• Aucun résultat trouvé

Concave and Convex photonic Barriers in Gradient Optics

N/A
N/A
Protected

Academic year: 2021

Partager "Concave and Convex photonic Barriers in Gradient Optics"

Copied!
29
0
0

Texte intégral

(1)

HAL Id: hal-00005556

https://hal.archives-ouvertes.fr/hal-00005556

Submitted on 23 Jun 2005

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Concave and Convex photonic Barriers in Gradient Optics

Alexandr Shvartsburg, Guillaume Petite

To cite this version:

Alexandr Shvartsburg, Guillaume Petite. Concave and Convex photonic Barriers in Gradient Optics.

The European Physical Journal D : Atomic, molecular, optical and plasma physics, EDP Sciences, 2005, 36, pp.111. �hal-00005556�

(2)

&21&$9($1'&219(;3+2721,&%$55,(56,1*5$',(17237,&6

%\

$OH[DQGHU%6KYDUWVEXUJDQG*XLOODXPH3HWLWH

&HQWUDO'HVLJQ%XUHDXIRU8QLTXH,QVWUXPHQWDWLRQRIWKH5XVVLDQ$FDGHP\RI6FLHQFHV

%XWOHURY6WU0RVFRZ5XVVLDQ)HGHUDWLRQ

/DERUDWRLUHGHV6ROLGHV,UUDGLpV805&($'60&156HW(FROH3RO\WHFKQLTXH )3DODLVHDX)UDQFH

$EVWUDFW

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

3$&6%V

(3)

,,QWURGXFWLRQ

7KLVSDSHULVGHYRWHGWRWKHWUDYHOLQJDQGWXQQHOLQJUHJLPHVRISURSDJDWLRQRIHOHFWURPDJQHWLFZDYHV WKURXJK WKLQ GLHOHFWULF OD\HUV ZLWK FRQWLQXRXV VSDWLDO GLVWULEXWLRQV RI GLHOHFWULF VXVFHSWLELOLW\ LQ WKH GLUHFWLRQRISURSDJDWLRQε]7KLVSUREOHPKDVDORQJKLVWRU\VWDUWLQJIURPWKHILUVWDQDO\WLFUHVXOWV REWDLQHG E\ 5D\OHLJK IRU ZDYHV ZKRVH YHORFLW\ LQVLGH WKH PHGLXP GHSHQGV OLQHDUO\ XSRQ WKH FRRUGLQDWH/DWHUWKHOLQHDUSURILOHIRUε]DVZHOODVDQH[SRQHQWLDODQGPRUHJHQHUDO(SVWHLQ SURILOHVZHUHXVHGIRUWKHDQDO\VLVRIUDGLRSURSDJDWLRQLQWKHLRQRVSKHUH6RPHPRUHFRPSOLFDWHG GLVWULEXWLRQVZHUHPRGHOHGE\SLHFHZLVHSURILOHVRIε]GHVFULEHGE\D:.%DSSUR[LPDWLRQRU WUHDWHGQXPHULFDOO\ 7KHVHUHVHDUFKHVIRFXVHGRQWKHSURSDJDWLRQRI (0ZDYHVLQKHWHURJHQHRXV PHGLDZLWKSRVLWLYHεDOWKRXJKWKHWXQQHOLQJSKHQRPHQDDULVLQJZKHQεZHUHWRXFKHGVRPHWLPHV WRRHJLQWKHFDVHRIUDGLRZDYHVSHUFRODWLRQQHDUE\WKHLRQRVSKHULFPD[LPD

7KHDGYHQWRIODVHUV DWWUDFWHGDWWHQWLRQXSRQOLJKWWXQQHOLQJ LQDVHULHVRI RSWRHOHFWURQLFVSUREOHPV VXFK DV HJ WKH HYDQHVFHQW PRGHV LQ GLHOHFWULF ZDYHJXLGHV VXUIDFH ZDYHV RQ PLFURVSKHUHV

*RRV ± +DQFKHQ HIIHFW IRU RSWLFDO FRDWLQJV $ QHZ EXUVW RI LQWHUHVW LQWR WKHVH SKHQRPHQD ZDV VWLPXODWHGE\WKHLQWULJXLQJSHUVSHFWLYHRIVXSHUOXPLQDOOLJKWSURSDJDWLRQWKURXJKRSDTXHEDUULHUV 7KH H[SHULPHQWV LQ PLFURZDYH UDQJH ZLWK ³XQGHUVL]HG´ ZDYHJXLGH DQG ELSULVP GHYLFH DV ZHOO DV WKH DQDO\VLV RI VSDWLDO GLVSODFHPHQW RI WKH SHDN RI D WXQQHOLQJ SXOVH DQG WKH GLUHFW PHDVXUHPHQWRISKRWRQV¶WXQQHOLQJWLPHZHUHFRQVLGHUHGE\VRPHDXWKRUVLQIDYRURIWKHFRQFHSW RI VXSHUOXPLQDO SKDVH WLPH IRU WKH WXQQHOLQJ (0 ZDYHV +RZHYHU WKLV FRQFHSW DURXVHG FRQWURYHUVLDOYLHZSRLQWV

7KHWKHRUHWLFDOEDFNJURXQGRIWKH DIRUHVDLGUHVHDUFKHVLVEDVHGRQWKH UHFWDQJXODU PRGHORIRSDTXH EDUULHUSLRQHHUHGE\*DPRYIRUWKHWKHRU\RIα±GHFD\DVORQJDJRDVLQ$QRWKHUPRGHO FRPELQLQJWKHZHOONQRZQUHFWDQJXODUDQGOLQHDUEDUULHUVWKH³WUDSH]RLGDO´EDUULHUZDVGHYHORSHGLQ

(4)

+RZHYHUWKHVHPRGHOVEXLOWIURPWKHEURNHQVWUDLJKWOLQHVIDLOWRLQWHUSUHWWKHVDOLHQWIHDWXUHVRI SKRWRQLFEDUULHUVIRUPHGE\FRQWLQXRXVVPRRWKYDULDWLRQVRIPDWHULDO¶VWUDQVSDUHQF\

2QWKHFRQWUDU\WKLVSDSHULVLQWHQGHGWREULGJHWKHJDSEHWZHHQWKHWUDYHOLQJDQGHYDQHVFHQWUHJLPHV RI ZDYH SURSDJDWLRQ WKURXJK FRQFDYH DQG FRQYH[ SKRWRQLF EDUULHUV IRUPHG E\ FRQWLQXRXVO\

GLVWULEXWHGKHWHURJHQHLWLHVRIε]7KHVHUHJLPHVDUHVKRZQWREHGHWHUPLQHGE\VWURQJKHWHURJHQHLW\

LQGXFHGGLVSHUVLRQ+,'GHSHQGHQWXSRQWKHILQLWHVSDWLDOVFDOHVRIKHWHURJHQHLW\SURILOH6XFKQRQ

±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

7KH WXQQHOLQJ RI OLJKW ZDYH WKURXJK DQ DUELWUDU\ DPRXQW RI WKLQ GLHOHFWULF QDQROD\HUV ZLWK FRQFDYH SURILOHε]LVFRQVLGHUHGLQ6HFWLRQ97KHSKDVHVKLIWRIWKHHYDQHVFHQWZDYHLVVKRZQWRWHQGWRVRPH FRQVWDQWOLPLWDIWHUSDVVDJHWKURXJKDVWDFNFRQWDLQLQJDILQLWHDPRXQWRIOD\HUV$VXSHUOXPLQDOSKDVH WLPHSURYLGHGE\WKLVVDWXUDWLRQRISKDVHVKLIWLVIRXQG,QVHFWLRQ9,ZHVWXG\WKHFDVHRIORVV\ILOPV DQGVKRZWKDWWKH\SUHVHQWVSHFLILFIHDWXUHV

(5)

,,+HWHURJHQHLW\±LQGXFHGGLVSHUVLRQRIWKLQGLHOHFWULFILOPV

:HUHFDOOKHUHVRPHUHVXOWVH[SRVHGLQDSUHYLRXVSDSHURQWKHDQWLUHIOHFWLRQSURSHUWLHVRIVXFK LQKRPRJHQHRXV ILOPV 7R YLVXDOL]H WKH HIIHFWV RI +,' OHW XV FRQVLGHU D VLPSOH SUREOHP RI QRUPDO LQFLGHQFHRIOLQHDUO\SRODUL]HG(0ZDYHZLWKFRPSRQHQWV([DQG+\SURSDJDWLQJLQWKH]±GLUHFWLRQ LQFLGHQWLQJRQWKHLQWHUIDFH] RIDKHWHURJHQHRXVQRQPDJQHWLFDQGORVVOHVVPDWHULDOGHVFULEHGE\

VRPHFRQWLQXRXVGLVWULEXWLRQRIGLHOHFWULFVXVFHSWLELOLW\LQWKHWUDQVSDUHQWUHJLRQε]!]! LQD IRUP

ε ] =Q 8 ] 8 ]= =

+HUHQ LV WKH YDOXH RI UHIUDFWLYH LQGH[ RI PDWHULDO RQ WKH ERXQGDU\] ([SUHVVLQJ WKH ILHOG FRPSRQHQWV([DQG+\ WKURXJKWKHYHFWRU±SRWHQWLDO$$[ ψ$\ $] RQHFDQUHGXFHWKH V\VWHPRI0D[ZHOOHTXDWLRQVUHODWHGWRWKLVJHRPHWU\WRRQHHTXDWLRQJRYHUQLQJWKHIXQFWLRQψ

ð ð

ð

=

W F

] 8 Q ]

ψ

ψ

:HZLOOH[DPLQHWKHVROXWLRQRIIRUWKHSURILOHV8]

V ] V ]

8 ] V V

/ /

§ ·

= +¨ + ¸ = ± = ±

© ¹

FRQWDLQLQJWZRVSDWLDOVFDOHV/DQG/,QWKHFDVHRIRSSRVLWHVLJQVRIVDQGVWKHSURILOHKDV HLWKHUDPD[LPXPV V RUDPLQLPXPV V ZLWKDYDOXH8P

(6)

(

)

8P = +V \ \ /= /

7KHVFDOHV/DQG/DUHOLQNHGLQWKHVHFDVHVZLWKWKHWKLFNQHVVGDQGWKHYDOXHVRI8P/ G\

/ G\7KHVROXWLRQRIWKHKHWHURJHQHRXVZDYHHTXDWLRQZLWKWKHSURILOH8FDQEHZULWWHQDV DVSDWLDOO\QRQVLQXVRLGDOZDYH

( ) ( ) ( )

H[S

]

8 ] L T W 8 ] G]

ψ =ª¬ º¼ ª¬ η ω º¼ η=

³

T Q 1 1

F ω

ω

= = −

7KHFKDUDFWHULVWLFIUHTXHQFLHVLQDUHGLIIHUHQWIRUFRQFDYHDQGFRQYH[SURILOHV

( ) ( )

F \ F \

Q / Q /

+

Ω = Ω =

:HQRWHWKDWηLVSURSRUWLRQDOWRWKHRSWLFDOSKDVHSDWKDWWKHSRVLWLRQ]DQGWKDWWKHTXDQWLW\Q1KDV WKHPHDQLQJRIDUHIUDFWLYHLQGH[,WLVUHPDUNDEOHWKDW1DVZHOODVWKHFKDUDFWHULVWLFIUHTXHQFLHV DQGDUHGHWHUPLQHGE\WKHVSDWLDOVFDOHV/DQG/RQO\DQGQRWE\WKHFKDUDFWHULVWLFIUHTXHQFLHVRI WKHPDWHULDOLWVHOI

7KHVDOLHQWIHDWXUHVRIWKLVKHWHURJHQHLW\±LQGXFHGGLVSHUVLRQ+,'DUH

6XEMHFWWRWKHLQWHUSOD\RIVFDOHV/DQG/IRUFRQFDYHRUFRQYH[SURILOHV8]SDUDPHWHUVLQ FDQ KDYH SRVLWLYH QHJDWLYH RU ]HUR YDOXHV +HUHLQ LQ D FDVH! WKH IDFWRU1 UHVHPEOHV WKH UHIUDFWLYHLQGH[IRUDSODVPDRUZDYHJXLGHZLWKFXW±RIIIUHTXHQF\GHSHQGHQWXSRQWKHSURILOHε]

(7)

7KH YDOXHV RI IRU D QDQROD\HU ZLWK WKLFNQHVVG RI DERXW QP UHIUDFWLYH LQGH[ Q DQG PRGXODWLRQ GHSWK RI DERXW PD\ EH DV KLJK DV UDG V ZKLFK UHODWHV WR D IUHH VSDFH ZDYHOHQJWKLQDQHDU,5UDQJH

7KHZDYHZLWKIUHTXHQF\OHVVWKHQFXWRIIIUHTXHQF\IRUWKHFRQFDYHSKRWRQLFEDUULHULVSDVVLQJ WKURXJKWKLVEDUULHULQDWXQQHOLQJUHJLPH

7KH YDULDWLRQV RI ZDYH YHORFLW\ LQVLGH WKH PHGLXP SURGXFHG E\ +,' FDQ H[FHHG WKH DQDORJRXV HIIHFWVRIPDWHULDOGLVSHUVLRQE\VHYHUDORUGHUVRIPDJQLWXGH

(OHFWULF DQG PDJQHWLF FRPSRQHQWV RI WKH (0 ILHOG FDQ EH IRXQG E\ PHDQV RI WKH YHFWRU±SRWHQWLDO FRPSRQHQW$[ ψ

[

L &

( = ω ψF

( )

]

\ ]

L Q &8 ] L8 G8

+ 8

F T8 G]

ω ª ºψ

= « + » =

¬ ¼

ZKHUH&LVDQDPSOLWXGHIDFWRU7KHVROXWLRQV±DUHYDOLGIRUDUELWUDU\ZDYHOHQJWKVDQGVSDWLDO VFDOHVRIKHWHURJHQHLW\,QWKHOLPLW/WKHSURILOHLVUHGXFHGWRWKHZHOONQRZQ5D\OHLJKSURILOH ZKLFKLVWKXVDOLPLWLQJFDVHRIWKHSURILOHVXQGHUGLVFXVVLRQ0RUHRYHUZKHQERWKVFDOHV/DQG/

DUHLQFUHDVLQJDQGWKXVWKHKHWHURJHQHLW\LVYDQLVKLQJ1VROXWLRQLVUHGXFHGLQWKLV OLPLWWRD:.%DSSUR[LPDWLRQZKHUHWKHVH+,'HIIHFWVDUHGURSSHG,QWKHSDUWLFXODUFDVHZKHUHWKH KHWHURJHQHLW\ VFDOHV/ DUH ILQLWH EXW2 WKLV DSSURDFK UHYHDOV WKH SHFXOLDU8] SURILOHV HJ

( )

8 ] = +] / IRUZKLFKWKH:.%VROXWLRQVUHPDLQH[DFW

(8)

,,,*URXSYHORFLWLHVRIWUDYHOLQJ(0ZDYHVLQFRQFDYHDQGFRQYH[SKRWRQLFEDUULHUV

7KHVSDWLDOZDYHIRUPVRIWKH(0ILHOGLQVLGHWKHKHWHURJHQHRXVPHGLXPDUHQRQ±VLQXVRLGDODQGWKLV ILHOGLVIRUPHGGXHWRLQWHUIHUHQFHRIIRUZDUGDQGEDFNZDUGZDYHVVRWKHJURXSYHORFLWLHVYJRIWKHVH ZDYHIRUPVKDYHWREHIRXQGE\PHDQVRIHQHUJ\IOX[3RLQWLQJYHFWRU3DQGHQHUJ\GHQVLW\:

5H

J

F

: π ª º

= 3 = ¬ ¼

Y 3 ( +

(

)

: ε

= π ( + +

7R VWUHVV RXW WKH SK\VLFDO SKHQRPHQD LQLWLDWHG E\ WKH KHWHURJHQHLW\ ZH ZLOO FRQVLGHU D ILOP LQ DLU ZLWKRXWVXEVWUDWH7KHVSDWLDOVWUXFWXUHRIWKH(0ILHOGLQVLGHWKHEDUULHULVIRUPHGE\WKHLQWHUIHUHQFH RIIRUZDUGZDYHSDVVLQJWKURXJKWKHSODQH] DQGEDFNZDUGRQHUHIOHFWHGIURPWKHSODQH] G 8VLQJIRUPXODH±RQHFDQSUHVHQWWKHVHZDYHVLQDIRUP

( )

( )

LT LT

[

LT LT LT LT

]

\

L &

( H 4H

F 8

+ LT& 8 L8 H 4H H 4H T8

η η

η η η η

ω

= +

ª º

= « + + »

¬ ¼

ZKHUH

]

8 ]V

8 \V

T/ /

§ ·

= − ¨ + ¸

© ¹

(9)

)RU VLPSOLFLW\ WKH WLPH ± GHSHQGHQW IDFWRU H[SLωW LV RPLWWHG KHUH DQG EHORZ 7KH GLPHQVLRQOHVV SDUDPHWHU4GHVFULEHVWKHUHIOHFWLYLW\RIWKHIDUERXQGDU\] G,QWURGXFLQJWKHUHIOHFWLRQFRHIILFLHQWRI WKHILOP5ZHFDQZULWHWKHFRQWLQXLW\FRQGLWLRQVRQWKHSODQH]

( ) ( )

L

L &

( 5 4

F

+ = ω +

( )

( )

L

L Q 1& LV

( 5 4 4

F T/

ω ª º

= « + + − »

¬ ¼

+HUH(LLVWKHHOHFWULFFRPSRQHQWRIWKHLQFLGHQWLQJZDYHVRWKDWWKHDPSOLWXGH&ZULWHV

(

)

LF(L 5

&

4

+

= +

DQGSDUDPHWHU4FDQEHIRXQGIURPWKHFRQWLQXLW\FRQGLWLRQVRQWKHLQWHUIDFH] G

( )

H[S

LT L V Q 1

4 L

V Q 1

η γ

γ

§ ·

¨© ¸¹

= +

γ Fω/

7KH YDOXHη η G ZLOO EH REWDLQHG IURP 7KH ILHOG FRPSRQHQWV([ DQG+\ FDQ EH UHZULWWHQXVLQJHTVDQG

( )

(

L

) (

LT LT

)

[

( 5

( H 4H

4 8

η η

= + +

+

(10)

( )

L LT LT

\

( 5 Q 1 L V ] L V ]

+ 8 H V \ 4H V \

4 T/ / T/ /

η η

­ ª º ª º½

+ ° § · § · °

= ® « ¨ + ¸» « + ¨ + ¸»¾

+ °¯ ¬ © ¹¼ ¬ © ¹¼°¿

6XEVWLWXWLRQRI±LQWREULQJVWKHH[SUHVVLRQIRUWKH(0HQHUJ\IORZ3]

( )

( )

( )

( )

( )

L ]

0 Q 1 ( 3 F

5 5 LV

0 Q 1

4 4

π

γ

=

+ +

= ∆ = +

+ +

:HWKXVQRWHWKDWWKHHQHUJ\IORZ3]LVFRRUGLQDWH±LQGHSHQGHQWZKLOHRQWKHFRQWUDU\WKH(0HQHUJ\

GHQVLW\ :SURYHVWREHFRRUGLQDWH±GHSHQGHQWVXEVWLWXWLRQRI±LQWRSHUPLWVRQHWR GHVFULEHWKLVGHSHQGHQFHE\PHDQVRIDGLPHQVLRQOHVVIXQFWLRQθ

( )

0 Q 1

: 8θ

π +

=

ZLWK

( )

( )

( )

( )

FRV

VLQ

Q Q 1 \ ]

1 T/ /

Q 1

] ]

T Q Q 1 \ \

1 T/ / T/ /

V T Q

1 T/

γ γ

θ

γ

γ γ

η η

γ γ

η η

+ +

+

+ + +

+

ª § · º

ª º

+ « »

= + + +«¬ + »¼«¬ + ¨© ¸¹ »¼+

­ + ª ºª § · º § ·½

° « » °

+ + + +

ª º ® « » ¨ ¸ ¨ ¸¾

¬ ¼°¯ ¬ ¼«¬ © ¹ »¼ © ¹°¿

+

ª º

¬ ¼

( )

] ]

Q 1 \ Q 1 \

/ γ T/ /

+ +

­ ª º§ · ª § · º½

° + « »°

® «¬ »¼ ©¨ ¸¹ « ¨© ¸¹ »¾

° ¬ ¼°

¯ ¿

(11)

+HUHWKHVXEVFULSWV³´LQθDQG1LQGLFDWHVWKDWZHDUHFRQVLGHULQJWKHFDVH1!)LQDOO\PDNLQJ XVHRIZHZLOOILQGWKHJURXSYHORFLW\YJLQVLGHWKHFXUYLOLQHDUEDUULHU

J

Y F

8θ+

=

,WLVZRUWKVWUHVVLQJRXWWKDWWKLVH[SUHVVLRQIRUWKHJURXSYHORFLW\LVYDOLGIRUERWKFRQYH[DQGFRQFDYH SURILOHVRIε]ZLWK1!

,9*URXSYHORFLW\RIHYDQHVFHQWZDYHVLQFRQFDYHEDUULHUV

,IWKHZDYHIUHTXHQF\LVOHVVWKDQWKHFXW±RIIIUHTXHQF\WKHUDGLDWLRQIOX[ZLOOEHWUDQVPLWWHGWKURXJK WKHILOPLQWKHWXQQHOLQJUHJLPH7KLVVLWXDWLRQFDQDULVHLQDWUDQVSDUHQWILOPGXHWRFRQFDYHSURILOH 8]HTV V ,QWURGXFLQJWKHQRWDWLRQV

S Q 1 1 X X

F ω

ω

= = = >

DQGSURFHHGLQJDVDERYHRQHFDQZULWHWKHILHOGFRPSRQHQWVIRUWKHHYDQHVFHQWZDYHV

( )

(L )

(

S S

)

[

( 5

( H 4H

4 8

η η

+

= +

+

( )

L S S

\

( 5 Q 1 ] ]

+ L 8 H \ 4H \

4 S/ / S/ /

η η

­ ª º ª º½

+ ° § · § · °

= ® « ¨ + ¸» « + ¨ ¸»¾

+ °¯ ¬ © ¹¼ ¬ © ¹¼°¿

(12)

3DUDPHWHU4FRQQHFWHGZLWKWKH³UHIOHFWLRQ´RIWKHHYDQHVFHQWZDYHRQWKHIDUVLGHRIILOPLV

( )

H[S

S Q 1 L

4

Q 1 L

η γ γ

§ ·

¨© + + ¸¹

= − −

,WLVUHPDUNDEOHWKDWWKHSUHVHQWDWLRQRI(0ILHOGWXQQHOLQJWKURXJKWKHKHWHURJHQHRXVILOPFDQEH IRXQGIURPWKHUHOHYDQWIRUPXODHIRUWKHWUDYHOLQJZDYHWKURXJKWKHIROORZLQJUHSODFHPHQWV

( ) ( ) ( ) ( )

FRV FK VLQ VK

T LS 1 L1

T η η S η η T η η L S η η

+

ª º ª º ª º ª º

¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼

7KXVHTDQGDUHPDSSHGWRHTDQGUHVSHFWLYHO\8VLQJWKLVUHSODFHPHQW RQHFDQZULWHHQHUJ\IOX[3]DQGGHQVLW\:LQWKHIRUPVVLPLODUWR±ZLWKWKHIXQFWLRQ

( )

( )

( )

( )

F

V

Q Q 1 \ ]

1 S/ /

Q 1

] ]

K S Q Q 1 \ \

1 S/ / S/ /

K S Q Q

1 S/

γ γ

θ

γ

γ γ

η η

γ γ

η η

ª § · º

ª º

+ « »

= + +« » ¨ ¸ +

« »

¬ ¼¬ © ¹ ¼

­ + ª ºª § · º § ·½

° « » °

+ + + + + +

ª º ® « » ¨ ¸ ¨ ¸¾

¬ ¼°¯ ¬ ¼«¬ © ¹ »¼ © ¹°¿

+ + +

ª º

¬ ¼

( )

] ]

1 \ Q 1 \

/ γ S/ /

­ ª º§ · ª § · º½

° + « + »°

® « »¨ ¸ ¨ ¸ ¾

« »

¬ ¼ © ¹ © ¹

° ¬ ¼°

¯ ¿

(13)

6XEVWLWXWLRQRIθLQVWHDGRIθLQWR\LHOGVWKHJURXSYHORFLW\YJRIWKHHYDQHVFHQWZDYH 7UDQVLWLRQIURPWKHWUDYHOLQJPRGHVω1WRWKHHYDQHVFHQWRQHVω1LV FRQWLQXRXVLQ1 ERWKIXQFWLRQVθDQGθKDYLQJLQWKLVSRLQWLGHQWLFDOYDOXHV

( )

( )

1 1

Q ] Q ] ] \

Q \ \ \ \

\ / / \ / / \

\ \ ]

/ /

θ θ

η η

η η γ

+ = = = =

­ ½

ª § · ° ª § · º § ·§ ·°

« + + ¨ ¸ ® « + +¨ ¸ »+¨ ¸¨ + ¸¾+

+ + +

« © ¹ ° « © ¹ » © ¹© ¹°

¬ ¯ ¬ ¼ ¿

­ ½º

° § · § ·°»

+ ®°¯ + +¨© ¸ ¨¹ © + ¸¹¾°¿¼»

0RUHRYHU WKH WUDQVLWLRQ EHWZHHQ WKH PRGHV LQ FRQFDYH DQG FRQYH[ EDUULHUV LQ LV SHUIRUPHG E\

FKDQJLQJWKHVLJQVRIVDQGVWRWKHRSSRVLWHRQHV)LQDOO\WKHWUDQVLWLRQWRWKHKRPRJHQHRXVILOP DULVLQJLQDOLPLW 1 8 η GLQERWKFDVHV1!DQG1EULQJVWKHYDOXHRIJURXS YHORFLW\LQDKRPRJHQHRXVILOPYJ YJR

J Y F

= Q

+

7KXV WKH H[SUHVVLRQV REWDLQHG IRUθ± SURYLGH WKH JHQHUDO IRUPXOD IRU W\SHV RI SKRWRQLF EDUULHUV FRQYH[1!FRQFDYH1!DQGFRQFDYHIRUHYDQHVFHQWZDYHV17RFDOFXODWHWKHYDOXHV RIYJRQHKDVWRGHILQHWKHYDOXHVRIYDULDEOHVη6XEVWLWXWLRQRILQWR\LHOGVIRUWKHFRQFDYH SURILOH

(14)

( )

( )

OQ

OQ

/ \ \ ] /

] \ \ \

\ \ ] /

\

/ \

G \ \

η

η η

+

±

+

+

§ + ·

= + ¨© ¸¹ = + ±

§ ·

= = ¨ ¸

+ © ¹

/LNHZLVHRQHILQGVWKHYDULDEOHηIRUWKHFRQYH[SURILOH

$UFWJ ð

$UFWJ ð

] / \

] / \

\ \] /

\ \

G /

\ \ I

I I

 ¬­

ž ­

žžžžŸ ­­­­®

 ¬­

ž ­

žžžžŸ ­­­­®

8VLQJDQGZHZLOOILQGWKHJURXSYHORFLWLHV7RYLVXDOL]HWKHHIIHFWVRIKHWHURJHQHLW\LWLV FRQYHQLHQWWRSUHVHQWWKHYDOXHVRIYJQRUPDOL]HGE\PHDQVRILWVYDOXHVLQKRPRJHQHRXVILOPV 9 YJYJ7KHVHJURXSYHORFLWLHVDUHSUHVHQWHGLQ)LJXUHIRUWKUHHW\SHVRIEDUULHUVZLWKWKLFNQHVVG DQGUHIUDFWLYHLQGH[Q DWDZDYHOHQJWKλ QPWUDQVSDUHQWFRQYH[G QP8P

WUDQVSDUHQWFRQFDYHG QP8P DQGWXQQHOLQJFRQFDYHG QP8P EDUULHUVIRUWKH GLIIHUHQW SRVLWLRQV LQVLGH WKH ILOP 1RWH WKDW WKH YDOXH9 DW WKH IDU VLGH RI WKH OD\HUV DULVH IRU WKH V\PPHWULFDOSURILOHV8 8G WKLVZRXOGQRWEHWKHFDVHIRUDV\PPHWULFDOSURILOHHJ8 8G 8P7KXVRQHFDQVHHWKDWWKHVHSKRWRQLFEDUULHUVIRUPKLJKO\GLVSHUVLYHILOPVWKHYHORFLW\RI HQHUJ\WUDQVIHUWKURXJKVXFKEDUULHUVFDQEHLQFUHDVHGRUGHFUHDVHGHVVHQWLDOO\GXHWRWKHε]SURILOH LQVLGH WKH EDUULHU 2QH QRWHV D UHPDUNDEOH VLPLODULW\ EHWZHHQ WKH JHQHUDO EHKDYLRXU RI WKH JURXS YHORFLWLHVIRUWKHWZRFRQFDYHSURILOHVGHVSLWHWKHIDFWWKDWRQHRIWKHPILJELVWUDQVSDUHQW1!

(15)

ZKLOHWKHRWKHUILJFLVRSDTXH17KH\ ERWKGLIIHUWRWDOO\IURPWKDWREVHUYHGLQWKHFRQYH[

FDVHILJD

93KDVHVKLIWRIZDYHVWXQQHOLQJDFRQFDYHGLHOHFWULFEDUULHU+DUWPDQHIIHFW 7KHJURXSGHOD\WJIRUZDYHVWUDYHUVLQJWKHILOPFDQEHIRXQGGLUHFWO\IURP

G

WJ 8 G]

F θ

=

³

7KH YDOXHV RIWJIRU W\SLFDO SDUDPHWHUV RI KHWHURJHQHRXV ILOPV GLVFXVVHG DERYH GR QRW H[FHHG WKH

³FDXVDOLW\ OLPLW´ W GF HJ WKH UDWLR WJW IRU WKH HYDQHVFHQW ZDYH ZLWKλ QP WXQQHOLQJ WKURXJKFRQFDYHEDUULHUIRUPHGE\WKHILOPZLWKWKLFNQHVVG QPDQG8P IRXQGIURP LV1RSKDVHYHORFLW\LV NQRZQWREH DWWULEXWHGWRWKHVHZDYHV+RZHYHUWKHUHLV DSKDVH VKLIW DFFXPXODWHGE\WKHVHZDYHVGXHWRWXQQHOLQJ6LGHE\VLGHZLWKGHOD\WLPHWJRQHFDQGHILQHWKH

³SKDVHWLPH´WSFRQQHFWHGZLWKWKHSKDVHVKLIWϕRIWXQQHOLQJZDYHV

WS ϕ ω

=

7KLV WLPHWS LV ZLGHO\ GLVFXVVHG LQ WKH OLWHUDWXUH GHYRWHG WR VXSHUOXPLQDO SURSDJDWLRQ RI HYDQHVFHQW ZDYHV WKURXJK WKLFN RSDTXH EDUULHUV NG!! ZKHUH WKH FDVHGWS!F ZDV DQDO\VHG DQG LQWHUSUHWHGDV³VXSHUOXPLQDO´SURSDJDWLRQ

8QOLNHWKHWUDGLWLRQDOVFKHPHVIRUREVHUYDWLRQRIRSWLFDOWXQQHOLQJEDVHGRQWKHLQFOLQHG LQFLGHQFH VXFKDVHJWKHIUXVWUDWHGWRWDOLQWHUQDOUHIOHFWLRQRU*RRV±+DQFKHQVKLIWZHZLOOH[DPLQHWKH DIRUHVDLGSKDVHHIIHFWIRUWKHVLPSOHJHRPHWU\RIQRUPDOLQFLGHQFHRIZDYHRQWKHFRQFDYHSKRWRQLF

(16)

EDUULHU XVLQJ WKH UHVXOWV RI 6HFWLRQ ,9 7KH TXDQWLW\ FDQ EH IRXQG IURP WKH FRPSOH[ WUDQVPLVVLRQ IXQFWLRQ7 VXFK WKDW(W (L7 IRU WKH HYDQHVFHQW ZDYHV ± 7KLV IXQFWLRQ GHSHQGV XSRQ WKH UHIOHFWLRQFRHIILFLHQW5ZKLFKFDQEHIRXQGIRUWKHVHZDYHVIURPWKHFRPSDULVRQRIHT±DW WKHLQWHUIDFH] HTEHLQJWDNHQLQWRDFFRXQW

( )

( ) ( )

WK ð

WK ð WK

S Q 1 Q 1

5 S Q 1 Q 1 L Q 1 S

η γ γ

η γ γ γ η

ª + + º

¬ ¼

= ª¬ º¼+ + ª¬ º¼

6XEVWLWXWLRQRILQWREULQJVWKHWUDQVPLVVLRQIXQFWLRQ7 _7_H[SLϕ

( )

{

( )

(

)

( )

}

WK ð WK

FK

7 Q 1 S Q 1 Q 1 Q 1 S

S η γ γ γ η

η

= ª¬ + º¼ +ª¬ º¼

( )

( )

( )

WK ð

$UFWJ

WK

S Q 1 Q 1

Q 1 S

η γ γ

ϕ γ η

ª + º

« »

= «¬ »¼

7KH³SKDVHWLPH´WSFDOFXODWHGE\PHDQVRIDQGLVWKHQ

( ) ( )

( )

( )

( )

( )

( )

FRV ð ð ð OQ

WK ð

WK FK ð

ð OQ

ð WK

FK ð

S

\ \

W S Q X Q 1

Q 1 S X1 S

\ \ Q X

Q 1 X WJ S

1 1 X1 S

ϕ η γ γ

ω γ η η

γ ϕ γ η γ

η

+

+

ª« § · § ·

= «« ¨© + ¸ ¨¹ ©− − ¸¹

¬

§ ·º

¨ ¸»

§ ·

¨ + ¸+ ¨ ¸»

© ¹ ¨¨© ¸¸»¹¼»

(17)

7KH SKDVH WLPH IRUλ QP FRUUHVSRQGLQJ WR WKH SURILOH XVHG IRU ILJ F IRU GLIIHUHQW WKLFNQHVVHVGWSVXSSRVLQJIRUDQHVWLPDWLRQWKHYHORFLW\YSWREHFRQVWDQWYS GWSRQHILQGV WKHVXEOXPLQDOYDOXHYS F

7RH[DPLQHWKHSKDVHVKLIWSURGXFHGE\PVLPLODUDGMDFHQWILOPVRQHKDVWRJHQHUDOL]HWKHWUDQVPLVVLRQ IXQFWLRQ7 ± IRU D VHW RIP DGMDFHQW SDUDOOHO OD\HUV WKH YDOXHP EHLQJ DUELWUDU\ 8VLQJ WKH FRQWLQXLW\ FRQGLWLRQV RQ HDFK ERXQGDU\ EHWZHHQ WZR DGMDFHQW OD\HUV RQH FDQ ILQG WKH ILHOG LQ HDFK OD\HU DWWULEXWLQJ WKH QXPEHUP IRU WKH OD\HU DW WKH IDU VLGH RI WKH VHW ZH ZLOO ILQG D VLPSOH UHFXUVLYHUHODWLRQIRUSDUDPHWHU4PIRUWKHPWKOD\HUP

4P H[S>PSη@4

7KHYDOXH4LVJLYHQLQ3URFHHGLQJDVDERYHZHZLOOEXLOGDJHQHUDOL]DWLRQRI7 \LHOGLQJWKH SKDVHRIWUDQVPLVVLRQIXQFWLRQIRUPOD\HUV

( )

( )

( )

WK ð

$UFWJ

WK

PS Q 1 Q 1

Q 1 PS

η γ γ

ϕ γ η

ª + º

« »

= «« »»

¬ ¼

$SORWRIWKHSKDVHWLPHDQGSKDVHVKLIWVDVDIXQFWLRQRIPLVVKRZQRQILJXUH7KHSKDVHLVJURZLQJ ZLWK WKH LQFUHDVH RI DPRXQW RI OD\HUV WKXV IRUP DQGP WKH YDOXHV DUH UDG DQG UDG UHVSHFWLYHO\+RZHYHUWKLVJURZWKLVGHFHOHUDWLQJZLWKWKHLQFUHDVHRIPWKHYDOXHVIRUP DQGP DUHUDGDQGUDGUHVSHFWLYHO\:KHQWKHDPRXQWRIOD\HUVLVJURZLQJVRWKDWPSη!!WKH SKDVHEHFRPHVLQGHSHQGHQWIURPP

Références

Documents relatifs

Les caractéristiques les plus intéressantes de cette approche résident dans l'utilisation de l'approche par scénarios qui aborde non seulement l’aspect séquentiel

La place de la faute dans le divorce ou le syndrome Lucky Luke Fierens, Jacques Published in: Droit de la famille Publication date: 2007 Document Version le PDF de l'éditeur Link

Furthermore, by using the interference technique combined with mass transport effect, we have demonstrated the fabrication of desired surface relief grating structures, with

Due to definition 4, M and P are respectively the first upper and the last lower leaning points of the same MS, assumed without loss of generality to have points of

In the first part of this thesis, apart from the previously used Z-scan method, we have also used the ultrafast Optical Kerr Effect method coupled to Optical Heterodyne De-

In the weakly resonant case the radiation is described by a transverse wave (dispersive) equation with moving source term localized where the main pulse is and which is responsible

7 La obra de On˜a ejercio´ gran influencia en Lope, pues el Fe´nix la utilizo´ como una de las fuentes principales de La Dragontea y, solamente un an˜o despue´s, como la base de

However, our results show that vortex splitting is accompanied by the generation of vortex-antivortex pairs, which wanders around the phase profile and are not distinguishable from