Introduction to Photonics Lecture 20-21-22 Guided Wave Optics

8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics

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Introduction to Photonics

Lecture 20/21/22: Guided-Wave OpticsDecember 1/3/8, 2014

Integrated optics Waveguide architectures

Photonic materials

Guided-wave theory

Planar-mirror waveguides

Planar dielectric waveguides

2D waveguides 1


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2

Why Integrated Optics?

ElectroabsorptionModulator Phase

Modulator

Single-FrequencyTunable Laser

OpticalSplitter/Combiner

Semiconductor

Optical Amplifier

Circuit Components

Optical filter Arrayed WaveguideGrating

ElectronicIntegrated Circuits


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Waveguides are Fundamental

Example of an integrated-optic device used as an optical receiver/transmitter Received light is coupled into a waveguide and directed to a photodiode where it is

detected. Light from a laser is guided, modulated, and coupled into a fiber for transmission.

www.intel.com


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Siliconize Photonics

Optical interconnect

Light sources Waveguides Modulators Photodetectors PassiveAlignment

CMOSprocessing

http://www.intel.com/technology/silicon/sp/index.htm

Neil Savage, IEEE Spectrum, Aug. 2004

Si ?

0.2 dB/cm< 10 GHzB 0.8 A/W

B 30 GHz

R

CMOS compatible- Easy integration with microelectronics

Low cost- Fully established process technology

- Scalability

- Inexpensive material

IEEE Spectrum, January 2004


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Potential Monolithic Integration with Electronics

Potential disruptive technology

CMOS compatible Low cost, mass production

Easy to integrate with electronics

Compatible with SOI technology

Guiding, splitting, switching, wavelength multiplexing,

and amplification of light on a single chip

Low loss (


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Integrated Passive Components:

Basic Photonic Interconnects

Passive components used to interconnect active components

But passive components also have other functionality Bend can filter higher order modes

MMI coupler splits incoming field and produces a phase shift

Active components are also constructed in waveguide structures

Curve or bend

S-bend

Y-branch

Directional coupler

Multimode interference couplerFlare

Taper


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Integrated Photonic Functions

Active Passive

Emission

Modulation

Detection

Amplification

Filtering

Coupling

CombiningSplitting

Routing

Tunable Coupling

Tunable Filtering

Variable Attenuation

Attenuation

Mode Transformation

Variable function that responds to

external actuation or control

Fixed and constant function

Optical Isolation


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Integrated Passive Components

1x8 MMI Coupler

Integrated DelayLine (IDL)

Arrayed Waveguide Grating(AWG)

Grating CouplerSpot Size Converter


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Integrated Active Components

Semiconductor laser Photodiode

Optical modulator Semiconductor optical amplifier

Photonic integration challenge: traditionally these (and passivecomponents) are made from different materials


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11

Photonic Materials

Indium Phosphide (InP)

= 5-10% Small devices (~m-mm)

Lasers, modulators,SOAs, photodetectors,passives

Silicon on Insulator(SOI)

= 40-45%

Small devices (~m)

Modulators,photodetectors,passives

Silica on Si (Dielectric) Ge, B, P:doped SiO2 Si3N4 (n=1.9)/SiO2 (n=1.5)

= 0.5-20%

Large devices (~mm-cm) Passives

Lithium Niobate(LiNbO3)

= 0.5-1%

Large devices (~mm-

cm) Modulators,

passives

Index contrast = = (ncore2 ncladding

2)/(2ncore2)

Typical architectures shown; we are not constrained to these architectures

Why is the lower cladding of Silica waveguide so thick? How can we drastically increase index contrast for InP-based waveguides?


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Passive Component Figures of Merit

Common figures of merit (which can have differentmeaning for each of many passive components

available) Loss (cm-1 or dB/cm)

Useful conversion 10 cm-1 = 4.34 dB/mm

Coupling loss (dB)

Insertion loss (dB)

Includes all losses: coupling loss, waveguide loss

Wavelength dependence or bandwidth

Wavelength dependent loss, coupling,

Polarization dependence

Polarization dependent loss, coupling,


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Passive Components in MZM

MZMs that we studied look like this:

Why use extra bends (source of loss) and not just build like this?


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Waveguide Architectures

Planar Slab Ridge Rib

Buried Rib Buried Channel Deep Ridge

Which of these has the highest optical confinement?

How can we increase the confinement without altering geometry?


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Planar-Mirror Waveguides

Concept of waveguide modes

Start with a simple planar-mirror waveguide Not realistic but a good introduction to dielectric waveguides to introduce the

concept of guided modes

Apply E&M analysis by assigning each ray a TEM plane wave The total field is then the sum of these plane waves Conditions:

Wave is polarized in xand lies in y-zplane

Each reflection induces a -phase shift but amplitude and polarization aremaintained (perfect mirror)


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Self-consistency condition: As wave reflects twice, it reproducesitself so as to yield only two distinct plane waves Original wave must interfere with itself constructively (only certain

fields satisfy this condition eigenfunctions or modes)

Modes are the fields that satisfy self-consistency condition Modes are the fields that maintain the same transverse distribution and

polarization at all locations along the waveguide axis



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= 2AC/ 2 2AB/ = 2q, q = 0, 1, 2, . . .

AC AB = 2d sin(2/)2dsin = 2m, m = 1, 2, . . .

where m = q + 1

)2)(cossin/(sin/ dABdAC ==

)(sin2-1)cos(22 =

Phase shift from A to B must equal (or differ by integer multiple of 2) phase shiftfrom A to C (which undergoes two reflections); recall: = kl

Reflected wave Original wave


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Self-consistency therefore satisfied for

certain values of = m (bounce angles)


(2/)2dsin= 2m, m = 1, 2, . . .

sinm = m/2d, m = 1, 2, . . .

- Each m corresponds to a mode- m = 1 first or fundamental mode

(has smallest angle)

ky = nkosin

kym

= nko

sinm

= (2/)sinm

kym = m/d, m = 1, 2, 3, . . .

y component of the propagation constant

- Therefore kym are spaced by /d

- Phase shift for one round trip (vertical distance

of 2d) must be multiple of 2- Note dependence on d(only confined in vertical)

quantized form:


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Propagation Constants

The propagation constant of the guided wave is kz = kcos.

Thus is quantized with values

m

= kcosm

Higher order modes travel with smaller propagation constants.

dm

k

m

mm

2sin

)sin1( 222

=

=

Dispersion relation

The sum (or difference) of the two distinct waves (that traveling at angle +and

that traveling at angle ) has component exp(-jkzz)


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Quantization

Note: Fundamental mode (m = 1) has smallest bounce angle and

largest propagation constant


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Recall: total field in waveguide is sum of upward and downward TEM plane waves

When the self-consistency condition is satisfied, the phases of the upward and

downward plane waves at points on the z axis differ by half the round-trip phase

shift q, q = 0, 1, . . . , or (m 1), m = 1, 2, . . .

So waves add for odd m and subtract for even mThere are therefore symmetric modes, for which the two plane-wave components are

added, and antisymmetric modes, for which they are subtracted.

exp(-jkyy z)

mm

exp[j(m-1)]exp(jkyy jz)

z

y

General principle: the modes of every symmetric structure can be classified as

ODD or EVEN with respect to a symmetry axis

The phase shift encountered when a wave travels a distance 2d(one round trip) in they

direction, with propagation constant kym, must be a multiple of 2. )22( mdkym =

Field Distributions


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TE Modes

Consider first TE modes, such that the electric field is in thex direction

Upward wave component:zjyjk

m

mymeA

Downward wave component:zyjk

m

mj mym

eAe )1(

Aty = 0 the two waves differ in phase by (m 1).

Symmetric modes (m odd) components add

Asymmetric modes (m even) components subtract

zj

ymmxmeykAzyE

= )cos(2),(

zj

ymmxmeykjAzyE

= )sin(2),(

Total field:


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Complex Field Amplitudes

Amplitude of the mode

ykmy

Transverse electric field,x-polarized

mm

mm

Adja

Ada

2

2

=

= Odd m

Even m

Write in this form:Transverse distributions

Transverse distributions have been normalized

And can be shown to be orthogonal


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m=1 m=2 m=3 m=4

y

0

d

z

Field distributions cos(my/d)exp(-jmz) m odd

sin(my/d)exp(-jmz) m even

m = 4



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Take Home Messages

Each mode can be viewed as a standing wave in they direction, traveling in the z

direction.

Modes of large m vary in the transverse plane at a greater rate, ky

, and travel with

a smaller propagation constant .

Field vanishes at mirror boundary (y = d/2) for all modes, so the boundary

conditions are always satisfied.

mm

ym

m

k

dmk

dm

m

cos

2sin

,...3,2,1

=

=

=

=


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H

E .

H

E

H

E.

H

E.

TE Modes

TM Modes

cos(my/d)exp(-jmz) m odd

sin(my/d)exp(-jmz) m evenEx

Ezcos(my/d)exp(-jmz) m odd

sin(my/d)exp(-jmz) m even

y

z

y

z

Thez component behaves exactly as thex component of a TE mode.Now there areEcomponents in they andz directions.

TE vs. TM Modes

.


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Number of Modes

Recall sinm = m/2d, m = 1, 2, . . .

Since sinm < 1, the maximum allowed value of m is the greatest integer

smaller than 1/(/2d)

2d/ is reduced to the nearest integer

Example: when 2d/ = 0.9, 1, and 1.1, we

haveM= 0, 0, and 1, respectively.

Mincreases with increasing ratio of the mirror separation to the wavelength

The wavelengthc = 2dis called the cutoff wavelength of the waveguide.

Under conditions such that 2d/1 (corresponding to d/2)Mis seen to be 0

Self-consistency condition cannot be met and the waveguide cannot supportany modes.

The actual number of modes

that carry the optical power

depends on the source of

excitation but the maximum

number isM


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Dispersion Relation

Dispersion relation is the relation between the propagation constant and

the angular frequency

c = 2c = c/d

The propagation constantfor mode m is:

zero at angular frequency = mc increases monotonically with angular frequency

approaches the linear relation = /c for sufficiently large values of

Below cutoff

Linear for large

(approaches

homogeneous

medium case where

= /c)


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Group Velocity

A pulse of light (wavepacket) travels with a velocity v = d/d(group velocity)

Take derivative of -relation (ignoring dispersion in waveguide material, i.e.

assume c independent of )

More oblique modes travel with smaller group velocities since

they are delayed by the longer paths (zigzag process)

For each mode, the group velocity increases

monotonically from 0 to c as the angular frequency

increases (above the mode cutoff frequency).

Group velocity of mode m


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Review (Planar-Mirror Waveguide)

Number of guided modes -relation Group velocity


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TM Modes

TM modes have magnetic field in thex direction; electric field has

components iny andz

Recall that thez-component of the electric field here, behaves exactly asx-

component for TE mode (both always parallel to mirrors)

z-component of TM mode This is the sum of the upward and

downward waves (that have equal

amplitude and phase difference (m 1) The , ky,are identical to those for TE

modes

Boundary conditions still satisfied

becauseEz vanishes at mirrors

y-component of TM mode


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Multimode Fields

Waveguide may support several modes (more than one mode satisfy boundary conditions)

Arbitrary field polarized in thex direction and satisfying the boundaryconditions can be written as a weighted superposition of the TE modes:

Optical power divided among modes

and power distribution is position

dependent

E1(y, z) = u1(y)exp(-j1z)


ETOT(y, z) = u1(y)exp(-j1z) +u2(y)exp(-j1z)


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Concept of waveguide modes

Start with a simple planar-mirror waveguide Not realistic but a good introduction to dielectric waveguides to introduce the

concept of guided modes

Apply E&M analysis by assigning each ray a TEM plane wave The total field is then the sum of these plane waves Conditions:

Wave is polarized in xand lies in y-zplane

Each reflection induces a -phase shift but amplitude and polarization aremaintained (perfect mirror)


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Self-consistency condition: As wave reflects twice, it reproducesitself so as to yield only two distinct plane waves Original wave must interfere with itself constructively (only certain

fields satisfy this condition eigenfunctions or modes)

Modes are the fields that satisfy self-consistency condition Modes are the fields that maintain the same transverse distribution and

polarization at all locations along the waveguide axis



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Self-consistency therefore satisfied for

certain values of = m (bounce angles)


(2/)2dsin= 2m, m = 1, 2, . . .

sinm = m/2d, m = 1, 2, . . .

- Each m corresponds to a mode- m = 1 first or fundamental mode

(has smallest angle)

ky = nkosin

kym = nkosinm = (2/)sinm

kym = m/d, m = 1, 2, 3, . . .

y component of the propagation constant

- Therefore kym are spaced by /d

- Phase shift for one round trip (vertical distance

of 2d) must be multiple of 2- Note dependence on d(only confined in vertical)

quantized form:


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The propagation constant of the guided wave is kz = kcos.

Thus is quantized with values

m = kcosm

Higher order modes travel with smaller propagation constants.

dm

k

m

mm

2sin

)sin1( 222

=

=

Dispersion relation

The sum (or difference) of the two distinct waves (that traveling at angle +and

that traveling at angle ) has component exp(-jkzz)


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Amplitude of the mode

ykmy

Transverse electric field,x-polarized

mm

mm

Adja

Ada

2

2

=

= Odd m

Even m

Write in this form:Transverse distributions

Transverse distributions have been normalized

And can be shown to be orthogonal


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Dispersion Relation

Dispersion relation is the relation between the propagation constant and

the angular frequency

c = 2c = c/d

The propagation constantfor mode m is:

zero at angular frequency = mc increases monotonically with angular frequency

approaches the linear relation = /c for sufficiently large values of

Below cutoff

Linear for large

(approaches

homogeneous

medium case where

= /c)


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Group Velocity

A pulse of light (wavepacket) travels with a velocity v = d/d(group velocity)

Take derivative of -relation (ignoring dispersion in waveguide material, i.e.

assume c independent of )

More oblique modes travel with smaller group velocities since

they are delayed by the longer paths (zigzag process)

For each mode, the group velocity increases

monotonically from 0 to c as the angular frequency

increases (above the mode cutoff frequency).

Group velocity of mode m


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TM M d


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TM Modes

TM modes have magnetic field in thex direction; electric field has

components iny andz

Recall that thez-component of the electric field here, behaves exactly asx-

component for TE mode (both always parallel to mirrors)

z-component of TM mode This is the sum of the upward and

downward waves (that have equal

amplitude and phase difference (m 1)

The , ky,are identical to those for TEmodes

Boundary conditions still satisfied

becauseEz vanishes at mirrors

y-component of TM mode

M l i d Fi ld


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Multimode Fields

Waveguide may support several modes (more than one mode satisfy boundary conditions)

Arbitrary field polarized in thex direction and satisfying the boundaryconditions can be written as a weighted superposition of the TE modes:

Optical power divided among modes

and power distribution is position

dependent



ETOT(y, z) = u1(y)exp(-j1z) +u2(y)exp(-j1z)

Pl Di l t i W id


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Dielectric slab waveguide

The dielectric waveguide has an inner medium (core or slab) with refractive index n1 largerthan that of the outer medium (cladding or cover/substrate) n2

The electromagnetic wave is trapped in the inner medium by total internal reflection at an

angle greater than the critical angle c = sin-1

(n2/n1) Waves making larger angles refract therefore losing a portion of power at each reflection(so eventually vanish)

n1 > n2

c = sin-1(n2/n1)

Planar Dielectric Waveguide

c>

or is smaller than the complement

of the critical angle = /2 sin1(n2/n1) = cos1 (n2/n1)

Guiding condition:

c


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Dielectric Waveguide Analysis Approach

To determine the waveguide modes, solutions to Maxwells equations can be

reached in the core and cladding regions where appropriate boundary

conditions are imposed (EC770 covers full-vector treatment)

Following the Photonics book, apply similar approach to that for planar-mirror waveguide

Write a solution in terms of TEM plane waves bouncing between

surfaces of the slab

Apply self-consistency condition to determine m,, um, vg

y B


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d

12

3

Self-consistency: Wave 1 at B has the same phase as wave 3 at C (wave reproduction)

y

A

z

C

r = phase introduced by total internal reflection(replaces from planar-mirror waveguide)

1

C two reflections

D t i TE M d


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Determine TE Modes

Guiding (self-consistency) condition: md r

22sin2

2=

Phase shift for TE

(from analysis of

reflection at boundary):1

sin

sin

2tan

2

2

=

cr

)2/tan(2

sintan rmd

=

cc

=

=

2/

2/1

Rewrite self-consistency

equation in this form:

This is a transcendental equation for sin plot both sides

Solutions yield the bounce angles

D t i TE M d


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tand

sinm

2

=

sin2 c

sin2 1

Self-consistency condition (TE modes):

LHS RHS

dc

28sin

=

M= 9

In this plot:

dmm

r

r

2/sin

)2/tan(

=

=

=

For planar-mirror:

Crossings yield the bounce angles m of the guided modes

even m (tan)

Determine TE Modes

odd m (cot)

m are between 0 and c



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Since cosm lies between 1 and cosc = n2/n1

m lies between n2k0 and n1k0

0102 knkn m


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Number of Modes


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When /2d> sinc or (2d/0)NA < 1 only one mode allowed(single-mode waveguide)

Dielectric waveguide has no absolute cutoff frequency, i.e. there is at

least one TE mode since fundamental mode (m = 0) always exists Cutoff frequency given by:

Single-mode operation when by > c M= /c

_

No forbidden region as for

planar-mirror waveguide

Number of Modes

.

Field Distributions


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Field Distributions

)(yum functions

Concept of internal and external fields

Higher order modes leak more into upper and lower cladding layers

Forward-looking observation:

TE Internal Fields


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TE Internal Fields The field inside the slab is composed of two TEM plane waves traveling at angles

m and -m with wavevector components (kx, ky, kz) = (0, n1k0sinm, n1k0cosm).

At the center of slab, these fields have same amplitude and phase shift (m, i.e. halfof a round trip)

Arbitrary field is superposition over all the modes:

where

Proportionality constant to be

determined by matching the fields at

the boundaries

Note: field distributions are harmonic but do

not vanish at boundaries

TE External Fields


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TE External Fields

The external field must match the internal field at all boundary pointsy = d/2.

Extinction coefficient/decay rate

Proportionality constants determined by matching

internal and external fields aty = d/2 and using

normalization.

As mode number increases, m decreases and

modes penetrate more into cladding and substrate

02 >mFor guided waves

Substitute

into

> om kn2 therefore exponential solutions

General Properties of the Modes


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General Properties of the Modes

Normalization: +

=1)(2 dyyum

Orthogonality:

ml

dyyuyulm

=

+

for

0)()(

Arbitrary TE field in the waveguide:

= m mmmx zjyuazyE )exp()(),(

Optical Confinement Factor


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Optical Confinement Factor

Ratio of power in slab to total power

Dispersion Relation


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Dispersion Relation

Dispersion relations for different modes lie

between the lines = c2and = c1 Dotted light lines represent propagation in

homogeneous media (with refractive indices

of the surrounding medium and the slab)

As frequency increases above mode cutoff

frequency, the dispersion relation moves fromthe light line of the surrounding medium

toward the light line of the slab

From expressing self-consistency equation

in terms of and

Rewrite in parametric form in terms of cand n and then plot

Waves of shorter wavelength are

more confined in high-index slab

Group Velocity


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G oup e oc ty

dk

dv =

(slope of the dispersion)

Group velocity

For each mode, asincreases above modecutoff frequency, v decreases

Maximum value of v is c2, minimum value

is below c1 v asymptotically returns back toward c1

The group velocities of the allowed modes

range from c2 to a value slightly below c1.

Note: modes have different group velocities modal dispersion

When v varies slightly as a function of , dispersion small so negligible pulse spreading

Rectangular Waveguide


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Rectangular Waveguide

2

22

4

NAd

M

For a waveguide with a square cross section,and ifMis large:

Two-dimensional waveguides confine light in the two transverse directions

(thex andy directions)

The number of modes is a measure of the degrees of

freedom. When we add a second dimension we simply

multiply the number of degrees of freedom.

Rectangular Mirror Waveguides


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Rectangular Mirror Waveguides

Start with square mirror waveguide of width d

As for planar case, light guided by multiple

reflections at all angles

For plane wave (with wavevector (kx, ky, kz)) to

satisfy self consistency, must have

(i.e. self-consistency in both dimensions)

Then determinefrom:

kx, ky, kz () therefore have discrete values Each mode identified by indices mx, my

As shown in plot, all integer values permitted aslong as kx2 + ky

2 n2ko2

Number of modes (per polarization) (Mlarge):

20

2222 knkk yx =++

2

22

4 NA

d

M

Compared to 1-D waveguide, we seemultiplication of degrees of freedom

Rectangular Dielectric Waveguide


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Rectangular Dielectric Waveguide

cyx knkk 22

0

2

1

22sin+

The components of the wavevector must satisfy:

1

21cosn

nc

=

Note: Unlike the mirror waveguide, kx and kyof modes are not uniformly spaced.

However, two consecutive values of kx (or ky)

are separated by an average value of /d(the

same as for the mirror waveguide)

2

2

2

1NA nn =

Number of modes (each polarization)

Now kx and ky lie within reduced area

Can determine values using phase shifts() as for planar case

Geometries of Channel Waveguides


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Geometries of Channel Waveguides

Basic waveguide geometries

Basic waveguide functions

The exact analysis of these geometries/devices is far from easy and approximations are needed

See: Fundamentals of Optical waveguides, K. Okamoto, Academic Press, 2000

Waveguide Coupling for Integration


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67

Waveguide Coupling for Integration

Butt-coupling from emission source to waveguide

Fiber-to-chip coupling

These are coupling problems mode matching problems

Introduction to Photonics Lecture 20-21-22 Guided Wave Optics

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Transcript of Introduction to Photonics Lecture 20-21-22 Guided Wave Optics