Ray optics & Optical beams

Zhipei Sun

Photonics Group Photonics ( ELEC-E3240 )

Ray optics & Optical beams

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Practical issues:

Once Upon a Time

Euclid's Optics (~300BC)

Rectilinear Propagation of Light

Lots of videos on Youtube to show you how to build your own pinhole camera.

Pinhole camera

Mozi (400BC) & Aristotle (400BC)

?

The first long-distance (>km) optical data transmission system

The Beacon Tower (~1000BC)

Capacity, Distance, Speed, Working condition

Reflection of Light

• Refractive index

– Values for n :

n_air = 1 n_H

2O 1,3 n_SiO

2  1,5

n_Si  3,5

2 2

1

1 ^

sin  ^{ n}

^

sin 

n

material vacuum

c n  c

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

=41.8

^o

• Total reflection can bind light into the high-n material

– Optical fibers and planar waveguides

1

sin

2

n n

c





Photonics

Total Internal Reflection

Bending light?

John Tyndall (1820-1893)

Photonics

Guiding light in a “water-fiber”

Optical Fiber for Light Guiding

Photonics

Optical Fiber for Light Guiding

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.

n t

 sin sin  ₁

2 2 2

1 n

n 



Apply Snell’s Law to light refracting into waveguide:

Assume critical angle TIR

1 2 1

2 1

2 2 2

1

2 n

n n

n n

n 

 





Index difference Numerical Aperture









 n sin n

₁²

n

₂²

n

₁

2 NA

_air



The refractive indices of the core and cladding therefore

determine a cone of acceptance.

A 3-layer Slab Waveguide:

The Ray-Optics Description

) ,

( n   

₀

  

₀

DOCUMENTTYPE 1 (1)

NOKIA RESEARCH CENTER TypeDateHere

Ari Tervonen

cover n_c

film n_f

d 

substrate n_s

c s

f

n n

n  

Total internal reflection rule: sin>n_c/n_f

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)

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.

Ray tracing for design

Simple cases in the real word

Wiki

Ray tracing for design

Complicated cases in the real word

Lens design Microsoft HoloLens

Is light a wave or particle?

Wave-particle duality

This question has puzzled scientist for >200 years

Light is a particle!

Isaac Newton (1643-1727)

Light is a wave!

Christiaan Huygens (1629-1695) Francesco Maria Grimaldi (1618-1663)

Augustin-Jean Fresnel (1788 to 1827) Thomas Young (1773 to 1829) James Clerk Maxwell (1831 to 1879)

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

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)

 ^{x , y , z , t} 

E

 ^{x , y , z , t} 

H

















 









 







D B

J D H

E B

0

t

t

Electromagnetic Model of Light Propagation

















 









 







D B

J D H

E B

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,

Constitutive relations:

E D

H

B   ,  

(1) (2) (3) (4)

 

 

  ^{ 0 0}













 







 







E H H E E H







t

t

Electromagnetic Model of Light Propagation

(5)

) (

)

( t

 













 H

E 

Take the curl of both sides of Equation 1:

) ( t

 













 H

E 

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

H E





 t

t

Using a vector identity

 

E E

E E

E   ²     ²







 ( )

* )

( 



, we get:

 

 



  



 

 

 

 E E  E

2 2 2

t

 

 

 ⁰⁰





























E H H E E H







t

t (1) (2) (3) (4)

















































E E

E E

E D

0

(*) 0

Wave Equations

2

0

2





 

 t

E E



2 2

⁰

2





 

 t

H H



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:

) , , (

2

0

2

z y x i

i

i

E E E E

t

E E  



 

 

²

and we get scalar wave equations:

ˆ , ˆ

ˆ

^{2 2}

2

2

  E

_x

x   E

_y

y   E

_z

z

 E

(6)

 (5)



 



 



 

 

 

 E E  E

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.

Helmholtz Equation

) , , (

2

0

2

z y x i

i

i

E E E E

t

E E  



 

 

²

(6)

] )

( Re[

) ,

(

^{i t}

i

t e

E r   r

^

Monochromatic waves:

Then from Equation (6)

we get: 

²

  

²

  0

) , ,

( E

_i

 E

_x

E

_y

E

_z

c

n 2

2 2

0





 







 _{   }

 c

n k n

With we get:

2 0

2  

  ^k  The Helmholtz Equation

2 0

2 



 

 t

E_i Eⁱ

 2

Cut-away of spherical wavefronts

Electromagnetic Model of Light Propagation Maxwell’s Equations

o

t



 





 H

E 

n t

o



 



 E

H 

²

 0









o

E 

 0



 H

 ^x , ^y , ^z , ^t   E ^   ^x , ^{y e}

^{jtz}

E

 ^{x , y , z , t}  ^{ H }   ^{x , y e}

^{jtz}

H

Electric and magnetic field amplitudes are functions of x, y, z and t.

 ^{x , y , z , t} 

E

 ^{x , y , z , t} 

H

2.2 Electromagnetic model for waveguides

Electromagnetic Model of Light Propagation



 











 



 











 



 











 





 

 

 



















 

 

y E x E z

E x E z

E y E

E E

y x

E E

z x E

E

z y

E E

E

z y x

t

y x x

y z z

y x

z x z

y

z y

x o

k j

i

k j

i

k j

i H E

   ^{ }

z y x z

y x

z y x z

y x

z t j

E t j

E

E z j

E

e y x E E

, , ,

,

, , ,

,

,









 





 



  ^

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

x z

y o

x y z

o

E x j

H E j

z E x E t

H









 









 







 



 

 

 

y E x H E

j

y E x E t

H

y x z

o

y x z

o







 









 



 







  ^{ }

z y x z

y x

z y x z

y x

z t j

H t j

H

H z j

H

e y x H H

, , ,

,

, , ,

,

,









 





 



  ^

Maxwell’s equations:  x E in Cartesian coordinates

(7) (8) (9)

Electromagnetic Model of Light Propagation



 











 



 











 



 











 





 

 

 



















 



y H x

H z

H x

H z

H y

H

H H

y x

H H

z x H

H

z y

H H

H

z y x

n t

y x x

y z z

y x

z x z

y

z y

x o

k j

i

k j

i

k j

i E H

 2

y z

x o

z y x

o

H y j

E H j n

z H y

H t

n E









 

 







 





2 2

x z

y o

x y z

o

H x j

E H j n

z H x

H t

n E









 







 











 

 



2 2

y H x

E H j n

y H x

H t

n E

y x z

o

y x z

o







 







 











2 2

Maxwell’s equations:  x H in Cartesian coordinates

  ^{ }

z y x z

y x

z y x z

y x

z t j

E t j

E

E z j

E

e y x E E

, , ,

,

, , ,

,

,









 





 



  ^   ^{ }

z y x z

y x

z y x z

y x

z t j

H t j

H

H z j

H

e y x H H

, , ,

,

, , ,

,

,









 





 



  ^

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

Electromagnetic Model of Light Propagation

x o y

z j E j H

y

E     





x o y

z j H j n E

y

H ₂



 

 



y o z

x j H

x E E

j  







y o z

x j n E

x H H

j   ²



 



 

x j E y j H

n

E_{x o o o} ^{z z}



 



 



  





 ^{2 2 2}

 

y j H x n E j n

H_{y o o o} ^{z z}



 



 



  





 ^{2 2 2 2}

 

x j H y n E j n

H_{x o o o} ^{z z}



 



 

  





 ^{2 2 2 2}

 

y j E x j H

n

E_{y o o o} ^{z z}



 



 



  





 ^{2 2 2} (7)

(10)

(8)

(11)

(13)

(14)

(15)

(16)

Eliminate H_y

Eliminate E_x

Eliminate E_y

Eliminate H_x

Key parameters: Ez, Hz

Electromagnetic Model of Light Propagation

(12) (14)

(15)

y H x

E H j

n _z ^{y x}

o 

 



 



 ²

 

y j H x n E j n

H_{y o o o} ^{z z}



 



 



  





 ^{2 2 2 2}

 

x j H y n E j n

H_{x o o o} ^{z z}



 



 

  





 ^{2 2 2 2}



^{2 2 2}

 ⁰

2 2 2

2





 

 





z o

o z

z

n E

y E x

E    

Electromagnetic Model of Light Propagation

z o y x

H y j

E x

E   









 

x j E y j H

n

E_{x o o o} ^{z z}



 



 



  





 ^{2 2 2}

 

y j E x j H

n

E_{y o o o} ^{z z}



 



 



  





 ^{2 2 2}

(9) (13)

(16)



^{2 2 2}

 ⁰

2 2 2

2





 

 





z o

o z

z

n H

y H x

H    

Electromagnetic Model of Light Propagation



^{2 2 2}

 ⁰

2 2 2

2

  



 





z o

o z

z

n E

y E x

E    



^{2 2 2}

 ⁰

2 2 2

2

  



 





z o

o z

z

n H

y H x

H    

2

^nd

order differential equations in E

_z

and H

_z

Solutions are Eigenfunctions / Eigenmodes

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

Naming Modes: TE, TM, HE and HM modes

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

Case 2, TE mode: E_z = 0 and H_z 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



^{2 2 2}

 ⁰

2 2 2

2

  



 





z o

o z

z

n E

y E x

E    



^{2 2 2}

 ⁰

2 2 2

2

  



 





z o

o z

z

n H

y H x

H    

Naming Modes: TE, TM, HE and HM modes

Case 4, HE/HM mode: E_z and H_z These types of waves are called hybrid-mode waves, and are denoted as HE or HM, depending upon if the

 0

 0



^{2 2 2}

 ⁰

2 2 2

2

  



 





z o

o z

z

n E

y E x

E    



^{2 2 2}

 ⁰

2 2 2

2

  



 





z o

o z

z

n H

y H x

H    

Case 3, TM mode: H_z = 0 and E_z 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

2.2 Electromagnetic model for waveguides

We will now solve the previous wave equations for the special case of a 2D slab waveguide, infinite in the y-z plane, thickness 2a in the x-direction, and with light propagation along the z-axis.

EM model solution for 2D infinite slab waveguide

A 2D slab waveguide, infinite in y, has EM wave solutions with no y- dependency.



^{2 2 2}



⁰

2 2 2

2   



 





z o

o z

z n E

y E x

E     ² ₂ ² ₂ ^



^{2 2  2}



^{ 0}



 





z o

o z

z n H

y H x

H    

0 0

If we now just consider only a TM wave solution (H

_z

= 0), then the wave equation we need to solve is just:



^{2 2 2}



⁰

2

2   





z o

o

z n E

x

E    

The general solution of this can be expressed in two parts, depending upon whether the wave is in the core or the cladding.

where n = n₁ ^for -a < x < a ^(core) n = n₂ ^for | x | > a ^(cladding)

EM model solution for 2D infinite slab waveguide

We also apply boundary conditions:

E_z and its derivative must be continuous at the

2.2 Electromagnetic model for waveguides

General solutions :

a x

a  



a x 

a x  

(i.e. in the core)

(i.e. in the cladding)

(i.e. in the cladding)

  ^{x z A}  ^{x C}   ^{j z} 

E

_z

,  sin 

₁

 exp  

  ^{x z B}  ^x   ^{j z} 

E

_z

,  exp  

₂

exp  

  ^{x z B}    ^{x j z} 

E

_z

,  exp 

₂

exp  

Therefore, within the core we have a propagating wave in the z-direction, with standing wave solutions in the x-direction.

Within the cladding, we also have a propagating wave in the z-direction, but with an exponentially decaying (evanescent) wave in the x-direction.

Cyclic nature of sinusoidal solution implies discrete wavefunctions (modes) .

EM model solution for 2D infinite slab waveguide

2 2

1 2 2

2 2

1     

  _{o o} n   k n 

2 2 2 2

2 2 2

2    _o_o n    k n



(i.e. ₁ real in core)

(i.e. ₂ imaginary in cladding)

EM model solution for 2D infinite slab waveguide

Mode numbering: Numbers correspond to # of nulls in

their intensity distribution.

Evanescent Wave

TM₀

TM₁

TM₂

TM₃

Note: A significant fraction of the energy propagates within the cladding layer as an evanescent wave.

Evanescent wave coupling is commonly used in photonic and nanophotonic devices as waveguide sensors

Evanescent wave

Claddinglayer (air)

Bufferlayer

P P

X Z

1 m  n

_core

n

_clad

n

_buffer

Evanescent field

Evanescent wave: Useful or Useless

 Evanescent waves are electromagnetic waves that penetrate tens of nm through a surface and propagate along the surface.

 They many be used to sample volume

“outside” a waveguide for sensing

 They may be used to locally excite a sample

on the other side of an interface for imaging

Evanescent wave

element air

buffer layer

evanescent field

gap d movement

light in light out

waveguide

Disc resonator as add/drop filter

G.J.M.Krijnen et.al., J MM, 9, 203-205 (1999).

Waveguide modes

•In optical waveguide, the radiation is restricted to travel in certain

“modes” (i.e., Only a finite number of modes will be guided. The spectrum of β for guided modes is discrete)

•Every eigenvalue β corresponds to a distinct mode of the system and every mode has a unique field distribution (profile).

•Most modes will not be guided and the values of β will lead to unguided modes (radiation modes). The spectrum of β for unguided modes is continuous (i.e. there are infinite number of unguided modes).

•All modes are orthogonal. Each mode is unique and cannot be described in terms of other modes.

•The modes of a given system form a complete set. Any

continuous distribution of field can be described as a

superposition of the appropriately weighted modes of the

waveguide.

Mode Characteristics

The field amplitude is related to the optical power carried in the waveguide. The power is calculated by integrating the z component of the Poynting vector over the cross sectional area:

Optical mode confinement:

ˆ ) 2 Re(

1 z

S

_z

 E  H 

The time-averaged power in a TE-mode is:





^











 



 

 E H dx E dx

P

z y x y

2

2

0

2 1





Fraction of the power contained in the slab is

P

_core

P

_total

= E

_y

(x)H

_x^*

-a

ò

a

^(x)dx

E

_y

(x)H

_x^*

-¥

ò

¥

^(x)dx

Summary about the next week work

 Your lectures & home assignments

1. Group 1: Prepare the home assignment (3-5 questions), share them within the team, and announce it on the mycourses website (or TEMAs) before 10AM 25

^th

, Jan

(Please have a group discussion about questions/answers) 2. Other groups: finish the home assignment

3. Collect all answers before 10AM 3

^th

, Feb

4. Evaluate & Exercise lecture Before/on 7

^th

, Feb. 14:15-16:00

Thank you!

Any questions?