Upload
lauren-stevenson
View
243
Download
1
Embed Size (px)
Citation preview
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
1/65
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
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
2/65
2
Why Integrated Optics?
ElectroabsorptionModulator Phase
Modulator
Single-FrequencyTunable Laser
OpticalSplitter/Combiner
Semiconductor
Optical Amplifier
Circuit Components
Optical filter Arrayed WaveguideGrating
ElectronicIntegrated Circuits
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
3/65
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
4/65
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
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
5/65
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
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
6/65
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 (
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
7/65
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
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
8/65
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
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
9/65
Integrated Passive Components
1x8 MMI Coupler
Integrated DelayLine (IDL)
Arrayed Waveguide Grating(AWG)
Grating CouplerSpot Size Converter
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
10/65
Integrated Active Components
Semiconductor laser Photodiode
Optical modulator Semiconductor optical amplifier
Photonic integration challenge: traditionally these (and passivecomponents) are made from different materials
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
11/65
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?
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
12/65
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,
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
13/65
Passive Components in MZM
MZMs that we studied look like this:
Why use extra bends (source of loss) and not just build like this?
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
14/65
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?
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
15/65
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)
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
16/65
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
Planar-Mirror Waveguides
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
17/65
Planar-Mirror Waveguides
= 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
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
18/65
Self-consistency therefore satisfied for
certain values of = m (bounce angles)
Planar-Mirror Waveguides
(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:
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
19/65
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)
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
20/65
Quantization
Note: Fundamental mode (m = 1) has smallest bounce angle and
largest propagation constant
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
21/65
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
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
22/65
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:
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
23/65
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
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
24/65
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
Complex Field Amplitudes
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
25/65
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
=
=
=
=
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
26/65
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
.
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
27/65
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
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
28/65
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
29/65
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)
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
30/65
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
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
31/65
Review (Planar-Mirror Waveguide)
Number of guided modes -relation Group velocity
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
32/65
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
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
33/65
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)
E2(y, z) = u2(y)exp(-j1z)
ETOT(y, z) = u1(y)exp(-j1z) +u2(y)exp(-j1z)
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
34/65
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)
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
35/65
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
Planar-Mirror Waveguides
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
36/65
Self-consistency therefore satisfied for
certain values of = m (bounce angles)
Planar-Mirror Waveguides
(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:
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
37/65
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)
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
38/65
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
39/65
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
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
40/65
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)
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
41/65
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
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
42/65
TM M d
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
43/65
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
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
44/65
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)
E2(y, z) = u2(y)exp(-j1z)
ETOT(y, z) = u1(y)exp(-j1z) +u2(y)exp(-j1z)
Pl Di l t i W id
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
45/65
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
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
46/65
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
47/65
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
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
48/65
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
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
49/65
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
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
50/65
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
Propagation Constants
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
51/65
Propagation Constants
Since cosm lies between 1 and cosc = n2/n1
m lies between n2k0 and n1k0
0102 knkn m
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
52/65
Number of Modes
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
53/65
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
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
54/65
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
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
55/65
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
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
56/65
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
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
57/65
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
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
58/65
60
Optical Confinement Factor
Ratio of power in slab to total power
Dispersion Relation
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
59/65
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
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
60/65
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
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
61/65
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
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
62/65
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
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
63/65
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
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
64/65
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
8/10/2019 Introduction to Photonics Lecture 20-21-22 Guided Wave Optics
65/65
67
Waveguide Coupling for Integration
Butt-coupling from emission source to waveguide
Fiber-to-chip coupling
These are coupling problems mode matching problems