Diffraction of Gaussian beam in a 3D smoothly inhomogeneous ... - arXiv

'LIIUDFWLRQ RI *DXVVLDQ EHDP LQ D ' VPRRWKO\ LQKRPRJHQHRXV PHGLDHLNRQDOEDVHGFRPSOH[JHRPHWULFDORSWLFVDSSURDFK 3%HUF]\QVNL .@UHIOHFWLRQIURPDOLQHDU SURILOHOD\HU>@DVZHOODVRWKHULVVXHV 

7KH HLNRQDOEDVHG IRUP RI WKH SDUD[LDO &*2 VHHPV WR EH DQ HYHQ PRUH SRZHUIXO LQVWUXPHQWRIWKHZDYHWKHRU\DVFRPSDUHGZLWKWKHUD\EDVHGYHUVLRQ,WUHGXFHV*% GLIIUDFWLRQ SUREOHP WR WKH VROXWLRQ RI RUGLQDU\ GLIIHUHQWLDO HTXDWLRQV RI 5LFFDWL W\SH ZKLFK DUH PRUH FRQYHQLHQWIRUQXPHULFDOVROXWLRQWKDQ+DPLOWRQLDQUD\HTXDWLRQV 'HYHORSPHQW RI WKH HLNRQDOEDVHG IRUP RI &*2 ZDV HVVHQWLDOO\ LQIOXHQFHG E\ %DELþ DSSURDFK>@ZKLFKGHDOVZLWKQDUURZEHDPVFRQFHQWUDWHGLQWKHYLFLQLW\RIWKHFHQWUDOUD\ ZKLFKLVWKHJHRGHVLFLQWKHVSDFHZLWKWKHFRUUHVSRQGLQJPHWULF %DELþDSSURDFKLVEDVHGRQ DQ DEULGJHGSDUDEROLFZDYHHTXDWLRQZKLFKSUHVHUYHVRQO\TXDGUDWLFWHUPVLQVPDOOGHYLDWLRQV IURPDFHQWUDOUD\7KHUHWKHVROXWLRQRIWKHDEULGJHGSDUDEROLFZDYHHTXDWLRQKDVEHHQREWDLQHG LQ WKH IRUP RI *% ZLWK SDUDPHWHUV REH\LQJ FRPSOH[YDOXHG 5LFFDWL HTXDWLRQ 7KHUHE\ %DELþ UHGXFHG *% GLIIUDFWLRQ SUREOHP WR VROXWLRQ RI RUGLQDU\ GLIIHUHQWLDO HTXDWLRQV RI 5LFFDWL W\SH %DELþ DSSURDFK JDYH ULVH WR WKH G\QDPLFDO UD\ WUDFLQJ PHWKRG ZKLFK DOVR UHGXFHV GLIIUDFWLRQ SUREOHPV WR VROXWLRQ RI 5LFFDWLW\SH HTXDWLRQV RQ WKH EDVLV RI GLIIHUHQW WHFKQLTXHV >@ %DELþDSSURDFKDQGG\QDPLFDOUD\WUDFLQJPHWKRGLQFRPELQDWLRQZLWK*%VXPPDWLRQPHWKRG >@ KDV IRXQG ZLGH DSSOLFDWLRQV ERWK LQ JHRSK\VLFV DQG LQ RWKHU EUDQFKHV RI SK\VLFV >@1RWZLWKRXWDQLQIOXHQFHRI%DELþDSSURDFKWKHSUREOHPRI*%GLIIUDFWLRQZDV UHGXFHGWRWKHVROXWLRQRIRUGLQDU\GLIIHUHQWLDOHTXDWLRQVRI5LFFDWLW\SHDOVRLQWKHIUDPHZRUNRI SDUD[LDO :.% DSSUR[LPDWLRQ GHYHORSHG LQ >@ 7KLV PHWKRG EHFDPH NQRZQ LQ SODVPD SK\VLFVDOVRDVEHDP WUDFLQJPHWKRG>@3DUD[LDO:.%VWHPVIURPUHSUHVHQWDWLRQRIWKHZDYH ILHOGLQWKHJHRPHWULFDORSWLFVIRUPZLWKDQDGGLWLRQDO*DXVVLDQIDFWRU (LNRQDOEDVHGSDUD[LDO&*2XVHVWKHTXDGUDWLFUHSUHVHQWDWLRQIRUWKHHLNRQDOVLPLODUWR %DELþDSSURDFK4XDGUDWLFDSSUR[LPDWLRQIRUWKHFRPSOH[HLNRQDOVWLPXODWHGE\%DELþVROXWLRQ RIWKHDEULGJHGSDUDEROLFZDYHHTXDWLRQDSSHDUHGLQVHYHUDOSXEOLFDWLRQVGHDOLQJZLWK&*2LQ H[SOLFLW RU LQH[SOLFLW IRUP >@ WR QDPH D IHZ $OO DERYHPHQWLRQHG PHWKRGV GLIIHU LQ KHXULVWLFDO SULQFLSOHVDQGLQWHFKQLTXHVRIGHULYDWLRQRIWKH5LFFDWLW\SHHTXDWLRQ&RPSDUDWLYH DQDO\VLV RI WKH %DELþ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

 )RUPXODWLRQRIWKHSUREOHP /HW XV FRQVLGHU WKH SURSDJDWLRQ RI D PRQRFKURPDWLF *DXVVLDQ EHDP LQ D VPRRWKO\ LQKRPRJHQHRXV LVRWURSLF PHGLXP FKDUDFWHUL]HG E\ GLHOHFWULF SHUPLWWLYLW\ H U LQ D VFDODU DSSUR[LPDWLRQ 7KH VPRRWK FKDQJH RI WKH PHGLXP SDUDPHWHUV LPSOLHV VPDOOQHVV RI WKH JHRPHWULFDORSWLFVSDUDPHWHU O  P*2 {      / 

S N { S F  Z  LV WKH ZDYHOHQJWK LQ YDFXXP Z LV WKH DQJXODU IUHTXHQF\ DQG

+HUH O / a ’H



LV WKH FKDUDFWHULVWLF VFDOH RI PHGLXP LQKRPRJHQHLW\ 'HVFULSWLRQ RI D QDUURZ EHDP

ZLWKLQ WKH IUDPHZRUN RI JHRPHWULFDO RSWLFV DOVR LPSOLHV VPDOOQHVV RI WKH IROORZLQJ WZR SDUDPHWHUV O P ',) {      Z Z  P 5() {      / ZKHUH Z LVWKHFKDUDFWHULVWLFEHDPZLGWK7KHGLIIUDFWLRQSDUDPHWHU GHWHUPLQHVWKHDQJOHRI WKH EHDP GLIIUDFWLRQZLGHQLQJZKHUHDVWKHUHIUDFWLRQSDUDPHWHU FKDUDFWHUL]HVWKHLQIOXHQFH RI WKH PHGLXP LQKRPRJHQHLW\ RQ WKH GLIIUDFWLRQ 3DUDPHWHUV   DQG   FKDQJH DV WKH EHDP SURSDJDWHVRZLQJWRWKHYDULDWLRQVRIWKHEHDP¶VZLGWK Z ZHDVVXPHWKHRWKHUTXDQWLWLHVWREH LQYDULDEOHLQRUGHURIPDJQLWXGH ,WIROORZVIURP(TV  WKDW  P*2  P ',)  P 5()    6PDOOQHVVRIWKUHHSDUDPHWHUV  DOORZVRQHWRXVHDSDUD[LDODSSUR[LPDWLRQDORQJWKH FHQWUDO UD\ GHVFULEHG E\ WKH UHDO JHRPHWULFDO RSWLFV ,Q SDUD[LDO DSSUR[LPDWLRQ WKH ZDYH ILHOG X U FDQEHSUHVHQWHGLQWKHIRUPDVIROORZV 

X   



$ H[S ª¬LN



 

\ W  [  [  \ F W  G\ W  [  [  

º¼ 

 

ZKHUH W LV WKH SDUDPHWHU DORQJ WKH FHQWUDO UD\ ZKLFK UHODWHV WR WKH UD\ DUF OHQJWK V DV GW GV  H F DQG [ DUHFRRUGLQDWHVRUWKRJRQDOWRWKHUD\ZKLFKZLOOEHLQWURGXFHGSURSHUO\ 



LQWKHQH[W6HFWLRQ+HUH H F W H  UF DQG UF 

W   

LVWKHUDGLXVYHFWRUIRUWKHFHQWUDOUD\

LQ W  [  [  FRRUGLQDWHV$PSOLWXGH $W LQ(T GHWHUPLQHVYDULDWLRQVRIWKHEHDPLQWHQVLW\ DORQJ WKH UD\ (LNRQDO \ LQ (T  LV VXSSRVHG WR FRQVLVW RI WZR VXPPDQGV \ F W LV WKH HLNRQDO RQ WKH FHQWUDO UD\ ZKLOH G\ W  [  [  LV D VPDOO GHYLDWLRQ IURP \ F  G\  UF {  

&RPSOH[ HLNRQDO G\ GHVFULEHV ERWK WKH FXUYDWXUH RI WKH SKDVH IURQW RI WKH EHDP DQG LWV DPSOLWXGHSURILOH :LWKLQ SDUD[LDO DSSUR[LPDWLRQ GHYLDWLRQ G\ IRU *% FDQ EH SUHVHQWHG DV D TXDGUDWLF IRUPDQDORJRXVO\WR%DELþDSSURDFK>@   G\ W  [  [  % W [ [     ,Q ZKDW IROORZV L   DQG VXPPDWLRQ RYHU UHSHDWHG LQGLFHV LV LPSOLHG % W DUH FRPSOH[ LM

L

M

YDOXHG IXQFWLRQV ZKLFK FRQVWLWXWH D V\PPHWULF WHQVRU ZLWK % { %  7KH UHDO SDUWV RI WKHVH IXQFWLRQV FKDUDFWHUL]H WKH FXUYDWXUHV RI WKH *% SKDVH IURQW ZKHUHDV WKH LPDJLQDU\ SDUWV GHWHUPLQH WKH HOOLSWLFDO FURVVVHFWLRQ RI WKH *% VHH (TV  DQG   EHORZ  ,Q YLHZ RI H[WUHPDO SURSHUWLHV RI WKH FHQWUDO UD\ WKH WHUPV OLQHDU LQ [ L GR QRW FRQWULEXWH WR (T  LQ DQ LVRWURSLFPHGLXP 7RWDOFRPSOH[HLNRQDO\ VDWLVILHVWKHHLNRQDOHTXDWLRQ LM



 ’\



H

ZKLOHDPSOLWXGH $ REH\VWKHWUDQVSRUWHTXDWLRQ GLY ¬ª $’\ ¼º

 



 

ZKLFKSURYLGHVFRQVHUYDWLRQRIHQHUJ\IOX[DORQJWKHFHQWUDOUD\RIWKHEHDP(TXDWLRQV  FRUUHVSRQG WR WKH ILUVW DSSUR[LPDWLRQ LQ JHRPHWULFDO RSWLFV SDUDPHWHU   DQG FRQWDLQ D GHVFULSWLRQ RI WKH GLIIUDFWLRQ SURFHVVHV LQ WKH ILUVW QRQYDQLVKLQJ DSSUR[LPDWLRQ LQ SDUDPHWHUV  DQG LHLQWKHSDUD[LDODSSUR[LPDWLRQ  

 &RPSOH[HLNRQDOHTXDWLRQ ,WLVNQRZQWKDWWKHFRRUGLQDWHV\VWHPZLWKXQLWYHFWRUVRIWKH)UHQHWQDWXUDOWULKHGURQRI WKH FHQWUDO UD\ O Q E  KHUH O  Q  DQG E DUH WKH XQLW YHFWRUV RI WKH UD\¶V WDQJHQW SULQFLSDO QRUPDODQGELQRUPDOUHVSHFWLYHO\ H[SHULHQFHVORFDOURWDWLRQZLWKUHVSHFWWRWKHFHQWUDOUD\ZLWK DQJXODU YHORFLW\ SURSRUWLRQDO WR WKH UD\¶V WRUVLRQ F  $V LW IROORZV IURP WKH )UHQHW6HUUHW HTXDWLRQVWKHFRRUGLQDWHV\VWHPDVVRFLDWHGZLWKWKHQDWXUDOWULKHGURQLVQRQRUWKRJRQDOVRWKDW WKHFRUUHVSRQGLQJPDWUL[RIWKH/DPHFRHIILFLHQWVKDSSHQVWREHQRQGLDJRQDO>@+HQFHLWLV ZRUWK GHDOLQJ ZLWK DQ RUWKRJRQDO FRRUGLQDWH V\VWHP ZKLFK ZRXOG EH ORFDOO\ URWDWLRQOHVV ZLWK UHVSHFWWRWKHFHQWUDOUD\7KHXQLWYHFWRUV  O H  H  RIVXFKV\VWHPDUHGHWHUPLQHGE\UHODWLRQV >@ 

H

Q FRV T  E VLQ T  H

E FRV T  Q VLQ T 

 

ZKHUH T LV WKH URWDWLRQ DQJOH REH\LQJ WKH HTXDWLRQ GT  GW  H F  8QLW YHFWRUV H DUH WUDQVIHUUHG DORQJ WKH FHQWUDO UD\ LQ DFFRUGDQFH ZLWK WKH /HYL&LYLWD SDUDOOHO WUDQVSRUW ODZ DQ H[DPSOHRIWKLVODZLQWKHJHRPHWULFDORSWLFVFRQWH[WLVWKH5\WRY9ODGLPLUVNLL%HUU\ODZIRUWKH HYROXWLRQRISRODUL]DWLRQRIDQHOHFWURPDJQHWLFZDYHDORQJWKHUD\VHH>@ ,QZKDWIROORZV [H  [ H  LVD ZH UHIHUWRWKHVHW  O H  H DVWKH/HYL&LYLWDRUSDUDOOHOWUDQVSRUW EDVLV,I 

L

WZRGLPHQVLRQDOYHFWRULQWKHSODQHRUWKRJRQDOWRWKHFHQWUDOUD\ UF W  WKHQWKHUDGLXVYHFWRU U 



LV FRQQHFWHG WR WKH /HYL&LYLWD FRRUGLQDWHV W  [  [  E\ UHODWLRQ U UF W  [ H  [  H   7KH

/DPHFRHIILFLHQWVIRUWKH/HYL&LYLWDFRRUGLQDWHV\VWHP W  [  [  HTXDO>@ 

KW { K

ª  ’A H º H «  » H ¼ ¬

 K[

K[



 

ZKHUH ’ A w  w  %\YLUWXHRI(TV DQG WKHHLNRQDOHTXDWLRQ LQWKH/HYL&LYLWDFRRUGLQDWHVWDNHV WKHIROORZLQJIRUP  ½°  ­°      § ·       \  \ % [ [  % [ [ H  ® F F LM L M LM L M ¸ ¾   %L[ L   %L[ L ¨  K ¯° © ¹ ¿° )URP KHUH RQ GRW VWDQGV IRU GLIIHUHQWLDWLRQ ZLWK UHVSHFW WR W  2QH FDQ H[SDQG GLHOHFWULF SHUPLWWLYLW\ H LQHTXDWLRQ LQD7D\ORUVHULHVLQVPDOOGHYLDWLRQ    H U H F   ’A H U U   ’A H  2  P 5()     U U :LWKLQWKHSDUD[LDODSSUR[LPDWLRQDOOWKHWHUPVKLJKHUWKDQWKHVHFRQGRUGHULQVPDOOSDUDPHWHUV   DQG   VKRXOG EH RPLWWHG ,Q WKLV ZD\ ZH QHJOHFW LQ (T  WKH WKLUG WHUP LQ WKH EUDFHV    ,Q IDFW WKLV ZKLFK LV SURSRUWLRQDO WR [  DQG GHULYDWLYHV % DQG LV RI RUGHU RI P ',) U

UF

F

F

DSSUR[LPDWLRQFRUUHVSRQGVWRWKHSDUDEROLFHTXDWLRQLQWKHTXDVLRSWLFVWKHRU\>@ %\VROYLQJ(T ZLWKWKHKHOSRISHUWXUEDWLRQWKHRU\LQVPDOOSDUDPHWHUV DQG DQG WDNLQJ(T LQWRDFFRXQWZHKDYHLQ]HURDSSUR[LPDWLRQ  \ F H F    (T   FRUUHVSRQGV WR WKH HLNRQDO HTXDWLRQ RI WKH UHDO JHRPHWULFDO RSWLFV 7KH +DPLOWRQLDQ DSSURDFKGHWHUPLQHVWKHFHQWUDOUD\¶VWUDMHFWRU\E\WKHIROORZLQJHTXDWLRQV>@ LM





G O H GW

 GU ’H   GW 

O H 

 

,Q YLHZ RI H[WUHPDO SURSHUWLHV RI WKH FHQWUDO UD\ WKH WHUPV OLQHDU LQ [ L YDQLVK DQG ZH

 REWDLQDGLIIHUHQWLDOHTXDWLRQIRUWKHTXDGUDWLFWHUPVRIRUGHURI P 5() DQG P '  , ) 

% [ [   % [ LM

L

M

L

  % [ 

L

L

ª  ’ A  H   ’ AH  º  « »   H «¬ »¼ U U



L

 

(TXDWLRQ LVDTXDGUDWLFIRUPLQ [  ,QRUGHUWRVDWLVI\WKLVHTXDWLRQRQHVKRXOGHTXDODOOWKH FRHIILFLHQWV LQ LWV OHIWKDQG DQG ULJKWKDQG VLGHV $V D UHVXOW ZH REWDLQ D WHQVRU 5LFFDWLW\SH HTXDWLRQIRU % W  %  % % D    F

L

LM

ZKLFKLVDV\VWHPRIWKUHHHTXDWLRQVIRUWKHFRPSOH[SDUDPHWHUV % W  %   %   %  D  LM

LN

NM

LM

LM

%  %  %  % D  %   %   %  D 

 

















D 

E  F 

+HUH E\ DQDORJ\ ZLWK >@ EXW ZLWK WKH RSSRVLWH VLJQV  ZH KDYH LQWURGXFHG WKH IROORZLQJ TXDQWLWLHV §  w H  wH wH ·  D W ¨    ¸  ¨  w[ w[ H w[ w[ ¸¹ © U U LM

L

M

L

M

7KHWHUPVTXDGUDWLFLQ % LQ(TV DUHUHVSRQVLEOHIRUGLIIUDFWLRQLQKRPRJHQHRXVPHGLXP F

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þ DSSURDFK >@ DV ZHOO DV IURP G\QDPLFDO UD\ WUDFLQJ PHWKRG>@WKHODWHUGHDOVZLWKµDVVRFLDWHG¶UD\HTXDWLRQVGHVFULELQJHYROXWLRQRIWKHVHFRQG GHULYDWLYHVRIWKHHLNRQDODORQJWKHUD\  7KH FRPSOH[ HTXDWLRQV   FDQ EH SUHVHQWHG DV D V\VWHP RI VL[ UHDO HTXDWLRQV IRU TXDQWLWLHV 5 { 5H % DQG , { ,P %  5  5 5  , , D    ,  5 ,  , 5  LM

LM

LM

LM

LM

LM

RULQWKHFRPSRQHQWV 



  

LN

LM

LN

NM

NM

LN

LN

NM

LM

NM

5  5  5  ,  ,

D 

5  5  5  5  ,  ,  ,  D  5   5  5  ,   , D   ,   5,   5 ,   ,  5  ,  ,  ,  5  5  

D 

E 

F  G  H 

,   5 ,    5 ,   I  /HW XVGLVFXVVKRZWKHDERYHLQWURGXFHGSDUDPHWHUVUHODWHWRWKHUHDOSK\VLFDOTXDQWLWLHV 7KHUHDODQGLPDJLQDU\SDUWVRIWKHTXDGUDWLFIRUP GHVFULEHWKHVKDSHVRIWKHSKDVHIURQWDQG WKHEHDPFURVVVHFWLRQUHVSHFWLYHO\,QWKHSDUD[LDODSSUR[LPDWLRQXQGHUFRQVLGHUDWLRQWKHSKDVH



















W c  [ [ 

IURQWLQWKHYLFLQLW\RIDSRLQW W

RU

D

VDGGOHW\SH

H F W c W  W c 

VXUIDFH

LQ

'

 UHSUHVHQWVDVHFRQGRUGHUVXUIDFHDSDUDERORLG

[  [  W

VSDFH

GHWHUPLQHG

E\

HTXDWLRQ

 5LM W c [ L[ M   ZKLOHWKHEHDPFURVVVHFWLRQLVWKHVHFRQGRUGHUFXUYHHOOLSVH  SODQH GHWHUPLQHGE\HTXDWLRQ N , W c [ [ FRQVW  1RWHWKDWLQWKHJHQHULFFDVHWKH

LQ [  [ 

SULQFLSDOD[HVRIWKHUHDODQGLPDJLQDU\SDUWVRI GRQRWFRLQFLGHZLWKHDFKRWKHU>@,IZH GHQRWH WKH HLJHQYDOXHV RI WHQVRUV 5 W DQG , W IRUPV DV 5 W DQG , W  UHVSHFWLYHO\ L

LM

M

WKHQLQDFFRUGDQFHZLWK>@WKH\FDQEHDVVRFLDWHGZLWKWKHSULQFLSDOFXUYDWXUHUDGLLRIWKH ZDYH SKDVH IURQW U W  DQG ZLWK WKH EHDP SULQFLSDO ZLGWKV LH SULQFLSDO VHPLD[HV RI WKH LM

LM

L

L

FURVVVHFWLRQHOOLSVH  Z W  LQWKHIROORZLQJZD\ L

HF   ,   UL N  Z ,Q WHUPVRIFRPSRQHQWVRI 5 DQG , LQHYHU\SRLQWWKHVHTXDQWLWLHVFDQEHH[SUHVVHGDV L

5L



L

 

L

LM



Z  





LM

  ¯ N ¡ , , o ,  , , ° °± ¢¡  HF  5  5 r  5  5  5 



U

















D 





E 



(TXDWLRQ D  LQWURGXFHV DQ DGGLWLRQDO UHVWULFWLRQ IRU &*2 GHVFULSWLRQ RI *% ,QGHHG LI RQH UHTXLUHV Z WREHSRVLWLYHWKLVOHDGVWRLQHTXDOLW\ 



L



,  ,  !



 ,  , 







  , 

 

ZKLFKHQVXUHVWKDW , !  LVSRVLWLYHLHWKHEHDPLVERXQGHG L

 7UDQVSRUWHTXDWLRQIRUDPSOLWXGHDQGFRQVHUYDWLRQRIHQHUJ\ IOX[

,QWKHUD\FHQWHUHGFRRUGLQDWHV W  [  [  WKHWUDQVSRUWHTXDWLRQ WDNHVWKHIRUP

 w\ G$ ª  w §  w\ · w \ w \ º «    ¨ ¸ »$  K  wW GW ¬ K wW © K wW ¹ w[ w[  ¼ %\XVLQJSDUD[LDODSSUR[LPDWLRQDQGLQWURGXFLQJQHZDPSOLWXGH $ H F $  WKHDERYHHTXDWLRQ FDQEHUHGXFHGWRWKHIROORZLQJIRUP    $  7U % $      ZKHUH 7U % { % %  %  ,WDGPLWVDQH[SOLFLWVROXWLRQ 

 

LM

ª  º   $  H[S «  ³ 7U % GW »  ¬  ¼ ZKHUH $  { $  LV WKH LQLWLDODPSOLWXGHRIWKHEHDP7KXVWHQVRU %  GHILQHGIURP(TV  LM



LL

$ W

LM

HQDEOHVRQHWRGHWHUPLQHWKHFRPSOH[DPSOLWXGH $ DVZHOO 7KHDEVROXWHYDOXHRI $ HTXDOV 

LM

ª  º $  H[S «  ³ 7U 5 GW »  ¬  ¼ ,W IROORZV IURP (TVG I  WKDW WKH FRPELQDWLRQ ' { 'HW  , $

HTXDWLRQ 

7KHUHIRUH

LM

'   7U 5 ' LM



³ 7U 5



 $

LM

GW

ZKHUH ' { '  7KHQ(TV DQG \LHOG

 OQ

LM

 



,,   , REH\V WKH



 

'  '

 

'  '

 $ 

 

1RWHWKDW  ' KDVWKHVHQVHRIWKHDUHDRIWKH*%FURVVVHFWLRQ  ' v ZZ  VRWKDW(T  WDNHVWKHIRUP Z Z  _ $ _ _ $  _    D  ZZ 7KHDERYHHTXDWLRQVH[SUHVVWKHFRQVHUYDWLRQRIWKHHQHUJ\IOX[WKURXJKWKH*%FURVVVHFWLRQ 7KHWRWDOHQHUJ\IOX[ 3 LQWKHEHDPFDQEHFDOFXODWHGE\LQWHJUDWLRQRI(T WRJHWKHUZLWK  DQG   $     3 { ³³ H X G[G[  v v $ ZZ FRQVW   ' ZKHUHWKHLQWHJUDWLRQLVSHUIRUPHGRYHUWKHZKROHSODQH [  [  WUDQVYHUVHWRWKHUD\7KHHQHUJ\

IOX[FRQVHUYDWLRQVLPLODUWR(T KDVEHHQREWDLQHGDQDO\WLFDOO\DOVRLQ>@ZLWKLQWKHIUDPH RI WKH SDUDEROLF HTXDWLRQ DSSURDFK IRU WKH FDVH RI VHSDUDWHG YDULDEOHV ZKHQ D {   ZKLFK FRUUHVSRQGV WR WKH WRUVLRQOHVV UD\ 7KHVH UHVXOWV DUH LQ DJUHHPHQW ZLWK WKH GHVFULSWLRQ RI *% GLIIUDFWLRQ LQ WKH IUDPHZRUN RI DEULGJHV SDUDEROLF ZDYH HTXDWLRQ >@ DV ZHOO DV ZLWK G\QDPLFDOUD\WUDFLQJPHWKRG>@DQGSDUD[LDO:.%DSSUR[LPDWLRQ>@ ,W LVZRUWKQRWLFLQJWKDWLQWKHSUHVHQFHRIZHDNDEVRUSWLRQ ,P H  5H H WKHGLVVLSDWLRQ IDFWRU H[S ª¬  N   ³ ,P H GO º¼ DSSHDUVLQ(T OLNHLQWKHWUDGLWLRQDOJHRPHWULFDORSWLFV>@

 $QDO\WLFDOVROXWLRQV 'LIIUDFWLRQLQDKRPRJHQHRXVPHGLXP ,Q DKRPRJHQHRXVPHGLXPZKHUH D {  

HTXDWLRQV   SRVVHVV DQ DQDO\WLFDO VROXWLRQ ,Q WKH JHQHUDO FDVH LW FDQ EH ZULWWHQ LQ D PDWUL[ IRUP>@  %Ö W ª¬%Ö    ,ÖW º¼     ZKHUH %Ö LV  u  PDWUL[ZLWKFRPSRQHQWV %  ZKLOH ,Ö LVWKHXQLWPDWUL[  ,QSDUWLFXODUZKHQWKH LM

SULQFLSDOD[HVRIWKHSKDVHIURQWDQGRIWKHEHDPFURVVVHFWLRQLHWKRVHRIWHQVRUV 5 DQG , 

LM

GRQRWFRLQFLGHVROXWLRQ GHVFULEHVHYROXWLRQRI*%ZLWKJHQHUDODVWLJPDWLVP>@ /HWXVFRQVLGHUWKHVSHFLDOFDVHZKHQWKHSULQFLSDOD[HVRI 5 DQG , FRLQFLGHZLWKHDFK LM

LM

RWKHUDW W   *LYHQWKLVRQHFDQFKRRVHD[HVRI [ FRRUGLQDWHVLQVXFKDZD\WKDW %    7KHQLWIROORZVIURP(TE WKDW % W {   % W { % W  DQG % W { % W  LHWKHFURVV LM

L



LM

VHFWLRQDQGWKHZDYHIURQWRIWKHEHDPXQGHUFRQVLGHUDWLRQFRQVHUYHWKHLURULHQWDWLRQDORQJWKH ZKROHWUDMHFWRU\7KHQVROXWLRQ \LHOGV % %       % W ZKHUH % { %   %\VXEVWLWXWLQJ(T LQ(T ZHREWDLQWKHHTXDWLRQIRUDPSOLWXGH $  L

L

L

L



%   § % $  ¨   ©   %W   %W

L

ZKLFK\LHOGV 

$

$ 

· ¸$  ¹

$ 

  %W   %W

 

% %  % %

 

/HWXVH[SUHVVVROXWLRQV DQG LQWHUPVRI(T $VVXPLQJIRUWKHVLPSOLFLW\VDNHWKH LQLWLDO SKDVH IURQW RI WKH EHDP WR EH SODQH U  { U  f  DQG VXEVWLWXWLQJ (T  ZLWK % 5  L, LQWR(T ZHFRPHWRWKHNQRZQH[SUHVVLRQVIRUWKHHYROXWLRQRI U DQG Z  L

L

L

L

L

L

L

§ W · ª § /5L ·  º U L W «  ¨   ¸¸  ¸ »  ZL ZL    ¨¨ ¬« © W ¹ »¼ © /5L ¹ ZKHUH /5L N ZL LVWKH5D\OHLJKGLVWDQFHFRUUHVSRQGLQJWRWKHLQLWLDOZLGWK Z  { Z   (TXDWLRQV GHPRQVWUDWHW\SLFDOEHKDYLRUIRU*%GLIIUDFWLRQLQHPSW\VSDFH,QWKHQHDU ]RQH W  /5L  *% ZLGWKV DUH DSSUR[LPDWHO\ FRQVWDQWZKHUHDVWKHFXUYDWXUHUDGLLRIWKH ZDYH IURQWGHFUHDVH /  Z | Z  FRQVW  U L | 5L  D  W ,Q WKHIDU]RQH W  /5L  ERWKWKHEHDPZLGWKVDQGWKHFXUYDWXUHUDGLLRIWKHZDYHIURQWLQFUHDVH OLQHDUO\ Z ZL | L  W  U | W   E  /5L $W WKH 5D\OHLJK GLVWDQFH W /5L  WKH L WK FXUYDWXUH UDGLXV UHDFKHV LWV PLQLPXP YDOXH 

L

L

L

L

L

L

UL

 /5L  ZLWKWKHFRUUHVSRQGLQJEHDP¶VZLGWKHTXDOLQJ Z   %\ VXEVWLWXWLQJ UHSUHVHQWDWLRQ   LQWR (T  ZH ILQG WKDW LQ WKH QHDU ]RQH ZLWK WKH SODQHLQLWLDOSKDVHIURQW WKHDEVROXWHYDOXHRIFRPSOH[DPSOLWXGH $ UHPDLQVDOPRVWFRQVWDQW  $ | $ FRQVW  D  L

PLQ

ZKLOHLQWKHIDU]RQH

N Z Z Z Z $ $  E  W ZZ 7KXV (TV DJUHHZLWKWKHFRQVHUYDWLRQRIWKHHQHUJ\IOX[WKURXJKWKHEHDPFURVVVHFWLRQ (TV DQG  

$|

 'LIIUDFWLRQLQDOHQVOLNHPHGLXP/HWXVWXUQWRWKHDQDO\VLVRIHTXDWLRQV LQDQ LQKRPRJHQHRXVPHGLXP$GHWDLOHGDQDO\VLVRIWKHLQIOXHQFHWHQVRU D RQWKHEHDPG\QDPLFV ZLWKLQWKHIUDPHZRUNRIDSDUDEROLFHTXDWLRQZDVSUHVHQWHGLQSDSHU>@:KHQWKHWHQVRU D LM

KDV QHJDWLYH HLJHQYDOXHV D L    WKHQ WKH LQKRPRJHQHRXV PHGLXP IRFXVHV WKH EHDP WKHUHE\ FRPSHQVDWLQJWRVRPHGHJUHHWKHGLIIUDFWLRQZLGHQLQJ2QWKHFRQWUDU\LI D SRVVHVVHVSRVLWLYH

HLJHQYDOXHV D L !   WKHQWKHLQKRPRJHQHRXVPHGLXPGHIRFXVHVWKHEHDP 

LM

LM

*%GLIIUDFWLRQLQ'OHQVOLNHPHGLXPZLWK H  B [   / ZDYHJXLGHDQGDQWLZDYHJXLGH UHVSHFWLYHO\  ZDV VWXGLHG LQ SDSHU >@ IRU D FDVH LQ ZKLFK WKH EHDP SURSDJDWHV DORQJ WKH PHGLXP V\PPHWU\ D[LV ,Q WKLV 6HFWLRQ ZH H[WHQG WKH UHVXOWV RI SDSHU >@ WR ' OHQVOLNH PHGLXPDQGFRQVLGHUDEHDPSURSDJDWLQJDORQJWKHD[LVRIDZDYHJXLGHRUDQDQWLZDYHJXLGH  ZLWK H   D[  D [   :H DVVXPH WKDW WKH SULQFLSDO D[HV RI WKH WHQVRUV D  5  DQG , LM

FRLQFLGH ZLWK HDFK RWKHU )RU OHQVOLNH PHGLD RI WKLV NLQG WKH ULJKWKDQG VLGHV RI (TV  EHFRPH FRQVWDQW D W { D W FRQVW  D  W { D  W FRQVW  DQG D W D  W {   7KHQ (TV WDNHWKHIRUPRIWZRLQGHSHQGHQWHTXDWLRQV %L  %L D L     ZKLFKDUHLGHQWLFDOWRWKHHTXDWLRQDQDO\]HGLQ>@IRU'PHGLXP7KHUHIRUHDOOWKHUHVXOWVRI >@FDQEHDSSOLHGWR'PHGLXPXQGHUFRQVLGHUDWLRQ6ROXWLRQRI(T FDQEHSUHVHQWHGLQ WKHIROORZLQJIRUP  DW D  % &H    ZKHUH    % D &  DW D %  &H LM

LM

L

L

L

L

L

L

L

L

ZKLFK HPEUDFHV ERWK WKH ZDYHJXLGH QHJDWLYH D DQG DQWLZDYHJXLGH SRVLWLYH D L

L

SURSDJDWLRQ)RUWKHSODQHLQLWLDOSKDVHIURQWZKHQ U 

L

L

f  RQHKDV %L 

L  N Z

 L

L  /5L 

/HW XV GHWHUPLQH WKH FKDUDFWHULVWLF LQKRPRJHQHLW\ VFDOHV DORQJ [ D[HV DV / L

D

L

 



7KHQLQWKHFDVHRIGHIRFXVLQJPHGLXP D !   DWWKHGLVWDQFHV W t / WKHGHIRFXVLQJSURFHVVHV VWDUWWRSUHYDLODQGWKHEHDPZLGHQVLQDQH[SRQHQWLDOZD\ L

L

L

L

L

§ / · Z  ZL | L    ¨ L ¸ HW /     © /5L ¹ $WDFHUWDLQVWDJHFRQGLWLRQ IDLOVDQGVROXWLRQV DQG EHFRPHLQDSSOLFDEOH)URPWKH DQDO\VLV RI (T  RQH FDQ GHULYH UHVWULFWLRQV IRU WKH GLVWDQFH RI &*2 DQG SDUD[LDO DSSUR[LPDWLRQDSSOLFDELOLW\    W d / PLQ  OQ P 5()  OQ P ',)    

L

,Q SUDFWLFHWKHORJDULWKPVRIVPDOOSDUDPHWHUV DQG FDQQRWEHWRRODUJHWKHUHIRUHLQWKH FDVH RI GHIRFXVLQJ PHGLXP WKH &*2 VROXWLRQV DV ZHOO DV WKRVH RI SDUDEROLF HTXDWLRQ  DUH DSSOLFDEOHDWGLVWDQFHVRIDIHZLQKRPRJHQHLW\VFDOHVRQO\ )RUDIRFXVLQJOHQVOLNHPHGLXP D    VROXWLRQ WDNHVWKHIRUP / L L  WDQ W  /L  /5L  %L    /L L /L WDQ W  /    L /5L 7KH FRUUHVSRQGLQJ EHDP ZLGWKV Z DQG WKH ZDYH IURQW FXUYDWXUH UDGLL U  (TV  DUH DV IROORZV L

L

L



ZL

ZL 

L

ª§ / ·  º   «¨ L ¸  » VLQ  W /L  «¬© /5L ¹ »¼

§ / ·   ¨ L ¸ WDQ  W /L © /5L ¹ /L  ª§ / ·  º «¨ L ¸  » WDQ W /L «¬© /5L ¹ »¼

D 





UL



E 

,I /L /5L  WKHEDODQFHEHWZHHQGLIIUDFWLRQEURDGHQLQJDQGIRFXVLQJSURFHVVWDNHVSODFH,QWKLV FDVHHTXDWLRQ SRVVHVVHVDVWDWLRQDU\VROXWLRQDFFRUGLQJWRZKLFK*%ZLWKIODWZDYHIURQW  U o f SURSDJDWHV ZLWK FRQVWDQW ZLGWK Z Z   7KH VWDWLRQDU\ VROXWLRQ FRUUHVSRQGV WR WKH UHJLRQZKHUH P 5() a P ',) DQG&*2PHWKRGLVDSSOLFDEOHIRUDQDUELWUDU\ODUJHGLVWDQFH W  ,QD JHQHUDO QRQVWDWLRQDU\  FDVH WKH EHDP SDUDPHWHUV RVFLOODWH DURXQG WKH HTXLOLEULXP YDOXHV DFFRUGLQJ WR(TV $FFRUGLQJWR>@WKHVHRVFLOODWLRQVDUHFDXVHGE\WKHEHDWLQJEHWZHHQ HLJHQPRGHVRIWKHZDYHJXLGH2VFLOODWLQJ&*2VROXWLRQVDUHDOVRDSSOLFDEOHIRUDUELWUDU\ODUJH GLVWDQFHV L

L

L

 *DXVVLDQEHDPGLIIUDFWLRQDORQJKHOLFDOUD\LQDQD[LDOO\ LQKRPRJHQHRXVPHGLXP ,QWKLV6HFWLRQZHFRQVLGHUDQRQWULYLDOH[DPSOHRIWKH*%SURSDJDWLRQDQGGLIIUDFWLRQLQ DQHVVHQWLDOO\'PHGLXPZLWKLQVHSDUDEOHYDULDEOHV1RQ]HURWRUVLRQRIWKHFHQWUDOUD\FDXVHVD QRQWULYLDO SDUDOOHO WUDQVSRUW (T  DQG SURYLGHV IRU HQWDQJOHPHQW RI HTXDWLRQV EHFDXVH WKH LQKRPRJHQHLW\LQIOXHQFHLHWKHULJKWKDQGVLGHVRI(TV LVOLQNHGXSWRQRUPDOELQRUPDO FRRUGLQDWHV\VWHP ’H OLHVLQWKH  O Q SODQHVHH>@DQG(TV ZKLOHWKHGLIIUDFWLRQE\

LWVHOILHWKHOHIWKDQGVLGHVRI(TV LVDWWDFKHGWRWKHSDUDOOHOWUDQVSRUWFRRUGLQDWHV\VWHP  7KXVGLIIUDFWLRQRIDUD\ZLWKWRUVLRQGHPRQVWUDWHVVRPHVRUWRIµFRPSHWLWLRQ¶EHWZHHQWKH QRUPDOELQRUPDO V\PPHWU\ RI LQKRPRJHQHLW\ DQG SDUDOOHO WUDQVSRUW V\PPHWU\ RI WKH SXUH GLIIUDFWLRQ /HW XV FRQVLGHU D[LDOO\ V\PPHWULFDO IRFXVLQJ PHGLXP LQ F\OLQGULFDO FRRUGLQDWHV  U  M  ] ZLWKGLHOHFWULFSHUPLWWLYLW\ U  H U H    !     / 6XFK D PHGLXP IRUPV D ' FLUFXODU ZDYHJXLGH ZKLFK PDQLIHVWV IRFXVLQJ SURSHUWLHV LQ UDGLDO GLUHFWLRQEHFDXVH H cc H c  U  DU {          H / H/ ZKHUH SULPHV VWDQG IRU WKH GHULYDWLYHV ZLWK UHVSHFW WR U  $W WKH VDPH WLPH WKH PHGLXP LV KRPRJHQHRXV LQ D]LPXWKDO DQG ORQJLWXGLQDO GLUHFWLRQV D M ] {   DQG FDQ EULQJ DERXW WKH XQERXQGHG GLIIUDFWLRQ ZLGHQLQJ LQ WKHVH GLUHFWLRQV ,Q WKH VLPSOHVW FDVH ZKHQ WKH EHDP SURSDJDWHV DORQJ WKH ZDYHJXLGH D[LV ]  RQH GHDOV ZLWK D OHQVOLNH PHGLXP RI F\OLQGULFDO FRQVW !   7KHEHDP¶VZLGWKVZLOOH[SHULHQFHSHULRGLFDOFRPSUHVVLRQVDQG V\PPHWU\ZLWK D 

VWUHWFKHVOLNHLQ(TD  2I PXFKPRUHLQWHUHVWLVWKHFDVHRIDEHDPSURSDJDWLQJDORQJWKHKHOLFDOUD\OD\LQJRQWKH F\OLQGULFDOVXUIDFHRIFRQVWDQWUDGLXV UF  ,QFRQWUDVWWRDOOWKHDERYHFRQVLGHUHGH[DPSOHVVXFKD UD\GHPRQVWUDWHVDQRQWULYLDOSDUDOOHOWUDQVSRUW )URP(TV VHHDOVR>@ RQHFDQILQG RXWWKDWUDGLXV UF HTXDOV LM



UF

 OM

H Hc



 

/

 

+HUH OM LVWKHD[LDOFRPSRQHQWRIWKHXQLWWDQJHQWYHFWRU O  6XEVWLWXWLRQRI(T LQWR(T  \LHOGV 

UF

OM H 

  OM



U UF

7KHWRUVLRQRIWKLVKHOLFDOUD\HTXDOV  ZKHUH O]

  OM VLQFH OU

  OM

O] OM

F



FRQVW 

H /

UF

 

7KH QRUPDO DQG ELQRUPDO WR WKH UD\ DUH FRQQHFWHGWRWKHXQLWYHFWRUVRIWKHF\OLQGULFDO FRRUGLQDWHVDV  Q { HU  E   O] HM  OM H ]   

%\ XVLQJ (T  ZH LQWURGXFH FRRUGLQDWH V\VWHP FRUUHVSRQGLQJ WR WKH /HYL&LYLWD SDUDOOHO WUDQVSRUW  H HU FRV T   O] HM  OM H ] VLQ T  H    O ] HM  OM H ] FRV T  HU VLQ T   

+HUH

T

  OM

W   / LVWKHURWDWLRQDQJOHRIWKHQRUPDOQ DURXQGWKHUD\ 7R VROYH HTXDWLRQV   LW LV QHFHVVDU\ WR GHWHUPLQH IXQFWLRQV D W LQ WKH FRRUGLQDWH





V\VWHP [  [  W ZLWKXQLWYHFWRUV 7DNLQJLQWRDFFRXQWWKDW w  w[

LM

D[LDOO\LQKRPRJHQHRXVPHGLXP ’ H U G  GU  RQHFDQUHZULWH(T DV L

H ’ L

DQGWKDWIRUDQ

ª H cc   H c  º D    » H H H H «  H »¼ «¬  ,Q YLHZRI(TV DQG WKHVHH[SUHVVLRQVWDNHWKHIRUP   OM   OM   OM    D    FRV T  D    VLQ T  D  D  VLQ T FRV T    / / / :H VHHWKDWWKHLQKRPRJHQHLW\LVOLQNHGXSWRWKHF\OLQGULFDORUWKHQRUPDOELQRUPDOFRRUGLQDWH V\VWHP (T  DQG LQ WKH SDUDOOHO WUDQVSRUW FRRUGLQDWHV (T  WKH WHQVRU D  (T  LM

L

U

M

U

URWDWHVUHVXOWLQJLQWKHHQWDQJOHPHQWRIHTXDWLRQVD F  (TXDWLRQV   ZLWK     KDYH EHHQ VROYHG QXPHULFDOO\ DQG WDNLQJ (TV  LQWR DFFRXQWZHFDQSUHVHQWWKHG\QDPLFVRIWKH*%SDUDPHWHUV)LJXUHGHSLFWVWKHHYROXWLRQRIWKH KHOLFDO*%¶VZLGWKV Z W  (TD ,WLVVHHQWKDWRQHRIWKHZLGWKVJURZVLQILQLWHO\ZKHUHDV LM

DQRWKHU RQH GHPRQVWUDWHV RVFLOODWLRQV W\SLFDO IRU IRFXVLQJ PHGLD 6XFK D EHKDYLRU FRQILUPV D SKHQRPHQRQWKDWKDVEHHQSUHGLFWHGLQ>@ZKHQWKHYDOXHRIWKHVTXDUHGEHDP¶VWRUVLRQWXUQV RXW EHWZHHQ WKH HLJHQYDOXHV RI WHQVRU D  WKH EHDP EHFRPHV SDUDPHWULFDOO\ XQVWDEOH ,Q RXU L

FDVHZHKDYH D M ]  F  D U DQGXQOLPLWHGJURZWKRIRQHRIWKHEHDP¶VZLGWKFDQEHUHJDUGHG LM



DVDFRQVHTXHQFHRIWKLVLQVWDELOLW\'XHWRVXFKLQVWDELOLW\&*2VROXWLRQVDUHDSSOLFDEOHRQO\DW ILQLWH GLVWDQFHV RI WKH RUGHU RI VHYHUDO 5D\OHLJK GLVWDQFHV DIWHU WKDW WKH FRQGLWLRQ   IDLOV )LJXUHVKRZVWKHHYROXWLRQRIWKHFXUYDWXUHUDGLLRIWKHEHDP¶VSKDVHIURQW%RWKUDGLLRVFLOODWH ZLWKWKHSHULRGRIVPDOOZLGWKRVFLOODWLRQV)LJD$WWKDWWKHUHDUHUHJLRQVRIHTXDOVLJQVRIWKH FXUYDWXUHV LQ WKLV FDVH SKDVH IURQW LV D SDUDERORLG  DV ZHOO DV UHJLRQV ZKHUH WKHLU VLJQV DUH RSSRVLWH SKDVH IURQW LV D VDGGOHW\SH VXUIDFH  )LJXUH  GHSLFWV H[SOLFLWO\ WKH HYROXWLRQ RI WKH EHDPFURVVVHFWLRQLWLVGHWHUPLQHGE\HTXDWLRQ N , [ [  DQGWKHFURVVVHFWLRQRILWVSKDVH 

IURQW ZKLFK ZH GHWHUPLQH E\ N 5 [ [

 LQ WKH /HYL&LYLWD FRRUGLQDWHV [  [  DVVRFLDWHG LM

L

M

ZLWKWKHXQLWYHFWRUV ZKHUHDV)LJVKRZVWKHVDPHHYROXWLRQVLQWKHFRRUGLQDWHV [ Q  [ E LM

L

M

FRQQHFWHG WR WKH QRUPDOELQRUPDO XQLW YHFWRUV   LQ IDFW LQ WKH FRRUGLQDWHV DWWDFKHG WR WKH 

F\OLQGULFDORQHV 2QHFDQVHHWKDWWKHEHDP¶VFURVVVHFWLRQURWDWHVLQ [  [  FRRUGLQDWHVZLWK

DQJXODUYHORFLW\WKDWDSSUR[LPDWHO\HTXDOV GT  GW

 H F  (TV  $WWKHVDPHWLPHLQ

WKH [ Q  [ E FRRUGLQDWHV WKH FURVVVHFWLRQ HOOLSVH DSSUR[LPDWHO\ FRQVHUYHV LWV RULHQWDWLRQ EHLQJ VWUHWFKHGDORQJ [ E D[LV7KLVUHVXOWPHDQVWKDWWKHGLIIUDFWLRQRIWKH*%XQGHUFRQVLGHUDWLRQLV GHWHUPLQHGPDLQO\E\WKHLQKRPRJHQHLW\$VZDVPHQWLRQHGDERYHWKHIRFXVLQJFKDUDFWHURIWKH UDGLDO LQKRPRJHQHLW\OHDGVWRILQLWHRVFLOODWLRQVRIWKHEHDP¶VZLGWKDORQJ U LH [ WRJHWKHU ZLWKLQILQLWHJURZVRIWKHEHDPDORQJ [ E D[HVDORQJZKLFKWKHPHGLXPLVORFDOO\KRPRJHQHRXV +RZHYHUWKHEHDP¶VFURVVVHFWLRQHOOLSVHLVQRWH[DFWO\RULHQWHGDORQJ [ E D[HVDVLWLVVHHQDW )LJLWLVVOLJKWO\LQFOLQHGLQWKHSRVLWLYHFORFNZLVH GLUHFWLRQ7KLVUHIOHFWVDQDWXUDOWHQGHQF\ RI WKH EHDP WR IROORZ WKH SDUDOOHO WUDQVSRUW ODZ ,QGHHG LQ [  [ FRRUGLQDWHV WKH SDUDOOHO Q

WUDQVSRUWFRRUGLQDWHV [  [



URWDWHLQWKHSRVLWLYHGLUHFWLRQDQGWKHLQFOLQHGRULHQWDWLRQRIWKH Q

E

FURVVVHFWLRQ HOOLSVH UHIOHFWV VRPH VRUW RI EDODQFH EHWZHHQ WKH GRPLQDWHG LQKRPRJHQHLW\ V\PPHWU\DQGWKHSDUDOOHOWUDQVSRUWWHQGHQF\$VIRUSKDVHIURQWHYROXWLRQLWGHPRQVWUDWHVUDWKHU FRPSOLFDWHGG\QDPLFVWKDWFDQEHDWWDFKHGQHLWKHUWRSDUDOOHOWUDQVSRUWFRRUGLQDWHV\VWHPQRUWR WKH QRUPDOELQRUPDORQH,Q)LJ DQG)LJ RQHFDQVHHH[SOLFLWO\WKHFDVHVRISDUDERORLG SKDVHIURQWHOOLSWLFDOGDVKHGFXUYHV DQGVDGGOHW\SHSKDVHIURQWK\SHUEROLFGDVKHGFXUYHV $OO PHQWLRQHGSLFWXUHVFRPSOHWHO\GHVFULEHWKHKHOLFDO*%¶VFRPSOH[HYROXWLRQWKXVGHPRQVWUDWLQJ WKHSRZHURIWKHSDUD[LDO&*2DSSURDFK 



 &RQFOXVLRQ ,Q VXPPDU\ZHKDYHFRQVLGHUHGDSUREOHPRI*DXVVLDQEHDP*% GLIIUDFWLRQLQD' LQKRPRJHQHRXVPHGLXP7KHPDLQUHVXOWRIWKHSDSHUFRQVLVWVLQWKHGHULYDWLRQRIWKHSDUD[LDO FRPSOH[ JHRPHWULFDO RSWLFV UHODWLRQV IURP WKH ILUVW SULQFLSOHV ZKLFK UHGXFHV WKH GLIIUDFWLRQ SUREOHPWRWKHVROXWLRQRIRUGLQDU\GLIIHUHQWLDOHTXDWLRQVVLPLODUWR%DELþ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

$FNQRZOHGJHPHQWV 7KH ZRUN ZDV SDUWLDOO\ VXSSRUWHG E\ ,17$6 JUDQW 1R  7KH DXWKRUV DUH LQGHEWHG WR 51RZDFN /.OLPHV DQG ,3VHQFLN IRU IULHQGO\ VXSSRUW DQG YDOXDEOH UHIHUHQFHV RQ*DXVVLDQEHDPVWKHRU\.%LVJUDWHIXOWR0DULWLPH8QLYHUVLW\RI6]F]HFLQIRUKRVSLWDOLW\ 

5HIHUHQFHV