Espectroscopia de campo-próximo em sistemas bidimensionais

❊s♣❡❝tr♦s❝♦♣✐❛ ❞❡ ❈❛♠♣♦✲Pró①✐♠♦ ❡♠ ❙✐st❡♠❛s ❇✐❞✐♠❡♥s✐♦♥❛✐s

❘♦❞♦❧❢♦ ❱✐❡✐r❛ ▼❛①✐♠✐❛♥♦

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s ✲ ❯❋▼● ■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ✲ ■❈❊① Pr♦❣r❛♠❛ ❞❡ Pós ●r❛❞✉❛çã♦ ❡♠ ❋ís✐❝❛

❊s♣❡❝tr♦s❝♦♣✐❛ ❞❡ ❈❛♠♣♦✲Pró①✐♠♦ ❡♠ ❙✐st❡♠❛s ❇✐❞✐♠❡♥s✐♦♥❛✐s

❘♦❞♦❧❢♦ ❱✐❡✐r❛ ▼❛①✐♠✐❛♥♦

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ▲✉✐③ ●✉st❛✈♦ ❞❡ ❖❧✐✈❡✐r❛ ▲♦♣❡s ❈❛♥ç❛❞♦ ❈♦✲♦r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❆❞♦ ❏ór✐♦ ❞❡ ❱❛s❝♦♥❝❡❧♦s

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ à ❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ▼■◆❆❙ ●❊❘❆■❙✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ❡♠ ❋ís✐❝❛✳

➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ Ó♣t✐❝❛✳

❆❣r❛❞❡❝✐♠❡♥t♦s

◗✉❡r♦ ❛❣r❛❞❡❝❡r✱ ❞❡ ❢♦r♠❛ ❡s♣❡❝✐❛❧✱ à ♠✐♥❤❛ q✉❡r✐❞❛ ♠ã❡✱ ❱❡r❛ ▲ú❝✐❛✱ q✉❡ s❡♠♣r❡ ♠❡ ♠♦t✐✈♦✉ ❛ ❡st✉❞❛r ❡ ❧✉t❛r ♣❛r❛ t❡r ✉♠❛ ✈✐❞❛ ♠❡❧❤♦r✳ ❊♠ t❡♠♣♦s ❢á❝❡✐s✱ ♦✉ ❞✐❢í❝❡✐s✱ ♦ s❡✉ ❛♣♦✐♦ ❢♦✐ s❡♠♣r❡ ♠❛✐♦r ❞♦ q✉❡ ♦s ❞❡s❛✜♦s✳ ▼❡s♠♦ t❡♥❞♦ s♦❢r✐❞♦ ❝♦♠ t♦❞❛s t✉r❜✉❧ê♥❝✐❛s ❞❡❝♦rr❡♥t❡s ❞♦ ❢❛❧❡❝✐♠❡♥t♦ ❞❡ ♠❡✉ ♣❛✐✱ ✈♦❝ê ♥✉♥❝❛ ❞❡s✐st✐✉ ♥❡♠ ♥♦s ❞❡✐①♦✉ ❞❡s❛♠♣❛r❛❞♦s✳ ❖❜r✐❣❛❞♦ ♠ã❡✳

❆❣r❛❞❡ç♦ t❛♠❜é♠✿

• à ♠✐♥❤❛ ❡s♣♦s❛✱ P❛✉❧❛✱ ❝♦♠♣❛♥❤❡✐r❛ ❞❡❞✐❝❛❞❛ ❡ ❛♠♦r♦s❛✳

• à ♠✐♥❤❛ q✉❡r✐❞❛ ✐r♠ã ◆❛tá❧✐❛✱ ❡ ✐r♠ã♦ ❈áss✐♦❀ ♥♦ss♦s s✉♣♦rt❡s ♠❡s♠♦ à ❞✐stâ♥❝✐❛✳ • ❛♦ ●✉st❛ ♣❡❧❛ ♦r✐❡♥t❛çã♦✱ ♣❛❝✐ê♥❝✐❛ ❡ ❜♦❛ ✈♦♥t❛❞❡✳

• ❛♦ ◆❡✇t✐♥❤♦ ❡ ▲✉q✉✐t❝❤❛ ♣❡❧❛ ❛♠✐③❛❞❡ ❡ ❞❡❞✐❝❛çã♦✱ ❡♠ ❧❛❜♦r❛tór✐♦ ❡ ❢♦r❛ ❞❡❧❡✳ • ❛♦ ❆❞♦✱ ♣❡❧❛s ✐♥str✉çõ❡s ❞❡ ❝♦♠♣♦rt❛♠❡♥t♦ ♣r♦✜ss✐♦♥❛❧ ❡ ❝♦✲♦r✐❡♥t❛çã♦✳ • ❛♦ ❏♦♥❛t❤❛♥ ♣❡❧♦ s✉♣♦rt❡ ✐♥❝♦♥❞✐❝✐♦♥❛❧ ❡ ❝♦♠♣❛♥❤❡✐r✐s♠♦✳

• ❛♦ ❆❜r❛❤❛♠✱ ❏❡♥❛✱ ▼❛r❝❡❧❧❛✱ P✐r❛❝❛✱ ❆❧✐ss♦♥✱ ♣❡❧❛ ❛♠✐③❛❞❡ ❡ ❜♦❛ ❝♦♥✈✐✈ê♥❝✐❛✳ • ❛♦s ❛♠✐❣♦s ♥❛ ❋ís✐❝❛✱ ♣♦r ❡st❛r❡♠ ❛♦ ♠❡✉ ❧❛❞♦ ♥❡st❛ ❥♦r♥❛❞❛✳

❘❡s✉♠♦

◆❡st❡ tr❛❜❛❧❤♦✱ ❞❡s❡♥✈♦❧✈❡♠♦s ✉♠ ♠♦❞❡❧♦ t❡ór✐❝♦ q✉❡ ❞❡s❝r❡✈❡ ♦ ❛✉♠❡♥t♦ ❞♦ s✐♥❛❧ ❘❛✲ ♠❛♥ ❡♠ s✐st❡♠❛s ✷❉ ♣❛r❛ ❡①♣❡r✐♠❡♥t♦s ❞❡ ❚✐♣✲❊♥❤❛♥❝❡❞ ❘❛♠❛♥ ❙♣❡❝tr♦s❝♦♣② ✭❚❊❘❙✮✱ ♦✉ ❊s♣❡❝tr♦s❝♦♣✐❛ ❘❛♠❛♥ ❞❡ ❝❛♠♣♦✲♣ró①✐♠♦✳ ❆ ❛♥á❧✐s❡ q✉❛♥t✐✜❝❛ ♦ ✈❛❧♦r ❞❛ ✐♥t❡♥s✐❞❛❞❡ ❘❛♠❛♥ ❡♠ r❡❣✐♠❡ ❞❡ ❝❛♠♣♦✲♣ró①✐♠♦ ❝♦♠♦ ❢✉♥çã♦ ❞❛ ❞✐stâ♥❝✐❛ ♣♦♥t❛✲❛♠♦str❛✱ ❞♦ t❡♥s♦r ♣♦❧❛r✐③❛❜✐❧✐❞❛❞❡ ❘❛♠❛♥✱ ❝♦♥✜❣✉r❛çã♦ ❞♦ ❧❛s❡r ✐♥❝✐❞❡♥t❡✱ ❡ ♦r✐❡♥t❛çã♦ ❞❛ ♣♦♥t❛ r❡❧❛t✐✈❛ ❛♦ ♣❧❛♥♦✳ ❆ ❛♥á❧✐s❡ ❧❡✈❛ ❡♠ ❝♦♥t❛ ❛♠❜♦s ♦s r❡❣✐♠❡s ❡s♣❛❝✐❛❧♠❡♥t❡ ❝♦❡r❡♥t❡s ❡ ✐♥❝♦❡r❡♥t❡s ❞❡ ❡s♣❛❧❤❛♠❡♥t♦✱ ❝✉❥❛s ✐♥t❡♥s✐❞❛❞❡s ✈❛r✐❛♠ ♣r♦♣♦r❝✐♦♥❛❧♠❡♥t❡ ❛♦ ✐♥✈❡rs♦ ❞❛ ✶✵❛ _{❡ ✽}❛ _{♣♦tê♥❝✐❛✱}

r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ◆ós ❛♥❛❧✐s❛♠♦s ♦s r❡s✉❧t❛❞♦s ♣❛r❛ ♦s ♠♦❞♦s ✈✐❜r❛❝✐♦♥❛✐s q✉❡ ♦❝♦rr❡♠ ❡♠ s✐st❡♠❛s ❜✐❞✐♠❡♥s✐♦♥❛✐s ✭♣♦r ❡①❡♠♣❧♦✱ ❣r❛❢❡♥♦ ❡ ♥✐tr❡t♦ ❞❡ ❜♦r♦✮ ❧❡✈❛♥❞♦ ❡♠ ❝♦♥t❛ ❛ ♣♦❧❛r✐✲ ③❛çã♦ ❞❛ ❧✉③ ✐♥❝✐❞❡♥t❡ ♥♦s ♠♦❞♦s ❧✐♥❡❛r ❡ r❛❞✐❛❧✳ ◆♦ss♦s r❡s✉❧t❛❞♦s ♠♦str❛♠ q✉❡✱ ♣❛r❛ ❝❛❞❛ ♠♦❞♦ ✈✐❜r❛❝✐♦♥❛❧✱ ❤á ✉♠❛ ❝♦♠♣❡t✐çã♦ ❡♥tr❡ ♦ ♠❡❧❤♦r â♥❣✉❧♦ ♣❛r❛ ❡①❝✐t❛r ♦ ❞✐♣♦❧♦ ❢♦r♠❛❞♦ ♥❛ ♣♦♥t❛ ❡ ♦ ❞✐♣♦❧♦ ❘❛♠❛♥ ♥♦ ♠❛t❡r✐❛❧✳ ❉❡t❡r♠✐♥❛♠♦s ♦s â♥❣✉❧♦s ót✐♠♦s ♣❛r❛ ❛ ♠❡❞✐❞❛ ❡♠ ❝❛❞❛ ✉♠ ❞❡ss❡s ❝❛s♦s✳ ❚♦❞♦s ❡ss❡s ♣❛râ♠❡tr♦s ❢♦r♠❛♠ ✉♠ ❣✉✐❛ ♣❛r❛ ❡①♣❡r✐♠❡♥t♦s ❞❡ ❚❊❘❙ ❡♠ ♠❛t❡r✐❛✐s ❜✐❞✐♠❡♥s✐♦♥❛✐s✱ ❝♦♠♦ ❣r❛❢❡♥♦ ♦✉ ❣❛s❡s ❞❡ ❡❧étr♦♥s ❜✐❞✐♠❡♥s✐♦♥❛✐s✱ ♣♦❞❡♥❞♦ s❡r ❡st❡♥❞✐❞♦ ♣❛r❛ ♠❛t❡r✐❛✐s ♦♣❛❝♦s ❝♦♠ s✉♣❡r❢í❝✐❡s ♣❧❛♥❛s✳

P❛❧❛✈r❛s✲❝❤❛✈❡s✿ ❝❛♠♣♦✲♣ró①✐♠♦✱ ❡s♣❡❝tr♦s❝♦♣✐❛ ❘❛♠❛♥✱ ♠❛t❡r✐❛✐s ❜✐❞✐♠❡♥s✐♦♥❛✐s

❆❜str❛❝t

❆ t❤❡♦r② ❞❡s❝r✐❜✐♥❣ t❤❡ ♥❡❛r✲✜❡❧❞ ❘❛♠❛♥ ❡♥❤❛♥❝❡♠❡♥t ✐♥ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ ✭✷❉✮ s②st❡♠s ✐s ♣r❡s❡♥t❡❞✳ ❚❤❡ ❛♥❛❧②s✐s q✉❛♥t✐✜❡s t❤❡ ♥❡❛r✲✜❡❧❞ ❘❛♠❛♥ ✐♥t❡♥s✐t② ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ t✐♣✲ s❛♠♣❧❡ ❞✐st❛♥❝❡✱ ❘❛♠❛♥ ♣♦❧❛r✐③❛❜✐❧✐t② t❡♥s♦r ❝♦♠♣♦♥❡♥ts✱ ✐♥❝✐❞❡♥t ❧❛s❡r ❜❡❛♠ ❝♦♥✜❣✉r❛t✐♦♥✱ ❛♥❞ t✐♣ ♦r✐❡♥t❛t✐♦♥ r❡❧❛t✐✈❡ t♦ t❤❡ s❛♠♣❧❡ ♣❧❛♥❡✳ ❖✉r r❡s✉❧ts s❤♦✇ t❤❛t t❤❡ ♥❡❛r✲✜❡❧❞ ❘❛♠❛♥ ✐♥t❡♥s✐t② ✐s ✐♥✈❡rs❡❧② ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ✶✵th _{❛♥❞ ✽}th _{♣♦✇❡r ♦❢ t❤❡ t✐♣✲s❛♠♣❧❡ ❞✐st❛♥❝❡ ✐♥} t❤❡ ✐♥❝♦❤❡r❡♥t ❛♥❞ ❝♦❤❡r❡♥t s❝❛tt❡r✐♥❣ r❡❣✐♠❡s✱ r❡s♣❡❝t✐✈❡❧②✳ ❖♣t✐♠❛❧ ❝♦♥❞✐t✐♦♥s ❢♦r t❤❡ t✐♣ ✐♥❝❧✐♥❛t✐♦♥ ❛♥❣❧❡ ❢♦r ❞✐✛❡r❡♥t ❝♦♥✜❣✉r❛t✐♦♥s ❛r❡ ❞❡t❡r♠✐♥❡❞✱ ❛♥❞ t❤❡ r❡s✉❧ts ❝❛♥ ❜❡ ✉s❡❞ ❛s ❛ ❣✉✐❞❡ ❢♦r ❚❊❘❙ ❡①♣❡r✐♠❡♥ts ✐♥ ✷❉ s②st❡♠s✱ s✉❝❤ ❛s ❣r❛♣❤❡♥❡ ❛♥❞ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❡❧❡❝tr♦♥ ❣❛s❡s✱ ❛♥❞ ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ ♦♣❛q✉❡ ❜✉❧❦ ♠❛t❡r✐❛❧s ✇✐t❤ ✢❛t s✉r❢❛❝❡s✳

❑❡②✇♦r❞s✿ ♥❡❛r✲✜❡❧❞✱ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ ♠❛t❡r✐❛❧s✱ ❘❛♠❛♥ s♣❡❝tr♦s❝♦♣②

❙✉♠ár✐♦

❘❡s✉♠♦ ■

❆❜str❛❝t ■■

✶ ■♥tr♦❞✉çã♦ ✶

✶✳✶ ❘❡♣r❡s❡♥t❛çã♦ ❡sq✉❡♠át✐❝❛ ❞❡ ✉♠ ❡①♣❡r✐♠❡♥t♦ ❚❊❘❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼

✷ ❊q✉❛çõ❡s ❞❡ ▼❛①✇❡❧❧ ❡ ❡q✉❛çã♦ ❞❡ ♦♥❞❛ ❞❡ ❍❡❧♠❤♦❧t③ ✾

✷✳✶ ❊q✉❛çã♦ ❞❡ ❍❡❧♠❤♦❧t③ ♣❛r❛ ♦ ❝❛♠♣♦ ❡❧étr✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵

✸ ❘❡♣r❡s❡♥t❛çã♦ ❞♦ ❊s♣❡❝tr♦ ❆♥❣✉❧❛r ❞♦s ❈❛♠♣♦s Ó♣t✐❝♦s ✶✸ ✸✳✶ ❈♦♥str✉çã♦ ❞♦ ❊s♣❡❝tr♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✸✳✷ Pr♦♣❛❣❛çã♦ ❞♦ ❝❛♠♣♦ ❡ ♣❡r❞❛ ❞❡ ✐♥❢♦r♠❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✸✳✷✳✶ ❖ ❧✐♠✐t❡ ❞❡ ❞✐❢r❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽

✹ ❚r❛♥s❢❡rê♥❝✐❛ ❞❡ ✐♥❢♦r♠❛çã♦ ✷✶

✺ ❆ ❛♣r♦①✐♠❛çã♦ ♣❛r❛①✐❛❧ ❡ ♦s ♠♦❞♦s ❞❡ ♣♦❧❛r✐③❛çã♦ ❞♦ ❢❡✐①❡ ✷✾ ✺✳✶ ❖❜t❡♥çã♦ ❞♦ ♠♦❞♦ r❛❞✐❛❧♠❡♥t❡ ♣♦❧❛r✐③❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✺✳✷ ❈♦♠♣♦♥❡♥t❡s ❧♦♥❣✐t✉❞✐♥❛✐s ♥❛ r❡❣✐ã♦ ❢♦❝❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺

✻ Pr✐♥❝í♣✐♦s ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ❘❛♠❛♥ ✹✷

✻✳✶ ❊s♣❡❝tr♦s❝♦♣✐❛ ❘❛♠❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✻✳✷ ❖s ♣r✐♥❝í♣✐♦s ❞♦ ❡s♣❛❧❤❛♠❡♥t♦ ❘❛♠❛♥ ❞❡ ❝❛♠♣♦✲♣ró①✐♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾

✼ ❘❡s✉❧t❛❞♦s ✺✺

✼✳✶ ❙t❛t❡♠❡♥t ♦❢ t❤❡ ♣r♦❜❧❡♠ ❛♥❞ t❤❡♦r❡t✐❝❛❧ ❜❛s✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✼✳✷ ■♥t❡♥s✐t② ♦❢ t❤❡ s❝❛tt❡r❡❞ s✐❣♥❛❧ ✐♥ t❤❡ ♥❡❛r✲✜❡❧❞ r❡❣✐♠❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾

❙❯▼➪❘■❖ ■❱

✼✳✸ Pr❛❝t✐❝❛❧ ❝♦♥s✐❞❡r❛t✐♦♥s ❢♦r ✈✐❜r❛t✐♦♥❛❧ ♠♦❞❡s ✐♥ ❣r❛♣❤❡♥❡ ❛♥❞ ❇◆ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ✼✳✹ ❚❤❡ ♠❡❛s✉r❛❜✐❧✐t② ♦❢ t❤❡ ♥❡❛r✲✜❡❧❞ s✐❣♥❛❧ ✐♥ ✷❉ s②st❡♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼

✽ ❈♦♥❝❧✉sõ❡s ✻✾

❆ ❋✉♥çã♦ ❞❡ ●r❡❡♥ ✼✶

❆✳✶ ❉❡s❝r✐çã♦ ❞♦ ♣r♦❜❧❡♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷ ❆✳✷ ❙♦❧✉çã♦ ♣❛r❛ ❢♦♥t❡s ♣✉♥t✉❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹ ❆✳✸ ❆♣❧✐❝❛çã♦ ♣❛r❛ ❢♦♥t❡s ❣❡r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻ ❆✳✹ ❊①t❡♥sã♦ ♣❛r❛ ❢✉♥çã♦ ❞❡ ●r❡❡♥ ❉✐á❞✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼ ❆✳✺ ❖❜t❡♥çã♦ ❞♦ ❝❛♠♣♦ ❡❧étr✐❝♦ ❞❡✈✐❞♦ ❛ ✉♠ ❞✐♣♦❧♦ ♣✉♥t✉❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✵ ❆✳✻ ❋♦r♠❛ ❡①♣❧í❝✐t❛ ❞❛ ❢✉♥çã♦ ❞❡ ●r❡❡♥ ❉✐á❞✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✶ ❆✳✼ ❚❡r♠♦s ❞♦♠✐♥❛♥t❡s ❞❛ ❢✉♥çã♦ ❞❡ ●r❡❡♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✺

❇ ❖ ▲❛♣❧❛❝✐❛♥♦ ❞❡ 1

R ✽✽

❈ ❚❡♦r❡♠❛ ❞❛ ❈♦♥✈♦❧✉çã♦ ✾✷

❈✳✶ ❊①❡♠♣❧♦ s✐♠♣❧❡s ❞❡ ❝♦♥✈♦❧✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✺

❈❛♣ít✉❧♦ ✶

■♥tr♦❞✉çã♦

❊s♣❡❝tr♦s❝♦♣✐❛ ❘❛♠❛♥ é ✉♠❛ ❞❛s ♣r✐♥❝✐♣❛✐s té❝♥✐❝❛s ✉t✐❧✐③❛❞❛s ♣❛r❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❡ ❣r❛❢❡♥♦ ❡ ❞❡ ♦✉tr♦s ♠❛t❡r✐❛✐s ❜✐❞✐♠❡♥s✐♦♥❛✐s✱ ❝♦♠♦ ♥✐tr❡t♦ ❞❡ ❜♦r♦ ✭BN✮ ❡ s✉❧❢❡t♦ ❞❡ ♠♦✲

❧✐❜❞ê♥✐♦ ✭MoS2✮✳ P❛r❛ ✐♥✈❡st✐❣❛r ❛❧❣✉♠❛s ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ss❡s s✐st❡♠❛s✱ ❝♦♠♦ ❞❡❢❡✐t♦s

♣✉♥t✉❛✐s ❡ ❡st✐r❛♠❡♥t♦ ❧♦❝❛❧✱ ❛ ♠✐❝r♦s❝♦♣✐❛ ❘❛♠❛♥✱ ❛ss✐♠ ❝♦♠♦ ♦✉tr❛s té❝♥✐❝❛s ó♣t✐❝❛s ❝♦♥✲ ✈❡♥❝✐♦♥❛✐s✱ ❛♣r❡s❡♥t❛ ✉♠❛ ❢♦rt❡ r❡str✐çã♦ q✉❛♥t♦ à r❡s♦❧✉çã♦ ❡s♣❛❝✐❛❧ ♣♦ssí✈❡❧ ❞❡ s❡r ♦❜t✐❞❛✳ P♦r s❡r ✐♠♣♦st❛ ♣❡❧♦ ❧✐♠✐t❡ ❞❡ ❞✐❢r❛çã♦ ❞❛ ❧✉③✱ ❛ r❡s♦❧✉çã♦ ❡s♣❛❝✐❛❧ ♦❜t✐❞❛ ❡♠ ❡①♣❡r✐✲ ♠❡♥t♦s ❞❡ ♠✐❝r♦s❝♦♣✐❛ ❝♦♥❢♦❝❛❧ é r❡❧❛t✐✈❛♠❡♥t❡ ❜❛✐①❛✿ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ 1

2 ❞♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ♦♥❞❛ ❞❛ ❧✉③ ✉s❛❞❛ ♣❛r❛ ❡①❝✐t❛çã♦✳ ❆ ❡s♣❡❝tr♦s❝♦♣✐❛ ❘❛♠❛♥ ❞❡ ❝❛♠♣♦✲♣ró①✐♠♦✱ ♦✉ ❚✐♣ ❊♥❤❛♥❝❡❞ ❘❛♠❛♥ ❙❝❛tt❡r✐♥❣✱ ❛❜r❡✈✐❛❞♦ ❝♦♠♦ ❚❊❘❙✱ ❢♦r♥❡❝❡ ✉♠❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ ✐r ❛❧é♠ ❞❡ss❡ ❧✐♠✐t❡✱ t♦r♥❛♥❞♦ ♣♦ssí✈❡❧ ❛ ♦❜t❡♥çã♦ ❞❡ r❡s♦❧✉çõ❡s ❡s♣❛❝✐❛✐s ♣ró①✐♠❛s ❞❡ 1

100 ❞♦ ❝♦♠✲ ♣r✐♠❡♥t♦ ❞❡ ♦♥❞❛ ❞❛ ❧✉③ ✐♥❝✐❞❡♥t❡✳ ❆ ♠❡❧❤♦r✐❛ ♥❛ r❡s♦❧✉çã♦ ❡s♣❛❝✐❛❧ é ♣♦ssí✈❡❧ ❞❡ s❡r ♦❜t✐❞❛ q✉❛♥❞♦ ♦ á♣✐❝❡ ❞❡ ✉♠❛ ♣♦♥t❛ ♠❡tá❧✐❝❛ é ♣♦s✐❝✐♦♥❛❞♦ ❛ ❛❧❣✉♥s ♥❛♥ô♠❡tr♦s ❞❛ s✉♣❡r❢í❝✐❡ ❞❛ ❛♠♦str❛✳ ❆ ❛çã♦ ❞❛ ♣♦♥t❛ s♦❜r❡ ❛ ❛♠♦str❛ é s❡♠❡❧❤❛♥t❡ ❛ ❞❡ ✉♠❛ ❛♥t❡♥❛ ó♣t✐❝❛✱ q✉❡ ✐rá ❣❡r❛r ❝❛♠♣♦s ❛♠♣❧✐✜❝❛❞♦s ♥❛ r❡❣✐ã♦ ❞♦ á♣✐❝❡✳ ❆❧é♠ ❞✐ss♦✱ ❞❡✈✐❞♦ à ♣r♦①✐♠✐❞❛❞❡ ❝♦♠ ❛ s✉♣❡r❢í❝✐❡ ❞❛ ❛♠♦str❛✱ ❡❧❛ s❡rá ❝❛♣❛③ ❞❡ ❛♠♣❧✐✜❝❛r s✐♥❛✐s ✈✐♥❞♦s ❞❛ ♠❡s♠❛ q✉❡ ♥ã♦ s❡r✐❛♠ ❞❡t❡❝t❛❞♦s ♣♦r ♥ã♦ s❡ ♣r♦♣❛❣❛r❡♠ ♥❛ ❛✉sê♥❝✐❛ ❞❛ ♣♦♥t❛✳ ❆ r❡s♦❧✉çã♦ ❡s♣❛❝✐❛❧ ♣❛ss❛ ❡♥tã♦ ❛ s❡r ❞❡t❡r♠✐♥❛❞❛ ♣❡❧♦ ❞✐â♠❡tr♦ ❞❛ ♣♦♥t❛✳

❈❛♣ít✉❧♦ ✶✳ ■♥tr♦❞✉çã♦ ✷

❈♦♠♦ ❡♠ ❞✐✈❡rs♦s r❛♠♦s ❞❛ ❝✐ê♥❝✐❛✱ ❛s ♣r✐♠❡✐r❛s ✐❞é✐❛s s♦❜r❡ ❛ ❡s♣❡❝tr♦s❝♦♣✐❛ ❞❡ ❝❛♠♣♦✲ ♣ró①✐♠♦ ❢♦r❛♠ t✐❞❛s ♣♦r ❞✐✈❡rs♦s ♣❡sq✉✐s❛❞♦r❡s✱ ❞❡ ♠❛♥❡✐r❛ ✐♥❞❡♣❡♥❞❡♥t❡✱ ❡ ❛té ♠❡s♠♦ ❡♠ é♣♦❝❛s ❞✐❢❡r❡♥t❡s✳ ❆ ♣r✐♠❡✐r❛ ♣✉❜❧✐❝❛çã♦ r❡❧❛❝✐♦♥❛❞❛ à ❡s♣❡❝tr♦s❝♦♣✐❛ ❞❡ ❝❛♠♣♦✲♣ró①✐♠♦ ❞❛t❛ ❞❡ ✶✾✷✽✱ ❡ tr❛t❛✲s❡ ❞❡ ✉♠ ❛rt✐❣♦ ❞❡ ❊✳ ❍✳ ❙②♥❣❡✶ _{❬✶❪✳ ❆ r❡♣r❡s❡♥t❛çã♦ ❡sq✉❡♠át✐❝❛}

❞♦ ❡①♣❡r✐♠❡♥t♦ ♣r♦♣♦st♦ ♣♦❞❡ s❡r ✈✐st♦ ♥❛ ✜❣✉r❛ ✶✳✶✳ ▼❡s♠♦ ♥ã♦ s❡♥❞♦ ❛♣❧✐❝❛❞❛ ❡♠ s✉❛ é♣♦❝❛ ✭❞❡✈✐❞♦ ❛ ó❜✈✐❛s ✐♠♣♦ss✐❜✐❧✐❞❛❞❡s ❡①♣❡r✐♠❡♥t❛✐s✮✱ ❛ ✐❞é✐❛ ♥♦ ❛rt✐❣♦ ❞❡ ❙②♥❣❡ ♣♦❞❡ s❡r ❝♦♥s✐❞❡r❛❞❛ ♦ ✐♥í❝✐♦ ❞❡st❡ r❛♠♦ ❞❛ ❢ís✐❝❛✳ ❱❛❧❡ r❡ss❛❧t❛r q✉❡ ❛ s✉❣❡stã♦ ❞♦ ❛rt✐❣♦ ❞❡ ❙②♥❣❡✱ ❡ ❛s ♣r✐♠❡✐r❛s ♠❡❞✐❞❛s q✉❡ s❡r✐❛♠ ❢❡✐t❛s ❢✉t✉r❛♠❡♥t❡✱ ♥ã♦ ✉t✐❧✐③❛✈❛♠ ✉♠❛ ♣♦♥t❛ ♠❡tá❧✐❝❛✱ ♠❛s s✐♠ s✉♣❡r❢í❝✐❡s ♦♣❛❝❛s✱ ♦✉ ❣✉✐❛s ❞❡ ♦♥❞❛✱ q✉❡ ♣♦ss✉íss❡♠ ✉♠ ♣❡q✉❡♥♦ ❢✉r♦ ❡♠ s✉❛ ❡①✲ tr❡♠✐❞❛❞❡ ❝♦♠♦ ❢♦♥t❡✴❝♦❧❡t❛ ❞❡ r❛❞✐❛çã♦✳

❋✐❣✉r❛ ✶✳✶✿ ❋✐❣✉r❛ r❡t✐r❛❞❛ ❞❡ ✉♠❛ ❝❛rt❛ ❞❡ ❊✳ ❙②♥❣❡ ❛ ❆✳ ❊✐♥st❡✐♥✳ ◆❡❧❛✱ ❙②♥❣❡ ❞❡t❛❧❤❛✈❛ s✉❛s ✐❞é✐❛s ♣❛r❛ r❡❛❧✐③❛çã♦ ❞❡ ♠✐❝r♦s❝♦♣✐❛ ❝♦♠ r❡s♦❧✉çã♦ ♠❛✐♦r q✉❡ ❛ ✐♠♣♦st❛ ♣❡❧♦ ❧✐♠✐t❡ ❞❡ ❞✐❢r❛çã♦✳ ❆♣ós ❛❧❣✉♠❛s ❝❛rt❛s ❡♠ q✉❡ ❞✐s❝✉t✐r❛♠ ❛s ❞✐✜❝✉❧❞❛❞❡s ❞♦ ❡①♣❡r✐♠❡♥t♦✱ ❊✐♥st❡✐♥ r❡❝♦♠❡♥❞♦✉ ❛ ♣✉❜❧✐❝❛çã♦ ❞♦ ❛rt✐❣♦✳ ❋✐❣✉r❛ ❛❞❛♣t❛❞❛ ❞❡ ❬✷❪✳

❈❛♣ít✉❧♦ ✶✳ ■♥tr♦❞✉çã♦ ✸

♣ró①✐♠♦ ♣❡❧♦ ❣r✉♣♦ ❞❡ ❊✳ ❆✳ ❆s❤ ❬✸❪✱ ♠❛s ✉t✐❧✐③❛♥❞♦ r❛❞✐❛çã♦ ❞❡ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ♦♥❞❛ ♥❛ ❢❛✐①❛ ❞❡ ♠✐❝r♦✲♦♥❞❛s✳ ❋♦✐ ♦❜t✐❞❛ ✉♠❛ r❡s♦❧✉çã♦ ❞❡ λ

60✱ ♠✉✐t♦ s✉♣❡r✐♦r à ✐♠♣♦st❛ ♣❡❧♦ ❧✐♠✐t❡ ❞❡ ❞✐❢r❛çã♦✳ P❛r❛ q✉❡ ♦ ♠❡s♠♦ ❡①♣❡r✐♠❡♥t♦ ❢♦ss❡ ❢❡✐t♦ ♥❛ r❡❣✐ã♦ ❞♦ ✈✐sí✈❡❧ ❡r❛ ♥❡✲ ❝❡ssár✐♦ ✉♠❛ r❡❞✉çã♦ ❞❡ ❡s❝❛❧❛ ♥♦ s✐st❡♠❛ ❞❡ q✉❛s❡ ❝✐♥❝♦ ♦r❞❡♥s ❞❡ ♠❛❣♥✐t✉❞❡✱ ❡♥✈♦❧✈❡♥❞♦ ♠✐❝r♦✲♣♦s✐❝✐♦♥❛♠❡♥t♦✱ r❡❣✉❧❛çã♦ ❞❛ ❞✐stâ♥❝✐❛ ❢♦♥t❡✲❛♠♦str❛✱ ❡t❝✳ P♦ré♠✱ t❛❧ ❛✈❛♥ç♦ ♥ã♦ ❢♦✐ ♣♦ssí✈❡❧ ❞❡ s❡r ♦❜t✐❞♦ ♥❛ é♣♦❝❛✳ ❙♦♠❡♥t❡ ❝♦♠ ♦ ❛✈❛♥ç♦ ❞❛ ár❡❛ ❞❡ ❙❝❛♥♥✐♥❣ Pr♦❜❡ ▼✐❝r♦s✲ ❝♦♣② ✭▼✐❝r♦s❝♦♣✐❛ ❞❡ ❱❛rr❡❞✉r❛ ♣♦r ❙♦♥❞❛✱ ❙P▼✮ q✉❡ t❡❝♥♦❧♦❣✐❛s ♥♦ r❛♠♦ ❞❡ ó♣t✐❝❛ ❡♠ ❡s❝❛❧❛ ♥❛♥♦♠étr✐❝❛ ✈✐r✐❛♠ ❛ s✉r❣✐r ❬✹❪✳ ❆ ♠❡❞✐❞❛ ❡♥✈♦❧✈❡♥❞♦ ❢r❡q✉ê♥❝✐❛s ó♣t✐❝❛s ❞♦ ❡s♣❡❝tr♦ ❡❧❡tr♦♠❛❣♥ét✐❝♦ ❢♦✐ ♣✉❜❧✐❝❛❞❛ ♣❡❧♦ ❣r✉♣♦ ❞❡ ♣❡sq✉✐s❛ ❞❡ ❉✳ ❲✳ P♦❤❧✱ ❡♠ ✶✾✽✹ ❬✻❪✱ ❡♥q✉❛♥t♦ ❛ ♣r✐♠❡✐r❛ ♣✉❜❧✐❝❛çã♦ ❡♠ q✉❡ ❢♦✐ ✉s❛❞❛ ✉♠❛ ♣❛rtí❝✉❧❛ ❡s♣❛❧❤❛❞♦r❛ ♥♦s ❡①♣❡r✐♠❡♥t♦s ❛♦ ✐♥✈és ❞❡ s✐st❡♠❛s ❝♦♠ ❢✉r♦s ♣❛r❛ ♣❛ss❛❣❡♠ ❞❡ r❛❞✐❛çã♦ ❢♦✐ ❢❡✐t❛ ♣♦r ❏✳ ❲❡ss❡❧✱ ❡ ❞❛t❛ ❞♦ ❛♥♦ ❞❡ ✶✾✽✺ ❬✺❪✳ ❊st❡s ❛rt✐❣♦s ❢♦r♠❛♠ ❛ ❜❛s❡ ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❡①♣❡r✐♠❡♥t❛❧ ❞♦ ❚❊❘❙✱ ❡ ❛ ❜✉s❝❛ ♣♦r ♠❛✐♦r r❡s♦❧✉çã♦ ❡s♣❛❝✐❛❧ ❝♦♥t✐♥✉♦✉ ❡♠ ár❡❛s ♣❛r❛❧❡❧❛s à ♠✐❝r♦s❝♦♣✐❛ ❞❡ ❝❛♠♣♦✲♣ró①✐♠♦✳ ❉✐✈❡rs♦s ❛♣❛r❛t♦s✱ t❡❝♥♦❧♦❣✐❛s✱ ❡ t✐♣♦s ❞❡ ❡①♣❡r✐♠❡♥t♦s ❥á ❢♦r❛♠ ✉t✐❧✐③❛❞♦s ❛♦ ❧♦♥❣♦ ❞♦s ❛♥♦s✱ ❡ ❛❧❣✉♥s ❝❛♠♣♦s ❞❡ ♣❡sq✉✐s❛ s❡ ❞❡s❡♥✈♦❧✈❡r❛♠ ❡ r❛♠✐✜❝❛r❛♠ ❛ ♣❛rt✐r ❞✐ss♦ ✭❝♦♠♦ s✐♥❣❧❡✲♠♦❧❡❝✉❧❡ s♣❡❝tr♦s❝♦♣②✮✳ ◆❡st❛ ❞✐ss❡rt❛çã♦ ♦ ♥♦ss♦ ❢♦❝♦ s❡rá ❡♠ ❚❊❘❙✱ ♠❛s ♦ ❧❡✐t♦r ♣♦❞❡rá ❡♥❝♦♥tr❛r ✐♥❢♦r♠❛çõ❡s s♦❜r❡ ❛ ❡✈♦❧✉çã♦ ❞❛ ♣❡sq✉✐s❛ ❡♠ ❝❛♠♣♦✲♣ró①✐♠♦ ❞❡ ❢♦r♠❛ ❛❜r❛♥❣❡♥t❡ ♥❛ r❡❢❡rê♥❝✐❛ ❬✼❪✱ ♦♥❞❡ é ♠❡♥❝✐♦♥❛❞♦ t❛♠❜é♠ q✉❡ ♦ ❛✈❛♥ç♦ ❡♠ ❞✐❢❡r❡♥t❡s ár❡❛s ❢♦r❛♠ tã♦ ❡ss❡♥❝✐❛✐s ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ❚❊❘❙ q✉❛♥t♦ ♦s ❞❡s❡♥✈♦❧✈✐♠❡♥t♦s ❡♠ ❙P▼✱ ♦s ♣r✐♥❝✐♣❛✐s s❡♥❞♦ r❡❧❛❝✐♦♥❛❞♦s à ♣❡sq✉✐s❛ ❞❡ ❢❡♥ô♠❡♥♦s ó♣t✐❝♦s ❡♠ ♠❡t❛✐s✱ ❡ ❙❊❘❙ ✭❙✉r✲ ❢❛❝❡ ❊♥❤❛♥❝❡❞ ❘❛♠❛♥ ❙♣❡❝tr♦s❝♦♣②✮ ❬✽❪ ✲ té❝♥✐❝❛ q✉❡ ❢♦r♥❡❝❡ ✉♠ ❛✉♠❡♥t♦ ❞♦ s✐♥❛❧ ❘❛♠❛♥ ❞❡ ❢♦r♠❛ ❛♥á❧♦❣❛ ❛♦ ❚❊❘❙✱ ♣♦ré♠ s❡♠ ✐♥❝r❡♠❡♥t♦ ❞❛ r❡s♦❧✉çã♦ ❡s♣❛❝✐❛❧✳

❈❛♣ít✉❧♦ ✶✳ ■♥tr♦❞✉çã♦ ✹

❋✐❣✉r❛ ✶✳✷✿ ❋✐❣✉r❛ ❛❞❛♣t❛❞❛ ❞❛ ♣❛t❡♥t❡ ❬✾❪✱ q✉❡ ❞❡t❛❧❤❛ ♦ ❛♣❛r❛t♦ ❡①♣❡r✐♠❡♥t❛❧ ♥❡❝❡ssár✐♦ ♣❛r❛ r❡❛❧✐③❛çã♦ ❞❛ ♠❡❞✐❞❛ ❞❡ ❚❊❘❙✳ ❖s ♥ú♠❡r♦s ❝♦rr❡s♣♦♥❞❡♠ ❛✿ á♣✐❝❡ ❞❛ ♣♦♥t❛ ✭✶✷✮✱ ♣♦♥t❛ ✭✶✹✮✱ ❢♦♥t❡ ó♣t✐❝❛ ✭✹✵✮✱ ♠♦❞✉❧❛r ❛❝úst✐❝♦✲ó♣t✐❝♦ ✭✹✷✮✱ ❧❡♥t❡s ✭✹✹ ❡ ✹✻✮✱ ❜❡❛♠✲s♣❧✐tt❡r ✭✹✽✮✱ ❡ ❢♦t♦❞✐♦❞♦ ✭✺✵✮✳

❛♠♦str❛s ❞❡ ❜r✐❧❧✐❛♥t ❝r❡s②❧ ❜❧✉❡ ✭❇❈❇✮ ❝♦♠ r❡s♦❧✉çã♦ ❡s♣❛❝✐❛❧ ❞❡ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ✺✺ ♥♠✳ ❉❡s❞❡ ❡♥tã♦✱ ❞✐✈❡rs❛s ♠❡❞✐❞❛s ❢♦r❛♠ ❢❡✐t❛s ❝♦♠ ❛ té❝♥✐❝❛✱ ❡♠ ❞✐✈❡rs♦s ♠❛t❡r✐❛✐s ❞✐❢❡r❡♥t❡s✱ ❡ ✉♠❛ ❞❛s ♠❡❧❤♦r❡s r❡s♦❧✉çõ❡s ❥á ♦❜t✐❞❛s ❢♦✐ ❡♠ ✷✵✵✾ ♥❛ r❡❢❡rê♥❝✐❛ ❬✶✷❪✱ ❝✉❥❛ ♣r✐♥❝✐♣❛❧ ✐♠❛❣❡♠ ❡stá ♥❛ ✜❣✉r❛ ✶✳✸ ❡ r❡♠❡t❡ ❛ ✉♠❛ r❡s♦❧✉çã♦ ❞❡ ❝❡r❝❛ ❞❡ ✶✺ ♥♠✳ ➱ ♣♦ssí✈❡❧ ♦❜t❡r ♠❛✐♦r r❡s♦❧✉çã♦ ❡s♣❛❝✐❛❧ ❛tr❛✈és ❞❡ ♦✉tr❛s té❝♥✐❝❛s✱ t❛✐s ❝♦♠♦ ❆t♦♠✐❝ ❋♦r❝❡ ▼✐❝r♦s❝♦♣② ✭❆❋▼✮ ♦✉ ❙❝❛♥♥✐♥❣ ❚✉♥♥❡❧✐♥❣ ▼✐❝r♦s❝♦♣② ✭❙❚▼✮✱ ♠❛s ❝♦♠♦ ♥ã♦ sã♦ té❝♥✐❝❛s ❞❡ ❡s♣❡❝tr♦s❝♦♣✐❛ ó♣t✐❝❛✱ ♥ã♦ ♣♦❞❡♠ ❢♦r♥❡❝❡r ❛s ♠❡s♠❛s ✐♥❢♦r♠❛çõ❡s ♦❜t✐❞❛s ♣♦r ❚❊❘❙✳ ❖✉tr♦ ❡①❡♠♣❧♦ ❞❡ ♠❡❞✐❞❛ ❘❛♠❛♥ ❝♦♠ ❛❧t❛ r❡s♦❧✉çã♦ ❡s♣❛❝✐❛❧ ❢♦✐ ♦❜t✐❞♦ ❡♠ ✉♠ ❡①♣❡r✐♠❡♥t♦ ❞❡ ❚✐♣ ❊♥❤❛♥❝❡❞ ❈♦❤❡r❡♥t ❛♥t✐✲❙t♦❦❡s ❘❛♠❛♥ ❙❝❛tt❡r✐♥❣ ✭❚❊✲❈❆❘❙✮✱ ♦♥❞❡ ✉♠❛ ✐♠❛❣❡♠ ❞❡ r❡s♦❧✉çã♦ ✐❣✉❛❧ ❛ ✶✺ ♥♠ ❢♦✐ ♦❜t✐❞❛ ❬✶✸❪✱ ❝♦♠♦ ♠♦str❛❞♦ ♥❛ ✜❣✉r❛ ✶✳✹✳ ❊♠ ♠❛t❡r✐❛✐s ❜✐❞✐♠❡♥s✐♦♥❛✐s✱ ♦s ú❧t✐♠♦s ❛✈❛♥ç♦s ❢♦r❛♠ ♣✉❜❧✐❝❛❞♦s r❡❝❡♥t❡♠❡♥t❡ ♣♦r ❏✳ ❙t❛❞❧❡r ❬✶✹❪✱ ♦♥❞❡ ♦❜t✐✈❡r❛♠ ✉♠❛ ✐♠❛❣❡♠ ❞❡ ❚❊❘❙ ❡♠ ❣r❛❢❡♥♦ ❝♦♠ r❡s♦❧✉çã♦ ❞❡ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ✶✷✲✶✺ ♥♠✳ ❙t❛❞❧❡r ❛✐♥❞❛ ❛♣♦♥t❛ q✉❡ ♦ s✐♥❛❧ ❘❛♠❛♥ ❡♠ r❡❣✐õ❡s ♥❛s q✉❛✐s ♦ ❣r❛❢❡♥♦ s❡ ❡♥❝♦♥tr❛ ♣❡r♣❡♥❞✐❝✉❧❛r à ♣♦♥t❛ é ❜❛✐①♦✱ ♦ q✉❡ ❡stá ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦ ♠♦❞❡❧♦ q✉❡ ❛♣r❡s❡♥t❛r❡♠♦s ♥❡st❡ tr❛❜❛❧❤♦✳

❈❛♣ít✉❧♦ ✶✳ ■♥tr♦❞✉çã♦ ✺

❋✐❣✉r❛ ✶✳✸✿ ❊♠ ✭❛✮✱ ♦♥❞❡ ❛ ❜❛rr❛ ❞❡ ❡s❝❛❧❛ ❝♦rr❡s♣♦♥❞❡ ❛ ✹ µ♠✱ t❡♠♦s ✉♠❛ ✐♠❛❣❡♠ ❞❡

♠✐❝r♦s❝♦♣✐❛ ❝♦♥❢♦❝❛❧ ❞❡ ✉♠ ♥❛♥♦t✉❜♦ ❞❡ ❝❛r❜♦♥♦ ❛tr❛✈és ❞♦ ♠❛♣❡❛♠❡♥t♦ ❞❛ ✐♥t❡♥s✐❞❛❞❡ ❞❛ ❜❛♥❞❛ ●✱ ❡♥q✉❛♥t♦ ❡♠ ✭❜✮✱ ♦♥❞❡ ❛ ❜❛rr❛ ❞❡ ❡s❝❛❧❛ ❝♦rr❡s♣♦♥❞❡ ❛ ✽✵✵ ♥♠✱ t❡♠♦s ❛ ✐♠❛❣❡♠ ❞❡ ❚❊❘❙ ❞❛ r❡❣✐ã♦ ❝♦♥t✐❞❛ ♥♦ q✉❛❞r✐❝✉❧❛❞♦ ❜r❛♥❝♦ ❞❡ ✭❛✮✳ ❆ r❡s♦❧✉çã♦ ♦❜t✐❞❛ é ❞❡ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ✶✺ ♥♠✳ ❊♠ ✭❝✮✱ ♦ ♣❡r✜❧ ❞❡ ✐♥t❡♥s✐❞❛❞❡ ❛♦ ❧♦♥❣♦ ❞❛ ❧✐♥❤❛ ♣♦♥t✐❧❤❛❞❛ ❞❡ ✭❜✮✳ ❊♠ ✭❞✮ ✉♠ ❡①❡♠♣❧♦ ❞❡ ❡s♣❡❝tr♦ ❛♠♣❧✐✜❝❛❞♦ ♣❡❧❛ té❝♥✐❝❛✱ ♦♥❞❡ ❛ ♣❛rt❡ ✈❡r♠❡❧❤❛ ❝♦rr❡s♣♦♥❞❡ ❛♦ ❡s♣❡❝tr♦ s❡♠ ❛ ♣r❡s❡♥ç❛ ❞❛ ♣♦♥t❛ ♠❡tá❧✐❝❛ ❡ ❛ ♣❛rt❡ ❡♠ ♣r❡t♦ ❝♦rr❡s♣♦♥❞❡ ❛♦ ❡s♣❡❝tr♦ ❛♠♣❧✐✜❝❛❞♦ ♣❡❧❛ ♣r❡s❡♥ç❛ ❞❡ t❛❧ ♣♦♥t❛ ♥❛ ♠❡s♠❛ r❡❣✐ã♦✳ ❋✐❣✉r❛ ❡①tr❛í❞❛ ❞❡ ❬✶✷❪✳

❈❛♣ít✉❧♦ ✶✳ ■♥tr♦❞✉çã♦ ✻

❋✐❣✉r❛ ✶✳✹✿ ❆ ✜❣✉r❛ ♠♦str❛ ✉♠❛ r❡❞❡ ❞❡ ❡str✉t✉r❛s ❞❡ ❉◆❆✱ ✐♠❛❣❡♠ ❝♦♥str✉í❞❛ ♣♦r ❚❊✲ ❈❆❘❙ ❝♦♠ r❡s♦❧✉çã♦ ❞❡ ✶✺ ♥♠✱ ❝♦♠♦ ✐♥❞✐❝❛❞♦✳ ❋✐❣✉r❛ ❡①tr❛í❞❛ ❞❡ ❬✶✸❪✳

✉♠ ❞♦s ♠ét♦❞♦s ♠❛✐s ❛♠♣❧❛♠❡♥t❡ ✉t✐❧✐③❛❞♦s ❬✼❪✳ ❊❧❡s sã♦ ✉t✐❧✐③❛❞♦s ♥❡st❡ tr❛❜❛❧❤♦✳ ❖✉tr❛ ❢❡rr❛♠❡♥t❛ q✉❡ s❡ t♦r♥♦✉ ❝♦♠✉♠ ❞❡✈✐❞♦ à s✐♠♣❧✐❝✐❞❛❞❡ ❡ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ❞❡s❝r❡✈❡r ♦s ❢❡♥ô✲ ♠❡♥♦s ❞❡ ❝❛♠♣♦✲♣ró①✐♠♦ é ❛ r❡♣r❡s❡♥t❛çã♦ ❞♦ ❡s♣❡❝tr♦ ❛♥❣✉❧❛r ❞♦s ❝❛♠♣♦s ó♣t✐❝♦s✱ ❛ss✐♠ ❝♦♠♦ ❞❡s❝r✐t❛ ♥❛ r❡❢❡rê♥❝✐❛ ❬✶✽❪✱ q✉❡ ✉t✐❧✐③❛ ❡①♣❛♥sõ❡s ❞❡ ❋♦✉r✐❡r ♣❛r❛ ❞❡s❝r✐çã♦ ❞♦ ❝❛♠♣♦ ❡❧étr✐❝♦✱ ❝♦♠♦ ✈❡r❡♠♦s ♥♦ ❈❛♣ít✉❧♦ ✸✳

❈❛♣ít✉❧♦ ✶✳ ■♥tr♦❞✉çã♦ ✼

❡①♣❡r✐♠❡♥t❛❧✳

✶✳✶ ❘❡♣r❡s❡♥t❛çã♦ ❡sq✉❡♠át✐❝❛ ❞❡ ✉♠ ❡①♣❡r✐♠❡♥t♦ ❚❊❘❙

❈❛♣ít✉❧♦ ✶✳ ■♥tr♦❞✉çã♦ ✽

Filtro Passa-Longa

❈❛♣ít✉❧♦ ✷

❊q✉❛çõ❡s ❞❡ ▼❛①✇❡❧❧ ❡ ❡q✉❛çã♦ ❞❡

♦♥❞❛ ❞❡ ❍❡❧♠❤♦❧t③

P❛r❛ ❝♦♠♣r❡❡♥❞❡r ❛ ❞✐s❝✉ssã♦ q✉❡ t❡r❡♠♦s ♥♦s ❝❛♣ít✉❧♦s s❡❣✉✐♥t❡s✱ ❛❧❣✉♠❛s ❝♦♥s✐❞❡r❛✲ çõ❡s ♣r❡❧✐♠✐♥❛r❡s sã♦ ♥❡❝❡ssár✐❛s✳ ❊❧❛s s❡rã♦ ❛♣r❡s❡♥t❛❞❛s ❛q✉✐ ❡ r❡♣r❡s❡♥t❛♠ ♦ ♣❛ss♦ ✐♥✐❝✐❛❧ ❞❡ ♥♦ss♦ ❡st✉❞♦✳ ❉❡ ❢♦r♠❛ r❡s✉♠✐❞❛✱ ♣r❡❝✐s❛♠♦s r❡❧❡♠❜r❛r ❛s ❡q✉❛çõ❡s ❞❡ ▼❛①✇❡❧❧ ❡ ❝♦♠♦ ♦❜t❡r ❛ ♣❛rt✐r ❞❡❧❛s ❛ ❡q✉❛çã♦ ❞❡ ♦♥❞❛ ❞❡ ❍❡❧♠❤♦❧t③ ♣❛r❛ ♦ ❝❛♠♣♦ ❡❧étr✐❝♦✳ ❊st❛ ú❧t✐♠❛ é ✐♠♣♦rt❛♥t❡ ♣♦✐s ❝♦♠ ❡❧❛ q✉❡ ❞❡s❝r❡✈❡♠♦s ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛ ❧✉③ ❝♦♠♦ ♦♥❞❛ ♣r♦♣❛❣❛♥t❡ ❡ ♣♦r ❡❧❛ q✉❡ ❞❡s❝♦❜r✐r❡♠♦s q✉❡ ❡①✐st❡♠ ❝♦♠♣♦♥❡♥t❡s ❡✈❛♥❡s❝❡♥t❡s ❞❛ ❧✉③ q✉❡ é ❡♠✐t✐❞❛ ♣♦r ✉♠ ♦❜❥❡t♦✳ ❆t❡♥t❡ ❛♦ ❢❛t♦ ❞❡ q✉❡ ❡st❡ ❝❛♣ít✉❧♦ t❡♠ ❝♦♠♦ ✜♠ ♦❜t❡r ❛ ❡q✉❛çã♦ ❞❡ ♦♥❞❛ ❞❡ ♦♥❞❛ ❞❡ ❍❡❧♠❤♦❧t③ ♥♦ ✈á❝✉♦✱ ❡ ❛ ❞✐s❝✉ssã♦ ❡ ❡q✉❛çõ❡s s❡rã♦ s✐♠♣❧✐✜❝❛❞❛s ❛✜♠ ❞❡ ❛t✐♥❣✐r ❡st❡ ♦❜❥❡t✐✈♦ s✉❝✐♥t❛♠❡♥t❡✱ ❛ss✐♠✱ ❛❧❣✉♠❛s ❣❡♥❡r❛❧✐❞❛❞❡s s❡rã♦ ♦♠✐t✐❞❛s ♦✉ ✐❣♥♦r❛❞❛s✳

❈♦♠❡ç❛r❡♠♦s ❝♦♠ ❛s ❡q✉❛çõ❡s ❞❡ ▼❛①✇❡❧❧✱ ❝♦♠♦ ✈✐st❛s ❡♠ ❬✷✵❪✳ ❈♦♠♦ ♣♦❞❡ s❡r ✈✐st♦ ♥❛ r❡❢❡rê♥❝✐❛✱ ❛s ❡q✉❛çõ❡s ❞❡ ▼❛①✇❡❧❧ ✜❝❛♠ ❞❡st❛ ❢♦r♠❛ ❛♦ tr❛❜❛❧❤❛r♠♦s ❝♦♠ ♦♥❞❛s ♣❧❛♥❛s ❤❛r♠ô♥✐❝❛s✱ ❝❛❞❛ ✉♠❛ ❞❛s ❝♦♠♣♦♥❡♥t❡s ❞❡ ❋♦✉r✐❡r ❞♦ ❝❛♠♣♦ ❡❧étr✐❝♦ ♦✉ ♠❛❣♥ét✐❝♦ ♦r✐❣✐♥❛❧✳ ❊st❛s ❡q✉❛çõ❡s✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ❛ ❧❡✐ ❞❛ ❢♦rç❛ ❞❡ ▲♦r❡♥t③ ❝♦♠♣õ❡♠ ♦ ❛❧✐❝❡r❝❡ ❞♦ ❡❧❡tr♦♠❛❣✲ ♥❡t✐s♠♦ ❝❧áss✐❝♦✳ ❘❡♣r❡s❡♥t❛♠ ✉♠❛ ❞❛s ♠❛✐♦r❡s ❝♦♥q✉✐st❛s ❞❛ ❝✐ê♥❝✐❛✱ ✉s❛❞❛s ♣❛r❛ ❞❡s❝r❡✈❡r s✐st❡♠❛s t❛♥t♦ ❡♠ ♠✐❝r♦ q✉❛♥t♦ ❡♠ ♠❛❝r♦ ❡s❝❛❧❛✱ ❡ q✉❡ ❞ã♦ ♦r✐❣❡♠ à t♦❞❛ ó♣t✐❝❛ ❝❧áss✐❝❛❀

❈❛♣ít✉❧♦ ✷✳ ❊q✉❛çõ❡s ❞❡ ▼❛①✇❡❧❧ ❡ ❡q✉❛çã♦ ❞❡ ♦♥❞❛ ❞❡ ❍❡❧♠❤♦❧t③ ✶✵

sã♦ ❡❧❛s✿

∇ ×E(r) = iωB(r), ✭✷✳✶✮

∇ ×H(r) = ₋iωD(r) + j(r), ✭✷✳✷✮

∇ ·D(r) = ρ(r), ✭✷✳✸✮

∇ ·B(r) = 0. ✭✷✳✹✮

■♥❝❧✉✐♠♦s ❛q✉✐ t❛♠❜é♠ ❛s r❡❧❛çõ❡s ❝♦♥st✐t✉t✐✈❛s✿

D=ǫ0ǫE, ✭✷✳✺✮

B=µ0µH, ✭✷✳✻✮

q✉❡ s❡rã♦ ♠✉✐t♦ ✉t✐❧✐③❛❞❛s✳ ❖ ❢♦r♠❛❧✐s♠♦ q✉❡ ❛s ❞❡✐①❛ ❞❡st❛ ❢♦r♠❛ é ❞❡t❛❧❤❛❞♦ ♥❛ ♣ró♣r✐❛ r❡❢❡rê♥❝✐❛ ❬✷✵❪✳ ❆q✉✐✱ E(r) = E(r, ω) r❡♣r❡s❡♥t❛ ♦ ❝❛♠♣♦ ❡❧étr✐❝♦✱ D(r) = D(r, ω) ♦ ❞❡s❧♦✲

❝❛♠❡♥t♦ ❡❧étr✐❝♦✱H(r) =H(r, ω) ♦ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦✱B(r) =B(r, ω) ❛ ✐♥❞✉çã♦ ♠❛❣♥ét✐❝❛✱

r ❞❡♥♦t❛ ❛ ♣♦s✐çã♦ ♥♦ ❡s♣❛ç♦✱ ω ❞❡♥♦t❛ ❛ ❢r❡q✉ê♥❝✐❛ ❞❡ ♦s❝✐❧❛çã♦✱ ǫ0 ❡ ǫ = ǫ(r, ω) sã♦ ❛s ♣❡r♠✐ss✐✈✐❞❛❞❡s ❡❧étr✐❝❛s ❞♦ ✈á❝✉♦ ❡ ❞♦ ♠❛t❡r✐❛❧✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ µ0 ❡ µ =µ(r, ω) sã♦ ❛s ♣❡r♠❡❛❜✐❧✐❞❛❞❡s ♠❛❣♥ét✐❝❛s ❞♦ ✈á❝✉♦ ❡ ❞♦ ♠❛t❡r✐❛❧✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡ ✜♥❛❧♠❡♥t❡✱ j(r) ❡

ρ(r) sã♦ ❛s ❞❡♥s✐❞❛❞❡s ❞❡ ❝♦rr❡♥t❡ ❡ ❝❛r❣❛✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❆♣❡♥❛s ♣❛r❛ ❝♦♠♣❛❝t❛r ❛ ♥♦✲

t❛çã♦✱ ♦ ❛r❣✉♠❡♥t♦ ω♥ã♦ é ✉t✐❧✐③❛❞♦ ♥❛s ❡q✉❛çõ❡s ✭✷✳✶✮✲✭✷✳✻✮✱ ♥❡♠ ♥♦ r❡st♦ ❞♦ t❡①t♦✳ ❆❧é♠

❞✐ss♦✱ ♥ã♦ tr❛t❛r❡♠♦s ❞❡ ♣♦ssí✈❡✐s ❞❡♣❡♥❞ê♥❝✐❛s ❞❡ j(r) ❡ ρ(r) ❝♦♠ ♣♦s✐çã♦ ♦✉ ❢r❡q✉ê♥❝✐❛✱

♣♦✐s ❡♠ ❜r❡✈❡ r❡str✐♥❣✐r❡♠♦s ♦ ❡st✉❞♦ ❛♦ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❡♠ q✉❡ ❛♠❜♦s sã♦ ③❡r♦✳ ■r❡♠♦s ♣❛rt✐r ❛❣♦r❛ ♣❛r❛ ❛ ❞❡❞✉çã♦ ❞❛ ❡q✉❛çã♦ ❞❡ ♦♥❞❛ ❞❡ ❍❡❧♠❤♦❧t③✳

✷✳✶ ❊q✉❛çã♦ ❞❡ ❍❡❧♠❤♦❧t③ ♣❛r❛ ♦ ❝❛♠♣♦ ❡❧étr✐❝♦

❈❛♣ít✉❧♦ ✷✳ ❊q✉❛çõ❡s ❞❡ ▼❛①✇❡❧❧ ❡ ❡q✉❛çã♦ ❞❡ ♦♥❞❛ ❞❡ ❍❡❧♠❤♦❧t③ ✶✶

❛♣❧✐❝❛r ✉♠ r♦t❛❝✐♦♥❛❧ ❞❡ ❛♠❜♦s ♦s ❧❛❞♦s✳ ❊♠ s❡❣✉✐❞❛ s✉❜st✐t✉í♠♦s ❛ ❡q✳ ✭✷✳✷✮ ♥♦ r❡s✉❧t❛❞♦✳

∇ ×E(r) = iω[µ0µH(r)],

∇ × ∇ ×E(r) = iωµ0µ[∇ ×H(r)],

= iωµ0µ[−iωǫ0ǫE(r) + j(r)],

= ω2µ0µǫ0ǫE(r) + i ωµ0µj(r),

∇ × ∇ × −k2

E(r) = i ωµ0µj(r), ✭✷✳✼✮

♦♥❞❡ k= [ω2µ0µǫ0ǫ]

1 2 = ω

c[µǫ]

1

2✱ s❡♥❞♦c= (µ₀ǫ₀)− 1

2 ❛ ✈❡❧♦❝✐❞❛❞❡ ❞❛ ❧✉③ ♥♦ ✈á❝✉♦✳

❯♠❛ ❡q✉❛çã♦ ♠✉✐t♦ s❡♠❡❧❤❛♥t❡ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ♣❛r❛ ♦ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦✱ ❢❛③❡♥❞♦ ♦ ♠❡s♠♦ ♣r♦❝❡❞✐♠❡♥t♦ ❞❡ s✉❜st✐t✉✐çã♦✳ ❆❣♦r❛ ❝♦♥s✐❞❡r❡♠♦s q✉❡ ♦ ♠❡✐♦ ♦♥❞❡ ♦ ❝❛♠♣♦ s❡ ❡♥❝♦♥tr❛ é ❤♦♠♦❣ê♥❡♦✱ ✐s♦tró♣✐❝♦✱ ❡ ❡❧❡tr✐❝❛♠❡♥t❡ ♥❡✉tr♦ ✭j = 0 ❡ ρ = 0✱ q✉❡ ✐♠♣❧✐❝❛ ❡♠ ∇ ·E= 0✮✶✳ ❆❧é♠ ❞✐ss♦✱ s❡ ✉s❛r♠♦s ❛ ✐❞❡♥t✐❞❛❞❡ ✭✷✳✽✮✱ q✉❡ é

∇ × ∇× = _−∇2 + _{∇ ∇·} , ✭✷✳✽✮

t❡r❡♠♦s ♣❛r❛ ❛ ❡q✳ ✭✷✳✼✮

∇ × ∇ × −k2

E(r) = 0,

−∇2E(r) +_∇[_∇E(r)]₋ k2E(r) = 0,

∇2+ k2

E(r) = 0, ✭✷✳✾✮

q✉❡ é ❛ ❡q✉❛çã♦ ❞❡ ♦♥❞❛ ❞❡ ❍❡❧♠❤♦❧t③ ♦♥❞❡ k = ω_c é ♦ ♠ó❞✉❧♦ ❞♦ ✈❡t♦r ❞❡ ♦♥❞❛ ❞❛ ❧✉③

♥♦ ✈á❝✉♦✳ ❊st❛ ❡q✉❛çã♦ ✷✳✾ s❡rá ❞❡ ❢✉♥❞❛♠❡♥t❛❧ ✐♠♣♦rtâ♥❝✐❛ ♥❛s ❞✐s❝✉ssõ❡s q✉❡ s❡❣✉❡♠✳ ❊❧❛ ❞❡s❝r❡✈❡ t❛♥t♦ ❛ ❡✈♦❧✉çã♦ ❞♦ ❝❛♠♣♦ ❡❧étr✐❝♦ q✉❛♥t♦ ❞♦ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ✲ q✉❡ t❡♠ s✉❛ ✶_{P♦r ❡①❡♠♣❧♦✱ ♦ ✈á❝✉♦ s❛t✐s❢❛③ t♦❞❛s ❡st❛s ❝❛r❛❝t❡rít✐❝❛s✳ ❈♦♠♦ ♣♦❞❡♠♦s ♦❜s❡r✈❛r ♣❡❧❛s ❡q✉❛çõ❡s ❝♦♥s✲} t✐t✉t✐✈❛s ✭✷✳✺✮ ❡ ✭✷✳✻✮✱ ♣❛r❛ ❡st❡ ♠❡✐♦ t❡r❡♠♦sµ= 1❡ǫ= 1✱ ♦ q✉❡ r❡s✉❧t❛rá ❡♠k= ω

❈❛♣ít✉❧♦ ✷✳ ❊q✉❛çõ❡s ❞❡ ▼❛①✇❡❧❧ ❡ ❡q✉❛çã♦ ❞❡ ♦♥❞❛ ❞❡ ❍❡❧♠❤♦❧t③ ✶✷

❈❛♣ít✉❧♦ ✸

❘❡♣r❡s❡♥t❛çã♦ ❞♦ ❊s♣❡❝tr♦ ❆♥❣✉❧❛r

❞♦s ❈❛♠♣♦s Ó♣t✐❝♦s

❯t✐❧✐③❛♠♦s ❛ r❡♣r❡s❡♥t❛çã♦ ❞♦ ❡s♣❡❝tr♦ ❛♥❣✉❧❛r ♥❡st❡ tr❛❜❛❧❤♦ ♣❛r❛ ❞❡s❝r❡✈❡r ♣r♦♣❛❣❛✲ çõ❡s ❞❡ ❝❛♠♣♦s ó♣t✐❝♦s✳ ◆❡st❡ ❝❛♣ít✉❧♦✱ ✈❡r❡♠♦s ❝♦♠♦ ♣r♦❝❡❞❡r ♣❛r❛ ❞❡s❝r❡✈❡r ♦ ❝❛♠♣♦ ❞❡st❛ ♠❛♥❡✐r❛✳ ❋❛r❡♠♦s ♦ ♣r♦❝❡ss♦ ❞❡t❛❧❤❛❞♦ ❛♣❡♥❛s ♣❛r❛ ♦ ❝❛♠♣♦ ❡❧étr✐❝♦✱ ✉♠❛ ✈❡③ q✉❡ ♦ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ♣♦❞❡ s❡r tr❛t❛❞♦ ❞❡ ❢♦r♠❛ t♦t❛❧♠❡♥t❡ ❛♥á❧♦❣❛✳ ❖ ♠ét♦❞♦ ♣♦❞❡ ♣❛r❡❝❡r ❝♦♠♣❧✐❝❛❞♦ ✐♥✐❝✐❛❧♠❡♥t❡✱ ♠❛s ❛ ❡ssê♥❝✐❛ é s✐♠♣❧❡s✿ ✐r❡♠♦s ❞❡❝♦♠♣♦r ♦ ❝❛♠♣♦ ❡❧étr✐❝♦ ❡♠ ❝♦♠♣♦♥❡♥t❡s✱ ❛ss✐♠ ❝♦♠♦ ♣♦❞❡♠♦s ❞❡❝♦♠♣♦r ✉♠ ✈❡t♦r ♥♦ ♣❧❛♥♦xy ❡♠ ❝♦♠♣♦♥❡♥t❡sxˆ ❡yˆ✳

◆❛ r❡♣r❡s❡♥t❛çã♦ q✉❡ ✉t✐❧✐③❛r❡♠♦s✱ ❛s ❝♦♠♣♦♥❡♥t❡s s❡rã♦ ❞✐t❛s ✧❝♦♠♣♦♥❡♥t❡s ❞❡ ❋♦✉r✐❡r✧✱ ♣♦r s❡r❡♠ ♦❜t✐❞❛s ❛tr❛✈és ❞❡ ✉♠❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ss❡ t✐♣♦✶_{✱ ❡ ❞✐r❡♠♦s q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞❡}

t♦❞❛s ❡❧❛s ❢♦r♠❛♠ ♦ ✧❊s♣❡❝tr♦ ❞❡ ❋♦✉r✐❡r ❞♦ ❝❛♠♣♦✧✳ ❈♦♠♦ ♥♦ ❡①❡♠♣❧♦ ❞♦ ✈❡t♦r ♥♦ ♣❧❛♥♦

xy✱ s❛❜❡♠♦s q✉❡ t❡♥❞♦ ♦ ✈❡t♦r s❡ ♦❜té♠ ❛s ❝♦♠♣♦♥❡♥t❡s✱ ♦✉ ❝♦♥❤❡❝❡♥❞♦✲s❡ ❛s ❝♦♠♣♦♥❡♥t❡s✱

é ♣♦ssí✈❡❧ ♦❜t❡r ♦ ✈❡t♦r✳ ❱❡r❡♠♦s q✉❡ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❝❛♠♣♦ ✲ ❝♦♠♣♦♥❡♥t❡s ❞♦ ❡s♣❡❝tr♦ ❛♥❣✉❧❛r é ❛♥á❧♦❣♦✳

✸✳✶ ❈♦♥str✉çã♦ ❞♦ ❊s♣❡❝tr♦

❈♦♠❡ç❛♠♦s ♥♦ss♦ tr❛t❛♠❡♥t♦ ❝♦♥s✐❞❡r❛♥❞♦ ✉♠❛ ❢♦♥t❡ ❡♠✐ss♦r❛ ❞❡ ❧✉③ ❝✉❥❛ ❢r❡q✉ê♥❝✐❛ é

ω✱ ❧♦❝❛❧✐③❛❞❛ ❡♠ r0 = 0✳ ◆♦ss♦ ♦❜❥❡t✐✈♦ é ❡♥t❡♥❞❡r ❛ ❡✈♦❧✉çã♦ ❞♦ ❝❛♠♣♦ ❡❧étr✐❝♦ ❡♠ ✉♠❛

♣♦s✐çã♦r= (x, y, z) ♠❡❞✐❞❛ ❛ ♣❛rt✐r ❞❛ ❢♦♥t❡✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ✐r❡♠♦s ❛♥❛❧✐③❛r ❛ ❡✈♦❧✉çã♦ ❞♦

✶_{P❛r❛ ♦s ♥ã♦ ❢❛♠✐❧✐❛r✐③❛❞♦s✱ ♦ ❛♣ê♥❞✐❝❡ ❈ ❝♦♥tê♠ ❛ ❞❡✜♥✐çã♦ ❞❡ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ✉s❛❞❛ ❛q✉✐✱} ❛ss✐♠ ❝♦♠♦ ❛ ❞❡✜♥✐çã♦ ❞❛ tr❛♥s❢♦r♠❛❞❛ ✐♥✈❡rs❛✳

❈❛♣ít✉❧♦ ✸✳ ❘❡♣r❡s❡♥t❛çã♦ ❞♦ ❊s♣❡❝tr♦ ❆♥❣✉❧❛r ❞♦s ❈❛♠♣♦s Ó♣t✐❝♦s ✶✹

❝❛♠♣♦ ó♣t✐❝♦ ❛♦ ❧♦♥❣♦ ❞❡ ✉♠❛ ❞✐r❡çã♦ ❛r❜✐trár✐❛ z✳ ❆s ❝♦♠♣♦♥❡♥t❡s ❞❡ ❋♦✉r✐❡r✱ t❛♠❜é♠

❝❤❛♠❛❞❛s ❝♦♠♣♦♥❡♥t❡s ❡s♣❡❝tr❛✐s✱ ❡♠ ✉♠ ♣❧❛♥♦ ❞❡✜♥✐❞♦ ♣♦r ✉♠ ✈❛❧♦r ❝♦♥st❛♥t❡ ❞❡ z sã♦

❞❡s❝r✐t❛s ❝♦♠♦✿

ˆ

E(kx, ky;z) =

1 4π2

Z Z +∞

−∞

E(x, y, z)e−i[kxx+kyy]_{dx dy,} ✭✸✳✶✮ ♦♥❞❡ k =

q

kx2+ky2+kz2 = ωn_c é ♦ ♠ó❞✉❧♦ ❞♦ ✈❡t♦r ❞❡ ♦♥❞❛ ❞❛ ❧✉③ ❡ n é ♦ í♥❞✐❝❡ ❞❡ r❡❢r❛çã♦ ❞♦ ♠❡✐♦✳ ❆ tr❛♥s❢♦r♠❛❞❛ ✐♥✈❡rs❛ ♥♦s ❢♦r♥❡❝❡ ♦ ❝❛♠♣♦ ❡❧étr✐❝♦ ♥♦ ♣❧❛♥♦z=cte❞❛

s❡❣✉✐♥t❡ ♠❛♥❡✐r❛

E(x, y, z) =

Z Z +∞

−∞ ˆ

E(kx, ky;z) ei[kxx+kyy]dkxdky. ✭✸✳✷✮ ❱❡❥❛♠♦s ❛❣♦r❛✱ q✉❛✐s ✐♥❢♦r♠❛çõ❡s ❛❞✐❝✐♦♥❛✐s ♣♦❞❡♠♦s ♦❜t❡r ❛ r❡s♣❡✐t♦ ❞❛s ❝♦♠♣♦♥❡♥t❡s ❞❡ ❋♦✉r✐❡r ❛♦ ❛♣❧✐❝❛r ♦ ❝❛♠♣♦ E(x, y, z)✱ ❞❡s❝r✐t♦ ❝♦♠♦ ❢✉♥çã♦ ❞❡❧❛s✱ ♥❛ ❡q✉❛çã♦ ❞❡ ♦♥❞❛ ❞❡

❍❡❧♠❤♦❧t③✳ P❛r❛ ✐ss♦✱ ✈❛♠♦s s✉❜st✐t✉✐r ❛ ❡q✳ ✭✸✳✷✮ ♥❛ ❡q✳ ✭✷✳✾✮✳ ❈♦♠♦ ❛ s♦♠❛ ❞❛ ✐♥t❡❣r❛❧ é ❢❡✐t❛ s♦❜r❡ ♦ ❡s♣❡❝tr♦ ❞❡ ❢r❡q✉ê♥❝✐❛s ❡s♣❡❝✐❛✐s kx ❡ ky ✭♥ã♦ ❛t✉❛ ♥❛s ❝♦♦r❞❡♥❛❞❛s x, y, z✮ ♣♦❞❡♠♦s ♣❛ss❛r ♦ ♦♣❡r❛❞♦r ▲❛♣❧❛❝✐❛♥♦ ♣❛r❛ ❞❡♥tr♦ ❞♦ ✐♥t❡❣r❛♥❞♦ ❢❛③❡♥❞♦✿

∇2+ k2

Z Z +∞

−∞ ˆ

E(kx, ky;z)ei[kxx+kyy]dkxdky = 0,

Z Z +∞

−∞

∂2

∂x2 +

∂2

∂y2 +

∂2

∂z2

+ k2

ˆ

E(kx, ky;z)ei[kxx+kyy]dkxdky = 0,

Z Z +∞

−∞

−k2_{x −} k2_y+ ∂

2

∂z2

+ k2

ˆ

E(kx, ky;z)ei[kxx+kyy]dkxdky = 0. ❊st❛ ✐♥t❡❣r❛❧ s❡rá s❛t✐s❢❡✐t❛ ♣❛r❛

∂2 ∂z2 + k

2 z

ˆ

E(kx, ky;z) = 0, ♦♥❞❡

kz =

q

k2 _{− k}2

x − k2y. ✭✸✳✸✮

♦ q✉❡ ♥♦s ❧❡✈❛ ❛

ˆ

E(kx, ky;z) = Eˆ(kx, ky; 0)e±ikzz. ✭✸✳✹✮ ❆♦ s✉❜st✐t✉✐r ❛ ❡q✳ ✭✸✳✹✮ ♥❛ ❡q✳ ✭✸✳✷✮ t❡r❡♠♦s

E(x, y, z) =

Z Z +∞

−∞ ˆ

❈❛♣ít✉❧♦ ✸✳ ❘❡♣r❡s❡♥t❛çã♦ ❞♦ ❊s♣❡❝tr♦ ❆♥❣✉❧❛r ❞♦s ❈❛♠♣♦s Ó♣t✐❝♦s ✶✺

❆❣♦r❛✱ ❛❧❣✉♠❛s ❞✐ss❝✉ssõ❡s ❛ r❡s♣❡✐t♦ ❞❡st❡ r❡s✉❧t❛❞♦✳ Pr✐♠❡✐r❛♠❡♥t❡✱ ❛ ❡q✳✭✸✳✹✮ s✐❣♥✐✜❝❛ q✉❡ ♦ ❝❛♠♣♦ ❡♠ ❞❛❞♦ ♣❧❛♥♦z=cte♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞♦ s❡ s✉❛s ❝♦♠♣♦♥❡♥t❡s ❞❡ ❋♦✉r✐❡r ❡♠ z= 0 ❢♦r❡♠ ❝♦♥❤❡❝✐❞❛s ❡ ✉s❛r♠♦s ♦ ♣r♦♣❛❣❛❞♦r e±ikzz✳ ◆♦t❡ q✉❡ ♦ s✐♥❛❧ ♣♦s✐t✐✈♦ ♥♦ ❡①♣♦✲ ❡♥t❡ s✐❣♥✐✜❝❛ ✉♠❛ ♦♥❞❛ q✉❡ s❡ ♣r♦♣❛❣❛ ♥❛ ❞✐r❡çã♦ ♣♦s✐t✐✈❛ ❞♦ ❡✐①♦z✱ ❡♥q✉❛♥t♦ ♦ ♥❡❣❛t✐✈♦

s✐❣♥✐✜❝❛ ✉♠❛ ♦♥❞❛ s❡ ♣r♦♣❛❣❛♥❞♦ ♥♦ s❡♥t✐❞♦ ♦♣♦st♦✱ s❡♥❞♦ ❝♦♠♣❧❡t❛♠❡♥t❡ ❛♥á❧♦❣♦ ♦ ❝♦♠✲ ♣♦rt❛♠❡♥t♦ ❞❛s ❞✉❛s✳ ■r❡♠♦s ❡♥tã♦✱ s❡♠ ♣❡r❞❛ ❞❡ ❝♦♥t❡ú❞♦✱ ❝♦♥t✐♥✉❛r ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ✉s❛♥❞♦ ❛ q✉❡ s❡ ♣r♦♣❛❣❛ ♥♦ s❡♥t✐❞♦ ♣♦s✐t✐✈♦✳ ❖✉tr♦ ❞❡t❛❧❤❡ ❛♦ q✉❛❧ ❞❡✈❡♠♦s ♥♦s ❛t❡♥t❛r é q✉❡✱ ♣❡❧❛ ❢♦r♠❛ ❝♦♠♦ ❢♦✐ ❞❡✜♥✐❞♦ ❡♠ ✭✸✳✸✮✱kz ♣♦❞❡ ♣♦ss✉✐r ✈❛❧♦r❡s ✐♠❛❣✐♥ár✐♦s ❞❡♣❡♥❞❡♥❞♦ ❞❛s ♠❛❣♥✐t✉❞❡s ❞❡kx ❡ky✳ ◗✉❛♥❞♦ ❢♦r ♦ ❝❛s♦✱ ❛ ♣❛rt❡ ✐♠❛❣✐♥ár✐❛ ❞❡✈❡ s❡r ♥❡❝❡ss❛r✐❛♠❡♥t❡ ♣♦s✐t✐✈❛ ♦✉ ♥✉❧❛ ♣❛r❛ q✉❡✱ ❛♦ ✉t✐❧✐③❛r♠♦s ✉♠❛ ❞❛s s♦❧✉çõ❡s ✭♣♦r ❡①❡♠♣❧♦✱ ❛ q✉❡ s❡ ♣r♦♣❛❣❛ ♥❛ ❞✐r❡çã♦ ❞❡z ♣♦s✐t✐✈♦✮✱ ♦ ❝❛♠♣♦ ♣❡r♠❛♥❡ç❛ ✜♥✐t♦ ♥♦ ❧✐♠✐t❡ z_→+_∞✳

❙❡q✉❡♥❝✐❛❧♠❡♥t❡✱ t❡♠♦s ❞❡ ❣❛r❛♥t✐r q✉❡ ❝❛❞❛ ✉♠❛ ❞❛s ❝♦♠♣♦♥❡♥t❡s ❞❡ ❋♦✉r✐❡r _Eˆ _❞♦ ❝❛♠♣♦ E s❡❥❛♠ t❛✐s q✉❡ s❛t✐s❢❛ç❛♠ ❛s ❡q✉❛çõ❡s ❞❡ ▼❛①✇❡❧❧✳ ❈♦♠♦ ❝♦♥str✉í❞❛s✱ ❡❧❛s ❥á s❛t✐s❢❛③❡♠ ❛s ❡qs✳ ✭✷✳✶✮ ❡ ✭✷✳✷✮✳ ❋❛❧t❛ ❣❛r❛♥t✐r q✉❡ ❛s ❡qs✳✭✷✳✸✮ ❡ ✭✷✳✹✮ t❛♠❜é♠ s❡❥❛♠ s❛t✐s❢❡✐✲ t❛s✳ ❋❛ç❛♠♦s ✐ss♦ ❧❡♠❜r❛♥❞♦ q✉❡ ❝♦♥s✐❞❡r❛♠♦s ✉♠ ♠❡✐♦ ♦♥❞❡ ♥ã♦ ❤á ❝❛r❣❛s ❧✐✈r❡s ✭ρ= 0✮✱

♣♦rt❛♥t♦ ∇ ·E(r) = 0✳ ❆♣❧✐❝❛♥❞♦ ❛ ❡q✉❛çã♦ ✭✸✳✺✮ t❡r❡♠♦s✿

∇ ·E(r) = 0,

=

Z Z +∞

−∞

∂ ∂xxˆ +

∂ ∂yyˆ+

∂ ∂zzˆ

· Eˆ(kx, ky; 0)ei[kxx+kyy+kzz]dkxdky,

=

Z Z +∞

−∞

[i kxxˆ + i kyyˆ + i kzzˆ]· Eˆ(kx, ky; 0)ei[kxx+kyy+kzz]dkxdky,

= i

Z Z +∞

−∞

k_· Eˆ(kx, ky; 0)ei[kxx+kyy+kzz]dkxdky. ✭✸✳✻✮ ❯♠❛ ❢♦r♠❛ ❞❡ ❣❛r❛♥t✐r q✉❡ ❡st❛ ❡q✉❛çã♦ s❡❥❛ s❡♠♣r❡ ✈❡r❞❛❞❡ é ❢❛③❡♥❞♦✿

ˆ

E(kx, ky; 0) ·k= 0, ✭✸✳✼✮

❈❛♣ít✉❧♦ ✸✳ ❘❡♣r❡s❡♥t❛çã♦ ❞♦ ❊s♣❡❝tr♦ ❆♥❣✉❧❛r ❞♦s ❈❛♠♣♦s Ó♣t✐❝♦s ✶✻

r✐♦ ♣❛r❛kz✱ ✐st♦ é✿

k2_≥k_x2+k2_{y ⇒} kz ∈Re ˆ

E_∝ei[kxx+kyy]_ei|kz|z

❖♥❞❛ ♣❧❛♥❛ ✭✸✳✽✮

k2 < k2_x+k_y2_⇒ kz ∈Im ˆ

E_∝ei[kxx+kyy]_e− |kz| |z|

❖♥❞❛ ❡✈❛♥❡s❝❡♥t❡ ✭✸✳✾✮

❆ ✐♥t❡r♣r❡t❛çã♦ ❛ s❡r ❛♣❧✐❝❛❞❛ ❛q✉✐ é q✉❡ ✉♠❛ ♦♥❞❛ ❡❧❡tr♦♠❛❣♥ét✐❝❛ s❡ ♣r♦♣❛❣❛♥❞♦ ♣♦r ✉♠ ♠❡✐♦ q✉❛❧q✉❡r ✭q✉❡ s❛t✐s❢❛ç❛ ❛s ❝♦♥❞✐çõ❡s ❞✐s❝✉t✐❞❛s ❛♥t❡r✐♦r♠❡♥t❡✮✱ ❡♠ ✉♠ ❞❛❞♦ ♣♦♥t♦ ❞♦ ❡s♣❛ç♦✱ s❡♠♣r❡ ♣♦❞❡ s❡r ❡♥t❡♥❞✐❞❛ ❝♦♠♦ ✉♠❛ s✉♣❡r♣♦s✐çã♦ ❞❡ ❞✐✈❡rs❛s ❝♦♠♣♦♥❡♥✲ t❡s ❡s♣❡❝tr❛✐s✳ ❚♦❞❛s ❡st❛s ❝♦♠♣♦♥❡♥t❡s t❛♠❜é♠ s❡rã♦ ♦♥❞❛s q✉❡ s❛t✐s❢❛③❡♠ ❛s ❡q✉❛çõ❡s ❞❡ ▼❛①✇❡❧❧ ♥❛q✉❡❧❡ ❞❛❞♦ ♣♦♥t♦✳ ❆❧❣✉♠❛s s❡rã♦ ♦♥❞❛s ♣r♦♣❛❣❛♥t❡s ❡♥q✉❛♥t♦ ♦✉tr❛s ❛♣r❡✲ s❡♥t❛♠ ✉♠ ❞❡❝❛✐♠❡♥t♦ ❡①♣♦♥❡♥❝✐❛❧ ❝♦♠ ♦ ❛✉♠❡♥t♦ ❞❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❛ ❢♦♥t❡ ❡ ♦ ♣♦♥t♦ ❞❡ ♦❜s❡r✈❛çã♦✳

P❛r❛ r❡❝♦♥str✉✐r ❡①❛t❛♠❡♥t❡E(x, y, z= 0)♣r❡❝✐s❛♠♦s ❞❡ t♦❞❛s ❛s ❝♦♠♣♦♥❡♥t❡sEˆ(kx, ky; 0) ♣❛r❛ r❡❛❧✐③❛r ❛ tr❛♥s❢♦r♠❛❞❛ ❡♠ ✭✸✳✺✮✳ ❙❡ ❧❡✈❛r♠♦s ❡♠ ❝♦♥t❛ q✉❡ ❝❛❞❛ ✉♠❛ ❞❛s ❝♦♠♣♦♥❡♥✲ t❡s ❡s♣❡❝tr❛✐s ❝❛rr❡❣❛ ♣❛rt❡ ❞❛ ✐♥❢♦r♠❛çã♦ ❞❡ ✉♠ s✐♥❛❧ ❡♠✐t✐❞♦ ♣❡❧❛ ❢♦♥t❡ ✭✉♠❛ ♠♦❧é❝✉❧❛✱ ♣♦r ❡①❡♠♣❧♦✮ ♣♦❞❡♠♦s ❡♥t❡♥❞❡r q✉❡✱ ❛ ♠❡❞✐❞❛ q✉❡ ♥♦s ❞✐st❛♥❝✐❛♠♦s ❞❛ ❢♦♥t❡✱ ♣❡r❞❡♠♦s ✐♥❢♦r♠❛çã♦ ♥❛ ❢♦r♠❛ ❞❡ ❝♦♠♣♦♥❡♥t❡s ❡✈❛♥❡s❝❡♥t❡s✳ ◗✉❛♥❞♦ ❛s ✐♥t❡♥s✐❞❛❞❡s ❞❛s ❝♦♠♣♦♥❡♥✲ t❡s ❡✈❛♥❡s❝❡♥t❡s s❡ ❛♣r♦①✐♠❛♠ ❞❡ ③❡r♦✱ ♥♦ss❛ ❝❤❛♥❝❡ ❞❡ ❝♦❧❡t❛r ❛ ✐♥❢♦r♠❛çã♦ ❝♦♥t✐❞❛ ♥❛s ♠❡s♠❛s s❡ t♦r♥❛ ✐rr✐sór✐❛✳

❈❛♣ít✉❧♦ ✸✳ ❘❡♣r❡s❡♥t❛çã♦ ❞♦ ❊s♣❡❝tr♦ ❆♥❣✉❧❛r ❞♦s ❈❛♠♣♦s Ó♣t✐❝♦s ✶✼

✸✳✷ Pr♦♣❛❣❛çã♦ ❞♦ ❝❛♠♣♦ ❡ ♣❡r❞❛ ❞❡ ✐♥❢♦r♠❛çã♦

❊♠ t❡♦r✐❛✱ ❡①✐st❡♠ ✐♥✜♥✐t❛s ❝♦♠♣♦♥❡♥t❡s ❡s♣❡❝tr❛✐s ❡♠ q✉❛❧q✉❡r ❝❛♠♣♦✱ s❡♥❞♦ ❛s q✉❡ s❡ ♣r♦♣❛❣❛♠ ❝♦♠♦ ♦♥❞❛s ♣❧❛♥❛s r❡str✐t❛s ❛ ✈❛❧♦r❡sk2 _≥k2_x+k2_y ❬✈❡r ❡qs✳✭✸✳✽✮ ❡ ✭✸✳✾✮❪✳ ❚♦❞❛s

❛s ♦✉tr❛s sã♦ ❡✈❛♥❡s❝❡♥t❡s✱ ❡ ❛ ✜❣✉r❛ ✸✳✶ ✐❧✉str❛✱ ❛♦ ❞❡❧✐♠✐t❛r ❝♦♠ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ q✉❛✐s ❛s ❝♦♠♣♦♥❡♥t❡s sã♦ ♣r♦♣❛❣❛♥t❡s ✭❡✈❛♥❡s❝❡♥t❡s✮ ♣♦r ♣♦ss✉ír❡♠ ✉♠ ✈❛❧♦r ❞❡qk2

x+ky2 ♠❡♥♦r ✭♠❛✐♦r✮ ❞♦ q✉❡ ♦ r❛✐♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ k = nω_c ✳ ◆♦ ❡♥t❛♥t♦✱ ♥ã♦ ❝♦♥s✐❞❡r❛♠♦s t♦❞❛s ❡❧❛s

♣❛r❛ ❞❡s❝r✐çã♦ ❞♦ s✐st❡♠❛ ❢ís✐❝♦✿ q✉❛♥t♦ ♠❛✐s ❛ s♦♠❛ ❞❡ k_x2 ❡k_y2 ✉❧tr❛♣❛ss❛k2✱ ♠❛✐s rá♣✐❞♦

é ♦ ❞❡❝❛✐♠❡♥t♦ ❝♦♠ ❛ ❞✐stâ♥❝✐❛✳ ❉❡ q✉❛❧q✉❡r ❢♦r♠❛✱ ❞❛❞❛ ✉♠❛ ❞✐stâ♥❝✐❛ ✐♥✜♥✐t❛ ❞♦ ♣❧❛♥♦

kx

ky

k₁

k₂ k₃

Circunferência (n ω/c) ² = k_x² + k_y²

❋✐❣✉r❛ ✸✳✶✿ ❘❡♣r❡s❡♥t❛çã♦ ❞♦s ♣♦ssí✈❡✐s ✈❛❧♦r❡s ❞❡ qk2

x+k2y ❡ s✉❛ ✐♥✢✉ê♥❝✐❛ ♥♦ ❝♦♠♣♦rt❛✲ ♠❡♥t♦ ❞❡ ✉♠❛ ❝♦♠♣♦♥❡♥t❡ ❡s♣❡❝tr❛❧✳ ◆❛ ✜❣✉r❛✱ ❛ q✉❛❧q✉❡r ✈❡t♦r k q✉❡ ❡st❡❥❛ ❞❡♥tr♦ ❞♦

❝ír❝✉❧♦✱ ❞❡ r❛✐♦k = nω_c ✭❡✳❣✳ k1✮✱ ❡st❛rá ❛ss♦❝✐❛❞❛ ✉♠❛ ❝♦♠♣♦♥❡♥t❡ ❡s♣❡❝tr❛❧ ❞♦ t✐♣♦ ♦♥❞❛

♣r♦♣❛❣❛♥t❡✳ P❛r❛ ✈❡t♦r❡sk❢♦r❛ ❞♦ ❝ír❝✉❧♦ ✭❝♦♠♦k2 ❡k3✮ t❡r❡♠♦s s❡♠♣r❡ ♦♥❞❛s ❡✈❛♥❡s❝❡♥✲ t❡s✱ s❡♥❞♦ q✉❡ ❛s ♠❛✐s ❞✐st❛♥t❡s ❞❛ ❜♦r❞❛ ❞♦ ❝ír❝✉❧♦ ♣♦ss✉✐rã♦ ✉♠ ❞❡❝❛✐♠❡♥t♦ ❡①♣♦♥❡♥❝✐❛❧ ♠❛✐s rá♣✐❞♦ ❞♦ q✉❡ ❛s ♣ró①✐♠❛s ✭✐✳❡✳ ❛ ❝♦♠♣♦♥❡♥t❡ r❡❧❛❝✐♦♥❛❞♦ ❛ k2 t❡♠ ❞❡❝❛✐♠❡♥t♦ ♠❛✐s s✉❛✈❡ ❞♦ q✉❡ ❛ ❞❡k3✮✳

❡♠✐ss♦r✱ t♦❞❛s ❛s ❝♦♠♣♦♥❡♥t❡s ❡✈❛♥❡s❝❡♥t❡s t❡rã♦ ❞❡❝❛í❞♦ ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❛ ③❡r♦✱ s❡❥❛ ✧rá✲ ♣✐❞❛✧♦✉ ✧❧❡♥t❛♠❡♥t❡✧✳ ❆ss✐♠✱ ❛s ✐♥❢♦r♠❛çõ❡s q✉❡ ♣♦❞❡♠♦s ♦❜t❡r ❞❡ ✉♠ ♦❜❥❡t♦ ❛ ❣r❛♥❞❡s ❞✐stâ♥❝✐❛s ❡st❛rã♦ ✜❧tr❛❞❛s ♣♦r ✉♠ ✜❧tr♦ ❞♦ t✐♣♦ ♣❛ss❛✲❜❛✐①❛ ✭❧♦✇ ♣❛ss✮ ❞❡ ❢r❡q✉ê♥❝✐❛s ❡s✲ ♣❛❝✐❛✐s✿ k_x2 +k_y2 < (nω_c )2✳ ❙❡♠♣r❡ q✉❡ ♦ t❡r♠♦ qk2

❈❛♣ít✉❧♦ ✸✳ ❘❡♣r❡s❡♥t❛çã♦ ❞♦ ❊s♣❡❝tr♦ ❆♥❣✉❧❛r ❞♦s ❈❛♠♣♦s Ó♣t✐❝♦s ✶✽

q✉❡ ❡❧❛ ❝❛rr❡❣❛ s❡rá ♣❡r❞✐❞❛✳ ❈♦♠♦ t❡♠♦s ❝♦♠♣♦♥❡♥t❡s q✉❡ ❞❡❝❛❡♠ ❡①tr❡♠❛♠❡♥t❡ rá♣✐❞♦✱ s❡♠♣r❡ ❤á ♣❡r❞❛ ❞❡ ✐♥❢♦r♠❛çã♦✱ ♥ã♦ ✐♠♣♦rt❛♥❞♦ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞❡t❡❝t♦r ❡ ♦❜❥❡t♦✱ s❡♥❞♦ ❡st❛ ♣❡r❞❛ ♠❛✐♦r ❛ ♠❡❞✐❞❛ q✉❡ ❛ ❞✐stâ♥❝✐❛ ❛✉♠❡♥t❛ ❡ s❛í♠♦s ❞♦ r❡❣✐♠❡ ❞❡ ❝❛♠♣♦✲♣ró①✐♠♦✳

✸✳✷✳✶ ❖ ❧✐♠✐t❡ ❞❡ ❞✐❢r❛çã♦

❖ ❧✐♠✐t❡ ❞❡ ❞✐❢r❛çã♦ ❡stá ❞✐r❡t❛♠❡♥t❡ r❡❧❛❝✐♦♥❛❞♦ à ♠á①✐♠❛ r❡s♦❧✉çã♦ ❡s♣❛❝✐❛❧ ♣♦ssí✈❡❧ ❞❡ s❡ ♦❜t❡r ❡♠ ✉♠ ❡①♣❡r✐♠❡♥t♦ ó♣t✐❝♦✳ ❊st❡ ❧✐♠✐t❡ é ❡st❛❜❡❧❡❝✐❞♦ ❛tr❛✈és ❞❡ ❝r✐tér✐♦s ✭♦s ♠❛✐s ❝♦♥❤❡❝✐❞♦s s❡♥❞♦ ♦ ❞❡ ❆❜❜é ❡ ♦ ❞❡ ❘❛②❧❡✐❣❤✮✳ ❈❛❞❛ ❝r✐tér✐♦ ✉t✐❧✐③❛ ✉♠ s✐st❡♠❛ ❢ís✐❝♦ ❞✐❢❡r❡♥t❡ ♣❛r❛ ♦s ❝á❧❝✉❧♦s✱ ❝♦♠♦ ❢❡♥❞❛s ♦✉ ❞✐♣♦❧♦s ❡❧étr✐❝♦s ❛❧✐♥❤❛❞♦s ❡♠ ✉♠❛ ❞❛❞❛ ❞✐r❡çã♦✱ ✐✳❡✳✱ ♦ ❝r✐tér✐♦ ❞❡ ❆❜❜é ✉t✐❧✐③❛ ❞✐♣♦❧♦s ❡❧étr✐❝♦s ❛❧✐♥❤❛❞♦s ♣❡r♣❡♥❞✐❝✉❧❛r♠❡♥t❡ à ❞✐r❡çã♦ ❞♦ ❡✐①♦ ó♣t✐❝♦ ❬✷✷❪✱ ❡♥q✉❛♥t♦ ❘❛②❧❡✐❣❤ ✉t✐❧✐③❛ ✉♠ ❡s♣❡❝trô♠❡tr♦ ❞❡ ❣r❛❞❡ ❞❡ ❞✐❢r❛çã♦ ❬✷✸❪✳ P♦r ❡st❛r❡♠ s✉❥❡✐t❛s ❛ t❛✐s ❛r❜✐tr❛r✐❡❞❛❞❡s✱ ♦s ✈❛❧♦r❡s ♦❜t✐❞♦s ♣❛r❛ ♦ ❧✐♠✐t❡ ❞❡ ❞✐❢r❛çã♦ ❡♠ ❝❛❞❛ ✉♠ ❞❡ss❡s ♠♦❞❡❧♦s sã♦✱ ❧♦❣✐❝❛♠❡♥t❡✱ ❞✐❢❡r❡♥t❡s✳ P❛r❛ ❛ r❡s♦❧✉çã♦ ❡s♣❛❝✐❛❧ ♠á①✐♠❛ ❞❡ ✉♠ s✐st❡♠❛ ❞❡ ♠✐❝r♦s❝♦♣✐❛ ó♣t✐❝❛ ❝♦♥✈❡♥❝✐♦♥❛❧✱ ♦♥❞❡ ✉s❛♠♦s ✉♠❛ ❧❡♥t❡ ♦❜❥❡t✐✈❛ ♣❛r❛ ❢♦❝❛❧✐③❛çã♦✴❞❡t❡❝çã♦ t❡♠♦s ♦ ❝r✐tér✐♦ ❞❡ ❆❜❜é

∆x= 0,61λ

N A ✭✸✳✶✵✮

♦♥❞❡λ❡N Asã♦ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ♦♥❞❛ ❞❛ ❧✉③ ✉t✐❧✐③❛❞❛ ♥♦ ❡①♣❡r✐♠❡♥t♦ ❡ ❛ ❛❜❡rt✉r❛ ♥✉♠é✲

r✐❝❛ ❞❡ t❛❧ ❧❡♥t❡✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ◆❛ ♣rát✐❝❛✱ ❡♠ s✐st❡♠❛s ó♣t✐❝♦s ❝♦♥✈❡♥❝✐♦♥❛✐s✱ ❝♦♥s❡❣✉❡✲s❡ r❡s♦❧✉çõ❡s ❡s♣❛❝✐❛✐s ❞❛ ♦r❞❡♠ ❞❡ λ

2(≈250nm ♣❛r❛ ❧✉③ ✈✐sí✈❡❧✮✱ ❡ q✉❛❧q✉❡r ❞❡✜♥✐çã♦ ♣❛r❛ ♦ ❧✐♠✐t❡ ❞❡ ❞✐❢r❛çã♦ é ✉♠❛ ❢♦r♠❛ ❞❡ ❡♥t❡♥❞❡r ♦ ♠❡s♠♦ ❢❡♥ô♠❡♥♦ ♣rát✐❝♦✿ ❛❧❣✉♠❛ ♣❛rt❡ ❞❛ ✐♥❢♦r♠❛çã♦ ❡♠✐t✐❞❛ ♣❡❧❛ ❢♦♥t❡ é ♣❡r❞✐❞❛ ♥♦ ❝❛♠♣♦ ❞✐st❛♥t❡✱ ♥♦ ❝❛♠✐♥❤♦ ❛té ♦ ❞❡t❡❝t♦r ✭♠❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ ❛ ✐♥❢♦r♠❛çã♦ ❝❛rr❡❣❛❞❛ ♣❡❧❛s ❝♦♠♣♦♥❡♥t❡s ❞❡ ❋♦✉r✐❡r ❡✈❛♥❡s❝❡♥t❡s✮✳

❈♦♠♦ ❞✐t♦ ♥❛ s❡çã♦ ❛♥t❡r✐♦r✱ ❛s ❝♦♠♣♦♥❡♥t❡s q✉❡ ❡✈❛♥❡s❝❡♠ ❞✉r❛♥t❡ ❛ ♣r♦♣❛❣❛çã♦ ❞♦ ❢❡✐①❡ sã♦ ✧✜❧tr❛❞❛s✧❡ só r❡st❛♠ ❛s r❡str✐t❛s ❛♦ ✐♥t❡r✈❛❧♦ [k_x2+k2_y]12 ≤ nω

c = 2πnλ ✱ ❝♦♠♦ ♠♦s✲ tr❛❞♦ ♥❛ ✜❣✉r❛ ✸✳✶✳ ❆♥❛❧✐s❛r❡♠♦s✱ ❛❣♦r❛✱ ❛ ♠❡❧❤♦r r❡s♦❧✉çã♦ q✉❡ ♣♦❞❡♠♦s ♦❜t❡r ❡♠ ✉♠ ❡①♣❡r✐♠❡♥t♦ ♥♦ ❝❛s♦ ❡♠ q✉❡ ✉♠ ❢❡✐①❡ s❡ ♣r♦♣❛❣❛ ❛té q✉❡ ♣❡r❝❛ t♦❞❛s s✉❛s ❝♦♠♣♦♥❡♥t❡s ❡✈❛♥❡s❝❡♥t❡s ✭♠✐❝r♦s❝♦♣✐❛ ❝♦♥✈❡♥❝✐♦♥❛❧✮✳ ❚❡r❡♠♦s✱ ♥❡ss❛ s✐t✉❛çã♦✱ ❛ ❜❛♥❞❛ ❞♦s k✬s✱ ♣❛r❛

❈❛♣ít✉❧♦ ✸✳ ❘❡♣r❡s❡♥t❛çã♦ ❞♦ ❊s♣❡❝tr♦ ❆♥❣✉❧❛r ❞♦s ❈❛♠♣♦s Ó♣t✐❝♦s ✶✾

❢❡✐①❡ t❛♠❜é♠ s❡rá r❡♣r❡s❡♥t❛❞♦ ♣♦r ✉♠❛ ❞✐str✐❜✉✐çã♦ ●❛✉ss✐❛♥❛✱ ❝✉❥❛ ❧❛r❣✉r❛ s❡rá ❞❛❞❛ ♣♦r

∆r_k = 1

∆k_k. ✭✸✳✶✶✮

■r❡♠♦s ❛ss♦❝✐❛r ❡st❛ ❧❛r❣✉r❛ ❞♦ ❝❛♠♣♦ r❡❛❧ ❝♦♠ ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ r❡s♦❧✈❡r ❛ ✐♠❛❣❡♠ ❞❡ ✉♠ ♦❜❥❡t♦✱ ♦✉ s❡❥❛✱ ❞✐r❡♠♦s q✉❡ ♦ ❧✐♠✐t❡ ❞❡ r❡s♦❧✉çã♦ ❡s♣❛❝✐❛❧ ✭❞❛❞♦ ♣❡❧♦ ❧✐♠✐t❡ ❞❡ ❞✐❢r❛çã♦✮✱ ❛ss♦❝✐❛❞♦ ❛♦ ❡①♣❡r✐♠❡♥t♦✱ é ♦❜t✐❞♦ ♣❡❧❛ ❧❛r❣✉r❛ ❛ ♠❡✐❛ ❛❧t✉r❛ ❞❛ ❞✐str✐❜✉✐çã♦ ❞♦ ❝❛♠♣♦ ❡❧étr✐❝♦ q✉❡ ❛t✐♥❣❡ ❛ ❛♠♦str❛ ♥❛q✉❡❧❡ ♣♦♥t♦❀ ✐✳❡✳✱ s❡ ❞♦✐s ♦❜❥❡t♦s ❡st✐✈❡r❡♠ s❡♣❛r❛❞♦s ♣♦r ✉♠❛ ❞✐stâ♥❝✐❛ ✐♥❢❡r✐♦r ❛ ❡st❡∆rk✱ ♥ã♦ s❡rá ♣♦ssí✈❡❧ ❞✐❢❡r❡♥❝✐❛r ❛s ♣♦s✐çõ❡s ❞❡❧❡s ♦♣t✐❝❛♠❡♥t❡✳

❆ss✐♠✱ ♦ ❧✐♠✐t❡ ❞❡ ❞✐❢r❛çã♦ s❡rá ❞❛❞♦ ♣♦r

∆r_k = λ

2πn. ✭✸✳✶✷✮

❖ ❝❛s♦ ❛❝✐♠❛ ❝♦♥t❛ ❝♦♠ ♦ ❢❛t♦ ❞❡ q✉❡ é s✉♣♦st❛♠❡♥t❡ ♣♦ssí✈❡❧ ♣r♦✈❛r t♦❞♦ ♦ ❡s♣❡❝tr♦ ❞❡ ❢r❡q✉ê♥❝✐❛s ❡s♣❡❝✐❛✐s q✉❡ ❝❤❡❣✉❡♠ ❛té ✉♠❛ ❧❡♥t❡ ♦❜❥❡t✐✈❛✱ ♦ q✉❡ ♥ã♦ é ✈❡r❞❛❞❡✱ ♣♦✐s s✉❛

N A=nsenθé ✉♠ ❢❛t♦r ❧✐♠✐t❛♥t❡✱ ♦♥❞❡θé ❞❡✜♥✐❞♦ ❝♦♠♦ ♠❡t❛❞❡ ❞♦ â♥❣✉❧♦ ❞❡ ❝♦❧❡t❛ ✭s❡r✐❛

♣❡r❢❡✐t❛ s❡ ♦ â♥❣✉❧♦ ❞❡ ❝♦❧❡t❛ ❢♦ss❡ π✱ ♦♥❞❡ t❡rí❛♠♦s nsenπ₂ = n✮✳ ◆♦ ♥♦ss♦ ❝❛s♦✱ ❧❡✈❛♥❞♦

❡♠ ❝♦♥t❛ ❛ ❝❛♣❛❝✐❞❛❞❡ ❞❛ ♦❜❥❡t✐✈❛✱ t❡r❡♠♦s✿

∆r_k= λ

2πN A , ✭✸✳✶✸✮

q✉❡✱ ❛♣❡s❛r ❞❡ s❡r ✉♠❛ ❜♦❛ ❛♣r♦①✐♠❛çã♦✱ ❢♦r♥❡❝❡ ✉♠❛ r❡s♦❧✉çã♦ ♠❛✐♦r ❞♦ q✉❡ ❛q✉❡❧❛ ❞❛❞❛ ♣❡❧♦ ❧✐♠✐t❡ ❞❡ ❆❜❜é✱ q✉❡ é ❛♣r♦①✐♠❛❞❛♠❡♥t❡ 3,8 ✈❡③❡s ♠❡♥♦r ❡ ♠❛✐s ♣ró①✐♠❛ ❞♦s ✈❛❧♦r❡s

♦❜t✐❞♦s ❡①♣❡r✐♠❡♥t❛❧♠❡♥t❡✳ ▼❛s ❡✈✐❞❡♥❝✐❛♠♦s ❝♦♠ ❡st❡ r❛❝✐♦❝í♥✐♦ q✉❡ ♦ ❧✐♠✐t❡ ❞❡ ❞✐❢r❛çã♦ ❡①✐st❡ ❞❡✈✐❞♦ à ♣❡r❞❛ ❞❛s ❝♦♠♣♦♥❡♥t❡s ❡✈❛♥❡s❝❡♥t❡s q✉❡ ♥ã♦ ❝♦♥s❡❣✉❡♠ ❛t✐♥❣✐r ♦ ❞❡t❡❝t♦r✳ ◆♦t❡ q✉❡ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❡q✉❛çã♦ ✭✸✳✶✶✮✱ s❡ t✐✈❡r♠♦s ✉♠❛ ❧❛r❣✉r❛ ∆k_k ❛r❜✐tr❛r✐❛♠❡♥t❡

❣r❛♥❞❡✱ ❝♦♥s❡❣✉✐rí❛♠♦s ❞❡✜♥✐r ♦♣t✐❝❛♠❡♥t❡ ❛ ♣♦s✐çã♦ ❞❡ q✉❛❧q✉❡r ♦❜❥❡t♦ ❝♦♠ ❡①❛t✐❞ã♦✳ ◆❛ ♣rát✐❝❛✱ ❝♦♠♦ ❞✐t♦✱ ❞✉r❛♥t❡ ❛ ♣r♦♣❛❣❛çã♦✱ ❛❧❣✉♠❛s ❝♦♠♣♦♥❡♥t❡s s❡♠♣r❡ s❡rã♦ ♣❡r❞✐❞❛s✱ s❡✉s

k✬s ❞❡✐①❛♥❞♦ ❞❡ s❡r ❛❝❡ssí✈❡✐s ♥♦ r❡❣✐♠❡ ❞❡ ❝❛♠♣♦✲❞✐st❛♥t❡✳ P❛r❛ ❝♦♥t♦r♥❛r ❡st❡ ❡♠♣❡❝✐❧❤♦

t❡♠♦s ❛ ❡s♣❡❝tr♦s❝♦♣✐❛ ❞❡ ❝❛♠♣♦✲♣ró①✐♠♦✱ té❝♥✐❝❛ ❞✐s❝✉t✐❞❛ ❛❞✐❛♥t❡✱ q✉❡ ❝♦♥s✐st❡ ❡♠ ✉t✐❧✐③❛r ♦ á♣✐❝❡ ❞❡ ✉♠❛ ♣♦♥t❛ ♠❡tá❧✐❝❛✱ q✉❡ s❡r✈✐rá ❝♦♠♦ ❛♥t❡♥❛ ó♣t✐❝❛✱ ❜❡♠ ♣ró①✐♠❛ ❞♦ ♦❜❥❡t♦ ❞❡ ❡st✉❞♦✱ ❛✜♠ ❞❡ s✉♣❡r❛r ♦ ❧✐♠✐t❡ ❞❡ ❞✐❢r❛çã♦✱ ❝♦❧❡t❛♥❞♦ ❡ss❛s ❝♦♠♣♦♥❡♥t❡s ❡✈❛♥❡s❝❡♥t❡s ❛♥t❡s q✉❡ s❡ ♣❡r❝❛♠✳

❈❛♣ít✉❧♦ ✸✳ ❘❡♣r❡s❡♥t❛çã♦ ❞♦ ❊s♣❡❝tr♦ ❆♥❣✉❧❛r ❞♦s ❈❛♠♣♦s Ó♣t✐❝♦s ✷✵

❈❛♣ít✉❧♦ ✹

❚r❛♥s❢❡rê♥❝✐❛ ❞❡ ✐♥❢♦r♠❛çã♦

❞♦ ❝❛♠♣♦✲♣ró①✐♠♦ ♣❛r❛ ♦ ❝❛♠♣♦✲❞✐st❛♥t❡

❆❝❛❜❛♠♦s ❞❡ ✈❡r ❝♦♠♦ ❛ ✜❧tr❛❣❡♠ ❞❛s ❝♦♠♣♦♥❡♥t❡s ❡✈❛♥❡s❝❡♥t❡s ♥❛ tr❛♥s✐çã♦ ❞❡ ❝❛♠♣♦✲ ♣ró①✐♠♦ ♣❛r❛ ❝❛♠♣♦✲❞✐st❛♥t❡ ❞❡✜♥❡ ♦ ❧✐♠✐t❡ ❞❡ ❞✐❢r❛çã♦ ❡ ❝♦♠♦ ❡ss❡ ❧✐♠✐t❡ ❞❡✜♥❡ ❛ ♠á①✐♠❛ r❡s♦❧✉çã♦ ❡s♣❛❝✐❛❧ ♣♦ssí✈❡❧ ❞❡ s❡ ♦❜t❡r ❡♠ ✉♠ ❡①♣❡r✐♠❡♥t♦ ❞❡ ♠✐❝r♦s❝♦♣✐❛ ó♣t✐❝❛ ❝♦♥✈❡♥❝✐✲ ♦♥❛❧✳ ◆❡st❡ ❝❛♣ít✉❧♦✱ ✈❡r❡♠♦s ♦ q✉❡ ♣♦❞❡ s❡r ❢❡✐t♦ ♣❛r❛ q✉❡ ❤❛❥❛ ✉♠❛ ♠❡❧❤♦r❛ ❞❛ r❡s♦❧✉çã♦ ♠á①✐♠❛ ✉s✉❛❧✱ ✉❧tr❛♣❛ss❛♥❞♦ ♦ ✈❛❧♦r ✐♠♣♦st♦ ♣❡❧♦ ❧✐♠✐t❡ ❞❡ ❞✐❢r❛çã♦ ❞❛ ❧✉③✳

❖ s✐st❡♠❛ ♠♦str❛❞♦ ♥❛ ✜❣✉r❛ ✭✹✳✶✮ é ❝♦♠♣♦st♦ ♣♦r ✉♠ ♣❧❛♥♦ ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ ❡✐①♦ z

s✐t✉❛❞♦ ❡♠ z = ₋z0✱ ❝♦♠ z0 ≪ λ✱ ♦♥❞❡ s❡ s✐t✉❛ ❛ ❢♦♥t❡ ❧✉♠✐♥♦s❛ ✭❡✳❣✳ ♦ á♣✐❝❡ ❞❡ ✉♠❛ ♣♦♥t❛ ♠❡tá❧✐❝❛ ❡♠ ✉♠ ❡①♣❡r✐♠❡♥t♦ ❞❡ ❚❊❘❙✮✱ ♥♦ q✉❛❧ t❡♠♦s ❛ ❡♠✐ssã♦ ❞❡ ❝♦♠♣♦♥❡♥t❡s t❛♥t♦ ❡✈❛♥❡s❝❡♥t❡s q✉❛♥t♦ ♣r♦♣❛❣❛♥t❡s✳ ❆ ❛♠♦str❛ ❜✐❞✐♠❡♥s✐♦♥❛❧ s❡ s✐t✉❛ ♥♦ ♣❧❛♥♦ z = 0

❡ s✉❛ ❞❡s❝r✐çã♦ ❡stá ❝♦♥t✐❞❛ ❡♠ ✉♠❛ ❢✉♥çã♦ tr❛♥s♠✐ssã♦T(x, y)✳ ❖ ♣❧❛♥♦ ❞❡ ❞❡t❡❝çã♦ ❡stá

s✐t✉❛❞♦ ❡♠ z∞ ≫λ✱ ♦♥❞❡ ❛ ❧✉③ s❡rá ❝♦❧❡t❛❞❛✳ ◆❡st❛ ❝♦♥✜❣✉r❛çã♦ ❛ ❢♦♥t❡ ❡stá tã♦ ♣ró①✐♠❛ ❞❛ ❛♠♦str❛ q✉❡ ❛❧❣✉♠❛s ❞❛s ❝♦♠♣♦♥❡♥t❡s ❡✈❛♥❡s❝❡♥t❡s ❞♦ ❝❛♠♣♦ ❝♦♥s❡❣✉❡♠ ✐♥t❡r❛❣✐r ❝♦♠ ♦ ♠❛t❡r✐❛❧✱ ❡♥q✉❛♥t♦ ♦ ❞❡t❡❝t♦r ❡stá ❛ ✉♠❛ ❞✐stâ♥❝✐❛ ✐♥✜♥✐t❛ ❞♦s ❞♦✐s✱ ♦✉ s❡❥❛✱ ♥❡♥❤✉♠❛ ❝♦♠♣♦♥❡♥t❡ ❡✈❛♥❡s❝❡♥t❡ ♣❛rt✐♥❞♦ ❞❛ ❛♠♦str❛ ❛❧❝❛♥ç❛rá ♦ ♣❧❛♥♦ ❞❡ ❞❡t❡❝çã♦✳

❉❛r❡♠♦s ✐♥í❝✐♦ à ❞❡s❝r✐çã♦ ❛♣r♦①✐♠❛❞❛✱ ❡s❝r❡✈❡♥❞♦ ♦ ❝❛♠♣♦ ❡♠✐t✐❞♦ ♣❡❧❛ ❢♦♥t❡ ♥❛ ♣♦s✐çã♦

z=₋z0 ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ r❡♣r❡s❡♥t❛çã♦ ❞♦ ❡s♣❡❝tr♦ ❛♥❣✉❧❛r✿

Ef onte(x, y;−z0) =

Z Z +∞

−∞ ˆ

❈❛♣ít✉❧♦ ✹✳ ❚r❛♥s❢❡rê♥❝✐❛ ❞❡ ✐♥❢♦r♠❛çã♦ ✷✷

T (x,y) z = 0

z0 <<

∞

z >> x

y

z

❋✐❣✉r❛ ✹✳✶✿ ❙✐st❡♠❛ ✉s❛❞♦ ♣❛r❛ ❞❡s❝r✐çã♦ ❞❛ tr❛♥s❢❡rê♥❝✐❛ ❞❡ ✐♥❢♦r♠❛çã♦ ❞♦ ❝❛♠♣♦✲♣ró①✐♠♦ ♣❛r❛ ♦ ❝❛♠♣♦✲❞✐st❛♥t❡✳ ❊♠z= 0 t❡♠♦s ♦ ♣❧❛♥♦ ❞♦ ♠❛t❡r✐❛❧✱ ❡♠z=₋z0 t❡♠♦s ♦ ♣❧❛♥♦ ❞❛ ❢♦♥t❡ ❧✉♠✐♥♦s❛ ❡ ❛ ✉♠❛ ❞✐stâ♥❝✐❛ ✐♥✜♥✐t❛ ❞♦s ❞♦✐s t❡♠♦s ♦ ♣❧❛♥♦ ❞❡t❡❝t♦r✳

P♦❞❡♠♦s ♦❜t❡r ♦ ❝❛♠♣♦ q✉❡ ❛t✐♥❣❡ ❛ ❛♠♦str❛ ❛♣ós t❡r ♣r♦♣❛❣❛❞♦ ♣❡❧❛ ❞✐stâ♥❝✐❛z0 ❝♦♠♦ Ef onte(x, y; 0) =

Z Z +∞

−∞ ˆ

Ef onte(kx, ky;−z0)ei kzz0ei(kxx+kyy)dkxdky, ✭✹✳✷✮ ♦♥❞❡ ✉s❛♠♦s ♦ ♣r♦♣❛❣❛❞♦reikzz✭❝♦♠_z=z0✮✱ ♣♦✐s ✜③❡♠♦s_Eˆ

f onte(kx, ky; 0) =Eˆf onte(kx, ky;−z0)eikzz0✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❡q✉❛çã♦ ✭✸✳✹✮✳ ◆♦t❡ q✉❡✱ ❞❡✈✐❞♦ à ♣r♦①✐♠✐❞❛❞❡ ❡♥tr❡ ❛ ❢♦♥t❡ ❡ ❛ s✉♣❡r❢í❝✐❡

❞❛ ❛♠♦str❛ ✭z0 ≪ λ✮✱ ♦ ❝❛♠♣♦ Ef onte(x, y; 0) é ❝♦♠♣♦st♦ ♣♦r ✉♠❛ s✉♣❡r♣♦s✐çã♦ ❞❡ ♦♥❞❛s ♣❧❛♥❛s ❡ ❡✈❛♥❡s❝❡♥t❡s✳

❆❣♦r❛✱ ❝♦♥s✐❞❡r❛♠♦s ❛ ✐♥t❡r❛çã♦ ❞❡ Ef onte(x, y; 0) ❝♦♠ ❛ ❛♠♦str❛ ❛tr❛✈és ❞❛ ❢✉♥çã♦ tr❛♥s♠✐ssã♦ T(x, y)✳ ❆♣ós ❡st❛ ✐♥t❡r❛çã♦ ♦ ❝❛♠♣♦ ❡❧étr✐❝♦ ❞❛ ❧✉③ ❡♠✐t✐❞❛ ♣❡❧♦ ♠❛t❡r✐❛❧

❜✐✲❞✐♠❡♥s✐♦♥❛❧ s❡rá ❝❛r❛❝t❡r✐③❛❞♦ ♣♦r