FTIR Manual Part1 Theory

International Laser Center of Moscow State University

Fourier-Transform Infrared Spectrometer Task Manual, Part 1, Theory

Moscow, Oulu 2005

FTIR Spectrometer Manual. University of Oulu, Finland Table of Contents

Table of Content Table of Content ................................................................................................................................... 2 0. Introduction....................................................................................................................................... 3

0.1 Electromagnetic radiation and units of measurements ............................................................... 3 0.2 Spectrum Representation ............................................................................................................ 5

1. FT-IR Spectrometer Basics............................................................................................................... 7 1.1 Sources of Infrared Radiation ..................................................................................................... 9

Thermal, unpolarized IR sources .................................................................................................. 9 Bright, pulsed, polarized, micro-IR sources ............................................................................... 11

1.2 The Michelson Interferometer .................................................................................................. 13 Obtaining the interferogram........................................................................................................ 14 Real Interferometers.................................................................................................................... 16 What are OPD and ZPD?............................................................................................................ 18

1.3 Infrared Transmitting Materials................................................................................................ 19 1.4 Detectors ................................................................................................................................... 21 1.4 Advantages of FT-IR Instruments over Dispersive Instruments .............................................. 23

Multiplex (Fellgett) Advantage................................................................................................... 24 The Throughput (Jacquinot) Advantage ..................................................................................... 24 High Resolution and Linearity (Connes) Advantage.................................................................. 24

1.5 Older Technology ..................................................................................................................... 25 2. Interferograms and Data Processing ............................................................................................... 26

2.1 Interferograms........................................................................................................................... 26 2.2 Finite Mirror Displacement. Resolution. .................................................................................. 31 2.3 Apodization function. ............................................................................................................... 32

2.3.1 Instrument line shape and Apodization function. .............................................................. 33 2.4 The IR Spectrum....................................................................................................................... 36

2.4.1 Single Sided and Double Sided Interferograms................................................................. 38 2.4.2 Instrument Resolution and Spectral Noise......................................................................... 39

2.5 Phase Correction ....................................................................................................................... 40 2.5.1 Double-sided Phase Correction.......................................................................................... 41 2.5.2 Single-sided Phase Correction ........................................................................................... 42

Literature............................................................................................................................................. 45

Laser Teaching Lab., International Laser Center of Moscow State University 2

FTIR Spectrometer Manual. University of Oulu, Finland Introduction

0. Introduction

FT-IR stands for Fourier Transform Infra Red, the preferred method of infrared spectroscopy.

In infrared spectroscopy, IR radiation is passed through a sample. Some of the infrared radiation is

absorbed by the sample and some of it is passed through (transmitted). The resulting spectrum

represents the molecular absorption and transmission, creating a molecular fingerprint of the sample.

Like a fingerprint no two unique molecular structures produce the same infrared spectrum. This

makes infrared spectroscopy useful for several types of analysis.

Fig. 0.1. Schematic drawing of the spectrometer.

• It can identify unknown materials

• It can determine the quality or consistency of a sample

• It can determine the amount of components in a mixture

Infrared spectroscopy is a widely used technique for both qualitative and quantitative analysis.

IR spectra are routinely used in identification of synthesis products for organic chemistry, as well as

for qualitative analysis work in environmental, forensic, and industrial applications. IR spectroscopy

can be used quantitatively for determination of the concentrations of both liquid and gas phase

solutions. In addition, reflective IR techniques can be employed for surface analysis.

0.1 Electromagnetic radiation and units of measurements

The electromagnetic spectrum is divided into different spectral regions depending on the

frequency: radio waves have wavelengths greater than 0.1 m and microwaves between 500 µm and

30 cm; the infrared region is divided into far-, mid- and near infrared. The visible (VIS) region

includes wavelengths from 400 to 800 nm and near ultraviolet (UV) and vacuum UV region extends



from 400 to 200 or 100 nm. Molecular spectroscopic analysis methods are based on interactions

between molecules and light quanta with different energy levels. They are named after spectral

region and imply different effects on the molecules (Table 0.1), depending on the energy, effects

such as electron excitation, molecular vibration and molecular rotation.

Type of transfer Molecular rotation Molecular

vibration Electron excitation

Spectroscopic methods Microwave

Absorption Infrared spectroscopy UV-VIS spectroscopy

Spectral region radio waves microwaves infrared (IR) far mid near VIS UV X-

rays Wavelength [m]

vc

=λ 10 1 10-1 10-2 10-3 10-4 10-5 10-6 10-7 10-8 10-9

Frequency [Hz]

λcv = 107 109 1011 1013 1015 1017

Wave number [cm-1]

λ1~ =v 10-3 10-1 10 103 105 107

Table 0.1. Electromagnetic radiation.

The nature of various radiation frequencies shown in Table 0.1 have been interpreted by

Maxwell’s classical theory of electro- and magneto- dynamics. According to this theory, radiation is

described as two mutually perpendicular electric and magnetic fields, oscillating in single planes at

right angles to each other. These fields are in phase and are being propagated as a sine wave

(Fig. 0.2).

Fig.0.2. An electromagnetic wave. The electric and magnetic vectors are presented by E and H.

A significant discovery made about electromagnetic radiation was that the velocity of

propagation in a vacuum was constant for all frequencies of spectrum. This is known as the velocity

of light c and it has the following value c = 2.997925 × 108 m/sec.



vcvc =⇒= λλ , (0.1)

where

c – velocity of light

λ – wavelength

ν – frequency (number of vibrations, [sec-1 or Hz])

Another unit, which is commonly used in infrared spectroscopy, is the wavenumber, which is

expressed as cm-1. This is the number of electromagnetic waves in a length of one centimeter and is

given by the following relationship:

cvv ==

λ1~ (0.2)

This unit has the advantage of being linear with photon energy.

0.2 Spectrum Representation

The infrared spectrum is a plot of transmission of infrared radiation as a function of

wavelength. The infrared spectrum results from the interaction of infrared radiation with sample

molecules. The wavenumber scale (x-axis) is used to present the spectrum, to achieve constant

distances between the data points. The majority of commercial infrared spectrometers present the

spectrum with wavenumber decreasing from left to right.

The infrared spectrum can be divided into three regions, which are the far infrared (400 –

0 cm-1), the mid infrared (4000-400 cm-1) and the near infrared (14285-4000 cm-1). Most infrared

applications employ the mid-infrared region, but the near- and far-infrared regions can also provide

information about certain materials.

The ordinate scale may be presented in transmittance or absorbance. The spectrum shows the

transmission of the infrared radiation through the sample as a function of wavelength. For each

wavelength, the transmittance T is the intensity of the infrared radiation that has passed through the

sample divided by the intensity of the infrared radiation that has entered the sample. When there is

no absorption, the value of transmittance T is 1 (or 100 %), which indicates that 100 % of the

infrared radiation at that wavelength goes through the sample. If the intensity of the radiation

entering the sample is I0 and the intensity of the radiation that has passed through the sample is I, the

transmittance T can be expressed as:

T = I/I0, (0.3)

where

T – transmittance,



I – intensity passed through the sample,

I0 – intensity entering the sample.

In addition to transmittance T, the absorption of the infrared radiation can be presented using

the absorbance scale. Absorbance A is given by the logarithm of the reciprocal transmittance

TLogT

LogA 10101

−=⎟⎠⎞

⎜⎝⎛= (0.4)

where

A – absorbance

T – transmittance

The Fig. 0.3 (a, b) illustrates the difference in appearance between absorbance and

transmittance spectra. The advantage of using the absorbance scale is that the value of absorbance is

directly proportional to the thickness of the sample (absorption path length), and the concentration of

the sample.

1000 1500 2000 2500 3000 3500 40000.0

0.2

0.4

0.6

0.8

1.0

Tran

smitt

ance

Wavenumber, cm-11000 1500 2000 2500 3000 3500 4000

0.0

0.2

0.4

0.6

0.8

1.0

Abs

orpt

ion,

OD

Wavenumber, cm-1

Fig. 0.3 a). Transmittance spectrum of a polymer sample

(pristine MEH-PPV film on BaF2 substrate).

Fig. 0.3 b). Absorbance spectrum of a polymer sample

(pristine MEH-PPV film on BaF2 substrate).


FTIR Spectrometer Manual. University of Oulu, Finland FTIR Basics

1. FT-IR Spectrometer Basics

The development of interferometry was initiated in 1880 when Dr. Albert A. Michelson

invented his interferometer to study the speed of light and to fix the standard meter with the

wavelength of a known spectral line. FT-IR method is based on the old idea of the interference of

two radiation beams to yield an interferogram. An interferogram is a signal produced by two

radiation beams. Interferogram is the interference intensity as a function of the change of optical

path difference. The two domains of distance and frequency are interconvertible by the mathematical

Fourier transformation.

Although the basic optical component of FT-IR spectrometers, the Michelson interferometer

has been known for almost a century, it was not until advantages in data acquisition and computing

in early 80´s and late 70´s that the technique could be successfully and widely applied. Only in

recent years Fourier-transform infrared (FT-IR) spectroscopy has found increasing use in industrial

applications. The number of applications of FT-IR spectrometers is increasing continuously.

Nowadays, there are different kinds of FT-IR spectrometers, especially interferometers that

are used. A typical FT-IR spectrometer consists of a radiation source, a modulator

(interferometer), a sample compartment, a detection unit and an electronic and computing unit

(Fig.1.1).

Light

Source Interferometer Sample Detector Signal and data

processing

Fig. 1.1. Basic components of a FT-IR spectrometer.

Each building block varies from manufacturer to manufacturer but basically they all obey the

same laws of physics.

The normal instrumental process is as follows:

1. The Source: Infrared energy is emitted from a glowing black-body source. This beam

passes through an aperture that controls the amount of energy presented to the sample

(and, ultimately, to the detector).

2. The Interferometer: The beam enters the interferometer where the “spectral encoding”

takes place. The resulting interferogram signal then exits the interferometer.

3. The Sample: The beam enters the sample compartment where it is transmitted through or

reflected off of the surface of the sample, depending on the type of analysis being


FTIR Spectrometer Manual. University of Oulu, Finland FTIR Basics

accomplished. This is where specific frequencies of energy, which are uniquely

characteristic of the sample, are absorbed.

4. The Detector: The beam finally passes to the detector for final measurement. The

detectors used are specially designed to measure the special interferogram signal.

5. The Computer: The measured signal is digitized and sent to the computer where the

Fourier transformation takes place. The final infrared spectrum is then presented to the

user for interpretation and any further manipulation.

Because there needs to be a relative scale for the absorption intensity, a background

spectrum must also be measured. This is normally a measurement with no sample in the beam. This

can be compared to the measurement with the sample in the beam to determine the “percent

transmittance”. This technique results in a spectrum, which has all of the instrumental characteristics

removed. Thus, all spectral features, which are present, are strictly due to the sample. A single

background measurement can be used for many sample measurements because this spectrum is

characteristic of the instrument itself (see Fig. 1.2).

Fig. 1.2. Obtaining an instrument independent spectrum.


FTIR Spectrometer Manual. University of Oulu, Finland Sources of IR Radiation

1.1 Sources of Infrared Radiation Thermal, unpolarized IR sources

All IR spectrometers have a source of infrared radiation which is usually some solid material

heated to incandescence by an electric current. The Nernst Glower is a source composed mainly of

oxides of rare earths such as zirconium, yttrium and thorium, and the Globar is a silicon carbide rod.

Other materials have been used as well. All these sources are fairly efficient emitters of infrared

radiation and approach the energy distribution of a theoretical black body.

Using the quantum hypothesis, Planck derived a distribution law for blackbody radiation,

which holds over all wavelengths. Planck's distribution law giving the radiant energy between the

wavelengths λ to λ + dλ may be expressed in the form

λλ

λ λλ de

cdE Tc 1)/(

51

2 −=

−

(1.1)

c1 = 2πc2h = 3.740 × 10–12 W cm2 (1.2)

c2 = ch/k = 1.438 cm °K (1.3)

where c is the speed of light, 2.998 x 1010 cm/sec, h is Planck's constant, 6.625 x 10–34W

sec2, and k is the Boltzmann constant, 1.380 x 10–23 W sec/oK. Eλ dλ is the radiant energy emitted

per unit area per unit increment of wavelength. With the units shown and dλ in µm, the units of Eλ dλ are

W/cm2/µm.

The wavelength λm at which the energy is at a maximum at any temperature is found by

differentiating Eλ with respect to λ and setting the derivative dEλ/dλ equal to zero. The solution, of

the resulting equation yields Wien's displacement law

Tb

m =λ (1.4)

where b is Wien's displacement constant, 2897 µm °K, and T is absolute temperature, °K.

The actual energy distribution of a blackbody radiator at three different temperatures is

shown in Fig. 7. The energy at any wavelength is given by the area of the vertical strip under the

curve at that wavelength. If the height of the strip is Eλ and the width is dλ, then the area is Eλ dλ

as indicated on the 1000 ºC curve in Fig. 7.



Fig. 1.3 also shows how the peak of the energy curve is temperature dependent. For

example, at a temperature of 1000ºC the wavelength at which the energy is at a maximum, Eλm, is

given by Eq. (1.4) and would be (2.897 x 103)/(1000 + 273) = 2.276 µm (see Fig. 1.3). At a lower

temperature of 600°C the energy peak shifts to a longer wavelength, namely, 3.318 µm.

0 1 2 3 4 5 6 7 80

1

2

3

4

5

E λ, W/c

m2 /µ

m

Wavelength, µm

λm = 2.276

600ºC

800ºC

1000ºC

AREA = Eλ dλ

Eλ

dλ

Fig.1.3. Energy distribution of a black body radiation at 600, 800, and 1000ºC.

The Stefan-Boltzman law states that the total radiation from a black body (fhe total area

under a curve in Fig.1.3) is proportional to the fourth power of its absolute temperature. It is also

important to note in Fig.1.3 that the energy output falls off very rapidly at long wavelengths. If one

tries to increase the long wavelength energy by increasing the temperature of the source, a small

increase in long wavelength energy will be accompanied by a huge energy increase in the

unwanted short wavelengths as seen in Fig.1.3. Most IR sources are operated at a temperature

where the energy maximum is near the short wavelength limit of the spectrum. Another graphical

representation of Planck's Law is showing the spectral radiance of a black body radiator versus

wavenumber (see Fig.1.4).

There is another problem with using IR sources at high temperature: heating air to high

temperatures can be problematic – generation of the well known (NO)x! The equilibrium N2 + O2 ↔

NO2 ↔N2O4 lies towards the reactants and moves towards NO2 and hence N2O4 at elevated

temperatures. Unfortunately, good modern sources can be hot enough to generate measurable

amounts of NO2 (and other nitrogen oxides like (NO)x ). In the atmosphere, this would be only a

trivial nuisance, one swamped by other sources of pollution but inside the sealed housing of an



FTIR, the (NO)x can well cause unwanted absorption and even corrosion of the metal and plastic

components. So, sources should be hot but not too hot. Temperatures vary but a little above 1000ºC

are typical. Use of 2000ºK is not realistic.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

T = 1600 K T = 1800 K T = 2000 K

Pow

er, A

.U.

Absolute Wavenumbers, cm-1

Fig.1.4. Blackbody curves for a variety of temperatures on an absolute wavenumber abscissa scale in power units. As the temperature is increased the overall intensity increases and the maximum of the curve shifts to higher frequencies.

Bright, pulsed, polarized, micro-IR sources

Accelerator-based and laser sources emit IR beams with high spectral radiance and high

spatial resolution (<1 µm to 10's µm) with coherent or polarized radiation. Such sources may be used

for a variety of applications (e.g., commercial and military purposes).

Bright, pulsed, polarized, micro-IR sources are ideal for small samples (e.g., samples relevant

to geomicrobiology, meteoritics, materials synthesized at high pressure), surface species (e.g.,

weathering rinds), and samples with low energy, weak absorption signatures (such as films). Another

advantage is that these high spectral radiance sources may be pulsed to study in situ chemical

reactions on mineral surfaces on rapid time-scales.

Accelerator-based sources have high spectral radiance IR radiation that is produced by

accelerating charged particles (e.g., electrons) in a magnetic field. In synchrotron-based IR, the

electrons are accelerated around a storage ring and may be pulsed at tens of picoseconds to

nanoseconds.

In free electron laser (FEL) sources, the micro-beams are added to create the radiation used

for analysis. These sources may be pulsed at 10-15 to 10-9 seconds. FELs provide high intensity

beams, however intensities fluctuate therefore they are inherently not as stable as synchrotron

sources.



Fig.1.5 a) Irradiation wavelengths of different light sources.

Fig.1.5 b) Irradiation intensity versus wavelengths for different light sources (by Bruker Optics).


FTIR Spectrometer Manual. University of Oulu, Finland Interferometers

1.2 The Michelson Interferometer

The principle of a Michelson interferometer which is used in a Fourier transform infrared

(FT-IR) spectrometer is illustrated in Fig.10. As seen in Fig.1.6 (a) the device consists of two plane

mirrors, one fixed and one moveable, and a beam splitter. One type of beam splitter is a thin layer of

germanium on an IR-transmitting support. The radiation from the source is made parallel and as seen

in Fig.1.6 (b), strikes the beam splitter at 45°. The beam splitter has the characteristic that it

transmits half of the radiation and reflects the other half. The transmitted and reflected beams from

the beam splitter strike two mirrors oriented perpendicular to each beam, and are reflected back to

the beam splitter.

Fig.1.6. The interferometer is shown in (a). In (b) the beam splitter transmits half the source radiation and reflects the other half. In (c), where only one or the other mirror is in place, the radiation from the mirror is split by the beam splitter. When both minors are in place, interference will occur where the beams from the two mirrors combine. In (d)-(g), monochromatic radiation is used. In (d), the minors are equidistant. In (e), the moveable mirror is moved by λ/4. In (f), the moveable mirror is moved by λ /2. In (g), an interferogram is shown for monochromatic radiation. The monochromatic radiation intensity going to the detector varies as a cosine function of the retardation, or twice the mirror displacement, and reaches a maximum every time the retardation equals a whole number of wavelengths of radiation.

If only one or the other mirror is in place, as seen in Fig.1.6 (c), the beam reflected from the

mirror returns to the beam splitter, which then sends half the radiation to the detector and half back

to the source. If both mirrors are in place, interference occurs at the beam splitter where the radiation

from the two mirrors combine.



The detailed action at the beam splitter is complex, but as shown in Fig.1.6 (d), when the two

mirrors are equidistant from the beam splitter, constructive interference occurs for the beam going to

the detector, and destructive interference occurs for the beam going back to the source, for all

wavelengths of radiation. The path length of the two beams in the interferometer are equal in this

case, and the path length difference, called the retardation, is zero.

Obtaining the interferogram.

Let the source of radiation be a laser that emits only monochromatic radiation of wavelength,

and let the moveable mirror be moved from the equidistant point by λ/4 as seen in Fig.1.6 (e). The

retardation is λ/2. This means that for the beam going to the detector, the radiation wave components

from the two mirrors combine at the beam splitter one-half a wavelength out-of-phase and

destructive interference occurs. Then, when the mirror is moved from the equidistant point by λ/2 as

seen in Fig.1.6 (f), the retardation is A and the wave components from the two mirrors combine in-

phase again for the beam going to the detector. The plot of detector response as a function of

retardation is an interferogram. As seen in Fig.1.6 (g), the interferogram of a monochromatic laser

source described above, is a cosine function (see also Fig.1.7 (a)). The Fourier transform of a single

cosine wave interferogram is a single wavelength of radiation emitted from the laser source.

Fig.1.7 a). Interference pattern from monochromatic light

The total radiation entering the beam splitter from the two mirrors is constant regardless of

the retardation, which means that the total radiation leaving the beam splitter to the detector and

back to the source must also be a constant. That part of the radiation leaving the beam splitter which

does not go to the detector, goes back to the source, as seen in Fig.1.6 (e).

If the laser source described previously is changed so that it emits monochromatic radiation

of a different wavelength with a different intensity, the interferogram will be a cosine wave with a

different maximum amplitude and a different retardation length for one detector signal cycle. If the

radiation from both laser sources described above enters the interferometer, the interferogram will be

the sum of the two individual cosine wave interferograms, see Fig.1.7 (b).



Fig.1.7 b) Interference pattern from two monochromatic sources.

A polychromatic source can be considered as a multitude of laser-like monochromatic

sources with closely spaced wavelengths covering the whole wavelength region of the. source. The

interferogram of the polychromatic source can be considered as a sum of all the cosine waves from

the laser-like monochromatic emission components. The polychromatic interferogram will have a

strong maximum intensity at zero retardation, where all the cosine components are in-phase, and it

extends, in principle, to a retardation of infinity. While the monochromator “generates” a spectrum

(selects a spectral region), an interferometer produces an interferogram from which a spectrum is

generated by a Fourier transform. This would be an extremely laborious task to do by hand, but FT-

IR instruments have computers that do this quickly. The computer can then be used for further

processing.

Fig.1.7 c) Interference pattern from continuous spectrum source.

The sample is normally placed between the interferometer and the detector. The spectrum

generated is a single-beam spectrum where the vertical coordinate is the radiation intensity from the

source after sample absorption, measured as a function of radiation wavelength or wavenumber. In

order to generate a percent transmission spectrum, the sample single beam spectrum is stored in the

computer and is later normalized by a reference single-beam spectrum generated without the sample

present.

A central part of a real interferogram is shown in Fig.1.8.

A single scan of the moveable mirror produces a complete single-beam spectrum. However,

usually a number of scans are taken and signal-averaged by the computer, which reduces the noise

by the square root of the number of scans. The resolution is constant over the whole spectrum and is

increased by increasing the length of travel of the moveable mirror. A truly monochromatic emission



line has an interferogram which is in infinitely long cosine function. Of necessity, the measured

retardation is less than infinite and this truncation has the effect of broadening the width of a spectral

line generated from a monochromatic source.

Fig.1.8 A central part of a real broadband source interferogram.

Real Interferometers

Early interferometers (and some persist to this day) use a system identical to that in

Fig.1.6 (a). To make the light parallel, mirror collimators are used and so the instrument looks like

Fig.1.9.

All components are firmly fixed to a base but mirror MM moves backward and forwards on

an axis normal to its surface. The movement must be smooth, reproducible and involve no wobble or

shake. These requirements are hard to achieve and lend to the use of high precision engineering.

Let we have our interferometer and hence, as we shall see our interferogram. The next

problem is to measure the path difference. A ruler is no good, the path difference has to be measured

to an accuracy limited only by the precision of the engineering in the interferometer itself. The way

this is done is to use the interferometer itself to do the job and to feed it with a source of precisely

known wavelength. The He-Ne laser has a wavelength near 632.8 nm and is known to very high

precision. If a He-Ne laser beam passes through the interferometer, its intensity at the output will

generate a superb simple cosine with a peak in intensity every 632.8 nm of change in the path

difference. How this can be done is shown below in Fig. 1.10.


Fig.1.9. A complete Michelson Interferometer. J – Jacquinot stop, M1= off-axis ellipsiod, M2 and M3 – off-axis paraboloids. Note, the Beamsplitter BS need not be at 45° to the incoming beam. It rarely is in real instruments.

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He-Ne Detector

Fig.1.10. In this scheme of FT-IR spectrometer the He-Ne laser is used for measuring the optical path difference.


The He-Ne laser generates a parallel beam reflected along the axis of the interferometer by

tiny mirror. The beam leaving the interferometer is picked up by another tiny mirror and passed to a

detector. The output of the detector is a cosine. As the signal switches from positive to negative

about the mean, a trigger circuit is operated providing a pulse twice every 632.8 nm of charge in path

difference. This pulse train is used to measure the path difference as shown in Fig.1.11.

IR Interferogram

He-Ne laser Interferogram

Primary IR Interferogram that was sampled

e

Fig. 1.11. Sampling

What are OPD and ZPD?

Optical Path Difference (OPD) is the o

through the two arms of an interferometer. OP

traveled by the moving mirror (multiplied by 2

number of reflecting elements used) and n, th

interferometer arms (air, Nitrogen for purged

number of (signal, OPD) pairs of values.

FT-IR has a natural reference point when

from the beam splitter. This condition is called

moving mirror displacement, ∆, is measured fro

the moving mirror travels 2∆ further than the bea

between optical path difference, and mirror displa

Laser Teaching Lab., International Laser Center of M

Optical path differenc

ZPD

an actual interferogram.

ptical path difference between the beams traveling

D is equal to the product of the physical distance

, 4, or other multiplier which is a function of the

e index of refraction of the medium filling the

systems, etc.). The raw FT-IR data consists of a

the moving and fixed mirrors are the same distance

Zero Path Difference or ZPD (see Fig.1.11). The

m the ZPD. In Fig.1.6 (a) the beam reflected from

m reflected from the fixed mirror. The relationship

cement, ∆, is: OPD = 2∆n.

oscow State University 18

FTIR Spectrometer Manual. University of Oulu, Finland IR Materials

1.3 Infrared Transmitting Materials

Cells for holding samples, or windows within the spectrometer must be made of infrared

transmitting material. Table 1.1 lists approximate low wavenuniber transmission limits for some

optical materials used in infrared spectroscopy. All the materials listed transmit in the mid-IR above

the transmission limit except for polyethylene, which is used below 600 cm-1 only, as it has

absorption bands at higher frequencies. The transmission limits are not sharply defined and depend

somewhat on the thickness of the material used. For example, a typical cell for liquid samples which

is made of NaCl has a transmission that starts to drop at about 700 cm-1, is roughly 50% at 600 cm-1,

and is nearly opaque at 500 cm-1. As seen in Table 1.1 glass and quartz can be used in the near-IR

region but not for most of the mid-IR region.

Some of the most useful window materials for the IR are quite soluble in water, which

include NaCl, KC1, KBr, CsBr and CsI. KRS-5 is a synthetic optical crystal, slightly soluble in

water, consisting of about 42% thallium bromide and 58% thallium iodide. The others listed in

Table 1.1 are mostly water insoluble.

glass 3000 quartz 2500 LiF 1500 CaF2 1200 BaF2 850 ZnS 750 Ge 600 NaCl 600 KC1 500 ZnSe 450 KBr 350 AgCl 350 KRS-5 250 CsBr 250 CsI 200 polyethylene (high density) 30

Table 1.1. Approximate transmission limits for optical materials (in cm-1)

The more detailed information (by Bruker optics) about IR transmitting materials is shown in

Table 1.2.


FTIR Spectrometer Manual. University of Oulu, Finland IR Materials

Table 1.2. Infrared transmitting materials (by Bruker optics)


FTIR Spectrometer Manual. University of Oulu, Finland Detectors

1.4 Detectors

The infrared detector is a device which measures the infrared energy of the source which has

passed through the spectrometer. One way or another these devices change radiation energy into electrical

energy which can be processed to generate a spectrum. There are two basic types, thermal detectors which

measure the heating effects of radiation and respond equally well to all wavelengths, and selective detectors

whose response is markedly dependent on the wavelength. Examples of thermal detectors include thermocouples, bolometers and pyroelectric detectors. A

thermocouple has a junction of two dissimilar metals . When incident radiation is absorbed at the junction,

the temperature rise causes an increase in the electromotive potential developed across the junction leads. A

bolometer is a detecting device which depends on a change of resistance with temperature. The pyroelectric detector consists of a thin pyroelectric crystal such as deuterated triglycine

sulfate (DTGS). If such a material is electrically polarized in an electric field, it retains a residual electric

polarization after the field is removed. The residual polarization is sensitive to changes in the temperature.

Electrodes on the crystal faces collect the charges so the device acts as a capacitor across which a voltage

appears, the amount of which is sensitive to the temperature of the device. The pyroelectric detector

operates at room temperature. Being a thermal device, it possesses essentially flat wavelength response

ranging from the near infrared through the far infrared. It can handle signal frequencies of up to several

thousand Hertz and hence is well suited for Fourier transform infrared spectrometers. The most important type of selective detector is the photoconductive cell which has a very rapid

response and a high sensitivity. An example is the mercury cadmium teiluride detector (MCT) which is

cooled with liquid nitrogen. These cells show an increase in electrical conductivity when illuminated by

infrared light. These detectors utilize photon energy to promote bound electrons in the detector material to

free states, which results in in-creased electrical conduction. There is a long wavelength limit to the

response however, because photons with wavelengths longer than a certain limit will have insufficient

energy to excite the electrons.

The normalized detectivity versus wavelength for different photodetectors is shown in

Fig.1.12.

The output from the detector goes to a preamplifier where it is converted into a voltage signal

varying with time. This signal has to be digitised and the job is usually done with a dedicated

'analogue to digital' converter chip. These devices will measure the input voltage in binary numbers

with 8, 16, 20 or 32 or more digits. Clearly, the number of useful digits is governed by the quality

(S:N ratio) of the signal. 16 and 20 bit devices are frequently used.


FTIR Spectrometer Manual. University of Oulu, Finland Detectors

Fig. 1.12. Normalized detectivity, D*, for different detectors as a function of optical radiation wavelength. The NEP (noise equivalent power) is the incident radiant power resulting in a signal-to-noise ratio of 1 within a given bandwidth of 1 Hz and at a given wavelength. The numbers given in italics are the equivalent number of light quanta (hν).

An A/D converter will carry out the conversion only when it is told to do so – it needs a

pulse (or ‘handshake’) to tell it to do the job. The pulse is provided by the He-Ne fringe detector and

trigger circuit. So, the A/D converter makes its measurement every 316.4 nm of path difference.

Thus, the He-Ne makes it possible to measure the interferogram at ultra precise path difference

intervals. If the temperature changes, or you lean on the FTIR, the interferometer will very slightly

distort and hence the interferogram will shift. However, the He-Ne beam will also shift helping to

correct for the distortion.

The digitized interferogram is now fed to the F-T processor. This can be a dedicated chip or a

PC. Obviously, a dedicated chip has its advantages (simplicity, lower cost etc) but the PC increases

the versatility enormously.


FTIR Spectrometer Manual. University of Oulu, Finland Advantages of FTIR Instruments

1.4 Advantages of FT-IR Instruments over Dispersive Instruments

Some of the major advantages of FT-IR over the dispersive technique include:

• Speed: Because all of the frequencies are measured simultaneously, most

measurements by FT-IR are made in a matter of seconds rather than several minutes.

This is sometimes referred to as the Felgett Advantage.

• Sensitivity: Sensitivity is dramatically improved with FT-IR for many reasons. The

detectors employed are much more sensitive, the optical throughput is much higher

(referred to as the Jacquinot Advantage) which results in much lower noise levels,

and the fast scans enable the coaddition of several scans in order to reduce the

random measurement noise to any desired level (referred to as signal averaging).

• Mechanical Simplicity: The moving mirror in the interferometer is the only

continuously moving part in the instrument. Thus, there is very little possibility of

mechanical breakdown.

• Internally Calibrated: These instruments employ a HeNe laser as an internal

wavelength calibration standard (referred to as the Connes Advantage). These

instruments are self-calibrating and never need to be calibrated by the user.

These advantages, along with several others, make measurements made by FT-IR extremely

accurate and reproducible. Thus, it a very reliable technique for positive identification of virtually

any sample. The sensitivity benefits enable identification of even the smallest of contaminants. This

makes FT-IR an invaluable tool for quality control or quality assurance applications whether it be

batch-to-batch comparisons to quality standards or analysis of an unknown contaminant. In addition,

the sensitivity and accuracy of FT-IR detectors, along with a wide variety of software algorithms,

have dramatically increased the practical use of infrared for quantitative analysis. Quantitative

methods can be easily developed and calibrated and can be incorporated into simple procedures for

routine analysis.

Thus, the Fourier Transform Infrared (FT-IR) technique has brought significant practical

advantages to infrared spectroscopy. It has made possible the development of many new sampling

techniques which were designed to tackle challenging problems which were impossible by older

technology. It has made the use of infrared analysis virtually limitless.

The main advantages are the Fellgett, Jacquinot, and Connes advantage.


FTIR Spectrometer Manual. University of Oulu, Finland Advantages of FTIR Instruments

Multiplex (Fellgett) Advantage In a dispersive spectrometer, wavenumbers are observed sequentially, as the grating is

scanned. In an FT-IR spectrometer, all the wavenumbers of light are observed at once. When spectra

are collected under identical conditions (spectra collected in the same measurement time, at the same

resolution, and with the same source, detector, optical throughput, and optical efficiency) on

dispersive and FT-IR spectrometers, the signal-to-noise ratio of the FT-IR spectrum will be greater

than that of the dispersive IR spectrum by a factor of √M, where √M is the number of resolution

elements. This means that a 2 cm-1 resolution 800 - 8000 cm-1

spectrum measured in 30 minutes on a

dispersive spectrometer would be collected at equal S/N on an FT-IR spectrometer in 1 second,

provided all other parameters are equal.

The multiplex advantage is also shared by array detectors (PDAs and CCDs) attached to

spectrographs. However, the optimum spectral ranges for these kinds of systems tend to be much

shorter than FT-IRs and therefore the two techniques are mostly complementary to each other.

The Throughput (Jacquinot) Advantage FT-IR instruments do not require slits (in the traditional sense) to achieve resolution.

Therefore, you get much higher throughput with an FT-IR than you do with a dispersive instrument.

This is called the Jacquinot Advantage. In reality there are some slit-like limits in the system, due to

the fact that one needs to achieve a minimum level of collimation of the beams in the two arms of

the interferometer for any particular level of resolution. This translates into a maximum useable

detector diameter and, through the laws of imaging optics, it defines a useful input aperture.

High Resolution and Linearity (Connes) Advantage Spectral resolution is a measure of how well a spectrometer can distinguish closely spaced

spectral features. In a 2 cm-1 resolution spectrum, spectral features only 2 cm-1 apart can be

distinguished. In FT-IR, the maximum achievable value of OPD, determines spectral resolution. The

interferograms of light at 2000 cm-1 and 2002 cm-1 can be distinguished from each other at values of

0.5 cm or longer.

The linearity advantage of FT-IR spectrometers is the fact that

νobserved = constant · νtrue

This advantage gives high wavenumber accuracy and makes an easy wavenumber calibration

possible. Calibration is accomplished by using a reference laser (HeNe) to measure optical path

difference.


FTIR Spectrometer Manual. University of Oulu, Finland Older Technologies

1.5 Older Technology

The original infrared instruments were of the dispersive type. These instruments separated

the individual frequencies of energy emitted from the infrared source. This was accomplished by the

use of a prism or grating. An infrared prism works exactly the same as a visible prism, which

separates visible light into its colors (frequencies). A grating is a more modern dispersive element

that better separates the frequencies of infrared energy. The detector measures the amount of energy

at each frequency, which has passed through the sample. This results in a spectrum, which is a plot

of intensity versus frequency.

Fourier transform infrared spectroscopy is preferred over dispersive or filter methods of

infrared spectral analysis for several reasons:

• It is a non-destructive technique

• It provides a precise measurement method which requires no external calibration

• It can increase speed, collecting a scan every second

• It can increase sensitivity – one second scans can be co-added together to ratio out

random noise

• It has greater optical throughput

• It is mechanically simple with only one moving part

* * *

Infrared spectroscopy has been a workhorse technique for materials analysis in the laboratory

for over seventy years. An infrared spectrum represents a fingerprint of a sample with absorption

peaks, which correspond to the frequencies of vibrations between the bonds of the atoms making up

the material. Because each different material is a unique combination of atoms, no two compounds

produce the exact same infrared spectrum. Therefore, infrared spectroscopy can result in a positive

identification (qualitative analysis) of every different kind of material. In addition, the size of the

peaks in the spectrum is a direct indication of the amount of material present. With modern software

algorithms, infrared is an excellent tool for quantitative analysis.


FTIR Spectrometer Manual. University of Oulu, Finland Data Processing

2. Interferograms and Data Processing

2.1 Interferograms During an infrared scan (see Fig.2.1), the interferometer sequentially (I) divides light emitted

from the IR source in two beams using a beam splitter, (II) changes the optical path of one beam

using a movable mirror, (III) recombines the two beams to create optical interference, and (IV)

passes the IR light through the sample for measurement of a single-beam spectrum. The ratio of the

single-beam spectra with and without the sample in the light path yields a sample spectrum in

percentage transmittance. As shown, the moving mirror cycles back and forth during an FTIR scan.

The optical path difference for any mirror displacement x is calculated by the equation |2(OM - OF)|

and is denoted optical retardation δ.

Fig. 2.1. Essential components of Michelson interferometer.

Understanding how interferograms encode spectral information is conveniently described by

first considering a monochromatic laser-light source entering a Michelson interferometer. When

laser light enters an interferometer, only a single wavelength of light can exit the interferometer.

However, at the detector, the laser light appears to alternate from light to dark at increments of

mirror movement x, producing a maximum signal at mirror positions eliciting maximum

constructive interference and a minimum signal at mirror positions eliciting maximum destructive

interference. At mirror positions between the maximum constructive interference and the maximum

destructive interference, optical interference causes the input laser light to appear at some

intermediate intensity. Although the incident laser-light intensity is constant, the interferometer

transformed the high-frequency IR beam into a modulated beam of varying intensity as the moving

mirror travels through one cycle. Thus the output of the interferometer is merely a beam of light,

oscillating in intensity. A plot of this oscillating beam intensity I(x) versus mirror movement x [or,



equivalently, I(δ) versus optical retardation δ] is shown in Fig.2.2 A. We note that the output of an

interferometer actually contains both an AC and a DC component. The DC component is omitted for

clarity because the spectral information is encoded in the AC component.

Fig. 2.2. Output from a Michelson interferometer as a function of mirror

displacement x. The interferograms I(x) are for a monochromatic IR source

(A) and a polychromatic IR source (B). Both interferograms are even

functions, I(—x) = I(x).

Fig. 2.2 A shows that the interferogram of a laser-light source is a pure cosine wave with

constant amplitude. This can be explained by considering the detector signal at discrete mirror

positions. The optical path difference between the light beams which recombine at the beam splitter

is calculated by the equation |2(OM - OF)| and is referred to as optical retardation δ. Starting from

the point where the fixed and moving mirrors have a zero path difference ZPD (i.e., OM = OF as

shown in Fig. 2.1), the two beams interfere constructively and the detector observes a maximum

signal intensity. As shown in Fig. 2.2, this mirror position is at the center of the interferogram. If the

movable mirror is displaced a distance 1/4 λ away from this ZPD, the optical retardation is now

1/2 λ. The laser beams in each arm of the interferometer are exactly 180° out of phase, and upon

recombination at the beam splitter, destructive interference causes a minimum detector response. A

further displacement of 1/4 λ makes the total optical retardation λ, and the two beams are once more

in phase and the condition of constructive interference exists. Therefore for a monochromatic IR

beam, there is no way to determine whether the signal maximum corresponds to ZPD or some

integral number of λ.



Actually modern FTIR spectrometers do not stop the mirror at individual positions of mirror

displacement as described above; rather the mirror moves at a constant velocity during a scan.

Consequently the signal intensity I(δ) oscillates during each 1/2 λ of optical retardation δ. In theory,

the interferogram of a monochromatic source is a simple cosine function described by (2.1)

,)2cos()(5.02cos)(5.0)( δνπνλδπνδ III =⎟

⎠⎞

⎜⎝⎛= (2.1)

where )(νI is the IR source intensity, δ is the optical path difference |2(OM - OF)|, λ is the

wavelength of light, and 1/ λ = v. The factor 0.5 in Eq. (2.1) results from a physical limitation of the

interferometer; due to the position of the optical components in the interferometer, only 50% of the

reflected light that recombines at the beam splitter travels toward the detector, whereas 50% returns

toward the source. Even for this trivial case using a monochromatic IR source, the experimentally

recorded interferogram is actually a function of several instrument variables, which include the

intensity of the light source, beam-splitter efficiency, detector response, and amplifier

characteristics. Because each of these instrument variables affects the optical throughput of )(δI

measured at the detector, Eq. (2.1) must be rewritten as

)2cos()()( πνδνδ BI = (2.2)

where B(v) is the intensity from the IR light source emitted at a specific wavenumber v as

modified by the instrument. The optical retardation t seconds after ZPD is given by

Vt2=δ (2.3)

where V is the velocity of the movable mirror in centimeters per second. During a scan, the

product Vt [in Eq. (2.3)] is the distance of mirror movement x (i.e., x = Vt), and since the reflected

light from the moving mirror travels twice the distance that the mirror has moved, the factor 2 in Eq.

(2.3) is needed. In other words, optical retardation δ is always twice the distance of mirror movement

(i.e., δ = 2x). Substituting Eq. (2.3) into the argument of the cosine function in Eq. (2.2) gives

)22cos()()( VtBtI νπν= (2.4)

Equations (2.2) and (2.4) are mathematically equivalent. However, the “new” dependent

variable I(t) is the optical throughput at the detector as a function of the new independent variable

time t. Equation (2.4) is a cosine wave and the amplitude at time t of any cosine wave of frequency f

can be described by the general equation



,)2cos()( 0 ftAtA π= (2.5)

where A0 is the maximum amplitude of the wave. Equation (2.4) is identical to Eq. (2.5),

where

)()( tItA = (2.6 a)

)(0 νBA = (2.6 b)

νVf 2= (2.6 c)

Thus the interferogram intensity I(x) [or I(t)] resulting from laser light entering an

interferometer, as shown in Fig. 2.2 A, is a pure cosine wave with a frequency f and maximum

amplitude )(νB . A real infrared source emits broadband IR radiation, thus each v emitted

contributes a pure cosine wave to the final interferogram and the amplitude of each cosine wave in

the experimentally measured interferogram depends in part on the intensity of the particular ν

emitted from the IR source. In fact, based on Eq. (2.6 c), each unique ν emitted from any IR source

is transformed by the interferometer into a unique cosine-wave interferogram whose modulation

frequency f is dependent on the mirror velocity V.

Equation (2.6 c) is a mathematical description of the critical purpose of all interferometers.

The critical function of the interferometer is to convert high-frequency radiation into low-frequency

signals that can be measured. High-frequency IR light (~1014 Hz) entering the interferometer is

modulated into a strobe-like fringe frequency f. From Eq. (2.6 c) this fringe frequency f is unique for

each ν emitted. The mirror velocity (typically near 0.16 cm/sec for the mid-IR range) of rapid-

scanning interferometers produce modulated frequencies f in the audio frequency range. Detector

response is dependent on f, and therefore the mirror velocity is a user-controlled parameter. In

general, the TGS detector response decreases with increasing mirror velocity (i.e., increasing f);

however, the MCT detector response increases with increasing mirror velocity (i.e., increasing f).

Therefore mirror velocity is set higher for the MCT detector when compared to the TGS detector.

In practice, typical interferograms rapidly decay as the mirror moves away from either side

of the ZPD as shown in Fig. 2.2 B. When the fixed and moving mirrors depicted in Fig. 2.1 are of

equal path length (i.e., zero optical retardation), all wavelengths of infrared light from the source are

in phase and the interferogram elicits a maximum amplitude and maximum constructive interference

occurs. Thus at ZPD the maximum optical throughput occurs for both polychromatic and

monochromatic IR sources and maximum amplitude is expected. However as the mirror travels

away from ZPD, individual ν cosine waves are increasingly out of phase with one another. This

results in greater destructive interference between the ν cosine waves and this causes the



interferogram to decay rapidly. Experimentally, there is usually a small optical phase shift near ZPD,

and consequently most measured interferograms are asymmetric in this region.

Equation (2.2) [or Eq. (2.4)] is a general function that describes interferograms for specific

wavelengths of light, and consequently when multiple v enter an interferometer, the measured

interferogram is described by the use of Eq. (2.2) [or Eq. (2.4)] for each ν emitted. In other words,

Eq. (2.2) applied to each ν emitted is summed to give the interferogram of a broadband IR source

(i.e., the superposition principle applies to the optical waves contributing to the interferogram).

Consider mathematically describing one data point in the actual interferogram produced by a

broadband IR source (e.g., Fig. 2.2 B) at a fixed mirror position x = L (the maximum optical

retardation). In order to describe numerically the intensity of light reaching the detector at the mirror

displacement equal to L cm [i.e., I(L)], Eq. (2.3) must be summed over all ν emitted by the source

and averaged over all wavelengths of light sampled,

)2cos()()(0

LBLI i

k

ii νπν∑

=

= (2.7)

We emphasize that the movable mirror is fixed in this hypothetical example, and only one

data point is being mathematically described in the total interferogram. In Eq. (2.7), L is constant and

iν , and )(νiB are the i-th wavelength and intensity of this wavelength being measured at the

detector. To describe completely the interferogram produced by a typical IR source during a

complete scan, one merely uses Eq. (2.7) for each mirror position associated with data acquisition. In

other words, instead of calculating I(L) at the maximum mirror displacement L (which is a constant),

experimental interferograms require I(x) for all possible mirror positions x (which is a variable).

Data acquisition occurs at multiple discrete mirror positions, and for each mirror position, Eq. (2.7)

can be used to calculate I(x). Thus the general form of Eq. (2.7), for any particular mirror position

associated with data acquisition, is

)2cos()()(0

xBxI i

k

ii νπν∑

=

= (2.8)

or, in integral form,

,)2cos()()(0

ννπν dxBxIv

v∫

∞=

=

= (2.9)

where the limits of integration reflect an infinite spectrum.

Unfortunately I(x) is experimentally measured and the integral given in Eq. (2.9) is merely a

mathematical description of what is actually measured. In fact because I(x) is experimentally



measured, the desired spectral information )(νB is mathematically obtained from this measured

interferogram using the cosine Fourier transform of Eq. (2.9), which is

,)2cos()()( dxxxIBx

x

νπν ∫∞=

−∞=

= (2.10)

where the limits are for infinite mirror displacement.

2.2 Finite Mirror Displacement. Resolution.

Consider again a monochromatic light source emitting a single IR wavelength )(νB . Only

one computer calculation using Eq. (2.10) would be required to describe )(νB (i.e., the entire

spectrum). However, it is not possible to calculate )(νB from Eq. (2.10) because the integration

limits require infinite mirror displacement. It is impossible to move the mirror infinite distances, and

consequently Eq. (2.10) must be rewritten

,)2cos()()()( dxxxDxIBx

xg νπν ∫

∞=

−∞=

= (2.11)

where Dg(x) has the value of 1 during the scan, |x| ≤ |L|, and the value of zero after the scan,

|x| > |L|, where L defines the maximum mirror displacement. In other words, before integration, the

function “ )2cos()( xxI νπ ” in the above integral is multiplied by 0 at distances corresponding to

|x| > |L|. This allows the integral to maintain the limits that approach infinity to satisfy the Fourier

transform requirement, and the Dg(x) function accounts for the physical limitation of finite mirror

movement. Since the interferogram I(x) is measured with finite mirror movement x = ± L cm, the

resulting IR spectrum will have a finite resolution of 1/2L cm-1.


FTIR Spectrometer Manual. University of Oulu, Finland Apodization Function

2.3 Apodization function.

The key concept in Eq. (2.11) is that multiplication of each measured interferogram data

point I(x) by Dg(x) occurs before integration, i.e., before calculating )(νB intensities. Dg(x) is

denoted as an apodization function. We emphasize that Dg(x) is a function that can depend on the

value of x and that Dg(x) need not have the value either 1 or 0 as described above, and in fact, Dg(x)

is usually a continuous function of x. Any one of approximately eight apodization functions are

routinely used in infrared spectroscopy during data acquisition. For condensed phase spectra the

choice of apodization function usually does not significantly influence the final band shape because

the natural IR bandwidth is much greater than the bandwidth associated with the apodization

function.

The distortion of IR spectral lineshapes caused by apodization functions is best illustrated

using the interferogram produced by a monochromatic light source, i.e., the cosine interferogram in

Fig. 2.3 A. The triangular apodization function described by the equation Dg(x) = 1 – x/L is shown in

Fig. 2.3 B. The triangular apodization function is a triangle with its apex located at x = 0 and the

base ranging from – L cm to L cm. The multiplication of Dg(x) with I(x) results in a triangular-

shaped interferogram I(x)Dg(x) shown in Fig. 2.3 C.

Fig. 2.3. An apodization function Dg(x) changes the interferogram profile and the lineshape of the corresponding spectrum. The interferogram I(x) of a monochromatic light source (A) and the triangular apodization function Dg(x) shape (B) are multiplied together, resulting in an interferogram (C) described by I(x)Dg(x). The Fourier transformation of this interferogram results in a spectral lineshape (D) described by the apodization function lineshape F[Dg(x)].

Thus multiplying the monochromatic light-source interferogram I(x) by an apodization

function Dg(x) causes the profile of the interferogram to resemble the shape of the apodization

function. In this example, Fourier transformation of the triangularly apodized monochromatic



interferogram (Fig. 2.3 C) using Eq. (2.11) results in an IR lineshape described by the equation

F[Dg(x)] = L sinc2(πν L) shown in Fig. 2.3 D. In other words, for this very narrow input frequency,

a monochromatic light source, the IR spectrum has bands with the line-shape dominated by the

Fourier transformation of the apodization function Dg(x). When the instrumental resolution is 2 cm-1

(i.e., 2L = 0.5 cm), the IR linewidth in Fig. 3D (calculated as ∆ν = 1.772/L for this apodization

function) is approximately 3.5 cm-1. Thus a monochromatic IR light source is always measured with

a bandwidth approximately equal to the instrumental resolution regardless of the apodization

function. Apodization functions differ significantly regarding side-lobe intensity and Fig. 3D shows

that the triangular apodization function has only positive side lobes. Other apodization functions,

e.g., boxcar and trapezoidal, have large negative side lobes.

2.3.1 Instrument line shape and Apodization function.

Thus, an apodization function is a function (also called a tapering function) used to bring an

interferogram smoothly down to zero at the edges of the sampled region. This suppresses sidelobes

which would otherwise be produced, but at the expense of widening the lines and therefore

decreasing the resolution.

Instrument function (or Instrument line shape, or Apparatus function) is the finite

Fourier cosine transform of an apodization function, also known as an apparatus function. The

instrument function corresponding to a given apodization function is )(kI )(xA

∫−

=a

a

ikx dxxAekI )()( 2π

(2.12)

which, upon expanding the complex exponential,

== ∫−

a

a

ikx dxxAekI )()( 2π

∫∫−−

+=a

a

a

a

dxxAkxidxxAkx )()2sin()()2cos( ππ (2.13)

For an even function, the left integrand is even (and hence is equal to twice its value

over half its interval) and the right integrand is odd (and hence equal to 0), so

)(xA

∫=a

dxxAkxkI0

)()2cos()( π (2.14)



The following are apodization functions for symmetrical (2-sided) interferograms, together

with the instrument functions (or apparatus functions) they produce and a blowup of the instrument

function sidelobes. The instrument function corresponding to a given apodization function

can be computed by taking the finite Fourier cosine transform.

)(kI

)(xA

Type Apodization function Instrument function

Instrument function FWHM

Bartlett (Triangle) a

x ||1− 2)sin(

⎟⎠⎞

⎜⎝⎛

kakaa

ππ 1.77179

Blackman )(xBA )(kBI 2.29880

Connes

(Biparabolic)

2

2

2

1 ⎟⎟⎠

⎞⎜⎜⎝

⎛−

ax 2/5

2/5

)2()2(

28ka

kaJa

ππ

π 1.90416

Cosine ⎟⎠⎞

⎜⎝⎛

ax

2cos π

)161()2cos(4

22kaaka

−ππ 1.63941

Hamming )(xHmA )(kHmI 1.81522



Hanning )(xHnA )(kHnI 2.00000

Uniform

(Rectangle

or Boxcar)

1 ka

kaaπ

π )2sin( 1.20671

Welch 2

2

1ax

− )(kWI 1.59044

where

)(xBA = ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛+

ax

ax ππ 2cos08.0cos5.042.0

)(kBI = )41)(1(

)2(sin)36.084.0(2222

22

kakaakckaa

−−− π

)(xHmA = ⎟⎠⎞

⎜⎝⎛+

axπcos46.054.0

)(kHmI = 22

22

41)2(sin)64.008.1(

kaakckaa

−− π

)(xHnA = ⎟⎠⎞

⎜⎝⎛

ax

2cos2 π

)(kHnI = 2241)2(sin

kaakca

−π

)(kWI = =2/32/3

)2()2(

22ka

kaJa

ππ

π

= 3332)2cos(2)2sin(

akkakakaa

ππππ −


FTIR Spectrometer Manual. University of Oulu, Finland IR Spectrum

2.4 The IR Spectrum

Obtaining an experimental infrared spectrum of a sample requires measuring two

interferograms (see Fig.2.4): (A) the background interferogram and (B) the sample interferogram.

These interferograms were Fourier transformed into their corresponding single-beam spectra shown

below each interferogram (C and D) through the use of Eq. (2.11). The inset to the background

interferogram (A) shows the spectral information in the wings of the interferogram. Sharp water

vapor and carbon dioxide bands are apparent in both single-beam spectra.

Fig. 2.4. Spectrum obtaining procedure

Dividing the sample spectrum (Fig. 2.4 D) by background spectrum (Fig. 2.4 C) we can

obtain transmission spectrum of the sample (see Fig. 2.5)

%100)(

)()[%]( ×=

background

sample

BB

Tν

νν (2.14)

or absorption spectrum

( 100/)[%](log)( )νν TA −= . (2.15)

In FTIR spectroscopy, plotting data in absorbance units is preferred because data

manipulation such as spectral subtraction is possible only in the absorbance scale.



Fig.2.5. Transmission spectrum (A) and corresponding absorbance spectrum (B)

obtained from the single-beam spectra in Fig. 2.4. The percentage transmittance

T(v)[%] is calculated using Eq. (2.14), which takes the ratio of the single-beam

spectra of a broadband source measured with (Fig. 2.4D) and without (Fig.

2.4C) the sample present. The absorbance spectrum A(v) is subsequently

calculated from the T(v)[%] spectrum using Eq. (2.15).

A few final comments on producing a single-beam spectrum by calculating each B(v) using

the numerical form of equation (2.11) are needed. Interferograms, in theory, are symmetrical on both

sides of ZPD and therefore it is convenient to expand Eq. (2.11),

dxxxDxIdxxxDxIB gg ∫∫∞

∞−

+=0

0

]2cos[)()(]2cos[)()()( νπνπν (2.16)

Based on the integral limits, these integrals merely calculate the area on each side of ZPD.

This is graphically shown in Fig. 2.2 for both interferograms. The area on each side of ZPD (i.e., x =

0) are equal because the cosine functions in the above integrals are even functions, that is, I(-x) =

I(x). Because the integrals are equal, Eq. (2.16) can be rewritten as

dxxxDxIB g∫∞

=0

]2cos[)()(2)( νπν (2.17)



where the integral limits are now the mirror displacement x for a single-sided scan.

2.4.1 Single Sided and Double Sided Interferograms

Based on Eq. (2.17), a single-sided scan (mirror movement from 0 to L cm) generates a

single-sided interferogram, whereas a double-sided scan (mirror movement from —L to +L cm)

generates a double-sided interferogram, but both interferograms provide the same spectral

information. Data collection for both double-sided interferograms and single-sided interferograms

occurs as the mirror moves away from the beam splitter. Thus as the movable mirror travels in the

forward direction, data collection occurs (see Fig. 2.1). For a fixed number of scans, double-sided

interferograms provide better signal-to-noise ratios than single-sided scans because the spectral

information is sampled twice (once on each side of ZPD). However, obtaining spectra from single-

sided interferograms allows a greater number of scans per second, which is useful for gas phase

experiments that require rapid data acquisition. Also, obtaining spectra from single-sided

interferogram allows increasing spectral resolution in the FTIR spectrometers with short optical path

difference. Single-sided interferograms begin data collection just to the left of ZPD to allow for

phase correction.

The Fig. 2.6 represents the real double sided and single sided interferograms.

a) b) Fig. 2.6 Typical interferograms for Double Sided processing (a) and for Single Sided processing (b).



2.4.2 Instrument Resolution and Spectral Noise

The instrumental resolution, δ0, of IR bands depends solely on the maximum distance of

mirror movement L,

L21

0 =δ (2.18)

where δ0 is the instrumental resolution in cm -1. As L increases the optical retardation

increases [Eq. (2.3)] and the resolution at which the IR data are obtained increases [Eq. (2.18)].

Unfortunately, as L increases, the amount of spectral noise increases. Spectral noise increases

because during a complete scan, electronic and instrument noise remains constant, whereas the

signal decays or decreases (Fig. 2.4 A); therefore the S/N of the interferogram decreases as the

mirror is displaced further from the ZPD. Equation (10) demonstrated that every I(x) value in the

interferogram is used to calculate each B(v) in the single beam spectrum. Thus as L increases, i.e.,

S/N decreases, more interferogram noise causes all IR frequencies to have increased spectral noise.

Measured interferograms decay rapidly with increasing mirror movement beyond ZPD, but at some

finite mirror position, little (or no) IR signal exists and the detector is recording only noise. If the

detector is recording mostly noise because the optical retardation is too high, the instrument

resolution should be decreased to acquire the data. In other words, the S/N depends on the

instrument resolution, and when the maximum mirror displacement causes the spectral signal

intensity to be equal to or less than the instrumental-noise intensity, IR data should not be measured.

It is important to avoid high-resolution IR scans when the actual condensed phase spectra

exhibit naturally wide IR bands. High resolution requires the movable mirror to travel excessive

distances to obtain sufficient optical retardations for the high resolution. For condensed phase

spectra, it is rarely necessary to measure spectra at resolution greater than 2 cm-1, and typically

4 cm-1 resolution is acceptable. To determine the optimum resolution for a particular condensed

phase sample, one merely obtains IR spectra at low, intermediate, and high resolution to determine

when the spectral noise is apparent in the broad IR bands. Alternatively, obtaining one high-

resolution interferogram can yield all the information needed to identify the desired resolution. Prior

to Fourier transformation to obtain the single-beam spectrum, the high-resolution interferogram is

merely truncated at mirror positions corresponding to lower instrument resolution. In other words,

the interferogram is measured with a constant amount (in fact an excessive number of) data points,

but the number of transform points, used to obtain the single-beam spectrum, is reduced to obtain a

low-resolution spectrum. Thus one high-resolution interferogram can be Fourier transformed several

times at sequentially lower resolutions by truncating the measured interferogram prior to Fourier

transformation. Obviously the high-resolution interferogram would have to be stored as a separate

computer file for this experiment.


FTIR Spectrometer Manual. University of Oulu, Finland Phase Correction

2.5 Phase Correction The phase correction procedure is applied to correct the interferograms for any phase that

may be caused missampling of the position of zero path difference or by the presence of dispersive

elements in the spectrometer (e.g. optics, electronics).

The even symmetry of the spectrometer theoretically implies that its calculated spectrum

should be a real function, as shown in the equation below.

)())(( νBxIFT EVEN = (2.19)

The presence of dispersive elements and particularly the possibility that the position of zero

path difference is not properly sampled can lead to a loss of this even symmetry and a resultant

spectrum that contains both real and imaginary components.

)(

ImRe )()()())(( νϕννν iEVEN eBiBBxIFT =+= (2.20)

An example of a spectrum containing both real and imaginary components is shown in

Fig. 2.7.

Fig. 2.7: Fourier Transform of an uneven double-sided interferogram. The red curve is the real

portion of the spectrum; the blue curve is the imaginary portion of the spectrum. The process of phase correction moves the components of the spectrum that are located in the

imaginary domain to the real domain.

Much like the Fourier Transform function, the Phase Correction function can be used to

correct either single-sided or double-sided interferograms.



2.5.1 Double-sided Phase Correction

If the interferograms are double-sided, that is symmetric about the ZPD position, then their

resultant spectra will contain phase information for each spectral element. As such, the phase

correction can take place in the spectral domain. Referring to the eq. (2.20) above, the calculated

spectrum, , can be corrected by way of multiplication with a phase correction function

(PCF) of the form . That is,

)()( νϕν ieB)(νϕiePCF −=

=×= PCFeBSpectrumCorrectedPhase i )()( νϕν

=×= − )()()( νϕνϕν ii eeB (2.21)

)(νB=

Note that the PCF is not simply equal to the negative of the measured phase. Rather, a fit is

made to the in-band phase of the calculated spectrum and it is the negative of this fit that is used as

the PCF (see Fig. 2.8). The basis for this is that by only correcting for the fitted phase rather than the

calculated phase, some of the noise associated with the measurement is left in the imaginary domain.

For random sources of measurement noise, this can lead to an increase in the signal-to-noise ratio by

a factor of √2.

Figure 2.8: Fitting a function to the measured phase. The blue curve is the spectral

phase derived from the double-sided interferogram; the red curve is a fitted function to

the in-band portion of the measured phase.



2.5.2 Single-sided Phase Correction

Phase correction for single-sided interferograms, interferograms that are not symmetric about

the ZPD position (see Fig. 2.9), while mathematically equivalent to phase correction of double-sided

interferograms, differs slightly from a procedural point of view. A different procedure must be

employed because, for single-sided interferograms, the uncorrected spectrum (magnitude and phase)

can only be calculated for the symmetric portion of the interferogram.

)(

ImRe )()()())(( νϕννν iDSDS eBiBBxIFT =+= (2.22)

The phase correction function computed from the symmetric portion of the single-sided

interferogram can still be used to correct the single-sided interferogram, however. Recall that for

Fourier Transforms, multiplication in one domain is equivalent to convolution in the other domain.

Thus, rather than multiplying the single-sided spectrum by a PCF, phase correction can be achieved

by convolving the single-sided interferogram with the inverse transform of the PCF. That is,

)()()()( )(11 νϕieFTxIPCFFTxICorrectionPhaseSidedSingle −−− ⊗=⊗= (2.23)

where the PCF is found from the phase of the symmetric portion of the single-sided

interferogram (see Fig. 2.10 and Fig. 2.11).

Fig. 2.9: Single-sided interferogram. Note the short symmetric portion about ZPD (x=0).



Fig. 2.10: Phase correction function, . The red curve is the real portion of the

PCF; the blue curve is the imaginary portion of the PCF.

)(νϕie−

Fig. 2.11. Inverse transform of the phase correction function. Phase correction of

the single-sided interferogram is achieved by convolution with this function.

The result of the convolution given in eq. (2.23) is the single-sided interferogram is rendered

symmetric about its position of zero path difference. Put another way, the single-sided interferogram

is converted to an even function (see Fig. 2.12). As such, only the portion of the single-sided

interferogram where the optical path difference is greater or equal to zero needs to be used to

compute the resultant spectrum (see Fig. 2.13).



Figure 2.12. Apodized, phase corrected, and butterflied single-sided interferogram

Fig. 2.13. Spectrum from a phase corrected single-sided interferogram.



Literature

• J.R.Bell, Introductory Fourier Transform Spectroscopy, New York, Academic, 1972

• N.B.Colthup, L.H.Daly, S.E.Wiberely, Introduction to Infrared and Raman

Spectroscopy, Academic Press, London, 1990

• P. Griffiths and J. deHaseth, Fourier Transform Infrared Spectrometry, In Chemical Analysis, Vol. 83, Wiley and Sons, New york, 1986

• R.J. Markovich, Ch.Pidgeon, Introduction to Fourier Transform Infrared

Spectroscopy and Applications in the Pharmaceutical Sciences, Pharmaceutical research, Vol.8, No.6, 1991, p.663-675

• Jasco FTIR Seminar, http://www.jascofrance.fr/pdf/ftir.pdf

• Introduction to Fourier Transform Infrared Spectroscopy, Thermo Nicolet Corp.

2001, http://mmrc.caltech.edu/mmrc_html/FTIR/FTIRintro.pdf

• P.L. King, M.S. Ramsey, P.F. McMillan, and G. Swayze, Chapter 3. Laboratory fourier transform infrared spectroscopy methods for geologic samples, in Infrared Spectroscopy in Geochemistry, Exploration, and Remote Sensing, P. King, M. Ramsey, G. Swayze (eds.), Mineral. Assoc. of Canada, London, ON, vol. 33, 57-91, 2004, http://ivis.eps.pitt.edu/ramsey/papers/GAC2.pdf

• Guide for Infrared Spectroscopy, Bruker Optics, www.brukeroptics.com

• Hue Phan, Fundamental infrared spectroscopy, MIDAC Corporation, Application

notes, http://www.midac.com/apnotes/Tn-100.PDF

• Oriel Instruments, Introduction to FT-IR Spectroscopy, http://www.lot-oriel.com/

• P. Davis, A. Rebolledo, T. Fulton, Function Guide for the Fourier Transformation Package, http://research.uleth.ca/spire/papers/FTpackageFunctionDescriptionRFP.pdf


http://www.jascofrance.fr/pdf/ftir.pdf

http://mmrc.caltech.edu/mmrc_html/FTIR/FTIRintro.pdf

http://ivis.eps.pitt.edu/ramsey/papers/GAC2.pdf

http://www.brukeroptics.com/

http://www.midac.com/apnotes/Tn-100.PDF

http://www.lot-oriel.com/

http://research.uleth.ca/spire/papers/FTpackageFunctionDescriptionRFP.pdf

FTIR Manual Part1 Theory

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