Propagation of magnetic dipole radiation through a medium HENK F. ARNOLDUS* AND ZHANGJIN XU Department of Physics and Astronomy, Mississippi State University, P.O. Drawer 5167, Mississippi State, Mississippi 39762-5167, USA *Corresponding author: hfa1@msstate.edu Received 29 January 2016; revised 31 March 2016; accepted 31 March 2016; posted 1 April 2016 (Doc. ID 258121); published 15 April 2016 An oscillating magnetic dipole moment emits radiation. We assume that the dipole is embedded in a medium with relative permittivity ε r and relative permeability μ r , and we have studied the effects of the surrounding material on the flow lines of the emitted energy. For a linear dipole moment in free space the flow lines of energy are straight lines, coming out of the dipole. When located in a medium, these field lines curve toward the dipole axis, due to the imaginary part of μ r . Some field lines end on the dipole axis, giving a nonradiating contribution to the energy flow. For a rotating dipole moment in free space, each field line of energy flow lies on a cone around the axis perpendicular to the plane of rotation of the dipole moment. The field line pattern is an optical vortex. When embedded in a material, the cone shape of the vortex becomes a funnel shape, and the windings are much less dense than for the pattern in free space. This is again due to the imaginary part of μ r . When the real part of μ r is negative, the field lines of the vortex swirl around the dipole axis opposite to the rotation direction of the dipole moment. For a near-single-negative medium, the spatial extent of the vortex becomes huge. We compare the results for the magnetic dipole to the case of an embedded electric dipole. © 2016 Optical Society of America OCIS codes: (260.2110) Electromagnetic optics; (260.2160) Energy transfer; (350.5500) Propagation; (350.5610) Radiation. http://dx.doi.org/10.1364/JOSAA.33.000882 1. INTRODUCTION The optical properties of a linear, homogeneous, isotropic material are represented by the relative permittivity ε r and the relative permeability μ r . Both parameters are complex, in gen- eral, with a nonnegative imaginary part. The index of refraction n is defined as n 2  ε r μ r ; Im n ≥ 0: (1) This leaves an ambiguity if ε r and μ r are both positive or both negative. Then we include small positive imaginary parts in these parameters, and consider the limit where these imaginary parts approach zero. We then find that n is positive when ε r and μ r are both positive (normal dielectric material) and n is neg- ative when ε r and μ r are both negative (negative index of re- fraction material). When a plane wave of light travels through a medium, the wavelength changes, as compared to propagation in free space, and this is due to the real part of n. The effect of the imaginary part of n is damping of the amplitudes of the electric and magnetic fields in the propagation direction, and this gives a corresponding damping of the intensity along the direction of propagation. The disappearing energy is ab- sorbed by the material. The flow lines of energy are the field lines of the Poynting vector. For propagation in free space, these field lines are straight, and they remain straight for propagation in a material. One could argue that the damping only affects the magnitude of the Poynting vector, and, since field lines are only determined by the direction of the Poynting vector, the field lines should remain unaltered for propagation in a medium. The damping, due to absorption by the material, diminishes the intensity along the propagation direction, but it does not affect the paths of energy flow. For a plane wave, this argument holds true, but in general the effect of damping is more intri- cate, as we shall show below. 2. MAGNETIC DIPOLE RADIATION We consider a magnetic dipole oscillating at angular frequency ω and located at the origin of coordinates. The dipole moment is given by pt  Repe -iωt ; (2) with p being the complex amplitude. The dipole is embedded in a medium with relative permittivity ε r and relative per- meability μ r . The emitted electric field is time harmonic, Er;t  ReEre -iωt ; (3) with Er being the complex amplitude, and a similar repre- sentation holds for the magnetic field Br;t . With a slight generalization of [1] we obtain, 882 Vol. 33, No. 5 / May 2016 / Journal of the Optical Society of America A Research Article 1084-7529/16/050882-05 Journal © 2016 Optical Society of America