MIT非线性光学讲义IV

E4 ( z ) = i
Κ4 E2 (0) ei (φ1 − φ 3 ) sin γ z Κ2
E2 ( z ) = E2 (0)cos γz
104
Phase matching
∆k = k − k p =
Assume
→ → → → k4 − ( k1 + k2 − k3 ) =0.
ω1 = ω 2 = ω 3 = ω 4
103
∂ E( ω1 ) iω1µ oc 3 (3) * = ε o χeff E2 E3 E4e i∆kz . ∂z 2n1 2
Assume -
E1 , E3 large and not depleted E2 varying phase matching ∆k = 0
(Similar to upconversion in p. 89)
Self Phase Modulation and Self Focusing
Self Phase Modulation ω ω ω ω
P(3) (ω ) =
3 (3) ε o χ eff ( ω : ω , −ω , ω ) E( ω ) E * ( ω ) E (ω ) 4 3 2 (3) = ε oχ eff ( ω : ω , − ω , ω ) E( ω ) E( ω ) . 4
iωµ o c NL −i (k − k p )z ∂ 1 ∂ E+ E= P e ∂z vg ∂t 2n P NL = 3 2 ε o χ (3) E (ω ) E( ω ) 4
kp − k = 0
Process is phase matched. Wave propagation equation becomes,
with D=6.
ω4 = ω 1 + ω2 − ω 3
D * (3) ε o χeff (ω 4 : ω1 , ω 2 , − ω 3 ) E1 E2 E3 4
Other equations of ω1 , ω 2 , ω 3 for power conservation:
3 (3) * ε o χeff (ω 3: ω1 , ω 2 , −ω 4 ) E1 E2 E4 2 3 (3) * P(3) (ω 2 ) = ε o χeff ( ω 2 : −ω1 , ω 3 , ω 4 ) E1 E3 E4 2 3 (3) * P(3) (ω 1 ) = ε o χeff ( ω1 : −ω 2 , ω 3 , ω 4 ) E2 E3 E4 2 P(3) (ω 3 ) =
conversion)
∂ E4 * = iΚ 4 E1E2 E3 ∂z ∂ E3 * = i Κ 4 E1E2 E4 ∂z
Trial solution e
±γ z * γz E4 ∝ E3 e
"Phase conjugate with gain" Counter propagation
k1 k3 k2 k4
113
after propagation
t
longer wavelength move slower Can be used to compensate self-phase modulation induced chirp (if n2 >0)
Now consider a short pulse,
after propagation t t
longer wavelength move faster
ii
∂ 2n <0 ∂λ2 ∂ vg ∂ω
"anomalous dispersion" or "negative dispersion"
>0
⇒
longer wavelength, vg smaller shorter wavelength, vg larger
L
Optical thickness will be increased in the center if n2 > 0 , or decreased in the center if n2 < 0 .
n2 > 0 ,
convex lens effect,
focusing
n2 < 0 ,
concave lens effect,
φ=
n ω nω ωz z = o z + n2 I c c c
linear phase term
ϕ (t)
I
t ϕ (t)
linear phase
t
"Self Phase Modulation" Phase modulated ⇒ new frequency
108
δω ∝ − ⇒
∂ϕ ( t ) ∂t
2 + 2 noδ n n 2 = ( no + δ n )2 ≅ no
Where
2 = 1 + χ (1) , no 3 2 2 noδ n = χ (3) E( ω ) 4 3 (3) 2 χ E (ω ) δn = 8no
n = no + δ n 3 (3) 2 = no + χ E (ω ) 8no = no + n2 I n With I = o 2 εo 2 E , µo
§ Third Order Nonlinearity
- allowed fro any materials and weaker than χ - for isotropic media, χ
(3) (2)
.
is the lowest order nonlinear process.
Four-Wave Mixing (FWM) process: ω1 ω2 ω3 P(3) (ω 4 ) =
(Degenerate FWM)
k1 , k2 , k3 , k4 has identical length (in isotropic media)
Collinear Geometry
k1 k2 k3 k4
Non-collinear Geometry
k1 k3 k4
→ k 4 wave is "phase conjugate" wave propagate in backward → direction of k3 wave.
109
- chirp (frequency shift in time)
δω t
linear chirp (can be compressed) - spectral broadening (see attachment)
⇒
pulse compression
If nonlinear ϕ = 2. 5π
Backward Parametric Amplifier
E2
idler
Ei E1
E
s
signal
ωs + ωi = ω1 + ω2
Similar to phase conjugate and parametric amplification. system can have gain.
This
106
E2 (0) ,
We get
E4 (0) = 0 .
E4 ( z ) = C sin γz , E2 ( z ) = E2 (0) cos γz .
We can determine C by substitution
C =i
Κ4 E2 (0) ei (φ1 −φ 3 ) Κ2
e iφ i ≡
Ei Ei
k2
This phase matching condition corresponds to 2 Bragg processes:
k1 k3 k4
k2
k1 k3 k4
k2
105
→ → If k1 , k2 waves large and not depleted. (similar to down
no gain, no absorption
If χ (3) real: only modify phase velocity
Physically, this implies an intensity dependent index of refraction.
D = εo E + P = ε o E + ε oχ (1) E + P NL 3 2 = ε o (1 + χ (1) + χ (3) E( ω ) ) E 4 2 = εon E
∂n (group velocity) ∂λ
112
−1 n λ ∂ n −1 ∂ω ∂ k = vg = = − c c ∂λ ∂k ∂ω c λ ∂n ≅ 1+ n n ∂λ
∂ 2n • Second order: (dispersion) ∂λ2
2. 5 π 2π 1. 5 π π 0. 5 π t δω ϕ (t )
t
I
⇒
δω
Self Focusing
110
A Gaussian beam incidents into a nonlinear material with n = no + n2 I and length L.
r I
n = no + n2 I
107
n2 =
I. If n2 > 0, If n2 < 0, II.
3 (3) µ o . 2 χ εo 4 no
Intensity stronger, move slower. Intensity stronger, move faster.
Phase depends on I, i.e. phase varies in time.
Different parts of the pulse have different frequencies.
ϕ ( t)
t δω
earlier time
t
later time frequency up shift
t
frequency down shift
"Chirp" Two major effects:
defocusing
111
Self Trapping: Balance between self focusing an diffraction
( n2 > 0 )
Nonlinear Propagation with n2 Wave propagation equation for envelope:
In collinear case, k p = k − k + k = k
1 ∂ i 3ω (3) 2 ∂ E+ E= χ E E z v t 8 cn ∂ g ∂
Nonlinear term ⇒ phase shift in propagation. Now consider dispersion ε (ω ) or n( ω ) in material: • First order:
E( z, t ) e −i(ω ot −k (ω o ))z
Envelope Carrier
Fourier transform to frequency domain centered at ω o
E( z, ω − ω o )e ik(ω o )z
∂ E4 * = iΚ 4 E1E2 E3 ∂z ∂ E2 * = i Κ2 E1 E3Biblioteka BaiduE4 ∂z
with Trial solution e
ω i µ oc 3 (3) ε o χeff . 2 ni 2
± iγ z 2 2
γ 2 = Κ 2 Κ 4 E1 E3 γ = Κ2 Κ 4 E1 E3
Boundary condition
Phase matching
→ → → → k4 − ( k1 + k2 − k3 )
∆k = k − k p =
Envelope equations:
∂ E( ω 4 ) iω 4µ oc 3 * − i∆kz = ε o χ (3) eff E1E2 E3 e 2n4 2 ∂z ∂ E( ω 3 ) iω 3µ o c 3 (3) * − i∆ kz E1 E2 E4 e = ε o χ eff 2 n3 2 ∂z ∂ E( ω 2 ) iω 2 µ oc 3 (3) * E1 E3 E4 e i∆kz = ε o χeff 2n2 2 ∂z
−1 ∂ 2k λ3 ∂ 2n ∂ vg 1 ∂ vg = = = − 2 ∂ω 2 2πc 2 ∂λ2 ∂ω ∂ω vg
i
∂ 2n >0 ∂λ2 ∂ vg ∂ω
<0
"normal dispersion" or "positive dispersion"
⇒
longer wavelength, vg larger shorter wavelength, vg smaller