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\title{Âèçóàëèçàöèÿ èçî÷àñòîòíûõ êîíòóðîâ ñèëüíî ëîêàëèçîâàííûõ âîëíîâîäíûõ ìîä â ïëàíàðíûõ äèýëåêòðè÷åñêèõ ñòðóêòóðàõ}

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\rtitle{Âèçóàëèçàöèÿ èçî÷àñòîòíûõ êîíòóðîâ\dots}

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\sodtitle{Âèçóàëèçàöèÿ èçî÷àñòîòíûõ êîíòóðîâ ñèëüíî ëîêàëèçîâàííûõ âîëíîâîäíûõ ìîä â ïëàíàðíûõ äèýëåêòðè÷åñêèõ ñòðóêòóðàõ}


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  Ä.\,Â.\,Ïåðìÿêîâ$^{a}$ \thanks{e-mail: d.permyakov@metalab.ifmo.ru},
  È.\,Ñ.\,Ñèíåâ$^{a}$
  Ñ.\,Ê.\,Ñû÷åâ$^{a}$
  A.\,C.\,Ãóäîâñêèõ$^{b}$
  À.\,À.\,Áîãäàíîâ$^{a}$
  À.\,Â.\,Ëàâðèíåíêî$^{a,c}$
  è À.\,Ê.\,Ñàìóñåâ$^{a}$
}

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%\rauthor{Ä.\,Â.\,Ïåðìÿêîâ, È.\,Ñ.\,Ñèíåâ, Ñ.\,Ê.\,Ñû÷åâ, À.\,À.\,Áîãäàíîâ, À.\,Â.\,Ëàâðèíåíêî and À.\,Ê.\,Ñàìóñåâ}
\rauthor{Ä.\,Â.\,Ïåðìÿêîâ, È.\,Ñ.\,Ñèíåâ, Ñ.\,Ê.\,Ñû÷åâ è äð.}

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$^a$ Óíèâåðñèòåò ÈÒÌÎ, 199034, Ñàíêò-Ïåòåðáóðã, Ðîññèÿ\\
$^b$ Ñàíêò-Ïåòåðáóðãñêèé Àêàäåìè÷åñêèé óíèâåðñèòåò ÐÀÍ, 194021, Ñàíêò-Ïåòåðáóðã, Ðîññèÿ\\
$^c$ DTU Fotonik, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark
}

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\abstract{
	Ïðåäëîæåíà ýêñïåðèìåíòàëüíàÿ ìåòîäèêà èçó÷åíèÿ äèñïåðñèîííûõ ñâîéñòâ îïòè÷åñêèõ ïîâåðõíîñòíûõ è âîëíîâîäíûõ ìîä â ïëàíàðíûõ ñòðóêòóðàõ. Ýêñïåðèìåíòàëüíàÿ óñòàíîâêà ïðåäñòàâëÿåò ñîáîé ìèêðîñêîï ñ âûñîêîàïåðòóðíûì îáúåêòèâîì è ëèíçîé òâåðäîé èììåðñèè â ôîðìå ïîëóñôåðû èç ñåëåíèäà öèíêà, ïðèâîäèìîé â êîíòàêò ñ ïîâåðõíîñòüþ îáðàçöà. Ïðè ýòîì ðåãèñòðàöèÿ îòðàæåíèÿ îò îáðàçöà ïðîâîäèòñÿ â çàäíåé ôîêàëüíîé ïëîñêîñòè ñèñòåìû. Òàêàÿ êîíôèãóðàöèÿ ïîçâîëÿåò èññëåäîâàòü â âèäèìîì è áëèæíåì èíôðàêðàñíîì ñïåêòðàëüíîì äèàïàçîíàõ ñèëüíî ëîêàëèçîâàííûå ñîñòîÿíèÿ ñ ýôôåêòèâíûì ïîêàçàòåëåì ïðåëîìëåíèÿ äî 2.25 â âèäèìîì è áëèæíåì èíôðàêðàñíîì ñïåêòðàëüíûõ äèàïàçîíàõ.
	Íà ïðèìåðå òîíêîãî ñëîÿ êðåìíèÿ, íàíåñåííîãî íà ñòåêëÿííóþ ïîäëîæêó, ïðîäåìîíñòðèðîâàíà âîçìîæíîñòü âèçóàëèçàöèè èçî÷àñòîòíûõ êîíòóðîâ ñ ïîëÿðèçàöèîííûì ðàçðåøåíèåì è âîññòàíîâëåíèÿ äèñïåðñèè ìîä âîëíîâîäà â çàâèñèìîñòè îò íàïðàâëåíèÿ èõ ðàñïðîñòðàíåíèÿ.
}

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%\section{Ââåäåíèå}

Ïëàíàðíûå ôîòîííûå ñòðóêòóðû ñîñòàâëÿþò îñíîâó ñîâðåìåííîé îïòîýëåêòðîíèêè. Ê èõ ÷èñëó îòíîñÿòñÿ  äèôðàêöèîííûå ðåøåòêè, ðàçëè÷íûå ôîòîííî-êðèñòàëëè÷åñêèå ïëàñòèíû, ìåìáðàíû, âîëíîâîäíûå ñòðóêòóðû è ìåòàïîâåðõíîñòè -- äâóìåðíûé àíàëîã ìåòàìàòåðèàëîâ~\cite{tamir2013guided,haus1984waves,bonaccorso2010graphene,ebeling2012integrated}. Îñíîâíîé õàðàêòåðèñòèêîé òàêèõ ñòðóêòóð ÿâëÿåòñÿ ñïåêòð èõ ñîáñòâåííûõ ìîä, êîòîðûé â îáùåì ñëó÷àå ñîñòîèò èç âîëíîâîäíûõ, ïîâåðõíîñòíûõ è ðåçîíàíñíûõ ñîñòîÿíèé (óòåêàþùèõ ìîä). Ðåçîíàíñíûå ñîñòîÿíèÿ ìîãóò áûòü èññëåäîâàíû â ýêñïåðèìåíòàõ ïî èçìåðåíèþ ñïåêòðîâ ïðîïóñêàíèÿ~\cite{lousse2004angular}, îòðàæåíèÿ~\cite{liu1998high}, ïîãëîùåíèÿ~\cite{christ2003waveguide} èëè ëþìèíåñöåíöèè~\cite{ganesh2007enhanced}. Âîëíîâîäíûå è ïîâåðõíîñòíûå ìîäû  \footnote{Äëÿ êðàòêîñòè äëÿ âîëíîâîäíûõ è ïîâåðõíîñòíûõ ìîä ìû áóäåì èñïîëüçîâàòü òåðìèí ëîêàëèçîâàííûå ñîñòîÿíèÿ.}   â òàêèõ ýêñïåðèìåíòàõ ñåáÿ íå îáíàðóæèâàþò, òàê êàê èõ äèñïåðñèîííûå êðèâûå ëåæàò íèæå ñâåòîâîé ëèíèè. Ïîýòîìó äëÿ èõ âîçáóæäåíèÿ èñïîëüçóþòñÿ ïðèçìû ñ âûñîêèì ïîêàçàòåëåì ïðåëîìëåíèÿ, äèôðàêöèîííûå ðåøåòêè, ðåçîíàíñíûå îïòè÷åñêèå àíòåííû èëè øåðîõîâàòîñòè~\cite{maier2007plasmonics}. 

Àíèçîòðîïèÿ ïëàíàðíûõ ôîòîííûõ ñòðóêòóð ïðèâîäèò ê çàâèñèìîñòè äèñïåðñèè ëîêàëèçîâàííûõ ñîñòîÿíèé îò íàïðàâëåíèÿ èõ ðàñïðîñòðàíåíèÿ. Ýòî ïîçâîëÿåò ðåàëèçîâàòü íåîáû÷íûå ðåæèìû ðàñïðîñòðàíåíèÿ âîëí, íàïðèìåð, êàíàëèðîâàíèå, îòðèöàòåëüíîå ïðåëîìëåíèå èëè îäíîíàïðàâëåííîå ðàñïðîñòðàíåíèå~\cite{high2015visible,takayama2017photonics}. Ýòè ýôôåêòû ìîæíî íàãëÿäíî îáúÿñíèòü â òåðìèíàõ èçî÷àñòîòíûõ êîíòóðîâ, òàê êàê îíè îïðåäåëÿþò çàâèñèìîñòü ãðóïïîâîé ñêîðîñòè âîëíû $\mathbf{v}_{\text{g}}=\nabla_\mathbf{k}\omega$ îò íàïðàâëåíèÿ ðàñïðîñòðàíåíèÿ è, êàê ñëåäñòâèå, ôîðìó âîëíîâûõ ôðîíòîâ. Ïîýòîìó, âàæíîé çàäà÷åé ýêñïåðèìåíòàëüíîé îïòèêè ÿâëÿåòñÿ âèçóàëèçàöèÿ èçî÷àñòîòíûõ êîíòóðîâ ñîñòîÿíèé, âîçáóæäàåìûõ â òàêèõ ñòðóêòóðàõ.

Îäíèì èç âîçìîæíûõ ðåøåíèé ÿâëÿåòñÿ èñïîëüçîâàíèå ìåòîäà ìèêðîñêîïèè çàäíåé ôîêàëüíîé ïëîñêîñòè (òàêæå íàçûâàåìîé Ôóðüå-ìèêðîñêîïèåé). Ðåãèñòðàöèÿ ñèãíàëà â çàäíåé ôîêàëüíîé ïëîñêîñòè îïòè÷åñêîé ñèñòåìû ïîçâîëÿåò âèçóàëèçèðîâàòü äèàãðàììû íàïðàâëåííîñòè èçëó÷åíèÿ îäèíî÷íûõ èñòî÷íèêîâ~\cite{zhang2014back} è õàðàêòåðèçîâàòü çîííóþ ñòðóêòóðó ôîòîííûõ êðèñòàëëîâ~\cite{wagner2012back}. Ïðè èñïîëüçîâàíèè èììåðñèîííûõ îáúåêòèâîâ ñòàíîâèòñÿ âîçìîæíûì íàáëþäàòü è äèàãðàììû íàïðàâëåííîñòè ëîêàëèçîâàííûõ ñîñòîÿíèé, íàõîäÿùèõñÿ ïîä ñâåòîâûì êîíóñîì âîçäóõà.  ÷àñòíîñòè, ìåòîä ìèêðîñêîïèè çàäíåé ôîêàëüíîé ïëîñêîñòè áûë óñïåøíî ïðèìåíåí â óñòàíîâêå ìèêðîñêîïèè óòå÷êè èçëó÷åíèÿ äëÿ íàáëþäåíèÿ äèàãðàìì íàïðàâëåííîñòè ïîâåðõíîñòíûõ ïëàçìîí-ïîëÿðèòîíîâ, âîçáóæäàåìûõ äåôåêòàìè íà òîíêèõ ïëåíêàõ ìåòàëëà~\cite{drezet2008leakage,sinev2017chirality}. Îäíàêî, â ýòîì ñëó÷àå ìàêñèìàëüíûé âîëíîâîé âåêòîð íàáëþäàåìûõ ïîâåðõíîñòíûõ âîëí îãðàíè÷åí ÷èñëîâîé àïåðòóðîé èñïîëüçóåìîãî èììåðñèîííîãî îáúåêòèâà, à ÷àùå - ïîêàçàòåëåì ïðåëîìëåíèÿ ïîäëîæêè, ÷åðåç êîòîðóþ ïðîèñõîäèò óòå÷êà èçëó÷åíèÿ.  ðåçóëüòàòå, ïîäîáíûå óñòàíîâêè íå ïîçâîëÿþò ïîëó÷àòü èíôîðìàöèþ î ëîêàëèçîâàííûõ ìîäàõ ñ âîëíîâûì âåêòîðîì âûøå (1.4-1.6)$k_0$, ãäå $k_0$ - âîëíîâîé âåêòîð ñâåòà â âàêóóìå. 
Äëÿ íàáëþäåíèÿ ñîñòîÿíèé ñ ñèëüíîé ëîêàëèçàöèåé (òî åñòü, ñ áîëüøèìè âîëíîâûìè âåêòîðàìè â ïëîñêîñòè ñòðóêòóðû) ìîæåò áûòü òàêæå èñïîëüçîâàí ìåòîä ñêàíèðóþùåé áëèæíåïîëüíîé ìèêðîñêîïèè~\cite{lin2013polarization,permyakov2014mapping}, êîòîðûé, îäíàêî, ãîðàçäî áîëåå ñëîæåí òåõíè÷åñêè. 

Óâåëè÷åíèå ýôôåêòèâíîé ÷èñëîâîé àïåðòóðû îïòè÷åñêîé ñèñòåìû âîçìîæíî çà ñ÷åò èñïîëüçîâàíèÿ òàê íàçûâàåìûõ ëèíç òâåðäîé èììåðñèè~\cite{mansfield1990solid}. Òàêàÿ ëèíçà ïðåäñòàâëÿåò ñîáîé ýëåìåíò èç ïðîçðà÷íîãî ìàòåðèàëà, îáû÷íî èìåþùèé ôîðìó ïîëóñôåðû, êîòîðûé ïîìåùàåòñÿ ìåæäó îáúåêòèâîì è èññëåäóåìûì îáðàçöîì è ïðèâîäèòñÿ â êîíòàêò ñ ïîâåðõíîñòüþ ïîñëåäíåãî, êàê ïðîèëëþñòðèðîâàíî íà Ðèñ.~\ref{FigExpSetupAndSample}.  Ïðè èñïîëüçîâàíèè ñ îáû÷íûì ñóõèì îáúåêòèâîì ëèíçà ìîæåò óâåëè÷èâàòü ýôôåêòèâíóþ ÷èñëîâóþ àïåðòóðó ñèñòåìû â $n$ ðàç, ãäå $n$ - ïîêàçàòåëü ïðåëîìëåíèÿ ìàòåðèàëà ëèíçû. Èäåÿ èñïîëüçîâàíèÿ òàêèõ ýëåìåíòîâ â ñèñòåìàõ ìèêðîñêîïèè áûëà âïåðâûå ïðåäëîæåíà â ðàáîòå~\cite{mansfield1990solid}. Ñ òåõ ïîð ëèíçû òâåðäîé èììåðñèè ïîëó÷èëè øèðîêîå ïðèìåíåíèå â ìèêðîñêîïèè.  ïåðâóþ î÷åðåäü, îíè èñïîëüçóþòñÿ êàê óñòðîéñòâà äëÿ óëó÷øåíèÿ ôîêóñèðîâêè ñâåòîâûõ ïó÷êîâ äëÿ ïðèëîæåíèé â ëàçåðíîé ëèòîãðàôèè~\cite{ghislain1999near} è ôîðìèðîâàíèè èçîáðàæåíèé ñî ñâåðõâûñîêèì ïðîñòðàíñòâåííûì ðàçðåøåíèåì~\cite{wu1999realization,mason2010enhanced,wang2011wide}, à òàêæå äëÿ íàáëþäåíèÿ ëþìèíåñöåíöèè îäèíî÷íûõ èñòî÷íèêîâ â ãåîìåòðèè ïîëíîãî âíóòðåííåãî îòðàæåíèÿ (ÏÂÎ)~\cite{wrigge2007efficient,rezus2012single}. Êðîìå òîãî, áûëî ïðåäëîæåíî èñïîëüçîâàòü ëèíçû òâåðäîé èììåðñèè â ñèñòåìàõ ìèêðîñêîïèè çàäíåé ôîêàëüíîé ïëîñêîñòè~\cite{zhang2004wide}, ÷òî ïîçâîëèëî íàáëþäàòü äèàãðàììû íàïðàâëåííîñòè ïîâåðõíîñòíûõ ïëàçìîí-ïîëÿðèòîíîâ, âîçáóæäàåìûõ íà ãðàíèöå òîíêèõ ìåòàëëè÷åñêèõ ïëåíîê, îäíàêî ÷èñëîâàÿ àïåðòóðà òàêèõ ñèñòåì îñòàâàëàñü ñðàâíèòåëüíî íåáîëüøîé.

 íàøåé ðàáîòå ìû ïðåäëàãàåì èñïîëüçîâàòü àíàëîãè÷íóþ ñõåìó, ñîâìåùàþùóþ ìèêðîñêîïèþ çàäíåé ôîêàëüíîé ïëîñêîñòè è òâåðäóþ èììåðñèþ, äëÿ âèçóàëèçàöèè èçî÷àñòîòíûõ êîíòóðîâ ëîêàëèçîâàííûõ ìîä, âîçáóæäàåìûõ â ïëàíàðíûõ âîëíîâîäàõ è ìåòàïîâåðõíîñòÿõ. Áëàãîäàðÿ èñïîëüçîâàíèþ ìèíèàòþðíîé (äèàìåòð 3~ìì) ëèíçû èç ZnSe ($n_{ZnSe}=2.54$ ïðè $\lambda=700$~íì~\cite{Marple1964znse}) â ïðåäëîæåííîé ñõåìå äîñòèãàåòñÿ áîëüøàÿ ýôôåêòèâíàÿ ÷èñëîâàÿ àïåðòóðà (îêîëî 2.25 â çàâèñèìîñòè îò äèàïàçîíà ÷àñòîò, â êîòîðîì ïðîâîäÿòñÿ èçìåðåíèÿ), ïîçâîëÿþùàÿ ïîëó÷àòü èíôîðìàöèþ î ìîäàõ ñ âûñîêèì ýôôåêòèâíûì ïîêàçàòåëåì ïðåëîìëåíèÿ. Âàæíîé îñîáåííîñòüþ óñòàíîâêè ÿâëÿåòñÿ âîçìîæíîñòü õàðàêòåðèçàöèè äèñïåðñèè ïîâåðõíîñòíûõ âîëí â çàâèñèìîñòè îò íàïðàâëåíèÿ ðàñïðîñòðàíåíèÿ. Ìû äåìîíñòðèðóåì ôóíêöèîíàëüíîñòü ðàçðàáîòàííîé íàìè óñòàíîâêè íà ïðèìåðå âîëíîâîäà èç àìîðôíîãî êðåìíèÿ. Ìû âèçóàëèçèðóåì èçî÷àñòîòíûå êîíòóðû âîëíîâîäíûõ ìîä â òàêîé ñèñòåìå ñ ïîëÿðèçàöèîííûì ðàçðåøåíèåì è èç ïîëó÷åííûõ äàííûõ âîññòàíàâëèâàåì èõ çàêîíû äèñïåðñèè. 

\begin{figure}[t]\centering
	\includegraphics[width=0.9\linewidth]{FigExpSetup}
	\caption{
		\label{FigExpSetupAndSample}
		Ðèñ.~\ref{FigExpSetupAndSample}.
		(à)~Îáùàÿ ñõåìà ýêñïåðèìåíòàëüíîé óñòàíîâêè äëÿ íàáëþäåíèÿ ïîâåðõíîñòíûõ âîëí.
		(á)~Ñõåìàòè÷íîå ïðåäñòàâëåíèå îáðàçöà, ïðèâåäåííîãî â êîíòàêò ZnSe ïðèçìîé. Âåëè÷èíà âîçäóøíîãî çàçîðà $h_{air}$ ìîæåò âàðüèðîâàòüñÿ ñ ïîìîùüþ ïüåçîïîçèöèîíåðà â äèàïàçîíå îò íåñêîëüêèõ ìêì äî $\approx$20~íì. Ñèñòåìà íàñòðàèâàåòñÿ òàêèì îáðàçîì, ÷òîáû ôîêóñ îáúåêòèâà íàõîäèëñÿ â öåíòðå ïëîñêîé ÷àñòè ïîëóñôåðè÷åñêîé ëèíçû.
		(â)~Ñõåìà, èëëþñòðèðóþùàÿ ðàñïðåäåëåíèå ïîëÿðèçàöèè âîçáóæäàþùåãî èçëó÷åíèÿ â çàäíåé ôîêàëüíîé ïëîñêîñòè îáúåêòèâà äëÿ ëèíåéíîé ïîëÿðèçàöèè ïó÷êà çàñâåòêè. Ðàçíûå òî÷êè â çàäíåé ôîêàëüíîé ïëîñêîñòè ñîîòâåòñòâóþò ðàçíûì óãëàì ïàäåíèÿ ëó÷åé íà îáðàçåö.
	}
\end{figure}

\section{Ìåòîäû}

\begin{figure*}[t]\centering
	\includegraphics[width=0.8\linewidth]{FigGaps}
	\caption{
		\label{FigGaps}
		Ðèñ.~\ref{FigGaps}.
		Âåðõíèé ðÿä: èçîáðàæåíèÿ îòðàæåíèÿ îò ïëîñêîé ãðàíèöû ëèíçû òâåðäîé èììåðñèè â çàäíåé ôîêàëüíîé ïëîñêîñòè îïòè÷åñêîé ñèñòåìû äëÿ ðàçëè÷íûõ âåëè÷èí âîçäóøíîãî çàçîðà ìåæäó ëèíçîé è ïëàíàðíûì êðåìíèåâûì âîëíîâîäîì. Äëèíà âîëíû âîçáóæäåíèÿ $\lambda=700$~íì, ïîëÿðèçàöèÿ îòìå÷åíà ñòðåëêîé íà ðèñóíêå (à).
		Íèæíèé ðÿä: ÷èñëåííûé	ðàñ÷åò îòðàæåíèÿ îò ñëîèñòîé ñòðóêòóðû (ZnSe/âîçäóøíûé çàçîð/Si âîëíîâîä/êâàðöåâàÿ ïîäëîæêà), ïðåäñòàâëåííûé â îñÿõ âîëíîâûõ âåêòîðîâ âîëíû, ïàäàþùåé èç ZnSe.}
\end{figure*}

Ñõåìà ýêñïåðèìåíòàëüíîé óñòàíîâêè ïðåäñòàâëåíà íà Ðèñ.~\ref{FigExpSetupAndSample}à. Ñèñòåìà ñî÷åòàåò â ñåáå âûñîêîàïåðòóðíûé îáúåêòèâ ñ äëèííûì ðàáî÷èì îòðåçêîì (Mitutoyo, M Plan Apo HR, 100$\times$, 0.9~NA) è ïîëóñôåðè÷åñêóþ ëèíçó èç ZnSe ìàëîãî ðàçìåðà (äèàìåòð îñíîâàíèÿ 3~ìì). Çà ñ÷åò ýòîãî ñòàíîâèòñÿ âîçìîæíûì ôîêóñèðîâàòü ñâåò íà ïëîñêîé ïîâåðõíîñòè ëèíçû, ÷òî ïîçâîëÿåò ïðîâîäèòü èçìåðåíèÿ îòðàæåíèÿ â ãåîìåòðèè íàðóøåííîãî ÏÂÎ. Ýôôåêòèâíàÿ ÷èñëîâàÿ àïåðòóðà ñèñòåìû ïðè ýòîì âîçðàñòàåò îòíîñèòåëüíî ÷èñëîâîé àïåðòóðû èñïîëüçóåìîãî îáúåêòèâà ïðîïîðöèîíàëüíî ïîêàçàòåëþ ïðåëîìëåíèÿ ZnSe ïîëóñôåðû.  íàøåì ñëó÷àå ýòà âåëè÷èíà ñîñòàâèëà ïðèáëèçèòåëüíî 2.25.

Âàæíûì ïàðàìåòðîì, îïðåäåëÿþùèì ýôôåêòèâíîñòü âîçáóæäåíèÿ (ñèëó ñâÿçè) ìîä îáðàçöà, íàáëþäàåìûõ â ãåîìåòðèè íàðóøåííîãî ÏÂÎ, ÿâëÿåòñÿ âåëè÷èíà âîçäóøíîãî çàçîðà ìåæäó ëèíçîé òâåðäîé èììåðñèè è îáðàçöîì.  ýêñïåðèìåíòå îíà èçìåíÿëàñü ñ ïîìîùüþ ïüåçîïîäâèæêè, ïðèæèìàþùåé îáðàçåö ê ëèíçå (Ðèñ.~\ref{FigExpSetupAndSample}à). Áëàãîäàðÿ ýòîìó, âîçäóøíûé çàçîð êîíòðîëèðîâàëñÿ ñ âûñîêîé òî÷íîñòüþ, è â óñëîâèÿõ ìàêñèìàëüíîãî ïðèæàòèÿ îáðàçöà ê ïðèçìå äîñòèãàë âåëè÷èíû ïîðÿäêà 20~íì, ÷òî îïðåäåëÿëîñü ïóòåì ñðàâíåíèÿ ýêñïåðèìåíòàëüíûõ êàðò îòðàæåíèÿ ñ ÷èñëåííûì ìîäåëèðîâàíèåì.

Äëÿ èññëåäîâàíèÿ çàâèñèìîñòè ôîðìû èçî÷àñòîòíûõ êîíòóðîâ îò äëèíû âîëíû âîçáóæäàþùåãî ïó÷êà ìû èñïîëüçîâàëè èñòî÷íèê ëàçåðíîãî ñóïåðêîíòèíóóìà (Fianium WhiteLase SC400-6), ñîïðÿæåííûé ñ àêóñòîîïòè÷åñêèì ôèëüòðîì Fianium AOTF. Ýòà ñèñòåìà ïîçâîëÿåò ïîëó÷àòü óçêîñïåêòðàëüíûé ëàçåðíûé ïó÷îê ñ ïåðåñòðàèâàåìîé äëèíîé âîëíû â äèàïàçîíå 600-1050 íì, êîòîðûé çàâîäèëñÿ â îïòè÷åñêóþ ñõåìó ÷åðåç íåïîëÿðèçóþùèé äåëèòåëü ïó÷êà (Ðèñ.~\ref{FigExpSetupAndSample}à). ×òîáû îáåñïå÷èòü ñèììåòðè÷íîå ðàñïðåäåëåíèå èíòåíñèâíîñòè ïó÷êà â ïëîñêîñòè âõîäíîãî çðà÷êà îáúåêòèâà, ëàçåðíûé ïó÷îê ïîäâåðãàëñÿ äîïîëíèòåëüíîé ïðîñòðàíñòâåííîé ôèëüòðàöèè ÷åðåç îäíîìîäîâîå îïòè÷åñêîå âîëîêíî. Ñâåò, îòðàæåííûé îò ïëîñêîé ïîâåðõíîñòè èììåðñèîííîé ëèíçû, ðåãèñòðèðîâàëñÿ â êàíàëå ñáîðà îäíîâðåìåííî â ïëîñêîñòè ðåàëüíîãî èçîáðàæåíèÿ è â ïëîñêîñòè, ñîïðÿæåííîé ñ çàäíåé ôîêàëüíîé ïëîñêîñòüþ îáúåêòèâà. 



Íà Ðèñ.~\ref{FigGaps} ïðèâåäåíû ïðèìåðû èçîáðàæåíèé çàäíåé ôîêàëüíîé ïëîñêîñòè ñèñòåìû ïðè äëÿ ðàçëè÷íûõ âåëè÷èí âîçäóøíûõ çàçîðîâ ìåæäó ëèíçîé è ïëàíàðíûì êðåìíèåâûì âîëíîâîäîì (210~íì àìîðôíîãî êðåìíèÿ íà ïîäëîæêå èç ïëàâëåíîãî êâàðöà).
Ïðè ïðèáëèæåíèè âîëíîâîäà ê ëèíçå ðàñïðåäåëåíèå èíòåíñèâíîñòè îòðàæåíèÿ â çàäíåé ôîêàëüíîé ïëîñêîñòè ìåíÿåòñÿ. Íà áîëüøîì óäàëåíèè íàáëþäàåòñÿ îáû÷íàÿ êàðòèíà íåâîçìóùåííîãî ÏÂÎ, ïðåäñòàâëÿþùàÿ ñîáîé êîëüöî ñ âíóòðåííèì ðàäèóñîì, ñîîòâåòñòâóþùèì âîëíîâîìó âåêòîðó ñâåòà â âîçäóõå (Ðèñ.~\ref{FigGaps}a). Çàòåì, íà ðàññòîÿíèÿõ ïîðÿäêà íåñêîëüêèõ ìèêðîí â îáëàñòè âíóòðè êîëüöà ÏÂÎ èç-çà èíòåðôåðåíöèè â çàçîðå ìåæäó ëèíçîé è îáðàçöîì íà÷èíàþò íàáëþäàòüñÿ êîëüöà Íüþòîíà (Ðèñ.~\ref{FigGaps}b). Ïðè äàëüíåéøåì óìåíüøåíèè ðàññòîÿíèÿ ïðîÿâëÿåòñÿ ðåæèì íàðóøåííîãî ÏÂÎ, â êîòîðîì íàáëþäàþòñÿ âîëíîâîäíûå ìîäû ñòðóêòóðû (Ðèñ.~\ref{FigGaps}c). Íàêîíåö, íà ðàññòîÿíèÿõ ïîðÿäêà íåñêîëüêèõ äåñÿòêîâ íàíîìåòðîâ âèäíî êîëüöî, ñîîòâåòñòâóþùåå ïîëíîìó âíóòðåííåìó îòðàæåíèþ ìåæäó ZnSe è ïîäëîæêîé (Ðèñ.~\ref{FigGaps}d).

Äëÿ ìîäåëèðîâàíèÿ íàáëþäàåìûõ êàðò îòðàæåíèÿ â çàäíåé ôîêàëüíîé ïëîñêîñòè îïòè÷åñêîé ñèñòåìû ìû èñïîëüçîâàëè ìåòîä ìàòðèö ïåðåíîñà~\cite{abeles1950investigations}.
Ðàñ÷åò ïðîèçâîäèëñÿ îòäåëüíî äëÿ TE-, è TM-ïîëÿðèçîâàííûõ ìîä. Èòîãîâûå êàðòû áûëè ïîëó÷åíû èç ôîðìóëû (1). Ñðàâíåíèå ðåçóëüòàòîâ ìîäåëèðîâàíèÿ äëÿ ïëàíàðíîãî êðåìíèåâîãî âîëíîâîäà (íèæíèé ðÿä Ðèñ.~\ref{FigGaps}) ñ ýêñïåðèìåíòàëüíûìè êàðòèíàìè îòðàæåíèÿ ïîêàçûâàåò îòëè÷íîå ñîîòâåòñòâèå.

\begin{figure}[t]\centering
	\includegraphics[width=0.9\linewidth]{FigLambda}
	\caption{
		\label{FigLambda}
		Ðèñ.~\ref{FigLambda}.
		Ðàñïðåäåëåíèÿ èíòåíñèâíîñòè îòðàæåíèÿ îò ïëàíàðíîãî êðåìíèåâîãî âîëíîâîäà â çàäíåé ôîêàëüíîé ïëîñêîñòè ñèñòåìû ñ ëèíçîé òâåðäîé èììåðñèè, ïîëó÷åííûå ïðè ôèêñèðîâàííîì âîçäóøíîì çàçîðå ($h_{air}\approx140$~íì) ìåæäó ëèíçîé è îáðàçöîì íà ðàçëè÷íûõ äëèíàõ âîëí. Ïîëÿðèçàöèÿ âîçáóæäàþùåãî èçëó÷åíèÿ îòìå÷åíà ñòðåëêîé íà ïàíåëè (a).
	}
\end{figure}

\section{Âîññòàíîâëåíèå äèñïåðñèè âîëíîâîäíûõ ìîä}

Áëàãîäàðÿ òîìó, ÷òî ïðè èçìåðåíèÿõ èñïîëüçîâàëñÿ ëèíåéíî ïîëÿðèçîâàííûé ñâåò, â êàðòàõ îòðàæåíèÿ â çàâèñèìîñòè îò äëèíû âîëíû íàáëþäàëèñü è TE-, è TM-ïîëÿðèçîâàííûå ìîäû (Ðèñ.~\ref{FigLambda}).  ïðîñòåéøåì ñëó÷àå ïëàíàðíîãî âîëíîâîäà äèñïåðñèÿ âîëíîâîäíûõ ìîä íå çàâèñèò îò íàïðàâëåíèÿ èõ ðàñïðîñòðàíåíèÿ, ïîýòîìó èíôîðìàöèþ î íåé ìîæíî ëåãêî èçâëå÷ü èç èçìåðåííûõ êàðòèí îòðàæåíèÿ, ïðîèçâåäÿ ñâåðòêó ðàñïðåäåëåíèÿ ñèãíàëà â çàäíåé ôîêàëüíîé ïëîñêîñòè ñ ôóíêöèåé ðàñïðåäåëåíèÿ ïîëÿðèçàöèè ïàäàþùåé âîëíû. Ôóíêöèÿ ðàñïðåäåëåíèÿ ïîëÿðèçàöèè $\ket{P}$ äëÿ ëèíåéíî ïîëÿðèçîâàííîãî ïó÷êà èìååò ïðîñòóþ ôîðìó
\begin{equation}
\ket{P} = \sin^2(\varphi)\ket{TE} + \cos^2(\varphi)\ket{TM},
\end{equation} 
ãäå $\varphi$ - óãîë ìåæäó íàïðàâëåíèåì ïîëÿðèçàöèè ïàðàëëåëüíîãî ïó÷êà çàñâåòêè è ðàäèóñ-âåêòîðîì äàííîé òî÷êè â çàäíåé ôîêàëüíîé ïëîñêîñòè ñèñòåìû. Ýòà çàâèñèìîñòü ïðîèëëþñòðèðîâàíà íà Ðèñ.~\ref{FigExpSetupAndSample}c. Äëÿ íàèáîëåå íàãëÿäíîãî ïðåäñòàâëåíèÿ äèñïåðñèîííûõ êðèâûõ ìû èñïîëüçîâàëè ñëåäóþùåå ïðåîáðàçîâàíèå èçìåðÿåìîãî ñèãíàëà îòðàæåíèÿ $I(\omega,k,\varphi)$:
\begin{equation}
F(\omega, k) = 4\frac{\int_0^{2\pi} I(\omega,k,\varphi) \cos^2(\varphi) \text{d}\varphi}{\int_0^{2\pi} I(\omega,k,\varphi)\text{d}\varphi} - 2.
\end{equation}
 ýòîì ñëó÷àå $F(\omega,k)=0$ ñîîòâåòñòâóåò îòñóòñòâèþ ìîäóëÿöèè ñèãíàëà îòðàæåíèÿ îò óãëà, $F(\omega,k)=1$ ñîîòâåòñòâóåò âîçáóæäåíèþ TM-ïîëÿðèçîâàííîé ìîäû, $F(\omega,k)=-1$ - âîçáóæäåíèþ TE-ïîëÿðèçîâàííîé ìîäû. Ðåçóëüòàòû îáðàáîòêè ýêñïåðèìåíòàëüíûõ äàííûõ îòðàæåíèÿ â çàâèñèìîñòè îò äëèíû âîëíû ïðåäñòàâëåíû íà Ðèñ.~\ref{FigModeMaps}a. ×òîáû âîññòàíîâèòü ñîîòâåòñòâèå ìåæäó íàáëþäàåìûìè ïðîâàëàìè â êàðòàõ îòðàæåíèÿ è ìîäàìè ïëàíàðíîãî âîëíîâîäà, àíàëîãè÷íûì îáðàçîì áûëè îáðàáîòàíû è äàííûå ÷èñëåííîãî ìîäåëèðîâàíèÿ îòðàæåíèÿ. Ðåçóëüòàò îáðàáîòêè ïðèâåäåí íà Ðèñ.~\ref{FigModeMaps}b è ïîêàçûâàåò õîðîøåå ñîîòâåòñòâèå ñ èçìåðåííûìè äàííûìè.  èññëåäóåìîì ñïåêòðàëüíîì äèàïàçîíå êàê â ýêñïåðèìåíòàëüíîé, òàê è â ÷èñëåííîé êàðòàõ $F(\omega, k)$ ÿðêî ïðîÿâëÿþòñÿ äâå âîëíîâîäíûå ìîäû ñ ðàçëè÷íûìè ïîëÿðèçàöèÿìè. Àíàëèòè÷åñêîå ìîäåëèðîâàíèå òàêæå ïîçâîëÿåò îïðåäåëèòü èíäåêñû íàáëþäàåìûõ âîëíîâîäíûõ ìîä, êîòîðûå îáîçíà÷åíû íà Ðèñ.~\ref{FigModeMaps} ïóíêòèðíûìè ëèíèÿìè. Íåáîëüøîå îòëè÷èå ýêñïåðèìåíòàëüíûõ çíà÷åíèé $F(\omega, k)$ îò íóëÿ â îáëàñòè âîëíîâûõ âåêòîðîâ âûøå $1.5k_0$ ñâÿçàíî ñ íåáîëüøîé àíèçîòðîïèåé ðàñïðåäåëåíèÿ èíòåíñèâíîñòè â ïó÷êå çàñâåòêè ïîñëå ïðîñòðàíñòâåííîé ôèëüòðàöèè îïòè÷åñêèì âîëîêíîì. Êðîìå òîãî, îñöèëëÿöèè, íàáëþäàåìûå â âîññòàíîâëåííîì ñèãíàëå íà íèçêî÷àñòîòíîé TM ìîäå, ñâÿçàíû ñ õðîìàòè÷åñêèìè àáåððàöèÿìè îáúåêòèâà, êîòîðûå âîçðàñòàþò ïî ìåðå óäàëåíèÿ îò åãî ðàáî÷åãî ñïåêòðàëüíîãî äèàïàçîíà (400-800~íì).

\begin{figure}[t]\centering
	\includegraphics[width=1.0\linewidth]{FigModeMaps}
	\caption{
		\label{FigModeMaps}
		Ðèñ.~\ref{FigModeMaps}.
		Äèñïåðñèîííûå çàâèñèìîñòè TE è TM ìîä ïëàíàðíîãî êðåìíèåâîãî âîëíîâîäà, âîññòàíîâëåííûå èç êàðò îòðàæåíèÿ â ïðîñòðàíñòâå âîëíîâûõ âåêòîðîâ èç ýêñïåðèìåíòàëüíûõ äàííûõ (a) è èç ðåçóëüòàòîâ ÷èñëåííîãî ìîäåëèðîâàíèÿ ìåòîäîì ìàòðèö ïåðåíîñà (b) äëÿ âåëè÷èíû âîçäóøíîãî çàçîðà 140~íì.
	}
\end{figure}

\section{Çàêëþ÷åíèå}

Òàêèì îáðàçîì, â íàøåé ðàáîòå ìû ïðîäåìîíñòðèðîâàëè âîçìîæíîñòè óñòàíîâêè ìèêðîñêîïèè çàäíåé ôîêàëüíîé ïëîñêîñòè, â êîòîðîé â êà÷åñòâå îñíîâíîãî îïòè÷åñêîãî ýëåìåíòà âûñòóïàåò âûñîêîàïåðòóðíûé îáúåêòèâ, ñîâìåùåííûé ñ ëèíçîé òâåðäîé èììåðñèè, äëÿ âèçóàëèçàöèè èçî÷àñòîòíûõ êîíòóðîâ ëîêàëèçîâàííûõ ñîñòîÿíèé â ïëàíàðíûõ ñòðóêòóðàõ.  îòëè÷èå îò ñòàíäàðòíûõ ìåòîäîâ íàáëþäåíèÿ ïîâåðõíîñòíûõ âîëí ìåòîäîì ñïåêòðîñêîïèè îòðàæåíèÿ â ãåîìåòðèè íàðóøåííîãî ïîëíîãî âíóòðåííåãî îòðàæåíèÿ, òàêàÿ óñòàíîâêà ïîçâîëÿåò îäíîâðåìåííî ïîëó÷àòü èíôîðìàöèþ î õàðàêòåðèñòèêàõ ïîâåðõíîñòíûõ ñîñòîÿíèé èëè âîëíîâîäíûõ ìîä, ðàñïðîñòðàíÿþùèõñÿ âî âñåõ íàïðàâëåíèÿõ. Ïðè ýòîì ýôôåêòèâíûå ïîêàçàòåëè ïðåëîìëåíèÿ íàáëþäàåìûõ ëîêàëèçîâàííûõ ñîñòîÿíèé ìîãóò äîñòèãàòü çíà÷åíèé 2.25 â âèäèìîì ñïåêòðàëüíîì äèàïàçîíå ïðè èñïîëüçîâàíèè ëèíçû èç ñåëåíèäà öèíêà, ÷òî äàåò ðåøàþùåå ïðåèìóùåñòâî â ñðàâíåíèè ñ ñèñòåìàìè ñ æèäêîé èììåðñèåé. Ðàçðàáîòàííàÿ óñòàíîâêà ïðåäëàãàåòñÿ êàê óíèâåðñàëüíûé èíñòðóìåíò äëÿ ýêñïåðèìåíòàëüíîé õàðàêòåðèçàöèè ëîêàëèçîâàííûõ ñîñòîÿíèé â áóðíî ðàçâèâàþùåìñÿ íàïðàâëåíèè ïëàíàðíîé ôîòîíèêè è ìåòàïîâåðõíîñòåé.

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%% Acknowledgements
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Ýêñïåðèìåíòàëüíûå èçìåðåíèÿ áûëè âûïîëíåíû ïðè ôèíàíñîâîé ïîääåðæêå Ðîññèéñêîãî íàó÷íîãî ôîíäà (ãðàíò \# 15-12-20028). ×èñëåííûé ðàñ÷åò áûë âûïîëíåí ïðè ôèíàíñîâîé ïîääåðæêå ÐÔÔÈ (ãðàíò \#17-02-01234).
%Ðàáîòû áûëè âûïîëíåíû ïðè ôèíàíñîâîé ïîääåðæêå Ðîññèéñêîãî íàó÷íîãî ôîíäà (ãðàíò \# 15-12-20028).
%Àâòîðû âûðàæàþò áëàãîäàðíîñòü Èâàíó Ìóõèíó çà ïðåäîñòàâëåíèå îáðàçöîâ ïëàíàðíûõ êðåìíèåâûõ âîëíîâîäîâ.


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