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rα

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 S(Rn ) = f ∈ C ∞ (Rn ) : D β f

β ∈ Nn0



T§ @MPNJsc>DOBHGNBESb\D9F>=,¬'J'SU;'F']Éc'BESŸ”$PVJa`Ÿ@5SU{E=,;'F'•$LUDOKMF BEFCJA@5? BEF‡` D9S¿{H`bBEDOBESUWOBEDuBESUSNBEDOPNJaL7>\SNGNLNBEF G¿•^˜MF'F BEF–` D9S GUDOPVJ BESxGNBED9FˆÆc @MGVGÉ=,¬'S n c>DOBC\)$DOGULNBEF'{ec'BES‡²žF‰LNBE]8SV@5WOBCD9F'F BESUJA@5W9?‹c'BES‡— KM;'SUDOBES¤|žˆSV@5F GU=-KMSUy‡@5LUDOKMF ]^BHGUDOPVJ BESULxDOGULƨGNKMWÀ@5F']^B kf ∈∈ RS(Rn) 7Ÿ³m`¨BED9LNBEF G–` DOGNGNBEFŒ` D9S‡FA@MPNJ ªg>6žÆ¿c @MGVG S(Rn) c>DOPNJ‰LsD9F L2 (Rn) DOGUL7)§­@MPVJ ªg8$g bDOGNL©c>DOB–— KM;'SUDOBES¤|žˆSV@5F GU=-KMSUy‡@5LUDOKMF‹BED9FÇ;'F'D9L 5SNBES‡€_ BESV@5LNKMS F : L2(Rn) → S(Rn) Æ)c'BES GNKMyzD9Ls@5;>= F : L2(Rn) → L2(Rn) BESU`¨BED9LNBESUL–`¨BESNc'BEFŒ•8@5F'Fˆ7¿™­DOBs=ÁKMW9]^BEF c'BC™TB¶ FaLUDOKMF c'BES‡— KM;'SUDOBES¤| ˆSV@5F GU=-KMSUy‡@5LUDOKMFÉd^KMy ¾BE•$LNKMS¤=ÁBEWOc u ∈ [L2 (Rn)]n DOGUL3@5WOGˆ•^KMyz KMF BEFaLNBEF‰`¨BEDOGNB<²žF‰LNBE]8SV@5LUDOKMF {H; d^BESNGULNBEJ BEFˆ7 %u>np  „ À.h -.  †ul‚ 4M+,!a4(q 1¿45 # (%žlA4M">n(+,l‚#   ?) u ∈ [L2(Rn)]n $'-P/0 A; *1&( .>?&<8Y0= :'& F : [L2(Rn)]n →

[L2 (Rn )]n

F S)9N'-$'17 0

ˆ (k) = Fu(r) = (2π)−n/2 u

Z

  .>?&<8Y0= :'& C * 60 N'-  ': >?)* *)*+ )gV'&<;RF4:')*[  *)/0I WY:'& @ N'-  P P*R7%N'>?&S$ )UV'&;RF4:')A[ VG0= :'& K


ˆ (k) u

Ág>7 5® 

dr u(r) e−ikr .

Rn




f

F : u → Fu



:G>S)* /)A+

™­DOB²&Fad^BESULUDOBESU;'F']_c'BES3— KM;'SUDOBES¤|žˆSV@5F G¤=ÁKMSUy‡@5LUDOKMFÉDOGULˆBE? EB F>=Á@5W9WOGu•^BED9F ISNKM?'WOBEyCÆ(c @Ÿc>DOB—AKM;'SUDOBES¤|žˆSV@5F G¤=ÁKMSUy‡@5LUDOKMF BED9F B‡œbD UBE•‰LUDOKMFd^KMF [L2(Rn)]n FA@MPNJ [L2(Rn)]n DOGNL7  EB D9LNBESUJ'D9FÇDOGUL[c>DOB‡:_?'?'D9WOc>;'F'] DOGVKMy›BELUSUDOGNPNJ ªg>6 ž7  D9Sš•^˜MF'F BEF GNKMyzD9L =ÁKMW9]^BEF c'BEFR”>@5LU{Z=ÁKMSUy[;'W9DOBESNBEFˆ  

5® 
























u>np „Àh.Àh  +,#‚!a45! †ulm345+,!‰4q 1¨4 #3%&lA4M">n(+,lm# %

[L2 (Rn )]n → [L2 (Rn )]n



u(r) = F

−1

7 V'> 0I/0 D  F4:'7 $L0

ˆ (k) = (2π) u

−n/2

Z

/)A&(/) $'17 0 N'  [j, #3t523!a45!‰jÀqHrjo!$+,t52u)# * hFf, FgiL2



 



 

 
 


   

 9 -#1&WT*)A;4  ': >?)* *)*+ )gV'&<;RF4:')*[ VG0==:G& 


ˆ (k) eikr . dk u

F −1 :

Ág>7 8~ 

Rn

∀f, g ∈ S(Rn ).

= hf, giL2

™­DOBɁ‚BESUDOK$c>DOGNPVJ B›Ì;'SNPVJkc>DOB  D9]8F BES¤|1”$BED9LU{E|1³‚BEW9WOB   ”'³ ­D9y IJ KMLNKMF'DOGNPVJ BEF–QTSUDOGNLV@5W9WAd^KMSU]^BE]^BE? BEFˆ7‰™­DOBHGNB_Ì;'{HDOBESULŸBED9F B<Ì_FaLNBESULNBED9W9;'F'] c'BES ¾BE•$LNKMS¤=,;'F'•$LUDOKMF u ∈ [L2(Rn)]n 7A”$KMyzD9L<]8D9W9L ˆ (k) = (2π) u

−n/2

= (2π)−n/2



Z

dr u(r) e−ikr Rn Z

X

dr u(r + R) e−ik(r+R) .

W SZ

9D S›]^BEJ BEF‹c @d^KMF 5@ ; G(Æ¿c @MGNG–— KMW9]^BEF‹D9F L2 ]8WOBEDOPNJ'y  8D9] •^KMFad^BESU]^BEF‰LCGUD9F cu7b”$KMyzD9Lx•^˜MF'F BEF“` D9Ssc>DOB ”‰;'yzy›BZ¬'? BES<@5W9WOB }­9D LULNBESUd^BE•$LNKMSNBEF R yzD9Lšc'BEy ²&FaLNBE]8SV@5W d^BESULV@5; GVPNJ BEFˆ7A\¿G ]8D9W9L R



ˆ (k) = (2π)−n/2 u = (2π)

−n/2

Z

Z

dr W SZ

X

u(r + R) e−ik(r+R)

R

dr W SZ

X |R

u(r + R) e−ikR e−ikr . {z

Ág>7 8µ 

}

:=F u(r,k)

—)¬'J'SNBEF ` D9S–c>DOBR]8WOBEDOPNJ BEFÌ_yÉ=ÁKMSUy[;'F']^BEF F K$PNJ BED9F'y‡@5WTyzD9Lsc'BESsD9F‰d^BESNGNBEF — KM;'SUDOBES¤|žˆSV@5F G¤=ÁKMSUy‡@5LUDOKMF c>;'SNPNJ ;'F cCBESUJA@5W9LNBEF −n/2

Z

ˆ (k) eikr dk u XZ −n/2 ˆ (k + G) ei(k+G)r = (2π) dk u

u(r) = (2π)

= (2π)−n/2

Rn

ZG

BZ

dk

BZ

= (2π)−n/2

Z

ˆ (k + G) ei(k+G)r u

G

dk BZ

X

X |G

ˆ (k + G) eiGr eikr . u {z

:=Fˆ u ˆ(k,r)

}

Ág>7 ^± 

'; F c Fˆ uˆ GUD9F c"c>DOB›—)WOK a; BEL¤|ž SV@5F G¤=ÁKMSUy‡@5LUDOKMF BEFÇD9y €_SULNGUSV@5;'y ;'F c D9y — KM;'SUDOBES¤| _@5;'yC7)™­DOB›™TB¶'| F'D9LUDOKMF c'BES­—)WOK a; BEL¤|ž SV@5F G¤=ÁKMSUy‡@5DOKMF D9y €_SULNGUSV@5;'y BEF‰LUF'D9yzy[LYy‡@5Ffc'BESZ£uD9LNBESV@5LU;'S 8¦$ž7m™­DOB©—)WOK ‰; BEL¤| ˆSV@5F GU=-KMSUy‡@5LUDOKMFÇD9y — KM;'SUDOBES¤| ­@5;'y DOGNL[F BE;ˆ7 —m@MGNGNBEFk` D9S ?‚BEDOc'BxˆSV@5F G¤=ÁKMSUy‡@5LUDOKMF BEF F K‰PVJBED9F'y‡@5Wb{H;>| GV@5yzy›BEFˆ 

Fu

 









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8~





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… !#)+Àn(+,l‚# „O„  † jol
ˆ3!^nEq 1¨45 # %&lA4M">n(+,lm#

ˆ (k) ∈ [L2 (Rn )]n r ∈ Rn u(r), u 
F

k

ˆ (k) u

u(r)

F u(r, k) :=

X

Ág>7 8ƒ 

u(r + R) e−ikR

Ág>7 8¹ 

R

Fˆ u ˆ(k, r) :=

NLV')A;*0I/7R7d*& K

X

ˆ (k + G) e+iGr u

G

à @5F •8@5F'F{BED9]^BEF 8¦$b=,¬'SÉGNPVJ'F BEW9W¨=¡@5W9W9WOBEF c'Bs—';'F'•$LUDOKMF BEF u(r) Əc @MGNGÉc>DOBs—)WOK ‰; BEL¤|žˆSV@5F GU=-KMSUy‡@5LUDOKMF R Ág>7 8ƒ –yzD9LCc'BEy BESUDOK‰c>DOGVPNJ BEF ™­D uBESNBEFaLUDÀ@5WOKMA@5SV@5LNKMS  >796g xd^BESULV@5; GNPNJ‰L7­”$KMyzD9LCJA@5? BEF ?‚BEDOc'Bk€_ BÂ| SV@5LNKMSNBEF c @MGR]8WOBEDOPNJ B\D9]^BEF GUÅ>GULNBEyC7­™Z@‘` D9S c>DOBǗ3WOK ‰; BEL¤|žˆSV@5F G¤=ÁKMSUy‡@5LUDOKMF¸@5WOGf BED9WzBED9F BESR—AKM;'SUDOBES¤| ˆSV@5F GU=-KMSUy‡@5LUDOKMF"¬'? BESZBED9F BEST‚BESUDOK$c>DOGNPNJ BEF”‰LUSU;'•$LU;'SZc @5SNGULNBEW9WOBEF •^˜MF'F BEF Ág>7 8µ ÂÆ DOGULTc @MGZ\D9]^BEF GUÅ>GULNBEy @5; G<\D9]^BEF>=,;'F'•$LUDOKMF BEF D9F [L2 (Rn)]n BEFaLUJA@5W9LNBEFˆ7 Ì= c>DOB‡\D9]^BEF GNPNJA@=-LNBEF c'BHG[²&FaLNBE]8SV@5F c'BEFÇd^KMF Ág>7 8µ ­;'F c Ág>7 ^± ZBED9F']^BEJ BEFk{H;•^˜MF'F BEFˆÆ =-¬'J'SNBEF"` D9S @5FA@5WOKM]C{H;'S[£ˆD9LNBESV@5LU;'S 8¦$¿c>DOBz{H;']^BEJ ˜MSNBEF c'BEF —';'F'•$LUDOKMF BEFk{H;'S[—)WOK a; BEL¤|ž SV@5F G¤=ÁKMSUy‡@5LUDOKMF BED9Fˆ  … ! #)+Àn(+,l‚# „ 9º  pa3*A!alm4Š#3!^n!"†ˆ)## " n(+,l‚# !$#    10 NM/& O<:')gV'>M;A;4/0324>S&?$M*& NM*) 7%/032Q0I*&   &10= :'&

$'17 0

φu(r, k) :=

X

Ág>7ªg8¦  Ág>7ªg>6!

uk (r + R) e−i(r+R)k

R

ˆu(k, r) := φˆ

X

ˆ (k + G, r) ei(k+G)r . u

G

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l ˆ !a2n 1¿45 #3%žlm45k>n(+ol‚#   -   $S*&<;/B/C?VUF/0I*& NM*) 7%:/>/0=+

)UV'&;RF4:')A[ VG0= :'& F?) ?> &bN ;A1&bNMH 1[  :'?> )* *)*+ V'>S[


k

r

0 ˆ (k − G0 , r) = Fˆ u ˆ (k, r) · eiG r Fˆ u ˆ (k, r + R), r) = Fˆ u ˆ (k, r) Fˆ u

1[ ` )/01;A+ V'>S[ F u(r, k + G) = F u(r, k) 0

F u(r − R0 , k) = F u(r, k) · e−iR k

    $S*&<;/B/C?VUF/0I*& NM/)24>'$Sg:')gN'&(/0I*&  >S&<8T0==:'& >S&bN NM*) A&0I $')gV'&bNS*&cWT:'&  K   >?&bN  K    F S) >?&bN *; R&(NMH r

k

8µ 











 








 



R[  ': >?)A-*)*+  VG>S[ 

 



 

 
 


   

1[ ` )*01;*+  VG>S[

ˆ (k − G0 , r) = φˆ u ˆ (k, r) φˆ u ˆ (k, r + R) = φˆ u ˆ (k, r) · eikR φˆ u

φ u(r, k + G) = φ u(r, k) · e−iGr

φ u(r − R0 , k) = φ u(r, k)

9D S •^˜MF'F EB Fe@5; Gš?'DOGUJ BESUD9]^BEFe\SU•^BEF'FaLUF'DOGNGVBEFec @MG<œbWOK$PNJ>|žŸJ BHKMSNBEy =,¬'S F>|ž™­D9y›BEF GUDOKMF BEF =-KMSUy '; W9DOBESNBEFˆ  1[23!alm4!$ „ 9º ,._ ! j,lutM2uq 1 2 !‰lA45!‰ +-#Ä#uq …  
gNM T;*>S&S$cNM*)   *)]*&0= V'7 $'7d*=B/CM>S&?$ I R[  VG7R7 &  @W*$'7 K ?K1J   EV'>S; P]*) NM*[ $MA;4V'[ 0I*&  C?:L0-:'&R;/B*C /&  )*R;*0-V'717 R$MV'&?2  8TV'&& NMV')A+ $MA;/0I*717 0D  /)UNM/& R&(NM*[

?1&+-5 :'&.  5 R&b0I $')* *)/0D1)UN K -  V') ;/0I*717 >?&S$ NS*) T;*>S&?$ K  ; $'R7 0 R;*0 *1&( PQP*17%N'>S&?$ [L2 (Rn )]n

Rn

f : [L2 (BZ)]n → [L2 (Rn )]n

ˆ (k, r) f u




= (2π)

−n/2

Z

Ág>7ªg 

dk Fˆ u ˆ(k, r) eikr .

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BZ

Fˆ u ˆ(k, r)

R

x

G

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Ek (r) =

X

ˆr (k + G) e+i(G+k)r . e

 D9STBESU•^BEF'F BEFkJ'DOBESYc'BEF"²žF‰LNBE]8SV@5F c'BEF"d^KMF Ág>7ªg ­ ` DOBHc'BES7u:_; GZc>DOBHGNBEy BESUy =-¬'S F K$PNJ c'BEFCIJA@MGVBEF>=Á@5•$LNKMS eikr @5? GUA@5W9LNBEFˆÆAGNKÉ]8D9W9L G

Ek (r) = ek (r) eik r

Ek (r)

•^˜MF'F BEF"` D9S

Ág>7ªg8g 

 



















 









^±



yzD9L|ž—';'F'•$LUDOKMF X



ek (r) :=

Ág>7ªg5® 

ˆr (k + G) e+iGr . e

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Divergenz

H

∇ eik r h(r) = eik r (∇ + ik) h(r) =: eik r ∇k h(r)

∇ × eik r h(r) = eik r (∇ + ik) × h(r) =: eik r ∇k × h(r)

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∇ eik r hk (r) = ∇ eik r · hk (r) + eik r · ∇ hk (r) = ik eik r · hk (r) + eik r · ∇ hk (r) = eik r (∇ + ik) hk (r).

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  eik r hx ∂y ik r    hk (r) = ∇ × e hy = ∂z eik r hz ∂x  iky hz − ikz hy = eik r ·  ikz hx − ikx hz ikx hy − iky hx = eik r · (∇ + ik) × hk (r)

 eik r hz − ∂z eik r hy eik r hx − ∂x eik r hz  eik r hy − ∂y eik r hx    ∂y hz − ∂ z hy  +  ∂z hx − ∂ x hz  ∂x hy − ∂ y hx

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⇔

Z

G

∇k ×

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Z h

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∇k = ∇ + ik

,® 796!

ω2 r ek · φ dV, c2

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(∇ + ik) × µ−1 r (∇ + ik) × ek · φ dV

=

i −1 −1 −1 ∇ × µ−1 ∇ × e + ∇ × µ ik × e + ik × µ ∇ × e + ik × µ ik × e · φ dV k k k k r r r r

=

G

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Z

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∇×

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∇×

µ−1 r

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ZG

µ−1 r ∇

× ek · ∇ × φ dV +

µ−1 r ik

Z

× ek · ∇ × φ dV +

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n × (µ−1 r ∇ × ek ) · φ dF, n × (µ−1 r ik × ek ) · φ dF

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G

∂G







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Z ∇ × φ dV + µ−1 µ−1 ∇ × e · k r r ik × ek · ∇ × φ dV + · · · G G Z Z −1 + ik × µr ∇ × ek · φ dV + ik × µ−1 r ik × ek · φ dV

=

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G



Z

Z

µ−1 r kn × ek · ∇ × φ dV + · · · Z Z 2 +iλ φ · kn × µ−1 ∇ × e dV − λ φ · kn × µ−1 k r r kn × ek dV G

µ−1 r ∇× G

ek · ∇ × φ dV + iλ

G

G

=

Z

G

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−1 φ · kn × µ−1 r kn × ek = −(µr kn × ek ) · (kn × φ)

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Z

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M := hkn × φ , µ−1 r kn × e k i

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Krot := h∇ × φ , µ−1 r ∇ × ek i KM

:= hr φ , ek i

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= 0.

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x = x1 + (x2 − x3 ) · ξ + (x3 − x1 ) · η y = y1 + (y2 − y3 ) · ξ + (y3 − y1 ) · η

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yzD9L

J=



x2 − x 3 x3 − x 1 y2 − y 3 y3 − y 1



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ˆ fˆ, ∇f = J −T ∇

f = J −T ˆf ,

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∇×u =

J ˆ ˆ, ∇×u |J|

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1 ˆ ˆ. ∇·v |J|

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ˆl1 = φˆ1 · ∇φˆ2 − φˆ2 · ∇ ˆl2 = φˆ2 · ∇φˆ3 − φˆ3 · ∇ ˆl3 = φˆ3 · ∇φˆ1 − φˆ3 · ∇

φˆ1 φˆ2 φˆ1

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—)¬'Sšc>DOBYÃf@5LUSUD ) ˆ = G



i,j=1,2,3

ˆ G

gˆij



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,

gˆij = ih∇ × ˆli , kn × ˆlj i − ihkn × ˆli , ∇ × ˆlj i.

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ij

ij

ij

ij



ω2 −1 ˆ −1 ˆ ˆ ˆ M a = 0. λ2 µ−1 M a − λ µ G a + µ K a + r K rot r r r c2

  




  






 



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ij



ij



−1 −1 |λ|2 µ−1 r M a − λ µr G a + µr Krot a +

ω2 r KM a = 0. c2

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Á~>796! c ™Z@MGGU•M@5WÀ@5SNBb— BEWOc u BEFaLNGU'SUDOPVJaL¿c @5? BED>c'BES){E|žQ_KMyz‚KMF BEF‰LNBšc'BESBEFaLNGU'SNBHPVJ BEF c'BEFx— BEWOc'BES@5; G  >796¹>Æ >7  7 ™­DOB<J'DOBES¨?‚BESNBHPVJ'F BELNBEFC£ˆ˜8GU;'F']^BEFsGUD9F c›d^KMW9WOGNL 5F c>D9]Éc>D9d^BESU]^BEF'{E=-SNBED‚;'F cxc>DOB­™­DOGU‚BESNGNDOKMF'SNBEWÀ@5LUDOKMF Ág>7ª¹  DOGUL ;'FaLNBES­c'BEy :_F GV@5LU{ Ág>7ȱ bW9D9F B@5S7 ÃeD9L|žŸJ BHKMSNBEy Ág>7ªg8g  X Á ~>7  u(r) = u ˆ(k + G) e , `¨KM? BED G c'BES©}­D9LULNBESUd^BE•$LNKMSY;'F c k c'BES  BEW9WOBEF‰d^BE•‰LNKMS DOGUL7uœŸBESNBHPNJ'F BEFÇ` D9SYc'BEFk£ @5'WÀ@MPHB‡d^KMy œbWOK‰PVJ>| −∆u(r) =

 ω 2

u(r),

r := (x, y, 0)T .

i (G+k) r

G

:_F GV@5LU{

∆u(r) = −

X G


k2 u ˆ + G2 u ˆ + 2k · G u ˆ ei(k+G)r .

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Á~>7ªg 

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”$BELU{BEFC` D9S Á ~>7 Ÿ ;'F c Á ~>7ªg bD9FC;'F GNBESšISNKM?'WOBEy Á ~>796!ŸBED9FC;'F cCBESUJA@5W9LNBEF X

Á~>7 ® 

 ω 2 X
k2 u ˆ + G2 u ˆ + 2k · G u ˆ ei(k+G)r = u ˆ(k + G) ei (G+k) r . c

­™ DOBT}TWOBEDOPNJa;'F']z]8D9W9L¨=-¬'S UBHc'BEFBED9F'{BEW9F BEFC”‰;'yzy‡@5F c'BEFˆ7  D9Sbd^BESUFA@MPNJ'W MGNGUD9]^BEFCc @5J BES c>DOBZ”‰;'yzy›B87  EB D·| LNBESUJ'D9Fc>D9d‰DOc>DOBESNBEF` D9S c>;'SNPVJ c>DOBT\)$ KMF BEFaLUDÀ@5W·=-;'F'•$LUDOKMFC;'F cC”‰•M@5WÀ@5S¤=-;'F'•‰LUDOKMF uˆ ;'F cBESUJA@5W9LNBEFCc>DOB ‰;A@ | c>SV@5LUDOGNPVJ B }­WOBEDOPNJ‰;'F']xD9F k G

G



k2 + 2k · G + G2 =

Á~>7ª~ 

 ω 2

\ G{BED9]8L¨GUDOPNJˆÆ^c @MGVGc>;'SNPNJ–c'BEF‡£ˆ˜8GU;'F']^GV@5F G¯@5LU{ Á~>7 c>DOB d^KMSUJ BESSNBED9F‡W9D9F B@5SNB<™­DOGU BESNGUDOKMF GUSNBEWÀ@5LUDOKMF UBELU{HL F'DOPNJ‰LZy›BEJ'STW9D9F B@5SZDOGUL7 ™­BESZœbWOK$PNJ>|ž:_F GV@5LU{zBESU`¨BED9LNBESUL[GNKMyzD9LZc'BEF £ˆ˜8GU;'F']^GNSV@5;'y ;'y BED9F BÉ:_F'{(@5J'Wd^KMF £ˆ˜8GU;'F']^BEFˆÆ8c>DOB @5; Gc'BES‚BESUDOK$c>DOGNPVJ BEF›ÌDOBHGNB d^BESUy›BED9FaLUW9DOPVJ Bɔ‰LUSU;'•$LU;'Sš]a@5SšF'DOPVJaL­B*)$DOGULUDOBESUL7 œbBESNBHPVJ'F BEF` D9S ¤BELU{HL 5J'F'W9DOPNJ;'F GVBESNBEy F‰;'y›BESUDOGNPNJ BEF“ISNKM?'WOBEy c @MG λ(ω, kn )|žISNKM?'WOBEy°@5WOGÉGU•8@5WÀ@5SNB }­WOBEDOPVJa;'F']Y@5;>=mBED9F BES)J KMy›KM]^BEF BEF›”‰LUSU;'•$LU;'S7M:<WOGVKš? BESNBHPNJ'F BEFÉ` D9S)c'BEF›”‰LUSNBHPN•$;'F']^G¤=¡@5•‰LNKMS =-¬'S3BED9F B ]^BE]^BE? BEF B—'SNB ‰; BEF'{ ω ;'F cYBED9F B¿]^BE]^BE?‚BEF B¨:<; GUSUDOPVJaLU;'F'] kn 7  D9SˆGNBELU{BEFÉ@5WOGNKŸ=-¬'Su;'F GNBESNBEF λ ∈BEW9CWOBEF‰d^BE•‰LNKMS =-KMW9]^BEF c'BHG­@5F k c





k = λ kn

yzD9L

kn =

k . kkk

:_; ^BESNc'BEy«]8D9W9L =-¬'Sšc'BEFR}­D9LULNBESUd^BE•$LNKMSšD9y«SNBE{HD9'SNKM•^BEF ­ @5;'y 

G = n 1 b1 + n 2 b2

Á~>7ªµ 

, n1,2 ∈ N , b1,2 ∈ R2 .

”$BELU{BEFC` D9SDOBHGšD9Fe}­WOBEDOPNJ‰;'F'] Á ~>7ª~ ŸBED9F

Á~>7ȱ 

 ω 2
λ2 + 2 · n1 kn · b1 + n2 kn · b2 λ + (n1 b1 + n2 b2 )2 − = 0. c

ÃeD9Lšw
yzD9L

r  p 2 p −q = − ∓ 2 2 p = 2kn (n1 b1 + n2 b2 ) q = −(n1 b1 + n2 b2 )2 +

 ω 2 c

WÀ@MGNGNBEFfGUDOPVJRc>DOB λ1,2 =-¬'S_]^BE]^BE? BEF B ω ;'F c n1,2 ? BESNBHPNJ'F BEFˆ7u™­DOB n1,2 ∈ Z ? BE{BEDOPNJ'F BEF"c>DOBz}­SN˜ ^B©;'F c c>DOB b1,2 c>DOBx}TBHKMy›BELUSUDOB–c'BHG[:_; GNGNPVJ'F'D9LULNG[c'BHG[;'F BEF c>W9DOPNJ BEFQTSUDOGULV@5W9WOGÆ3c'BESYd^KMFk;'F G[?‚BELUSV@MPVJaLNBEL©` D9SNc -:_?'?ˆ7 ~>79!6 Â7  DOB›GVPNJ KMF BESU` 5J'FaLÆ)•^KMyzy›BEF`¨BED9LNBESNB–y›˜M]8W9DOPNJ B–£ˆ˜8GU;'F']^BEF c>;'SNPNJ c'BEFœbWOK$PNJ>|ž:_F GV@5LU{‡J'D9F'{H;ˆ7³m;'S ¾BESNBED9F>=¡@MPNJa;'F'] c'BES \SU]^BE?'F'DOGNGNB?‚BELUSV@MPVJaLNBEF¢` D9SR@5; GNGNPVJ'W9DOB 8W9DOPNJ ‰;A@Mc>SV@5LUDOGNPNJ Bœ¨SUD9W9WOKM;'D9F>|1³‚KMF BEF @5;>= ‰;A@Mc>SV@5LUDOGNPNJ BEF}­D9LULNBESUFˆ7  BED9LNBESUJ'D9F“? BELUSV@MPNJ‰LNBEF ` D9S–@5; PNJ‹Fa;'Ssc>DOB \SU]^BE?'F'DOGNGNBe?‚BE{H¬']8W9DOPVJ‹c'BES  BEW·| WOBEFad^BE•$LNKMSNBEFkD9F x| šDOPNJ‰LU;'F'] 7  D9SZdM@5SUD9DOBESNBEFÇ@5WOGNKsFa;'S[c'BEF }­D9LULNBESUd^BE•‰LNKMS G c>;'SNPNJc>DOB


     

 





 




 



 






 

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(0, 1) n2

(−1, 1)

(0, 2)

(1, 1)

(2, 1)

(−1, 0)

(0, 0)

(1, 0)

(2, 0)

(−1, 0) (−1, −1) (0, −1) (1, −1) (2, −1)

n1

_: ?'?'D9WOc>;'F']–~>7968  w­KMy›KM]^BEF BES_IJ KMLNKMF'DOGVPNJ BESTQTSUDOGULV@5W9WˆyzD9LDOBZd^KMF c'BES  5@ J'Wu=-¬'S ;'F c n2 @5?'J5F']8L7

n1

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n1,2 = 0

n1 = [−1, 1]

0.5



1

0.5

n1,2 = 0 Γ

n1 = [−2, 2]

n1,2 = 0

n1 = [−1, 1] X <λ

Γ0

0

Γ

n2 = 0

1.5

a2 ω π·c

1

0

n2 = 0

1.5

a2 ω π·c

a2 ω π·c

1.5

1

0.5

n2 = 0 X <λ

Γ0

0

Γ

X <λ

Γ0

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n1 = [−2, 2] Γ Γ0 1.5 Γ X 0 Γ <(λ) X 1 <(λ)

1

k = q1,2 x 0.5

Γ

2.5

X Γ0 <(λ) 
  
   =(λ) <(λ)

2

4

n1,2 = [−2, 2]



2

=(λ)

0

Γ

k = q1,2 x

2.5 2

a ω 2π·c

a ω 2π·c

1.5

1

1

0.5

0.5

0

Γ

X <(λ)

Γ0

0 15

10

=(λ)

5

0

Γ

Γ0

X

< (λ )

< (λ )

6

X

1.5

=(λ)

0

Γ0

X

< (λ )

0

X

k = q1,2 x

0.5

Γ Γ0 X

< (λ )

Γ Γ0 X <(λ)



a ω 2π·c

a ω 2π·c

1.5

n2 = [−1, 1]

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ω(k)

 T UV587EKHTSFGM œ,/&€õ„&)9<31G{%'& P C'(SHSGA#1.09<(-5K#1%^;=u120Nd&)#^jŒõ5K90&)(<,s5K;,/.<9“DU(0oA~U90&)#{90&),=;/.,=WtWa&)(X#1GO203„öX&)C'(<,/.<9<,=NbT `XC1.ƒ=‡Š‰'i1…^‡k‹&)(<365K;=90&)#p*-,=(X%^,/&7vGA;=Dd&)#1%'& öX&)C'(<,/.<9<,=Nbï ó·C1.<94|K#1%'&a%'&_.}˜:C'#{90&)(0&_.<90&)#·š ^T>-5K#1%'&_.°NU5K#'#rW}5K#gig5K;/>. W(


RS=)M UQ79? UVKHRS=J5;KH=>5;M 4 TS=)B>F >: S5;= 















ω



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1 hH, ΘH Hi 2 hH, Hi H

 ω 2

⇒

c  ω 2 c

= =

⇒

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=

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è˜ ™^T +{š ”O&)9 P &)#p*-,=(S%^,/& WJ;/&),/203BC'#'Dr˜è™^T +{š–,=#˜è™^T=‡›š@&),=#gi1.4Gt&)(<365K;=90&)#p*-,=(’7vGA;=Dd&)#1%'&€`SC1.0.Ø5KDd&7:u'(“%^,/&°#1&)(
=

1 2hH, Hi

Z

dr

2 1 ω D(r) . (r) c

™


  
    !"#$%&'  ( )  *  

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0.8

0.7

0.6

ω 2πa c

0.5

0.4

0.3

0.2

0.1

Γ

X

–> (<,=;=;/GAC',=#^j¯ólGA#1& >-5K#1%'.<9<(
X



K

Γ



0.8

0.7

0.6

ω 2πa c

0.5

0.4

0.3

0.2

0.1

–> (<,=;=;/GAC',=#^j¯ólGA#1& `X]']',=;/%^C'#'D‚™^T†…^ï1xw,=(“.0&)31&)# %^,/&°]b&)(0&_203'#1&)90&)#egoU.@,=;/%îC'#{90&)#·šL%^,/& cdGA;=;/.094|K#1%^,=Dd&„>-5K#1%^;=u124Nd&„7:u'(q%^,/&”{9<(-5K#1%^;=u120Nd&÷˜v>@,=;/%IGA]l&)#·št]l&),X%'&)(L”{9<(@,=;/%C'#B90&)#·škiˆ%^,/&‚,=WYõ5K90&)(<,s5K;=]65K#1%?;=,/&)Dd&)#gT(<*’5K(<9
Γ

k

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Homo TM TE −2

∆ω

10

−3

10

−4

10

−5

10

1

2

3

4

5

 10   10   10

10

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|ω−ωa | ωa

∆ω =



ωa



ω

D

D

E

M

E



f

R

RG

=

E ∗ (r) · D(r)dr

GE

∗ (r)

· D(r)dr



f

.

G

G

G

I G

=

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G

= =

Z

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E∗ (r) · D(r)dr  e(r) eik r · e(r) eik r dr  e(r) · e(r) e−ik r eik r dr

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kn

ω

λ

kn

λ

X

kn

k = λ · kn

™Uð

  8    S "S &   8   8  @ 






      
         !  

0.5

0.5

λ∈R λ∈C

0.49

0.49

0.47

0.47

0.46

0.46 a2 ω π·c

0.48

a2 ω π·c

0.48

0.45

0.45

0.44

0.44

0.43

0.43

0.42

0.42

0.41

0.41

0.4

0.4 0


   X 
 

 0.1  0.2  
 

0.3

    ""   
 $#       "! 

0.68

0.68

0.66

0.66

0.64

0.64

0.62

0.62 a2 ω π·c

0.7

a2 ω π·c

0.7

0.6

0.6

0.58

0.58

0.56

0.56

0.54

0.54

0.52

λ∈R λ∈C

0.5

 M
    
  

0.52 0.5 0

0.25 0.5 0.75 1    


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X

M



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ω = [0.58, 0.6]













M



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M







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W}5K#÷&),=#1&‚%^(0&),/&_24N{,=Dd& ”{9<(’5K#1%^;=u124Nd&˜v`X]']gT ™^T†…dškT ôu'(‘%^,/&\]b&),/%'&)#a”{ø{WtWa&)9<(<,/&)M'C'#'NO90& C'#1% .0&)31&)#€*-,=(•,=W>’&)(0&),/243F%'&_.‘õ5K90&)(<,s5K;=]65K#1%'&_.&),=#1&–.0945K(J%^T ,/ý &-šô1*-&);/,/%^&_%'(<,/&)20(-3{9<%1C'5A#'.@Dd;=&),=#1#‚&ŠC'5K(0#1&X%tZ\ô1&)&)(<;/36%^5K]b;=&)909<&)(4#L|KDd,=Wû&UTdô•%^,/u'&)(\;/&)%1NO5A9<.(<,/&).0;/20&)3[NO9<.0(<24,/3B.0*@20315A&-2431ô1&)&)# ;/%}õ%'5K&_90.&)e(<,sC^5K7:;Î9JTòñUš&),=#q&);/&)NO9<(<,/.02031&_.’ô1&);/%iO%15A.–.0945K(-5K#1%^;=u120Nd& 5KC1.ŠT œJ5A.-%^C'(0243'Dd&)31&)#1%[]b&)9<(45KDd.<W}|K~U,=Dt31GA31&°ô1&);/%p5KW GA]b&)(0&)#LC'#1%qC'#{90&)(0&)# QX5K#1%q.<,=#1%q`X]']',=;/%^C'#'Dd.Ë7è&)3';/&)( ,=#aô1GA;=Dd&“%'&_.ˆô1&)3';/&)#1.\cdGA#L”{9’&)9<(45KDWt,=9•%'&)Wœ,s5KDU(45KWtW%'&_.Z\&)NO90GA(Ë7v&);/%'&_.•cd&)(-5K#1%^;=u120Nd& ˜v`X]']gT8>JT†™dš *-,/.02031&)#tõ5K90&)(<,s5K;ŸjbC'#1%+eC^7v9<]65K#1%F&)(’&)9<(45A243B9


X

M

K

M

y

λ(ω, kn )



™ý

  8    S "S &   8   8  @ 






   ""   
  !    
   M !              

λ∈R λ∈C

0.55

0.55

0.45

0.45

a2 ω π·c

0.5

a2 ω π·c

0.5

0.4

0.4

0.35

0.35

0.3

0.3

0.25

 
M   
 

0.25 0

 0.25  0.5  
 0.75 

      
  !    
   K !              

0.5

0.5 a2 ω π·c

0.55

a2 ω π·c

0.55

0.45

M

0.45

0.4

0.4

0.35

0.35

λ∈R λ∈C

0.3


K   
 

0.3 0

0.25 0.5 0.75 1    


`X]']',=;/%^C'#'D™^T ý ï -jŒ>’5K#1%'%^,s5KDU(45KWtWY%'&)(F”{9<(’&)(0&),/203%'&_.€õ5K90&kj (<,s5K;=]65K#1%'&_. 5K#r]b&),/%'&)#”{øOWtWa&)9<(<,/&)M'C'#'N{90&)#n˜v`S]']gTGA]b&)#gibC'#B90&)#·š°%15A.&);/&)N{9<(<,/.42031&aô1&);/% .<,/203r,=W.<945K(’&)(0&),/203 %'&_.-eC^7v9<]65K#1%'&_.S%15A.-&);/&)NO9<(<,/.02431&°ô1&);/%q,=W .0243B*@5A243p%^,/&);/&)N{9<(<,/.42031&)#põ5K90&)(<,s5K;b5KC^7v36|K;=9ŠT^”OGAWt,=9@*–&)(0%'&)#L7vu'(–%15A.@eC^7v9<]65K#1%‚31oA31&)(0&XC'#1%}7:u'(@%15A.@õ58j 90&)(<,s5K;=]65K#1%LN{;/&),=#1&)(0&Xô'(0& -{C1&)# P &)#LWaoADU;=,/243gT{œ,/&_.–,/.<9ˆ&),=#1&>’&)DU(-5K#1%^;=u124Nd& ,=# %^,/&_.0&)(X”{9<(’&),/%'&\(
K

M



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    !"#$%&'  ( )  *  
      
 
 R

−1

10

#  # 

−2

∆λ

10

C R C

−3

10

−4

10

−5

10

2

10

3

4

5

   10   10



10

Wt,=9–%'&)W­Wt,=9 ýUý …UüU™VX#B]b&)NU5K#'#^j X` ]']',=;/%^C'#'D+™^T†™^ïdxw,=(\]b&_.0203'(0&),=]b&)#‚%'&)#‚(0&);s5K9<,=cd&)#‚ô1&)3';/&)( 90&)#}Dd&)#6|K31&)(<90&)#[QX&k7v&)(0&)# P *@&)(<9 Tdxw,=(ˆ]b&)9<(45A203{90&)#[%15K]l&),l%^,/&XHXGA#Bcd&)(JT†™dški'%15A.0. %^,/&@Z\&)N{90GA(Ë7è&);/%'&)(\]b&)9<(45KDd.<W}|K~U,=D€5K]'NO;=,=#'Dd&)#gTdœ,/&_.‘NdGAWtW€9ˆ%^C'(0243}%'&)#a>@;/G{243^jŒ`S#1.45K9 P-P C1.0945K#1%'&UTUxf,=(‘]b&kj (0&_203'#1&)#LWt,=9ˆöX,=;Ÿ7v&“C'#1.0&)(0&)(ˆ#{C'Wa&)(<,/.02031&)#[õ„&)9<31G{%'&)#L%15A.ˆQ-jÎMb&)(<,/GO%^,/.02031&Xô1&);/% ] P *T TBœ,/&_.0& GA(<90.<Mb&)(<,/G{%^,/.42031&)#îô1&);/%'&)(‚*-,/&_%'&)(<31GA;/&)#f.<,/243?7vu'( Ë&_%'&qxw,=DU#1&)(Ëj¯”O&),=9 P j¯ól&);=;/&UT’œJ5A.acdGA;=;/.<94|K#1%^,=Dd&p&);/&)NO9<(<,Ÿj .02431&+ô6&);/%r&)(<36|K;=9°W}5K#gib*–&)#'#zW}5K#r&),=#1&)#z&)#B90.0M'(0&_2031&)#1%'&)#E\365A.0&)#^7è5KNO90GA(°Dd&)W}|K~p%'&)W>@;/G{243^jŒ`S#1.Ø5K9 P W€C';=,=9<M';=, P ,/&)(<9ŠTl\.-DU,=;=9SWt,=9“%'&)W xz&);=;/&)#Bcd&)NO90GA( ∆λ =



λa TM −C

TE −R







|λ−λa | λa

TM −R 

TE −C





M

ek (r)



hk (r)

k = λ · kn

E = ek (r) ei<(k)r e=(k)r .

`X#65K;/GAD€DU,=;=9’%^,/&_.–&)]b&)#^7è5K;=;/.\7:u'(@%15A.ˆW}5KDU#1&)9<,/.42031&Jô1&);/%T{xf,=(–*–&)(0%'&)#qC'#1.\3',/&)(-5K]l&)(’5KC^7—%15A.–&);/&)NO9<(<,/.02431& ô1&);/%}C'#1%}%15KWt,=9@5KC^7g%^,/& -‘jŒEGA;s5K(<,/.Ø5K9<,/GA#L]b&_.0243'(4|K#'Nd&)#gT{ô•u'(–&),=#1&)#‚NdGAWtM';/&kú^&)#L,=Dd&)#B*@&)(<9-&)(-5K#1%^;=u124Nd&‚&),=#1&tW}58ú^,=W}5K;/&}\,=#1%^(<,=#'DU9<,/&k7v&L%'&)(€&);/&)NO9<(0GAW}5KDU#1&)9<,/.02031&)# xz&);=;/& ˜v`X]']gT—™^TòñOi GA]b&)#·škTxf,=(°NdoA#'#1&)#÷5K;/.0G[&)(-5K#1%'.<9<(@(<,=;=;/GAC',=#^j¯ólGA#1&a%'&)Wxz&)D P *-,/.02431&)# 365K]b&)C'#a#1Nd% oA#'#1&)T·#gVXidW;s5A.45K.0&);=;/#a& *-eg,=oU(’.0C'”{9<#'(0Dd&_&)24# N{C'P #'CDd.<&)7è(Ë5K7Î5ANO.090.0GA&)(0#g&)#il%^,/& u']b&),=# (ˆ1%'C1&).4#‚.E5KC'C^#'7ˆNO%^9 ,/&+œ°3'|K,=Wt#65KM^C17vC'.–#']'D‚,/. ,=#C'Q“W ,/24E3BC'9
Γ→M

a

K K



K

Γ→K M

λ



kn

K



M

λ

K

™dñ

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ω = 0.418 



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`X]']',=;/%^C'#'D+>JT†™^ïB\;/&)N{9<(<,/.02431&Xô6&);/%t7vu'(ˆ%^,/&Sô'(0&-{C1&)# P 5KWû”{øOWtWa&)9<(<,/&)M'C'#'N{9 &)#B90.<M'(0&_2431&)#1% %'&)(`S]']',=;/%^C'#'Da˜è™^T ý škTUœJ&)(•Z\&)(<90&),=;=C'#'Dd.Ë7è5KNO90GA(]b&)9<(4|KDU9 ^T†ð +1iA.0GAWt,=9•]b&kž1#1%'&)#F.0,/203að +:¹%'&_.•&);/&)NO9<(<,/.02431&)# ô1&);/%'&_.S,=# %'&)(S.<945K(’&)9<(45KD ˜vGA]b&)#gi']l&), š@,/.<9S&),=#pœ°5K(0.<90&);=;=C'#'Dd.<7v&)3';/&)(ŠT ω = 0.43 

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y = −0.4

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