12 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Fourier-transform spectroscopy | 2/2 | https://en.wikipedia.org/wiki/Fourier-transform_spectroscopy | reference | science, encyclopedia | 2026-05-05T10:04:36.785085+00:00 | kb-cron |
where
I
(
ν
~
)
{\displaystyle I({\tilde {\nu }})}
is the spectrum to be determined. Note that it is not necessary for
I
(
ν
~
)
{\displaystyle I({\tilde {\nu }})}
to be modulated by the sample before the interferometer. In fact, most FTIR spectrometers place the sample after the interferometer in the optical path. The total intensity at the detector is
I
(
p
)
=
∫
0
∞
I
(
p
,
ν
~
)
d
ν
~
=
∫
0
∞
I
(
ν
~
)
[
1
+
cos
(
2
π
ν
~
p
)
]
d
ν
~
.
{\displaystyle {\begin{aligned}I(p)&=\int _{0}^{\infty }I(p,{\tilde {\nu }})d{\tilde {\nu }}\\&=\int _{0}^{\infty }I({\tilde {\nu }})[1+\cos(2\pi {\tilde {\nu }}p)]\,d{\tilde {\nu }}.\end{aligned}}}
This is just a Fourier cosine transform. The inverse gives us our desired result in terms of the measured quantity
I
(
p
)
{\displaystyle I(p)}
:
I
(
ν
~
)
=
4
∫
0
∞
[
I
(
p
)
−
1
2
I
(
p
=
0
)
]
cos
(
2
π
ν
~
p
)
d
p
.
{\displaystyle I({\tilde {\nu }})=4\int _{0}^{\infty }\left[I(p)-{\frac {1}{2}}I(p=0)\right]\cos(2\pi {\tilde {\nu }}p)\,dp.}
== Pulsed Fourier-transform spectrometer == A pulsed Fourier-transform spectrometer does not employ transmittance techniques. In the most general description of pulsed FT spectrometry, a sample is exposed to an energizing event which causes a periodic response. The frequency of the periodic response, as governed by the field conditions in the spectrometer, is indicative of the measured properties of the analyte.
=== Examples of pulsed Fourier-transform spectrometry === In magnetic spectroscopy (EPR, NMR), a microwave pulse (EPR) or a radio frequency pulse (NMR) in a strong ambient magnetic field is used as the energizing event. This turns the magnetic particles at an angle to the ambient field, resulting in gyration. The gyrating spins then induce a periodic current in a detector coil. Each spin exhibits a characteristic frequency of gyration (relative to the field strength) which reveals information about the analyte. In Fourier-transform mass spectrometry, the energizing event is the injection of the charged sample into the strong electromagnetic field of a cyclotron. These particles travel in circles, inducing a current in a fixed coil on one point in their circle. Each traveling particle exhibits a characteristic cyclotron frequency-field ratio revealing the masses in the sample.
=== Free induction decay === Pulsed FT spectrometry gives the advantage of requiring a single, time-dependent measurement which can easily deconvolute a set of similar but distinct signals. The resulting composite signal, is called a free induction decay, because typically the signal will decay due to inhomogeneities in sample frequency, or simply unrecoverable loss of signal due to entropic loss of the property being measured.
=== Nanoscale spectroscopy with pulsed sources === Pulsed sources allow for the utilization of Fourier-transform spectroscopy principles in scanning near-field optical microscopy techniques. Particularly in nano-FTIR, where the scattering from a sharp probe-tip is used to perform spectroscopy of samples with nanoscale spatial resolution, a high-power illumination from pulsed infrared lasers makes up for a relatively small scattering efficiency (often < 1%) of the probe.
== Stationary forms of Fourier-transform spectrometers == In addition to the scanning forms of Fourier-transform spectrometers, there are a number of stationary or self-scanned forms. While the analysis of the interferometric output is similar to that of the typical scanning interferometer, significant differences apply, as shown in the published analyses. Some stationary forms retain the Fellgett multiplex advantage, and their use in the spectral region where detector noise limits apply is similar to the scanning forms of the FTS. In the photon-noise limited region, the application of stationary interferometers is dictated by specific consideration for the spectral region and the application.
== Fellgett advantage ==
One of the most important advantages of Fourier-transform spectroscopy was shown by P. B. Fellgett, an early advocate of the method. The Fellgett advantage, also known as the multiplex principle, states that when obtaining a spectrum when measurement noise is dominated by detector noise (which is independent of the power of radiation incident on the detector), a multiplex spectrometer such as a Fourier-transform spectrometer will produce a relative improvement in signal-to-noise ratio, compared to an equivalent scanning monochromator, of the order of the square root of m, where m is the number of sample points comprising the spectrum. However, if the detector is shot-noise dominated, the noise will be proportional to the square root of the power, thus for a broad boxcar spectrum (continuous broadband source), the noise is proportional to the square root of m, thus precisely offset the Fellgett's advantage. For line emission sources the situation is even worse and there is a distinct `multiplex disadvantage' as the shot noise from a strong emission component will overwhelm the fainter components of the spectrum. Shot noise is the main reason Fourier-transform spectrometry was never popular for ultraviolet (UV) and visible spectra.
== Measurement of real-time periodic dynamics == Fourier-transform spectroscopy can also be used for measuring real-time periodic spectral intensity dynamics. Consider an electric field component
E
(
t
)
=
E
(
t
+
2
π
/
ω
m
)
{\displaystyle E(t)=E(t+2\pi /\omega _{m})}
being a periodic function of modulation frequency
ω
m
{\displaystyle \omega _{m}}
, its information content measured by a Fourier-transform spectrometer is given by:
|
E
(
t
)
|
2
cos
(
ω
D
t
)
=
∑
n
f
n
exp
(
i
ω
m
t
)
cos
(
ω
D
t
)
{\displaystyle {\begin{aligned}|E(t)|^{2}\cos {(\omega _{D}t)}&=\sum _{n}f_{n}\exp {(i\omega _{m}t)}\cos {(\omega _{D}t)}\\\end{aligned}}}
where
ω
D
{\displaystyle \omega _{D}}
is the Doppler frequency, and the field dynamics has been expanded into a Fourier series. The field dynamics in
E
(
t
)
{\displaystyle E(t)}
can be measured at the frequency domain at isolated RF frequencies of
n
ω
m
±
ω
D
{\displaystyle n\omega _{m}\pm \omega _{D}}
. By encoding the broadband coherent light source with common and fixed
ω
m
{\displaystyle \omega _{m}}
, periodic spectral intensity dynamics can be simultaneously readout for light components at different
ω
D
{\displaystyle \omega _{D}}
.
== See also ==
== References ==
== External links == Description of how a Fourier transform spectrometer works The Michelson or Fourier transform spectrograph Internet Journal of Vibrational Spectroscopy – How FTIR works Fourier Transform Spectroscopy Topical Meeting and Tabletop Exhibit