ch 11-integrated optics - san jose state university 11-integrate… · document info ch 11....
Post on 01-May-2018
222 Views
Preview:
TRANSCRIPT
Document info ch 11.
Integrated OpticsChapter 11
Physics 208, Electro-opticsPeter Beyersdorf
1
ch 11.
Dielectric Waveguides
Optical waveguides “patterned” onto class or crystals form the basis for optical integrated circuits that can perform most of the functions of free space optics, and some functions that free space optics cannot.
2
ch 11.
xy
Wave Equation in a Waveguide
Consider an isotropic material with translational symmetry along z and a refractive index profile in the x-y plane such as
Ampere’s law and Faraday’s law can be written in the form
and will have solutions of the form
! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! where β≡kz 3
!!" !H = i"#0n2(x, y) !E
!!" !E = #i"µ0!H
!E = !E(x, y)ei(!t!"z)
!H = !H(x, y)ei(!t!"z)
ns
nc
ch 11.
Solutions in the Substrate
The wave equation !! ! ! ! ! ! can be written as
where α2(x,y)≡n2(x,y)[k0x2+k0y2] is the transverse
component of the propagation vector such that
For a guided mode the fields should go to zero far from the waveguide core, thus α must be imaginary making E an exponentially decaying function in x and y. Thus
4!2 >
"2
c2n2
s
!2E " µ!"2E
"t2= 0
!!2
c2n2(x, y)! "2(x, y)! #2
"$E = 0
!E(x, y) =!
!0
!E("0)ein(x,y)!0!
x2+y2
ch 11.
Solutions in the Core
Since the fields decay exponentially to zero far from the core in the substrate, they must have a maximum value in the core (i.e. a point where the field gradient is zero, ∇E=0, and the laplacian is negative, ∇2E<0)
Thus with our phasor notation where ∇2→α2+β2 and with the constraint on α(x,y) and β from the wave equation
in the core we must have
5
!2 <"2
c2n2
c
!!2
c2n2(x, y)! "2(x, y)! #2
"$E = 0
ch 11.
Orthogonality of Modes
The longitudinal propagation constant β must satisfy
any number of modes, each with a different β satisfying this condition can propagate simultaneously, without interacting (i.e. without exchanging power).
6
!2
c2n2
s < "2 <!2
c2n2
c
ch 11.
Slab Waveguide Example
Consider a one dimensional slab waveguide, as is commonly used in solid state lasers. Taking n2>n3≥n1 i.e. we have a high index core for guiding the radiation
What do the waveguide modes look like?
7
ch 11.
Slab Waveguide Example
8
in material 1!2
!x2E(x, y) + (k2
0n21 ! "2)E(x, y) = 0
in material 2!2
!x2E(x, y) + (k2
0n22 ! "2)E(x, y) = 0
in material 3!2
!x2E(x, y) + (k2
0n23 ! "2)E(x, y) = 0
in material 1, 2 and 3 respectively
β larger than nsk0
is not physicalβ too small for field to decay at ±∞
ch 11.
Guided Modes
Consider a symmetric slab waveguide (ns≡n3=n1) with nc=n2 with thickness d
For guided modes
In the substrate the field is exponentially decaying and has the form
9
!2
c2n2
s < "2 <!2
c2n2
c
E(x > d) = E+e!q(x!d)
E(0 < x < d) = Ec [A cos hx + B sinhx]E(x < 0) = E!eqx
h≡α(x,y) in the core
q≡iα(x,y) in the substrate
ch 11.
TE Modes
If you consider a ray zig-zagging through the slab, the plane of zig-zagging defines a polarization direction. For a mode with an electric field transverse to this plane (a TE mode) the field can be expressed as
and must be continuous at the interfaces giving
10
Ey(x, y, z, t) = Ey(x)ei(!t!"z)
E(x > d) = C!cos hd +
q
hsinhd
"e!q(x!d)
E(0 < x < d) = C!cos hx +
q
hsinhx
"
E(x < 0) = Ceqx
ch 11.
TE Modes
Plugging each of
into the corresponding wave equations
in the three regions gives
11
h =!
n2ck
20 ! !2
q =!
!2 ! n2sk
20
E(x > d) = C!cos hd +
q
hsinhd
"e!q(x!d)
E(0 < x < d) = C!cos hx +
q
hsinhx
"
E(x < 0) = Ceqx
!2
!x2E(x, y) + (k2
0n2s ! "2)E(x, y) = 0
!2
!x2E(x, y) + (k2
0n2c ! "2)E(x, y) = 0
!2
!x2E(x, y) + (k2
0n2s ! "2)E(x, y) = 0
ns
nsnc
ch 11.
TE Modes
The H field must also be continuous across the interfaces. From Faraday’s law
with
gives
12
!!" !E = #i"µ0!H
!E = Ey(x)ei(!t!"z)j
This is the magnitude of the E field which is
already continuous across the boundaries
!H =i
"µ0
!i#
"Ey(x)ei(!t!"z)
#i +
dEy(x)dx
ei(!t!"z)k
$
The magnetic field has a component along the
direction of propagation
ns
nsnc
ch 11.
TE Modes
13
h sinhd! q cos hd = q cos hd +q2
hsinhd
Requiring H(z) be continuous across the boundaries gives
where h and q are functions of β. Thus guided modes can only exist for discrete values of β which satisfy this mode condition
unconfined modes, not propagating along z
non physical (kz>k)
ns
nsnc
nsnc
β
nck0
nsk0β
θi
θt
For light to leak out of coreβ<nsko
ch 11.
TM Modes
Analogous analysis of TM modes gives similar results. In terms of the magnetic field amplitude:
subject to the mode constraint
14
h sinhd! n2c
n2s
q cos hd =n2
c
n2s
q cos hd +!
n2c
n2s
"2q2
hsinhd
Hy(x > d) = C
!n2
sh
n2cq
cos hd +q
hsinhd
"e!q(x!d)
Hy(0 < x < d) = C
!n2
sh
n2cq
cos hx +q
hsinhx
"
Hy(x < 0) =n2
sh
n2cq
Ceqx
ch 11.
Cutoff Frequencies
Waves below a “cutoff” frequency will not be guided by the waveguide. Roughly speaking the waveguide dimensions must be larger than a wavelength. The requirements for guiding the mth mode is
15
ωc
d
!! m
2!
n2c " n2
s
For!! ! ! ! ! ! only a single mode can be guided d
!<
1!n2
c ! n2s
ch 11.
Mode Structure
Mode profiles in a symmetric 1D waveguide
Given a mode profile, how do you determine its mode number?
16
ch 11.
Integrated EOM devices
Two issues with free-space EOM devices can be largely eliminated by integrating the devices into waveguides
Half wave voltage can be much lower due to the small gap between electrodes allowing a much larger field
Interaction length can be much longer because the beam does not spread out as it propagates
17
waveguide
electrode
ch 11.
Dielectric Tensor Perturbation
Consider a perturbation to the dielectric tensor created by an externally applied field E.
this perturbation affects the propagation of modes in the waveguide. In an isotropic material
18
!! = ! + !! = "0""1 + r #E
!! = ! + !! = "0""1 + r #E = "0(" + !")"1
!! = !!0n4(x)(rE)
!0!!1(1 + !!!!1)!1 = !0!
!1 + r "E
!0!!1(1!!!!!1) = !0!
!1 + r "E
!!0!!1!!!!1 = r "E
!! = !!(rE)!!0
ch 11.
Mode Coupling
Consider the coupling between two modes (m and n)introduced by this perturbation. For efficient coupling the modes must have similar propagation constants (βm≈βn) thus they must be orthogonally polarized with the same mode number (m=n). Considering only these two modesthe filed can be written as
where Am and Bn will be functions of z in the presence of mode coupling introduced by the external electric field
19
!E(!r, t) =!AmETE
m (x)e!i!T Em z + BnETM
n (x)e!i!T Mn z
"ei"t
ch 11.
Mode Coupling Equations
Coupled mode analysis (see slide 9.21) leads to a relation between Am(a) and Bm(z)
with
and
20
!! = !TEm ! !TM
n
“overlap integral”
dAm
dz= !i!mnBnei!!z
dBn
dz= !i!!mnAme"i!!z
!E(!r, t) =!AmETE
m (x)e!i!T Em z + BnETM
n (x)e!i!T Mn z
"ei"t
!mn ="
4
! !
"!E#TE
m · #ETMn dx
ch 11.
Overlap Integral
With the overlap integral
and the fields for the slab waveguide already studied where ETE is along y and ETE is along x it is the ε6 term that couples the modes:
if the modes are well confined so that they see an index n(x) that is primarily nc then
21
!6 = !!0n4(x)(r6kEk)
!mn ! "12n3
c(x)k0r6kEk
!mn ="
4
! !
"!E#TE
m · #ETMn dx
ch 11.
Phase Matched Coupling
The solutions to the mode amplitudes (subject to Am(0)=A0 and Bn(0)=0) when the modes are phase matched (Δβ=0) is
as with phase matching for acousto-optic interactions or second harmonic generation, imperfect phase matching reduces the maximum amount of power coupling between the modes
22
Am = A0 cos(!mnz)Bn = !iA0 sin(!mnz)
ch 11.
Phase Mismatched Coupling
If Δβ≠0 the modes drift out of phase reducing (and eventually inverting) the transfer of energy between modes, resulting in less than 100% conversion efficiency
Periodically reversing the sign of the perturbation can “quasi-phase match” the interaction allowing 100% conversion efficiency
23
ch 11.
Directional Couplers
Consider the coupling of two modes from different waveguides that are parallel to each other but fully separated. Power can be coupled form one to another in a process called “Directional coupling”
24
ch 11.
Waveguide Coupling
The electric field in a structure with two parallel waveguides (a and b) can be written
and we express square of the refractive index as the sum of three parts, that of the surrounding cladding (ns), that of core a (na), and core b (nb)
25
n2(x, y) = n2s(x, y) + !n2
a(x, y) + !n2b(x, y)
!E(x, y, z, t) = !A(z)Ea(x, y)ei(!t!"az) + !B(z)Eb(x, y)ei(!t!"bz)
ch 11.
Coupled Modes
The eigenmodes in each uncoupled waveguide obey the wave equation
giving
introducing the coupling between the waveguides this becomes (for waveguide a)
and likewise for waveguide bwith a b
26
!2Ea,b =n2
c2
!2
!t2Ea,b
Coupling term
0
!!2
!x2+
!2
!y2! "2
a,b
"Ea,b(x, y) = !#2
c2
#n2
s(x, y) + !n2a,b(x, y)
$Ea,b(x, y)
!!2
!x2+
!2
!y2!
!"2
a + 2i"adA
dz+
d2A
dz2
""Ea(x, y) = !#2
c2
#A n2
s(x, y) + A !n2a(x, y) + B !n2
b(x, y)$Ea(x, y)
ch 11.
Differential Equations
This coupled mode analysis leads to the following equations for the mode amplitudes A and B
with
27
dA
dz= !i!abBei(!a!!b)z ! i!aaA
dB
dz= !i!baAe!i(!a!!b)z ! i!bbB
!ab ! 14"#0
!E!
a · !n2a(x, y)Ebdxdy
!ba ! 14"#0
!E!
b · !n2b(x, y)Eadxdy
!aa ! 14"#0
!E!
a · !n2b(x, y)Eadxdy
!bb ! 14"#0
!E!
b · !n2a(x, y)Ebdxdy
Modification to βa due to Δnb
Modification to βb due to Δna
ch 11.
Alternative Form
28
Letting β’a→βa+κaa and β’b→βb+κbb gives a simpler form of coupled differential equations
For a symmetric waveguide (κ=κab=κba) this has the same form as the equations governing the acoustooptic interaction or second harmonic generation
dA
dz= !i!abBei(!!
a!!!b)z
dB
dz= !i!baAe!i(!!
a!!!b)z
ch 11.
Directional Coupler Solutions
With only a single input (A(0)=A0, B(0)=0) the power in the waveguides at z is is Pa(z)=A(z)*A(z) and Pb(z)=B*(z)B(z) and has the form
29
Pb(z) = P0!2
!2 +!
!!!
2
"2 sin2
#
$%
!2 +&
!"!
2
'2
z
(
)
Pa(z) = P0 ! Pb(z)
ch 11.
Practical Applications
For typical parameters, κ-1≈2mm
Uses for this type of coupler include
Amplitude modulator (Vary output of waveguide a or b by varying Δβ via EO effect)
EO switch (adjust Δβ via EO effect to change power coupling from 0 to 100%)
Wideband frequency filter (based on λ dependence of κ)
Narrow band frequency filter (based on resonant enhancement [as described in Pradeep’s talk])
Frequency division multiplexer
Integrated “beam splitter” 30
ch 11.
References
Yariv & Yeh “Optical Waves in Crystals” chapter 11
31
top related