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University Physics Volume 3

4.4 Diffraction Gratings

University Physics Volume 34.4 Diffraction Gratings
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  1. Preface
  2. Unit 1. Optics
    1. 1 The Nature of Light
      1. Introduction
      2. 1.1 The Propagation of Light
      3. 1.2 The Law of Reflection
      4. 1.3 Refraction
      5. 1.4 Total Internal Reflection
      6. 1.5 Dispersion
      7. 1.6 Huygens’s Principle
      8. 1.7 Polarization
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    2. 2 Geometric Optics and Image Formation
      1. Introduction
      2. 2.1 Images Formed by Plane Mirrors
      3. 2.2 Spherical Mirrors
      4. 2.3 Images Formed by Refraction
      5. 2.4 Thin Lenses
      6. 2.5 The Eye
      7. 2.6 The Camera
      8. 2.7 The Simple Magnifier
      9. 2.8 Microscopes and Telescopes
      10. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    3. 3 Interference
      1. Introduction
      2. 3.1 Young's Double-Slit Interference
      3. 3.2 Mathematics of Interference
      4. 3.3 Multiple-Slit Interference
      5. 3.4 Interference in Thin Films
      6. 3.5 The Michelson Interferometer
      7. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    4. 4 Diffraction
      1. Introduction
      2. 4.1 Single-Slit Diffraction
      3. 4.2 Intensity in Single-Slit Diffraction
      4. 4.3 Double-Slit Diffraction
      5. 4.4 Diffraction Gratings
      6. 4.5 Circular Apertures and Resolution
      7. 4.6 X-Ray Diffraction
      8. 4.7 Holography
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
  3. Unit 2. Modern Physics
    1. 5 Relativity
      1. Introduction
      2. 5.1 Invariance of Physical Laws
      3. 5.2 Relativity of Simultaneity
      4. 5.3 Time Dilation
      5. 5.4 Length Contraction
      6. 5.5 The Lorentz Transformation
      7. 5.6 Relativistic Velocity Transformation
      8. 5.7 Doppler Effect for Light
      9. 5.8 Relativistic Momentum
      10. 5.9 Relativistic Energy
      11. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    2. 6 Photons and Matter Waves
      1. Introduction
      2. 6.1 Blackbody Radiation
      3. 6.2 Photoelectric Effect
      4. 6.3 The Compton Effect
      5. 6.4 Bohr’s Model of the Hydrogen Atom
      6. 6.5 De Broglie’s Matter Waves
      7. 6.6 Wave-Particle Duality
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    3. 7 Quantum Mechanics
      1. Introduction
      2. 7.1 Wave Functions
      3. 7.2 The Heisenberg Uncertainty Principle
      4. 7.3 The Schrӧdinger Equation
      5. 7.4 The Quantum Particle in a Box
      6. 7.5 The Quantum Harmonic Oscillator
      7. 7.6 The Quantum Tunneling of Particles through Potential Barriers
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    4. 8 Atomic Structure
      1. Introduction
      2. 8.1 The Hydrogen Atom
      3. 8.2 Orbital Magnetic Dipole Moment of the Electron
      4. 8.3 Electron Spin
      5. 8.4 The Exclusion Principle and the Periodic Table
      6. 8.5 Atomic Spectra and X-rays
      7. 8.6 Lasers
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    5. 9 Condensed Matter Physics
      1. Introduction
      2. 9.1 Types of Molecular Bonds
      3. 9.2 Molecular Spectra
      4. 9.3 Bonding in Crystalline Solids
      5. 9.4 Free Electron Model of Metals
      6. 9.5 Band Theory of Solids
      7. 9.6 Semiconductors and Doping
      8. 9.7 Semiconductor Devices
      9. 9.8 Superconductivity
      10. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    6. 10 Nuclear Physics
      1. Introduction
      2. 10.1 Properties of Nuclei
      3. 10.2 Nuclear Binding Energy
      4. 10.3 Radioactive Decay
      5. 10.4 Nuclear Reactions
      6. 10.5 Fission
      7. 10.6 Nuclear Fusion
      8. 10.7 Medical Applications and Biological Effects of Nuclear Radiation
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    7. 11 Particle Physics and Cosmology
      1. Introduction
      2. 11.1 Introduction to Particle Physics
      3. 11.2 Particle Conservation Laws
      4. 11.3 Quarks
      5. 11.4 Particle Accelerators and Detectors
      6. 11.5 The Standard Model
      7. 11.6 The Big Bang
      8. 11.7 Evolution of the Early Universe
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
  4. A | Units
  5. B | Conversion Factors
  6. C | Fundamental Constants
  7. D | Astronomical Data
  8. E | Mathematical Formulas
  9. F | Chemistry
  10. G | The Greek Alphabet
  11. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
  12. Index

Learning Objectives

By the end of this section, you will be able to:
  • Discuss the pattern obtained from diffraction gratings
  • Explain diffraction grating effects

Analyzing the interference of light passing through two slits lays out the theoretical framework of interference and gives us a historical insight into Thomas Young’s experiments. However, most modern-day applications of slit interference use not just two slits but many, approaching infinity for practical purposes. The key optical element is called a diffraction grating, an important tool in optical analysis.

Diffraction Gratings: An Infinite Number of Slits

The analysis of multi-slit interference in Interference allows us to consider what happens when the number of slits N approaches infinity. Recall that N2N2 secondary maxima appear between the principal maxima. We can see there will be an infinite number of secondary maxima that appear, and an infinite number of dark fringes between them. This makes the spacing between the fringes, and therefore the width of the maxima, infinitesimally small. Furthermore, because the intensity of the secondary maxima is proportional to 1/N21/N2, it approaches zero so that the secondary maxima are no longer seen. What remains are only the principal maxima, now very bright and very narrow (Figure 4.12).

Figure a shows a graph of I versus sine theta. It has two vertical lines at sine theta equal to lambda by D and minus lambda by D. Figure b shows a bright red spot on a black background in the center. This is surrounded on either side by progressively dimmer spots, going outwards.
Figure 4.12 (a) Intensity of light transmitted through a large number of slits. When N approaches infinity, only the principal maxima remain as very bright and very narrow lines. (b) A laser beam passed through a diffraction grating. (credit b: modification of work by Sebastian Stapelberg)

In reality, the number of slits is not infinite, but it can be very large—large enough to produce the equivalent effect. A prime example is an optical element called a diffraction grating. A diffraction grating can be manufactured by carving glass with a sharp tool in a large number of precisely positioned parallel lines, with untouched regions acting like slits (Figure 4.13). This type of grating can be photographically mass produced rather cheaply. Because there can be over 1000 lines per millimeter across the grating, when a section as small as a few millimeters is illuminated by an incoming ray, the number of illuminated slits is effectively infinite, providing for very sharp principal maxima.

Figure shows a rectangular flat block with thin, parallel grooves. The grooves are cut at regular spacings d.
Figure 4.13 A diffraction grating can be manufactured by carving glass with a sharp tool in a large number of precisely positioned parallel lines.

Diffraction gratings work both for transmission of light, as in Figure 4.14, and for reflection of light, as on butterfly wings and the Australian opal in Figure 4.15. Natural diffraction gratings also occur in the feathers of certain birds such as the hummingbird. Tiny, finger-like structures in regular patterns act as reflection gratings, producing constructive interference that gives the feathers colors not solely due to their pigmentation. This is called iridescence.

Figure shows a vertical line on the left. This has five grooves. A ray enters from the left and five rays emerge from the right, one from each groove. These point to squares which are labeled, from top to bottom: second order rainbow, first order rainbow, central white, first order rainbow, second order rainbow. The first order rainbows shown in the squares are brighter than the second order rainbows.
Figure 4.14 (a) Light passing through a diffraction grating is diffracted in a pattern similar to a double slit, with bright regions at various angles. (b) The pattern obtained for white light incident on a grating. The central maximum is white, and the higher-order maxima disperse white light into a rainbow of colors.
Figure a is a photograph of an opal pendant reflecting various colours. Figure b is the photograph of a butterfly.
Figure 4.15 (a) This Australian opal and (b) butterfly wings have rows of reflectors that act like reflection gratings, reflecting different colors at different angles. (credit a: modification of work by "Opals-On-Black"/Flickr; credit b: modification of work by “whologwhy”/Flickr)

Applications of Diffraction Gratings

Where are diffraction gratings used in applications? 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 fringes are narrower and brighter while their dark regions are darker. 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 wavelength 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 wavelengths for such use.

Example 4.5

Calculating Typical Diffraction Grating Effects Diffraction gratings with 10,000 lines per centimeter 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, respectively). (b) What is the distance between the ends of the rainbow of visible light produced on the screen for first-order interference? (See Figure 4.16.)

A vertical line on the left is labeled grating and one on the right is labeled screen. They are a distance x equal to 2 meters apart. Four arrows radiate from the grating to the screen. The first and second from the top make angles theta R and theta V respectively with the central axis. The points where they fall on the screen are at distances yR and yV respectively from the central axis. Rainbows are formed on the screen between the first and second arrow and between the third and fourth arrow.
Figure 4.16 (a) The diffraction grating considered in this example produces a rainbow of colors on a screen a distance x=2.00mx=2.00m from the grating. The distances along the screen are measured perpendicular to the x-direction. In other words, the rainbow pattern extends out of the page.
(b) In a bird’s-eye view, the rainbow pattern can be seen on a table where the equipment is placed.

Strategy Once a value for the diffraction grating’s slit spacing d has been determined, the angles for the sharp lines can be found using the equation

dsinθ=mλform=0,±1,±2,....dsinθ=mλform=0,±1,±2,....

Since there are 10,000 lines per centimeter, each line is separated by 1/10,000 of a centimeter. Once we know the angles, we an find the distances along the screen by using simple trigonometry.

Solution

  1. The distance between slits is d=(1cm)/10,000=1.00×10−4cm or1.00×10−6m.d=(1cm)/10,000=1.00×10−4cm or1.00×10−6m. Let us call the two angles θVθV for violet (380 nm) and θRθR for red (760 nm). Solving the equation dsinθV=mλforsinθV,dsinθV=mλforsinθV,
    sinθV=mλVd,sinθV=mλVd,

    where m=1m=1 for the first-order and λV=380nm=3.80×10−7m.λV=380nm=3.80×10−7m. Substituting these values gives
    sinθV=3.80×10−7m1.00×10−6m=0.380.sinθV=3.80×10−7m1.00×10−6m=0.380.

    Thus the angle θVθV is
    θV=sin−10.380=22.33°.θV=sin−10.380=22.33°.

    Similarly,
    sinθR=7.60×10−7m1.00×10−6m=0.760.sinθR=7.60×10−7m1.00×10−6m=0.760.

    Thus the angle θRθR is
    θR=sin−10.760=49.46°.θR=sin−10.760=49.46°.

    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).
  2. The distances on the secreen are labeled yVandyRyVandyR in Figure 4.16. Notice that tanθ=y/x.tanθ=y/x. We can solve for yVandyR.yVandyR. That is,
    yV=xtanθV=(2.00m)(tan22.33°)=0.815myV=xtanθV=(2.00m)(tan22.33°)=0.815m

    and
    yR=xtanθR=(2.00m)(tan49.46°)=2.338m.yR=xtanθR=(2.00m)(tan49.46°)=2.338m.

    The distance between them is therefore
    yRyV=1.523m.yRyV=1.523m.

Significance 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.

Check Your Understanding 4.4

If the line spacing of a diffraction grating d is not precisely known, we can use a light source with a well-determined wavelength to measure it. Suppose the first-order constructive fringe of the HβHβ emission line of hydrogen (λ=656.3nm)(λ=656.3nm) is measured at 11.36°11.36° using a spectrometer with a diffraction grating. What is the line spacing of this grating?

Interactive

Take the same simulation we used for double-slit diffraction and try increasing the number of slits from N=2N=2 to N=3,4,5...N=3,4,5.... The primary peaks become sharper, and the secondary peaks become less and less pronounced. By the time you reach the maximum number of N=20N=20, the system is behaving much like a diffraction grating.

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