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Recovery based error estimation for the Method of Moments

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❘❡❝♦✈❡r② ❇❛s❡❞ ❊rr♦r ❊st✐♠❛t✐♦♥ ❢♦r t❤❡

▼❡t❤♦❞ ♦❢ ▼♦♠❡♥ts

❜②

❲✐❧❧❡♠ ❏❛❝♦❜✉s ❙tr②❞♦♠

❚❤❡s✐s ♣r❡s❡♥t❡❞ ✐♥ ♣❛rt✐❛❧ ❢✉❧✜❧♠❡♥t ♦❢ t❤❡ r❡q✉✐r❡♠❡♥ts ❢♦r

t❤❡ ❞❡❣r❡❡ ♦❢ ▼❛st❡r ♦❢ ❊♥❣✐♥❡❡r✐♥❣ ✐♥ t❤❡ ❋❛❝✉❧t② ♦❢

❊♥❣✐♥❡❡r✐♥❣ ❛t ❙t❡❧❧❡♥❜♦s❝❤ ❯♥✐✈❡rs✐t②

❉❡♣❛rt♠❡♥t ♦❢ ❊❧❡❝tr✐❝❛❧ ❛♥❞ ❊❧❡❝tr♦♥✐❝ ❊♥❣✐♥❡❡r✐♥❣✱ ❯♥✐✈❡rs✐t② ♦❢ ❙t❡❧❧❡♥❜♦s❝❤✱ Pr✐✈❛t❡ ❇❛❣ ❳✶✱ ▼❛t✐❡❧❛♥❞ ✼✻✵✷✱ ❙♦✉t❤ ❆❢r✐❝❛✳ ❙✉♣❡r✈✐s♦r✿ Pr♦❢❡ss♦r ▼❛tt❤②s ▼✳ ❇♦t❤❛

▼❛r❝❤ ✷✵✶✺

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❉❡❝❧❛r❛t✐♦♥

❇② s✉❜♠✐tt✐♥❣ t❤✐s t❤❡s✐s ❡❧❡❝tr♦♥✐❝❛❧❧②✱ ■ ❞❡❝❧❛r❡ t❤❛t t❤❡ ❡♥t✐r❡t② ♦❢ t❤❡ ✇♦r❦ ❝♦♥t❛✐♥❡❞ t❤❡r❡✐♥ ✐s ♠② ♦✇♥✱ ♦r✐❣✐♥❛❧ ✇♦r❦✱ t❤❛t ■ ❛♠ t❤❡ s♦❧❡ ❛✉t❤♦r t❤❡r❡♦❢ ✭s❛✈❡ t♦ t❤❡ ❡①t❡♥t ❡①♣❧✐❝✐t❧② ♦t❤❡r✇✐s❡ st❛t❡❞✮✱ t❤❛t r❡♣r♦❞✉❝t✐♦♥ ❛♥❞ ♣✉❜❧✐❝❛t✐♦♥ t❤❡r❡♦❢ ❜② ❙t❡❧❧❡♥❜♦s❝❤ ❯♥✐✈❡rs✐t② ✇✐❧❧ ♥♦t ✐♥❢r✐♥❣❡ ❛♥② t❤✐r❞ ♣❛rt② r✐❣❤ts ❛♥❞ t❤❛t ■ ❤❛✈❡ ♥♦t ♣r❡✈✐♦✉s❧② ✐♥ ✐ts ❡♥t✐r❡t② ♦r ✐♥ ♣❛rt s✉❜♠✐tt❡❞ ✐t ❢♦r ♦❜t❛✐♥✐♥❣ ❛♥② q✉❛❧✐✜❝❛t✐♦♥✳ ▼❛r❝❤ ✷✵✶✺ ❈♦♣②r✐❣❤t ➞ ✷✵✶✺ ❙t❡❧❧❡♥❜♦s❝❤ ❯♥✐✈❡rs✐t② ❆❧❧ r✐❣❤ts r❡s❡r✈❡❞ ✐

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❆❜str❛❝t

❚❤❡ ▼❡t❤♦❞ ♦❢ ▼♦♠❡♥ts ✭▼♦▼✮ ✐s r♦✉t✐♥❡❧② ✉s❡❞ ❢♦r t❤❡ ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥ ♦❢ ❡❧❡❝✲ tr♦♠❛❣♥❡t✐❝ s✉r❢❛❝❡ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s✳ ❙♦❧✉t✐♦♥ ❡rr♦rs ❛r❡ ✐♥❤❡r❡♥t t♦ ❛♥② ♥✉♠❡r✐❝❛❧ ❝♦♠♣✉t❛t✐♦♥❛❧ ♠❡t❤♦❞✱ ❛♥❞ ❡rr♦r ❡st✐♠❛t♦rs ❝❛♥ ❜❡ ❡✛❡❝t✐✈❡❧② ❡♠♣❧♦②❡❞ t♦ r❡❞✉❝❡ ❛♥❞ ❝♦♥tr♦❧ t❤❡s❡ ❡rr♦rs✳ ■♥ t❤✐s t❤❡s✐s✱ ❣r❛❞✐❡♥t r❡❝♦✈❡r② t❡❝❤♥✐q✉❡s ♦❢ t❤❡ ❋✐♥✐t❡ ❊❧❡♠❡♥t ▼❡t❤♦❞ ✭❋❊▼✮ ❛r❡ ❢♦r♠✉❧❛t❡❞ ✇✐t❤✐♥ t❤❡ ▼♦▼ ❝♦♥t❡①t✱ ✐♥ ♦r❞❡r t♦ r❡❝♦✈❡r ❛ ❤✐❣❤❡r✲♦r❞❡r ❝❤❛r❣❡ ♦❢ ❛ ❘❛♦✲❲✐❧t♦♥✲●❧✐ss♦♥ ✭❘❲●✮ ▼♦▼ s♦❧✉t✐♦♥✳ ❋✉rt❤❡r♠♦r❡✱ ❛ ♥❡✇ r❡❝♦✈❡r② ♣r♦✲ ❝❡❞✉r❡✱ ❜❛s❡❞ s♣❡❝✐✜❝❛❧❧② ♦♥ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❘❲● ❜❛s✐s ❢✉♥❝t✐♦♥s✱ ✐s ✐♥tr♦❞✉❝❡❞ ❜② t❤❡ ❛✉t❤♦r✳ ❚❤❡s❡ r❡❝♦✈❡r❡❞ ❝❤❛r❣❡ ❞✐str✐❜✉t✐♦♥s ❛r❡ ✉s❡❞ ❢♦r ❛ ♣♦st❡r✐♦r✐ ❡rr♦r ❡s✲ t✐♠❛t✐♦♥ ♦❢ t❤❡ ❝❤❛r❣❡✳ ■t ✇❛s ❢♦✉♥❞ t❤❛t t❤❡ ♥❡✇❧② ♣r♦♣♦s❡❞ ❝❤❛r❣❡ r❡❝♦✈❡r② ♠❡t❤♦❞ ❤❛s t❤❡ ❤✐❣❤❡st ❛❝❝✉r❛❝② ♦❢ t❤❡ ❝♦♥s✐❞❡r❡❞ r❡❝♦✈❡r② ♠❡t❤♦❞s✱ ❛♥❞ ✐s t❤❡ ♠♦st s✉✐t❡❞ ❢♦r ❛♣♣❧✐❝❛t✐♦♥s ✇✐t❤✐♥ r❡❝♦✈❡r② ❜❛s❡❞ ❡rr♦r ❡st✐♠❛t✐♦♥✳ ■♥ ❛❞❞✐t✐♦♥ t♦ ❝❤❛r❣❡ r❡❝♦✈❡r②✱ t❤❡ ♣♦ss✐❜✐❧✐t② ♦❢ r❡❝♦✈❡r② ♣r♦❝❡❞✉r❡s ❢♦r t❤❡ ▼♦▼ s♦❧✉t✐♦♥ ❝✉rr❡♥t ❛r❡ ❛❧s♦ ✐♥✈❡st✐❣❛t❡❞✳ ❆ t❡❝❤♥✐q✉❡ ✐s ❡①♣❧♦r❡❞ ✇❤❡r❡❜② ❛ r❡❝♦✈❡r❡❞ ❝❤❛r❣❡ ✐s ✉s❡❞ t♦ ✜♥❞ ❛ ❤✐❣❤❡r✲♦r❞❡r ❞✐✈❡r❣❡♥t ❝✉rr❡♥t r❡♣r❡s❡♥t❛t✐♦♥✳ ❚✇♦ ♥❡✇❧② ❞❡✈❡❧♦♣❡❞ ♠❡t❤♦❞s ❢♦r t❤❡ s✉❜s❡q✉❡♥t r❡❝♦✈❡r② ♦❢ t❤❡ s♦❧❡♥♦✐❞❛❧ ❝✉rr❡♥t ❝♦♠♣♦♥❡♥t✱ ❛s ❝♦♥t❛✐♥❡❞ ✐♥ t❤❡ ❘❲● s♦❧✉t✐♦♥ ❝✉rr❡♥t✱ ❛r❡ ❛❧s♦ ✐♥tr♦❞✉❝❡❞ ❜② t❤❡ ❛✉t❤♦r✳ ❆ ♣♦st❡r✐♦r✐ ❡rr♦r ❡st✐♠❛t✐♦♥ ♦❢ t❤❡ ▼♦▼ ❝✉rr❡♥t ✐s ❛❝❝♦♠♣❧✐s❤❡❞ t❤r♦✉❣❤ t❤❡ ✉s❡ ♦❢ t❤❡ r❡❝♦✈❡r❡❞ ❝✉rr❡♥t ❞✐str✐❜✉t✐♦♥s✳ ❆ ♠✐①❡❞ s❡❝♦♥❞✲♦r❞❡r r❡❝♦✈❡r❡❞ ❝✉rr❡♥t✱ ❜❛s❡❞ ♦♥ ❛ ✈❡❝t♦r r❡❝♦✈❡r② ♣r♦❝❡❞✉r❡✱ ✇❛s ❢♦✉♥❞ t♦ ♣r♦❞✉❝❡ t❤❡ ♠♦st ❛❝❝✉r❛t❡ r❡s✉❧ts✳ ❚❤❡ ❡rr♦r ❡st✐♠❛t✐♦♥ t❡❝❤♥✐q✉❡s ❞❡✈❡❧♦♣❡❞ ✐♥ t❤✐s t❤❡s✐s ❝♦✉❧❞ ❜❡ ✐♥❝♦r♣♦r❛t❡❞ ✐♥t♦ ❛♥ ❛❞❛♣t✐✈❡ s♦❧✈❡r s❝❤❡♠❡ t♦ ♦♣t✐♠✐s❡ t❤❡ s♦❧✉t✐♦♥ ❛❝❝✉r❛❝② r❡❧❛t✐✈❡ t♦ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❝♦st✳ ✐✐

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❖♣s♦♠♠✐♥❣

❉✐❡ ▼♦♠❡♥t ▼❡t♦❞❡ ✭▼♦▼✮ ✈✐♥❞ ❛❧❣❡♠❡♥❡ t♦❡♣❛ss✐♥❣ ✐♥ ❞✐❡ ♥✉♠❡r✐❡s❡ ♦♣❧♦ss✐♥❣ ✈❛♥ ❡❧❡❦tr♦♠❛❣♥❡t✐❡s❡ ♦♣♣❡r✈❧❛❦ ✐♥t❡❣r❛❛❧✈❡r❣❡❧②❦✐♥❣s✳ ◆✉♠❡r✐❡s❡ ❢♦✉t❡ ✐s ✐♥❤❡r❡♥t t♦t ❞✐❡ ♣r♦s❡❞✉r❡✿ ❢♦✉t❜❡r❛♠✐♥❣st❡❣♥✐❡❦❡ ✐s ❞✉s ♥♦❞✐❣ ♦♠ ❞✐❡ ❜❡tr♦❦❦❡ ❢♦✉t❡ t❡ ❛♥❛❧✐s❡❡r ❡♥ t❡ r❡❞✉s❡❡r✳ ●r❛❞✐ë♥t ✈❡r❤❛❧✐♥❣st❡❣♥✐❡❦❡ ✈❛♥ ❞✐❡ ❊✐♥❞✐❣❡ ❊❧❡♠❡♥t ▼❡t♦❞❡ ✇♦r❞ ✐♥ ❤✐❡r❞✐❡ t❡s✐s ✐♥ ❞✐❡ ▼♦▼ ❦♦♥t❡❦s ❣❡❢♦r♠✉❧❡❡r✳ ❍✐❡r❞✐❡ t❡❣♥✐❡❦❡ ✇♦r❞ ✐♥❣❡s♣❛♥ ♦♠ ❞✐❡ ♦♣♣❡r✲ ✈❧❛❦❧❛❞✐♥❣ ✈❛♥ ✬♥ ❘❛♦✲❲✐❧t♦♥✲●❧✐ss♦♥ ✭❘❲●✮ ▼♦▼ ♦♣❧♦ss✐♥❣ ♥❛ ✬♥ ✈❡r❜❡t❡r❞❡ ❤♦ër✲♦r❞❡ ✈♦♦rst❡❧❧✐♥❣ t❡ ♥❡❡♠✳ ❱❡r❞❡r ✐s ✬♥ ♥✉✇❡ ❧❛❞✐♥❣ ✈❡r❤❛❧✐♥❣st❡❣♥✐❡❦ ❞❡✉r ❞✐❡ ♦✉t❡✉r ✈♦♦r❣❡st❡❧ ✇❛t s♣❡s✐✜❡❦ ♦♣ ❞✐❡ ❡✐❡♥s❦❛♣♣❡ ✈❛♥ ❞✐❡ ❘❲● ❜❛s✐s ❢✉♥❦s✐❡s ❣❡❜❛s❡❡r ✐s✳ ❉✐❡ ✈❡r❤❛❛❧❞❡ ❧❛❞✐♥❣s✈❡rs♣r❡✐❞✐♥❣s ✐s ❣❡ï♠♣❧❡♠❡♥t❡❡r ✐♥ ❛ ♣♦st❡r✐♦r✐ ❢♦✉t ❜❡r❛♠✐♥❣ ✈❛♥ ❞✐❡ ❧❛❞✐♥❣✳ ❉✐❡ ♥✉✉t ✈♦♦r❣❡st❡❧❞❡ t❡❣♥✐❡❦ ❤❡t ❞✐❡ ❛❦❦✉r❛❛tst❡ r❡s✉❧t❛t❡ ❣❡❧❡✇❡r✱ ✉✐t ❞✐❡ ❣r♦❡♣ ✈❡r❤❛❧✐♥❣st❡❣♥✐❡❦❡ ✇❛t ♦♥❞❡rs♦❡❦ ✐s✳ ❆❞❞✐s✐♦♥❡❡❧ t♦t ❧❛❞✐♥❣s✈❡r❤❛❧✐♥❣✱ ✐s ❞✐❡ ♠♦♦♥t❧✐❦❤❡✐❞ ✈❛♥ ▼♦▼✲str♦♦♠ ✈❡r❤❛❧✐♥❣st❡❣✲ ♥✐❡❦❡ ♦♦❦ ♦♥❞❡rs♦❡❦✳ ✬♥ ▼❡t♦❞❡ ✈✐r ❞✐❡ ✈❡r❤❛❧✐♥❣ ✈❛♥ ✬♥ ❤♦ër✲♦r❞❡ ❞✐✈❡r❣❡♥t❡ str♦♦♠ ❦♦♠♣♦♥❡♥t✱ ❣❡❜❛s❡❡r ♦♣ ❞✐❡ ✈❡r❤❛❛❧❞❡ ❧❛❞✐♥❣✱ ✐s ❣❡ï♠♣❧❡♠❡♥t❡❡r✳ ❱❡r❞❡r ✐s t✇❡❡ ♥✉✇❡ ♠❡t♦❞❡s ✈✐r ❞✐❡ ✈❡r❤❛❧✐♥❣ ✈❛♥ ❞✐❡ s♦❧❡♥♦❞✐❛❧❡ ❦♦♠♣♦♥❡♥t ✈❛♥ ❞✐❡ ❘❲● str♦♦♠ ❞❡✉r ❞✐❡ ♦✉t❡✉r ✈♦♦r❣❡st❡❧✳ ❆ ♣♦st❡r✐♦r✐ ❢♦✉t❜❡r❛♠✐♥❣ ✈❛♥ ❞✐❡ ▼♦▼✲str♦♦♠ ✐s ♠❡t ❜❡❤✉❧♣ ✈❛♥ ❞✐❡ ✈❡r❤❛❛❧❞❡ str♦♦♠ ✈❡rs♣r❡✐❞✐♥❣s ❣❡r❡❛❧✐s❡❡r✱ ❡♥ ❞❛❛r ✐s ❣❡✈✐♥❞ ❞❛t ✬♥ ❣❡♠❡♥❣❞❡ t✇❡❡❞❡✲♦r❞❡ ✈❡r❤❛❛❧❞❡ str♦♦♠✱ ❣❡❜❛s❡❡r ♦♣ ✬♥ ✈❡❦t♦r ♠❡t♦❞❡✱ ❞✐❡ ❜❡st❡ r❡s✉❧t❛t❡ ❧❡✇❡r✳ ❉✐❡ ❢♦✉t❜❡r❛♠✐♥❣st❡❣♥✐❡❦❡ ✇❛t ✐♥ ❤✐❡r❞✐❡ t❡s✐s ♦♥❞❡rs♦❡❦ ✐s✱ ❦❛♥ ✐♥ ✬♥ ❛❛♥♣❛s❜❛r❡ s❦❡♠❛ ♦♣❣❡♥❡❡♠ ✇♦r❞ ♦♠ ❞✐❡ ❛❦❦✉r❛❛t❤❡✐❞ ✈❛♥ ✬♥ ♥✉♠❡r✐❡s❡ ♦♣❧♦ss✐♥❣✱ r❡❧❛t✐❡❢ t♦t ❞✐❡ ❜❡r❡❦❡♥✐♥❣s❦♦st❡✱ t❡ ♦♣t✐♠❡❡r✳ ✐✐✐

(5)

❆❝❦♥♦✇❧❡❞❣❡♠❡♥ts

❚❤❡ ❢♦❧❧♦✇✐♥❣ ♣❡♦♣❧❡ ❛♥❞ ♦r❣❛♥✐s❛t✐♦♥s ♠✉st ❜❡ ❛❝❦♥♦✇❧❡❞❣❡❞ ❢♦r t❤❡✐r s✉♣♣♦rt ❛♥❞ s♣❡❝✐❛❧ ❝♦♥tr✐❜✉t✐♦♥ t♦ t❤✐s ♣r♦❥❡❝t✳ ❼ ▼② ❧♦✈✐♥❣ ❢❛♠✐❧②✱ ❛♥❞ ❡s♣❡❝✐❛❧❧② ♠② ❣✐r❧❢r✐❡♥❞✱ ❙t❛❝②✲▲❡✐❣❤✱ ❢♦r t❤❡✐r s✉♣♣♦rt ❛♥❞ ♣❛t✐❡♥❝❡ t❤r♦✉❣❤ t❤❡ ②❡❛rs ♦❢ ♠② st✉❞✐❡s✳ ❼ Pr♦❢❡ss♦r ▼❛tt❤②s ▼✳ ❇♦t❤❛✱ ❢♦r ✐♥tr♦❞✉❝✐♥❣ ♠❡ t♦ t❤❡ ✜❡❧❞ ♦❢ ❈♦♠♣✉t❛t✐♦♥❛❧ ❊❧❡❝tr♦♠❛❣♥❡t✐❝s ❛♥❞ ❢♦r ❤✐s ♦✉tst❛♥❞✐♥❣ ❛❝❛❞❡♠✐❝ ❣✉✐❞❛♥❝❡✳ ❼ ❆❧t❛✐r ❉❡✈❡❧♦♣♠❡♥t ❙✳❆✳ ✭Pt②✮ ▲t❞✳ ❢♦r t❤❡ ✜♥❛♥❝✐❛❧ s✉♣♣♦rt t❤❡② ♣r♦✈✐❞❡❞ ♠❡ ✇✐t❤✳ ❼ ❚❤❡ ♦t❤❡r ♦✣❝❡ ♠❡♠❜❡rs ♦❢ t❤❡ ❘❛❞❛r ▲❛❜✱ ❢♦r s❤❛r✐♥❣ t❤❡ ♣♦st❣r❛❞✉❛t❡ ❥♦✉r♥❡② ✇✐t❤ ♠❡ ❛♥❞ ❛❧✇❛②s ♦✛❡r✐♥❣ t❤❡✐r s✉♣♣♦rt ❛♥❞ ✐♥s✐❣❤ts ❞✉r✐♥❣ ♠② r❡s❡❛r❝❤✳ ❙♣❡❝✐❛❧ t❤❛♥❦s t♦ ❘②♥♦ ❛♥❞ ❆♥♥❡❦❡ ❢♦r ❦❡❡♣✐♥❣ t❤❡ ❝♦✛❡❡✲❜❡❛♥ s✉♣♣❧② st❡❛❞②✦ ❼ ❚❤❡ ❯♥✐✈❡rs✐t② ♦❢ ❙t❡❧❧❡♥❜♦s❝❤ ❛♥❞ t❤❡ ❉❡♣❛rt♠❡♥t ♦❢ ❊❧❡❝tr✐❝❛❧ ❛♥❞ ❊❧❡❝tr♦♥✐❝ ❊♥❣✐♥❡❡r✐♥❣✱ ❢♦r t❤❡ ❡①❝❡❧❧❡♥t q✉❛❧✐t② ♦❢ ❡❞✉❝❛t✐♦♥ t❤❡② ♣r♦✈✐❞❡❞ ♠❡ ✇✐t❤✳ ✐✈

(6)

❈♦♥t❡♥ts

❉❡❝❧❛r❛t✐♦♥ ✐ ❆❜str❛❝t ✐✐ ❖♣s♦♠♠✐♥❣ ✐✐✐ ❆❝❦♥♦✇❧❡❞❣❡♠❡♥ts ✐✈ ▲✐st ♦❢ ❋✐❣✉r❡s ✈✐✐✐ ▲✐st ♦❢ ❚❛❜❧❡s ①✐ ✶ ■♥tr♦❞✉❝t✐♦♥ ✶ ✷ ❊rr♦r ❊st✐♠❛t✐♦♥✿ ❚❤❡ ❙❡tt✐♥❣ ✸ ✷✳✶ ❊rr♦r ❡st✐♠❛t✐♦♥ ✐♥ t❤❡ ▼❡t❤♦❞ ♦❢ ▼♦♠❡♥ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✷✳✶✳✶ ❘❡s✐❞✉❛❧ ❜❛s❡❞ ❡rr♦r ❡st✐♠❛t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✷✳✶✳✷ ❘❡❝♦✈❡r② ❜❛s❡❞ ❡rr♦r ❡st✐♠❛t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✷✳✷ ❊rr♦r ❡st✐♠❛t✐♦♥ ✐♥ t❤❡ ❋✐♥✐t❡ ❊❧❡♠❡♥t ▼❡t❤♦❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✷✳✷✳✶ ❘❡s✐❞✉❛❧ ❜❛s❡❞ ❡rr♦r ❡st✐♠❛t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✷✳✷✳✷ ●r❛❞✐❡♥t r❡❝♦✈❡r② ❜❛s❡❞ ❡rr♦r ❡st✐♠❛t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✷✳✷✳✸ ❆♣♣❧✐❝❛❜✐❧✐t② t♦ t❤❡ ▼❡t❤♦❞ ♦❢ ▼♦♠❡♥ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✸ ❈♦♥❝❧✉s✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✸ ❋♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ▼❡t❤♦❞ ♦❢ ▼♦♠❡♥ts ✾ ✸✳✶ ❚❤❡ ❘❛♦✲❲✐❧t♦♥✲●❧✐ss♦♥ ✭❘❲●✮ ▼♦▼ ❢♦r♠✉❧❛t✐♦♥ ❢♦r ❛r❜✐tr❛r② s✉r❢❛❝❡s ✳ ✾ ✸✳✶✳✶ ❊❧❡❝tr✐❝ ✜❡❧❞ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥ ✭❊❋■❊✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✸✳✶✳✷ ❘❲● ❜❛s✐s ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✸✳✶✳✸ ●❛❧❡r❦✐♥ t❡st✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✸✳✶✳✹ ▼♦▼ ♠❛tr✐① ❡q✉❛t✐♦♥ ❞❡r✐✈❛t✐♦♥ ❛♥❞ s♦❧✉t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✸✳✶✳✺ ❱❡r✐✜❝❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✸✳✷ ❆♥ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ t❤❡ ❘❲● ▼♦▼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✸✳✷✳✶ ●❡♦♠❡tr② s❡t✲✉♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✸✳✷✳✷ ❈❛❧❝✉❧❛t✐♦♥ ♦❢ ✐♥t❡❣r❛❧s ❛♥❞ s✐♥❣✉❧❛r✐t② ❝❛♥❝❡❧❧❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✈

(7)

❈❖◆❚❊◆❚❙ ✈✐ ✸✳✸ ❈♦♥❝❧✉s✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✹ ◆✉♠❡r✐❝❛❧ ■♥t❡❣r❛t✐♦♥ ✷✶ ✹✳✶ ❖♥❡✲❞✐♠❡♥s✐♦♥❛❧ q✉❛❞r❛t✉r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✹✳✶✳✶ ❚❤❡ r❡❝t❛♥❣❧❡ ♠❡t❤♦❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✹✳✶✳✷ ●❛✉ss✐❛♥ q✉❛❞r❛t✉r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✹✳✷ ❚✇♦✲❞✐♠❡♥s✐♦♥❛❧ q✉❛❞r❛t✉r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✹✳✷✳✶ ❈♦♠♣♦s✐t❡ ♠❡t❤♦❞s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✹✳✷✳✷ ●❛✉ss✐❛♥ q✉❛❞r❛t✉r❡ ❢♦r tr✐❛♥❣✉❧❛r ❞♦♠❛✐♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✹✳✸ ❙✐♥❣✉❧❛r✐t② ❝❛♥❝❡❧❧❛t✐♦♥ q✉❛❞r❛t✉r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✹✳✸✳✶ ❈♦♥str✉❝t✐♦♥ ♦❢ s✉❜✲❞♦♠❛✐♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✹✳✸✳✷ ❚❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✹✳✸✳✸ ❆❝❝✉r❛❝② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✹✳✹ ❈♦♥❝❧✉s✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✺ ❇❛s✐s ❋✉♥❝t✐♦♥s ✸✷ ✺✳✶ ❖♥❡✲❞✐♠❡♥s✐♦♥❛❧ s❝❛❧❛r ❜❛s✐s ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✺✳✶✳✶ ▲❛❣r❛♥❣❡ ❜❛s✐s ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✺✳✷ ❙✐♠♣❧❡① ❝♦♦r❞✐♥❛t❡s ❢♦r tr✐❛♥❣✉❧❛r ❞♦♠❛✐♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✺✳✸ ❚✇♦✲❞✐♠❡♥s✐♦♥❛❧ s❝❛❧❛r ❜❛s✐s ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✺✳✸✳✶ ❚r✐❛♥❣✉❧❛r ♣✉❧s❡ ❢✉♥❝t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✺✳✸✳✷ ❋✐rst✲♦r❞❡r ▲❛❣r❛♥❣❡ ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✺✳✸✳✸ ❙❡❝♦♥❞✲♦r❞❡r ▲❛❣r❛♥❣❡ ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✺✳✹ ❱❡❝t♦r ❜❛s✐s ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✺✳✹✳✶ ❉✐✈❡r❣❡♥❝❡✲❝♦♥❢♦r♠✐♥❣ ✜rst✲♦r❞❡r ❜❛s✐s ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✺✳✹✳✷ ❙♦❧❡♥♦✐❞❛❧ ③❡r♦✲♦r❞❡r ❜❛s✐s ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✺✳✹✳✸ ❙♦❧❡♥♦✐❞❛❧ ✜rst✲♦r❞❡r ❜❛s✐s ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✺✳✹✳✹ ❉✐✈❡r❣❡♥t s❡❝♦♥❞✲♦r❞❡r ❜❛s✐s ❢✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✺✳✺ ❚r❡❡✲❝♦tr❡❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✺✳✻ ❈♦♥❝❧✉s✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✻ ❘❡❝♦✈❡r② ♦❢ ❛ ▼♦▼ ❈❤❛r❣❡ ✹✻ ✻✳✶ ❚❤❡ ▼♦▼ ❝❤❛r❣❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✻✳✷ ❆✈❡r❛❣✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✻✳✷✳✶ ❘❡s✉❧ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✻✳✷✳✷ ❈♦♥❝❧✉s✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✻✳✸ ❩✐❡♥❦✐❡✇✐❝③✲❩❤✉ ♣❛t❝❤ r❡❝♦✈❡r② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✻✳✸✳✶ ❙♣❡❝✐❛❧ ❝♦♥s✐❞❡r❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✻✳✸✳✷ ❘❡s✉❧ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✻✳✸✳✸ ❈♦♥❝❧✉s✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✻✳✹ ❘❲●✲❜❛s❡❞ r❡❝♦✈❡r② ♠❡t❤♦❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹

(8)

❈❖◆❚❊◆❚❙ ✈✐✐ ✻✳✹✳✶ ❆♥ ❘❲●✲♣❛t❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✻✳✹✳✷ ❚❤❡ r❡❝♦✈❡r② ♣r♦❝❡❞✉r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✻✳✹✳✸ ❙♣❡❝✐❛❧ ❝♦♥s✐❞❡r❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✻✳✹✳✹ ❘❡s✉❧ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✻✳✹✳✺ ❈♦♥❝❧✉s✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✻✳✺ ❆ ♣♦st❡r✐♦r✐ ❡rr♦r ❡st✐♠❛t✐♦♥ ♦❢ t❤❡ ▼♦▼ ❝❤❛r❣❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✻✳✺✳✶ ❊❧❡♠❡♥t✲✇✐s❡ ❡rr♦r ❡st✐♠❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✻✳✺✳✷ ●❧♦❜❛❧ ❡rr♦rs ❛♥❞ ❣❧♦❜❛❧ ❡rr♦r ❡st✐♠❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✻✳✻ ❈♦♥❝❧✉s✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ✼ ❘❡❝♦✈❡r② ♦❢ ❛ ▼♦▼ ❈✉rr❡♥t ✻✸ ✼✳✶ ❊①tr❛❝t✐♦♥ ♦❢ s♦❧❡♥♦✐❞❛❧ ❝♦♠♣♦♥❡♥ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✼✳✷ ❆ r❡❝♦✈❡r② ♣r♦❝❡❞✉r❡ ❢♦r t❤❡ ✜rst✲♦r❞❡r ❞✐✈❡r❣❡♥t ❝✉rr❡♥t ❝♦♠♣♦♥❡♥ts ✳ ✳ ✳ ✻✼ ✼✳✸ ❘❡❝♦✈❡r② ♣r♦❝❡❞✉r❡s ❢♦r t❤❡ ③❡r♦✲♦r❞❡r s♦❧❡♥♦✐❞❛❧ ❝✉rr❡♥t ❝♦♠♣♦♥❡♥ts ✳ ✳ ✳ ✻✾ ✼✳✸✳✶ ❆ ♣♦t❡♥t✐❛❧✲❜❛s❡❞ r❡❝♦✈❡r② ♣r♦❝❡❞✉r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾ ✼✳✸✳✷ ❆ ✈❡❝t♦r✲❜❛s❡❞ r❡❝♦✈❡r② ♣r♦❝❡❞✉r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹ ✼✳✹ ❆ ❢✉❧❧ ✜rst✲♦r❞❡r r❡❝♦✈❡r❡❞ ❝✉rr❡♥t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✽ ✼✳✺ ❆ ♣♦st❡r✐♦r✐ ❡rr♦r ❡st✐♠❛t✐♦♥ ♦❢ t❤❡ ▼♦▼ ❝✉rr❡♥t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✵ ✼✳✺✳✶ ●❧♦❜❛❧ ❡rr♦rs ❛♥❞ ❣❧♦❜❛❧ ❡rr♦r ❡st✐♠❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✵ ✼✳✺✳✷ ❊❧❡♠❡♥t✲✇✐s❡ ❡rr♦r ❡st✐♠❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✷ ✼✳✻ ❈♦♥❝❧✉s✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✹ ✽ ❈♦♥❝❧✉s✐♦♥s ❛♥❞ ❘❡❝♦♠♠❡♥❞❛t✐♦♥s ✽✺ ❆ ▲✐♥❡❛r ❋✉♥❝t✐♦♥ ❋✐t ✽✼ ❇✐❜❧✐♦❣r❛♣❤② ✽✾

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▲✐st ♦❢ ❋✐❣✉r❡s

✸✳✶ ❚r✐❛♥❣❧❡ ♣❛✐r ❛ss♦❝✐❛t❡❞ ✇✐t❤ ✐♥t❡r✐♦r ❡❞❣❡ n✱ ❛❧♦♥❣ ✇✐t❤ ❣❡♦♠❡tr✐❝❛❧ ♣❛r❛♠❡✲ t❡rs ✉s❡❞ ❞✉r✐♥❣ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ❘❲● ❜❛s✐s ❢✉♥❝t✐♦♥s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✸✳✷ ❚✇♦ sq✉❛r❡ P❊❈ ♣❧❛t❡s ♦❢ ❞✐♠❡♥s✐♦♥s 0.15λ ❛♥❞ 1.0λ r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡ ♠❡s❤❡s ❛♥❞ ♣r✐♥❝✐♣❛❧ ❝✉ts✱ ❆❆ ❛♥❞ ❇❇✱ ❛s ✉s❡❞ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡ ❛r❡ ✐❧❧✉str❛t❡❞✳ ✳ ✳ ✳ ✳ ✶✺ ✸✳✸ ❉♦♠✐♥❛♥t ❝✉rr❡♥t ❝♦♠♣♦♥❡♥t✱ JX✱ ♦♥ ❛ 0.15λ sq✉❛r❡ ✢❛t ♣❧❛t❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✸✳✹ ❉♦♠✐♥❛♥t ❝✉rr❡♥t ❝♦♠♣♦♥❡♥t✱ JX✱ ♦♥ ❛ 1.0λ sq✉❛r❡ ✢❛t ♣❧❛t❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✸✳✺ ❆ 1.5λ × 0.5λ P❊❈ ♣❧❛t❡ tr✐❛♥❣✉❧❛t❡❞ ✇✐t❤ ✭❛✮ t❤❡ ❉❡❧❛✉♥❛② ❢✉♥❝t✐♦♥ ✐♥ ●◆❯ ❖❝t❛✈❡✱ ✭❜✮ ❛ ❢❛❝t♦r✲✽ ✉♥✐❢♦r♠ h✲r❡✜♥❡❞ ✈❡rs✐♦♥ ♦❢ ❛ t✇♦ tr✐❛♥❣❧❡ ♠❡s❤✱ ✭❝✮ t❤❡ ❢r❡❡ s♦❢t✇❛r❡ ♣❛❝❦❛❣❡ ●♠s❤ ❛♥❞ ✭❞✮ t❤❡ s♦❢t✇❛r❡ ♣❛❝❦❛❣❡ ❋❊❑❖✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✸✳✻ ❙❛♠♣❧❡ ♣♦✐♥ts ♦♥ t✇♦ r♦t❛t✐♦♥s ♦❢ ❛ tr✐❛♥❣❧❡ ❢♦r ❛ s✐①✲♣♦✐♥t ●❛✉ss✐❛♥ q✉❛❞r❛✲ t✉r❡ s❝❤❡♠❡ ✇✐t❤ ❞♦t s✐③❡s ✐♥❞✐❝❛t✐✈❡ ♦❢ ✇❡✐❣❤t✐♥❣ ❝♦❡✣❝✐❡♥ts✳ ❚r✐❛♥❣❧❡ ❝❡♥✲ tr♦✐❞s ✐♥❞✐❝❛t❡❞ ❢♦r r❡❢❡r❡♥❝❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✹✳✶ ❙❛♠♣❧❡ ♣♦✐♥ts ❛♥❞ ✐♥t❡❣r❛t✐♦♥ s✉❜✲❞♦♠❛✐♥s ❢♦r ❛ ❢♦✉r✲♣♦✐♥t ❛♥❞ s✐①t❡❡♥✲♣♦✐♥t ❝♦♠♣♦s✐t❡ ❝❡♥tr❡✲♣♦✐♥t q✉❛❞r❛t✉r❡ s❝❤❡♠❡ ✇✐t❤ ❞♦t s✐③❡s ✐♥❞✐❝❛t✐✈❡ ♦❢ ✇❡✐❣❤t✲ ✐♥❣ ❝♦❡✣❝✐❡♥ts✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✹✳✷ ❙❛♠♣❧❡ ♣♦✐♥ts N ❢♦r t❤r❡❡ ❞✐✛❡r❡♥t ●❛✉ss✐❛♥ q✉❛❞r❛t✉r❡ s❝❤❡♠❡s ♦❢ ♦r❞❡r p ✇✐t❤ ❞♦t s✐③❡s ✐♥❞✐❝❛t✐✈❡ ♦❢ ✇❡✐❣❤t✐♥❣ ❝♦❡✣❝✐❡♥ts✳ ❚r✐❛♥❣❧❡ ❝❡♥tr♦✐❞s ✐♥❞✐❝❛t❡❞ ❢♦r r❡❢❡r❡♥❝❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✹✳✸ ❚r✐❛♥❣✉❧❛r ❞♦♠❛✐♥ ❞✐✈✐❞❡❞ ✐♥t♦ t❤r❡❡ s✉❜✲tr✐❛♥❣❧❡s ❛t t❤❡ ❧♦❝❛t✐♦♥ ♦❢ ❛ s✐♥❣✉✲ ❧❛r✐t②✳ ❊❛❝❤ s✉❜ tr✐❛♥❣❧❡ ❤❛s ❧♦❝❛❧ ❝♦♦r❞✐♥❛t❡s ❞❡✜♥❡❞ ❛s s❤♦✇♥ r✐❣❤t✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✹✳✹ ❙❛♠♣❧❡ ♣♦✐♥ts ♦❢ ❛ 6 × 6 ●❛✉ss✐❛♥ ♣r♦❞✉❝t r✉❧❡ ♦♥ tr❛♥s❢♦r♠❡❞ ❛♥❞ ♦r✐❣✐♥❛❧ s✉❜✲tr✐❛♥❣❧❡ ✐♥t❡❣r❛t✐♦♥ ❞♦♠❛✐♥s✳ ❉♦t s✐③❡s ✐♥❞✐❝❛t✐✈❡ ♦❢ ✇❡✐❣❤t✐♥❣ ❝♦❡✣❝✐❡♥ts✳ ✷✾ ✹✳✺ ❙❛♠♣❧❡ ♣♦✐♥ts ♦❢ ❛ 3 × 6 × 6 ❘❛❞✐❛❧✲❆♥❣✉❧❛r✲R1✲❙qrt q✉❛❞r❛t✉r❡ s❝❤❡♠❡✳ ✳ ✳ ✳ ✸✵ ✹✳✻ ◆♦r♠❛❧✐s❡❞ ❡rr♦r ♦❢ t❤r❡❡ q✉❛❞r❛t✉r❡ r✉❧❡s ❛s ♥✉♠❜❡r ♦❢ q✉❛❞r❛t✉r❡ ♣♦✐♥ts ✐♥❝r❡❛s❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✺✳✶ ❖♥❡✲❞✐♠❡♥s✐♦♥❛❧ s❝❛❧❛r ❜❛s✐s ❢✉♥❝t✐♦♥s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✺✳✷ P✐❡❝❡✇✐s❡✲❝♦♥st❛♥t ❛♥❞ ♣✐❡❝❡✇✐s❡✲❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s ♦❢ ❛♥ ❛r❜✐tr❛r② tr❛♥✲ s❝❡♥❞❡♥t❛❧ ❢✉♥❝t✐♦♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✺✳✸ ❙❡❝♦♥❞✲♦r❞❡r ▲❛❣r❛♥❣❡ ❢✉♥❝t✐♦♥s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✺✳✹ ❙✐♠♣❧❡① ❝♦♦r❞✐♥❛t❡ L1 ❞❡✜♥❡❞ ♦♥ ❛ tr✐❛♥❣✉❧❛r ❡❧❡♠❡♥t✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✈✐✐✐

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▲■❙❚ ❖❋ ❋■●❯❘❊❙ ✐① ✺✳✺ ❈♦♥st❛♥t ❜❛s✐s ❢✉♥❝t✐♦♥ B0✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✺✳✻ ▲✐♥❡❛r ▲❛❣r❛♥❣❡ ❜❛s✐s ❢✉♥❝t✐♦♥ B1✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✺✳✼ ▲✐♥❡❛r ♣②r❛♠✐❞ ❜❛s✐s ❢✉♥❝t✐♦♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✺✳✽ ◗✉❛❞r❛t✐❝ ▲❛❣r❛♥❣❡ ❜❛s✐s ❢✉♥❝t✐♦♥s Bijk✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✺✳✾ ▲✐♥❡❛r ❞✐✈❡r❣❡♥❝❡✲❝♦♥❢♦r♠✐♥❣ ✈❡❝t♦r ❜❛s✐s ❢✉♥❝t✐♦♥s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✺✳✶✵ ❊❞❣❡ ❜❛s❡❞ ❧✐♥❡❛r ❞✐✈❡r❣❡♥❝❡✲❝♦♥❢♦r♠✐♥❣ ✈❡❝t♦r ❜❛s✐s ❢✉♥❝t✐♦♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✺✳✶✶ ◆♦❞❛❧ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ ③❡r♦✲♦r❞❡r s♦❧❡♥♦✐❞❛❧ ❢✉♥❝t✐♦♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✺✳✶✷ ❊❞❣❡ ❜❛s❡❞ ❧✐♥❡❛r s♦❧❡♥♦✐❞❛❧ ✈❡❝t♦r ❜❛s✐s ❢✉♥❝t✐♦♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✺✳✶✸ ◗✉❛❞r❛t✐❝ ❞✐✈❡r❣❡♥t ✈❡❝t♦r ❜❛s✐s ❢✉♥❝t✐♦♥s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✺✳✶✹ ❙♣❛♥♥✐♥❣ tr❡❡ ♦♥ ❛ tr✐❛♥❣✉❧❛t❡❞ r❡❝t❛♥❣✉❧❛r ♣❛t❝❤✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✺✳✶✺ ❩❡r♦✲♦r❞❡r s♦❧❡♥♦✐❞❛❧ ❢✉♥❝t✐♦♥ ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ❞♦♠❛✐♥ ❜♦✉♥❞❛r② ♦❢ ❛ r❡❝t✲ ❛♥❣✉❧❛r ♣❛t❝❤✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✻✳✶ P✐❡❝❡✇✐s❡✲❝♦♥st❛♥t ❝❤❛r❣❡ ♠❛❣♥✐t✉❞❡ ♦❢ ❛ ✜rst✲♦r❞❡r ▼♦▼ s♦❧✉t✐♦♥ ♦♥ ❛ r❡❝t✲ ❛♥❣✉❧❛r P❊❈ ♣❧❛t❡ ✭❧♦❣ s❝❛❧❡✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✻✳✷ ❋❛❝t♦r✲❢♦✉r ✉♥✐❢♦r♠ ❤✲r❡✜♥❡♠❡♥t ♦❢ ❛ tr✐❛♥❣❧❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✻✳✸ ❘❡❢❡r❡♥❝❡ ❝❤❛r❣❡ ♠❛❣♥✐t✉❞❡ ❞✐str✐❜✉t✐♦♥ ♦❜t❛✐♥❡❞ ♦♥ ❛ ❢❛❝t♦r✲❢♦✉r ❤✲r❡✜♥❡❞ ✈❡rs✐♦♥ ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ♠❡s❤ ♦❢ ❛ r❡❝t❛♥❣✉❧❛r P❊❈ ♣❧❛t❡ ✭❧♦❣ s❝❛❧❡✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✻✳✹ ▲♦❝❛❧ ♣❛t❝❤ ❢♦r♠❡❞ ❜② t❤❡ ✉♥✐♦♥ ♦❢ ❛❧❧ ❡❧❡♠❡♥ts ❛r♦✉♥❞ ✈❡rt❡① ♥♦❞❡ k✳ ✳ ✳ ✳ ✳ ✹✾ ✻✳✺ ❘❡❝♦✈❡r❡❞ ❝❤❛r❣❡ ♠❛❣♥✐t✉❞❡ ❞✐str✐❜✉t✐♦♥ ♦❜t❛✐♥❡❞ ✇✐t❤ ♥♦❞❛❧ ❛✈❡r❛❣✐♥❣ ✭❧♦❣ s❝❛❧❡✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✻✳✻ ❉✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ r❡❢❡r❡♥❝❡ ❝❤❛r❣❡ ♠❛❣♥✐t✉❞❡ ❛♥❞ ♥♦❞❛❧ ❛✈❡r❛❣✐♥❣ r❡❝♦✈❡r❡❞ ❝❤❛r❣❡ ♠❛❣♥✐t✉❞❡ ✭❧♦❣ s❝❛❧❡✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✻✳✼ ❆ ❜♦✉♥❞❛r② ♥♦❞❡ t❤❛t ❞♦❡s ♥♦t ❢❛❧❧ ✇✐t❤✐♥ ❛♥② ✐♥t❡r✐♦r ♣❛t❝❤✱ ❛♥❞ ❛ ❜♦✉♥❞❛r② ♥♦❞❡ t❤❛t ❢❛❧❧s ✇✐t❤✐♥ t✇♦ ✐♥t❡r✐♦r ♣❛t❝❤❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✻✳✽ ❘❡❝♦✈❡r❡❞ ❝❤❛r❣❡ ♠❛❣♥✐t✉❞❡ ❞✐str✐❜✉t✐♦♥ ♦❜t❛✐♥❡❞ ✇✐t❤ t❤❡ ❩❩✲r❡❝♦✈❡r② ♠❡t❤♦❞ ✭❧♦❣ s❝❛❧❡✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✻✳✾ ❉✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ r❡❢❡r❡♥❝❡ ❝❤❛r❣❡ ♠❛❣♥✐t✉❞❡ ❛♥❞ ❩❩✲r❡❝♦✈❡r② ❝❤❛r❣❡ ♠❛❣✲ ♥✐t✉❞❡ ✭❧♦❣ s❝❛❧❡✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✻✳✶✵ ❈❤❛r❣❡ ❞❡♥s✐t② ♦❢ t❤❡ ❘❲●✲r❡❝♦✈❡r② ♣r♦❝❡❞✉r❡ ♦♥ ❛ ❧♦❝❛❧ ♣❛t❝❤✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✻✳✶✶ ❘❡❝♦✈❡r② ♦❢ t❤❡ ❝❤❛r❣❡ ❛t t❤❡ ♠✐❞♣♦✐♥t ♦❢ t❤❡ nth ❡❞❣❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✻✳✶✷ ❆ s✉r❢❛❝❡ ❥✉♥❝t✐♦♥ ✇❤❡r❡ t❤r❡❡ ❡❧❡♠❡♥ts ♠❡❡t✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✻✳✶✸ ❘❡❝♦✈❡r❡❞ ❝❤❛r❣❡ ♠❛❣♥✐t✉❞❡ ❞✐str✐❜✉t✐♦♥ ♦❜t❛✐♥❡❞ ✇✐t❤ t❤❡ ❘❲●✲❜❛s❡❞ r❡✲ ❝♦✈❡r② ♠❡t❤♦❞ ✭❧♦❣ s❝❛❧❡✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✻✳✶✹ ❉✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ r❡❢❡r❡♥❝❡ ❝❤❛r❣❡ ♠❛❣♥✐t✉❞❡ ❛♥❞ ❘❲●✲r❡❝♦✈❡r❡❞ ❝❤❛r❣❡ ♠❛❣♥✐t✉❞❡ ✭❧♦❣ s❝❛❧❡✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✻✳✶✺ ❊❧❡♠❡♥t✲✇✐s❡ r❡❢❡r❡♥❝❡ ❝❤❛r❣❡ ❡rr♦r ❞✐str✐❜✉t✐♦♥ ηr❡❢ ρ ✭❧♦❣ s❝❛❧❡✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✻✳✶✻ ❊❧❡♠❡♥t✲✇✐s❡ ❝❤❛r❣❡ ❡rr♦r ❡st✐♠❛t✐♦♥ η∗ ρ ❝❛❧❝✉❧❛t❡❞ ✇✐t❤ t❤❡ t❤r❡❡ ❝❤❛r❣❡ r❡✲ ❝♦✈❡r② ♠❡t❤♦❞s ✭❧♦❣ s❝❛❧❡✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵

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▲■❙❚ ❖❋ ❋■●❯❘❊❙ ① ✼✳✶ ❈✉rr❡♥t ♠❛❣♥✐t✉❞❡ ♦❢ ❛ ♠✐①❡❞ ✜rst✲♦r❞❡r ▼♦▼ s♦❧✉t✐♦♥ ♦♥ ❛ P❊❈ ♣❛t❝❤ ✭❧♦❣ s❝❛❧❡✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ✼✳✷ ❘❡❢❡r❡♥❝❡ ❝✉rr❡♥t ♠❛❣♥✐t✉❞❡ ♦❜t❛✐♥❡❞ ♦♥ ❛ ❢❛❝t♦r✲❢♦✉r ❤✲r❡✜♥❡❞ ✈❡rs✐♦♥ ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ♠❡s❤ ✭❧♦❣ s❝❛❧❡✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✼✳✸ ■♥st❛♥t❛♥❡♦✉s ❝✉rr❡♥t ❝♦♠♣♦♥❡♥ts✱ ωt = 0o ❛♥❞ ωt = 90o✱ ♦❢ ③❡r♦✲♦r❞❡r s♦❧❡♥♦✐❞❛❧ ❝✉rr❡♥t ˜Jtr❡❡ ✭❧♦❣ s❝❛❧❡✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ✼✳✹ ■♥st❛♥t❛♥❡♦✉s ❝✉rr❡♥t ❝♦♠♣♦♥❡♥ts✱ ωt = 0o❛♥❞ ωt = 90o✱ ♦❢ r❡❢❡r❡♥❝❡ s♦❧✉t✐♦♥✱ ♦❜t❛✐♥❡❞ ♦♥ ❛ ❢❛❝t♦r✲❢♦✉r ❤✲r❡✜♥❡❞ ✈❡rs✐♦♥ ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ♠❡s❤ ✭❧♦❣ s❝❛❧❡✮✳ ✳ ✳ ✻✻ ✼✳✺ ❇❛s❡ ♦❢ ❛ tr❡❡ ❣r❛♣❤✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼ ✼✳✻ ❘❡❝♦✈❡r❡❞ ❞✐✈❡r❣❡♥t ❝✉rr❡♥t ✇✐t❤ s♦❧❡♥♦✐❞❛❧ ❝♦♠♣♦♥❡♥t ♦❢ ♦r✐❣✐♥❛❧ s♦❧✉t✐♦♥ ✭❧♦❣ s❝❛❧❡✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽ ✼✳✼ ❉✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ r❡❢❡r❡♥❝❡ ❝✉rr❡♥t ❛♥❞ r❡❝♦✈❡r❡❞ ❞✐✈❡r❣❡♥t ❝✉rr❡♥t ✇✐t❤ s♦❧❡♥♦✐❞❛❧ ❝♦♠♣♦♥❡♥t ♦❢ ♦r✐❣✐♥❛❧ s♦❧✉t✐♦♥ ✭❧♦❣ s❝❛❧❡✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾ ✼✳✽ P✐❡❝❡✇✐s❡✲❧✐♥❡❛r ❝♦♥t✐♥✉♦✉s ♣♦t❡♥t✐❛❧ ✜❡❧❞ ♦♥ ❛ ❧♦❝❛❧ ♣❛t❝❤✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵ ✼✳✾ P✐❡❝❡✇✐s❡✲❧✐♥❡❛r x✲✈❛r✐❛t✐♦♥ ♦❢ ❛ ♣♦t❡♥t✐❛❧ ✜❡❧❞ ♦♥ ❛ ❧♦❝❛❧ ♣❛t❝❤✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶ ✼✳✶✵ ◗✉❛❞r❛t✐❝ x✲✈❛r✐❛t✐♦♥ ♦❢ ❛ ♣♦t❡♥t✐❛❧ ✜❡❧❞ ♦♥ ❛ ❧♦❝❛❧ ♣❛t❝❤✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶ ✼✳✶✶ ❘❡❝♦✈❡r❡❞ q✉❛❞r❛t✐❝ ♣♦t❡♥t✐❛❧ ✜❡❧❞ ♦♥ ❛ ❧♦❝❛❧ ♣❛t❝❤✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷ ✼✳✶✷ ■♥st❛♥t❛♥❡♦✉s ❝✉rr❡♥t ❝♦♠♣♦♥❡♥ts✱ ωt = 0o ❛♥❞ ωt = 90o✱ ♦❢ ♣♦t❡♥t✐❛❧✲❜❛s❡❞ r❡❝♦✈❡r❡❞ ✜rst✲♦r❞❡r s♦❧❡♥♦✐❞❛❧ ❝✉rr❡♥t ˜Js♦❧ ✭❧♦❣ s❝❛❧❡✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸ ✼✳✶✸ P♦t❡♥t✐❛❧✲❜❛s❡❞ r❡❝♦✈❡r❡❞ ♠✐①❡❞ s❡❝♦♥❞✲♦r❞❡r ❝✉rr❡♥t ♠❛❣♥✐t✉❞❡ ✭❧♦❣ s❝❛❧❡✮✳ ✳ ✼✹ ✼✳✶✹ ❉✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ r❡❢❡r❡♥❝❡ ❝✉rr❡♥t ❛♥❞ ♠✐①❡❞ s❡❝♦♥❞✲♦r❞❡r ♣♦t❡♥t✐❛❧✲❜❛s❡❞ r❡❝♦✈❡r❡❞ ❝✉rr❡♥t ✭❧♦❣ s❝❛❧❡✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹ ✼✳✶✺ ❘❡❝♦✈❡r❡❞ ✈❡❝t♦r ❝♦♠♣♦♥❡♥ts ♦♥ ❛ ❧♦❝❛❧ ♣❛t❝❤✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺ ✼✳✶✻ ■♥st❛♥t❛♥❡♦✉s ❝✉rr❡♥t ❝♦♠♣♦♥❡♥ts✱ ωt = 0o ❛♥❞ ωt = 90o✱ ♦❢ r❡❝♦✈❡r❡❞ ✈❡❝t♦r✲ ❜❛s❡❞ ✜rst✲♦r❞❡r s♦❧❡♥♦✐❞❛❧ ❝✉rr❡♥t ˜Js♦❧ ✭❧♦❣ s❝❛❧❡✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼ ✼✳✶✼ ❱❡❝t♦r✲❜❛s❡❞ r❡❝♦✈❡r❡❞ ♠✐①❡❞ s❡❝♦♥❞✲♦r❞❡r ❝✉rr❡♥t ♠❛❣♥✐t✉❞❡ ✭❧♦❣ s❝❛❧❡✮✳ ✳ ✳ ✳ ✼✽ ✼✳✶✽ ❉✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ r❡❢❡r❡♥❝❡ ❝✉rr❡♥t ❛♥❞ ♠✐①❡❞ s❡❝♦♥❞✲♦r❞❡r ✈❡❝t♦r✲❜❛s❡❞ r❡✲ ❝♦✈❡r❡❞ ❝✉rr❡♥t ✭❧♦❣ s❝❛❧❡✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✽ ✼✳✶✾ ❱❡❝t♦r✲❜❛s❡❞ r❡❝♦✈❡r❡❞ ❢✉❧❧ ✜rst✲♦r❞❡r ❝✉rr❡♥t ♠❛❣♥✐t✉❞❡ ✭❧♦❣ s❝❛❧❡✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✾ ✼✳✷✵ ❉✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ r❡❢❡r❡♥❝❡ ❝✉rr❡♥t ❛♥❞ ❢✉❧❧ ✜rst✲♦r❞❡r ✈❡❝t♦r✲❜❛s❡❞ r❡❝♦✈❡r❡❞ ❝✉rr❡♥t ✭❧♦❣ s❝❛❧❡✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✵ ✼✳✷✶ ❊❧❡♠❡♥t✲✇✐s❡ r❡❢❡r❡♥❝❡ ❝✉rr❡♥t ❡rr♦r ❞✐str✐❜✉t✐♦♥ ηr❡❢ J ✭❧♦❣ s❝❛❧❡✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✷ ✼✳✷✷ ❊❧❡♠❡♥t✲✇✐s❡ ❝✉rr❡♥t ❡rr♦r ❡st✐♠❛t✐♦♥s η∗ J ✭❧♦❣ s❝❛❧❡✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✸ ❆✳✶ ▲♦❝❛❧ ❝♦♦r❞✐♥❛t❡s ❢♦r ❛♥ ❛r❜✐tr❛r② tr✐❛♥❣❧❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✼

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▲✐st ♦❢ ❚❛❜❧❡s

✸✳✶ ❚❤❡ ♦♥❡✲✱ t❤r❡❡✲ ❛♥❞ s✐①✲♣♦✐♥t ●❛✉ss✐❛♥ q✉❛❞r❛t✉r❡ r✉❧❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✹✳✶ ●❛✉ss✐❛♥ q✉❛❞r❛t✉r❡ r✉❧❡s ♦❢ ♦r❞❡r p ✇✐t❤ ✇❡✐❣❤ts w∗ i ❛♥❞ ❛❜s❝✐ss❛s x∗i✳ ✳ ✳ ✳ ✳ ✷✸ ✹✳✷ ●❛✉ss✐❛♥ q✉❛❞r❛t✉r❡ r✉❧❡s ♦❢ ♦r❞❡r p ✇✐t❤ ✇❡✐❣❤ts wi ❢♦r tr✐❛♥❣✉❧❛r ❞♦♠❛✐♥s✳ ✳ ✷✻ ✻✳✶ ●❧♦❜❛❧ ❝❤❛r❣❡ ❡rr♦rs ✐♥ r❡❧❛t✐♦♥ t♦ t❤❡ r❡❢❡r❡♥❝❡ s♦❧✉t✐♦♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✻✳✷ ●❧♦❜❛❧ ❡st✐♠❛t❡s ♦❢ t❤❡ ❝❤❛r❣❡ ❡rr♦r ✐♥ t❤❡ ♦r✐❣✐♥❛❧ s♦❧✉t✐♦♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✼✳✶ ●❧♦❜❛❧ ❝✉rr❡♥t ❡rr♦rs ✐♥ r❡❧❛t✐♦♥ t♦ t❤❡ r❡❢❡r❡♥❝❡ s♦❧✉t✐♦♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✶ ✼✳✷ ●❧♦❜❛❧ ❡st✐♠❛t❡s ♦❢ t❤❡ ❝✉rr❡♥t ❡rr♦r ✐♥ t❤❡ ♦r✐❣✐♥❛❧ s♦❧✉t✐♦♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✶ ①✐

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❈❤❛♣t❡r ✶

■♥tr♦❞✉❝t✐♦♥

❊❧❡❝tr♦♠❛❣♥❡t✐❝ ♣r♦❜❧❡♠s ♦❢t❡♥ r❡✈♦❧✈❡ ❛r♦✉♥❞ t❤❡ ♣r❡❞✐❝t✐♦♥ ♦❢ ✜❡❧❞s ✐♥ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❝♦♠♣❧✐❝❛t❡❞ str✉❝t✉r❡s✳ ❙♦❧✈✐♥❣ t❤❡s❡ ♣r♦❜❧❡♠s ✐♥✈♦❧✈❡s ❡①t❡♥s✐✈❡ ♥✉♠❡r✐❝❛❧ ♣r♦❝❡❞✉r❡s t❤❛t ❛✐♠ t♦ ❛♣♣r♦♣r✐❛t❡❧② ❛♣♣r♦①✐♠❛t❡ ▼❛①✇❡❧❧✬s ❡q✉❛t✐♦♥s✳ ❙✉r❢❛❝❡ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥ ❢♦r♠✉❧❛t✐♦♥s ❛r❡ ❞✐s❝✉ss❡❞ ✐♥ s❡✈❡r❛❧ t❡①ts ❬✶✱ ✷❪✳ ❚②♣✐❝❛❧❧②✱ ❛ ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥ ✐s ❢♦✉♥❞ ♦♥ ❛ s✉❜✲s❡❝t✐♦♥❡❞ ♠♦❞❡❧ ✭❛❧s♦ r❡❢❡rr❡❞ t♦ ❛s ❛ ♠❡s❤❡❞ ♠♦❞❡❧✮ t❤❛t r❡♣r❡s❡♥ts t❤❡ s✉r❢❛❝❡ ♦❢ ❛ str✉❝t✉r❡✳ ❚❤❡ ♥✉♠❡r✐❝❛❧ ❢♦r♠✉❧❛t✐♦♥ r❡♣r❡s❡♥ts s✉r❢❛❝❡ ❝✉rr❡♥ts ♦♥ t❤❡ str✉❝t✉r❡ ✇✐t❤ ♣✐❡❝❡✇✐s❡✲♣♦❧②♥♦♠✐❛❧ ❜❛s✐s ❢✉♥❝t✐♦♥s✱ ❛♥❞ ❛ ❧❛r❣❡ ❧✐♥❡❛r s②st❡♠ ♦❢ ❡q✉❛t✐♦♥s ✐s ❝♦♥str✉❝t❡❞ ❛♥❞ s♦❧✈❡❞✱ ❜② ✐♠♣♦s✐♥❣ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ♦♥ t❤❡ ✜❡❧❞s✳ ❖♥❡ s✉❝❤ ♣r♦❝❡ss✱ ✇❤✐❝❤ ✐s ❝♦♥s✐❞❡r❡❞ ✐♥ t❤✐s ✇♦r❦✱ ✐s ❦♥♦✇♥ ❛s t❤❡ ▼❡t❤♦❞ ♦❢ ▼♦♠❡♥ts ✭▼♦▼✮✳ ❚❤❡ ▼♦▼ ✐s ❛ ♠❡t❤♦❞ ❢♦r s♦❧✈✐♥❣ s✉r❢❛❝❡ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s ♦❢ r❛❞✐❛t✐♦♥ ❛♥❞ s❝❛tt❡r✐♥❣ ❛♥❞ ✐s ♦♥❡ ♦❢ t❤❡ ♠♦st ✇✐❞❡❧② ✉s❡❞ ♥✉♠❡r✐❝❛❧ ♠❡t❤♦❞s ✐♥ t❤❡ ✜❡❧❞ ♦❢ ❝♦♠♣✉t❛t✐♦♥❛❧ ❡❧❡❝tr♦♠❛❣♥❡t✐❝s ✭❈❊▼✮✳ ❚❤❡ ✉s❡ ♦❢ ▼♦▼ ✐s ♣r❡✈❛❧❡♥t ✐♥ s❝❛tt❡r✐♥❣ ❛♣♣❧✐❝❛t✐♦♥s✱ s✉❝❤ ❛s r❛❞❛r ❝r♦ss s❡❝t✐♦♥ ♣r❡❞✐❝t✐♦♥s✱ ❛s ✇❡❧❧ ❛s r❛❞✐❛t✐♦♥ ❛♣♣❧✐❝❛t✐♦♥s✱ s✉❝❤ ❛s ❛♥t❡♥♥❛ ❛♥❛❧②s❡s ❛♥❞ ❡❧❡❝tr♦♠❛❣♥❡t✐❝ ✐♥t❡r❢❡r❡♥❝❡ ❛♥❞ ❝♦♠♣❛t✐❜✐❧✐t② ❛♥❛❧②s❡s✳ ❚❤❡ ❡❧❡❝tr✐❝ ✜❡❧❞ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥ ✭❊❋■❊✮ ❢♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ▼♦▼✱ s✉✐t❡❞ t♦ ❛r❜✐tr❛r② s✉r❢❛❝❡ ♠♦❞❡❧❧✐♥❣ ♦❢ ♦♣❡♥ ❛♥❞ ❝❧♦s❡❞ ♣❡r❢❡❝t ❡❧❡❝tr✐❝ ❝♦♥❞✉❝t✐♥❣ ✭P❊❈✮ ❜♦❞✐❡s✱ ✐s ❝♦♥s✐❞❡r❡❞ ✐♥ t❤✐s ✇♦r❦✳ ❆❧t❤♦✉❣❤ s✐♠✉❧❛t✐♦♥ ❢♦r♠s ❛ ❝❡♥tr❛❧ ♣❛rt ♦❢ ❛♥② ❞❡s✐❣♥ ♦r ❛♥❛❧②s✐s ♣r♦❝❡ss✱ t❤❡ ❛♣✲ ♣r♦①✐♠❛t❡ ♥❛t✉r❡ ♦❢ t❤❡ ▼♦▼ s♦❧✉t✐♦♥ s❤♦✉❧❞ ❛❧✇❛②s ❜❡ ❝♦♥s✐❞❡r❡❞✳ ■♥❤❡r❡♥t t♦ t❤❡ ▼♦▼ ♣r♦❝❡❞✉r❡ ✐s ❛ s♦❧✉t✐♦♥ ❡rr♦r ❞✉❡ t♦ t❤❡ ❞✐s❝r❡t✐s❛t✐♦♥ ❛♣♣r♦①✐♠❛t✐♦♥✳ ❆♥ ✐♥❝r❡❛s❡ ✐♥ t❤❡ ♦r❞❡r ♦❢ t❤❡ ♣♦❧②♥♦♠✐❛❧ ❜❛s✐s ❢✉♥❝t✐♦♥s ❡♠♣❧♦②❡❞✱ ❦♥♦✇♥ ❛s ♣✲❛❞❛♣t❛t✐♦♥✱ ♦r ❛ ❞❡❝r❡❛s❡ ✐♥ t❤❡ ❡❧❡♠❡♥t s✐③❡ ♦❢ t❤❡ s✉r❢❛❝❡ ❞✐s❝r❡t✐s❛t✐♦♥✱ ❦♥♦✇♥ ❛s ❤✲❛❞❛♣t❛t✐♦♥✱ ❝❛♥ ❜❡ ❛♣♣❧✐❡❞ t♦ r❡❞✉❝❡ t❤❡ s♦❧✉t✐♦♥ ❡rr♦r ❬✸❪✳ ■t s❤♦✉❧❞ ❤♦✇❡✈❡r ❜❡ ❛♣♣❛r❡♥t t❤❛t t❤❡ ❡rr♦r ❝❛♥ ♦♥❧② ❜❡ ♠✐♥✐♠✐s❡❞✱ ❛♥❞ ♥❡✈❡r ❝♦♠♣❧❡t❡❧② r❡♠♦✈❡❞✳ ❈♦♥tr♦❧ ♦✈❡r t❤❡ s♦❧✉t✐♦♥ ❡rr♦r ✐s st✐❧❧ ❞❡s✐r✲ ❛❜❧❡✱ ❛♥❞ ❛s s✉❝❤✱ r❡❧✐❛❜❧❡ ❡st✐♠❛t✐♦♥ ♦❢ t❤❡ ❡rr♦r ✐s ♥❡❡❞❡❞✳ ❚❤❡ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ ❛❞❛♣t✐✈❡ r❡✜♥❡♠❡♥t ♣r♦❝❡❞✉r❡s ❝❛♥ ❛❧s♦ ❜❡ s❡❧❡❝t✐✈❡❧② ❛♣♣❧✐❡❞ t♦ r❡❣✐♦♥s ♦❢ ❤✐❣❤ ❧♦❝❛❧ ❡rr♦r ✐♥ t❤❡ ♠❡s❤✱ ❡✛❡❝t✐♥❣ ❛ ❞❡s✐r❡❞ r❡❞✉❝t✐♦♥ ✐♥ t❤❡ ❣❧♦❜❛❧ ❡rr♦r ✇❤✐❧❡ ♠✐♥✐♠✐s✐♥❣ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❝♦sts ✐♥✈♦❧✈❡❞✳ ❚❤❡s❡ ♣r♦❝❡❞✉r❡s ❛r❡ ❞❡s✐r❛❜❧❡ s✐♠✉❧❛t✐♦♥ t♦♦❧s t❤❛t ❛r❡ ❤❡❛✈✐❧② r❡❧✐❛♥t ♦♥ t❤❡ ❛✈❛✐❧❛❜✐❧✐t② ♦❢ ❛ ❧♦❝❛❧ ❡st✐♠❛t❡ ♦❢ t❤❡ s♦❧✉t✐♦♥ ❡rr♦r✳ ✶

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❈❍❆P❚❊❘ ✶✳ ■◆❚❘❖❉❯❈❚■❖◆ ✷ ❲✐t❤ t❤✐s ✐♥ ♠✐♥❞✱ t❤❡ ❛✐♠ ♦❢ t❤✐s r❡s❡❛r❝❤ ✐s t♦ ❢♦r♠✉❧❛t❡ ❛♥ ❛ ♣♦st❡r✐♦r✐ ♠❡t❤♦❞ ❢♦r ❣❧♦❜❛❧ ❛♥❞ ❧♦❝❛❧ q✉❛♥t✐t❛t✐✈❡ ❡rr♦r ❡st✐♠❛t✐♦♥ ✇✐t❤✐♥ t❤❡ ❘❛♦✲❲✐❧t♦♥✲●❧✐ss♦♥ ✭❘❲●✮ ▼♦▼✳ ❈❤❛♣t❡r ✷ ♦✛❡rs ❛♥ ♦✈❡r✈✐❡✇ ♦❢ ▼♦▼ ❡rr♦r ❡st✐♠❛t✐♦♥✱ ❛s ✇❡❧❧ ❛s s♦♠❡ r❡❧❛t❡❞ ✇♦r❦ ✇✐t❤✐♥ t❤❡ ❋✐♥✐t❡ ❊❧❡♠❡♥t ▼❡t❤♦❞ ✭❋❊▼✮ ❝♦♥t❡①t✳ ■t ❛❧s♦ ♣r♦✈✐❞❡s t❤❡ ♠♦t✐✈❛t✐♦♥ ❢♦r t❤❡ ❢♦❝✉s ♦❢ t❤✐s r❡s❡❛r❝❤✱ r❡❝♦✈❡r② ❜❛s❡❞ ❡rr♦r ❡st✐♠❛t✐♦♥✱ ✇❤✐❝❤ ✐s ❛ ✇❡❧❧✲❦♥♦✇♥ t♦♣✐❝ ✐♥ t❤❡ ❋❊▼✲❝♦♥t❡①t ❜✉t ❛♥ ✉♥❡①♣❧♦r❡❞ t♦♣✐❝ ✇✐t❤✐♥ t❤❡ ▼♦▼✲❝♦♥t❡①t✳ ❚❤❡ ❊❋■❊✲❜❛s❡❞ ▼♦▼ ❢♦r♠✉❧❛t✐♦♥ ✐s ♣r❡s❡♥t❡❞ ✐♥ ❈❤❛♣t❡r ✸✳ ❚❤❡ str✉❝t✉r❡ ♦❢ ❈❤❛♣t❡r ✸ ✐s s✐♠✐❧❛r t♦ t❤❡ ♣❛♣❡r ♦❢ ❘❛♦✱ ❲✐❧t♦♥✱ ❛♥❞ ●❧✐ss♦♥ ❬✹❪✱ ✇❤❡r❡ ❛ ♠✐①❡❞ ✜rst✲♦r❞❡r ▼♦▼ ❢♦r♠✉❧❛t✐♦♥ ✐s ♣r♦✈✐❞❡❞✳ ❘❡s✉❧ts ♦❜t❛✐♥❡❞ ✐♥ t❤✐s st✉❞② ❛r❡ ❛❧s♦ ❝♦♠♣❛r❡❞ ✇✐t❤ t❤♦s❡ ♣r❡s❡♥t❡❞ ✐♥ t❤❛t ♣❛♣❡r✳ ■♥ ❈❤❛♣t❡rs ✹ ❛♥❞ ✺✱ t❤❡ t♦♣✐❝s ♦❢ ♥✉♠❡r✐❝❛❧ ✐♥t❡❣r❛t✐♦♥ ❛♥❞ ❜❛s✐s ❢✉♥❝t✐♦♥s ❛r❡ ❡①♣❧♦r❡❞✳ ❚❤❡s❡ t❤❡♠❡s ❛r❡ ❝❡♥tr❛❧ t♦ ♥✉♠❡r✐❝❛❧ ♠❡t❤♦❞s ❛♥❞ t❤✉s ❝r✉❝✐❛❧ t♦ t❤❡ ❞❡✈❡❧♦♣♠❡♥t ❛♥❞ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ❛ r✐❣♦r♦✉s ❛♥❞ ❛❝❝✉r❛t❡ ▼♦▼ ❢♦r♠✉❧❛t✐♦♥✳ ❈❤❛♣t❡r ✻ ✐♥✈❡st✐❣❛t❡s s♦♠❡ ♣♦ss✐❜❧❡ ♠❡t❤♦❞s ♦❢ ❝❤❛r❣❡ r❡❝♦✈❡r② ❛♥❞ ❧♦♦❦s ❛t ❝❤❛r❣❡ ❡rr♦r ❡st✐♠❛t♦rs ❜❛s❡❞ ♦♥ ❛ r❡❝♦✈❡r❡❞ ♥✉♠❡r✐❝❛❧ ❝❤❛r❣❡✳ ❆ ♥❡✇ ♠❡t❤♦❞ ❢♦r ❝❤❛r❣❡ r❡❝♦✈❡r②✱ ❛s ❞❡✈❡❧♦♣❡❞ ❜② t❤❡ ❛✉t❤♦r✱ ✐s ❛❧s♦ ✐♥tr♦❞✉❝❡❞ ✐♥ t❤✐s ❝❤❛♣t❡r✳ ❚❤❡ ❝♦♥❝❡♣t ♦❢ ❝❤❛r❣❡ r❡❝♦✈❡r② ✐s t❤❡♥ ❡①t❡♥❞❡❞ t♦ ❝✉rr❡♥t r❡❝♦✈❡r② ✐♥ ❈❤❛♣t❡r ✼✳ ❚✇♦ ❝✉rr❡♥t r❡❝♦✈❡r② ♠❡t❤♦❞s✱ ❞❡✈❡❧♦♣❡❞ ❜② t❤❡ ❛✉t❤♦r✱ ❛r❡ ✐♥tr♦❞✉❝❡❞ ✐♥ t❤✐s ❝❤❛♣t❡r✱ ❛♥❞ ❡rr♦r ❡st✐♠❛t✐♦♥s ♦❢ t❤❡ s♦❧✉t✐♦♥ ❝✉rr❡♥t ❛r❡ ✐♥✈❡st✐❣❛t❡❞ ❛♥❞ ❞✐s❝✉ss❡❞✳ ❚❤❡ ❞♦❝✉♠❡♥t ✐s ❝♦♥❝❧✉❞❡❞ ✐♥ ❈❤❛♣t❡r ✽ ✇✐t❤ ❛ s✉♠♠❛r② ♦❢ t❤❡ r❡s❡❛r❝❤✳ ❘❡❝♦♠♠❡♥❞❛t✐♦♥s ❢♦r ❢✉t✉r❡ r❡s❡❛r❝❤ ❛r❡ ❛❧s♦ ♣r♦✈✐❞❡❞✳

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❈❤❛♣t❡r ✷

❊rr♦r ❊st✐♠❛t✐♦♥✿ ❚❤❡ ❙❡tt✐♥❣

❈❊▼ ♠❡t❤♦❞s ✐♥✈♦❧✈❡ t❤❡ ❞✐s❝r❡t✐s❛t✐♦♥ ♦❢ ❛ ❝♦♥t✐♥✉✉♠ ♠♦❞❡❧ ♦❢ ❡❧❡❝tr♦♠❛❣♥❡t✐❝ ❜❡✲ ❤❛✈✐♦✉r ✐♥ ♦r❞❡r t♦ ♦❜t❛✐♥ ❛ ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥✳ ■♥❤❡r❡♥t t♦ ❛❧❧ t❤❡s❡ ♠❡t❤♦❞s ❛r❡ ♥✉✲ ♠❡r✐❝❛❧ s♦❧✉t✐♦♥ ❡rr♦rs✳ ❈♦♥tr♦❧✱ ❛♥❛❧②s✐s ❛♥❞ ❡✛❡❝t✐✈❡ ♠✐♥✐♠✐s❛t✐♦♥ ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r ♦❢ ❛ ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥ r❡st ❤❡❛✈✐❧② ♦♥ t❤❡ ❛✈❛✐❧❛❜✐❧✐t② ♦❢ ❛ r❡❧✐❛❜❧❡ ❡st✐♠❛t❡ ♦❢ s✉❝❤ ❛♥ ❡rr♦r✳ ❇r♦❛❞❧② s♣❡❛❦✐♥❣✱ ❡rr♦r ❡st✐♠❛t♦rs ♠❛② ❜❡ ❣r♦✉♣❡❞ ❛❝❝♦r❞✐♥❣ t♦ ❛ ♣r✐♦r✐ ❛♥❞ ❛ ♣♦s✲ t❡r✐♦r✐ ❡st✐♠❛t♦rs ❬✸❪✳ ❚❤❡ ❢♦r♠❡r ✐s ❜❛s❡❞ ♦♥ ❛ t❤❡♦r❡t✐❝❛❧ ❛♣♣r♦❛❝❤ t❤❛t ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ♣r♦❜❧❡♠ ❞❛t❛ ❛♥❞ ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥✱ ✇❤✐❧❡ t❤❡ ❧❛tt❡r ✐s ❞❡r✐✈❡❞ ❞✐r❡❝t❧② ❢r♦♠ t❤❡ ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥✳ ❚❤✐s st✉❞② ❢♦❝✉s❡s ♦♥ ❛ ♣♦st❡r✐♦r✐ ❡rr♦r ❡st✐♠❛t✐♦♥✱ ❛s ♥❡❡❞❡❞ ❢♦r ❛❞❛♣t✐✈❡ r❡✜♥❡♠❡♥t ✐♥ t❤❡ ▼♦▼✳

✷✳✶ ❊rr♦r ❡st✐♠❛t✐♦♥ ✐♥ t❤❡ ▼❡t❤♦❞ ♦❢ ▼♦♠❡♥ts

❚✇♦ ❛♣♣r♦❛❝❤❡s ❢♦r ❞❡❛❧✐♥❣ ✇✐t❤ t❤❡ ♥✉♠❡r✐❝❛❧ ❡rr♦r ✐♥ t❤❡ ▼♦▼ ❛r❡ ♦❢t❡♥ ❡♥❝♦✉♥t❡r❡❞✳ ❚❤❡ ✜rst ✐♥✈♦❧✈❡s s✐♠♣❧❡ ✈❛❧✐❞❛t✐♦♥ ❛♥❞ ✈❡r✐✜❝❛t✐♦♥ ♣r♦❝❡❞✉r❡s t❤❛t ❡♠♣❧♦② t❡st ❝❛s❡s ❛♥❞ ❛♥❛❧②t✐❝❛❧ s♦❧✉t✐♦♥s✳ ❚❤❡ s❡❝♦♥❞ ✐s t❤❡ t❤❡♦r❡t✐❝❛❧ ❢✉♥❝t✐♦♥❛❧ ❛♥❛❧②s✐s ❛♣♣r♦❛❝❤ ♦❢ t❤❡ ❛♣♣❧✐❡❞ ♠❛t❤❡♠❛t✐❝s ❝♦♠♠✉♥✐t②✳ ❋♦r t❤❡ ❡♥❣✐♥❡❡r ✉s✐♥❣ ❛ ❈❊▼ t♦♦❧✱ ✐♥❞✐r❡❝t ♠❡t❤♦❞s s✉❝❤ ❛s ❜❡♥❝❤♠❛r❦✐♥❣ ♣r♦❜❧❡♠s ✇✐t❤ q✉❛s✐✲❛♥❛❧②t✐❝❛❧ s♦❧✉t✐♦♥s ♦r ♠❡❛s✉r❡❞ ❞❛t❛ ❛r❡ ♦❢ ❧✐tt❧❡ ✉s❡✳ ❇❡♥❝❤♠❛r❦✐♥❣ ♦❢ ❛ ▼♦▼ s♦❧✈❡r ✇♦✉❧❞ ♦♥❧② s❡r✈❡ t♦ ✈❡r✐❢② ❛ ♥✉♠❡r✐❝❛❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ❛❣❛✐♥st ♣r❡✲✈❛❧✐❞❛t❡❞ s♦❧✉t✐♦♥s✳ ❚❤❡ ✐ss✉❡ ♦❢ ❛ ♠❡❛s✉r❡ ♦❢ t❤❡ s♦❧✉t✐♦♥ ❛❝❝✉r❛❝② ♦❢ ❛ ❣❡♥❡r❛❧ ♣r♦❜❧❡♠ ✐s ♥♦t ❛❞❞r❡ss❡❞✳ ❚❤❡ ❛ ♣r✐♦r✐ t❤❡♦r❡t✐❝❛❧ ❛♣♣r♦❛❝❤ t♦ t❤❡ ♣r♦❜❧❡♠ ♦❢ ❡rr♦r ❛♥❛❧②s✐s s❡❡♥ ✐♥ t❤❡ ❛♣♣❧✐❡❞ ♠❛t❤❡♠❛t✐❝s ❝♦♠♠✉♥✐t② ❛❧s♦ ❞♦❡s ♥♦t ❞❡❛❧ ✇✐t❤ t❤✐s ♣r♦❜❧❡♠✳ ◆✉♠❡r✐❝❛❧ ❛♥❛❧②sts st✉❞②✐♥❣ ❝♦♠♣✉t❛t✐♦♥❛❧ ♠❡t❤♦❞s ♦❢t❡♥ ❛tt❡♠♣t t♦ q✉❛♥t✐❢② t❤❡ r❛t❡ ♦❢ r❡❞✉❝t✐♦♥ ✐♥ t❤❡ ❡rr♦r ♦❢ t❤❡ s♦❧✉t✐♦♥ ♦❢ ❛ ❣✐✈❡♥ ♣r♦❜❧❡♠ ❛s t❤❡ ♠❡s❤ ✐s r❡✜♥❡❞✳ ❚❤❡ ❛s②♠♣t♦t✐❝ ❝♦♥✈❡r❣❡♥❝❡ r❛t❡ ♦❢ t❤❡ ▼♦▼✱ ❛♥❞ ✇❤❡t❤❡r t❤❡ s♦❧✉t✐♦♥ ❝♦♥✈❡r❣❡s t♦ t❤❡ ❝♦rr❡❝t ❛♥s✇❡r ❢♦r ❛ ❣✐✈❡♥ ♣r♦❜❧❡♠✱ ❛r❡ ✇❡❧❧ ❞♦❝✉♠❡♥t❡❞ ❜② ♠❛♥② ❛✉t❤♦rs✱ ✐♥❝❧✉❞✐♥❣ ❬✺❪✳ ❲❤❡♥ q✉❛♥t✐✲ t❛t✐✈❡❧② ❡✈❛❧✉❛t✐♥❣ t❤❡ s♦❧✉t✐♦♥ ❛❝❝✉r❛❝② ♦❢ ❛ ♣❛rt✐❝✉❧❛r ♣r♦❜❧❡♠✱ ♦r ❞r✐✈✐♥❣ ❛♥ ❛❞❛♣t✐✈❡ ✸

(16)

❈❍❆P❚❊❘ ✷✳ ❊❘❘❖❘ ❊❙❚■▼❆❚■❖◆✿ ❚❍❊ ❙❊❚❚■◆● ✹ ♠♦❞❡❧ r❡✜♥❡♠❡♥t✱ ✐t ✐s ❤♦✇❡✈❡r ❞❡s✐r❛❜❧❡ t♦ ❤❛✈❡ ❛ ♣r♦❜❧❡♠✲s♣❡❝✐✜❝ ♠❡❛s✉r❡ ♦❢ t❤❡ ❡rr♦r✳ ❛ ♣♦st❡r✐♦r✐ ❡rr♦r ❡st✐♠❛t✐♦♥✱ ✇❤❡r❡ t❤❡ ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥ ✐ts❡❧❢ ✐s ✉s❡❞ t♦ ❝❛❧❝✉❧❛t❡ ❛ ♠❡❛s✉r❡ ♦❢ t❤❡ ❡rr♦r✱ ✐s r❡q✉✐r❡❞ ✐♥ s✉❝❤ ❝❛s❡s✳

✷✳✶✳✶ ❘❡s✐❞✉❛❧ ❜❛s❡❞ ❡rr♦r ❡st✐♠❛t♦rs

■♥ ❛ ❜❛s✐❝ ▼♦▼ s♦❧✈❡r✱ t❤❡ s✉r❢❛❝❡ ♦❢ ❛ ❝♦♥❞✉❝t✐♥❣ str✉❝t✉r❡ ✐s ♠♦❞❡❧❧❡❞ ❜② ✢❛t ❡❧❡♠❡♥ts ♦♥ ✇❤✐❝❤ t❤❡ s✉r❢❛❝❡ ❝✉rr❡♥t ❞❡♥s✐t② ✐s r❡♣r❡s❡♥t❡❞ ❜② ❛♥ ❡①♣❛♥s✐♦♥ ♦❢ ♣✐❡❝❡✇✐s❡✲❧✐♥❡❛r ❜❛s✐s ❢✉♥❝t✐♦♥s✳ ❊❛❝❤ ❜❛s✐s ❢✉♥❝t✐♦♥ ✐s t❛♥❣❡♥t✐❛❧ ♦♥ t❤❡ ❧♦❝❛❧ ❡❧❡♠❡♥ts t❤❛t ❢♦r♠ ✐ts s✉♣♣♦rt✳ ❚❤❡ ❡❧❡❝tr✐❝ ✜❡❧❞ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥ ✭❊❋■❊✮ r❡❧❛t❡s t❤✐s s✉r❢❛❝❡ ❝✉rr❡♥t t♦ t❤❡ s❝❛tt❡r❡❞ ❡❧❡❝tr✐❝ ✜❡❧❞ Es ❛❝❝♦r❞✐♥❣ t♦ Es= L J, ✭✷✳✶✮ ✇❤❡r❡ L ✐s ❛ ♠❛t❤❡♠❛t✐❝❛❧ ♦♣❡r❛t♦r t♦ ❜❡ ♣❡r❢♦r♠❡❞ ♦♥ t❤❡ s✉r❢❛❝❡ ❝✉rr❡♥t J✳ ❇② ✉t✐❧✐s✐♥❣ t❤❡ ❦♥♦✇♥ ✐♥❝✐❞❡♥t ❡❧❡❝tr✐❝ ✜❡❧❞ Ei✱ t❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ❢♦r t❤❡ t♦t❛❧ ❡❧❡❝tr✐❝ ✜❡❧❞ ♦♥ t❤❡ ❝♦♥❞✉❝t✐♥❣ s✉r❢❛❝❡✱ ˆn × (Ei + Es) = 0✱ ✐s ❡♥❢♦r❝❡❞ ✐♥ ❛ ✇❡✐❣❤t❡❞✲r❡s✐❞✉❛❧ ♠❛♥♥❡r ✇✐t❤ ❛ s❡❝♦♥❞ s❡t ♦❢ t❛♥❣❡♥t✐❛❧ t❡st✐♥❣ ❢✉♥❝t✐♦♥s✳ ❋♦r ❛ ❢✉❧❧ ❞✐s❝✉ss✐♦♥ ♦❢ t❤❡ ❢♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ▼♦▼✱ s❡❡ ❈❤❛♣t❡r ✸✳ ❙✐♥❝❡ t❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ✐s ♦♥❧② s❛t✐s✜❡❞ ✐♥ ❛ ✇❡✐❣❤t❡❞ ♠❛♥♥❡r✱ ❛ ❧♦❝❛❧ ♥♦♥✲③❡r♦ r❡s✐❞✉❛❧ ❡①✐sts ❛❝❝♦r❞✐♥❣ t♦ R(r) = (L ˜J)tan+ Eitan, ✭✷✳✷✮ ✇❤❡r❡ ˜J ✐s t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ tr✉❡ s✉r❢❛❝❡ ❝✉rr❡♥t J✳ ■t ✐s t❤✐s ❧♦❝❛❧ r❡s✐❞✉❛❧ ✈❛❧✉❡ t❤❛t ❝❛♥ ❜❡ ✉t✐❧✐s❡❞ t♦ ❞❡t❡r♠✐♥❡ t❤❡ ♥✉♠❡r✐❝❛❧ ❡rr♦r ♦❢ ❛ ▼♦▼ s♦❧✉t✐♦♥✳ ❆ ❝♦rr❡❧❛t✐♦♥ ♦❢ t❤❡ tr✉❡ ❡rr♦r e = J − ˜J ❛♥❞ t❤❡ r❡s✐❞✉❛❧ ❡rr♦r ❤❛s ❜❡❡♥ ❡st❛❜❧✐s❤❡❞ ❛♥❞ ✐❧❧✉str❛t❡❞ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡ ❬✻✱ ✼✱ ✽❪✳ ❆ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ❡rr♦r ♠❡tr✐❝ ✐s ✐♥tr♦❞✉❝❡❞ ✐♥ ❬✻❪ ❛❧♦♥❣ ✇✐t❤ ❛❝❝✉r❛t❡ ❡①♣r❡ss✐♦♥s ♦❢ s❝❛tt❡r❡❞ ✜❡❧❞s t❤❛t ❛r❡ ✈❛❧✐❞ ❛t ♦❜s❡r✈❛t✐♦♥ ♣♦✐♥ts ♦♥ t❤❡ s✉r❢❛❝❡ ♦❢ t❤❡ str✉❝t✉r❡✳ ❚❤✐s ❡rr♦r ♠❡tr✐❝ ❛❧❧♦✇s ❛♥ ❡st✐♠❛t❡ ♦❢ t❤❡ ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥ ❡rr♦r t♦ ❜❡ ❝♦♥str✉❝t❡❞ ❢r♦♠ ❊q✉❛t✐♦♥ ✷✳✷✳ ■t ✐s ❢✉rt❤❡r ♥♦t❡❞ ✐♥ ❬✽❪✱ t❤❛t ❝❛r❡ s❤♦✉❧❞ ❜❡ t❛❦❡♥ ✇❤❡♥ ❝♦♠♣✉t✐♥❣ t❤❡ ✜❡❧❞ ✈❛❧✉❡s ♦❢ t❤❡ r❡s✐❞✉❛❧ ♦❢ ❊q✉❛t✐♦♥ ✷✳✷✱ ❛s ❛ s✐♠✐❧❛r ♠❡t❤♦❞ t♦ t❤❛t ✉s❡❞ ✐♥ t❤❡ ▼♦▼ ❢♦r♠✉❧❛t✐♦♥ ✇♦✉❧❞ ♥♦t ②✐❡❧❞ ✐♥s✐❣❤t❢✉❧ ✐♥❢♦r♠❛t✐♦♥✳ ■t ✐s t❤✉s ♥❡❝❡ss❛r② t♦ ❝♦♠♣✉t❡ t❤❡ r❡s✐❞✉❛❧ ❡rr♦r ✇✐t❤ ❛♥ ❛❧t❡r♥❛t✐✈❡ ❢♦r♠✉❧❛t✐♦♥ t♦ ♦❜t❛✐♥ ❛ ✉s❡❢✉❧ ♠❡❛s✉r❡ ♦❢ t❤❡ ❡rr♦r✳ ■t ✐s ❛❧s♦ ♣♦ss✐❜❧❡ t♦ ❝❛❧❝✉❧❛t❡ t❤❡ ❝✉rr❡♥t ❛♣♣r♦①✐♠❛t✐♦♥ ˜J ❢r♦♠ ❛♥ ♦✈❡r❞❡t❡r♠✐♥❡❞ s②st❡♠ ♦❢ ❡q✉❛t✐♦♥s ❬✾❪✱ ❢♦r ✇❤✐❝❤ t❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ❝❛♥ ♥♦t ❜❡ ❡①❛❝t❧② s❛t✐s✜❡❞ ❛t ❡✈❡r② t❡st✐♥❣ ♣♦✐♥t✳ ❆ ❧❡❛st✲sq✉❛r❡s ❛♣♣r♦❛❝❤ ✇♦✉❧❞ ②✐❡❧❞ ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ❢♦r ✇❤✐❝❤ t❤❡ ❡rr♦r ✐♥ t❤❡ r❡s✐❞✉❛❧ ✐s ♠✐♥✐♠✐s❡❞✳ ❋♦r s✉❝❤ ❛♥ ❛♣♣r♦❛❝❤✱ t❤❡ ❡rr♦r ✐s ❡①tr❛❝t❡❞ ❢r♦♠ t❤❡ ❢♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ♦r✐❣✐♥❛❧ s②st❡♠ ♦❢ ❡q✉❛t✐♦♥s✳ ❆ r❡s✐❞✉❛❧ ❡rr♦r ❡st✐♠❛t♦r ❝❛♥ ❛❧s♦ ❜❡ ❜❛s❡❞ ♦♥ ❛♥ ❡✈❛❧✉❛t✐♦♥ ♦❢ t❤❡ ❜♦✉♥❞❛r② ❝♦♥✲ ❞✐t✐♦♥ ♦❢ t❤❡ ♥♦r♠❛❧ ❝♦♠♣♦♥❡♥t ♦❢ t❤❡ t♦t❛❧ ❡❧❡❝tr✐❝ ✜❡❧❞ ♦♥ t❤❡ s✉r❢❛❝❡ ♦❢ ❛ ❝❧♦s❡❞ P❊❈

(17)

❈❍❆P❚❊❘ ✷✳ ❊❘❘❖❘ ❊❙❚■▼❆❚■❖◆✿ ❚❍❊ ❙❊❚❚■◆● ✺ str✉❝t✉r❡ ❬✽❪✳ ❚❤❡ ♥♦r♠❛❧ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ✐s ❣✐✈❡♥ ❛s ˆ n· (Ei+ Es ) = ρs ǫ , ✭✷✳✸✮ ✇✐t❤ ρs t❤❡ s✉r❢❛❝❡ ❝❤❛r❣❡ ❞❡♥s✐t② ❛t ❛ ♣♦✐♥t ♦♥ t❤❡ s✉r❢❛❝❡ ❛♥❞ ǫ t❤❡ ♣❡r♠✐tt✐✈✐t② ♦❢ t❤❡ s✉rr♦✉♥❞✐♥❣ s♣❛❝❡✳ ❆ ♠❡❛s✉r❡ ♦❢ t❤❡ ❛❝❝✉r❛❝② ✇✐t❤ ✇❤✐❝❤ t❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ✐s s❛t✐s✜❡❞ ❝❛♥ ❜❡ ❝❛❧❝✉❧❛t❡❞ ✐♥ ❛ ❧♦❝❛❧ ♦r ❣❧♦❜❛❧ ♠❛♥♥❡r s♦ ❛s t♦ ♣r❡❞✐❝t t❤❡ ❡rr♦r ♦❢ t❤❡ ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥✳ ❨❡t ❛♥♦t❤❡r ✐♠♣❧✐❝✐t ❛ ♣♦st❡r✐♦r✐ r❡s✐❞✉❛❧ ♠❡t❤♦❞ ❝❛♥ ❜❡ ❝♦♥str✉❝t❡❞ ❢r♦♠ ❧♦❝❛❧ ❝❛❧❝✉✲ ❧❛t✐♦♥s t❤❛t ✐♥❝♦r♣♦r❛t❡ ❤✐❣❤❡r✲♦r❞❡r ❜❛s✐s ❢✉♥❝t✐♦♥s✳ ■♥ ❬✶✵❪✱ s✉❝❤ ❛ ♠❡t❤♦❞ ✐s s✉❣❣❡st❡❞ ❛♥❞ s❤♦✇♥ t♦ ❜❡ ❝♦♠♣✉t❛t✐♦♥❛❧❧② ✐♥❡①♣❡♥s✐✈❡✱ ❝♦♥s✐st❡♥t ❛♥❞ s✉✐t❛❜❧❡ ❢♦r ✉s❡ ✇✐t❤✐♥ ❛♥ ❛❞❛♣t✐✈❡ r❡✜♥❡♠❡♥t s❝❤❡♠❡✳

✷✳✶✳✷ ❘❡❝♦✈❡r② ❜❛s❡❞ ❡rr♦r ❡st✐♠❛t♦rs

❚❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ ❛♥ ❛ ♣♦st❡r✐♦r✐ ❡rr♦r ❡st✐♠❛t♦r ❜❛s❡❞ ♦♥ ❛ ♣♦st✲♣r♦❝❡ss✐♥❣ r❡❝♦✈❡r② ♣r♦❝❡❞✉r❡ ✐s t❤❡ ♣r✐♠❛r② ❣♦❛❧ ♦❢ t❤✐s t❤❡s✐s✳ ❚❤✐s ✐s ❛ t♦♣✐❝ t❤❛t ❤❛s ♥♦t r❡❝❡✐✈❡❞ ♠✉❝❤ ❛tt❡♥t✐♦♥ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡✱ ✇✐t❤ ♠♦st ❛ ♣♦st❡r✐♦r✐ ❡rr♦r ❡st✐♠❛t♦rs ❜❡✐♥❣ ❞❡✈❡❧♦♣❡❞ ❛r♦✉♥❞ t❤❡ r❡s✐❞✉❛❧ ❝♦♥❝❡♣t✳ ❚❤❡ ♥♦t✐♦♥ ❜❡❤✐♥❞ ❛ r❡❝♦✈❡r② ❜❛s❡❞ ❡rr♦r ❡st✐♠❛t♦r ✐s ❜✉✐❧t ♦♥ ❛ ♣♦st✲♣r♦❝❡ss✐♥❣ st❡♣ t❤❛t s♦♠❡❤♦✇ ♣r♦✈✐❞❡s ❛♥ ✐♠♣r♦✈❡❞ ✈❡rs✐♦♥ ♦❢ ❛ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ s♦❧✉t✐♦♥ ✜❡❧❞✳ ❆ ❞❡t❛✐❧❡❞ ❞✐s❝✉ss✐♦♥ ♦❢ s✐♠✐❧❛r ♠❡t❤♦❞s ❡♠♣❧♦②❡❞ ✐♥ t❤❡ ❋✐♥✐t❡ ❊❧❡♠❡♥t ▼❡t❤♦❞ ✭❋❊▼✮✱ ✇❤❡r❡ t❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ s♦❧✉t✐♦♥ ✜❡❧❞ ✐s r❡❝♦✈❡r❡❞✱ ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❬✶✶❪✳ ❲✐t❤✐♥ t❤❡ ▼♦▼ ❝♦♥t❡①t t❤❡ ✈❛❧✉❡ ❜❡✐♥❣ r❡❝♦✈❡r❡❞ ✐s t❤❡ s✉r❢❛❝❡ ❝❤❛r❣❡ ❞❡♥s✐t②✱ ♦❜t❛✐♥❡❞ ❛s ρs = − 1 jω∇s· J, ✭✷✳✹✮ ✇✐t❤ ω = 2πf t❤❡ ❛♥❣✉❧❛r ❢r❡q✉❡♥❝②✳ ❚❤❡ ❡rr♦r ✐♥ t❤❡ ❝❤❛r❣❡ ♠❛② t❤❡♥ ❜❡ ❡✈❛❧✉❛t❡❞ ❛s t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ tr✉❡ ❝❤❛r❣❡ ρs✱ ❛♥❞ t❤❡ ❝❤❛r❣❡ ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ s♦❧✉t✐♦♥ ❝✉rr❡♥t ❛s eρ= ρs− ˜ρs, ✭✷✳✺✮ ✇❤❡r❡ ˜ρs ✐s t❤❡ ❞✐s❝♦♥t✐♥✉♦✉s ❝❤❛r❣❡ r❡❧❛t❡❞ t♦ ❛ ✜rst✲♦r❞❡r ▼♦▼ s♦❧✉t✐♦♥✳ ❆ r❡❝♦✈❡r② ♣r♦❝❡❞✉r❡ s❡❡❦s t♦ ✜♥❞ ❛♥ ❡st✐♠❛t❡ ♦❢ t❤❡ tr✉❡ ❝❤❛r❣❡✱ ✇✐t❤ ✇❤✐❝❤ ❛♥ ❡st✐♠❛t❡ ♦❢ t❤❡ ❝❤❛r❣❡ ❡rr♦r ❝❛♥ ❜❡ ❝♦♥str✉❝t❡❞ ❛s eρ∗ = ˜ρs∗− ˜ρs, ✭✷✳✻✮ ✇❤❡r❡ ˜ρs∗ ✐s t❤❡ r❡❝♦✈❡r❡❞ ❝❤❛r❣❡ ❞✐str✐❜✉t✐♦♥✳ ❉♦❝✉♠❡♥t❡❞ r✐❣♦r♦✉s r❡❝♦✈❡r② ❜❛s❡❞ ❛ ♣♦st❡r✐♦r✐ ❡rr♦r ❡st✐♠❛t♦rs ❢♦r t❤❡ ▼♦▼ ❛r❡ ♥♦t ❛✈❛✐❧❛❜❧❡ ✐♥ t❤❡ ♠❛✐♥str❡❛♠ ❧✐t❡r❛t✉r❡✳ ❙♦♠❡ ❡rr♦r ❡st✐♠❛t✐♦♥ t❡❝❤♥✐q✉❡s t❤❛t r♦✉❣❤❧② r❡s❡♠❜❧❡ r❡❝♦✈❡r② ❜❛s❡❞ ❡rr♦r ❡st✐♠❛t✐♦♥ ❛r❡ ❝♦♥str✉❝t❡❞ ❛r♦✉♥❞ t❤❡ ✐❞❡❛ t❤❛t s✐❣♥✐✜❝❛♥t ❞✐s❝♦♥t✐♥✉✐t✐❡s ✐♥ t❤❡ ❝❤❛r❣❡✱ ♦r ❡✈❡♥ ❛ r❛♣✐❞ ❝❤❛♥❣❡ ✐♥ t❤❡ s✉r❢❛❝❡ ❝✉rr❡♥t✱ ❛r❡ ✐♥❞✐❝❛t✐♦♥s ♦❢ ❧♦❝❛❧ ❡rr♦rs✳ ❚❤❡s❡ ♠❡t❤♦❞s✱ ❛❧t❤♦✉❣❤ s♦♠❡✇❤❛t ✉s❡❢✉❧✱ ❛r❡ ♥♦t ✇❡❧❧ ❞♦❝✉♠❡♥t❡❞ ❛♥❞ ❝❛rr② s✐❣♥✐✜❝❛♥t ❧✐♠✐t❛t✐♦♥s✳

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