Exact solution of Helmholtz equation for the case of non-paraxial Gaussian beams.

Transcription

1 Exct solton o Helmholtz eton o the cse o non-xl Gssn bems. Sege V. Eshov Insttte o Tme Nte Exlotons M.V. Lomonosov's Moscow Stte Unvest Lennse go - Moscow 999 ss e-ml: segej-eshov@nex. Kewos: Helmholtz eton cct eton Gssn bem. new te o exct soltons o the ll 3 mensonl stl Helmholtz eton o the cse o non-xl Gssn bems s esente hee. We conse ote eesentton o the solton o Gssn bems n shecl coonte sstem b sbstttng t to the ll 3 mensonl stl Helmholtz Eton. nlzng the stcte o the nl eton we obtn tht govenng etons o the comonents o o solton e eesente b the oe cct etons o comlex vle whch hs no nltcl solton n genel cse. Bt we n one o the ossble exct soltons whch s ove to sts to sch n etons o Gssn bems.

2 . Intocton. The ll 3-mensonl stl Helmholtz eton oves soltons tht escbe the ogton o wves ove sce e.g. electomgnetc wves ne oe bon contons; t shol be esente n shecl coonte sstem s below [-]: 0. - whee Δ - s the Llcn s the wvenmbe n s the mlte. So the evton vnce n ths mnsct stts wth the scl Helmholtz eton exesse n shecl co-ontes. Beses n shecl coonte sstem [3]: sn. Secl soltons to ths eton hve genete contnng nteest n the otcl hscs commnt snce the scove o nsl non-ctng wves sch s Bessel n bems [4-6]. Let s sech o soltons o E.. n clsscl om o Gssn bems [7-9] whch col be esente n tesn coonte sstem s below [0]: w0 ex w w - whee w e the el nctons escbng the ote metes o bem; w s the bem wst sze s wveont s o cvte n s the Go s hse sht oel [0].

3 The clsscl om o Gssn bems bove col be lso eesente s below - hee s the comlex hse-sht o the wves ng the ogton long the xs; s the oe comlex mete o bem whch s etemnng the Gssn ole o wve n the tnsvese e t oston. Beses let s lso note tht t the let t o the exesson bove we exess the tem /w n om o Gssn bems s ex ² w = ex - w. The ght t o the exesson bove col be tnsome n shecl coonte sstem to the om below: The solton * s tonll ssme to be neenent o the zmthl coonte to obseve t ne well-nown xl oxmton [0] lso. Then hvng sbsttte the exesson * nto E.. we shol obtn 0: * sn ex ex ex w w. sn sn sn. sn sn sn

4 . Exct soltons. Let s e-esgnte ote tem n * s below: In sch cse E.. col be tnsome s below 0: Ths ll ossble soltons o eesentng o Gssn bems n om * e escbe b the Eton.. Bt we shol esecll note tht ng the ocess o the obtnng o solton o exmle we e sml ssmng secl eonl solton [0-] to the Helmholtz eton some o mn etes o the solton col be ece; so sch solton nee not hve n elton to Gssn om *. Beses one o the obvos soltons o PDE-etons.: = ₁ + ₂ ** - whee ₁ ₂ e the nctons o comlex vle.. sn 0.

5 Let s ssme s below: - hee s constnt o comlex vle. Fo sch cse E.. col be ece s below 0: 3. Pesentton o exct solton. Une ssmton ** E.. col be eesente s below: - whee the lst eton s nown to be the cct ODE [3] whch hs no solton n genel cse. Bt = 0 E. 3. hs oe solton ₀ = const:..3 0 sn csc csc 3. csc 0 csc csc 3. csc csc 0 0 0

6 Beses E..3 col be esente s below = 0: - whee the lst cct ODE 3.3 hs oe solton below = 0 see [3] the cse.04. Inee let s ssme 0 0: - then we obtn:. 3.3 tn h tn tnh / snh 4 / h

7 Tng nto conseton the exesson ** o the solton s well s let s nll esent new te o non-xl solton whch s ove to sts to the Helmholtz eton. s below: h csc snh csc / 4 / 4 - o tn sn tn / 4 / whee Dscssons & conclson. new te o exct soltons o the ll 3 mensonl stl Helmholtz eton o the cse o non-xl Gssn bems s esente hee. We conse ote eesentton o the solton o Gssn bems n shecl coonte sstem b sbstttng t to the ll 3 mensonl stl Helmholtz Eton. nlzng the stcte o the nl eton we obtn tht govenng etons o the comonents o o solton e eesente b the oe cct etons o comlex vle whch hs no nltcl solton n genel cse. We shol note

8 tht moen metho exsts o obtnng o the nmecl solton o cct etons wth goo oxmton []. Bt we n one o the ossble exct soltons 3.5 whch s ove to sts to the Helmholtz eton. o bems *. Inee snce the nctons g = sn/ o g = / n 3.5 e tsel n exct soltons o the ll Helmholtz eton. [-] the oml o the Llcn n shecl coontes gves o = hg h = tg/: - whch s obvosl vl o the nge o mete 0. s o the ote exmle o xl oxmton o sch non-xl exct solton 3.5 o the ll Helmholtz eton. t col be esl obtne n the cse +0 see the exesson 3.5 bove. Let s exess the el t o solton 3.5 n the tesn co-ontes s below < /4 : 0 sn sn 0 tn 4. tn

9 s we now shecl-wve solton g = / col be schemtcll mgne n the tesn co-onte sstem s below [-]: Fg.. schemtc lot o shecl-wve te o the soltons. - whee ech o shecl wves s ssme to be concentc shee evenl enlgng om xe ont soce o wves see Fg.-. The solton 3.5 es om the shecl-wves on cto tn/ bt the totl eneg o bem shol not excee the totl eneg o the ote shecl-wves solton o Helmholtz eton. The eneg o the bem s o cose essentll the bsolte mgnte o the solton shecll ntegte ove sce.

10 Fg.. schemtc lot o the e shecl-wve soltons. So we shol estct the nge o mete 0 to the nge [₀ ₁] {whee ₀ = ctn/e 044 e = ₁ = ctne 07756} o the eson tht nelt: tn/ < tn/ shol be vl o ll menngs o ncton tn/ n tht nge o esecll > /4. So these nsl bems wth lmte mlte col be comng to the shecl-wves solton whch s mch moe close to ech othe thn othe exotc bems onl t the nge o mete [~404 ~396]. Fg.3. schemtc lot o the ncton x/*/ hee we esgnte: x = = ²+² 0 =

11 s o the ote exmles o xl oxmton = ²+² exesse b E. 4. n tesn co-ontes see Fg.34. Fg.4. schemtc lot o the ncton x/*/ hee we esgnte: x = = ²+² = Let s lso schemtcll mgne the shecl-wve solton to come t wth the solton bove: Fg.5. schemtc lot o the ncton / hee we esgnte: =

12 Fg.6. schemtc lot o the ncton / hee we esgnte: = lso let s note tht these nsl bems col be comng to the Bessel bem soltons [] t ll the nge o mete 0. To obtn the eneg o the bem we shol shecll ntegte the bsolte mgnte o the solton ove sce so sch clcltons shol oce the nnte eneg o bem e to the stcte o the solton: = tn//. s o the ont o clng the hscl content o the eve solton Fg. 3-6 col esent wve tvelg on the ocen sce o exmle. Sch solton s sose to be el enhncng the s ng ogton n -ecton bt t s molte b the ncton tn/ so tht the totl mlte = tn//. Jmng o hse-ncton o solton * n om 3.5 o n mlte beng el to zeo t the menng o mete = / col be ssocte wth

13 the exstence o n otcl votex [0] t ths ont. Otcl votex lso nown s scew slocton o hse snglt s zeo o n otcl el ont o zeo ntenst. esech nto the oetes o votces hs thve snce comehensve e [3] escbe the bsc oetes o "sloctons n wve tns". eeences:..sommeel 949. Ptl Deentl Etons n Phscs. cemc Pess New o.. Sew Moses n Moe 004. Moen Phscs 3 e.. Boos ole. ISBN E. Kme 97. Hn-boo o ODE. Scence Moscow. 4. Mgel. lonso Mgel. Bnes 04. Geneton o nonxl cceletng els thogh mos. II: Thee mensons. Otcs Exess Vol. Isse Mgel. lonso Mgel. Bnes 04. Geneton o nonxl cceletng els thogh mos. I: Two mensons. Otcs Exess Vol. Isse Mgel. lonso Mgel. Bnes 0. Shecl els s nonxl cceletng wves. OPTIS LETTES Vol. 37 No Qng ho Go-Qn n Wng o-gng 03. Nonxl ogton o Hemte-Lgee-Gssn bems n nxl cstl othogonl to the otcl xs. Ioscence hnese Phs. B o:0.088/ //6/ M. Tgzhnov. S. Blgovestchens &. P. Kselev 0. "omlex soce" wve els: soces n el sce. J.Phs.: Mth. Theo hen et l. 00. nlses o vecto Gssn bem ogton n the vlt o xl n shecl oxmtons. JOS vol Svelto Ozo 00. Pncles o Lses 5th e.. See lso:

14 htt://en.we.og/w/gssn_bem see Mthemtcl om.. P. W. Mlonn n J. H. Ebel 00. Lse Phscs n e.. Secton l M. Bene Steven. Oszg 999. vnce Mthemtcl Methos o Scentsts n Engnees. Ogn.blshe b McGw Hll 978 IV J. F. Ne M. V. Be 974. Dsloctons n wve tns. Poceengs o the ol Socet o Lonon Sees : 65. See lso: htt://en.we.og/w/otcl_votex enx checng o the exct solton. The ect sbsttton o the nl exesson 3.5 nto the Helmholtz eton. s n es mtte showng tht ths s ell n exct solton. Let s begn to chec the solton 3.5 om the -st t o sch solton: Fo the eson some o ees m be hve no the scent tme to execte the clcltons oel t hs been me ste-b-ste below: 0 4 / sn tn 0 sn sn sn 0

15 - whee the lst entt s obvosl vl o the nge o mete : < /4. The checng o -n t o solton 3.5 col be execte n the sme w. 0 sn sn 0 sn ] sn [ ] sn sn [ 0 sn sn 0 sn sn sn sn 3 3 4