Transformations in Science: Catch the Waves!
Stephen M. Kuebler
Associate Professor of Chemistry and Optics
University of Central Florida
Interim Assistant VP for Research
Office of Research and Commercialization
kuebler@ucf.edu
http://npm.creol.ucf.edu
Are these the same?
How about these?
Or these?
Now compare these as well? See any similarities?
These images are related through the
Fourier transform (FT)
FT
FT
FT
FT
FT
FT
FT
FT
Jean!Baptiste!Joseph!Fourier
21!March!1768!" 16!May!1830
So what is a Fourier transform?
Just kidding!
A Fourier transform is this:
Ouch !!!
Let's try a different approach...
A Fourier transform (FT) is...
• A powerful mathematical tool...
• It let's us express a set of data (a signal) in terms of a series of "waves".
• The signal can be:
- 1D (like an MP3 audio track)
- 2D (like the images)
FT
FT
- or nth-dimensional...
A Fourier transform (FT) is...
• One of many mathematical "transformations" used across STEM that make it easier or
faster to:
• represent,
• transmit,
• process, and
• compare...
INFORMATION !
Salvador!Dali:!Book!Transforming!Itself!into!a!Nude!Woman.
1940!oil!on!canvas,!16¼"!x!20""!(pvt.!collec#on).
This!painting!was!hidden!from!public!view!from!1941!until!2004.!
How is FT used?
Noise reduction
Image compression
Microchip manufacturing
Holography
Anandtech
Wikipedia
How is FT used?
Medical imaging
Drug design
Wikipedia
...and!also:
• Optics
• Sound!analysis
• Image!analysis
• Spectroscopy
How does FT work?
First, we need to learn how to speak "wave"...
Wave talk
The wave's period (T)
and frequency (!) are
reciprocal:
!=1/T
So a frequency of
! = 100 waves / sec
corresponds to...
T = 0.01 sec.
Waves can add
Constructive interference
Waves of same phase combine
"
Destructive interference
Wave of opposite phase combine "
Amplitude becomes larger
Amplitude becomes smaller
A Fourier transform "pulls out" the waves
Frequency"Domain!Spectrum
Time"Domain!Signal
FT of a signal (in the time domain) gives a spectrum (in the frequency domain).
A spectrum is:
• The set of waves that comprise the signal.
• A frequency-domain representation of the time-domain
signal (note the inverse relationship).
A spectrum shows:
• Amplitude and phase
of each wave, at each
frequency.
Let's HEAR an example
http://en.wikipedia.org/wiki/Harmonic_series_(music)
http://www.falstad.com/fourier/
A Fourier transform "pulls out" the waves
• The waves identify repeated patterns in the signal.
• 1D -- In an MP3 audio track...
The pattern could be the beat of a drum.
You can see that pattern
with a graphic equalizer.
A Fourier transform "pulls out" the waves
• 2D -- In an image the pattern could be a repetive series of dots...
Note the inverse relationship
between fringe-period and
spread of dots in the FT.
FT
FT
The FT in spectroscopy:
Example of Nuclear Magnetic Resonance (NMR)
The FT in spectroscopy:
Example of Nuclear Magnetic Resonance (NMR)
"CH3
"CH2"
FT
"OH
A simple lens performs an FT
x1
k1
k2
x2
k3
Positions,!x
x3
k$vectors
(direction)
• For a time-domain signal, f and t are inverse parameters (f = 1 / Period).
• For a spatial-domain signal (an image), k and x are inverse parameters (k = 2#/$).
This is one approach for ultrafast all-optical computing.
Using FT to compare images:
The "art" of image correlation
Multiply image FTs here!
• Finger print recognition
• Facial identification
• Target acquisition
Let's see a device that does this...
Image correlation by optical FT and nonlinear optics
Kippelen!et!al.,
J.!Mater.!Chem.!(2009)
cordis.europa.eu/esprit
Nemoli!et!al.,!"Correlation!and!image!moment!approaches!to!
analyze!the!Glagolitic!script!carved!in!stone!tablets!"!Optic!(2012)
Image correlation by FT and nonlinear optics
Tariq!Manzur!et!al.!Appl.!Opt.!(2012)
Is there any connection between FT and Quantum
Mechanics?
Universe!Today
Heisenberg's uncertainty principle:
How do we explain it?
A!particles!position!
and!momentum!
cannot!be!measured!
with!infinite!precision.
Equivalently,!nor!can!
its!energy!and!time!
when!it!had!that!
energy.
Heisenberg's uncertainty principle:
How do we explain it?
h × %!
E=h×!
so
%E = h × %!
and
%E%t ! h/2
Harvard!Nat.!Sci.!Lecture!Demos
Use of FT in the
Kuebler Group!
Multi-Photon 3D Direct Laser Writing: Capabilities
Enabling capabilities
! Free-form sub-micron 3D fabrication
! Direct preparation of integrated photonic devices
! Compatible with other fab processes
! Versatile, low-cost, and scalable
Materials for DLW
! Polymers
! Metallization
! SiO2 inversion
! Si double-inversion
! Semiconductors
(As2S3)
! Pre-ceramics
Resolution is limited by the focal spot intensity
distribution
I2
Intensity across
focal plane
Polymerization
threshold
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-2
-1.5
-1
-0.5
0
Focal
Spot
How can we re-shape
the intensity profile at
the focus ?
0.5
1
1.5
2
Beam-shaping with Diffractive Optical Elements
(DOEs)
Beam splitting
• High speed drilling
• High speed welding &
cutting
Beam shaping
• Pattern lump
processing
SEI Technical Review, no. 53 (2002)
Beam homogenizing
• Surface treatment
• Laser soldering
Challenges in designing DOEs for high-NA focusing
ik
!
e(P) 1 0 //2
2#
Must account for vector
character of field under
high-NA.
!
ik
h(P) 1 0 //2
2#
DOE " Intensity profile
• Easy!
• Compute profile using
&
!
a sx , s y
!
b sx , s y
Op
tic
al
Ax
is
• Difficult!
• Target profile is not
necessarily a solution to
the wave equation.
• No theory existed for
axial profile shaping
under high-NA.
sz
x
y
Diffractive Optical
Element
x
y
' exp*ik(.(s , s ) - s! , r!(P))+ds ds
x
y
Wolf , Proc. R. Soc. Lond. A, 53, 349-357 (1959)
vector diffraction theory.
Intensity profile " DOE
&
sz
' exp*ik(.(s , s ) - s! , r!(P))+ds ds
x
y
Convergence to axially super-resolving DOE
• 30% axial superresolution (G = 0.70)
• Side-lobes < 52% of peak intensity
MGP results for an axially superresolving DOE
PSF with DOE
before 1.4-NA lens
G = 0.71
M = 0.52
Diffraction-limited
PSF
Intensity along
optical axis
Jabbour, Petrovich, and Kuebler,
Optics Commun., 281, 2002 (2008)
Transverse field synthesis
3D wave synthesis was implemented in the scalar regime based on MGP
Projection of constrains
is applied on each plane
separately in the focal volume
Example of
3D field synthesis
Piestun and Shamir, Proc. IEEE, 90, 222-244 (2002)
Vectorial beam shaping problem
Flat-top square
at focal plane (scaled by 7)
Flat-top circular input
Focus
NA = 1.4
$ = 800 nm
MGP constraints
I t 1 {1 for &9 , 8 ' 3 2 ;
0 for &9 , 8 ' 3 2 c }
Analog phase-only DOE
Diffraction integrals evaluated as Fourier- and Inverse-Fourier transform
E x ( x, y ) 1
(
)
2
5I t ( x, y ) 0 I y ( x, y ) 0 I z ( x, y ) exp[i4 x ( x, y )]
nc6 0
for ( x, y ) 3 2
Particle swarm optimization (PSO)
• Developed by Kennedy and Eberhart in 1995.
• Stochastic optimization technique inspired
by natural systems such as flocking birds
and schooling fish.
• Population-based method, like genetic algorithm.
• “Social learning” terms guide population to an optimum solution.
• Finds global solutions more
readily than MGP.
Li et al., Phys. Med. Biol., 50, 3491-3514 (2005)
Sites where you can explore Fourier transform
Hearing!the!waves!
http://www.falstad.com/fourier/
YouTube!video
http://www.youtube.com/watch?v=ObklYbQaX24
The!Math!
http://mathworld.wolfram.com/FourierTransform.html
Doing!some!2D!FTs!on!images:
http://cns$alumni.bu.edu/~slehar/fourier/fourier.html
http://en.wikipedia.org/wiki/Fourier_transform
http://www.brainflux.org/java/classes/FFT2DApplet.html
http://homepages.inf.ed.ac.uk/rbf/HIPR2/fourier.htm
http://www.thefouriertransform.com/
Thanks for listening!