Chemistry 301 Mathematics 251 Chapter 5

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Chemistry 301/ Mathematics 251Chapter 5

• Complex Transforms

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Power Series
Consider power series
Expanded the function into an infinite
polynomial Used the powers of x as a basis
set Can consider other expansions
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Fourier Series
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Fourier Series
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Fourier Series
Will also work if
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Fourier Series
consider
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Fourier Series
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Fourier Series
Can also show
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Fourier Series
Change of interval from 2p to 2L
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Fourier Transform
L ? ?
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Fourier Transform
Use Euler relations
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Fourier Transform
x, y conjugate variables Eg. Time
frequency Distance wavelength
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Fourier Transform
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Fourier Transform
Shine laser (n0) on detector, measure signal as
function of time
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Fourier Transform
Beamsplitter
Fixed Plane Mirror
If no optical path difference then beams will
simply recombine If there is an optical path
difference then beams will be out of phase -
interference
Moving Plane Mirror
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Michelson Interferometer
a/2
Monochromatic beam
(ad)/2
Signal on detector will vary with optical path
difference
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Michelson Interferometer
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Michelson Interferometer
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Michelson Interferometer
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Michelson Interferometer
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Michelson Interferometer
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Michelson Interferometer
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Michelson Interferometer
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Michelson Interferometer
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Fourier Transform
Interferogram is of finite length
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Fourier Transform
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Apodization

• Lineshape is due to fact that interferogram is
  not infinite.
• Multiply experimental interferogram by
  apodization function to force it to zero (or
  close to zero)
• Apodization functions
• Triangular H(d) 1- d/X
• quarter wave H(d) cos (0.5pd/X)
• Happ Genzel H(d) 0.54 0.46cos(pd/X)

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Apodization
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Applications of FT to Spectrometry

• IR
• Raman
• UV-Vis (not popular)
• NMR
• FT-ICR Mass Spectrometry

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Advantages of FT - Spectroscopy

• Multiplex (Fellgett) advantage
• signal averaging increases SNR
• same time, FT gives better SNR (10 to 1000 x)
• Throughput (Jacquinot) advantage
• no slits
• ³ 100 x
• Connes Accuracy (IR Raman)
• Use He-Ne Laser to determine mirror position
  (very reproducible)
• Wavenumber are very precise and accurate
• Do not need to calibrate spectrometer with a
  standard.

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Discrete Fourier Transform

• Digital data
• Interferogram sampled at discrete points (N)
• Will have N points in the spectrum
• Zero fill
• Add N zeros to end of interferogram
• Total 2N points
• Interferogram does not contain any new
  information
• Spectrum will have 2N points

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Hilbert/ Kramers - Kronig Transformations
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Hilbert/ Kramers - Kronig Transformations
Cauchy Goursat integral
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Hilbert/ Kramers - Kronig Transformations
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Hilbert/ Kramers - Kronig Transformations
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Hilbert/ Kramers - Kronig Transformations
X
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Hilbert/ Kramers - Kronig Transformations
Cauchy Principal Value
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Hilbert/ Kramers - Kronig Transformations
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Hilbert/ Kramers - Kronig Transformations
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Hilbert/ Kramers - Kronig Transformations
Special Case
Hilbert Transforms
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Hilbert/ Kramers - Kronig Transformations
Special Case
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Hilbert/ Kramers - Kronig Transformations
Special Case
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Hilbert/ Kramers - Kronig Transformations
Special Case
Kramers - Kronig (KK) Transforms
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Optical properties

• Non-absorbing medium

Speed of light in vacuum
Speed of light in medium
Refractive index

• Absorbing medium

Equations for non-absorbing hold for absorbing
medium if use complex refractive index
Complex Refractive index
Imaginary Refractive index (absorption index)
Real Refractive index
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Optical properties

• Non-absorbing medium (Dielectric Constant)

• Absorbing medium

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Optical properties
Kramers - Kronig (KK) Transforms
Hilbert Transforms
C. D. Keefe, J. Mol. Spec., 205, 261-268 (2001).
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Transmission of Sample

• Non-absorbing medium
• Not all of light will reach detector due to
  reflection
• Absorbing medium
• Some of light will be absorbed
• Measured intensity will depend on reflection and
  absorption
• i.e will depend on n and k

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Transmission of cell
Liquid- absorbing- reflection and absorption
(complex refractive index)
One measurement (amount of light lost) Two
unknowns (n k) Use iterative procedure with KK
Windows - non-absorbing- reflection only (real
refractive index)
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Iterative procedure to determine optical constants

• Determine n?
• Measure experimental transmission
• Assume k 0, n n?
• Calculate transmission of system (using Fresnel
  equations)
• Compare 4 with 2
• Adjust k
• Calculate n from k (KK transform)
• Repeat 4 7 until calculated transmission agrees
  with exp (within tolerance)

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Benzene absorption spectrum
Wavenumber / cm-1
J. E. Bertie C. D. Keefe, J. Mol. Struct.,
695-696, 39-57 (2004).
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Benzene k spectrum
Wavenumber / cm-1
J. E. Bertie C. D. Keefe, J. Mol. Struct.,
695-696, 39-57 (2004).
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Benzene n spectrum
Wavenumber / cm-1
J. E. Bertie C. D. Keefe, J. Mol. Struct.,
695-696, 39-57 (2004).
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Benzene e? spectrum
Wavenumber / cm-1
J. E. Bertie C. D. Keefe, J. Mol. Struct.,
695-696, 39-57 (2004).
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Benzene e' spectrum
Wavenumber / cm-1
J. E. Bertie C. D. Keefe, J. Mol. Struct.,
695-696, 39-57 (2004).
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Necessary Approximations

• Experimental data is digital
• Calculate numerical integral
• MacLaurin approx (sum over every second point)
• K. Ohta H. Ishida, Appl. Spectrosc. 42, 952
  (1998)
• Experimental measurement is finite
• Eg. IR will have contributions from UV/Vis
• J. E. Bertie Z. Lan, J. Chem. Phys., 103, 10152
  (1995)
• Absorption may not be zero at low limit
• J. E. Bertie S. L. Zhang, Can. J. Chem., 70,
  520 (1992)
• Absorptions below limit of region

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Hadamard Transform

• Series of orthogonal symmetric transforms of
  order 2m, m 0, 1, 2,
• Defined by recursive formula

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Hadamard Transform

• Used in data compression, video compression
• Offers multiplexing advantage
• Same number of measurements ? greater SNR

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Hadamard Transform

• M 2 case (i.e want to measure x1 x2)
• Instead of measuring directly measure the
  combinations s1 s2

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Hadamard Transform

• Consider x1 50g x2 40g
• Assume our balance has a systematic error of 5g
• Measure x1 x2 directly get x1 55g x2 45g
• Measure s1 s2 get s1 67g s2 11g

Improved value
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Hadamard Transform

• Consider x1 50g x2 40g
• Assume our balance has a random error of d
• Measure x1 x2 directly get x1 50 d g x2
  40 d g
• Measure s1 s2 get s1 (90 d)g/?2 s2 (10
  d)g/?2

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Hadamard Transform

• Must measure positive and negative signals
• Requires 2 detectors
• Not always practical
• Use alternate form based on Sylvester (S-type)
  matrix

Omit normalization factor Omit 1st row and
column Replace 1 with 0 Replace -1 with 1
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Hadamard Transform
Omit normalization factor Omit 1st row and
column Replace 1 with 0 Replace -1 with 1
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Hadamard Transform
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Hadamard Transform
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Hadamard Transform - Example
Wrong zero on balance
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Hadamard Transform
Systematic error (ds)
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Hadamard Transform
Random error (dr)
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Hadamard Transform