Fourier Transform

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Fourier Transform as Used by Chemists Literatur
e Seminar by Michael Curry
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Applications of Fourier Transform in Spectroscopy

• Infrared Spectroscopy (FT-IR)
• Raman Spectroscopy (FT-Raman)
• Nuclear Magnetic Resonance Spectroscopy (FT-NMR)
• Mass Spectrometry (FT-MS)

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Fourier Transform Spectrometers

• First commercial FT spectrometer 1962
  (Grubb-Parson infrared spectrometer)
• Suffered from drawbacks (slow, sensitive to
  vibrations, limited sample analysis)
• Advancements in computers
• Development of the Cooley-Tukey algorithm (Fast
  Fourier Transform, 1965)

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Fourier Series

• In 1805, Joseph Fourier introduced Fourier series
• Mathematical process that can be used to
  decompose any waveform into a sum of sine waves,
  or harmonics.
• The sum of the harmonics makes up the original
  waveform.

Reproduced from Johnston, S. F. Fourier Transform
Infrared 1991
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Fourier Transform (FT)

• is the signal (t-domain)
• is the spectrum (f-domain)

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Development of Optical Spectroscopy

• In early 1800s, quest for monochromatic light
• In 1752, spectroscope invented by Thomas Melvill
• In 1814, telescope added to view light by Joseph
  Fraunhofer
• In 1850, the introduction of chemical analysis by
  Bunsen and Kirchoff (professors of chemistry and
  physics, respectively)

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Interference Patterns of Waves

• In 1802, Thomas Youngs light interference
  discovery observed bright and dark fringes
• Hippolyte Fizeau (ca. 1840) studied interference
  patterns (large optical path differences)
• In 1862, fringe counting revealed the ratio
  between slightly different wavelengths

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Wavelength Comparison

• Fringe counting (principle used by Fizeau)
• Wavelengths in phase (bright) and out of phase
  (dark)
• The first variable path interferometer

Reproduced from Johnston, S. F. Fourier Transform
Infrared 1991
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Interferential Spectroscopy

• In 1880, Albert A. Michelson (University of
  Berlin) developed the Michelson interferometer
• Study of luminiferous aether (medium allowing
  propagation of light through space)
• Essential elements Beam splitter, fixed and
  moveable mirror, and a hand-turned screw

Reproduced from Johnston, S. F. Fourier Transform
Infrared 1991
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The Interferogram

• Lack of sensitive detectors (Michelson used his
  eye as the detector)
• Mechanical harmonic analyzer invented by
  Michelson and Stratton
• Investigation of the infrared region (Heinrich
  Rubens)
• Exploration of the region between light waves to
  radio waves
• Employment of an energy detector and broad band
  measurements (observed a wide band of wavelength
  simultaneously)
• In 1911, first recorded interferogram (Rubens and
  R. W. Wood)

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Visible Spectrum

• Interferogram (top) and visible curve (bottom)
• Dotted line relates visibility curve to
  interferogram
• Recorded overall characteristics of the
  interferogram

Reproduced from Kauppinen, J. et al Fourier
Transform Spectroscopy 2001
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Harmonic Analyzer

• Consist of rods, levers, springs, drums, and a
  pen
• Constructed using 20 mechanical elements (later
  80 elements)
• Simulated light waves of particular frequency

Reproduced from Kauppinen, J. et al Fourier
Transform Spectroscopy 2001
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First Published Interferogram

• Heinrich Rubens and S. W. Wood (1911)
• Output intensity from interferometer versus the
  optical path difference of two beams

Reproduced from Kauppinen, J. et al Fourier
Transform Spectroscopy 2001
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The Spectrum

• Difficulties in Fourier transforms computation
  prevented calculation of spectrum
• Spectrum shapes were guessed and compared to the
  interferogram
• In 1949, first spectrum computed by Peter
  Fellgett
• Scanned interference pattern formed by two wedge
  plates
• Performed the Fourier transform by hand

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Fourier Transform Infrared Spectroscopy

• Source generates radiation
• Collimated and impinged upon a beam splitter
• Half reflected to the fixed and half to moveable
  mirror
• Both halves reflected by to beam splitter
• Recombined and detected

Reproduced from Skoog, D. A. Principles of
Instrumental Analysis 1992
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Interferograms and Optical Spectra

• A beat frequency is observed as waves go in and
  out of phase

Reproduced from Skoog, D. A. Principles of
Instrumental Analysis 1992
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Discrete Fourier Transform (DFT)

• Digital processing of the data
• Finite limits (-T to T-1)
• ?t is the sample interval

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Cooley-Tukey Algorithm

• Advancements in computers (vacuum tube to
  transistors)
• Algorithm published in 1965 by James Cooley and
  John Tukey
• Reduced the number of operations needed to
  compute an n-point spectrum
• FFT requires 2n data points

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Truncation of the Signal

• Truncation function ?2T(t) times the original
  signal h(t)
• The Boxcar function

Reproduced from Kauppinen, J. et al Fourier
Transform Spectroscopy 2001
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Distortion of the Spectrum

• Truncating the signal in the time domain give
  rise to distortion of the spectrum in the
  frequency domain.
• The signal is convolved by a sinc function
  (Ringing pattern).
• The sinc function is defined as H (?) 2T
  sinc(2p? T) H(?).

h(t)
Reproduced from Kauppinen, J. et al Fourier
Transform Spectroscopy 2001
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Apodization

• Multiplying the time domain signal by a
  triangular function to minimize ringing pattern
  (foot removal)
• Smooth the interferogram to zero intensity

Reproduced from Faires, M. L. Analytical
Chemistry 1986, 58, 1023A-1033A
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Apodization Functions

• Apodization function (left hand side) and
  corresponding line shape (right hand side)

Reproduced from Davis, S. P. et al Fourier
Transform Spectrometry 2001
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Sampling of the Frequency

• In order to represent the signal correctly it
  must be sampled at the Nyquist Frequency
• A 1600 Hz cosine wave (dash line) sampled at 2000
  data per second

Reproduced from Skoog, D. A. Principles of
Instrumental Analysis 1992
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Zero-Filling

• Zeros added to the interferogram before
  transformation

Reproduced from Cassady, C. J. Personal
Communication
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Advantages of FT-IR Compared to Dispersive
Instruments

• Multiplex or Fellgett advantage
• More information obtained about each frequency
• Rapid scanning
• S/N increase with increased signal averaging
• Throughput or Jacquinot advantage
• Power of radiation reaching the detector is
  greater
• Increased S/N and sensitivity
• High wavelength accuracy and precision
• Signal averaging possible
• Improved S/N

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Summary I

• Advancements in mathematics
• Joseph Fourier (Fourier series)
• Discovery of interference patterns
• Developments in optical spectroscopy
• Michelson Interferometer
• Advancements in computers
• smaller, cheaper, faster
• Cooley-Tukey algorithm
• Fast Fourier Transform

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Ion Cyclotron Resonance

• Origin of ion cyclotron resonance mass
  spectrometry begins with E. O. Lawrences
  Cyclotron
• In 1950, Thomas and Hipple introduced first ion
  cyclotron spectrometer (Omegatron)
• Beginning in 1974, Marshall and Comisarow applied
  Fourier transform methods
• In 1981, first commercial FT-ICR

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Analyzer Cell

• Trapping plates left and right
• Excitation plates front and back
• Detection plates top and bottom

Reproduced from Asamoto, B. FT-ICR/MS 1991.
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Ionization Methods

• Ions formed internally or externally
• Some methods include
• Electrospray Ionization (ESI)
• Matrix Assisted Laser Desorption Ionization
  (MALDI)
• Chemical Ionization (CI)
• Electron Ionization (EI)

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Ion Excitation

• Figure a and b impulse excitation and chirp
  excitation, respectively

Reproduced from Asamoto, B. FT-ICR/MS 1991.
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Signal Generation

• Ionization of the sample
• Ions trapped in magnetic field are accelerated
• Image current detected by detector plates

Reproduced from Asamoto, B. FT-ICR/MS 1991.
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Signal Acquisition

• Signal acquired in full spectrum mode (broadband)
  or heterodyne mode (narrowband)

Reproduced from Asamoto, B. FT-ICR/MS 1991.
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Data Processing

• Discrete Fourier Transform is used
• Fast Fourier Transform
• Truncation
• Distortion
• Apodization of the time domain signal
• Minimizes artifacts
• Zero-Filling
• S/N increased

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Advantages of FT-ICR compared to Conventional
Instruments

• Multiplex or Fellgett advantage
• More information obtained about each frequency
• Rapid scanning
• S/N increase with increased signal averaging
• Superconducting Magnets in FT-ICR/MS
• Higher magnetic field strengths
• Less expensive to operate

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Summary II

• FT revolutionized data processing
• Optical spectroscopy
• Non optical spectroscopy
• Advancements in technology
• Bench top instrument
• Rapid scanning

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Acknowledgments
Dr. John Doe Dr. Jane Doe The University of
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