Please do not remove this page RF phase sensitive amplifier with 70 dB phase sensitive dynamic range Davies, Paul; Bui, Lam; Scott, James https://researchrepository.rmit.edu.au/esploro/outputs/journalArticle/RF-phase-sensitive-amplifier-with-70/9921861058201341/filesAndLinks?index=0 Davies, P., Bui, L., & Scott, J. (2019). RF phase sensitive amplifier with 70 dB phase sensitive dynamic range. Optics Communications, 433, 115–119. https://doi.org/10.1016/j.optcom.2018.10.002 Document Version: Accepted Manuscript Published Version: https://doi.org/10.1016/j.optcom.2018.10.002 Repository homepage: https://researchrepository.rmit.edu.au © 2018 Elsevier B.V. All rights reserved. Downloaded On 2023/09/23 00:13:13 +1000 Please do not remove this page Thank you for downloading this document from the RMIT Research Repository. The RMIT Research Repository is an open access database showcasing the research outputs of RMIT University researchers. RMIT Research Repository: http://researchbank.rmit.edu.au/ Citation: Davies, P, Bui, L and Scott, J 2019, 'RF phase sensitive amplifier with 70 dB phase sensitive dynamic range', Optics Communications, vol. 433, pp. 115-119. 6HHWKLVUHFRUGLQWKH50,75HVHDUFK5HSRVLWRU\DW https://researchbank.rmit.edu.au/view/rmit:55026 9HUVLRQ Accepted Manuscript &RS\ULJKW6WDWHPHQW ‹ 2019. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/ /LQNWR3XEOLVKHG9HUVLRQ https://dx.doi.org/10.1016/j.optcom.2018.10.002 PLEASE DO NOT REMOVE THIS PAGE Accepted Manuscript RF phase sensitive amplifier with 70dB phase sensitive dynamic range Paul Davies, Lam Bui, James Scott PII: DOI: Reference: S0030-4018(18)30869-1 https://doi.org/10.1016/j.optcom.2018.10.002 OPTICS 23519 To appear in: Optics Communications Received date : 21 June 2018 Revised date : 30 September 2018 Accepted date : 1 October 2018 Please cite this article as: P. Davies, et al., RF phase sensitive amplifier with 70dB phase sensitive dynamic range, Optics Communications (2018), https://doi.org/10.1016/j.optcom.2018.10.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. *Manuscript Click here to view linked References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 ❘❋ P❤❛s❡ ❙❡♥s✐t✐✈❡ ❆♠♣❧✐✜❡r ✇✐t❤ ✼✵❞❇ P❤❛s❡ ❙❡♥s✐t✐✈❡ ❉②♥❛♠✐❝ ❘❛♥❣❡ P❛✉❧ ❉❛✈✐❡s❛✱∗✱ ▲❛♠ ❇✉✐❜ ✱ ❏❛♠❡s ❙❝♦tt❛ ❛ ❘▼■❚ ❯♥✐✈❡rs✐t②✱ ✸✼✻✲✸✾✷ ❙✇❛♥t♦♥ ❙tr❡❡t✱ ▼❡❧❜♦✉r♥❡ ❱■❈ ✸✵✵✶✱ ❆✉str❛❧✐❛ ❜ ❈❡♥tr❛❧ ◗✉❡❡♥s❧❛♥❞ ❯♥✐✈❡rs✐t②✱ ✶✷✵ ❙♣❡♥❝❡r ❙t✱ ▼❡❧❜♦✉r♥❡✱ ❱■❈ ✸✵✵✵✱ ❆✉str❛❧✐❛ ❆❜str❛❝t ■♥ t❤✐s ♣❛♣❡r✱ ❛ ♥♦✈❡❧ ♠✐❝r♦✇❛✈❡ ♣❤♦t♦♥✐❝ ♣❤❛s❡ s❡♥s✐t✐✈❡ ❛♠♣❧✐✜❡r ✐s ♣r♦♣♦s❡❞ ❛♥❞ ❞❡♠♦♥str❛t❡❞✳ ❆ t❤❡♦r❡t✐❝❛❧ ♠♦❞❡❧ ✐s ❞❡✈❡❧♦♣❡❞ ❛♥❞ ❝♦♠♣❛r❡❞ t♦ t❤❡ r❡s✉❧ts ♦❜t❛✐♥❡❞ ❢r♦♠ ❛ s✐♠✉❧❛t✐♦♥✳ ❯s✐♥❣ ❡✣❝✐❡♥t ♠✐①✐♥❣ ✐♥ s❡❝♦♥❞ ♦r❞❡r ♥♦♥❧✐♥❡❛r ♠❛t❡r✐❛❧s s✉❝❤ ❛s ♣❡r✐♦❞✐❝❛❧❧② ♣♦❧❡❞ ❧✐t❤✐✉♠ ♥✐♦❜❛t❡✱ t❤❡ r❡s✉❧t❛♥t ❛♠♣❧✐✜❡r ❡①❤✐❜✐ts ♦✈❡r ✼✵❞❇ ♦❢ ♣❤❛s❡ ❞❡♣❡♥❞❡♥t ❣❛✐♥ ✈❛r✐❛t✐♦♥✳ ❚❤✐s ❘❋ ♣❤❛s❡ s❡♥s✐t✐✈✐t② ❝❛♥ ❜❡ ✉s❡❞ t♦ r❡❞✉❝❡ ♣❤❛s❡ ♥♦✐s❡ ❛♥❞ t❤❡r❡❢♦r❡ ✜♥❞s ❛♣♣❧✐❝❛t✐♦♥s ✐♥ P❙❑ ♣❤❛s❡ r❡❣❡♥❡r❛t✐♦♥ ❛♥❞ ♣❤❛s❡ ♥♦✐s❡ r❡❞✉❝t✐♦♥✳ ❑❡②✇♦r❞s✿ ❘❋ P❤❛s❡ ❙❡♥s✐t✐✈❡ ❆♠♣❧✐✜❝❛t✐♦♥✱ ▼✐❝r♦✇❛✈❡ P❤♦t♦♥✐❝✱ PP▲◆✱ ◆♦♥❧✐♥❡❛r ❲❛✈❡❣✉✐❞❡s✱ ❘❛❞✐♦ ♦✈❡r ❋✐❜r❡ P❆❈❙✿ ✼✶✳✸✺✳✲②✱ ✼✶✳✸✺✳▲❦✱ ✼✶✳✸✻✳✰❝ ✷✵✵✽ ▼❙❈✿ ✷✸✲✺✺✼ ✶✳ ■♥tr♦❞✉❝t✐♦♥ ❘❛❞✐♦✲♦✈❡r✲✜❜r❡ ✭❘♦❋✮ ❬✶❪ ✐s ❛♥ ❛ttr❛❝t✐✈❡ t❡❝❤♥✐q✉❡ t♦ tr❛♥s♠✐t ❘❋ s✐❣♥❛❧s t❤r♦✉❣❤ ✜❜r❡✳ ❘♦❋ ❝❛♥ ❜❡ ✉s❡❞ ❢♦r ❝❛❜❧❡ t❡❧❡✈✐s✐♦♥ ✭❈❆❚❱✮ ❞✐str✐❜✉t✐♦♥ ♥❡t✇♦r❦s ❬✷❪✱ ❧✐♥❦ ❜❛s❡ st❛t✐♦♥s t♦ ❝❡♥tr❛❧ ♦✣❝❡s ❢♦r ❝❡❧❧✉❧❛r ❛♥❞ ♠♦❜✐❧❡ ❞❛t❛ ♣r♦✈✐❞❡rs✱ ✇✐r❡❧❡ss ❧♦❝❛❧ ❛r❡❛ ♥❡t✇♦r❦s ✭❲▲❆◆✮ ❬✸❪✱ ❛♥✲ t❡♥♥❛ r❡♠♦t✐♥❣ ❬✹❪ ❛♥❞ s✐❣♥❛❧ ♣r♦❝❡ss✐♥❣ ❬✺❪✳ ❈♦♠♣❛r❡❞ t♦ ❝♦❛①✐❛❧ ❝❛❜❧❡✱ ❛❞✈❛♥t❛❣❡s s✉❝❤ ❛s ✈❡r② ✇✐❞❡ ❜❛♥❞✇✐❞t❤✱ ❧♦✇ ❧♦ss✱ ❧✐❣❤t ✇❡✐❣❤t ❛♥❞ ✐♠♠✉♥✐t② t♦ ❘❋ ✐♥t❡r❢❡r❡♥❝❡ ❛r❡ ♦❜t❛✐♥❡❞✳ ❘♦❋ ♠♦❞✉❧❛t✐♦♥ t❡❝❤♥✐q✉❡s ✐♥❝❧✉❞❡ ❛♠♣❧✐t✉❞❡ s❤✐❢t ❦❡②✐♥❣ ✭◆❘❩✴❘❩✮✱ ♣❤❛s❡ s❤✐❢t ❦❡②✐♥❣ ✭P❙❑✮ ❬✻❪ ❛♥❞ ◗❆▼ ❬✼❪✳ ❖♣t✐❝❛❧ s②st❡♠s ❡♠♣❧♦②✐♥❣ P❙❑ ❤❛✈❡ ❢♦✉♥❞ s✐❣♥❛❧ r❡✲ ❣❡♥❡r❛t✐♦♥ ❛♥❞ ♣❤❛s❡ ♥♦✐s❡ r❡❞✉❝t✐♦♥ ♣♦ss✐❜❧❡ ✉s✐♥❣ ♣❤❛s❡ s❡♥s✐t✐✈❡ ❛♠♣❧✐✜❝❛t✐♦♥ ✭P❙❆✮✿ t❤❡ s✐❣♥❛❧✬s ✐♥✲♣❤❛s❡ ❝♦♠♣♦✲ ♥❡♥t ✐s ❛♠♣❧✐✜❡❞ ✇❤✐❧❡ ❛tt❡♥✉❛t✐♥❣ t❤❡ q✉❛❞r❛t✉r❡ ❝♦♠♣♦✲ ♥❡♥t ❬✽✕✶✶❪✱ t❤✐s ✐♥❝❧✉❞❡s ❉P❙❑ ❬✶✷❪✱ ◗P❙❑ ❬✶✸❪ ❛♥❞ ◗❆▼ ❬✶✹❪✳ ❚♦ ❛❝❤✐❡✈❡ t❤✐s✱ ❛ ❝♦♠♣❛r✐s♦♥ ✐s ♠❛❞❡ ❜❡t✇❡❡♥ t❤❡ ♣❤❛s❡ ♦❢ t❤❡ ✐♥♣✉t s✐❣♥❛❧ ❛♥❞ ❛ ❦♥♦✇♥ r❡❢❡r❡♥❝❡✳ ■t ✐s t❤❡r❡✲ ❢♦r❡ ❝❧❡❛r t❤❛t ❛ ❘❋✲P❙❆ ❝♦✉❧❞ ❞❡❧✐✈❡r s✐♠✐❧❛r ❜❡♥❡✜ts ❢♦r ❛ ❘♦❋ s②st❡♠✳ ❚❤❡r❡ ❛r❡ r❡♣♦rts ♦❢ ❘❋✲P❙❆s ♦♣❡r❛t✐♥❣ ✐♥ t❤❡ ❘❋ ❞♦✲ ♠❛✐♥ ❬✶✺✕✶✼❪✳ ❚❤❡ ❏♦s❡♣❤s♦♥ ♣❛r❛♠❡tr✐❝ ❛♠♣❧✐✜❡r ✭❏P❆✮ ❣❡♥❡r❛t❡s ♣❤❛s❡ sq✉❡❡③❡❞ st❛t❡s ❬✶✻❪✳ ❍♦✇❡✈❡r t❤❡s❡ ❞❡✲ ✈✐❝❡s s✉✛❡r ❜❛♥❞✇✐❞t❤ ❧✐♠✐t❛t✐♦♥s✱ ✐♠♣❡❞❛♥❝❡ ♠✐s♠❛t❝❤❡s✱ ✐♥st❛❜✐❧✐t✐❡s ❛♥❞ ❣❛✐♥ ❞❡♣❡♥❞❛♥t ♥♦✐s❡ ❜✉t ❞❡♠♦♥str❛t❡ ❛♠♣❧✐✜❝❛t✐♦♥ ❛♥❞ ❛tt❡♥✉❛t✐♦♥ ♦❢ t❤❡ ♣❤❛s❡s ❬✶✺✱ ✶✻❪✳ ❆ r❡❝❡♥t ❈▼❖❙ ❝✐r❝✉✐t ❡①♣❡r✐♠❡♥t❛❧❧② s❤♦✇❡❞ ♣❤❛s❡ s❡♥s✐✲ t✐✈❡ ❛♠♣❧✐✜❝❛t✐♦♥ ♦❢ ❛♥ ❘❋ s✐❣♥❛❧ ❛♥❞ r❡♣♦rt❡❞ ❛ ❣❛✐♥ ❞✐❢✲ ❢❡r❡♥❝❡ ♦❢ ✾❞❇ ✇✐t❤ ❛ ❜❛♥❞✇✐❞t❤ ♦❢ ✻✵✵▼❍③ ❛t ✽✳✼✺●❍③ ❬✶✼❪✳ ❚❤❡ ✉s❡ ♦❢ ♣❤♦t♦♥✐❝s t♦ ❛❝❤✐❡✈❡ P❙❆ ✐s ✇❡❧❧ ❦♥♦✇♥✱ ❛tt❛✐♥✐♥❣ ❣❛✐♥ ✈❛r✐❛t✐♦♥s ♦❢ ✷✵❞❇ ♦r ♠♦r❡ ❬✶✽❪✳ ■t ✐s t❤❡r❡✲ ❢♦r❡ r❡❛s♦♥❛❜❧❡ t♦ ❡①♣❡❝t t❤❛t ❛ ♠✐❝r♦✇❛✈❡ ♣❤♦t♦♥✐❝ P❙❆ ❝♦✉❧❞ ❛❝❤✐❡✈❡ s✐♠✐❧❛r r❡s✉❧ts✳ ❚❡❝❤♥✐q✉❡s ❡st❛❜❧✐s❤❡❞ ❢♦r P❙❆ ✐♥ ♣❡r✐♦❞✐❝❛❧❧② ♣♦❧❡❞ ❧✐t❤✐✉♠ ♥✐♦❜❛t❡ ✭PP▲◆✮ ✇❛✈❡❣✲ ✉✐❞❡s ❬✶✽✱ ✶✾❪ ♦✛❡r ❛❞✈❛♥t❛❣❡s ♦✈❡r t❤❡ ❤✐❣❤❧② ♥♦♥ ❧✐♥❡❛r ✜❜❡r ❛♣♣r♦❛❝❤✳ ▼✉❧t✐♣❧❡ ❢✉♥❝t✐♦♥s ❝❛♥ ❜❡ ✐♥t❡❣r❛t❡❞ ❛♥❞ t❤❡ ✇❛✈❡❣✉✐❞❡s ❛r❡ ❡✛❡❝t✐✈❡❧② ❢r❡❡ ❢r♦♠ ♣r♦❜❧❡♠❛t✐❝ st✐♠✲ ✉❧❛t❡❞ ❜r✐❧❧♦✉✐♥ s❝❛tt❡r✐♥❣ ✭❙❇❙✮ ❛♥❞ ✐♥t❡r♠♦❞✉❧❛t✐♦♥ ❞✐s✲ t♦rt✐♦♥ ❬✷✵❪✳ ❲✐t❤ s✉✐t❛❜❧❡ ♣♦✇❡r✱ s❡❝♦♥❞ ❤❛r♠♦♥✐❝ ❣❡♥✲ ❡r❛t✐♦♥ ✭❙❍●✮ ❛♥❞ ❞✐✛❡r❡♥❝❡ ❢r❡q✉❡♥❝② ❣❡♥❡r❛t✐♦♥ ✭❉❋●✮ ♣r♦❝❡ss❡s s❤♦✇♥ ✐♥ ❬✷✶❪ ❝❛♥ ❜❡ ✉t✐❧✐③❡❞✱ ✇❤✐❝❤ ❛r❡ ❡ss❡♥t✐❛❧ ❢✉♥❝t✐♦♥s ❢♦r ❛ PP▲◆ P❙❆✳ ■♥ t❤✐s ♣❛♣❡r ✇❡ ♣r♦♣♦s❡ ❛ ♥♦✈❡❧ PP▲◆ ❜❛s❡❞ ❘❋✲P❙❆ ✉s✐♥❣ ❛ ❞❡❣❡♥❡r❛t❡ s✐♥❣❧❡ ♣✉♠♣ s❝❤❡♠❡ ✐♥ ❛ ❙❍●✴❉❋● ❝♦♥✜❣✉r❛t✐♦♥ s✐♠✐❧❛r t♦ ❬✷✷❪✳ ❚❤✐s ✇♦r❦ ❜✉✐❧❞s ✉♣♦♥ ♣r❡✲ ❧✐♠✐♥❛r② ✇♦r❦ ♣✉❜❧✐s❤❡❞ ✐♥ ❬✷✸❪✳ ❆ s✐♠♣❧✐✜❡❞ ♣r✐♥❝✐♣❧❡ ♦❢ ♦♣❡r❛t✐♦♥ ✐s ❞✐s❝✉ss❡❞ ✐♥ s❡❝t✐♦♥ ✷ ❢♦❧❧♦✇❡❞ ❜② ❛ ❞❡t❛✐❧❡❞ t❤❡♦r❡t✐❝❛❧ ❛♥❛❧②s✐s t❤❛t ❢♦r♠s t❤❡ ❤②♣♦t❤❡s✐s ❛♥❞ ♠❡t❤♦❞✲ ♦❧♦❣② ✐♥ s❡❝t✐♦♥ ✸✳ ❙❡❝t✐♦♥ ✹ ❞❡s❝r✐❜❡s t❤❡ ❱P■♣❤♦t♦♥✐❝s s✐♠✉❧❛t✐♦♥ ❬✷✹❪ s❡t✉♣✱ s❡❝t✐♦♥ ✺ ❝♦♠♣❛r❡s t❤❡ r❡s✉❧ts t♦ t❤❡ ❤②♣♦t❤❡s✐s ♣r❡❞✐❝t✐♦♥ ❛♥❞ ✐♥✈❡st✐❣❛t❡s t❤❡ s❡♥s✐t✐✈✐t② ♦❢ t❤❡ s②st❡♠ t♦ ♣✉♠♣ ♣♦✇❡r✳ ❚❤❡ s✐♠✉❧❛t♦r ✐s t❤❡♥ ✉s❡❞ t♦ ❡①❛♠✐♥❡ t❤❡ ♣❤❛s❡ sq✉❡❡③✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❘❋✲P❙❆✳ ❙❡❝t✐♦♥ ✻ ❝♦♥❝❧✉❞❡s t❤✐s ✇♦r❦✳ ✷✳ Pr✐♥❝✐♣❧❡ ♦❢ ❖♣❡r❛t✐♦♥ ∗ ❈♦rr❡s♣♦♥❞✐♥❣ ❛✉t❤♦r P❛✉❧✳❉❛✈✐❡s❅r♠✐t✳❡❞✉✳❛✉ ✭P❛✉❧ ❉❛✈✐❡s✮✱ ▲✳❇✉✐❅❝q✉✳❡❞✉✳❛✉ ✭▲❛♠ ❇✉✐✮✱ ❏❛♠❡s✳❙❝♦tt❅r♠✐t✳❡❞✉✳❛✉ ✭❏❛♠❡s ❊♠❛✐❧ ❛❞❞r❡ss❡s✿ ❙❝♦tt✮ Pr❡♣r✐♥t s✉❜♠✐tt❡❞ t♦ ❖♣t✐❝s ❈♦♠♠✉♥✐❝❛t✐♦♥s ❚❤✐s s❡❝t✐♦♥ ♣r♦✈✐❞❡s ❛ ❜❛s✐❝ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ s②st❡♠ ❜❧♦❝❦ ❞✐❛❣r❛♠ s❤♦✇♥ ✐♥ ❋✐❣✳ ✶ ❛♥❞ ❞❡♠♦♥str❛t❡s t❤❡ ❛♣✲ ♣r♦❛❝❤ ✉s❡❞ t♦ ♦❜t❛✐♥ ❘❋✲P❙❆✳ ❚❤❡ ❘❋ ✐♥♣✉t s✐❣♥❛❧ ❛t ❙❡♣t❡♠❜❡r ✷✼✱ ✷✵✶✽ ✷ RF PSA 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 PIN,RF POUT,RF 2π φS 0 (4) MZM RF (1) in EΩ S EΩ EΩ R -Ω S ωS Ω S (2) RF Ref AS BS π Optical Amplifier 2π ωS (5) Optical Amplifier φS (4) ω ωP AP χ 2ωP 0 (7) (6) (7) vDC vDC PM A S,OUT ω (3) ω S 0 Voltage processing EΩ 1553.6nm RF Out ωS φS Laser RF In π 2π φS 2ωP χ ωi 2 PPLN2 2ωP χ 2 AP 2 PPLN1 ωP ΩS ω (8) 2 (9) 2 (10) PD ωS=ωi=ω EΩ RF out S,OUT AP , AS BS PPLN2 -Ω S ω Ω S 2ωP ❋✐❣✉r❡ ✶✿ ❙②st❡♠ ❜❧♦❝❦ ❞✐❛❣r❛♠ ♦❢ t❤❡ ♥♦♥❧✐♥❡❛r s❡❝♦♥❞ ♦r❞❡r PP▲◆ ❜❛s❡❞ ❘❋ ♣❤❛s❡ s❡♥s✐t✐✈❡ ❛♠♣❧✐✜❡r✳ ❢r❡q✉❡♥❝② ΩS ✐s ✐♥t❡♥s✐t② ♠♦❞✉❧❛t❡❞ ✭■▼✮ ❜② t❤❡ ▼❛❝❤ ❩❡❤♥❞❡r ▼♦❞✉❧❛t♦r ✭▼❩▼✮ ♦♥t♦ ❛ ❝♦♣② ♦❢ t❤❡ ♦♣t✐❝❛❧ ❝❛r✲ r✐❡r ❞❡r✐✈❡❞ ❢r♦♠ t❤❡ ❧❛s❡r✳ ❆ s❡❝♦♥❞ ❝♦♣② ❦♥♦✇♥ ❛s t❤❡ ❚❛❜❧❡ ✶✿ P❤❛s❡ r❡❧❛t✐♦♥s❤✐♣ ♦❢ t❤❡ ❘❋ ✐♥♣✉t s✐❣♥❛❧✱ ✈♦❧t❛❣❡ ♣r♦❝❡ss✐♥❣ ❜❧♦❝❦ ❛♥❞ ♦♣t✐❝❛❧ ♣❤❛s❡✳ ❘❋ r❡❢❡r❡♥❝❡ ♣❤❛s❡✱ φR = 0✳ ■♥t❡❣❡rs ✐♥ ❜r❛❝❦❡ts ❛r❡ s✐❣♥❛❧ ✐❞❡♥t✐✜❡rs ✉s❡❞ ✐♥ ❋✐❣✳ ✶✳ ✏♣✉♠♣✑ ❛❞✈❛♥❝❡s t♦ t❤❡ ♣❤❛s❡ ♠♦❞✉❧❛t♦r ✭P▼✮✳ ❚❤❡ ♣✉♠♣ ✐s ♣❤❛s❡ s❤✐❢t❡❞ ❜② ❛ ✈♦❧t❛❣❡ ♦❜t❛✐♥❡❞ ❢r♦♠ t❤❡ ✈♦❧t❛❣❡ ♣r♦✲ ❘❋ ♣❤❛s❡ ❱♦❧t❛❣❡ ❖♣t✐❝❛❧ ♣❤❛s❡ φS ♣r♦❝❡ss✐♥❣ θRS ❝❡ss✐♥❣ ❜❧♦❝❦✱ ✇❤❡r❡ t❤❡ ✈♦❧t❛❣❡ ❝❤❛r❛❝t❡r✐st✐❝ ✐s ❞❡r✐✈❡❞ ❜② ❝♦♠♣❛r✐♥❣ t❤❡ ♣❤❛s❡s ♦❢ t❤❡ ❘❋ ✐♥♣✉t s✐❣♥❛❧ ❛♥❞ ❘❋ vDC ❘❋ ✭✶✮ r❡❢❡r❡♥❝❡ ✭s❡❡ ❚❛❜❧❡ ✶✮✳ ❚❤❡ ♣✉♠♣ ♣❛ss❡s t❤r♦✉❣❤ t❤❡ ✜rst ✭✺✮ P▼ ✭✻✮ PP▲◆✶ ✵ ✵ ✵ ✵ t✐♦♥ ✭❙❍●✮ ❤❛❧✈❡s t❤❡ ✇❛✈❡❧❡♥❣t❤ ❛♥❞ ❞♦✉❜❧❡s t❤❡ ♣❤❛s❡ ✾✵ ✵✳✺ ✾✵ ✶✽✵ ❛♥❣❧❡✳ ❆❢t❡r ✜❧t❡r✐♥❣✱ t❤❡ s❡❝♦♥❞ ❤❛r♠♦♥✐❝ ♣✉♠♣ ❛♥❞ ■▼✲ ✶✽✵ ✶✳✵ ✶✽✵ ✸✻✵ ✷✼✵ ✵✳✺ ✾✵ ✶✽✵ ♦❢ t✇♦ PP▲◆ ✇❛✈❡❣✉✐❞❡s✱ ✇❤❡r❡ s❡❝♦♥❞ ❤❛r♠♦♥✐❝ ❣❡♥❡r❛✲ ❘❋ s✐❣♥❛❧ ❝♦♥♥❡❝t ✐♥t♦ t❤❡ s❡❝♦♥❞ PP▲◆ ✇❛✈❡❣✉✐❞❡✳ ❚❤❡ ♥♦♥❧✐♥❡❛r ♣r♦❝❡ss ♦❢ t❤❡ ✇❛✈❡❣✉✐❞❡ ❛❝ts ❛s ❛ ❞✐✛❡r❡♥❝❡ ❢r❡✲ q✉❡♥❝② ❣❡♥❡r❛t♦r ✭❉❋●✮✱ ✇❤❡r❡ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ■▼✲❘❋ s✐❣♥❛❧ ❛♥❞ t❤❡ s❡❝♦♥❞ ❤❛r♠♦♥✐❝ ♣✉♠♣ ❢♦r♠ ❛ ♥❡✇ ♣r♦❞✉❝t ❦♥♦✇♥ ❛s ❛♥ ✐❞❧❡r✳ ❆s t❤❡ ✇❛✈❡❧❡♥❣t❤ ♦❢ t❤❡ ✐❞❧❡r ❛♥❞ ■▼✲❘❋ s✐❣♥❛❧ ❛r❡ t❤❡ s❛♠❡✱ ✈❡❝t♦r s✉♠♠❛t✐♦♥ ✇✐❧❧ r❡✲ ✐♥❢♦r❝❡ ♦r ❝❛♥❝❡❧ t❤❡ ■▼✲❘❋ s✐❣♥❛❧✳ ■t ✐s t❤✐s ✇❛✈❡❧❡♥❣t❤ ❛♥❞ ♣❤❛s❡ r❡❧❛t✐♦♥s❤✐♣ t❤❛t ♣r♦❞✉❝❡s ❘❋ ♣❤❛s❡ s❡♥s✐t✐✈❡ ❛♠♣❧✐✜❝❛t✐♦♥✳ ■❢ t❤❡ ✐❞❧❡r ✐s ✶✽✵ ❞❡❣r❡❡s ♦✉t ♦❢ ♣❤❛s❡ ✇✐t❤ t❤❡ ■▼✲❘❋ s✐❣♥❛❧✱ t❤❡♥ ❝❛♥❝❡❧❧❛t✐♦♥ ♦❢ t❤❡ ■▼✲❘❋ s✐❣♥❛❧ ✇✐❧❧ ♦❝❝✉r✳ ❚❤✐s ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ❘❋ ✐♥♣✉t s✐❣♥❛❧ ♣❤❛s❡ ❜❡✐♥❣ ✾✵ ♦r ✷✼✵ ❞❡❣r❡❡s ❛s s❤♦✇♥ ✐♥ ❚❛❜❧❡ ✶✳ ❙✐♠✐❧❛r❧② t❤❡ ✐❞❧❡r ✇✐❧❧ ❜❡ ✐♥✲♣❤❛s❡ ✐❢ t❤❡ ■▼✲❘❋ s✐❣♥❛❧ ✐s ✵ ♦r ✶✽✵ ❞❡✲ s✐❞❡ ✐s t❤❡ ❘❋ ✐♥♣✉t s✐❣♥❛❧ ✭✶✮ ❛t ❢r❡q✉❡♥❝② ΩS ✭✐♥t❡❣❡rs ✐♥ ❜r❛❝❦❡ts ❛r❡ s✐❣♥❛❧ ✐❞❡♥t✐✜❡rs ✉s❡❞ ✐♥ t❤❡ ✜❣✉r❡✮✳ ❊q✉❛t✐♦♥ ✭✶✮ s❤♦✇s t❤❡ ❝♦♠♣❧❡① ✜❡❧❞ ❡①♣r❡ss✐♦♥ ♦❢ t❤❡ ❘❋ ✐♥♣✉t s✐❣♥❛❧ EΩS ❛♥❞ ✐ts t✐♠❡ ✐♥❞❡♣❡♥❞❡♥t ♣❤❛s❡ s❤✐❢t ❝♦♠♣❧❡① ❝♦♥❥✉❣❛t❡ ✐s ❛♥♥♦t❛t❡❞ ❜② ❙ ❈❈ φS ✳ ❚❤❡ ❛♥❞ t❤❡ s✉❜s❝r✐♣t ❞❡♥♦t❡s t❤❛t ✐t ✐s t❤❡ s✐❣♥❛❧✳ EΩS = ej(ΩS t+φS ) + CC ✭✶✮ ❚❤❡ ❝♦♠♣❧❡① ❡❧❡❝tr✐❝ ✜❡❧❞ ♦❢ t❤❡ ❘❋ r❡❢❡r❡♥❝❡ ✭✷✮ ✐s ❣r❡❡s✳ ❆❢t❡r ❛ ❜❛♥❞♣❛ss ✜❧t❡r t❤❡ ❘❋ s✐❣♥❛❧ ✐s r❡❝♦✈❡r❡❞ ❜② s❤♦✇♥ ❜② ❡q♥✳ t❤❡ ♣❤♦t♦❞❡t❡❝t♦r✳ ❊①❛♠✐♥✐♥❣ t❤❡ ❞❡♠♦❞✉❧❛t❡❞ ❘❋ ♦✉t✲ r❛❞✐❛♥s✳ ♣✉t ♣♦✇❡r r❡✈❡❛❧s ✐t ✐s ❝♦♥tr♦❧❧❡❞ ❜② t❤❡ ♣❤❛s❡ ♦❢ t❤❡ ❘❋ t❤❡ ❘❋ ✐♥♣✉t s✐❣♥❛❧ ✐♥♣✉t s✐❣♥❛❧✱ ✐✳❡✳ ✇❡ ❤❛✈❡ ♣❤❛s❡ s❡♥s✐t✐✈❡ ❛♠♣❧✐✜❝❛t✐♦♥✳ ❜❡ ❛❝❤✐❡✈❡❞ ❜② ✉s✐♥❣ ❛ ♣❤❛s❡ ❧♦❝❦❡❞ ❧♦♦♣ s✉❝❤ ❛s ❛ ❈♦st❛s ✭✷✮ ❛♥❞ ✐s ❛ss✉♠❡❞ t♦ ❤❛✈❡ ❛ ♣❤❛s❡ ♦❢ ✵ ❚❤❡ ❘❋ r❡❢❡r❡♥❝❡ ❛t ΩS ΩR ✐s ❢r❡q✉❡♥❝② ❧♦❝❦❡❞ t♦ ΩR = ΩS ✳ ❚❤✐s ❝♦✉❧❞ s♦ t❤❛t ❧♦♦♣ ❬✷✺❪ ♦r ❡✈❡♥ ✐♥❥❡❝t✐♦♥ ❧♦❝❦✐♥❣ t❡❝❤♥✐q✉❡s ❬✷✻❪✳ ❋✉rt❤❡r✱ ✸✳ ❚❤❡♦r❡t✐❝❛❧ ▼♦❞❡❧ ❉❡✈❡❧♦♣♠❡♥t ✐t ✐s ❛♥t✐❝✐♣❛t❡❞ t❤❛t ♥♦♥✲✐❞❡❛❧ ❧♦❝❦✐♥❣ ✇✐❧❧ r❡s✉❧t ✐♥ ❝♦♠✲ ♣r♦♠✐s❡ ♦r r❡❞✉❝t✐♦♥ ♦❢ t❤❡ ✐♥✲♣❤❛s❡ ❣❛✐♥ ❛♥❞ t❤✉s ✇♦✉❧❞ ❞❡❣r❛❞❡ t❤❡ ❛❝❤✐❡✈❡❞ ♣❤❛s❡ ❞❡♣❡♥❞❡♥t ❣❛✐♥ ✈❛r✐❛t✐♦♥✳ ❚❤✐s s❡❝t✐♦♥ ✐s ❛♥ ✐♥✲❞❡♣t❤ t❤❡♦r❡t✐❝❛❧ ✐♥✈❡st✐❣❛t✐♦♥ ♦❢ t❤❡ s②st❡♠ ❜❧♦❝❦ ❞✐❛❣r❛♠ s❤♦✇♥ ❜② ❋✐❣✳ ✶✳ ❖♥ t❤❡ ❧❡❢t EΩR = ejΩR + CC ✭✷✮ ✸ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 ❚❤❡ ❧❛s❡r ✭✸✮ ♣r♦✈✐❞❡s t❤❡ ♦♣t✐❝❛❧ ❝❛rr✐❡r ❣✐✈❡♥ ✐♥ ❡q♥✳ ✭✸✮ ✇❤❡r❡ ❆ ❞❡✜♥❡s t❤❡ ♣❡❛❦ ♠❛❣♥✐t✉❞❡ ♦❢ t❤❡ ✜❡❧❞ ❛t r❛❞✐❛♥ ❢r❡q✉❡♥❝② ω ✳ EL = A jωt e + CC 2 ✭✸✮ ❚❤❡ ♦✉t♣✉t ♦❢ t❤❡ ❧❛s❡r ✐s ❞✐✈✐❞❡❞ ❡q✉❛❧❧② ❤❛✈✐♥❣ ❛♥ ✐❞❡♥t✐❝❛❧ ❛♠♣❧✐t✉❞❡ ❛♥❞ ♣❤❛s❡ ❛t t❤❡ ❡♥❞ ♦❢ ❡❛❝❤ ♣❛t❤✳ ❚❤✐s ❝♦✉❧❞ ❜❡ ❛tt❛✐♥❡❞ ❜② ♣r❡❝✐s❡❧② ❡♥❣✐♥❡❡r✐♥❣ t❤❡ ♦♣✲ t✐❝❛❧ s♣❧✐tt❡r ❛♥❞ t❤❡ ♦♣t✐❝❛❧ ♣❛t❤s ✇♦✉❧❞ ❤❛✈❡ ✐❞❡♥t✐❝❛❧ ❧❡♥❣t❤s ❛♥❞ ❧♦ss❡s ✉s✐♥❣ ❛♠♣❧✐t✉❞❡ ❛♥❞ ♣❤❛s❡ tr✐♠♠❡rs✳ ❈♦♥♥❡❝t❡❞ t♦ t❤❡ t♦♣ r♦✉t❡ ✐s ❛ ▼❩▼✳ ■t ✐s ❜✐❛s❡❞ ❛t q✉❛❞r❛t✉r❡ ❛♥❞ ♠♦❞✉❧❛t❡s t❤❡ ❘❋ ✐♥♣✉t s✐❣♥❛❧ EΩS ♦♥t♦ t❤❡ ♦♣t✐❝❛❧ ❝❛rr✐❡r ✭✹✮✳ ❈♦♥s✐❞❡r t❤❡ ❝♦♠♣❧❡① ❡❧❡❝tr✐❝ ✜❡❧❞ ♦❢ t❤❡ ■▼✲❘❋ s✐❣♥❛❧ ❣✐✈❡♥ ✐♥ ❡q♥✳ ✭✹✮✱ ✇❤❡r❡ AS ❞❡♥♦t❡s t❤❡ ❝❛rr✐❡r✱ BS ✐s t❤❡ ✜rst ♦r❞❡r s✐❞❡❜❛♥❞ ♣❛✐rs ❛♥❞ ✇❤♦s❡ ♣❤❛s❡ ✐s ❣♦✈❡r♥❡❞ ❜② t❤❡ ❘❋ ✐♥♣✉t s✐❣♥❛❧ ♣❤❛s❡ φS ✳ ES = AS jωS t BS j[(ωS +ΩS )t+φS ] e + e 2 2 BS j[(ωS −ΩS )t−φS ] + e + CC 2 = 1 arcsin [cos (φS )] − 2 π vDC ✭✹✮ ✭✺✮ ■t ❤❛s ❜❡❡♥ ❢♦✉♥❞ t❤❛t ❛ s❤❛r♣❡r ❣❛✐♥ ✈❛r✐❛t✐♦♥ ✇✐t❤ ♣❤❛s❡ ❝❛♥ ❜❡ ❛❝❤✐❡✈❡❞✳ ❚❤❡ ✈♦❧t❛❣❡ vDC s❤♦✇♥ ✐♥ ❡q♥✳ ✭✻✮ ✐s ❜❛s❡❞ ♦♥ t❤❡ s✐♠✉❧❛t✐♦♥ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ❋✐❣✳ ✷❜ ❛♥❞ ❛♣♣❧✐❡s t❤❡ sq✉❛r❡ r♦♦t t♦ s❤❛r♣❡♥ ✉♣ t❤❡ r❡s♣♦♥s❡ ❛t t❤❡ ❧♦✇❡r ✈♦❧t❛❣❡s ❜② ♣r♦❞✉❝✐♥❣ ❛ ♥♦♥❧✐♥❡❛r s❧♦♣❡✳ ❚❤❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡ ❛♥❞ ❝♦♥st❛♥ts ♥♦r♠❛❧✐③❡ t❤❡ ✈♦❧t❛❣❡ ♣r✐♦r t♦ ❛♣♣❧②✐♥❣ t❤❡ sq✉❛r❡ r♦♦t✳ P❧♦ts ♦❢ ❡q♥s✳ ✭✺✮ ❛♥❞ ✭✻✮ ❛r❡ ❣✐✈❡♥ ✐♥ ❋✐❣✳ ✸✳ ■t ✇✐❧❧ ❜❡ s❤♦✇♥ t❤❛t ❡q♥✳ ✭✻✮ ✐♠♣r♦✈❡s t❤❡ ♣❤❛s❡ sq✉❡❡③✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❘❋✲P❙❆✳ ❚❤❡ ✈♦❧t❛❣❡ ♣r♦❝❡ss✐♥❣ ❜❧♦❝❦ ❝♦✉❧❞ ❜❡ r❡❛❧✐③❡❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ♠❛♥✲ ♥❡r✿ t❤❡ s✉♠ ♦❢ t❤❡ ❘❋ ✐♥♣✉t s✐❣♥❛❧ ❛♥❞ ❘❋ r❡❢❡r❡♥❝❡ ❝❛♥ 1 2 = r 2 arcsin [cos (φS )] −1 π ✭✻✮ ❆ ❝♦♣② ♦❢ t❤❡ ❧❛s❡r✬s ❝♦♠♣❧❡① ❡❧❡❝tr✐❝ ✜❡❧❞ EL ✐s ❛♣♣❧✐❡❞ t♦ t❤❡ P▼✳ ❆t ✐ts ♦✉t♣✉t ✐s t❤❡ ♣✉♠♣ ✭✼✮ ❛s ❞❡s❝r✐❜❡❞ ❜② ❡q♥✳ ✭✼✮✱ ✇❤❡r❡ AP ✐s t❤❡ ❛♠♣❧✐t✉❞❡✱ ωP ✐s t❤❡ r❛❞✐❛♥ ❢r❡q✉❡♥❝② ❛♥❞ θRS ✐s t❤❡ ♦♣t✐❝❛❧ ♣❤❛s❡ ❛♥❞ ❞❡♥♦t❡s t❤❡ ♣❤❛s❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡ ❘❋ ✐♥♣✉t s✐❣♥❛❧ ❛♥❞ ❘❋ r❡❢❡r❡♥❝❡✳ EP = ❚❤❡ ✈♦❧t❛❣❡ ♣r♦❝❡ss✐♥❣ ❜❧♦❝❦ ✭✺✮ ❝♦♠♣❛r❡s t❤❡ ❘❋ ✐♥✲ ♣✉t s✐❣♥❛❧ ❛♥❞ t❤❡ ❘❋ r❡❢❡r❡♥❝❡ t♦ ♦❜t❛✐♥ t❤❡ ♣❤❛s❡ ❞✐❢✲ ❢❡r❡♥❝❡✳ ❉❡t❛✐❧ ♦❢ t❤❡ s✐♠✉❧❛t❡❞ ✈♦❧t❛❣❡ ♣r♦❝❡ss✐♥❣ ❜❧♦❝❦ ✐s s❤♦✇♥ ✐♥ ❋✐❣✳ ✷❛ ❛♥❞ ❛ s✐♠♣❧✐✜❡❞ ❡q✉❛t✐♦♥ ❡①♣r❡ss❡❞ ❜② ❡q♥✳ ✭✺✮ ❞❡s❝r✐❜❡s t❤❡ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❜❧♦❝❦✱ ✇❤❡r❡ EΩS ❛♥❞ EΩR ❛r❡ t❤❡ ❘❋ ✐♥♣✉t ❛♥❞ r❡❢❡r❡♥❝❡ s✐❣♥❛❧s r❡s♣❡❝✲ t✐✈❡❧②✱ φS ❛♥❞ φR ❛r❡ t❤❡✐r ♣❤❛s❡s r❡s♣❡❝t✐✈❡❧② ❛♥❞ VDC ✐s t❤❡ ❉❈ ♦✉t♣✉t ✈♦❧t❛❣❡✳ ❚❤❡ s✐♠✉❧❛t❡❞ ✈♦❧t❛❣❡ ♣r♦❝❡ss✐♥❣ ❜❧♦❝❦ s❤♦✇s t❤❛t t❤❡ s✉♠♠❡❞ ✐♥♣✉ts ❛r❡ ❞❡❧❛②❡❞✱ ♥♦r♠❛❧✲ ✐③❡❞ ❜② ❝♦♥st❛♥ts ❛♥❞ ❧✐♥❡❛r✐③❡❞ ❜② ❛♥ ❛r❝s✐♥ ❢✉♥❝t✐♦♥✳ ❚❤❡ ❉❈ ♦✉t♣✉t ✈♦❧t❛❣❡ ❝♦♥♥❡❝ts t♦ t❤❡ P▼ ✭✻✮ ✇❤♦s❡ ♣❤❛s❡ ❞❡✈✐❛t✐♦♥ ✐s s❡t t♦ π r❛❞✐❛♥s ♣❡r ✈♦❧t✳ ❚❤❡ ❡q✉❛t✐♦♥ s❤♦✇s t❤❛t ✐❢ t❤❡ ♣❤❛s❡ ♦❢ t❤❡ ❘❋ ✐♥♣✉t s✐❣♥❛❧ ✐s ✵ r❛❞✐❛♥s ✭✐✳❡✳ ✐♥ ♣❤❛s❡ ✇✐t❤ t❤❡ ❘❋ r❡❢❡r❡♥❝❡✮ t❤❡♥ t❤❡ ❉❈ ✈♦❧t❛❣❡ ✐s ✵❱✳ ❲❤❡♥ t❤❡ ♣❤❛s❡ ✐s π/2 ♦r 3π/2 r❛❞✐❛♥s✱ t❤❡ ❉❈ ✈♦❧t❛❣❡ ✐s ✵✳✺❱✱ ♦r π t❤❡♥ t❤❡ ✈♦❧t❛❣❡ ✐s ✶✳✵❱ ✭s❡❡ ❚❛❜❧❡ ✶✮✳ ❚❤❡ ✈♦❧t❛❣❡ ♣❤❛s❡ s❤✐❢ts t❤❡ ♣✉♠♣ r❡s✉❧t✐♥❣ ✐♥ ❛♥ ♦♣t✐❝❛❧ ♣❤❛s❡ ✈❛❧✉❡ θRS s❤♦✇♥ ✐♥ ❚❛❜❧❡ ✶✳ vDC ❜❡ ❛♣♣❧✐❡❞ t♦ ❛♥ ❘❋ ❙❝❤♦tt❦② ❞✐♦❞❡ ❞❡t❡❝t♦r✳ ❚❤❡ ♦✉t♣✉t ❝✉rr❡♥t ♦❢ t❤❡ ❞❡t❡❝t♦r ✇✐❧❧ ❜❡ ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ✐♥st❛♥✲ t❛♥❡♦✉s ♣♦✇❡r ♦❢ t❤❡ s✉♠♠❡❞ s✐❣♥❛❧s✱ ✇❤✐❝❤ ✇✐❧❧ ✈❛r② ✇✐t❤ t❤❡ ♣❤❛s❡ ♦❢ t❤❡ ❘❋ ✐♥♣✉t s✐❣♥❛❧✳ ❆ ❧♦♦❦✉♣ t❛❜❧❡ ❝❛♥ t❤❡♥ ❜❡ ✉s❡❞ t♦ ❝♦♥✈❡rt t❤❡ ❞❡t❡❝t♦r ♦✉t♣✉t ✐♥t♦ t❤❡ ❝♦♥tr♦❧ ✈♦❧t❛❣❡ vDC t♦ ❞r✐✈❡ t❤❡ P▼✳ ❚❤❡ ❧♦♦❦✉♣ t❛❜❧❡ ❡♥tr✐❡s ❝❛♥ ❜❡ s❡❧❡❝t❡❞ t♦ ❛❝❤✐❡✈❡ t❤❡ ❞❡s✐r❡❞ ✈♦❧t❛❣❡ ♣r♦❝❡ss✐♥❣ ❢✉♥❝t✐♦♥✳ AP j(ωP t+θRS ) e + CC 2 ✭✼✮ ❚❤❡ ♣✉♠♣ ❡♥t❡rs t❤❡ ✜rst PP▲◆ ✇❛✈❡❣✉✐❞❡ ✭PP▲◆✶✮ ✇❤♦s❡ s❡❝♦♥❞ ♦r❞❡r ♥♦♥❧✐♥❡❛r✐t② χ(2) ♣r♦❞✉❝❡s ❛ ✜❡❧❞ ❝♦♠✲ ♣♦♥❡♥t ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ sq✉❛r❡ ♦❢ t❤❡ ❡❧❡❝tr✐❝ ✜❡❧❞✳ ■❣✲ ♥♦r✐♥❣ ❧♦ss❡s✱ t✇♦ ♣r♦❞✉❝ts ❛♥❞ ❛ ❉❈ t❡r♠ ❢♦r♠✿ EP 2 =
A2P ej(2ωP t+2θRS ) + e−j(2ωP t+2θRS ) + 2 ✳ ❚❤❡ r❛❞✐❛♥ ❢r❡✲ 4 q✉❡♥❝② ♦❢ t❤❡ ♣✉♠♣ ❤❛s ❞♦✉❜❧❡❞✱ ❛s ❤❛s ✐ts ♣❤❛s❡ ❝♦♠♣♦✲ ♥❡♥t θRS ❛♥❞ t❤❡ ❛♠♣❧✐t✉❞❡ ✐s ❤❛❧✈❡❞ r❡❧❛t✐✈❡ t♦ t❤❡ ✐♥♣✉t✳ ❆ ❜❛♥❞ ♣❛ss ✜❧t❡r r❡♠♦✈❡s t❤❡ ❉❈ ❝♦♠♣♦♥❡♥t ❧❡❛✈✐♥❣ t❤❡ ✜❡❧❞ ♦❢ t❤❡ s❡❝♦♥❞ ❤❛r♠♦♥✐❝ ♣✉♠♣ ✭✽✮ ❛s s❤♦✇♥ ❜② ❡q♥✳ ✭✽✮✳ EP 2 = A2P j(2ωP +2θRS ) e + CC 4 ✭✽✮ ❆t t❤❡ ✐♥♣✉t t♦ t❤❡ s❡❝♦♥❞ ✇❛✈❡❣✉✐❞❡ ✭PP▲◆✷✮ ❛r❡ t✇♦ ♦♣t✐❝❛❧ ✇❛✈❡s✱ t❤❡ s❡❝♦♥❞ ❤❛r♠♦♥✐❝ ♣✉♠♣✱ ❡q♥✳ ✭✽✮ ❛♥❞ t❤❡ ■▼✲❘❋ s✐❣♥❛❧✱ ❡q♥✳ ✭✹✮✳ ❚❤❡ q✉❛❞r❛t✐❝ ♥♦♥❧✐♥❡❛r✲ ✐t② ♦❢ t❤❡ s❡❝♦♥❞ ✇❛✈❡❣✉✐❞❡ ❜❡❤❛✈❡s ❞✐✛❡r❡♥t❧② ❜❡❝❛✉s❡ ♦❢ t❤❡ ✇❛✈❡s ❞✐ss✐♠✐❧❛r ❢r❡q✉❡♥❝✐❡s✳ ❚❤❡ ✇❛✈❡❣✉✐❞❡ sq✉❛r❡s t❤❡ s✉♠ ♦❢ t❤❡ t✇♦ ✐♥♣✉ts (EP 2 + ES )2 ♣r♦❞✉❝✐♥❣ t❤❡ ❞✐❢✲ ❢❡r❡♥❝❡ ❢r❡q✉❡♥❝② 2ωP − ωS ✳ ❚❤❡ ❉❋● ②✐❡❧❞s ❛ ♥✉♠❜❡r ♦❢ ♠❛t❤❡♠❛t✐❝❛❧ ♣r♦❞✉❝ts✱ ❤♦✇❡✈❡r ♦♥❧② s♦♠❡ ❛r❡ ♦❢ ✐♥t❡r❡st✳ ❚❤❡ ✇❛♥t❡❞ ♣r♦❞✉❝ts ❛r❡ r❡❢❡rr❡❞ t♦ ❛s t❤❡ ✐❞❧❡r Ei ❛♥❞ ❛r❡ s❤♦✇♥ ✐♥ t❤❡ t♦♣ ♣❛r❡♥t❤❡s❡s ♦❢ ❡q♥✳ ✭✾✮✳ Ei =    A2P 4 AS ej[ωS t+2θRS ] + A2P j[(ωS +ΩS )t+2θRS +φS ] + 4 BS e A2P j[(ωS −ΩS )t+2θRS −φS ] B e S 4  + AS jωS t + 2 e j[(ωS +ΩS )t+φS ] BS + 2 e BS j[(ωS −ΩS )t−φS ] e 2      + CC ✭✾✮ ❆❧❧ ♣r♦❞✉❝ts ♦❜t❛✐♥❡❞ ❢r♦♠ t❤❡ ❉❋● ♣❛ss t❤r♦✉❣❤ ❛ ❜❛♥❞ ♣❛ss ✜❧t❡r ✭✾✮ t❤❛t str✐♣s ♦✉t t❤❡ ♥♦♥✲✐❞❧❡r ♣r♦❞✉❝ts ❧❡❛✈✐♥❣ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ ❢r❡q✉❡♥❝②✳ ❚❤❡ ✐❞❧❡r ✐s t❤❡ ❡①❛❝t ✹ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 EΩ S EΩ S Linear voltage processing − EΩ R π 1 2 π 2 1 ArcSin Nonlinear voltage processing v DC − EΩ R 0.5 π 1 2 π 2 ✭❛✮ ▲✐♥❡❛r ✈♦❧t❛❣❡ ♣r♦❝❡ss✐♥❣ s❝❤❡♠❛t✐❝✳ ArcSin 1 ABS v DC ABS 1 2 0.5 ✭❜✮ ◆♦♥❧✐♥❡❛r ✈♦❧t❛❣❡ ♣r♦❝❡ss✐♥❣ s❝❤❡♠❛t✐❝✳ ❋✐❣✉r❡ ✷✿ ❙✐♠✉❧❛t✐♦♥ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ t❤❡ ✈♦❧t❛❣❡ ♣r♦❝❡ss✐♥❣ ❜❧♦❝❦s✳ s❛♠❡ r❛❞✐❛♥ ❢r❡q✉❡♥❝② ❛s t❤❡ ❧❛s❡r✱ ✇❤✐❝❤ ✐♥ t✉r♥ ✐s t❤❡ s❛♠❡ ❢r❡q✉❡♥❝② ❛s t❤❡ ■▼✲❘❋ s✐❣♥❛❧✳ ❆s t❤❡ ✐❞❧❡r ✇❛✈❡ ❛♥❞ ♦t❤❡rs ❛r❡ t❤❡ s❛♠❡ ❢r❡q✉❡♥❝②✱ ✐✳❡✳ ωi = (2ωP − ωS ) = ωS = ωP = ω ✱ t❤❡ s②st❡♠ ✐s t❤❡r❡❢♦r❡ ❞❡❣❡♥❡r❛t❡ ❛♥❞ ✈❡❝t♦r s✉♠♠❛t✐♦♥ ♦❢ t❤❡ t✇♦ ✇❛✈❡s ✇✐❧❧ ♦❝❝✉r✳ ❚❤✐s ✐s r❡♣r❡s❡♥t❡❞ ❜② ❛❞❞✐♥❣ t❤❡ ■▼✲❘❋ s✐❣♥❛❧ t♦ t❤❡ ✐❞❧❡r ❛s s❡❡♥ ✐♥ t❤❡ ❜♦tt♦♠ ♣❛rt ♦❢ ❡q♥✳ ✭✾✮✳ ❚❤❡ ♦♣t✐❝❛❧ ✜❡❧❞ ♣❛ss❡s ✐♥t♦ t❤❡ ♣❤♦t♦❞❡t❡❝t♦r ✇❤❡r❡ t❤❡ s✐❣♥❛❧ EΩS ✐s r❡❝♦✈❡r❡❞✳ ❊q✉❛t✐♦♥ ✭✶✵✮ s❤♦✇s t❤❡ ♠❛❣✲ ♥✐t✉❞❡ ♦❢ t❤❡ ❘❋ ♦✉t♣✉t s✐❣♥❛❧ EΩS,OU T ✐s ♣❤❛s❡ ❞❡♣❡♥✲ ❞❡♥t✳ ❋r♦♠ t❤❡ ❡q✉❛t✐♦♥ ✐t ✐s ❡✈✐❞❡♥t t❤❛t t❤❡ ♦✉t♣✉t ♣♦✇❡r ✈❛r✐❡s ✇✐t❤ t❤❡ ♣❤❛s❡ ♦❢ t❤❡ ❘❋ ✐♥♣✉t s✐❣♥❛❧ φS ✳ 1 Linear voltage function Nonlinear voltage function 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 30 60 90 120 150 180 210 240 270 300 330 360 S ❋✐❣✉r❡ ✸✿ ❱♦❧t❛❣❡ ❝♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ❧✐♥❡❛r ❛♥❞ ♥♦♥❧✐♥❡❛r ❢✉♥❝t✐♦♥s✳ EΩS,OU T = AS BS cos (ΩS t)  2 cos (φS ) A4  + 2P cos (φS )   +A2 cos (2θRS + φS ) P +A2P cos (−2θRS + φS )     ✭✶✵✮ ✹✳ ❙✐♠✉❧❛t✐♦♥ ❙❡t✉♣ ❚❤❡ s✐♠✉❧❛t✐♦♥ ✇❛s ✐♠♣❧❡♠❡♥t❡❞ ✐♥ t❤❡ ❱P■♣❤♦t♦♥✐❝s s✐♠✉❧❛t✐♦♥ s♦❢t✇❛r❡ ❬✷✹❪ ❛♥❞ t❤❡ s❝❤❡♠❛t✐❝ ✇❛s ❜❛s❡❞ ✉♣♦♥ ❋✐❣✳ ✶✳ ❚❤❡ ✈♦❧t❛❣❡ ♣r♦❝❡ss✐♥❣ ❜❧♦❝❦s ❛r❡ s❤♦✇♥ ✐♥ ❋✐❣✳ ✷❛ ❛♥❞ ✷❜ ❛♥❞ ♣r♦✈✐❞❡ t❤❡ str✉❝t✉r❡ ♦❢ t❤❡ ❧✐♥❡❛r ❛♥❞ ♥♦♥❧✐♥✲ ❡❛r ❢✉♥❝t✐♦♥s r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡ s✐♠✉❧❛t✐♦♥ ❤❛s t✇♦ ✶✵●❍③ s✐♥✉s♦✐❞❛❧ ✐♥♣✉ts✱ t❤❡ ❘❋ r❡❢❡r❡♥❝❡ EΩR ❛♥❞ t❤❡ ❘❋ ✐♥♣✉t s✐❣♥❛❧ EΩS ✱ ✇❤♦s❡ ❛♠♣❧✐t✉❞❡s ❛r❡ ✵✳✺vpk−pk ✳ ❚❤❡ ❧❛s❡r ✇❛s s❡t ❛t ✶✾✸✳✶❚❍③ ✭✶✺✺✸✳✻♥♠✮ ✇✐t❤ ✶✵❞❇♠ ♦❢ ♦♣t✐❝❛❧ ♣♦✇❡r✳ ❆♥ ❛♠♣❧✐✜❡r ✐♥❝r❡❛s❡❞ t❤❡ ✐♥t❡♥s✐t② ❢r♦♠ t❤❡ ▼❩▼ t♦ ✷✵❞❇♠ ❛♥❞ ❛ s❡❝♦♥❞ ❞❡✈✐❝❡ ❛♠♣❧✐✜❡❞ t❤❡ ♣✉♠♣ ♣♦✇❡r ❛✈❛✐❧❛❜❧❡ ❛t t❤❡ ✐♥♣✉t t♦ t❤❡ ✜rst PP▲◆ ✇❛✈❡❣✉✐❞❡✳ ▲✐♠✲ ✐t❛t✐♦♥s ♦❢ t❤❡ s✐♠✉❧❛t✐♦♥ r❡q✉✐r❡❞ t❤❡ ✉s❡ ♦❢ t❤❡ ❱P■ ❝♦✲ s✐♠✉❧❛t✐♦♥ ❡♥✈✐r♦♥♠❡♥t✱ ❛s PP▲◆ ✇❛✈❡❣✉✐❞❡ ♠♦❞❡❧s ✇❡r❡ ♥♦t ❛✈❛✐❧❛❜❧❡ ❛t t❤❡ t✐♠❡✳ ❚❤❡ sq✉❛r✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ ❡❛❝❤ ✇❛✈❡❣✉✐❞❡ ✇❛s ✐♠♣❧❡♠❡♥t❡❞ ✐♥ ▼❆❚▲❆❇ ❬✷✼❪✳ ❈♦♥♥❡❝t❡❞ t♦ t❤❡ ♦✉t♣✉t ♦❢ t❤❡ PP▲◆ ✇❛✈❡❣✉✐❞❡s ❛r❡ t✇♦ ❜❛♥❞♣❛ss ✜❧t❡rs t❤❛t ❤❛✈❡ ❛ ❜❛♥❞✇✐❞t❤ ♦❢ ✶♥♠ ❛♥❞ ❛ ✸✵❞❇ ❝✉t✲♦✛ ❛t ❛ ❜❛♥❞✇✐❞t❤ ♦❢ ✷♥♠✳ ❚❤❡ s✐♠✉❧❛t♦r ✈❛r✐❡❞ t❤❡ ♣❤❛s❡ ♦❢ t❤❡ ❘❋ ✐♥♣✉t s✐❣♥❛❧ φS ♦✈❡r ❛ r❛♥❣❡ ❢r♦♠ ✵ t♦ 2π ❛♥❞ ♠❡❛s✉r❡❞ t❤❡ ♦✉t♣✉t ♣♦✇❡r ❛t t❤❡ ♣❤♦t♦❞❡t❡❝t♦r✳ ✺✳ ❘❡s✉❧ts ❛♥❞ ❉✐s❝✉ss✐♦♥ ❋✐❣✉r❡s ✷❛ ❛♥❞ ✷❜ r❡♣r❡s❡♥t t❤❡ ✈♦❧t❛❣❡ ♣r♦❝❡ss✐♥❣ ❜❧♦❝❦ ❢✉♥❝t✐♦♥s❀ t❤❡ ❞❡r✐✈❡❞ ♦✉t♣✉t ✈♦❧t❛❣❡ vDC ❝♦rr❡❧❛t❡s t♦ t❤❡ ♣❤❛s❡ ♦❢ t❤❡ ❘❋ ✐♥♣✉t s✐❣♥❛❧ ❛♥❞ ❜✐❛s❡s t❤❡ P▼✳ ❚❤❡ ♦✉t✲ ♣✉t ✈♦❧t❛❣❡ ❢r♦♠ t❤❡s❡ ♦♣❡r❛t✐♦♥s ✐s s❤♦✇♥ ✐♥ ❋✐❣✳ ✸ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❘❋ s✐❣♥❛❧ ♣❤❛s❡ φS ✳ ❚❤❡ ❧✐♥❡❛r ❢✉♥❝t✐♦♥ s❤♦✇s ❛ ♣❡❛❦ ✐♥ ✈♦❧t❛❣❡ ❛t ✶ ✈♦❧t ✇❤❡♥ t❤❡ ♣❤❛s❡ ✐s ✶✽✵ ❞❡❣r❡❡s✳ ❚❤❡ s❧♦♣❡ ♦❢ t❤❡ ❧✐♥❡ ✐s ✺✳✺✻❡✲✸ ✈♦❧ts ♣❡r ❞❡❣r❡❡ ❛♥❞ ❡q✉❛t❡s t♦ ❛♣♣r♦①✐♠❛t❡❧② ✵✳✺ ✈♦❧t ✇❤❡♥ t❤❡ ❘❋ ♣❤❛s❡ ✐s ✾✵ ♦r ✷✼✵ ❞❡❣r❡❡s✳ ❚❤❡ ✈♦❧t❛❣❡ ❝❤❛r❛❝t❡r✐st✐❝ ♦❜t❛✐♥❡❞ ❢r♦♠ t❤❡ ♥♦♥❧✐♥❡❛r ❢✉♥❝t✐♦♥ s❤♦✇s ♣❡❛❦s ♦❢ ✵✳✺ ✈♦❧t ❛t ✾✵ ❛♥❞ ✷✼✵ ❞❡❣r❡❡s ❛♥❞ ❛ st❡❡♣❡r ❞❡❝❧✐♥❡ t♦ ✵ ✈♦❧t ❞✉❡ t♦ t❤❡ ♥♦♥❧✐♥❡❛r✐t② ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳ ❚❤❡ r❡s✉❧t ♦❜t❛✐♥❡❞ ✐♥ ❋✐❣✳ ✹ ❝♦♠♣❛r❡s t❤❡ ♠♦❞❡❧ ❛♥❞ s✐♠✉❧❛t✐♦♥ ❛t ✐♥✲♣❤❛s❡ ❛♥❞ q✉❛❞r❛t✉r❡ ❛s t❤❡ ♣✉♠♣ ♣♦✇❡r ✐♥❝r❡❛s❡❞ ❢r♦♠ ✷✺❞❇♠ t♦ ✸✺❞❇♠✳ ❉✉r✐♥❣ t❤❡ ✐♥✲ ♣❤❛s❡ st❛t❡ t❤❡ ❘❋ ♦✉t♣✉t ♣♦✇❡r ✐♥❝r❡❛s❡s st❡❛❞✐❧②✱ ✇❤✐❧❡ ❛t q✉❛❞r❛t✉r❡ t❤❡ ♣♦✇❡r ❞❡❝❧✐♥❡s t♦ ❢♦r♠ ❛ ❞❡❡♣ ♥✉❧❧ ✇❤❡♥ t❤❡ ♣✉♠♣ ♣♦✇❡r ✐s ❛❞❥✉st❡❞ t♦ ✸✸❞❇♠ ♣r♦✈✐❞✐♥❣ ❛♣♣r♦①✐✲ ♠❛t❡❧② ✼✵❞❇ ♦❢ ❘❋ ♣❤❛s❡ s❡♥s✐t✐✈❡ ❞②♥❛♠✐❝ r❛♥❣❡ ✭P❙❉❘✮ ✇❤❡♥ ❝♦♠♣❧❡t❡ ❝❛♥❝❡❧❧❛t✐♦♥ ♦❝❝✉rs✳ ❚❤❡ ♠♦❞❡❧ ✇❛s ♥♦r✲ ♠❛❧✐③❡❞ t♦ t❤❡ s✐♠✉❧❛t✐♦♥ ❛s ✸✷✵❞❇ ❞✐✛❡r❡♥❝❡ ❡①✐sts ✇✐t❤ t❤❡ ❞✐s♣❛r✐t② ❞✉❡ t♦ ❝♦♠♣✉t✐♥❣ t❤❡ ❝♦s✐♥❡ ❛t q✉❛❞r❛t✉r❡✱ ❡q♥✳ ✭✶✵✮ ✇❤❡r❡ t❤❡ ✈❛❧✉❡ ❛♣♣r♦❛❝❤❡s ✵✳ ❖✈❡r❛❧❧ t❤❡ ♠♦❞❡❧ ❛♥❞ s✐♠✉❧❛t✐♦♥ ❛❣r❡❡ r❡❛s♦♥❛❜❧② ✇❡❧❧✳ ❚❤❡ ♦✉t❝♦♠❡ ❢r♦♠ t❤✐s ❝♦♠♣❛r✐s♦♥ s❤♦✇s ❛♥ ♦♣t✐♠✐③❡❞ ❵s✇❡❡t s♣♦t✬ ✇❤❡r❡ ✇❡ ❝❛♥ ♦❜t❛✐♥ ❣♦♦❞ q✉❛❞r❛t✉r❡ ❛tt❡♥✉❛t✐♦♥ ♦❢ t❤❡ ❘❋ s✐❣♥❛❧ ❜② ♦♣❡r❛t✐♥❣ t❤❡ ♣✉♠♣ ❛t ✸✸❞❇♠ ❣✐✈✐♥❣ t❤❡ ❧❛r❣❡st P❙❉❘✳ ❈♦♠♣❛r❡❞ t♦ ❬✶✼❪ ✇❤♦s❡ P❙❉❘ ✐s ✾❞❇✱ ♦✉r ❘❋✲P❙❆ ♣r♦✲ ✈✐❞❡s ❛♥ ❡①❝❡❧❧❡♥t r❡s✉❧t ✇✐t❤ ❛ P❙❉❘ ♦❢ ✼✵❞❇✳ ❙✐♠✉❧❛t✐♦♥ ❝♦♠♣♦♥❡♥ts s✉❝❤ ❛s t❤❡ ▼❩▼ ❤❛✈❡ s♠❛❧❧ ♣❤❛s❡ ✐♠♣❡r❢❡❝✲ t✐♦♥s✱ t❤❡ ♣❤♦t♦❞❡t❡❝t♦r✱ ❧❛s❡r ❛♥❞ ♦✈❡r❛❧❧ s②st❡♠ ❧♦ss❡s ❛t✲ tr✐❜✉t❡ t♦ ♥♦✐s❡✱ t❤❡s❡ ❢❛❝t♦rs ❝♦♥tr✐❜✉t❡ t♦ t❤❡ ❞✐✛❡r❡♥❝❡s ❜❡t✇❡❡♥ t❤❡ s✐♠✉❧❛t✐♦♥ ❛♥❞ ♠♦❞❡❧ r❡s✉❧ts ❛t q✉❛❞r❛t✉r❡✳ ❚❤❡ ♣❧♦t ✐♥ ❋✐❣✳ ✺ ❝♦♠♣❛r❡s t❤❡ t❤❡♦r❡t✐❝❛❧ ♠♦❞❡❧ t♦ t✇♦ s✐♠✉❧❛t✐♦♥s ✐♠♣❧❡♠❡♥t✐♥❣ t❤❡ ✈♦❧t❛❣❡ ♣r♦❝❡ss✐♥❣ ❢✉♥❝✲ ✺ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 20 In-phase (S) Quad. (S) In-phase (M) Quad. (M) 0 -20 -40 -60 -80 -100 -120 25 26 27 28 29 30 31 32 33 34 35 P ❋✐❣✉r❡ ✹✿ ❖✉t♣✉t ♣♦✇❡r ❛t t❤❡ ✐♥✲♣❤❛s❡ ❛♥❞ q✉❛❞r❛t✉r❡ ♣❤❛s❡ st❛t❡s ✇❤✐❧❡ t❤❡ ♣✉♠♣ ♣♦✇❡r ✐♥❝r❡❛s❡❞ ❢r♦♠ ✷✺❞❇♠ t♦ ✸✺❞❇♠✳ ❙✐♠✉❧❛t✐♦♥ ✭❙✮ ❛♥❞ ♠♦❞❡❧ ✭▼✮ ♠❡❛s✉r❡♠❡♥ts✳ 20 0 -20 0 -50 -40 -100 -150 Sim. Nonlinear Sim. Linear Theoretical model -60 0 30 60 -200 90 0 90 180 270 360 120 150 180 210 240 270 300 330 360 S ❋✐❣✉r❡ ✺✿ P❤❛s❡ s❡♥s✐t✐✈❡ ❘❋ ♦✉t♣✉t ♣♦✇❡r r❡s♣♦♥s❡✳ ❚❤❡ ♠♦❞❡❧ ♣r❡❞✐❝t✐♦♥ ✐s ❝♦♠♣❛r❡❞ t♦ t✇♦ s✐♠✉❧❛t✐♦♥s✿ t❤❡ ❧✐♥❡❛r ❛♥❞ ♥♦♥❧✐♥❡❛r ♣r♦❝❡ss✐♥❣ ✈♦❧t❛❣❡s✳ t✐♦♥s✱ ❡q♥✳ ✭✺✮ ❛♥❞ ❡q♥✳ ✭✻✮✳ ❚❤❡ ♣❤❛s❡ s❡♥s✐t✐✈❡ ❘❋ ♦✉t✲ ♣✉t ♣♦✇❡r r❡s♣♦♥s❡s ✇❡r❡ ♦❜t❛✐♥❡❞ ❜② s❡tt✐♥❣ t❤❡ ♣✉♠♣ ♣♦✇❡r t♦ ✸✸❞❇♠ ❛t t❤❡ ✐♥♣✉t t♦ PP▲◆✶ ❛♥❞ s✇❡❡♣✐♥❣ t❤❡ ♣❤❛s❡ ♦❢ t❤❡ ❘❋ ✐♥♣✉t s✐❣♥❛❧ ✉♣ t♦ 2π ✳ P❡❛❦s ❢♦r♠ ❛t t❤❡ ✐♥✲♣❤❛s❡ st❛t❡ ❛♥❞ ♥✉❧❧s ❛t q✉❛❞r❛t✉r❡ ❝♦♥✜r♠✐♥❣ ♦✈❡r ✼✵❞❇ ♦❢ P❙❉❘✳ ❚❤❡ s✐♠✉❧❛t❡❞ ❘❋ ♦✉t♣✉t ♣♦✇❡r ♣❡❛❦ ♠❡❛s✉r❡❞ ✶✸✳✾❞❇♠✱ ✇❤✐❝❤ ✐s ✵✳✶❞❇ ❣r❡❛t❡r t❤❛♥ t❤❡ ♠♦❞❡❧✳ ❚❤❡ ♠♦❞❡❧ ❣❛✈❡ ♣❡r❢❡❝t ❝❛♥❝❡❧❧❛t✐♦♥ ❛t q✉❛❞r❛t✉r❡ ❛♥❞ s♦ ❛♥ ✐♥✜♥✐t❡ P❙❉❘✳ ❚❤❡ ♥♦♥❧✐♥❡❛r ♣r♦❝❡ss ❤❛s ❛ s❤❛r♣❡r ❘❋ ♦✉t♣✉t ♣♦✇❡r r❡s♣♦♥s❡ ❛t ♣❤❛s❡ ❛♥❣❧❡s ❝❧♦s❡r t♦ t❤❡ ❵✐♥✲ ♣❤❛s❡✬✱ t❤❡r❡❜② ♦✛❡r✐♥❣ ❣r❡❛t❡r ❛tt❡♥✉❛t✐♦♥ ❛t t❤❡s❡ ♣❤❛s❡ ❛♥❣❧❡s✳ ❚❤✐s ♥♦✈❡❧ ❛ttr✐❜✉t❡ ✐s ✉s❡❢✉❧ ❢♦r ✐♠♣r♦✈✐♥❣ t❤❡ ♣❤❛s❡ sq✉❡❡③✐♥❣ ❝❤❛r❛❝t❡r✐st✐❝ ♦❢ t❤❡ ❘❋✲P❙❆✳ ❚❤❡ r❡s✉❧ts ✐♥ ❋✐❣✳ ✻❛ ✇❡r❡ ♦❜t❛✐♥❡❞ ❜② s✇❡❡♣✐♥❣ t❤❡ ♣❤❛s❡ ♦❢ t❤❡ ❘❋ s✐❣♥❛❧ ❢r♦♠ ✵ t♦ 2π ❛♥❞ ♣❧♦tt✐♥❣ t❤❡ ♥♦r✲ ♠❛❧✐③❡❞ ♦✉t♣✉t ♣♦✇❡r ♦♥ ❛ ♣♦❧❛r ♣❧♦t✳ ❚❤❡ ♦✉t❡r ♠♦st ❝✐r❝❧❡ ✭❜❧❛❝❦✮ ✐s t❤❡ ✐♥♣✉t ♣❤❛s❡ ♦❢ t❤❡ ❘❋ s✐❣♥❛❧ ❛s ✐t s✇❡❡♣s ♦✈❡r t❤❡ r❛♥❣❡✳ ❚❤❡ ♠✐❞❞❧❡ ♣❧♦t ✭❜❧✉❡✮ s❤♦✇s t❤❡ ♦✉t♣✉t ♣❤❛s❡ r❡s♣♦♥s❡ ♦❢ t❤❡ ❘❋✲P❙❆ ❛s ♣r♦❝❡ss❡❞ ❜② t❤❡ ❧✐♥❡❛r ❢✉♥❝t✐♦♥✳ ❚❤❡ ✐♥♥❡r✲♠♦st ♦❜❧♦♥❣ ♦✈❛❧ ❝✉r✈❡ ✭❣r❡❡♥✮ ✐s t❤❡ ♣❤❛s❡ r❡s♣♦♥s❡ ✇❤❡♥ t❤❡ ♥♦♥❧✐♥❡❛r ❢✉♥❝t✐♦♥ ✐s ❛♣✲ ♣❧✐❡❞✳ ❚❤❡ ♥♦♥❧✐♥❡❛r ❢✉♥❝t✐♦♥ s❤♦✇s ❛ s❤❛r♣❡r ♣❤❛s❡ ❝❤❛r✲ ❛❝t❡r✐st✐❝ ❛r♦✉♥❞ t❤❡ ✐♥✲♣❤❛s❡ st❛t❡s✳ ❋r♦♠ t❤❡ ✜❣✉r❡ ✐t ✐s ❛♣♣❛r❡♥t t❤❡ ♥♦♥❧✐♥❡❛r ✈♦❧t❛❣❡ ♣r♦❝❡ss✐♥❣ ♣r♦❞✉❝❡s s✉♣❡✲ r✐♦r ♣❤❛s❡ s❡♥s✐t✐✈✐t②✳ ■♥ t❤❡ ✜♥❛❧ s✐♠✉❧❛t✐♦♥✱ ●❛✉ss✐❛♥ ♥♦✐s❡ ✐s ❛❞❞❡❞ ♦♥t♦ t❤❡ ✶✵●❍③ s✐♥✉s♦✐❞❛❧ ❘❋ ✐♥♣✉t s✐❣♥❛❧ ❛♥❞ t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ t❤❡ ❘❋✲P❙❆ ✐s ♠❡❛s✉r❡❞ ❢♦r ✐ts ♣❤❛s❡ ❝♦♠♣r❡ss✐♦♥ ❝❤❛r✲ ❛❝t❡r✐st✐❝✳ ❚❤❡ ❞✉r❛t✐♦♥ ♦❢ t❤❡ ❘❋ ✐♥♣✉t s✐❣♥❛❧ ✐s ✶✵✷✹ s✐✲ ♥✉s♦✐❞❛❧ ❝②❝❧❡s ❛♥❞ t❤❡ s✐♠✉❧❛t✐♦♥ ❡♥s✉r❡❞ t❤❛t t❤❡ ●❛✉s✲ s✐❛♥ ♥♦✐s❡ ♣❛tt❡r♥ ✇❛s ✐❞❡♥t✐❝❛❧ ❢♦r t❤❡ ❧✐♥❡❛r ❛♥❞ ♥♦♥❧✐♥❡❛r ❢✉♥❝t✐♦♥s✳ ❚❤❡ s✐♥✉s♦✐❞❛❧ ❘❋ ♦✉t♣✉t s✐❣♥❛❧ ✇❛s s❛♠♣❧❡❞ ❛♥❞ ♠❡❛s✉r❡❞ ❛♥❞ t❤❡ r❡s✉❧ts ❣✐✈❡♥ ✐♥ t❤❡ ♥♦r♠❛❧✐③❡❞ ❝♦♥✲ st❡❧❧❛t✐♦♥ ♣❧♦ts ♦❢ ❋✐❣✳ ✻❜ ❛♥❞ ✻❝✳ ❚❤❡ r❡s✉❧ts ✐❧❧✉str❛t❡ t❤❡ ❘❋ ✐♥♣✉t s✐❣♥❛❧ ❛❢t❡r ●❛✉ss✐❛♥ ♥♦✐s❡ ✐s ❛♣♣❧✐❡❞ ❛♥❞ ❝♦♠✲ ♣❛r❡s t❤❡ ♦✉t♣✉t ♣❤❛s❡ r❡s♣♦♥s❡ ♦❢ t❤❡ ❧✐♥❡❛r ❛♥❞ ♥♦♥❧✐♥❡❛r ♣r♦❝❡ss✐♥❣ ❢✉♥❝t✐♦♥s✳ ❚❤❡ r❡s✉❧t s❤♦✇ ❛ s♠❛❧❧ ❛♠♦✉♥t ♦❢ ♣❤❛s❡ ❝♦♠♣r❡ss✐♦♥ ✉s✐♥❣ t❤❡ ❧✐♥❡❛r ❢✉♥❝t✐♦♥✱ ❜✉t t❤❡ ♥♦♥✲ ❧✐♥❡❛r ❢✉♥❝t✐♦♥ ✐♠♣r♦✈❡s t❤❡ ♣❤❛s❡ ❝♦♠♣r❡ss✐♦♥✱ ✐✳❡✳ t❤❡ ♣❤❛s❡ ♥♦✐s❡ ❤❛s ❜❡❡♥ ❢✉rt❤❡r r❡❞✉❝❡❞✳ ❚❤❡ ❝♦♠♣r❡ss✐♦♥ t♦✲ ✇❛r❞ t❤❡ ①✲❛①✐s ✐❧❧✉str❛t❡s ♣❤❛s❡ sq✉❡❡③✐♥❣✳ ❚❤✐s ♦✉t❝♦♠❡ ✐s s✐♠✐❧❛r t♦ ❛♥ ♦♣t✐❝❛❧ P❙❆ ✇❤❡♥ ❇P❙❑✴❉P❙❑ ♠♦❞✉✲ ❧❛t✐♦♥ ✇❛s ✉s❡❞ ❬✷✽❪✳ ■t ✐s ❡✈✐❞❡♥t ❢r♦♠ ❋✐❣✳ ✻ t❤❛t t❤❡ ♣r♦♣♦s❡❞ t❡❝❤♥✐q✉❡ ♦❢ ❝♦♠♣❛r✐♥❣ t❤❡ ✐♥✲♣❤❛s❡ ❛♥❞ q✉❛❞r❛✲ t✉r❡ ♣❤❛s❡ ❞✉r✐♥❣ ❛♠♣❧✐✜❝❛t✐♦♥ ❝❛♥ ❜❡ ✉s❡❞ t♦ r❡❣❡♥❡r❛t❡ t✇♦ ♣❤❛s❡ st❛t❡s ❛♥❞ t❤✉s ✐t ✐s s✉✐t❛❜❧❡ ❢♦r ❇P❙❑ ❛♥❞ ❉P❙❑✳ ■♥ ♦r❞❡r t♦ ❡①t❡♥❞ t❤✐s t❡❝❤♥✐q✉❡ t♦ ♠♦❞✉❧❛t✐♦♥ s❝❤❡♠❡s ✇✐t❤ ❤✐❣❤❡r ♥✉♠❜❡r ♦❢ ♣❤❛s❡ st❛t❡s✱ ✐t ✐s ❛♥t✐❝✐✲ ♣❛t❡❞ t❤❛t ❛❞❞✐t✐♦♥❛❧ ♣❤❛s❡ r❡❢❡r❡♥❝❡s ❛♥❞ PP▲◆ ✇❛✈❡❣✲ ✉✐❞❡s ✇♦✉❧❞ ❜❡ r❡q✉✐r❡❞✳ ❚❤✐s ❝♦✉❧❞ ❜❡ ❛❝❤✐❡✈❡❞ ❜② ❛❞❞✐♥❣ ♣✉♠♣ s✐❣♥❛❧s ❤❛✈✐♥❣ ❞✐✛❡r❡♥t r❡❢❡r❡♥❝❡ ♣❤❛s❡✱ ❢♦r ❡①❛♠✲ ♣❧❡ ❜♦t❤ q✉❛❞r❛t✉r❡ ❛♥❞ ✐♥✲♣❤❛s❡s t♦ ❛❧❧♦✇ r❡❣❡♥❡r❛t✐♦♥ ♦❢ ◗P❙❑ ♦r ✹✲◗❆▼ s✐❣♥❛❧s ✇❤✐❝❤ ✐s s✐♠✐❧❛r t♦ t❤❡ ❛♣♣r♦❛❝❤ ❞❡♠♦♥str❛t❡❞ ✐♥ ❬✷✾❪✳ ❲❤✐❧❡ t❤❡ s✐♠✉❧❛t✐♦♥s ❡①❛♠✐♥❡❞ ♦♥❡ ❘❋ ✐♥♣✉t ❢r❡q✉❡♥❝②✱ t❤❡ ❘❋ ❜❛♥❞✇✐❞t❤ ✇♦✉❧❞ ❜❡ ❧✐♠✐t❡❞ ❜② t❤❡ ✾✵ ❞❡❣r❡❡ ❞❡❧❛② ❛♣♣❧✐❡❞ ✐♥ t❤❡ ✈♦❧t❛❣❡ ♣r♦❝❡ss✐♥❣ ❜❧♦❝❦ ❛♥❞ t❤❡ ❧♦✇ ♣❛ss ✜❧t❡r ❛❢t❡r t❤❡ ♣❤♦t♦❞❡t❡❝t♦r✳ ✻✳ ❈♦♥❝❧✉s✐♦♥ ■♥ s✉♠♠❛r②✱ ❛ ♠✐❝r♦✇❛✈❡ ♣❤♦t♦♥✐❝ ❘❋✲P❙❆ ✐s ♣r♦✲ ♣♦s❡❞ t❤❡♦r❡t✐❝❛❧❧② ❛♥❞ ❞❡♠♦♥str❛t❡❞ t❤r♦✉❣❤ s✐♠✉❧❛t✐♦♥✳ ❯s✐♥❣ ❛ s✐♥❣❧❡ ♣✉♠♣ ❞❡❣❡♥❡r❛t❡ s❝❤❡♠❡ ✇✐t❤ t✇♦ PP▲◆ ✇❛✈❡❣✉✐❞❡s✱ t❤❡ ♣❤❛s❡ s❡♥s✐t✐✈❡ ❞②♥❛♠✐❝ r❛♥❣❡ ♦❢ t❤❡ ❘❋✲ P❙❆ ✐s ♠❡❛s✉r❡❞✳ ❚❤❡ ✉s❡ ♦❢ ❧✐♥❡❛r ❛♥❞ ♥♦♥❧✐♥❡❛r ♣r♦✲ ❝❡ss✐♥❣ ❢✉♥❝t✐♦♥s ✇❡r❡ ✐♥✈❡st✐❣❛t❡❞ ❛♥❞ ✐t s❤♦✇❡❞ t❤❛t t❤❡ ❧❛tt❡r ✐♠♣r♦✈❡❞ t❤❡ ♣❤❛s❡ sq✉❡❡③✐♥❣ ❝❤❛r❛❝t❡r✐st✐❝ ♦❢ t❤❡ ❘❋✲P❙❆✳ ❚❤❡ ♣r♦♣♦s❡❞ ❘❋✲P❙❆ ❞❡♠♦♥str❛t❡❞ ❛ ♣❤❛s❡ s❡♥s✐t✐✈❡ ❣❛✐♥ r❛♥❣❡ ♦❢ ✼✵❞❇✱ s❤♦✇✐♥❣ t❤❛t ✐t ❝♦✉❧❞ ❜❡ ♣r♦♠✐s✐♥❣ ❢♦r ♣❤❛s❡ ♥♦✐s❡ r❡❞✉❝t✐♦♥ ♦r r❡❣❡♥❡r❛t✐♦♥ ✐♥ ❘♦❋ P❙❑ t❡❝❤♥♦❧♦❣✐❡s✳ ❘❡❢❡r❡♥❝❡s ❬✶❪ ❏✳ ❨❛♦✱ ▼✐❝r♦✇❛✈❡ ♣❤♦t♦♥✐❝s✱ ■❊❊❊ ❏✳ ▲✐❣❤t✇❛✈❡ ❚❡❝❤♥♦❧✳ ✷✼ ✭✸✮ ✭✷✵✵✾✮ ✸✶✹✕✸✸✺✳ ❬✷❪ ●✳ ❚r✐♣❛❧❞✐✱ ❙✳ ❉✳ ❇❛rt♦❧♦✱ ❙✳ ❇❡tt✐✱ ❆✳ P✐③③♦❧❡♦✱ ❋✳ ▼❛t❡r❛✱ ❊①✲ ♣❡r✐♠❡♥t❛❧ ✐♥✈❡st✐❣❛t✐♦♥ ♦♥ ❉❱❇✲❚ ♦✈❡r ❈❆❚❱✲❘♦❋ ❝♦♥str❛✐♥ts ✐♥ ❧♦♥❣ ❤❛✉❧ ♣❛ss✐✈❡ ♦♣t✐❝❛❧ ♥❡t✇♦r❦s✱ ✐♥✿ ■❊❊❊ ■❈❚❖◆✱ ✷✵✶✹✱ ♣♣✳ ✶✕✹✳ ❬✸❪ ❙✳ ❘❛❥♣❛❧✱ ❘✳ ●♦②❛❧✱ ❆ r❡✈✐❡✇ ♦♥ r❛❞✐♦✲♦✈❡r✲✜❜❡r t❡❝❤♥♦❧♦❣②✲ ❜❛s❡❞ ✐♥t❡❣r❛t❡❞ ✭♦♣t✐❝❛❧✴✇✐r❡❧❡ss✮ ♥❡t✇♦r❦s✱ ❏✳ ❖♣t✳ ❈♦♠♠✳ ✸✽ ✭✶✮ ✭✷✵✶✼✮ ✶✾✕✷✺✳ ✻ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 ✭❛✮ ◆♦r♠❛❧✐③❡❞ ♣♦❧❛r ♣❧♦ts ♦❢ t❤❡ ♦✉t♣✉t ♣♦✇❡r ❝♦♠♣❛r✐♥❣ t❤❡ ❧✐♥❡❛r ❛♥❞ ♥♦♥❧✐♥✲ ❡❛r ✈♦❧t❛❣❡ ♣r♦❝❡ss✐♥❣✳ ✭❜✮ ▲✐♥❡❛r ✈♦❧t❛❣❡ ♣r♦❝❡ss✐♥❣✳ ✭❝✮ ◆♦♥❧✐♥❡❛r ✈♦❧t❛❣❡ ♣r♦❝❡ss✐♥❣✳ ❋✐❣✉r❡ ✻✿ ◆♦r♠❛❧✐③❡❞ ♣♦❧❛r ❛♥❞ ❝♦♥st❡❧❧❛t✐♦♥ ♣❧♦ts ❝♦♠♣❛r❡ t❤❡ ❡✛❡❝t t❤❛t ❧✐♥❡❛r ❛♥❞ ♥♦♥❧✐♥❡❛r ✈♦❧t❛❣❡ ♣r♦❝❡ss✐♥❣ ❤❛s ♦♥ ♣❤❛s❡ ❝♦♠♣r❡ss✐♦♥✳ ❬✹❪ ❆✳ ❙❡❡❞s✱ ▼✐❝r♦✇❛✈❡ ♣❤♦t♦♥✐❝s✱ ❚r❛♥s❛❝t✐♦♥s ♦♥ ▼✐❝r♦✇❛✈❡ ❚❤❡♦r② ❛♥❞ ❚❡❝❤♥✐q✉❡s✳ ✺✵ ✭✸✮ ✭✷✵✵✷✮ ✽✼✼✕✽✽✼✳ ❬✺❪ ❏✳ ❈❛♣♠❛♥②✱ ❏✳ ▼♦r❛✱ ■✳ ●❛s✉❧❧❛✱ ❏✳ ❙❛♥❝❤♦✱ ❏✳ ▲❧♦r❡t✱ ❙✳ ❙❛❧❡s✱ ▼✐❝r♦✇❛✈❡ ♣❤♦t♦♥✐❝ s✐❣♥❛❧ ♣r♦❝❡ss✐♥❣✱ ■❊❊❊ ❏✳ ▲✐❣❤t✇❛✈❡ ❚❡❝❤✲ ♥♦❧✳ ✸✶ ✭✹✮ ✭✷✵✶✸✮ ✺✼✶✕✺✽✻✳ ❬✻❪ ❉✳ ❩✐❜❛r✱ ❳✳ ❨✉✱ ❈✳ P❡✉❝❤❡r❡t✱ P✳ ❏❡♣♣❡s❡♥✱ ■✳ ▼♦♥r♦②✱ ❉✐❣✐t❛❧ ❝♦❤❡r❡♥t r❡❝❡✐✈❡r ❢♦r ♣❤❛s❡✲♠♦❞✉❧❛t❡❞ r❛❞✐♦✲♦✈❡r✲✜❜❡r ♦♣t✐❝❛❧ ❧✐♥❦s✱ ■❊❊❊ P❤♦t♦♥✐❝ ❚❡❝❤♥♦❧✳ ▲❡tt✳ ✷✶ ✭✸✮ ✭✷✵✵✾✮ ✶✺✺✕✶✺✼✳ ❬✼❪ ❑✳ ●❤♦✉♠✐❞✱ ❙✳ ▼❡❦❛♦✉✐✱ ❆✳ ❊❧ ▼♦✉ss❛t✐✱ ❏✳ ❩❛✐❞♦✉♥✐✱ ✶✻✲ ◗❆▼ ♠♦❞✉❧❛t✐♦♥ ❜❛s❡❞ ♦♥ q✉❛❞✲♣❛r❛❧❧❡❧ ♠❛❝❤✲③❡❤♥❞❡r ♠♦❞✉✲ ❧❛t♦r ❞❡❞✐❝❛t❡❞ ❢♦r r❛❞✐♦✲♦✈❡r✲✜❜❡r s②st❡♠✱ ■♥t❡r♥❛t✐♦♥❛❧ ❏✳ ■♥✲ ❢♦r♠❛t✐♦♥ ❊❧❡❝tr♦♥✳ ❊♥❣✳ ✸ ✭✺✮ ✭✷✵✶✸✮ ✺✵✷✕✺✵✺✳ ❬✽❪ ❘✳ ❙❧❛✈✐❦✱ ❋✳ P❛r♠✐❣✐❛♥✐✱ ❏✳ ❑❛❦❛♥❞❡✱ ❈✳ ▲✉♥❞str♦♠✱ ▼✳ ❙❥♦❞✐♥✱ P✳ ❆✳ ❆♥❞r❡❦s♦♥✱ ❘✳ ❲❡❡r❛s✉r✐②❛✱ ❙✳ ❙②❣❧❡t♦s✱ ❆✳ ❉✳ ❊❧✲ ❧✐s✱ ▲✳ ●r✉♥❡r✲◆✐❡❧s❡♥✱ ❉✳ ❏❛❦♦❜s❡♥✱ ❙✳ ❍❡rstr♦♠✱ ❘✳ P❤❡❧❛♥✱ ❏✳ ❖✬●♦r♠❛♥✱ ❆✳ ❇♦❣r✐s✱ ❉✳ ❙②✈r✐❞✐s✱ ❙✳ ❉❛s❣✉♣t❛✱ P✳ P❡tr♦♣♦✉✲ ❧♦s✱ ❉✳ ❏✳ ❘✐❝❤❛r❞s♦♥✱ ❆❧❧✲♦♣t✐❝❛❧ ♣❤❛s❡ ❛♥❞ ❛♠♣❧✐t✉❞❡ r❡❣❡♥❡r❛✲ t♦r ❢♦r ♥❡①t✲❣❡♥❡r❛t✐♦♥ t❡❧❡❝♦♠♠✉♥✐❝❛t✐♦♥s s②st❡♠s✱ ◆❛t✳ P❤♦✲ t♦♥✐❝s ✹ ✭✶✵✮ ✭✷✵✶✵✮ ✻✾✵✕✻✾✺✳ ❬✾❪ ▼✳ ❊✳ ▼❛r❤✐❝✱ ❈✳✲❍✳ ❍s✐❛✱ ❖♣t✐❝❛❧ ❛♠♣❧✐✜❝❛t✐♦♥ ❛♥❞ sq✉❡❡③❡❞✲ ❧✐❣❤t ❣❡♥❡r❛t✐♦♥ ✐♥ ✜❜r❡ ✐♥t❡r❢❡r♦♠❡t❡rs ♣❡r❢♦r♠✐♥❣ ❞❡❣❡♥❡r❛t❡ ❢♦✉r✲✇❛✈❡ ♠✐①✐♥❣✱ ◗✉❛♥t✉♠ ❖♣t✳✿ ❊✉r♦♣❡❛♥ ❖♣t✐❝❛❧ ❙♦❝✐❡t② P❛rt ❇ ✸ ✭✻✮ ✭✶✾✾✶✮ ✸✹✶✕✸✺✽✳ ❬✶✵❪ ❉✳ ▲❡✈❛♥❞♦✈s❦②✱ ▼✳ ❱❛s✐❧②❡✈✱ P✳ ❑✉♠❛r✱ ❆♠♣❧✐t✉❞❡ sq✉❡❡③✐♥❣ ♦❢ ❧✐❣❤t ❜② ♠❡❛♥s ♦❢ ❛ ♣❤❛s❡✲s❡♥s✐t✐✈❡ ✜❜❡r ♣❛r❛♠❡tr✐❝ ❛♠♣❧✐✜❡r✱ ❖♣t✳ ▲❡tt✳ ✷✹ ✭✶✹✮ ✭✶✾✾✾✮ ✾✽✹✕✾✽✻✳ ❬✶✶❪ ❈✳ ▲✉♥❞strö♠✱ ❩✳ ❚♦♥❣✱ ▼✳ ❑❛r❧ss♦♥✱ P✳ ❆♥❞r❡❦s♦♥✱ P❤❛s❡✲ t♦✲♣❤❛s❡ ❛♥❞ ♣❤❛s❡✲t♦✲❛♠♣❧✐t✉❞❡ tr❛♥s❢❡r ❝❤❛r❛❝t❡r✐st✐❝s ♦❢ ❛ ♥♦♥❞❡❣❡♥❡r❛t❡✲✐❞❧❡r ♣❤❛s❡✲s❡♥s✐t✐✈❡ ❛♠♣❧✐✜❡r✱ ❖♣t✳ ▲❡tt✳ ✸✻ ✭✷✷✮ ✭✷✵✶✶✮ ✹✸✺✻✕✹✸✺✽✳ ❬✶✷❪ ❑✳ ❈r♦✉ss♦r❡✱ ●✳ ▲✐✱ P❤❛s❡ ❛♥❞ ❛♠♣❧✐t✉❞❡ r❡❣❡♥❡r❛t✐♦♥ ♦❢ ❞✐❢✲ ❢❡r❡♥t✐❛❧ ♣❤❛s❡✲s❤✐❢t ❦❡②❡❞ s✐❣♥❛❧s ✉s✐♥❣ ♣❤❛s❡✲s❡♥s✐t✐✈❡ ❛♠♣❧✐✲ ✜❝❛t✐♦♥✱ ■❊❊❊ ❏✳ ❙❡❧✳ ❚♦♣✳ ◗✉❛♥t✉♠ ❊❧❡❝tr♦♥✳ ✶✹ ✭✸✮ ✭✷✵✵✽✮ ✻✹✽✕✻✺✽✳ ❬✶✸❪ ❏✳ ❑❛❦❛♥❞❡✱ ❆✳ ❇♦❣r✐s✱ ❘✳ ❙❧❛✈✐❦✱ ❋✳ P❛r♠✐❣✐❛♥✐✱ ❉✳ ❙②✈r✐❞✐s✱ P✳ P❡tr♦♣♦✉❧♦s✱ ❉✳ ❘✐❝❤❛r❞s♦♥✱ ▼✳ ❲❡st❧✉♥❞✱ ▼✳ ❙❦ö❧❞✱ ◗P❙❑ ♣❤❛s❡ ❛♥❞ ❛♠♣❧✐t✉❞❡ r❡❣❡♥❡r❛t✐♦♥ ❛t ✺✻ ❣❜❛✉❞ ✐♥ ❛ ♥♦✈❡❧ ✐❞❧❡r✲❢r❡❡ ♥♦♥✲❞❡❣❡♥❡r❛t❡ ♣❤❛s❡ s❡♥s✐t✐✈❡ ❛♠♣❧✐✜❡r✱ ✐♥✿ ❖❋❈✱ ◆❋❖❊❈✱ ✷✵✶✶✱ ♣✳ ❖▼❚✹✳ ❬✶✹❪ ❚✳ ❘♦❡t❤❧✐♥❣s❤♦❡❢❡r✱ ●✳ ❖♥✐s❤❝❤✉❦♦✈✱ ❇✳ ❙❝❤♠❛✉ss✱ ●✳ ▲❡✉❝❤s✱ ❆❧❧✲♦♣t✐❝❛❧ s✐♠✉❧t❛♥❡♦✉s ♠✉❧t✐❧❡✈❡❧ ❛♠♣❧✐t✉❞❡ ❛♥❞ ♣❤❛s❡ r❡❣❡♥✲ ❡r❛t✐♦♥✱ ■❊❊❊ P❤♦t♦♥✐❝ ❚❡❝❤♥♦❧✳ ▲❡tt✳ ✷✻ ✭✻✮ ✭✷✵✶✹✮ ✺✺✻✕✺✺✾✳ ❬✶✺❪ ❇✳ ❨✉r❦❡✱ ▲✳ ❘✳ ❈♦rr✉❝❝✐♥✐✱ P✳ ●✳ ❑❛♠✐♥s❦②✱ ▲✳ ❲✳ ❘✉♣♣✱ ❆✳ ❉✳ ❙♠✐t❤✱ ❆✳ ❍✳ ❙✐❧✈❡r✱ ❘✳ ❲✳ ❙✐♠♦♥✱ ❊✳ ❆✳ ❲❤✐tt❛❦❡r✱ ❖❜s❡r✈❛t✐♦♥ ♦❢ ♣❛r❛♠❡tr✐❝ ❛♠♣❧✐✜❝❛t✐♦♥ ❛♥❞ ❞❡❛♠♣❧✐✜❝❛t✐♦♥ ✐♥ ❛ ❥♦s❡♣❤s♦♥ ♣❛r❛♠❡tr✐❝ ❛♠♣❧✐✜❡r✱ P❤②s✳ ❘❡✈✳ ❆ ✸✾ ✭✺✮ ✭✶✾✽✾✮ ✷✺✶✾✕✷✺✸✸✳ ❬✶✻❪ ▼✳ ❆✳ ❈❛st❡❧❧❛♥♦s✲❇❡❧tr❛♥✱ ❑✳ ❉✳ ■r✇✐♥✱ ●✳ ❈✳ ❍✐❧t♦♥✱ ▲✳ ❘✳ ❱❛❧❡✱ ❑✳ ❲✳ ▲❡❤♥❡rt✱ ❆♠♣❧✐✜❝❛t✐♦♥ ❛♥❞ sq✉❡❡③✐♥❣ ♦❢ q✉❛♥t✉♠ ♥♦✐s❡ ✇✐t❤ ❛ t✉♥❛❜❧❡ ❥♦s❡♣❤s♦♥ ♠❡t❛♠❛t❡r✐❛❧✱ ◆❛t✳ P❤②s✐❝s ✹ ✭✶✷✮ ✭✷✵✵✽✮ ✾✷✾✕✾✸✶✳ ❬✶✼❪ ❲✳ ▲❡❡✱ ❊✳ ❆❢s❤❛r✐✱ ❆ ❈▼❖❙ ♥♦✐s❡✲sq✉❡❡③✐♥❣ ❛♠♣❧✐✜❡r✱ ■❊❊❊ ❚r❛♥s✳ ▼✐❝r♦✇✳ ❚❤❡♦r② ❚❡❝❤♥✐q✳ ✻✵ ✭✷✮ ✭✷✵✶✷✮ ✸✷✾✕✸✸✾✳ ❬✶✽❪ ❑✳ ❏✳ ▲❡❡✱ ❋✳ P❛r♠✐❣✐❛♥✐✱ ❙✳ ▲✐✉✱ ❏✳ ❑❛❦❛♥❞❡✱ P✳ P❡tr♦♣♦✉❧♦s✱ ❑✳ ●❛❧❧♦✱ ❉✳ ❘✐❝❤❛r❞s♦♥✱ P❤❛s❡ s❡♥s✐t✐✈❡ ❛♠♣❧✐✜❝❛t✐♦♥ ❜❛s❡❞ ♦♥ q✉❛❞r❛t✐❝ ❝❛s❝❛❞✐♥❣ ✐♥ ❛ ♣❡r✐♦❞✐❝❛❧❧② ♣♦❧❡❞ ❧✐t❤✐✉♠ ♥✐♦❜❛t❡ ✇❛✈❡❣✉✐❞❡✱ ❖♣t✳ ❊①♣r❡ss ✶✼ ✭✷✷✮ ✭✷✵✵✾✮ ✷✵✸✾✸✕✷✵✹✵✵✳ ❬✶✾❪ ❇✳ ❏✳ P✉tt♥❛♠✱ ❉✳ ▼❛③r♦❛✱ ❙✳ ❙❤✐♥❛❞❛✱ ◆✳ ❲❛❞❛✱ P❤❛s❡✲ sq✉❡❡③✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ ♥♦♥✲❞❡❣❡♥❡r❛t❡ P❙❆s ✉s✐♥❣ PP▲◆ ✇❛✈❡❣✉✐❞❡s✱ ❖♣t✳ ❊①♣r❡ss ✶✾ ✭✷✻✮ ✭✷✵✶✶✮ ❇✶✸✶✕❇✶✸✾✳ ❬✷✵❪ ❚✳ ❯♠❡❦✐✱ ▼✳ ❆s♦❜❡✱ ❍✳ ❚❛❦❡♥♦✉❝❤✐✱ ■♥✲❧✐♥❡ ♣❤❛s❡ s❡♥s✐t✐✈❡ ❛♠✲ ♣❧✐✜❡r ❜❛s❡❞ ♦♥ PP▲◆ ✇❛✈❡❣✉✐❞❡s✱ ❖♣t✳ ❊①♣r❡ss ✷✶ ✭✶✵✮ ✭✷✵✶✸✮ ✶✷✵✼✼✕✶✷✵✽✹✳ ❬✷✶❪ ❈✳ ▲❛♥❣r♦❝❦✱ ❙✳ ❑✉♠❛r✱ ❏✳ ▼❝●❡❡❤❛♥✱ ❆✳ ❲✐❧❧♥❡r✱ ▼✳ ❋❡❥❡r✱ ❆❧❧✲♦♣t✐❝❛❧ s✐❣♥❛❧ ♣r♦❝❡ss✐♥❣ ✉s✐♥❣ χ✭✷✮ ♥♦♥❧✐♥❡❛r✐t✐❡s ✐♥ ❣✉✐❞❡❞✲ ✇❛✈❡ ❞❡✈✐❝❡s✱ ■❊❊❊ ❏✳ ▲✐❣❤t✇❛✈❡ ❚❡❝❤♥♦❧✳ ✷✹ ✭✼✮ ✭✷✵✵✻✮ ✷✺✼✾✕ ✷✺✾✷✳ ❬✷✷❪ ❚✳ ❯♠❡❦✐✱ ❖✳ ❚❛❞❛♥❛❣❛✱ ❆✳ ❚❛❦❛❞❛✱ ▼✳ ❆s♦❜❡✱ P❤❛s❡ s❡♥s✐t✐✈❡ ❞❡❣❡♥❡r❛t❡ ♣❛r❛♠❡tr✐❝ ❛♠♣❧✐✜❝❛t✐♦♥ ✉s✐♥❣ ❤✐❣❤❧② ❡✣❝✐❡♥t PP▲◆ r✐❞❣❡ ✇❛✈❡❣✉✐❞❡s✱ ✐♥✿ ❖❋❈✱ ◆❋❖❊❈✱ ✷✵✶✶✱ ♣♣✳ ✶✕✸✳ ❬✷✸❪ P✳ ❉❛✈✐❡s✱ ❏✳ ❙❝♦tt✱ ❘❋ ♣❤❛s❡ s❡♥s✐t✐✈❡ ❛♠♣❧✐✜❡r✱ ✐♥✿ ■❊❊❊ ❆▼❙✱ ✷✵✶✻✱ ♣♣✳ ✸✼✕✸✽✳ ❬✷✹❪ ❱P■♣❤♦t♦♥✐❝s✱ ❱♣✐tr❛♥s♠✐ss✐♦♥♠❛❦❡r ♦♣t✐❝❛❧ s②st❡♠s ✭✷✵✶✽✮✳ ❯❘▲ ❤tt♣✿✴✴✇✇✇✳✈♣✐♣❤♦t♦♥✐❝s✳❝♦♠✴✐♥❞❡①✳♣❤♣ ❬✷✺❪ ▼✳ ❑❛r✐♠✐✲●❤❛rt❡♠❛♥✐✱ ❍✳ ❑❛r✐♠✐✱ ▼✳ ❘✳ ■r❛✈❛♥✐✱ ❆ ♠❛❣♥✐t✉❞❡✴♣❤❛s❡✲❧♦❝❦❡❞ ❧♦♦♣ s②st❡♠ ❜❛s❡❞ ♦♥ ❡st✐♠❛t✐♦♥ ♦❢ ❢r❡q✉❡♥❝② ❛♥❞ ✐♥✲♣❤❛s❡✴q✉❛❞r❛t✉r❡✲♣❤❛s❡ ❛♠♣❧✐t✉❞❡s✱ ■❊❊❊ ❚r❛♥s✳ ■♥❞✳ ❊❧❡❝tr♦♥✳ ✺✶ ✭✷✮ ✭✷✵✵✹✮ ✺✶✶✕✺✶✼✳ ❬✷✻❪ ❇✳ ❘❛③❛✈✐✱ ❆ st✉❞② ♦❢ ✐♥❥❡❝t✐♦♥ ❧♦❝❦✐♥❣ ❛♥❞ ♣✉❧❧✐♥❣ ✐♥ ♦s❝✐❧❧❛t♦rs✱ ■❊❊❊ ❏✳ ❙♦❧✐❞✲❙t❛t❡ ❈✐r❝✉✐ts ✸✾ ✭✾✮ ✭✷✵✵✹✮ ✶✹✶✺✕✶✹✷✹✳ ❬✷✼❪ ▼❛t❤✇♦r❦✱ ▼❆❚▲❆❇ ✭✷✵✶✽✮✳ ❯❘▲ ❤tt♣✿✴✴✇✇✇✳♠❛t❤✇♦r❦s✳❝♦♠ ❬✷✽❪ ❑✳ ❈r♦✉ss♦r❡✱ ❈✳ ❑✐♠✱ ●✳ ▲✐✱ ❆❧❧✲♦♣t✐❝❛❧ r❡❣❡♥❡r❛t✐♦♥ ♦❢ ❞✐✛❡r✲ ❡♥t✐❛❧ ♣❤❛s❡✲s❤✐❢t ❦❡②✐♥❣ s✐❣♥❛❧s ❜❛s❡❞ ♦♥ ♣❤❛s❡✲s❡♥s✐t✐✈❡ ❛♠♣❧✐✲ ✜❝❛t✐♦♥✱ ❖♣t✳ ▲❡tt✳ ✷✾ ✭✷✵✮ ✭✷✵✵✹✮ ✷✸✺✼✕✷✸✺✾✳ ❬✷✾❪ ❩✳ ❩❤❡♥❣✱ ▲✳ ❆♥✱ ❩✳ ▲✐✱ ❳✳ ❩❤❛♦✱ ❳✳ ▲✐✉✱ ❆❧❧✲♦♣t✐❝❛❧ r❡❣❡♥❡r❛✲ t✐♦♥ ♦❢ ❉◗P❙❑✴◗P❙❑ s✐❣♥❛❧s ❜❛s❡❞ ♦♥ ♣❤❛s❡✲s❡♥s✐t✐✈❡ ❛♠♣❧✐✲ ✜❝❛t✐♦♥✱ ❖♣t✳ ❈♦♠♠✉♥✳ ✷✽✶ ✭✶✵✮ ✭✷✵✵✽✮ ✷✼✺✺✕✷✼✺✾✳