experimental optics course - universiteit twenteedu.tnw.utwente.nl/optprac/manual experimental...
TRANSCRIPT
10/24/2012
1
Experimental Optics
Course
Part II
Final Assignment (140702)
Dr ir J. S. Kanger
Dr ir F. A. van Goor
Oktober 2012
10/24/2012
3
CONTENTS
1 General Information .................................................................................................. 5
1.1 Content ................................................................................................................... 5
1.2 Reporting and evaluation ....................................................................................... 5
1.3 Practical information .............................................................................................. 6
1.4 Computer and software .......................................................................................... 6
2 Safety regulations ....................................................................................................... 7
3 Description of experiments ........................................................................................ 8
3.1 Coherence ............................................................................................................. 10
3.2 Fourier & Schlieren optics ................................................................................... 14
3.3 The Michelson interferometer .............................................................................. 17
3.4 The Mach-Zehnder interferometer ....................................................................... 23
3.5 Grating monochromator ....................................................................................... 25
3.6 Fluorescence microscopy ..................................................................................... 27
3.7 Holography, Spatial light modulator .................................................................... 29
3.8 The Helium-Neon laser ........................................................................................ 37
4 Techniques and instruments 48
4.1 Examples of construction with micro-optic bench 48
4.2 Lightsources 50
4.3 Photodiode 53
4.4 Spectral analysis devices 54
4.5 Amplification of small signals (phase-sensitive detection) 57
10/24/2012
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1 Introduction. This manual provides information about the experimental optical course (PART II) code 140702. This
course is a continuation from the course “Inleiding Optica” 146012 given during the first quarter of B2.
Rather than several small experiments, students perform a single experiment during 5 afternoons and
give a presentation and demonstration of the achieved results during the sixth and last afternoon. In
contrast to the optics experiments in the first quarter, this practicum gives much freedom to the
students. The instructions to the experiments are only a guidance. Setting up an experimental plan,
carrying out experiments, and interpreting the results is the main focus of this practicum course.
1.1 Content
Each team of two students carry out one assignment and can choose out of the following experiments
(each experiment can be chosen by only one team ):
1. Coherence
2. Fourier & Schlieren optics
3. The Michelson interferometer
4. The Mach-Zehnder interferometer
5. Grating monochromator
6. Fluorescence microscopy
7. Holography
8. The HeNe laser
For a description of the experiments see chapter 3 of this manual. During day 1-5 the students carry out
the assignment, during the last afternoon each team gives a presentation and demonstration of their
results.
1.2 Reporting and evaluation
The experiments are performed by teams of two students. Each team does one of the 8 assignments.
Students report on their experimental results in the following way:
During the assignments students are expected to use a laboratory notebook (“journaal”). The
notebook is checked on a daily basis by the teacher.
A report is delivered describing the experimental results.
A powerpoint presentation (15 minutes) and a demonstration of the experiment is scheduled
on the last (6th
) afternoon.
Important aspects of the evaluation of the assignments are:
Experimental skills (preparation, planning, knowledge of the required theory and the
application, “journalisering”, experimental results) (40%)
Report (clarity, structure, conclusions) (35%)
Presentation skills (25%)
1.3 Practical Information
* PART II is given in the second quarter.
* The experiments are performed during 6 afternoons (from 13:15 to 17:00 hrs)
(see schedule TN B2 for details).
* Place: building Carré CR 4620. See also: http://www.utwente.nl/tnw/onderwijs/
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* In the practicum room it is NOT allowed to eat or drink.
* Jackets can be put outside the practicum room
* Keep the alleys in the practicum room empty (put bags etc. under the tables)
* IMPORTANT: It is required to subscribe to the experimental course.
(see website: http://www.utwente.nl/tnw/onderwijs).
1.4 Computer and Software
During the practicum you will use software to control and/or record data from the experiment. To this
end you will use your OWN LAPTOP. Required software can be downloaded from the website of the
notebook service center (http://www.nsc.utwente.nl).
For DAY 1 you require the software: NI Scope
1) Log in to the website of the Notebook Service Center with your student number and ITBE password
https://www.nsc.utwente.nl/index.php?p=login
2) Click on “Software manager”
Educational UT Software (29)
Opleiding gerelateerde software
voor studenten
Bekijk software...
3) Click on “Bekijk software…”
4) Go to Kwartiel 1 (TN B2 146012). Doubleclick “NI scope”
5) Unpack the Zipfile and install. (NOTE: This software requires the Labview runtime engine. If you
have not already installed this software earlier please do so. You find it on the same page Kwartiel 1
(Algemeen)).
From Day 3 to Day 6 you need software to read images from a Lucam CCD camera. For the Lucam
camera you require the LuCam software:
Repeat steps 1 - 3 above
3) Double click on “Lumenera LuCam Install”
4) Unpack Zipfile and install
IMPORTANT NOTES ON THE LUCAM SOFTWARE:
If you plug in the USB Lucam camera in your LAPTOP the computer tries to install proper drivers. For
this you may require administrator rights. Make sure you have this.
Make sure you plug in the USB camera always in the SAME usb port.
By default the Lucam software looks at the internal (if present) LAPTOP webcam. If this is the case
you have to disable this camera in the “device manager”.
7
2 Safety Regulations
USING LASERS IN THE OPTICS PRACTICAL COURSE
The lasers used in this practical course are Helium-Neon lasers (wavelength: 632 nm)
and laser diodes (wavelength ~ 670 nm) with a maximum power of 5 mW.
To use these lasers in a safe and responsible way the following regulations must be
taken into account:
1. NEVER look into the laser beam directly and BE CAREFUL with direct or
diffuse reflections (chains, watches, screw drivers, pens, paper).
2. Only use lasers that are rigidly connected to a ground plate.
3. Point the laser beam towards the wall when you‟re building your laser set-up.
4. Keep the path of the laser beam well below working height.
5. Use screens between your set-up and those of your fellow students to prevent your laser
beam to cause inconveniences for others.
6. When aligning the set-up, use the lowest possible laser power. The power of HeNe lasers
can be reduced with grey-filters.
7. Before turning on the laser: check the surroundings of the set-up for reflecting objects that
don‟t belong there.
8. Never leave the laser on when you leave it: close the shutter or turn the laser off.
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3 Description of experiments
Eight different experiments are available. Each team of two students chooses one
experiment. Below you find a short description of the available experiments. A more
detailed description is found in section 3.1-3.8.
1. Coherence (chapter 3.1)
In this experiment you will use a Young‟s interferometer to study the
coherence properties (both spatial and temporal) of various light sources: a
HeNe laser, diode laser and a Halogen lamp.
2. Fourier optics (chapter 3.2)
In this experiment you will explore aspects of optical filtering and the
Schlieren setup. The latter will be used to visualize e.g. a stream of CO2 gas
3. The Michelson interferometer (chapter 3.3)
In this experiment the coherence length of several light sources are measured.
In addition the refractive index of a glass slide is determined and the splitting
of the Na D lines are measured.
4. The Mach-Zehnder interferometer (chapter 3.4)
The interferometer is used to measure refractive indices of gasses and glass
plates. Also the temperature dependence of the refractive index of water is
determined.
5. The Monochromator (chapter 3.5)
A grating monochromator is constructed and characterized. The
monochromator is used to measure emission spectra of mercury and sodium.
6. Fluorescence microscopy (chapter 3.6)
A microscope is constructed and characterized. Dichroic mirrors are explored
and fluorescent particles in solution are imaged.
7. Holography, The Spatial light modulator (chapter 3.7)
In this experiment a diffractive optical element is used to modify the
wavefront of a lightbeam. Holography is one of the applications studied.
8. The Helium Neon laser (chapter 3.8)
An Helium-Neon laser is constructed and aligned. Important aspects of lasers,
such as mode profiles and laser stability, are explored.
10
3.1 Coherence.
Aspects:
Young's interferometer, fringes, temporal and spatial coherence, visibility, degree of
coherence.
Theory to be studied before the practical session:
Study the theory described in Pedrotti chapter 9 and/or Hecht, 4th
edition, „Optics‟:
chapter 12, (minimal preparation: 12.1 - 12.3), chapter 13.1.1 – 13.1.3, recall chapter
9 (interference). Make yourself familiar with the spectral properties of the light
sources available for this experiment. Which observations do you expect?
Propose a number of experiments to the assistant at the first day of the practicum.
Experiments:
Study the coherence properties of several light sources: a HeNe laser, a diode laser,
which can be operated below and above threshold, and a Halogen lamp. Compare the
experimental results with the theory.
Material:
HeNe laser.
Diode laser. The output can be adjusted with a modulation voltage (0 - 5V).
Halogen lamp + 12V transformer + variac.
Band pass filter to be used with halogen lamp.
Adjustable slit (Very delicate device, treat very careful! ).
Microscope to measure the hole diameters, distance between the holes and the slit-
widths.
Screen with two holes.
Several positive lenses.
Thick + thin glass plates. These plates can be placed in front of one or both the
holes.
Optical rail and micro-optic bench with mounting equipment.
CCD camera + computer.
Micro-optic bench system.
MathCAD, Matlab and MS Word for the evaluation.
11
3.1.1 Young's experiment (as introduction) and temporal coherence.
A. Setup with HeNe laser:
1. Use the micro-optic bench system.
2. Adjust f1 and f2 for a plane wave falling on the screen with the two holes. f1 - f2 at
210 mm distance.
3. Adjust f3 for a focus on the CCD camera (remove the screen with the two holes)
4. Use the polarizer to attenuate the beam.
5. Place the screen with the two holes and eventually fine-adjust f1 or f2 for nice
Young's fringes.
6. Calibrate the CCD camera with a mm transparency.
7. Measure the distance between the fringes and compare it with theory: a
fY
3
.
8. Place the glass plates before one hole. Effects: a very little displacement of the
fringe pattern can be observed because of the extra phase shift due to the glass, no
effect on fringe visibility because of the long coherence length of the HeNe laser.
9. Place a non-transparent screen before one of the holes. Effect: fringes disappear,
one-hole Fraunhofer diffraction pattern. Measure the diameter of the first dark
ring and compare with theory: D
fq
3
122.1 .
CCD camera HeNe laser
Polarizer
f1 = 10 mm f2 = 200 mm screen with two
holes, D=0.3 mm
at a=1.0 mm
f3 = 300 mm
glass plate: 1.5 mm or
0.1 mm (cover plate)
12
B. Setup with diode laser:
1. The setup with the diode laser is almost similar to the setup with the HeNe laser.
Use a stronger lens for f2 instead.
2. The diode laser beam diverges more, so that f1 -f2 must be a little larger than 50
mm for a plane wave (about 55 mm).
3. Place the thin (0.1 mm) glass before one hole. Effect: the visibility reduces
because of the shorter coherence length of the diode laser The effect is hard to
see!.
4. Place the thin plate before both holes and the visibility increases to old value.
5. Place the thick glass (1.5 mm) before one hole. Effect: Visible decreases more,
Fraunhofer diffraction pattern of round hole. Thick glass before both holes,
fringes visible again.
6. Operate the laser above threshold (modulation connection = open circuit) and
below circuit (modulation connection = short circuit). Effect: Below threshold:
fringes disappear completely with 1.5 mm glass plate . Above threshold the
visibility increases. With the thin glass the same effect can be observed but less
dramatic.
diode laser
Polarizer f1 = 10 mm
f2 = 40 mm screen with two
holes, D=0.3 mm
at a=1.0 mm
f3 = 300 mm
glass plate: 1.5 mm or
0.1 mm (cover plate)
CCD camera
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3.1.2 Spatial coherence.
Setup with halogen lamp:
1. Use the optical rail system.
2. Place the halogen lamp (12V) just before the adjustable slit. Use the 12V
transformer and the variac to adjust the lamp intensity. (Be aware not to put the
banana plugs directly into the 220V mains, it happens!!!)
3. Place a lens, f=150 mm, at about 25 cm from the slit. Use an achromatic lens to
reduce spherical and chromatic aberration.
4. Place the CCD camera at about 37.5 cm behind the lens for a sharp image of the
slit.
5. Place the screen with the two holes just before the lens, and observe a diffraction
pattern with a narrow enough slit. The imaginary line between the two holes must
be perpendicular to the slit, otherwise no pattern will be visible.
6. Increase the slit-width to observe the change of the visibility. Notice the shift of
the fringe pattern when the slit width a
lb
.
7. Measure the visibility as a function of the slit width and compare with theory:
b
l
a
II
IIbV
sinc)(
minmax
minmax . This can be done using the CCD camera and
Mathcad or Matlab. You could average in the vertical direction to reduce the noise
and you can record the background intensity by taking a picture with the slit
blocked (with your finger). Subtract the background from the measured fringe
pattern. This must be done each time you change the lamp intensity! As an
alternative you could scan with a photo diode with a pinhole (100m) in front of
it. Use phase-sensitive detection if you do this.
8. Study the effect of a band-pass filter on the fringe pattern. Use an interference
filter (yellow or green). Effect: with narrow slit width you should see more than
three fringes.
adjustable slit
halogen lamp
screen with two
holes, D=0.3 mm at
a=1.0 mm
f = 150 mm
l = 250 mm s = 375 mm
CCD
camera
b
filter
14
3.2 Fourier Optics
Aspects:
Optical Fourier transformer, spatial filtering, Schlieren method.
Theory to be studied before the practical session:
Study the theory described in Pedrotti chapter 21 and/or Hecht, „Optics‟, 4th
edition:
chapter 13, (minimal preparation: 13.2.1 - 13.2.5, 13.2.4 briefly). Propose a number of
experiments about spatial filtering and the Schlieren method to the assistant at the first
day of the practicum.
Experiments:
1) Study optical filtering with a HeNe laser as the light source and a number of
transparencies. Make a clean laser beam with an optical filter: (Use a
microscope objective and a pin hole for this). Use the micro-optic bench
system for the set-up.
2) Demonstrate the Schlieren method for the observation of a phase plate and for
observing CO2 gas flowing out of a tube. Use a Halogen lamp – collimator -
slit combination as the light source. Build the Schlieren set-up with the optical
rail system. Optimize the set-up for maximum contrast.
Material:
HeNe laser.
Polarizer as attenuator.
Microscope objective (20x/0,35) (See: Pedrotti 3.6) and 50m pin hole for beam
expansion and beam clean-up.
Several positive lenses.
Transparency holder.
Two adjustable knife-edges for optical filtering experiments.
Halogen lamp + 12V transformer + variac.
Adjustable slit.
Several positive lenses.
Adjustable knife-edge.
Optical rail and micro-optic bench with mounting equipment.
CCD camera + computer.
MathCAD, Matlab and MS Word for the evaluation and presentation of the
results.
15
3.2.1 Optical filtering.
Set up:
1. Use the optical rail system
2. Align the HeNe laser at the proper height and in the good direction using 2
diaphragms.
3. Use the microscope objective (20x/0,35) and f1 as a beam expander. The
microscope objective must be mounted in a translation stage for getting the focus in
the pin-hole. The position of the pin-hole is just after the objective. Position can be
done by hand (no micro transducer needed)
4. Verify that the beam from f1 is still aligned with the breadboard. Slightly adjust the
HeNe and the objective when necessary.
5. Use several transparencies to demonstrate Fourier optics. The grid-transparency is
very illustrative to show the disappearance of the vertical (or horizontal) lines. Use
the two adjustable razor blades as the spatial filter. As an alternative the student can
make a transparency where the spatial filter is drawn on it with a black pen:
16
3.2.2 Schlieren method.
Set up:
1. Use the optical rail system to build the Schlieren set-up. 2. Use a large collimator lens in front of the halogen lamp. And focus a little unsharp
on the adjustable slit. 3. Use two large achromats for the f=200 mm lenses. 4. The phase object is a transparency with a few characters. The characters are not
visible as it should be for a phase object. 5. Place the slit in the focus and screen the central spot. 6. The f=50 mm lens must be adjusted such that a sharp image of the phase plate is
visible on the CCD camera. The image of the phase plate is made with the second
f=200 mm lens as well as with the f=50 mm lens. 7. Optimize the positions of the lenses and the slit for maximum contrast and a sharp
image. 8. Replace the transparency with a flow of CO
2 gas and observe the distribution of
the flow.
17
3.3 Michelson Interferometer
Aspects:
Interference, coherence length
Theory to be studied before the practical session:
Pedrotti chapter 8.1, 8.2 and 9.3 and/or Hecht, Optics, 4th
edition, chapter 7.4.3 and
9.4.2, recall chapter 9; see also below in this document.
Experiments
In this experiment the coherence length of several light sources are measured. In
addition the refractive index of a glass slide is determined and the splitting of the Na
D lines are measured.
Equipment:
Laser-diode
Several LEDs
Detector: photodiode BPW34
Screen
Michelson interferometer + piezo-crystal
Halogen lamp (via fiber)
Glass plates + rotation unit
Band-filters (for Hg green line at 546nm)
Osciloscope
NI interface box
Colour glass filters
Na and Hg lamps
18
Introduction (set-up and coherence length)
Fig. 1 Michelson interferometer
On scanning the path-length of one of the branches of a Michelson interferometer one
can obtain information about the coherence length of a light source (see above
sections in Hecht). An example of a few experiments is given in figure 2.
B
A
Low coherence source
Laser
0
O ptica l path length d ifference (L1-L
2)
Lcoh
50%
Figure 2. Interference patterns as a typical output of Michelson’s interferometer
for two different light sources.
Spiegel (S1)
Spiegel (S2)
(piezo)
Lichtbron (L)
Beamsplitter (BS)
Detector/Scherm (D)
L1
L2
1
M1
M2
D
L
19
Fine adjustment (0 – 0.1 mm)
Coarse adjustment(0 – 12 mm)
Assignments
Warning: be careful with the piezo-crystal (both in treatment of it and in applying
voltages); never touch the piezo-crystal or the mirrors.
Alignment of the interferometer
Use the laser diode as a source, and turn it on using the adaptor. The rest of
the equipment is off.
Remove the photo-diode from the equipment and put a white screen far
behind it.
Use for the alignment only the screws of mirror 1, and arrange things such
that the two beams have a good spatial overlap in the output section: an
interference pattern becomes visible on the screen.
Improve the alignment such that the fringes become as wide as possible.
Use of the set-up
In case of using a source with a short
coherence length on should try to find the
position corresponding to nearly equal path
lengths for both branches. For this the mm-
adjustment (see figure) has to be used; the
screen is placed on the position of the photo-
diode. (large outside knob for fine
adjustment: 1 Div = 0.5 m, 1 turn = 25 Div
= 12.5m; Coarse adjustment: 1 Div = 0.01
mm, 1 turn = 50 Div = 0.5 mm)
After adjustment the screen is removed, and the photo-diode is put in place. Connect
it to the power supply and connect the output to both an oscilloscope and the DIG-57
(data retrieval).
For an accurate scan (maximum range 100m) one uses the piezo-crystal as follows.
1. Check the DC-offset on the power supply, it should be zero: the button on the
piezo-controller should be turned fully anti clock wise.
2. The swich „servo‟ should be „ON‟.
3. Switch the piezo controller on; switch on the backside (ask the supervisor for a
check!)
4. Start the piezo-controller program:
Start programma‟s PI PI PZT control
The following screen can be seen:
20
Check the entries, PC interface should be “NI IEEE 488 Interface SetUp” with Board
ID „0‟ en Device Address „14‟.
Klick „OK‟
Next, start the „waveeditor‟ by klicking the „waveeditor‟ button . This facility is to
adjust the modulation profile for the piezo-crystal. The following screen should
appear.
Check if the entries are approximately as in the above picture.
If so, klick “SEND” and thereafter “START”. The piezo-crystal causes a periodic
movement of the mirror. Look on the scope and optimize the pattern with the fine-
adjustment of mirror 2.
The result can be measured via the pc with the NI interface box and the NIscope
software.
21
Note that the program for the piezo crystal should not be stopped before the piezo
movement is stopped (do this when it is at a position of ~0micrometer, use the stop
button on the PC screen next to the „wave-editor‟ button).
Also note that the movement of the mirror can be monitored via the connection
„sensor motion‟ on the power supply for the piezo-crystal. The voltage can be
measured via another channel of the NI interface box.
Experiments (to be performed in agreement with the supervisor)
Measure the coherence length of a number of LEDs
Calibrate the equipment using the Hg green line at 546nm; use the band filter.
Measure the coherence length of white light, i. e. the light of a halogen lamp
without filter (should be about 1.5 m), and with different coloured glass
filters and interference filters. The spectral bandwidth of those filters can be
determined from that.
Place the small plate of glass (thickness 145 15 m ) in one of the arms, such
that half of the light goes through it, and the other half not. Determine the
glass refractive index.
Place the glass plate in one of the arms (all the light goes through it) and rotate
the glass plate. Measure the effective path length of the glass. Compare the
result with theoretical calculations (see Appendix).
Measure the intensity at the center of the interference pattern as a function of
the mirror position. Determine the wavelength for a number of spectral lines.
If required: perform a Fourier analysis on the recorded spectra.
Determine experimentally the splitting of the Na D lines (average wavelength
of 589nm); perform the scan of the mirror position manually.
Propose one or more experiments for the equipment; discuss these with the
supervisor.
22
Appendix. Effect of a glass plate on the optical path length
We consider a parallel beam of light impinging on a glass plate in air (index 1) under
a certain angle, say .
A
B
C
d
D
index n
gF
G
We will consider the ray that apparently travels through A and C (see figure). The
phase difference between light in points A and B (see figure) corresponds to an
optical path length of cosnd . The optical path length DB is given by
sin sin tanDB BC d . The phase in points D and C are the same (same phase
front). So the optical path length difference along AC as introduced by the glass plate
is given by:
2cos sin / cos / cos ( cos cos )
g gOPL nd d d d n .
Where we have used that 2 21 sin cos .
An alternative derivation of the above can be done by considering refraction of the ray
at A, and at the bottom glass-air interface, at F. Then, in a similar derivation as above
the same result for the optical path length difference is obtained. The optical path
length difference then follows from:
/ cos sin / cosg
O PL nAF FG AC nd C F d
Using that (tan tan )g
C F d and Snell‟s law sin sing
n it follows:
2 sin sinsin
( cos cos )cos cos cos cos
g
g
g g
dnd d dO PL d n
Note that in a Michelson set up the beam travels twice through such a glass plate, and
transverse displacements are compensated.
23
3.4 The Mach-Zehnder interferometer. Aspects:
Fringe patterns, Interference, Coherence length.
Preparation:
Pedrotti chapter 8.3 and/or Hecht: 9.4.2;9.5
Set up:
Required parts/equipment
2 adjustable 45o beam steering mirrors in holders.
2 beam splitter cubes in holders.
Lenses
HeNe laser.
Polariser.
CCD camera + computer.
Breadboard, mechanical parts S&H‟s microbench
Transparent tube for gasses, CO2 or He source
Vessel for liquids/piece of Perspex
thermo couple
object
Beam
splitter
Beam
splitter
Screen
or CCD
camera
24
Experiments.
1. Build the Mach-Zehnder interferometer using the components provided. Use a
beam expander to make a plane wave with large diameter from the HeNe laser
output beam. Use the polarizer to attenuate the HeNe laser beam. Consider both
linear and circular fringe patterns. Build the MZI in such a way that the glass
vessel can be put in one arm.
2. Put a tube (filled with air) in one of the two branches. Fill the tube with some
other gas (CO2 or He) and determine the index difference between the gas and air.
3. Heat a transparent vessel filled with water (or a piece of Perspex) and study the
effect of (inhomogeneous) heating of it, using an electric hair dryer. Determine the
thermo-optic coefficient of water (dn/dT), and compare the obtained value with
that of literature.
4. Determine the index of a rotatable glass plate.
5. Rebuild the MZI such that it allows to measure small vibrations/displacements of
a thin glass slide that contains a small mirror (ask assistant). Use the setup to
record sound picked up by the glass slide.
Alignment of the MZI
The students should first make a rough sketch of the set-up, taking into account the
length of the glass vessel. A well aligned MZI is not easy for most students, in
particular if they start to try all kind of things without having a strategy. A working
strategy is the following:
position the laser and the metal beams for the complete set up, also place the beam
splitters, polarizer and mirrors
align the laser such that the laser beam is horizontal and well positioned above the
iron beams of the first branch (=top branch of figure above), use 2 diaphragms for
this.
Adjust the 2 mirrors such that the laser beam is well positioned for the rest of the
set-up (use diaphragms), and directed to the CCD camera.
Normally the above does not lead to a nice interference pattern because the beams
are not nearly parallel. So, remove the camera and shift the upper mirror (for
example) and use the adjustments screws such that the 2 beams coincide both at
the second beamsplitter and a few meters further away.
Place the beam expander, the CCD camera and do some fine tuning while
watching the interference pattern.
Comments to the experiments
Ad 2. Replacing air by CO2 leads for the present glass vessel (~15cm long) to a shift
of 20 fringes.
Ad 3. This experiment is a bit nasty as there will often be inhomogeneous heating
(thermo-couple required)
25
3.5 The grating monochromator
Aspects:
Resolving power, measurement of spectra.
Theory to be studied before the practical session:
Pedrotti chapter 12 and/or Hecht, 4th
edition 10.2.8.
Required parts/equipment
Optical rail
Various spectral lamps
Lenses
Various gratings
Rotation table
Adjustable slit
CCD camera /PC
Hg, Cd, Na, He spectral lamps.
Halogen lamp
Building of a grating monochromator
Set-up:
Experiments
1. Build a grating monochromator
2. Calibrate the relation between wavelength and position on the CCD camera;
also take into account the zero-order diffraction. Use the Cd and Hg lines.
3. Investigate the relation between spectral linewidth and slit width.
Spectral
lamp
slit grating lens CCD
lens
26
4. Investigate, for a small slit width, the effect of covering part of the grating
on the spectral line width.
5. Repeat experiment 2 with a different grating.
6. Resolve the yellow mercury lines and the yellow sodium lines. Study the
influence of the slit-width.
7. Determine from the measurements the distance between the two yellow
sodium lines with given mercury lines (see figure). Compare the result with
the literature value (0.597 nm)
8. Measure the transmission spectrum of an interference filter using a „white
light‟ source (halogen lamp).
27
3.6 Fluorescence Microscopy
Aspects:
Microscope design, fluorescence excitation and detection, dichroic filters
Theory to be studied:
Pedrotti 3.6; 11.4; 7.4 and/or Hecht chapter 5.7.5 The compound microscope. See the
appendix for more information on fluorescence microscopy.
Equipment:
Standard “Linos” components
Objective lenses
Lenses with different focal distances
CCD camera
Xyz-translation stage
Ruler (10 m) on glass slide
Different samples with fluorescent beads in water
Filtercubes
Power LED‟s
Various samples
Introduction:
Fluorescence microscopy is a widely used technique to study the spatial distributions
of specific molecules/structures. A major application is in the field of cell biology.
The molecules that make up a living cell show no contrast under a normal white light
microscope. But labeling specific molecules (e.g. DNA) inside a cell allows to
measure the distribution of these molecules in the cells because they can now be
distinguished from the non-labeled molecules by means of a fluorescence microscope.
In this experiment you will build a fluorescence microscope and will use it to observe
various fluorescent samples.
The experiments take approximately 5 afternoons.
IT IS IMPORTANT to carefully read the theory about microscopy and florescence
(see above)
Experiments:
A The microscope
1. (Re)design a microscope consisting of 1 objective lens and one or more
normal lenses that gives an image on the computer screen with a
magnification of 5000 times. Have the design checked by the supervisor
before to proceed.
2. Realize the designed microscope. Determine the magnification of the
microscope. Calibrate the microscope in terms of nanometers/pixel.
28
3. Make images of polystyrene beads with different diameters (see samples).
Why do the beads move around in the solution?
4. Determine the diameters of the polystyrene beads.
5. Make some images of red blood cells. Determine the average diameter.
6. What details can still be observed with the microscope. Explain why the
resolution is limited.
B Filter characterization
Several filters (interference filters) and dichroic mirrors are available. Here you will
characterize these filters.
1. Design a setup that allows to measure the filter (both transmission and
reflection) characteristics (using white light source and fiberspectrometer).
2. Measure the transmission spectra of the available filters. (in other words
measure T as function of wavelength, where T is the transmission coefficient)
3. Measure the reflectioncoefficient R and transmission coefficient T as function
of the wavelength of the dichroic mirror (at an angle of 45 degrees)
4. For one filter measure the transmission spectrum at 0 degrees and at a small
angle. Explain the (difference between) observed spectra.
C Fluorescence microscopy
1. Use the fluorescence filter cube and the power led to rebuild the microscope
into an epi-fluorescence microscope..
2. Test the fluorescence microscope by making images of fluorescent polystyrene
beads.
3. Optimize the fluorescence intensity as much as possible.
4. Use different beads, filtercubes and light sources to image beads and compare
the observed images.
29
3.7 Experiments with the HoloEye LCD spatial light modulator
Aspects:
Spatial phase distribution, axicon, lenses.
Theory to be studied before the practical session:
Pedrotti, Introduction to Optics, 3rd
edition, chapter 17-5: especially Liquid-Crystal
Displays(LCD)
Equipment:
• Optical rail
• HeNe laser
• Several lenses from the lensboxes in the lab
• HoloEye LC2002 spatial light modulator + power supply
• Extra VGA monitor
• VGA splitter + power supply
• 2 ”thick” VGA cables to connect the extra monitor and the student‟s
laptop to the splitter
• 1 “thin” VGA cable to connect the HoloEye LC2002 to the splitter
• White screen + alignment screen with hole.
• Ruler + caliper
• Several Matlab programs
• LightPipes for Matlab (practicum edition) optical toolbox
Introduction
The HoloEye.
HoloEye model LC2002 spatial light modulator
The HoloEye (http://www.holoeye.com/spatial_light_modulator_lc_2002.html) LC
2002 Spatial Light Modulator (SLM) contains a Sony SVGA (800x600 pixels) liquid
crystal microdisplay and drive electronics. It can be used with a standard personal
computer or laptop by plugging the device to a VGA graphics connector. The phase
distribution of a beam input to the SLM can be modulated by a voltage on the
electrodes (800x600 pixels) of the LCD. The phase modulation is a side-effect of the
30
liquid crystal material and is caused by its birefringence property. It yields a more or
less linear phase shift with voltage of maximum 2 radians at 532 nm. This phase
shift is used to introduce a phase modulation of a coherent input beam by setting a
voltage on each pixel of the modulator. In this exercise three different phase
distributions are used leading to a ring-shape intensity distribution, a diffractive
optical element (DOE) acting as a positive lens and a phase hologram respectively.
These phase distributions are calculated with Matlab programs provided. Also,
simulations can be performed using LightPipes for Matlab (Practicum version).
Diffractive Optics.
Diffractive Optical Elements (DOE‟s) differ from the classical optical elements like
lenses and mirrors because they are based on their diffractive properties rather than
reflection and refraction. When dealing with the traditional optical elements,
diffraction phenomena are considered as undesirable features, which influences the
performance of an optical system build with these elements and hence should be
minimized. DOE‟s on the other hand, make use of diffraction to manipulate the
waveform of an incoming beam of light. Because of the nature of diffraction mostly
highly monochromatic and coherent light is used with DOE‟s. The first application of
wavefront manipulation was holography. An interference pattern of two light beams, a
reference beam and a beam coming from an object, is recorded on a photographic
plate after which the recorded information of the object in three dimensions can be
extracted from the hologram by illumination with a laser beam. Holography inspired
people to wavefront processing in which the surface of a substrate was processed to
change an input wavefront into another form. In principle lenses and other classical
optical elements are all wavefront processors, but their functionality is limited to
relative simple actions. DOE‟s on the other hand allow for more complex wavefront
manipulations and resulted in modern optical applications as „holographic head up
displays‟ in fighter jets and recently in automobiles.
Nowadays the complex recording techniques using holography are more and more
replaced by direct writing micro-structures on substrates using computer-calculated
diffraction patterns, or as we will apply in this practicum: using electronically
addressed spatial light modulators (SLM).
3.7.1 Initial instructions:
1. Complex amplitude. A coherent beam of light can be described by its complex amplitude
distribution: )),(exp(),(),(0
~
yxiyxEyxE . E0(x,y) and x,y) are the
amplitude and phase as a function of the transverse coordinates. The amplitude
can be determined easily with screens or CCD cameras by taking the square
root of the intensity distribution. The phase distribution is much more difficult
to measure. The phase, however, plays an important role and can never be
neglected in optics. A lens, for example, will change the phase distribution but
leaves the amplitude distribution unaltered.
2. Phase distribution of a lens. The phase distribution of a coherent monochromatic light beam after passing a
lens can be calculated by considering the optical paths of rays through the
lens.
31
The phase can be found by calculating the optical path of a ray parallel to and
at r above the optical axis. It is (glass + air):
)()1()()()(000000
rnkkrkrnkr , with:
22
2202
22
1101)()( rRRrRRr , and
0
0
2
k ,
the propagation constant in vacuum. This can be approximated (by expanding
the square roots) to: f
rk
r
RRnkr
22
11)1()(
2
0
2
21
0
, with
21
1 11)1(
RRnf , the „lens-makers formula‟ for the focal length of a
thin lens. Note the sign conventions for the lens radii. (The constant phase,
00nk is not relevant and can be neglected.) This formula is used in the
Matlab programs and is executed by the LightPipes for Matlab
„F=LPLens(f,0,0,F)‟ command, which inserts a lens at the optical axis in the
field (complex amplitude) F .
The ray-angle with the optical axis is: f
rr )( . So, as expected, all rays
point to a single point on the optical axis, the focal point of the lens. The
approximation made holds for rays close to the optical axis (small angles) and
is called the „paraxial approximation‟. If the approximation is not valid, we are
dealing with (spherical) aberration and the rays to far from the axis will not
pass through the paraxial focal point.
3. Phase distribution of an axicon. An axicon is a conical optical element as shown in the figure below.
r
R2<0
01 02
R1>0
0
Focal point
32
To find the phase distribution and the ray-angles, I, after passage through the
axicon, a similar calculation as with the lens can be performed. Calculate the
phase distribution (paraxial approximation) and show that the ray-angles are
given by: 1)( nr , with 2
90
, the sharp angle of the axicon
and n the refractive index of the axicon material.
4. Connection of the HoloEye spatial light modulator (SLM) to your laptop. The HoloEye SLM can be operated by your laptop by simply connecting it to
the extra VGA port on your laptop (normally used for beamers). The set-up
you use is shown in figure 1.
Fig. 1. Set-up of the experiments with the SLM.
You have to set your laptop in „dual monitor‟ mode. This is probably done by
pressing the „F8‟ key a few times. Furthermore the second monitor must be set
to 800 x 600 pixels with 256 bit color depth. This can be done by running your
graphics driver software on your PC. Probably it can be accessed by left-
screen f=400mm f=300mm
f=10mm
VGA cable laptop
HoloEye LC 2002
spatial light modulator
HeNe laser
r
r
n
33
clicking on your desktop. The voltage on the LCD pixels is proportional to the
gray levels (0-255) of the image displayed on the HoloEye „second monitor‟.
Once the HoloEye has been connected to your laptop you should plug-in the
power supply cable of the HoloEye. Run the Matlab „Axicon.m‟ program to
test the SLM. A ring should appear in the focus of the 400 mm lens placed
behind the SLM. Sometimes it is necessary to un-plug the power cable and to
plug it in again to reset the SLM.
3.7.2 Assignment 1. Determination of the pixel spacing of the LC2002.
Build the setup shown in figure 1.
1. Start with the HeNe laser. It must be aligned parallel to the optical rail.
2. Place the 10mm lens and adjust its position such that the expanded beam does
not deflect.
3. Place the 300mm lens and verify that the expanded beam does not deflect and
is not diverging or converging. Check the distance between the 300 and 10
mm lenses. What distance do you expect?
4. Place the HoloEye modulator such that the beam fills the whole aperture of the
device.
5. Place the 400mm lens and place a white or mm screen in its focal plane.
6. You should observe an array of bright spots.
What is the cause of the array of bright spots? Determine the spacing of the pixels of
the SLM.
3.7.3 Assignment 2. Holographic Optical Element (HOE).
In this assignment we are going to make a holographic optical element which has the
same function as that of a positive lens. This hologram can be made in practice by
letting coherent light originating from a point source and that of a plane wave to
interfere as shown in figure 2. Of course the initial phases of the two beams must be
stationary, which is best organized by originating them somehow from the same
source.
Fig. 2. Recording of a hologram acting as a HOE lens.
Once the interference pattern (intensity distribution) on the photographic plate has
been transformed into a phase pattern (this can be done by etching the surface of a
Plane wave
spherical
wave
Photographic plate
34
glass plate, i.e. transforming the gray levels of the intensity pattern into hills and
valleys in the glass, causing fluctuations in the local optical path lengths and hence in
phase fluctuations of an incoming beam of light).
As an alternative, the phase distribution of the interfering beams can also be
programmed in the SLM. The hologram can be compared to a Fresnel zone phase
plate in (see Hecht 4th
ed., chapter 10.3.5) in which the rings are replaced by
smooth cosine functions. Figure 3 shows a cross section of such a phase plate.
3 2 1 0 1 2 30
1
2
3
4
radius [mm]
Ph
ase
[ra
dia
ns]
Fig. 3. Cross section of a cosine phase plate (red). The blue curve is a Fresnel zone
plate. (Here the plot must be considered as a set of concentric annular screens
blocking the odd Fresnel zones and passing the even zones or annular structures
producing a /2 phase jump from zone to zone.) f = 1 m, = 632.8 nm.
The phase as a function of the radius is given by:
1
22cos
2)(
2
f
rr
,
where f is the desired focal length of the HOE lens.
1. Use the set-up shown in figure 1 but with the f=400mm lens removed.
2. Use the Matlab „HOElens.m‟ program to verify the theory.
3. If you like, play with the LightPipes simulation.
4. A disadvantage of the holographic lens is that it has low efficiency because
there is not only a converging beam, but also a diverging and a plane wave
emerging from the holographic lens. Modify the Matlab program such that the
phase distribution of a thin lens is transferred to the SLM.
3.7.4 Assignment 3. Hologram of a 2D picture.
Here you learn how to make a computer generated 2D hologram using the so-called
Gerchberg-Saxton algorithm. This an iterative process in which Fourier and inverse
Fourier transforms are taken after substituting the intensity pattern of the input wave
and the desired pattern at each iteration step keeping the phase distribution
unchanged.
35
Fig. 4. Gerchberg-Saxton algorithm to calculate a phase hologram. See the Matlab
‘Hologram.m’ document for details.
Try several examples included and see what happens when you increase the number
of iterations. If the HoloEye SLM is connected, you have to perform the 2-D Fourier
transform (line 58 in the program) optically to see the image on a screen. How can
you do that?
3.7.5 Assignment 4. Ring focus with an axicon and a positive lens.
A ring focus can be made using an axicon followed by a positive lens. In the focal
plane of the lens a ring shaped intensity distribution will appear. An axicon is a cone-
shaped optical element as depicted in figure 4. In this assignment we are
programming the phase distribution of an axicon and a positive lens in the SLM. This
will be done using (part) of the LightPipes optical toolbox „LightPipes for Matlab‟.
The Matlab program calculates the propagation of a plane monochromatic wave
through an axicon and a lens. The phase distribution is extracted from the complex
amplitude and displayed in a Matlab figure. This figure, with gray values from 0
(black) to 255 (white) is loaded into the SLM and a collimated beam will be
transformed into a ring at the focal plane of the lens.
1. Use the set-up build in assignment 1.
2. Remove the 400 mm lens because a lens will be programmed in the SLM.
3. Run the „Axicon.m‟ Matlab program.
Input wave (here plane wave:
amplitude = 1, phase = 0)
Take 2-D Fourier transform
Substitute desired intensity
distribution, leave phase
unchanged
Take 2-D Inverse Fourier
transform
Substitute input intensity
distribution, leave phase
unchanged
36
4. Measure for various values of the top-angle of the axicon and focal lengths of
the lens the diameter of the ring.
5. Derive an expression for the ring diameter for an axicon with top-angle, , and
refractive index, n, and a lens with focal length, f.
6. Compare the measurements with the calculations. (Use a graph to present the
results).
7. Place the 400 mm lens back in the set-up and remove the lens in the Matlab
program (just put a comment (%) in front of the command).
8. Repeat the measurements and compare again with the calculations.
Remarks:
1. If the top angle of the axicon is too small, the ring will be larger than the
spacing between the orders of the grating due to the pixels of the SLM and
overlap of rings will occur. Can you derive an expression of the minimum
allowed axicon top angle for our 600 x 600 / 19.2mm x 19.2mm grid?
2. You can also simulate the set-up and compare with the experiments using
LightPipes for Matlab. Un-comment the appropriate lines in the „Axicon.m‟
program. You must increase the focal length of the lens (f~1000mm) and the
top angle (~179 deg) in the program because of the limitation of the grid.
3. Besides a phase modulation, our SLM also changes the local polarization of
the beam. You can use a polarizer to enhance the contrast. Also rotate the laser
to change and optimize its polarization. Where must the polarizer be placed?
Can you give an explanation?
4. For those who are interested in modern optics: It is possible to simulate with
the „Axicon.m‟ program a so-called Bessel-beam. (See Hecht Optics 4th
edition, chapter 10.2.7. The axicon method is even „more elegant‟ than the
annular slit described in Hecht, because no beam power is wasted by the
annular slit). These Bessel beams do not diffract and have a constant beam
size during propagation over very long distances (several meters) in free
space. Simply remove the lens in the simulation by commenting the
appropriate command line (line 30). Now f is the distance where the Bessel
intensity profile has to be observed. Use large top angles (>179.7 deg) to get a
reasonable resolving power with our 600 x 600 grid. Try to observe the Bessel
beam experimentally as well by removing the 400 mm lens from the set-up.
37
3.8 Helium Neon Laser
Aspects:
Laser resonator, stability conditions, high order modes, wavelength tuning.
Theory to be studied before the practical session:
Pedrotti chapter 26 and/or Hecht, Optics, 4th
edition, chapter 13.1: especially 13.1.3,
chapter 8.
Equipment:
Helium Neon discharge unit with Brewster windows and power supply
Concave high reflecting mirrors:
radius of curvature: 500mm, 750mm, 1000mm, plane
Concave 98% reflecting outcoupling mirrors:
radius of curvature: 500mm, 750mm
Adjustable horizontal and vertical mounted wires for transverse mode
selection
Quartz plate for wavelength selection
HeNe laser for resonator alignment
Spectrometer for measuring the laser wavelength.
CCD camera to monitor the beam profile.
Power meter to monitor the laser power (not available at the moment).
Optical rail 2m length, mirror holders, alignment diaphragms
Matlab program to calculate resonator properties
Introduction (set-up)
Figure 1 Helium Neon laser set-up
In general, a LASER (Light Amplification by Stimulated Emission Radiation)
consists of a gain medium and two mirrors at each side of it. The mirrors are aligned
precisely such that an optical resonator similar to a Fabry Perot interferometer is
formed. The mirrors can be concave, plane or convex and one of the mirrors is made
High reflecting concave mirror M1
Alignment HeNe laser
98% reflecting concave mirror (outcoupler) M2
HeNe discharge tube with Brewster windows
Quartz plate
Thin wires
L
Power meter
CCD camera
Spectrometer
M1
L
38
partially transparent to allow light to propagate to the outside world as a well
collimated, coherent, almost monochromatic beam. The gain medium can amplify
light based on the principle of stimulated emission. Most, well designed, lasers emit
the radiation as a beam concentrated around the optical axis with a Gaussian intensity
profile, the fundamental TEM00 transversal mode. This mode is desired for most
applications using laser light. Under some conditions, however, a laser can be forced
to operate on higher order transverse modes, indicated by TEMm,n. One of the
conditions is that the gain medium has a sufficient large bore compared to the beam
diameter because these modes have a larger cross section than the fundamental mode,
TEM0,0.
Normally a laser will oscillate at the wavelength having the largest gain. By
introducing wavelength depended losses in the resonator, however, a laser can be
forced to oscillate on weaker lines.
With the HeNe laser discharge laser tube the student can study a number of
fundamental laser properties: resonator stability conditions, high-order transversal
modes and wavelength selection. By using mirror combinations with different radii of
curvature and varying the length of the resonator the stability criterion can be
investigated. Also by selecting mirrors and distance such that a very small beam size
is obtained inside the discharge tube, a condition can be obtained where multi-
transversal mode oscillation is possible (See Hecht p. 592). A particular mode can be
selected by adjusting a vertical and horizontal thin wire into the resonator.
Other wavelengths than the strong 632.8 nm transition can be forced to oscillate by
the insertion of a birefringent quartz plate in the resonator. The plate is positioned at
the Brewster angle to minimize its losses and then rotated to vary the optical length of
the resonator. In this way the frequency of a longitudinal mode (see Hecht p. 591) can
be tuned such that is coincides with the centre of a particular (weak) transition while
other (stronger) transitions are off-resonance.
The discharge tube has Brewster windows at both ends. Because the laser will
oscillate at resonator modes with the lowest losses the output beam is linear polarized.
Figure 2 HeNe discharge tube with Brewster windows
39
Assignments
Warnings: The laser will emit at 632 nm less than 5 mW (under best conditions
about 3.5 mW) which means that it is a class 3A laser. USE PROTECTIVE
EYEWEAR.
NEVER look directly into the beam even while wearing protecting glasses. Be
careful with reflections from mirrors and other reflecting objects. Arrange your
set-up such that all reflections are in the horizontal plane.
Be very careful with the mirrors and other optics. Do not touch them with your
fingers or un-mount them from the mirror holders.
Alignment of the laser
Verify that the Helium Neon discharge unit has been placed on the rail at
about 1 m from the side (left side) closest to the wall. Never remove the
discharge unit from the rail, only shifting on the rail is permitted. Leave the
discharge unit off for the moment.
Do not place the adjustable wires and quartz plate.
Place the alignment laser as far as possible at the opposite side (right side) of
the rail. The laser beam should be directed to the wall and NOT to the
laboratory room.
Adjust the alignment laser in vertical and horizontal direction as well as its
angle such that the beam goes through the discharge tube without hitting its
inner wall. Use the CCD camera.
Place one of the high reflecting mirrors (start with the R=750mm mirror) at
the left side of and close to the discharge tube, adjust it such that the laser
spot of the alignment laser is at the centre of the mirror and let the beam
reflect through the tube and coincide precisely with the beam leaving the
alignment laser. Placing a diaphragm in front of the alignment laser will
help.
Place one of the outcoupling mirrors (start with the R=750 mm outcoupler)
close to the right side of the discharge tube. Adjust this mirror such that the
beam reflected from the back side of this mirror goes exactly through the
diaphragm in front of the alignment laser. Switch-on the discharge unit. If
you are lucky the laser starts emitting immediately. If not, carefully adjust
the mirror with small steps in vertical and horizontal direction until the laser
starts. The laser oscillates if bright spots are visible on the left and right
mirrors. These spots are caused by scattering of the intense intra cavity
radiation at the mirror surfaces. Optimize the mirror alignment until these
spots have maximum brightness.
If you cannot get the laser to oscillate you can try the following: loose the
screw which locks the slider to the rail. Then while rotating the mirror
horizontally by rotating the slider, change the vertical mirror adjuster in
small steps until you see a red spot blinking on the mirror surfaces. Lock the
slider and vary the horizontal adjuster until the bright red spot appears.
Now you can change the length of the resonator by moving one or both
mirrors. Do this with small steps. Readjustment may be necessary,
depending on how good you did the initial alignment.
40
At optimum conditions, the laser will produce about 3.5mW output power.
This means that, using an outcoupler with 98% reflectivity, there is a power
of 3.5/(1-0.98)mW=175mW inside the resonator. If you look at a small
angle to the laser you can see this high power beam between the mirrors due
to scattering from small dust particles. Sometimes people perform
experiments inside the resonator to make use of the high power.
Experiments (to be performed in agreement with the supervisor)
Check the stability conditions (see appendix) of the laser for the given sets of
mirrors. Finish the table below and try a few mirror pairs (two) to test the
stability condition.
Use the Matlab program (appendix) to find out which resonator configuration
will produce the most output power in the Gaussian TEM00 mode. This will be
the case if there is optimum overlap of the gain medium (in our case about 1
mm diameter) with the mode. Also choose a resonator configuration that will
result in low output power. Build both resonators and compare them.
The laser can oscillate on a high order transversal mode if the mode fits in the
gain medium. A well-designed laser will only oscillate on the TEM00 mode.
This is the case if the diameter of the gain medium is just large enough for the
TEM00 mode. The higher order modes have larger sizes, will not fit and as a
consequence will have higher losses. Multi mode oscillation can be obtained
with our set-up by choosing a pair of mirrors and separate them at a distance
such that the TEM00 mode will have an as small as possible diameter (or beam
waist, see appendix) in the gain medium. In that case the high order modes
R1 [mm]
refl. ~100%
sticker
color
R2 [mm]
refl = 98%
sticker
color
stable if L [cm] is:
1000 blue 500 red 0 < L < 50 or 100 < L < 150
750 orange 500 red
500 yellow 500 red
∞ yellow-
white
500 red
1000 blue 750 green
750 orange 750 green
500 yellow 750 green
∞ yellow-
white
750 green
Figure 3 With adjustable, thin wires high-order modes can be
selected.
41
will experience sufficient gain to oscillate. If so, the spot size of the beam is
larger than normal and dark and, bright spots in the mode pattern are visible
because of interference. The mode pattern is very sensitive for the alignment
of the mirrors. Usually a few higher order modes seem to oscillate
simultaneously. In most cases the laser hops continuously from mode to mode
due to small variations in the alignment (due to vibrations, temperature
changes, etc). As you can see in Hecht, a high order mode has dark and bright
spots. A trick to select one particular high order transversal mode is to place a
thin obstacle in such a dark area. Try this by first building a resonator which
permits oscillation on multi transversal modes and select a high order mode by
adjusting a thin wire placed inside the resonator.
Wavelength tuning
The HeNe can, in principle, be tuned to the wavelengths: 640.1 nm(4,3), 635.2
nm(1,0), 632.8 nm(10,0), 629.4 nm(1,9) and 611.8 nm (1,7). The relative strengths of
the transitions is given between the brackets. Without any tuning element the laser
will generate output at the strongest transition, at 632.8 nm. With a tuning element
inside the resonator, losses can be introduced for all wavelengths except one. Tuning
elements could be reflection gratings, prisms, mirrors with narrow-band reflection
coatings and birefringent materials like calcite and natural quartz (Hecht chapter 8). In
this exercise we tune the laser with a ~1 mm thick natural quartz plate positioned at
the Brewster angle (also called: “polarization angle”) for minimum loss.
The optical axis is parallel to the plate. With such a plate the polarization of the light
entering the plate can be changed by rotating the plate. However, at certain angles the
polarization will not be changed and the losses will be minimal. Because the
refractive indices depend on the wavelength the angle can be used to tune the laser.
Follow the steps below to set-up a tunable HeNe laser.
o Align the laser with the two 750mm radius mirrors.
o Remove the outcoupler and rotate the alignment laser around its axis until the
reflection from the Brewster windows is zero. Now the linear polarization of
the alignment laser is horizontal.
Figure 4 With a birefringent crystal (natural quartz) wavelength
tuning is possible.
42
o Place the quartz plate at the right side of the discharge tube. You can see two
spots reflected from this plate. One is from the front surface, the other from its
back surface. The spots are very close to each other. Rotate the quartz plate
around the vertical axis until the reflection from the front surface is zero. Now
the quartz plate is at the Brewster angle. This angle should be about 57o.
o Next rotate the quartz plate around its horizontal axis until the reflection from
the back surface is zero too. The reflection from the Brewster plate on the
discharge tube will be zero as well. Now the quartz plate causes minimum
losses for the wavelength of the alignment laser, which is 632.8 nm. The
losses are minimal because the polarization is rotated 360o and as a
consequence, the polarization of the beam exiting the plate is again horizontal.
You can find a large number of angles where this is the case.
o Place the outcoupling mirror back and you should get laser output again at
632.8 nm.
o Optimize the output by rotating the quartz plate around the horizontal axis and
the vertical axis. Find the optimum by iteration of both rotations. Maybe you
must also do some adjustments of the mirror mount, because the beam is a
little tilted by the plate.
o Use the spectrometer to measure the spectrum.
o Now you can tune the laser to another wavelength by rotating the quartz plate
around the horizontal axis. The second strongest transition: 640.1 nm can be
selected by this. Unfortunately the wavelengths of the weaker transitions
cannot oscillate with our set-up.
43
Appendix. Resonator theory
Here a brief overview of the resonator theory is given. More details can be found in:
“Lasers”, A.E. Siegman and “Optics”, E. Hecht.
A Gaussian TEM00 beam at its beam waist can be characterized by its amplitude, u0,
and beam size (defined as the 1/e radius), w0. At the beam waist the size of the beam
is smallest and the radius of the wavefront is infinitive (i.e. it has a plane wavefront).
In formula:
2
0
0)(
w
r
euru
The intensity of the beam is proportional to the square of the amplitude and can be
written as:
2
0
2
0)(
w
r
eIrI
When a Gaussian beam propagates in the z direction from its waist at z = 0 to a point
z, it can be shown that the radius of its wavefront is given by:
z
zzzR
R
2
)(
Where ZR is the Rayleigh range (See Hecht: 13.18)
In a resonator with two mirrors positioned at z=z1 and z=z2 respectively, the radii of
the wavefronts at the mirrors must be equal to the radii of curvatures, R1 and R2
respectively, of the mirrors. The size of the beam inside the resonator can thus be
found by solving:
L
Z1 Z2
Z
R1 R2
44
12
2
2
2
22
1
1
2
11
)(
)(
zzL
Rz
zzzR
Rz
zzzR
R
R
(1)
The minus sign in the first equation arises because of the sign convention used for
mirror- and wavefront radii. We define the so-called g-parameters of the resonator:
2
2
1
1
1
1
R
Lg
R
Lg
(2)
It can be found by solving (1) and using (2), that:
2
2
2121
21212
2
1L
gggg
ggggz
R
,
L
gggg
ggz
2121
12
12
1
,
L
gggg
ggz
2121
21
22
1
R
z
gggg
ggggLw
2
2121
21212
0
2
1
211
22
11 ggg
gLw
,
212
12
21 ggg
gLw
Where is the laser wavelength, w1 and w2 are the beam sizes at the mirrors, L, is the
distance between the mirrors, and w0 is the beam waist at z = 0.
From the equations it can be seen that real solutions only exist if:
1021 gg
This is called the stability range of the resonator. Outside this range no beam can exist
between the mirrors and hence no radiation will be emitted by the laser.
45
A Matlab program is available, which calculates the beam size inside the resonator for
given laser wavelength, mirror radii and resonator length.
-60 -40 -20 0 20 40-1
-0.5
0
0.5
1
z [cm]
w [
mm
]TEM
00 beam size inside the resonator
-5 0 5-5
0
5
g1
g2
Stability diagram
L = 1.2 m, R1 = 75 cm, R
2 = 50 cm
g1 = -0.6, g
2 = -1.4
g1 * g
2 = 0.84
z1 = -73 cm, z
2 = 47 cm
w1 = 0.961 mm, w
2 = 0.629 mm
w0 = 0.155 mm, Z
R = 12 cm
The width of the Gaussian TEM00 mode inside the resonator is plotted as well as the
stability diagram in which the two red curves are for g1g2 = 1. The M1 mirror is at the
left side at z = z1 and the M2 mirror is at the right at z = z2. The beam waist is at z = 0.
The blue dot in the stability diagram indicates the resonator configuration given by L,
R1 and R2. The blue dot should be inside the area defined by the red curves and the g1
= 0 and the g2 = 0 lines. If it is outside this area, the resonator is unconfined and no
mode can exist.
Copy-paste the m-file listing below into your Matlab editor and run it.
%####################################################################
###### % %Calculation of the size of the fundamental Gauss mode, TEM00, inside
a %stable resonator. See: "Lasers", A.E. Siegman, Chapter 19. % %Fred van Goor, 7-8-2008 %####################################################################
######
%definition of units: m=1; mm=1E-3*m; cm=1E-2*m; micron=1E-6*m; s=1;
c=2.9979E8*m/s; %velocity of light in vacuum lambda=0.6328*micron; %laser wavelength (HeNe laser) L=120*cm; %resonator length
46
% radii of curvatures of concave mirrors. (For a convex mirror use a
negative value): R1=750*mm; %radius of curvature of mirror M1 R2=500*mm; %radius of curvature of mirror M2
%resonator g-parameters: g1=1-L/R1; g2=1-L/R2;
%Rayleigh range ZR=(sqrt(g1*g2*(1-g1*g2)/((g1+g2-2*g1*g2)^2)))*L;
w0=sqrt(lambda*ZR/pi); %Beam size at waist
at z = 0 z1=-g2*(1-g1)*L/(g1+g2-2*g1*g2); %position of mirror
M1 z2=g1*(1-g2)*L/(g1+g2-2*g1*g2); %position of mirror
M2 w1=sqrt(lambda*L/pi*sqrt(g2/(g1*(1-g1*g2)))); %Beam size at mirror
M1 w2=sqrt(lambda*L/pi*sqrt(g1/(g2*(1-g1*g2)))); %Beam size at mirror
M2
q0=1i*pi*w0^2/lambda; %q-parameter at beam waist z = 0, see
"Lasers", A.E. Siegman
N=1000; %number of points for the plot z_inc=(z2-z1)/(N-1); %increment of z z=zeros(1,N); %initiation of z array w=zeros(1,N); %initiation of w array for i=1:1:N; z(i)=z1+(i-1)*z_inc; %filling z array q = q0 + z(i); %q-parameter after propagation a distance
z(i) from z = 0 w(i)=sqrt(-lambda/pi/imag(1/q)); %beam size at z(i) end
%Plot the results in a 2D plot: subplot(211); if isreal(w) %plot only if w is real plot(z/cm,w/mm,'r',z/cm,-w/mm,'r'); %Also plot the lower edge of
the beam xlim([z1/cm,z2/cm]);ylim([-1,1]); grid on; else %if w is not real the mode cannot exist in the resonator text(L/3/cm,0.5, '!! not confined !!','FontSize',14); text(L/3/cm,0, sprintf('g_1 * g_2 = %0.3g',g1*g2),'FontSize',12);
%inform the user of the g1g2 product text(L/3/cm,-0.5, sprintf('g_1g_2 must be: 0 < g_1 * g_2 <
1'),'FontSize',12); xlim([0,L/cm]);ylim([-1,1]); grid off; end xlabel('z [cm]');ylabel('w [mm]'); title('TEM_0_0 beam size inside the resonator');
gg1l=zeros(1,N); gg2l=zeros(1,N); gg1r=zeros(1,N); gg2r=zeros(1,N);
47
gg1b=-5; gg1e=0; gg1_inc=(gg1e-gg1b)/(N-1); for i=1:1:N; gg1l(i)=gg1b+(i-1)*gg1_inc; gg2l(i)=1/gg1l(i); end gg1b=0; gg1e=5; gg1_inc=(gg1e-gg1b)/(N-1); for i=1:1:N; gg1r(i)=gg1b+(i-1)*gg1_inc; gg2r(i)=1/gg1r(i);
end;
%Plot the stability diagram: subplot(212); plot(gg1l,gg2l,'r',gg1r,gg2r,'r',g1,g2,'.','MarkerSize',20); axis equal; xlim([-5,5]); ylim([-5,5]); xlabel('g_1');ylabel('g_2'); title('Stability diagram'); text(6,4,sprintf('L = %0.3g m, R_1 = %0.3g cm, R_2 = %0.3g
cm',L/m,R1/cm, R2/cm),'VerticalAlignment','bottom','FontSize',8); text(6,2,sprintf('g_1 = %0.3g, g_2 =
%0.3g',g1,g2),'VerticalAlignment','bottom','FontSize',8); text(6,0,sprintf('g_1 * g_2 =
%0.3g',g1*g2),'VerticalAlignment','bottom','FontSize',8); if isreal(w) text(6,-2,sprintf('z_1 = %0.3g cm, z_2 = %0.3g cm',z1/cm,
z2/cm),'VerticalAlignment','bottom','FontSize',8); text(6,-4,sprintf('w_1 = %0.3g mm, w_2 = %0.3g mm',w1/mm,
w2/mm),'VerticalAlignment','bottom','FontSize',8); text(6,-6,sprintf('w_0 = %0.3g mm, Z_R = %0.3g cm',w0/mm,
ZR/cm),'VerticalAlignment','bottom','FontSize',8); end grid on;
49
4 techniques and instruments
4.1 Examples of constructions with the micro-optic bench:
50
4.2 Light sources
Light sources can be distinguished according to one of the following properties:
1 Total intensity of the light;
2 Clarity (light flux per m per sterad) as a function of the direction of the light;
3 Spectral distribution of the transmitted light.
The latter property will be described in more detail in the following subsections.
Continuous spectrum
A material that is heated to a very high temperature will emit light with a continuous spectrum. The
wavelength distribution of the transmitted radiation can be calculated with Planck‟s radiation law (see
Alonso & Finn, part 4, chapter 4.1etc), which can be written in the following form:
M T TC
eC T
, , . .
/
11
125
In this equation, the parameters represent the following quantities:
M(λ,T): The power of the emitted radiation per unit of radiating surface, per unit of
wavelength interval over a space angle of 2π.
ε(λ,T): The emission coefficient of the radiating surface. This coefficient is slightly
dependent on λ and T. If the radiating surface is Wolfram, ε ≈ 0.42 in the
visible spectrum at a temperature of T = 2500 – 3000 K.
C1: 3.741 * 10-16
Wm.
C2: 1.439 * 10-2
mK.
Figure 3.1 M() curves at
two different temperatures.
The maximum of the curve shown above changes according to the Wien Displacement Law:
maxT = const. (const. = 2,90.10-3
mK)
The total radiation power of a radiating surface of 1 m2 is defined by the Stefan-Boltzmann Law:
M = T4 ; = 5,67.10
-8 Wm
-2 K
-4
Only a part of the power emitted by a thermal source is available as visible light (between 400 and 800
nm). An ideal light source exclusively emitting light of λ = 550 nm (the maximum of the sensitivity
curve of the human eye), would have a yield of 687 lumen/Watt. An ordinary light bulb only reaches a
fraction of this value.
51
Discrete spectrum
A discrete spectrum or line spectrum occurs in a gas discharge tube which is shown in the following
figure. Such a lamp is filled with low-pressure gas or vapour. The gas is conductive at a certain
combination of voltage and distance between electrodes. The light emitted has a line spectrum that is
characteristic for the gas in the tube.
I
X
VR- +
A
a
PF DK D
K
k
D
NA D
Figure 3.2
0
1
2
E(eV)
Hg3
4
4 ,7
4 34 8
5 02 65 79 1
4 07 8
4 04 7
4 35 8
5 46 1
5 77 05 79 0
4 91 6
4 10 8
8 ,88 ,8
4 ,9
5 ,5
6 ,7
5
6
7
8
9
10
Figure 3.3
As an example, the figure 3.3 shows the emission of mercury. (Complete energy level diagram with
transitions in the visible spectrum; numbers in Angstrom = 10-10
m).
In the spectrum of a gas discharge tube, the sharpness of the lines depends on the temperature and
pressure in the gas mixture. When both temperature and pressure are low, the spectral lines are sharp
and the energy levels are well defined.
At a higher temperature, the Doppler effect causes an observer to also see light with higher and lower
frequencies than the frequency that was actually emitted. This causes a broadening of the lines in the
spectrum. This broadening is larger for light atoms than for heavy atoms.
If there is a higher pressure, the energy levels of separate atoms or molecules will interfere with each
other. This causes the lines to be less sharp. It‟s also possible that it causes a continuous background
spectrum.
Low pressure gas discharge tubes are applied as spectral light sources when light of a well defined
wavelength is required. The clarity (lumen per m2 radiating surface per sterad) of these light sources is
rather low. High pressure lamps can reach very high clarity.
52
In semi-conductor material one can also observe energy transitions in which light is emitted. An
example is the Light Emitting Diode, or LED. The most monochromatic light source is the LASER.
Notations
Wavelengths are usually given in nm (1 nm = 10-9
m). An older notation that is still used now and then
is the Ångstrom (1 Å = 10-10
m = 0.1 nm).
It is often convenient to use energy units, because energy level diagrams directly show the energy of
the radiation emitted. Figure 3.4 gives some frequently used scales for the spectrum of EM waves.
The EM-spectrum:
-scale: 10-7
10-6
10-5
m
102 10
3 10
4 nm
Energie scales :
frequency (c
) 3.1015
3.1014
3.1013
Hz
wavenumber (1
c
) 10
5 10
4 10
3 cm
-1
energy ( E h ) 1
1.24 10 1.24 1
1.24 10
eV
(N.B. 1 eV = 1,6.10-19
J)
Figure 3.4
53
4.3 Photo-diode
A light sensitive diode consists of a pn diode in which incoming light frees electrons close to the pn
junction, i.e. light causes the electrons to jump from the valance band to the conductance band. This
results in a current that is proportional to the intensity of the incoming light.
Light causes the IV characteristic of the diode to change (see figure 4.1). A photo-diode can be used in
two different modes, which will be described below. More details can be found in Wilson and Hawkes.
- V
- I
+ V
+ I
10 0 Am
20 Am
-0,25 -0,20 -0,15 -0,10 -0,05
80 Am
60 Am
m40 A
0,05 0,1 0,15 0,20 0,25
R = 10 kL W
R = 500L W
R =dV /d ISH
P C
P VIp,5
Ip,4
Ip,3
Ip,2
Ip,1
Ip,0
Figure 4.1
Photovoltaic mode
When the photo-diode is used in photovoltaic mode, no help voltage is applied (the 4th quadrant in
figure 4.1). The signal current is runs in the back direction and is accurately proportional to the
illumination strength. There is practically no current in the dark.
Photoconductive mode
In the photoconductive mode a supplemental voltage is applied in the back direction (the 3rd
quadrant
in figure 4.1). The dynamic behaviour is better than in a photovoltaic cell: frequencies up to GHz can
be reached.
Quantum efficiency (QE)
When every photon adds 1 electron to is, the quantum efficiency is 1. In a normal photo-diode QE is
0.5 to 0.8 at the maximum of the spectral sensitivity curve.
54
4.4 Spectral analysis devices
Spectral analysis devices (or spectrometers) are used to measure the distribution energy over different
wavelengths in a beam of light. Such a device must contain a dispersing element, i.e. an element in
which the refraction or the reflection of light is dependent on the wavelength. The dispersing element is
usually a prism or a grating.
Devices based on a prism
A prism is used in the following simple set-up:
Figure 4.2
The set-up can be used as a spectrograph. In that case the whole spectrum is either projected on a
screen (at E) or is recorded photographically.
The device can also be constructed such that E contains a moveable slit that only allows light in a small
range of wavelengths to pass. In that case we have an adjustable filter or a monochromator. However, it
is not very convenient to use this device as a monochromator in a laboratory set-up because the
location and the direction of the axis of the emerging light beam change when the device is set to a
different wavelength.
Devices based on a grating
In a grating monochromator the axes of the incoming and the emerging beam coincide. The direction of
the light rays is shown in figure 4.3. The monochromator can be set to different wavelengths by turning
the grating. The grating used in such a monochromator is a blazed grating. A blazed grating reflects
light with a high intensity in a distinct direction (e.g. the –1 or the +1 order) due to the form of its
grooves.
Figure 4.3
Chromatic resolving power
The chromatic resolving power is defined as:
55
where Δλ is the wavelength difference of two light beam of equal intensity that can just be observed
separately.
The chromatic resolving power of a spectral analysis is limited by:
1. Imaging errors of the mirrors;
2. Imperfection of the grating or the prism;
3. Mirrors that aren‟t perfectly clean (causing light scattering);
4. A slit width that is too large;
5. Improper positioning of the slit (e.g. the slit is not in the focus of the mirror or the image of
the entrance slit is not parallel to the exit slit);
6. Diffraction.
Unlike the other problems, the latter is fundamental and will be discussed below.
Diffraction
When purely monochromatic light hits a very thin entrance slit, its image in the plane of the exit slit is
determined by the deflection pattern of the parallel beam coming from the grating. If this beam is larger
(broader), the image is sharper and the chromatic resolving power larger.
Literature gives us formulas for calculating the maximum resolving power that can be reached
theoretically.
For instruments based on a prism (Longhurst, ch. 11-5) this is:
dnb
d
where b is the length of the basis of the prism.
For instruments based on a grating (Hecht, 10.2.7; Longhurst, ch.12-6) this is:
pN
where N is the number of grooves in the grating.
Alignment
To actually obtain the maximum resolving power it is important to adjust the entrance slit width
properly. In general, there is an optimum value for the slit width: ,slit opt
w . Lower values would hardly
improve the resolution and only lower the transmitted power (see figure 4.4). The producer of the
device usually indicates the optimum slit width. Furthermore, the light beam that hits the entrance slit
should be large enough to completely fill the prism or the grating (thus the angle θ in figure 4.4 should
be sufficiently large). To facilitate this, the aperture of the instrument is given in the documentation.
The set-up drawn in figure 4.4 can be used at the entrance of the spectrometer:
2
source
L L chopper L entrance slit
56
Figure 4.4
If we consider the width of the source, 1
w , and the width of the different images, , 2, 3...i
w i , it
can be shown from the theory of lenses that ~i i
w constant (i=1,2…). Here we have assumed that the
lenses are sufficiently large andi
is the angle between the outermost rays and the optic axis. Knowing
the optimum slit width the required acceptance angle at the light source can be estimated. Normally, if
a condenser lens is used the acceptance angle at the position of the source is sufficiently large.
If a chopper is used in the set-up above, you must be sure that the source is imaged on the plane of the
chopper blades.
The optimum slit width
We consider a source emitting a single wavelength, , as the input of a grating or prism spectral
analyser. Even for a zero entrance slit width the output beam will have a certain width corresponding to
a wavelength difference of:
/( )mN grating, m=order, N=number of grooves
/( / )b n prism; b=length prism base, /n =dispersion.
The non-zero width of the entrance slit, w, will introduce an extra spread, say i
, of the input angle
and so extra broadening of the output beam.
We will first consider the grating spectrometer. Using the grating equation, i.e.,
(sin sin )m i
a m it follows:
cos / /( )slit i i
a m aw mL
where L is the distance between the slit and the lens (or spherical mirror). Consequently, the width of
the outgoing beam corresponds to a spread in wavelength according to:
/( ) /( ).mN aw mL
The optimum slit width is such that it introduces a broadening of the same order of magnitude as that of
the theoretical limit. For a grating, this is:
/( ) /opt
w m L am N L b
where b is the width of the grating.
For a prism the following can be derived for the angular dispersion:
/ /i
C n
Here i
is the input angle (defined with respect to the normal of the prism surface). The parameter C
depends on the shape of the prism and the angle of incidence, we may assume for the order of
magnitude: 1C . With the above it follows:
57
./ni .
Then,
0/( / ) /( / )
w slitb n w L n
Now proceeding as for the grating we arrive at the optimum slit width for a prism:
/ .opt
w L b
where b is the width of the prism base.
4.5 Amplification of small signals
The text below presents a detailed description of phase-sensitive detection and related topics. For the
Practical Course Optics it is sufficient to study and understand figure 4.13. Note that the short-hand
notation used there 1 /( 1)pt (low frequency pass filter) should be understood as 1 /( 1)i , with
a parameter (units second) describing the response of the circuit.
If small signals are to be measured, the detection limit and accordingly the differential sensitivity
(amplification factor) are constrained by a few effects:
1. zero deviation/residual deflection;
2. Noise and fluctuation phenomena.
The influence of these effects can be decreased by modulation and demodulation, thus increasing the
detection limit. The possibilities for (de)modulation can be deduced from two broad and universal
statements on noise and fluctuations:
1. Just like signals, noise and fluctuations can be localised in h frequency domain. Some frequencies
are „contaminated‟, e.g. 50 Hz (frequency of the electricity network) and a few multiples of 50 Hz
(the higher harmonics of the network frequency). Noise is proportionally larger at lower
frequencies (e.g. due to mechanical vibrations). The same goes for electrical fluctuations in semi-
conductors, which is important because in order to measure small signals they very often have to
be amplified electronically. The zero deviation/residual deflection of several measuring
instruments is considered as very low frequency noise.
2. The amplitude of the resultant of all noise and fluctuations increases when the bandwidth of the
amplifiers increases.
Usually, the entrance signal of the measurement chain is a DC voltage, i.e. in the frequency domain
0≤ω<ωmax. There are a few alternative methods for the amplification and measurement of a small DC
signal. These methods are described in the following subsections.
AC amplification with a limited bandwidth
The easiest method of reducing the influence of noise and fluctuations is to make the measurement
system slow by giving it a transfer function and by decreasing the bandwidth. Some examples of this
method are:
1. A galvano meter;
2. Filtering the fluctuating signal;
3. Using a low-pass filter.
The latter reduces the maximum frequency ωmax of the signal as well. This method is far from ideal.
Although the second property of noise and fluctuations mentioned above will cause the amplitude of
the noise to reduce, the most inconvenient noise mentioned in 1 and the zero deviation/residual
deflection remain present in the exit signal and will still influence the detection limit.
58
Increasing the detection limit by modulation and selective amplification
This method is based on the following. The entrance signal is tcos.x̂tx
M11
, where ωM is the
maximum frequency of the signal. By modulation the signal tcos.tcosx̂x̂ty
DM211
is
formed in which ωD is the carrier frequency. The signal y1(t) is the sum of two components with
respective frequencies of ωD-ωM and ωD+ωM. After that y1(t) is amplified selectively by a selective
amplifier with bandwidth B that is tuned to a frequency of ωD.
X (t)1
y (t)1
y (t)2
x cos( t)2 D
w
wD
se l. verst.:
bandbreedte B
lin .
dem odulator
m odula tie
dem odulatie
x,y
0 wM
wD-w
Mw
D+w
M
wD
w
B
x1
^
y (t)3
|G |1
G1
Figure 4.10
For simplicity we assume that the following statements hold for the selective amplifier:
D D
1 1G K for - B B
2 2
and
G 0 for a ll o ther frequencies
In that case only frequencies that follow ωM < ½ B are amplified with a factor K and other frequencies
aren‟t amplified.
Linear demodulation of the signal tcos.tcosx̂x̂ty
DM212
results in signal y3(t), which
envelops the positive part of y2(t):
3 1 2 M
M
ˆ ˆy t K .x .x .cos t
1fo llow ing the condition: B
2
If ωD is chosen correctly, this method only amplifies signals in a frequency domain that contains no
extra noise. When the modulation occurs in the beginning of the measurement chain, the effect of
the zero deviation/residual deflection is also eliminated. The choice of ωD can sometimes cause
problems because the amplification process has some constructional restrictions at the entrance. For
example, a carrier frequency - ωD – of 1000 Hz is already rather high when a light interrupter is to be
used. The bandwidth of the system taken from signal x1(t) or y3(t) is ½ B.
59
When it is used as described above, selective amplification is better than AC amplification. However,
the combination of selective amplification and linear demodulation is not used very frequently due to
some disadvantages:
1. The sign of x1(t) is lost after linear demodulation;
2. The bandwidth ½ B cannot be decreased as much as is desirable for improvement of the
detection limit.
The latter can be explained with the following example. Suppose the carrier frequency is 1 kHz and the
signal has a maximum frequency of 1 Hz. A selective amplifier bandwidth of B = 2 Hz would in that
case be sufficient and maximise the detection limit. In practice however, it is very difficult to build a
selective amplifier with such a small relative bandwidth. Furthermore, even a very small displacement
of the carrier frequency ωD relative to the pass curve of the amplifier would considerably influence the
resulting amplification factor. Thus, for all variables that can cause such a displacement the sensitivity
to fluctuations is large. The most important variables in this respect are: the supply voltages of the
selective amplifier and the oscillator that generates ωD and the ambient temperature.
Phase-sensitive detection
The following set-up is generally used for modulating, selectively amplifying and demodulating small
signals:
x (t)1
x (t)2
wD
y (t)1
y (t)2
y (t)3 1
p +1t1
K
Figure 4.11. Note that p (in pt+1) is a short hand notation for i
This set-up has a bandwidth limiting effect that can be explained I the following way. The signals x1
cos(M t) en x2 cos(D t) form
tcos.tcosx̂x̂tyDM211
which is amplified to form y2(t) = K . y1 (t). This signal is demodulated by amplification with the
carrier x2(t) to form:
t2cos1tcosx̂x̂K2
1ty
DM213
The low-pass filter behind the demodulator is implemented to repress components with a double
frequency. Furthermore, it determines the bandwidth of the combination. If the low-pass filter has a
bandwidth B, the amplification of y1(t) has a bandwidth of 2B, which can be easily deduced. This is
because in the multiplication of y2(t) and x2(t) frequencies of ωN > B result in differential frequencies
that are unable to pass the low-pass filter.
The filter in figure 4.11 is a first order low-pass filter that is commonly used. This filter has a
completely different purpose than the RC circuit in diode demodulation. In the underlying case the
time-constant can be increased to decrease the bandwidth. Unlike with selective amplification and
linear demodulation there is no tracking problem when the carrier frequency drifts off from the pass
band. The system is very selective, but stays „tuned‟ to the carrier frequency, no matter how much it
drifts off. This kind of detection is often called synchronous or coherent detection. The bandwidth is
determined by a simple low-pass filter instead of by a band filter that is difficult to construct for small
bandwidths. With this set-up a bandwidth of e.g. 1 Hz at a carrier frequency of 10kHz is a normal
specification that is reached without difficulties.
60
Applying phase-sensitive detection in practice The principle of phase-sensitive detection is described above. However, the practical application of this
principle asks for some further remarks.
First of all, it is necessary to multiply the signal with the carrier as early in the measuring chain as
possible, preferably directly at the source. After this multiplication, there is of course a considerable
amplification of the small signal y1(t) in what is called the „signal channel‟. The complex transfer
function1 2
G G G of the complete signal channel will generally not be real at the frequencies that
are in the channel:
Sj-.KpH.KG
The phase shift in the signal channel, S, must be taken into account. A phase shift R must be
introduced in the „reference channel‟ – which is the signal chain that transfers the carrier to the
demodulator. This can be done with the adjustable phase shifter that is implemented in the reference
channel.
In order to investigate the consequences of the occurrence of phase shifts, we must write down an
expression of y1(t) with all its spectral components:
tcostcosx̂x̂2
1ty
MDMD211
Assume that S is the same for both side-band frequencies (D - M) and (D + M). This is a
reasonable assumption because M << D. In that case:
SS
tcostcosx̂x̂K2
1ty
MDMD212
Then the reference signal is:
RD23
tcosx̂tx
And the demodulated signal is:
RSMD
RSM
RSMD
RSM
2
21
RDSMD
RDSMD
2
213
t2cos
tcos
t2cos
tcosx̂x̂K4
1
tcostcos
tcostcosx̂x̂K2
1ty
In this signal the terms with the double frequency 2D are filtered out by the low-pass filter. After
passing this filter, the signal can be expressed as follows:
RSM
2
214cos.tcosx̂.x̂K
2
1ty
61
This shows that it is necessary to set S = R; it is not enough to have the signal y2(t) within the pass-
band of width 2B and central frequency D because only components with the right phase are detected.
This is where the method derives its commonly used name phase-sensitive detection from, although the
process is described more accurately by the term synchronous detection or coherent detection.
The instruments that are most commonly used demodulate the signal by multiplication with a square
function derived from x3(t). The time functions corresponding with this are sketched in figure 4.13 for
two cases. In the first case, the phase in the reference channel is properly set (S = R) and in the second
case / 2S R
. It is clear that in the first case y3(t) consists of an „DC component‟ of low
frequency M, which is passed by the low-pass filter for 0 < 1/π. The other components resulting in
the ripple are not passed. In the second case, the average is zero and there is no exit signal.
Another important detail of the practical implementation of phase-sensitive detection can also be seen
in figure 4.11 . It is common to put a selective amplifier in the signal channel. This is not necessary for
the final bandwidth limitation, because that is obtained by the low-pass filter after demodulation.
However the method of phase-sensitive detection offers such an effective tool for the oppression of
noise and fluctuations that the amplitudes of these fluctuations are allowed to be very large before the
oppression takes place, i.e. in the complete signal channel up until the low-pass filter. As a result, it is
possible that the amplifiers are overloaded which would cause y4(t) to be completely out of proportion
compared to x1(t). Therefore, one usually implements several overload-indicators in the signal channel.
Thus the bandwidth is limited already while the incoming signal including noise is amplified. This is
the function of the selective – or at least somewhat bandwidth limiting – amplifier stage. An extra
advantage is that the shape of y1(t) is not subject to high requirements: in a light beam interrupter y1(t)
doesn‟t need to be a perfect square function because the selective amplifier only amplifies the
fundamental harmonic.
Some final remarks on phase-sensitive detection 1. In the application of phase-sensitive detection it is very important that the carrier is only
modulated by the measuring signal, thus it must not be partly modulated by noise. Therefore
modulation must take place as early in the measuring chain as possible, as was also outlined in
the preceding section. For example, it is wrong to first multiply a DC voltage with a DC
amplifier and only multiply it with the carrier after that. In that case the zero deviation/residual
deflection of the DC amplifier also modulates the carrier and one of the advantages of phase-
sensitive detection is lost. In optical systems, modulation can very well be used to discriminate
against „false light‟ as long as one makes sure that this false light is not taken into the
modulation. This sounds like an easy requirement, but it‟s one that takes care and
inventiveness to meet. The best method is to engineer the measuring process such that the
measuring signal is modulated right after it is produced.
2. The term „modulation‟ is ambiguous. In this course text we consequently called the measuring
signal the „modulating signal‟ and we called the carrier the „modulated signal‟. In
measurement technique amplitude demodulation with repressed carrier frequency is often
used, which in practice means multiplication of signal and carrier. One often calls this
“modulation of the measuring signal with 1 kHz” or “modulation of the measuring signal with
a square function”. Only in this type of modulation these terms mean the same as “modulation
of the 1 kHz carrier signal with the measuring signal” or “modulation of a square function”.
However it is preferable to use the definitions that are also commonly used in literature, i.e.
always to refer to the carrier as the modulated signal.
3. The fact that synchronous detection is phase-sensitive, offers several opportunities in
measurement technique. If the phase shifter in the reference channel is calibrated, a
measurement system as the one illustrated in figure 4.13 can be used to measure the argument
of complex transfer functions. In a some applications phase-sensitive detection can be used as
a zero instrument to measure both modulus and argument of an out-of-balance signal
independently. In a deflecting instrument an out-of-balance signal that is π/2 out of phase with
the supply voltage is not detected. This is used in resistance strain gauge instruments to keep
imbalance caused by parasitic capacities out of the measurement results.
4. Phase-sensitive detection is an example of a set of measurement methods that are based on the
existence of a correlation between signals. For use in measurement set-ups one can purchase
complete instruments that include the amplifiers, the reference channel with phase shifter, the
62
demodulator and the low-pass filter as sketched in figure 4.13. Such a complete system is
called phase-sensitive detector or lock-in amplifier. The user is supposed to take care of the
modulation in setting up his measurement set-up (the method for this in different in every case
and the carrier frequency must be sent to the reference channel).
63
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