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Beamwidth

Beamwidth of an Eye

Objective:  To understand the concept of beamwidth by measuring the beamwidth of your eye.

Introduction:  (To be read by students or presented as lecture)  When we point the SRT at the sky, it does not act like our eye.  No image is formed - the telescope measures the intensity of radio waves of a certain frequency coming from one direction.  This is like a single pixel image that changes depending on where we point the telescope. 

Imagine what it would be like if your eye worked this way.  To see a person’s face you would have to look at each point on the face, and slowly build up an image.  You would have to look at a few thousand points just to be sure it was a face!  And a few million points to get as clear an image as you see with your eye.

Sitting a few meters from a whiteboard, you can see fairly sharply a dot made by a marker.  (If  you cannot you should get glasses!)  The beamwidth of your eye depends on the smallest dot you can see sharply.  Since  you can see different sized dots sharply depending on how far you are from the board, we measure beamwidth as an angle.

It may surprise you that the best the beamwidth of your eye (or any optical instrument) can be is limited by the wave nature of light.  The equation for the diffraction limit (the best possible beamwidth) is given by

q = 1.22(l/d),

 where l is the wavelength of the light and d is the diameter of the aperture the light is entering. 

Procedure 

1.       One person should make a very small, filled in circle on the whiteboard, and a very small “+” next to it.

2.       The second person should start as far from the board as possible, and walk toward the board until it is possible to tell for sure which is which. 

3.       Measure the distance from the board to the observer, and the diameter of the circle. 

Analysis

1.        Make a carefully labeled sketch of  your experiment, showing what  you measured.  The sketch need not be to scale.

2.        Find the beamwidth of your eye from the data you gathered.  For small angles, tan q » sin q » q.  Look up the approximate wavelength of the color light you were seeing.

Discussion Questions

1.       We chose the width of the object on the board to be the diameter of the circle.  Is this appropriate?  What other choices could you make?

2.       What ways are there to improve the image  you see?  What devices are used to see smaller or farther away objects?  How do these devices improve the beamwidth of your eye?

3.       Find the diameter of a telescope that could see sharply a star the size of the sun, a distance of 10 light-years away, using 700 nm light.

4.       When an object is very small or far away, but is visible because it is emitting its own light, how large    does the object appear?

The answer is that the apparent size of the object is mainly determined by the beamwidth of  your eye.  The brightness of the object also plays some part.   When you look up at the sky, all the stars are much, much smaller than the beamwidth of your eye.  So they appear as dots smeared out to the beamwidth of  your eye.

 

5.       If only the beamwidth of your eye determines the apparent size of distant stars, why do brighter stars appear larger than dimmer stars?

 

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