''' Bloch eigenmode solver/operators This module contains functions for generating and solving the 3D Bloch eigenproblem. The approach is to transform the problem into the (spatial) fourier domain, transforming the equation 1/mu * curl(1/eps * curl(H)) = (w/c)^2 H into conv(1/mu_k, ik x conv(1/eps_k, ik x H_k)) = (w/c)^2 H_k where: - the _k subscript denotes a 3D fourier transformed field - each component of H_k corresponds to a plane wave with wavevector k - x is the cross product - conv denotes convolution Since k and H are orthogonal for each plane wave, we can use each k to create an orthogonal basis (k, m, n), with k x m = n, and |m| = |n| = 1. The cross products are then simplified with k @ h = kx hx + ky hy + kz hz = 0 = hk h = hk + hm + hn = hm + hn k = kk + km + kn = kk = |k| k x h = (ky hz - kz hy, kz hx - kx hz, kx hy - ky hx) = ((k x h) @ k, (k x h) @ m, (k x h) @ n)_kmn = (0, (m x k) @ h, (n x k) @ h)_kmn # triple product ordering = (0, kk (-n @ h), kk (m @ h))_kmn # (m x k) = -|k| n, etc. = |k| (0, -h @ n, h @ m)_kmn k x h = (km hn - kn hm, kn hk - kk hn, kk hm - km hk)_kmn = (0, -kk hn, kk hm)_kmn = (-kk hn)(mx, my, mz) + (kk hm)(nx, ny, nz) = |k| (hm * (nx, ny, nz) - hn * (mx, my, mz)) where h is shorthand for H_k, (...)_kmn deontes the (k, m, n) basis, and e.g. hm is the component of h in the m direction. We can also simplify conv(X_k, Y_k) as fftn(X * ifftn(Y_k)). Using these results and storing H_k as h = (hm, hn), we have e_xyz = fftn(1/eps * ifftn(|k| (hm * n - hn * m))) b_mn = |k| (-e_xyz @ n, e_xyz @ m) h_mn = fftn(1/mu * ifftn(b_m * m + b_n * n)) which forms the operator from the left side of the equation. We can then use a preconditioned block Rayleigh iteration algorithm, as in SG Johnson and JD Joannopoulos, Block-iterative frequency-domain methods for Maxwell's equations in a planewave basis, Optics Express 8, 3, 173-190 (2001) (similar to that used in MPB) to find the eigenvectors for this operator. === Typically you will want to do something like recip_lattice = numpy.diag(1/numpy.array(epsilon[0].shape * dx)) n, v = bloch.eigsolve(5, k0, recip_lattice, epsilon) f = numpy.sqrt(-numpy.real(n[0])) n_eff = norm(recip_lattice @ k0) / f v2e = bloch.hmn_2_exyz(k0, recip_lattice, epsilon) e_field = v2e(v[0]) k, f = find_k(frequency=1/1550, tolerance=(1/1550 - 1/1551), direction=[1, 0, 0], G_matrix=recip_lattice, epsilon=epsilon, band=0) ''' from typing import List, Tuple, Callable, Dict import logging import numpy from numpy import pi, real, trace from numpy.fft import fftn, ifftn, fftfreq import scipy import scipy.optimize from scipy.linalg import norm import scipy.sparse.linalg as spalg from .eigensolvers import rayleigh_quotient_iteration from . import field_t logger = logging.getLogger(__name__) def generate_kmn(k0: numpy.ndarray, G_matrix: numpy.ndarray, shape: numpy.ndarray ) -> Tuple[numpy.ndarray, numpy.ndarray, numpy.ndarray]: """ Generate a (k, m, n) orthogonal basis for each k-vector in the simulation grid. :param k0: [k0x, k0y, k0z], Bloch wavevector, in G basis. :param G_matrix: 3x3 matrix, with reciprocal lattice vectors as columns. :param shape: [nx, ny, nz] shape of the simulation grid. :return: (|k|, m, n) where |k| has shape tuple(shape) + (1,) and m, n have shape tuple(shape) + (3,). All are given in the xyz basis (e.g. |k|[0,0,0] = norm(G_matrix @ k0)). """ k0 = numpy.array(k0) Gi_grids = numpy.meshgrid(*(fftfreq(n, 1/n) for n in shape[:3]), indexing='ij') Gi = numpy.stack(Gi_grids, axis=3) k_G = k0[None, None, None, :] - Gi k_xyz = numpy.rollaxis(G_matrix @ numpy.rollaxis(k_G, 3, 2), 3, 2) m = numpy.broadcast_to([0, 1, 0], tuple(shape[:3]) + (3,)).astype(float) n = numpy.broadcast_to([0, 0, 1], tuple(shape[:3]) + (3,)).astype(float) xy_non0 = numpy.any(k_xyz[:, :, :, 0:1] != 0, axis=3) if numpy.any(xy_non0): u = numpy.cross(k_xyz[xy_non0], [0, 0, 1]) m[xy_non0, :] = u / norm(u, axis=1)[:, None] z_non0 = numpy.any(k_xyz != 0, axis=3) if numpy.any(z_non0): v = numpy.cross(k_xyz[z_non0], m[z_non0]) n[z_non0, :] = v / norm(v, axis=1)[:, None] k_mag = norm(k_xyz, axis=3)[:, :, :, None] return k_mag, m, n def maxwell_operator(k0: numpy.ndarray, G_matrix: numpy.ndarray, epsilon: field_t, mu: field_t = None ) -> Callable[[numpy.ndarray], numpy.ndarray]: """ Generate the Maxwell operator conv(1/mu_k, ik x conv(1/eps_k, ik x ___)) which is the spatial-frequency-space representation of 1/mu * curl(1/eps * curl(___)) The operator is a function that acts on a vector h_mn of size (2 * epsilon[0].size) See the module-level docstring for more information. :param k0: Bloch wavevector, [k0x, k0y, k0z]. :param G_matrix: 3x3 matrix, with reciprocal lattice vectors as columns. :param epsilon: Dielectric constant distribution for the simulation. All fields are sampled at cell centers (i.e., NOT Yee-gridded) :param mu: Magnetic permability distribution for the simulation. Default None (1 everywhere). :return: Function which applies the maxwell operator to h_mn. """ shape = epsilon[0].shape + (1,) k_mag, m, n = generate_kmn(k0, G_matrix, shape) epsilon = numpy.stack(epsilon, 3) if mu is not None: mu = numpy.stack(mu, 3) def operator(h: numpy.ndarray): """ Maxwell operator for Bloch eigenmode simulation. h is complex 2-field in (m, n) basis, vectorized :param h: Raveled h_mn; size (2 * epsilon[0].size). :return: Raveled conv(1/mu_k, ik x conv(1/eps_k, ik x h_mn)). """ hin_m, hin_n = [hi.reshape(shape) for hi in numpy.split(h, 2)] #{d,e,h}_xyz fields are complex 3-fields in (1/x, 1/y, 1/z) basis # cross product and transform into xyz basis d_xyz = (n * hin_m - m * hin_n) * k_mag # divide by epsilon e_xyz = fftn(ifftn(d_xyz, axes=range(3)) / epsilon, axes=range(3)) # cross product and transform into mn basis b_m = numpy.sum(e_xyz * n, axis=3)[:, :, :, None] * -k_mag b_n = numpy.sum(e_xyz * m, axis=3)[:, :, :, None] * +k_mag if mu is None: h_m, h_n = b_m, b_n else: # transform from mn to xyz b_xyz = (m * b_m[:, :, :, None] + n * b_n[:, :, :, None]) # divide by mu h_xyz = fftn(ifftn(b_xyz, axes=range(3)) / mu, axes=range(3)) # transform back to mn h_m = numpy.sum(h_xyz * m, axis=3) h_n = numpy.sum(h_xyz * n, axis=3) return numpy.hstack((h_m.ravel(), h_n.ravel())) return operator def hmn_2_exyz(k0: numpy.ndarray, G_matrix: numpy.ndarray, epsilon: field_t, ) -> Callable[[numpy.ndarray], field_t]: """ Generate an operator which converts a vectorized spatial-frequency-space h_mn into an E-field distribution, i.e. ifft(conv(1/eps_k, ik x h_mn)) The operator is a function that acts on a vector h_mn of size (2 * epsilon[0].size) See the module-level docstring for more information. :param k0: Bloch wavevector, [k0x, k0y, k0z]. :param G_matrix: 3x3 matrix, with reciprocal lattice vectors as columns. :param epsilon: Dielectric constant distribution for the simulation. All fields are sampled at cell centers (i.e., NOT Yee-gridded) :return: Function for converting h_mn into E_xyz """ shape = epsilon[0].shape + (1,) epsilon = numpy.stack(epsilon, 3) k_mag, m, n = generate_kmn(k0, G_matrix, shape) def operator(h: numpy.ndarray) -> field_t: hin_m, hin_n = [hi.reshape(shape) for hi in numpy.split(h, 2)] d_xyz = (n * hin_m - m * hin_n) * k_mag # divide by epsilon return [ei for ei in numpy.rollaxis(ifftn(d_xyz, axes=range(3)) / epsilon, 3)] return operator def hmn_2_hxyz(k0: numpy.ndarray, G_matrix: numpy.ndarray, epsilon: field_t ) -> Callable[[numpy.ndarray], field_t]: """ Generate an operator which converts a vectorized spatial-frequency-space h_mn into an H-field distribution, i.e. ifft(h_mn) The operator is a function that acts on a vector h_mn of size (2 * epsilon[0].size) See the module-level docstring for more information. :param k0: Bloch wavevector, [k0x, k0y, k0z]. :param G_matrix: 3x3 matrix, with reciprocal lattice vectors as columns. :param epsilon: Dielectric constant distribution for the simulation. Only epsilon[0].shape is used. :return: Function for converting h_mn into H_xyz """ shape = epsilon[0].shape + (1,) k_mag, m, n = generate_kmn(k0, G_matrix, shape) def operator(h: numpy.ndarray): hin_m, hin_n = [hi.reshape(shape) for hi in numpy.split(h, 2)] h_xyz = (m * hin_m + n * hin_n) return [ifftn(hi) for hi in numpy.rollaxis(h_xyz, 3)] return operator def inverse_maxwell_operator_approx(k0: numpy.ndarray, G_matrix: numpy.ndarray, epsilon: field_t, mu: field_t = None ) -> Callable[[numpy.ndarray], numpy.ndarray]: """ Generate an approximate inverse of the Maxwell operator, ik x conv(eps_k, ik x conv(mu_k, ___)) which can be used to improve the speed of ARPACK in shift-invert mode. See the module-level docstring for more information. :param k0: Bloch wavevector, [k0x, k0y, k0z]. :param G_matrix: 3x3 matrix, with reciprocal lattice vectors as columns. :param epsilon: Dielectric constant distribution for the simulation. All fields are sampled at cell centers (i.e., NOT Yee-gridded) :param mu: Magnetic permability distribution for the simulation. Default None (1 everywhere). :return: Function which applies the approximate inverse of the maxwell operator to h_mn. """ shape = epsilon[0].shape + (1,) epsilon = numpy.stack(epsilon, 3) k_mag, m, n = generate_kmn(k0, G_matrix, shape) if mu is not None: mu = numpy.stack(mu, 3) def operator(h: numpy.ndarray): """ Approximate inverse Maxwell operator for Bloch eigenmode simulation. h is complex 2-field in (m, n) basis, vectorized :param h: Raveled h_mn; size (2 * epsilon[0].size). :return: Raveled ik x conv(eps_k, ik x conv(mu_k, h_mn)) """ hin_m, hin_n = [hi.reshape(shape) for hi in numpy.split(h, 2)] #{d,e,h}_xyz fields are complex 3-fields in (1/x, 1/y, 1/z) basis if mu is None: b_m, b_n = hin_m, hin_n else: # transform from mn to xyz h_xyz = (m * hin_m[:, :, :, None] + n * hin_n[:, :, :, None]) # multiply by mu b_xyz = fftn(ifftn(h_xyz, axes=range(3)) * mu, axes=range(3)) # transform back to mn b_m = numpy.sum(b_xyz * m, axis=3) b_n = numpy.sum(b_xyz * n, axis=3) # cross product and transform into xyz basis e_xyz = (n * b_m - m * b_n) / k_mag # multiply by epsilon d_xyz = fftn(ifftn(e_xyz, axes=range(3)) * epsilon, axes=range(3)) # cross product and transform into mn basis crossinv_t2c h_m = numpy.sum(e_xyz * n, axis=3)[:, :, :, None] / +k_mag h_n = numpy.sum(e_xyz * m, axis=3)[:, :, :, None] / -k_mag return numpy.hstack((h_m.ravel(), h_n.ravel())) return operator def find_k(frequency: float, tolerance: float, direction: numpy.ndarray, G_matrix: numpy.ndarray, epsilon: field_t, mu: field_t = None, band: int = 0, k_min: float = 0, k_max: float = 0.5, ) -> Tuple[numpy.ndarray, float]: """ Search for a bloch vector that has a given frequency. :param frequency: Target frequency. :param tolerance: Target frequency tolerance. :param direction: k-vector direction to search along. :param G_matrix: 3x3 matrix, with reciprocal lattice vectors as columns. :param epsilon: Dielectric constant distribution for the simulation. All fields are sampled at cell centers (i.e., NOT Yee-gridded) :param mu: Magnetic permability distribution for the simulation. Default None (1 everywhere). :param band: Which band to search in. Default 0 (lowest frequency). return: (k, actual_frequency) The found k-vector and its frequency """ direction = numpy.array(direction) / norm(direction) def get_f(k0_mag: float, band: int = 0): k0 = direction * k0_mag n, _v = eigsolve(band + 1, k0, G_matrix=G_matrix, epsilon=epsilon) f = numpy.sqrt(numpy.abs(numpy.real(n[band]))) return f res = scipy.optimize.minimize_scalar(lambda x: abs(get_f(x, band) - frequency), (k_min + k_max) / 2, method='Bounded', bounds=(k_min, k_max), options={'xatol': abs(tolerance)}) return res.x * direction, res.fun + frequency def eigsolve(num_modes: int, k0: numpy.ndarray, G_matrix: numpy.ndarray, epsilon: field_t, mu: field_t = None, tolerance = 1e-20, ) -> Tuple[numpy.ndarray, numpy.ndarray]: """ Find the first (lowest-frequency) num_modes eigenmodes with Bloch wavevector k0 of the specified structure. :param k0: Bloch wavevector, [k0x, k0y, k0z]. :param G_matrix: 3x3 matrix, with reciprocal lattice vectors as columns. :param epsilon: Dielectric constant distribution for the simulation. All fields are sampled at cell centers (i.e., NOT Yee-gridded) :param mu: Magnetic permability distribution for the simulation. Default None (1 everywhere). :param tolerance: Solver stops when fractional change in the objective trace(Z.H @ A @ Z @ inv(Z Z.H)) is smaller than the tolerance :return: (eigenvalues, eigenvectors) where eigenvalues[i] corresponds to the vector eigenvectors[i, :] """ h_size = 2 * epsilon[0].size kmag = norm(G_matrix @ k0) ''' Generate the operators ''' mop = maxwell_operator(k0=k0, G_matrix=G_matrix, epsilon=epsilon, mu=mu) imop = inverse_maxwell_operator_approx(k0=k0, G_matrix=G_matrix, epsilon=epsilon, mu=mu) scipy_op = spalg.LinearOperator(dtype=complex, shape=(h_size, h_size), matvec=mop) scipy_iop = spalg.LinearOperator(dtype=complex, shape=(h_size, h_size), matvec=imop) y_shape = (h_size, num_modes) prev_E = 0 d_scale = 1 prev_traceGtKG = 0 prev_theta = 0.5 D = numpy.zeros(shape=y_shape, dtype=complex) y0 = None if y0 is None: Z = numpy.random.rand(*y_shape).astype(complex) else: Z = y0 while True: Z2 = Z.conj().T @ Z Z_norm = numpy.sqrt(real(trace(Z2))) / num_modes Z /= Z_norm Z2 /= Z_norm * Z_norm try: U = numpy.linalg.inv(Z2) except numpy.linalg.LinAlgError: Z = numpy.random.rand(*y_shape).astype(complex) continue trace_U = real(trace(U)) if trace_U > 1e8 * num_modes: Z = Z @ scipy.linalg.sqrtm(U).conj().T prev_traceGtKG = 0 continue break def rtrace_AtB(A, B): return real(numpy.sum(A.conj() * B)) def symmetrize(A): return (A + A.conj().T) * 0.5 max_iters = 10000 for iter in range(max_iters): U = numpy.linalg.inv(Z.conj().T @ Z) AZ = scipy_op @ Z AZU = AZ @ U ZtAZU = Z.conj().T @ AZU E = real(trace(ZtAZU)) sgn = numpy.sign(E) E = numpy.abs(E) G = (AZU - Z @ U @ ZtAZU) * sgn if iter > 0 and abs(E - prev_E) < tolerance * 0.5 * (E + prev_E + 1e-7): logging.info('Optimization succeded: {} - 5e-8 < {} * {} / 2'.format(abs(E - prev_E), tolerance, E + prev_E)) break KG = scipy_iop @ G traceGtKG = rtrace_AtB(G, KG) gamma_numerator = traceGtKG reset_iters = 100 if prev_traceGtKG == 0 or iter % reset_iters == 0: print('RESET!') gamma = 0 else: gamma = gamma_numerator / prev_traceGtKG D = gamma * d_scale * D + KG d_scale = numpy.sqrt(rtrace_AtB(D, D)) / num_modes D /= d_scale AD = scipy_op @ D DtD = D.conj().T @ D DtAD = D.conj().T @ AD ZtD = Z.conj().T @ D ZtAD = Z.conj().T @ AD symZtD = symmetrize(ZtD) symZtAD = symmetrize(ZtAD) U_sZtD = U @ symZtD dE = 2.0 * (rtrace_AtB(U, symZtAD) - rtrace_AtB(ZtAZU, U_sZtD)) S2 = DtD - 4 * symZtD @ U_sZtD d2E = 2 * (rtrace_AtB(U, DtAD) - rtrace_AtB(ZtAZU, U @ S2) - 4 * rtrace_AtB(U, symZtAD @ U_sZtD)) # Newton-Raphson to find a root of the first derivative: theta = -dE/d2E if d2E < 0 or abs(theta) >= pi: theta = -abs(prev_theta) * numpy.sign(dE) # ZtAZU * ZtZ = ZtAZ for use in line search ZtZ = Z.conj().T @ Z ZtAZ = ZtAZU @ ZtZ.conj().T def Qi_func(theta, memo=[None, None]): if memo[0] == theta: return memo[1] c = numpy.cos(theta) s = numpy.sin(theta) Q = c*c * ZtZ + s*s * DtD + 2*s*c * symZtD try: Qi = numpy.linalg.inv(Q) except numpy.linalg.LinAlgError: logger.info('taylor Qi') # if c or s small, taylor expand if c < 1e-4 * s and c != 0: Qi = numpy.linalg.inv(DtD) Qi = Qi / (s*s) - 2*c/(s*s*s) * (Qi @ symZtD.conj().T @ Qi.conj().T) elif s < 1e-4 * c and s != 0: Qi = numpy.linalg.inv(ZtZ) Qi = Qi / (c*c) - 2*s/(c*c*c) * (Qi @ symZtD.conj().T @ Qi.conj().T) else: raise Exception('Inexplicable singularity in trace_func') memo[0] = theta memo[1] = Qi return Qi def trace_func(theta): c = numpy.cos(theta) s = numpy.sin(theta) Qi = Qi_func(theta) R = c*c * ZtAZ + s*s * DtAD + 2*s*c * symZtAD trace = rtrace_AtB(R, Qi) return numpy.abs(trace) #def trace_deriv(theta): # Qi = Qi_func(theta) # c2 = numpy.cos(2 * theta) # s2 = numpy.sin(2 * theta) # F = -0.5*s2 * (ZtAZ - DtAD) + c2 * symZtAD # trace_deriv = rtrace_AtB(Qi, F) # G = Qi @ F.conj().T @ Qi.conj().T # H = -0.5*s2 * (ZtZ - DtD) + c2 * symZtD # trace_deriv -= rtrace_AtB(G, H) # trace_deriv *= 2 # return trace_deriv * sgn ''' theta, new_E, new_dE = linmin(theta, E, dE, 0.1, min(tolerance, 1e-6), 1e-14, 0, -numpy.sign(dE) * K_PI, trace_func) ''' #theta, n, _, new_E, _, _new_dE = scipy.optimize.line_search(trace_func, trace_deriv, xk=theta, pk=numpy.ones((1,1)), gfk=dE, old_fval=E, c1=min(tolerance, 1e-6), c2=0.1, amax=pi) result = scipy.optimize.minimize_scalar(trace_func, bounds=(0, pi), tol=tolerance) new_E = result.fun theta = result.x improvement = numpy.abs(E - new_E) * 2 / numpy.abs(E + new_E) logger.info('linmin improvement {}'.format(improvement)) Z *= numpy.cos(theta) Z += D * numpy.sin(theta) prev_traceGtKG = traceGtKG prev_theta = theta prev_E = E ''' Recover eigenvectors from Z ''' U = numpy.linalg.inv(Z.conj().T @ Z) Y = Z @ scipy.linalg.sqrtm(U) W = Y.conj().T @ (scipy_op @ Y) eigvals, W_eigvecs = numpy.linalg.eig(W) eigvecs = Y @ W_eigvecs for i in range(len(eigvals)): v = eigvecs[:, i] n = eigvals[i] v /= norm(v) eigness = norm(scipy_op @ v - (v.conj() @ (scipy_op @ v)) * v ) f = numpy.sqrt(-numpy.real(n)) df = numpy.sqrt(-numpy.real(n + eigness)) neff_err = kmag * (1/df - 1/f) logger.info('eigness {}: {}\n neff_err: {}'.format(i, eigness, neff_err)) order = numpy.argsort(numpy.abs(eigvals)) return eigvals[order], eigvecs.T[order] #def linmin(x_guess, f0, df0, x_max, f_tol=0.1, df_tol=min(tolerance, 1e-6), x_tol=1e-14, x_min=0, linmin_func): # if df0 > 0: # x0, f0, df0 = linmin(-x_guess, f0, -df0, -x_max, f_tol, df_tol, x_tol, -x_min, lambda q, dq: -linmin_func(q, dq)) # return -x0, f0, -df0 # elif df0 == 0: # return 0, f0, df0 # else: # x = x_guess # fx = f0 # dfx = df0 # isave = numpy.zeros((2,), numpy.intc) # dsave = numpy.zeros((13,), float) # x, fx, dfx, task = minpack2.dsrch(x, fx, dfx, f_tol, df_tol, x_tol, task, # x_min, x_max, isave, dsave) # for i in range(int(1e6)): # if task != 'F': # logging.info('search converged in {} iterations'.format(i)) # break # fx = f(x, dfx) # x, fx, dfx, task = minpack2.dsrch(x, fx, dfx, f_tol, df_tol, x_tol, task, # x_min, x_max, isave, dsave) # return x, fx, dfx