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Transcript Waveguide modes - EECS: www 

Agenda for today:
Waveguides
β€’ Today we will use Lumerical MODE and FDTD to examine
the properties of optical waveguides.
1) Basic waveguide theory – Slab waveguide
2) Waveguide mode simulation using Lumerical MODE
a. Calculate 1D slab waveguide modes
b. Calculate silicon photonic waveguide modes (demo)
3) Using mode port source in Lumerical FDTD (demo)
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Optical waveguides
Slab waveguide
Ridge waveguide
Rib waveguide
π‘›π‘π‘™π‘Žπ‘‘
π‘›π‘π‘™π‘Žπ‘‘
π‘›π‘π‘™π‘Žπ‘‘
π‘›π‘π‘œπ‘Ÿπ‘’
π‘›π‘π‘œπ‘Ÿπ‘’
π‘›π‘π‘œπ‘Ÿπ‘’
π‘›π‘π‘™π‘Žπ‘‘
π‘›π‘π‘™π‘Žπ‘‘
π‘›π‘π‘™π‘Žπ‘‘
An optical waveguide has the general
property that there is always a material
with high refractive index (π‘›π‘π‘œπ‘Ÿπ‘’ ) surrounded
by a material with lower refractive index (π‘›π‘π‘™π‘Žπ‘‘ )
π‘›π‘π‘™π‘Žπ‘‘
π‘›π‘π‘œπ‘Ÿπ‘’
Light is confined by total internal reflection
in ray optics picture
Step-index
optical fiber
Need wave optics to solve for guided modes
Ref: Chuang, Ch. 7
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What is a waveguide mode?
β€’ A mode (in general) is a time-harmonic solution to
Maxwell’s equations (∝ eiΟ‰t)
β€’ A waveguide mode is a stable propagating mode with the
special property that its spatial field distribution does not
change with propagation (in the absence of loss)
β€’ Waveguide modes are determined by the cross-sectional
refractive index profile of the waveguide
β€’ Waveguide modes are wavelength dependent
Wang et al. Opt. Express 20, 15547-15558 (2012)
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Waveguide mode properties /
figures-of-merit
β€’ Effective index: 𝑛𝑒𝑓𝑓 οƒ  π‘˜π‘§ = 𝑛𝑒𝑓𝑓
2πœ‹
πœ†
– This number depends upon waveguide design and mode.
Larger number means mode is more tightly confined to
waveguide core.
β€’ Modal dispersion: 𝑛𝑒𝑓𝑓 πœ”
β€’ Modal group velocity: 𝑣𝑔 =
πœ•πœ”
πœ•π‘˜π‘§
– Velocity at which energy flows
β€’ Group velocity dispersion: 𝐷 =
πœ•(π‘£π‘”βˆ’1 )
πœ•πœ†
– Different wavelengths travel at different speeds down
waveguide! Units are time per unit wavelength per unit length.
Particularly important for fiber optic cable.
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Effective index
β€’ Each mode has an effective index that can be defined by:
2πœ‹
π‘˜π‘§ = 𝑛𝑒𝑓𝑓
πœ†
β€’ The effective index tells you how tightly the mode is
confined to the waveguide core
– Guided mode οƒ  π‘›π‘π‘™π‘Žπ‘‘ < 𝑛𝑒𝑓𝑓 < π‘›π‘π‘œπ‘Ÿπ‘’
– Tightly confined to the core οƒ  𝑛𝑒𝑓𝑓 ~ π‘›π‘π‘œπ‘Ÿπ‘’
– Weakly confined to the core οƒ  𝑛𝑒𝑓𝑓 ~ π‘›π‘π‘™π‘Žπ‘‘
– Unguided or radiating modes οƒ  𝑛𝑒𝑓𝑓 < π‘›π‘π‘™π‘Žπ‘‘
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Effective index
Small effective index
Large effective index
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Lumerical MODE
β€’ Lumerical MODE is a finite difference frequency-domain
solver that can be used to find waveguide modes
β€’ 1D or 2D cross-section of a waveguide is discretized and
simulated. MODE will solve for each mode and display the
mode properties (e.g. 𝑛𝑒𝑓𝑓 , etc)
Single-mode optical fiber
EE232 Discussion 02/02/2017
Photonic crystal fiber
7
Basic waveguide theory
β€’ Time-harmonic electromagnetic fields in a source-free
region must satisfy the vector Helmholtz equation
(𝛻 2 +πœ”2 πœ‡πœ–)𝐄 = 0
(𝛻 2 +πœ”2 πœ‡πœ–)𝐇 = 0
β€’ The electric-field and magnetic-field vectors in general will
have components in the x, y, and z directions.
β€’ Writing out the equation for Ex:
(𝛻 2 +πœ”2 πœ‡πœ–)𝐸π‘₯ = 0
πœ• 2 𝐸π‘₯ πœ• 2 𝐸π‘₯ πœ• 2 𝐸π‘₯
+
+
+ (πœ”2 πœ‡πœ–)𝐸π‘₯ = 0
2
2
2
πœ•π‘₯
πœ•π‘¦
πœ•π‘§
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Basic waveguide theory
β€’ Solve by separation of variables
πœ• 2 𝐸π‘₯ πœ• 2 𝐸π‘₯ πœ• 2 𝐸π‘₯
+
+
+ (πœ”2 πœ‡πœ–)𝐸π‘₯ = 0
2
2
2
πœ•π‘₯
πœ•π‘¦
πœ•π‘§
𝐸π‘₯ = 𝑋 π‘₯ π‘Œ 𝑦 𝑍(𝑧)
β€’ Plug back into scalar Helmholtz equation to find:
𝑋 = 𝐴π‘₯ 𝑒 ±π‘—π‘˜π‘₯ π‘₯ π‘Œ = 𝐴𝑦 𝑒 ±π‘—π‘˜π‘¦ 𝑦 𝑍 = 𝐴𝑧 𝑒 ±π‘—π‘˜π‘§ 𝑧
π‘˜π‘₯2 + π‘˜π‘¦2 + π‘˜π‘§2 = π‘˜ 2 = πœ”2 πœ‡πœ–
Important result!
β€’ The same can be done for the other electric and magnetic field
components. Usually some field components are either exactly
zero or very small and can be ignored.
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Solving for waveguide modes:
General approach
1) Assume propagation in the z-direction: E, H ∝ eπ‘—π‘˜π‘§ 𝑧
2) Make an educated guess for the form of the solution in
each dielectric region
Traveling wave: 𝐸π‘₯ = 𝐸0 𝑒 π‘—π‘˜π‘₯ π‘₯ eπ‘—π‘˜π‘§ 𝑧
Standing wave: 𝐸π‘₯ = 𝐸0 cos π‘˜π‘₯ π‘₯ eπ‘—π‘˜π‘§ 𝑧
Decaying wave: 𝐸π‘₯ = 𝐸0 𝑒 βˆ’π›Όπ‘₯ eπ‘—π‘˜π‘§ 𝑧
3) Plug educated guess back into the Helmholtz equation
and apply boundary conditions at interfaces to find
characteristic equations that will allow you to solve for π‘˜π‘§
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Planar slab waveguide
x
π‘›π‘π‘™π‘Žπ‘‘
π‘›π‘π‘œπ‘Ÿπ‘’
z
d/2
y
Slab waveguide has two types of modes
transverse electric (TE) and
transverse magnetic (TM)
π‘›π‘π‘™π‘Žπ‘‘
For TE mode the E-field only has a component in the y-direction. Assuming we are trying to
solve for a wave guided along the z-direction we also recognize that the structure is y-invariant
therefore we can simplify the scalar Helmholtz equation as:
πœ• 2 𝐸𝑦 πœ• 2 𝐸𝑦 πœ• 2 𝐸𝑦
+
+
+ (πœ”2 πœ‡πœ–)𝐸𝑦 = 0
2
2
2
πœ•π‘₯
πœ•π‘¦
πœ•π‘§
πœ• 2 𝐸𝑦 πœ• 2 𝐸𝑦
+
+ (πœ”2 πœ‡πœ–)𝐸𝑦 = 0
2
2
πœ•π‘₯
πœ•π‘§
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Planar slab waveguide
x
π‘›π‘π‘™π‘Žπ‘‘
π‘›π‘π‘œπ‘Ÿπ‘’
z
d/2
y
Slab waveguide has two types of modes
transverse electric (TE) and
transverse magnetic (TM)
π‘›π‘π‘™π‘Žπ‘‘
πœ• 2 𝐸𝑦 πœ• 2 𝐸𝑦
+
+ (πœ”2 πœ‡πœ–)𝐸𝑦 = 0
2
2
πœ•π‘₯
πœ•π‘§
The general solution will take the form:
𝐸𝑦 =
𝑒 π‘—π‘˜π‘§ 𝑧
𝐴𝑦0 𝑒 ±π‘—π‘˜π‘₯1π‘₯
π‘₯ β‰₯𝑑 2
𝐴𝑦1 𝑒 ±π‘—π‘˜π‘₯2π‘₯ π‘₯ ≀ 𝑑 2
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Planar slab waveguide
x
π‘›π‘π‘™π‘Žπ‘‘
π‘›π‘π‘œπ‘Ÿπ‘’
z
d/2
y
Slab waveguide has two types of modes
transverse electric (TE) and
transverse magnetic (TM)
π‘›π‘π‘™π‘Žπ‘‘
πœ• 2 𝐸𝑦 πœ• 2 𝐸𝑦
+
+ (πœ”2 πœ‡πœ–)𝐸𝑦 = 0
2
2
πœ•π‘₯
πœ•π‘§
Further simplify as (even solutions):
𝐸𝑦 =
𝑒 π‘—π‘˜π‘§ 𝑧
𝐴𝑦0 𝑒 βˆ’π›Ό( π‘₯ βˆ’π‘‘
𝐴𝑦1 cos π‘˜π‘₯ π‘₯
2)
π‘₯ β‰₯𝑑 2
π‘₯ ≀𝑑 2
Field decays into cladding layers
Standing wave solution in core
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Planar slab waveguide
x
π‘›π‘π‘™π‘Žπ‘‘
π‘›π‘π‘œπ‘Ÿπ‘’
z
d/2
y
Slab waveguide has two types of modes
transverse electric (TE) and
transverse magnetic (TM)
π‘›π‘π‘™π‘Žπ‘‘
πœ• 2 𝐸𝑦 πœ• 2 𝐸𝑦
+
+ (πœ”2 πœ‡πœ–)𝐸𝑦 = 0
2
2
πœ•π‘₯
πœ•π‘§
Further simplify as (odd solutions):
𝐸𝑦 = 𝑒 π‘—π‘˜π‘§ 𝑧
𝐴𝑦0 𝑒 βˆ’π›Ό(π‘₯βˆ’π‘‘
𝐴𝑦1 sin π‘˜π‘₯ π‘₯
2)
βˆ’π΄π‘¦0 𝑒 𝛼(π‘₯+𝑑
2)
π‘₯β‰₯𝑑 2
π‘₯ ≀𝑑 2
Field decays into cladding layers
Standing wave solution in core
π‘₯≀𝑑 2
Field decays into cladding layers
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Planar slab waveguide
x
π‘›π‘π‘™π‘Žπ‘‘
π‘›π‘π‘œπ‘Ÿπ‘’
z
d/2
y
Slab waveguide has two types of modes
transverse electric (TE) and
transverse magnetic (TM)
π‘›π‘π‘™π‘Žπ‘‘
Plug back into the wave equation and apply boundary conditions to find:
Even solutions
Odd solutions
π‘˜π‘₯2 + π‘˜π‘§2 = πœ”2 πœ‡πœ–1
π‘˜π‘₯2 + π‘˜π‘§2 = πœ”2 πœ‡πœ–1
βˆ’π›Ό 2 + π‘˜π‘§2 = πœ”2 πœ‡πœ–2
𝛼 2 + π‘˜π‘§2 = πœ”2 πœ‡πœ–2
𝛼 = π‘˜π‘₯ tan(π‘˜π‘₯ 𝑑/2)
𝛼 = βˆ’π‘˜π‘₯ cot(π‘˜π‘₯ 𝑑/2)
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TE modes
TE0
TE2
TE1
π‘›π‘π‘™π‘Žπ‘‘
π‘›π‘π‘™π‘Žπ‘‘
π‘›π‘π‘™π‘Žπ‘‘
π‘›π‘π‘œπ‘Ÿπ‘’
π‘›π‘π‘œπ‘Ÿπ‘’
π‘›π‘π‘œπ‘Ÿπ‘’
π‘›π‘π‘™π‘Žπ‘‘
π‘›π‘π‘™π‘Žπ‘‘
π‘›π‘π‘™π‘Žπ‘‘
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Simulating slab waveguide
with Lumerical MODE
β€’ Open MODE software
β€’ Create new rectangle: Structures > Rectangle
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Create waveguide core
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Create waveguide core
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Create 1-D FDE simulation
region
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Create 1-D FDE simulation
region
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Create 1-D FDE simulation
region
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Calculate modes
β€’ Click Run
β€’ Set wavelength to 1.55 and click Calculate Modes
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Mode list
Field profile
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Mode analysis
β€’ There are a lot of modes here! But remember, not all of
these modes are guided. Only modes with 𝑛𝑒𝑓𝑓 > π‘›π‘π‘™π‘Žπ‘‘ =
3.1 are guided modes
Guided modes
Unguided
radiating
modes
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Mode analysis (guided modes)
TE0
TM0
TE1
TM1
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Mode analysis (unguided modes)
Notice: There is no exponential decay in the cladding! Power will be radiated
into the cladding as these unguided β€œmodes” propagate down waveguide
Example #1
Example #2
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Waveguide cutoff
β€’ Let’s run a parameter sweep to plot effective index as a
function of slab core thickness
β€’ You should set your minimum mesh spacing to 10nm for
this analysis
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Waveguide cutoff
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Waveguide cutoff
β€’ Click run button:
β€’ When done, select sweep, select all results in Result View
(bottom left), right click > Visualize > New Visualizer
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Waveguide cutoff
ncore
No cutoff for
TE0, TM0
TE1, TM1 cutoff
ncladding
See Chuang 7.1.2
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Rectangular waveguide
𝑛1
𝑛2 > 𝑛1
𝑛1
x
Cross-section
z
y
β€’ Pure TE and TM modes do
not exist instead we have
hybrid modes (all field
components exist for each
mode).
β€’ BUT, the transverse
components of fields
dominate and so they are
often called quasi-TE and
quasi-TM (often we even
drop the term quasi)
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Silicon photonic ridge waveguide
silicon
220nm
Buried oxide (BOX)
3πœ‡m
Silicon substrate
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Waveguide modes
β€’ Silicon thickness of 220nm for silicon-on-substrate (SOI)
substrate is somewhat standardized
β€’ We often desire waveguide to contain only a single mode
β€’ Our goal is to determine the silicon ridge width such that
that waveguide is single mode
β€’ Open Silicon_Photonic_Ridge_Waveguide.lms from
the bcourses website
β€’ You will see a silicon photonic ridge waveguide with
220nm height and 900nm width.
β€’ Use MODE to calculate the guided modes.
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900nm width ridge waveguide
𝐸
2
Quasi-TE00
Ex dominates,
1 lobe in x,
1 lobe in y
𝐸
2
Quasi-TE10
Ex dominates,
2 lobes in x,
1 lobe in y
EE232 Discussion 02/02/2017
|𝐻|2
Quasi-TM00
Hx dominates,
1 lobe in x,
1 lobe in y
35
Single-mode waveguide
β€’ Run the parametric sweep on the silicon ridge width and
plot 𝑛eff for each of the modes on the previous slide as a
function of 𝑛eff
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Single-mode waveguide
Single mode Multimode
Mode crossing:
Mode1 switches from
TE to TM
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Single-mode waveguide
Single mode Multimode
Mode crossing:
Mode1 switches from
TE to TM
Note: Lumerical indexes modes by effective index, while we
usually want to sort by spatial distribution
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Single-mode waveguide
Index mode by eigenmode
Index mode by spatial distribution
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Mode source in FDTD
β€’ When running FDTD simulation we often want to excite a
particular waveguide mode
β€’ This can be done using a mode source in Lumerical
FDTD.
β€’ Open the file
Silicon_Photonic_Waveguide_Port_Source.fsp from
bcourses
β€’ This Lumerical FDTD simulation file contains 3D silicon
photonic ridge waveguide that we just simulated with ridge
height of 220nm and ridge width of 500nm to ensure
single mode operation
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Mode source in FDTD
β€’ Click Sources οƒ  Mode
β€’ Select the geometry tab first
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Mode source in FDTD
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Mode source in FDTD
β€’ Click Visualize Data
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Mode source in FDTD
β€’ A movie monitor has already been added at the x = 0
plane so that we can see the wave propagate down the
waveguide
β€’ Click Run
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Movie monitor
air
ridge
y
BOX
z
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Limitation to port source
β€’ Caution should be used when applying port source
β€’ Port source mode is only calculated at the center
frequency therefore there will be some mode mismatch
(i.e. reflections) at other frequencies.
β€’ Keep this in mind if you are running a broadband
simulation.
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