karkoV. Ha Abstract -Itiswell-known, that theFresnel -Kirchhoff diff-raction theory imposes boundary conditions onafield strength andanormal derivative simultaneously. Itisalso knownthat these requirements contradict eachother. A. Sommerfeld hasshown that forelimination ofthese conflicts ofthediffiraction theory, itisnecessary andenough to select aGreen function sothat thefunction orits normal derivative onaboundary surface areequal tozero. Butsuch Green function isknown, however, forpractically unique caseofadiffraction onaplane screen. According tothe Sommerfeld, this Green function isformed byapoint source andits mirror image inaplane screen. Ifthepoint source andits image areantiphased, theGreen function isidentically equal tozeroonall surface ofthescreen. Ifthe point source andits image arecophased, thenormal derivative oftheGreen function isequal tozeroonthescreen. Theconflicts oftheFresnel -Kirchhoff theory areeliminated ineach casc. Inthework, wepresent oneofpossible versions ofaGreen function forinterior field ofcylindrical elliptic cavities orscreens. Atthesolution ofdiffraction problems insuchfields forsimplification oftheequations everyone usually transfers Cartesian coordinates to elliptical coordinates. Theinterrelations between coordinates arefeatured bymultivalued functions, therefore are available realfield oftheexistence ofcoordinates andset"dummy" fields overlapped onrealfield. This circumstance allows todesign foranelliptic cavity aGreen function satisfying totheSommerfeld conditions. For this purpose, inaninterior point ofreal field ofacavity weputthepoint source featured bytheHankel function HI10 (kr) ofthefirst kindandofthezeroorder, andintherelevant conjugate points "dummy" field weputthepoint source featured bytheliankel function H20(kr) ofthesecond kind andofthezeroorder. TheGreen function is formed asthetotal sumoftheaforesaid cophased point sources.We notethat argument rfortheHlankel function in "dummy" field isnecessary totake bynegative. Itiseasy toverify that such Green function isequal tozeroandits normal derivative isnotequal tozero onaboundary surface ofanelliptic cavity. Anumerous investigations during 40years haveledtoagoodunderstanding oftheeigen- modestructure andlosses both unstable (1-3) andstable laser resonators (4-6). However, satisfac- tory analytical solution formarginally unstable resonators islacking; therefore, itispossible that suchresonators donothavetheapplied utility. Inthis paper, wepresent approximate analytical expression foreigenmodes oftwo-dimensional elliptic emptyresonator withsmall instability which canbeutilised inmetrology forevaluation oftheFresnel phase shift atreproduction ofthe unit ofalength -meter. Foranalysing ofmarginally unstable resonator, weusethemethod ofintegral equations in two-dimensional elliptical coordinates. Suchanapproach hasfollowing advantages: -theelliptic mirror surface hastheleast deviation fromspherical form; -theelliptic mirror surface coincides acoordinate surface andintegral equations havethe mostsimple fonn, thereby; -inelliptical coordinates, wecananalyse both unstable (prolate ellipse) andstable (oblate ellipse) resonators bythesamewayanditisconvenient for thedifferent resonator comparison; -themethod ofintegral equations isgood forusing ofthediverse simplification; -thewaveequation isseparated inelliptical coordinates andwecanusesimple one- dimensional integral equations; -intwo-dimensional coordinates, wecanusethescalar theory ofdiffraction; -intwo-dimensional coordinates, weemploy thepowerful system ofcomplex functions. Weintroduce thesystem ofprolate elliptical coordinates associated withCartesian coordi- nates byformulas oftransformations

http://ieeexplore.ieee.org/iel5/10411/33072/01553868.pdf

K. I. Muntean

Papers

Eigenmodesofatwo-dimensional elliptical laserresonator