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ECEE 5606 Advanced Optics Lab Robert R. McLeod, University of Colorado 1 Bragg holography and holographic polymers Outline Thin holography Holographic photopolymers Bragg holography Efficiency Selectivity

Bragg holography and holographic polymersecee.colorado.edu/~ecen5606/2014/Bragg holography... · • Bragg holography –Efficiency . ECEE 5606 Advanced Optics Lab Robert R. McLeod,

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Page 1: Bragg holography and holographic polymersecee.colorado.edu/~ecen5606/2014/Bragg holography... · • Bragg holography –Efficiency . ECEE 5606 Advanced Optics Lab Robert R. McLeod,

ECEE 5606 Advanced Optics Lab

Robert R. McLeod, University of Colorado 1

Bragg holography and

holographic polymers Outline

• Thin holography

• Holographic photopolymers

• Bragg holography

– Efficiency

– Selectivity

Page 2: Bragg holography and holographic polymersecee.colorado.edu/~ecen5606/2014/Bragg holography... · • Bragg holography –Efficiency . ECEE 5606 Advanced Optics Lab Robert R. McLeod,

ECEE 5606 Advanced Optics Lab

Robert R. McLeod, University of Colorado 2

Basics of holography Recording the optical phase

**22

2

RORORO

ROWRITE

EEEEEE

EEI

• Recording medium must physically respond to intensity (not field)

• We wish to record the complete optical field (“whole drawing”)

• Must use interference

Reference

Object Conjugate

Object

Ambiguity Reference

Write Read

2*22

**22

RORRORRO

RRORORO

RREAD

EEEEEEEE

EEEEEEE

ETE

Interfere object with reference

Film transmission proportional to IWRITE

Ambiguity Conjugate Object

• Reconstructed object weighed by intensity of reference

• Must separate terms in angle

• Bragg holography and materials

–Basics via thin holograms

Page 3: Bragg holography and holographic polymersecee.colorado.edu/~ecen5606/2014/Bragg holography... · • Bragg holography –Efficiency . ECEE 5606 Advanced Optics Lab Robert R. McLeod,

ECEE 5606 Advanced Optics Lab

Robert R. McLeod, University of Colorado

Hologram types A brief glossary

3

• Object beam: the light from the object being recorded

• Reference beam: the light that is interfered with the

object beam during recording. Also the light used to

reconstruct the hologram during playback.

• Thin vs Thick: Thin holograms are most common in the

commercial world. They exhibit wide angular and

spectral replay tolerance. Thick or “Bragg” holograms are

used in many research settings. They exhibit narrow

angular and/or spectral replay tolerance. This is exploited

in holographic data storage.

• Phase vs amplitude: Holograms can be made via spatial

modulation of the phase or the amplitude of the field.

Typically amplitude is modulated via varying absorbance

(loss).

• Phase: Phase holograms can be formed by spatial

modulation of the index of refraction, by spatial

modulation of the surface profile or both.

• Transmission vs reflection: The object can be

reconstructed on the same side from which the reference

is incident (reflection) or the opposite (transmission).

Security holograms are a common form of reflective, thin,

phase, surface-relief hologram.

• Bragg holography and materials

–Basics via thin holograms

Page 4: Bragg holography and holographic polymersecee.colorado.edu/~ecen5606/2014/Bragg holography... · • Bragg holography –Efficiency . ECEE 5606 Advanced Optics Lab Robert R. McLeod,

ECEE 5606 Advanced Optics Lab

Robert R. McLeod, University of Colorado

Simple absorption hologram and maximum efficiency

4

Write Read Object

Reference

Object

Reference

Conjugate

Ambiguity

A perfect (100% to 0% transmission) amplitude (absorption)

hologram would have a transmission function like:

x

E

ET

xin

out

2cos

2

1

2

1

%25.64

12

inref

outobj

I

I

x

When illuminated by a reference, the electric field just after the

hologram would be:

xjxjinin

inoutxx

EEETE

22

expexp42

Ambiguity Conjugate Object

Note that this is expressed as a Fourier transform. Each term

corresponds to diffraction into a particular angle, only one of

which is the object reconstruction. It’s power efficiency is

• Bragg holography and materials

–Basics via thin holograms

Page 5: Bragg holography and holographic polymersecee.colorado.edu/~ecen5606/2014/Bragg holography... · • Bragg holography –Efficiency . ECEE 5606 Advanced Optics Lab Robert R. McLeod,

ECEE 5606 Advanced Optics Lab

Robert R. McLeod, University of Colorado 5

Allocation of bandwidth Separation of terms

• Let the spatial-frequency bandwidth of the object field EO be BO

• Let the spatial-frequency bandwidth of the object field ER be BR

•The bandwidth of is = to the bandwidth of

which is 2 BO

2

xEO xOxO fEfE

xfT

mxf 1

Max[2BO, 2BR] BO + BR

Ambiguity Conj Obj

RO ff RO ff

xRx

xREAD

fEfT

fE

BR + Max[2BO, 2BR] BO +2 BR

Ambiguity Conj Obj

RORORRO BBBBBff 22,2Max2

1

ORO Bff2

3

To separate terms

Special case for plane-wave reference (BR=0)

mxf 1

BO + BR

RO ff RO ff

BO +2 BR

• Bragg holography and materials

–Basics via thin holograms

Page 6: Bragg holography and holographic polymersecee.colorado.edu/~ecen5606/2014/Bragg holography... · • Bragg holography –Efficiency . ECEE 5606 Advanced Optics Lab Robert R. McLeod,

ECEE 5606 Advanced Optics Lab

Robert R. McLeod, University of Colorado 6

Conjugate playback Time reversal

Reference

Object

Write Conjugate read

Object

Conjugate

reference

RRORORRO

RRORORO

RCONJ

EEEEEEEE

EEEEEEE

ETE

2*22

**22

Illuminate with phase-conjugate reference

Ref Object

Ref

Reconstruction

Write Read

Phase

perturbation

Use in imaging through phase (not amplitude!) perturbation:

• Bragg holography and materials

–Basics via thin holograms

Page 7: Bragg holography and holographic polymersecee.colorado.edu/~ecen5606/2014/Bragg holography... · • Bragg holography –Efficiency . ECEE 5606 Advanced Optics Lab Robert R. McLeod,

ECEE 5606 Advanced Optics Lab

Robert R. McLeod, University of Colorado 7

Angular bandwidth Replay with tilted reference beam

Reference

Object

Tilted

object

Tilted

reference

Write Tilted read

2*22

R

xjk

ORR

xjk

O

xjk

RRO

xjk

RREAD

EeEEEeEeEEE

eETE

xxx

x

Ok

Rk

GK

mkz

1

mkx

1

Ok

Rk

GK

• Add spatial frequency to reconstruction at plane of hologram.

• Similar to tilt – precisely equivalent to grating deflection.

• Only limitation is TIR = wide angular bandwidth.

Real space

Fourier space

mkz

1

mkx

1

• Bragg holography and materials

–Basics via thin holograms

Page 8: Bragg holography and holographic polymersecee.colorado.edu/~ecen5606/2014/Bragg holography... · • Bragg holography –Efficiency . ECEE 5606 Advanced Optics Lab Robert R. McLeod,

ECEE 5606 Advanced Optics Lab

Robert R. McLeod, University of Colorado 8

Spectral bandwidth Replay with different color reference beam

Red

object

Red

reference

Red read

Blue

object

Blue

reference

Blue read

Ok

Rk

GK

Ok

Rk

GK

Red

2

Blue

2

redR blueR

blueRredReff

blueR

blue

redR

red

blueRredR

n

kxkx

sinsin

sin2

sin2

ˆˆ

To reconstruct object at same angle

where

red

blueeffn

• Reconstruction is equivalent to refracting into a material of index neff

• Only limitation is available angular bandwidth for ref and obj beams

• Thus, wide spectral bandwidth.

Real space

Fourier space

mkz

1

mkx

1

mkz

1

mkx

1

• Bragg holography and materials

–Basics via thin holograms

Page 9: Bragg holography and holographic polymersecee.colorado.edu/~ecen5606/2014/Bragg holography... · • Bragg holography –Efficiency . ECEE 5606 Advanced Optics Lab Robert R. McLeod,

ECEE 5606 Advanced Optics Lab

Robert R. McLeod, University of Colorado

Glossary of polymer chemistry

9

• Bragg holography and materials

–Holographic photopolymers

• Monomer “single part”: small building blocks reacted to become…

• Oligomer “few parts”: medium-sized molecule of 2 or more monomers

• Polymer “many parts”: macromolecule built from many monomers

• Photolysis “light cutting” : photoinitiator molecule breaks into two radicals

• Photoinitiator : species which absorbs light to start initiation reaction

• Initiation : radical reacts with monomer to start propagation reaction

• Propagation or polymerization : Chain reaction which adds monomer units

• Termination: Reaction which stops propagation, including radical dimerization

• Inhibitor: A species such as O2 which reacts with and terminates radicals

• O2 threshold: Optical dose required to create sufficient radicals to consume O2

• Hydrogen abstraction: Transfer of radical by exchange of a hydrogen nucleus (proton)

• Chain transfer: termination of one propagating polymer by release of radical

Page 10: Bragg holography and holographic polymersecee.colorado.edu/~ecen5606/2014/Bragg holography... · • Bragg holography –Efficiency . ECEE 5606 Advanced Optics Lab Robert R. McLeod,

ECEE 5606 Advanced Optics Lab

Robert R. McLeod, University of Colorado

Typical “two component” chemistry

10

Chemical component Chemical function wt %

Diphenyle(2,4,6-

trimethylbenzoyl)

phosphineoxide (TPO)

Photo initiator 0.067

Tribromophenyl Acrylate Writing monomer 6.00

Butyl phthalate 99% Plasticizer 0.50

Polyester Block Polyether Matrix component 1 55.61

Desmodur 3900 Matrix component 2 37.81

Dibutyltin Dilaurate, 95% Catalyst 0.01

Matrix polymer

• Urethane

• Catalyzed at room T

• Low shrinkage stress

• Phase uniform

• Low scatter

• Low refractive index

Photopolymer

• 405 nm initiation

• 1 photon = high speed

• High refractive index

• Small, mobile

• Low shrinkage

• Low scatter

Initiating radicals attach

to the matrix and react

with monomer .

“Sponge” model

• Bragg holography and materials

–Holographic photopolymers

Page 11: Bragg holography and holographic polymersecee.colorado.edu/~ecen5606/2014/Bragg holography... · • Bragg holography –Efficiency . ECEE 5606 Advanced Optics Lab Robert R. McLeod,

ECEE 5606 Advanced Optics Lab

Robert R. McLeod, University of Colorado

Initiation

Basic mechanisms

Photons create mobile radicals, some of which react with matrix

to create fixed pattern.

• Bragg holography and materials

–Holographic photopolymers

Page 12: Bragg holography and holographic polymersecee.colorado.edu/~ecen5606/2014/Bragg holography... · • Bragg holography –Efficiency . ECEE 5606 Advanced Optics Lab Robert R. McLeod,

ECEE 5606 Advanced Optics Lab

Robert R. McLeod, University of Colorado

Initiation Reaction/diffusion

Immobile polymer chains grow from these radicals.

Replacement monomer diffuses into reaction region.

Basic mechanisms

• Bragg holography and materials

–Holographic photopolymers

Page 13: Bragg holography and holographic polymersecee.colorado.edu/~ecen5606/2014/Bragg holography... · • Bragg holography –Efficiency . ECEE 5606 Advanced Optics Lab Robert R. McLeod,

ECEE 5606 Advanced Optics Lab

Robert R. McLeod, University of Colorado

Initiation Reaction/diffusion

Matrix swells, decreasing matrix concentration in bright regions.

Basic mechanisms

• Bragg holography and materials

–Holographic photopolymers

Page 14: Bragg holography and holographic polymersecee.colorado.edu/~ecen5606/2014/Bragg holography... · • Bragg holography –Efficiency . ECEE 5606 Advanced Optics Lab Robert R. McLeod,

ECEE 5606 Advanced Optics Lab

Robert R. McLeod, University of Colorado

Initiation Reaction/diffusion Cure

Optical flood exposure consumes remaining initiator

and monomer.

Basic mechanisms

• Bragg holography and materials

–Holographic photopolymers

Page 15: Bragg holography and holographic polymersecee.colorado.edu/~ecen5606/2014/Bragg holography... · • Bragg holography –Efficiency . ECEE 5606 Advanced Optics Lab Robert R. McLeod,

ECEE 5606 Advanced Optics Lab

Robert R. McLeod, University of Colorado

Initiation Reaction/diffusion Index profile

Segregation of writing polymer and matrix creates index

distribution.

Images courtesy of Robert McLeod

Cure

Basic mechanisms

• Bragg holography and materials

–Holographic photopolymers

Page 16: Bragg holography and holographic polymersecee.colorado.edu/~ecen5606/2014/Bragg holography... · • Bragg holography –Efficiency . ECEE 5606 Advanced Optics Lab Robert R. McLeod,

ECEE 5606 Advanced Optics Lab

Robert R. McLeod, University of Colorado

Source of refractive index

16

n = nMatrixPolymerjMatrixPolymer + nWritingPolymerjWritingPolymer

= nMatrixPolymer 1-jWritingPolymer( ) + nWritingPolymerjWritingPolymer

= nMatrixPolymer + nWritingPolymer - nMatrixPolymer( )jWritingPolymer= n0 +dn

Mobile writing monomer swells matrix polymer, replacing it 1:1

Validation with confocal Raman spectroscopy:

• Bragg holography and materials

–Holographic photopolymers

Page 17: Bragg holography and holographic polymersecee.colorado.edu/~ecen5606/2014/Bragg holography... · • Bragg holography –Efficiency . ECEE 5606 Advanced Optics Lab Robert R. McLeod,

ECEE 5606 Advanced Optics Lab

Robert R. McLeod, University of Colorado

Typical grating dynamics

17

i: = 0.7 m, I = 100 mW cm-2 t = 3 sec

ii: = 5 m, I = 100 mW cm-2, t = 1 sec

iii: = 0.7 m, I = 10 mW cm-2, t = 3 sec.

This material is unusually slow. Commercial materials will develop in seconds.

• Bragg holography and materials

–Holographic photopolymers

Page 18: Bragg holography and holographic polymersecee.colorado.edu/~ecen5606/2014/Bragg holography... · • Bragg holography –Efficiency . ECEE 5606 Advanced Optics Lab Robert R. McLeod,

ECEE 5606 Advanced Optics Lab

Robert R. McLeod, University of Colorado

Coupled modes derivation (1/2) Homogeneous = Fourier optics

18

Start with the vector wave equation with the material described by a homogeneous

and an inhomogeneous permittivity

The solutions to the homogeneous equation (uniform material, RHS=0) are two

polarized plane waves. Since these plane waves are complex exponentials in time

and space, an incident field on the half-space boundary z=0 can be expressed as a

sum of the eigenfunction plane waves via a Fourier transform in time, x and y (here

written in the scalar form for simplicity):

e w,kx ,ky,z = 0( ) = Ftxy E t,x,y,0( )éë ùû

Since these eigenfunctions propagate in z via accumulation of phase, the fields can

be calculated at any distance z via the transfer function:

E t,x, y,z( ) = Ftxy-1 e w ,kx ,ky,z = 0( ) e- jkz w ,kx ,ky( )z{ }

= Ftxy-1 Ftxy E t,x,y,0( )éë ùû e

- jkz w ,kx ,ky( )z{ }Which is the foundational equation of Fourier optics. Note that the amplitudes of

the plane waves are constant with z. e

• Bragg holography

–Efficiency

Page 19: Bragg holography and holographic polymersecee.colorado.edu/~ecen5606/2014/Bragg holography... · • Bragg holography –Efficiency . ECEE 5606 Advanced Optics Lab Robert R. McLeod,

ECEE 5606 Advanced Optics Lab

Robert R. McLeod, University of Colorado

Coupled modes derivation (2/2) Inhomogeneous

19

As an approximate solution to the inhomogeneous case, assume that the plane

wave amplitudes can exchange energy via coupling and thus vary in z.

e w,kx ,ky,z = 0( ) = Ftxy E t,x,y,0( )éë ùû

E t,x,y,z( ) = Ftxy-1 e w ,kx ,ky,z( ) e- jkz w ,kx ,ky( )z{ }

…and substitute into the wave-equation. The product of two functions with z

dependence will generate multiple terms via the chain rule. All terms on the right

hand side which don’t involve z-derivatives of the envelope must sum to zero

because they are solutions to the homogeneous wave equation, leaving an ordinary

DE in z for the envelope which is unfortunately complicted by the RHS:

where the second derivative has been dropped by the slowly-varying envelope

approximation (SVEA). To deal with the RHS, we now assume that, for the

problem of interest, only a finite number of modes present. This reduces the

continuous Fourier transform integrals to a sum over a discrete number of “modes”

= plane waves in this case:

E t,x,y,z( ) = ui z( )i

å e- j kz ,i z = e

iz( )e

j wt-kxx-kyy( )

i

å e- j kz ,i z

which reduces the equation to a coupled set of first-order ODEs:

dum

dz= - jkz,mum z( ) - j km,i z( )ui z( )

i¹m

å

where κ is the coupling between modes caused by the perturbation. Note that if the

coupling is zero, the solution is uncoupled plane waves with unchanging

amplitude.

• Bragg holography

–Efficiency

Page 20: Bragg holography and holographic polymersecee.colorado.edu/~ecen5606/2014/Bragg holography... · • Bragg holography –Efficiency . ECEE 5606 Advanced Optics Lab Robert R. McLeod,

ECEE 5606 Advanced Optics Lab

Robert R. McLeod, University of Colorado

Example:

Two coupled modes

20

d

dzu0 = - jb0u0 - jk u1

d

dzu1 = - jb1u1 - jk u0

2

22

11

2

222

00

1

sin

1

cos

zgzuzI

zgzuzI

2

2

10

2

1022

g

Coupling vs. distance

1,22,1

The equations for the field amplitudes are um z( ) º em z( )e- jbm z

which have simple solutions:

where

where

0 1 2 3 z g

0

1

I /

I 0(0

)

I0

10

1 0

I1

Zc

• Bragg holography

–Efficiency

Page 21: Bragg holography and holographic polymersecee.colorado.edu/~ecen5606/2014/Bragg holography... · • Bragg holography –Efficiency . ECEE 5606 Advanced Optics Lab Robert R. McLeod,

ECEE 5606 Advanced Optics Lab

Robert R. McLeod, University of Colorado

Application to volume

holography

21

h ºIdiff L( )Iinc 0( )

= sin2 p L dn

cosqdiffl0

æ

èç

ö

ø÷

Thus the efficiency of the hologram (Koglenik solution) is

h =p L dn

cosqdiffl0

æ

èç

ö

ø÷

2

For weak diffraction, diffracted field grows linearly with L (k-space solution)

uinc

udif

f

kinc

kdiff

x

z

kx

K

diffinc

kx k =p dn

l0 cosqdiff

If the incident wave is Bragg matched such that Δβ = 0,

Iinc z( ) º uinc z( )2

= cos2 k z( )

Idiff z( ) º udiff z( )2

= sin2 k z( )

If M holograms divide the available Δn into equal fractions,

hM =p L Dn

Mcosqdiffl0

æ

èçç

ö

ø÷÷

2

ºM #

M

æ

èçö

ø÷

2

, where M# ºp L Dn

cosqdiffl0

Thus the efficiency of multiplexed holograms falls quadratically with number.

• Bragg holography

–Efficiency

100% efficiency possible

Page 22: Bragg holography and holographic polymersecee.colorado.edu/~ecen5606/2014/Bragg holography... · • Bragg holography –Efficiency . ECEE 5606 Advanced Optics Lab Robert R. McLeod,

ECEE 5606 Advanced Optics Lab

Robert R. McLeod, University of Colorado 22

Angular and wavelength selectivity

L

4

zk

xk

0k

1k

zk

0k

1k

Wavelength selectivity

Angular selectivity

xk

Write in blue, shift towards

red until reach first null of

diffraction efficiency

sin2kK

L2

2p

L= 2Dk sinj( ) tanj

= DkK

k

æ

èçö

ø÷tanj

Dk

k=

Dl

l=

L

L tanj

L

4

sin2kK

A2

A

L

sin2 k

What tilt of the incident wave causes kz

phase mismatch equal to the first null?

Dkz = 2dj k sinj

h Dkz( ) = hBraggMatched sinc2 DkzL

2p

æ

èçö

ø÷

Dkz,Null =2p

L= 2djNull k sinj =

2p

LdjNull

djNull =L

L

What wavelength shift of the incident wave

causes kz phase mismatch equal to the first null?

2p

L

A2

k

Write at ϕ, tilt until

reach first null of

diffraction efficiency

Consider a hologram of thickness L written

by two plane waves, then read out by one of

them.

• Bragg holography

–Selectivity