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Transcript
```Dunlap Institute Summer School:
Fourier Transform Spectroscopy Lab
Facilitators: Rachel Friesen∗, Laura Newburgh∗, David Naylor†,
Suresh Sivanandam∗, Kieran MacLeod∗
1
Objective
The purpose of this lab is to understand the basic principle and operation of the Fourier transform
interferometer.
In this lab you will:
1. Set up and align a Michelson interferometer using optical alignment techniques you learned
in previous labs
2. Find the zero path difference (ZPD), observe the interference fringes, and determine the
wavelength of a laser light source by measuring the fringe spacing
3. Using incoherent LED and white light sources, observe the fringe envelope, and determine the
wavelength and bandwidth of the sources
4. Use the white light fringes to determine the thickness of a polymer film and (if time allows)
the speed of light in air
Section 2 gives a brief review of the theory and history of the Michelson interferometer. Your
instructions on setting up and aligning the interferometer and investigating the Fourier transform.
The main hardware is listed in Section 4.
1
2
Dunlap Institute for Astronomy and Astrophysics
University of Lethbridge
1
2
The Michelson interferometer and the FTS
A beam of light can be modelled as a wave of oscillating electric and magnetic fields. When two
beams of light meet in space, these fields add according to the principle of superposition. At each
point in space, the resultant electric and magnetic fields are the vector sum of the fields of the
separate beams. If the two beams of light originate from separate sources, there is generally no
fixed relationship between the frequencies and phases of the individual beams; the beams are said
to be incoherent. If two such beams meet, at any instant in time there will be points in space where
the fields add to produce a maximum field strength. Since uncorrelated light doesn’t constructively
or destructively interfere on timescales we can perceive, the human eye averages these results and
perceives a uniform intensity of light. However, if the two beams of light originate from the same
source, there is generally some degree of coherence between the two beams. At one point in space
the light from the beams may be continually in phase. In this case, the combined field will always
be a maximum and a bright spot will be seen. At another point the light from the two beams may
be continually out of phase and a minimum, or dark spot, will be seen.
The coherence time, τc , is the time over which a propagating wave (e.g., from a laser) may be
considered coherent. It is equal to the reciprocal of the temporal bandwidth:
τc =
1
,
∆ν
(1)
where the frequency bandwidth ∆ν is simply νmax − νmin . The coherence length, lc , is the propagation distance over which a coherent wave (e.g. from a laser) maintains a specified degree of
coherence. Interference is strong when the paths taken by all of the interfering waves differ by less
than the coherence length:
lc =
c
λmax λmin
=
∆ν
∆λ
(2)
Thomas Young was one of the first to design a method for producing such an interference pattern.
He allowed a single, narrow beam of light to fall on two narrow, closely spaced slits. Opposite the
slits he placed a viewing screen. Where the light from the two slits struck the screen, a regular
pattern of dark and bright bands became visible. When first performed, Young’s experiment offered
important evidence for the wave nature of light. Young’s slits function as a simple interferometer.
If the spacing between the slits is known, the spacing of the maxima and minima can be used to
2
fixed mirror
beamsplitter
screen
mirror &
translation stage
lens
light source
Figure 1: Diagram of a Michelson interferometer. A beam of light from the laser travels through a lens and
strikes the beam-splitter. The beam-splitter is designed to reflect 50% of the incident light and transmit the
other 50%. The incident beam therefore splits into two beams; one beam is reflected toward the translating
mirror, the other is transmitted toward the fixed mirror. Both mirrors reflect the beams back toward the
beam-splitter. Half the light from the translating mirror is transmitted through the beam-splitter to the
viewing screen, and half the light from the fixed mirror is reflected by the beam-splitter to the viewing
screen. Since the two beams are from the same coherent source their phases are highly correlated and
interference occurs.
determine the wavelength of the light. Conversely, if the wavelength of the light is known, the
spacing of the slits could be determined from the interference patterns.
In 1881, some 78 years after Young introduced his two-slit experiment, A.A. Michelson designed
and built an interferometer using a similar principle. The Michelson interferometer consists of two
mirrors set at right angles, and a beamsplitter that divides an input beam of light into two beams of
approximately equal intensity. The two beams of light travel along separate paths, and are eventually
recombined at the beamsplitter. Michelson designed his interferometer as a method to test for the
existence of the ether, a hypothesized medium through which light propagated. The null result of
this experiment, arguably the most important null measurement in the history of science, heralded
the development of Einstein’s special theory of relativity. Since then Michelson’s interferometer has
become a widely used instrument for measuring the wavelength of light (spectroscopy), and for using
the wavelength of a known light source to measure extremely small distances (metrology).
3
A Fourier transform spectrometer uses the same basic configuration of mirrors and beamsplitter as
a Michelson interferometer, but one of the mirrors can be moved back and forth (Figure 1). In astronomical instruments, the translation stage for the movable mirror is automatic and precise. Here,
we will use a manual translation stage to understand how the FTS works and gain an appreciation
for the response of the observed fringe pattern to fine movements.
In our experiment, we will set up and align an FTS using a laser light source. For monochromatic
light, split equally into two beams, we can write the amplitude of the combined beam as
ET = E0 eiωt + E0 ei(ωt+δ)
(3)
where δ is the phase difference between the two beams resulting from the different path lengths
∆ for a path difference ∆, or
traveled. This phase difference δ = 2π
λ
δ=
2π
[2(l1 − l2 )cosθ],
λ
(4)
where l1 − l2 is the difference in the lengths of the arms l1 and l2 , and θ is the incident angle of
the light on the interferometer mirrors away from normal. Using wavenumber notation, where the
wavenumber σ = 1/λ, the intensity of the combined beam is then
I = |ET ET∗ | = I0 (1 + cosδ) = I0 (1 + cos(2πσ∆)).
(5)
For a non-monochromatic source, the total intensity is given by the sum of intensities at different
σ:
Z
I(∆) =
∞
I(σ)(1 + cos(2πσ∆))dσ,
(6)
0
which is within a constant of the cosine form of the Fourier transform of I(σ). Thus, as you learned
in the FTS lecture, the intensity I(∆) of the recombined beam as a function of the path difference
for light from the two arms, ∆, is the Fourier transform of the intensity of the light source, I(σ).
The Fourier transform of a Gaussian is another Gaussian, with narrow Gaussians transformed to
wider features, and vice versa. If the light source is a single Gaussian spectral line (i.e., for the
laser light source used initially here) with σ0 the wavenumber at the centre of the distribution, the
intensity of the combined beam can be given as:
I(∆) = I0 (1 + e−
(πσ1/2 ∆)2
4ln2
4
cos(2πσ0 ∆))
(7)
Figure 2: A single Gaussian line and its Fourier transform. Note that the line width δσD in the Figure is
equivalent to σ1/2 in this introduction.
The cosine term gives a rapid oscillation in intensity as the mirror is moved as the combined beams
constructively and destructively interfere (the interference fringe pattern you will see here). For light
incident normal to the mirrors, when ∆ = 2(l1 − l2 ) = mλ, we have constructive interference, and
when ∆ = (m + 1/2)λ, we have destructive interference. The exponential term varies more slowly,
and describes an ‘envelope’ that defines the relative amplitude and contrast of the light and dark
fringes. Both the rapid oscillation and slow envelope effects are shown in Figure 2.
The exponential term has a maximum at the zero path difference, and drops to half the maximum
value at a path difference
∆1/2 =
2ln2
πσ1/2
(8)
where σ1/2 is the width of the envelope in wavenumbers, and can be related to the actual width of
the bandpass ∆λ via σ1/2 = ∆λ/λ2 . This is simply the coherence length from Equation 2.
5
3
Exploring the Fourier transform spectrometer
First, you need to set up and align the FTS. Figure 3 shows the FTS assembled with the laser
light source, while Figure 4 shows the construction and heights of the interferometer
components. Refer to these diagrams as you go. Note that the different spacers are
used to put the optical components at the same level - check this before you screw things
down!
3.1
Initial setup and alignment of the mirrors
Place the beamsplitter-attached 1-inch thick spacer on top of a 2-inch thick spacer (see Figure 4a),
and secure this near the centre of your optical plate, with the spacers aligned parallel to the long
axis of the optical plate. The beamsplitter has a dot on the top surface that shows which sides of
the cube should face the incident laser. Use two 3 1/4-inch screws and washers, placed at diagonal
corners. Do not fully tighten the screws yet, as we need the beamsplitter to determine the
correct height of the laser, but will remove it before aligning the fixed mirror.
Be careful not to touch the mirror surfaces or the beamsplitter.
Next, add the base plates and post holders for the light source and the lens (omit the actual lens
for now) facing one of the preferred cube sides. Secure these base plates, with long sides parallel
to the beamsplitter mount, to the optical plate beside the beamsplitter mount using four 1/4”-20,
0.5-inch screws (see Figure 3). Secure the laser light source in the further post holder.
Do not look directly at the laser! Use the white cards provided to test the
interferometer alignment.
Turn the laser on and adjust the height of the light source until it hits the centre of the beamsplitter.
The laser mount also has fine adjustment screws to find a more precise alignment. Note where the
second beam goes; this is where you will place the second mirror. Once you have centred
the laser beam in the beamsplitter cube, turn off the laser, and remove the beamsplitter and mount.
6
Figure 3: Overhead view of the completed interferometer setup.
Next, we will add the fixed mirror. Keep the tissue paper on the mirrors as you attach them
to the spacer or translation stage to avoid touching or scratching their surfaces. Place a
2-inch thick spacer on the other side of the beamsplitter, directly opposite the laser mount. The
spacer should be ∼ 2 inches from the beamsplitter mount (or two holes on the optical plate), aligned
parallel to the beamsplitter mount. Secure to the optical plate with two 2 1/4-inch screws, again
placed at diagonal corners. Then secure one mirror mount to the 2-inch spacer (see Figure 4b),
using two 0.5-inch screws, with the mirror facing the beamsplitter and laser.
Align the mirror so that the reflected laser beam returns along the same path. With the laser turned
on, hold a white card to one side of the laser to find where the reflected beam hits the laser mount.
For large offsets, you can shift the spacer slightly before completely tightening the screws. Then
adjust the mirror’s fine resolution screws until the reflected beam is directed back to the centre of
the original beam.
Next we will set up the moving mirror. We need the path lengths of the beams to be approximately equal to obtain fringes. When attaching the stage and mirror assembly, check
7
Figure 4: Diagrams showing the construction and heights of three of the interferometer components: a)
beamsplitter mounted on two spacers; b) mirror mount attached to single spacer; c) mounted lens on a
post/post holder.
with a ruler that the mirror is approximately the same distance from the beamsplitter
face as the fixed mirror. Mount a 1-inch spacer in the direction of the second laser path, parallel
to the long side of the optical plate. Next add the translation stage using 0.5-inch screws - this will
only work if it is right-side-up! Finally, mount the mirror to the translation stage using two 0.5-inch
screws and two flat washers. Adjust the micrometer on the translation stage so it is approximately
at the centre of its range of motion.
Important! The translation stage has a maximum range of motion (travel) of 0.500
(12.7 mm) and two adjustment dials: coarse and fine. The coarse adjustment provides 0.500 of
travel, which is lockable via a thumbscrew located on the mounting barrel. The fine adjustment
dial moves 25 µm per one complete revolution of the graduated knob. Each graduation of the
fine control knob is therefore 0.5 µm. There is a white line that marks the furthest extent
the stage should be extended - if you see this white line STOP and call a facilitator!
3.2
Adding the beamsplitter and aligning the second mirror
Remount the beamsplitter and 1-inch spacer on the 2-inch spacer in the centre of your optical plate
so that the beam again passes through the beamsplitter. Do not tightly secure the spacers yet.
8
Place cards in front of both mirrors, so that the laser light is not reflecting back from them. The
beamsplitter reflects a very small fraction of light back from the incident surface, and we will use
this to align the cube more precisely. Again, use a white card to find where the reflected beam hits
the laser mount, and shift the beamsplitter mount until the reflected beam is directed back to the
centre of the original beam. Then secure the mount tightly, checking afterwards that your cube is
still aligned well.
Next, remove both cards from the mirrors, and hold one a small distance away from where the
recombined beam exits the beamsplitter. You will likely see two laser dots, hopefully not too far
distant! Since we have already aligned the first, fixed mirror, we don’t want to adjust
it again. Adjust the fine resolution screws on the translating mirror until the laser dots line up.
When you have good alignment, you should see some dark lines - the fringes - across the laser dot.
The diode laser beam that we use has a small spot size (1 mm) and the interference can be difficult
to see.
3.3
Adding the lens and circular fringes
Next we will place the lens between the light source and the beam splitter. Add the lens mount to
the empty post in front of the laser. Adjust the height of the post in the holder until the laser beam
travels through the centre of the lens.
Look at the fringes again. What has changed? By placing a lens between the laser and the
beamsplitter the beam diverges and an interference pattern of dark and bright rings, or fringes, is
readily seen on the viewing screen. This divergence means that only the central ray of the laser
beam is travelling on a straight line through the interferometer. The surrounding rays, travelling
at different angles and thus different distances through the interferometer, depending on how close
they are to the centre of the beam. The result is a bullseye pattern of bright and dark fringes instead
of just one spot (see Figure 5).
3.4
Finding the zero path difference
If your mirrors are exactly perpendicular to each other, and the path length difference is near zero,
you should see centred, circular fringes. If the mirrors are not perpendicular, you may see some
straight-line fringes. Try adjusting the fine resolution screws on the translating mirror to obtain
curved fringes, and finally a circularly-symmetric pattern. Test how sensitive the fringes are to your
alignment.
With the ring pattern, find the zero path difference (ZPD) point by moving the translation stage
9
Figure 5: Two source representation showing how bullseye pattern is formed. Bright rings (i.e.
constructive interference) occur when 2(L2 − L1 ) cos θ = nλ.
using the coarse adjustment dial until the fringes are as widely spaced as you can get them. If
you have not placed your mirrors approximately equidistant from the beamsplitter, this
may take a LONG time. If you are having troubles, use the ruler to check your mirror
distances. Do NOT move the translation stage more than 1 cm! If you see a white line
on the fine control knob, STOP and get a facilitator.
With this setup, at ZPD the dark or light features should largely fill the cube outline. Lock the
coarse adjustment dial with the pin. Then move the translation stage forward and back by small
amounts using the fine adjustment screw to see how the fringe spacings change. Are you confident
you have found the ZPD? Next, move the translation stage further in one direction (remembering
in what direction you are moving the stage!). Does the intensity or contrast in the fringes change?
3.5
Experiment 1: Spectroscopy with a Michelson interferometer
We will now determine the wavelength of your light source. Find again the zero path difference,
with the fringes as widely spaced as you can (unlock and use the coarse translation dial if necessary).
Have one person watch the micrometer reading while another slowly moves the translation stage and
10
counts the number of fringes observed (use the dark central spot of the bullseye pattern, counting
each dark fringe). After the translation stage has moved a set distance d, stop and record d and
the number of fringes. Swap roles so that everyone makes a measurement.
QUESTION 1: When the mirror moves a distance d, the optical path has changed by 2d. The
wavelength of light is simply 2d divided by the number of fringes. Compute the average wavelength
of your measurements - we will determine the standard deviation between all the lab stations later and estimate the error based on your own assessment of how accurately you can read the micrometer
(see Question 2 below). Record these values for discussion with the lab group.
QUESTION 2: Assuming you can read the micrometer to one-half of its smallest scale division,
and can count fringe shifts with complete certainty, how much should the arm length be changed
to guarantee a wavelength result with an accuracy of 1 percent?
3.6
Experiment 2: Fringes from broadband sources
So far, we have looked at the fringe pattern obtained for a very narrow spectrum light source (the
laser). It is also possible to see fringes for a broad-spectrum, or bandwidth (the wavelength range
over which it is emitting light) light source, such as an LED. Remember that due to the nature of the
Fourier transform, narrow spectrum sources have very broad fringe envelopes, while broader spectrum
sources will have more narrow fringe envelopes (see Figure 2 and the introduction). Because the
LEDs have a broader bandwidth than the laser light source, you will see that the amplitude of the
fringes drops as you move the translation stage away from the zero path difference. We will again
measure the central wavelength of the light source, and will also determine the bandwidth of two
LED light sources.
QUESTION 3: Based on what you’ve learned about Fourier transforms, discuss with your group
what you expect the fringe envelope to look like in the narrow-line limit of a delta function, such as
the laser light source we use in this lab. Then calculate the coherence length of a laser of wavelength
640 nm and bandwidth 100 MHz. Does the result agree with your discussion?
QUESTION 4(a): What is the coherence length of an LED of wavelength 640 nm and bandwidth
20 nm? Compare this with the result from the laser. What does this mean for our experiment?
3.6.1
Red LED fringes
Bring the translation stage back to your recorded ZPD position. If you feel you can improve your
estimate of the ZPD position, do so! Once at the ZPD, lock the coarse adjustment dial, and move
the translation stage away slightly from the ZPD position with the fine adjustment dial, recording
11
the direction you moved the stage. This way, you should know which direction to move the
stage to again find the ZPD for the LED light source.
Swap the laser light source for the LED source. Using the fine adjustment dial, slowly move the
stage toward your recorded ZPD position until you are able to see the interference fringes. This
may take some time; let the facilitators know if you haven’t found the ZPD within ∼ 10 minutes.
Record your new ZPD position. How close was your original ZPD estimate to the value you
have now?
QUESTION 4(b): Repeat the fringe-counting experiment from the earlier section and record the
wavelength of your LED. Next, slowly move the stage until you are unable to see the fringes anymore,
and record the distance moved. Have each person in the group repeat the measurement. From these
measurement and Equation 8, calculate the bandwidth of your LED light source.
3.6.2
White light fringes
QUESTION 5(a): What is the coherence length of the super bright white LED (whose output
ranges from 400 - 700 nm)? Think back to question 2. What does this mean for our experiment?
The answers to questions 3-5 show that in order to see white light fringes the path difference between
the two interfering beams must be set very close to zero and provide a method of doing so.
Return the stage to the ZPD position you found for the red LED source. Then again adjust the
micrometer slightly in one direction, away from the zero path difference location. This way, to find
the white light fringes, you know which way to move the stage to find the zero path difference again.
Describe the pattern as the translation stage is adjusted to either side of ZPD.
QUESTION 5(b): Measure the central wavelength of the light source and bandwidth as you did for
the previous light sources. Is this possible? What are your uncertainties? Record your measurements
for discussion with the lab group.
Why do rainbow colours appear? Discuss with your group.
3.7
Experiment 3: Metrology with a Michelson interferometer
You will be provided with two samples (green and yellow) of thin polymer film of different thicknesses
and of refractive index n = 1.4. You are to use the interferometer as a metrology tool to determine
the thickness of the two films.
Record the micrometer reading with the interferometer aligned to display the white light fringes.
Place one film sample in the fixed arm of the interferometer. What happens to the white light fringes?
12
Carefully increase the optical path difference until white light fringes are restored and record the
change in the ZPD. Use this information to determine the thickness of each film. Remember that
by adding material with a different refractive index n, we change the effective path length of the
light by 2 × n × df ilm , where df ilm is the film width.
Remove the film and restore the white light fringes. Place one film in the fixed arm of the interferometer. Next place the same type sample in the moving arm. Describe what you see.
4
Hardware
Optical plate and optical mounts
The optical components are laid out on a 12 × 18 × 0.5-inch optical plate fabricated from black
anodized aluminum. The plate is tapped with 1/4 − 20 screw holes on 1−inch centers. This
plate provides a stable, reconfigurable mounting platform for optical components. Components are
mounted on posts via 8 − 32 or 1/4 − 20 screws held in post holders (the light sources and lens),
or mounted on 2” × 3” spacers (mirrors and beamsplitter) for stability.
Non-polarizing cube beamsplitter
Here, we use a cube beamsplitter to split the incident light into two separate beams, and recombine
the beams as they return from the stationary and
translating mirror. Using a cube beamsplitter ensures the light from both beams travels the same
distance through the glass. This beamsplitter is
made of a pair of precision right angle prisms cemented together with a metallic dielectric coating
on the hypotenuse of one of the prisms. A dot on
the top surface indicates the preferred incident light
cube face.
13
Differential translation stage
The translation stage holds one of the small angle
mirror mounts (see next subsection) to allow for
movement of the mirror, changing the light path
length along one axis. The translation stage is
equipped with a micrometer, allowing you to move
the stage precise distances.
Small angle mount with mirror
This mirror mount has two stainless steel fine resolution adjustment screws allow precise tilting of the
mounting surface in two directions to align the light
paths. One mirror is fixed in place on the provided
spacers, while the second mirror is attached to the
translation stage.
14
```
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