indico.ictp.itindico.ictp.it/event/a13188/session/5/contribution/19/material/0/0.pdf · winter...

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

[email protected]

Institute of Photonics and Electronics AS CR, v.v.i.,Prague, Czech Republic


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


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


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


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


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= − = −


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 = −−


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


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>


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.


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


-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


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


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


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


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)


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


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”)


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)


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


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!)


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′ ′′= −


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!


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


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.


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)



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


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


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,


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


ν

− − − +++ +−

+ −−+ −−

= +

= +


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


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:


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


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 =


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


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


( )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−= =


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


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


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)


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


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 =


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 =


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 =


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


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


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


“Canonic” (elementary) waveguide devices


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


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


2. Excitation into a single arm

se

se


1e


Without the second arm, the transmittance is 100 %


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≤


1e

2e

2ϕΔ


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

≈ +

≈ −

≈ +

≈ −


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 ≈


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


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


( )( )

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


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


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


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


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.


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:


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.


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


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


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


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


Surface plasmons in guided-wave optics


Surface plasmon(-polariton)Mutually coupled electromagnetic and charge wave

localized at the interface between a dielectric and a metal

mdSP

md

Nm

d md


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= = ≈

+ −

>


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)


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


Glass substrate

2 mm

40 nm AuTM0

SPW TM0 TM1

K+↔Na+ ion-exchanged

waveguide

AnalyteTa2O5 layer

Integrated-optic sensorbased on surface plasmon resonance


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)


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).


“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


“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


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)


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);


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


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


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).


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…


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”)


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


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…


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


Numerical modelling (in-house 2D Fourier Modal Method)Forward propagation Backward propagation


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”:


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”)


guided waves

surfacewave

guidedwaves

surfacewave

(TM)modefields

“Rigorous” 2D analysis (planar waveguides)


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


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


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:


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"


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



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− =


“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′


Is there really anything newunder the Sun?

Yes! (hopefully…)

45 years of Integrated Optics:

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