Fourier Transforms and Atomic Physics
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Fourier Transforms Fourier Transforms and and Atomic Physics Atomic Physics Dallin S. Durfee Dallin S. Durfee Presented to Math 303 Presented to Math 303 Winter 2007 Winter 2007
description
Fourier Transforms and Atomic Physics. Dallin S. Durfee Presented to Math 303 Winter 2007. Some transforms we already know. Laplace transform. Some transforms we already know. Taylor series. Fourier’s Theorem. Any periodic function can be written as a sum of sines and cosines. - PowerPoint PPT Presentation
Transcript of Fourier Transforms and Atomic Physics
Fourier TransformsFourier Transformsandand
Atomic PhysicsAtomic Physics
Dallin S. DurfeeDallin S. Durfee
Presented to Math 303Presented to Math 303
Winter 2007Winter 2007
Some transforms we already knowSome transforms we already know
Laplace transformLaplace transform
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Some transforms we already knowSome transforms we already know
Taylor seriesTaylor series
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Fourier’s TheoremFourier’s Theorem
Any periodic function can be written as a Any periodic function can be written as a sum of sines and cosines.sum of sines and cosines.
If a function If a function ff((tt) is periodic in ) is periodic in tt with a with a period period TT, then, then
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mm tmbtmatf
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Building a Square WaveBuilding a Square Wave
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Fourier Transform
Inverse Fourier Transform
Waves on a Finite StringWaves on a Finite String
Waves on a Finite StringWaves on a Finite String
Plucked StringPlucked String
Whacked StringWhacked String
Fourier Transforms of Non-Periodic Fourier Transforms of Non-Periodic FunctionsFunctions
A non-periodic function is simply periodic A non-periodic function is simply periodic with with TT==∞∞
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““Hello”Hello”
A Few Fun Applications of F.T.A Few Fun Applications of F.T.MP3MP3
A Few Fun Applications of F.T.A Few Fun Applications of F.T.
A Few Fun Applications of F.T.A Few Fun Applications of F.T.MP3MP3
JPGJPG
A Few Fun Applications of F.T.A Few Fun Applications of F.T.
A Few Fun Applications of F.T.A Few Fun Applications of F.T.MP3MP3JPGJPGMPG / AVI / MOV / WMVMPG / AVI / MOV / WMVDSPDSP– AutotunersAutotuners
Brittany Spears and Boy Bands Brittany Spears and Boy Bands – Music Effects ProcessorsMusic Effects Processors
Cheap high-quality guitar effects Cheap high-quality guitar effects Really funky special effects Really funky special effects
– Active Sound ControlActive Sound Control
Made possible by a slick numerical technique Made possible by a slick numerical technique called the “Fast Fourier Transform” (FFT)called the “Fast Fourier Transform” (FFT)
But What About Atomic Physics?But What About Atomic Physics?
Fabry-Perot EtalonFabry-Perot Etalon
Calcium SpectroscopyCalcium Spectroscopy
The Uncertainty PrincipleThe Uncertainty Principle
2
1 t
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1 xk
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k
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In Quantum Mechanics, every object is represented by a wave.
kp 2
xp
Calcium SpectroscopyCalcium Spectroscopy
Random Phase KicksRandom Phase Kicks
y=sin(5 x) + random phase kicks Power Spectrum
Making an atomic clockMaking an atomic clock
Making an atomic clockMaking an atomic clock
4s2 1S0
4s4p 1P14s3d 1D2
4s4p 3P
The Continuum (49,304)
423
nm
657 nm
4s5p 1P1
672 nm410 H
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Atomic Beam ClockAtomic Beam Clock
Pure Sine WavePure Sine Wave
y=sin(5 x) Power Spectrum
““Shuttered” Sine WaveShuttered” Sine Wave
y=sin(5 x)*shutter(x) Power Spectrum
2
1 t
Off on a tangent...Off on a tangent...
Fraunhofer Diffraction and Fraunhofer Diffraction and Fourier TransformsFourier Transforms
Fraunhofer Diffraction and Fraunhofer Diffraction and Fourier TransformsFourier Transforms
Imaging a Star with a TelescopeImaging a Star with a Telescope
Apodization Apodization
The Uncertainty PrincipleThe Uncertainty Principle
Getting to the Natural Linewidth Getting to the Natural Linewidth with Ramsey Spectroscopywith Ramsey Spectroscopy
Nobel Prize, 1989
NBS-1NBS-1
NIST-F1NIST-F1
Our DesignOur Design
Edge Mirrors
Detectors
PrecisionBeam Splitter
Right AnglePrismPenta Prism
Atomic Beam
AtomicBeam
CollimationApertures
Spatial Filter
Detector657 nm
689 nmBeam Splitter
1 2 3 4AtomicBeamExit
AperturesTransverseLaser
Cooling
SrOven
CaOven
MixingChamber
StrontiumFluorescence
Probe
CalciumFluorescence
Probe
Spatial Filter
Some transforms we already knowSome transforms we already know
Taylor seriesTaylor series
– ff((xx) represents a finite set of continuous data.) represents a finite set of continuous data.
– aaii is an infinite set of discrete values. is an infinite set of discrete values.
Some transforms we already knowSome transforms we already know
Laplace transformLaplace transform– ff((tt) is an infinite set of continuous data) is an infinite set of continuous data– FF((pp) is also...) is also...
Fourier’s TheoremFourier’s Theorem
– ff((tt) represents a finite set of continuous data.) represents a finite set of continuous data.
– aaii and and bbii are infinite sets of discrete values. are infinite sets of discrete values.
– Like the Laplace transform, it is an integral Like the Laplace transform, it is an integral transform.transform.
0
00 )cos()sin()(m
mm tmbtmatf
T
20
Fourier Transforms of Non-Periodic Fourier Transforms of Non-Periodic FunctionsFunctions
A non-periodic function is simply periodic A non-periodic function is simply periodic with with TT==∞∞
– ff((tt) is an infinite set of continuous data) is an infinite set of continuous data– aa((ωω) and ) and bb((ωω) are also...) are also...
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