Interference. Electric fields from two different sources at a single location add together. The same is true for magnetic fields at a single location.

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1 Interference Electric fields from two different sources at a single location add together. The same is true for magnetic fields at a single location. Thus, interacting electromagnetic waves also add together.

2 Interference The addition of waves is called interference and may be constructive or destructive. Constructive:

3 Interference The addition of waves is called interference and may be constructive or destructive. Destructive:

4 Interference Regular patterns of interference can be produced by adding waves of the same wavelength and amplitude.

5 Interference Regular patterns of interference can be produced by adding waves of the same wavelength and amplitude. Each color represents a snapshot at a different moment in time.

6 Interference Regular patterns of interference can be produced by adding waves of the same wavelength and amplitude. Maxima Minima

7 Interference Consider two waves traveling in the same direction: Waves in phase (Wave 1, Wave 2, Sum) Waves out of phase by π (Wave 1, Wave 2, Sum)

8 Interference For simultaneous waves, path difference must be an integer number of wavelengths for constructive interference. L=m λ

9 Interference For simultaneous waves, path difference must be an half-integer number of wavelengths for destructive interference. L=(m+ 1 2 )λ

10 Young's Double Slit Experiment For intensity maximum, paths differ by integer number of wavelengths. L d =sinθ θ d L

11 Young's Double Slit Experiment For intensity maximum, paths differ by integer number of wavelengths. L d =sinθ θ y tan θ= y R

12 Small Angle Approximations R θ R R tan θ R R sin θ For small angles, sin θ θ tan θ

13 Young's Double Slit Experiment L d =sinθ θ y tan θ= y R For small angles, L d y R

14 Young's Double Slit Experiment For constructive interference, L=m λ For destructive interference, L=( m+1 2) λ θ y θ L y d R d (small angle) L

15 Young's Double Slit Experiment Phase difference as a function of angle, ϕ= 2π λ d sin θ θ y Intensity as a function of phase, I=I 0 cos 2 ( ϕ 2)

16 Example: a viewing screen is separated from the double-slit source by 1.2 m. The distance between the two slits is mm. The second-order bright fringe (m = 2) is 4.5 cm from the center line. Determine the wavelength of the light.

17 Example: a viewing screen is separated from the double-slit source by 1.2 m. The distance between the two slits is mm. The second-order bright fringe (m = 2) is 4.5 cm from the center line. Determine the distance between adjacent bright fringes.

18 Example: a viewing screen is separated from the double-slit source by 1.2 m. The distance between the two slits is mm. The second-order bright fringe (m = 2) is 4.5 cm from the center line. Determine the width of the bright fringes.

19 Diffraction Light can bend around edges. Each point of a wave front behaves as an independent source of light. Produces no surprises for broad wave fronts without obstacles.

20 Diffraction Light can bend around edges. Each point of a wave front behaves as an independent source of light. Produces no surprises for broad wave fronts without obstacles. Produces bend around obstacles. Single Edge Narrow Gap

21 Diffraction Light can bend around edges. Significant when object dimensions are comparable to wavelength. Single Edge Narrow Gap

22 Single Slit Diffraction Light passing through a narrow gap. Consider ray from bottom of gap and ray from just above middle of gap. θ y θ

23 Single Slit Diffraction Destructive Interference θ y θ

24 Single Slit Diffraction For every ray from the lower half there is a corresponding ray from the upper completely out of phase with the first ray at the screen.

25 Single Slit Diffraction The effect can be repeated for any even division of the gap. Destructive Interference:

26 Single Slit Diffraction Phase difference as a function of angle: Intensity as a function of phase difference:

27 Example: 633 nm laser light is passed through a narrow slit and a diffraction pattern is observed on a screen 6.0 m away. The distance on the screen between the centers of the first minima outside the central bright fringe is 32 mm. What is the slit width?

28 Single Slit Diffraction In Each of the Double Slits? If the slit width is comparable to wavelength instead of much smaller then one must also consider single slit diffraction in the double slit experiment.

29 Diffraction Gratings Derivation of maxima and minima similar to double slit. More intense and sharper maxima. Constructive Interference: θ d L

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32 Example: the wavelengths of visible light are from approximately 400 nm (violet) to 700 nm (red). Find the angular width of the first-order visible spectrum produced by a plane grating with 600 slits per millimeter when white light falls normally on the grating. 700 nm* 400 nm angle?

33 Example: the wavelengths of visible light are from approximately 400 nm (violet) to 700 nm (red). Find the angular width of the first-order visible spectrum produced by a plane grating with 600 slits per millimeter when white light falls normally on the grating.

34 X-ray Diffraction Regular spacing of atoms in a material can yield pattern that can be used to determine the spacing. Complex regular patterns in three dimensions can be determined.

35 Resolving Power If the minimum wavelength difference that can be resolved by a diffraction grating is given by then the resolving power is where is the average of the two wavelengths. It can be shown that where is the number of slits and refers to the th-order maxima.

36 Example: Light from mercury vapor lamps contain several wavelengths in the visible region of the spectrum including two yellow lines at 577 and 579 nm. What must be the resolving power of a grating to distinguish these two lines? mercury

37 Example: how many lines of the grating must be illuminated if these two wavelengths are to be resolved in the first-order spectrum? mercury

38 Finding Maxima and Minima Double slit maxima (constructive interference) Single slit minima (destructive interference) Diffraction grating maxima (constructive interference)

39 Intensity Maximum intensity corresponds to constructive interference (bright fringes). Double slit Phase difference: Intensity: Single slit Phase difference: Intensity: