Download Principles of Fourier Transform Optical Measurements

Principles of Fourier Transform
Optical Measurements
Chapter 7
Time Domain Spectroscopy
• Encode spectral information about a source in the
form of a time dependent electrical signal called
interferogram.
• The interferogram is the time dependent function of
the intensity of the source.
– Time domain signal: Intensity versus Time/Distance
• The spectrum- the interferogram, when appropriately
analyzed, yields the spectrum of the source.
– Frequency Domain Signal: Signal versus Frequency.
1
The Michelson interferometer
• Converts a high frequency signal to a measurable low
frequency signal
– Signal/frequency modulation
• Interferogram is analyzed to obtain the spectrum
Components of a Michelson Interferometer
•
Two plane mirrors at right
angle
–
–
•
•
Fixed mirror
Movable mirror
Beam splitter at an angle
of 45° to the mirrors:
divides incoming light
(ideally 50 % transmitted
and 50 % reflected)
Compensator: equalizes
the optical path lengths in
both arms
2
Simple Case of Idealized Monochromatic
Radiation
• Assume for simplicity that there is a monochromatic input to the
interferometer adjusted so that the optical path lengths in both arms
are identical.
• The two beams will be in phase when they return to the beam
splitter and, as such, they will constructively interfere.
• Looking into the interferometer at the output, the field will appear
bright.
• Move one mirror back 1/4 of a wavelength. The two beams will be
180° out of phase and they will destructively interfere, the field will
appear dark.
• If the mirror is continuously moved, the field will oscillate from light to
dark for each quarter-wavelength movement of the mirror.
• One cycle of the interferogram occurs when the mirror moves a
distance equal to 1/2 λ.
Relationship between the Frequency of
Interferogram and Optical Frequency
Let τ be the time required for the mirror to move 1/2 wavelength,
VM the constant mirror velocity, f
the frequency of the
inteferogram, and ν the optical frequency.
λ
2
= VM × τ
f =
1
τ
=
2VM
λ
(1)
=
2VM
ν = 2VM ν
c
(2)
(τ is the period of the interferogram / time required to complete a
cycle)
if the mirror velocity is 1 cm/s, the frequency of the interferogram
is,
f = 6.67 × 10 −11ν
3
The equation for the interferogram
P (δ ) =
1
P (ν ) cos(2πft )
2
It is a simple cosine wave, the amplitude of which depends on the
intensity of the monochromatic source
δ :retardation: difference in path travelled by the two beams
P (δ ) : radiant power of the output as a function of retardation
P(ν ) :
radiant power of the source as a function of optical
frequency (source spectrum)
Modified interferogram equation
P(δ ) = B (ν ) cos(2πft )
The modified equation accounts for unequal splitting of the source
power and the frequency dependence of the detector response
Substitute
f
by equation 2, then substitute VM by
P(δ ) = B (ν ) cos(2πδ ν )
δ /2
t
The frequency of the oscillation depends on two factors:
1) the frequency of the incoming electromagnetic radiation
2) the velocity of the mirror
Thus, using an interferometer a very high optical frequency can
be uniquely encoded in the form of a low-frequency oscillation.
4
Simple Interferograms and Spectra
What is the nature of a broadband spectral input?
• Each input can be treated independently
and hence the output will be the
summation of cosine functions.
• At zero path difference, all the waves are
in phase
• As the mirror is moved away from zero
position the waves rapidly sum out to a
steady average value.
• The resulting ac signal/ the interferogram
can be expressed mathematically as:
+∞
(1) P (δ ) =
∫ B(ν ) cos(2πδν )dν
−∞
5
Fourier Transformation
• The Fourier transform of the integral (1) is integral
(2)
• Integrals (1) and (2) form a cosine Fourier transform
pair.
+∞
(1) P(δ ) =
∫ B(ν ) cos(2πδν )dν
−∞
FT
+∞
(2) B (ν ) =
∫ P(δ ) cos(2πδν )dδ
−∞
Effects of finite mirror movement and digitization
• To reconstruct the exact spectrum of the source, the
interferogram must be measured from - ∞ to ∞+.
However, the mirror movement is finite.
• Digitization of the interferogram requires use of
finite-sized sampling interval.
• These two practical constraints limit the resolution
and the frequency range.
• The width of the signal is inversely related to the
maximum extent of the interferogram, hence the
mirror movement.
6
Resolution
• Resolution depends on the width of the
interferogram (mirror travel distance)
• The minimum mirror travel distance required
for two lines to be resolved is given by:
∆ν = ν 2 − ν1 =
1
δ
Advantages of FT Methods
•
•
•
•
•
•
Multiplex or Fellgets advantage
Throughput advantage
High accuracy and reproducibility of frequency
measurements
High resolution
Computerization
Controlled resolution function
7
Multiplex or Fellgets advantage
•
•
In scanning each resolution element is "seen"
during a fraction of the total scanning time (T).
If M is the number of resolution elements, then
Number of resolution elements =(λmax − λmin )
∆λ
T
Signal ∝
Noise ∝
S
∝
N
M
T
M
T
M
Fellgets advantage
• With the interferometer, each resolution element is "seen" all
the time (all optical frequencies are incident on the detector at
once). The spectral information is said to be multiplexed.
Signal ∝ T
Noise ∝ T
S
∝ T
N
• Superior by a factor of
M
8
Throughput advantage
•
•
•
•
Throughput is the amount of light that one can get through
the spectrometer.
One of the factor that limits the throughput of a grating or
prism instrument is the necessity for an entrance slit.
The interferometer has a large circular entrance aperture,
roughly the size of the mirrors and, as such, has greater
throughput.
The improvement is a factor of 100.
9