光电子技术基础5

1 n2 x12 x22 x32 1 (Isotropic Crystal)
When a strain S is applied in x direction, the index ellipsoid becomes:
1 n2 p11S x12 1 n2 p13S x22 1 n2 p12S x32 1 (Biaxial Crystal)
n1 n 1 2n3 p11S, n2 n 1 2n3 p13S, n3 n 1 2n3 p12S
Acousto-Optic Effects
➢ Strain Induced by An Acoustic Wave
An acoustic wave propagated in k direction can be expressed as: u(r,t) k u0 cos( t k r)
or Bii xi xi 1
Bii
1 r,ii
0 ii
Result in:
ε r＝B－1 ,
or
B
ε
1 r
ห้องสมุดไป่ตู้
or
equivalently: Bij r, jk
ik
(valid for any case)
➢ Perturbation to The Medium Polarization
When an acoustic wave applied, the medium polarization P=ε0 χ•E (χ = εr-1) will be changed to P+ΔP with ΔP= ε0 (Δχ)•E, and Δχ= Δεr. That is:
An acoustic wave generate a dynamic grating, so a input field Ein will be diffracted by the grating. In the grating region, the total field is (assume the
p12 p11 p13 p14 0
p31 p41
p31 p33 p41 0
0 p44
0 0
0
0 0
p66
p11 2
p12
0 0 0 0
00 00
p55 p14
p41 p66
Examples
➢ Fused Silica (Optical Fiber)
The index ellipsoid of fused silica without strain is:
Bij xi x j 1
When a strain S is applied, the index ellipsoid of the medium becomes:
（Bij＋Bij )xi x j 1 Bij pijklSkl (Linear Photoelastic Effect)
Both ΔB and S are symmetric tensors, so the photoelastic Eq. is simplified to:
e j(int kinr)
jkdif
dEdif ,i e j(dif tkdif r) c.c. drdif
1 2 0 4 t 2
e (0)
ii
j (t k r )
c.c.
E e E e c.c. j(intkinr) in ,i
j (dif t k dif r) dif ,i
Acousto-Optic Diffraction: Summery
then the coupled wave Eqs. can be reduced to:
Ai z
ni c
Ai t
e
j (
t ki
z)
j0
ki
2 (Pi ) t 2
Acousto-Optic Diffraction: The Wave Equations
➢ The Total Field in The Grating Region
The strain tensor S induced by the wave is:
Sij S ji 1 2 ui x j u j xi
➢ Dynamic Index Gratings Induced by Acoustic Waves
Consider an acoustic wave propagated in z direction in an isotropic medium:
Coupled Wave Equations
➢ The Wave Equations
With an acoustic wave applied, the medium polarization P will be changed to P+ΔP. The corresponding wave eqs. are:
2 Ei t 2
0
2 (Pi ) t 2
with Pi 0ij E j ni2n2j 0 pijkl Skl E j
The wave equations for the three field components are coupled together.
➢ The Slowly Varying Envelope Approximation
n1 n 1 2n3 p11S, n2 n3 n 1 2n3 p12S
Applications: polarization controller; polarization maintained fiber.
➢ Ⅲ-ⅤCompound Semiconductors
The index ellipsoid without strain is:
Bi pij S j , (i, j 1, 2, 3, 4, 5, 6)
pij is the photoelastic coefficient matrix of the medium. The rule for the indices arrangement is as following:
Ai Aii (i 1, 2, 3), A4 A23 A32, A5 A13 A31, A6 A12 A21 ➢ Photoelastic Coefficient Matrix for Some Materials
dE dE in,i
j(kdif kkin )r
dif ,i
j(kdif kkin )r
j E e , j E e in dif ,i
dif in,i
dr dr in
dif
with:
1
2
in(dif ),i 2 0 ii in(dif )
For this case, the phase matching condition is:
Consider a optical field propagating along z, and it’s amplitude is varied in z:
Ei (z, t) Ai (z, t)e j( tkiz) (i 1, 2) If Ai(z,t) varies slowly, i.e.: 2 Ai z2 ki Ai z , and 2 Ai t 2 Ai t
system can be written as:
Pi
1 2
e (0) j( tkr) ii
c.c. Ei
➢ The Wave Equations
For quasi CW waves (∂E/ ∂t ≈ 0), the wave Eqs. under SVEA is:
jkin
dEin,i drin
kin k dif k
When ωin= ωdif-ω, the phase matching condition is: kin k dif k
u3 (x3, t) Acos( t ks x3 )
The induced strain is:
S3 (x3, t) ks Asin( t ks x3 ), and S j 0 for j 3
Due to the acoustic wave, the index ellipsoid becomes:
P P P
r,ij r,ij r,ij , ij ij ij , ij r,ij
From
Bij r, jk
ik
and
＝n2
r,ij i ij
we obtained:
ij ni2n2j Bij ni2n2j pijkl Skl
And finally:
Pi 0ij E j ni2n2j 0 pijkl Skl E j
medium is isotropic without strain) :
E Ein Edif
1 2
ei
E e E e c.c. j(intkinr) in,i
j(dif t kdif r) dif ,i
➢ Perturbation in The Medium Polarization
An acoustic wave propagated in k direction can be expressed as:
u(r, t )
1 2
k
u0e
j (
t k r )
c.c.
induced strain: Sij
1 2
Si(j0)e j( tkr) c.c.
The perturbation in the medium polarization in the principal dielectric coordinate
➢ The Relationship Between εij and Bij