Sergio B. Mendes

University of Louisville

May 2015 Perm State University

Introduction to Integrated Optics A Short Course

Major References:

Optical Integrated Circuits by Nishihara, Haruna, and Suhara

Fundamentals of Photonics by Saleh and Teich

Guided-Wave Optoelectronics edited by Tamir

Classnotes available at

http://www.physics.louisville.edu/sbmendes/

Early days …

Why integrate ?

A Somewhat “Recent” (2008) Retrospect

Light Guiding Geometries

2D (slab) and 3D (channel & optical fiber)

𝑛𝑓 > 𝑛𝑐

𝑛𝑓 > 𝑛𝑠

graded refractive index

step refractive index

𝑇 > 𝑡0

Plane Waves

discrete set of modes

continuous set of modes

continuous set of modes

𝜃𝑠 > 𝜃 > 𝜃𝑐

𝜃 > 𝜃𝑠 > 𝜃𝑐

𝜃𝑠 > 𝜃𝑐 > 𝜃

𝜃𝑐 ≡ 𝑠𝑖𝑛−1𝑛𝑐𝑛𝑓

𝜃𝑠 ≡ 𝑠𝑖𝑛−1𝑛𝑠𝑛𝑓

Critical angles:

Guiding Condition

𝜃 > 𝜃𝑠 = 𝑠𝑖𝑛−1𝑛𝑠𝑛𝑓

𝜃 > 𝜃𝑐 = 𝑠𝑖𝑛−1𝑛𝑐𝑛𝑓

𝑁 ≡ 𝑛𝑓 𝑠𝑖𝑛 𝜃 > 𝑛𝑠

𝑁 ≡ 𝑛𝑓 𝑠𝑖𝑛 𝜃 > 𝑛𝑐

𝜓 𝑥, 𝑦, 𝑧, 𝑡 = 𝐴 𝑒𝑗 𝜔𝑡−𝑘.𝑟

𝑧

𝑥

𝑘 =𝑘𝑥0𝛽= 𝑛𝑓

𝜔𝑐 cos 𝜃0sin 𝜃

= 𝜔𝑐 𝑛𝑓

2 −𝑁2

0𝑁

𝑛𝑓 > 𝑁 > 𝑛𝑠,𝑐

𝑘 𝑁 ≡ 𝑒𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑖𝑛𝑑𝑒𝑥

Maxwell’s Equations (isotropic, linear, lossless, non-magnetic)

𝛻 × 𝑬 = −𝜇0 𝜕𝑯

𝜕𝑡

𝛻 × 𝑯 = 𝑛2 𝜖0 𝜕𝑬

𝜕𝑡

Faraday’s law

Ampere’s law

𝑬 → − 𝑯

𝑯 → 𝑬

Note:

𝜖 = 𝑛2 𝜖0 ↔ 𝜇0

𝛻 × 𝛻 × 𝑬 = −𝜇0 𝜕𝑯

𝜕𝑡

𝛻 × 𝛻 × 𝑯 = 𝑛2 𝜖0 𝜕𝑬

𝜕𝑡

𝛻2𝑬 = 𝑛2

𝑐2𝜕2𝑬

𝜕𝑡2

𝛻2𝑯 = 𝑛2

𝑐2𝜕2𝑯

𝜕𝑡2

Wave Equations

Plane-Wave Solution along the Guide

𝑬 𝑥, 𝑦, 𝑧, 𝑡 = 𝐸 𝑥, 𝑦 𝑒𝑗 𝜔 𝑡 − 𝛽 𝑧

𝑯 𝑥, 𝑦, 𝑧, 𝑡 = 𝐻 𝑥, 𝑦 𝑒𝑗 𝜔 𝑡 − 𝛽 𝑧

𝜕2

𝜕𝑡2= −𝜔2

𝛻2 =𝜕2

𝜕𝑥2+𝜕2

𝜕𝑦2− 𝛽2

𝜕2𝐸 𝑥, 𝑦

𝜕𝑥2+𝜕2𝐸 𝑥, 𝑦

𝜕𝑦2+𝑛2𝜔2

𝑐2 − 𝛽2 𝐸 𝑥, 𝑦 = 0

𝜕2𝐻 𝑥, 𝑦

𝜕𝑥2+𝜕2𝐻 𝑥, 𝑦

𝜕𝑦2+𝑛2𝜔2

𝑐2 − 𝛽2 𝐻 𝑥, 𝑦 = 0

2D Optical Waveguides

By considering the symmetry along y-axis: (slab case)

𝐸 𝑥, 𝑦 = 𝐸 𝑥

𝐻 𝑥, 𝑦 = 𝐻 𝑥

𝑑2𝐸 𝑥

𝑑𝑥2+𝑛2𝜔2

𝑐2 − 𝛽2 𝐸 𝑥 = 0

𝑑2𝐻 𝑥

𝑑𝑥2+𝑛2𝜔2

𝑐2 − 𝛽2 𝐻 𝑥 = 0

Transverse Electric (TE)

𝐸 𝑥 =0𝐸𝑦 𝑥

0

𝑑2𝐸𝑦 𝑥

𝑑𝑥2+𝑛2𝜔2

𝑐2 − 𝛽2 𝐸𝑦 𝑥 = 0

𝛻 × 𝑬 𝑥, 𝑦, 𝑧, 𝑡 = −𝜇0 𝜕𝑯 𝑥, 𝑦, 𝑧, 𝑡

𝜕𝑡 𝐻 𝑥 =

−𝛽 𝐸𝑦 𝑥

𝜔 𝜇00

− 1

𝑗 𝜔 𝜇0 𝑑𝐸𝑦 𝑥

𝑑𝑥

Transverse Magnetic (TM)

𝐻 𝑥 =0𝐻𝑦 𝑥

0

𝑑2𝐻𝑦 𝑥

𝑑𝑥2+𝑛2𝜔2

𝑐2 − 𝛽2 𝐻𝑦 𝑥 = 0

𝛻 × 𝑯 𝑥, 𝑦, 𝑧, 𝑡 = 𝑛2 𝜖0𝜕𝑬 𝑥, 𝑦, 𝑧, 𝑡

𝜕𝑡 𝐸 𝑥 =

𝛽 𝐻𝑦 𝑥

𝜔 𝑛2 𝜖00

1

𝑗𝜔 𝑛2 𝜖0 𝑑𝐻𝑦 𝑥

𝑑𝑥

Guided TE Solution 𝑑2𝐸𝑦 𝑥

𝑑𝑥2+𝜔2

𝑐2𝑛2 𝑥 − 𝑁2 𝐸𝑦 𝑥 = 0

𝑥

𝑧

𝑁

𝑛𝑠

𝑛𝑓

𝑛𝑐

𝑥 > 0 → 𝑛 𝑥 = 𝑛𝑐 < 𝑁

−𝑇 < 𝑥 < 0 → 𝑛 𝑥 = 𝑛𝑓 > N

𝑥 < −𝑇 → 𝑛 𝑥 = 𝑛𝑠 < N

𝐸𝑦 𝑥 = 𝐸𝑐 𝑒−𝛾𝑐 𝑥

𝐸𝑦 𝑥 = 𝐸𝑠 𝑒𝛾𝑠 𝑥+𝑇

𝑇

𝛾𝑐 =𝜔

𝑐𝑁2 − 𝑛𝑐

2

𝛾𝑠 =𝜔

𝑐𝑁2 − 𝑛𝑠

2

𝐸𝑦 𝑥 = 𝐸𝑓 𝑐𝑜𝑠 𝑘𝑥 𝑥 + 𝜙𝑐

𝑘𝑥 =𝜔

𝑐𝑛𝑓2 − 𝑁2

Boundary Condition at Cladding-Film Interface

𝐸𝑐 = 𝐸𝑓 𝑐𝑜𝑠 𝜙𝑐

𝑥 = 0

𝐸𝑦

𝐻𝑧 =− 1

𝑗 𝜔 𝜇0 𝑑𝐸𝑦 𝑥

𝑑𝑥 𝛾𝑐𝐸𝑐 = 𝑘𝑥 𝐸𝑓 sin 𝜙𝑐

tan 𝜙𝑐 =𝛾𝑐𝑘𝑥

Boundary Condition at Substrate-Film Interface

𝐸𝑠 = 𝐸𝑓 𝑐𝑜𝑠 −𝑘𝑥 𝑇 + 𝜙𝑐

𝑥 = −𝑇

𝐸𝑦

𝐻𝑧 =− 1

𝑗 𝜔 𝜇0 𝑑𝐸𝑦 𝑥

𝑑𝑥 𝛾𝑠𝐸𝑠 = −𝑘𝑥 𝐸𝑓 sin −𝑘𝑥 𝑇 + 𝜙𝑐

tan 𝑘𝑥 𝑇 − 𝜙𝑐 =𝛾𝑠𝑘𝑥

Dispersion Relation for TE Modes

tan 𝜙𝑐 =𝛾𝑐𝑘𝑥

tan 𝑘𝑥 𝑇 − 𝜙𝑐 =𝛾𝑠𝑘𝑥

𝑘𝑥 𝑇 = 𝑡𝑎𝑛−1𝛾𝑠𝑘𝑥+ 𝑡𝑎𝑛−1

𝛾𝑐𝑘𝑥+𝑚 𝜋

&

2 𝜋

𝜆𝑇 𝑛𝑓

2 − 𝑁2 = 𝑡𝑎𝑛−1𝑁2 − 𝑛𝑠

2

𝑛𝑓2 − 𝑁2

+ 𝑡𝑎𝑛−1𝑁2 − 𝑛𝑐

2

𝑛𝑓2 − 𝑁2

+𝑚 𝜋

b-V diagram

2 𝜋

𝜆𝑇 𝑛𝑓

2 − 𝑁2

𝑉 ≡2 𝜋

𝜆𝑇 𝑛𝑓

2 − 𝑛𝑠2

𝑏𝐸 ≡𝑁2 − 𝑛𝑠

2

𝑛𝑓2 − 𝑛𝑠

2

𝑎𝐸 ≡𝑛𝑠2 − 𝑛𝑐

2

𝑛𝑓2 − 𝑛𝑠

2

𝑉 1 − 𝑏𝐸 = 𝑡𝑎𝑛−1

𝑏𝐸1 − 𝑏𝐸

+ 𝑡𝑎𝑛−1𝑎𝐸 + 𝑏𝐸1 − 𝑏𝐸

+𝑚 𝜋

cut-off:

𝑁 𝑛𝑠

0 𝑏𝐸

𝑉𝑚 = 𝑉0 +𝑚 𝜋

𝑉0 ≡ 𝑡𝑎𝑛−1 𝑎𝐸

asymmetry factor

Field Profile of Guided Modes Discrete Set of Solutions

evanescent field

oscillatory behavior

m = mode order

Intensity Profile along the Guide

pure mode: 0

pure mode: 1

mixed modes: 0 & 1

Guided TM Solution 𝑑2𝐻𝑦 𝑥

𝑑𝑥2+𝜔2

𝑐2𝑛2 𝑥 − 𝑁2 𝐻𝑦 𝑥 = 0

𝑥

𝑧

𝑁

𝑛𝑠

𝑛𝑓

𝑛𝑐

𝑥 > 0 → 𝑛 𝑥 = 𝑛𝑐 < 𝑁

−𝑇 < 𝑥 < 0 → 𝑛 𝑥 = 𝑛𝑓 > N

𝑥 < −𝑇 → 𝑛 𝑥 = 𝑛𝑠 < N

𝐻𝑦 𝑥 = 𝐻𝑐 𝑒−𝛾𝑐 𝑥

𝐻𝑦 𝑥 = 𝐻𝑠 𝑒𝛾𝑠 𝑥+𝑇

𝑇

𝛾𝑐 =𝜔

𝑐𝑁2 − 𝑛𝑐

2

𝛾𝑠 =𝜔

𝑐𝑁2 − 𝑛𝑠

2

𝐻𝑦 𝑥 = 𝐻𝑓 𝑐𝑜𝑠 𝑘𝑥 𝑥 + 𝜙𝑐

𝑘𝑥 =𝜔

𝑐𝑛𝑓2 − 𝑁2

Boundary Condition at Cladding-Film Interface

𝐻𝑐 = 𝐻𝑓 𝑐𝑜𝑠 𝜙𝑐

𝑥 = 0

𝐻𝑦

𝛾𝑐𝑛𝑐2𝐻𝑐 =𝑘𝑥𝑛𝑓2 𝐻𝑓 sin 𝜙𝑐

tan 𝜙𝑐 =𝛾𝑐𝑛𝑐2

𝑛𝑓2

𝑘𝑥

𝐸𝑧 =1

𝑗 𝜔 𝑛2 𝜖0 𝑑𝐻𝑦 𝑥

𝑑𝑥

Boundary Condition at Substrate-Film Interface

𝐻𝑠 = 𝐻𝑓 𝑐𝑜𝑠 −𝑘𝑥 𝑇 + 𝜙𝑐

𝑥 = −𝑇

𝐻𝑦

𝐸𝑧 =1

𝑗 𝜔 𝑛2 𝜖0 𝑑𝐻𝑦 𝑥

𝑑𝑥

𝛾𝑠𝑛𝑠2𝐻𝑠 = −

𝑘𝑥𝑛𝑓2𝐻𝑓 sin −𝑘𝑥 𝑇 + 𝜙𝑐

tan 𝑘𝑥 𝑇 − 𝜙𝑐 =𝛾𝑠𝑛𝑠2

𝑛𝑓2

𝑘𝑥

Dispersion Relation for TM Modes

𝑘𝑥 𝑇 = 𝑡𝑎𝑛−1𝛾𝑠𝑛𝑠2

𝑛𝑓2

𝑘𝑥+ 𝑡𝑎𝑛−1

𝛾𝑐𝑛𝑐2

𝑛𝑓2

𝑘𝑥+𝑚 𝜋

&

2 𝜋

𝜆𝑇 𝑛𝑓

2 − 𝑁2 = 𝑡𝑎𝑛−1𝑛𝑓2

𝑛𝑠2

𝑁2 − 𝑛𝑠2

𝑛𝑓2 − 𝑁2

+ 𝑡𝑎𝑛−1𝑛𝑓2

𝑛𝑐2

𝑁2 − 𝑛𝑐2

𝑛𝑓2 − 𝑁2

+𝑚 𝜋

tan 𝜙𝑐 =𝛾𝑐𝑛𝑐2

𝑛𝑓2

𝑘𝑥 tan 𝑘𝑥 𝑇 − 𝜙𝑐 =

𝛾𝑠𝑛𝑠2

𝑛𝑓2

𝑘𝑥

Dispersion Relation

2 𝜋

𝜆𝑇 𝑛𝑓

2 − 𝑁2 = 𝑡𝑎𝑛−1𝑛𝑓

𝑛𝑠

2𝜌𝑁2 − 𝑛𝑠

2

𝑛𝑓2 − 𝑁2

+ 𝑡𝑎𝑛−1𝑛𝑓

𝑛𝑐

2𝜌𝑁2 − 𝑛𝑐

2

𝑛𝑓2 − 𝑁2

+𝑚 𝜋

𝜌 = 0

𝜌 = 1

TE

TM

𝑣𝑝 =𝜔

𝛽=𝑐

𝑁

𝑣𝑔 =𝑑𝜔

𝑑𝛽

phase velocity:

group velocity:

Propagating Power along the Waveguide

𝑆 = 1

2Re 𝐸 × 𝐻∗ 𝑃𝑧 =

1

2𝑆𝑧 𝑑𝑥

∞

−∞

Power/unit-width:

TE mode:

𝑃𝑧 = −1

2𝐸𝑦 𝐻𝑥

∗𝑑𝑥∞

−∞

𝐻𝑥 =−𝛽 𝐸𝑦 𝑥

𝜔 𝜇0

𝑃𝑧 =𝛽

2 𝜔 𝜇0 𝐸𝑦

2 𝑑𝑥

∞

−∞

Poynting vector:

𝑃𝑧 =𝛽

2 𝜔 𝜇0 𝐸𝑦

2 𝑑𝑥

∞

−∞=𝛽

4 𝜔 𝜇0𝐸𝑓2 𝑇𝑒𝑓𝑓

𝑇𝑒𝑓𝑓 ≡ 𝑇 + 𝜆

2𝜋 𝑁2 − 𝑛𝑠2

+𝜆

2𝜋 𝑁2 − 𝑛𝑐2

effective thickness or mode size wavelength dependent

Easier Route to Dispersion Relation:

Phase-change under total internal reflection

𝑟𝑐 = 𝑒𝑗𝜙𝑐

𝑟𝑐 𝑟𝑠

phase-change at film/substrate interface

phase-change at film/cladding interface

𝜙𝑐 = −2 𝑡𝑎𝑛−1𝑛𝑓

𝑛𝑐

2𝜌𝑁2 − 𝑛𝑐

2

𝑛𝑓2 − 𝑁2

𝑟𝑠 = 𝑒𝑗𝜙𝑠

𝜙𝑠 = −2 𝑡𝑎𝑛−1𝑛𝑓

𝑛𝑠

2𝜌𝑁2 − 𝑛𝑠

2

𝑛𝑓2 − 𝑁2

𝑁 = 𝑛𝑓 𝑠𝑖𝑛𝜃

Phase Change due to Propagation

𝜙𝑝𝑟 = 𝑛𝑓 𝜔𝑐 𝐴𝐵 + 𝐵𝐶 = 𝑛𝑓

𝜔𝑐 2 𝑇 𝑐𝑜𝑠𝜃 = 2 𝑇 𝑘𝑥

𝑛𝑠

𝑛𝑐

𝑛𝑓 𝜃 𝐴

𝐶

𝐵

𝑇

2 𝑇

Resonant Condition:

𝜙𝑝𝑟 +𝜙𝑠 + 𝜙𝑐 = 2 𝜋 𝑚

2 𝑘𝑥𝑇 − 2 𝑡𝑎𝑛−1𝑛𝑓

𝑛𝑠

2𝜌𝑁2 − 𝑛𝑠

2

𝑛𝑓2 − 𝑁2

−2 𝑡𝑎𝑛−1𝑛𝑓

𝑛𝑐

2𝜌𝑁2 − 𝑛𝑐

2

𝑛𝑓2 − 𝑁2

= 2 𝜋 𝑚

Guiding Light with Graded Refractive Index

𝑑2𝐸𝑦 𝑥

𝑑𝑥2+𝜔2

𝑐2𝑛2 𝑥 − 𝑁2 𝐸𝑦 𝑥 = 0

TE polarization

Methodologies: • Ray Optics • WKB • Multilayer Modelling

Solution requires: • some knowledge of index profile 𝑛2 𝑥

Phase-change due to propagation

𝑘𝑥 𝑇 = 𝑡𝑎𝑛−1𝛾𝑠𝑘𝑥+ 𝑡𝑎𝑛−1

𝛾𝑐𝑘𝑥+𝑚 𝜋

=𝜔

𝑐 𝑛 𝑥 𝑐𝑜𝑠 𝜃 𝑥 =

=𝜔

𝑐 𝑛2 𝑥 − 𝑛 𝑥 𝑠𝑖𝑛 𝜃 𝑥 2 =

=𝜔

𝑐 𝑛2 𝑥 − 𝑁2

𝑘𝑥 𝑥

𝜃 𝑥𝑖

𝜃 𝑥𝑖+1

𝑘𝑥 𝑥 𝑇

𝑘𝑥 𝑥𝑖 Δ𝑥𝑖𝑖

𝜔

𝑐 𝑛2 𝑥 − 𝑁2 𝑑𝑥𝑥𝑡

0

𝑛 𝑥𝑖+1

𝑛 𝑥𝑖

𝑁

Cladding-Film Interface

𝑘𝑥 𝑇 = 𝑡𝑎𝑛−1𝛾𝑠𝑘𝑥+ 𝑡𝑎𝑛−1

𝛾𝑐𝑘𝑥+𝑚 𝜋

𝑎𝑡 𝑥 = 0 𝛾𝑐 =𝜔

𝑐𝑁2 − 𝑛𝑐

2

𝑘𝑥 =𝜔

𝑐𝑛2 𝑥 = 0 −𝑁2=

𝜔

𝑐𝑛𝑓2 − 𝑁2

𝑡𝑎𝑛−1𝛾𝑐

𝑘𝑥= 𝑡𝑎𝑛−1

𝑁2−𝑛𝑐2

𝑛𝑓2−𝑁2

≅𝜋

2

𝑁

Turning Point “Interface”

𝑘𝑥 𝑇 = 𝑡𝑎𝑛−1𝛾𝑠𝑘𝑥+ 𝑡𝑎𝑛−1

𝛾𝑐𝑘𝑥+𝑚 𝜋

𝑎𝑡 𝑥 = 𝑥𝑡

𝛾𝑠 =𝜔

𝑐𝑁2 − 𝑛2 𝑥 = 𝑥𝑡 − ∆𝑥

𝑘𝑥 =𝜔

𝑐𝑛2 𝑥 = 𝑥𝑡 + ∆𝑥 −𝑁

2

𝑁

𝑥𝑡

𝑡𝑎𝑛−1𝛾𝑐

𝑘𝑥= 𝑡𝑎𝑛−1

𝑁2−𝑛2 𝑥=𝑥𝑡−∆𝑥

𝑛2 𝑥=𝑥𝑡+∆𝑥 −𝑁2≅ 𝑡𝑎𝑛−1 1 =

𝜋

4

𝑛 𝑥 = 𝑥𝑡 = 𝑁

Bringing all the pieces together:

𝑘𝑥 𝑇 = 𝑡𝑎𝑛−1𝛾𝑠𝑘𝑥+ 𝑡𝑎𝑛−1

𝛾𝑐𝑘𝑥+𝑚 𝜋

𝜔

𝑐 𝑛2 𝑥 − 𝑁2 𝑑𝑥𝑥𝑡

0

=3

4+𝑚 𝜋

𝜔

𝑐 𝑛2 𝑥 − 𝑁2 𝑑𝑥𝑥𝑡

0

=𝜋

4+𝜋

2+𝑚 𝜋

𝑁

𝑥𝑡

dispersion relation for a graded-refractive index waveguide

Symmetric Graded Refractive Index

𝜔

𝑐 𝑛2 𝑥 − 𝑁2 𝑑𝑥𝑥𝑡

−𝑥𝑡

=1

2+𝑚 𝜋

𝑥

𝑁

−𝑥𝑡

𝑥𝑡

In this case we have a 𝜋

4 phase-shift at

both turning points ± 𝑥𝑡

Ray Optics Approach: • provides dispersion equation • helpful for determination of refractive-index profile • lacks information on field profile