ACTIVITY4

In Kit #1 of this series, you learned about the technique of spectroscopy, the diagnostic tool that enables astronomers to measure physical conditions, velocity, and distances of astronomical objects. Here you will learn about the principles on which the actual instrument is based.

An Introduction to Spectrometers

In the 1870's, a Johns Hopkins University professor named Henry Rowland invented a machine called a "ruling engine" that etched thousands of closely spaced lines per centimeter onto glass or metal plates, thus producing diffraction gratings. Diffraction gratings are fundamental components of spectrometers, scientific instruments that have lead to many astounding discoveries of modern astronomy.

Astronomers study light that comes from extraterrestrial objects. When modern astronomers use the word light, they do not mean only visible light; they mean electromagnetic radiation of all wavelengths. Most naturally occurring light is polychromatic. For example, light from our closest star, Sun, appears to be yellowish white. However when sunlight is passed through a prism, it is dispersed (spread out) into a rainbow of red, orange, yellow, green, blue, and violet colors. Not only is the yellowish white light of Sun composed of these visible colors, it also contains longer wavelength infrared light and shorter wavelength ultraviolet light. To learn more about how astronomers use light to learn about the universe visit: http://violet.pha.jhu.edu/~wpb/spectroscopy/spec_home.html.

The Far Ultraviolet Spectroscopic Explorer (FUSE), as its name implies, will contribute important clues to further our understanding of the universe by observing astronomical objects in a band of far ultraviolet light of wavelengths 91 - 120 nanometers. (Note that far ultraviolet refers to the short wavelength end of the portion of the electromagnetic spectrum called ultraviolet.) Therefore the telescope within the satellite must not only "catch" light, but the science instrument must organize the light in its wavelength components. The spectrometer's diffraction gratings perform this function.

How Light Behaves: Some Basic Optics

To understand how the diffraction gratings perform, you must know that light interacts differently with different sized objects. If the wavelength of light is very much smaller than the dimensions of the objects in its path, the light can be modeled as a stream of particles. For example, when light is reflected from a smooth pocket mirror, it acts like a stream of tiny elastic balls. If the light is aimed directly at the mirror, it all bounces straight back., or reflects. However, if we could shrink the mirror so that its width is almost as small as the wavelength of the light, we would observe one of the wave-like properties of light: diffraction. The light does not all bounce back in the direction it came from. Instead, it spreads out, or diffracts, as shown in the figure below.

In this activity, you will learn that both light phenomena: reflection and diffraction, play an important role in the FUSE spectrometer. The branch of physics that describes the behavior of light is called “optics”. Optics has many useful applications in everyday life as well as astronomy.

Figure 1. How light reflects depends on the size of the reflector.

Observing Diffraction of Water Waves (Optional)

The wave phenomenon of diffraction is observed when water waves bend around a corner or spread out when passing through a narrow opening. Partially fill a flat, glass baking pan with water. Dip a straight edge, like the flat side of your protractor, up and down to make waves in the pan so that the wave fronts look like evenly spaced straight lines. Now hold a second protractor still in the water so that it blocks part of the straight waves you are making. Do you see the water waves bending around the barrier as shown in the figure below? Ask a classmate to hold two barriers in the water so that there is a small opening for the straight wave fronts to pass through. If you again make straight waves, you will see curved wave fronts spreading out beyond the opening. The wave comes to the barrier heading in one direction and, at a corner, spreads into many directions.

Light that passes through a very narrow opening also exhibits this spreading, as does light reflected from a very thin mirror. This is the phenomena called “diffraction.” Diffraction gratings are designed to either spread the transmitted light (transmission grating) or the reflected light (reflection grating). In Step 4 of this exercise, you will learn how a diffraction grating, a set of regularly spaced openings or reflecting surfaces, separates light into its component wavelengths.

Figure 2. The diffraction of water waves showing: a) bending around the corner of a barrier and b) spreading through a small opening.

Tracing Light Through FUSE

In this worksheet, you will trace light from a distant astronomical object through the FUSE instrument until it lands on the detector, where it is then recorded as a digital signal. Light may take one of four paths through FUSE, each geometrically similar. For simplicity, we will focus on only one path. In each path there exist four basic devices: a mirror, a few slits, a diffraction grating, and a detector. First, light falls on the mirror that serves to collect light and then reflect it into the spectrometer. Light reflected from the mirror enters through a narrow slit, falls on the grating, and is diffracted into a spectrum. Each photon lands on the detector at a specific location according to its wavelength.

To complete this activity, you need a calculator and a protractor. Use the protractor as a straightedge to draw straight lines as well as to measure angles. You will be introduced to relevant concepts and equations as needed, and you must use these tools to deduce how the FUSE spectrometer works. You will accomplish this task in four steps.

Step 1 - Reflection

Light from a distant astronomical object, say a star, is incident on a mirror, which reflects it. The Law of Reflection for mirrors states that the angle at which light is reflected from a mirror equals the angle at which light is incident on the mirror. In symbols, the Law of Reflection is:

q _i = q _r

where q represents the measure of an angle between a light ray and an imaginary line perpendicular to the mirror surface, the normal.
q _i is the measure of the angle between the incident ray and the normal.
q _r is the measure of the angle between the reflected ray and the normal.

In order to enter the FUSE spectrometer, the reflected light must pass through a slit. This limits the light analyzed by the spectrometer to that emitted by the target star. In only one of the drawings in Figure 3 can light from the star obey the Law of Reflection and reach the slit.

a) Circle the letter of of the drawing that allows star light to pass through the slit.
b) Draw a line representing a light ray from the star to the mirror and draw another line to represent the reflected ray.
c) Draw the normal and label and record the measures of angles q _i and q _r.

Figure 3. It is only possible to trace light from the star to the slit in one of these sketches.

Step 2 – Derive the Grating Equation

Now that you understand how light passes through the slit, you must determine what governs the light's path inside the spectrograph. The three components of the spectrograph, the slit, grating, and detector are all positioned on a circle as shown in Figure 4. This circle has a special name, the Rowland circle, named for the professor mentioned earlier. In fact, the grating has a curved surface with radius of curvature equal to the diameter of the Rowland circle while the detector face lies on the circle. It is very useful to spectrometer designers to use this configuration since the curved grating then focuses the spectrum onto the curved detector. Recall that the "image" that falls on the detector is not a picture of the star, but a spectrum of its light since the light has been spread according to wavelength by the diffraction grating.

Figure 4. When the specially curved grating, the slit and the detector are positioned on the Rowland Circle, light entering the slit is focussed on the detector.

How does the diffraction grating separate light into its components at different wavelengths? You will now derive an equation that describes how different wavelengths of light are reflected by the grating at different reflection angles. Note that in order to do this you must think of light not as a stream of particles, but as a collection of waves of varying wavelengths. First, consider a grating illuminated only by monochromatic (single colored) light. For simplicity, also suppose the light is incident normal to the plane of the grating.

Consider the light reflected at a certain angle, q , as shown in Figure 5. The two arrows represent light rays reflected from two consecutive grating surfaces separated by spacing distance, d. Note that light ray A will travel a longer distance than light ray B. If this longer distance is a multiple of a wavelength, A and B together will add as shown in Figure 6. In this case A and B are said to be “in phase” and they will combine to give a wave on double amplitude and maximum brightness. The opposite extreme, destructive interference, would be the combination of waves A and C in Figure 6. These waves cancel.

Figure 5. Three rays of light reflect from adjacent reflecting surfaces of a diffraction grating.

Figure 6. Waves A and B are "in phase" and add "constructively." Waves A and C are "out of phase and will cancel.

Apply triangle trigonometry to the shaded triangle in Figure 5 to derive the grating equation:

d sin q = m l

d = groove separation in grating
q = angle between incident and reflected light ray
m = 0, ± 1, ± 2,… spectral order (an integer)
l = wavelength of light

HINT: Note that the shaded triangle is a right triangle and one of its angles has measure q.

Step 3 – Use the Grating Equation

Now that you understand how the grating equation applies to monochromatic light, you can apply the equation to determine how polychromatic light falls on the detector.

a) Using the specification that the FUSE gratings have approximately 5500 lines/mm and the wavelength region over which the instrument is sensitive is about 91 - 120 nm, calculate the angle at which the first order (m = 1) spectral line of light with l = 91 nm is diffracted.

b) At what angle is light of 120 nm diffracted?

Step 4 – Test Your Understanding

Putting together all of your knowledge of how the FUSE spectrograph works, on Figure 7 below:

a) Draw a mirror in an appropriate location and angle so that light from the star reflects through the slit onto the grating along the normal to the grating.

b) Use the results of your calculations in Step 3 and a protractor to draw two rays leaving the grating and arriving at the detector at a point of constructive interference, one representing 91 nm wavelength light and one representing a 120 nm wavelength light.
Note that in the actual FUSE spectrograph, light is incident on the grating at angles greater than 0^o, so the equation that predicts the angle of diffraction is a little more complicated. If you are interested in the more general grating equation, look at the Diffraction Grating Handbook.