Some new forms of Huygens' principle

Abstract

Several theorems are derived by using the reaction concept. They apply to single frequency sources of finite extent, and to fields which are finite and continuous on a hypothetical closed surface <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</tex> whose inside or outside is source-free. 1) The field on the source-free side of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</tex> is uniquely determined by the normal components <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E_{n}</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H_{n}</tex> on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</tex> . 2) For measurements on the source-free side of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</tex> , the primary source can be replaced by any of the following secondary sources on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</tex> : (a) Normal electric and magnetic dipoles in free space. (b) Normal electric and magnetic quadrupoles backed by a medium which makes <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E_{n} = 0</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H_{n} = 0</tex> . (c) A certain combination of electric and magnetic normal dipoles and quadrupoles which gives zero field on the source side of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</tex> . The prescription for (a) requires the solution for the field of the primary source when it is inside a cavity on whose walls <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E_{n} = 0</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H_{n} = 0</tex> . The surface densities of the various components of secondary sources (b) and (c) are given in terms of the free-space primary field. The formulas are comparatively simple when <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</tex> is plane. Many applications are cited including a combination of multipoles, consisting of vertical electric and magnetic dipoles, which is equivalent to a horizontal electric dipole.