A practical guided-mode resonance lter operating in the visible band of the electromagnetic spectrum is numerically designed in this paper.
❱❖▲❯▼❊✿ ✸ | ■❙❙❯❊✿ ✷ | ✷✵✶✾ | ❏✉♥❡ ●✉✐❞❡❞✲♠♦❞❡ r❡s♦♥❛♥❝❡ ❢✐❧t❡r ✇✐t❤ ✉❧tr❛✲♥❛rr♦✇ ❜❛♥❞✇✐❞t❤ ♦✈❡r t❤❡ ✈✐s✐❜❧❡ ❢r❡q✉❡♥❝✐❡s ❢♦r ❧❛❜❡❧✲❢r❡❡ ♦♣t✐❝❛❧ ❜✐♦s❡♥s♦r 1,2,∗ P❤✉❝ ❚♦❛♥ ❉❆◆● 1,2 ✱ ❑❤❛✐ ◗✳ ▲❊ 3,4 ✱ ◗✉❛♥❣ ▼✐♥❤ ◆●❖ ✱ ❍✳ P✳ ❚✳ ◆●❯❨❊◆ ✱ 1,2,∗ ❚r✉♦♥❣ ❑❤❛♥❣ ◆●❯❨❊◆ ❉✐✈✐s✐♦♥ ♦❢ ❈♦♠♣✉t❛t✐♦♥❛❧ P❤②s✐❝s✱ ■♥st✐t✉t❡ ❢♦r ❈♦♠♣✉t❛t✐♦♥❛❧ ❙❝✐❡♥❝❡✱ ❚♦♥ ❉✉❝ ❚❤❛♥❣ ❯♥✐✈❡rs✐t②✱ ❍♦ ❈❤✐ ▼✐♥❤ ❈✐t②✱ ❱✐❡t♥❛♠ ❋❛❝✉❧t② ♦❢ ❊❧❡❝tr✐❝❛❧ ❛♥❞ ❊❧❡❝tr♦♥✐❝s ❊♥❣✐♥❡❡r✐♥❣✱ ❚♦♥ ❉✉❝ ❚❤❛♥❣ ❯♥✐✈❡rs✐t②✱ ❍♦ ❈❤✐ ▼✐♥❤ ❈✐t②✱ ❱✐❡t♥❛♠ ❯♥✐✈❡rs✐t② ♦❢ ❙❝✐❡♥❝❡ ❛♥❞ ❚❡❝❤♥♦❧♦❣② ♦❢ ❍❛♥♦✐✱ ❱✐❡t♥❛♠ ❆❝❛❞❡♠② ♦❢ ❙❝✐❡♥❝❡ ❛♥❞ ❚❡❝❤♥♦❧♦❣②✱ ✶✽ ❍♦❛♥❣ ◗✉♦❝ ❱✐❡t✱ ❈❛✉ ●✐❛②✱ ❍❛♥♦✐✱ ❱✐❡t♥❛♠ ●r❛❞✉❛t❡ ❯♥✐✈❡rs✐t② ♦❢ ❙❝✐❡♥❝❡ ❛♥❞ ❚❡❝❤♥♦❧♦❣②✱ ❱✐❡t♥❛♠ ❆❝❛❞❡♠② ♦❢ ❙❝✐❡♥❝❡ ❛♥❞ ❚❡❝❤♥♦❧♦❣②✱ ✶✽ ❍♦❛♥❣ ◗✉♦❝ ❱✐❡t✱ ❈❛✉ ●✐❛②✱ ❍❛♥♦✐✱ ❱✐❡t♥❛♠ ❉❡♣❛rt♠❡♥t ♦❢ ❊❧❡❝tr✐❝❛❧ ❛♥❞ ❈♦♠♣✉t❡r ❊♥❣✐♥❡❡r✐♥❣✱ ◆❡✇ ❏❡rs❡② ■♥st✐t✉t❡ ♦❢ ❚❡❝❤♥♦❧♦❣②✱ ❯♥✐✈❡rs✐t② ❍❡✐❣❤ts✱ ◆❡✇❛r❦✱ ◆❏ ✵✼✶✵✷✱ ❯❙❆ ✯❈♦rr❡s♣♦♥❞✐♥❣ ❆✉t❤♦rs✿ P❤✉❝ ❚♦❛♥ ❉❛♥❣ ✭❡♠❛✐❧✿ ❞❛♥❣♣❤✉❝t♦❛♥❅t❞t✉✳❡❞✉✳✈♥✮❀ ❚r✉♦♥❣ ❑❤❛♥❣ ◆❣✉②❡♥ ✭❡♠❛✐❧✿ ♥❣✉②❡♥tr✉♦♥❣❦❤❛♥❣❅t❞t✉✳❡❞✉✳✈♥✮ ✭❘❡❝❡✐✈❡❞✿ ✶✽✲❋❡❜r✉❛r②✲✷✵✶✾❀ ❛❝❝❡♣t❡❞✿ ✶✸✲▼❛②✲✷✵✶✾❀ ♣✉❜❧✐s❤❡❞✿ ✸✵✲❏✉♥❡✲✷✵✶✾✮ ❉❖■✿ ❤tt♣✿✴✴❞①✳❞♦✐✳♦r❣✴✶✵✳✷✺✵✼✸✴❥❛❡❝✳✷✵✶✾✸✷✳✷✷✸ ❆❜str❛❝t✳ ❆ ♣r❛❝t✐❝❛❧ ❣✉✐❞❡❞✲♠♦❞❡ r❡s♦♥❛♥❝❡ ✜❧t❡r ♦♣❡r❛t✐♥❣ ✐♥ t❤❡ ✈✐s✐❜❧❡ ❜❛♥❞ ♦❢ t❤❡ ❡❧❡❝✲ tr♦♠❛❣♥❡t✐❝ s♣❡❝tr✉♠ ✐s ♥✉♠❡r✐❝❛❧❧② ❞❡s✐❣♥❡❞ ✐♥ t❤✐s ♣❛♣❡r✳ ❚❤❡ ✜❧t❡r ♣r♦✈✐❞❡s ❤✐❣❤ ❜❛❝❦❣r♦✉♥❞ tr❛♥s♠✐ss✐♦♥ ✭>✾✵✪✮ ✇✐t❤ ❛❧♠♦st ♣❡r❢❡❝t r❡✢❡❝✲ t✐♦♥ ❛t r❡s♦♥❛♥❝❡ ✇❛✈❡❧❡♥❣t❤s ♦❢ ✻✷✸ ♥♠ ❛♥❞ ✻✹✶ ♥♠ ❢♦r ❚❊ ❛♥❞ ❚▼ ♠♦❞❡s✱ r❡s♣❡❝t✐✈❡❧②✳ ❖✉r ✜❧t❡r ✐s ❛❧s♦ ❝❤❛r❛❝t❡r✐③❡❞ ❜② ✐ts s❡♥s✐t✐✈✐t② t♦ ✐♥❝✐❞❡♥t ❛♥❣❧❡s✱ ♣♦❧❛r✐③❛t✐♦♥s✱ ❛♥❞ ❛ r❡❢r❛❝t✐✈❡ ✐♥❞❡① ♦❢ t❤❡ s✉rr♦✉♥❞✐♥❣ ❡♥✈✐r♦♥♠❡♥t ✇❤✐❝❤ ❛r❡ ✉t✐❧✐③❡❞ ✐♥ ♣r❛❝t✐❝❛❧ ❛♣♣❧✐❝❛t✐♦♥s s✉❝❤ ❛s t✉♥❛❜❧❡ ♦♣t✐❝❛❧ ✜❧t❡rs✱ ✐♠❛❣✐♥❣ ♦r ❞❡t❡❝t✐♦♥✳ ❲❡ s❤♦✇ t❤❛t t❤❡ r❡s♦♥❛♥t tr❛♥s♠✐ss✐♦♥ s♣❡❝tr❛❧ r❡s♣♦♥s❡ ❝❛♥ ❜❡ ✉s❡❞ ❢♦r ❤✐❣❤❧② s❡♥s✐t✐✈❡✱ ❛ ♣♦t❡♥t✐❛❧ ❧❛❜❡❧✲ ❢r❡❡ r❡❢r❛❝t✐✈❡ ✐♥❞❡① ❜✐♦s❡♥s♦r ❤❛✈✐♥❣ s❡♥s✐t✐✈✐✲ t✐❡s ♦❢ ✾✵ ♥♠✴❘■❯ ❛♥❞ ✶✵✸ ♥♠✴❘■❯✱ ❛♥❞ ✜❣✉r❡ ♦❢ ♠❡r✐ts ♦❢ ✶✳✾✸ ❛♥❞ ✷✳✶✸ ❢♦r ❚▼ ❛♥❞ ❚❊ ♣♦✲ ❧❛r✐③❛t✐♦♥s✱ r❡s♣❡❝t✐✈❡❧②✳ ✹✵✻ ❑❡②✇♦r❞s ❣✉✐❞❡❞✲♠♦❞❡ r❡s♦♥❛♥❝❡✱ ✜❧t❡r✱ ✈✐s✐❜❧❡✱ ♥❛rr♦✇ ❜❛♥❞✳ ✶✳ ■◆❚❘❖❉❯❈❚■❖◆ ●✉✐❞❡❞✲♠♦❞❡ r❡s♦♥❛♥❝❡ ✭●▼❘✮ ♦r ✇❛✈❡❣✉✐❞❡✲ ♠♦❞❡ r❡s♦♥❛♥❝❡ ✐s ❦♥♦✇♥ ❛s ❛ ♣❤❡♥♦♠❡♥♦♥ ✐♥ ✇❤✐❝❤ t❤❡ r❡s♦♥❛♥t ✇❛✈❡❣✉✐❞❡ ♠♦❞❡s ❛r❡ ❡①✲ ❝✐t❡❞ ✐♥ ♣❤❛s❡✲♠❛t❝❤✐♥❣ ❡❧❡♠❡♥ts s✉❝❤ ❛s s❧❛❜ ✇❛✈❡❣✉✐❞❡ ❣r❛t✐♥❣s ❛♥❞ ♣❤♦t♦♥✐❝ ❝r②st❛❧ s❧❛❜s ❬✶❪✳ ●▼❘ ❣r❛t✐♥❣s ❛♥❞ ♣❤♦t♦♥✐❝ ❝r②st❛❧ s❧❛❜s ❛r❡ ✉s✉❛❧❧② ✉s❡❞ ❢♦r ♦♣t✐❝❛❧ ✜❧t❡r✐♥❣ ❛♣♣❧✐❝❛t✐♦♥ t❤❛♥❦s t♦ t❤❡✐r ✉♥✐q✉❡ s♣❡❝tr❛❧ r❡s♣♦♥s❡✳ ❆ t②♣✲ ✐❝❛❧ ●▼❘ ❣r❛t✐♥❣ ✜❧t❡r ✐♥❝❧✉❞❡s ❛ st❛❝❦ ♦❢ t❤✐♥ ❞✐❡❧❡❝tr✐❝ ♠❛t❡r✐❛❧ ❧❛②❡rs ✇✐t❤ ❣r❛t✐♥❣s✴♣❤♦t♦♥✐❝ ❝r②st❛❧s ✐♥s❝r✐❜❡❞ ♦♥ t❤❡ ✇❛✈❡❣✉✐❞✐♥❣ ❧❛②❡r t♦ ❝ ✷✵✶✾ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ❱❖▲❯▼❊✿ ✸ s✉♣♣♦rt ❣✉✐❞❡❞ ♠♦❞❡s ✇❤✐❝❤ r❡s♦♥❛♥t❧② r❡s✉❧ts ✐♥ ❤✐❣❤ r❡✢❡❝t✐♦♥ ❛♥❞ ♥❡❛r✲③❡r♦ tr❛♥s♠✐ss✐♦♥ ❛t t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ r❡s♦♥❛♥t ✇❛✈❡❧❡♥❣t❤s ❬✷✕✻❪✳ ●▼❘ ❡✛❡❝t ❛r✐s❡s ❛s ❛♥ ❡✈❛♥❡s❝❡♥t ❞✐✛r❛❝t✐♦♥ ♣❤❡♥♦♠❡♥♦♥ ♦❝❝✉rr✐♥❣ ❛t ❛♥ ✐♥t❡r❢❛❝❡ ❜❡t✇❡❡♥ ❣r❛t✐♥❣s ❛♥❞ ❢r❡❡✲s♣❛❝❡ ✇❤❡♥ ❛♥ ✐♥❝✐❞❡♥t ❧✐❣❤t ✐s ❝♦✉♣❧❡❞ ✐♥t♦ t❤❡ ❣✉✐❞❡❞ ♠♦❞❡ ♦❢ t❤❡ ✇❛✈❡❣✉✐❞❡ ❝♦♠♣♦♥❡♥t ❛♥❞ ♣r♦♣❛❣❛t❡s ✐♥ ✐t ❛t s♣❡❝✐✜❝ ♦♣t✐✲ ❝❛❧ ♣❛r❛♠❡t❡rs ♦❢ ✇❛✈❡❧❡♥❣t❤✱ ❛♥❣❧❡ ❛♥❞ ♣♦❧❛r✲ ✐③❛t✐♦♥ ♠♦❞❡s ♦❢ t❤❡ ✐♥❝✐❞❡♥t ❧✐❣❤t ❬✺✱ ✻❪✳ ●▼❘ ✜❧t❡rs ♠✐❣❤t ❤❛✈❡ ♠❛♥② ✉s❡❢✉❧ ❝❤❛r❛❝t❡r✐st✐❝s ✇❤✐❝❤ ✐♥❝❧✉❞❡ ♥❛rr♦✇ ❜❛♥❞✱ ❤✐❣❤ ♣❡❛❦ ❡✣❝✐❡♥❝②✱ ✢❡①✐❜❧❡ str✉❝t✉r❡s ❬✷❪✱ ❬✼✕✶✶❪✱ ❡t❝✳ ❚❤❡r❡❢♦r❡✱ t❤❡② ❤❛✈❡ ❜❡❡♥ ✇✐❞❡❧② st✉❞✐❡❞ ❢♦r ✜❧t❡r ❛♣♣❧✐✲ ❝❛t✐♦♥s ✇✐t❤ ♣r❛❝t✐❝❛❧ ❞❡♠❛♥❞s s✉❝❤ ❛s ♥❛rr♦✇ ❜❛♥❞✱ t♦t❛❧ r❡✢❡❝t✐♦♥✱ ❛♥❞ t❤❡ ♦t❤❡rs ❬✶✷✱ ✶✸❪✳ ◆♦r♠❛❧❧②✱ ❛ ❤✐❣❤✲✐♥❞❡① ❝♦♥tr❛st ❣r❛t✐♥❣ str✉❝✲ t✉r❡ ✐s ✉s❡❞ ❢♦r t❤❡s❡ ❛♣♣❧✐❝❛t✐♦♥s s✐♥❝❡ ✐t ❤❛s ❧♦✇✲❧♦st ❞✐❡❧❡❝tr✐❝ t❤❛♥❦s t♦ ❛ ❝♦♠❜✐♥❡❞ ❛r❝❤✐✲ t❡❝t✉r❡ ❜❡t✇❡❡♥ ❛ ❤✐❣❤ ✐♥❞❡① ♠❛t❡r✐❛❧ ❣r❛t✐♥❣ ❛♥❞ ❧♦✇ ✐♥❞❡① ♠❛t❡r✐❛❧s ❬✶✹❪✳ ❇❡s✐❞❡s✱ ♣❤♦t♦♥✐❝ ❝r②st❛❧ str✉❝t✉r❡s ✐♠♣❧❡♠❡♥t❡❞ ✐♥ ♣❧❛♥❛r ✇❛✈❡❣✲ ✉✐❞❡s ✐s ❛❧s♦ ♣r❡❢❡rr❡❞ ❜❡❝❛✉s❡ ♦❢ ✐ts ❤✐❣❤ q✉❛❧✐t② ✭◗ ✮ ❢❛❝t♦r ♣r♦♣❡rt② ❬✶✺❪✳ ●▼❘ ✜❧t❡rs ♣r❡s❡♥t ❛ ❤✐❣❤❧②✲s❡♥s✐t✐✈❡ ♣r♦♣✲ ❡rt② t♦ ♦♣t✐❝❛❧ ♣❛r❛♠❡t❡rs ♦❢ t❤❡✐r str✉❝t✉r❛❧ ❣❡✲ ♦♠❡tr② ❛♥❞ ❝♦♥❞✐t✐♦♥s ♦❢ t❤❡ ✐♥❝✐❞❡♥t ❧✐❣❤t✳ P❛r✲ t✐❝✉❧❛r❧②✱ t❤❡ ❛♥❣✉❧❛r s❡♥s✐t✐✈✐t② ✇✐❧❧ ❧❡❛❞ t♦ t❤❡ s♣❡❝tr❛❧ ❧♦❝❛t✐♦♥ s❡♥s✐t✐✈✐t② ✐♥ t❤❡ ❜❛♥❞✇✐❞t❤ r❛♥❣❡ s♦ t❤❛t ✐t ❝❛♥ ❜❡ ✉s❡❞ ❡✛❡❝t✐✈❡❧② t♦ ❛❞❥✉st t❤❡ ❝❡♥tr❛❧ tr❛♥s♠✐ss✐♦♥ ❞✐♣s ♦❢ t❤❡ ✜❧t❡r t♦ t❤❡ ❞❡s✐r❡❞ ✇❛✈❡❧❡♥❣t❤✳ ❚❤❡r❡❢♦r❡✱ ✐t ❝❛♥ ❜❡ ✉s❡❞ t♦ ❞❡s✐❣♥ t✉♥❛❜❧❡ ♦♣t✐❝❛❧ ✜❧t❡rs ✐♥ ❜♦t❤ t❤❡✐r r❡s♦♥❛♥t ✇❛✈❡❧❡♥❣t❤s ❛♥❞ ◗ ✲❢❛❝t♦rs ❬✷❪✱ ❬✶✻❪✳ ❆♣❛rt ❢r♦♠ ✜❧t❡r✐♥❣ ❛♣♣❧✐❝❛t✐♦♥s✱ ✇❡ ❤❛✈❡ r❡✲ ❝❡♥t❧② ❡♠♣❧♦②❡❞ ●▼❘✲❜❛s❡❞ ❣r❛t✐♥❣s✴♣❤♦t♦♥✐❝ ❝r②st❛❧ s❧❛❜s ❢♦r ♦♣t✐❝❛❧ s✇✐t❝❤✐♥❣✴❜✐st❛❜✐❧✐t② ❛♣✲ ♣❧✐❝❛t✐♦♥s✳ ❲❡ ✐♥tr♦❞✉❝❡❞ ✐♥♥♦✈❛t✐✈❡ ❛❧❧✲♦♣t✐❝❛❧ s✇✐t❝❤✐♥❣ ❞❡✈✐❝❡s ✇✐t❤ ❧♦✇ s✇✐t❝❤✐♥❣ ♣♦✇❡r ❛♥❞ ❤✐❣❤ ❜✐st❛❜✐❧✐t② ❡✣❝✐❡♥❝② t❤❛♥❦s t♦ t❤❡ ✐♥❞✉❝❡❞ ●▼❘ ✐♥ t❤❡ ❣r❛t✐♥❣s ❛♥❞ ♣❤♦t♦♥✐❝ ❝r②st❛❧ s❧❛❜ ✇❛✈❡❣✉✐❞❡s ❬✶✼❪✳ ■♥ ❛❞❞✐t✐♦♥✱ ♦♥❡ ♦❢ t❤❡ ❡ss❡♥t✐❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ♦❢ ●▼❘ ✜❧t❡rs ✐s ✐ts ❤✐❣❤ s❡♥s✐t✐✈✲ ✐t② t♦ r❡❢r❛❝t✐✈❡ ✐♥❞❡① ❝❤❛♥❣❡s ✐♥ t❤❡ s✉rr♦✉♥❞✲ ✐♥❣ ❡♥✈✐r♦♥♠❡♥t ♦❢ t❤❡ ❤✐❣❤✲✐♥❞❡① ✇❛✈❡❣✉✐❞✐♥❣ ❧❛②❡r✳ ❚❤❡r❡❢♦r❡✱ ❣✉✐❞❡❞✲♠♦❞❡ r❡s♦♥❛♥❝❡ ✜❧t❡rs ❤❛✈❡ ❜❡❡♥ ✐♥❝r❡❛s✐♥❣❧② ✉t✐❧✐③❡❞ ❢♦r s❡♥s✐♥❣ ❛♣♣❧✐✲ ❝❛t✐♦♥s ❬✶✽❪✱ ❬✶✾❪✳ | ■❙❙❯❊✿ ✷ | ✷✵✶✾ | ❏✉♥❡ ❛t✐♥❣ ✐♥ t❤❡ ✈✐s✐❜❧❡ ❜❛♥❞ ♦❢ t❤❡ ❡❧❡❝tr♦♠❛❣♥❡t✐❝ s♣❡❝tr✉♠✳ ❆♥❣✉❧❛r ❛♥❞ ♣♦❧❛r✐③❛t✐♦♥ ♦❢ ✐♥❝✐❞❡♥t ❧✐❣❤t ❛♥❞ s✉rr♦✉♥❞✐♥❣ ❡♥✈✐r♦♥♠❡♥t ✐♥✢✉❡♥❝❡ ♦♥ t❤❡ ●▼❘ ✜❧t❡r ✇✐❧❧ ❝♦♠♣r❡❤❡♥s✐✈❡❧② ❜❡ ❛♥❛❧②③❡❞ ✐♥ t❤✐s st✉❞②✳ ❲❡ ♦❜s❡r✈❡ ❛ st♦♣✲❜❛♥❞ ✇❤✐❝❤ ❜❧♦❝❦s t❤❡ ✐♠♣✐♥❣✐♥❣ ❧✐❣❤t ♦✈❡r ❛ ♥❛rr♦✇ ❜❛♥❞✲ ✇✐❞t❤ ♦❢ ❢r❡q✉❡♥❝✐❡s ✐♥ t❤❡ ✈✐s✐❜❧❡ ❛♥❞ ♣❛ss❡s ❛❧❧ r❡♠❛✐♥✐♥❣ ❢r❡q✉❡♥❝✐❡s ❜❡②♦♥❞ t❤❡ ❜❛♥❞✲st♦♣✱ ✇❤✐❝❤ ♣❧❛②s ❛♥ ✐♠♣♦rt❛♥t r♦❧❡ ✐♥ ✈❛r✐♦✉s ✐♠❛❣✐♥❣ ♦r ❞❡t❡❝t✐♦♥ ❛♣♣❧✐❝❛t✐♦♥s ❬✷✵❪✳ ■♥ ❛❞❞✐t✐♦♥✱ ✇❡ ✐♥✈❡st✐❣❛t❡ t❤❡ r❡❢r❛❝t✐✈❡ ✐♥❞❡① s❡♥s✐♥❣ ♣❡r❢♦r✲ ♠❛♥❝❡ ♦❢ t❤❡ ●▼❘ ✜❧t❡r ❢♦r ❜♦t❤ tr❛♥s✈❡rs❡ ❡❧❡❝✲ tr✐❝ ✭❚❊✮ ❛♥❞ tr❛♥s✈❡rs❡ ♠❛❣♥❡t✐❝ ✭❚▼✮ ♠♦❞❡s✳ ❚❤❡ r❡s✉❧t✐♥❣ ❤✐❣❤ s❡♥s✐t✐✈✐t② ❛♥❞ s❡❧❡❝t✐✈✐t② ♦❢ t❤❡ ✜❧t❡r t♦ ❛ r❡❢r❛❝t✐✈❡ ✐♥❞❡① ❝❤❛♥❣❡ ♦❢ t❤❡ s✉rr♦✉♥❞✐♥❣ ❡♥✈✐r♦♥♠❡♥t ♣r♦✈✐❞❡ ♣♦ss✐❜✐❧✐t✐❡s t♦ r❡❛❧✐③❡ ❤✐❣❤✲❡✣❝✐❡♥❝② ✐♥t❡❣r❛t❡❞ ♦♥✲❝❤✐♣ ❧❛❜❡❧✲ ❢r❡❡ ♦♣t✐❝❛❧ ❜✐♦s❡♥s♦rs✳ ❆❧❧ s✐♠✉❧❛t✐♦♥s ❛r❡ ♣❡r✲ ❢♦r♠❡❞ ❜② ✉s✐♥❣ t❤❡ ❝♦♠♠❡r❝✐❛❧ ❡❧❡❝tr♦♠❛❣♥❡t✐❝ s✐♠✉❧❛t✐♦♥ ❈❙❚ ▼■❈❘❖❲❆❱❊ ❙❚❯❉■❖ ✭❈❙❚ ▼❲❙✮ ♣❛❝❦❛❣❡ ❬✷✶❪✳ ✷✳ ●▼❘ ❋■▲❚❊❘ ❉❊❙■●◆ ❋✐❣✳ ✶ s❤♦✇s ❛ s❝❤❡♠❛t✐❝ ✐❧❧✉str❛t✐♦♥ ♦❢ t❤❡ ♣r♦♣♦s❡❞ ●▼❘ ✜❧t❡r ✇❤✐❝❤ ❝♦♥s✐sts ♦❢ ❛ ❚❛2 ❖5 ✇❛✈❡❣✉✐❞✐♥❣ ❧❛②❡r ✇✐t❤ ♣❛tt❡r♥❡❞ ❣r❛t✐♥❣s ♣♦s✐✲ t✐♦♥❡❞ ♦♥ ❛ ❣❧❛ss s✉❜str❛t❡ ✈✐❛ ❛ ✶✵✲♥♠✲t❤✐❝❦ ❛❞✲ ❤❡s✐♦♥ ❙✐❖2 ❧❛②❡r✳ ❆❧❧ ♠❛t❡r✐❛❧s ✇✐t❤ ❞✐s♣❡rs✐✈❡ ♣r♦♣❡rt✐❡s ❛r❡ ❡①tr❛❝t❡❞ ❢r♦♠ t❤❡ ♠❛t❡r✐❛❧ ❧✐✲ ❜r❛r② ♦❢ t❤❡ s✐♠✉❧❛t✐♦♥ s♦❢t✇❛r❡ ❬✷✶❪✳ ❚❤❡ ✇❛✈❡✲ ❧❡♥❣t❤ ❞❡♣❡♥❞❡♥t r❡❢r❛❝t✐✈❡ ✐♥❞✐❝❡s ♦❢ t❤❡ ♠❛t❡✲ r✐❛❧s ✇❡r❡ t❛❦❡♥ ❢r♦♠ t❤❡ ❧✐t❡r❛t✉r❡✳ ❲❡ st❛rt ♦✉r ✐♥✈❡st✐❣❛t✐♦♥ ✇✐t❤ ❛ ❞❡s✐❣♥❡❞ ●▼❘ ✜❧t❡r ❤❛✈✐♥❣ ❛ tr❛♥s♠✐ss✐♦♥ r❡s♦♥❛♥❝❡ ✐♥ t❤❡ ✈✐s✐❜❧❡✳ ❲❡ ❤❛✈❡ st✉❞✐❡❞ ♠❛♥② ♣❛r❛♠❡t❡rs ❛♥❞ ❢♦✉♥❞ t❤❛t t❤❡ ♣❡r✐♦❞✐❝✐t② ❛✛❡❝t❡❞ t❤❡ r❡s♦♥❛♥❝❡ ✇❛✈❡✲ ❧❡♥❣t❤s s✐❣♥✐✜❝❛♥t❧②✳ ❖t❤❡r ♣❛r❛♠❡t❡rs s✉❝❤ ❛s ❤✱ ❲✱ ❞g ✇❡r❡ ✉s❡❞ t♦ ♦♣t✐♠✐③❡ t❤❡ r❡✢❡❝t✐♦♥ ❞✐♣ ❛♥❞ tr❛♥s♠✐ss✐♦♥ ❜❛❝❦❣r♦✉♥❞ ❛t t❤❡ r❡s♦♥❛♥❝❡ ✇❛✈❡❧❡♥❣t❤✳ ❚❤❡ ❞❡s✐❣♥ ✉t✐❧✐③❡s ❛ t♦t❛❧ ❚❛2 ❖5 t❤✐❝❦♥❡ss ♦❢ ❤ ❂ ✵✳✶ µ♠✱ ❛ ❣r❛t✐♥❣ ❞❡♣t❤✱ ♣✐t❝❤ ❛♥❞ ✇✐❞t❤ ♦❢ ❞g ❂ ✵✳✵✽ µ♠✱ P ❂ ✵✳✹✾ µ♠✱ ❛♥❞ ❲ ❂ ✵✳✶✻ µ♠✱ r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡ ✇❛✈❡❧❡♥❣t❤ ❞❡✲ ♣❡♥❞❡♥t r❡❢r❛❝t✐✈❡ ✐♥❞✐❝❡s ♦❢ ❣❧❛ss✱ ❛❞❤❡s✐♦♥✱ ❛♥❞ ❤♦♠♦❣❡♥❡♦✉s ❚❛2 ❖5 ❧❛②❡rs ❛r❡ ❣✐✈❡♥ ❛s ♥s ✱ ♥a ✱ ❛♥❞ ♥w ✱ r❡s♣❡❝t✐✈❡❧②✳ ■♥ t❤✐s ♣❛♣❡r✱ ✇❡ ♥✉♠❡r✐❝❛❧❧② ❞❡s✐❣♥ ❛♥❞ ❝❤❛r✲ ❚♦ ❛❝❤✐❡✈❡ r❡s♦♥❛♥❝❡✱ ✇❛✈❡❣✉✐❞❡ ♠♦❞❡s ❤❛✈❡ ❛❝t❡r✐③❡ ❛♥ ✉❧tr❛✲♥❛rr♦✇❜❛♥❞ ●▼❘ ✜❧t❡r ♦♣❡r✲ t♦ ❜❡ ❣❡♥❡r❛t❡❞ ✇✐t❤ t❤❡ ✐♥❝✐❞❡♥t ✇❛✈❡ s❛t✐s❢②✲ ❝ ✷✵✶✾ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ✹✵✼ ndwidth over the visible frequencies for label-free optical biosensor) wave to exist in the grating structure can be represented [2] as 𝑚𝑎𝑥[𝑛𝑐 , 𝑛𝑠 ] ≤ 𝑛𝑒𝑓𝑓 < 𝑛𝑤 , (2) ✸✳ ❱❖▲❯▼❊✿ ✸ | ■❙❙❯❊✿ ✷ | ✷✵✶✾ | ❏✉♥❡ ❙P❊❈❚❘❖❙❈❖P■❈ P❘❖P❊❘❚❨ ❖❋ ●▼❘ ❋■▲❚❊❘ ❚❤❡ ♠❛✐♥ ❣♦❛❧ ♦❢ t❤❡ ♣r♦♣♦s❡❞ ✜❧t❡r ✐s t♦ ❜❧♦❝❦ ❛ ❧✐❣❤t✇❛✈❡ ❛t ❛ s✐♥❣❧❡ ✇❛✈❡❧❡♥❣t❤ ✇❤✐❧❡ ♣❛ss✐♥❣ ✐t ❛t t❤❡ ♦t❤❡rs ✐♥ t❤❡ ✈✐s✐❜❧❡ ❜❛♥❞ ♦❢ t❤❡ ❡❧❡❝tr♦✲ ♠❛❣♥❡t✐❝ s♣❡❝tr✉♠✳ ■♥ t❤✐s ❝❛s❡✱ t❤❡ tr❛♥s♠✐s✲ s✐♦♥ s♣❡❝tr♦s❝♦♣② ❤❛s ❛♥ ✉❧tr❛✲s❤❛r♣ tr❛♥s♠✐s✲ s✐♦♥ ❞✐♣ ❛t t❤❡ r❡s♦♥❛♥❝❡ ❛♥❞ ❜r♦❛❞❜❛♥❞ tr❛♥s✲ ♠✐ss✐♦♥ ❛t t❤❡ ✇❛✈❡❧❡♥❣t❤s ❛✇❛② ❢r♦♠ t❤❡ r❡s♦✲ Fig 1.❋✐❣✳ Sketch of the proposed filter.❚❤❡ The❣r❛t✐♥❣ grating♥❛♥❝❡✳ ❚❤❡ r❡s♦♥❛♥t ✇❛✈❡❧❡♥❣t❤ ✐s t✉♥❛❜❧❡ ✉♣♦♥ ✶✿ ❙❦❡t❝❤ ♦❢ t❤❡ ♣r♦♣♦s❡❞GRM ●❘▼ ✜❧t❡r✳ ♣♦❧❛r✐③❛t✐♦♥ st❛t❡s ❛♥❞ ❛♥❣❧❡s ♦❢ ✐♥❝✐❞❡♥❝❡✳ ❉✉r✲ ❧❛②❡r ✐s ❛ r❡❝t❛♥❣✉❧❛r ♣r♦✜❧❡ ✇✐t❤ ❞ ❂ t❤✐❝❦✲ layer is a rectangular profile with dg = thickness g of grating, ♥❡ss ♦❢ ❣r❛t✐♥❣✱ ❲ ❂ ❣r❛t✐♥❣ ✇✐❞t❤✱ P ❂ ❣r❛t✐♥❣ W = grating width, P = grating period ns, na, nw are✐♥❣ t❤❡ ✐♥✈❡st✐❣❛t✐♦♥✱ ✇❡ ♦❜s❡r✈❡❞ t❤❛t t❤❡ ♣❡r✐✲ ♣❡r✐♦❞✳ ♥s ✱ ♥a ✱ ♥w ❛r❡ r❡❢r❛❝t✐✈❡ ✐♥❞✐❝❡s ♦❢ t❤❡ ♦❞✐❝✐t② ✭♦r ♣✐t❝❤✮ ♦❢ t❤❡ str✉❝t✉r❡ ♣❧❛②s ❛♥ ✐♠✲ refractive indices the glass the ❧❛②❡r✱ SiO2 ❛♥❞ adhesion ❣❧❛ss of s✉❜str❛t❡✱ t❤❡substrate, ❙✐❖2 ❛❞❤❡s✐♦♥ t❤❡ layer, and the❤♦♠♦❣❡♥❡♦✉s homogeneous layer,♣♦rt❛♥t r♦❧❡ ✐♥ ♣♦s✐t✐♦♥✐♥❣ t❤❡ tr❛♥s♠✐ss✐♦♥ r❡s✲ ❚❛2 ❖5Ta ✇❛✈❡❣✉✐❞✐♥❣ ❧❛②❡r✱ r❡s♣❡❝✲ 2O5 waveguiding ♦♥❛♥❝❡ ❛t t❤❡ ♥♦r♠❛❧ ✐♥❝✐❞❡♥❝❡✳ t✐✈❡❧②✳ respectively ❋♦r s✐♠♣❧✐✜❝❛t✐♦♥ ♣✉r♣♦s❡s✱ ✐♥ ❋✐❣✳ ✷✱ ✇❡ ♦♥❧② s❤♦✇ t❤❡ tr❛♥s♠✐ss✐♦♥ ❢❡❛t✉r❡s ❢♦r ✈❛r✐♦✉s ♣❡r✐✲ The equation describes the regions of resonance for♦❞✐❝✐t✐❡s ♦❢ t❤❡ str✉❝t✉r❡✳ ❚❤❡ r❡s♦♥❛♥t tr❛♥s✲ ✐♥❣ t❤❡ ♣❤❛s❡✲♠❛t❝❤✐♥❣ ❝♦♥❞✐t✐♦♥ wavelengths, ♦❢ t❤❡ ♣❡r✐♦❞✐❝part♠✐ss✐♦♥ ✇❛✈❡❧❡♥❣t❤ ✐s ❧✐♥❡❛r❧② s❤✐❢t❡❞ ✇✐t❤ r❡✲ guided‐mode resonance At resonance str✉❝t✉r❡ ❬✷❪ of the applied wave is coupled into a guided mode whichs♣❡❝t t♦ t❤❡ ♣❡r✐♦❞✐❝✐t② ♦❢ t❤❡ str✉❝t✉r❡ ❢♦r t❤❡ gradually leaks out from the waveguide The leaky-waves❛♠❡ str✉❝t✉r❛❧ ♣❛r❛♠❡t❡rs✳ ❋♦r t❤❡ ❝❛s❡ ♦❢ P combines with the applied wave to generating a filtering❂ ✵✳✹✹✶ µ♠✱ t❤❡ s♣❡❝tr❛ s❤✐❢t❡❞ ❧✐♥❡❛r❧② t♦✇❛r❞ response in the spectrum why λtransmission✭✶✮ dipss❤♦rt ✇❛✈❡❧❡♥❣t❤ ❛♥❞ ❛ s❡❝♦♥❞ ❞✐♣ s✐♠✉❧t❛♥❡✲ nef f = nThat's c sinθi − m , P appear at resonance wavelengths ♦✉s❧② ❛♣♣❡❛rs ♥❡❛r t❤❡ ✇❛✈❡❧❡♥❣t❤ ♦❢ ✵✳✻✼ µ♠ ✐♥ ❚❊ ♠♦❞❡✳ ❋✐❣✳ ✸ s❤♦✇s t❤❡ tr❛♥s♠✐ss✐♦♥ s♣❡❝✲ III SPECTROSCOPIC ✇❤❡r❡ ♥ef f ✐s t❤❡ ❡✛❡❝t✐✈❡ ✐♥❞❡① ♦❢ t❤❡ ❡q✉✐✈✲ tr✉♠ ♦✈❡r t❤❡ ✵✳✺ ✕ ✵✳✽ µ♠ ✇❛✈❡❧❡♥❣t❤ r❛♥❣❡ ❛t OF GMR ❛❧❡♥t PROPERTY ❤♦♠♦❣❡♥❡♦✉s ✇❛✈❡❣✉✐❞❡✱ ♥c FILTER ✐s r❡❢r❛❝t✐✈❡ ♥♦r♠❛❧ ✐♥❝✐❞❡♥❝❡ ❢♦r ❜♦t❤ ❚▼ ❛♥❞ ❚❊ ♣♦❧❛r✐③❛✲ The main♦❢ goal of ✐sthet❤❡ proposed is λto ✐sblock ✐♥❞❡① ❛✐r✱ P ❣r❛t✐♥❣ filter ♣❡r✐♦❞✱ t❤❡ at✐♦♥ st❛t❡s ♦❢ t❤❡ ♦♣t✐♠✐③❡❞ ✜❧t❡r str✉❝t✉r❡ ✇❤✐❝❤ lightwave a single wavelength passing ❛♥❣❧❡✱ it at the❤❛✈✐♥❣ t❤❡ ❣r❛t✐♥❣ ♣❡r✐♦❞ ♦❢ ✵✳✹✾ µ♠✱ ❣r❛t✐♥❣ ❢r❡❡ at s♣❛❝❡ ✇❛✈❡❧❡♥❣t❤✱ θi ✐s while t❤❡ ✐♥❝✐❞❡♥t th others ❛♥❞ in the band of the electromagnetic spectrum.✇✐❞t❤ ♦❢ ✵✳✶✻ µ♠✱ t❤❡ ❣r❛t✐♥❣ t❤✐❝❦♥❡ss ♦❢ ✵✳✵✽ t❤❡visible ✐♥t❡❣❡r ♠ r❡♣r❡s❡♥ts t❤❡ ♠ ❞✐✛r❛❝t❡❞ In this♦r❞❡r✳ case, the transmission spectroscopy has ❣✉✐❞❡❞ an ultra-µ♠ ❛♥❞ t❤❡ ❚❛2 ❖5 ✇❛✈❡❣✉✐❞❡ ❧❛②❡r ♦❢ ✵✳✶ µ♠✳ ▼♦r❡♦✈❡r✱ t❤❡ ❝♦♥❞✐t✐♦♥ ❢♦r t❤❡ sharp ✇❛✈❡ transmission resonance and broadband t♦ ❡①✐stdip ✐♥ at t❤❡the ❣r❛t✐♥❣ str✉❝t✉r❡ ❝❛♥ ❜❡ ❋♦r ❚▼✲♣♦❧❛r✐③❡❞ ✐♥❝✐❞❡♥❝❡✱ ❛ s✐♥❣❧❡ ❞✐♣ ✐♥ t❤❡ transmission at the❬✷❪ wavelengths away from the resonance.tr❛♥s♠✐ss✐♦♥ s♣❡❝tr✉♠ ✐s ♦❜s❡r✈❡❞ ❛t ✵✳✻✹✶ µ♠ r❡♣r❡s❡♥t❡❞ ❛s The resonant wavelength is tunable upon polarization❛s ❛ st♦♣❜❛♥❞ ✇❤✐❝❤ ❝♦rr❡s♣♦♥❞s t♦ ❛ ❧♦✇ tr❛♥s✲ states and angles of incidence During the investigation,♠✐ss✐♦♥ ❧❡ss t❤❛♥ ✷✪✱ s❤♦✇♥ ✐♥ ❋✐❣✳ ✸✭❛✮✳ ❋♦r we observed thatmax the periodicity (or [nc , ns ] ≤ n < nwof, the structure ✭✷✮ ❚❊✲♣♦❧❛r✐③❡❞ ✐♥❝✐❞❡♥❝❡✱ t❤❡ s♣❡❝tr❛❧ r❡s♣♦♥s❡ ♦❢ ef fpitch) plays an important role in positioning the transmissiont❤❡ ✜❧t❡r s♣❧✐ts ✐♥t♦ t✇♦ s♣❡❝tr❛❧ ❞✐♣s ❝♦♠♣r✐s✐♥❣ resonance at the normal incidence ♦❢ ❛ ♣r✐♠❛r② ❞✐♣ ❛♥❞ ❛ s❡❝♦♥❞❛r② ❞✐♣ ❧♦❝❛t❡❞ ❛t For simplification purposes, in Fig 2, we only showt❤❡ r❡s♦♥❛♥t ✇❛✈❡❧❡♥❣t❤s ♦❢ ✵✳✻✷✸ µ♠ ❛♥❞ ✵✳✼✸✼ ❚❤❡ ❡q✉❛t✐♦♥ ❞❡s❝r✐❜❡s t❤❡ periodicities r❡❣✐♦♥s ♦❢ r❡s♦✲ the transmission features for various of theµ♠✱ r❡s♣❡❝t✐✈❡❧②✱ s❤♦✇♥ ✐♥ ❋✐❣✳ ✸✭❜✮✳ ❚❤❡ ♣r✐✲ ♥❛♥❝❡ ❢♦r ❣✉✐❞❡❞✲♠♦❞❡ r❡s♦♥❛♥❝❡✳ ❆t r❡s♦♥❛♥❝❡ structure The resonant transmission wavelength is linearly♠❛r② ❞✐♣ ♣r♦❞✉❝❡s ❛ tr❛♥s♠✐ss✐♦♥ ♦❢ ❧❡ss t❤❛♥ ♦❢ periodicity t❤❡ ❛♣♣❧✐❡❞ of ✇❛✈❡ ❝♦✉♣❧❡❞for shifted✇❛✈❡❧❡♥❣t❤s✱ with respect♣❛rt to the the ✐sstructure ✶✪✳ ✐♥t♦ ❛ ❣✉✐❞❡❞ ♠♦❞❡ ✇❤✐❝❤ ❣r❛❞✉❛❧❧② ❧❡❛❦s the same structural parameters For the case of P = ♦✉t 0.441 ❢r♦♠ t❤❡ ✇❛✈❡❣✉✐❞❡✳ ❚❤❡ toward ❧❡❛❦②✲✇❛✈❡ µm, the spectra shifted linearly short ❝♦♠❜✐♥❡s wavelength ❋✐❣✳ ✹ ♣r♦✈✐❞❡s t❤❡ tr❛♥s♠✐ss✐♦♥ ❛s ❛ ❢✉♥❝t✐♦♥ ✇✐t❤ t❤❡ ❛♣♣❧✐❡❞ ✇❛✈❡ t♦ ❣❡♥❡r❛t✐♥❣ ❛ and a second dip simultaneously appears ✜❧t❡r✐♥❣ near the♦❢ ❛♥❣❧❡s ♦❢ ✐♥❝✐❞❡♥❝❡✳ ■♥ ❞❡t❛✐❧✱ ❋✐❣s✳ ✹✭❛✮ ❛♥❞ r❡s♣♦♥s❡ ✐♥ t❤❡ s♣❡❝tr✉♠✳ ❚❤❛t✬s ✇❤② tr❛♥s♠✐s✲ wavelength of 0.67 μm in TE mode Fig shows the✹✭❜✮ r❡s♣❡❝t✐✈❡❧② s❤♦✇ t❤❡ ❝❛❧❝✉❧❛t❡❞ tr❛♥s♠✐s✲ s✐♦♥ ❞✐♣s ❛♣♣❡❛r over ❛t r❡s♦♥❛♥❝❡ transmission spectrum the 0.5 –✇❛✈❡❧❡♥❣t❤s✳ 0.8 μm wavelengths✐♦♥ s♣❡❝tr❛ ❢♦r ✈❛r✐♦✉s ✐♥❝✐❞❡♥t ❛♥❣❧❡s ✐♥ ❚▼ range at normal incidence for both TM and TE polarization states of the optimized filter structure which having✹✵✽ the grating period of 0.49 μm, grating width ❝ ✷✵✶✾of ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ 0.16 μm, the grating thickness of 0.08 μm and the Ta 2O5 URNAL OF ADVANCED ENGINEERING AND COMPUTATION JOURNAL OF ADVANCED ENGINEERING AND COMPUTATION http://dx.doi.org/ http://dx.doi.org/ ISSN (online):ISSN …-…(online): ∙ ISSN (print): …-… ∙ …-… ISSN (print): …- L 0, NO 0, VOL 0-0, DEC 0000 0, NO 0, 0-0, DEC 0000 ❱❖▲❯▼❊✿ ✸ veguide layer of 0.1 μm waveguide layer of 0.1 μm (a) (a) (a) | ■❙❙❯❊✿ ✷ | ✷✵✶✾ | ❏✉♥❡ (a) (b) (b) (b) (b) Fig The spectral transmission response of the GMR Fig Transmission of the GMR The spectral❋✐❣✳ transmission response of the GMR Fig Transmission spectra of spectra the GMR filter Thefilter ✷✿ ❚❤❡ s♣❡❝tr❛❧ tr❛♥s♠✐ss✐♦♥ r❡s♣♦♥s❡ ♦❢ t❤❡ ●▼❘ filter for various periodicities for TM mode (a) and TE resonance is due to the excitation of a TM-pola ❋✐❣✳ ✸✿ ❚r❛♥s♠✐ss✐♦♥ s♣❡❝tr❛ ♦❢ t❤❡ ●▼❘ ✜❧t❡r✳ ❚❤❡ er for various periodicities TM mode (a) and ✜❧t❡r for ❢♦r ✈❛r✐♦✉s ♣❡r✐♦❞✐❝✐t✐❡s ❢♦r TE ❚▼ ♠♦❞❡resonance ✭❛✮ is r❡s♦♥❛♥❝❡ due to ✐sthe❞✉❡excitation of a TM-polarized t♦ t❤❡ ❡①❝✐t❛t✐♦♥ ♦❢ guided ❛ ❚▼✲ mode (b) mode (b) at ❛♥❞ normal incidence The filter ❚❤❡ profile is guided mode (a) and a TE-polarized ❚❊ ♠♦❞❡ ✭❜✮ ❛t ♥♦r♠❛❧ ✐♥❝✐❞❡♥❝❡✳ ✜❧t❡r de (b) at normal incidence The filter profile is guided mode ♣♦❧❛r✐③❡❞ (a) and a❣✉✐❞❡❞ TE-polarized mode (b) ♠♦❞❡ ✭❛✮ guided ❛♥❞ ❛ ❚❊✲♣♦❧❛r✐③❡❞ ♣r♦✜❧❡ ✐s ❞❡s❝r✐❜❡❞ ✐♥ ❋✐❣✳h✶=✇✐t❤ ❤ described in Fig with parameters: 0.1♣❛r❛♠❡t❡rs✿ μm, dg = 0.08 ❣✉✐❞❡❞ ♠♦❞❡ ✭❜✮✳ cribed in Fig with parameters: h = 0.1 μm, d = 0.08 g ❂ ✵✳✶ µ♠✱ ❞g ❂ ✵✳✵✽ µ♠✱ ❛♥❞ ✇ ❂ ✵✳✶✻ µ♠✳ μm, and w = 0.16 μm To highlight transmission features through our , and w = 0.16 μm To highlight transmission features through our filter structure, we plot spatial distributions of the electric structure, we plot spatial distributions of the electric field For TM-polarized incidence, a single dip in the at these resonant wavelengths for both polarization s For TM-polarized incidence, a single dip in the at these resonant wavelengths for both polarization states transmission spectrum is observed at 0.641 μm as a The amplitude magnitudes of the electric field |Ey| a ❛♥❞ ❚❊ ♣♦❧❛r✐③❛t✐♦♥s✳ ❆s t❤❡ ❛♥❣❧❡s ♦❢ ✐♥❝✐❞❡♥❝❡ nsmission spectrum is observed at 0.641 μm as a The amplitude magnitudes of the electric field |Ey| and of stopband which❢r♦♠ corresponds to a lowt❤❡ transmission less the magnetic field |Hy| for the both TE and TM mode ✐♥❝r❡❛s❡ ✵ t♦ ✶✺ ❞❡❣r❡❡s✱ tr❛♥s♠✐ss✐♦♥ pband which corresponds to a low transmission less the magnetic field |Hy| for the both TE ❛t and TM ❂ mode in 𝑦̂ ❢♦r ❚▼ ♥♦r♠❛❧❧② ✻✹✶shows than 2%, incidence, direction are ✐♥❝✐❞❡♥t shown in❧✐❣❤t Fig 5.λres Fig 5(a) the ❞✐♣shown s♣❧✐ts in❛tFig λ ❂3(a) ✻✹✶ For ♥♠ TE-polarized ✇❤✐❝❤ ✐s ❝❧❡❛r❧② ♦❜✲ n 2%, shown in Fig 3(a) For TE-polarized incidence, direction are shown in Fig Fig 5(a) shows the total ♥♠✳ ■t ✐s ♦❜✈✐♦✉s t❤❛t t❤❡ ✜❡❧❞ ❞✐str✐❜✉t✐♦♥✱ ✐♥ the spectral of the splits into❧✐❣❤t✳ two spectral magnetic field distribution for TM normally incident s❡r✈❡❞response ✐♥ ❋✐❣✳ ✹✭❛✮ ❢♦rfilter ❚▼✲♣♦❧❛r✐③❡❞ ❙✐♠✲ spectral response of the filtera splits intodip twoand spectral t❤✐s field ❝❛s❡✱distribution ✐s ❧♦❝❛t❡❞ ✐♥ t❤❡TM ❣r❛t✐♥❣ ❛♥❞ ❞✐s♣❡rs❡s magnetic normally dips comprising primary a secondary at λres = 641 nm.for It is obvious that incident the fieldlight distributio ✐❧❛r❧②✱ t❤❡ of tr❛♥s♠✐ss✐♦♥ ❢❡❛t✉r❡ s♣❧✐tt✐♥❣ ❛t λ ❂dip ✐♥ t❤❡ s✉❜str❛t❡✳ ❙✐♠✐❧❛r❧②✱ ❋✐❣✳ ✺✭❜✮ distribution, s❤♦✇s t❤❡ in s comprising of a primary dip and a secondary dip at λ = 641 nm It is obvious that the field res located ✼✸✼ at the wavelengths 0.623 μm and❧✐❣❤t 0.737 this case, is located in the grating and disperses i ♥♠resonant ❛♥❞ λ ❂ ✻✷✸ ♥♠ ❢♦rof❚❊✲♣♦❧❛r✐③❡❞ t♦t❛❧ is❡❧❡❝tr✐❝ ❢♦r ♥♦r♠❛❧❧② ated at theμm, resonant wavelengths of 0.623 μm3(b) and 0.737 thisdip case, located✜❡❧❞ in ❞✐str✐❜✉t✐♦♥ the grating and❚❊ disperses in the respectively, shown in Fig The primary substrate Similarly, Fig 5(b) shows the total electric ✐s ❛❧s♦ ❝❧❡❛r❧② ♦❜s❡r✈❡❞ ✐♥ ❋✐❣✳ ✹✭❜✮✳ ✐♥❝✐❞❡♥t ❧✐❣❤t ❛t λres ❂ shows ✻✷✸ ♥♠✳ ❚❤❡ ✜❡❧❞ ❞✐s✲ field , respectively, shown in Fig 3(b) The primary dip substrate Similarly, Fig 5(b) the total electric produces a transmission of less than 1% distribution for TE normally incident light at λres = 62 ❚♦of❤✐❣❤❧✐❣❤t tr❛♥s♠✐ss✐♦♥ ❢❡❛t✉r❡s t❤r♦✉❣❤ tr✐❜✉t✐♦♥ ❝♦♥❝❡♥tr❛t❡s ♦♥light t❤❡ at ❣r❛t✐♥❣ ✐♥✲ nm duces a transmission less than 1% distribution for ♠❛✐♥❧② TE normally incident λres = 623 The field distribution mainly concentrates on the gr ♦✉r ✜❧t❡r str✉❝t✉r❡✱ ✇❡ ♣❧♦t s♣❛t✐❛❧ ❞✐str✐❜✉t✐♦♥s t❡r❢❛❝❡ ❛♥❞ t❤❡ s✉❜str❛t❡✳ ❚❤❡ ❡❧❡❝tr✐❝on✜❡❧❞ ❡♥✲ The field distribution mainly concentrates the grating Fig provides the transmission as a function of angles interface and the substrate The electric field enhance ♦❢ t❤❡ ❡❧❡❝tr✐❝ ✜❡❧❞ ❛t t❤❡s❡ r❡s♦♥❛♥t ✇❛✈❡❧❡♥❣t❤s ❤❛♥❝❡♠❡♥t t❤❡ str✉❝t✉r❡ ✐s ❞✉❡ t❤❡ s✉r✲ Fig provides the transmission asFigs a function of angles interface and the ✐♥s✐❞❡ substrate The electric fieldt♦toenhancement of incidence In detail, 4(a) and 4(b) respectively inside the structure is due the surface e ❢♦r ❜♦t❤ st❛t❡s✳ ❚❤❡ ❛♠♣❧✐t✉❞❡ ✐♥t❡r❢❡r❡♥❝❡ t♦ t❤❡ s✉❜str❛t❡ ❢r♦♠ ❡✈✲ incidence.show In detail, Figs.♣♦❧❛r✐③❛t✐♦♥ 4(a) transmission and 4(b) respectively inside ❢❛❝❡ the❡♥❡r❣② structure is due to the surface energy the calculated spectra for various interference to▼♦r❡♦✈❡r✱ the substrate from every grating pe ♠❛❣♥✐t✉❞❡s ♦❢ t❤❡spectra ❡❧❡❝tr✐❝ for ✜❡❧❞various ⑤❊②⑤ ❛♥❞ ♦❢ interference t❤❡ ❡r② ❣r❛t✐♥❣ ♣❡r✐♦❞✳ t❤❡ ❝♦✉♣❧❡❞ ✇❛✈❡s w the calculated transmission to the substrate fromwaves every grating period incident♠❛❣♥❡t✐❝ angles in✜❡❧❞ TM and TE polarizations As the angles Moreover, the coupled also take part in the su ⑤❍②⑤ ❢♦r t❤❡ ❜♦t❤ ❚❊ ❛♥❞ ❚▼ ❛❧s♦ t❛❦❡ ♣❛rt ✐♥ t❤❡ s✉r❢❛❝❡ ❡♥❡r❣② ✐♥t❡r❢❡r❡♥❝❡ ident angles TM and TE polarizations As the angles Moreover, the coupled waves also take part ingreater the surface of inincidence increase from to 15 degrees, the energy interference and it has a value tha ♠♦❞❡ from ✐♥ yˆ ❞✐r❡❝t✐♦♥ ❛r❡ s❤♦✇♥ ✐♥ ❋✐❣✳ ❋✐❣✳ ❛♥❞ ✐t ❤❛s ❛ ❣r❡❛t❡r ✈❛❧✉❡ t❤❛♥ t❤❡ ♠❛①✐♠✉♠ incidencetransmission increase to 15 degrees, the ✺✳ energy interference and it has a [22] greater value than the dip splits at λ = 641 nm which is clearly maximum electric field ✺✭❛✮ s❤♦✇s t❤❡ t♦t❛❧ ♠❛❣♥❡t✐❝ ✜❡❧❞ ❞✐str✐❜✉t✐♦♥ ❡❧❡❝tr✐❝ ✜❡❧❞ ❬✷✷❪✳ nsmission observed dip splits = 641 nm which is light clearly maximum electric field [22] in at Fig.λ 4(a) for TM-polarized Similarly, the erved in Fig 4(a) for TM-polarized light Similarly, the transmission feature splitting at λ = 737 nm and λ = 623 nsmission nm feature splitting at λlight = 737 nm and λ =observed 623 for TE-polarized is also clearly in Fig ❝ ✷✵✶✾ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ✹✵✾ for TE-polarized light is also clearly observed in Fig 4(b) ) ❱❖▲❯▼❊✿ ✸ | ■❙❙❯❊✿ ✷ | ✷✵✶✾ | ❏✉♥❡ (Guided-mode resonance filter with ultra-narrow bandwidth over the visible frequencies for label-free optical biosensor) (Guided-mode resonance filter with ultra-narrow bandwidth over the visible frequencies for label-free optical biosensor) (a) (a) (a) (b) (b) Simulated transmission as a function incident of incident Fig Simulated transmission as aoffunction ❙✐♠✉❧❛t❡❞ ❛ ❢✉♥❝t✐♦♥ ✐♥❝✐❞❡♥t e and wavelength for (a) TMtr❛♥s♠✐ss✐♦♥ polarization, and (b) TE♦❢and angle❋✐❣✳ and✹✿wavelength for (a) TM ❛s polarization, (b) TE ❛♥❣❧❡ ❛♥❞ ✇❛✈❡❧❡♥❣t❤ ❢♦r ✭❛✮ ❚▼ ♣♦❧❛r✐③❛t✐♦♥✱ rization The filter ❛♥❞ is sensitive to angles of incidence polarization The filter is sensitive to angles of incidence ✭❜✮ ❚❊ ♣♦❧❛r✐③❛t✐♦♥✳ ❚❤❡ ✜❧t❡r ✐s s❡♥s✐t✐✈❡ a strong splitting oft♦splitting the transmission features with a strong the transmission features ❛♥❣❧❡s ♦❢of✐♥❝✐❞❡♥❝❡ ✇✐t❤ ❛ str♦♥❣ s♣❧✐tt✐♥❣ ♦❢ t❤❡ tr❛♥s♠✐ss✐♦♥ ❢❡❛t✉r❡s✳ (a) (b) (b) ❋✐❡❧❞ ❞✐str✐❜✉t✐♦♥ ♣r♦✜❧❡s ❢♦r ❚▼ ❛♥❞ ❚❊ ❛s✲ s♦❝✐❛t❡❞ ✇✐t❤ t❤❡ tr❛♥s♠✐ss✐♦♥ ❢❡❛t✉r❡s ❛t Fig Field distribution profiles forprofiles TM and TE Fig Field distribution fort❤❡ TM and TE r❡s♦♥❛♥t ✇❛✈❡❧❡♥❣t❤s✳ ✭❛✮ ⑤❍②⑤ ❛t λthe ❂ ✵✳✻✹✶ µ♠ IV SURROUNDING REFRACTIVE associated with the transmission features at resonant IV SURROUNDING REFRACTIVE associated with the transmission features at the resonant ✇✐t❤ ✐♥❝✐❞❡♥t ❚▼✲♣♦❧❛r✐③❡❞ ✐♥ yˆ ❞✐r❡❝t✐♦♥✳ ✭❜✮ ✹✳ ❙❯❘❘❖❯◆❉■◆● wavelengths (a) |Hy| at λ(a) = |Hy| 0.641 incident TMINDEX EFFECT wavelengths atμm λ✐♥❝✐❞❡♥t =with 0.641 μm with incident TMINDEX EFFECT ⑤❊②⑤ ❛t λ ❂ ✵✳✻✷✸ µ♠ ✇✐t❤ ❚❊✲♣♦❧❛r✐③❡❞ polarized inpolarized 𝑦 ̂ yˆdirection |Ey| at λ(b)= |Ey| 0.623at μm Owing to Owing the GMR filter’s narrow bandwidth and in 𝑦̂ (b) direction λ =with 0.623 μm with ✐♥ ❞✐r❡❝t✐♦♥✳ to ❘❊❋❘❆❈❚■❱❊ the GMR filter’s narrow■◆❉❊❳ bandwidth and ❋✐❣✳ ✺✿ incident in 𝑦̂ direction sensitivity to the incidence environment, it is incident TE-polarized in 𝑦̂ direction high sensitivity to the incidence environment, it is TE-polarized ❊❋❋❊❈❚ ntial for sensing applications In this section, we potential for sensing applications In this section, we Generally, as the refractive of theindex surrounding stigateinvestigate a bulk refractive sensing application of Generally, as theindex refractive of the surrounding a bulk index refractive index sensing application of ❚▼ ❛♥❞ ❚❊ ♣♦❧❛r✐③❛t✐♦♥s✳ ❇❡s✐❞❡ t❤❡ s❡♥s✐t✐✈✲ environment increases, the resonant wavelength at the proposed GMR filter in the visible band Many works environment increases, the resonant wavelength at the the proposed GMR the visible band.❜❛♥❞✇✐❞t❤ Many works ❖✇✐♥❣ t♦ t❤❡ filter ●▼❘in ✜❧t❡r✬s ♥❛rr♦✇ ✐t② ❙ ✱ ❛♥♦t❤❡r ♠♦st ✐♠♣♦rt❛♥t ❢❛❝t♦r ❢♦r s❡♥s✐♥❣ transmission dip increases along with a corresponding ed to optical sensing applications have been proposed transmission dip increases along with a corresponding related to optical sensing applications have been proposed ❛♥❞ ❤✐❣❤ s❡♥s✐t✐✈✐t② t♦ t❤❡ ✐♥❝✐❞❡♥❝❡ ❡♥✈✐r♦♥✲ increase of increase the quality factor (Q-factor λres/FWHM) In/FWHM) In ❛♣♣❧✐❝❛t✐♦♥s ✐soft❤❡ ✜❣✉r❡ ♦❢ factor ♠❡r✐t= (Q-factor ✭❋❖▼✮ ✇❤✐❝❤ eviousinpublications [23-29] Fig shows the quality = λres previous [23-29] Fig.a ❛♣♣❧✐❝❛t✐♦♥s✳ 6resonant shows a resonant ♠❡♥t✱ ✐tpublications ✐s ♣♦t❡♥t✐❛❧ ❢♦r s❡♥s✐♥❣ ■♥ detail, when the refractive index of the surrounding ✐s ❞❡✜♥❡❞ ❛s ❛ r❛t✐♦ ❜❡t✇❡❡♥ s❡♥s✐t✐✈✐t② ❛♥❞ ❢✉❧❧✲ with respect to a change in the cover refractive index detail, when the refractive index of the surrounding shift t❤✐s with s❡❝t✐♦♥✱ respect to change in the coverr❡❢r❛❝t✐✈❡ refractive ✐♥✲ index ✇❡a ✐♥✈❡st✐❣❛t❡ ❛ ❜✉❧❦ increases, the resonant wavelength gradually ✇✐❞t❤environment ❛t ❤❛❧❢✲♠❛①✐♠✉♠ ✭❋❲❍▼✮ ❝❡♥t❡r❡❞ ❛t n = 1from to n n= = 1.31 in the wavelength range of 0.6range – 0.7of 0.6environment increases, the resonant wavelength gradually to n = 1.3 in the wavelength – 0.7 ❞❡① s❡♥s✐♥❣ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ ♣r♦♣♦s❡❞ ●▼❘ shifts t❤❡ to near-infrared and the Q-factor of the transmission the both TM and TE polarizations Beside the r❡s♦♥❛♥t ✇❛✈❡❧❡♥❣t❤✳ shifts to near-infrared and the Q-factor of the transmission μm ✜❧t❡r the both TM and TE polarizations Beside the ✐♥ t❤❡ ✈✐s✐❜❧❡ ❜❛♥❞✳ ▼❛♥② ✇♦r❦s r❡❧❛t❡❞spectrum t♦ increases For refractive indices higher than 1.3, itivity sensitivity S, another S,most important for sensing spectrum increases For refractive indices higher than 1.3, another most factor important factor for sensing ❚❤✐s ♦♣t✐❝❛❧ s❡♥s✐♥❣ ❛♣♣❧✐❝❛t✐♦♥s ❤❛✈❡ ❜❡❡♥ ♣r♦♣♦s❡❞ ❢❛❝t♦r ✐s ❛♣♣❧✐❡❞ t♦ ❢✉rt❤❡r ❡✈❛❧✉❛t❡ t❤❡ almost all mediums are solids and liquids, therefore Q-therefore Qications is the figure of merit (FOM) which is defined almost all mediums are solids and liquids, applications is the figure of merit (FOM) which is defined ✐♥ ♣r❡✈✐♦✉s ♣✉❜❧✐❝❛t✐♦♥s ❬✷✸✲✷✾❪✳at ❋✐❣✳ ✻ s❤♦✇s ♣❡r❢♦r♠❛♥❝❡ ❛s of t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥✱ factors❡♥s✐♥❣ will increase with falling FWHM The calculated ratioasbetween sensitivity and full-width halffactor will increase with falling of ✴FWHM The calculated a ❛ ratio between sensitivity andt♦ full-width half-❋❖▼❂ r❡s♦♥❛♥t s❤✐❢t ✇✐t❤ r❡s♣❡❝t ❛ ❝❤❛♥❣❡ ✐♥att❤❡ ❙ FOM ✴❋❲❍▼ ❬✸✵❪✱ 90nm/RIU ✇❤❡r❡ ❙ ❂and δλ δ ♥ for ✐s t❤❡ sensitivity and are about 1.93 TM1.93 for TM imum maximum (FWHM) centered atcentered the resonant wavelength sensitivity and FOM are about 90nm/RIU and (FWHM) at the resonant wavelength ❝♦✈❡r r❡❢r❛❝t✐✈❡ ✐♥❞❡①evaluate ❢r♦♠ ♥ the ❂ ✶sensing t♦ ♥ ❂ ✶✳✸polarization ✐♥ r❡❢r❛❝t✐✈❡ ✭✐✳❡✳✱ and s♣❡❝tr❛❧ s❤✐❢t while✐♥❞❡① abouts❡♥s✐t✐✈✐t② 103.33 nm/RIU 2.13 for TE 2.13 for TE This factorThis is applied to applied further polarization while about 103.33 nm/RIU and factor is evaluate the ❜♦t❤ sensing♣❡r r❡❢r❛❝t✐✈❡ t❤❡ ✇❛✈❡❧❡♥❣t❤ r❛♥❣❡to♦❢further ✵✳✻ ✕ ✵✳✼ µ♠ t❤❡ ✐♥❞❡①✮✱forδλ the ❛♥❞ δsame ♥ ❛r❡ structure, t❤❡ ✈❛r✐❛❜✐❧✲ polarization.polarization Obviously, TE ormance as the following relation, FOM=S/FWHM Obviously, for the same structure, TE performance as the following relation, FOM=S/FWHM polarizationpolarization results in higher FOM and In FOM In where S =where δλ/δn Sis =the refractive sensitivity results sensitivity in higher and sensitivity [30], δλ/δn is the index refractive index sensitivity comparison between the spectral properties of biosensing spectral shift per refractive index), δλ and δn are the comparison between the spectral properties of biosensing (i.e., spectral shift per refractive index), δλ and δn are the for the both TM and TE polarizations at ability of wavelength and refractive index, respectively ❝ applications for the both TM and TE polarizations at ✹✶✵ ✷✵✶✾ ❏♦✉r♥❛❧applications ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ variability of wavelength and refractive index, respectively normal incidence, the state of TE polarized light has a spectral properties of TM- and TE-polarized incident normal incidence, the state of TE polarized light has a The spectral properties of TM- and TE-polarized incident better performance in terms of sensitivity and FOM in the with respect to the surrounding medium refractive better performance in terms of sensitivity and FOM in the light with respect to the surrounding medium refractive same wavelength range Each of polarization states will x are presented in Table and Table 2, respectively same wavelength range Each of polarization states will index are presented in Table and Table 2, respectively JOURNAL OF ADVANCED ENGINEERING AND COMPUTATION http://dx.doi.org/ VOL 0, NO 0, 0-0, DEC 0000 ISSN (online): …-… ∙ ISSN (print): … primarily works as a wavelength selective polarizer Depending on the state of ❱❖▲❯▼❊✿ ✸ | ■❙❙❯❊✿ ✷ | ✷✵✶✾ | ❏✉♥❡ Refractive index, n ❚❛❜✳ ✶✿ 1.0 1.1 1.2 Resonant ❙♣❡❝tr❛❧ ♣r♦♣❡rt✐❡s ♦❢ ❜✐♦s❡♥s✐♥❣ ❛♣♣❧✐❝❛t✐♦♥s wavelength 623 632 ❢♦r ❚▼ ♣♦❧❛r✐③❛t✐♦♥✳ 642 (nm) ❘❡❢r❛❝t✐✈❡ ✐♥❞❡①✱ ❆t ❘❡s♦♥❛♥t ♥Q-factor ✶✳✵ ✶✳✶ ✻✹✶ ✻✹✽ At ✇❛✈❡❧❡♥❣t❤ Sensitivity S dip ✭♥♠✮ ◗ ✲❢❛❝t♦r (nm/RIU) ✶✸✳✷✹ ✶✸✳✺ ❞✐♣ ❙❡♥s✐t✐✈✐t② ❙ ✭♥♠✴❘■❯✮ Figure-of❋✐❣✉r❡✲♦❢✲merit ✶✳✷ 12.71 ✶✳✸ 12.98 ✻✺✼ ✻✻✽ ✶✸✳✾✽ 13.24 Saverage = 103.33 ✶✺✳✺✸ ❙average ❂ ✾✵ ❋❖▼average (FOM) FOMaverage = 2.13 ❂ ✶✳✾✸ ♠❡r✐t ✭❋❖▼✮ Table Spectral properties of biosensing applicatio TE polarization (a) ❚❛❜✳ ✷✿ ❙♣❡❝tr❛❧ ♣r♦♣❡rt✐❡s ♦❢ ❜✐♦s❡♥s✐♥❣ ❛♣♣❧✐❝❛t✐♦♥s ❢♦r ❚❊ ♣♦❧❛r✐③❛t✐♦♥✳ V CONCLUSIONS We✐♥❞❡①✱ have ❘❡❢r❛❝t✐✈❡ (b) an filter ope ♥presented ✶✳✵ ✶✳✶ultra-narrowband ✶✳✷ ✶✳✸ ❆t in ❘❡s♦♥❛♥t the visible ❞✐♣ ✇❛✈❡❧❡♥❣t❤ filter almost ✻✷✸ ✻✸✷ ✻✹✷ ✻✺✹ band of the electromagnetism spectrum blocks the lightwave transmission a ✭♥♠✮ resonant wavelength while passing it over the rem ◗ ✲❢❛❝t♦r ✶✷✳✼✶ ✶✷✳✾✽ ✶✸✳✷✹ ✶✸✳✺✼ wavelengths out of the resonance for both TM an ❙❡♥s✐t✐✈✐t② ❙average ❂ ✶✵✸✳✸✸ polarization ❙ ✭♥♠✴❘■❯✮states The transmission features ca amended by adjusting theaverage incident angles, the polari ❋✐❣✉r❡✲♦❢✲ ❋❖▼ ❂ ✷✳✶✸ of♠❡r✐t incident light, and the refractive index o ✭❋❖▼✮ surrounding environment Such selective characteris the proposed GMR filter meet demands for pra applications such as tunable optical filters, v imaging or detection, and refractive index sensing ❛❧❧ ♠❡❞✐✉♠s ❛r❡ s♦❧✐❞s ❛♥❞ ❧✐q✉✐❞s✱ t❤❡r❡❢♦r❡ ❋✐❣✳ ✻✿ ❚r❛♥s♠✐ss✐♦♥ ❞✐♣ s❤✐❢t ✇✐t❤ t❤❡ r❡❢r❛❝t✐✈❡ ✐♥❞❡① Fig Transmission dip shift with the refractive index of ♦❢ t❤❡ s✉rr♦✉♥❞✐♥❣ ❡♥✈✐r♦♥♠❡♥t ✈❛r✐❡❞ ❢r♦♠ ✶ t♦ ◗✲❢❛❝t♦r ✇✐❧❧ ✐♥❝r❡❛s❡ ✇✐t❤ ❢❛❧❧✐♥❣ ♦❢ ❋❲❍▼✳ the surrounding ✶✳✸ environment to ✭❜✮✳ 1.3 for TM ❚❤❡ ❝❛❧❝✉❧❛t❡❞ ❢♦r ❚▼ ♠♦❞❡varied ✭❛✮ ❛♥❞from ❚❊ ♠♦❞❡ References:s❡♥s✐t✐✈✐t② ❛♥❞ ❋❖▼ ❛r❡ ❛❜♦✉t mode (a) and TE mode (b) ✾✵♥♠✴❘■❯ ❛♥❞ ✶✳✾✸ ❢♦r ❚▼ ♣♦❧❛r✐③❛t✐♦♥ ✇❤✐❧❡ R MAGNUSSON, WANG, S “New princip ❛❜♦✉t[1] ✶✵✸✳✸✸ ♥♠✴❘■❯ ❛♥❞ ✷✳✶✸ ❢♦r ❚❊ S ♣♦❧❛r✐③❛✲ polarization of incident light and the refractive index of t✐♦♥✳ ❖❜✈✐♦✉s❧②✱ optical filters”, Lett., ❚❊ vol.♣♦✲ 61, no 9, pp ❢♦r t❤❡ Appl s❛♠❡ Phys str✉❝t✉r❡✱ ♦❢ ✇❛✈❡❧❡♥❣t❤ r❡❢r❛❝t✐✈❡ r❡s♣❡❝✲ the cover✐t② medium, the ideal❛♥❞ sensor should ✐♥❞❡①✱ be based on TE ❧❛r✐③❛t✐♦♥ 1992 r❡s✉❧ts ✐♥ ❤✐❣❤❡r s❡♥s✐t✐✈✐t② ❛♥❞ ❋❖▼✳ t✐✈❡❧②✳ ❚❤❡ s♣❡❝tr❛❧ ♣r♦♣❡rt✐❡s ♦❢ ❚▼✲ ❛♥❞ ❚❊✲ [2] WANG, S S., R.t❤❡ MAGNUSSON “Theory and appli mode with greater Saverage and FOMaverage to fabricate ■♥ ❝♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ s♣❡❝tr❛❧ ♣r♦♣❡rt✐❡s ♣♦❧❛r✐③❡❞ ✐♥❝✐❞❡♥t ❧✐❣❤t ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ s✉r✲ r♦✉♥❞✐♥❣ ♠❡❞✐✉♠ r❡❢r❛❝t✐✈❡ ✐♥❞❡① ❛r❡ ♣r❡s❡♥t❡❞ Refractive✐♥index, 1.2 1.3 ❚❛❜❧❡n✶ ❛♥❞ 1.0 ❚❛❜❧❡ ✷✱ 1.1 r❡s♣❡❝t✐✈❡❧②✳ At dip of guided-mode filters,” ♦❢ ❜✐♦s❡♥s✐♥❣ ❛♣♣❧✐❝❛t✐♦♥sresonance ❢♦r t❤❡ ❜♦t❤ ❚▼Appl ❛♥❞ Opt., vol 14, pp 2606–2613, ❚❊ ♣♦❧❛r✐③❛t✐♦♥s ❛t ♥♦r♠❛❧ 1993 ✐♥❝✐❞❡♥❝❡✱ t❤❡ st❛t❡ ♦❢ ❚❊ ♣♦❧❛r✐③❡❞ ❧✐❣❤t ❤❛s ❛ ❜❡tt❡r ♣❡r❢♦r♠❛♥❝❡ [3] E POPOV, L MASHEV, and D MAYSTRE “Theo Resonant ●❡♥❡r❛❧❧②✱ ❛s t❤❡ r❡❢r❛❝t✐✈❡ ✐♥❞❡① ♦❢ t❤❡ s✉r✲ ✐♥ t❡r♠s ♦❢ s❡♥s✐t✐✈✐t② ❛♥❞ ❋❖▼ ✐♥ t❤❡ s❛♠❡ study of the anomalies of coated dielectric gratings,” wavelength 641 648 657t❤❡ r❡s♦♥❛♥t 668 r♦✉♥❞✐♥❣ ❡♥✈✐r♦♥♠❡♥t ✐♥❝r❡❛s❡s✱ ✇❛✈❡❧❡♥❣t❤ r❛♥❣❡✳ ❊❛❝❤ ♦❢ ♣♦❧❛r✐③❛t✐♦♥ st❛t❡s no 5, pp.tr❛♥s♠✐ss✐♦♥ 607–619, 1986 (nm) ✇❛✈❡❧❡♥❣t❤ ❛t t❤❡ tr❛♥s♠✐ss✐♦♥ ❞✐♣ ✐♥❝r❡❛s❡s ✇✐❧❧ ❝r❡❛t❡Opt., ✐ts vol ♦✇♥33,r❡s♦♥❛♥t ❞✐♣s ❛❧♦♥❣ ✇✐t❤ ❛ ❝♦rr❡s♣♦♥❞✐♥❣ ✐♥❝r❡❛s❡ ♦❢ t❤❡ q✉❛❧✲ ♦❝❝✉rr✐♥❣ ❛t ❞✐✛❡r❡♥t ✇❛✈❡❧❡♥❣t❤s✳ ❚❤❡r❡❢♦r❡✱ [4] D LACOUR, G GRANET, PLUMEY, J P., a Q-factor 13.24 13.5 13.98 15.53 ✐t② ❢❛❝t♦r ✭◗✲❢❛❝t♦r ❂ λres ✴❋❲❍▼✮✳ ■♥ ❞❡✲ t❤❡ ♣r♦♣♦s❡❞ ●▼❘ ✜❧t❡r ♣r✐♠❛r✐❧② ✇♦r❦s ❛s ❛ MURE-RAVAUD “Polarization independence of t❛✐❧✱ ✇❤❡♥ t❤❡ r❡❢r❛❝t✐✈❡ ✐♥❞❡① ♦❢ t❤❡ s✉rr♦✉♥❞✲ ✇❛✈❡❧❡♥❣t❤ s❡❧❡❝t✐✈❡ ♣♦❧❛r✐③❡r✳ ❉❡♣❡♥❞✐♥❣ ♦♥ dimensional grating in conical mounting,” J Opt So Sensitivity S Saverage 90 ✐♥❣ ❡♥✈✐r♦♥♠❡♥t ✐♥❝r❡❛s❡s✱ t❤❡= r❡s♦♥❛♥t ✇❛✈❡✲ t❤❡ st❛t❡ ♦❢ ✐♥❝✐❞❡♥t ❧✐❣❤t 2003 ❛♥❞ t❤❡ A.,♣♦❧❛r✐③❛t✐♦♥ vol 20, no 8,♦❢pp 1546–1552, (nm/RIU) ❧❡♥❣t❤ ❣r❛❞✉❛❧❧② s❤✐❢ts t♦ ♥❡❛r✲✐♥❢r❛r❡❞ ❛♥❞ t❤❡ r❡❢r❛❝t✐✈❡ ✐♥❞❡① ♦❢ t❤❡ ❝♦✈❡r ♠❡❞✐✉♠✱ t❤❡ ✐❞❡❛❧ G NIEDERER, NAKAGAWA, HERZIG, H P., ◗✲❢❛❝t♦r ♦❢ t❤❡ tr❛♥s♠✐ss✐♦♥ s♣❡❝tr✉♠ ✐♥❝r❡❛s❡s✳ s❡♥s♦r[5]s❤♦✉❧❞ ❜❡ ❜❛s❡❞ ♦♥W ❚❊ ♠♦❞❡ ✇✐t❤ ❣r❡❛t❡r Figure-of❋♦r r❡❢r❛❝t✐✈❡ ✐♥❞✐❝❡s FOM ❤✐❣❤❡raverage t❤❛♥ ✶✳✸✱ ❛❧♠♦st ❙average ❛♥❞ ❋❖▼average Thiele “Designt♦ ❢❛❜r✐❝❛t❡✳ and characterization of a t = 1.93 merit (FOM) polarization-independent resonant grating filter,” Opt vol 13, no 6, pp 2196–2200, 2005 ❝ ✷✵✶✾ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ applications ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✹✶✶J R., KELL Table Spectral properties of biosensing for✭❏❆❊❈✮[6] PETERS, D W., BOYE, R R., WENDT, TM polarization R A., KEMME, S A., CARTER, T R., and S SAM “Demonstration of polarization-independent re ❱❖▲❯▼❊✿ ✸ ✺✳ ❈❖◆❈▲❯❙■❖◆❙ ❲❡ ❤❛✈❡ ♣r❡s❡♥t❡❞ ❛♥ ✉❧tr❛✲♥❛rr♦✇❜❛♥❞ ✜❧t❡r ♦♣❡r❛t✐♥❣ ✐♥ t❤❡ ✈✐s✐❜❧❡ ❜❛♥❞ ♦❢ t❤❡ ❡❧❡❝tr♦♠❛❣✲ ♥❡t✐s♠ s♣❡❝tr✉♠✳ ❚❤❡ ✜❧t❡r ❛❧♠♦st ❜❧♦❝❦s t❤❡ ❧✐❣❤t✇❛✈❡ tr❛♥s♠✐ss✐♦♥ ❛t t❤❡ r❡s♦♥❛♥t ✇❛✈❡✲ ❧❡♥❣t❤ ✇❤✐❧❡ ♣❛ss✐♥❣ ✐t ♦✈❡r t❤❡ r❡♠❛✐♥✐♥❣ ✇❛✈❡✲ ❧❡♥❣t❤s ♦✉t ♦❢ t❤❡ r❡s♦♥❛♥❝❡ ❢♦r ❜♦t❤ ❚▼ ❛♥❞ ❚❊ ♣♦❧❛r✐③❛t✐♦♥ st❛t❡s✳ ❚❤❡ tr❛♥s♠✐ss✐♦♥ ❢❡❛✲ t✉r❡s ❝❛♥ ❜❡ ❛♠❡♥❞❡❞ ❜② ❛❞❥✉st✐♥❣ t❤❡ ✐♥❝✐❞❡♥t ❛♥❣❧❡s✱ t❤❡ ♣♦❧❛r✐③❛t✐♦♥ ♦❢ ✐♥❝✐❞❡♥t ❧✐❣❤t✱ ❛♥❞ t❤❡ r❡❢r❛❝t✐✈❡ ✐♥❞❡① ♦❢ t❤❡ s✉rr♦✉♥❞✐♥❣ ❡♥✈✐r♦♥✲ ♠❡♥t✳ ❙✉❝❤ s❡❧❡❝t✐✈❡ ❝❤❛r❛❝t❡r✐st✐❝s ♦❢ t❤❡ ♣r♦✲ ♣♦s❡❞ ●▼❘ ✜❧t❡r ♠❡❡t ❞❡♠❛♥❞s ❢♦r ♣r❛❝t✐❝❛❧ ❛♣♣❧✐❝❛t✐♦♥s s✉❝❤ ❛s t✉♥❛❜❧❡ ♦♣t✐❝❛❧ ✜❧t❡rs✱ ✈❛r✲ ✐♦✉s ✐♠❛❣✐♥❣ ♦r ❞❡t❡❝t✐♦♥✱ ❛♥❞ r❡❢r❛❝t✐✈❡ ✐♥❞❡① s❡♥s✐♥❣✳ | ■❙❙❯❊✿ ✷ | ✷✵✶✾ | ❏✉♥❡ ♦❢ ♣♦❧❛r✐③❛t✐♦♥✲✐♥❞❡♣❡♥❞❡♥t r❡s♦♥❛♥t s✉❜✲ ✇❛✈❡❧❡♥❣t❤ ❣r❛t✐♥❣ ✜❧t❡r ❛rr❛②s✳ ❖♣t✳ ▲❡tt✳✱ ✸✺ ✭✻✮✱ ✸✷✵✶✕✸✷✵✸✳ ❬✼❪ ❲❛♥❣✱ ❙✳❙✳✱ ✫ ▼❛❣♥✉ss♦♥✱ ❘✳ ✭✶✾✾✹✮✳ ❉❡✲ s✐❣♥ ♦❢ ✇❛✈❡❣✉✐❞❡✲❣r❛t✐♥❣ ✜❧t❡rs ✇✐t❤ s②♠✲ ♠❡tr✐❝❛❧ ❧✐♥❡ s❤❛♣❡s ❛♥❞ ❧♦✇ s✐❞❡❜❛♥❞s✳ ❖♣t✳ ▲❡tt✳✱ ✶✾ ✭✶✷✮✱ ✾✶✾✕✾✷✶✳ ❬✽❪ ❙❛♥❣✱ ❚✳✱ ❈❛✐✱ ❙✳✱ ✫ ❲❛♥❣✱ ❩✳ ✭✷✵✶✶✮✳ ●✉✐❞❡❞✲♠♦❞❡ r❡s♦♥❛♥❝❡ ✜❧t❡r ✇✐t❤ ❛♥ ❛♥✲ t✐r❡✢❡❝t✐✈❡ s✉r❢❛❝❡ ❝♦♥s✐st✐♥❣ ♦❢ ❛ ❜✉✛❡r ❧❛②❡r ✇✐t❤ r❡❢r❛❝t✐✈❡ ✐♥❞❡① ❡q✉❛❧ t♦ t❤❛t ♦❢ t❤❡ ❣r❛t✐♥❣✳ ❏✳ ▼♦❞✳ ❖♣t✱ ✺✽ ✭✶✹✮✱ ✶✷✻✵✕ ✶✷✻✽✳ ❬✾❪ ▲✐✉✱ ❩✳❙✳✱ ❚✐❜✉❧❡❛❝✱ ❙✳✱ ❙❤✐♥✱ ❉✳✱ ❨♦✉♥❣✱ P✳P✳✱ ✫ ▼❛❣♥✉ss♦♥✱ ❘✳ ✭✶✾✾✽✮✳ ❍✐❣❤✲ ❡✣❝✐❡♥❝② ❣✉✐❞❡❞✲♠♦❞❡ r❡s♦♥❛♥❝❡ ✜❧t❡r✳ ❖♣t✳ ▲❡tt✳✱ ✷✸ ✭✶✾✮✱ ✶✺✺✻✕✶✺✺✽✳ ❬✶✵❪ ◗✉❛♥✱ ❍✳ ✫ ●✉♦✱ ❩✳ ✭✷✵✵✺✮✳ ❙✐♠✉❧❛t✐♦♥ ♦❢ ✇❤✐s♣❡r✐♥❣✲❣❛❧❧❡r②✲♠♦❞❡ r❡s♦♥❛♥❝❡ s❤✐❢ts ❢♦r ♦♣t✐❝❛❧ ♠✐♥✐❛t✉r❡ ❜✐♦s❡♥s♦rs✳ ❏✳ ◗✉❛♥t✳ ▼❛❣♥✉ss♦♥✱ ❘✳✱ ✫ ❲❛♥❣✱ ❙✳❙✳ ✭✶✾✾✷✮✳ ◆❡✇ ❙♣❡❝tr♦s❝✳ ❘❛❞✐❛t✳ ❚r❛♥s❢✳✱ ✾✸ ✭✶✮✱ ✷✸✶✕✷✹✸✳ ♣r✐♥❝✐♣❧❡ ❢♦r ♦♣t✐❝❛❧ ✜❧t❡rs✳ ❆♣♣❧✳ P❤②s✳ ❬✶✶❪ ▲❡❡✱ ❑✳❏✳✱ ✫ ▼❛❣♥✉ss♦♥✱ ❘✳ ✭✷✵✶✶✮✳ ❙✐♥❣❧❡✲ ▲❡tt✳✱ ✻✶ ✭✾✮✱ ✶✵✷✷✳ ▲❛②❡r ❘❡s♦♥❛♥t ❍✐❣❤ ❘❡✢❡❝t♦r ✐♥ ❚❊ P♦✲ ❧❛r✐③❛t✐♦♥✿ ❚❤❡♦r② ❛♥❞ ❊①♣❡r✐♠❡♥t✳ ■❊❊❊✳ ❲❛♥❣✱ ❙✳❙✳✱ ✫ ▼❛❣♥✉ss♦♥✱ ❘✳ ✭✶✾✾✸✮✳ ❚❤❡✲ P❤♦t♦♥✳ ❏✱ ✸ ✭✶✮✱ ✶✷✸✕✶✷✾✳ ♦r② ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s ♦❢ ❣✉✐❞❡❞✲♠♦❞❡ r❡s✲ ♦♥❛♥❝❡ ✜❧t❡rs✳ ❆♣♣❧✳ ❖♣t✳✱ ✸✷ ✭✶✹✮✱ ✷✻✵✻✕ ❬✶✷❪ ❩❤♦♥❣✱ ❨✳✱ ●♦❧❞❡♥❢❡❧❞✱ ❩✳✱ ▲✐✱ ❑✳✱ ❙tr❡②❡r✱ ✷✻✶✸✳ ❲✳✱ ❨✉✱ ▲✳✱ ◆♦r❞✐♥✱ ▲✳✱ ▼✉r♣❤②✱ ◆✳✱ ✫ ❲❛ss❡r♠❛♥✱ ❉✳ ✭✷✵✶✼✮✳ ▼✐❞✲✇❛✈❡ ✐♥❢r❛r❡❞ P♦♣♦✈✱ ❊✳✱ ▼❛s❤❡✈✱ ▲✳✱ ✫ ▼❛②str❡✱ ❉✳ ♥❛rr♦✇ ❜❛♥❞✇✐❞t❤ ❣✉✐❞❡❞ ♠♦❞❡ r❡s♦♥❛♥❝❡ ✭✶✾✽✻✮✳ ❚❤❡♦r❡t✐❝❛❧ st✉❞② ♦❢ t❤❡ ❛♥♦♠❛❧✐❡s ♥♦t❝❤ ✜❧t❡r✳ ❖♣t✳ ▲❡tt✳✱ ✹✷ ✭✷✮✱ ✷✷✸✕✷✷✻✳ ♦❢ ❝♦❛t❡❞ ❞✐❡❧❡❝tr✐❝ ❣r❛t✐♥❣s✳ ❏✳ ▼♦❞✳ ❖♣t✳✱ ✸✸ ✭✺✮✱ ✻✵✼✕✻✶✾✳ ❬✶✸❪ ❩❡♥❣✱ ●✳✱ ❩♦✉✱ ❳✳✱ ❳✉✱ ▲✳✱ ✫ ❲❛♥❣✱ ❏✳ ✭✷✵✶✼✮✳ ❙✐♥❣❧❡ ❧❛②❡r ♥❛rr♦✇ ❜❛♥❞✇✐❞t❤ ▲❛❝♦✉r✱ ❉✳✱ ●r❛♥❡t✱ ●✳✱ P❧✉♠❡②✱ ❏✳P✳✱ ✫ ❛♥❣❧❡✲✐♥s❡♥s✐t✐✈❡ ❣✉✐❞❡❞✲♠♦❞❡ r❡s♦♥❛♥❝❡ ▼✉r❡✲❘❛✈❛✉❞✱ ❆✳ ✭✷✵✵✸✮✳ P♦❧❛r✐③❛t✐♦♥ ✐♥✲ ❜❛♥❞st♦♣ ✜❧t❡rs✳ ❖♣t✐❦✱ ✶✸✵✱ ✶✾✕✷✸✳ ❞❡♣❡♥❞❡♥❝❡ ♦❢ ❛ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ❣r❛t✐♥❣ ✐♥ ❝♦♥✐❝❛❧ ♠♦✉♥t✐♥❣✳ ❏✳ ❖♣t✳ ❙♦❝✳ ❆♠✳ ❆✱ ❬✶✹❪ ❈❤❛♥❣✲❍❛s♥✐❛♥✱ ❈✳❏✳✱ ✫ ❨❛♥❣✱ ❲✳ ✭✷✵✶✷✮✳ ❍✐❣❤✲❝♦♥tr❛st ❣r❛t✐♥❣s ❢♦r ✐♥t❡❣r❛t❡❞ ♦♣t♦✲ ✷✵ ✭✽✮✱ ✶✺✹✻✕✶✺✺✷✳ ❡❧❡❝tr♦♥✐❝s✳ ❆❞✈✳ ❖♣t✳ P❤♦t♦♥✳✱ ✹ ✭✸✮✱ ✸✼✾✕ ✹✹✵✳ ◆✐❡❞❡r❡r✱ ●✳✱ ◆❛❦❛❣❛✇❛✱ ❲✳✱ ❍❡r③✐❣✱ ❍✳P✳✱ ❘❡❢❡r❡♥❝❡s ❬✶❪ ❬✷❪ ❬✸❪ ❬✹❪ ❬✺❪ ✫ ❚❤✐❡❧❡✱ ❍✳ ✭✷✵✵✺✮✳ ❉❡s✐❣♥ ❛♥❞ ❝❤❛r✲ ❬✶✺❪ ❆❦❛❤❛♥❡✱ ❨✳✱ ❆s❛♥♦✱ ❚✳✱ ❙♦♥❣✱ ❇✳❙✳✱ ✫ ❛❝t❡r✐③❛t✐♦♥ ♦❢ ❛ t✉♥❛❜❧❡ ♣♦❧❛r✐③❛t✐♦♥✲ ◆♦❞❛✱ ❙✳ ✭✷✵✵✸✮✳ ❍✐❣❤✲◗ ♣❤♦t♦♥✐❝ ♥❛♥♦❝❛✈✲ ✐♥❞❡♣❡♥❞❡♥t r❡s♦♥❛♥t ❣r❛t✐♥❣ ✜❧t❡r✳ ❖♣t✳ ✐t② ✐♥ ❛ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ ♣❤♦t♦♥✐❝ ❝r②st❛❧✳ ❊①♣r✳✱ ✶✸ ✭✻✮✱ ✷✶✾✻✕✷✷✵✵✳ ◆❛t✉r❡✱ ✹✷✺✱ ✾✹✹✕✾✹✼✳ ❬✻❪ P❡t❡rs✱ ❉✳❲✳✱ ❇♦②❡✱ ❘✳❘✳✱ ❲❡♥❞t✱ ❏✳❘✳✱ ❬✶✻❪ ●♦❧✉❜❡♥❦♦✱ ●✳❆✳✱ ❙✈❛❦❤✐♥✱ ❆✳❙✳✱ ❙②❝❤✉❣♦✈✱ ❑❡❧❧♦❣❣✱ ❘✳❆✳✱ ❑❡♠♠❡✱ ❙✳❆✳✱ ❈❛rt❡r✱ ❱✳❆✳✱ ✫ ❚✐s❤❝❤❡♥❦♦✱ ❆✳❱✳ ✭✶✾✽✺✮✳ ❚♦t❛❧ r❡✲ ❚✳❘✳✱ ✫ ❙❛♠♦r❛✱ ❙✳ ✭✷✵✶✵✮✳ ❉❡♠♦♥str❛t✐♦♥ ✢❡❝t✐♦♥ ♦❢ ❧✐❣❤t ❢r♦♠ ❛ ❝♦rr✉❣❛t❡❞ s✉r❢❛❝❡ ✹✶✷ ❝ ✷✵✶✾ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ❱❖▲❯▼❊✿ ✸ ♦❢ ❛ ❞✐❡❧❡❝tr✐❝ ✇❛✈❡❣✉✐❞❡✳ ❙♦✈✳ ❏✳ ◗✉❛♥t✉♠✳ ❊❧❡❝tr♦♥✳✱ ✶✺✱ ✽✽✻✕✽✽✼✳ | ■❙❙❯❊✿ ✷ | ✷✵✶✾ | ❏✉♥❡ ❲❛♥❣✱ ❳✳✱ ❏✐♥✱ ❈✳✱ ✫ ❲❛♥❣✱ ❏✳ ✭✷✵✶✸✮✳ P❧❛s✲ ♠♦♥✐❝ ❣♦❧❞ ♠✉s❤r♦♦♠ ❛rr❛②s ✇✐t❤ r❡❢r❛❝✲ t✐✈❡ ✐♥❞❡① s❡♥s✐♥❣ ✜❣✉r❡s ♦❢ ♠❡r✐t ❛♣♣r♦❛❝❤✲ ✐♥❣ t❤❡ t❤❡♦r❡t✐❝❛❧ ❧✐♠✐t✳ ◆❛t✳ ❈♦♠♠✉♥✳✱ ✹✱ ✷✸✽✶✳ ❬✶✼❪ ◆❣♦✱ ◗✳▼✳✱ ▲❡✱ ❑✳◗✳✱ ✫ ❱✉✱ ❉✳▲✳ ✭✷✵✶✷✮✳ ❖♣t✐❝❛❧ ❜✐st❛❜✐❧✐t② ❜❛s❡❞ ♦♥ ❣✉✐❞❡❞✲♠♦❞❡ r❡s♦♥❛♥❝❡s ✐♥ ♣❤♦t♦♥✐❝ ❝r②st❛❧ s❧❛❜s✳ ❏✳ ❖♣t✳ ❬✷✻❪ ❊❧✲●♦❤❛r②✱ ❙✳❍✱ ❈❤♦✐✱ ❏✳▼✱ ❑✐♠✱ ◆✳❍✳✱ ❙♦❝✳ ❆♠✳ ❇✳✱ ✷✾ ✭✻✮✱ ✶✷✾✶✕✶✷✾✺✳ ✫ ❇②✉♥✱ ❑✳▼✳ ✭✷✵✶✹✮✳ P❧❛s♠♦♥✐❝ ♠❡t❛❧✕ ❞✐❡❧❡❝tr✐❝✕♠❡t❛❧ st❛❝❦ str✉❝t✉r❡ ✇✐t❤ s✉❜✲ ❬✶✽❪ ▲❡✱ ❑✳◗✳✱ ◆❣♦✱ ◗✳▼✳✱ ✫ ◆❣✉②❡♥✱ ❚✳❑✳ ✇❛✈❡❧❡♥❣t❤ ♠❡t❛❧❧✐❝ ❣r❛t✐♥❣s ❢♦r ✐♠♣r♦✈✐♥❣ ✭✷✵✶✼✮✳ ◆❛♥♦str✉❝t✉r❡❞ ♠❡t❛❧✲✐♥s✉❧❛t♦r✲ s❡♥s♦r s❡♥s✐t✐✈✐t② ❛♥❞ s✐❣♥❛❧ q✉❛❧✐t②✳ ❆♣♣❧✳ ♠❡t❛❧ ♠❡t❛♠❛t❡r✐❛❧s ❢♦r r❡❢r❛❝t✐✈❡ ✐♥❞❡① ❖♣t✳✱ ✺✸ ✭✶✵✮✱ ✷✶✺✷✕✷✶✺✼✳ ❜✐♦s❡♥s✐♥❣ ❛♣♣❧✐❝❛t✐♦♥s✿ ❉❡s✐❣♥✱ ❢❛❜r✐✲ ❝❛t✐♦♥✱ ❛♥❞ ❝❤❛r❛❝t❡r✐③❛t✐♦♥✳ ■❊❊❊✳ ❏✳ ❙❡❧✳ ❚♦♣✐❝s✳ ◗✉❛♥t✉♠✳ ❊❧❡❝tr♦♥✳✱ ✷✸ ✭✷✮✱ ❬✷✼❪ ❲❛♥❣✱ ❨✳✱ ❑❛r✱ ❆✳✱ P❛t❡rs♦♥✱ ❆✳✱ ❑♦✉r❡♥t③✐✱ ❑✳✱ ▲❡✱ ❍✳✱ ❘✉❝❤❤♦❡❢t✱ P✳✱ ❲✐❧❧s♦♥✱ ❘✳✱ ✫ ✻✾✵✵✺✵✻✳ ❇❛♦✱ ❏✳ ✭✷✵✶✹✮✳ ❚r❛♥s♠✐ss✐✈❡ ◆❛♥♦❤♦❧❡ ❆r✲ r❛②s ❢♦r ▼❛ss✐✈❡❧②✲P❛r❛❧❧❡❧ ❖♣t✐❝❛❧ ❇✐♦s❡♥s✲ ❬✶✾❪ ▲❡✱ ❑✳◗✳✱ ❇❛✐✱ ❏✳✱ ◆❣♦✱ ◗✳▼✳✱ ✫ ❈❤❡♥✱ P✳❨✳ ✐♥❣✳ ❆❈❙✳ P❤♦t♦♥✐❝s✱ ✶ ✭✸✮✱ ✷✹✶✕✷✹✺✳ ✭✷✵✶✻✮✳ ❋❛❜r✐❝❛t✐♦♥ ❛♥❞ ◆✉♠❡r✐❝❛❧ ❈❤❛r✲ ❛❝t❡r✐③❛t✐♦♥ ♦❢ ■♥❢r❛r❡❞ ▼❡t❛♠❛t❡r✐❛❧ ❆❜✲ s♦r❜❡rs ❢♦r ❘❡❢r❛❝t♦♠❡tr✐❝ ❇✐♦s❡♥s♦rs✳ ❏✳ ❊❧❡❝tr♦♥✳ ▼❛t❡r✳✱ ✹✻ ✭✶✮✱ ✻✻✽✕✻✼✻✳ ❬✷✽❪ ▼❛②❡r❤ö❢❡r✱ ❚✳●✳✱ ❑♥✐♣♣❡r✱ ❘✳✱ ❍ü❜♥❡r✱ ❯✳✱ ❈✐❛❧❧❛✲▼❛②✱ ❉✳✱ ❲❡❜❡r✱ ❑✳✱ ▼❡②❡r✱ ❍✳●✳✱ ✫ P♦♣♣✱ ❏✳ ✭✷✵✶✺✮✳ ❯❧tr❛❙❡♥s✐♥❣ ❜② ❈♦♠✲ ❜✐♥✐♥❣ ❊①tr❛♦r❞✐♥❛r② ❖♣t✐❝❛❧ ❚r❛♥s♠✐ss✐♦♥ ❬✷✵❪ ❱♦✲❉✐♥❤✱ ❚✳✱ ●r✐✣♥✱ ●✳✱ ❙t♦❦❡s✱ ❉✳▲✳✱ ✇✐t❤ P❡r❢❡❝t ❆❜s♦r♣t✐♦♥✳ ❆❈❙✳ P❤♦t♦♥✐❝s✳✱ ❙tr❛t✐s✲❈✉❧❧✉♠✱ ❉✳◆✳✱ ❆s❦❛r✐✱ ▼✳✱ ✫ ❲✐♥✲ ✷ ✭✶✶✮✱ ✶✺✻✼✕✶✺✼✺✳ t❡♥❜❡r❣✱ ❆✳ ✭✷✵✵✹✮✳ ❖♣t✐❝❛❧ s❡♥s♦rs✿ ■♥❞✉s✲ tr✐❛❧ ❡♥✈✐r♦♥♠❡♥t❛❧ ❛♥❞ ❞✐❛❣♥♦st✐❝ ❛♣♣❧✐❝❛✲ ❬✷✾❪ ▲✉✱ ❳✳✱ ❲❛♥✱ ❘✳✱ ✫ ❩❤❛♥❣✱ ❚✳ ✭✷✵✶✺✮✳ t✐♦♥s✳ ❇❡r❧✐♥✿ ❙♣r✐♥❣❡r✳ ▼❡t❛❧✲❞✐❡❧❡❝tr✐❝✲♠❡t❛❧ ❜❛s❡❞ ♥❛rr♦✇ ❜❛♥❞ ❛❜s♦r❜❡r ❢♦r s❡♥s✐♥❣ ❛♣♣❧✐❝❛t✐♦♥s✳ ❖♣t✳ ❬✷✶❪ ❈❙❚ ●♠❜❍✱ ❈❙❚ ▼✐❝r♦✇❛✈❡ ❊①♣r✳✱ ✷✸ ✭✷✸✮✱ ✷✾✽✹✷✕✷✾✽✹✼✳ ❙t✉❞✐♦✳ ✭✷✵✶✻✮✳ ❘❡tr✐❡✈❡❞ ❢r♦♠ ❤tt♣✿✴✴✇✇✇✳❝st✳❝♦♠ ❬✸✵❪ ▲❡✱ ❑✳◗✳✱ ❆❧ù✱ ❆✳✱ ✫ ❇❛✐✱ ❏✳ ✭✷✵✶✺✮✳ ▼✉❧t✐✲ ♣❧❡ ❋❛♥♦ ✐♥t❡r❢❡r❡♥❝❡s ✐♥ ❛ ♣❧❛s♠♦♥✐❝ ♠❡t❛✲ ❬✷✷❪ ❙✉♥✱ ❚✳✱ ▼❛✱ ❏✳✱ ❲❛♥❣✱ ❏✳✱ ❏✐♥✱ ❨✳✱ ❍❡✱ ❍✳✱ ♠♦❧❡❝✉❧❡ ❝♦♥s✐st✐♥❣ ♦❢ ❛s②♠♠❡tr✐❝ ♠❡t❛❧✲ ❙❤❛♦✱ ❏✳✱ ✫ ❋❛♥✱ ❩✳ ✭✷✵✵✽✮✳ ❊❧❡❝tr✐❝ ✜❡❧❞ ❞✐s✲ ❧✐❝ ♥❛♥♦❞✐♠❡rs✳ ❏✳ ❆♣♣❧✳ P❤②s✳✱ ✶✶✼ ✭✷✮✱ tr✐❜✉t✐♦♥ ✐♥ r❡s♦♥❛♥t r❡✢❡❝t✐♦♥ ✜❧t❡rs ✉♥❞❡r ✵✷✸✶✶✽✳ ♥♦r♠❛❧ ✐♥❝✐❞❡♥❝❡✳ ❏✳ ❖♣t✳ ❆✿ P✉r❡✳ ❆♣♣❧✳ ❖♣t✳✱ ✶✵✱ ✶✷✺✵✵✸✳ ❬✷✸❪ ❇♦tt❛③③✐✱ ❇✳✱ ❋♦r♥❛s❛r✐✱ ▲✳✱ ❋r❛♥❣♦❧❤♦✱ ❆✳✱ ●✐✉❞✐❝❛tt✐✱ ❙✳✱ ▼❛♥t♦✈❛♥✐✱ ❆✳✱ ▼❛r❛❜❡❧❧✐✱ ❋✳✱ ▼❛r❝❤❡s✐♥✐✱ ●✳✱ P❡❧❧❛❝❛♥✐✱ P✳✱ ❚❤❡r✐s♦❞✱ ❘✳✱ ✫ ❱❛❧s❡s✐❛✱ ❆✳ ✭✷✵✶✹✮✳ ▼✉❧t✐♣❧❡①❡❞ ❧❛❜❡❧✲❢r❡❡ ♦♣t✐❝❛❧ ❜✐♦s❡♥s♦r ❢♦r ♠❡❞✐❝❛❧ ❞✐✲ ❛❣♥♦st✐❝s✳ ❏✳ ❇✐♦♠❡❞✳ ❖♣t✳✱ ✶✾ ✭✶✮✱ ✶✼✵✵✻✳ ❆❜♦✉t ❆✉t❤♦rs r❡❝❡✐✈❡❞ ❇❛❝❤❡❧♦r✬s ❞❡❣r❡❡ ✐♥ P❤②s✐❝❛❧ ♣❡❞❛❣♦❣② ❢r♦♠ ❆♥ ●✐❛♥❣ ❯♥✐✈❡rs✐t②✱ ❱✐❡t♥❛♠✱ ✐♥ ✷✵✶✷✱ ❛♥❞ ▼❛st❡r✬s ❞❡❣r❡❡ ✐♥ ❚❤❡♦✲ r❡t✐❝❛❧ ❛♥❞ ♠❛t❤❡♠❛t✐❝❛❧ ♣❤②s✐❝s ❢r♦♠ ❈❛♥ ❚❤♦ ❯♥✐✈❡rs✐t②✱ ❱✐❡t♥❛♠✱ ✐♥ ✷✵✶✻✳ ❋r♦♠ ✷✵✶✻ t♦ ❬✷✹❪ ▲✉✱ ❳✳✱ ❩❤❛♥❣✱ ▲✳✱ ✫ ❩❤❛♥❣✱ ❚✳ ✭✷✵✶✺✮✳ ✷✵✶✾✱ ❤❡ ❥♦✐♥❡❞ t❤❡ ■♥st✐t✉t❡ ❢♦r ❈♦♠♣✉t❛t✐♦♥❛❧ ◆❛♥♦s❧✐t✲♠✐❝r♦❝❛✈✐t②✲❜❛s❡❞ ♥❛rr♦✇ ❜❛♥❞ ❙❝✐❡♥❝❡✱ ❚♦♥ ❉✉❝ ❚❤❛♥❣ ❯♥✐✈❡rs✐t②✱ ❛s ❛♥ ❛ss✐s✲ ❛❜s♦r❜❡r ❢♦r s❡♥s✐♥❣ ❛♣♣❧✐❝❛t✐♦♥s✳ ❖♣t✳ t❛♥t r❡s❡❛r❝❤❡r✳ ❙✐♥❝❡ ✷✵✶✾ t♦ ♣r❡s❡♥t✱ ❤❡ ❤❛s ❜❡❡♥ ❛ P❤❉✳ s❞t✉❞❡♥t ✐♥ ❉✐s♣❧❛② ❊♥❣✐♥❡❡r✐♥❣ ❊①♣r✳✱ ✷✸ ✭✶✻✮✱ ✷✵✼✶✺✕✷✵✼✷✵✳ ▲❛❜✱ ❋❛❝✉❧t② ♦❢ ❊❧❡❝tr♦♥✐❝s ❛♥❞ ■♥❢♦r♠❛t✐♦♥ ❬✷✺❪ ❙❤❡♥✱ ❨✳✱ ❩❤♦✉✱ ❏✳✱ ▲✐✉✱ ❚✳✱ ❚❛♦✱ ❨✳✱ ❏✐❛♥❣✱ ❊♥❣✐♥❡❡r✐♥❣✱ ❈❤♦♥❜✉❦ ◆❛t✐♦♥❛❧ ❯♥✐✈❡rs✐t②✳ ❍✐s ❘✳✱ ▲✐✉✱ ▼✳✱ ❳✐❛♦✱ ●✳✱ ❩❤✉✱ ❏✳✱ ❩❤♦✉✱ ❩✳❑✳✱ ❝✉rr❡♥t r❡s❡❛r❝❤ ✐♥t❡r❡sts ✐♥❝❧✉❞❡ s♦❧❛r ❝❡❧❧s✱ P❤✉❝ ❚♦❛♥ ❉❆◆● ❝ ✷✵✶✾ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ✹✶✸ ❱❖▲❯▼❊✿ ✸ | ■❙❙❯❊✿ ✷ | ✷✵✶✾ | ❏✉♥❡ ❧✐q✉✐❞ ❝r②st❛❧✱ ❞✐❡❧❡❝tr✐❝ s♣❡❝tr♦s❝♦♣②✱ ♦♣t✐❝s ❯♥✐✈❡rs✐t②✱ ❈❛♥❛❞❛ ✭✷✵✶✷✮✳ ❍❡ ✐s t❤❡ ❢♦✉♥❞❡r ❛♥❞ ♣❤♦t♦♥✐❝s✳ ❛♥❞ ❞✐r❡❝t♦r ♦❢ t❤❡ ♠♦❧❡❝✉❧❛r ❜❡❛♠ ❡♣✐t❛①② ❢❛❝✐❧✐t② ❛t ◆❏■❚✳ ❍✐s ❝✉rr❡♥t r❡s❡❛r❝❤ ✐♥t❡r❡sts ❑❤❛✐ ◗✳ ▲❊ ❤❛s ♠♦r❡ t❤❛♥ ✶✵ ②❡❛rs ♦❢ ✐♥❝❧✉❞❡ ♠♦❧❡❝✉❧❛r ❜❡❛♠ ❡♣✐t❛①✐❛❧ ❣r♦✇t❤✱ ❢❛❜r✐✲ ❡①♣❡r✐❡♥❝❡ ✐♥ ♥❛♥♦♣❤♦t♦♥✐❝s✱ ✇❤♦ ✐s ✐♥ ❝❤❛r❣❡ ♦❢ ❝❛t✐♦♥✱ ❛♥❞ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ■■■✲❱ ♥❛♥♦✇✐r❡ ❞❡s✐❣♥✱ ❢❛❜r✐❝❛t✐♦♥ ❛♥❞ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ s✉❜✲ ❤❡t❡r♦str✉❝t✉r❡s ❢♦r ❤✐❣❤✲♣❡r❢♦r♠❛♥❝❡ ♥❛♥♦✲ ✇❛✈❡❧❡♥❣t❤ ♦♣t✐❝❛❧ ❝♦♠♣♦♥❡♥ts✳ ❍❡ r❡❝❡✐✈❡❞ ♦♣t♦❡❧❡❝tr♦♥✐❝ ❞❡✈✐❝❡s ✐♥❝❧✉❞✐♥❣ ▲❊❉s✱ ❧❛s❡rs✱ ❤✐s P❤✳❉✳ ✐♥ P❤♦t♦♥✐❝s ❊♥❣✐♥❡❡r✐♥❣ ❛t ●❤❡♥t ♣❤♦t♦❞❡t❡❝t♦rs✱ s♦❧❛r ❢✉❡❧s✱ ❛♥❞ s♦❧❛r ❝❡❧❧s✳ ❯♥✐✈❡rs✐t②✱ ❇❡❧❣✐✉♠ ✐♥ ✷✵✶✶✳ ❍❡ ❝♦♥❞✉❝t❡❞ ❉r✳ ◆❣✉②❡♥ ✐s t❤❡ ❛✉t❤♦r✴❝♦❛✉t❤♦r ♦❢ ♠♦r❡ ❤✐s ♣♦st❞♦❝t♦r❛❧ r❡s❡❛r❝❤ ✐♥ s❡✈❡r❛❧ ❧❡❛❞✐♥❣ t❤❛♥ ✺✺ ❥♦✉r♥❛❧ ❛rt✐❝❧❡s ❛♥❞ ✾✵ ❝♦♥❢❡r❡♥❝❡ ❛❝❛❞❡♠✐❝ ✐♥st✐t✉t✐♦♥s ✐♥❝❧✉❞✐♥❣ ❯♥✐✈❡rs✐t② ♦❢ ♣r❡s❡♥t❛t✐♦♥s✳ ❍❡ ✇❛s ❛ r❡❝✐♣✐❡♥t ♦❢ t❤❡ ❙P■❊ ❚❡①❛s ❛t ❆✉st✐♥✱ ❯❙❆✱ ❯♥✐✈❡rs✐t② ♦❢ ❚♦r♦♥t♦✱ s❝❤♦❧❛rs❤✐♣ ✐♥ ♦♣t✐❝s ✭✷✵✶✷✮✱ t❤❡ ❜❡st st✉❞❡♥t ❈❛♥❛❞❛✱ ❯♥✐✈❡rs✐t② ♦❢ ▼✐♥♥❡s♦t❛ ❉✉❧✉t❤✱ ❯❙❆ ♣❛♣❡r ❛✇❛r❞ ✭✷♥❞ ♣❧❛❝❡✮ ❛t t❤❡ ■❊❊❊ P❤♦✲ ❛♥❞ ■♥st✐t✉t❡ ❢♦r ▼♦❧❡❝✉❧❛r ❙❝✐❡♥❝❡✱ ❏❛♣❛♥✳ ❍❡ t♦♥✐❝s ❝♦♥❢❡r❡♥❝❡ ✭✷✵✶✶✮✱ ❛♥❞ t❤❡ ♦✉tst❛♥❞✐♥❣ ❤❛s r❡❝❡♥t❧② s✇✐t❝❤❡❞ t♦ t❤❡ ✐♥❞✉str② ✇♦r❦✐♥❣ st✉❞❡♥t ♣❛♣❡r ❛✇❛r❞ ❛t t❤❡ ✷✽t❤ ◆♦rt❤ ❆♠❡r✲ ❛s ❛ ♥❛♥♦❢❛❜r✐❝❛t✐♦♥ ❡♥❣✐♥❡❡r ❛t t❤❡ ❛❇❡❛♠ ✐❝❛♥ ▼♦❧❡❝✉❧❛r ❇❡❛♠ ❊♣✐t❛①② ❝♦♥❢❡r❡♥❝❡ ✭✷✵✶✶✮✳ ❚❡❝❤♥♦❧♦❣✐❡s✱ ■♥❝✳ ❍✐s ♠❛✐♥ t❛s❦s ❛r❡ r❡❧❛t❡❞ t♦ t❤❡ ♣❤♦t♦♥✐❝ ❞❡s✐❣♥✱ ♥❛♥♦❢❛❜r✐❝❛t✐♦♥ ❛♥❞ ♦♣t✐❝❛❧ ❚r✉♦♥❣ ❑❤❛♥❣ ◆●❯❨❊◆ r❡❝❡✐✈❡❞ t❤❡ ♠❡❛s✉r❡♠❡♥ts✳ ❍❡ ❤❛s ♣✉❜❧✐s❤❡❞ ✻✺ s❝✐❡♥t✐✜❝ ❇✳❙✳ ❞❡❣r❡❡ ✐♥ ❈♦♠♣✉t❛t✐♦♥❛❧ P❤②s✐❝s ❢r♦♠ ♣❛♣❡rs ✭❝✐t❡❞ ♦✈❡r ✶✵✽✵ t✐♠❡s✮ ✐♥ ♣❡❡rr❡✈✐❡✇❡❞ t❤❡ ❯♥✐✈❡rs✐t② ♦❢ ❙❝✐❡♥❝❡✱ ❱✐❡t♥❛♠ ◆❛t✐♦♥❛❧ ❤✐❣❤✲✐♠♣❛❝t ❢❛❝t♦r ❥♦✉r♥❛❧s✳ ❍❡ ✐s ❝✉rr❡♥t❧② ❛❧s♦ ❯♥✐✈❡rs✐t②✱ ❍♦ ❈❤✐ ▼✐♥❤ ❈✐t② ✐♥ ✷✵✵✻✱ ❛♥❞ ❛✣❧✐❛t❡❞ ✇✐t❤ ▲❛✇r❡♥❝❡ ❇❡r❦❡❧❡② ◆❛t✐♦♥❛❧ ▲❛❜ t❤❡ ▼✳❙✳ ❛♥❞ P❤✳❉✳ ❞❡❣r❡❡s ✐♥ ❊❧❡❝tr✐❝❛❧ ❛♥❞ ✭▲❇◆▲✮✱ ❯❙❆ ❛♥❞ ❚♦♥ ❉✉❝ ❚❤❛♥❣ ❯♥✐✈❡rs✐t②✱ ❈♦♠♣✉t❡r ❊♥❣✐♥❡❡r✐♥❣ ❢r♦♠ ❆❥♦✉ ❯♥✐✈❡rs✐t② ❱✐❡t♥❛♠✳ ✐♥ ❙✉✇♦♥✱ ❑♦r❡❛ ✐♥ ✷✵✶✸✳ ❋r♦♠ ❖❝t✳ ✷✵✶✸ t♦ ❉❡❝✳ ✷✵✶✹✱ ❤❡ ✇♦r❦❡❞ ❛t ❉✐✈✐s✐♦♥ ♦❢ ❊♥❡r❣② ◗✉❛♥❣ ▼✐♥❤ ◆●❖ r❡❝❡✐✈❡❞ ❤✐s P❤❉ ✐♥ ❙②st❡♠s ❘❡s❡❛r❝❤✱ ❆❥♦✉ ❯♥✐✈❡r✐s✐t②✱ ❑♦r❡❛ ❛s ❡❧❡❝tr✐❝❛❧ ❡♥❣✐♥❡❡r✐♥❣ ❛t ❆❥♦✉ ❯♥✐✈❡rs✐t②✱ t❤❡ ❛ ♣♦st❞♦❝t♦r❛❧ ❢❡❧❧♦✇✳ ❍❡ ✐s ❝✉rr❡♥t❧② ❍❡❛❞ ♦❢ ❘❡♣✉❜❧✐❝ ♦❢ ❑♦r❡❛ ✐♥ ✷✵✶✶✳ ❋r♦♠ ▼❛r❝❤ ✷✵✶✷ ❉✐✈✐s✐♦♥ ♦❢ ❈♦♠♣✉t❛t✐♦♥❛❧ P❤②s✐❝s ❛t ■♥st✐t✉t❡ t♦ ❏❛♥✉❛r② ✷✵✶✾✱ ❤❡ ❤❛❞ ✇♦r❦❡❞ ❛s t❤❡ ❧❡❛❞❡r ❢♦r ❈♦♠♣✉t❛t✐♦♥❛❧ ❙❝✐❡♥❝❡✱ ❚♦♥ ❉✉❝ ❚❤❛♥❣ ♦❢ ♠✐❝r♦✲ ❛♥❞ ♥❛♥♦♣❤♦t♦♥✐❝s r❡s❡❛r❝❤ ❣r♦✉♣ ❛t ❯♥✐✈❡rs✐t② ✐♥ ❍♦ ❈❤✐ ▼✐♥❤ ❈✐t②✱ ❱✐❡t♥❛♠✱ ❛♥❞ ■♥st✐t✉t❡ ♦❢ ▼❛t❡r✐❛❧s ❙❝✐❡♥❝❡ ✭■▼❙✮✱ ❱✐❡t♥❛♠ ❛❧s♦ ▼❛♥❛❣✐♥❣ ❊❞✐t♦r ♦❢ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❆❝❛❞❡♠② ♦❢ ❙❝✐❡♥❝❡ ❛♥❞ ❚❡❝❤♥♦❧♦❣② ✭❱❆❙❚✮✱ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥✳ ❍❡ ❤❛s ❛✉✲ ❍❛♥♦✐✱ ❱✐❡t♥❛♠✳ ❙✐♥❝❡ ❋❡❜r✉❛r② ✷✵✶✾✱ ❤❡ t❤♦r❡❞ ❛♥❞ ❝♦✲❛✉t❤♦r❡❞ ✻✵ ♣❡❡r✲r❡✈✐❡✇❡❞ ■❙■ ❤❛s ✇♦r❦❡❞ ❛s t❤❡ ❞✐r❡❝t♦r ♦❢ ❆❞♠✐♥✐str❛t✐♦♥✱ ❥♦✉r♥❛❧ ❛rt✐❝❧❡s ❛♥❞ ✹✵ ❝♦♥❢❡r❡♥❝❡ ♣❛♣❡rs✳ ❯♥✐✈❡rs✐t② ♦❢ ❙❝✐❡♥❝❡ ❛♥❞ ❚❡❝❤♥♦❧♦❣② ♦❢ ❍❛♥♦✐ ❍❡ ❤❛s ✇r✐tt❡♥ ♦♥❡ ❜♦♦❦ ❝❤❛♣t❡r ✐♥ t❤❡ ❛r❡❛ ✭❯❙❚❍✮✱ ❱❆❙❚✱ ❍❛♥♦✐✱ ❱✐❡t♥❛♠✳ ❍❡ ✐s ❛♥ ♦❢ t❡r❛❤❡rt③ ❛♥t❡♥♥❛ ❛♥❞ ✜❧❡❞ ♦♥❡ ♣❛t❡♥t ♦♥ ❛✉t❤♦r✴❝♦❛✉t❤♦r ♦❢ ✺✵ ■❙■ ♣❛♣❡rs✳ ❍✐s ♠❛✐♥ t❡r❛❤❡rt③ str✐♣❧✐♥❡ ❛♥t❡♥♥❛✳ ❍✐s ❝✉rr❡♥t r❡s❡❛r❝❤ r❡s❡❛r❝❤ ❢♦❝✉s❡s ♦♥ ❞❡s✐❣♥✱ s✐♠✉❧❛t✐♦♥✱ ❢❛❜✲ ✐♥t❡r❡sts ✐♥❝❧✉❞❡ ▼✐❝r♦✇❛✈❡ ❆♥t❡♥♥❛ ❢♦r ❲✐r❡✲ r✐❝❛t✐♦♥ ❛♥❞ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ♠✐❝r♦✲ ❛♥❞ ❧❡ss ❈♦♠♠✉♥✐❝❛t✐♦♥❀ ❚❡r❛❤❡rt③ ❆♥t❡♥♥❛ ❢♦r ♥❛♥♦♣❤♦t♦♥✐❝s ✐♥ t❤❡ ✈✐s✐❜❧❡ ❛♥❞ ♥❡❛r✲✐♥❢r❛r❡❞ ❈♦♠♣❛❝t ❛♥❞ ❊✣❝✐❡♥t ❙♦✉r❝❡❀ ◆❛♥♦ ❙tr✉❝t✉r❡s s♣❡❝tr❛❧ r❡❣✐♦♥s ❢♦r ♦♣t✐❝❛❧ ❞❡✈✐❝❡s✳ ❛♥❞ ◆❛♥♦ ❆♥t❡♥♥❛ ❢♦r ❖♣t✐❝❛❧ ❆♣♣❧✐❝❛t✐♦♥s❀ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥❛❧ ▼✐❝r♦✴◆❛♥♦ ❋❧✉✐❞✐❝s✳ P✳ ❚✳ ❍✳ ◆●❯❨❊◆ ✐s ❛♥ ❛ss✐st❛♥t ♣r♦❢❡ss♦r ✐♥ t❤❡ ❉❡♣❛rt♠❡♥t ♦❢ ❊❧❡❝tr✐❝❛❧ ❛♥❞ ❈♦♠♣✉t❡r ❊♥❣✐♥❡❡r✐♥❣ ❛t ◆❡✇ ❏❡rs❡② ■♥st✐t✉t❡ ♦❢ ❚❡❝❤✲ ♥♦❧♦❣② ✭◆❏■❚✮✱ ❯✳❙✳ ❍❡ r❡❝❡✐✈❡❞ ❇✳❙✳ ❞❡❣r❡❡ ✐♥ P❤②s✐❝s ❢r♦♠ ❱✐❡t♥❛♠ ◆❛t✐♦♥❛❧ ❯♥✐✈❡rs✐t② ✐♥ ❍♦ ❈❤✐ ▼✐♥❤ ❈✐t②✱ ❱✐❡t♥❛♠ ✭✷✵✵✺✮✱ t❤❡ ▼✳❙✳ ❞❡❣r❡❡ ✐♥ ❊❧❡❝tr♦♥✐❝s ❊♥❣✐♥❡❡r✐♥❣ ❢r♦♠ ❆❥♦✉ ❯♥✐✈❡rs✐t②✱ ❙♦✉t❤ ❑♦r❡❛ ✭✷✵✵✾✮✱ ❛♥❞ t❤❡ P❤❉✳ ❞❡❣r❡❡ ✐♥ ❊❧❡❝tr✐❝❛❧ ❊♥❣✐♥❡❡r✐♥❣ ❢r♦♠ ▼❝●✐❧❧ ✹✶✹ ✧❚❤✐s ✐s ❛♥ ❖♣❡♥ ❆❝❝❡ss ❛rt✐❝❧❡ ❞✐str✐❜✉t❡❞ ✉♥❞❡r t❤❡ t❡r♠s ♦❢ t❤❡ ❈r❡❛t✐✈❡ ❈♦♠♠♦♥s ❆ttr✐❜✉t✐♦♥ ▲✐❝❡♥s❡✱ ✇❤✐❝❤ ♣❡r♠✐ts ✉♥r❡str✐❝t❡❞ ✉s❡✱ ❞✐str✐❜✉t✐♦♥✱ ❛♥❞ r❡♣r♦❞✉❝t✐♦♥ ✐♥ ❛♥② ♠❡❞✐✉♠ ♣r♦✈✐❞❡❞ t❤❡ ♦r✐❣✐♥❛❧ ✇♦r❦ ✐s ♣r♦♣❡r❧② ❝✐t❡❞ ✭❈❈ ❇❨ ✹✳✵✮✳✧ ... ❏✉♥❡ (Guided- mode resonance filter with ultra- narrow bandwidth over the visible frequencies for label- free optical biosensor) (Guided- mode resonance filter with ultra- narrow bandwidth over the visible. .. ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ✹✵✼ ndwidth over the visible frequencies for label- free optical biosensor) wave to exist in the grating structure can be represented [2] as