Guided waves

7. Guided-wave optics

Optical waveguides

slab strip fiber

strip embedded strip rib or ridge strip loaded

Symmetric & Asymmetric waveguides

Cladding (or, cover) : n_c

Core (or, film) : n_f

Cladding (or, substrate) : n_s x

y

Guided waves

A waveguide

Electromagnetic modes of waveguides

A mode of a waveguide is a stable, propagating pattern of electric and magnetic fields that is periodic along the axis of the waveguide, apart from attenuation

7.1 Planar perfect-mirror waveguides

Waveguide modes

Condition of self-consistency

2 2

cos 2 (1 2 sin ) - 2 sin 2 sin AB AC  AC    AC AB AC   d 

Bounce angles

(Transverse Component)

: The ray-optics picture of light guidance by multiple reflections

2d k _ym  2



Propagation constants

Bounce angle Propagation constant

z

2 2 2 2 2

( / )

m kzm k kym k m d





k  nk

k

m d

 



sin ^m m 2 d

 _{ }^  ^_

 



0 sin

ym m

k  nk 

2 2 2

( / )

m k m d





Field distributions : TE modes

The complex amplitude of the total field in the waveguide is the superposition of the two bouncing TEM plane waves :

upward wave + downward wave

= +

: symmetric modes, odd modes

: antisymmetric modes, even modes

are normalized

are orthogonal in [-d/2. d/2] interval

Assume that the bouncing TEM plane wave is polarized in the x direction,

the guided wave is a transverse-electric (TE) wave.

TE mode

(

, 0, 0); (0,

,

)

E  E H  H H H

Each mode can be view as a standing waves in the y direction, traveling in the z direction.

Modes of large m vary in the transverse plane at a greater rate k_y, and travel with a smaller propagation constant .

The field vanishes at y = +d/2 for all modes, so that the boundary conditions at the surface of the mirrors are always satisfied.

[ TE guided waves ]

Number of modes

( d < /2 ) 

 ( /2 < d <  )  single-mode waveguide

Group velocities

Group velocity of mode m :

More oblique modes travel with a smaller group velocity

since they are delayed by the longer path of the zigzaging process.

Geometrically,

Field distributions : TM modes

upward wave + downward wave Magnetic field is in the x direction,

the guided wave is a transverse-magnetic (TM) wave.

Since the z component of the electric field is parallel to the mirror, it must behave like the x component of the TE mode :

y components of the electric field:

_m E_z

E_y E

z components of the electric field:

E

(0,

,

); (

0, 0) E  E E H  H H

TM mode

Multimode fields

( m = 1 )

( m = 2 )

( m = 1 & 2 )

7.2 Planar dielectric waveguides

Cladding (or, cover) : n_c

Core (or, film) : n_f

Cladding (or, substrate) : n_s x

y

n

Let’s first consider a symmetric waveguide.

Transverse Component of the wavevector

: The ray-optics picture of light guidance by multiple reflections

Self-consistency condition

2 2 sind  2m

 ^

(Perfect mirror)

2d k _ym  2m (Perfect mirror)

Self-consistency condition : TE modes

Self-consistency condition (TE mode)

From the TIR of TE modes, 

Self-consistency condition : TE modes

(open circles)

0,1, 2,

m  

Number of modes : TE modes

Single-mode waveguide (TE mode) In a dielectric waveguide, there is at least one TE mode, since the fundamental mode m = 0 is always allowed.

No cutt-off frequency

Propagation constants : TE modes

: propagation constant ( the z-component of wavevector)

n

1 2 M

Field distributions : TE modes

: Extinction coefficient

Confinement factor

 the ratio of power in the slab to the total power

Dispersion relation

2 2 2 2

1 2 1 2

2 2

2 2

1 1

2 tan

c

n n n n

m n n

n n



 





 

         

0 0

/

N c Nk

   

/ 2

/(2 )

c

c d NA

  

(N : effective index)

Single-mode waveguide (TE mode)

Nk

 



3 modes

1 mode

2 modes

c2

  

c1

  

Group velocities

 The group velocities lie between c

and c

(the phase velocities in the slab and substrate).

 At a given ,

 the lowest-order mode (the least oblique mode, m = 0) travels with a group velocity closest to c

.

 The most oblique mode (m = M) has a group velocity ~ c

.

The higher mode is faster than the lower!

Group velocities

1 GH cos

c

 z

 

  



 More oblique modes travel this lateral distance at a fast speed than less oblique modes

 증명 !!!

 The higher mode is faster than the lower!

 Lateral shift

 Time delay

Goose-Hanchen shift

Goose-Hanchen effect  Evanescent field

Now, consider asymmetric waveguides.

( Slab )

For TE modes :

Guided TE modes :

Evanescent waves (x > d, x < 0)

Penetration distances

Guided waves ( d > x > 0)

 Transverse wavevector ( d > x > 0)

Effective thickness of the waveguide

Dispersion relation for TE modes in a planar dielectric waveguide

Dispersion relation & Effective thickness

Normalized frequency : V

Dispersion relation for TE modes in terms of V , b and a:

Dispersion relation for TM modes in terms of V , b and a:

(TE)

(TE)

Summary of planar dielectric waveguides

TE :

TM :

has two modes .

7.3 Two-dimensional waveguides

Comparison : the number of modes

0

M 2d NA l

æ ö÷

ç ÷

» çççè ø÷÷÷

0 0

2 / 2

d d

M = l = l

2 2

0 0

2 1

4 4 / 2

d d

M p

l p l

æ ö÷ æ ö÷

ç ÷ ç ÷

» çççè ÷÷÷ø = çççè ÷÷÷ø

2 2

0

2 4

M p d NA l

æ ö÷

ç ÷

» çççè ø÷÷÷

1-d Mirror Guide

1-d Dielectric Guide

2-d Mirror Guide

2-d Dielectric Guide

(

)

0

2 d

V p NA k d NA

= l =

For the mirror guide the number of modes is just the number of ½ wavelengths that can fit.

For dielectric guides

it is the number that can fit but now limited by the angular cutoff characterized by the NA of the guide

7.4 Optical coupling in waveguides

A. Mode excitation

Input couplers

Coupling by focusing beam

End butt Coupling

Prism coupling

B. Coupling between waveguides

In the n₁ slab waveguide, In the n₂ slab waveguide, When a is very large (no-coupling)

Derivation of coupled wave equations

n₁ n₂

: coupling coefficient.

Coupled wave equations

Coupling coefficients

Phase mismatch per unit length

Exchange of power

Exchange of power when the guides are identical (phase-matched)

: transfer distance

: 3-dB coupling

Power transfer ratio in a small phase mismatched case

EO, TO, MO waveguide switches