Document info ch 11.

Integrated OpticsChapter 11

Physics 208, Electro-opticsPeter Beyersdorf

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ch 11.

Dielectric Waveguides

Optical waveguides “patterned” onto class or crystals form the basis for optical integrated circuits that can perform most of the functions of free space optics, and some functions that free space optics cannot.

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ch 11.

xy

Wave Equation in a Waveguide

Consider an isotropic material with translational symmetry along z and a refractive index profile in the x-y plane such as

Ampere’s law and Faraday’s law can be written in the form

and will have solutions of the form

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! where β≡kz 3

!!" !H = i"#0n2(x, y) !E

!!" !E = #i"µ0!H

!E = !E(x, y)ei(!t!"z)

!H = !H(x, y)ei(!t!"z)

ns

nc

ch 11.

Solutions in the Substrate

The wave equation !! ! ! ! ! ! can be written as

where α2(x,y)≡n2(x,y)[k0x2+k0y2] is the transverse

component of the propagation vector such that

For a guided mode the fields should go to zero far from the waveguide core, thus α must be imaginary making E an exponentially decaying function in x and y. Thus

4!2 >

"2

c2n2

s

!2E " µ!"2E

"t2= 0

!!2

c2n2(x, y)! "2(x, y)! #2

"$E = 0

!E(x, y) =!

!0

!E("0)ein(x,y)!0!

x2+y2

ch 11.

Solutions in the Core

Since the fields decay exponentially to zero far from the core in the substrate, they must have a maximum value in the core (i.e. a point where the field gradient is zero, ∇E=0, and the laplacian is negative, ∇2E<0)

Thus with our phasor notation where ∇2→α2+β2 and with the constraint on α(x,y) and β from the wave equation

in the core we must have

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!2 <"2

c2n2

c

!!2

c2n2(x, y)! "2(x, y)! #2

"$E = 0

ch 11.

Orthogonality of Modes

The longitudinal propagation constant β must satisfy

any number of modes, each with a different β satisfying this condition can propagate simultaneously, without interacting (i.e. without exchanging power).

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!2

c2n2

s < "2 <!2

c2n2

c

ch 11.

Slab Waveguide Example

Consider a one dimensional slab waveguide, as is commonly used in solid state lasers. Taking n2>n3≥n1 i.e. we have a high index core for guiding the radiation

What do the waveguide modes look like?

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ch 11.

Slab Waveguide Example

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in material 1!2

!x2E(x, y) + (k2

0n21 ! "2)E(x, y) = 0

in material 2!2

!x2E(x, y) + (k2

0n22 ! "2)E(x, y) = 0

in material 3!2

!x2E(x, y) + (k2

0n23 ! "2)E(x, y) = 0

in material 1, 2 and 3 respectively

β larger than nsk0

is not physicalβ too small for field to decay at ±∞

ch 11.

Guided Modes

Consider a symmetric slab waveguide (ns≡n3=n1) with nc=n2 with thickness d

For guided modes

In the substrate the field is exponentially decaying and has the form

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!2

c2n2

s < "2 <!2

c2n2

c

E(x > d) = E+e!q(x!d)

E(0 < x < d) = Ec [A cos hx + B sinhx]E(x < 0) = E!eqx

h≡α(x,y) in the core

q≡iα(x,y) in the substrate

ch 11.

TE Modes

If you consider a ray zig-zagging through the slab, the plane of zig-zagging defines a polarization direction. For a mode with an electric field transverse to this plane (a TE mode) the field can be expressed as

and must be continuous at the interfaces giving

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Ey(x, y, z, t) = Ey(x)ei(!t!"z)

E(x > d) = C!cos hd +

q

hsinhd

"e!q(x!d)

E(0 < x < d) = C!cos hx +

q

hsinhx

"

E(x < 0) = Ceqx

ch 11.

TE Modes

Plugging each of

into the corresponding wave equations

in the three regions gives

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h =!

n2ck

20 ! !2

q =!

!2 ! n2sk

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E(x > d) = C!cos hd +

q

hsinhd

"e!q(x!d)

E(0 < x < d) = C!cos hx +

q

hsinhx

"

E(x < 0) = Ceqx

!2

!x2E(x, y) + (k2

0n2s ! "2)E(x, y) = 0

!2

!x2E(x, y) + (k2

0n2c ! "2)E(x, y) = 0

!2

!x2E(x, y) + (k2

0n2s ! "2)E(x, y) = 0

ns

nsnc

ch 11.

TE Modes

The H field must also be continuous across the interfaces. From Faraday’s law

with

gives

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!!" !E = #i"µ0!H

!E = Ey(x)ei(!t!"z)j

This is the magnitude of the E field which is

already continuous across the boundaries

!H =i

"µ0

!i#

"Ey(x)ei(!t!"z)

#i +

dEy(x)dx

ei(!t!"z)k

$

The magnetic field has a component along the

direction of propagation

ns

nsnc

ch 11.

TE Modes

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h sinhd! q cos hd = q cos hd +q2

hsinhd

Requiring H(z) be continuous across the boundaries gives

where h and q are functions of β. Thus guided modes can only exist for discrete values of β which satisfy this mode condition

unconfined modes, not propagating along z

non physical (kz>k)

ns

nsnc

nsnc

β

nck0

nsk0β

θi

θt

For light to leak out of coreβ<nsko

ch 11.

TM Modes

Analogous analysis of TM modes gives similar results. In terms of the magnetic field amplitude:

subject to the mode constraint

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h sinhd! n2c

n2s

q cos hd =n2

c

n2s

q cos hd +!

n2c

n2s

"2q2

hsinhd

Hy(x > d) = C

!n2

sh

n2cq

cos hd +q

hsinhd

"e!q(x!d)

Hy(0 < x < d) = C

!n2

sh

n2cq

cos hx +q

hsinhx

"

Hy(x < 0) =n2

sh

n2cq

Ceqx

ch 11.

Cutoff Frequencies

Waves below a “cutoff” frequency will not be guided by the waveguide. Roughly speaking the waveguide dimensions must be larger than a wavelength. The requirements for guiding the mth mode is

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ωc

d

!! m

2!

n2c " n2

s

For!! ! ! ! ! ! only a single mode can be guided d

!<

1!n2

c ! n2s

ch 11.

Mode Structure

Mode profiles in a symmetric 1D waveguide

Given a mode profile, how do you determine its mode number?

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ch 11.

Integrated EOM devices

Two issues with free-space EOM devices can be largely eliminated by integrating the devices into waveguides

Half wave voltage can be much lower due to the small gap between electrodes allowing a much larger field

Interaction length can be much longer because the beam does not spread out as it propagates

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waveguide

electrode

ch 11.

Dielectric Tensor Perturbation

Consider a perturbation to the dielectric tensor created by an externally applied field E.

this perturbation affects the propagation of modes in the waveguide. In an isotropic material

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!! = ! + !! = "0""1 + r #E

!! = ! + !! = "0""1 + r #E = "0(" + !")"1

!! = !!0n4(x)(rE)

!0!!1(1 + !!!!1)!1 = !0!

!1 + r "E

!0!!1(1!!!!!1) = !0!

!1 + r "E

!!0!!1!!!!1 = r "E

!! = !!(rE)!!0

ch 11.

Mode Coupling

Consider the coupling between two modes (m and n)introduced by this perturbation. For efficient coupling the modes must have similar propagation constants (βm≈βn) thus they must be orthogonally polarized with the same mode number (m=n). Considering only these two modesthe filed can be written as

where Am and Bn will be functions of z in the presence of mode coupling introduced by the external electric field

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!E(!r, t) =!AmETE

m (x)e!i!T Em z + BnETM

n (x)e!i!T Mn z

"ei"t

ch 11.

Mode Coupling Equations

Coupled mode analysis (see slide 9.21) leads to a relation between Am(a) and Bm(z)

with

and

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!! = !TEm ! !TM

n

“overlap integral”

dAm

dz= !i!mnBnei!!z

dBn

dz= !i!!mnAme"i!!z

!E(!r, t) =!AmETE

m (x)e!i!T Em z + BnETM

n (x)e!i!T Mn z

"ei"t

!mn ="

4

! !

"!E#TE

m · #ETMn dx

ch 11.

Overlap Integral

With the overlap integral

and the fields for the slab waveguide already studied where ETE is along y and ETE is along x it is the ε6 term that couples the modes:

if the modes are well confined so that they see an index n(x) that is primarily nc then

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!6 = !!0n4(x)(r6kEk)

!mn ! "12n3

c(x)k0r6kEk

!mn ="

4

! !

"!E#TE

m · #ETMn dx

ch 11.

Phase Matched Coupling

The solutions to the mode amplitudes (subject to Am(0)=A0 and Bn(0)=0) when the modes are phase matched (Δβ=0) is

as with phase matching for acousto-optic interactions or second harmonic generation, imperfect phase matching reduces the maximum amount of power coupling between the modes

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Am = A0 cos(!mnz)Bn = !iA0 sin(!mnz)

ch 11.

Phase Mismatched Coupling

If Δβ≠0 the modes drift out of phase reducing (and eventually inverting) the transfer of energy between modes, resulting in less than 100% conversion efficiency

Periodically reversing the sign of the perturbation can “quasi-phase match” the interaction allowing 100% conversion efficiency

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ch 11.

Directional Couplers

Consider the coupling of two modes from different waveguides that are parallel to each other but fully separated. Power can be coupled form one to another in a process called “Directional coupling”

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ch 11.

Waveguide Coupling

The electric field in a structure with two parallel waveguides (a and b) can be written

and we express square of the refractive index as the sum of three parts, that of the surrounding cladding (ns), that of core a (na), and core b (nb)

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n2(x, y) = n2s(x, y) + !n2

a(x, y) + !n2b(x, y)

!E(x, y, z, t) = !A(z)Ea(x, y)ei(!t!"az) + !B(z)Eb(x, y)ei(!t!"bz)

ch 11.

Coupled Modes

The eigenmodes in each uncoupled waveguide obey the wave equation

giving

introducing the coupling between the waveguides this becomes (for waveguide a)

and likewise for waveguide bwith a  b

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!2Ea,b =n2

c2

!2

!t2Ea,b

Coupling term

0

!!2

!x2+

!2

!y2! "2

a,b

"Ea,b(x, y) = !#2

c2

#n2

s(x, y) + !n2a,b(x, y)

$Ea,b(x, y)

!!2

!x2+

!2

!y2!

!"2

a + 2i"adA

dz+

d2A

dz2

""Ea(x, y) = !#2

c2

#A n2

s(x, y) + A !n2a(x, y) + B !n2

b(x, y)$Ea(x, y)

ch 11.

Differential Equations

This coupled mode analysis leads to the following equations for the mode amplitudes A and B

with

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dA

dz= !i!abBei(!a!!b)z ! i!aaA

dB

dz= !i!baAe!i(!a!!b)z ! i!bbB

!ab ! 14"#0

!E!

a · !n2a(x, y)Ebdxdy

!ba ! 14"#0

!E!

b · !n2b(x, y)Eadxdy

!aa ! 14"#0

!E!

a · !n2b(x, y)Eadxdy

!bb ! 14"#0

!E!

b · !n2a(x, y)Ebdxdy

Modification to βa due to Δnb

Modification to βb due to Δna

ch 11.

Alternative Form

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Letting β’a→βa+κaa and β’b→βb+κbb gives a simpler form of coupled differential equations

For a symmetric waveguide (κ=κab=κba) this has the same form as the equations governing the acoustooptic interaction or second harmonic generation

dA

dz= !i!abBei(!!

a!!!b)z

dB

dz= !i!baAe!i(!!

a!!!b)z

ch 11.

Directional Coupler Solutions

With only a single input (A(0)=A0, B(0)=0) the power in the waveguides at z is is Pa(z)=A(z)*A(z) and Pb(z)=B*(z)B(z) and has the form

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Pb(z) = P0!2

!2 +!

!!!

2

"2 sin2

#

$%

!2 +&

!"!

2

'2

z

(

)

Pa(z) = P0 ! Pb(z)

ch 11.

Practical Applications

For typical parameters, κ-1≈2mm

Uses for this type of coupler include

Amplitude modulator (Vary output of waveguide a or b by varying Δβ via EO effect)

EO switch (adjust Δβ via EO effect to change power coupling from 0 to 100%)

Wideband frequency filter (based on λ dependence of κ)

Narrow band frequency filter (based on resonant enhancement [as described in Pradeep’s talk])

Frequency division multiplexer

Integrated “beam splitter” 30

ch 11.

References

Yariv & Yeh “Optical Waves in Crystals” chapter 11

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