Bragg holography and holographic polymersecee.colorado.edu/~ecen5606/2014/Bragg holography... ·...
Transcript of Bragg holography and holographic polymersecee.colorado.edu/~ecen5606/2014/Bragg holography... ·...
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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