Coupled wave theory

By Daniel Marks

September 10, 2009

ECE 299 Holography and Coherence Imaging Lecture 7

Duke University

What is coupled wave theory?

Interactingmedium

(e.g. holographic emulsion)

Interactingmedium

(e.g. holographic emulsion)

Two or more waves interact in a medium (e.g. holographic emulsion) altering each other.

In holography, waves are coupled by a pattern recorded in the emulsion.

Volume holography

Two plane waves interfere inside a medium.

2

2010 )exp()exp()( xikExikExI Intensity of two superimposed plane waves

Spatial frequency k1/2

Spatial frequency k2/2

Spatial frequency (k2- k1)/2

Wave #1

Wave #2

Interference pattern

The hologram records a periodic pattern which has a spatial frequency given by the difference between the spatial frequencies of the interacting waves.

The pattern recorded by two plane waves (consider x direction only)

2

2010 )exp()exp()( xikExikExI

)exp()exp(Re22)( 212

02

0 xikxikEExI

xkkiEExI )(expRe22)( 212

02

0

xkkEExI )(cos22)( 212

02

0

This is the periodic pattern recorded in the emulsion.

The emulsion electric permittivity changes in proportion to the intensity dose on the film.

xkkxIx m )(cos)()( 21

There are many mechanisms for this photosensitivity (photochemical change, trapped charge, etc.)

The wave equation in a periodic hologram

0)(22 UxU Wave equation in

inhomogeneous medium (scalar approximation).

0)(cos 2122 UxkkU

Wave equation in periodically modulated

permittivity medium

We use coupled wave theory to approximately solve this equation for two incoming plane waves.

Assumption of this derivation: the incoming plane waves vary spatially on a length scale much bigger than a wavelength (slowly varying envelope approximation).

Coupled wave theory.

)exp()()exp()(),( zikxikzSzikxikzRzxU szsxrzrx

Express the field U(x,z) as a sum of two slowly varying plane waves R & S.

)exp( zikxik rzrx

)(zR )(zSandSlowly varying amplitudes in z direction of R & S waves.

)exp( zikxik szsx

X component of spatial frequency of plane waves

X component of spatial frequency of plane waves

Coupled wave theory (contd).

Insert U(x,z) into the wave equation…

…expand out all of the derivatives…

…omit of the terms proportional to and2

2

dz

Rd2

2

dz

Sd

because R(z) & S(z) are slowly varying.

Some terms are proportional to exp(ikrzz) and some are proportional to exp(ikszz). We separate these into two equations because the spatial oscillations at these frequencies are “out of phase” and interact very little.

Yet more coupled wave theory

We also remove a common propagation phase exp[ik(krz+ksz)z/2] and we get the two coupled differential equations:

0)(exp)(2

2

zkkizSk

dz

dRik rzszrz

0)(exp)(2

2

zkkizRk

dz

dSik szrzsz

Note krx-ksx=K to make the x plane wave

components cancel the hologram phase.

22 k 222 )( kKkk rzsz

222 )( kKkk rzsz

Phase matching condition

zkxk szsx ˆˆ

zkzk rzsz ˆˆ zkxk rzrx ˆˆ

xKxkxk rxsx ˆˆˆ

How to solve these equations

Define zkkizSzS rzsz )(exp)()(~

and substitute…

0)(~

2

2

zSk

dz

dRikrz

0)(exp)(2

~)(

~ 2

zkkizRk

Skkkdz

Sdik szrzszrzszrz

0)0( RR

0)0(~

S

0z dz Initial conditions, no reference, a signal wave

?)(zR

?)(~

zSHologram

How to solve these equations (contd.)

Use guess of sum of complex exponentials and solve the indicial equation. Back substitute and you find

Insert boundary conditions, solve for constants of integration, and you get…

)sin(2

)cos(2

exp)( 0 zki

zzki

RzR zz

)sin(2

exp2

)(2

0 zzki

k

kiRzS z

sz

szrz kkk )/(2

1 2242 szrzz kkkk

So what does this solution mean?

S(z) is proportional to -1

2/12242 )/()(~

szrzz kkkkzS

The larger kz, the more phase mismatch and the less power exchanged from the reference to the signal beam. )(

~zS

zk

1

2/1

)/(2szrzkkk

Diffraction efficiency

For the ideal case kz=0 with no phase mismatch, we find

)cos()( 0 zRzR )sin()( 0 ziRzS

Maximum of power to the signal occurswhen sin(z)=0 or z=/2

Efficiency at power transfer (diffraction efficiency)

2

0R

S

Diffraction efficiency is 1 when

k

k

k

k szrz

And now for the simulations….

I performed a simulation of coupled wave theory.

Simulated forming a hologram, reconstructinga hologram, Bragg diffraction.

Instead of these equations which are approximatebut analytically tractable, I used the FiniteDifference Time Domain (FDTD) method.