1

4. Integrated Photonics (or optoelectronics on a flatland)

Benefits of 𝒊𝒏𝒕𝒆𝒈𝒓𝒂𝒕𝒊𝒐𝒏𝒙

−∞

in Electronics:

4

Mach-Zehnder modulator made from Indium Phosphide (InP) designed for 128 Gbs.

Are we experiencing a

similar transformation in Photonics ?

5

“Photonic Integrated Circuits are the next logical step in the world of optics!”, Infinera Corporation.

“Waveguide Integrated Optics involves the control of light analogous to integrated circuits in electronics. Processing and routing of data in the optical domain can offer advantages compared to electronic solutions, especially at increasing data rates”, Optical Society of America, 2015.

6

lasers photodetectors

A Few Examples of Integrated Photonic Components

optical fibers planar waveguides

7

modulators

add/drop filters

optical amplifiers

8

wavelength division multiplexing (WDM) couplers

optical isolator

Bragg gratings

9

M. Liu et al., Nature 474, 64 (2011)

A graphene-based electro-absorption modulator: In a device such as the one demonstrated by Liu et al. in 2011, electrically connected graphene is coupled to a SiO2

waveguide carrying a CW photon stream.

Driving Fundamental Research on Novel Materials and Devices

Early Days …

A Somewhat Recent (2008) Retrospect

A Crucial Element: Light Guiding Geometries

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

𝑛𝑓 > 𝑛𝑐

𝑛𝑓 > 𝑛𝑠

graded refractive index

step refractive index

𝑇 > 𝑡0

Requirements

Plane Waves

discrete set of modes

continuous set of modes

continuous set of modes

𝜃𝑠 > 𝜃 > 𝜃𝑐

𝜃 > 𝜃𝑠 > 𝜃𝑐

𝜃𝑠 > 𝜃𝑐 > 𝜃

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

A Propagating Wave 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

𝛻 × 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

Faraday’s law

𝛽 ≡𝜔

𝑐 𝑁

𝑑2𝐸𝑦 𝑥

𝑑𝑥2+𝜔2

𝑐2𝑛2 𝑥 − 𝑁2 𝐸𝑦 𝑥 = 0

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

Transverse Magnetic (TM)

𝐻 𝑥 =0𝐻𝑦 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

Ampere’s law

𝑑2𝐻𝑦 𝑥

𝑑𝑥2+𝜔2

𝑐2𝑛2 𝑥 − 𝑁2 𝐻𝑦 𝑥 = 0 𝛽 ≡

𝜔

𝑐 𝑁

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

Overall 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

v𝑔𝑟𝑜𝑢𝑝 =𝑑𝜔

𝑑𝛽𝑚 𝜔

phase velocity:

group velocity:

v𝑝ℎ𝑎𝑠𝑒 =𝜔

𝛽𝑚 𝜔=𝑐

𝑁𝑚 𝜔

Different Types of Dispersion in a Waveguide

• modal dispersion • material dispersion • waveguide dispersion

Field Profile of Guided Modes Discrete Set of Solutions

evanescent field

oscillatory behavior

m = mode order

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

How much power can we put on each mode of a guide

from an incoherent blackbody source?

𝑆 =𝑐

𝜆2 ℎ 𝜈 𝜖

𝑒ℎ 𝜈𝐾 𝑇 − 1

, 𝑆 = 𝑊

𝑚 ×𝑚𝑜𝑑𝑒

𝑆 = 19𝑝𝑊

𝑛𝑚 ×𝑚𝑜𝑑𝑒= −77

𝑑𝐵𝑚

𝑛𝑚 ×𝑚𝑜𝑑𝑒

𝜆 = 550 𝑛𝑚 𝑇 = 3,000 𝐾 𝜖 = 0.33

Intensity Profile propagating in Multimode Guides

pure excitation of mode 0

pure excitation of mode 1

mixed excitation of modes 0 & 1

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

Waveguide Couplers: injecting light into waveguides

• End couplers (usually used for channel and optical fibers)

• Transverse couplers (prism-coupler or grating-coupler) (typically used for slab waveguides)

End Coupler

𝔼𝑖𝑛 𝑥, 𝑦 = 𝑎𝛼

𝛼

𝑬𝛼 𝑥, 𝑦

𝜂𝛼 = 𝔼𝑖𝑛 . 𝑬𝛼

∗𝑑𝐴2

𝔼𝑖𝑛2𝑑𝐴 𝑬𝛼

2𝑑𝐴

𝔼𝑖𝑛

𝑬𝛼

input field decomposed into modes of the guide

Overlap Integral: fraction of coupled power into each mode

Prism Coupler

𝒏𝒑 𝒔𝒊𝒏 𝜽𝒑 = 𝑵

Grating Coupler

𝒏𝟎 𝒔𝒊𝒏 𝜽𝟎 + 𝒒𝝀

𝚲= 𝑵