Faisceaux gaussiens: propagation, propriétés, manipulation

~
k
d∼ ±λ/d
∆ψ−1
c2
∂2ψ
∂t
2

∆ψ+k2ψ= 0
z= 0
ψ(x, y, z = 0, t) = ψ0exp(−iωt) exp(−r2/w2
0)
r=p(x2+y2)
∝ψ2e2= 7,4r=w0
w0
ψ z = 0
ψ z
z ψ
ψ(r, z) = u(r, z) exp(ikz) exp(−iωt)

k=ω/c u
exp(ikz)
|du/dz
u|<< |d(eikz )/dz
eikz |
|du
dz |<< k|u|
u
|d2u
dz2|<< k|du dz |
xOy
1
r
d
dr rdψ dr +d2ψ dz2+k2ψ= 0
ψ
1
r
d
dr rdu dr −k2u+ 2ik du dz +d2u dz2+k2u= 0
u z
d2u/dz2
1
r
d
dr rdu dr + 2ik du dz = 0

u
u=A(z) exp(ikr2
2R(z)).exp(−r2 w2(z))
R(z)w(z)A(z)
R ψ
w
1
Q(z)=1 R(z)+i2 kw2(z)
u=A(z) exp(ikr2
2Q)
{A
Q+dA dz +i[kr2A 2Q2(1 −dQ dz )]}exp(ikr2 2Q) = 0
u
dQ
dz = 1
dA
dz =−A Q
Q=z+C
C ψ(r, z)z=
0w=w01/R(0) = 0
C=Q(0) = −ikw2
0
2=−izR
zR=kw2
0
2

k=ω/c = 2π/λ
zR=πw2
0
λ
1
Q=1 (z−izR)=1 R(z)+i2 kw2(z)
z
(z2+z2
R)=1
R(z)
izR
(z2+z2
R)=i2
kw2(z)
R(z) = z(1 + z2
R
z2)
w2(z) = 2(z2+z2
R)
kzR
w2=w2
0(1 + z2/z2 R)
z
w z z << zR∝z
z >> zR
z= 0 R∝z z >> zR
zRw0
z << zR
z >> zR
θ=w0
zR
=2
kw0
=2λ
π.2w0