19: The analysis shown in the figure below also applies to diffraction gratings with lines separated by a distance d. What is the distance between fringes produced by a diffraction grating having 125 lines per centimetre for 600-nm light, if the screen is 1.50 m away?

Infrared radiation (IR), sometimes referred to simply as infrared, is a region of the electromagnetic radiation spectrum where wavelengths range from about 700 ...

Diffractiongrating diagram

14: Show that a diffraction grating cannot produce a second-order maximum for a given wavelength of light unless the first-order maximum is at an angle less than 30.0 degrees.

5: Calculate the wavelength of light that has its second-order maximum at 45.0 degrees  when falling on a diffraction grating that has 5000 lines per centimetre.

12: An opal such as that shown in Figure 2 acts like a reflection grating with rows separated by about 8 μm If the opal is illuminated normally, (a) at what angle will red light be seen and (b) at what angle will blue light be seen?

If you add TeleVue eyepieces to your telescope eyepiece selection you not only will have a smile on your face every time you look in the eyepiece, you will have invested in eyepieces for a lifetime that will hold their value if you ever need to sell them or trade them in.

4: What is the distance between lines on a diffraction grating that produces a second-order maximum for 760-nm red light at an angle of 60.0degrees?

18: A He–Ne laser beam is reflected from the surface of a CD onto a wall. The brightest spot is the reflected beam at an angle equal to the angle of incidence. However, fringes are also observed. If the wall is 1.50 m from the CD, and the first fringe is 0.600 m from the central maximum, what is the spacing of grooves on the CD?

It’s not all about magnification: The best way to understand magnification is to use your telescope regularly for several nights on several types of celestial objects - the moon, planets, nebulas, star clusters and galaxies. Manufacturers include a telescope eyepiece or two that are usable - start with the eyepiece marked with the higher number - often 25 or 20. Unfortunately if you have a “department store trash scope” it may have come with one or two unusable, high power eyepieces in addition to a useless barlow lens (these are not sold by All-Star Telescope). While the conjunction of Saturn and Jupiter in December 2020 drew a lot of attention to viewing the planets which can benefit from higher magnification, 75% of your viewing likely will be at lower magnification. Why? Many celestial objects such as the Pleiades star cluster, the double cluster in Perseus, the Orion nebula, the twin galaxies M81 & M82 and Andromeda galaxy are large. The Andromeda galaxy is 4 or 5 times the diameter of the full moon. For best views you will want low magnification and large light gathering.For most amateur astronomers, it is not about “how far” you can see. It’s about “how bright” those deep sky objects (nebulas, galaxies and star clusters) and how much detail you can see. While 75% of your viewing will be at the lowest power, 10% may be at high power for viewing the planets and lunar features with 15% of your viewing at medium power. Each telescope, telescope eyepiece and nightly sky conditions will determine how much you can magnify. If you double the magnification you spread the same amount of light over a larger area and the object will be dimmer. This is not noticeable on brighter objects like the moon and planets but will be apparent on many deep sky objects. As you increase magnification you also magnify imperfections in our atmosphere and the celestial object can begin to look like a hockey puck at the bottom of a swimming pool. You will almost always have a sharper, clearer view of Saturn at a lower magnification and can increase the magnification until the planet begins to “swim” and offers a blurry view. To determine magnification, you divide the focal length of the telescope by the focal length of the eyepiece. Thus the 2000mm focal length of Celestron’s popular C-8 - (NexStar 8SE, etc.) comes with a 25mm eyepiece that results in 80X magnification. A 10mm telescope eyepiece results in 200X magnification. Ignore the manufacturer’s “maximum magnification” which might apply if you are on the space station or moon where there is no atmosphere to contend with.

7: (a) What do the four angles in the above problem become if a 5000-line-per-centimetre diffraction grating is used? (b) Using this grating, what would the angles be for the second-order maxima? (c) Discuss the relationship between integral reductions in lines per centimetre and the new angles of various order maxima.

The large distance between the red and violet ends of the rainbow produced from the white light indicates the potential this diffraction grating has as a spectroscopic tool. The more it can spread out the wavelengths (greater dispersion), the more detail can be seen in a spectrum. This depends on the quality of the diffraction grating—it must be very precisely made in addition to having closely spaced lines.

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Diffractiongrating pattern

(c) Decreasing the number of lines per centimeter by a factor of x means that the angle for the x-order maximum is the same as the original angle for the first – order maximum.

Consider a spectrometer based on a diffraction grating. Construct a problem in which you calculate the distance between two wavelengths of electromagnetic radiation in your spectrometer. Among the things to be considered are the wavelengths you wish to be able to distinguish, the number of lines per meter on the diffraction grating, and the distance from the grating to the screen or detector. Discuss the practicality of the device in terms of being able to discern between wavelengths of interest.

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An interesting thing happens if you pass light through a large number of evenly spaced parallel slits, called a diffraction grating. An interference pattern is created that is very similar to the one formed by a double slit (see Figure 1). A diffraction grating can be manufactured by scratching glass with a sharp tool in a number of precisely positioned parallel lines, with the untouched regions acting like slits. These can be photographically mass produced rather cheaply. Diffraction gratings work both for transmission of light, as in Figure 1, and for reflection of light, as on butterfly wings and the Australian opal in Figure 2 or the CD pictured in the opening photograph of this chapter,  Figure 1. In addition to their use as novelty items, diffraction gratings are commonly used for spectroscopic dispersion and analysis of light. What makes them particularly useful is the fact that they form a sharper pattern than double slits do. That is, their bright regions are narrower and brighter, while their dark regions are darker. Figure 3 shows idealized graphs demonstrating the sharper pattern. Natural diffraction gratings occur in the feathers of certain birds. Tiny, finger-like structures in regular patterns act as reflection gratings, producing constructive interference that gives the feathers colours not solely due to their pigmentation. This is called iridescence.

This comment from someone in northern Alberta is comparing views with someone in southern Arizona where the planet will be higher in the sky, experiencing less turbulence and less water vapor in the air. The solution is not a better telescope eyepiece but better viewing location where the celestial object will be higher in the sky with less atmospheric interference. A moonless night, no light pollution and higher elevation with thinner air will contribute to better viewing. If you have the opportunity to visit Hawaii’s Mauna Kea or Haleakala you will quickly experience the advantage of higher elevation with less atmosphere for stargazing. In fact we’ve been confused by the abundance of stars that are not normally visible from home.

The distances on the screen are labeled  yV  and yR in Figure 5. Noting that  for small angles, sin θ = tanθ = y/x , we can solve for yV and yR. That is,

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3: Can the lines in a diffraction grating be too close together to be useful as a spectroscopic tool for visible light? If so, what type of EM radiation would the grating be suitable for? Explain.

Diffraction gratings with 10,000 lines per centimetre are readily available. Suppose you have one, and you send a beam of white light through it to a screen 2.00 m away. (a) Find the angles for the first-order diffraction of the shortest and longest wavelengths of visible light (380 and 760 nm). (b) What is the distance between the ends of the rainbow of visible light produced on the screen for first-order interference? (See Figure 5.)

Diffractiongrating experiment

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Unfortunately this question is difficult or impossible to answer. However, getting out with the telescope under the nighttime sky can become a lifelong hobby and passion. In his book, Seeing in the Dark, Timothy Ferris says, “The universe is accessible to all, and can inform one’s existence with a sense of beauty, reason and awe as enriching as anything to be found in music, art of poetry.”

Diffraction gratingsexamples

15: If a diffraction grating produces a first-order maximum for the shortest wavelength of visible light at 30.0o, at what angle will the first-order maximum be for the longest wavelength of visible light?

OK, I can’t afford the TeleVue eyepieces that Terence Dickinson recommends. What should I purchase? In the lower power eyepieces, from 30mm to 40mm you can consider the Baader Hyperion 36mm and 31mm as a good second choice to the top choice TeleVue eyepieces. In the 20mm to 30mm range you may be able to afford the TeleVue Panoptic eyepieces or consider the Baader Hyperion 24mm and 21mm eyepieces. From the 17mm and higher power, you can consider the Morpheus17, TeleVue Delos, or Baader Morpheus telescope eyepieces. These are excellent. And in the higher power eyepieces, if you can’t afford the TeleVue 13mm, 9mm or 7mm or the TeleVue Delos 12mm, 10mm, 8mm or 6mm, take a look at the Baader Morpheus as a good second choice.

2: Find the angle for the third-order maximum for 580-nm-wavelength yellow light falling on a diffraction grating having 1500 lines per centimetre.

The distance between slits is d = (1 cm) /10,000 = 1.00 x 10-6 m. Let us call the two angles θV for violet (380 nm) and θR for red (760 nm). Solving the equation  d sinθV = mλ for sinθV:

Conclusion: Cheaper eyepieces may offer similar specifications to the better and best eyepieces. But they definitely will not offer similar viewing.

(a) What visible wavelength has its fourth-order maximum at an angle of 25.0o when projected on a 25,000-line-per-centimeter diffraction grating? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

But some brands offer the same magnification and “field of view” of the TeleVue eyepieces. Yes, but there are other qualities in an eyepiece to consider.

This comment expresses the hope that a telescope eyepiece with more magnification and better optical qualities will produce a better view of planets. Unfortunately sky conditions and the position of the celestial body in the sky will determine the “sharpness” or “clarity” of the view. When viewing objects closer to the horizon, you are looking through two or three times as much atmosphere as higher in the sky. That atmosphere is often turbulent as the air warms and cools from the heat of the earth and may have more haze or low cloud. Amateur astronomers use the term, “steady skies” for the best viewing.

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Red light of wavelength of 700 nm falls on a double slit separated by 400 nm. (a) At what angle is the first-order maximum in the diffraction pattern? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

Where are diffraction gratings used? Diffraction gratings are key components of monochromators used, for example, in optical imaging of particular wavelengths from biological or medical samples. A diffraction grating can be chosen to specifically analyze a wavelength emitted by molecules in diseased cells in a biopsy sample or to help excite strategic molecules in the sample with a selected frequency of light. Another vital use is in optical fiber technologies where fibers are designed to provide optimum performance at specific wavelengths. A range of diffraction gratings are available for selecting specific wavelengths for such use.

11: Structures on a bird feather act like a reflection grating having 8000 lines per centimetre. What is the angle of the first-order maximum for 600-nm light?

This comment illustrates how better telescope eyepieces will be the single most important upgrade to almost any telescope. All manufacturers mostly include a basic eyepiece or two in order to keep the cost of the telescope down and often a set of good eyepieces will exceed the original cost of the telescope. Photographers understand this about cameras and camera lenses. If you're trying to learn about telescope eyepieces and how to pick the best telescope eyepiece or eyepieces for your needs, read on.

6: An electric current through hydrogen gas produces several distinct wavelengths of visible light. What are the wavelengths of the hydrogen spectrum, if they form first-order maxima at angles of 24.2o, 25.7o, 29.1o, and 41.0o when projected on a diffraction grating having 10,000 lines per centimetre? Explicitly show how you follow the steps in Chapter Problem-Solving Strategies for Wave Optics

Diffractiongrating PDF

There’s a saying in astronomy, “Don’t look through an eyepiece more expensive than what you can afford.” Yes, quality telescope eyepieces make a huge difference in your viewing.

Diffraction gratingsin physics

7: It is possible that there is no minimum in the interference pattern of a single slit. Explain why. Is the same true of double slits and diffraction gratings?

17: (a) Show that a 30,000-line-per-centimetre grating will not produce a maximum for visible light. (b) What is the longest wavelength for which it does produce a first-order maximum? (c) What is the greatest number of lines per centimetre a diffraction grating can have and produce a complete second-order spectrum for visible light?

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where d is the distance between slits in the grating, λ is the wavelength of light, and m is the order of the maximum. Note that this is exactly the same equation as for double slits separated by d. However, the slits are usually closer in diffraction gratings than in double slits, producing fewer maxima at larger angles.

8: What is the maximum number of lines per centimetre a diffraction grating can have and produce a complete first-order spectrum for visible light?

9: The yellow light from a sodium vapour lamp seems to be of pure wavelength, but it produces two first-order maxima at 36.093o and 36.129o when projected on a 10,000 line per centimetre diffraction grating. What are the two wavelengths to an accuracy of 0.1 nm?

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once a value for the slit spacing d has been determined. Since there are 10,000 lines per centimetre, each line is separated by 1/10,000 of a centimetre. Once the angles are found, the distances along the screen can be found using simple trigonometry.

10: What is the spacing between structures in a feather that acts as a reflection grating, given that they produce a first-order maximum for 525-nm light at a 30.0o angle?

The spacing d of the grooves in a CD or DVD can be well determined by using a laser and the equation d sinθ = m λ  for m = 0, 1,  -1, 2, -2, 3, -3 … (constructive).  However, we can still make a good estimate of this spacing by using white light and the rainbow of colours that comes from the interference. Reflect sunlight from a CD onto a wall and use your best judgment of the location of a strongly diffracted colour to find the separation d.

Planediffractiongrating

Douglas College Physics 1207 Copyright © August 22, 2016 by OpenStax is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

The number of slits in this diffraction grating is too large. Etching in integrated circuits can be done to a resolution of 50 nm, so slit separations of 400 nm are at the limit of what we can do today. This line spacing is too small to produce diffraction of light.

Diffractiongrating formula

16: (a) Find the maximum number of lines per centimetre a diffraction grating can have and produce a maximum for the smallest wavelength of visible light. (b) Would such a grating be useful for ultraviolet spectra? (c) For infrared spectra?

4: If a beam of white light passes through a diffraction grating with vertical lines, the light is dispersed into rainbow colours on the right and left. If a glass prism disperses white light to the right into a rainbow, how does the sequence of colours compare with that produced on the right by a diffraction grating?

3: How many lines per centimetre are there on a diffraction grating that gives a first-order maximum for 470-nm blue light at an angle of 25.0 degrees?

“The only way you are going to get sharp performance edge-to-edge across the field is by spending big bucks for top-of-the line glass.” He continues, “As for eyepieces, TeleVue Panoptics are superb, as are TeleVue Naglers and Ethos.” And, “Anything else is second rate... Expensive? Yes. But if you want to smile every time you look in the eyepiece, that's the answer. For more, see my book Backyard Astronomer’s Guide, 3rd Edition (2008), which has an entire chapter on eyepieces with comments about many brands.”

13: At what angle does a diffraction grating produces a second-order maximum for light having a first-order maximum at 20.0o?

While it is tempting to buy a telescope eyepiece offering higher magnification, we recommend firstly buying one with lower magnification since 75% of your viewing will be done with that eyepiece. Terence Dickinson, author of NightWatch, co-author of Backyard Astronomer’s Guide and author of dozens of astronomy books once asked what he recommended. His answer was:

5: Suppose pure-wavelength light falls on a diffraction grating. What happens to the interference pattern if the same light falls on a grating that has more lines per centimetre? What happens to the interference pattern if a longer-wavelength light falls on the same grating? Explain how these two effects are consistent in terms of the relationship of wavelength to the distance between slits.

The analysis of a diffraction grating is very similar to that for a double slit (see Figure 4). As we know from our discussion of double slits in Chapter Young’s Double Slit Experiment, light is diffracted by each slit and spreads out after passing through. Rays traveling in the same direction (at an angle θ relative to the incident direction) are shown in the figure. Each of these rays travels a different distance to a common point on a screen far away. The rays start in phase, and they can be in or out of phase when they reach a screen, depending on the difference in the path lengths traveled. As seen in the figure, each ray travels a distance dsinθ different from that of its neighbour, where d is the distance between slits. If this distance equals an integral number of wavelengths, the rays all arrive in phase, and constructive interference (a maximum) is obtained. Thus, the condition necessary to obtain constructive interference for a diffraction grating is

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Notice that in both equations, we reported the results of these intermediate calculations to four significant figures to use with the calculation in part (b).

1: A diffraction grating has 2000 lines per centimetre. At what angle will the first-order maximum be for 520-nm-wavelength green light?