ray optics & optical beams - aalto€¦ · 4b11 photonic systems - optical waveguides 2.2...

Zhipei Sun

Photonics GroupDepartment of Micro- and Nanosciences

Aalto University

Photonics

(ELEC-E3240 )

Ray optics & Optical beams

Lecture Topics

• Course introduction

This course

Arrangement

We & You

Syllabus

Poster presentation

/ Group works / home

assignments

Evaluation system

General introduction of

photonics

Did You Select Your Group?

Group A

Group Murata

Group MR JEM

Lecture Topics• Course introduction

• Ray optics & optical beams

• Lab work

• Printable photonics

• Waveguides / optical fibers

• Optical amplifiers

• Structural coloration

• Plasmonics

• Quantum photonics

• Silicon photonics

• Poster Presentation & discussion

Rectilinear Propagation of Light

Youtube

Pinhole camera

?

The first long-distance optical data

transmission system

The Beacon Tower (~1000BC)

Distance, Speed, Working condition

Reflection of Light

• Refractive index

– Values for n :

nair = 1

nH2O 1,3

nSiO2 1,5

nSi 3,5

2211 sinsin nn

material

vacuum

c

cn

At the interfaces, light is refracted

according to the Snell’s law

Photonics

Reflections

Total Internal Reflection

• Total reflection when entering low-n medium

Critical angle

For glass/air, c=41.8o

• Total reflection can bind light into the high-n material– Optical fibers and planar

waveguides

1

2sinn

nc

Photonics

Total Internal Reflection

YoutubePhotonics

Bending light?

John Tyndall

(1820-1893)

Photonics

Guiding light in a “water-fiber”

Optical Fiber for Light Guiding

Photonics

Ray Optical Picture of Light Confinement in Optical Fiber

2.1 Ray optics model for waveguides

To successfully launch light into a waveguide, the angle of

incidence to the end facet must be less than or equal to the

acceptance angle of the waveguide.

tn sinsin 12

2

2

1 nn

Apply Snell’s Law to light refracting into waveguide:

Assume critical angle TIR

1

21

2

1

2

2

2

1

2 n

nn

n

nn

Index differenceNumerical Aperture

2sin 12

2

2

1 nnnnNA air

The refractive indices of the core and cladding therefore

determine a cone of acceptance.

A 3-layer Slab Waveguide:

The Ray-Optics Description

),( 00 n

DOCUMENTTYPE 1 (1)

NOKIA RESEARCH CENTER TypeDateHere Ari Tervonen

cover nc

film nf

d

substrate ns

csf nnn

Total internal reflection rule: sin>nc/nf

Photonics

Ray Optics

Rays are used to model the propagation of

light through an optical system, by dividing the

real light field up into discrete rays that can be

computationally propagated through the

system by the techniques of ray tracing.

Wiki

Pierre de Fermat

(1601-1665)

Sir Isaac Newton

(1642-1726) Ray tracing for design

Further Reading: Ray Optics (& Matrix Optics)

Chapter 2: A. Yariv, Optical Electronics in Modern Communications, 5th edition, Oxford University Press, 1997.

Chapter 1: B.E.A. Saleh, M.C. Teich, Fundamentals of Photonics, Wiley, 2007

Chapters 5-6, E. Hecht, Optics, Addison Wesley, 2002

R. K. Verma, Ray Optics, Discovery Publishing, 2006.

Photonics

Wave Optics

Christiaan Huygens

(1629-1695)Thomas Young

(1773-1829)

Wave optics (Electromagnetic theory) studies

interference, diffraction, polarization, and other

phenomena (e.g., light-matter interaction) for

which the ray approximation of geometric optics

is not valid.

3D Diagram of Optical Wave

Wiki

2.2 Electromagnetic model for waveguides

Electromagnetic Model of Light Propagation

Maxwell’s Equations

Electric and magnetic field

amplitudes are functions of

x, y, z and t.

James Clerk Maxwell

(1831–1879)

tzyx ,,,E

tzyx ,,,H

D

B

DJH

BE

0

t

t



D

B

DJH

BE

0

t

t

Maxwell’s equations

(J = 0)

(ρ = 0)

For waveguides (including fibres), we assume operation in a source free medium: ρ = 0,

J=0. For now, we assume a linear and isotropic medium: permittivity, ε, and permeability,

, are independent of E and its orientation.

Constitutive relations:

EDHB ,

(1)

(2)

(3)

(4)

00

E

H

EH

HE

t

t


(5)

)()(t

HE

Take the curl of both sides of Equation 1:

)(t

HE With μ(r,t) independent of time and position, we get:

Reversing the order of the curl and time derivative

operators and assuming that ε is time invariant:

.

)(

2

2

t

E

t

E

HE

t

t

Using a vector identity

EEEEE22

)(

*

)(

, we get:

E

EE

2

2

2

t

00

E

H

EH

HE

t

t(1)

(2)

(3)

(4)

EE

EEE

D

0

0(*)

Wave Equations

02

2

t

EE

2

022

t

HH

2

For most structures in guided-wave optics the term is negligible,

and the wave equation (5) reduces to its homogeneous form:

Similarly for H:

In Cartesian coordinates:

) , ,( 02

2

zyxii

i EEEEt

EE

2

and we get scalar wave equations:

,ˆˆˆ 2222 zEyExE zyx E

(6)

(5)

E

EE

2

2

2

t

Solutions of this equation describe propagation of disturbances out from the region at a

fixed speed in one or in all spatial directions, as do physical waves (e.g., sound / light /

wave waves) from plane or localized sources.

A pulse traveling through a string with fixed

endpoints as modeled by the wave equation.Spherical waves coming from

a point source.

A solution to the 2D wave equation

Helmholtz Equation

) , ,( 02

2

zyxii

i EEEEt

EE

2

(6)

])(Re[),( tii etE rr Monochromatic waves:

Then from Equation (6)

we get: 022

) , ,( zyxi EEEE

c

n222

0

c

nnkWith we get:

022 k The Helmholtz Equation

02

2

t

EE ii

2

Cut-away of spherical

wavefronts



Maxwell’s Equations

to

HE

tno

EH

2

0o

E

0 H

ztjeyxtzyx ,,,, EE

ztjeyxtzyx ,,,, HH

Electric and magnetic field

amplitudes are functions of

x, y, z and t.

Propagating wave in z, tStationary wave in x, y

tzyx ,,,E

tzyx ,,,H



y

E

x

E

z

E

x

E

z

E

y

E

EEyx

EEzx

EEzy

EEEzyx

t

xyxzyz

yxzxzy

zyx

o

kji

kji

kji

EH

zyx

zyx

zyx

zyx

ztj

Ejt

E

Ejz

E

eyxEE

,,

,,

,,

,,

,

x-components: y-components: z-components:

xz

yo

xzy

o

Ejx

EHj

z

E

x

E

t

H

y

E

x

EHj

y

E

x

E

t

H

xy

zo

xyzo

zyx

zyx

zyx

zyx

ztj

Hjt

H

Hjz

H

eyxHH

,,

,,

,,

,,

,

Maxwell’s equations: x E in Cartesian coordinates

(7) (8) (9)



y

H

x

H

z

H

x

H

z

H

y

H

HH

yxHH

zxHH

zy

HHH

zyx

tn

xyxzyz

yxzxzy

zyx

o

kji

kji

kji

HE2

yz

xo

yzxo

Hjy

HEjn

z

H

y

H

t

En

2

2

xz

yo

xzy

o

Hjx

HEjn

z

H

x

H

t

En

2

2

y

H

x

HEjn

y

H

x

H

t

En

xy

zo

xyzo

2

2

Maxwell’s equations: x H in Cartesian coordinates

zyx

zyx

zyx

zyx

ztj

Ejt

E

Ejz

E

eyxEE

,,

,,

,,

,,

,

zyx

zyx

zyx

zyx

ztj

Hjt

H

Hjz

H

eyxHH

,,

,,

,,

,,

,

(10) (11) (12)

x-components: y-components: z-components:


xoyz HjEj

y

E

xoyz EnjHj

y

H 2

yoz

x Hjx

EEj

yoz

x Enjx

HHj 2

x

Ej

y

HjnE zzooox

222

y

Hj

x

EnjnH zzoooy

2222

x

Hj

y

EnjnH zzooox

2222

y

Ej

x

HjnE zzoooy

222

(7)

(10)

(8)

(11)

(13)

(14)

(15)

(16)

Eliminate Hy

Eliminate Ex

Eliminate Ey

Eliminate Hx

Key parameters: Ez, Hz


(12)

(14)

(15)

y

H

x

HEjn x

y

zo

2

y

Hj

x

EnjnH zzoooy

2222

x

Hj

y

EnjnH zzooox

2222

02222

2

2

2

zoo

zz Eny

E

x

E


zo

xy Hjy

E

x

E

x

Ej

y

HjnE zzooox

222

y

Ej

x

HjnE zzoooy

222

(9)

(13)

(16)

02222

2

2

2

zoo

zz Hny

H

x

H

4B11 Photonic Systems - Optical waveguides 2.2 Electromagnetic model for waveguides


02222

2

2

2

zoo

zz Eny

E

x

E

02222

2

2

2

zoo

zz Hny

H

x

H

2nd order differential equations in Ez and HzSolutions are Eigenfunctions / Eigenmodes


Naming Modes: TE, TM, HE and HM modes

Case 1: Ez and Hz are both zero. A trivial solution (no wave propagates).

Case 2, TE mode: Ez = 0 and Hz In this type of wave, the electric field is

transverse to the direction of propagation and the magnetic field is parallel to

the direction of propagation. We call this solution the Transverse Electric

(TE) mode.

0

02222

2

2

2

zoo

zz Eny

E

x

E

02222

2

2

2

zoo

zz Hny

H

x

H


Naming Modes: TE, TM, HE and HM modes

Case 4, HE/HM mode: Ez and Hz These types of waves are called

hybrid-mode waves, and are denoted as HE or HM, depending upon if the

transverse electric or magnetic field is larger respectively. These waves are

examples of helical (skew) rays.

00

02222

2

2

2

zoo

zz Eny

E

x

E

02222

2

2

2

zoo

zz Hny

H

x

H

Case 3, TM mode: Hz = 0 and Ez In this type of wave, the magnetic

field is transverse to the direction of propagation and the electric field is

parallel to the direction of propagation. We call this solution the

Transverse Magnetic (TM) mode, and it is an example of a meridional ray.

0

This afternoon!

Weight: 5%Lab work• Content: optical fiber splicing (Discuss with Yadong Wang if you already know

how to splice fibers)

• Lab room: Room 4166 In Micronova

• Assistant: Yadong Wang ( Office number: 4184; Email: [email protected])

13, Apr.: 12:30-13:15 Group 1

13, Apr.: 13:15-14:00 Group 2

13, Apr.: 14:00-14:45 Group 3

Calendar(Please follow the Weboodi/Mycourses)

Week Day Time Teacher

18 11, May (Thu.) 10:15-12 John Ronn

11, May (Thu.) 12:30-14 Poster preparation

19 16, May (Tue.) 10:15-12 Poster presentation

18, May (Thu.) 10:15-12 Poster presentation

Exam (3 hours) 24, May (Wednesday)

Poster presentation

Exam preparation

Exam preparation

ZHIPEISUNPencil

Summary about the next week work

Your lectures & home assignments

1. Group A: Prepare the home assignment (3

questions) and announce it on the mycourses

website before 10AM 18th, April

2. Groups Murata/Mr. Jem: finished the home

assignment 10AM 25th, April

Select a poster topic (DL: 10AM, 18th April,

Group leaders email me).

ray optics & optical beams - aalto€¦ · 4b11 photonic systems - optical waveguides 2.2...

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