In an isotropic material, for example a cubic crystal, or an amorphous material, light vibrates equally easily in all directions. These materials do not affect polarised light. If an isotropic material is examined between crossed polars, extinction occurs, and the image appears dark.

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?

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The two transmitted rays interfere, and the effect produced depends on the phase difference between the O-rays and E-rays and their amplitudes at the analyser. Extinction occurs when the optical path difference between the O-ray and the E-ray is a whole wavelength.

Diffractiongrating pattern

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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.)

A polarised light microscope has a polariser and analyser fitted at 90� to each other in an illuminating system. The arrangement also allows for the insertion of plates at 45� to the planes of polarisation. These can be used to enhance the contrast in a specimen. For further effects, it is also often possible to rotate one of the polarisers if crossed polars are not to be used.

Diffractiongrating diagram

The series of photos below shows the difference in the appearance of some glass ceramic specimens as different plates are inserted.

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?

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?

The rays travel with different velocities through the crystal. The ordinary ray travels with the same velocity in all directions and the extraordinary ray travels with a direction-dependent velocity. When the O-rays and E-rays emerge from the crystal the phase of one set of rays is retarded with respect to the other. This retardation depends on the difference in velocities of the two rays and the thickness of the specimen. Such a crystal is said to exhibit birefringence .

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.

If light is polarised in one direction and then passed through a polariser at a different angle to the original polariser, only the component of the polarised light which is in the same direction as the new polariser will be transmitted. If the second polarisation direction is at 90� to the original polarisation direction, the arrangement is known as "crossed polars" and the second polariser is referred to as the analyser. In this arrangement extinction usually occurs, i.e. no light is transmitted, because there is no component of the polarised light which can pass through the second polariser.

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When white light is used, anisotropic crystals may appear coloured when viewed between crossed polars, due to interference effects between rays emerging from the analyser. Certain wavelengths, and therefore certain colours, will be extinguished due to destructive interference. The colours seen depend on the birefringence of the crystal, its thickness, and the orientation of the section relative to the optic axis. Colour variations are observed within each grain as the stage is rotated.

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?

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.

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.

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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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

Planediffractiongrating

(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.

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?

Glass ceramic transmission microscope image made with unpolarised light (Click on image to view larger version)

Glass ceramic transmission microscope image made with polarised light and quarter wave plate (Click on image to view larger version)

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.

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.

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?

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.

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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?

Diffraction gratingsexamples

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.

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.

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.

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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?

Diffractiongrating experiment

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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?

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).

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?

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:

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?

When a light ray enters an optically anisotropic crystal (other than along an optic axis ), it is resolved into two rays - an ordinary ray (or O-ray) and an extraordinary ray (or E-ray). These rays vibrate in fixed planes at right angles to each other. When the rays arrive at the analyser, those components of their vibration directions which are parallel to the polarisation of the analyser are transmitted, while those components which are perpendicular are absorbed.

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?

Diffractiongrating formula

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.

(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?

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

Diffractiongrating PDF

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

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.

Diffraction gratingsin physics

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Glass ceramic transmission microscope image made with polarised light (Click on image to view larger version)

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.

Glass ceramic transmission microscope image made with polarised light and full wave plate (Click on image to view larger version)

A quartz wedge viewed between crossed polars shows how the colour of the light changes as the retardation increases. In the photo below, the wedge increases in thickness from left to right. As the thickness increases, the retardation also increases. The relation between retardation, birefringence and thickness can be seen on a Michel-Levy chart.

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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?

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 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,

When observing a specimen, differences in birefringence allow phases and grains to be identified. For example, different grain orientations may exhibit differences in birefringence and this will cause them to appear a different colour. Enhanced colouration of the image observed under crossed polars can be obtained by insertion of a full wave sensitive tint plate (also known as a red tint plate).

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

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.

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?