2572-3
Winter College on Optics: Fundamentals of Photonics - Theory, Devices and Applications
Jiri Ctroky
10 - 21 February 2014
Institute of Photonics and Electronics AS CR, v.v.i., Prague Czech Republic
Introduction to Waveguide Optics
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Introduction to Waveguide OpticsJi í tyroký
Institute of Photonics and Electronics AS CR, v.v.i.,Prague, Czech Republic
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Where I come from…
Institute of Photonicsand Electronics AS CR, v.v.i.
Academy of Sciences of the Czech Republic is a non-university institution for basic and applied research consisting of 54 independent institutes
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
optical fibre channel optical waveguide
Examples of waveguide structures
photonic crystalwaveguide
etc…
loss
gain
“ARROW” (antiresonant reflecting OW)
“gain/loss” waveguide
subwavelength grating waveguide
longitudinally uniform
longitudinallyperiodic
metal
plasmonic waveguidemicroringresonator angularly
uniform
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Theoretical fundamentals of optical waveguides
• Planar waveguides; waveguide modes, their properties. Guided and leaky modes.Other types of waveguiding, guiding by a single interface
• Waveguide bends, whispering-gallery modes, circular resonators• Channel waveguides, approximate analytical methods.• More complex waveguide structures. Fundamentals of a rigorous coupled-mode theory• Introduction to modal methods; transfer matrix method, basics of the film mode matching• Periodic media, Bloch modes, origin of the bandgap, SWG waveguides, (photonic crystals)• “Canonical” waveguide structures: Y-junctions, directional coupler, two- and multimode
interference couplers, microresonators• Plasmonic waveguides and structures. Surface plasmon sensing. Hybrid dielectric-plasmonic
slot waveguide, plasmonic devices• Waveguide structures with loss and gain; asymmetric grating couplers, (“PT-symmetric”
waveguide structures)• …
Basic requirements: Theory of electromagnetic field, Maxwell equations
Tentative list of topics
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Basic math & phys backgroundDielectric (possibly also metallic) non-magnetic linear source-free medium,time-harmonic dependence of electromagnetic field:
02
02
0
0
, 0
, 0
i
i n
n
ωμ
ωε
εμ
∇ × = ∇ ⋅ =
∇ × = − ∇⋅ =
==
E H D
H E B
D EB H
{ }{ }
( , ) Re ( )exp( )
( , ) Re ( )exp( )
t i t
t i t
ωω
= −
= −
r E r
r H r
E
H2 20 0 0
2 20 0
2
1 (phase velocity)
1 (group velocity)
( ) group index
, typically larger than
g
g
g
k
k k k nk n
v cndk
v d c
d n dnn nd d
dnn nd
πω μ ελ
ε
ω
ωω ωω ω
λλ
= =
= =
= =
= =
= = +
= −
Plane-wave solution:0 0
0 02
2 2 2020
,
2
i i
i i
e e e
e e e
i n i
k
k
ε ε
ε
ε
′ ′′⋅ ⋅ − ⋅
′ ′′⋅ ⋅ − ⋅
= =
= =
′ ′′ ′ ′′= + = +
′ ′′ ′− =
′ ′′ ′′⋅ =
k r k r k r
k r k r k r
E E E
H H E
k k k
k k
k kcomplexwave vector
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Field at the interface between two media
iikrk
tk
z
x
iH
iE
TE polarization
rE
rH
tE
tH
, ,y x zE H H
iikrk
tk
z
x
iH
iE
TM polarization
rErH
tE
tH, ,y x zH E E
1n
2n
1n
2n
1 2n n
0 0 0 0 0 0e , e , e , e , e , ei t i tr ri i i ii ii i r r t t i i r r t t
⋅ ⋅ ⋅ ⋅⋅ ⋅= = = = = =k r k r k r k rk r k rE E E E E E H H H H H H
( ) ( )0 0 0 0 2 2 2 2 2 2, 0 1 , 0 2 1 , 1 2, , , i r i r t t i r t tk N k N N n N nγ γ γ γ= ± + = + + = + =k x z k x z
Plane wave incident on a planar interface
0 :x =Field continuity conditions at0 00 0 0 0
0 0 0 0 0 0, i ir r r ri z i zi z i z i z i zi r t i r tE e E e E e H e H e H e⋅ ⋅⋅ ⋅ ⋅ ⋅+ = + =k z k zk z k z k z k z
1 sini r t iN N N N n θ= = = = 2 2 2 21 1 2 2, n N n Nγ γ= − = −
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Fresnel coefficients1 2
1 2
2 2 2 21 22 2 2 21 2
TE r
i
ERE
n N n N
n N n N
γ γγ γ
−= =+
− − −=
− + −
2 21 1 2 2
2 21 1 2 2
2 2 2 2 2 22 1 1 22 2 2 2 2 22 1 1 2
TM r
i
H n nRH n n
n n N n n N
n n N n n N
γ γγ γ
−= =+
− − −=
− + −
total reflection
Brewster angle
critical angle
|RTE
|,|R
TM|
Angle of incidence (deg)
“region of interest”of waveguide optics
Angle of incidence (deg)
arg(
RTE
), ar
g(R
TM)
2 22
2 21
;
2arctan .
TETE i
TE
R e
N nn N
Φ=
−Φ = −−
2 2 21 2
2 22 1
;
2arctan .
TMTM i
TM
R e
n N nn n N
Φ=
−Φ = −−
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
The simplest waveguide: two (perfect) conductorsx
z
perfect conductors
d
0k k
R
RTwo equivalent physical pictures:1. a series of successive reflections of a plane wave2. two plane waves propagating upwards and downwardsIn both cases, the “nonzero wave” in the waveguide can exist only underthe “condition of transverse resonance”
0 00
,
,
,
,
( ),
1
1
y refTE
y inc
y refTM
y inc
k NE
RE
HR
H
γ± = ± +
= = −
= =
k x z
20 0exp(2 ) 1, or 2 2 , 0, 1, 2, R ik d k d m mγ γ π= = =
Waves can thus propagate only as discrete waveguide modes with propagation constants
( )22 2 2 2 2 20 0
TE TMm m m mk N k k k m dβ β γ π= = = − = −
The transverse field distributions have the forms ( )( )
0
0
( , ) sin exp( )
( , ) cos exp( )y m
y m
E x z E m x d i z
H x z H m x d i z
π β
π β
=
=
y
(for TM only)
… polarization degeneracy(except for m = 0)
2 m d λ>for … evanescent modes
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Dielectric slab waveguide
01 1( ) arg arg2 2tot w s cN k d R R mγ πΦ = + + =
0exp(2 ) 1,c s wR R ik dγ =
x
z0
d cR
sR
Condition of transverse resonance:
c s wn n n≤ ≤
snwn
cn 2 2 2 2, ,,w w c s c sn N i N nγ γ= − = −
Total reflection at both interfaces: ,w c sn N n> >
or
2 2 2 22 2
0 2 2 2 2arctan arctan ,w m s w m Cw m
s cw m w m
n N n n N nk d n N mn nn N n N
ν ν
π− −− = + +− −
0 (TE)2 (TM)
ν =
Dispersion equation for Nm shows polarization birefringence, TE TMm mN N>
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Effective thickness & “period of propagation”x
z0
deffd
GHaL
GHsL
pL
Total internal reflection is linked upwith the Goos-Hänchen shift, 0
1 (arg )GH
d d RLd k dNβ
Φ= − = −
( )2 2
,,c 2 2 2 2 2 20 0 ,c
2 2arctan ,( )
s cTEGHs
w s w
N nd NLk dN n N k N n n N
−= =
− − −
( )( ) ( ) ( )
2 22,
,c 2 2 20 ,
2 2 2 2, ,
4 2 2 4 2 22 2 2 2, ,0 ,
2 arctan
( )2 .
s cTM wGHs
s c w
w s c w s c
s c w w s cs c w
N nndLk dN n n N
n n n nNn n N n N nk N n n N
−= =
−
−=
− + −− −
0 sk γ
0 ak γ
( )0TEeff s cd d k γ γ= + +
( )02 20
2 2wtotp GSs GSc GSs GSc
w
d k dd NL L L d L Ld k dN n N
γβ
Φ= − = − + + = + +
−
(and a similar, somewhat more complicated expression for TM) – effective thickness
… “period of propagation”
These “ray-optic concepts” are useful also for inherently “wave-optic” phenomenon of optical waveguiding.
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Dispersion diagram (examples)
1,50
1,52
1,54
1,56
1,58
1,60
0 2 4 6 8 10
109
87
65
43
1
TM0
d/λ
Nef
f
TE0
2
1.50
1.52
1.54
1.56
1.58
1.60
0 2 4 6 8 10
109
87
65
43
2
Nef
f
d/λ
TE0 TM0
1
Asymmetric waveguide1 1.5 1.6c s wn n n= < = < =
Symmetric waveguide
1 5 1 6. .c s wn n n
Number of TE and TM modes identical,TE0 and TM0 always exist
All modes exhibit cut-off
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2
-0.2
-0.1
0.0
0.1
0.2
nc = 1,0
nw = 1,6
TE0
TE1
TE2
TE3
TE4
Mod
e fie
ld a
mpl
itude
x coordinate (μm)
ns = 1,5
substrate
guiding layer
cover
Distribution of modes of a slab waveguide
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Electromagnetic theory of a slab waveguide
1. TE polarization: , ,y x zE H H 2. TM polarization: , ,y x zH E E
0
0
20
( )( ) ,
( ) ( ),
( ) ( ) ( ) ( )
yz
x y
zx y
dE xiH xdx
H x E x
dH x i H x i n x E xdx
ωμβ
ωμ
β ωε
= −
=
− =
20
20
0
( )( ) ,
( )
( ) ( ),( )
( ) ( ) ( )
yz
x y
zx y
dH xiE xdxn x
E x H xn x
dE x i E x i H xdx
ωεβ
ωε
β ωμ
=
=
− = −
Planar waveguide as a structure with 1D permittivty distribution: ( ); 0xy
ε ∂ ≡∂
Eigenmode field: ( , ) ( )e , etc.i zy yE x z E x β=
( ) ( ) ( )2
2 2 202 y y
d k n x E x E xdx
β+ = ( )( )
( )2 2 2 202
1 ( ) ( )y yd dn x k n x H x H xdx dxn x
β+ =
Eigenvalue equations for eigenfunctions and eigenvalues ( ) or ( )y yE x H x 2β
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Analogy between a planar waveguideand a potential well in quantum mechanics
x
2 ( )n x
guided modes
substrate mode
radiation mode
20N
21N
x( )V x
0W1W bound states
reflection fromthe potential barrier
free particle
Field equation for a TE modein a planar waveguide
Schrödinger equation for a particle in a potential well
2 2 22 2
2 2 20
1 ( )( ) ( ) ( ) ( )2
yy y
d E d xn x E N E V x x W xmk dx dx
ψ ψ ψ+ = ⇔ − + =
0
2
2
( )
( )
( )
2
( )
y x
m
E x
k
n x
N
V x
W
ψ⇔
⇔
⇔
−⇔
−
There is not such an exact analogyfor TM polarization, but its behaviouris very similar
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Guiding of optical radiation in a dielectric waveguide
cn
wn
sn
0 0 0c s gk n k n k nβ≤ < <Guided modes:
Radiation (substrate) modes: 0 0 0c s gk n k n k nβ< < <
Radiation modes (superstrate): 0 0 0c s gk n k n k nβ < ≤ <
0
cn
wn
sn
guidedmode
substratemode
superstratemode
x
( )n x
guided mode
“substrate” mode
radiation mode
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Orthogonality of eigenmodesIt can be shown that fields of guided modes (from the discrete spectrum) are orthogonal,
00
1 ( ) ( ) , .2
mm n mn m m
mx x dx k Nβ δ β
β
∞
−∞
× ⋅ = =E H z
For lossless waveguides, E Hand of guided modes are real, and thus
012
*( ) ( ) .mm n mn
m
x x dxE H z
(Only) in this case, eigenmodes are also power-orthogonal:power carried by a superposition of (guided and non-evanescent radiation) modes
is given by the sum of powers of individual modes
For radiation and evanescent modes, the orthogonality condition sounds01 ( , ) ( , ) ( )
2 nx x dx ββ β δ β ββ
∞
−∞
′ ′× ⋅ = −E H z
Radiation and evanescent modes are always orthogonal to discrete guided modes:01 ( , ) ( ) 0,
2 nx x dxβ∞
−∞
× ⋅ =E H z
(in the senseof the principal valueof the integral)
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Leaky modesLeaky modes are not “true” eigenmodes of the waveguide structure but often allow fora simple description and physically understandable interpretation of wave propagationin waveguide structures with (weak) radiation loss
SOI planar waveguideradiating into the Si wafer
[ ]2 20
1 1arg ( ) arg (2
) ln (2 2
)S si c sk d n N R N R N i R N mπ− = − − + +
( )cR Nd
( )sR NSin
2SiOn
2c SiO Sin n n≈ <
Sin
tunelling
“Standard” dispersion relation(transverse resonance condition)
( )sR N …“frustrated” reflection coefficient , 1sR <
2 202( ) ( ) 1Siik d n N
c sR N R N e − =
SOI waveguide supports only leaky modes with complex effective indices, N N iN′ ′′= +
Waveguide with coupling gratingas a leaky-mode structure
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Another type of waveguiding – ARROW
M. A. Duguay et al., Appl. Phys. Lett. vol. 49, 13, 1986.
Antiresonant reflecting optical waveguide – ARROW)
Differences between an ARROW and a conventional index-guiding waveguide1. refractive index of the guiding layer can be lower than those of the grating2. the gratings must operate in the Bragg regime
(the 1D photonic crystal has to exhibit a bandgap)3. theoretically, ARROW with lower index guiding layer supports only leaky modes;
the number of periods has to be sufficient to suppress power leakage through the grating
waveguiding layer (“defect in the photonic crystal”)
Bragg grating (“1D photonic crystal”)
}}
Bragg grating (“1D photonic crystal”)
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Another type of waveguiding – ARROW
M. A. Duguay et al., Appl. Phys. Lett. vol. 49, 13, 1986.
Antiresonant reflecting optical waveguide – ARROW)
Differences between an ARROW and a conventional index-guiding waveguide1. refractive index of the guiding layer can be lower than those of the grating2. the gratings must operate in the Bragg regime
(the 1D photonic crystal has to exhibit a bandgap)3. theoretically, ARROW with lower index guiding layer supports only leaky modes;
the number of periods has to be sufficient to suppress power leakage through the grating
waveguiding layer (“defect in the photonic crystal”)
Bragg grating (“1D photonic crystal”)
}}
Bragg grating (“1D photonic crystal”)
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Guiding by a single interfaceAre two interfaces essential for waveguiding? (Typically yes, but…)
2 2 1 22 1 1 2 1 1 2 2
1 20; , .N N N ε εε γ ε γ γ ε γ ε
ε ε+ = = − = − =
+
d mε ε<
1. Surface plasmon-polariton at dielectric-metal interface
{ } { }, ImRe 0 0m mε ε< > Supported if dε
mε
power flow
2. Interface between media with a balance of loss and gain
{ } 22 Reg l l l
g l l
Nε ε ε ε
ε ε ε= = ≈
+
gε
lε
power flow
Pole of the reflection coefficient: possible only for TM polarization
Wave is confined if { } { }1 2Im 0, Im 0.γ γ> >
*, g l g lε ε ε ε≈complex,
3. Zenneck wave at the interface of a lossy and lossless media (1907)
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Circularly bent waveguide
M. Heiblum and J. H. Harris, IEEE JQE, vol. QE-11, pp. 75-83, 1975.
Bent waveguide:
Straight waveguide: 02 2
2 202 2 ( ) 0, ( , ) ( ) ik NyE E k n x E E x y E x e
x y∂ ∂+ + = =∂ ∂
( )( )
, ln
ln ,
iz x iy re w u iv R z R
u R r R r R v R
ϕ
ϕ= + = = + =
= ≈ − =
Conformal mapping:
y
x
gn
sn
sn
R
r( )
2 22 202 2 0, ( ) ( )r
E E k n r R E n r n r Rx y
∂ ∂+ + − = = −∂ ∂
x
( )n x
sngn
y
2 22 202 2 ( ) 0,eq
E E k n u Eu v
∂ ∂+ + =∂ ∂
Transformed equation:
( ) 0( ) 1 ( ), ( , ) ( ) ( )b bi v ik N Ru R u Req r
un u e n Re n u E u v E u e E u eR
β ϕ= ≈ + = =
r
( )rn r
sngn
R
Very large radius
u
( )eqn u
sngn
0
0N
0 u
( )eqn u
sn
gn
0
opticaltunelling
„whisperinggallery“ mode
0N 0N
internal interface“not necessary”bent guide
strong bend
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Ring resonator, or bent waveguide?
y
z
1r2r
1n
2n3n 0, ,xr E rE x
x
0; 0,x
∂ ≡∂
E xPolarization:
2 20 0( ) ,x xE k n r EHelmholz equation:
2 2 2 20 0( )
( )d d rr r k n r rdr dr Bessel equation
2 1 3,n n n>
( , ) ( )expxE r r i
angular propagation constant0 1J ( )A k n rν
0 2 0 2( ) ( )BJ k n r CY k n rν ν+(1)0 3H ( )D k n rν
Field continuity conditions: continuous at :( ), dr Hdr ϕψψ 1 2,r r
1 0 1 1 2 0 2 1 2 0 2 1
0 1 1 0 2 1 0 2 1
(1)2 0 2 2 2 0 2 2 3 0 3 2
(1)0 2 2 0 2 2 0 3 2
J ( ) J ( ) Y ( ) 0 0J ( ) J ( ) Y ( ) 0 0
00 J ( ) Y ( ) H ( )00 J ( ) ( ) H ( )
n k n r n k n r n k n r Ak n r k n r k n r B
Cn k n r n k n r n k n rDk n r Y k n r k n r
ν ν ν
ν ν ν
ν ν ν
ν ν ν
′ ′ ′− −− −
⋅ =′′ ′− −− −
.
Separation of variables:
(We are not going to solve this equation!)
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Ring resonator, or bent waveguide?( )det ( ) , 0 ;ν ω⋅ = Φ =Dispersion equation: Frequency introduced “on purpose”
We have two basic possibilities how to proceed:1. fix and seek for (complex) azimuthal propagation constant for a bent waveguide, or 2. fix (as an integer) and seek for (complex) resonant frequency of a ring resonator.
ω νν ω
1 3
3
1
2
1.6,1.7,10 μm,11 μm
n nnrr
= ====
bent waveguide ring resonator
[ ]0 1 (2 )i Qω ω= −exp( ) exp( )exp( )i iνϕ ν ϕ ν ϕ′ ′′= −
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Examples of field distributions1.6, 1.7, 10 μm, 1.55 μm s wn n r λ= = = =
diskring(Lossy) resonators
Eigenmodes of dielectric resonators are leaky modes!
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Channel waveguides
( )20 ,k x yε× × − =E E 0
( ) ( )10 lnε ε εε ⊥⋅ = ⋅ = − ⋅ = − ⋅E E E E
( ) 20ln kε ε⊥Δ + ⋅ + =E E E 0
Let’s separate transversal and longitudinal field components of an eigenmode:
( ) ( ) ( )2
02, , , , , i z i z i z
zx y e x y e x y ez z
β β β⊥ ⊥ ⊥
∂ ∂= = + ∇ = ∇ + Δ = Δ +∂ ∂
E e e e z
We obtain a 2D eigenvalue equation2 2 00( , ) ln ( , ) , z
ix y k x ye e e e e z e
Modes of channel waveguides are hybrid – all field components are generally nonzero
General considerations: vector equationx
y
z
( , )x yε
Weakly guiding waveguide: term is negligibly small; thenln e
2 20( , ) ( , )x y k x ye e e …essentially scalar equation, small.ze
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Marcatili method – separation of variables
( ) ( ) ( )!
2 2 2, x yn x y n x n y const= + +
Then , and( ) ( ) ( ), x ye x y e x e y=
( ) ( )
( ) ( )
22 2 202
22 2 202
2 2 2
0,
0,
xx x x
yy y y
x y
d e xk n N e x
dxd e x
k n N e xdy
N N N const
+ − =
+ − =
= + +
2
2 2
2
,
, 0 ,
, 0
a
x g
s
n x b
n n x b
n x
>
= < <
<
2gn 2
sn2sn
2sn
2an 2 22
s ga nn n+ −
222 gsn n−222 gsn n−
2 22s ga nn n+ −
x
ya0
b
2
2 2
2
, 0
, 0 ,
,
s
y g
s
n y
n n y a
n y a
<
= < <
>2gconst n= −
( ) ( ) ( )2 2 20, , , 0e x y k n x y N e x y⊥Δ + − =
A very simple approximate mode solving method for 2D waveguides
E.A.J. Marcatili, Bell System Tech Journal 48, 2071-2102, 1969
Approximation of weak guiding,
Simple solution if the profile were separable:
Subtracting from the permittivity in the corners makes the profile separable,2 2g sn n−
The task is reduced to solving 2 simple 1D equations, but results close to cut-off are questionable.
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Effective-index method for 2D profileSemi-intuitive method – reduction to a 1D problem
1) planar waveguide with a vertical profile,effective index N1 and mode field e1(x)
2) planar waveguide with a vertical profile,effective index N2 and mode field e2(x)
3) planar waveguide with a vertical profile,effective index N3 and mode field e3(x)
4) „planar“ waveguide with a lateral profileN1 , N2, N3 and the field e4(y)
The total field is approximately given by the product e(x,y) e2(x)e4(x).
Advantage: simplicity, clear physical interpretation. Disadvantage: inaccurate near cut-off, accuracy difficult to assess;
Well-applicable to shallow ridges and diffused channel waveguides, in many other cases very useful as a first guess. Commonly used method.
2sn
2cn
x
y
1) 2) 3)
2wn 4)
1( )e x 2( )e x 3( )e x
4( )e y
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x
y1y 2y ly
Effective-index method for diffused channel waveguides
( ) ( ) ( )2 2 20, , , 0e x y k n x y N e x y⊥Δ + − =
( ) ( ) ( ), ;x ye x y e x y e y≅
( ) ( ) ( ) ( )2
2 2 202
;; ; 0x
x x
d e x yk n x y N y e x y
dx+ − =
( )2xN y
22 2 202 0.y
x yd e y
k N y N e ydy
Efficient approximate method based on physical intuition.Suitable for graded-index waveguides and complex structures
Weak guiding, lateral dependence weaker:
Non-separable, but let us trywhere “the strong” y-dependence is concentrated in :( )ye y
is a parameter
Then we solve this equation for several values of and get as an effective lateral profile.In the next step, we solve the “lateral equation” to get and
y
y
( )y
N ( )ye y
What if there is no guided mode for some ? Take the substrate value instead of y sn ( )xN y
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More complex waveguide structures:“rigorous” formulation of the coupled-mode theory
( ) ( ) ( ) ( ), , , , , , ,i z i zx y z A x y e x y z A x y eμ μβ βμ μ μ μ μ μ= =E e H h
Mode orthogonality: 1 12 2S S
d d μμ ν μ ν μν
μ
βδ
β⊥ ⊥× ⋅ = × ⋅ =e h S e h S
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
, , , , , ( , , ) , , ,
, , , , , , , , , .
z z z
z z z
x z y a z x y b z x y E x y z a z e x y b z e x y
x z y a z x y b z x y H x z y a z h x y b z h x y
μ μ μ μ μ μ μ μμ μ
μ μ μ μ μ μ μ μμ μ
⊥ ⊥ ⊥
⊥ ⊥ ⊥
= + = −
= − = +
E e e
H h h
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
,
.
da zi a z K z a z K z b z
dzdb z
i b z K z a z K z b zdz
μμ μ μν ν μν ν
ν
μμ μ μν ν μν ν
ν
β
β
++ +−
−+ −−
= + +
= − + +
( )(0) , :x yεWave propagation in the permittivity distribution can be analyzed using the completenessand orthogonality of the eigenmodes of a waveguide with the permittivity profile
( , , )x y zε
D. Marcuse, Theory of dielectric optical waveguides, 2nd ed. Academic Press, 1991
Eigenmode fields:
Wave in can be expressed as ( , , )x y zε
From Maxwell equations we get the set of first order linear equations for complex amplitudes,
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Slowly varying amplitudes
( ) ( ) ( ) ( )
( )( ) ( )( ) ( ) ( ) ( )
00
000
*
1 for +, , ,
1 for
, , , ,4
,, , , ,
4 , ,
pq
S
z zS
K pK qk p q
iK z x z y x y dxdy
x yik z x z y x y dxdyx y z
μν μν μν
μμν μ ν
μ
μμν μ ν
μ
βωε ε εβ
β εωε ε εβ ε
⊥ ⊥
= + =− −
= − ⋅
= − ⋅
e e
e e
The coupling constants are given by overlap integrals
Introducing slowly varying amplitudes we obtain( ) ( ) ( ) ( ),i z i zA z a z e B z b z eμ μβ βμ μ μ μ
−= =
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
,
.
i z i z
i z i z
dAK z e A z K z e B z
dzdB
K z e A z K z e B zdz
μ ν μ ν
μ ν μ ν
β β β βμμν ν μν ν
ν
β β β βμμν ν μν ν
ν
− − − +++ +−
+ −−+ −−
= +
= +
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First-order (Born) approximation
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
00
00
,
,
z zi z i z
z zi z i z
dAdz K z e A z K z e B z dz
dz
dBdz K z e A z K z e B z dz
dz
μ ν μ ν
μ ν μ ν
β β β βμμν ν μν ν
ν
β β β βμμν ν μν ν
ν
− − − +++ +−
+ −−+ −−
≈ +
≈ +
( ) ( )
( ) ( )
0 0
0 0
( ) (0) (0) ( ) (0) ( ) ,
( ) (0) (0) ( ) (0) ( ) .
z zi z i z
z zi z i z
A z A A K z e dz B K z e dz
B z B A K z e dz B K z e dz
μ ν μ ν
μ ν μ ν
β β β βμ μ ν μν ν μν
ν
β β β βμ μ ν μν ν μν
ν
− − − +++ +−
+ −−+ −−
≈ + +
≈ + +
The integrals are “importantly non-zero” only if the integrands do not rapidly oscillate.Thus, if K is slowly vaying, the modes with close propagation constants are strongly coupled .Coupling of modes with significantly different ’s requires that this difference is compensatedby rapidly varying K(z).
Supposing slow variation of amplitudes and for small z
Let us formally integrate the set of equations:
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Two simple applications
( )( ) ( )i zdAK z e A z
dzμ νβ βμ
μν ν− −++≈
Let us preserve only “slow” terms satisfying the phase-matching condition:
For slowly-varying amplitudes w can approximately take 0 .( ) ( )A z A
Next, we will apply the Taylor expansion for and take only the first two terms:
( )0 0 0 0( ) ( ) ( ) ( ) ( ) ( ) ( ).g gN Ndd c
μ νμ ν μ ν μ ν μ νβ β β ω β ω β ω β ω ω ω β ω β ω ω ω
ω−
− ≈ − + − − = − + −
( ) ( )0
0 0
( )( ) ( ) ( ) .
(0)
g gN Nz z i zi z cA zT z K z e dz K z e dz
A
μ ν
μ νω ωβ βμ
μν μνν
−′− −′− −++ ++′ ′ ′ ′= ≈ ≈
μ νβ β−
The spectral dependence of the “transmission coefficient” is approximately given by the Fourier expansion of the spatial dependence of the coupling constant.
We define the “transmission coefficient” from mode to mode and get ν μ
1. Coupling of two forward modes
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Waveguide Bragg grating mirror
*
, ,1
( ),
( ), .
i zi d i d
i zd i d i
dA dz i e B z KdB dz i e A z iK
β
β
κ β β βκ κ
− Δ
Δ ++
= Δ = + −
= − =
iβ Λ
d iβ β≈ −
z L=
Boundary conditions:
00 0, .i i dA A B L
( ) ( )1 22,0
1* 2
,0
( ) cosh / 2 sinh , / 2 .
( ) coth2
i i
i z
d i
A z A z i z
B z i A e z iβ
δ δ δ β δ δ κ β
βκ δ δ
−
−Δ−
= − Δ = − Δ
Δ= −
222
0
02
( ) sinh
cosh / sinhd
i
B LR
A L i L
For 0,βΔ = 2 2tanh .R L
2. Coupling of forward and backward modes by a waveguide grating
2 , 0 2 , 2i d i iK K K
Nπ λβ β β= + − ΛΛ
Coupled equations:
Solution:
Grating reflectance:
0z =
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Spectral dependence of the reflectance
2B iNλ = Λ0.96 0.98 1.00 1.02 1.04
0.0
0.2
0.4
0.6
0.8
1.0
κ = 2,5×10-3k0
κ = 1×10-3k0
λ−λB/λB
Mod
al re
flect
ance
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Numerical methods for modelling waveguide structuresThree main classes of modelling methods:
1. Frequency-domain mode solvers for calculation of eigenmodes and propagation constants of straight and bent uniform waveguides; 1D, 2D, approximate, “rigorous”;scalar, semivectorial, full-vector; modal methods (Fourier modal method),discretization-based methods like FD, FE. BE, etc.
2. Frequency-domain “beam propagation” methods (BPM); in principle, scattering methodscalculating optical field distribution within a waveguide structure for a given excitation field; modal, FFT-BPM, FD-BPM, FE-BPM; 2D, 3D, scalar, full-vector; unidirectional, bi-directional (in fact, omnidirectional), etc.
3. Time-domain methods (FDTD , FETD,…): numerical model of optical field generatedby a given distribution of sources. Essentially, direct numerical solution of Maxwell equations.
… and many other special methods
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( )0
212
0 22 0
( ) ( ) 0,
0,00( )( ) ( ) ( ).
L
L L
f x f x
M Ng xg x g x M g x
= =
== ⋅
=M
For
The transfer matrix methodx
z
layer l
layer 1
layer L
0x
1x
lx
Lx
Matrices are even functions of !lγ
0
0 1
( )( ), .
( )( )
LL
lL l
f xf xg xg x =
= ⋅ = ∏M M M
Transformation over the whole multilayer structure:
dispersionequation
From Maxwell equations applied to a layer l we get
01
10
cos sin( ) ( )( ) ( )sin cos
l ly l y ll
z l z ll l j
ZiE x E xH x H xiY
ϕ ϕγ
γ ϕ ϕ
−
−
−= ⋅
−
2 20 1( ), ,l l l l l lk x x n Nϕ γ γ−= − = −
2
01
10 2
cos sin( ) ( )( ) ( )
sin cos
l ly l y ll
z l z lll l
niYH x H xE x E x
iZn
ϕ ϕγ
γ ϕ ϕ
−
−= ⋅
TE
TM
10 0 0 0Z Y μ ε−= =
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2D vectorial mode solver
Film mode matching (FMM)(straight guides: Sudbø 1993, 1994, bent guides Prkna 2004)
0
maxx
y
x
1y 2y 1Sy
1s s S2s
1n
2n
3n4n
n N
minx
waveguide cross-sectionTransversal refractive index distributionpiecewise constant
• Subdivide the cross-section intolaterally uniform “slices”; each slicerepresents a multilayer
• Find TE and TM modes of each slice• Express total field as superposition
of slice modes• Match fields at the boundaries of slices
Stable (impedance or scattering matrix)formalism; fully vectorial solution
Method of Lines (MoL) - requires 1D discretization, FD methodschool of prof. Reinhold Pregla, Fern-Universität Hagen, Germany
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An example: quasi-TE mode of a rib waveguide2.2, 1.9, 1, = 0.5 μmg s cn n n= = = =thickness height
Re{ }xE Re{ }yE Im{ }zE
Re{ }xH Re{ }yH Im{ }zH
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Film mode matching for circularly bent waveguides
1. subdivision of the structure into radially uniform “slices”,each “slice” forms a multilayer;
2. in each “slice”, mode field is expanded into TE and TM modes of a multilayer3. field matching at the interfaces between “slices”.
No (or minimum) discretizationField within the slice described analytically
Treatment analogous to the rectangular case;radial instead of lateral dependence.
Problem:Cylindrical functions of complex order instead of trigonometric functions.
0k r
0k x
max
min
2
1s2s 3s
1
L. Prkna, M. Hubalek, J. tyroký, IEEE Phot. Technol. Lett. vol. 16, 2057-2059 (2004).L. Prkna et al., IEEE J. Sel. Topics in Quantum Electr. vol. 11, 217-223 (2005)
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High-contrast SOI microresonator
2
Si
SiO
360 nm500 nm3.481.45
11.55 μm,
2 μm
c
hw
nn
n
R
λ
==
=
=
=
=
=
Quasi-TE00
Quasi-TM00
Quasi-TM10
rExE E
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Wave propagation in a periodic structure
2n 1n
z
x2 2 2 , 1,2.l lN n lγ= − =
Photonic analogy of the “Kronig – Penney” model of an electronic crystal
yHxE
zETM
yE
xHzHTE
1L2L tangential propagationconstant (given )
1 1 2 2sin sinn nγ θ θ= =1 2θ θor
longitudinalprop. constant
0
0
cos sin( ) (0)( ) (0)sin cos
l ly l yl
x l zl l j
ZiE L ENH L HiY N
ϕ ϕ
ϕ ϕ
−= ⋅
−
2
0
0 2
cos sin( ) (0)( ) (0)
sin cos
l ly l yl
x l xll j
niYH L HNE L ENiZ
n
ϕ ϕ
ϕ ϕ= ⋅
TE
TM
0l l lk N Lϕ =
lM
0, TE2, TM
ν =
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Electromagnetic Floquet – Bloch modesTransmission through layers 1 and 2 is described by matrices
The matrix for transitin over a single period is evidently 2 1Λ = ⋅M M M
Floquet-Bloch mode is determined as an eigenfunction and eigenvector of ,M
1 1 1 1
1 1 1 1
(0) (0) (0) (0), , ,
(0) (0) (0) (0)exp( )
F F F Fy y y yTE TMF F F F
x x x x
FE E H H
H H E Es s s iβΛ Λ⋅ = ⋅ = = ΛM Mor
is the prop. const.of the FB mode.FF is determined up to an additive constant 2 / ,K
it is sufficient to determine F in the interval 2 2/ /FK K first Brillouin zone.
1 2M Mand
2 1 1 21 2 1 2 0 1 2 2 1
1 21 2
2 1 1 20 2 1 1 2 1 2 1 2
2 1 2 1
cos cos sin sin sin cos sin cos
sin cos sin cos cos cos sin sin
n N n niYN Nn N
N N n NiZn n n N
ν ν ν
ν
ν
ν ν ν
ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ
ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ
Λ
− +
=
+ −
M
0, TE2, TM
ν =
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Eigenvalues and the photonic bandgap1 2 1 0 1 1 2 0 2 2, , ,L L k N L k N Lϕ ϕΛ = + = =Let us denote
22 2
1 2 1 2 1 2 1 22 21 1 1 1cos cos sin sin cos cos sin sin 1 .2 2
s ϕ ϕ ρ ϕ ϕ ϕ ϕ ρ ϕ ϕρ ρ
= − + ± − + −
1,sFB mode is propagating only if i.e., if
21 2 1 22
1 1cos cos sin sin 1.2
ϕ ϕ ρ ϕ ϕρ
− + ≤
The normalized wavenumber can be explicitly expressed as
21 1 2 2 2 1 1 2 2
1 1 12 2
arccos cos cos sin sin ./
FF N L N L N L N L
K c c c c
If 221 2 1 2
1 1 12
cos cos sin sin ,
F is complex, the wave is attenuated, and the photonic bandgap is created.
eigenvalues of the matrix Λ M are then
outside the bandgap,
within the bandgap
Fβ real
Fβ complex
1 2
2 1
n Nn N
ν
νρ =
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Band diagrams of a “1D crystal”
1
2
1
13.50
nnθ
=== º
1
2
1
11.50
nnθ
=== º
1
TE74θ = º 1
TM74θ = º
“valence”
“conduction”
“dielectric”
“air”
band
band
Brewster angle
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1
2
13.5
/ 0.6B
nn
λ λ
===
/ 1.0Bλ λ =
/ 1.75Bλ λ =
“dielectr.”band
“air”band
insidebandgap
near edgeof the BZ
Electromagnetic Floquet – Bloch modes
/ 1.5Bλ λ =
yE
xH
1n 2n 1n 2n 1n 2n 1n 2n
1n 2n 1n 2n 1n 2n 1n 2n
xH
yE
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3D analogue – subwavelength grating waveguides
, ,2lnB m mm B m
i Nπβλ
= − Γ =Λ
Propagation constant and the effective refractive index of the m-th Bloch mode:
Propagating modes in SWGW are Bloch modes
Grating constant of the SWGW: 2K π=Λ
“First Brillouin zone” of the SWGW as a 1D photonic crystal: 2B Kβ π< = Λ
For lossless propagation, 2s B BZn N nλ< < =
Λ
Group effective index: ,B g B BN N dN dλ λ= −In analogy with the behaviour of photonic crystals we expect that , 2B g BN N λ→ ∞ →
Λas
Region of represents the slow light region " 2BN λ
Λclose to"
(the “bandgap edge”)
( ),exp ,mm B miβΓ = Λ
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“Canonic” (elementary) waveguide devices
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Elementary waveguide structuresSymmetric Y-junction (1×2 power splitter)
1. Excitation into the single mode common arm
inP
1, / 2out inP P≤
2, / 2out inP P≤
Power is equally divided between the two output arms due to symmetry
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2. Excitation into a single arm
se( )
( )
( )
( )
1
2
1 2
1 2
1 ,21 ,2
1 ,2
1 .2
s a
s a
s
a
e e e
e e e
e e e
e e e
≈ +
≈ −
≈ +
≈ −
se ae
se
Symmetric Y-junction excited in the opposite direction
1e
2e
Basic waveguide structures
The antisymmetric mode canot propagate in the single-modeoutput arm and its power is radiated into the substrate
ae
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2. Excitation into a single arm
se
se
Symmetric Y-junction excited in the opposite direction
1e
Basic waveguide structures
Without the second arm, the transmittance is 100 %
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3. Excitation into both arms with an arbitrary phase shift between the armsse
se ae
se
( ) ( )/2 /2 /2 /21 2
1 2
1 1e e e e2 2
2 cos 2 sin ( )cos cos2 2 2 2
i i i iout s a s a
s a in
E e e e e e e
e i e e e E
ϕ ϕ ϕ ϕ
ϕ ϕ ϕ ϕ
Δ − Δ Δ − Δ≅ + = + + − =
Δ Δ Δ Δ= + → + =
2cos2out inP P ϕΔ≤
Symmetric Y-junction excited in the opposite direction
1e
2e
2ϕΔ
Basic waveguide structures
ae
2ϕ−Δ
1 2inE e e= +
( )
( )
( )
( )
1
2
1 2
1 2
1 ,21 ,2
1 ,2
1 .2
s a
s a
s
a
e e e
e e e
e e e
e e e
≈ +
≈ −
≈ +
≈ −
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antisymmetricmode
2 2
1, 0.1,
eff
s eff
N
n N θΔ >
<−
asymmetric Y, mode splittersymmetric Y, power splitter
Asymmetric Y-junction as a mode splitter
symmetricmode
two-mode waveguidesingle-mode arms
,"s"effN
," "eff aN
Adiabatic splitter: negligibly small mode coupling along the propagation length.The fundamental mode of the two-mode section remains “fundamental”, etc.
Criterion of asymmetry:
Since cannot be made too large, the decisive role is overtaken by θ.Typically, for and , the Y-junction behaves “asymmetrically”, for and , the Y-junction behaves as power splitter.
effNΔ0.2θ ≤
1θ ≥0effNΔ >
0effNΔ ≈
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Symmetric Y-junction(power divider)
Asymmetric Y-junction(mode splitter)
inP
2inP
Spectral bandwidth is limited by the requirements regarding the number of modes;the input and output ports must be single-mode, the central junction double-moded.
The bandwidth 1.2 – 1.6 μm is rather easily attainable.
Application: spectrally independent 2×2 (3-dB) splitter
2inP
2inP
2inP
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inP
inP
Application: spectrally independent 2×2 (3-dB) splitter
Symmetric Y-junction(power divider)
AsymmetricY-junction(mode splitter)
Spectral bandwidth is limited by the requirements regarding the number of modes;the input and output ports must be single-mode, the central junction double-moded.
The bandwidth 1.2 – 1.6 μm is attainable.
Reversed direction of propagation
2inP
2inP
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( )( )
23, 1,
24, 1,
cos ,
sin ,( ) 2,
( ).
out in
out in
s a
c s a
P P z
P P z
L
κ
κκ β β
π β β
=
== −= −
se ae si zse e β e ai z
ae β
( ) ( ) ( )( ) ( )
1
1 2 1 2
/21 2
1(0) ( ),2
1 1( )22
e cos sin .
s a s a
s a
s a
i z i z i z i zs a
i z
E e e e
E z e e e e e e e e e e
e z ie z
β β β β
β β κ κ+
= = +
= + = + + −
≈ +
Directional coupler1,inP
3,outP
4,outP
0 1 2 3 4 50.0
0.2
0.4
0.6
0.8
1.0
P4,out
P3,out
κ(λ)L
P3,
out /
P1,
in, P
4,ou
t /P
1,in
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Spectral dependence of the directional coupler
Refractive index profile (eff.index)
Distribution of optical intensity1 3 μm.
1 55 μm.
Directional coupler can be used as a (coarse) wavelength demux
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Coupled-mode theory of the directional coupler1,inP
3,outP
4,outP
1 1, eβ
2 2, eβ
Approximate description of the field:
1 1 2 2( , , ) ( ) ( , ) ( ) ( , )E x y z a z e x y a z e x y≈ +
Coupled-mode equations:
( ) ( )
( ) ( )
11̀ 1 12 2
22 2 21 1
( ),
( );
da zi a z i a z
dzda z
i a z i a zdz
β κ
β κ
= +
= +
Power conservation condition:
( )2 2 *1 2 12 210 ; d a a
dzκ κ+ = =
- normalized field
Without lost of generality we choose 12 21 .κ κ κ= = Boundary conditions: 1
2
(0) 1,(0) 0.
aa
==
( ) ( )
( ) ( )
1 2
1 2
21 1 2 1
222
2 1
0 cos sin , .2
0 sin , ,2
i z
i z
a z a e z i z
a z ia e z
β β
β β
βδ δ β β β
κ βδ δ κδ
+
+
Δ= − Δ = −
Δ= = +
Solution:
23, 1,
24, 1,
2 2
cos ( ), sin ( ),
( ) 2.
ou in
out in
P P zP P z
κβ κ
κ β β
=Δ =
= −For = 0
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Two-mode interference coupler1,inP
3,outP
4,outPSingle-mode I/O ports, two-mode central section
0 1 2 3 4 50.0
0.2
0.4
0.6
0.8
1.0
P4,out
P3,out
P3,
out /
P1,
in, P
4,ou
t /P
1,in
πz/(2Lt)
( )( )
23, 1,
24, 1,
cos ,
sin ,( ) 2,
( ).
out in
out in
s a
t s a
P P L
P P L
L
κ
κκ β β
π β β
=
== −= −
( ) ( ) ( )
( )
1
1 2 1 2
/21 2
1(0) ( ),2
1 1( ) e e22
e cos sin .2 2
s a s a
s a
s a
i z i z i z i zs a
i z
t t
E e e e
E z e e e e e e e e
z ze ieL L
β β β β
β β π π+
= = +
= + = + + −
≈ +
Similar to directional coupler,shorter “transfer length” Lt
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Imaging properties of an optical waveguideTwo-conductor waveguide: Propagation constants in a multimode waveguide:
222 20 0
0
0
11 , ,2
2(0),1, 2, , , .
mm mk n k n m Md k nd
k nd ndm M M
π πβ
π λ
= − ≈ −
= = =
If2 2
02
0
4 82 4 , 2
k ndL ndL Mdk ndπ
π λ= ≈ = ≈, i.e., then
2 22
0 020
1 2 .2m
m LL k nL k nL mk nd
πβ π≈ − = −
After propagation along a distance L , all modes meet (approximately) with the same phase.The field distribution at z = 0 is then reproduced at z = L. At the halfway, the “image” is reversed.
1L2L3L4L
(inverted image)
O. Bryngdahl and W.-H. Lee, J. Opt. Soc. Am. vol. 68, pp. 310-315, 1978.
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Multimode interference coupler (MMI)
1L2L3L4L
(inverted image)
M.T.Hill, J. Lightwave Technol. 21, 2305-2313, 2003
1 4×
1 3×
1 2×
Advantage:splitters with smallfootprint;
Problems:( )mβ dependence in dielectric waveguides is “less parabolic”
phase error for higher-order modes; error is largerfor higher refractive-index contrast;
transversal spatial resolution depends on the number of guided modes with small enough phase errors
Solution:
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Microring resonators
inthrough
out inthrough
outout of resonance
at resonance
microresonator
A. Driessen et al.:"Microresonators as building blocks for VLSI photonics," AIP Proceedings Vol.709, 2003.
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
in through
drop
Basic theory of microring resonatorsina thrua
dropa
t
κκ
t
1a
2a3a
4a
(Lossless) couplers:
31 4 2
0, dropthru in aa at i t i
aa a ai t i tκ κ
κ κ= ⋅ = ⋅
2 2 1;t κ+ =
0
Ring with the perimeter and the (complex) propagation constant :
2 22 1 4 3, ,r ri d i da e a a e aβ β= =
After elementary manipulations we obtain
( ) 22
2 2
2
3 12 2
1, ,
1 1
, .1 1
r r
r r
r
r r
i d i d
thru in drop ini d i d
i d
in ini d i d
t e ea a a at e t e
i te ia a a at e t e
β β
β β
β
β β
κ
κ κ
−= = −
− −
= =− −
drβ
At resonance, 2 , rd q qβ π= integer,
1 20, , ,1thru drop in in
ia a a a at
κ= = − =−
Off resonance, 2 1,rd qβ π= +2
2 2
22 2
2 , ,1 1
.
thru in drop in
thru drop in
t ia a a at t
a a a
κ= = −+ +
+ =For minimum cross-talk, ,drop ina a 1.κ
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
in through
drop
Spectral properties of a microresonator
Free spectral range2 / ( )q gFSR N dλ π≈
Resonant wavelength,qNd q qπ λ= 2 3integer (10 - 10 )
“Finesse”/F FSR λ= Δ
depends on the group index
Q qF=Quality factor
inte
nsity FSR
frequency
drop
thru
Lossless microresonator
gN
(1 )λ
inte
nsity
frequency
drop
thru
(1 )λ
Lossy microresonator
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Coupling between straight and bent waveguideProblems:• coupling between guided (lossless) and
leaky (radiating) modes• Role of the phase synchronism?
(variable relative phase velocities)Possible approach: linear superposition of mode
fields of a straight and bent waveguides:( ) ( ) ( ) ( ), ( , )w w b ba z x y a z x rϕ≈ +E r e e
( ) ( )2 20j j j ji n nωε∇ ⋅ × − × = − ⋅E H E H E E ( , ) ( , ) ,wi z i
j w bx y e x r eβ νϕ=E e ewith or
Finally we obtain a set of first-order coupled-mode equations for complex amplitudes
( )( )
( ) ( )( ) ( )
( )( )
w ww wb w
b bw bb b
a z z z a zd ia z z z a zdz
κ κκ κ
= ⋅
+ application of some general theorem, e.g., Lorentz-Lorenz reciprocity theorem:
R. Stoffer et al., Opt. Commun., vol. 256, pp. 46-67, Dec 2005.
then successive multiplication by followed by surface integration over S w be eand
with coupling “constants” givenby overlap integrals of mode fields at .
( )zκz
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Some results of the 3D CMT for the couplingof straight and bent waveguides
R. Stoffer, K. R. Hiremath, M. Hammer,L. Prkna, and J. tyroký, "Cylindrical integrated optical microresonators:Modeling by 3-D vectorial coupled mode theory,“ Optics Communications, vol. 256, pp. 46-67, Dec 2005.
Cooperation within 6FP “NAIS”, University of Twente & IPE AS CR
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Surface plasmons in guided-wave optics
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Surface plasmon(-polariton)Mutually coupled electromagnetic and charge wave
localized at the interface between a dielectric and a metal
mdSP
md
Nm
d md
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
surface plasmon
light line
ωp (εd + 1)–1/2
Wave number kSP
Freq
uenc
y ω
Surface plasmon is a slow wave
Re{ } SP d dN nε> =
SP is a slow wave
Lossless approximation: 0, ( 1)p dγ ω ω ε= < +
factor
It cannot be excitedfrom the dielectric!!
0
1
m dm dSP SP
m d m dk k N
c cε εε εω ω
ε ε ε ε= = ≈
+ −
>
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Waveguide polarization filter basedon resonant excitation of a surface plasmon
Silica block
Core of a fiberCladding
Low- buffer layernMetal (Al, Ag)
BK7 glass substrate
TM0 SP
TM0K+↔Na+
ion-exchanged waveguide
Al MgF2 buffer layer
0,5 0,6 0,7 0,8 0,9 1,0-50
-40
-30
-20
-10
0TE
TM
Atte
nuat
ion
b (d
B)
Wavelength λ (μm) J. tyroký et al., Proc.10th ECOC’84, pp. 44-45, 1984. (glass, LiNbO3)J. tyroký, H. J. Henning, Electron. Lett. vol.22, 756-757, 1986. (LiNbO3)
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Surface plasmons on a metal layer
J. tyroký et al. : Sensors and Actuators B 54, 66–73, 1999.
Mutually coupled surface plasmons long-range (magn. symmetric)short-range (magn. antisymm.){
Slightly asymmetricstructure
short-range
long-range
short-range
long-range
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Glass substrate
2 mm
40 nm AuTM0
SPW TM0 TM1
K+↔Na+ ion-exchanged
waveguide
AnalyteTa2O5 layer
Integrated-optic sensorbased on surface plasmon resonance
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Field distribution in the waveguidewith a section supporting SP
"analyte"na = 1.395
doped SiO2guide
SiO2 "substrate"
5.26 μm
1.5 mm
50 nm Au
-25 -20 -15 -10 -5 0 5
0,03
0,06
0,09
0,12
0,15
0
1
2
3
λ = 820 nm
Hy f
ield
(a.u
.)
z (mm)
y (μm)
-25 -20 -15 -10 -5 0 5
0,05
0,10
0,15
0
1
2
3
4
λ = 900 nm
Hy f
ield
(a.u
.)
z (mm)
x (μm)
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Experimental arrangementof a SPR integrated-optic sensor
0 5 10 15 20 25
790
795
800
805
810
815
Ref
ract
ive
inde
x
1.3410
1.3372
1.3333
1.3295
Res
onan
t wav
elen
gth
[nm
]
Time [min]
Refractive index resolutionbetter than 1.2×10-6
J. Dostalek, J. tyroký, J. Homola et al., Sensors and Actuators B-Chemical 76, 8-12 (2001).
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
“Bulk” surface plasmon sensor
`
φ , λ
optic
al p
ower
Refractive index resolution of about 1×10-7
prism
Input light Photo-detector
analyte
metal
θ
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
“Plasmonics”(“photonics” using surface plasmons instead of photonics)
2D guiding of surface plasmons
90° bendSP enables localization of radiation well below the diffraction limit.However, strong attenutation due to ohmic loss in metal allows for propagation at very short distances of the orders of 1-100 μm
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Plasmonic waveguides
Re(Ey) Re(Hx) Symmetric Asymmetric
Dielectrically supported Metal stripe SLOT (gap)(=LOADED)
Metal–dielectric interface IMI (metal film) structure MIM structures
Low-index hybrid Wedge Channel(groove)
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Novel types of plasmonic waveguidesSOI “slot waveguide”
Hybrid dielectric-plasmonicslot waveguide (HDPSW)
PIROW – plasmonic invertedrib optical waveguide
C. Koos & al., Nat. Photonics3(4), 16–219 (2009) H. Benisty and M. Besbes,
J. Appl. Phys. 108(6), 063108 (2010).H.-S. Chu & al., J. Opt. Soc. Am. B 28(12), 2895 (2011) (others, too)
xE
zP
air
Si
400 nm
260 nmTE
2SiO
nonlinearpolymer
yE
xE
R. F. Oulton & al., New J. Phys. 10,105018 (2008)
Si2S iO
Au (Ag)
w
h
d
??
00T E
00T M
R. F. Oulton & al., Nat. Photonics 2, 496 (2008);
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Hybrid dielectric-plasmonic slot waveguide
xE
0 500 1000 1500 2000
1.5
1.6
1.7
1.8
1.9
2.0
TM10
Si bar width w (nm)
Re
{Nef
f }
TM00
h = 30 nmd = 120 nm
0
20
40
60
80
100
120
L prop
(μm
)
Basic geometric parameters
A: d = 100 nm B: d = 200 nm
C: d = 300 nm D: d = 400 nm
w = 150 nm
0 100 200 300 400 500
1.5
2.0
2.5
B
w = 150 nm
Re{
Nef
f}
Si bar height d (nm)
w = 300 nmh = 30 nm
A C D0
50
100
150
200
250
300
L prop
(μm
)
xE xE xE
A: d = 100 nmw = 150 nm
B: d = 200 nm
Si
2SiO
Au
w
h
d
1 55 μm.
A: d = 100 nmw = 150 nm
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Multimode interference coupler1x2 MMI – simple configuration 1x2 MMI – improved configuration
11
21
24 dB ,6 dB
S
S11
21
51 dB ,5 5 dB.
S
S
xE12
31
2
3
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Mach-Zehnder interferometer
11
21
37 dB6 dB
S
S
“On” state “Off” state
11
21
25 dB21 dB
S
S
Si
2SiO
Au
w
h
dSi
0 15.n 0 15.n
J. tyroký et al.,, JEOS-RP vol. 8, 13021-13026 (2013).
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Waveguide structures with loss and gainAsymmetric complex grating-assisted coupler
Basic theory:L. Poladian, Phys. Rev. E, 54, 2963-2975, (1996).M. Greenberg, M. Orenstein, Opt. Express, 12, 4013-4018, 2004.M. Greenberg, M. Orenstein, Opt. Lett., 29, pp. 451-453, 2004.M. Greenberg, M. Orenstein, IEEE JQE, 41, 1013-1023, 2005.
Application proposals:M. Greenberg and M. Orenstein, PTL, 17, 1450-1452, 2005.M. Kulishov et al, Optics Express, 13, 3567-3578, 2005.
Photonic analogues of quantum-mechanical “PT-symmetric” systemsFirst works:
H.-P. Nolting, M. Sztefka, J. tyroký, Proc. of IPR, Boston, 76-79, 1996.G. Guekos, Ed., Photonic Devices for telecommunications, Springer,
1998, pp. 76-78. (“COST 240 Book”)
From the recent avalanche of papers:R. El-Ganainy et al., Optics Letters, vol. 32, pp. 2632-2634, 2007.K. G. Makris et al., Phys. Rev. Lett. vol. 100, pp. 103904(1-4), 2008.J. tyroký et al., Optics Express, vol. 18, pp. 21585-21593, 2010.C. E. Rüter et al., Nature Physics, vol. 6, pp. 192-195, 2010.H. Benisty et al., Optics Express, vol. 19, pp. 18004-18019, Sep 2011.A. A. Sukhorukov et al., Optics Letters, vol. 37, pp. 2148-2150, 2012.J. tyroký, Opt. Quantum Electron. vol.46, 465-475, 2014.
…and many others…
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Asymmetric complex grating-assisted coupler
1 31
2 42
z
x
Grating-assisted directional coupler using asymmetric complex grating
( )
( )
1 2
1 2
111
1 2
2
12
21 1 22 2
( ) ( ) ( ),
( ) ( )
( )
( )
( )
( ) ( ),
i z
i z
dA z i A z i e A zdz
dA z i e A z i z A zd
zz
zz
β β
β β
κ κ
κ κ
− −
−
≅ +
≅ +
1 1 1 2 2 2( , ) ( ) ( )exp( ) ( ) ( )exp( );yE x z A z e x i z A z e x i zβ β≈ +
1( )e x
2 ( )e x
0
((1) 2) 2
( ) ( , ) ( ) ( )2
, 2
mn m nS
iKzmn
iKzmn
kz x z e x e x dS
e Ke
κ ε
κ πκ
= Δ
= Λ+ + =
i i i
i i i i
or alternatively
complex, periodic in z
Fourier expansion contains only positive exponentials (“SSB modulation”)
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
0
Re{κ(1)12(z)}
z - coordinate
Im{κ(1)12(z)}
0
Re{
κ(1)
12(z
)}, I
m{κ
(1)
12(z
)}
Asymmetric complex gratingComplex permittivity perturbation in individual grating segments:
or
Re{ }
Im{ }
Re{ }
Im{ }
1
2
3
4
12
3 4
The second option seems technologically simpler since it requires only two different valuesof and ε ′Δ ε ′′Δ
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Let us consider the following ideal case of the grating at synchronism,
(1)11 12 12
21 22
1 2
( ) 0, ( ) exp( ),( ) 0, ( ) 0,
( ) 0.
z z iKzz z
K
κ κ κκ κ
β β β
= == =
Δ = − − =
Then, the coupled equations read
12
2
(1)12
( ) ( ),
( ) 0.
dA z i A zdz
dA zdz
κ≅
≅
1 2(0) 0, (0) 0A A= ≠
we get the solutionFor
2(1) (1) 21 2 1 212 12
2 2 2 2
( ) (0) , ( ) (0) ,
( ) (0) ., ( ) (0) .
A z i A z P z P z
A z A const P z P const
κ κ= =
= = = =
for
1 2(0) 0, (0) 0A A≠ =
1 1 1 1
2 2
( ) (0) ., ( ) (0),( ) 0, 0.
A z A const P z PA z P
= = == =
1 31
2 42
z
x
1 31
2 42
z
x
Some more coupled-mode theory…
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
ACGC is a reciprocal device!
1 31
2 42
z
x
1 31
2 42
z
x
1 31
2 42
z
x
1 31
2 42
z
x
Forward propagation Backward propagation
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Numerical modelling (in-house 2D Fourier Modal Method)Forward propagation Backward propagation
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Straightforward ACGC applicationsWideband ADD multiplexor
1 31
2 42
z
x1
1
2
2
2
M. Greenberg and M. Orenstein,PTL 17, 1450-1452, 2005
Light trapping in a ring resonator(a “dynamic memory cell”)M. Kulishov et al., OE 13, 3567-3578, 2005
grating “switched on”: grating “switched off”:
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Coupled waveguides with loss & gain: historical remarks
3.169355sn =
z
x
G wn n i n′′= −
L wn n in′′= +
w
4 1
1
1 μm,
10 [–; μm, cm ],4
1.55 μm, "loss/gain coefficient" [cm ]
w
n λ απ
λα
− −
−
=
′′ = ×
=0 2x103 4x103 6x103 8x103 1x104
3.17
3.18
3.19
3.20
3.21
3.22
Loss/gain coefficient α cm–1
loss
αbranchα1
TE1
TE0
Re{
Nef
f}
nG = 3.252398 - in''nL = 3.252398 + in''
nS = 3.169355
α = 4π n'' / λ
Two-mode region,NTE
1 = (NTE0 )*
Single-moderegion,real NTE
0
Two-moderegion,
real NTE0 , NTE
1
–4×103
4×103
2×103
Mod
e lo
ss α
m [c
m-1
]
0
–2×103
gain
[exp( )]i tω−
3.252398wn =
H.-P. Nolting, M. Sztefka, J. tyroký, Proc. of IPR, Boston, 76-79, 1996.G. Guekos, Ed., Photonic Devices for telecommunications, Springer, 1998, pp. 76-78. (“COST 240 Book”)
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
guided waves
surfacewave
guidedwaves
surfacewave
(TM)modefields
“Rigorous” 2D analysis (planar waveguides)
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Eigenmode equation for TE modesof a planar waveguide
Schrödinger equation for a particle ina 1D potential well
2 2 22
2 2 20
1 ( ) ( )( ) ( ) ( ) ( ) ( ) ( )2
d E x d xx E x N E x V x x E xk dx m dx
ψε ψ ψ+ = − + =
0
2
( )
2
( )
( ) ( )
x
m
E
V x
E
x
kx
N
ε
ψ⇔
⇔
⇔
−⇔
−
wave function
Formal analogy between a photonic waveguide and quantum-mechanical potential well
mode field distribution
wave number mass; Planck constant
relative permittivity profile
effective refractive index
potential
particle energy
Loss/gain structure: *( ) ( )x xε ε− = “PT symmetry”: complex potential, *( ) ( )V x V x− =
x
( )xε ( )V xx2
sN
2aN
sE
aE
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
10.89, 11.56s gε ε ′= =Balanced loss/gain“switching”:
1.5 μm, 0.75 μm,1 μm, 1.55 μm.
w hg λ
= == =
1g g giε ε ε′ ′′= +
w wg
hsε 2g g giε ε ε′ ′′= −
0.000 0.005 0.010 0.015 0.020 0.025-0.003
-0.002
-0.001
0.000
0.001
0.002
0.003
TE TM
Im{N
eff}
εg"0.000 0.005 0.010 0.015 0.020 0.025
3.3355
3.3360
3.3365
3.3370
3.3375
3.3380
TM
Re{
Nef
f}
εg"
TE
Belowexceptionalpoint
Aboveexceptionalpoint
gain channel
loss channel
C. E. Rüter et al. "Observation of parity–time symmetry in optics," Nature Physics, vol. 6, pp. 192-195, 2010.
*( , ) ( , )x y x yε ε− =
Coupled waveguides with loss/gain
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
0.000 0.005 0.010 0.015 0.020 0.025
3.3355
3.3360
3.3365
3.3370
3.3375
3.3380
TM
TE, ε"g0 = 0.008
TE TM
Re{
Nef
f}
εg2"0.000 0.005 0.010 0.015 0.020 0.025
-0.0025
-0.0020
-0.0015
-0.0010
-0.0005
0.0000
0.0005
ε"g = 0.010
TE TM
Im{N
eff}
εg2"
w wg
hsε 2g g giε ε ε′ ′′= −
Coupled waveguides with loss/gain
1.5 μm, 0.75 μm,1 μm, 1.55 μm.
w hg λ
= == =
10.89, 11.56s gε ε ′= =01g g giε ε ε′= + ′′
Belowexceptionalpoint
Aboveexceptionalpoint
gain channel
loss channel
*( , ) ( , )x y x yε ε− ≠
Fixed loss/variable gainswitching:
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Plasmonic loss/gain structureHybrid dielectric-plasmonic slot waveguide directional coupler with gain section
Strongly unbalanced structure!
Si
2SiO
Au
w
hd
s
Si
gain section 300 nm,120 nm,30 nm,1000 nm
wdhs
====
Only gain ( g” ) in the gain section is changed:2gain SiO giε ε ε ′′= −
0.00 0.05 0.10 0.15 0.20-0.03
-0.02
-0.01
0.00
0.01
Im{N
eff}
εg"
"Symmetric"
"Antisymmetric"
*( , ) ( , )x y x yε ε− ≠
0.00 0.05 0.10 0.15 0.201.660
1.662
1.664
1.666
1.668
1.670
1.672
Re{
Nef
f}
εg"
"Symmetric"
"Antisymmetric"
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Linear arrays of coupled waveguides with loss and gain
0.000 0.005 0.010 0.015 0.020 0.0253.3355
3.3360
3.3365
3.3370
3.3375
3.3380
3.3385
Im{N
eff}
εg"0.000 0.005 0.010 0.015 0.020 0.025
-0.002
-0.001
0.000
0.001
0.002
Re{
Nef
f}
εg"
0.000 0.005 0.010 0.015 0.020 0.025
3.3355
3.3360
3.3365
3.3370
3.3375
3.3380
3.3385
εg"
Re{
Nef
f}0.000 0.005 0.010 0.015 0.020 0.025
-0.002
-0.001
0.000
0.001
0.002
Im{N
eff}
εg"
0.000 0.005 0.010 0.015 0.020 0.0253.335
3.336
3.337
3.338
3.339
Re{
Nef
f}
εg"0.000 0.005 0.010 0.015 0.020 0.025
-0.002
-0.001
0.000
0.001
0.002
Im{N
eff}
εg"
8 coupled channel waveguides
6 coupled channel waveguides
4 coupled channel waveguides
h
sw
1.5 μm,0.75 μm,1 μm
whs
===
1 211.56 , 11.56 , 10.89g gg sg iiε ε εεε= = =′−+ ′′′
(quasi-TE polarization)*( , ) ( , )x y x yε ε− =
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
“Circular” arrays of coupled waveguides with loss and gain
r
ww
0.00 0.05 0.10 0.15 0.203.300
3.302
3.304
3.306
3.308
3.310
3.312
Re
{ Nef
f }
εg"0.00 0.05 0.10 0.15 0.20
-0.010
-0.005
0.000
0.005
0.010
Im {
Nef
f }
εg"
0.00 0.01 0.02 0.03 0.04 0.05
3.332
3.334
3.336
3.338
Re{
Nef
f}εg"
0.00 0.01 0.02 0.03 0.04 0.05-0.006
-0.004
-0.002
0.000
0.002
0.004
0.006
Im{N
eff}
εg"
0.00 0.01 0.02 0.03 0.04 0.05
3.332
3.333
3.334
3.335
3.336
3.337
3.338
3.339
R
e{N
eff}
εg"0.00 0.01 0.02 0.03 0.04 0.05
-0.004
-0.002
0.000
0.002
0.004
Im{N
eff}
εg"
1 μm1.5
wr w
==
1
2
11.56 ,
11.56 ,
10.89
g
gg
s
g
iiε
εε
εε=
=
=
′′
′′−
+
r
2r w=
8 waveguides
6 waveguides
4 waveguides(TE-like)
2.55r w=r
*
*
( , ) ( , )( , ) ( , )
x y x yx y x y
ε εε ε
′ ′ ′ ′− =′ ′ ′ ′− =
x′y′
WINTER COLLEGE on OPTICS: Fundamentals of Photonics - Theory,Devices and Applications, 10 - 21 February 2014, Trieste - Miramare, Italy
Is there really anything newunder the Sun?
Yes! (hopefully…)
45 years of Integrated Optics: