Comparison of Methods for Measuring Focal Lengths
By Bobby Rohde
3-15-99

Abstract

In this experiment I measured the focal length of a concave mirror, a concave lens and a convex lens, each through use of more than one method.  All three items were measured by virtue of finding the image position of an object under their effect.  The mirror was also studied by means of measuring the angle of reflection of a laser beam, and the lens were directly measured with a spherometer.  The focal length of the mirror was found by the two methods to be 39.9 +/- 0.24 cm and 42.9 +/- 3.29 cm.  The lenses were found indirectly to have length, 19.4 +/- 2.29 cm and 21.9 +/- 1.20 cm for the convex and concave lens respectively.  These were then measured directly to have lengths, 21.0 +/- 0.331 cm and 23.3 +/- 0.328 cm respectively.  The virtues of different methods were then compared.

Introduction

This experiment was fundamentally divided into three different sections, each of which used a different method in order to measure focal length of the given objects.  In the first section the focal length of a concave mirror was computed by measuring the displacement of a laser beam reflected off the surface of the mirror, and then relating that to the distance from the mirror.  In the second section all three objects, i.e. the mirror, a doubly concave lens and a doubly convex lens, were measured my determining the position of the image of a fixed object created by each.  This was found by means of observing the parallax motion between the object and its image.  A marker was positioned such that its position corresponded with the apparent position of the object's image as measured by having a zero parallax shift in the relative speeds of the marker's motion versus that of the image.  Once the image distance and the object distance are thus determined it is a simple matter to compute the focal length.  The third section of this experiment was simply the use of a spherometer to directly measure the curvature of the two lenses in order that one might have data with which to compare results.

Experiment

Section 1:  As shown below (Fig. 1) an optic track was setup containing a laser, a beam adjuster, screen with thin slit and the concave mirror.  The screen with thin slit also had a measured grid incremented in millimeter.  The beam adjuster was used to position the laser beam such that it passed through the center of the slit in the screen and then reflected off the center of the mirror and retraced its path back into the laser.  This was used to determine the zero point on the screen for purposes of measurement.  The beam adjuster was then used to displace the beam vertically and parallel to its original position so that it hit the beam higher up on its surface.  The beam then reflected back at some angle off vertical and impact the slit in the screen at a position other than that demarking the incoming beam.  The height of the parallel displacement, the height of the reflected beam and the distance between the screen and the mirror were then recorded.  When measuring the the distance from the screen to the mirror, it was taken in two parts, the distance along the track between the base of the screen and the base of the stand holding the mirror and then the depth of the mirror as measured by calipers to both the center of the mirror and the edge.  The total distance L was measured several times, and values were recorded for several different displacements of the beam.

Section 1 Setup (Fig 1)

Section 2:  In this part of the experiment the radius of curvature of the lenses and mirror was found by locating the image position for a given object position.  This was done by arranging the the mirror or lens on the optics track with two markers (Fig 2).  The lens and one marker, designated the object, were setup arbitrarily and then the position of the third marker was determined by looking down the track at the image of the object and then positioning the second marker so that it appeared to occupy the same space as the image of the object (Note the relative position of the object image and lens/mirror can change).  This effect what determined qualitatively by moving ones head out of line with the optics track and watching to see that the image and the marker moves at the same speed, i.e. had zero parallax shift.  This was repeated for each object for multiple positions, with both real and virtual images (when possible).  Additional the error in trying to determine parallax zero position was found through repeated determination of the at which there was zero shift for a fixed position of the object and lens.

Section 2 Setup (fig 2)

Section 3:  In this part of the experiment a spherometer was used to directly measure the curvature of of the two lenses.  It was first used on a flat surface in order to determine its zero point, multiple measurements were taken.  Also it was depressed onto some paper in order to leave an impression which allowed the distance between the prongs to be determined.  Then the spherometer was placed on the surface of each lens and used to measure its depth.  This was repeated multiple times and from this measurement one can determine the curvature of the lens.

Section 3 Setup (Fig 3, taken from Phys 375 Lab manual, published Univ. of Maryland)

Results/Analysis

Section 1:  When measuring the displacement of the laser beam from being reflected by the mirror one finds that the radius of curvature is determined by : R = (2*a)/(a-b)*L, where a is the displacement of the incoming laser beam, b is the height of the reflected beam, L is the distance between the screen at which a and b are measured and the surface of the mirror, and R is the radius of curvature.  The results for this part of the experiment were as follows : L was measured to be 17.81 +/- 0.12 cm and was not changed during this part of the experiment.  This value was gotten through repeated measurements taken in stages and the error represents the standard deviation.  The values for a, b, and R and there respective errors were :
 

Table 1 - Results from Section 1

Here the error in a/b is taken merely to be the limitation of measuring the distances on the grid provided.  Error in R is the computational extension of the error in L and the error in a and b.  For obvious reasons the fifth data point appears to be erroneous and was there for discarded in the calculation of best focal length.  Focal length is of course one half the radius of curvature and from this and waiting in terms of relative error we obtain the best approximation for focal length given by this data as 39.9 +/- 0.24 cm.

Section 2:  Here of course the experiment consisted of three part one dealing with each piece of equipment.  With the mirror (same as in section one) the data was broken down and placed onto a convenient graph shown here :

Mirror Data (figure 4)

With so denoting object distance, si denoting image distance and each in units of cm.  The line represents the best Chi^2 fit to the data.  Since 1/si + 1/so = 1/f where f is the focal length, it then follows that would be equal to the inverse of both the inverses of the x and y intercepts of the line shown.  These values were then computed and the error was computed based on the error weighted average of the distance of the points to the line of best fit.  Thus the best value of f was found to be 42.9 +/- 3.21 cm.  This is in reasonable agreement with the result reported for section 1, with this value less than 1 error term from that reported in section 1.

In the second part of this section the focal length of the convex lens was found through similar means, with the data tabulated as follows:

Convex Lens Data (Figure 5)

Using the same style of analysis as above the results for f were found to be 20.0 +/- 4.9 cm. However it was seen that the fifth point (see fig 5) seemed to be skewing the graph downward. By preforming some back calculations it was found that if the image distance of this point was adjusted by -2.7 cm all the points fell in line and the resulting error was significantly reduced. The results are thus found to be f = 19.4 +/- 2.3 cm.

In the last portion of this section the same analysis was done on a concave lens with the data as follows :

Concave Lens Data (Figure 6)

From this data and the same type of analysis as before the value of f was extracted to be 21.9 +/- 1.2 cm.

Error in Section 2: While I have already explained that the error for f was computed as a function of the distance of the points from the line of best fit, it is important to mention where the error bars on the data shown here was arrived at.  The ability to measure parallax shift = 0 was determined by means of multiple measurements on specific object/lens position and the standard deviation of this was used to determine the error for finding an image distance.  This was then translated into the 1/si and 1/so error values. However as another measure of accuracy we can look at the slopes of the data on the various graphs which should theoretically be -1 in all cases. For the mirror, convex lens and concave lens respectively the slopes were, -1.072, -1.050, -1.231. It may be important to note that all three of these values is less than -1, this may indicate that the error in parallax was not sufficiently compensated for with only one set of test runs, and may vary in an unknown way as one adjusts the object and image distances. This is the primary reason why I chose to report the error on f as a function of the scatter of the points.

Section 3:  In the last portion of this experiment there were three pieces of information directly measured.  The 0 point of the spherometer (p0), the height of the curvature surface (h') of the two lenses and the distance between the prongs of the spherometer (b).  Let h be the height of the lens surface minus the reading for the zero point of the spherometer.  We know therefore that R = ((b^2)/3 + h^2)/(2*h), where R is the radius of curvature.  Since the focal length is half the radius of curvature AND these lens are double sided we can take f, focal length, to be equal ((b^2)/3 + h^2)/(2*h).  Therefore the data looks as follows :

p0 = 10.2 +/- 0.036 mm, b = 3.893 +/- 0.0227 cm

Convex Lens (Table 2)

Concave Lens (Table 3)

In both cases the error of f is reported to be the error extrapolated from uncertainty in p0, b and h.

Thus by error weighted average the best value of f for the convex lens is 23.32 +/- 0.328 cm and the convex lens 21.04 +/- 0.332 cm.

By comparing to the results obtained in section 2 we see that these values are reasonably comparable, with the value for concave less than one standard error different and the value for convex just over a standard error.  However as these values are far more precise and can be obtained easier it clearly suggests that using a spherometer is the preferably method for calculating the curvature of a lens.

Conclusion

After performing multiple experiments one can be reassured by the fact that the values calculated of focal length come out similar when computed by a variety of means.  Since this is true it seems that the best method for calculating curvature is that of using a spherometer as this procedure is both quick and yields higher accuracy than any of the other methods.