Chapter 2:
Properties of a Spectrum
It is important to have a general understanding of a diffraction grating when operating the spectrometer and examining spectra. In this chapter, I introduce the grating equation and the intensity equation. The chapter is not math intensive, but the equations are introduced in order to demonstrate how the characteristics of the grating affect the spectrum.
2.1
Diffraction and a Spectrum
When a wave passes through an entrance smaller than its wavelength, it spreads out, producing a redistributed wave. This property is know as diffraction and is characteristic of all types of wave phenomena. The redistribution of the wave shape is a result of the wave interfering with itself as it passes through the opening. Along the opening, every infinitesimal point acts as its own wave source, which then interferes with its neighboring wave source. There is no difference between interference and diffraction, but the term “interference” is generally used when only two waves interfere and “diffraction” is used when many waves interfere. Figure 2.1 shows the redistribution of a plane wave into a circular wave as it passes through a small opening.
Figure 2.1
The new wave front in Figure 2.1 is circular and has an evenly distributed intensity or radiation field. Another opening can be made in the barrier to produce a similar circular wave front. This new opening is the same width d, and is separated by a distance s from the first opening. This second wave front also has an evenly distributed radiation field. However, if the original wave is monochromatic, the two circular wavefronts interact and the resultant radiation field develops minima and maxima. Constructive and destructive interference between the two wave fronts creates dark and light fringes, or intensity minima and maxima respectively. This may sound familiar as Young’s double slit experiment which he used to demonstrate that light behaves like a wave. When Young examined the intensity pattern on a screen, he saw light and dark fringes similar to Figure 2.2.
Figure 2.2
Young found that the positions of the maxima were determined by the separation of the slits and the wavelength of the light. These values are related by
s sinq = nl. (2.1)
The term n refers to the order number of the maxima and the angle q is taken from a line perpendicular to the equidistant point between the two slits. The zero order maximum occurs at an angle of zero and is an angle common to all wavelengths. If the fringes are observed on a screen at a distance much greater than s, the small angle approximation can be used and q can be used to measure the position along the screen in radians. This equation also holds true for the position of maxima when multiple slits are used, and therefore it is also known as the grating equation.
The intensity of the resulting field is modeled by the intensity equation:
(2.2)
Within this equation is buried the grating equation; however, the intensity equation gives a more accurate picture of the intensity distribution. The term N refers to the number of slits and Io is the intensity of the original wave. In Figure 2.3, we see the intensity distribution of the two-slit barrier. This image shows the maxima out to the third order. The equation is not linear, so the maxima are not evenly spaced; there is a trend for greater spacing and wider maxima at higher orders.
Figure 2.3
If two waves of different wavelengths strike the barrier, each slit then produces two wavefronts, which in turn produces two different interference patterns with apparently no effect on each other. Figure 2.4 shows the interference patterns. The blue line is the interference pattern with the shorter wavelength and the red line is from the longer wavelength. The only place where the maxima of the different frequencies overlap is in the zero order; at other orders the maxima are isolated. We can tell exactly where the maxima will occur for each frequency through the grating equation. We can see that lower frequencies, or longer wavelengths, have maxima farther from the zero order.
If the sources were producing many frequencies, we would see a corresponding pattern for each frequency or color. If we examined the maxima from one particular order, we would see the full spectrum of all the frequencies present. The position of the maximum tells us exactly the wavelength to which it corresponds. In addition, the intensity of that maximum tells us about the intensity of the individual frequency. If light of that wavelength was being strongly produced, its maximum would be intense as well. In the same respects, a weak maximum, or even the absence of a maximum, tells us that the frequency is being weakly produced. In Figure 2.5, another wavelength of light is added to show a spectrum of three colors. The isolation of these wavelengths is exactly what a diffraction grating does. Even a two-slit barrier serves as a crude diffraction grating, but it has the disadvantage that the orders are wide and the different colors would be difficult to distinguish.
2.2
Resolution of a Spectrum
The property of distinguishing different colors is referred to as spectral resolution. The property of the grating that affects the resolution of the spectrum is the number of slits. Figure 2.6 shows plots of the intensity equation for three wavelengths from a grating with 4, 8, and 24 slits. Notice that the maxima are no longer separated by smooth minima, but small subsidiary minima. The maxima are also sharper and more intense. Generally, as more sources are added, the maxima become sharper, more intense, and the subsidiary maxima within the minima regions become smaller and flatter. The sharper maxima make it easier to distinguish between lines.
Figure 2.6
Another important part of a spectrum’s resolution is the entrance slit of the spectrometer. The grating separates an image produced by a telescope into different wavelengths. If there is no entrance slit, the grating will produce multiple images of the object, all at different wavelengths. These images will overlap each other and make it difficult to distinguish the different wavelengths. Figure 2.7 is an example of a spectrum taken without an entrance slit. This is a spectrum of the Ring Nebula, which emits at only a few known wavelengths, but still it is difficult to separate them. Having an entrance slit produces a narrow image that does not overlap into the next wavelength. A narrower entrance slit generally means better resolution; however, it also means that less light is examined and longer exposures are required. Depending on the object to be examined, the observer has to reach a compromise.
Figure 2.7
(Buil 92).
2.3
Dispersion of a Spectrum
The dispersion of a spectrum is the rate at which the wavelength changes with relation to the observed angle. This is dependent on the spectral order and the spacing between the slits, s. By differentiating the grating equation, we see that
dl/dq = s cosq/n. (2.3)
Clearly, a smaller spacing s produces a greater dispersion. The dispersion also plays into the resolution. By increasing the spacing of the spectrum, there is less likelihood that the wavelengths will overlap.
It is important to remember that the dispersion equation is not linear and the dispersion increases at longer wavelengths. The change in dispersion is small and is usually ignored when analyzing a narrow region of the spectrum. However, for sensitive measurements, such as measuring radial velocities of stars, it is important to take this non-linearity into consideration.
2.4
Blazing
We have described gratings in terms of slits or apertures, also known as transmission gratings. It is possible to produce gratings in this fashion, but the majority of gratings produced today are reflection gratings that use small reflective grooves to act as wave sources. Reflection gratings have the advantages that they are easier to make and that it is possible to blaze them. The majority of the light from a transmission grating goes into the zero order and the rest is evenly distributed to the other maxima. Only 10% of the light ends up in either of the first order maxima. By tilting the reflective groves of a reflection grating, the majority of the light can be focused into one order. This is known as blazing and it can focus nearly 90% of the incident light on one order. Figure 2.8 shows a cross section of a reflection grating.
Figure 2.8
(Kitchin 127)
2.5 The SGS Gratings
There are two gratings in the SGS spectrometer. The high-resolution grating has 600 rulings per millimeter and the low-resolution grating has 150 rulings per millimeter. Their dispersions are 1.07 angstroms per pixel for the high and 4.3 angstroms per pixel for the low. Both are blazed to the first order and centered at 5000 angstroms, roughly the center of the visible spectrum. There are two entrance slits, 18 and 72 microns in width. These two gratings and two entrance slits can be used in combination and allow for four possible resolutions. Chart 2.1 shows the resolutions of each combination. These measurements are true only for certain pixel size; the ST7 camera has pixels that are 9x9 microns. Notice that because the larger entrance slit is four times as wide, the resolution is reduced four times.
Chart 2.1
(Alan Holmes of SBIG)