7
7.1
What   is   a   Fourier   transform?
A   Fourier   transform   (FT)   is   not   an   analytical   instrument   in   its   own   right;   in   fact   it   isn’t   even   a   real   object,   but   rather   a   mathematical   equation!
It   is   a   mathematical   procedure   developed   by   a   French   mathematician   by   the   name   of   Fourier   for   converting   complex   waveforms   into   a   combination   of   sine   waves   (see   Figure   7.1),   which   are   distinguished   by   their   intensity   and   frequency.
This   may   not   sound   terribly   relevant   to   chemical   instrumentation:   making   the   connection   is   what   this   chapter   is   about.
Combination
Wave 1
Wave 2
FIGURE   7.1
Fourier   transform   conversion   of   combination   waveform   to   separate   waves
To   explain   how   the   FT   works,   it   is   easiest   to   use   a   simple   emission   spectrum   of   three   lines,   which   would   look   like   Figure   7.2(a).
This   type   of   spectrum   is   known   as   a   frequency   domain   spectrum ,   since   it   is   a   plot   of   intensity   versus   frequency/wavelength.
This   is   the   normal   spectrum   that   chemical   analysis   is   based   on.
There   is   another   way   of   expressing   this   spectrum:   in   the   form   of   a   time   domain   spectrum ,   where   intensity   is   a   function   of   time   (as   in   Figure   7.2(b).
This   shows   the   oscillation   of   intensity   over   time,   and   is   not   a   smooth   sine   wave,   because   it   is   the   combination   of   three   different   waveforms   –   the   three   emission   wavelengths.
(b)
(a)
I
1
sin f
1 t
+
I
2
sin f
2 t
Frequency Time
FIGURE   7.2
Two   emission   lines   as   a   (a)   frequency   domain   and   (b)   time   domain   spectrum

7.
Fourier   Transform   Techniques
The   time   domain   spectrum   is   of   no   practical   use   for   chemical   analysis,   because   it   is   the   individual   lines   that   we   need.
However,   the   Fourier   transform   calculation   processes   a   time   domain   spectrum   and   delivers   the   information   needed   for   a   frequency   domain   spectrum:   frequency/wavelength   and   intensity   of   the   individual   waveforms   (i.e.
emission   lines).
Time   Domain   Spectrum   FT Frequency   Domain   Spectrum
Is   there   an   advantage   of   measuring   the   time   domain   spectrum?
The   time   variations   in   intensity   can   be   measured   without   the   need   to   scan   through   individual   wavelengths,   because   the   combined   waveform   contains   all   the   necessary   information   for   the   FT   to
produce   a   frequency   domain   spectrum.
But   there’s   a   problem!
To   measure   the   time   domain   spectrum,   we   need   a   detector   capable   of   responding   to   the   variations   in   intensity   of   electromagnetic   radiation.
If   we   choose   a   visible   wavelength   (e.g.
500   nm)   as   an   example,   then   it   has   a   frequency   of   6   x   10
14
Hz,   or   600,000,000,000,000   oscillations   per   second!
This
means   a   detector   would   have   to   have   a   mind ‐ boggling   response   time   to   see   these   oscillations.
C LASS   E XERCISE   7.1
Most   detectors   have   a   response   time   of   100   milliseconds.
This   means   they   only   see   an   average   of   what   occurs   each   100   ms.
(a)   How   many   oscillations   will   occur   while   the   detector   responds?
(b)   What   will   be   the   output   from   the   detector?
So   we   are   stuck   with   a   good   idea   and   no   way   to   make   it   work!
7.2
How   to   make   the   Fourier   transform   technique   workable
In   the   late   1800s   two   scientists,   Michelson   and   Morley,   built   a   device   which   was   intended   to   prove   that   light   moved   at   different   speeds   in   different   directions.
The   purpose   of   this   was   to   show   that   a   substance   known   as   an   “ether”   existed,   through   which   the   waveform   of   light   was   transmitted.
The   device   was   known   as   an   interferometer ,   and   is   based   on   the   idea   of   constructive   and   destructive   interference.
Waveforms   when   they   overlap   “combine”   and   the   resulting   intensity   depends   on   the   level   of   overlap.
As   it   turned   out,   Michelson   and   Morley   found   that   they   could   not   prove   what   they   hoped   (because   the   ether   didn’t   exist   and   light   moves   the   same   speed   in   all   directions).
The   interferometer   was   consigned   to   the   museum   of   scientific   curiosity,   until   some   very   bright   person   realised   that   the
Fourier   transform   and   the   interferometer   could   be   combined   to   make   a   working   spectrometer.
AIT 7.2

7.
Fourier   Transform   Techniques
How   does   an   interferometer   work?
Figure   7.3
shows   the   basic   components.
Fixed   Mirror
Interferometer
Moveable   Mirror
Radiation
Source
beam   splitter
Detector
FIGURE   7.3
Schematic   diagram   of   an   interferometer
How   does   the   interferometer   work?
Radiation   from   the   source   hits   a   beam   splitter   and   50%   is   reflected   at   right   angles   towards   the   fixed   mirror,   and   50%   passes   through   to   the   moveable   mirror.
The   two   beams   bounce   from   the   mirrors   and   are   recombined   at   the   beam   splitter   and   are   diverted   to   the   detector.
So   far,   so   good.
If   the   distance   that   the   two   beams   travel   after   splitting   is   exactly   the   same   then   they   will   recombine   perfectly   in   phase,   meaning   that   the   overall   intensity   is   not   decreased.
If   the   second   mirror   is   moved,   the   paths   will   be   of   different   lengths,   and   the   recombined   beams   will   be   out   of   phase,   causing   interference   that   reduces   the   intensity   of   the   beam.
Thus,   the   detector   sees   a   reduction   in   intensity.
As   the   mirror   moves   steadily   along   a   path   of   a   few   centimetres,   the   intensity   at   the   detector   varies   due   to   the   varying   interference,   producing   an   interferogram .
Now   comes   the   miracle!
The   interferogram   is   an   exact   replica   of   the   waveform   of   the   radiation   form   the   source,   with   a   frequency   that   is   directly   proportional   to   the   real   frequency   of   the   radiation .
It   does   not   matter   what   shape   the   incoming   waveform   is,   the   interferogram   will   replicate   it.
Now,   we   have   the   means   for   producing   a   waveform   that   can   be   detected,   and   then   processed   by   the   Fourier   transform   calculation.
But   we   need   a   way   of   determining   what   the   relationship   between   the   source   and   interferogram   frequencies   is.
Well,   a   simple   equation   exists   that   links   the   two   by   using   a   source   of   exactly   known   wavelength:   a   laser.
So   at   the   moment   we   have   a   way   of   determining   the   spectrum   of   a   source.
But   in   the   case   of   an   absorption   instrument   such   as   infrared,   we   want   to   measure   the   spectrum   after   it   has   been   absorbed   by   the   sample.
Thus,   the   sample   cell   must   be   located   into   the   configuration   shown   in   Figure   7.3.
In   all   reference   books,   the   sample   cell   is   shown   placed   between   the   interferometer   and   the   detector .
(It   must   be   said   that   the   author   of   these   notes   cannot   see   how   this   can   work,   and   that   the   logical   position   is   between   the   source   and   the   interferometer.)    The   Fourier   transform   technique,   together   with   the   interferometer,   has   been   used   in   spectroscopic   instruments   such   as   infrared,   NMR,   Raman   and   MS   (though   how   the   latter   works   without   radiation   being   involved   is   a   bit   of   a   mystery!).
AIT 7.3

7.
Fourier   Transform   Techniques
With   real   spectra,   the   combination   of   wavelengths   is   exceedingly   complex   because   we   are   dealing   with   polychromatic   radiation.
While   the   Fourier   transform   can   process   any   waveform   into   any   number   of   sine   waves,   computer   technology   was   not   sufficient   until   about   20   years   ago   to   make   it   viable.
7.3
Advantages   of   Fourier   transform   instruments
A   Fourier   transform ‐ based   spectrometer   has   all   the   advantages   of   a   multi ‐ channel   instrument,   without   the   extra   cost   of   multiple   detectors,   and   also   has   extra   ones   of   its   own.
These   advantages   are:
 speed   –   the   only   moving   part   in   the   instrument   is   the   mirror,   and   its   travel   only   occupies   a   few   seconds;   spectra   can   be   recorded   in   milliseconds   with   reduced   mirror   travel   (this   is   expanded   on   below)
 wavelength   accuracy   –   the   Fourier   transform   and   the   laser ‐ calibrated   interferometer   achieve   spectral   accuracy   undreamt   of   in   conventional   instruments   (eg   0.01
cm ‐ 1 )
 greater   sensitivity   –   because   there   are   fewer   optics   inside   the   instrument,   compared   to   a   monochromator ‐ based   one,   more   radiation   is   passing   through   the   sample,   allowing   improved   sensitivity
 better   quantitative   performance   –   the   wavelength   accuracy   and   sensitivity   combine   to   allow   far   better   quantitative   accuracy,   with   absorbances   of   greater   2   still   giving   linear   performance
The   speed   advantage   of   FT   instrument   leads   to   two   other   advantages:   noise   reduction   and   time ‐ resolved   spectra.
Background   noise   is   a   problem   for   low   concentration   samples,   and   obscures   the   peak.
This   is   the   known   as   the   signal ‐ to ‐ noise ‐ ratio   (S/N),   and   where   the   signal   is   low,   the   S/N   ratio   will   be   low   as   well.
Because   the   FT   instrument   is   so   fast,   we   can   record   multiple   spectra   in   less   than   the   time   normally   taken   for   a   monochromator   scan.
The   computer   then   averages   these   spectra,   and   the   noise   will
“disappear”   because   at   a   given   frequency,   in   some   spectra   it   will   be   high,   and   in   others   it   will   be   low,   the   average   being   somewhere   in   the   middle.
The   peak   will   also   be   the   same   intensity,   so   it   will   remain   unchanged   by   the   averaging   (see   Figure   7.4).
FIGURE   7.4
Multiple   spectra   (pale   lines)   and   average   spectrum   (dark   heavy   line)
AIT
7.4

7.
Fourier   Transform   Techniques
All   of   the   advantages   so   far   described   are   improvements   on   the   older   monochromator ‐ based   instruments.
There   is   one   advantage   of   FT   instruments   that   older   instruments   cannot   do:   record   spectra   so   quickly   that   you   can   see   a   chemical   reaction   actually   occurring.
Figure   7.5
shows   an   example   of   this:   the   sequence   of   spectra   are   recorded   at   400   microsecond   intervals .
The   FT   processing   occurs   afterwards.
FIGURE   7.5
Time ‐ resolved   infrared   spectra   (spectra   of   acetone   undergoing   a   photochemical   reaction,   recorded   at   400
 s   intervals,   unknown   source)
What   You   Need   To   Be   Able   To   Do
 define   important   terminology
 explain   what   the   Fourier   transform   does
 explain   how   an   interferometer   works
 draw   a   schematic   diagram   of   a   FT   instrument
 explain   why   an   interferometer   is   required
 list   instruments   using   Fourier   transform   techniques
 outline   advantages   of   FT   instruments
AIT 7.5

7.
Fourier   Transform   Techniques
Revision   Questions
1.
Draw   a   diagram   showing   the   difference   between   a   time ‐ domain   and   frequency ‐ domain   spectrum   for   monochromatic   radiation.
2.
Why   is   the   time ‐ domain   spectrum   superior?
Why   can’t   it   be   measured   for   normal   radiation?
3.
What   does   the   Fourier   transform   do   in   terms   of   the   two   types   of   domain   spectra?
4.
What   is   the   name   of   the   special   component   in   a   FT   instrument?
What   does   it   replace   in   conventional   instruments?
5.
Give   three   advantages   of   FT   instruments   over   conventional   equivalents?
6.
Give   one   application   of   FT   instruments   that   is   not   possible   at   all   with   a   conventional   equivalent.
Answers   on   the   next   page.
AIT 7.6

7.
Fourier   Transform   Techniques
Answers   to   revision   questions
Where   the   answer   can   be   found   directly   in   your   notes,   a   reference   to   them   will   be   provided.
1.
Figure   7.2
2.
No   need   to   scan.
Detectors   can’t   respond   quickly   enough.
3.
Converts   from   time   to   frequency   domain.
4.
Interferometer.
Monochromator.
5.
Section   7.3
6.
Figure   7.5
and   associated   text
AIT 7.7