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ω
2ω
LG
1
0
E(r, t)
B(r, t)
∇.
E(r, t) = 0,
∇ ×
E(r, t) = −
∂
B(r, t)
∂t
,
∇.
B(r, t) = 0,
∇ ×
B(r, t) =
1
c
2
∂
E(r, t)
∂t
,
c
E(r, t)
B(r, t)
∇.
A(r, t) = 0
B(r, t) = ∇ ×
A(r, t),
E(r, t) =
∂
A(r, t)
∂t
.
∇×(∇×
A) = ∇(∇.
A)−∇
2
A
A(r, t)
∇
2
A(r, t) =
1
c
2
∂
2
A(r, t)
∂
2
t
,
A(r, t)
L
L
L −→ ∞
A(r, t) =
1
√
0
L
3
k
A
k
(r, t)e
i
k·r
,
0
k
k =
2π
L
(n
x
x, n
y
y, n
z
z),
n
j
= 0, ±1, ±2, . . . (j = x, y, z)
A(r, t) ω
A
∗
k
(r, t) =
A
−k
(r, t)
A
k
= c
k
e
−iωt
+ c
∗
k
e
iωt
.
∇.
A(r, t) = 0
k ·
A
k
(r, t) = 0
c
k
k,s
s = 1, 2
k.
k,s
= 0,
∗
k,s
.
k,s
= δ
s,s
,
∗
k,s
×
k,s
=
k
k
,
c
k
= c
k,1
k,1
+ c
k,2
k,2
.
A(r, t) =
1
√
0
L
3
k,s
[c
k,s
k,s
e
−iωt
+ c
∗
−k,s
∗
k,s
e
iωt
]e
i
k.r
=
1
√
0
L
3
k,s
[c
k,s
k,s
e
i(
k.r−ωt)
+ c
∗
k,s
∗
k,s
e
−i(
k.r−ωt)
]
=
1
√
0
L
3
k,s
[u
k,s
(t)
k,s
e
i
k.r
+ u
∗
k,s
(t)
∗
k,s
e
−i
k.r
].
u
k,s
(t) = c
k,s
e
−iωt
k,s
e
i
k.r
E(r, t) =
i
√
0
L
3
k,s
ω[u
k,s
(t)
k,s
e
i
k.r
− u
∗
k,s
(t)
∗
k,s
e
−i
k.r
],
B(r, t) =
i
√
0
L
3
k,s
[u
k,s
(t)(
k ×
k,s
)e
i
k.r
− u
∗
k,s
(t)(
k ×
∗
k,s
)e
−i
k.r
].
E(r, t)
B(r, t)
H =
1
2
L
3
[
0
E
2
(r, t) +
1
µ
0
B
2
(r, t)] dv.
L
3
u
H
= 2
k,s
ω
2
|u
k,s
(t)|
2
.
u
k,s
(t)
q
k,s
(t) p
k,s
(t)
q
k,s
(t) = [u
k,s
(t) + u
∗
k,s
(t)],
p
k,s
(t) = −iω[u
k,s
(t) − u
∗
k,s
(t)].
H =
1
2
k,s
[p
2
k,s
(t) + ω
2
q
2
k,s
(t)].
k, s
q
k,s
(t)
p
k,s
(t) q
k,s
(t) p
k,s
(t)
q
k,s
(t), q
k,s
(t)
= 0,
p
k,s
(t), p
k,s
(t)
= 0,
q
k,s
(t), p
k,s
(t)
= iδ
3
k,
k
δ
s,s
.
H =
1
2
k,s
[p
2
k,s
(t) + ω
2
q
2
k,s
(t)].
q
k,s
(t) p
k,s
(t)
a
k,s
(t)
a
†
k,s
(t)
a
k,s
(t) =
1
2ω
ωq
k,s
(t) + ip
k,s
(t)
,
a
†
k,s
(t) =
1
2ω
ωq
k,s
(t) − ip
k,s
(t)
,
a
k,s
(t), a
k,s
(t)
= 0,
a
†
k,s
(t), a
†
k,s
(t)
= 0,
a
k,s
(t), a
†
k,s
(t)
= δ
3
k,
k
δ
s,s
.
a
k,s
(t) a
†
k,s
(t)
u
k,s
(t) u
∗
k,s
(t)
a
k,s
(t) = a
k,s
(0)e
−iωt
, a
†
k,s
(t) = a
†
k,s
(0)e
iωt
.
H =
k,s
ω
a
†
k,s
(t)a
k,s
(t) +
1
2
.
ω
2
E(r, t)
B(r, t)
E(r, t) =
i
L
3
2
k,s
ω
2
0
a
k,s
k,s
e
i(
k·r−ωt)
−a
†
k,s
∗
k,s
e
−i(
k·r−ωt)
,
B(r, t) =
i
L
3
2
k,s
2ω
0
a
k,s
(
k ×
k,s
)e
i(
k·r−ωt)
−a
†
k,s
(
k ×
∗
k,s
)e
−i(
k·r−ωt)
.
n
k,s
= a
†
k,s
(t)a
k,s
(t).
H =
k,s
ω
n
k,s
+
1
2
.
n
k,s
|n
k,s
= n
k,s
|n
k,s
.
|n
k,s
n
k,s
a
k,s
|n
k,s
=
n
k,s
|n
k,s
− 1,
a
†
k,s
|n
k,s
=
n
k,s
+ 1|n
k,s
+ 1.
k, s
|n
k,s
|{n} =
N
k,s
|n
k,s
.
{n} = n
k1,s1
, n
k2,s2
, n
k3,s3
, ...
|0 |vac |{n}
|{n} =
N
k,s
(a
†
)
n
k,s
n
k,s
!
|vac.
|{n}
H|{n} =
k,s
ω
n
k,s
+
1
2
|{n}.
ε =
k,s
ω
n
k,s
+
1
2
,
n
k,s
= 0
a
k,s
|α
k,s
= α
k,s
|α
k,s
,
α
k,s
|α
k,s
= e
−|α
k,s
|
2
2
∞
n
k,s
=0
α
n
k,s
k,s
n
k,s
!
|n
k,s
.
n = α|n|α = |α|
2
,
α
∆
2
n =
n
2
− n
2
= |α|
2
.
E (r, t) = E
0
cos
k ·r − ωt + ϕ
= E
P
cos
k ·r − ωt
+ E
Q
sen
k ·r − ωt
,
E
0
ϕ E
P
E
Q
P =
a + a
†
√
2
,
Q =
−i
a −a
†
√
2
,
P
Q
E (r, t) =
1
L
3
2
k,s
ω
2
0
P
k,s
cos
k ·r − ωt
+
Q
k,s
sen
k ·r − ωt

.
P ,
Q
= i,
V (
P ) V (
Q)
V
P
V
Q
≥ 1.
V (
P ) = V (
Q) = 1
D
(
α
)
D (α) = e
−
|α|
2
2
e
α
b
a
†
e
−α
∗
ba
.
|α =
D (α) |vac
= e
−
|α|
2
2
e
α
b
a
†
|vac.
a)
b)
Q
P
Q
P
a
D(α)
|i
|f
E (r, t) =
E
(+)
(r, t) +
E
(−)
(r, t) ,
E
(+)
=
i
L
3
2
k,s
ω
2
0
a
k,s
k,s
e
i
(
k·r−ωt
)
,
E
(−)
=
−i
L
3
2
k,s
ω
2
0
a
†
k,s
∗
k,s
e
−i
(
k·r−ωt
)
.
r
p
f
(r, t) ∝ |f|
E
(+)
(r, t) |i|
2
.
p (r, t) =
f
|f|
E
(+)
(r, t) |i|
2
=
f
i|
E
(−)
(r, t) |f f |
E
(+)
(r, t) |i
= i|
E
(−)
(r, t)
E
(+)
(r, t) |i,
|f
r
i
i
= e
k,s
n
k,s
.
n
k,s
e
93%
η
i
= ηe
k,s
n
k,s
.
T η
: n
2
d
:
= η
2
: n
2
a
:
,
d a
: o :
1
P
Q
V (
P ) V (
Q)
P
V
P
< 1,
V
Q
> 1.
S (ξ) = e
1
2
ş
ξ
∗
ba
2
−ξ
(
ba
†
)
2
ť
,
ξ = re
iθ
S (ξ)
|α, ξ =
S (ξ) |α
=
S (ξ)
D (α) |vac.
S (−ξ) =
S
−1
(ξ) =
S
†
(ξ)
S
†
(ξ) a
S (ξ) = acosh (r) − a
†
e
iθ
senh (r) ,
S
†
(ξ) a
†
S (ξ) = a
†
cosh (r) −ae
−iθ
senh (r) ,
Q
P
a)
c)
b)
Q
P
Q
P
θ = 0
∆
2
P = e
−r
,
∆
2
Q = e
r
.
θ = 0 ∆
2
P 1 ∆
2
Q
P r
V
P
−
=
V
P
1
−
P
2
2
.
1
V
Q
+
=
V
Q
1
+
Q
2
2
,
dj1
dI1
dj
2
dI
2
dj = - dj
dI = dI
1 2
1 2
ρ
ρ =
i
p
i
ρ
i1
⊗ ρ
i2
,
ρ
i1
ρ
i2
p
i
Σ
Σ =
V
P
1
−
P
2
2
+
V
Q
1
+
Q
2
2
,
Σ ≥ 1.
A
1
A
2
A
+
=
A
1
+ A
2
√
2
, A
−
=
A
1
− A
2
√
2
,
V
Q
+
=
V
Q
1
+
Q
2
2
, V
P
−
=
V
P
1
−
P
2
2
.
Q
+
A
+
A
−
P
−
A
1
A
2
V
Q
+
V
P
−
Σ
A
+
A
−
Q
P
P
Q
A1
A2
A+
A-
∆ν
1KHz ∆t
c
l
c
= c∆t
c
c
P
1
2
R = c t
1
1
R = c t
2
2
r
I (r, t) = |K
1
|
2
I
1
(r
1
, t −t
1
) + |K
2
|I
2
(r
2
, t − t
2
) + 2Re [K
∗
1
K
2
|Γ (r
1
, r
2
, τ )] ,
K
i
τ
Γ(r
1
, r
2
, τ )
Γ(r
1
, r
2
, τ ) = E
∗
1
(r
1
, t + τ) E
2
(r
2
, t)
T
.
γ(r
1
, r
2
, τ) =
Γ (r
1
, r
2
, τ )
√
I
1
I
2
.
γ(r
1
, r
2
, τ )
υ =
I
max
− I
min
I
max
+ I
min
= |γ(r
1
, r
2
, τ )|.
M
1
M
2
M
1
M
2
γ(r
1
, r
2
, τ ) τ = 0
γ
M
1
M
2
I
1
I
2
Γ
(2)
(r
1
, r
2
, t
1
, t
2
) = I
1
(r
1
, t
1
) I
2
(r
2
, t
2
)
T
,
γ
(2)
(r
1
, r
2
, t
1
, t
2
) =
Γ
(2)
I
1
I
2
.
γ
(2)
(r
1
, r
2
, t
1
, t
2
) = 1 + |γ (r
1
, r
2
, t
1
, t
2
) |
2
,
γ (r
1
, r
2
, t
1
, t
2
)
Γ
(2)
γ
p ∝
i
P
i
i|
E
(−)
(r, t)
E
(+)
(r, t) |i
P
i
|i
ρ =
i
P
i
|ii|,
p ∝ T r
ρ
E
(−)
(r, t)
E
(+)
(r, t)
.
G
(1)
(r
1
, t
1
, r
2
, t
2
) = T r
ρ
E
(−)
(r
1
, t
1
)
E
(+)
(r
2
, t
2
)
=
E
(−)
(r
1
, t
1
)
E
(+)
(r
2
, t
2
)
.
τ = t
2
− t
1
,
G
(1)
(r
1
, t
1
, r
2
, t
2
) ≡ G
(1)
(r
1
, r
2
, τ ) .
p ∝ G
(1)
(r, r, 0) .
r
1
r
2
t
1
t
2
p
2
∝
i
P
i
f
|f|
E
(+)
(r
2
, t
2
)
E
(+)
(r
1
, t
1
) |i|
2
=
i
P
i
i|
E
(−)
(r
1
, t
1
)
E
(−)
(r
2
, t
2
)
E
(+)
(r
1
, t
1
)
E
(+)
(r
2
, t
2
) |i,
p
2
= T r
ρ
E
(−)
(r
1
, t
1
)
E
(−)
(r
2
, t
2
)
E
(+)
(r
2
, t
2
)
E
(+)
(r
1
, t
1
)
.
G
(1)
G
(2)
(r
1
, t
1
, r
2
, t
2
) = T r
ρ
E
(−)
(r
1
, t
1
)
E
(−)
(r
2
, t
2
)
E
(+)
(r
2
, t
2
)
E
(+)
(r
1
, t
1
)
=
E
(−)
(r
1
, t
1
)
E
(−)
(r
2
, t
2
)
E
(+)
(r
2
, t
2
)
E
(+)
(r
1
, t
1
)
,
r
1
r
2
τ = t
2
− t
1
n
2n
G
g
(1)
(r
1
, r
2
, τ ) =
E
(−)
(r
1
, t)
E
(+)
(r
2
, t + τ)
E
(−)
(r
1
, t)
E
(+)
(r
1
, t)
E
(−)
(r
2
, t + τ)
E
(+)
(r
2
, t + τ)
,
g
(2)
(r
1
, r
2
, τ ) =
E
(−)
(r
2
, t + τ)
E
(−)
(r
1
, t)
E
(+)
(r
2
, t + τ)
E
(+)
(r
1
, t)
E
(−)
(r
1
, t)
E
(+)
(r
1
, t)
E
(−)
(r
2
, t + τ)
E
(+)
(r
2
, t + τ)
.
Γ G
S (r, t) = c
2
0
E (r, t) ×
B (r, t) .
p =
0
E (r, t) ×
B (r, t) .
p
P =
0
V
E (r, t) ×
B (r, t) d
3
r .
l = r × p =
0
r ×
E (r, t) ×
B (r, t)
.
A
E (r, t) ×
B (r, t) =
E (r, t) ×
∇ ×
A (r, t)

.
∇ ·
E = 0
L =
0
V
E ×
A d
3
r +
0
V
E
i
[r × ∇] A
i
d
3
r
=
L
S
+
L
O
,
i
r
k
S
(a) (b)
HG
mn
1
r
∂
∂r
r
∂ψ
∂r
+
1
r
2
∂
2
ψ
∂φ
2
+ 2ik
∂ψ
∂z
= 0.
LG
l
p
(r, z, φ) =
2p!
πw
2
(z) (p + |l|)!
√
2r
w (z)
|l|
exp
−
r
2
w
2
(z)
L
l
p
2r
2
w
2
(z)
×exp
i
kz − (2p + |l| + 1) arctan
z
z
R
+
kr
2
2R (z)
+ lφ

,
R(z) z
R
w(z) =
w
0
1 +
z
2
z
2
R
z w
0
L
l
p
HG
m,n
lφ φ
l = 0
l
l
(σ ± l) σ = ±1
p
p = l = 0 T EM
00
HG
00
= LG
0
0
p l = 0
l l
l
p
(a)
(b) (c)
(d) (c) (f)
(p, l) = (0, 0)
(p, l) = (0, 1) (p, l) = (0, −1)
(p, l) = (0, 2) (p, l) = (1, 0) (p, l) = (0, −2)
LG
±1
0
LG
±1
0
=
1
√
2
(HG
1,0
± iHG
0,1
) .
−→
−→
LG
HG
θ HG
1,0
HG
0,1
45
o
135
o
HG
45
o
0,1
=
1
√
2
(HG
1,0
+ HG
0,1
) ,
HG
135
o
0,1
=
1
√
2
(HG
1,0
− HG
0,1
) .
HG LG
s
1
=
I
0
o
− I
90
o
I
0
o
+ I
90
o
,
s
2
=
I
45
o
− I
135
o
I
45
o
+ I
135
o
,
s
3
=
I − I
I + I
,
I
j
j = 0
o
, 45
o
, ...
S
2
1
+ S
2
2
+ S
2
3
= 1,
0
o
, 90
o
, +45
o
, −45
o
(135
o
)
LG
±1
0
HG
1,0
HG
0,1
HG
45
o
1,0
HG
135
o
0,1
(a) (b)
p
1
=
I
HG
0
o
1,0
− I
HG
90
o
1,0
I
HG
0
o
1,0
+ I
HG
90
o
1,0
,
p
2
=
I
HG
45
o
1,0
− I
HG
135
o
1,0
I
HG
45
o
1,0
+ I
HG
135
o
1,0
,
p
3
=
I
LG
1
0
− I
LG
−1
0
I
LG
1
0
+ I
LG
−1
0
.
l l
l
l = 0 l = 1
LG
±1
0
(l = 2)
Kodalith
R
(a) (b) (c)
(
d
)
(LG
l
0
)
l
l = 1
l ±1
LG
±1
0
(a) (b)
m
l
LG
l
mW
LG
HG
LG HG
HG
HG
n,m
ϕ
n,m
= (n + m + 1) ϕ (z) ,
ϕ (z) = arctan(
z
z
R
) z = 0
HG
HG
(x, z) (y, z)
ϕ
n,m
=
n +
1
2
ϕ
x
(z) +
m +
1
2
ϕ
y
(z) ,
ϕ
x
(z) = arctan(
z
z
R
x
),
ϕ
y
(z) = arctan(
z
z
R
y
).
z
R
x
z
R
y
HG
0,1
45
o
±π/2
LG
±1
0
LG
±1
0
HG
0,1
±45
o
D
f π/2 D = f
√
2
π/2 HG
LG
π
LG
l
0
LG
−l
0
D = 2f π
HG LG
π/2
HG
1,0
HG
0,1
LG
±1
0
π
LG
+1
0
(LG
−1
0
)
HG LG
HG
HG
1,0
HG
0,1
HG
0,1
√
2W (z)
W (z)
π
HG 01
LG 01
HG
1,0
45
o
LG
1
0
l
l l
l
LG
+1
0
LG
−1
0
LG
+1
0
LG
−1
0
(a) (b)
(c) (d)
|l| = 1 |l| = 2
(a) (b) (c)
l = 1 l = 2
t(x, y)
t (x, y) = p [g (x, y)] ,
p(x) x = g(x, y)
g :
2
→
p
g(x, y) = α(x
2
+ y
2
) α
g(r, φ) = αr
2
+ lφ,
α l
φ l = 0
p[g(r, φ)]
t (r, φ) =
+∞
n=−∞
c
n
e
2iπνng(r,φ)
,
n
t (r, φ) = t
1
(r, φ) t
2
(r, φ) ,
t
1
(r, φ) t
2
(r, φ) =
+∞
n,n=−∞
c
1
n
c
2
n
e
2iνπ[ng
1
(r,φ)+ng
2
(r,φ)]
.
(n, n) = m(k
1
, k
2
) k
1
k
2
m
f
k
1
,k
2
(r, φ) =
+∞
m=−∞
c
1
mk
1
c
2
mk
2
e
2iνπm
[
g
k
1
,k
2
(r,φ)
]
,
g
k
1
,k
2
(r, φ) = k
1
g
1
(r, φ) + k
2
g
2
(r, φ) ,
f
k
1
,k
2
(k
1
, k
2
)
f
1,−1
k
1
= 1 k
2
= −1
l
1
l
2
x
g
1
(r, φ) = α
x +
2
2
+ y
2
+ l
1
φ,
g
2
(r, φ) = α
x −
2
2
+ y
2
+ l
2
φ,
1
g
1,−1
(r, φ) = 2αx + ∆lφ,
= 2αr cos φ + ∆lφ,
∆l = l
1
− l
2
g
1,−1
= qπ q = 0, ±1, ±2, ... l
1
= l
2
= 0
2αx = qπ
l l
1
= l
2
∆l l
1
= l
2
g
1,−1
φ
∆l
l
1
l
2
g
1
(r, φ) = α
1
r
2
+ l
1
φ,
g
2
(r, φ) = α
2
r
2
+ l
2
φ.
g
1,−1
(r, φ) = ∆αr
2
+ ∆lφ,
∆l =
l
1
− l
2
∆α = α
1
− α
2
|l| = 1 |l| = 2
L
2
l = 0
638.2nm 7mW
DP
L
2
l = 0
LG X −Y
X
l = 1
DP L
2
l = 1
∆l = 0
l = 0 l = 1
L
2
∆l = 1
l
1
= l
2
= 1
l
1
= 1, l
2
= 0 l
1
= 1, l
2
= −1
l
1
= l
2
= 1 l
1
= 1, l
2
= 0 l
1
= 1, l
2
= −1
l = 1 l = −1
l = −1
DP
∆l = 2
l = 0 l =
1
L
2
l
l = 1
l = 0 l = 1
∆l = 2
∆l = 0, 1
l
1
= l
2
= 1 l
1
= 1, l
2
= 0 l
1
= 1, l
2
= −1
l
1
= l
2
= 1 l
1
= 1, l
2
= 0 l
1
= 1, l
2
= −1
|l| = 2
l
1
= l
2
= 2
l
1
= 2, l
2
= −2 ∆l = 4
∆l = 0
l
1
= 2
, l
2
=
−
2
l
1
= l
2
= 2 l
1
= 2, l
2
= −2
l
1
= l
2
= 2 l
1
= 2, l
2
= −2
P (t) =
↔
χ
(1)
E (t) +
↔
χ
(2)
E (t)
E (t) +
↔
χ
(3)
E (t)
E (t)
E (t) + ...
≡
P
(1)
+
P
(2)
+
P
(3)
+ ...,
↔
χ
(n)
n
ω
0
ω
1
ω
2
= ω
0
− ω
1
∇ ×
H =
J +
∂
D
∂t
,
∇ ×
E = −
∂
B
∂t
.
-
x
_
y
-
x
_
y
Cristal
Não Linear
w
b
w
b
w
i
w
s
wa
D =
0
E +
P ,
B = µ
0
H,
J
P
D
∇ ×
H =
J +
∂
∂t
0
E +
P
,
∇ ×
E = −
∂
∂t
µ
0
H
.
P =
0
↔
χ
(1)
E +
P
NL
,
↔
χ
(1)
↔
χ
(1)
J = σ
E σ
∇ ×
H = σ
E +
∂
∂t
E
+
∂
∂t
P
NL
= (1+
↔
χ
(1)
)
0
∇ × ∇ ×
E = −µ
0
∂
∂t
∇ ×
H
,
∇·
E = 0
∇
2
E = µ
0
σ
∂
E
∂t
+ µ
0
∂
2
E
∂t
2
+ µ
0
∂
2
P
NL
∂t
2
.
ω
n
E (r, t) =
n
ε
ω
n
e
i
(
k
n
·r−ω
n
t
)
,
ε
ω
n
E(r, t) ε
−ω
n
ε
∗
ω
n
(z)
ω
0
, ω
1
, ω
2
(i, j, k)
(x, y)
(z)
E
(ω
0
)
i
=
1
2
ε
(ω
0
)
i
e
(k
0
z−ω
0
t)
+ c.c.
,
E
(ω
1
)
j
=
1
2
ε
(ω
1
)
j
e
(k
1
z−ω
1
t)
+ c.c.
,
E
(ω
2
)
k
=
1
2
ε
(ω
2
)
k
e
(k
2
z−ω
2
t)
+ c.c.
.
P
NL
P
(2)
i
= χ
2
ijk
E
j
E
k
k ω
2
P
(ω
2
)
NL
k
= χ
2
ijk
ε
(ω
0
)
i
(z) ε
(ω
1
)∗
j
(z) e
i[(ω
0
−ω
1
)t−(k
0
−k
1
)z]
+ c.c. .
k
∇
2
E
ω
2
k
=
∂
2
E
k
∂z
2
=
−
k
2
2
ε
(ω
2
)
k
+ ik
2
dε
(ω
2
)
k
dz
2
e
i(k
2
z−ω
2
t)
+ c.c.
d
2
ε
k
dz
2
ik
2
dε
k
dz
.
d
dt
→ iω
dε
ω
2
k
dz
= −σ
2
µ
0
2
ε
(ω
2
)
k
(z) − i
ω
2
2
µ
0
2
χ
(2)
kij
ε
(ω
0
)
i
ε
(ω
1
)∗
j
e
i(k
0
−k
1
−k
2
)z
,
ω
2
= ω
0
− ω
1
k
2
2
= µ
0
2
ω
2
2
σ
2
0 1 2
2
ε
(ω
2
)
k
=
ω
2
n
2
A
(ω
2
)
k
,
n
2
2
dA
(ω
2
)
k
dz
= −
σ
2
2
µ
0
2
A
(ω
2
)
k
(z) − i
1
2
µ
0
ω
0
ω
1
ω
2
0
n
0
n
1
n
2
χ
(2)
kij
A
(ω
0
)
i
A
(ω
1
)∗
j
e
i(k
0
−k
1
−k
2
)z
dA
(ω
1
)
j
dz
= −
σ
1
2
µ
0
1
A
(ω
2
)
j
(z) − i
1
2
µ
0
ω
0
ω
1
ω
2
0
n
0
n
1
n
2
χ
(2)
jik
A
(ω
0
)
i
A
(ω
2
)∗
k
e
i(k
0
−k
1
−k
2
)z
,
dA
(ω
0
)
i
dz
= −
σ
0
2
µ
0
1
A
(ω
2
)
j
(z) − i
1
2
µ
0
ω
0
ω
1
ω
2
0
n
0
n
1
n
2
χ
(2)
ijk
A
(ω
1
)
j
A
(ω
2
)
k
e
−i(k
0
−k
1
−k
2
)z
.
dA
0
dz
= −iκA
1
A
2
e
−i∆kz
,
dA
1
dz
= −iκA
0
A
∗
2
e
i∆kz
,
dA
2
dz
= −iκA
0
A
∗
2
e
i∆kz
,
0, 1, 2 ∆k =
k
0
− k
1
− k
2
κ =
µ
0
ω
0
ω
1
ω
2
0
n
0
n
1
n
2
χ
(2)
.
A
1
A
2
A
0
∆k = 0
A
0
(0) = A
0
≡ cte,
A
1
(0) = A
1
≡ cte,
A
2
(0) = 0.
A
1
(z) = A
1
cosh (A
0
κz) ,
A
2
(z) = iA
1
senh (A
0
κz) .
A
1
A
2
d
dz
|A
1
|
2
ω
1
=
d
dz
|A
2
|
2
ω
2
= −
d
dz
|A
0
|
2
ω
0
.
A
1
A
2
A
0
A
1
A
2
ω
0
ω
1
ω
2
ω
0
= ω
1
+ω
2
A
1
A
0
A
2
A
1
(0) = 0
A
1
= A
2
= 0
A
0
ω
b
ω
s
ω
c
-
x
_
y
-
x
_
y
Cristal
Não Linear
w
b
w
i
w
s
w
b
(0, 1, 2) (b, s, c)
ω
b
= ω
s
+ ω
c
,
k
b
=
k
s
+
k
c
,
ω
s
sen (β
s
) = ω
c
sen (β
c
) ,
β
s
β
c
P
i
(r, t) =
χ
(1)
ijk
(t
) E
i
(r, t − t
) dt
+
χ
(2)
ijk
(t
, t

) E
j
(r, t − t
) E
k
(r, t − t

) dt
dt

,
i, j, k
H
I
=
1
2
V c
E ·
P
NL
dr
=
1
2
V
c
drE
i
(r, t)
χ
(2)
ijk
(t
, t

) E
j
(r, t − t
) E
k
(r, t − t

) dt
dt

,
V
c
H = H
0
+ H
I
.
H =
H
0
+
H
I
.
E =
E
(+)
+
E
(−)
E
(+)
=
E
(−)
†
=
1
√
V
k,s
l (ω)
k,s
a
k,s
e
i
(
k·r−ωt
)
.
V
k,s
a
k,s
l (ω) = i
ω
k, s
2
0
n
2
k, s
1
2
.
H
I
=
1
V
3
2
k
b
,s
b
k
s
,s
s
k
c
,s
c
l (ω
b
) l
∗
(ω
s
) l
∗
(ω
c
) a
†
k
s
,s
s
a
†
k
c
,s
c
a
k
b
,s
b
e
i(ω
s
+ω
c
−ω
b
)
×
×
χ
(2)
ijk
k
b
,s
b
i
k
s
,s
s
∗
j
k
c
,s
c
∗
k
V
dre
−i
(
k
s
+
k
c
−
k
b
)
·r
+ h.c.,
h.c.
χ
(2)
ijk
= χ
(2)
ijk
(ω
b
= ω
c
+ ω
s
) + χ
(2)
ijk
(ω
s
= ω
b
− ω
c
) + χ
(2)
ijk
(ω
c
= ω
b
− ω
s
) ,
χ
(2)
ijk
(ω = ω
+ ω

) ≡
dt
dt

χ
(2)
ijk
(t
, t

) e
i(ω
t
+ω

t
)
.
b, s, c
t = t
0
U (t, t
0
) = exp
1
i
t
t
0
dτ
H
I
(τ)
.
t
|ψ (t) =
1 +
U
1
(t, t
0
)
|ψ (t
0
),
U
1
(t, t
0
) =
1
i
t
t
0
dτ
H
I
(τ) .
U
1
|υ(
k
b
)
υ(
k
b
)
ω
b
, ω
s
, ω
c
ω
b
= ω
s
+ ω
c
1
V
3
2
k
b
,s
b
k
s
,s
s
k
c
,s
s
−→
1
(2π)
3
2
d
k
b
d
k
s
d
k
c
,
U
1
=
χ
(2)
ijk
(2π)
3
2
d
k
s
d
k
c
Φ
k
s
,
k
c
×
sen [(ω
s
+ ω
c
− ω
b
) (t − t
0
) /2]
(ω
s
+ ω
c
− ω
b
) /2
a
†
k
s
a
†
k
c
,
Φ
k
s
,
k
c
=
d
k
b
υ
k
b
l (ω
b
) l
∗
(ω
s
) l
∗
(ω
c
)
3
m=1
sinc
k
s
+
k
c
−
k
b
m
L
m
2
,
L
m
υ
k
b
|ψ (t) ≈
1 + c
d
k
s
d
k
c
Φ
k
s
,
k
c
×
sen [(ω
s
+ ω
c
− ω
b
) (t − t
0
) /2]
(ω
s
+ ω
c
− ω
0
) /2
a
†
k
s
a
†
k
c
|vac,
c
|ψ (t) = α|vac+β
d
k
s
d
k
c
Φ
k
s
,
k
c
×
sen [(ω
s
+ ω
c
− ω
b
) (t − t
0
) /2]
(ω
s
+ ω
c
− ω
0
) /2
|1,
k
s
|1,
k
c
.
α β |ψ |α|
2
|β|
2
β
q
k
z
z
k = q + k
z
z.
|q| |
k|
k
bz
d
k
b
−→
dq
b
,
υ
k
b
≡ υ (q
b
) .
(L
x
, L
y
)
L
z
sinc
k
s
+
k
c
−
k
b
x
L
x
2
= sinc
k
s
+
k
c
−
k
b
y
L
y
2
≈ δ (q
s
+ q
c
− q
b
)
sinc
k
s
+
k
c
−
k
b
z
L
z
2
≈ 1.
Φ
k
s
,
k
c
υ (q
s
+ q
c
) .
|ψ = α|vac + β
dq
s
dq
c
υ (q
s
+ q
c
) |1, q
s
|1, q
c
,
r
s
r
c
G
(2)
= ψ|
E
(−)
(r
s
)
E
(−)
(r
c
)
E
(+)
(r
c
)
E
(+)
(r
s
) |ψ.
E
(+)
(r) =
dq a (q) e
i
ş
q·ρ+
√
k
2
−q
2
z
ť
.
r
k
(r = ρ + zz) (
k = q + k
z
z)
C (r
s
, r
c
) ∝ |
dq
c
dq
s
υ (q
s
+ q
c
) ×
exp
i
q
s
· ρ
s
−
q
2
s
ak
s
z
s

exp
i
q
c
· ρ
c
−
q
2
c
ak
c
z
c

|
2
.
C.C
C o m p l e m e ntar
Sinal
ns
W (ρ)
C (r
s
, r
c
) ∝ |
dρ W (ρ) exp
−i
k
b
2Z
0
|
R − ρ|
2
|
2
,
1
Z
0
=
k
s
k
b
1
z
s
+
k
C
k
b
1
z
c
,
R =
Z
0
z
s
k
s
k
b
ρ
s
+
Z
0
z
c
k
c
k
b
ρ
c
.
z = Z
0
C (r
s
, r
c
) ∝ |W
R, Z
0
|
2
.
z = Z
0
A
s
|ψ = α|vac + β
dq
s
dq
c
υ
b
(q
s
+ q
c
) |1, q
c
a
†
s
|υ
s
(q
s
),
υ
s
(q
s
)
q
s
υ
s
(q
s
)
|ψ
I(r
c
) r
c
I (r
c
) = ψ|
E
(−)
(r
c
)
E
(+)
(r
c
) |ψ,
I(r
c
) ∝
dρ|W (ρ) |
2
+ |
dρW (ρ) W
∗
s
(ρ) e
i
(
| ρ
c
−ρ|
2
k
c
2z
)
|
2
.
r
c
I(r
c
) ∝ |
dρW (ρ) e
i
(
| ρ
c
−ρ|
2
k
c
2z
)
|
2
,
z
I(r
c
) ∝ |
dρW
∗
s
(ρ) e
i
(
| ρ
c
−ρ|
2
k
c
2z
)
|
2
,
I(r
c
) ∝ |
dρ
LG
l
p
(ρ)
b
e
i
(
| ρ
c
−ρ|
2
k
c
2z
)
|
2
,
I(r
c
) ∝ |
dρ
LG
l
p
(ρ)
∗
s
e
i
(
| ρ
c
−ρ|
2
k
c
2z
)
|
2
.
LG
l
p
(ρ)
β mm
200 mW
442 nm
845 nm 925 nm
5ns
nm
µm
2
845 nm 150 mW
LG
l=+1
0
l = +1
LG LG
LG
1
0
200×200µm
2
LG
l=+1
0
30 × 30
LG
l=+1
0
LG
LG
l
b
= +1
×
LG LG
l
b
= l
s
+ l
c
l
b
= +1 l
s
= 0 l
c
= +1
LG
l
s
= +1 l
b
= 0 l
c
= −1
l
c
= −1
LG
l
c
=+1
0
LG
l
c
=−1
0
l
c
= −l
s
LG
C.C
BBO
Z1s
Z4i
Z2s
Z3s
Bombeio
Abertura
L
351, 1nm
(x, y)
z
3s
z
1s
+z
2s
+z
4i
f
-
x
_
y
Cristal
Não Linear
w
b
-
x
_
y
C
wC
wS
A1
A2
A1
A2
A1 A2
L2
L1
Z3s
Z0b
Z2c
Z4s
Z1c
C.C
G2
G1
G
1
z
0
G
2
G
(2)
(ρ
s
, ρ
i
) = ψ|
E
(−)
(ρ
s
)
E
(−)
(ρ
c
)
E
(+)
(ρ
c
)
E
(+)
(ρ
s
) |ψ,
= |vac|
E
(+)
(ρ
c
)
E
(+)
(ρ
s
) |ψ|
2
,
|ψ
|ψ = α|vac + β
dq
s
dq
c
υ
z
0
b
(q
s
+ q
c
) |1, q
c
a
†
s
|1, q
s
.
υ
z
0
b
(q
s
+ q
c
)
z
0
υ
z
0
b
(q
s
+ q
c
) ∝ υ
0
b
(q
s
+ q
c
) exp
−i
q
2
b
z
0
2k
b
.
E
(+)
ρ
c
, z
1c
+ z
+
2c
= G
2
(ρ
c
)
E
(+)
ρ
c
, z
1c
+ z
−
2c
.
− + z
G
2
(ρ
c
)
E
(+)
c
(ρ
c
) = G
2
(ρ
c
)
d
ρ

c
d
ρ
c
d
q
c
a
c
q
c
exp
i
q
c
·
ρ
c

× exp
i
|
ρ

c
−
ρ
c
|
2
k
c
2z
1c

τ
1
ρ

c
× exp
i
|ρ
c
−
ρ

c
|
2
k
c
2z
2c

,
k
c
τ
1
ρ

c
L
1
f
τ
1
ρ

c
= exp
−i
ρ
2
c
k
c
2f
.
E
(+)
s
(ρ
s
) =
d
ρ

s
d
ρ
s
d
q
s
a
s
q
s
exp
i
q
s
·
ρ
s

× exp
i
|
ρ

s
−
ρ
s
|
2
k
s
2z
3s

τ
2
ρ

s
× exp
i
|ρ
s
−
ρ

s
|
2
k
s
2z
4s

.
|ψ
vac|
E
(+)
s
(ρ
s
)
E
(+)
c
(ρ
c
) |ψ =
G
2
(ρ
c
)
d
ρ

s
d
ρ
s
d
ρ

c
d
ρ
c
d
q
s
d
q
c
υ
z
0
b
(q
s
+ q
c
)
×exp
i
|
ρ

s
−
ρ
s
|
2
k
s
2z
3s

exp
−i
ρ
2
s
k
s
2f
×exp
i
|ρ
s
−
ρ

s
|
2
k
s
2z
4s

exp
i
q
s
·
ρ
s

×exp
i
|
ρ

c
−
ρ
c
|
2
k
c
2z
1c

exp
−i
ρ
2
c
k
c
2f
×exp
i
|ρ
c
−
ρ

c
|
2
k
c
2z
2c

exp
i
q
c
·
ρ
c

.
ρ

c
ρ
c
ρ

s
ρ
s
vac|
E
(+)
s
(ρ
s
)
E
(+)
c
(ρ
c
) |ψ ∝
G
2
(ρ
c
)
dq
s
dq
c
υ
z
0
b
(q
s
+ q
c
)
×exp
iq
2
s
1
4α
s
−
z
3s
2k
s

exp
iq
2
c
1
4α
c
−
z
1c
2k
c

×exp
−i
k
s
2z
4s
q
s
· ρ
s
exp
−i
k
c
2z
2c
q
c
· ρ
c
,
α
j
, j = s, c
α
j
=
k
j
2f
−
k
j
2z
2j
.
ρ
s
ρ
s
q
z
ms
= z
mc
, m = 1, 2
k
s
= k
c
= k
α
s
= α
c
= α.
1
f
=
1
O
+
1
I
O = z
0
+ z
1
I = z
2
vac|
E
(+)
s
(ρ
s
)
E
(+)
c
(ρ
c
) |ψ ∝ G
2
(ρ
c
)
dq
s
dq
c
υ
z
0
b
(q
s
+ q
c
)
×exp
iB
q
2
s
+ q
2
c

exp
−i
O
I
(q
s
· ρ
s
+ q
c
· ρ
c
)
,
B
B =
1
4α
−
z
1
k
.
u = q
s
+ q
c
,
v = q
s
− q
c
,
vac|
E
(+)
s
(ρ
s
)
E
(+)
c
(ρ
c
) |ψ ∝ G
2
(ρ
c
)
×
dv exp
i
B
2
v
2
exp
−i
O
I
v ·
ρ
c
− ρ
c
2

×
du υ
0
b
(u) exp
−i
O
I
u ·
ρ
c
+ ρ
c
2

u
v
C ( ρ
s
, ρ
c
) ∝ |G
2
(ρ
c
) W
b
−( ρ
s
+ ρ
c
)
2
|
2
,
W
b
G
1
W
b
= ε
0
G
1
ε
0
C ( ρ
s
, ρ
c
) ∝ |G
2
(ρ
c
) G
1
−(ρ
s
+ ρ
c
)
2
|
2
.
L2
L1
L3
Z5s
Z1c
Z2c
Z3c
Z6s
Z7s
Z4c
C.C
G1
G2
G
1
G
2
E
(+)
c
(ρ
c
, z
1c
+
) =
d
ρ
c
d
ρ

c
d
ρ

c
d
ρ

c
d
q

c
a (q

c
) exp [iq

c
· ρ

c
]
×exp
i|ρ
c
− ρ
c
|
2
k
c
2z
4c
τ
2
(ρ
c
) exp
i|ρ
c
− ρ

c
|
2
k
c
2z
3c
G
1
(ρ

c
)
×exp
i|ρ

c
− ρ

c
|
2
k
c
2z
2c
τ
1
(ρ

c
) exp
i|ρ

c
− ρ

c
|
2
k
c
2z
1c
,
τ
1
(ρ
c
) τ
2
(ρ

c
) L
1
L
2
G
1
ρ
c
, ρ

c
, ρ

c
z
4c
= z
1c
= f z
2c
= z
3c
= 2f f
E
(+)
c
(ρ
c
, z
1c
+
) ∝
d
q

c
a
q

d
ρ

c
G
1
ρ

c
exp
i
q
 2
c
2k
c
×exp
i
ρ

c
ρ
c
f
+
q

c

.
E
(+)
s
(ρ
s
, z
7s
+
) =
d
ρ
s
d
ρ

s
d
ρ

s
d
q

s
a (q

s
) exp [iq

c
· ρ

s
]
×exp
i|ρ
s
− ρ
s
|
2
k
c
2z
5s
G
2
(ρ
s
) exp
i|ρ
s
− ρ

s
|
2
k
c
2z
6s
τ
3
(ρ

s
)
×exp
i|ρ

s
− ρ

c
|
2
k
c
2z
7s
,
ρ
s
, ρ

s
, ρ

s
z
5s
= f
z
6s
= z
7s
= 2f
E
(+)
s
(ρ
s
, z
7s
+
) ∝ G
2
−ρ
s
×
d
q

s
a (q

s
) exp [−iq

s
· ρ

s
] exp
−i
q
 2
s
z
5s
2k
s
.
ρ
c
ρ
s
vac|
E
(+)
s
(ρ
s
)
E
(+)
c
(ρ
c
) |ψ =
G
2
(−ρ
s
)
d
ρ

c
G
1
(ρ

c
)
d
q

s
d
q

c
exp
−i
q
 2
s
z
5s
2k
s
exp [−iq

s
· ρ
s
]
×exp
−i
ρ

c
ρ
c
k
c
f
+ q

c

υ (q
s
+ q
c
)
υ (q

s
+ q

c
)
vac|
E
(+)
s
(ρ
s
)
E
(+)
c
(ρ
c
) |ψ =
G
2
(−ρ
s
)
d
ρ

c
G
1
(ρ

c
) exp
−i
ρ

c
k
2f
exp
i
ρ

c
·
ρ
c
k
f
×
du
dv exp
−i
ρ

c
·
u + v
2

exp
i
f
2k
u · u + v
exp
−iρ
s
·
u −v
2

υ (u) .
u v
vac|
E
(+)
s
(ρ
s
)
E
(+)
c
(ρ
c
) |ψ =
G
2
(−ρ
s
)
d
ρ

c
G
1
(ρ

c
) υ
k
f
ρ

c
− ρ
s
exp
−i
k
ρ

c
ρ
c
f
.
υ
k
f
ρ

c
− ρ
s
−→ δ
ρ

c
− ρ
s
C ( ρ
s
, ρ
c
) ∝ |G
2
(−ρ
s
) G
1
(ρ
s
) |
2
.
L
1
(LiIO
3
) mm
nm
nm
nm
G
1
G
2
G
1
mm G
2
mm
G
1
mm G
2
mm
f = 15cm
G
1
f
p G
1
p
2
G
1
L
1
L
2
f
2f G
2
D
2
2f
G
1
×G
2
mm
2
G
1
mm
mm G
2
mm
mm s
mm
mm mm
mm
G
1
G
2
F = F
0
cos
2
πx
1.6mm
cos
2
πx
1.2mm
.
G
1
mm
G
1
mm
G
2
mm
mm
mm
mm
mm
s
mm
F = F
0
cos
2
πx
7.8mm
,
mm
mm
D
2
L
1
f G
2
2f
L
2
G
1
G
2
D
1
L
3
L
1
L
2
G
1
G
2
G
1
mm G
2
mm
mm
s
mm
mm
mm
G
1
mm G
2
mm
A0
In
A j
out
c
((2)
R
T
j
j
Rmax
Rmax
(a)
c
((2)
(b)
ω
0
= ω
1
+ ω
2
,
k
0
=
k
1
+
k
2
.
0, 1, 2
P
(ω
0
)
0
(r) =
0
χ
2
ε
1
(z) ε
2
(z) e
−
(
x
2
+y
2
)
w
2
0
e
−i∆kz
,
P
(ω
1
)
1
(r) =
0
χ
2
ε
0
(z) ε
∗
2
(z) e
−
(
x
2
+y
2
)
w
2
1
e
i∆kz
,
P
(ω
2
)
2
(r) =
0
χ
2
ε
0
(z) ε
∗
1
(z) e
−
(
x
2
+y
2
)
w
2
2
e
i∆kz
.
w
j
1
w
2
j
=
1
w
2
k
+
1
w
2
l
,
j = k k = l {j, k, l}{0, 1, 2} ε
j
(z)
z
0
χ
2
∆k
w
dA
0
dz
= −κA
1
A
2
e
−i∆kz
,
dA
1
dz
= κA
0
A
∗
2
e
i∆kz
,
dA
2
dz
= κA
0
A
∗
1
e
i∆kz
,
A
j
(ω
j
, r) =
n
j
0
cπw
2
j
4ω
j
ε (ω
j
, r) ,
κ = χ
(2)
w
0
w
1
w
2
w
2
0
w
2
1
+ w
2
0
w
2
2
+ w
2
2
w
2
1
ω
0
ω
1
ω
2
π
0
c
3
n
0
n
1
n
2
|A
j
|
2
l
A
0
(l) = A
0
(0) − gA
1
A
2
e
−i∆kl
sinc
∆kl
2
,
A
1
(l) = A
1
(0) + gA
0
A
∗
2
e
i
∆kl
2
sinc
∆kl
2
,
A
2
(l) = A
2
(0) + gA
0
A
∗
1
e
i
∆kl
2
sinc
∆kl
2
.
g = κl sinc
r
j
t
j
j ∈ {0, 1, 2}
A
j
A
0
= r
0
e
iϕ
0
[A
0
− g
∗
(∆k) A
1
A
2
] + t
0
A
in
0
,
A
1
= r
1
e
iϕ
1
[A
1
+ g (∆k) A
0
A
∗
2
] + t
1
A
in
1
,
A
2
= r
2
e
iϕ
2
[A
2
− g (∆k) A
0
A
∗
1
] + t
2
A
in
2
.
g (∆k)
g (∆k) = g sinc
∆k
2
e
i
∆k
2
.
(∆k = 0)
ϕ
j
= k
j
(n
j
l + L) ,
k
j
=
ω
j
c
n
j
j L
δ
j
ϕ
j
= 2p
j
π + δ
j
; δ
j
2π .
A
0
1 − r
0
e
iδ
0
= −r
0
gA
1
A
2
+ t
0
A
in
0
,
A
1
1 − r
1
e
iδ
1
= r
1
gA
0
A
∗
2
,
A
2
1 − r
2
e
iδ
2
= r
2
gA
0
A
∗
1
,
r
j
= 1 − γ
j
.
r
j
= 1 − γ
j
+ µ
j
= 1 − γ
,
γ
= γ + µ
(∆
j
=
δ
j
γ
j
)
A
0
γ
0
[1 − i∆
0
] = −gA
1
A
2
+
√
γ
0
A
in
0
,
A
1
γ
1
[1 − i∆
1
] = gA
0
A
∗
2
,
A
2
γ
2
[1 − i∆
2
] = gA
0
A
∗
1
,
A
1
= A
2
= 0
A
1
A
2
γ
1
γ
2
[1 − i∆
1
] [1 −i∆
2
] = g
2
|A
0
|
2
.
∆
1
= ∆
2
= ∆ ,
γ
1
|A
1
|
2
= γ
2
|A
2
|
2
.
T
j
= 2γ
j
γ
j
j = 1
j = 2
I
j
= 2γ
j
|A
j
|
2
j
I
1
I
2
=
γ
1
γ
2
γ
2
γ
1
.
µ
1
= µ
2
|A
0
|
2
=
γ
1
γ
2
(1 + ∆
2
)
g
2
.
g
|A
0
|
2
lim
=
γ
2
0
γ
1
γ
2
(1 + ∆
2
0
) (1 + ∆
2
)
2g
2
γ
0
.
|A
0
|
2
limres
=
γ
2
0
γ
1
γ
2
2g
2
γ
0
.
σ =
|A
in
0
|
2
|A
0
|
2
limres
,
A
1
σ =
1 − ∆∆
0
+
g
2
|A
1
|
2
γ
0
γ
2
2
+ (∆ + ∆
0
)
2
,
|A
1
|
2
|A
1
|
2
=
γ
0
γ
2
g
2
σ − (∆ + ∆
0
)
2
+ ∆∆
0
− 1
.
|A
1
|
2
σ > (∆ + ∆
0
)
2
≡
σ
a
σ = (1 + ∆
2
)(1 + ∆
2
0
) ≡ σ
b
∆∆
0
< 1 σ > σ
b
∆∆
0
> 1 σ
a
< σ < σ
b
A
2
|A
2
|
2
=
γ
0
γ
1
g
2
σ − (∆ + ∆
0
)
2
+ ∆∆
0
− 1
.
LG
+1
0
M
1
M
2
R
e
= 13 mm
532 − 1064 nm
Z
(x, y) φ = 23.5
o
x
(R
IR
= 99, 8%@1064nm) (R
V
= 92%@532nm)
R = 0.1%@1064nm
R = 0.5%@532nm
nm
l = 1 LG
1
0
60 mW
M
1
M
2
D
G
M
2
D
IR
z
D
G
D
IR
mW
T EM
01
k
ef
∂
2
y
U (x, y) = 2ik
ef
∂
x
U (x, y) ,
U (x, y) =
k
ef
x
R
π2
2n
n!
2
(x
2
+ x
2
R
)
1
4
H
n
y
k
ef
x
R
x
2
+ x
2
R
×exp
−i
k
ef
y
2
2 (x + ix
R
)
− i
n +
1
2
arctan
x
x
R

,
x
R
H
n
(x)
n ≥ 0
n +
1
2
arctan
x
x
R
,
n
w
0
=
2x
R
/k
ef
R(x) = x(1 + x
2
R
/x
2
)
k
ef
1064nm
n
x
= 1.7404, n
y
= 1.7479, n
z
= 1.8296,
532nm
n
x
= 1.7797, n
y
= 1.7897, n
z
= 1.8877.
n
o
n
z
n
e
= 1.7467 nm n
e
= 1.7881 nm
L
0
= 17.4mm
l = 10mm
L = L
0
− l
k
ef
− k
0
k
ef
,
k
ef
L
o
y
= 12.87mm, L
o
z
= 13.40mm, L
e
y

= 13.17mm, L
e
z
= 13.12mm.
o(e)
x
2
R
= L
2
(2R
e
−L)/4
Φ = 4arctan
√
L
√
2R
e
− L
,
Φ
o
y
= 3.122rad, Φ
o
z
= 3.204rad, Φ
e
y

= 3.167rad, Φ
e
z
= 3.161rad.
T EM
mn
Φ =
m +
1
2
Φ
z
+
n +
1
2
Φ
y
.
T EM
01
T EM
10
nm mrad
mrad
nm
T EM
01
T EM
10
T EM
mrad
mrad
mrad
E
i
(r, z) =
m,n
A
i
m,n
(z) u
m,n
(r, z)
i
e
ik
i
z
,
A
i
m,n
(z) z ω
i
u
m,n
(r, z)
r
l = ±1
T EM
01
T EM
10
u
m,n
(r, z)
∇
2
⊥
u
m,n
(r, z) + 2ik
i
∂
∂z
u
m,n
(r, z) = 0
∇
2
⊥
∇
2
⊥
E
i
(r, z) + 2ik
i
∂
∂z
E
i
(r, z) = −ω
i
µ
0
P
i
NL
e
ik
i
z
,
P
i
NL
ω
i
d
dz
A
i
m,n
=
iω
i
µ
0
2k
i
d
2
r u
∗
m,n
(r, z) P
i
NL
e
−ik
i
z
.
P
0
NL
=
0
χ
2
0
E
1
(r, z) E
2
(r, z) e
i(k
1
+k
2
)
,
P
1,2
NL
=
0
χ
2
1,2
E
0
(r, z) E
∗
1,2
(r, z) e
i(k
0
−k
1,2
)
,
χ
2
i
i
0, 1, 2
E
0
=
p,l
A
0
p,l
(z) u
0
p,l
(r, z) ,
E
1
=
q,m
A
1
q,m
(z) u
1
q,m
(r, z) ,
E
2
=
p,l
A
2
r,n
(z) u
2
r,n
(r, z) ,
d
dz
A
0
p,l
=
iω
0
χ
2
n
0
c
qm,rn
Λ
lmn
pqr
∗
A
1
qm
(z) A
2
rn
(z) e
i∆kz
,
d
dz
A
1
q,m
=
iω
1
χ
2
n
1
c
pl,rn
Λ
lmn
pqr
A
1
qm
(z) A
2
rn
(z) e
−i∆kz
,
d
dz
A
2
r,n
=
iω
2
χ
2
n
2
c
pl,qm
Λ
lmn
pqr
A
1
qm
(z) A
2
rn
(z) e
−i∆kz
,
∆k
Λ
lmn
pqr
=
d
2
ru
0
p,l
(r, z) u
1∗
q,m
(r, z) u
2∗
r,n
(r, z) .
Λ
lmn
pqr
T EM
p,l
T EM
q,m
T EM
r,n
LG
0,1
T EM
0
0,1
T EM
0
1,0
u u
jk
(r, z) j = b, s, c b
s c k = 0, h, v
0 T EM
00
h T EM
10
v T EM
01
Λ
lmn
=
d
3
r u
bl
(r, z) u
∗
sm
(r, z) u
∗
cn
(r, z) ,
l, m, n 0, v, h
(v, 0, 0) (v, v, v)
(v, v, 0) (h, h, 0)
T EM
01
T EM
10
T EM
00
τ
da
bv
dt
= −[γ
b
+ i (∆
b
+ σ
b
)] a
bv
− igΛ
∗
v0v
a
s0
a
cv
+
E
in
√
2
,
τ
da
bh
dt
= −[γ
b
+ i (∆
b
− σ
b
)] a
bh
− igΛ
∗
h0h
a
s0
a
ch
+ i
E
in
√
2
,
τ
da
s0
dt
= −(γ + i∆
s
) a
s0
+ igΛ
v0v
a
bv
a
∗
cv
+ igΛ
h0h
a
bh
a
∗
ch
,
τ
da
cv
dt
= −[γ + i (∆
c
+ σ
c
)] a
cv
+ igΛ
v0v
a
bv
a
∗
s0
,
τ
da
ch
dt
= −[γ + i (∆
c
− σ
c
)] a
ch
+ igΛ
h0h
a
bh
a
∗
s0
.
Λ
lmn
σ
j
E
in
a
bv
a
bh
π/2
τ
b
jk
= gΛ
000
τa
jk
, x
in
= gΛ
000
τ
2
E
in
, η
lmn
=
Λ
lmn
Λ
000
,
˜γ
j
= γ
j
τ,
˜
∆
j
= ∆
j
τ, ˜σ
j
= σ
j
τ.
∆
s
=
∆
c
= 0 σ
b
= σc = 0
η
v0v
= ηh0h = η
0 = −
˜γ
b
+ i
˜
∆
b
b
bv
− iηb
s0
b
cv
+
x
in
√
2
,
0 = −
˜γ
b
+ i
˜
∆
b
b
bh
− iηb
s0
b
ch
+ i
x
in
√
2
,
0 = −˜γb
s0
+ iηb
bv
b
∗
cv
+ iηb
bh
b
∗
ch
,
0 = −˜γb
cv
+ iηb
bv
b
∗
s0
,
0 = −˜γb
ch
+ iηb
bh
b
∗
s0
.
l = ±1
b
j±
=
b
jv
± ib
jh
√
2
,
j = b, s, c
I
b−
= I
c−
= 0, I
b+
=
˜γ
2
η
2
,
I
s0
= I
c+
= I
0
≡
˜γ
η
2
η
2
x
2
in
˜γ
2
−
˜
∆
2
b
− ˜γ
b
,
I
jk
= |b
jk
|
2
I
0
= 0
x
2
L
=
˜γ
2
η
2
˜γ
b
+
˜
∆
b
.
η
vv0
= η
hh0
η
v0v
= η
h0h
= 0.71
η
vv
0
= 0.70 η
hh
0
= 0.60 η
v
0
v
≈ η
h
0
h
=≈ 0.71 η
hh
0
γ
b
= 145mrad γ = 5mrad
σ
b
= 4mrad σ
c
= 3mrad
I
0
∆
p
= 0.071γ
p
∆
s
= 0 ∆
i
= 1 mrad γ
p
= 145 mrad γ = 5 mrad σ
p
= 4 mrad σ
i
= 3 mrad
E
in
= 3E
L
η
v0v
= η
h0h
= 0.71
T EM
00
τ
da
bv
dt
= −[γ
b
+ i (∆
b
+ σ
b
)] a
bv
− igΛ
∗
vv0
a
c0
a
sv
+
E
in
√
2
,
τ
da
bh
dt
= −[γ
b
+ i (∆
b
− σ
b
)] a
bh
− igΛ
∗
hh0
a
c0
a
sh
+ i
E
in
√
2
,
τ
da
c0
dt
= −(γ + i∆
c
) a
c0
+ igΛ
vv0
a
bv
a
∗
sv
+ igΛ
hh0
a
bh
a
∗
sh
,
τ
da
sv
dt
= −[γ + i (∆
s
+ σ
s
)] a
sv
+ igΛ
vv0
a
bv
a
∗
c0
,
τ
da
sh
dt
= −[γ + i (∆
s
− σ
s
)] a
sh
+ igΛ
hh0
a
bh
a
∗
c0
.
σ
s
σ
s
= 41mrad
γ = 5mrad
h v
∆
s
= −σ
s
v a
sh
= 0
η
hh0
η
vv0
= η
I
sh
= 0, I
bh
=
x
2
in
/2
˜γ
2
+
˜
∆
2
b
I
bv
=
˜γ
2
η
2
,
I
c0
= I
sv
= I
0
≡
˜γ
η
2
η
2
x
2
in
2˜γ
2
−
˜
∆
2
b
− ˜γ
b
.
h
I
0
v
I
0
I
0
= 0
x
2
L
=
2γ
2
η
2
(γ
b
+ ∆
b
) .
∆
p
= 0.28γ
p
∆
s
=
−41 mrad ∆
i
= 0 γ
p
= 145 mrad γ = 5 mrad σ
p
= 4 mrad σ
s
= 41 mrad
E
in
= 1.5E
L
η
vv0
= η
hh0
= 0.70
T EM
01
T EM
10
λ/4 nm
λ/2
λ
ρ
(γ
− i∆
1
) A
1
= gA
0
A
∗
1
+ i2ρe
i(θ−ψ)
A
2
,
(γ
− i∆
2
) A
2
= gA
0
A
∗
2
+ i2ρe
−i(θ−ψ)
A
1
,
θ ψ
A
1
A
2
ρ
ρ = 0
R = 95% R
MAX
R
MAX
T = 5%
nm
mW nm
mW nm
C.D
YAG
Servo
12 Mhz
A.E
l/4
A.E
l / 2
A.E
l / 4
PBS1
l / 2
Espectrômetro
A.E
A.EA.EA.E
l / 2
FD Rápido
l / 2
λ/4
0, 01
o
mK
a )
b)
-4
-3
-2
-1
0
1
2
3
0 10 20 30 40 50
Temperatura(°C)
Sep aração entr e sinal e com plem ntar
(nm )
T ~33 Cdeg
o
T
DEG
T
DEG
33
0
C
T
DEG
3
o
C
1 GHz
A
1
A
2
A
+
=
A
1
+ A
2
√
2
, A
−
=
A
1
− A
2
√
2
.
A
+
A
−
Z.T.
a )
b )
λ/4
λ/4
45
o
λ/2 λ/4
a )
b )
A
+
A
−
A
+
A
−
MHz
dB
A
+
Q
+
A
−
P
−
A
1
A
2
Σ =
V (P
−
)
2
+
V (Q
+
)
2
≥ 1 ,
P
−
= P
1
−P
2
A
1
A
2
Q
+
= Q
1
+ Q
2
Σ = 0, 5 ,
Σ = 0, 3
dB
a )
b)
R
e
= 13mm
R = 92% @ 532 nm R = 99, 8% @ 1064 nm
mW
mW
T EM
00
s
i
i
s
T EM
00
LG
1
0
mW mW
LG
1
0
l
c
+ l
a
= l
b
mm
dB
Σ = 0.5
(x, y, z)
n
x
∼ n
y
= n
z
n
x(y,z)
x(y, z)
D
∇ ·
D = 0
E(r, t) =
E(r) e
iωt
∇
2
E −
∇
∇ ·
E −
∇ ·
D = 0
α
+ k
0
↔
ε
·
E = 0,
∇ ·
D = 0 k
0
= ω/c
ω α
D =
↔
ε
·
E
↔
ε
↔
ε
=
n
2
x
0 0
0 n
2
y
0
0 0 n
2
z
.
D
E α = n
2
y
n
2
x
n
2
y
∂
2
x
E
x
+ ∂
2
y
E
x
+ ∂
2
z
E
x
−
1 −
n
2
z
n
2
y
∂
x
∂
z
E
z
+ k
2
0
n
2
x
E
x
= 0,
∂
2
x
E
y
+ ∂
2
y
E
y
+ ∂
2
z
E
y
−
1 −
n
2
x
n
2
y
∂
y
∂
x
E
x
−
1 −
n
2
z
n
2
y
∂
y
∂
z
E
z
+ k
2
0
n
2
y
E
y
= 0,
∂
2
x
E
z
+ ∂
2
y
E
z
+
n
2
z
n
2
y
−
1 −
n
2
x
n
2
y
∂
z
∂
x
E
x
+ k
2
0
n
2
z
E
z
= 0.
n
x
= n
y
x
x, y φ x
φ = 23, 5
o
x
, y
x
= x cosφ + y senφ, y
= −x senφ + y cosφ, z
= z.
φ
S x
z E
x
= E
y
= 0 E
z
= 0
z
E
z
= E
0z
e
in
z
k
0
x
.
x, y
E
z
= 0
E =
E
0
e
i(k
x
x+k
y
y)
,
E
0
= E
0x
x + E
0y
y E
x
k
2
x
n
2
y
+
k
2
y
n
2
x
= k
2
0
,
x, y
k
x
= n k
0
cosφ k
y
= n k
0
senφ
cos
2
φ
n
2
y
+
sen
2
φ
n
2
x
=
1
n
2
,
n x
D =
↔
ε
·
E
∇ ·
D = 0
n
2
x
k
x
E
x
+ n
2
y
k
y
E
y
= 0.
n
x
= n
y
E
k
S
E
k φ
S
k φ
φ
tanφ
=
senφ cosφ
n
2
y
− n
2
x
n
2
x
cos
2
φ + n
2
y
sen
2
φ
.
z
z
∂
2
x
+ ∂
2
y
+
n
2
z
α
∂
2
z
+ k
0
n
z
E
z
−
α − n
2
x
α
∂
x
∂
z
E
x
−
α − n
2
y
α
∂
y
∂
z
E
y
= 0.
z
|n
x
−n
y
| |n
z
−n
y
|
α = n
2
|α − n
j
|
α
10
−2
, j = x, y,
n
x
→ n
y
E
z
= u
z
(x
, y
, z)e
−in
z
k
0
x
z
∂
2
y
u
z
+
n
2
z
n
2
∂
2
z
u
z
= 2in
z
k
0
∂
x
u
z
.
y
z
z
y
z u
z
u
z
(x
, y
, z) =
U
z
(x
, y
)V
z
(x
, z)
∂
2
y
U
z
= 2in
z
k
0
∂
x
U
z
,
n
2
z
n
2
∂
2
z
V
z
= 2in
z
k
0
∂
x
V
z
.
l
U
z
l/n
z
V
z
ln
z
/n
2
(x, y)
E
z
α = n
2
x
E
y
E
y
=
u
y
(x
, y
, z)e
−in
z
k
0
x
2ik
0
n
y
cos
2
φ +
n
2
y
n
2
x
sen
2
φ
1
2
[∂
x
u
y
+ tanφ
∂
y
u
y
] =
sen
2
φ +
n
2
y
n
2
x
cos
2
φ
∂
2
y
u
y
+∂
2
z
u
y
,
φ
y

= y
− tanφ
x
.
(x, y)
ξ
2
= sen
2
φ +
n
2
x
n
2
y
cos
2
φ,
ξ
2
∂
2
y

u
y
+ ∂
2
z
u
y
= 2ik
0
n
2
y
n
∂
x
u
y
,
y

/ξ
ξ n
x
n
y
z
z
(x, y)
(x, y) z
u
y
ξ
2
∂
2
y

U
y
= 2ik
0
n
2
y
n
∂
x
U
y
,
∂
2
z
V
y
= 2ik
0
n
2
y
n
∂
x
V
y
.
x
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