import unittest import numpy as np import meep as mp class TestLDOS(unittest.TestCase): @classmethod def setUp(cls): cls.resolution = 25 # pixels/μm cls.dpml = 0.5 # thickness of PML cls.dair = 1.0 # thickness of air padding cls.L = 6.0 # length of non-PML region cls.n = 2.4 # refractive index of surrounding medium cls.wvl = 1.0 # wavelength (in vacuum) cls.fcen = 1 / cls.wvl # source properties (cylindrical) cls.df = 0.05 * cls.fcen cls.cutoff = 10.0 cls.src = mp.GaussianSource(cls.fcen, fwidth=cls.df, cutoff=cls.cutoff) # termination criteria cls.tol = 1e-8 def bulk_ldos_cyl(self): """Computes the LDOS of a point dipole in a homogeneous dielectric medium in cylindrical coordinates. """ sr = self.L + self.dpml sz = self.L + 2 * self.dpml cell_size = mp.Vector3(sr, 0, sz) pml_layers = [mp.PML(self.dpml)] sources = [ mp.Source( src=self.src, component=mp.Er, center=mp.Vector3(), ) ] sim = mp.Simulation( resolution=self.resolution, cell_size=cell_size, boundary_layers=pml_layers, sources=sources, dimensions=mp.CYLINDRICAL, m=-1, default_material=mp.Medium(index=self.n), ) sim.run( mp.dft_ldos(self.fcen, 0, 1), until_after_sources=mp.stop_when_fields_decayed( 20, mp.Er, mp.Vector3(), self.tol ), ) return sim.ldos_data[0] def cavity_ldos_cyl(self, sz): """Computes the LDOS of a point dipole in a planar cavity with lossless metallic walls in cylindrical coordinates. Args: sz: thickness of cavity. """ sr = self.L + self.dpml cell_size = mp.Vector3(sr, 0, sz) pml_layers = [mp.PML(self.dpml, direction=mp.R)] sources = [ mp.Source( src=self.src, component=mp.Er, center=mp.Vector3(), ) ] sim = mp.Simulation( resolution=self.resolution, cell_size=cell_size, boundary_layers=pml_layers, sources=sources, dimensions=mp.CYLINDRICAL, m=-1, default_material=mp.Medium(index=self.n), ) sim.run( mp.dft_ldos(self.fcen, 0, 1), until_after_sources=mp.stop_when_fields_decayed( 20, mp.Er, mp.Vector3(), self.tol ), ) return sim.ldos_data[0] def bulk_ldos_3D(self): """Computes the LDOS of a point dipole in a homogeneous dielectric medium in 3D Cartesian coordinates. """ s = self.L + 2 * self.dpml cell_size = mp.Vector3(s, s, s) pml_layers = [mp.PML(self.dpml)] sources = [ mp.Source( src=mp.GaussianSource(self.fcen, fwidth=0.2 * self.fcen), component=mp.Ex, center=mp.Vector3(), ) ] symmetries = [ mp.Mirror(direction=mp.X, phase=-1), mp.Mirror(direction=mp.Y), mp.Mirror(direction=mp.Z), ] sim = mp.Simulation( resolution=self.resolution, cell_size=cell_size, boundary_layers=pml_layers, sources=sources, symmetries=symmetries, default_material=mp.Medium(index=self.n), ) sim.run( mp.dft_ldos(self.fcen, 0, 1), until_after_sources=mp.stop_when_fields_decayed( 20, mp.Ex, mp.Vector3(), self.tol ), ) return sim.ldos_data[0] def cavity_ldos_3D(self, sz): """Computes the LDOS of a point dipole in a planar cavity with lossless metallic walls in 3D Cartesian coordinates. Args: sz: thickness of cavity. """ sxy = self.L + 2 * self.dpml cell_size = mp.Vector3(sxy, sxy, sz) boundary_layers = [ mp.PML(self.dpml, direction=mp.X), mp.PML(self.dpml, direction=mp.Y), ] sources = [ mp.Source( src=mp.GaussianSource(self.fcen, fwidth=0.2 * self.fcen), component=mp.Ex, center=mp.Vector3(), ) ] symmetries = [ mp.Mirror(direction=mp.X, phase=-1), mp.Mirror(direction=mp.Y), mp.Mirror(direction=mp.Z), ] sim = mp.Simulation( resolution=self.resolution, cell_size=cell_size, boundary_layers=boundary_layers, sources=sources, symmetries=symmetries, default_material=mp.Medium(index=self.n), ) sim.run( mp.dft_ldos(ldos=mp.Ldos(self.fcen, 0, 1)), until_after_sources=mp.stop_when_fields_decayed( 20, mp.Ex, mp.Vector3(), self.tol ), ) return sim.ldos_data[0] def purcell_enh_theory(self, c): """Computes the Purcell enhancement factor of a point dipole in a planar dielectric cavity with lossless metallic walls using equation 7 of: I. Abram et al., IEEE J. Quantum Electronics, Vol. 34, pp. 71-76 (1998). Args: c: cavity thickness in units of wavelength in the cavity medium. """ return 3 * np.fix(c + 0.5) / (4 * c) + ( 4 * np.power(np.fix(c + 0.5), 3) - np.fix(c + 0.5) ) / (16 * np.power(c, 3)) def ext_eff_cyl(self, dmat, h): """Computes the extraction efficiency of a point dipole embedded within a dielectric layer above a lossless ground plane in cylindrical coordinates. Args: dmat: thickness of dielectric layer. h: height of dipole above ground plane as fraction of dmat. """ sr = self.L + self.dpml sz = dmat + self.dair + self.dpml cell_size = mp.Vector3(sr, 0, sz) boundary_layers = [ mp.PML(self.dpml, direction=mp.R), mp.PML(self.dpml, direction=mp.Z, side=mp.High), ] src_cmpt = mp.Er src_pt = mp.Vector3(0, 0, -0.5 * sz + h * dmat) sources = [mp.Source(src=self.src, component=src_cmpt, center=src_pt)] geometry = [ mp.Block( material=mp.Medium(index=self.n), center=mp.Vector3(0, 0, -0.5 * sz + 0.5 * dmat), size=mp.Vector3(mp.inf, mp.inf, dmat), ) ] sim = mp.Simulation( resolution=self.resolution, cell_size=cell_size, dimensions=mp.CYLINDRICAL, m=-1, boundary_layers=boundary_layers, sources=sources, geometry=geometry, ) flux_air = sim.add_flux( self.fcen, 0, 1, mp.FluxRegion( center=mp.Vector3(0.5 * self.L, 0, 0.5 * sz - self.dpml), size=mp.Vector3(self.L, 0, 0), ), mp.FluxRegion( center=mp.Vector3(self.L, 0, 0.5 * sz - self.dpml - 0.5 * self.dair), size=mp.Vector3(0, 0, self.dair), ), ) sim.run( mp.dft_ldos(self.fcen, 0, 1), until_after_sources=mp.stop_when_fields_decayed( 20, src_cmpt, src_pt, self.tol ), ) out_flux = mp.get_fluxes(flux_air)[0] dV = np.pi / (self.resolution**3) total_flux = -np.real(sim.ldos_Fdata[0] * np.conj(sim.ldos_Jdata[0])) * dV ext_eff = out_flux / total_flux print(f"extraction efficiency (cyl):, " f"{dmat:.4f}, {h:.4f}, {ext_eff:.6f}") return ext_eff def ext_eff_3D(self, dmat, h): """Computes the extraction efficiency of a point dipole embedded within a dielectric layer above a lossless ground plane in 3D Cartesian coordinates. Args: dmat: thickness of dielectric layer. h: height of dipole above ground plane as fraction of dmat. """ sxy = self.L + 2 * self.dpml sz = dmat + self.dair + self.dpml cell_size = mp.Vector3(sxy, sxy, sz) symmetries = [mp.Mirror(direction=mp.X, phase=-1), mp.Mirror(direction=mp.Y)] boundary_layers = [ mp.PML(self.dpml, direction=mp.X), mp.PML(self.dpml, direction=mp.Y), mp.PML(self.dpml, direction=mp.Z, side=mp.High), ] src_cmpt = mp.Ex src_pt = mp.Vector3(0, 0, -0.5 * sz + h * dmat) sources = [ mp.Source( src=mp.GaussianSource(self.fcen, fwidth=0.1 * self.fcen), component=src_cmpt, center=src_pt, ) ] geometry = [ mp.Block( material=mp.Medium(index=self.n), center=mp.Vector3(0, 0, -0.5 * sz + 0.5 * dmat), size=mp.Vector3(mp.inf, mp.inf, dmat), ) ] sim = mp.Simulation( resolution=self.resolution, cell_size=cell_size, boundary_layers=boundary_layers, sources=sources, geometry=geometry, symmetries=symmetries, ) flux_air = sim.add_flux( self.fcen, 0, 1, mp.FluxRegion( center=mp.Vector3(0, 0, 0.5 * sz - self.dpml), size=mp.Vector3(self.L, self.L, 0), ), mp.FluxRegion( center=mp.Vector3( 0.5 * self.L, 0, 0.5 * sz - self.dpml - 0.5 * self.dair ), size=mp.Vector3(0, self.L, self.dair), ), mp.FluxRegion( center=mp.Vector3( -0.5 * self.L, 0, 0.5 * sz - self.dpml - 0.5 * self.dair ), size=mp.Vector3(0, self.L, self.dair), weight=-1.0, ), mp.FluxRegion( center=mp.Vector3( 0, 0.5 * self.L, 0.5 * sz - self.dpml - 0.5 * self.dair ), size=mp.Vector3(self.L, 0, self.dair), ), mp.FluxRegion( center=mp.Vector3( 0, -0.5 * self.L, 0.5 * sz - self.dpml - 0.5 * self.dair ), size=mp.Vector3(self.L, 0, self.dair), weight=-1.0, ), ) sim.run( mp.dft_ldos(self.fcen, 0, 1), until_after_sources=mp.stop_when_fields_decayed( 20, src_cmpt, src_pt, self.tol ), ) out_flux = mp.get_fluxes(flux_air)[0] dV = 1 / (self.resolution**3) total_flux = -np.real(sim.ldos_Fdata[0] * np.conj(sim.ldos_Jdata[0])) * dV ext_eff = out_flux / total_flux print(f"extraction efficiency (3D):, " f"{dmat:.4f}, {h:.4f}, {ext_eff:.6f}") return ext_eff def test_ldos_cyl(self): """Verifies that the Purcell enhancement factor of a parallel dipole in a planar dielectric cavity with lossless metallic walls computed in cylindrical coordinates agrees with the analytic result. """ ldos_bulk = self.bulk_ldos_cyl() # not a Van Hove singularity cavity_thickness = 1.63 gap = cavity_thickness * self.wvl / self.n ldos_cavity = self.cavity_ldos_cyl(gap) # Purcell enhancement factor (relative to bulk medium) pe_meep = ldos_cavity / ldos_bulk pe_theory = self.purcell_enh_theory(cavity_thickness) rel_err = abs(pe_meep - pe_theory) / pe_theory print( "ldos-cyl:, {:.6f} (Meep), {:.6f} (theory), " "{:.6f} (error)".format(pe_meep, pe_theory, rel_err) ) self.assertAlmostEqual(pe_meep, pe_theory, delta=0.1) def test_ldos_3D(self): """Verifies that the Purcell enhancement factor of a parallel dipole in a planar dielectric cavity with lossless metallic walls computed in 3D Cartesian coordinates agrees with the analytic result. """ ldos_bulk = self.bulk_ldos_3D() # not a Van Hove singularity cavity_thickness = 0.75 gap = cavity_thickness * self.wvl / self.n ldos_cavity = self.cavity_ldos_3D(gap) # Purcell enhancement factor (relative to bulk medium) pe_meep = ldos_cavity / ldos_bulk pe_theory = self.purcell_enh_theory(cavity_thickness) rel_err = abs(pe_meep - pe_theory) / pe_theory print( "ldos-3D:, {:.6f} (Meep), {:.6f} (theory), " "{:.6f} (error)".format(pe_meep, pe_theory, rel_err) ) self.assertAlmostEqual(pe_meep, pe_theory, delta=0.1) def test_ldos_ext_eff(self): """Verifies that the extraction efficiency of a point dipole in a dielectric layer above a lossless ground plane computed in cylindrical and 3D Cartesian coordinates agree. """ layer_thickness = 0.5 * self.wvl / self.n dipole_height = 0.5 ext_eff_cyl = self.ext_eff_cyl(layer_thickness, dipole_height) ext_eff_3D = self.ext_eff_3D(layer_thickness, dipole_height) self.assertAlmostEqual(ext_eff_cyl, ext_eff_3D, places=2) if __name__ == "__main__": unittest.main()