Fourier-Transform Infrared Spectroscopy(FTIR)

Principles of the Fourier Transform Spectroscopy

FTIR Spectrometer

Mostly based on:

Introductory Fourier Transform Spectroscopy Robert John Bell &

Michelson Interferometer

rktEE cos0

c

k

22

1~

xtEE ~2cos0

101~2cos xtEtrE

X1/2

X1/2

1

2

(r,t)

102~2cos xtEtrE

~cos~2cos2 1021 xtEtrEEER

~2cos1~)( Bi

Polychromatic source

~~2cos~~~~~2cos1~)(maxmaxmax

~

0

~

0

~

0

dBdBdBi

max

~

0

~~2)0(

dBi 02

1~~)(max

~

0

idBi

3.0

2.5

2.0

1.5

1.0

0.5

0.0

i

-3 -2 -1 0 1 2 3

i

i

~~2cos~)0(2

1)()()(

max~

0

dBiiii

Consider a source of monochromatic wave:

2

Ei time

iThe detector measures where

0

Fourier-Transform

~~2cos~)0(2

1)()()(

max~

0

dBiiii diiB ~2cos)0(2

1)(~

0

FT

max

Two obvious technical difficulties:

1. The retardation is always finite, max

Consequence: Signal (in frequency space) appears convoluted with an Instrumental Line Shape (ILS) function Limits the measuring resolution

Solution: Apodization

2. The zero retardation =0 (ZPD) is needed for calculating the spectrumConsequence: Asymmetry of the interferogram, phase error

Solution: Double-sided interferogram

~)()(~22 ~

BfBtVolttii scannervscanner VfFFTDetectortV

The FTIR spectrometer:

Bruker 113v/IR98

Advantages of the FTIR spectrometer

mmrc.caltech.edu/FTIR/FTIRintro.pdf

http://www.thermo.com/eThermo/CMA/PDFs/Product/productPDF_21615.pdf

Important Properties of Fourier-Transform

dxexfg xi

~2)(~

~~)(~2 degxf xi

dxxxfidxxxfg ~2sin)(~2cos)(~

~~2sin~~~2cos~)( dxgidxgxf

1. If f(x) is even (f(x)=f(-x) ) FT(f(x)) is real and even

If f(x) is odd (f(x)=-f(-x) ) FT(f(x)) is imaginary and odd

http://dagsaw.sdsu.edu

Convolution: dtgftgtf

)(

2. Convolution Theorem: The FT of the product of two function is the convolution of their individual FTs

)()()()(

)()(

)()(tgFTtfFTtgtfFT

tgFTtg

tfFTtf

~2cos~)( BiFT

max

diB

~2cos~

-max

)()()(~2cos~

iFTdiB

(x)=1 if -max max

0 if > max

Finite retardation Multiplication of the real space interferogram with a “boxcar function”

4

3

2

1

0

-4 -2 0 2 4

1.0

0.0-4.0-3.5-3.0-2.5-2.0-1.5-1.0-0.50.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

4

3

2

1

0

-1

Am

plit

.

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0Freq.

11~2cos~)( Bi

FT )~(~2cos~

111

diB

)~(~)(~2cos~11 GBdiB

Finite retardation the real spectrum is the “true spectrum” B(1) convoluted with the ILS

Monochromatic radiation of frequency 1 :

(x)=1 if -max max

0 if > max

FT maxmax

~2sin2)~( cG

1.0

0.0

-5 -4 -3 -2 -1 0 1 2 3 4 5

4

3

2

1

0

-5 -4 -3 -2 -1 0 1 2 3 4 5

max)

max)

1=3

Instrumental Line Shape (ILS)

1.0

0.5

0.0

-5 -4 -3 -2 -1 0 1 2 3 4 5

2max

1/max

0.605/max

max=0.5

ILS

Broadening of the spectral line

max11max~~sin~2~ cBB

Delta Function

4

3

2

1

0

5.04.54.03.53.02.52.0

The resolution of an instrument:

Two adjacent lines of equal intensity, having sinc2(x) shape, are resolved when the center of one line coincides with the first zero of the other

1 Rayleigh criterion :

Intuitively, the resolution min=1/max

2. FWHH criterion: Two triangularly shaped lines of equal intensity are resolved when the spacing between lines is grater than the FWHH of either line.

6

5

4

3

2

1

0

-1

6543210

0.5/max

1.0

0.0

6543210

6

5

4

3

2

1

0

-1

6543210

0.73/max

1.0

0.0

6543210

Apodization

If we use the triangular function (x) instead of the boxcar function (x) ILS sinc2(x)

2.0

1.0

0.0

-4 -3 -2 -1 0 1 2 3 4

1.0

0.0

-2 -1 0 1 2

1.0

0.0

-4 -3 -2 -1 0 1 2 3 4

max

0.88/max

2/max

1.0

0.5

0.0

-5 -4 -3 -2 -1 0 1 2 3 4 5

2max

1/max

0.605/max

Other Apodization Functions:

Particularly common in FTIR spectrometers:

Happ-Genzel

Sampling Phase Errors & Two-Side Interferogram

diiB ~2cos)()(~

0

Phase errors may arrive from misalignments of the interferometer (optical effects), from filters (electronic effects) and/or from sampling errors

Let’s assume that the ZPD is not correctly determined: 0 )()( ii

diiBmeas~2cos)()(~

0

The effect: ~2exp~~ iBBmeas

Consider the complex FT on both sides of =0, from -max to max :

diiiBmeas~2exp)()(~

Compute both real (cos) part, C(), and imaginary (sine) part, S() of the FT:

~~~~ 22 BSCBmeas

Mathematical proof for phase correction : TO BE ADDED

Sampling interval

Nyquist Criterion: The sampling frequency (wavenumber) should be equal or larger than twice the highest measured frequency (wavenumber).

The effect of undersampling is aliasing, or folding:

max2 ffsamp

http://zone.ni.com/devzone/cda/tut/p/id/5509

ff

ffffff

fsamp

realsampimagsamp

samp

real 22

alias

FTIR Spectrometer:

~2cos~)( Bi

~)(~22 ~

BfBtii scannervscanner VfFFTtV

tVBti scanner 2~2cos~)(

tfEtEtE 2coscos)( 00

~2~ scannerVf

cms

cm

sHz

11

It is better to sample equal intervals of retardation () rather than equal intervals of time (t)

If we want to measure up to max~

max

maxmaxmaxmax ~22

1~22~2~

scanner

sampscannersamp

Nyquist

scannerV

tVfVf

1

max

2

max~2

1~22

1

cmcm

Vt samp

tV

scanner

sampscanner

The Scanning Process & Duty Cycle Efficiency

Vsc

ann

er

time

Duty Cycle Efficiency = cycle wholea of Time

measured is raminterferog whileTime

When the interferograms are sampled coherently, i.e. between same retardation limits around ZPD:

)(N Scans ofNumber Signal scans

)(N Scans ofNumber Noise scans

The noise varies continuously (randomly) with time, it is not expected to have the scanning cycle periodicity

scansNSNR

Different “flavors” of scanning cycles:

Rapid-Scan Interferometer

Step-Scan Interferometer

Requirements:

Data collection needs to start at the same retardation from the ZPD each cycle

Sampling needs to take place at the same retardations each cycle

tVscanner

2

http://www.chem.uic.edu/tak/chem524/notes16/figureIR_3.gif

~)(~22 ~

BfBtii scannervscanner VfFFTtV

How the interferogram is sampled or why we need He-Ne laser and white light ?

The He-Ne frequency sets the sampling frequency (for one sample/cycle):

115800~~

cmsampNeHe 1

maxmax 7900~5.0~ cm

The white light determines when the first data point is taken (t=0), i.e. initiates data collection

Vtrig

3.0

2.5

2.0

1.5

1.0

0.5

0.0

-4 -2 0 2 4

Initiate Data Collection

White Light

He-Ne

IR

time

Sub-sequential data collection is triggered by the zero crossings of the He-Ne laser interferogram

NeHescannerNeHe VHzf ~2

Example: 13.02 scmVscanner

KHzf NeHe 74.4158003.0 (This is how OPUS labels scanner velocity)

Vtrig

3.0

2.5

2.0

1.5

1.0

0.5

0.0

-4 -2 0 2 4

Initiate Data Collection

White Light

He-Ne

IR

time

The sampling electronics (ADC board) starts measuring when a thresold signal Vtrig is detected

It is important that the white light interferogram occurs before the IR interferogram

DAQ can only take place when scanning mirror moves in one direction single-sided interferogram and poor duty cycle efficiency

Position of the white light interferogram must remain constant and cannot be adjusted with maximum retardation

How is the phase correction achieved? – To be discussed