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Transcript
10.2 Fourier Transform
Infrared Spectroscopy
• overview of the infrared including measurement
difficulties
• spectrometer throughput
• instead of a t ↔ f transform use an x ↔ ν 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
10.2 : 1/22
Overview of the Infrared
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.
10.2 : 2/22
Measurement Difficulties
• infrared photons are difficult to distinguish from thermal noise,
with the result that the measurement is detector noise limited
PN = 4kT ∆f
for ∆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
10.2 : 3/22
Spectrometer Optical Throughput
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/#
percent
1
6.2
2
1.5
5
0.25
10
0.06
20
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.
10.2 : 4/22
Impossibility of a t ↔ f Transform
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,
∞
φ( f ) = ∫ F (t )e − i 2 πft dt
−∞
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.
10.2 : 5/22
Possibility of an x ↔ ν Transform
Fortunately, an infrared wave can be written in an alternative
form using distance and wave number, where ν = 1/λ.
⎛ x⎞
cos ⎜ 2π ⎟ = cos(2πνx)
⎝ λ⎠
The Fourier transform equation takes the same form, simply
exchanging ν for f and x for t. It is numerically solved using the
same computer program as that used for t to f transforms.
∞
φ ( ν ) = ∫ F ( x)e − i 2 πνx dx
−∞
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!
10.2 : 6/22
The Michelson Interferometer
• light from an infrared source is
fixed mirror
collimated and directed to a beam
movable
mirror
splitter
• at the beam splitter, half is reflected
source
toward the fixed mirror and half is
transmitted toward the movable mirror
• when light reaches each mirror it is
beam
reflected back to the beam splitter
splitter
sample
• at the beam splitter, half the light
from the fixed mirror is reflected back
to the source and half is transmitted to
detector
the detector
• at the beam splitter, half the light from the movable mirror is
transmitted 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 measuring
the resultant intensity at the detector
10.2 : 7/22
Optical Autocorrelator (1)
Light traveling toward the detector can be described as the sum of
two cosines,
1
1
cos(2πνx) + cos(2πν[ x + ξ])
2
2
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.
10.2 : 8/22
Optical Autocorrelator (2)
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.
1
x0
x0
2
0.5cos(2
)
0.5cos(2
(
))
π
vx
+
π
v
x
+
ξ
dx
(
)
∫
0
Expansion of the square yields three integrals.
0.25
x0
x0
x0
0
0
0.25
2
cos(2
)
πν
x
dx
+
∫
x0
∫ cos ( 2πν ( x + ξ ) )
2
0.50
dx +
x0
x0
∫ cos(2πνx)cos ( 2πν ( x + ξ) ) dx
0
By letting x0 → ∞, the first two integrals each reduce to 0.125.
0.50
0.125 + 0.125 +
x0
10.2 : 9/22
x0
∫ cos(2πνx) cos ( 2πν ( x + ξ) ) dx
0
Optical Autocorrelator (3)
The third integral on the previous slide is the autocorrelation function,
1
C1,1 ( ξ ) =
x0
x0
∫ cos(2πνx) cos(2πν ( x + ξ))dx
0
C1,1 ( ξ ) = 0.5cos(2πνξ)
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(2πνξ)
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 detector
output. 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 .
10.2 : 10/22
Instrument Block Diagram
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
10.2 : 11/22
mirror piston
with air bearing
source
resolution
controlling
aperture
movable
mirror
beam
splitter
optical
filter
fixed
mirror
TGS pyroelectric
detector
mirrors flip 90 deg
between sample
and reference
reference
collimated
white
alignment
light
mirror swings
out of the way
for alignment
sample
Mirror Movement & Data Collection
mirror design
• supported by a near frictionless air
bearing
helium:neon
• driven back and forth by an
single-frequency
laser
@ 0.6328 µm
electromagnetic solenoid
15,802 cm
• the range of motion is determined
by two light emitting
visible beam splitter
made from two prisms
diode/photodiode pairs
• when the position paddle
photodiode detectors
interrupts the travel transducers,
the polarity of the electromagnet is
changed and the mirror travel
moving
direction changes
visible
mirror
• zero delay is about where the
"position paddle" is shown
-1
moving
infrared
mirror
10.2 : 12/22
piston
travel
electromagnetic
solenoid driver
white
light
"frictionless"
air bearing
fixed
visible
mirror
start of travel
transducer
position indicating
paddle
end of travel
transducer
mirror
travel
infrared
beam
to infrared
beam splitter
Measuring ξ
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 infrared
beam 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
10.2 : 13/22
Analog-to-Digital Conversion
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
10.2 : 14/22
Example Data: Infrared Source
The two figures below show the source (reference) interferogram
(distance vs. amplitude) and the spectrum (frequency vs. amplitude).
10.2 : 15/22
Example Data: K2ReCl6
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.
10.2 : 16/22
Optical Resolution
Consider a cosine signal. The process of digitizing the
interferogram involves starting and stopping, thus the cosine is
multiplied by a rectangle with width ξ0. The result is convolution
by a sinc function (with ν 0 = 1/ξ0 ).
ν0
8
4
2
1
0.1
ξ0 (cm)
0.125
0.25
0.5
1
10
mirror
(cm)
0.0625
0.125
0.25
0.5
5
The spectral resolution (bandpass) depends only upon how far the
mirror is moved!
10.2 : 17/22
Gas-Phase Ammonia Spectrum (1)
10.2 : 18/22
Gas-Phase Ammonia Spectrum (2)
10.2 : 19/22
Apodization
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 make
spectral 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
10.2 : 20/22
Signal-to-Noise Enhancement
multiplexing (Fellgett advantage)
• 0.1 cm-1 resolution:
3,500 − 400
= 31,000
decrease in measurement time:
0.1
increase in SNR: 31,000 = ~ 176
• 5 cm-1 resolution
3,500 − 400
= 620
decrease in measurement time:
5
increase in SNR: 620 = 25
optical throughput (Jacquinot advantage)
• 0.1 cm-1 resolution
scanning f/# = ~12 for ~0.05% throughput
interferometer f/# = ~1 for ~6.2% throughput
result = ×124 more light reaches the detector
• 5 cm-1 resolution
scanning f/# = ~5 for 0.25% throughput
result = ×25 more light reaches the detector
sum of the two advantages
• 0.1 cm-1 resolution = ×21,824
• 5 cm-1 resolution = ×625
10.2 : 21/22
Original Drivers for FTIR
• 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 allows
a precise absorption)
• far infrared spectra (interferometer instead of grating - tough to
rule appropriate gratings)
• high sample throughput (fast measurement time)
10.2 : 22/22