Scattering on finite rectangular lattice
Former-commit-id: cb5a9d855e8f1cd22390cf589669fc3621bf8936
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Marek Nečada
parent
232fd3146e
commit
9c0e2b2f28
215
qpms/qpms_p.py
215
qpms/qpms_p.py
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@ -887,3 +887,218 @@ def loadScuffTMatrices(fileName):
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# IMPORTANT: now we are going from Reid's/Kristensson's/Jackson's/whoseever convention to Taylor's convention
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# IMPORTANT: now we are going from Reid's/Kristensson's/Jackson's/whoseever convention to Taylor's convention
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TMatrices[:,:,:,:,:] = TMatrices[:,:,:,:,:] * np.sqrt(ny*(ny+1))[ň,ň,ň,ň,:] / np.sqrt(ny*(ny+1))[ň,ň,:,ň,ň]
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TMatrices[:,:,:,:,:] = TMatrices[:,:,:,:,:] * np.sqrt(ny*(ny+1))[ň,ň,ň,ň,:] / np.sqrt(ny*(ny+1))[ň,ň,:,ň,ň]
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return (TMatrices, freqs, freqs_weirdunits, lMax)
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return (TMatrices, freqs, freqs_weirdunits, lMax)
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# misc tensor maniputalion
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def apply_matrix_left(matrix, tensor, axis):
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"""
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TODO doc
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Apply square matrix to a given axis of a tensor, so that the result retains the shape
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of the original tensor. The summation goes over the second index of the matrix and the
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given tensor axis.
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"""
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tmp = np.tensordot(matrix, tensor, axes=(-1,axis))
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return np.moveaxis(tmp, 0, axis)
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####################
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# Array simulations
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####################
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def nelem2lMax(nelem):
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"""
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Auxiliary inverse function to nelem(lMax) = (lMax + 2) * lMax. Returns 0 if
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it nelem does not come from positive integer lMax.
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"""
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lMax = round(math.sqrt(1+nelem) - 1)
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if ((lMax < 1) or ((lMax + 2) * lMax != nelem)):
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return 0
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else:
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return lMax
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def scatter_plane_wave(omega, epsilon_b, positions, Tmatrices, k_dirs, E_0s, #saveto = None
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):
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"""
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Solves the plane wave linear scattering problem for a structure of "non-touching" particles
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for one frequency and arbitrary number K of incoming plane waves.
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Parameters
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----------
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omega : positive number
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The frequency of the field.
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epsilon_b : complex number
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Permittivity of the background medium (which has to be isotropic).
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positions : (N,3)-shaped real array
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Cartesian positions of the particles.
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TMatrices : (N,2,nelem,2,nelem) or compatible
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The T-matrices in the "Taylor convention" describing the scattering on a single nanoparticle.
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If all the particles are identical and equally oriented, only one T-matrix can be given.
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nelems = (lMax + 2) * lMax, where lMax is the highest multipole order to which the scattering
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is calculated.
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k_dirs : (K,3)-shaped real array or compatible
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The direction of the incident field wave vector, normalized to one.
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E_0s : (K,3)-shaped complex array or compatible
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The electric intensity amplitude of the incident field.
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Returns
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-------
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ab : (K, N, 2, nelem)-shaped complex array
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The a (electric wave), b (magnetic wave) coefficients of the outgoing field for each particle
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# Fuck this, it will be wiser to make separate function to calculate those from ab:
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# sigma_xxx : TODO (K, 2, nelem)
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# TODO partial (TODO which?) cross-section for each type of outgoing waves, summed over all
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# nanoparticles (total cross section is given by the sum of this.)
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"""
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nelem = TMatrices.shape[-1]
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if ((nelem != TMatrices.shape[-3]) or (2 != TMatrices.shape[-2]) or (2 != TMatrices.shape[-4])):
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raise ValueError('The T-matrices must be of shape (N, 2, nelem, 2, nelem) but are of shape %s' % (str(TMatrices.shape),))
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lMax = nelem2lMax(nelem)
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if not lMax:
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raise ValueError('The "nelem" dimension of T-matrix has invalid value (%d).' % nelem)
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# TODO perhaps more checks.
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raise Error('Not implemented.')
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pass
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def scatter_plane_wave_rectarray(omega, epsilon_b, xN, yN, xd, yd, TMatrices, k_dirs, E_0s, return_xy = False):
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"""
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Solves the plane wave linear scattering problem for a rectangular array of particles
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for one frequency and arbitrary number K of incoming plane waves.
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Parameters
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----------
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omega : positive number
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The frequency of the field.
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epsilon_b : complex number
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Permittivity of the background medium (which has to be isotropic).
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xN, yN : positive integers
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Particle numbers in the x and y dimensions
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xd, yd : positive numbers
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Periodicities in the x and y direction
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TMatrices : (xN, yN,2,nelem,2,nelem) or compatible or (2,nelem,2,nelem)
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The T-matrices in the "Taylor convention" describing the scattering on a single nanoparticle.
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If all the particles are identical and equally oriented, only one T-matrix can be given.
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nelems = (lMax + 2) * lMax, where lMax is the highest multipole order to which the scattering
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is calculated.
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Electric wave index is 0, magnetic wave index is 1.
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k_dirs : (K,3)-shaped real array or compatible
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The direction of the incident field wave vector, normalized to one.
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E_0s : (K,3)-shaped complex array or compatible
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The electric intensity amplitude of the incident field.
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Returns
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-------
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ab : (K, xN, yN, 2, nelem)-shaped complex array
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The a (electric wave), b (magnetic wave) coefficients of the outgoing field for each particle
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"""
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nelem = TMatrices.shape[-1]
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if ((nelem != TMatrices.shape[-3]) or (2 != TMatrices.shape[-2]) or (2 != TMatrices.shape[-4])):
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raise ValueError('The T-matrices must be of shape (N, 2, nelem, 2, nelem) but are of shape %s' % (str(TMatrices.shape),))
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lMax = nelem2lMax(nelem)
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if not lMax:
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raise ValueError('The "nelem" dimension of T-matrix has invalid value (%d).' % nelem)
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# TODO perhaps more checks.
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k_out = omega * math.sqrt(epsilon_b) / c # wave number
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my, ny = get_mn_y(lMax)
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N = yN * xN
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J_scat=3
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J_ext=1
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# Do something with this ugly indexing crap
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xind, yind = np.meshgrid(np.arange(xN),np.arange(yN), indexing='ij')
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xind = xind.flatten()
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yind = yind.flatten()
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xyind = np.stack((xind, yind, np.zeros((xind.shape),dtype=int)),axis=-1)
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cart_lattice=xyind * np.array([xd, yd, 0])
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x=cart_lattice[:,0]
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y=cart_lattice[:,1]
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xyind = xyind[:,0:2]
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# Lattice speedup
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Agrid = np.zeros((nelem, 2*xN, 2*yN, nelem),dtype=np.complex_)
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Bgrid = np.zeros((nelem, 2*xN, 2*yN, nelem),dtype=np.complex_)
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for yl in range(nelem): # source
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for xij in range(2*xN):
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for yij in range(2*yN):
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for yj in range(nelem): #dest
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if((yij != yN) or (xij != xN)):
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d_l2j = cart2sph(np.array([(xij-xN)*xd, (yij-yN)*yd, 0]))
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Agrid[yj, xij, yij, yl] = Ã(my[yj],ny[yj],my[yl],ny[yl],kdlj=d_l2j[0]*k_out,θlj=d_l2j[1],φlj=d_l2j[2],r_ge_d=False,J=J_scat)
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Bgrid[yj, xij, yij, yl] = B̃(my[yj],ny[yj],my[yl],ny[yl],kdlj=d_l2j[0]*k_out,θlj=d_l2j[1],φlj=d_l2j[2],r_ge_d=False,J=J_scat)
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# Translation coefficient matrix T
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transmat = np.zeros((xN* yN, 2, nelem, xN* yN, 2, nelem),dtype=np.complex_)
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for l in range(N):
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xil, yil = xyind[l]
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for j in range(N):
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xij, yij = xyind[j]
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if (l!=j):
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transmat[j,0,:,l,0,:] = Agrid[:, xij - xil + xN, yij - yil + yN, :]
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transmat[j,0,:,l,1,:] = Bgrid[:, xij - xil + xN, yij - yil + yN, :]
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transmat[j,1,:,l,0,:] = Bgrid[:, xij - xil + xN, yij - yil + yN, :]
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transmat[j,1,:,l,1,:] = Agrid[:, xij - xil + xN, yij - yil + yN, :]
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Agrid = None
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Bgrid = None
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# Now we solve a linear problem (1 - M T) A = M P_0 where M is the T-matrix :-)
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MT = np.empty((N,2,nelem,N,2,nelem),dtype=np.complex_)
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TMatrices = np.broadcast_to(TMatrices, (xN, yN, 2, nelem, 2, nelem))
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for j in range(N): # I wonder how this can be done without this loop...
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xij, yij = xyind[j]
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MT[j] = np.tensordot(TMatrices[xij, yij],transmat[j],axes=([-2,-1],[0,1]))
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MT.shape = (N*2*nelem, N*2*nelem)
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leftmatrix = np.identity(N*2*nelem) - MT
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MT = None
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if ((1 == k_dirs.ndim) and (1 == E_0s.ndim)):
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k_cart = k_dirs * k_out # wave vector of the incident plane wave
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pq_0 = np.zeros((N,2,nelem), dtype=np.complex_)
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p_y0, q_y0 = plane_pq_y(nmax, k_cart, E_0s)
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pq_0[:,0,:] = np.exp(1j*np.sum(k_cart[ň,:]*cart_lattice,axis=-1))[:, ň] * p_y0[ň, :]
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pq_0[:,1,:] = np.exp(1j*np.sum(k_cart[ň,:]*cart_lattice,axis=-1))[:, ň] * q_y0[ň, :]
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MP_0 = np.empty((N,2,nelem),dtype=np.complex_)
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for j in range(N): # I wonder how this can be done without this loop...
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MP_0[j] = np.tensordot(TMatrices[xij, yij],pq_0[j],axes=([-2,-1],[-2,-1]))
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MP_0.shape = (N*2*nelem,)
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ab = np.linalg.solve(leftmatrix, MP_0)
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ab.shape = (xN, yN, 2, nelem)
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else:
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# handle "broadcasting" for k, E
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if 1 == k_dirs.ndim:
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k_dirs = k_dirs[ň,:]
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if 1 == E_0s.ndim:
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E_0s = E_0s[ň,:]
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K = max(E_0s.shape[-2], k_dirs.shape[-2])
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k_dirs = np.broadcast_to(k_dirs,(K,3))
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E_0s = np.broadcast_to(E_0s, (K,3))
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# А ну, чики-брики и в дамки!
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lupiv = scipy.linalg.lu_factor(leftmatrix, overwrite_a=True)
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leftmatrix = None
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ab = np.empty((K,N*2*nelem), dtype=complex)
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for ki in range(K):
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k_cart = k_dirs[ki] * k_out
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pq_0 = np.zeros((N,2,nelem), dtype=np.complex_)
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p_y0, q_y0 = plane_pq_y(nmax, k_cart, E_0s[ki])
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pq_0[:,0,:] = np.exp(1j*np.sum(k_cart[ň,:]*cart_lattice,axis=-1))[:, ň] * p_y0[ň, :]
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pq_0[:,1,:] = np.exp(1j*np.sum(k_cart[ň,:]*cart_lattice,axis=-1))[:, ň] * q_y0[ň, :]
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MP_0 = np.empty((N,2,nelem),dtype=np.complex_)
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for j in range(N): # I wonder how this can be done without this loop...
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MP_0[j] = np.tensordot(TMatrices[xij, yij],pq_0[j],axes=([-2,-1],[-2,-1]))
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MP_0.shape = (N*2*nelem,)
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ab[ki] = scipy.linalg.lu_solve(lupiv, MP_0)
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ab.shape = (K, xN, yN, 2, nelem)
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if (return_xy):
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return (ab, x, y)
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else:
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return ab
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