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Experiments In Optics

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					               First Year Laboratory


                   Experiments
                     in optics
                                     2009 - 2010




Physics, Imperial College London   v. 09/2007 (Damzen)




                                                     1
                                    Experiments in Optics
Introduction
Welcome to Experiments in Optics. For the next 2 weeks, you will do a variety of experiments that
illustrate the ray and wave nature of light. Recent years have seen a revolution in optical technology
(e.g. lasers in CD- and DVD-players, telecommunications and medicine), and optics also remains
one of the main diagnostic measurement techniques in a wide range of industrial and research fields.
However, the aims of this part of the lab go beyond learning some optics - in particular, we want you
to appreciate that measurements are never exact and therefore that experiment and "theory" will
never agree precisely. The best we can hope for is that experiment and theory are consistent, bearing
in mind the errors of the experiment and the approximations of the theory.
Everyone knows that measurements have errors but it is easy to forget that theories have their "error"
- i.e. approximations - and this is illustrated in the present topic. In optics, we consider light to be
described by rays or waves or photons, and sometimes by all three in the same experiment!

                   Physical Model of Light                         Name given to model

                            Rays                                   Geometrical Optics

                           Waves                                     Physical Optics

                          Photons                                    Quantum Optics

One important thing to learn in any branch of science is what level of approximation is required to
solve a particular problem. For example, quantum optics is meant to provide a complete description
of light but if you tried to use it to design a simple lens you would (probably) never get an answer. It
turns out that an extremely high quality lens can be designed using the concept of rays (geometrical
optics) and some knowledge of the wave model (physical optics). On the other hand, the basic
principle of operation of a laser requires the quantum model - no amount of clever playing with rays
or waves can fully explain the light emission in a laser. In Experiments in Optics, we shall do
experiments that use the models of light that are called Geometrical (Ray) Optics and Physical
(Wave) Optics. Ray and wave pictures of light are fairly straightforward, and sufficiently
comprehensive, to account for most optical phenomena and devices
In Experiments in Optics you will work with a partner. There are 4 sessions (2 per week) and the
following table summarises the schedule of experiments:


                                Lab                Experiment
                               Session

                                1              Single Slit Diffraction

                                2        Two-slit Interference and Gratings

                                3                      Lenses

                                4                   Polarisation




                                                   2
                                      Working Practices
Safety
Laser Safety: During the course of this series of experiments, you will use a compact diode laser of
wavelength 670nm (red). The power of this laser is approximately 1 mW. Lasers produce a highly
collimated (parallel) beam of light: the eye could focus this to a very small spot as small as 10 m in
diameter, giving a power density of 1 kW per square cm (but of course only over a tiny area). Retinal
damage may occur before the "blink reflex" closes the eye. Therefore,
                         NEVER LOOK DIRECTLY INTO THE LASER
                    NEVER POINT THE LASER AT ANY OTHER PERSON
         DO NOT USE MIRRORS TO REFLECT LASER BEAMS AROUND THE LAB
Electrical Safety: In this experiment you will also use various items of mains-powered electrical
equipment: this should not be tampered with in any way. If you are in any doubt regarding the safety
of any piece of equipment or any experimental procedure, you should consult a demonstrator before
proceeding.
Trip Hazards: You are often working in darkened conditions, so it is especially important that bags
and coats are stowed thoughtfully and passageways around benches are kept clear.

Your Laboratory Notebook
Start by writing the day and time, and title of the experiment. As you do each part of the lab, it is
essential to keep a clear written record in your lab notebook. However, do not spend a long time
engrossed in your lab book - never more than 10 minutes at a time - remember this is a practical
laboratory, not an exercise in writing. Every 10 or 15 minutes, record what you have done in your
book. Write clearly, draw lots of clearly labelled sketches, write down any conclusions you have
drawn or decisions you have made. It is vital that you describe or draw what you actually see. If a
trace on a screen has lots of bumps, wiggles and spots then draw them. Don't draw what you think
you might get from a perfect experiment: you might be throwing away important details.
Working Practice
We expect you to work hard in the laboratory. Concentrate on the experiments and work
methodically. If you get stuck, first talk the problem through with your partner; if you cannot resolve
the issue after 5 minutes, ask a demonstrator.
Keep Optical Components Clean!
Do not touch the surface of an optical component with your bare hands. If lenses, gratings, polarisers
etc get covered with fingerprints and dirt, they will scatter light and, when quantitative measurements
are being made, may produce quite spurious effects.




                                                   3
                         INTERFERENCE and DIFFRACTION
When two or more waves are superimposed, the resultant amplitude (and hence intensity) depends on
the amplitudes and phases of the component waves. This process is known either as interference or
diffraction, depending upon the geometry of the particular experiment. When there are a finite
number of waves (e.g. light from two very narrow slits or two distinct optical paths) we talk of
interference but if there is a continuum of waves (e.g. from light passing through a single slit of non-
negligible width) then we talk of diffraction.
The aim of this set of experiments is to observe the interference and diffraction patterns obtained
when a laser illuminates various apertures and to compare them with those predicted according to the
wave theory of 1ight. Analysis of diffraction close to the diffracting aperture, known as near-field
diffraction, can become quite mathematically complex. A much simpler case is the so-called far-
field diffraction regime in which the plane of observation is very far from the aperture (ideally at
infinity) – as seen in Figure 1a. Due to the physical size of optical bench this is not always practical
and an equivalent effect is achieved by making all observations at the focal plane of a lens – as seen
in Figure 1b. From Figure 1b it is noted that rays diffracted at angle  are observed on the viewing
screen at a distance x from the lens axis where  = x/f (for small angles).
              diffracted rays at angle                       parallel rays meet at a
       a)                                            b)
               meet at infinity                             point in focal plane
    input                                        x                                                     x
    light                                                                       
                  
                                ~                             
                                                                                       f
                           L      very distant                     Lens, f                 observing
    diffracting object            observing                                                screen
                                  screen                  =x/f (small angles)

Figure 1: Input light beam (e.g. from laser) is diffracted into rays at new angles. The far-field diffraction
pattern can be observed a) on a very distant screen (ideally at infinity) or b) in the focal plane of a lens.

     Experiment 1: Diffraction by a Single Slit & by a Strand of Hair
There are basically two parts to this 3 hour session. First, you will investigate the diffraction pattern
of a single slit. Secondly you will examine the diffraction pattern of a strand of hair and based on
single slit diffraction and Babinet‟s principle determine its diameter.
List of Equipment
   1.5 m triangular profile optical bench, with 5 saddles
   Diode laser, post-mounted, with stabilised 5 V power supply
   bi-convex lens f =500 mm, bi-convex lens f=1000 mm
   holder for slides
   set of slides (diffracting slits and gratings)
   holder for observing screen, post-mounted, with white card
   precision translation stage with photodiode (+power supply and portable voltmeter)
   metre rule
Single Slit Diffraction Theory: The theoretical derivation of the intensity of light in the “far-field”
diffraction pattern of a single slit is found by summing the amplitudes of light from each elementary
point on the slit. This procedure follows from Huygens' Principle and the amplitude in this case is the
electric field of light and has both magnitude and phase. The intensity pattern is found by squaring
the amplitude pattern. For a slit of width a, the intensity I(x) of the diffraction pattern as a function
of distance x in the observation plane is given, for small angles  = x/f, by



                                                      4
                                                        sin  ax 
                                                                         2

                                          I( x )  I 0  ax f 
                                                         f  
                                                                                                                      (1)

where I0 is the intensity at the centre,  is the
wavelength and f is the focal length of the focussing                            1.0

lens. Note that the quantities a, x,  and f in Eq. (1) all
                                                                                 0.8
have dimensions of length and should all be expressed




                                                                     Intensity
in the same units, e.g. mm. Figure 2 shows the single                            0.6
slit diffraction pattern.
                                                                                 0.4
Figure 2: The single slit diffraction pattern of Eq.(1), with
the scaling constant I0=1. The horizontal axis is in units of
the quantity  x  .
               a                                                                 0.2
                 f

                                                                                 0.0
                                                                                    -10        -5        0        5         10

                                                                                                    Distance
From equation (1), the zeros of intensity occur at x-
coordinates xm given by
                                           axm 
                                           f   m
                                                                                        m=  1,  2,  3,...
                                                
                                                     f
which can be re-arranged as               xm  m                                          m  1,  2,  3 ....       (2)
                                                     a
EXPERIMENT 1a: SINGLE-SLIT DIFFRACTION.
Figure 3 shows the basic set-up for the experiments on interference and diffraction. In these
experiments, the light source is a red diode laser emitting at a mean wavelength of  = 670  1 nm.
This laser is powered by a special stable 5 volt power supply. [This is the type of laser device used in
bar-code readers at Supermarket check-out counters and within DVD players. The basic diode laser
inside the housing is an ultra-compact device and it is easy to forget that it may cause permanent
damage if pointed at the eye. Therefore - NEVER LOOK DIRECTLY INTO THE LASER -
NEVER POINT THE LASER AT ANY OTHER PERSON.]
                                       lens,                                     Photodiode/slit on
                                       focal length f                            translation stage

                LASER

                                                                 f
                  diffracting object                                                                    voltmeter
   Figure 3: Basic arrangement for measurement of the intensity of interference and diffraction patterns.
   The object is placed close to the laser output. Use the f = 0.5 m lens for measurements on the
   photodiode assembly. Use the f=1m lens for visual observations on a white screen.
For this experiment, use the slide marked "Single Slit". The exact width of this slit is not known but
it is about 60 m. To measure the intensity a photodiode is used together with a narrow slit (=100
m) in front of it. The photodiode output is a voltage that is linearly proportional to the light
intensity. The slit/photodiode are provided as a single unit that attaches to a precision translation
slide (the slit/photodiode can be rotated by loosening the screw on the mounting bracket of the
translation slide). The translation slide has a total range of 25 mm. Make sure that the photodiode slit
is vertical (loosen the screw and rotate the slit/photodiode if necessary). Use the bi-convex lens with
                                                          5
f=500mm and ensure the distance between the centre of the lens and the photodiode slit is accurately
located at the focal length of the lens to observe the far-field diffraction pattern. Looking by eye,
check that the centre of the diffraction pattern is approximately coincident with the photodiode slit
when the translation stage is set at its mid-point (about 12.5 mm) (make minor adjustments by tilting
the laser and re-centring the “single slit” in its holder, if necessary). You should then measure the
intensity over the full range of the micrometer movement.
   First, find the approximate centre of the diffraction pattern (the largest intensity) and choose the
    scale on the voltmeter (either 2 V FSD or 200 mV FSD).
   Second, get a feel for how often you will have to sample the diffraction pattern to get a good
    representation of it: every 1 mm, every 0.5 mm, every 0.25mm...? A little preliminary work here
    will save you a lot of effort later. [Make sure you know how to read the micrometer: ask a
    demonstrator if you have any doubt].
   Finally, make your set of measurements. You will have probably already noticed that background
    illumination affects your measurements so try to ensure that this remains constant and/or is very
    much reduced during the complete run. You may notice that the meter reading fluctuates more-
    or-less randomly. This is called "noise": what are the causes of this noise? How might you reduce
    the effects of noise physically or mathematically in your results?
Plot your measured intensity profile (by hand, on graph paper) as a function of x for x = 0 to 25 mm.
Find the position of the first zeroes (m =  1) and hence estimate the slit width.
Experiment 1b: DETERMINING THE DIAMETER OF A HAIR
Babinet's Principle. Suppose that the amplitude (electric field) of the light wave at a certain point x
in a diffraction pattern of a certain object is A1(x). Now, replace the object by its complementary
form: the complementary form can be pictured as the (photographic) "negative", i.e., the
complementary object is transmitting where the origina1 is opaque, and is opaque where the original
is transmitting. For example, the complementary object corresponding to a slit is a thin opaque strip
or wire. (i.e. a slit has transmission function T1(x‟)=1 for |x‟|<a/2 and T1=0 otherwise, the
complementary wire has T2(x‟)=0 for |x‟|<a/2 and T2=1 otherwise). Let the complementary screen
produce an amplitude A2(x) at point x in the diffraction pattern. Babinet's principle states that
                                        A1(x) + A2(x) = A(x)                                          (3)
where A(x) is the amplitude of the wave at point x in the diffraction pattern when there is no
diffraction screen present at all. This theorem results from the reasoning that no screen at all (T=1
everywhere) = sum of screen (T1(x‟))+ complement of screen(T2(x‟)) and hence the sum of the two
complementary screens‟ diffraction patterns = the diffraction pattern with no screen.         Hence,
according to Babinet, in regions where A(x) = 0, A1(x) = -A2(x). Since intensity is proportional to
the square of the amplitude (i.e. I(x)=|A(x)|2), the intensity diffraction pattern of the object is
(surprisingly!?) the same as the complementary object in these regions.
In the case of the set-up shown in Fig.3, observe the distribution of light at the observing screen
when there is no diffracting object (i.e. I(x)=|A(x)|2) with the 1m focal length lens in place and record
this in your labbook. Now tape a hair over one of the blank slides provided and observe its
diffraction pattern. Use Eqs.(1-3) to explain the form of the pattern you see and also to find the
diameter of the hair. Quote an error value on the diameter. Try using a micrometer to measure the
diameter of the hair and compare the answer with your diffraction-based estimates. Have you made
any assumptions to produce your estimate of the diameter of the hair using the diffraction method? If
so, make sure these are stated clearly in your lab book.




                                                   6
    Experiment 2: YOUNG’S DOUBLE SLIT INTERFERENCE &
                   DIFFRACTION GRATINGS
In this 3-hour laboratory session, you will explore more complex features of interference and
diffraction. In the first part you will investigate a two-slit diffraction pattern. In the later part you
will explore how gratings, made from multiple slits, diffract light.
Use the double slit slide, labelled "Slit 2" as the object and place it immediately after the laser output
as in Figure 3. The two slits are each supposedly of the same width (nominally that of the single slit
in Expt.l), separated by roughly 4 times the width. Observe the pattern on the screen using the 1.0m
focal length lens and observing on the screen placed 1m from the lens. Draw a labelled sketch (with
a scale marked) of this diffraction pattern. Using the scale on a ruler, or otherwise, make a crude
estimate of the spacing between successive minima of intensity.

                                                                            P
                                         r1
                                                                                   x
             S1
                                                    r2
           d                
                                                                                       I
             S2           dsin
                                                L

      Figure 4: Geometry for Young’s double slit interference phenomenon: for large L, the path
      difference to point P is approximately d sin , where d is the slit separation.
(ii) Figure 4 shows the geometry used to derive the form of the interference pattern produced by two
infinitely narrow (i.e. idealised) slits. If the slits are infinitely narrow, then the difference between
the paths S2P and S1P is (for a distant screen) given by
                                           = r2 - r1  d sin 
Since the total light pattern at point P is found by summing the contributions from the two slits (with
due account of the phases) the positions of the bright and dark-fringes are given by constructive or
destructive interference conditions:
                                       MAXIMA: d sin  = m
                                                           1
                                     MINIMA: d sin  =  m  
                                                           2
where m= 0,  1,  2,  3,....
In the experiment, we observe in the focal plane of the focussing lens f, and for small angles, sin 
  = x/f , so we can write the position of the bright and dark fringes as:
                                                             f
                                       MAXIMA: x m  m                                               (4a)
                                                             d
                                                         1  f
                                       MINIMA: x m   m                                           (4b)
                                                         2 d
where m= 0,  1,  2,  3,....

                                                    7
The spacing between fringes (between successive maxima or successive minima) is therefore f/d.
For slits of negligible width, the form of the intensity distribution of the interference pattern is, for
small angles:
                                                                     dx 
                                                I( x )  4I 0 cos 2                                       (5)
                                                                     f 
where I0 is the intensity due to a single source and is assumed to be uniform across the screen.
   Does equation (5) give a good description of the pattern you observed in part (i)? Describe any
    differences in your lab book.
In fact, when the slits have a finite width a, the intensity is equal to the product of the distribution of
the single-slit equation (1) with equation (5) and may be written as:

                                                       sin ax 
                                                                 2
                                                                      dx 
                                         I( x )  4I0  axf  cos2      
                                                        f  
                                                                                                            (6)
                                                                      f 
where a is the slit width and d is the separation of the slits. The pattern is sketched in Figure 5 for the
case where d = 4a.

                                   1.0


                                   0.8
                       Intensity




                                   0.6


                                   0.4


                                   0.2

      Figure 5: Young’s double slit fringes for the case where the separation of the slits equals 4 times
                         0.0 to
      their widths (assumed -10 be equal). -5          0           5           10
                                                         Distance
(iii) In this section you are asked to verify equation (4b) for the positions of the minima. You should
use the slit/photodiode assembly to locate the minima of intensity. Use the arrangement of Figure 3
with the double slit as the diffracting object and the f = 0.5 m focussing lens. Make sure that the
distance from the lens to the measuring slit on the photodiode stage is as close as possible to 500
mm, to within a tolerance of  2mm. Start near the centre of the pattern and work your way out,
searching for each minimum: note the micrometer reading x for as many orders as you can (at least
5). You will therefore end up with a table with the following headings:

                                         Order m                         Value of xm

Use the computer program CurveExpert to draw a graph of the values of xm against order m. The
graph should be a straight line whose slope is the separation between orders and according to the
theory it should equal f/d. Compute the slope and standard error of the slope of the least squares
best-fit straight line through the experimental data. Instructions for the use of this program are given
on colour cards near the computers.
Using  = 670  1nm, d =230  20m and f = 500  1 mm, calculate the theoretical value of the
gradient and its standard error. Compare the experimentally determined and calculated values and
their standard errors. Have you verified the theory to within the limits of random experimental error?
Experiment 2(b): N-slit diffraction and diffraction gratings
You are provided with slides marked “multiple slits 3”, “multiple slits 4”, etc up to 6 slits. Take the
3-slit slide and observe the diffraction pattern formed by it carefully. You will probably find it a lot

                                                             8
easier to see the pattern if you use the eyepiece provided to magnify the patterns as shown in Fig. 6.
Make a clear record in your lab book. Do the same for the slide marked 4-slits.
                                             bi-convex lens                                observing
                                             f=1000mm                           eyepiece   screen

          LASER

                                                                  f
            diffracting object
      Figure 6: Arrangement for observation of magnified interference and diffraction patterns using
      an eyepiece: observations are made on the screen.

Figure 7 shows the interference patterns for 2, 3 and 4 infinitely narrow slits, all on the same scale. Is
this what you see (be honest!)?

                                   18

                                   16

                                   14
                                                                  N=4
                                   12
              Intensity (I /I0 )




                                   10                             N=3
                                    8

                                    6

                                    4
                                                                      N=2
                                    2

                                    0
                                        -4       -2           0         2   4
                                                       Distance

 Figure 7: The intensity interference patterns for 2, 3 and 4 infinitely narrow slits of equal separation.
Now quickly look at the interference patterns for 5-slits and then 6-slits. Can you guess what the
pattern will look like as the number of slits N   ?
A diffraction grating consists of a very large number of slits. You are provided with three coarse
gratings, labelled “1”, “2” and “3”. Observe the diffraction pattern from each one carefully. You
might find it useful to compare the patterns with that from the two slit object. For each grating find
  1. The width a of each slit (they are nominally all the same).
  2. The centre-to-centre separation d of adjacent slits.




                                                                  9
                       Experiment 3. IMAGING WITH LENSES
The vast majority of instruments that incorporate some optics have simple items such as lenses and
mirrors. The function of these elements is usually to provide even, or controlled, illumination of an
area (the reflector behind a torch bulb must be one of the "simplest" examples of this) or to relay an
image of an object to a specified plane or surface (e.g. a camera lens). In this series of experiments,
you will study the imaging properties of lenses using the concepts of geometrical optics. First we
start by imaging with a single lens and test the lens formulae that predict perfect images, free of any
kind of defect or aberration. Finally we will examine the quality of images using a resolution test
chart.
Simple Theory of Thin Lens Imaging. Figure 8 is a ray diagram illustrating the imaging action of a
lens. The object distance s, the image distance s‟ are all measured from the centre of a thin
converging lens. (A similar diagram can be made for a diverging lens but we will not be using one in
this experiment so will not consider it here). The diagram is an oversimplification and we will later
consider some of the features of real lenses, including the quality of the image formation.




                                                                     F‟            I
                           O             F


                                         s                           s‟
                                                              f

     Figure 8: A lens forms an image (I) of an object (O) where s is the object distance and s’ is the
image distance.
For small angles, the "thin lens formula" can be derived from this diagram:
                                                  1 1 1
                                                                                                        (7)
                                                  s s f
with the following sign convention:
   Light travels from left to right: s is positive (real) on the left, negative (virtual) on the right: s‟ is
    positive on the right, negative on the left: f is positive for a converging lens, negative for a
    diverging lens. (You should be aware that there are other sign conventions).
The linear magnification m is defined, in this sign convention, as
                                                       s
                                                 m      .                                      (8)
                                                       s
List of Equipment
   Lathe-type optical bench with 4 standard saddles and 1 adjustable saddle
   Incandescent light source, post-mounted, with power supply. Maximum voltage is 5 V!
   Plano-convex lens, nominal focal length 100 mm
   Plano-convex lens, nominal focal length 160 mm
   Achromatic doublet lens, nominal focal length 160 mm
   Ground glass viewing screen, 40 mm diameter
   Set of object slides including a resolution test chart slide and green filter slide
   CCD camera
                                                     10
   Plane mirror, 40 mm diameter
Experiment 3a: Testing the lens Formula
The lens formula, Eq.(7), can be tested and used to find the focal length, by plotting the reciprocal of
the image distance, 1/s‟, against the reciprocal of the object distance, 1/s. The intercept of this graph
on the 1/s‟ axis is the reciprocal of the focal length (1/f).

                                               f                            image, observed on
                 object                                                     ground-glass screen

           Light
           source


                                 s                               s‟


                      Figure 8: Experimental system for imaging with a positive lens.
Figure 8 shows a simple set-up for imaging with a positive focal length lens. Switch on the light
source – the voltage on the supply should be at voltage 5V (max.) - and use the slide with a pin-hole
as an object placed in the slot-holder in front of the light source. Use the plano-convex lens of
nominal focal length 100mm. Observe the image of the object slide on the ground-glass screen by
looking at the back of the screen and adjusting its position for sharpest image. It may be difficult to
see the image until you are close to the image plane and you may need to centre the image on the
screen by adjusting the height of the lens.
Use your ruler to measure the s and s‟ values over as wide a range as the length of the bench allows.
Start with the smallest possible separation (s+s‟) between the object and image planes (how is this
distance related to f?) and then use 5 or 6 more object-to-image distances, using the whole length of
the optical bench. For each object-to-image distance, there will be two positions of the lens that yield
a sharp image, so your graph will consist of 10-12 points.
Now use computer program CurveExpert to plot a graph of 1/s‟ versus 1/s and also to determine the
focal length and its standard error. Instructions on how to use this program are given in Appendix l.
Print out a plot of the graph and paste it in your lab-book.

Experiment 3b: Image Quality and Resolution of Lenses

Previously, we considered imaging with “thin” lenses and within a small-angle geometrical optics
(ray) approximation. For real lenses a fuller treatment needs to consider also the physical optics
(wave) picture. There are two key reasons for loss of quality by imaging with a real lens: i) the
finite diameter D of the lens and ii) lens aberrations due to details of a real (“thick”) lens. Consider
light from a point object (e.g. point object A in Figure 9) that expands as a spherical wave (only the
centre ray is illustrated in Figure 9). Only the part of this wave, that is within the lens diameter, is
imaged. The lens, therefore, acts as a limiting (circular) aperture that will diffract light as well as
imaging it. The result is that the point object A is not imaged to a point but to an image spread
function A‟. For a lens with no aberrations the characteristic angular size of this image is:
                                                  1.22  D                                         (9)
for light of wavelength . (The 1.22 is a geometrical factor due to the circular shape of the
diffracting object). As a consequence, if two point objects (e.g. A and B) subtend an angle of <
the two images will be merged and will not be resolved as two separate objects. They are said to be
unresolved. (Resolution is commonly defined as the Rayleigh criterion and you may find it helpful
to read about this in a physics textbook). The objects may be said to be just resolved when =. The
minimum resolvable object size (x) can be found by using the (small-angle) relationship = x/L.
                                                    11
Equation 9 is the “diffraction-limited” resolution of a lens, but lens aberrations further degrade and
spread out the image, and hence lead to poorer resolution. A full discussion of lens aberration is
outside the scope of this experiment but some details of types of aberrations are made at the end of
the experimental section.
                   A                                                                        B‟

              x                                                                    

                                                                                               A‟
                   B                        L                                L

                  Fig.9 Resolution limit of a lens due to its finite diameter D (and aberrations)

The purpose of this experiment is to give you some idea of the imaging quality and resolving power
of lenses. In this experiment, we measure imaging quality using a resolution test based on the three
bar chart. This expresses resolution in line pairs per millimetre. For a line in a three-bar element to
be just resolved gives the relationship: smallest resolvable detail (in mm) = (0.5).(line pairs per mm)-
1
  .

                        One of the slides provided is the three-bar chart. Examine it visually. It consists of 8
                        groups numbered 0 - 7, with 6 elements per group, an element consisting of two
                        target patterns of three lines at 90 degrees to each other, the line-to-space ratio being
                        equal. The number of line pairs per mm equals 1.00 for the largest element on group
                        0 and increases by 6 2  1.12 per element. The table gives the line pairs per mm for
                        all elements in groups 1 to 7 in the chart. Look at the chart and try to identify the
                        positions of the largest groups.
  Element Number        Group 1      Group 2     Group 3      Group 4     Group 5       Group 6     Group 7

  1                     2.00         4.00        8.00         16.0        32.0          64.0        128.

  2                     2.24         4.49        8.98         17.95       36.0          71.8        144.

  3                     2.52         5.04        10.1         20.16       40.3          80.6        161.

  4                     2.83         5.68        11.3         22.62       45.3          90.5        181.

  5                     3.17         6.35        12.7         25.39       50.8          102.        203.

  6                     3.56         7.13        14.3         28.51       57.0          114.        228.



The following two experiments illustrate the use of this resolution chart for assessing the quality of
imaging systems.
(i) Resolution of the eye.
Put the resolution chart slide into the slide holder on the light source and sit in front of the source
with the slide at approximately 250mm, the near distance of the eye. Decide which is the finest
group you can resolve (i.e. just see the bars on the target). You might wish to do two measurements,
one for the vertical bars and a second for the horizontal bars. Your partner should also do this
experiment. Take 3 or 4 independent readings each and find the average resolution, expressed in line
pairs per millimetre.
(If you wear glasses or contact lenses, you might want to try doing the experiment with and without
these).
                                                        12
Is your result consistent with equation 9? (Take eye‟s pupil diameter D=2mm and wavelength
=550nm).
(ii) Resolution of simple lenses.
Using the three-bar resolution chart as an object, form an image of the chart directly onto the CCD1
camera with the doublet lens provided (f = 160mm), making sure its flatter surface is towards the
object. The CCD camera should be mounted on the adjustable saddle so that it can be moved easily
in both the vertical and horizontal directions. The magnification required is about 4 (or more), giving
an image of the central groups (2 and higher) that fills the TV screen. [You will need to use eqns 7
and 8]. It can be quite tricky to align and focus the system: in particular, make sure that the doublet is
not twisted and that it is exactly perpendicular to the optica1 axis. It is also important to make sure
there is no stray light from desk lamps, etc. When everything is a1igned properly, you should be able
to see targets in Group 6. Make careful observations of the appearance of the image, both its
resolution and contrast, for the following lenses, keeping the overall object-to-image distance
approximately constant and making sure the lens is perpendicular to the optical axis, i.e. it is imaging
the object on axis.
    (a) Doublet, f  160mm, flatter side towards object.
    (b) Doublet, f  160mm, more curved side towards object.
    (c) Plano-convex singlet, f  160mm, flat side towards object.
    (d) Plano-convex singlet, f  160mm, curved side towards object.
Rank the four cases in order of image quality.


An aside about lens aberrations - You will have noted that all lenses have the same diameter so
that the difference in each lens‟ imaging quality must be due to additional aberrations, over and
above the finite diameter of the lens. Full discuss of lens aberration is beyond the scope of this
script, however it is noted that two key types of aberration, spherical and chromatic, will be
important in these cases. The lens formula of eqn 7 assumes thin lenses and small angles.
Spherical aberration arises from „errors‟ due to rays far from axis (and hence larger angles) not
imaging to the same point as rays close to the axis (small angle, or paraxial, rays). Since refraction
occurs at both lens surfaces, the strength of this aberration is more severe when a plano-convex lens
is placed one way around compared to the other.
                                                                                     „Image point is spread‟




                                                                                       Larger spread



Chromatic aberration arises from the wavelength dependence of the refractive index (e.g. this
produces the rainbow-like dispersion by a prism). Since the focal length of the lens depends on the
1
  CCD stands for Charge Coupled Device and refers to the readout mechanism in the detector chip. The CCD camera has a total image
area of 6.4 by 4.4mm (H x V), within which there are 542 by 582 pixels (i.e. the pixels on the chip are approximately 12.2 by 7.6 m).
                                                                 13
refractive index it is clear that different wavelengths will image at slightly different positions – and
this can be quite severe for a white light source. For example, if n(blue)>n(red) then f(blue)<f(red)


                                          Red          „White-light‟ image is spread
                                Blue


The doublet lens is made of two glass materials with refractive index properties that partially
compensate for this aberration.


                                            Red
                                 Blue




                                                  14
                           Experiment 4. POLARISATION OF LIGHT
The purpose of this session is to give you an introduction to the concept of the polarisation of light.
Light from a normal incandescent or fluorescent lamp is usually "unpolarised" whereas laser light is
usually linearly polarised - to understand what these terms mean, we have to consider in some detail
the nature of light waves. Light is a transverse electro-magnetic vector wave. For light travelling in
the +z direction (say) there is a time- and space-varying electric field E(x,t)  E 0 cos(t  kz) and
magnetic field B(x,t)  B 0 cos(t  kz) , where E and B are vector quantities, transverse to the
propagation direction (i.e. they lie in the x-y plane). In free space, there is a simple relation between
E and B, so that specifying one of them - usually we choose E - is still a complete description of the
field.
We can think of E as a two-dimensional vector, with components- Ex(z, t) and Ey(z, t) along mutually
perpendicular x and y directions at each point in space (z) and time t. These components oscillate
sinusoidally in space and time, with amplitudes E0,x and E0,y and, in the general case with a phase
difference . Different states of polarisation are the result of different values of E 0,x, E0,y and
particularly , the phase difference. A simple mathematical representation of the field (at z=0,
say) is:
                                           E0, x 
                                      E  E  cos(t )
                                                                                                    (10)
                                             0, y 

Since the field components oscillate sinusoidally, the resultant field is simply the sum of two simple
harmonic motions of the same frequency applied in perpendicular directions. Consider the case when
Ex and Ey oscillate in phase (=0) and we get linearly polarised light. Figure 10 shows the case of
light linearly polarised (a) along the x-axis, and (b) when it is polarised at angle  to the x-axis.
Figure 10 also shows the case in which E0,x = E0,y and (c)  = 45 deg (elliptically-polarised light), and
(d)  = 90 deg (circularly-polarised light). [In the last case, the circle can be seen to be
circumscribed by summing the perpendicular components Ex=E0cos(t) and Ey=E0cos(t-
900)=E0sin(t)].

 a                           b                            c                             d
                                                



            1                      cos( )                 E 0  cos(t )               E0  cos(t ) 
     E  E0   cos(t )                                  E       
                                                                                   
                                                                                       E                
            0                      sin( )  cos(t )
                             E  E0 
                                                                2  cos(t   / 4) 
                                                                                              2  sin(t ) 
                                              
                                                                                                     

   Figure 10. Representation of polarised light states. (a) linearly-polarised along x-axis, (b) linearly-
   polarised angle  to x-axis (c) elliptical polarisation,  = 450, (d) circular polarisation,  = 900.
In natural light (e.g. light from a blackbody), the amplitudes E0,x and E0,y and the phase difference 
all vary randomly in time and therefore no fixed state of polarisation is observed: we say it is
"unpolarised". Laser light is produced in a cavity whose geometry usually forces a single polarisation
state, typically linear polarisation. Light that is initially unpolarised can be made linearly polarised by
a variety of means. In the laboratory experiment, we use sheet "Polaroid" in which long-chain
molecules are aligned in such a way as to transmit only the component of the electric field along one
direction. We refer to this device as a polariser.
Law of Malus. If linearly polarised light of unit amplitude (and hence unit intensity) is incident
on a polariser whose transmission axis is the same as that of the incident light, then the amplitude
                                                          15
and intensity of the transmitted wave is also unity. In practice, there are bound to be losses from true
absorption or from surface reflections and the transmitted intensity will be less than unity. A
polariser used to study already polarised light is usually called an analyser (obviously, the same
device can be viewed as a polariser or analyser depending on its use). On the other hand, if the
linearly polarised light has its electric field component perpendicular to the transmission axis of the
analyser, then in principle the amplitude and intensity of the transmitted light is zero. Again, in
practice, it will have some small, non-zero value because of imperfections in the analyser.
If the angle between the direction of the electric field of the linearly polarised light and the
transmission axis of the analyser is , then the transmitted amplitude (for a unit amplitude incident
wave) is cos (consider for example the input field has polarisation as shown in Figure 10b and that
the analyser polariser is aligned to x-axis). The transmitted intensity is cos2 (since intensity is
proportional to square of electric field). This is usually written in the form,
                                                I=I0 cos2                                 
where I0 is the incident intensity of the linearly polarised incident beam. Eq. (20) is known as the
Law of Malus.
List of Equipment
   Lathe-type optical bench and fittings used for lenses section
   Red LED light source, with power supply
   1 linear polariser ("Polaroid') 40 x 40 mm
   1 analyser (i.e. another linear polariser) in rotation post mount
   Photodiode (as used in diffraction experiments with slit removed) on translation stage and power
    supply.
Experiment 7. Verification of Malus' Law
                                                 bi-convex lens          Photodiode on
                                                 f=160mm                 translation stage

               Red LED
             light source
                                                             linear         analyser in
                                                             polariser    rotation mount

                             Figure 11: Set up for verification of Malus' Law.
Check that the power supply to the photodiode is switched on before taking any measurements. [This
is essential as the photodiode will give a reading even without power supply on but it will not be
linear with intensity]. Before placing the polariser and analyser in position, use the f = +160mm
focal length singlet lens to focus the central bright spot of the red LED light source onto the
photodiode at a magnification of approximately 0.25 (i.e. use eqn 7 and 8). Place the 40 x 40mm
linear polariser and the analyser in its rotation mount immediately in front of the detector, as shown
in Figure 11. Be careful with the alignment: make sure that the light from the LED is passing
centrally through both the polariser and analyser.
If you rotate the analyser, you will find that the signal on the photodiode varies, and in principle
should be varying as cos2. In order to reduce any background illumination you can make sensible
use of black card and switch off your "reading" light if this is practical (but do not keep switching it
on and off as this is going to affect your neighbours' results). Once you have minimised the
background illumination, record the intensity I as a function of the angle  marked on the rotation
mount, for increments of  of say 10deg between 0 and 360deg. The angle  =  + , where  is an

                                                    16
unknown offset which depends on angle of the transmission axis of the linear polariser (this depends
on how you mounted the polariser).
Plot a graph of I versus . Your graph should be cos2 in form (but not necessarily equal to a
maximum at  = 0 deg) on top of a background level. Determine the offset angle  and the mean
background level Iback.
Plot a second graph of I – Iback versus cos2 () using the computer and determine the least squares
straight line fit. Have you verified Malus law?




                                                 17

				
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