EOP4086 Optical Signal Processing - Lab 2 Abbe’s Theory of Imaging
Lab 2: Abbe’s Theory of Imaging
PRECAUTIONS
All the equipments/components are extremely sensitive to their conditions as well as
orientations. So, handle with great care. Keep the components away from dust as it would
greatly affect the experiment, and the results may not be obtained that easily. Avoid
fingerprints on mirrors and lenses. Never put your eyes directly in front of the laser
beam. It is always advisable to put the rings and wrist watches off whenever one works with
lasers.
1. Objective
To investigate the shape and quality of an image by the effect of diffraction phenomenon
2. Apparatus
Part
OB
LA
BSA-I
LCA
LCA*
TA-II
LBE
Qty
1
1
2
1
2
23
21
Description
Optical Breadboard
He-Ne Laser Assembly
Beam Steering Assembly with Plane Mirror
Lens Chuck Assembly
Lens Chuck Assembly without B-2 Base
Modified Target Assembly (Slide/Card Holder)
Lenses for Beam Expander
(f = 25.4 mm and f = +200 mm)
TL (KPX100)
1
Transform Lens (f = 150mm)
QI
1
Index Card with 2 mm Hole
QM
1
Microscope Slide (for making spatial filters)
QS
1
Square Mesh (as object)
Items prepared/made by student:
(It is your choice of how to make them. Refer to Section 4. Procedure)
Tools: e.g. scissors, transparent tape (Loytape), black drawing pen
Masking materials for making spatial filters: e.g. paper, cardboard, toothpicks/
needles/pins, ink
Transparency/slide as object: e.g. picture printed/fine drawing drawn on
transparent plastic
3. Theory
This experiment touches on the subject of spatial frequency content of objects and
how they could be used to control the shape and quality of an image. This subject is similar to
finding the frequency harmonic content of a waveform such as that produced by a musical
instrument. For example, a musical instrument may produce both a low pitch tone and a high
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EOP4086 Optical Signal Processing - Lab 2 Abbe’s Theory of Imaging
pitch tone. We can control the quality of sound by filtering out one of the two frequency
harmonics with a low-pass or a high-pass filter.
Objects have certain intensity profiles which can be translated into corresponding
spatial frequency distributions. In this project the frequency distribution of an object
illuminated with a collimated laser beam will be examined with a single lens placed after the
object. The light distribution formed at the focal plane tells us the frequency content of the
object, and by manipulating the light in the plane, we gain control of the quality and content
of the image to be displayed.
The laser beam to be used in the experiment has a smooth profile, i.e. the intensity
distribution does not have any wiggles, and when it is focused, it produces a single small
spot, i.e. the original beam contains only low spatial frequencies. On the other hand, if we
pass this beam through a grating or specimen which introduces many variations on the laser
profile, then in the focal plane of the lens we will have several spots indicating that additional
spatial frequency components have been added, as shown in Figure 1.
Abbe’s Theory (of Image Formation) is based on the necessity for the light rays
diffracted by a specimen to be collected by the objective and allowed to contribute to the
image of the specimen; if these diffracted rays are not included, the fine details which give
rise to the image cannot be resolved.
Figure 1: Abbe’s theory of image formation
Figure 1 above illustrates a schematic drawing of a microscope optical system
consisting of a condenser iris diaphragm, a condenser and an objective with a periodic grating
representing the specimen. The periodic specimen diffracts a collimated beam (arising from
each point of the condenser aperture), giving rise to the first-order, second-order, and higher
order diffracted rays on both sides of the undeviated zeroth-order beam; if the specimen is
removed, there is only the zeroth-order beam on the focal plane. The diffracted rays occur by
constructive interference at a specific angle (). Each diffracted-order ray (including the
zeroth) is focused at the rear focal plane of the objective. The period (s) between the focused
diffraction orders is proportional to the numerical aperture (f/D) of the ray entering the
objective. The situation is governed by the below equation (based on the diffraction equation,
dsinm = mλ with very small angle of m):
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EOP4086 Optical Signal Processing - Lab 2 Abbe’s Theory of Imaging
s/f /d = sin() ----------------------- Equation 1
where f is the focal length of the aberration-free objective that fulfils the Abbe sine condition,
D is the effective aperture diameter of the objective,  is the wavelength of light in the
specimen plane, d is the grating spacing (distance between grating lines) and  is the angle
between two adjacent diffracted-order rays.
The most surprising fact about Abbe's experiments is that when the first-order
diffraction pattern is masked at the objective rear aperture, so that only the zeroth and secondorder diffraction patterns are transmitted, the image of the specimen appears with twice the
spatial frequency, or with only half the spacing between the image lines.
4. Procedure
1.
2.
3.
4.
5.
6.
7.
Set up an imaging system on an optical breadboard as shown in Figure 2. (Hints: The
index card with 2 mm hole is used to help to align an optical component which
reflects laser beam back to the index card. The concave and convex lenses form a
laser beam expander/collimator. All the optical components are located so that the
laser beam intersects at the centers of the optical components. The expanded laser
beam should have symmetrical/uniform intensity profile.
Determine the distances u and v by using the thin lens equation so that the real image
is twice the size of the object.
Carry out experiments in sequence as shown in Table 1. Use a paper to block the laser
beam before the transform lens. Record the observations at the back focal and real
image planes (if your handphone has camera, taking pictures will help in the
discussion section).
Carry out experiments in sequence with square mesh object as shown in Table 2.
Place the square mesh in vertical orientation. Mark the locations of the Fourier image
dots which are on the x- and y-axis with a white paper pasted on an index card. These
locations will be useful to make various spatial filters in the following experiments.
Determine the period (s) and calculate the square mesh spacing (d).
Carry out experiments in sequence with square mesh object as shown in Table 3. The
long slit should be wide and long enough to allow the Fourier image dots on the x- or
y-axis to pass through. The small center-hole should only allow the center Fourier
image dot (zeroth-order dot) to pass through while the larger center-hole should only
allow zeroth-, first- and second-order dots to pass through. It is the same for the small
and larger center-blocks. The hollow square block should only block the first-order
dots. You may make any sizes of center-hole and center-block to investigate the
quality of the real image.
Place a transparency or slide at the object plane. The transparency can be any partially
transparent picture or drawing (2 mm x 2 mm size) printed/drawn on a transparent
plastic. The picture is intentionally superimposed (added) with many horizontal lines.
Place the picture so that the added lines are in horizontal orientation. Observe and
record the Fourier image and real image Table 4. Now, use
toothpicks/needles/pins/paper strips pasted on a microscope slide such that the Fourier
y-axis dots are blocked (the center dot should not be blocked). Observe and record the
real image in Table 4. You may try various center-holes and center-blocks to
investigate the quality of the real image as the experiments with square mesh object.
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EOP4086 Optical Signal Processing - Lab 2 Abbe’s Theory of Imaging
Optical breadboard
4”
4”
4”
4”
Index card with
2 mm hole
v
4”
u
f
3”
5”
Object
plane
5”
Transform
lens
Back focal plane
(Fourier image)
Real
image
plane
Figure 2: Schematic view of imaging experiment (drawing is not in scale)
5. Tables of Observations
Table 1: Observations without any object at the object plane
No Percentage of blocking Observation at back focal
Observation at real image
laser beam before the
plane
plane
transform lens (%)
1
0
2
50
3
75
Table 2: Observations with square mesh at the object plane
No Experiment
Observation at back focal
Observation at real image
plane
plane
1 Square mesh in
vertical orientation
2 50% of blocking laser
beam before the
transform lens
3 Rotating the square
mesh
Measured period value Fourier image, s = __________ mm
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EOP4086 Optical Signal Processing - Lab 2 Abbe’s Theory of Imaging
Calculated square mesh spacing value, d = __________ mm
Table 3: Observations with spatial filtering for square mesh object
No Experiment: Spatial filter at back
Observation at real image plane
focal plane
1 Long slit in vertical orientation to
allow the y-axis dots to pass through
2 Long slit in horizontal orientation to
allow the x-axis dots to pass through
3 Small center-hole to allow the zerothorder dot to pass through
4 Larger center-hole to allow the 0th-,
1st- and 2nd-order dots to pass through
5 Small center-block to block the 0thorder dot
6 Larger center-block to block the 0th-,
1st- and 2nd-order dots
7 Hollow square block to only block
the 1st-order dots
Table 4: Observations with transparency at the object plane
No Experiment
Observation at back focal
Observation at real image
plane
plane
1 Transparency with
superimposed lines in
horizontal orientation
2 Needle block in
vertical orientation
No observation involved
(refer to procedure for
the details)
6. Lab Report
The laboratory report should include the following.
1. Title and objective
2. Experimental procedures
3. Experimental observations and results
4. Discussion on the questions in Section 8 (including the observations and results)
5. Conclusion/Summary
The report must be submitted to the lab staff within 10 days from the date of
experiment.
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EOP4086 Optical Signal Processing - Lab 2 Abbe’s Theory of Imaging
7. Assessment (follow Lab Rubrics Structure, to be mapped to)
1. Preparation before coming for experiment – 5 marks
2. Skills to set up and carry out experiment (including recording observations and
results) – 10 marks
3. Discussion (illustration/explanation/analysis) – 5 marks
4. Conclusion/Summary – 5 marks
5. Report writing ability (grammar, organization and tidiness) – 5 marks
8. Questions for Discussion
1. Explain the physical significance of diffraction phenomena.
2. Discuss on the experimental observations and results.
3. Analyse the concept of spatial frequency in this experiment.
Updated May13, WO Siew
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