A　new　boundary　integral　equation　method　for　solving　the　near　field　in　three-dimensional　vector　form　in　scanning　near-field　optical　microscopy　(SNOM)　using　Borgnis　potentials　as　auxiliary　functions　is　presented.　A　boundary　integral　equation　of　the　electromagnetic　fields,　expressed　by　Borgnis　potentials,　is　derived　based　on　Green＇s　theorem.　The　harmonic　expansion　in　rotationally　symmetric　SNOM　probe--sample　systems　is　studied,　and　the　three-dimensional　electromagnetic　problem　is　partly　simplified　into　a　two-dimensional　one.　The　boundary　conditions　of　Borgnis　potentials　both　on　dielectric　boundaries　and　on　perfectly　conducting　boundaries　are　derived.　Relevant　algorithms　were　studied,　and　a　computer　program　was　written.　As　an　example,　a　SNOM　probe-sample　system　composed　of　a　round　metal-covered　probe　and　a　sample　with　a　flat　surface　has　been　numerically　studied,　and　the　computational　results　are　given.　This　new　method　can　be　used　efficiently　for　other　electromagnetic　field　problems　with　round　subwavelength　structures.