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Submitted on 20 May 2014

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Kirigami Auxetic Pyramidal Core: MechanicalProperties and Wave Propagation Analysis in Damped

LatticeFabrizio Scarpa, Morvan Ouisse, Manuel Collet, K. Saito

To cite this version:Fabrizio Scarpa, Morvan Ouisse, Manuel Collet, K. Saito. Kirigami Auxetic Pyramidal Core: Mechani-cal Properties and Wave Propagation Analysis in Damped Lattice. Journal of Vibration and Acoustics,American Society of Mechanical Engineers, 2013, 135, pp.041001-1 - 041001-11. 10.1115/1.4024433.hal-00993364

r♠ ①t ♣②r♠ ♦r ♠♥ ♣r♦♣rts ♥

♣r♦♣t♦♥ ♥②ss ♥ ♠♣ tt

r③♦ r♣∗

♥ ♦♠♣♦sts ♥tr ♦r ♥♥♦t♦♥ ♥ ♥

❯♥rst② ♦ rst♦ rst♦ ❯

♦r♥ ss ♥ ♦t†

♥sttt é♣rt♠♥t é♥q ♣♣qé

❯ s♥ç♦♥ r♥

③② t♦‡

♥sttt ♦ ♥str ♥ ❯♥rst② ♦ ♦②♦ ♣♥

strt

♦r srs t ♠♥tr♥ ♠♥ ♣r♦♣rts ♥ ♣r♦♣t♦♥ r

trsts ♦ ♣②r♠ tt ♠ ①t♥ ♥ ①t ♥t P♦ss♦♥s rt♦ ♦r

♦♥trr② t♦ s♠r tt tsst♦♥s ♣r♦ s♥ ♠t ♦rs t ♣②r♠ tt sr

♥ ts ♦r s ♠♥tr s♥ r♠ r♠ ♣s tt♥ ♣ttr♥ t♥q ♥

♣♣ t♦ r rt② ♦ tr♠♦st ♥ tr♠♦♣st ♦♠♣♦sts t♦ t ♣rtr

♦♠tr② rt tr♦ ts ♠♥tr♥ t♥q t r♠ ♣②r♠ tt s♦ ♥

♥rs♦♥ t♥ ♥♣♥ ♥ ♦t♦♣♥ ♠♥ ♣r♦♣rts ♦♠♣r t♦ ss ♦♥②♦♠

♦♥rt♦♥s ♦♥ ♥t ♣♣r♦①♠t♦♥s r s t♦ t t s♦♥ss rs s♦♥

♥s ③r♦rtr ♣♦♥♦♥ ♣r♦♣rts ♥ t tr♥srs ♣♥ ♥♦ ♣r♦♣t♦♥

t♥q s ♦♥ ♦ s ♦r ♠♣ strtrs s s♦ ♣♣ t♦ t t s♣rs♦♥

♦r ♦ ♦♠♣♦st r♣♦①② tts t ♥tr♥s ②strt ♦ss ♣r♦♣

t♦♥ ♥②ss s♦s ♥s♥ rtt② t r♥t rq♥② ♥ts ♥ ♦♠♣① ♠♦

♦r t♦ ♥s ♦r♠t♦♥ ♠♥s♠ ♦ t tt

∗tr♦♥ rss sr♣rst♦†tr♦♥ rss ♠♦r♥♦ss♥♦♠tr‡tr♦♥ rss st♦st♦②♦♣

❯

tt ♣②r♠ ♦rs ♥ ♦♣ ①t♥s② r♥ t st ②rs ②

rt ♦ ♥♦ ♠t st♥ t♥qs tt ♦ ♣r♦♥ t rt② ♦ ♦st r

♥trt s♥ ♣♥s ❬ ❪ ♥ ♦ t ♠♦r trs ♦ ts tt ♦rs s t

s♣ str♥t t♦ r ♥ ♥♣♥ st♥ss ♥ s t♦ ♦♣ ♥♦

♥r② s♦r♥ ♥trt r strtrs ♦r stt ♥ ②♥♠ ♦♥ ♥ tr♠

♠♥♠♥t ❬ ❪

♣②r♠ tt ♦♥rt♦♥ ♥ tsst ♥ r ②♦t r

s ♣rtr tsst♦♥ s ①t ♥ t ♣♥ ①ts ♥ts r ss ♦ s♦s

♥ strtrs ①t♥ ♥t P♦ss♦♥s rt♦ t s♦ ①♣♥s rtr t♥ ♦♥trts

♥ ♣ ♦♥ ♦♥ rt♦♥ ①ts ❬❪ ♥ ①t♥s② st ♦r tr ♠

t♥t♦♥ ♣r♦r♠♥ ♥ ♣rtr ♦r tr t♥t② ♥ ♥♣ ♦r ❬❪ ♥

r♦st t♦r♥ ♥ ♠♦r♣♥ strtrs ❬ ❪ r tsst♦♥ s ♥ ♥

t ② r♠ t s ♣♦t♥t ♦r♠t♦♥ ♠♥s♠ ♣r♦♥ ①tt② ♥ ♣♦②♠r

♥t♦rs t r♦♠ ①❬❪r♥ ♦s ❬❪ s♠ tsst♦♥ s ♥ ♥t s

①st♥ ♥ ②♣♣rs t s♥ ♥ ♠t r♦♥ ♥♥♦ts ①t♥ s♥

♥rs♦♥ r♦♠ ♣♦st t♦ ♥t P♦ss♦♥s rt♦ ♥ st t♦ ♠♥ ♦♥

❬ ❪

r♠♥s♦♥ ♦♠♣① r s♦s ♥ s♦ ♣r♦ s♥ ♦tr ♠trs

♥ ♣rtr tr♠♦st ♥ tr♠♦♣st ♦♠♣♦sts ② ♠♥s ♦ r♠ t♥qs

r♠ s ♥ ♥♥t ♣♥s rt ♦ ♦♠♥♥ ♣②♦♥ t♥qs r♠ t t

t♥ ♣ttr♥s t♦ ♦t♥ ♥r tsst ♥ ♦♠♣① strtrs r♠ s ♥ s

sss② t♦ ♣r♦ ♦♥②♦♠s t ♥r s♣s s♥ r♦♥ ♥ r ♦♥

rs ❬❪ ①t r♥tr♥t strtrs ❬❪ ♥ ♥♦ ♠♦r♣♥ ♥♦① ♦♥rt♦♥s

❬❪ r s♦s ♦ r♠ rs♦♥ ♦ t ♣②r♠ tt ♥ rt s♥

st♥r r ♣♦①② ♣r♣r♣s ① ♦♠♣♦sts t ❯ sts t

t ♦ ♥ r♠ ♣ttr♥ r t r♦♠ s♥ ♣② ♦ t ♣r♣r ② ♥ t♦♠t ♣②

ttr r sts t ♣♦②t②♥ rs ♠ r ♦ ♥ ♦r s♣s ②

s♥ sqr r♦s ① ① ❬♠♠❪ ♠♥♠ s r r♦s r ♣t

♦♥ t r♦ss ♣♦♥ts ♦ r r♦♠ t ♣♣r s ♥ ♦tr r♦s r ♣ ♦♥

r ♦♥s r♦♠ t s ♦ r tr ♣♥ t♦tr ts r♦s ♥ r

t st s ① ♥ t s♣ ♦ t ①t ♣②r♠ tt ss♠② s ②

t♣s ♥ ♥srt ♥t♦ ♠ rs rr♥♠♥t s ♣t ♥ ♥

t♦ ♦r ♠♥ts t Co t♦ r t ♣r♣r ♥ r s♠♣ s s♦♥ ♥

r

♦r sr ♥ ts ♣♣r s ♦♥r♥ t t t♦♥ ♦ t ♦♥ ♥t

♣r♦♣rts rst ♦rr ♦♠♦♥st♦♥ ♥ t t♦♠♥s♦♥ ♣r♦♣t♦♥ ♦ t

r♠ ♣②r♠ tt ♦♥ ♥t ♣♣r♦①♠t♦♥ s s t♦ r t t

strssstr♥ t♥s♦r ♦ t ♦♠♦♥s tt ♥ t♦ t t s♦♥ss rs r♦♠ t

♥ ♣r♦♠ ss♦t t♦ t rst♦s qt♦♥ ♥ r♦s ♣♥s rt♦♥

♥ t st② ♦ t s♦♥ss rs s t ♥tt♦♥ ♦ ♣♦ss ♥♦♠♦s ♦r

♦ tr rtr rtr ♦ t s♦♥ss sr s ♥ st s ♠trs t♦

♥t② sts ♥ ♣♦♥♦♥ ♠♥ r t sts r ♠sr s ♣r♦ ♥s ♦♥

t s♦♥ss sr ❬❪ ❩r♦s♦♥ss rs r tr♦r s②♠♣t♦♠ ♦ ♣♦ss s♦t♦♥s

rt t t♦r s ♦r♥t ♦♥ ♥s ♦rrs♣♦♥♥ t♦ ts ♥stt②

♣♦♥ts ♦ t st ♦ t t♦rs ♥♦ ♥ ①t♥s st② ♦ t ♦♠♦♥s

♠♥ ♣r♦♣rts rst ♦rr ♦♠♦♥st♦♥ ♥ s♦t♦♥s ♦ t rst♦ qt♦♥

♦ t ♣rtr ♣②r♠ tsst♦♥ s♦♥ ♥ r s ♥♦t ♥ ♣r♦r♠ ②t ❲

s♦ tt t tt s ①t ♥ t ♣♥ t s♦ ♣♦ssss♥ ♣r ♠♥

♦r r t ♥♣♥ st♥ss s r t♥ t tr♥srs ♦♥ ♦♥trr② t♦ ss

♥trs②♠♠tr ♦♥②♦♠ ♦♥rt♦♥s ①st♥ ♥ ♦♣♥ trtr r♠ tt

♦s s♦ ♥ ③r♦ rtr s♦♥ss rs ♥ t tr♥srs ♣♥ ♣♥♥ ♦♥ t

♦♠tr ♣r♠trs ♥♥ t tt ♦r♦r ♦♥sr s♦ t s s ♥ t

r ♥♥r♥ ♣♣t♦♥ r ②strt ♠♣♥ s ♣rs♥t ♥ t s②st♠ t♦

t ♥tr♥s ♦ss t♦r ♦ t ♣r♣r s t♦ ♠♥tr ts tts ♥♦

♣r♦♣t♦♥ t♥q s ♣rs♥t ♥ ts ♦r t♦ ♦♥sr t t ♦ t ♠♣♥ ♥ t

tt ♥ ♥t② t rtt② ♦ ♥s♥ ♠♦s ♣rs♥t t r♥t rq♥s

❲ t ♦♥ ♥t ♣♣r♦①♠t♦♥ ♦s ♥♦t r ♥② ♥s ♣r♦♣t♦♥

♣ttr♥ ♥ t ♣♥ ♦ t tt t ♠♣ ♣r♦♣t♦♥ t♥qs t

t ♦♠♣① ♠♦ ♦r ①st♥ t♥ t ♠♥ts ♦ t ♥t strtr ♥

t ♥s♥ rtt② t♦ s♣ ♠♦s ss♣t♥ ♠♦r ♥r② t♥ t ♦trs

r Pr♦t♦♥ ♦ ♣②r♠ tt ♦r s♥ ♣r♦rt♦♥r♦♥♦♥ t♥q

♦♥ ♠ts r♦♠ ❬❪

r t♣s ♦ t ♠♥tr♥ ♦r t r♠ ①t ♣②r♠ tt ♦r

♥ ♦♠♦♥st♦♥

q♥t ♠♥ ♣r♦♣rts ♦ t r♠ tt strtr ♥ ♦♠

♣t s♥ ♥t♠♥t s ♦♠♦♥st♦♥ t♥qs ♦r ♣r♦ ♠ ❬❪ ♥

♣r♦ ♠♠ t ♥rs ♥♦ s♣♠♥t t♦r u ♥ ①♣rss s

u = ǫx+ u′ r ǫ s ♥ ♣♣ str♥ ♥ u′ s♦② tt♥ ♣r♦ ♥t♦♥

♦ u rt♦♥ tt♦♥ t♦ ♥t② u′ s ①♣rss ② ❬❪

v′T [K] u′ = v′T F

❲r v′ s ♥♦tr ♣r♦ tt♥ ♥t♦♥ s♦♥ qt♦♥ [K] s t

st♥ss ♠tr① ♦ t s②st♠ ♥ F s t♦r ♦ ♥rs ♥♦ ♦rs ♦r♥ t♦

st♥r t♥qs t st♥ss ♠tr① s ♥ s

[K] =∑

e

[k]e =1

V

ˆ

e

[B]T [C] [B] dV

♥ [B] ♦♥t♥s t s♣t rts ♦ t s♣ ♥t♦♥s [C] t s♦♥ ♦rr

strssstr♥ t♥s♦r ♦ t ♠tr ss♦t t♦ t eth ♠♥t ♥ V t ♦♠ ♦ t

♦r ♠♦ ❲♥ ♣rsr str♥ ǫ s ♠♣♦s t ♥rs ♦r t♦r F♥ ①♣rss s

F = −[

K]

ǫ

❲r

[

K]

=∑

e

[

k]

e=

1

V

ˆ

e

[B]T [C] dV

♥srt♥ ♥ ♥ ♦♥ ♦t♥s t ♦♠♦♥s strssstr♥ t♥s♦r rt♦♥s♣

r♣rs♥t♥ t q♥t ♠tr ♥♦s ♥ t ♦♠ V

[

C]

ǫ = σ

❲r σ s t ♦♠ r t♦r ♦ t strsss σij, i, j = 1 . . . 6 ♦r t r♣

rs♥tt ♥t ♦ t ♣r♦ ♠r♦strtr ♦♠♦♥s ♦♠♣♥ ♠tr①

[

S]

=[

C]−1

s q ♦r tr♥srs s♦tr♦♣ ♠tr s

[

S]

=

1Ex

−νyxEx

−νzxEx

0 0 0

−νxyEy

1Ey

−νzyEy

0 0 0

−νxzEz

−νyzEz

− 1Ez

0 0 0

0 0 0 − 1Gxz

0 0

0 0 0 0 1Gyz

0

0 0 0 0 0 1Gxy

r♣rs♥tt ♥t ♠♥t ❱ s ♦r t r♠ ♣②r♠ tt s

s♦♥ ♥ r ♣②r♠ tt s sr s♥ t ♥♦♥♠♥s♦♥ ♣r♠

trs α = a/b β = t/l δ = l/b ♥ ♥tr♥ ♥ θ ♥ s♦tr♦♣ ♥♣♥ ♠♥

♦♥rt♦♥ s ♦r α = 1 ♣r♠tr δ ♥ts ♦ r t ♦r ♦ t

♠♥t l ♣rts r♦♠ ♠ strtr ❲t♥ t ♦♥t①t ♦ ts ♦r ♦♥② ♥

♣♥ s♦tr♦♣ ♦♥rt♦♥s α = 1 ♥ ♦♥sr ♠♦ ♦ t ♥t

s ♥ ♦♣ s♥ t ♦♠♠r ♦ ❨ r r♠♥s♦♥

strtr ①r ♠♥ts t r♥♥ ♥tr♣♦t♦♥ ♥t♦♥s ♥♦s

♥ s ♥ s t♦ r♣rs♥t t ♣②r♠ strtr t ♥♦r♠ ♠s s③

q♥t t♦ t/2 ss♠ tr ♠s ♦♥r♥ tst ♣r♦ ♦♥r② ♦♥t♦♥s

❬❪ ♥ ♣♣ t♦ t s xy ♦♥ t ♥t t ♠♥♠♠ ♥ ♠①♠♠ z

♦♦r♥ts ♥ ♣t r ♦ ♠♣♦s s♣♠♥ts ♦rrs♣♦♥♥ t♦ ♥♦r♠ str♥s

s t♦ r♣rs♥t ♥ ♥♥t ♣r♦ ♦r ♠tr ♦r s♥ strtrs ♥ tr♦r

♣r♦t② ♥ t xy♣♥ ♦♥② ♥t str♥s t t ♦♥r②

ǫx ǫy ǫz γxz γyz γxy

T

♥ ♠♣♦s s s♣♠♥t s ♥ t♦ stt ♥r ♣r♦♠ s♦ t

t♦♥♣s♦♥ s♦r ♣♥♥ ♦♥ t s♣ ♦♠♥t♦♥ ♦ ♥♦♥♠♥s♦♥ ♦♠

tr② ♣r♠trs s t ♠♦s s③s r♥♥ t♥ t♦ s s

♠♥ ♣r♦♣rts ♦ t ♦r r r♣rs♥tt ♦ r ♦♥ r ❬❪

t ❨♦♥s ♠♦s Ec = 30GPa P♦ss♦♥s rt♦ νc = 0.4 ♥ ♥st② ρc = 1600 kgm−3

♦r t s♠t♦♥s t ♥t ♦ t rs l s ss♠ ♦♥st♥t t ♠♠ t

rt ♥st② β s ♠♣♦s q t♦ t♦ ♦ sr ♦r♠t♦♥ ♦♥trt♦♥s r♦♠

t r♦ssst♦♥ ♦ t rs ♥ ss♠ t♥ ♠ ♦r ♣t strtrs ♥ ♣rs♥t

♥ t r♠ tt ♦♥sst♥t t♦ s♠♣s ♣r♦ t t ♠♥tr♥ ♣r♦ss

s♦♥ ♥ r

rst♦s qt♦♥s

s♦♥ss rs ♥ s♠t r♦♠ t s♦t♦♥ ♦ rst♦s qt♦♥s ♦r

♥ k ♣r♦♣t♦♥ rt♦♥ t qt♦♥s t ♦rrs♣♦♥♥ ♥ ♣r♦♠

❬ ❪

[Γ] A = γ A

❲r Γ = kicijklkl ♥ γ = ρv2 cijkl ♥ts t ♦♠♦♥s ♦rt♦rr strss

str♥ t♥s♦r ♦ t q♥t ♠tr ♦ t ♣②r♠ ①t tt tr♠♥♥t

♦ qt t♦ ③r♦ ②s tr ♥s ♥ tr♠s ♦ ρv2 (k) ♥ t♦ tr s♦♥ss

rs s (k) = 1/v (k) ss♦t t♦ |k| = 1 ❬❪ ♦♥sr♥ t t♦r k ♣r♦♣t♥

♥ t xz ♣♥ t k = cos (φ)k+sin (φ) i i,k ♥ t ♥t rs♦rs ♦ t x ♥ z ①s

rs♣t② t ♠tr① Γ♥ ♥ rst ♦r t tr♥srs s♦tr♦♣ ♠r♦strtr s

❬❪

[Γ] =

c55 + (c11 − c55) sin2 φ 0 (c13 + c55) cosφ sinφ

0 c44 + (c66 − c44) sin2 φ 0

(c13 + c55) cosφ sinφ 0 c33 + (c55 − c33) sin2 φ

❲♥ t st ♦♥st♥ts r ♥♦r♠s ♥st c11/√ρ ρ ♥ t ♥st② ♦ t

♦♠♦♥s ♠tr t tr ♥s γi t s♦♥ss rs si (φ) = v−1i (φ) =

γ−1/2i (φ) ♥ t xz ♣♥

♦r ♣r♦♣t♦♥ ♥ t xy ♣♥ k = cos (φ) j+ sin (φ) i t i, j t ♥t

rs♦rs ♦♥ t x ♥ y rt♦♥s rs♣t② t Γ ♠tr① ss♠s t ♦♦♥ ♦r♠

❬❪

[Γ] =

c11 cos2 φ+ c66 sin

2 φ (c12 + c66) cosφ sinφ 0

(c12 + c66) cosφ sinφ c66 cos2 φ+ c22 sin

2 φ 0

0 0 c55 cos2 φ+ c44 sin

2 φ

s♦♥ss rs si (φ) = γ−1/2i (φ) ♥ s♦ t ♥♦r♠s♥ t ♦♥ts

♥ ② c11/√ρ ♥s ♦ ♥ t s♥ t tt r♦t♥

t t t♥s♦r ♦♥ts cij ♦t♥ r♦♠ t ♦♠♦♥st♦♥ ♣r♦r sr

♦

♦qt ♥ ♦ t♦r♠s ♦r st♦②♥♠ s♣rs♦♥ ♥②ss

t♠t r♠♦r

t s ♦♥sr ♥ ♥♥t ♣r♦ st♦②♥♠ ♣r♦♠ s ♣rs♥t ♥ r

r♠♦♥ ♦♠♦♥♦s ②♥♠ qr♠ ♦ s②st♠ s r♥ ② t ♦♦♥ ♣rt

rt qt♦♥

ρ(x)ω2w(x) +∇C(x)∇sym(w(x)) = 0 ∀x ∈ R3

r w(x) ♥ R3 s t s♣♠♥t t♦r C(x) st♥s ♦r t ♦♦ stt② t♥s♦r

♥ ε(x) = ∇sym(w(x)) = 12(∇wT (x) + w(x)∇T ) s t str♥ t♥s♦r ② ♦♥sr♥

♣r♠t ♦ t ♣r♦ ♣r♦♠ ΩR ♥ ② s♥ t ♦ t♦r♠ t ss♦t

♦ ♥♠♦s ♥ t s♣rs♦♥ ♥t♦♥s ♥ ♦♥ ② sr♥ t ♥ s♦t♦♥s

♦ t ♦♠♦♥♦s ♣r♦♠

w(x) = wn,k(x,k)eik.x

r wn,k(x,k) r ΩR♣r♦ ♥t♦♥s ♥ tt s wn,k(x,k) ♥ ωn(k) r t

s♦t♦♥s ♦ t ♦♦♥ ♥r③ ♥s ♣r♦♠ ♦ t st ♦♣rt♦r s

sr ♥

ρ(x)ωn(k)2wn,k(x) +∇C(x)∇sym(wn,k(x))

−iC(x)∇sym(wn,k(x)).k − i∇C(x)12(wn,k(x).k

T + k.wTn,k(x))

+C(x)12(wn,k(x).k

T + k.wTn,k(x)).k = 0 ∀x ∈ ΩR

wn,k(x−R.n)−wn,k(x) = 0 ∀x ∈ ΓR

t s ♦♥sr t ♣rt rt qt♦♥s ♦♥ ♥t ΩR t ①♣rss♦♥

st♥s ♦r ♥r③ ♥ ♣r♦♠ ♥ t♦ ♦♠♣tt♦♥ ♦ t s♣rs♦♥ rs

ωn(k) ♥ ♦rrs♣♦♥♥ ♦qt ♥t♦rs wn,k(x) ♦r ♣♣t♦♥s t

t♦rs r s♣♣♦s t♦ ♦♠♣① ♠♣♥ tr♠s r ♥t♦ qt♦♥ ♥ ♥

tr♦r rtt♥ s k = k

sin(θ) cos(φ)

sin(θ) sin(φ)

cos(θ)

= kΦ r θ ♥ φ r♣rs♥t t rt♦♥

♥s ♥t♦ t r♣r♦ tt ♦♠♥ ♥ Φ s t rt♦♥ t♦r s ♦♠♣♦st♦♥

ss♠s tt r ♥ ♠♥r② ♣rts ♦ t t♦r k r ♦♥r

tr t♥ ♥t♦ ♦♥srt♦♥ t ♣r♦t② ♦♥t♦♥s ♥ rr♥♠♥ts

♦r♠t♦♥ ♦ t ♣r♦♠ ♥ rtt♥ s

∀wn,k(x) ∈ H1(ΩR,C3)/wn,k(x−Rn) = wn,k(x) ∀x ∈ ΓR ,

´

ΩRρ(x)ω2

n(k)wn,k(x)wn,k(x)− εn,k(x)C(x)εn,k(x)

+ikκn,k(x)C(x), εn,k(x)− ikεn,k(x)C(x)κn,k(x)

+k2κn,k(x)C(x)κn,k(x)dΩ = 0

♥♠r ♠♣♠♥tt♦♥ s ♦t♥ ② s♥ st♥r ♥t ♠♥ts ♠t♦ t♦

srts t ♦r♠t♦♥ ss♠ ♠tr① qt♦♥ s ♥ ②

(K + λL(Φ)− λ2H(Φ)− ω2n(λ, (Φ)M )wn,k(Φ) = 0

r λ = ik M ♥ K r t st♥r s②♠♠tr ♥t ♠ss ♥ s②♠♠tr s♠

♥t st♥ss ♠trs rs♣t② L s ss②♠♠tr ♠tr① ♥ H s s②♠♠tr

s♠♥t ♣♦st ♠tr①

M →ˆ

ΩR

ρ(x)ω2n(k)wn,k(x)wn,k(x)dΩ,

K →ˆ

ΩR

εn,k(x)C(x)εn,k(x)dΩ,

L →ˆ

ΩR

−κn,k(x)C(x)εn,k(x) + εn,k(x)C(x)κn,k(x)dΩ,

H →ˆ

ΩR

κn,k(x)C(x)κn,k(x)dΩ

❲♥ k ♥ Φ r ① t s②st♠ s ♥r ♥ ♣r♦♠ t s♦t♦♥ ♦

♦s ♦♠♣t♥ t s♣rs♦♥ ♥t♦♥s ω2n(k,Φ) ♥ t ss♦t ♦ ♥

t♦r wn,k(Φ)

s ♣♣r♦ s ss② s ♦r ♦♠♦♥③t♦♥ s♣tr s②♠♣t♦t ♥②ss ❬❪

♥ t ♦♥t①t ♦ ♣r♦♣t♦♥ s♣rs♦♥ rtrsts ♥ r ♥ ♦qt

♣r♦♣t♦rs r ♣rrr ♦r ♦r qs ♦♠♣tt♦♥ s ♥t ② ❬❪ ❬❪ ♦r

❬❪ ♦st ♦ t t♠ ts t♥qs r ♣♣ ♦r ♥♠♣ ♠♥ s②st♠s

r♣rs♥t ② st ♦ ♦♥st♥t ♥ r ♠trs ss ♣♣r♦ ♦♥ssts ♥ s♥

♠s ♦ t r ks♣ k ♦r λ ♥ Φ ♥s t rst r♦♥ ③♦♥ ♦r ♦t♥♥

t ♦rrs♣♦♥♥ rq♥② s♣rs♦♥ r♠s ♥ t ss♦t ♦qt t♦rs ♦r

♥♠♣ s②st♠s ♦♥② ♣r♦♣t♥ ♦r ♥s♥t s ①st ♦rrs♣♦♥♥ t♦ ♠s ♦

♥s♦t♦♥s ♣r② r ♦r ♠♥r② sr♠♥t♦♥ t♥ ss ♦ s s qt

strt♦rr ♠♣ s②st♠ s ♦♥sr ♠trs K, L, H r ♦♠♣①

t ♥s♥t ♣rt ♦ t ♣r♦♣t♥ s ♣♣r s t ♠♥r② ♣rt ♦ ω2n(λ,Φ) ♥

rs t t♥ ♦♠s r② t t♦ st♥s t t♦ ♠s ♦ s t s♦ t♦

♦♠♣t t ♦rrs♣♦♥♥ ♣②s s ♠♦♠♥ts ② ♣♣②♥ s♣t ♦♥♦t♦♥

t s ♣♦ss t♦ rrr♥ t s②st♠ ♥ ♦rr t♦ t ♥t♦ ♦♥t ♠♣♥ ts ❬❪

s ♦r♠t♦♥ s♦ ♣r♦s st ♦♥t♥ts ♦r t♠s♣ ♦♥♦t♦♥ ♥ ♦r t

♦♠♣tt♦♥ ♦ s♦♥ ♣r♦♣rts s ♥ ② ❬❪ ♦r ❬❪ t ♦♥ssts ♥ ♦♥sr♥ t

♦♦♥ ♥rs ♥ ♣r♦♠

(K − ω2M ) + λn(ω,Φ)L(Φ)− λ2n(ω,Φ)H(Φ))wn,k(Φ) = 0

♥ ts ♣r♦♠ t ♣st♦♥ ω ♥ t ♣r♦♣t ♥ Φ r ① r ♣r♠trs

❲s ♥♠rs λn = ikn ♥ ss♦t ♦qt t♦rs wn,k r t♥ ♦♠♣t ② s♦♥

t qrt ♥ ♣r♦♠ s ♣♣r♦ ♦s t♦ ♥tr♦ rq♥②♣♥♥t

♠trs ♦rrs♣♦♥♥ t♦ ♥r③ ♠♣♥ tr♠s s♦stt② ♠t♣②ss ♦♣♥

tr♦♠♥ t tr♦♥ ♦r♥r② r♥t qt♦♥ ♦♠ ♦tr ♠♦

♦r ♦♣♥ ♦♠♥ ♦♥r② ♦♥t♦♥s ♦♠♠r ♦♥t♦♥ s ♦♥ ts ♣♣r♦ ♥

♥rs ♦rr tr♥s♦r♠t♦♥ ♥ t ks♣ ♦♠♥ s t♦ t t ♣②s s

s♣♠♥ts ♥ ♥r② s♦♥ ♦♣rt♦r ♥ t ♣r♦ strt♦♥ s ♦♥♥t t♦

♥♦tr s②st♠ ❬❪ ♥♦tr t♠♣♦r ♥rs ♦rr tr♥s♦r♠t♦♥ ♥ ♣r♦ ②

t♦ ss s♣t♠♣♦r rs♣♦♥s ♦r ♥♦♥♦♠♦♥♦s ♥t ♦♥t♦♥s s L s s

s②♠♠tr t rst♥ ♥s r qr♣ λ, λ,−λ,−λ ♥ ♦♣s♥ ♥t♦ r

♦r ♠♥r② ♣rs ♦r s♥ ③r♦ ♥ ♠trs r r ♦r ♥♠♣ s②st♠s ♥

t ♥r ♠♣ s ♦♥ ♦t♥s ♣r ♦ r ♥s ♦rrs♣♦♥♥ t♦ ♥s♥t

♠♦s ♦r♥t ♥ t♦ ♦♣♣♦st rt♦♥s ♦♥ t ks♣ ♥ ♠♥r② s t♦ t♦

tr♥ s ♣r♦♣t♥ ♥ ♦♣♣♦st rt♦♥ ♥s♦t♦♥s r s♠r t♦ t♦s

♥ ② t ♠t♦ ♥ ♦r ♦♠♦♥♦s ♠tr ♥♦♥ ♣r♦ ♥ t♦♥

♠♣♦rt♥t ♣r♦♣rts ♥ ①tr♣♦t r♦♠ ❬❪ t② ♦ t ♠t♦ ♣r♦♣♦s

♥ ts ♦r s ♥ ♣r♦ t sr tst ss ♥ ❬❪

s ♣r♦s② ♠♥t♦♥ t r ♣rt ♦ k = kΦ t♦r s ss② rstrt t♦ t

rst r♦♥ ③♦♥ ♥ t qrt ♥ ♣r♦♠ t t② ♦ t ♦♠♣t

s♦t♦♥ s ♥♦ ♦♥r ♦♥♥ t♦ s♣ r♦♥ ③♦♥ ♦r rt♦♥ t♦rΦ ♦rt♦♦♥

t♦ t tt ts ♦r Φp1 = [1, 0]T ♥ Φp2 = [0, 1]T ♥ rt♥r t

♣r♦ ♦♥t♦♥s ①♣rss ♦r ♦♥ ♠♥s♦♥ r st λj(Φp) s

♥ ♥ ss♦t t♦ wj,k(Φp) t♥ ∀m ∈ Z3, λ + i.ΦTp (G.m) s s♦ ♥ ♥

ss♦t t♦ wj,k(Φp)e−i.ΦT

p (G.m)x s ♦r ♥♠♣ s②st♠s t ♥s r

♣r♦② strt ♥ t ks♣ ♦♥ ts ♣r♥♣ rt♦♥s

♦♠♣tt♦♥ ♦ ♥s♥ ♥ ♠♣ ♣♦r ♦ rtr

♥ ♠ ♦ ts ♦r s t♦ ♣r♦ ♥♠r ♠t♦♦♦② t♦ sr t ♣rtr

♦r ♦ t ♥r② ♦ ♥t♦ t ♣r♦② r♠ ①t ♦r ♥ tr♦r ♥s

t♦ ♥ st ♥t♦r ♦r st♥s♥ ♣r♦♣t ♥ ♥s♥t ♦r s♣②

♥ ♠♣ s②st♠ s ♦♥r♥ ♣t② ♦ ♥ ♦ t♦ tr♥s♣♦rt ♥r②

s ♥ ② ts r♦♣ ♦t② ♥ts ♦ ♥r② s tr♥s♣♦rt ♥t♦ t ♦♥sr

s②st♠ ♦♥ s♦ t♦ st♥s t♥ ♣r♦♣t ♥ ♥s♥t s ♦

♥ s♦t♦♥ un(ω, φ), kn(ω) s ♦♥sr t ss♦t r♦♣ ♦t② t♦r ❬❪

s ♥ ②

Cgn(ω, φ) = ∇kω =〈〈S〉〉〈〈etot〉〉

=〈I〉

〈Etot〉

❲r 〈〈:〉〉 s t s♣t ♥ t♠ r rs♣t② ♦♥ ♦♥ ♥ ♦♥ ♣r♦ ♦

t♠ s t ♥st② ♦ ♥r② ♦ I t ♠♥ ♥t♥st② ♥ etot Etot t t♦t ♥r② ♥

ts t♠ r ♦♥ ♣r♦ s ❭t④②s♥♦r⑥ ♦r ts

♥t♥st② t♦r I s ①♣rss s

〈In〉 = −ω

2Re

(ˆ

Ωx

C(εn(x) + ikΞn(x)).(w∗n(x))

dΩ

Vol

)

❲r .∗ s t ♦♠♣① ♦♥t Re st♥s ♦r r ♣rt ♥ Vol ♦r t ♦

♠♥ ♦♠ s t s♣t♠♣♦r r ♦ t s②st♠ r♥ s ♥ s

❭t④②s♥♦r⑥ t t♦t ♥r② r s ♣♣r♦①♠t ② ♦♥② ♦♠♣t♥

t ♥t ♥r② r

〈Etot〉 =1

2Vol

(

ˆ

Ωx

ρω2wn(x, ω, φ).w∗n(x, ω, φ)dΩ)

r♦♣ ♦t② t♦rs Cgn(ω, φ) r ♦♠♣t ♦r ♥♠rs t rq♥②

st s ♦r t r♠ ♣②r♠ tt

♣r♦♣♦s ♠t♦♦♦② s s t♦ st② t s♣rs♦♥ ♥t♦ ♠♥s♦♥

r♠ ①t tt t ♦♥ssts ♦ ♥ ♥♥t ♣r♦ ♠♥s♦♥ ♠

♦ ♣r♦ strt♦♥ ♦ t ♥tr② ♣rs♥t ♥ r s②st♠ s ♠ ♦

2.5mm t ♣t ss♠② ♠ ♦ s♦tr♦♣ ♠♣ ♥♦♥♦♥ r ♣♦②♠r t t

s♠ ♠♥ rtrsts s ♦r t ♦♠♦♥st♦♥ ♦ ♦♥sr t ♥tr♥s

♠tr ♠♣♥ ♦ t ♦r ♠tr ♥ ②strt ♠♣♥ t♦r ♦ 0.001 s ♥ s

s♠ ♦♥sst♥t t♦ ②strt ♦ss t♦rs ♦ tr♠♦st♣♦①② ♣r♣rs ❬❪ s③

s 120mm2 ♠t♦ ♦s s t♦ ♦♠♣t ♥ rq♥s ♦rrs♣♦♥♥ t♦ ♥② k

t♦r sr ♥ ②♥r ♦♦r♥ts s②st♠ ② ts rs k ♥ ts ♥ φ ♥ t ♦

rst r♦♥ ♦♠♥ ♥♠r ♠♣♠♥tt♦♥ s s ♦♥ t ♦r♠

t♦♥ s♥ ♠♥s♦♥ ♦r♥tt♦♥ ♥ t ks♣ ② ♠♣♦s♥ Φ =

cos(φ)

sin(φ)

0

♣♣ ♦♥r② ♦♥t♦♥s r qts ♦ s♣♠♥ts ♦♥ t t♦ ♣rs

♦ tr s Γr+1Γr−

1andΓr+

2Γr−

2♦♥ ♦♥rs ♦ ♠♣♦s s rt ♦♥

r② ♦♥t♦♥s ♥ ①trs♦♥ ♦♣♥ r ♠♣s ♦ s♣♠♥ts r♦♠ t s♦r

s ①♣♦rt t♦ t st♥t♦♥ ♦rrs♣♦♥♥ t♦ t ♦♣♣♦st ♦♥ r♦♠ Γr+1toΓr−

1

s t ♦♠♥s r ♦ t s♠ s♣ ♠♥s♦♥ t②♣② s ♣♦♥ts ♠♣♣♥

①♣♦rt ♠♣♣♥ s s♦ ♦♣ t♦ t st♥t♦♥ s♣♠♥t ② s♥ t

r♥ ♠t♣rs ♠♣♠♥tt♦♥ s ♠ t t♣②ss ♣t♦r♠

♥ ♣r♠tr ♦♠♣tt♦♥ t♦ ♦t♥ k(ω, φ) s rr ♦t t t r♦t♥s ♦r

♣r♠trs ω andφ t qrt ♥ ♣r♦♠ ♥ r♦r♠t s rst

♦rr ♦♥ ② ♦♥ t stt ♠♥s♦♥ tr ♦♥str♥t ♥♥ t s ♣♦ss t♦ rt

t s②st♠ ♥ t ♦r♠ Ax = λBx ♦rt♠ ♦♠♣ts t rst ♥

s ♦ t ♠tr① C = A−1B ♦ ♦ ts t s♦r ss t P

r♦t♥s ♦r rs ♥ ♣r♦♠s s sr ② ❬❪ s ♦ s s

♦♥ r♥t ♦ t r♥♦ ♦rt♠ t ♠♣t② rstrt r♥♦ ♠t♦

P r♦t♥s ♠st ♣r♦r♠ sr ♠tr①t♦r ♠t♣t♦♥s Cv

r ♦♠♣s r ② s♦♥ t ♥r s②st♠ Ax = Bv s♥ t P

s♦r ♦♣ ② ❬❪ s ♣r♦r ss ♦ ♣rs♦♥ ♦t♥ ♣♦♥t ♥♠rs ♥ s

♠♣♠♥t s♥ ♦t ♦ ♦r ♠♠♦r② ♠♥♠♥t ♥ ♦rr t♦ ♦ ♥② ♠♠♦r② ♣r♦♠

♥ ♥ ♥s ♥ ♦♥r ♠s s ♦♥sr r rst ♠s s ♦♥ssts

♦ ttrr r♥ qrt ♠♥ts ♦r rs ♦ r♦♠ ♥ t r♥

♦♥ ♦ ttrr r♥ qrt ♠♥ts ♦r rs ♦ r♦♠ ♥ t

tst ss ♦♥sr t♥ ts ♦r t rq♥② r♥ s ω = 2π.[1000 : 1000 : 10000]

t♥ ③ ♥ ③ ♥ φ = [0 : π20

: π2]

❯

♥ ♣r♦♣rts

r s♦s t rt♦♥ ♦ t ♥♣♥ P♦ss♦♥s rt♦ νxy q t♦ νyx ♦r s②♠♠tr②

rs♦♥s ♦r r♥t δ s t r②♥ ♥tr♥ ♥s β = 0.05 ①t ♥t

P♦ss♦♥s rt♦ ♦r ♦ t r♠ tt s ♥t ♦r t θ r♥ ♦♥sr ♣rt

r♦♠ r② s♠ θ < 2o ♥ r θ > 78o ♠ ♥s ♥t ♦♥rt♦♥s r t

♠♥ts l t♥ t♦ ♠♦st t ♦r③♦♥t ♥s r rt t♦ ♦♥rt♦♥s

r t ♠♥ts t♥ t♦ st♥ ♣rt rt ♥ ♦t ss t ♠♥ts l t♥ t♦

♦♠♥t ② ① ♦r♠t♦♥ ♠♥s♠s ♦♥trt s♥♥t② t♦ ♦r♥

♦♥ t P♦ss♦♥s rt♦ s ♥ ♥trs②♠♠tr ♦♥②♦♠ ss♠s t rs ♥

①r ♦r ❬❪ t♥ 10o ♥ 30o t P♦ss♦♥s rt♦ t♥s t♦ ts st

♠♥t ∼ −0.9 ♦r δ = 5.0 t♥s t♦ rs s s♦♦♥ s t strt♥ ts ♦

t ♠♥ts strt t♦ ♣rs♥t ♥ t ♠r♦strtr ♠♦r t ♠♥t s♦ ss♠s

♣t s♣ ♦r♥ δ s t ♦r t ♠♥t ♦ t ♥♣♥ P♦ss♦♥s

rt♦ t t ♠♥♠♠ ♥♦ r t ♦r ♥tr♥ ♥ νxy = −0.6 ♦r

δ = 1.0 ♥ t ♦♥trr② t ♦t♦♣♥ P♦ss♦♥s rt♦ νxz r s ②s ♣♦st

r♥♥ t♥ ♥ t s♠♠♦♥♦t♦♥ rs t t ♥tr♥ ♥

♠♦r ♣r♦♥♦♥ t ♣t rs l δ = 1.0

r s♦s t rt♦♥ ♦ t ♥♣♥ Ex = Ey ♥ ♦t♦♣♥ Ez ❨♦♥s

♠♦s rss t ♥tr♥ ♥ ♦r t♦ r s♣t rt♦s δ = 1.0 ♥ δ = 3.0 ♥

♠♣♦rt♥t s♣t t♦ t s t t tt t tr♥srs st♥ss s ♦r t♥ ♥ t

♥♣♥ ♦♥ ♦r r② s♠ ♥s ♥st r♠ ♦ r t rt♦ Ex/Ez s ♦r

δ = 1.0 ♦r θ > 20o t rt♦ ♥ r② t♥ ♥ t♠s ♦r st♥ss rt♦s r

♦sr ♦r ♥rs♥ δ s s ♦r s ♥s ♦r ss r strtrs

♦ s♦ ♥r s♥ rss β ♦ t tr♥srs ♠♥ ♣r♦♣rts t ♥

♣♥ ♦♥s s t β3 ❬❪ ♦r t r♠ ♣②r♠ tt ♥ t♦t s

♣rs♠t ♦♥②♦♠ st♦♥ r t rs ♦r♠ ♥r ♥♥ t xy♣♥

♣r♦♣rts r ♦♠♥t ② ♠♠r♥ ♥ sr ♦r♠t♦♥s ♦ t rs l ♥ tt s♥s

t ♣②r♠ tt ♦♥sr ♥ ts ♦r s ♥trs②♠♠tr ♦♥②♦♠

t ts ♣rs♠t r♦ssst♦♥ s s tr♦tt♥ss r♥♦r♠♥t ♥ ♠♦r

♣r♦♥♦♥ rt♦ t♥ ♥♣♥ ♥ ♦t♦♣♥ sr ♥ ♦sr ♥ r

r Gxy/Gxz rt♦s ♣ t♦ ♥ ♦sr ♦r θ < 3o ♥ δ = 1.0 ♥trst♥② ♦r

♦t δ ♦♥rt♦♥s ♦♥ ♥ ♦sr tr♥srs sr st♥ss ♥rs♦♥ ♦ θ = 55o

r t ♦t♦♣♥ sr ♠♦s ♦♠s r t♥ t ♥♣♥ ♦♥ s ♦r

s ② t♦ ♦♥t t♦ ♥rs♥ ♥trt♦♥ t♥ t ♠♠r♥ st♥ss t♥

t rs l ♥ t ♥♥ ♦♥ ♥ t ♣ts b t t s ♦ t ♥t r

t♥s t♦ qs ♦r s♣ ♥ ♥ δ t ♦♥st♥t α

♦♥ss rs

r s♦s t s♦♥ss rs ♥ t xz ♣♥ q ♦r s②♠♠tr② ♥ t yz ♣♥

♦r δ = 1.0 r ♥ δ = 5.0 r t ♦♥st♥t ♥tr♥ ♥ θ =

20o t tr s♦♥ss r s3 (φ) s♦s r② str♦♥ rtt② t φ = 0, π/2, π, 3π/2

♦rrs♣♦♥♥ s♦ t♦ t ♠♥ ①s ♦ t rs l rst t♦ s♦♥ss rs s1 (φ) ♥

s2 (φ) ♦r ♠♦r ♥trst♥ ♦r t ♥r ③r♦s♦♣ t♥ 55o <

φ < 135o ③r♦s♦♥ss s ♦r ♥ tr♠s ♦ r♥ ♦r t s♦♥ s♦♥ss r ♥

♣♥s ♦♥ t r s♣t rt♦ δ ❲ r♥ ♥ tr♠s ♦ ♥ φ r♥ ♦r t

③r♦s♦♣ ①sts ♦r t tt t δ = 1.0 t s1 (φ) r s♦s t s♦♣ ①t♥♥

♠♦st t♥ 20o < φ < 160o ♦r rs l ss♠♥ ♠♦r ♠ strtr δ = 5.0

♥ ♥♦r♠s tr♠s t s♦♥ s♦♥ss r ♥ δ = 1.0 ♦rrs♣♦♥s t♦ r ♦ts

v s♦t♦♥s ♦ t rst♦s qt♦♥ t♥ t ♦♥s ♦r t tt ♦♥rt♦♥

t δ = 5.0 ①st♥ ♦ ③r♦s♦♣ s♦♥ss rs ♥ ♥trs②♠♠tr ♠♦♥♦♥

tts s ♥ sst s ♣♦ss ♠♥s♠ t♦ ♥rt s♦t♦♥s ♦♥ s♣ s♣t

rt♦♥s ♦♥ t ♠r♦strtr ❬❪ ♥ tt s♥s t r♠ ♣②r♠ ①t tt

s♦ s♦♠ ♥trst♥ ③r♦s♦♣ rtrsts ♦r t rst ♥ s♦♥ s♦♥ss rs

t♦ ♥♦t ♦r t rst ♦♥ s ♥t ♥ s♣ ♠♦♥♦♥ ♦♥rt♦♥s ② ❲♥

♥ ②♥ ❬❪ ♦ ③r♦s♦♣ s♦♥ss r ♥ ♦sr ♦r t ♣r♦♣t♦♥ ♥

t xy ♣♥ t ♥ s♦tr♦♣ rtt② ♦r s1 (φ) ♥ s3 (φ) ♥ str♦♥ rt♦♥t② ♥

s2 (φ) ♦r φ = 0, π/2, π, 3π/2 r s ♥ t s ♦ t xz ♣♥ ♣r♦♣t♦♥

t ♦ts ♦rrs♣♦♥♥ t♦ t s♦♥ s♦♥ss r t♥ t♦ t♠s ♦r ♥

♥♦r♠s tr♠s ♦r t ♠ tt rs δ = 5.0 t♥ t q s♣t rt♦ ♥

♣t ♦♥rt♦♥s t δ = 1.0

❲ s♣rs♦♥

♥ ts st♦♥ q♥tts ♦ ♥trst r ♥ ♥ ♥♦♥♠♥s♦♥ ♥ts rr♥

♦t② s ♦♥sr s

cT =

√

Ec

2(1− νc)ρc

r Ec νc ♥ ρc r t ❨♦♥s ♠♦s P♦ss♦♥s rt♦ ♥ ♥st② ♦ t s ♠tr

♥ t ss♦t rr♥ rq♥② s

fref =πcTL

s♣rs♦♥ ♦♥ Γ−X rt♦♥ ♦ t s②st♠

r s♦s t r ♣rt ♦ t s♣rs♦♥ rs Re(kn) ♥ t s ♦ ♥r♥

♠ss ♥♥ t s ♥♠r ♦t♥ rsts ♥t r ♦♠

♣①t② ♦ t r♦♦st ♦r ♦ t r♠ ♣②r♠ tt s②st♠ ♥♦r♣♦rt♥

♥♠r♦s ♥s♥t ♠♦s ♥ ♦♠♣① s♣rs♦♥ rs ♦r ♣r♦♣t ♦♥s s st

♠♦ s t ② s♥ ♥r② ♦t② rtr♦♥ s♦ tt Cgn(ω, φ).Φ > τcT

t τ = 0.5% rt♥ ♣r♦♣t s r ♣rs♥t ♥ r t s ♣♦ss

t♦ t t s②st♠ rq♥② ♥ ♣s t t rst ♦♥ t♥ ♥ ♦

t rr♥ rq♥② t s♦♥ t♥ ♥ ♥rr♦ ♥♣ ♦♥ ♥

♥t r♦♥ ③ t st ♦♥ s ♣rs♥t t♥ ♥ t♥r

♠ s ♠♦s A0 So ♥ ♦sr ♥ r② ♦ rq♥② ♥ ♦rr

s♣♦♥♥ r♦♣ ♦ts r♠ s s♦♥ ♥ r r ♥ ♣ strtr ♥

♣rtr ♣r♦♣t rtrsts ② ttrt t♦ t ♥s ①t ♦r ♥

♦sr t♥ ♥ ♦ t rr♥ rq♥②

s♣rs♦♥ ♦ t s②st♠ ♥ t ♦ ♠♥s♦♥ s♣

♣r♦♣♦s ♦♠♣tt♦♥ ♠t♦ ♦s s t♦ ♦♠♣t ♠t ♠♦ s ♣r♦♣

t♦♥ ♥ t ♦♠♣t ♠♥s♦♥ ks♣ ♦ t rst r♦♥ ③♦♥ ♣r♦♣♦s

♠t♦♦♦② s s ♦♥ t ♦♠♣tt♦♥ ♦ rq♥②♣♥♥t ♦♠♣① ♥♠rs

♦ t♦r♠ s ①♣♥ ♥ t s ♦ ♠♣ s②st♠s ♥ t rsts ♦t♥

♦♠ ♦♠♣① ♥trt♥ ♣s ♦t② ♥ ♥s♥t ♣rt ♦r ♦♠♣t

♥♠r ss♦t t♦ t r ♥ ♠♥r② ♣rts ♦ t ♦t♥ ♥s ♦ qt♦♥

♠♣♥ ♦r s ♥tr♦ ② ss♠♥ ♦♠♣① ♦♦ stt② t♥s♦r

s♠ ♠t♦♦♦② ♦ ♥ r③ ② ♥tr♦♥ ♥② ♥ ♦ ♥r s♦st

♠♦♥ s s s♦s ♦r ♦r ♥② ♦tr ♦♠♣① rq♥②♣♥♥t tr♠s

r strts t t②♣ rsts ♦ t ♥②ss Pr♦♣t ♥♠rs ♦ t

♠♣ s②st♠ r s♦♥ ♦r r♥t ♥s r♦♠ φ = 0 t♦ φ = 45o t st♣ ♦ 9o t

♥ ♦sr tt φ = 0 t s②♠♠tr② strt ♥ r st ①sts s

s♦♦♥ s ♦tr rt♦♥s r ♦♥sr t ♣♣rs t t♦ rs♣ t ♦t♥ rsts

t t rt ♦t♦♥ ♥t♦ t rst r♦♥ ③♦♥ s ♥ ①♣♥ ② t t tt t

♣r♦t② ♦ t ♥t ♣ttr♥ s ♦st ♥ t ♦r♥tt♦♥ s ♥♦t ♣r t♦ ♦♥ ♦ t ss

♦ t ♥t ♥ s t♦ ♠♥s♦♥ ♦♥sst♥t tr♥s♦r♠t♦♥ ♦sr ♥

♣ strtr s♦ ♣♥s ♦♥t♦ t ♣r♦♣t ♥s r♦r t♦ t t ♥♣

♦ t ♣r♦ s②st♠ ♥ ♥t♦r ♦ ♠♥♠ ♥s♥ rt♦ ♦ t ♦♠♣t s

♦r ♦♥sr rq♥② ♥ s ♥ s

Ind(ω, φ) = minn

∣

∣

∣

∣

Real(λn)

|(λn)|

∣

∣

∣

∣

r s♦s t ♣♦t ♦ ts ♥t♦r ♥ t ♦ ♦♠♥ ♦t♦♥ ♦ t

♥ ♣s ♥ ♦sr r② ♥ ♥t tt s♣ rtt② ♥ ①sts t rt♥

rq♥s s t ♥ ♣s r ♥♦t ①s②♠♠tr ♥ ♥ ♦t t♥ s♣

♥s ①♠♠ ♥s♥ s ♦sr ♦♥ t rt♦♥s ♦ t r♠ ♣②r♠

rs ♦ ♥t l ♦r ♦ rq♥② r♥s ♦ ③ t♥ ♥ ③ t

♠①♠♠ ♥s♥ s♦s ♥ s♦tr♦♣ rtt② ♦ ③ t ♠①♠♠

rtt② s ♥ ♦sr ♦♥ t ①s ♦ t s♠♥ts l ♦ t ①t tt

❱ ❯

r♠ ♣②r♠ tt sr ♥ ts ♦r s s♦♥ t ♦♦♥ r

trsts

tt s ♥ ♥♣♥ ①t ♦r t s ♦ t t P♦ss♦♥s

rt♦ ♣♥♥ ♦r t ♦♠tr② ♣r♠trs ♦ t r ♦♥rt♦♥s Pt

rs t♥ t♦ rs t ①tt② ♦ t ♦♥②♦♠ ♦t♦♣♥ P♦ss♦♥s

rt♦ s ♣♦st ♦r t ♦♥rt♦♥s ♦♥sr

♥♣♥ ♥① st♥ss ❨♦♥s ♠♦s Ex s r ♦♠♣r t♦ t tr♥s

rs s r♠ ♣②r♠ tt s ♣rs♠t ♦♥②♦♠ t

t ♦r♥t tr♦ t t♥ss tr♦r ①♣♥♥ ts rtr ♣r

♠♥ ♦r s♠r tr♥ s ♦sr s♦ ♦r t sr ♠♦s

t♦ t s ♣♦ss t♦ ♥t② ♦♠♥t♦♥ ♦ ♥tr♥ ♥ ♥ r s♣t

rt♦ δ tt s t♦ Gxy = Gxz stt♦♥

❩r♦s♦♣ s♦♥ss rs ♥ ♦sr ♦r t ♣r♦♣t♦♥ ♥ t xz ♣♥

♦r t rst ♥ s♦♥ rs rs♣t② sst♥ ♣♦ss rt♦♥s ♦r t

t♦r r s♦t♦♥s ♦ rt s♥ ts tt s ♣t♦r♠ ♦ ③r♦rtr

♦r t s♦♥ss rs s ♥ ♥t ♥ t xy ♣♥

❲♥ ♦♥sr♥ ♠♥s♦♥ ♣r♦♣t♦♥ t♥ ♥t♦ ♦♥t t t

s ♦ t ♠r♦strtr ♥♦t ♥ ♦♥♥ t♦ t ♦♥ ♥t ♣♣r♦①♠

t♦♥ ♦ ♣♦♥ts t r♠ tt s♦s ♦♠♣① ♣ttr♥ ♥ ♥ t xy

♣♥ ♥ s♠ s ♦ ②strt ♠♣♥ s ♦♥sr ❯s♥ ♥ ♥s♥

♥① rt ♦r ts tt t s ♣♦ss t♦ ♥t② ♥s♥ rtt② ♣t

tr♥s t rt♦♥t② ♣♥♥ ♦ t ♠t♠♦ rtrsts ♦ t r♠

①t ♣②r♠ tt

♥♦♠♥ts

s ♦r s ♥ ♣rt② s♣♣♦rt ② r♥t♥ ♦r P ♦s ② t ♣♥

♦t② ♦r Pr♦♠♦t♦♥ ♦ ♥

❬❪ t ❲② P②r♠ tt trss strtrs t ♦♦ trsss t

rs ♥ ♥ ♥♥r♥ ♦❭♥♦④♦⑥④

♠s⑥

❬❪ t ❲② t♥♠ ♦② tt trss strtrs trs ♠♣

s♥ ♦❭♥♦④♦⑥④♠ts

⑥

❬❪ ❲② t♥t♦♥ ♣r♦ r ♠ts P♦s♦♣ r♥st♦♥s ♦ t ♦②

♦t② t♠t P②s ♥ ♥♥r♥ ♥s ♦

❭♥♦④♦⑥④rst⑥

❬❪ P♥ ❱ s♣♥ ❲② ♦♣s ♠♥s♠ ♠♣s ♦r

♦♦ ♣②r♠ tt Pr♦♥s ♦ t ♦② ♦t② t♠t P②s ♥

♥♥r♥ ♥ ♦❭♥♦④♦⑥④rs♣⑥

❬❪ rs♦♥ rs♦♥ ①t trs Pr♦♥s ♦ t ♥sttt♦♥ ♦ ♥

♥♥rs Prt ♦r♥ ♦ r♦s♣ ♥♥r♥

❬❪ ♦♥ ③③♥ ♥②ss ♦ ♥♣♥ ♣r♦♣t♦♥ ♥ ①♦♥ ♥ r♥tr♥t

tts ♦♥ ❱rt

❬❪ ♣♦♥ ③③♥ ♠r ♥ ①♣r♠♥t ♥②ss ♦ tt ♦♠♣♥ ♦

r rss ♦r r♦s ♦r♥ ♦ ♥s ♦ trs ♥ trtrs

❬❪ ♣♦♥ r♣ ③③♥ ❲ ♣r♦♣t♦♥ ♥ ①t ttrr ♦♥②

♦♠s ♦r♥ ♦ ❱rt♦♥ ♥ ♦sts

❬❪ rt♥ ②rr♥r ♠t r♣ P♦ttr ③③♥ ①r

♣rs♠t ♥♦① ♦♥♣t P②s tts ♦

❬❪ tt♥ P r♦ ♥r♦ ③③♥ ♣♦♥ ♦♠♣♦st r str

trs ♦r ♠♦r♣♥ r♦s ♠r ♥②ss ♥ ♦♣♠♥t ♦ ♠♥tr♥ ♣r♦ss

♦♠♣♦sts Prt ♥♥r♥

❬❪ r♠ ❲♠s tt ♥s ♦♥ ♦ ①t ♥t♦r ♣♦②♠rs

t r♦♠ ①❬❪r♥ ♥ ♦s ♦r ♠t♦♥ ♦

❭♥♦④♦⑥④⑥

❬❪ ❱ ♦ ♦ ♦③♦ ❩♥ ♥ts

♠♥ ♥ ♥ ♦ P♦ss♦♥s t♦ ♦r r♦♥ ♥♦t ts ♥

♦❭♥♦④♦⑥④s♥⑥

❬❪ ❱ ♦ ♦③♦ ❩♥ ♥ts ã♦ ♠♥

♦♥ t ①t tr♥st♦♥ ♦r r♦♥ ♥♥♦t sts P②s

♦❭♥♦④♦⑥④P②s⑥

❬❪ ♦♠ t♦ ♦♣♠♥t ♦ ② s♥ ❯trt ♦r trtrs

♥tr♥t♦♥ ♦r♥ rs

❬❪ t♦ r♣ r♠ ♦♠♣♦st ①t ♦♥②♦♠ ♥ t ♥tr♥t♦♥

♦♥r♥ ♦♥ ♦♠♣♦st trtrs

❬❪ t♦ ♥s r♣ r r♠ ♦r♣♥ ❲♥♦① ♦♥♣t ♦r♥ ♦

♥t♥t tr ②st♠s ♥ trtrs ♦❭♥♦④♦⑥④

❳⑥

❬❪ P ❲♦ ♠♥ P♦♥♦♥s ♠r ❯♥rst② Prss ♠r ❯

❬❪ ♦r♥rt rt P ♦r♠♥ ♦♠♦é♥ést♦♥ ♥ ♠é♥q s ♠tér①

♥ r♦♣ t

❬❪ r ♦♥sttt ♠♦♥ ♦ ♣③♦tr ♣♦②♠r ♦♠♣♦sts t tr

♦❭♥♦④♦⑥④t♠t

⑥

❬❪ ② ❲ ♣r♦♣t♦♥ ♥ ②r ♥s♦tr♦♣ ♠ t ♣♣t♦♥s t♦ ♦♠♣♦sts

♦rt ♦♥ sr ♥ ❱

❬❪ ❲♥ ②♥ ①st♥ ♦ ①tr♦r♥r② ③r♦rtr s♦♥ss r ♥ ♥s♦tr♦♣

st ♠ ♦r♥ ♦ t ♦st ♦t② ♦ ♠r

♦❭♥♦④♦⑥④⑥

❬❪ r ♦♥s ♦ ❲s ♦♠♦♥③t♦♥ ♥ ♣tr s②♠♣t♦t ♥②ss ♦r

♥ té♠tqs Prs t ♣♣qés

❬❪ ♦ r♦t ♥ s r♦♣ ♥ ♥r② ♦ts ♥t

♠♥ts ♦r♥ ♦ ♦♥ ♥ ❱rt♦♥ ❳ ♦

❭♥♦④♦⑥④④s⑥⑥

❬❪ ♥ ♦ t♠♦ ♣r♦♣t♦♥ ♥ s♦♥ ♥ strtrs tr♦ ♥t

♠♥ts r♦♣♥ ♦r♥ ♦ ♥s ♦s

♦❭♥♦④♦⑥④④r♦♠s♦⑥⑥

❬❪ ♦♦♥ ♦ ③q ❲ ♠♦t♦♥ ♥ t♥ strtrs ♦r♥ ♦ ♦♥

♥ ❱rt♦♥ ❳ ♦❭♥♦④♦⑥④④s

⑥⑥

❬❪ ♦t ss ③③♥ ♦ ♦qt♦ ♦♠♣♦st♦♥ ♦ t st♦

②♥♠ qt♦♥s ♣♣t♦♥ t♦ ♠♥s♦♥ s s♣rs♦♥ ♦♠♣tt♦♥ ♦ ♠♣

♠♥ s②st♠ ♥tr♥t♦♥ ♦r♥ ♦ ♦s ♥ trtrs

❬❪ ♦t ♥r ♦ ❲ ♦t♦♥ ♣t♠③t♦♥ ♥ Pr♦② strt

♥t P③♦♦♠♣♦st ♠ trtrs ♦r♥ ♦ ♥t t ②st ♥ trt

❬❪ r ♦♠♣tt♦♥ ♦ ♣r♦♣t s ♥ r r s♥ ♥t ♠♥t t♥q

♦r♥ ♦ ♦♥ ♥ ❱rt♦♥

❬❪ ❲ ②s♥ör ör♣rs♥r r♥♥ ③r r♥♥ ♦♥ ♥rt♥ ♥

♥t♥stät♥ ❲ss♥st ❱rssst tttrt

❬❪ rt♦t ssrr ❨ r♥ ♠♣♥ ♥②ss ♦ ♦♠♣♦st ♠

trs ♥ strtrs ♦♠♣♦st trtrs ♦

❭♥♦④♦⑥④♦♠♣strt⑥

❬❪ ♦q ♦r♥s♥ ❨♥ P srs s♦t♦♥ ♦ rs ♥

♣r♦♠s t ♠♣t② rstrt r♥♦ ♠t♦s ♠

❬❪ ♥ ärt♥r ♦♥ ♥s②♠♠tr s♣rs s②st♠s ♦ ♥r qt♦♥s t P

tr ♥rt♦♥ ♦♠♣tr ②st♠s ❳

❬❪ r♣ P P♥②♦t♦ ♦♠♥s♦♥ ♠r ♥ ①♣r♠♥t ♥① ♦♥ ♦♥ ♥

♣♥ ①t ♦♥②♦♠s ♦r♥ ♦ tr♥ ♥②ss ♦r ♥♥r♥ s♥

♦❭♥♦④♦⑥④⑥

❬❪ s♦♥ s② ♥s ♦ r♠♥s♦♥ r trs Pr♦

♥s ♦ t ♦② ♦t② ♦ ♦♥♦♥ t♠t ♥ P②s ♥s

♦❭♥♦④♦⑥④rs♣⑥

r ❱ ♣ t ♣②r♠ r♠ ♦r t ts ♥♦♥♠♥s♦♥ ♦♠tr②

♣r♠trs ❱ t α = 1, β = 0.05, δ = 6, θ = 20o

r ♥r ♣r♦ s

r ②♦t ♦ t ♠♣ r♠ tt ♥t ♦♥sr ♦r t

♣r♦♣t♦♥ st②

r ❯♥r♥ ♥ r♥ ♠s ss

0 10 20 30 40 50 60 70 80−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

θ [o]

ν xy

δ = 1.0

δ = 3.0

δ = 5.0

0 10 20 30 40 50 60 70 800.2

0.25

0.3

0.35

0.4

0.45

0.5

θ [o]

ν xz

δ = 1.0

δ = 3.0

δ = 5.0

r strt♦♥ ♦ t P♦ss♦♥s rt♦s νxy ♥ νxz rss t ♥ θ ♦r

r♥t δ ♣r♠trs ♥ α = 1

20 30 40 50 600

0.5

1

1.5

2

2.5

3

3.5

4

4.5

θ [o]

2 4 6 80

50

100

150

200

θ

Ex/E

c X 103, δ = 1.0

Ez/E

c X 103, δ = 1.0

Ex/E

c X 103, δ = 3.0

Ez/E

c X 103, δ = 3.0

30 40 50 60 70 800

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

θ [o]

2 4 6 8 100

10

20

30

40

θG

xy/G

c X 103, δ = 1.0

Gxz

/Gc X 103, δ = 1.0

Gxy

/Gc X 103, δ = 3.0

Gxz

/Gc X 103, δ = 3.0

r ♦♥♠♥s♦♥ ♥♣♥ ♥ tr♥srs ❨♦♥s ♠♦s rss t ♥tr♥

♥ θ ♦r r♥t s♣t rt♦s δ ♠r ♣r♠tr rs ♦r ♥♦♥♠♥s♦♥

♥♣♥ sr ♥ tr♥srs ♠♦s ♦r t♦♥s α = 1

0.5

1

1.5

30

210

60

240

90

270

120

300

150

330

180 0

θ = 20o

s1(φ)

s2(φ)

s3(φ)

δ = 1.0

θ = 20o

s1(φ)

s2(φ)

s3(φ)

δ = 5.0

r ♦♥ss r ♥ t xz ♣♥ ♦r α = 1, θ = 20o

0.1 0.2 0.3

θ = 20o

s1(φ)

s2(φ)

s3(φ)

δ = 1.0

θ = 20o

s1(φ)

s2(φ)

s3(φ)

δ = 5.0

r ♦♥ss rs ♥ t xy ♣♥ ♦r α = 1.0, θ = 20o

0 0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Real part of reduced wave number

Re

du

ce

d f

req

ue

ncy

r s♣rs♦♥ rs ♦ ♠♦s ♦ t st s②st♠ ♠♥r② ♣rt ♦ λn(ω)

♦r r ♣rt ♦ kn(ω)

0 0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Real part of reduced wave number

Re

du

ce

d f

req

ue

ncy

r s♣rs♦♥ rs ♦ ♣r♦♣t ♠♦s ♦ t st s②st♠ ♠♥r② ♣rt

♦ λn(ω) ♦r r ♣rt ♦ kn(ω)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.140

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Real part of reduced group velocity

Re

du

ce

d f

req

ue

ncy

r s♣rs♦♥ rs ♦ ♣r♦♣t ♠♦s ♦ t st s②st♠ ♠♥r② ♣rt

♦ λn(ω) ♦r r ♣rt ♦ kn(ω)

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

0.08

0.1

Reduced fre

quency

φ=0

0 0.5 1 1.50

0.02

0.04

0.06

0.08

0.1

φ=π/20

0 0.5 1 1.50

0.02

0.04

0.06

0.08

0.1

φ=π/10

0 0.5 1 1.50

0.02

0.04

0.06

0.08

0.1

Reduced fre

quency

Propagative part of reduced wave number

φ=3π/20

0 0.5 1 1.50

0.02

0.04

0.06

0.08

0.1

Propagative part of reduced wave number

φ=π/5

0 0.5 1 1.50

0.02

0.04

0.06

0.08

0.1

Propagative part of reduced wave number

φ=π/4

r Pr♦♣t ♥♠rs ♦ ♠♣ s②st♠ Im(λn(ω) ♦r r♥t ♥s

r rtt② s♥ t ♥s♥ ♥① strt t ♥t