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10.2 Fourier TransformInfrared Spectroscopy

• overview of the infrared including measurement difficulties• spectrometer throughput• instead of a t ↔ f transform use an x ↔ transform• introduction to the Michelson interferometer, and its behavior as an optical autocorrelator• block diagram of an FTIR spectrometer, including details of mirror movement and controlling when the ADC takes data• example far infrared spectrum of K2ReCl6• optical resolution including an example of gas-phase ammonia• apodization to eliminate sinc function oscillations• overall signal-to-noise enhancement that can be expected• historic types of measurements that drove the development of FTIR

ν

Overview of the Infrared

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The highest frequency vibration is the H2 stretching motion which occurs at 4,400 cm-1. Although this transition is not infrared active (it is Raman active), ~4,000 cm-1 is usually taken to be the upper limit of the infrared. The lower limit is determined by instrumental considerations and is often near 50 cm-1.

The mid infrared extends from 4,000 to 200 cm-1. This region is usually composed of characteristic bond frequencies or normal modes of molecules.

The far infrared extends from 200 to 50 cm-1. This region is usually composed of characteristic metal bond frequencies, or normal modes of large molecules. The end of the range overlaps with high energy rotations.

The resolution of spectra depends upon their use. Solution phase spectra are usually taken with bandwidths of 5-10 cm-1, while gas phase spectra of small molecules usually require a bandwidth of 0.1 cm-1.

Measurement Difficulties

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• infrared photons are difficult to distinguish from thermal noise, with the result that the measurement is detector noise limited

PN = 4kT ∆ffor ∆f = 1 Hz and T = 298 K

PN = 828 cm-1

• infrared transitions have very low molar absorptivities, ε ~ 1 L mol-1 cm-1, requiring high concentrations to obtain a reasonable level of absorption• high resolution spectrometers have a very low optical throughput, which limits the signal-to-noise ratio of the absorption measurement• the experiment would benefit enormously from multiplexed detection

Spectrometer Optical Throughput

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The standard measure of optical collection efficiency is the f/#(pronounced f-number). An f/# is the focal length of a lens or mirror, divided by the diameter of the lens or mirror. Collection efficiency is inversely proportional to the square of the f/#.

f/# 1 2 5 10 20 percent 6.2 1.5 0.25 0.06 0.02

A grating or prism bends different wavelengths at different angles. To obtain high resolution the wavelengths are separated by moving as far as possible away from the focusing lens or mirror. That is, long focal lengths are required.

It is very difficult to construct a grating or prism monochromator with a low f/# because the size of the grating or prism would need to increase in proportion to the focal length. Large gratings and prisms are either impossible to manufacture or are very expensive. A typical f/# for a 0.1 cm-1 resolution grating spectrometer is 12, which is about a 0.05% throughput.

Impossibility of a t ↔ f Transform

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To obtain an infrared spectrum via a Fourier transform, the signal would be recorded (digitized) as a function of time to obtain F(t). Then the spectrum would be computed using the following expression,

This will not work in the infrared because the frequencies are too fast. An infrared spectrum has frequencies from 1.2×1014 Hz (4,000 cm-1) to 6.0×1012 Hz (200 cm-1). The fastest electronic temporal resolution with commercial instrumentation is 7×10-12 s or ~840-42 times too slow.

2( ) ( ) i ftf F t e dt∞ − π

−∞φ = ∫

Possibility of an x ↔ Transform

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Fortunately, an infrared wave can be written in an alternative form using distance and wave number, where = 1/λ.

The Fourier transform equation takes the same form, simply exchanging for f and x for t. It is numerically solved using the same computer program as that used for t to f transforms.

The infrared spectral range from 4,000 cm-1 to 250 cm-1

corresponds to wavelengths of 2.5 µm to 40 µm. These lengths are easy to measure, all one has to do if figure out how to "stop" the wave from moving!

cos 2 cos(2 )x x⎛ ⎞π = πν⎜ ⎟λ⎝ ⎠

( ) 2( ) i xF x e dx∞ − πν

−∞φ ν = ∫

ν

ν

ν

The Michelson Interferometer

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source

fixed mirrormovablemirror

sample

detector

beamsplitter

• light from an infrared source is collimated and directed to a beam splitter• at the beam splitter, half is reflected toward the fixed mirror and half is transmitted toward the movable mirror• when light reaches each mirror it is reflected back to the beam splitter• at the beam splitter, half the light from the fixed mirror is reflected back to the source and half is transmitted to the detector• at the beam splitter, half the light from the movable mirror istransmitted back to the source and half is reflected to the detector• half the light reaches the detector, 0.25 from each path• light of every wavelength is always traveling through the sample to the detector (multiplexed)• an interferogram is obtained by moving the mirror and measuringthe resultant intensity at the detector

Optical Autocorrelator (1)

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Light traveling toward the detector can be described as the sum of two cosines,

where ξ is the extra distance (nearer or farther) traveled by light along the path involving the movable mirror.

• moving the mirror a distance ξ/2 farther from the beam splitter delays light by the distance +ξ. • the beam splitter divides intensity, not amplitude. Thus, 1/4 of the total photons reach the detector from each path. This corresponds to 1/2 the amplitude from each path. • the detector responds to intensity, which is the square of the electric field.

1 1cos(2 ) cos(2 [ ])2 2

x xπν + πν + ξ

Optical Autocorrelator (2)

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Although the detector output is averaged as a function of time, determination of a functional form for the signal requires that integration be over distance. In the FTIR instrument the optical signal (square of the amplitude) is averaged for 100 µs or longer. This corresponds to averaging the waveform over a distance of x0 = 3×104 meters.

( )0

2

0 0

1 0.5cos(2 ) 0.5cos(2 ( ))x

vx v x dxx

π + π + ξ∫

( )( ) ( )0 0 022

0 0 00 0 0

0.25 0.25 0.50cos(2 ) cos 2 cos(2 )cos 2 ( )x x x

x dx x dx x x dxx x x

πν + πν + ξ + πν πν + ξ∫ ∫ ∫

Expansion of the square yields three integrals.

By letting x0 → ∞, the first two integrals each reduce to 0.125.

( )0

0 0

0.500.125 0.125 cos(2 )cos 2 ( )x

x x dxx

+ + πν πν + ξ∫

Optical Autocorrelator (3)

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The third integral on the previous slide is the autocorrelation function,

( )

( )

0

1,10 0

1,1

1 cos(2 )cos(2 ( ))

0.5cos(2 )

xC x x dx

x

C

ξ = πν πν + ξ

ξ = πνξ

∫

where Fourier transforms were used to evaluate the integral. The detector output is then given by C1,1(ξ) plus the dc offset.

0.25 0.25cos(2 )+ πνξ

As the mirror is moved, the output of the detector will be an offset cosine. Each individual wavelength of light will have its own offset cosine. These offset cosines are added together in the detectoroutput. As the mirror is moved the output of the detector is digitized with sufficient resolution to reproduce all optical wavelengths. After collecting the data a Fourier transform is computed to extract the spectrum - amplitude versus . ν

Instrument Block Diagram

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movablemirror

fixedmirror

beamsplitter

sourceresolutioncontrolling

aperture

opticalfilter

collimatedwhite

alignmentlight

mirror swingsout of the wayfor alignment

sample

reference

TGS pyroelectricdetector

mirrors flip 90 degbetween sample

and reference

mirror pistonwith air bearing

source/aperture• the aperture controls the maximum possible resolution• optical filters minimize aliasing of non-infrared radiation

sample chamber• has a sample and reference so that transmission can be computed• the collimated white light source is used to align the sample and reference with the infrared beam

detector• output is electronically amplified• electronically low pass filtered• digitized at specified mirror displacements from zero

Mirror Movement & Data Collection

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pistontravel

electromagneticsolenoid driver

"frictionless"air bearing

start of traveltransducer

position indicatingpaddle

end of traveltransducer

mirrortravel

whitelight

helium:neonsingle-frequency

laser@ 0.6328 µm15,802 cm-1

photodiode detectors

fixedvisiblemirror

movingvisiblemirror

movinginfraredmirror

infraredbeam

visible beam splittermade from two prisms

to infraredbeam splitter

mirror design• supported by a near frictionless air bearing• driven back and forth by an electromagnetic solenoid• the range of motion is determined by two light emitting diode/photodiode pairs• when the position paddle interrupts the travel transducers, the polarity of the electromagnet is changed and the mirror travel direction changes• zero delay is about where the "position paddle" is shown

Measuring ξ

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zero delay determination• determined by white light and a visible interferometer constructed on the backside of the moving mirror• the interferogram of white light is a "spike" when the delay is zero• an electronic circuit monitoring the output of the white light photodiode can easily detect this spike• the fixed visible mirror is factory adjusted so that zero delay with the white light beam corresponds to zero delay with the infraredbeam in the infrared interferometer

distance determination• distance is measured by single frequency light from a He:Ne laser• the wavelength of the laser is 0.6328 µm, meaning that the output of the photodiode is an offset cosine with a maximum every 0.6328 µm• the analog-to-digital converter is set so that it only takes data whenever the He:Ne cosine is at a maximum

Analog-to-Digital Conversion

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mirror velocity• since the velocity of the mirror can be experimentally controlled, the apparent frequency of the cosine can be controlled to match the speed of the detector electronics• in this instrument the mirror velocity is adjusted to make the He:Ne laser produce a 5 kHz cosine at the detector• the highest un-aliased frequency that can be measured by sampling every peak of the cosine is 15,802/2 = 7,901 cm-1

• the spectrum then appears at frequencies from 2.5 kHz to dc

undersampling• since the upper end of the infrared spectrum is near 4,000 cm-1, lower digitizing rates can be used - this is called under sampling • for the far infrared the data will be under sampled by 16 to produce a highest optical frequency of 493.8 cm-1 corresponding to a 313 Hz• for a fixed resolution, under sampling decreases the size of the data set that needs to be numerically processed• under sampling permits the use of low pass filters with lower f3dB frequencies, thus a better SNR

Example Data: Infrared Source

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The two figures below show the source (reference) interferogram (distance vs. amplitude) and the spectrum (frequency vs. amplitude).

Example Data: K2ReCl6

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The three figures below show the sample interferogram, the sample intensity, and the sample transmission. The very sharp line at ~375 cm-1 is aliasing of an electric interference. The 330 cm-1 band is the Re-Cl stretch and the 180 cm-1 band is a Cl-Re-Cl bend. The peak at 80 cm-1 is an artifact due to low source intensity.

Optical Resolution

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Consider a cosine signal. The process of digitizing the interferogram involves starting and stopping, thus the cosine ismultiplied by a rectangle with width ξ0. The result is convolution by a sinc function (with 0 = 1/ξ0 ). ν

50.50.250.1250.0625mirror(cm)

1010.50.250.125ξ0 (cm)

0.112480ν

The spectral resolution (bandpass) depends only upon how far themirror is moved!

Gas-Phase Ammonia Spectrum (1)

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Gas-Phase Ammonia Spectrum (2)

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Apodization

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effect of abruptly starting and stopping the mirror movement •the process of starting and stopping the digitization of the cosine produces a spectrum convolved with a sinc function • a sinc function is said to have "feet," and the feet often makespectral interpretation difficult • apodization is literally "no feet making," and involves multiplication of the collected data by some function other than a rectangle

apodization functions• when computers were slow and had little memory, simple functions were used for apodization, i.e. trapezoid and triangle• with modern computers any desirable function can be used, e.g. Gaussian or exponential • note that apodization will decrease resolution via convolution with the spectral peaks

Signal-to-Noise Enhancement

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multiplexing (Fellgett advantage)• 0.1 cm-1 resolution:

decrease in measurement time: increase in SNR:

• 5 cm-1 resolutiondecrease in measurement time: increase in SNR:

optical throughput (Jacquinot advantage)• 0.1 cm-1 resolution

scanning f/# = ~12 for ~0.05% throughputinterferometer f/# = ~1 for ~6.2% throughputresult = ×124 more light reaches the detector

• 5 cm-1 resolutionscanning f/# = ~5 for 0.25% throughputresult = ×25 more light reaches the detector

sum of the two advantages• 0.1 cm-1 resolution = ×21,824• 5 cm-1 resolution = ×625

3,500 400 31,0000.1

− =31,000 ~ 176=

3,500 400 6205− =

620 25=

Original Drivers for FTIR

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• low concentrations (large amount of light on the detector, allows precise measurement of transmission)• high resolution spectra (large mirror movement)• unstable or transient species (fast measurement time)• gas chromatography detector (fast measurement time)• mixture spectra using Beer's Law (a precise transmission allowsa precise absorption)• far infrared spectra (interferometer instead of grating - tough to rule appropriate gratings) • high sample throughput (fast measurement time)