Transition de Fréedericksz

1
2²E21 2µH2
−→
D=¯ ¯²−→ E f =1
2−→
E .−→ D
Ez=−→ E .−→ n
²kExEy²⊥
f=1
2[²⊥E2
x+²⊥E2 y+²kE2 z]
=1
2[²⊥E2
x+²⊥E2 y+²⊥E2 z−²⊥E2 z+²kE2 z]
=1
2[²⊥E2+ (²k−²⊥)E2
z]
E2
−→
E
−→
n
f=−∆²
2[−→
E .−→ n]2−∆χ
2[−→
H .−→ n]2
−→
E .−→ n

E2
²k, ²⊥
∆χ=NS(ξk−ξ⊥)
∆²=NS(²k−²⊥)
−d/2d/2−→
E
θ=θ(x)
K22 =K33 =K K

fE=1
2¡K11.0 + K22(−→
n .−→−→ n)2+K33(−→ n×−→−→ n)2¢=K
2[∂θ(x) ∂x ]2
f=−∆²
2[−→
E .−→ n]2=−∆²
2[Esin θ(x)]2
f=K
2[∂θ(x) ∂x ]2−∆² 2[Ecos θ(x)]2
θ(x)F=RV ol f(x)dx
θ(x)
ξ2
E=K
∆²E2
0 = ξ2
E
∂2θ
∂x2−cos θsin θ
I[y(x)] = Rb
aF(y, ∂y
∂x , x)dx
y(x)∂F
∂y −d dx
∂F
∂(dy
dx )= 0

dθ/dx = 0 x= 0 θEx= 0
θ(x)
θ=θ(x) 2
(dθ
dx)2=C−cos2θ ξ2
E
dθ
dx =±1 ξEpcos2θE−cos2θ
x=±d/2θ=π/2x= 0 θ=θE
x= 0
θE
Zx
−d
2
dx
ξE
=Zπ
2
θ(x)
dθ0
√cos2θE−cos2θ0
d
2ξE
cos θE=Zπ
2
θE
dθ0
q1−(cos θ0
cos θE)2
θE
−→
E θ(x) = 0π/2 0 = 0
d/(2ξE)> π/2
θE6=π
2
Ec=π
drK ∆²
E > Ecθ(x)
E < Ec
ξE
E > Ec
ξE

Ec
θEx= 0
Ec
θ(x) = θE=π/2
θE
Vc=Ec∗d d