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## Angular resolution of our eyes

Introduction
Fig. 7 shows two patterns. The first pattern has many vertical lines while the other is simply gray in color. In this activity, we will see how good our eyes are in resolving the vertical lines in the former pattern.

 Fig. 7 Two image patterns consist of (upper) vertical lines and (lower) gray color.Download printable size Fig. 8 Looking the line pattern at a distance.

At a certain distance L from the patterns, you see an angular separation q between adjacent lines in the pattern (Fig. 8):

 q = a L

where a is the line separation. When you move away from the figure, the angular separation between adjacent lines decreases (i.e., the lines appear closer). You will soon discover that the lines can no longer be resolved by your eyes when you move beyond a certain distance Lm . In this case, the line pattern will look just identical to the gray pattern below. By determining the observation distance Lm at which the lines are barely resolvable, you can estimate the angular resolution of your eyes.

Procedure

1.

Download the printable version of the figure and print a hardcopy with a printer. The separation between adjacent lines in the hardcopy should be a = 1 mm.

2.

Post the figure at more than 5 m away from your eyes. The two patterns should look identical with the same shade of gray.

3.

Walk closer to the figure until you can barely resolve the lines and tell the difference between the two patterns. From the distance Lm (measured in metres), calculate the angular resolution of your eyes:

 angular resolution = 10-3 Lm

4.

Compare the angular resolution you estimate in this experiment with that found from the Rayleigh's criterion. How do you explain the difference?

Remarks: The actual angular resolution of our eyes depends on many factors including the optical defects in the eyes and external conditions. In a dim environment, for example, the pupil of our eyes is widen, and the resolving power of our eyes will increase slightly. In other words, our eyes are biological optical systems with adjustable resolving power.

## Laser experiment on Rayleigh's criterion

Introduction
Reyleigh's criterion states the theoretical angular resolution of an optical system is given by

 angular resolution = 1.22l D

Therefore an instrument with a large aperture diameter D will have a higher resolving power. In this activity, we will direct two laser beams through a small aperture and form two bright spots on the screen. For a small aperture diameter D, the diffraction pattern is large in size, making the two bright spots not resolvable. When the aperture is widen, the bright spots will become sharper and finally resolvable. This demonstrates how the resolving power of an optical instrument is limited by its aperture.

Apparatus1
 1 Two laser pointers 2 Three adjustable apertures/irises 3 Adjustable mirror 4 Adjustable reflective beam splitter 5 Variable intensity filter 6 Screen

 Fig. 9 Top view of the experimental setup.

Procedure

1.

The two identical lasers (can be laser pointers) are mounted at the same level and side-by-side to emit two parallel laser beams (Fig. 9).

2.

Two apertures (C and D) are placed in front of the laser to ensure the laser beams are almost circular.

3.

A mirror is to steer one laser beam B to hit on the beam splitter almost at the same position as the other beam A.

4.

By using a reflective beam splitter placed at around 45o, part of the laser beam A will pass through the beam splitter while part of the laser beam B will be reflected from the beam splitter. Adjust the mirror and beam splitter so that two beams are hitting on the beam splitter at the same horizontal level but at a small distance a apart, where a is typically of several millimeters. Adjust the variable intensity filter so that two beams have almost the same intensity.

5.

Further adjust the mirror and beam splitter so that the beams are overlapped with each other at aperture E which is at a distance d from the beam splitter. Aperture E mimics the aperture of an optical instrument (or the pupil of our eye). Two bright spots are formed on the screen which is placed a fixed distance from the aperture.

6.

When the diameter D of aperture E is made very small, the bright spots projected on the screen appear fuzzy or simply not distinguishable. Slowly widen the aperture until the bright spots are barely resolvable. Measure the diameter D of the aperture with a ruler.

7.

On the other hand, the angular separation of two laser beams can be calculated as:

 q = a d

According to Rayleigh's criterion, when the two bright spots are barely resolved, the aperture diameter Dm is

 Dm = 1.22l = 1.22l × d q a

Measure a and d with a ruler, calculate Dm and compare with the result you found in step 6. What is your conclusion?

 Fig. 10 Part of the apparatus used in the experiment. The beam splitter is shown on the right of the figure. Fig. 11 The two bright spots appearing on the screen. When the aperture diameter decreases, the spots become less resolvable.

1The materials can be purchased from suppliers of scientific instruments.

## Comparing optical microscopes with electron microscopes

 1 What are the wavelengths for blue, green and red lights? 2 What is the de Broglie wavelength of an electron accelerated from rest through a potential difference of 20 kV? 3 According to the answers in Q1 and Q2, which one can be used to achieve a higher resolving power, electrons or visible light? Why?