Introduction To Photonics Lecture 20-21-22 Guided Wave Optics [PDF]

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Introduction to Photonics Lecture 20/21/22: Guided-Wave Optics December 1/3/8, 2014

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Integrated optics Waveguide architectures Photonic materials Guided-wave theory –Planar-mirror waveguides –Planar dielectric waveguides –2D waveguides

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Why Integrated Optics?

Single-Frequency Tunable Laser

Electroabsorption Modulator

Phase Modulator

Semiconductor Optical Amplifier Electronic Integrated Circuits

Optical Splitter/Combiner

Circuit Components Optical filter

Arrayed Waveguide Grating

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Optical Waveguides and Integrated Photonics Cladding

Core Strip

Optical Fiber Substrate

Strip Waveguide

Cover Slab Substrate

Slab Waveguide

Laser diode

Integrated Photonic Transmitter/Receiver

Modulator

Coupler

Coupler Photodiode Fiber

Waveguides are Fundamental

www.intel.com

• • •

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.

Siliconize Photonics IEEE Spectrum, January 2004

• CMOS compatible - Easy integration with microelectronics

• Low cost

Optical interconnect Neil Savage, IEEE Spectrum, Aug. 2004

Si ?

Light sources

Waveguides

α < 0.2 dB/cm

- Fully established process technology - Scalability - Inexpensive material

Modulators

B ≈ 10 GHz

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

Photodetectors

R ≈ 0.8 A/W B ≈ 30 GHz

Passive Alignment

CMOS processing

Potential Monolithic Integration with Electronics Guiding, splitting, switching, wavelength multiplexing, and amplification of light on a single chip D. J. Paul, Advanced Materials, 11, 3, 191, 1999.

• Potential disruptive technology • CMOS compatible • Low cost, mass production • Easy to integrate with electronics • Compatible with SOI technology MISSING INGREDIENT Si-based light emitters

• Low loss ( n2

θc = sin-1(n2/n1)

• The dielectric waveguide has an inner medium (core or slab) with refractive index n1 larger than 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 θ greater than the critical angle θc = sin-1(n2/n1) • Waves making larger angles refract therefore losing a portion of power at each reflection (so eventually vanish) Guiding condition:

θ > θc

or θ is smaller than the complement of the critical angle θ c = π/2 − sin−1(n2/n1) = cos−1 (n2/n1)

Review on Total Internal Reflection Total internal reflection is accompanied by a phase shift ϕx = arg{rx} given by:

TE-reflection phase shift

TM-reflection phase shift

The phase shifts depend on incidence angle and on polarization

Dielectric Waveguide Analysis Approach • To determine the waveguide modes, solutions to Maxwell’s 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 planarmirror waveguide − Write a solution in terms of TEM plane waves bouncing between surfaces of the slab − Apply self-consistency condition to determine θm, β, um, vg

y

B

λ A

d

θ1

θ

2

1

3

z

θ C

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

ΑC − ΑΒ

two reflections

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

Determine TE Modes Guiding (self-consistency) condition:

Phase shift for TE (from analysis of reflection at boundary):

Rewrite self-consistency equation in this form:

2π

λ

2d sin θ − 2ϕ r = 2πm

sin 2 θ c tan = −1 2 2 sin θ

ϕr

θ1 = π / 2 − θ θc = π / 2 − θc

π  d tan  π sin θ − m  = tan(ϕ r / 2) 2  λ • This is a transcendental equation for sinθ •
plot both sides • Solutions yield the bounce angles

Determine TE Modes Self-consistency condition (TE modes):

 d π sin 2 θc tan π sinθ − m  = −1 2  λ 2 sin θ In this plot:

sin θ c = 8

LHS

λ

2d

RHS

Crossings yield the bounce angles θm of the guided modes even m (tan)

M=9

odd m (cot)

For planar-mirror: ϕr = π

⇒ tan(ϕ r / 2) = ∞ sin θ m = mλ / 2d

θm are between 0 and θ c

Propagation Constants z-components of wavevectors are the propagation constants

_ Since cosθm lies between 1 and cosθc = n2/n1 → βm lies between n2k0 and n1k0 Guiding condition

n2 k0 < β m < n1k0

Number of Modes _

Modes exist for all sinθ ≤ sinθc where there is a mode for each interval of λ/2d

sinθ c M= λ 2d .

cos θ c = n2 / n1

smallest integer greater than

Rewrite in terms of numerical aperture

.

M= .

2d

λ0

sin θ c = 1 − cos 2 θ c

NA

λ = λ0 / n1 NA = n12 − n 22

Number of Modes _

• When λ/2d > sinθc or (2d/λ0)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 ν > νc . •
M = ν /νc No forbidden region as for planar-mirror waveguide

Field Distributions Concept of internal and external fields

um ( y )

functions

Forward-looking observation: Higher order modes leak more into upper and lower cladding layers

TE Internal Fields • The field inside the slab is composed of two TEM plane waves traveling at angles θm and -θm with wavevector components (kx, ky, kz) = (0, ±n1k0sinθm, n1k0cosθm). • At the center of slab, these fields have same amplitude and phase shift (mπ, i.e. half of 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 The external field must match the internal field at all boundary points y = ±d/2. Substitute into

→ γ m2 > 0 For guided waves β m > n2 ko 

therefore exponential solutions

Extinction coefficient/decay rate

Proportionality constants determined by matching internal and external fields at y = d/2 and using normalization.

As mode number increases, γm decreases and modes penetrate more into cladding and substrate

General Properties of the Modes +∞

Normalization:

2 u ∫ m ( y)dy = 1

−∞

+∞

Orthogonality:

∫u

m

( y )ul ( y )dy = 0

−∞

for l ≠ m Arbitrary TE field in the waveguide:

E x ( y, z ) = ∑ amum ( y ) exp(− jβ m z ) m

Optical Confinement Factor Ratio of power in slab to total power

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Dispersion Relation From expressing self-consistency equation in terms of ω and β Rewrite in parametric form in terms of ωc and n and then plot

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Dispersion relations for different modes lie between the lines ω = c2β and ω = c1β 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 from the light line of the surrounding medium toward the light line of the slab

Waves of shorter wavelength are more confined in high-index slab

Group Velocity Group velocity

dω v= dk (slope of the dispersion) • For each mode, as ω increases above mode cutoff 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 ω, dispersion small so negligible pulse spreading

Rectangular Waveguide Two-dimensional waveguides confine light in the two transverse directions (the x and y directions)

For a waveguide with a square cross section, and if M is large:

π  2d  M ≈   NA2 4 λ  2

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 • 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 determine β from:

k x2 + k y2 + β 2 = n 2 k02 • kx, ky, kz (β) therefore have discrete values • Each mode identified by indices mx, my − As shown in plot, all integer values permitted as long as kx2 + ky2 ≤ n2ko2

• Number of modes (per polarization) (M large):

π  2d  M ≈   NA2 4 λ  2

Compared to 1-D waveguide, we see multiplication of degrees of freedom

Rectangular Dielectric Waveguide The components of the wavevector must satisfy:

k x2 + k y2 ≤ n1 k02 sin 2 θ c 2

θ c = cos −1

n2 n1

• Now kx and ky lie within reduced area • Can determine values using phase shifts (ϕ) as for planar case Number of modes (each polarization)

Note: Unlike the mirror waveguide, kx and ky of modes are not uniformly spaced. However, two consecutive values of kx (or ky) are separated by an average value of π/d (the same as for the mirror waveguide)

NA = n12 − n22

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 Fiber-to-chip coupling

These are coupling problems
mode matching problems

Butt-coupling from emission source to waveguide

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