 'b-+', label="miepython")
plt.semilogx(x, Qabs_adt(m,x), 'r', label="ADT")
plt.semilogx(x, Qabs_madt(m,x), 'g+:', label="MADT")

plt.ylabel("$Q_{abs}$")

plt.xlabel("Size Parameter")
plt.title("Absorption Efficiency for m=%.3f-%.3fi" % (m.real,abs(m.imag)))
plt.legend()
plt.grid()
plt.show()



    


















    
























In [19]:



    


plt.semilogx(x, qext, 'b-+', label="miepython")
plt.semilogx(x, Qext_adt(m,x), 'r', label="ADT")
plt.semilogx(x, Qext_madt(m,x), 'g+:', label="MADT")

plt.ylabel("$Q_{ext}$")

plt.xlabel("Size Parameter")
plt.title("Extinction Efficiency for m=%.3f-%.3fi" % (m.real,abs(m.imag)))
plt.legend()
plt.grid()
plt.show()



    


















    
























In [20]:



    


Qsca_adt = Qext_adt(m,x)-Qabs_adt(m,x)
Qsca_madt = Qext_madt(m,x)-Qabs_madt(m,x)

plt.semilogx(x, qsca, 'b', label="miepython")
plt.semilogx(x, Qsca_adt, 'r:', label="ADT")
plt.semilogx(x, Qsca_madt, 'g--', label="MADT")

plt.xlabel("Size Parameter")
plt.ylabel("$Q_{sca}$")
plt.title("Scattering Efficiency for m=%.3f-%.3fi" % (m.real,abs(m.imag)))
plt.legend()
plt.grid()
plt.show()



    


















    
























In [21]:



    


Qpr_adt = Qext_adt(m,x)-g*Qsca_adt
Qpr_madt = Qext_madt(m,x)-g*Qsca_madt

plt.semilogx(x, qext - g * qsca, 'b', label="miepython")
plt.semilogx(x, Qpr_adt, 'r:', label="ADT")
plt.semilogx(x, Qpr_madt, 'g--', label="MADT")


plt.xlabel("Size Parameter")
plt.ylabel("$Q_{pr}$")
plt.title("Radiation Pressure for m=%.3f-%.3fi" % (m.real,abs(m.imag)))
plt.legend()
plt.grid()
plt.show()



    


















    
























In [22]:



    


plt.semilogx(x, g, 'b', label="miepython")

plt.xlabel("Size Parameter")
plt.ylabel("g")
plt.title("Anisotropy for m=%.3f-%.3fi" % (m.real,abs(m.imag)))
plt.legend()
plt.grid()
plt.show()



    


















    
























In [23]:



    


Qbacks = abs(m-1)**2/abs(m+1)**2
Qback = Qbacks * np.ones_like(x)

plt.semilogx(x, qback, '+', label="miepython")
plt.semilogx(x, Qback, ':', label="limit")

plt.xlabel("Size Parameter")
plt.ylabel("g")
plt.title("Backscattering Efficiency for m=%.3f-%.3fi" % (m.real,abs(m.imag)))
plt.legend()
plt.grid()
plt.show()

A lot of absorption m=1.5-1j





In [24]:



    


m = 1.5-1j
x = np.logspace(-1, 5, 50)  # also in microns
qext, qsca, qback, g = mp.mie(m,x)
qabs = qext-qsca



    











In [25]:



    


plt.semilogx(x, qabs, 'b-+', label="miepython")
plt.semilogx(x, Qabs_adt(m,x), 'r', label="ADT")
plt.semilogx(x, Qabs_madt(m,x), 'g+:', label="MADT")

plt.ylabel("$Q_{abs}$")

plt.xlabel("Size Parameter")
plt.title("Absorption Efficiency for m=%.3f-%.3fi" % (m.real,abs(m.imag)))
plt.legend()
plt.grid()
plt.show()



    


















    
























In [26]:



    


plt.semilogx(x, qext, 'b-+', label="miepython")
plt.semilogx(x, Qext_adt(m,x), 'r', label="ADT")
plt.semilogx(x, Qext_madt(m,x), 'g+:', label="MADT")

plt.ylabel("$Q_{ext}$")

plt.xlabel("Size Parameter")
plt.title("Extinction Efficiency for m=%.3f-%.3fi" % (m.real,abs(m.imag)))
plt.legend()
plt.grid()
plt.show()



    


















    
























In [27]:



    


Qsca_adt = Qext_adt(m,x)-Qabs_adt(m,x)
Qsca_madt = Qext_madt(m,x)-Qabs_madt(m,x)

plt.semilogx(x, qsca, 'b', label="miepython")
plt.semilogx(x, Qsca_adt, 'r:', label="ADT")
plt.semilogx(x, Qsca_madt, 'g--', label="MADT")

plt.xlabel("Size Parameter")
plt.ylabel("$Q_{sca}$")
plt.title("Scattering Efficiency for m=%.3f-%.3fi" % (m.real,abs(m.imag)))
plt.legend()
plt.grid()
plt.show()



    


















    
























In [28]:



    


Qpr_adt = Qext_adt(m,x)-g*Qsca_adt
Qpr_madt = Qext_madt(m,x)-g*Qsca_madt

plt.semilogx(x, qext - g * qsca, 'b', label="miepython")
plt.semilogx(x, Qpr_adt, 'r:', label="ADT")
plt.semilogx(x, Qpr_madt, 'g--', label="MADT")


plt.xlabel("Size Parameter")
plt.ylabel("$Q_{pr}$")
plt.title("Radiation Pressure for m=%.3f-%.3fi" % (m.real,abs(m.imag)))
plt.legend()
plt.grid()
plt.show()



    


















    
























In [29]:



    


plt.semilogx(x, g, 'b', label="miepython")

plt.xlabel("Size Parameter")
plt.ylabel("g")
plt.title("Anisotropy for m=%.3f-%.3fi" % (m.real,abs(m.imag)))
plt.legend()
plt.grid()
plt.show()



    


















    
























In [30]:



    


Qbacks = abs(m-1)**2/abs(m+1)**2
Qback = Qbacks * np.ones_like(x)

plt.semilogx(x, qback, '+', label="miepython")
plt.semilogx(x, Qback, ':', label="limit")

plt.xlabel("Size Parameter")
plt.ylabel("g")
plt.title("Backscattering Efficiency for m=%.3f-%.3fi" % (m.real,abs(m.imag)))
plt.legend()
plt.grid()
plt.show()

Content source: scottprahl/miepython

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