EXPERIMENT - I
1- OBJECTIVE AND BASIC THEORY
Mathematically, the paraxial behaviour of an optical system can be summarized by the socalled equivalent system , a set of geometrical points and planes, also called cardinal elements , of special significance: the principal points and planes, H and H’, and the focal points and planes, F and F’. One of the most important parameters of an optical system is the image focal length , f’ , defined as the distance H’F’. We shall determine these elements for a real system: a thick converging doublet.
By using the equivalent system, the position of a paraxial image O’ can be obtained from the position of the object O (see Fig.1), either referred to their locations with respect to the focal points
( z, z’ positions) or measured from the principal points ( a, a’ positions). Each system produces a different, though equivalent- mathematical equation. If light goes from left to right, and if the sign of all distances are positive to the right and negative to the left, these equations are: z z’ = - f’ 2 ; - (1/ a ) + (1/ a’ ) = (1/ f’ )
O
 a z
F
 f
H H´ f´

F´ z´ a´
O´

Fig. 1 . Location of the principal and focal points and planes. The position of an object and that of its image is referred to either pair of cardinal elements. The focal lengths and the first and last surfaces of the system are also indicated.
The problem may be reduced to: i) Determining the focal length of the system ( f´ =  f ); ii) Measurement of the FF' distance. iii) Measurement of the distances from each focal point to the nearest real surface.
Lets remind that the system will be analyzed in air, i.e the incoming and outcoming index are n=1, and we may use the corresponding equations for these conditions (with equal extreme indices the modulus of the focal distances in the object and image spaces are the same): f´ =  f =  HF = HF´.
2- MATERIALS
• Optical bench with a millimeter scale.

• Conventional bulb lamp.
• Point-source: Scattering paper stuck to a small orifice made in a thin metallic slide.
• Collimating lens (plus a card with a small hole in the centre).
• Unknown optical system (plus a card with a small hole in the centre)
• Half-cut top reference-post
• Low magnification telescope.
• Long focal-length microscope..
• Holders with a vernier scale to get 0,1mm precision in their position on the bench.
3- ALIGNMENT AND SYSTEM PREPARATION
First of all, it is important to identify correctly the working materials. Check that the source is properly illuminating the point source.
As a starting point, all elements must be put on the bench in the correct order (see Fig 2):
Lamp  Orifice  Collimating lens  System  Microscope. Try to set all the elements in the same approximate height and transverse position than the orifice (the alignment that is described later pursues a centred system with a point-source on its axis). We will take the orifice as a fixed reference (not moving it again) and try to centre more precisely the rest of elements according to it.
Lamp
Orifice,
Point-Source
Collimating
Lens
System
-Unknown-
Microscope
T
T
L
Fig.2
Scheme of the material location on the bench. The horizontal+vertical transversal movements are indicated with
T, while the longitudinal movements along the bench are indicated with L.
The next step is focusing the microscope on the orifice (L-movement). To do this we can put apart or remove temporarily the rest of elements in the bench, if we prefer. We shall focus the orifice in such a way that it appear centred in the field of view (T-movements). Now the microscope will act as a transversal reference for the alignment of other elements on the bench.
For centring the collimating lens, we shall fix the circular cardboard on the lens. The cardboard has the size of the lens, and can be either held alone or helped with a sticky tape. We now focus our microscope on the collimating lens and try to find the (irregularly shaped) central orifice of the cardboard (approximately coincident with the centre of the collimating lens). By T-moving the positioners of the lens, not the microscope’s, we put the image of the orifice on the centre of the reference cross of the microscope. After this process, the cardboard must be removed.
The same procedure is carried out for the unknown system, focusing on its corresponding cardboard and T-moving the lens so that it get centred in the field of view of the microscope. After

this, the point source and the centres of the collimating and unknown lenses (where the orifices in the cardboards were located) are in a straight line along the bench: the optic axis of the system.
4- PROCEDURE
In an experiment, it is advisable to carry out every step, checking that everything is correct, before actually reading and taking the values.
1.-Produce a collimated beam: The point-source must be placed in the object-focus of the collimating lens. We first put the lens far from the point-source (half a meter or more), and find the point-like image of the point-source. Then, we approach the lens to the object, step by step, tracking the position of the image farther and farther from the lens (and with an increasing size which allows us to recognize the details of the orifice and diffuser). Making the steps smaller we end up with an image as far as it is possible to produce in the lab (from five to ten meters away). Now, the lens is very close to the correct position, and we must have a beam that approximately keeps its transversal size constant. We may increase the lens another small step and check the image (ideally in the infinity) with a small magnification telescope. [In order to prepare the telescope we may look through a window and move the eyepiece till we can distinctly observe very distant objects. These operations are better carried out by two persons]. After this process we shall consider that the collimating lens is producing a beam of rays parallel to the optic axis. By using a diaphragm, we limit the transversal size of the beam to a few cm, removing those rays that suffer the strongest spherical aberration. In this way, we rely on the paraxial behaviour of the system, and the criterion of maintaining the transversal size of the beam is more reliable.
2.-Place the unknown system in a fixed position, so that the collimated beam is incident on it. This position will not be varied during the experiment (the holder has a fixing accessory).
G
Light
F

O
1

O
1
´

H H´

O
2
F´
 z
1 z´
1
Light
F´
 z
2
O
2
O
2
´
 
H´
G
H

O
1 z´
2
F


Fig.3
Top : Starting situation for the unknown system, O
1
and O
2 faces). Positions of O
1
and its image O
1 system around the G axis, with light entering the O the points after the turn). The positions of O
2
2
face (we keep the notation for F and F´ for easier identification of
and its image O
2
representing the outer limits of the system (external
´, measured from the focal points, are shown. Bottom: Situation after turning the
´ with respect to the focal points are indicated.
The microscope is placed after the system, and must be moved longitudinally so that the observer can see through it the image of the point source. This position of the microscope, x
1
, must be read and written down in the laboratory note-book.
3.- We now cut two fine stripes of sticky tape and put them well centred on the faces of the unknown lens, one horizontally and the other vertically, for better distinguish them. These thin objects will represent points O
1
and O
2
(see Fig. 3, Top ).
4.- Move the microscope so that the tape representing O
2
is well focused. The position of the microscope on the bench, x
2
, must be read and registered. The difference ( x
1
-x
2
) represents the distance O
2
F'. After this, we move the microscope till the other peace of tape is focused. We are observing the image of O
1
through the system, O'
1
. We take the position of the microscope, x
3
.
5.- Now, we must carefully rotate the system 180°, trying to keep the height of the lens (see
Fig. 3, bottom ). By repeating the former process (steps 2 and 4) we focus on the image, and on the points O
1
and O'
2
taking three new positions of the microscope that we shall denote, respectively, x'
1 , x'
2
and x'
3
.
6.- We have two pairs of conjugate points (O
1
, O'
1
) y (O
2
, O'
2
) and some sets of values to characterize their positions. The magnitudes z and z´ can be obtained as follows:
 z
1
 =  FO
1
 =  x’
1
- x’
2
 >0 ; z
 z’
 z
 z’
2
 =  F’O
2
1
 =  F’O’
 =  FO’
2
1
 =  x
1
 =  x’
1
1
> 0  z
1
=  x’
1
- x’
2

- x
3
 >0 ; z’
1
2
 =  x
1
- x
2
 >0 ; z
2
- x’
3
 >0 ; z’
2
< 0  z’
1
> 0  z
2
< 0  z’
2
= -  x
1
- x
3

=  x
1
- x
2

= -  x’
1
– x’
3

[Note the care with the signs. Obviously, the result cannot depend on the positive direction of the scale on the bench. We know the real sign of these coordinates
(Fig. 3), crucial for applying Newton’s equations (1) and (2).]
According to Newton’s equation for a system with identical external indices, we have two equations from which to obtain the image focal length of the system, f´ : one for the pair (O
1
, O'
1
) and the second for (O
2
, O'
2
): z
1
z’
1
= f’ 2 (1) z
2 z’
2
= f’ 2 (2)
6.- By repeating six times the measurement of the sets ( x
1
, x
2
, x
3
) and ( x’
1
, x’
2
, x’
3
), we obtain the corresponding values for z
1
, z'
1
, z
2
, z'
2
. From Eq. (1) and (2) we can calculate f´ , averaging the twelve values (grouped in two families that should be very similar), and we may also obtain its corresponding variance 
f’
2 . Typical deviation , 
f ’
, will indicate the precision of this f´ measurement.
7.- Distances from the focal points to the actual limits of the system are  FO
1
 =  x’
1
- x’
2
 and  F’O
2
 =  x
1
- x
2
 . We can obtain the average value and standard deviation from our set of measurements.

8.- Finally, take the unknown system out of its holder and, taking care of not moving the carrier, put in the reference post (The one with the half-cut top). Focus the microscope on the planar face of the reference post, trying to see clearly the central groove. We may need external illumination to perform this part (for instance, with the torch). The position of the microscope on the bench, x
4
, is read and written down six times, obtaining its average value and standard deviation.
With this value we can calculate the distance from F to F´ and its standard deviation:
 FF´  =  FG  +  GF´  =  x´
1
- x
4
 +  x
1
- x
4

9.- From the results obtained so far, we can calculate the distance between the principal planes, H H’, and their position with respect to the actual limits (external faces) of the system. The thickness of the system can also be obtained through a simple sum. As a help to this part you can draw an schematic graph of the system in which all the real and cardinal points as well as the distances are represented (H, H´, F, F´, G, O
1
, O
2
, O
1
´, O
2
´, z
1
, z
2
, z
1
´, z
2
´, f , f´ ) therefore having a representation of the equivalent system on the real one.
5- QUESTIONS
1.
Would you say that this method can be applied to a diverging lens? Why?
2.
What is the consequence of a poor collimation, by producing either a slightly convergent or diverging beam?
3.
What do you think about the collimating method you have applied? Do you know of a way to improve it?
4.
After all the time and effort used in the collimating process, why do you think it is so necessary? In other words, would it be possible to use a point source instead of a collimated beam? Could we work directly with the point source as and object and work with its position?
5.
From the measurements taken in this experiment, can you predict the exact position on the bench of the image of the point source if the collimating lens is removed? (suppose you took the position of the microscope when focusing on the point source, say x
5
)