Lecture 13 Diffraction

Instructor: Kazumi Tolich Lecture 13

¨ Reading chapter 33-4 & 33-6 to 33-7 ¤ Single slit diffraction ¤ Two slit interference-diffraction ¤ Fraunhofer and Fresnel diffraction ¤ Diffraction and resolution Diffraction of waves

¨ Diffraction is bending of wavefronts due to an obstacle partially blocking the waves.

Plane waves Circular waves

¨ If a (slit width) >> λ, light emerging from the slit is mostly in the original forward direction. ¨ If a << λ, light emerging from the slit spreads out evenly in all directions. Demo: 1

¨ Ripple Tank ¤ Demonstration of water wave diffraction Single slit diffraction

¨ When a ~ λ, but a > λ, light emerging from the slit forms a diffraction pattern. Intensity ¨ Most of the light intensity is concentrated in the broad central diffraction maximum. ¨ There are minor secondary maxima bands on either side of the central maximum. ¤ Newton thought that light consisted of particles. ¤ But, diffraction pattern of light is another piece of evidence that light is a wave. ¤ The light as wave vs. particle controversy has persisted from ancient times until the 20th century. Analyzing single slit diffraction

¨ For an open slit of width a, subdivide the opening into equal segments and imagine a Hyugen wavelet originating from the center of each segment. Single slit diffraction central maximum

¨ The wavelets going forward (θ = 0) all travel the same distance to the screen and interfere constructively to produce the central maximum. Quiz: 1 & 2

8 Conditions for diffraction minima

θ = 0 sinθ = λ/a sinθ = 2λ/a Bright Dark Dark Distance from the central maximum

¨ The distance ym from the central maximum to the mth diffraction minimum is related to the angle θm and the distance L from the slit to the screen by Width of a single-slit diffraction pattern

¨ The width of the central maximum is the distance between the centers of the first minima. For small angle

-y1 0 y1 y2 y3

w Quiz: 3 Example 1

¨ Light from a helium-neon laser (λ = 633 nm) passes through a narrow slit and is seen on a screen 2.0 m behind the slit. The first minimum of the diffraction pattern is observed to be located 1.2 cm from the central maximum. How wide is the slit?

1.2 cm Demo: 2

¨ Single Slit (Variable Width) ¤ Demonstration of single slit diffraction ¤ The width of the central maximum is inversely proportional to the slit width. Combined diffraction and interference

¨ So far, we have treated d diffraction and interference a a independently. Interference Only ¨ However, in a two-slit system

both phenomena should be θ (degrees) present together. Diffraction Only

θ (degrees) Both

θ (degrees) Fraunhofer diffraction of a circular aperture

¨ Diffraction pattern is also observed when light passes through a circular aperture instead of a vertical slit.

¨ The angle θ subtended by the first diffraction minimum is related to the wavelength and the diameter of the opening D by

¤ the factor 1.22 arises because of the circular geometry.

Circular aperture of diameter D Fraunhofer and Fresnel diffraction patterns

¨ Diffraction patterns formed when the viewing screen is far from the apertures so that the rays from the apertures are parallel are called Fraunhofer diffraction patterns. ¤ They can be observed using a lens to focus parallel rays on a viewing screen placed in the focal plane of the lens.

¨ The diffraction pattern observed at any distance from an aperture or an obstacle is called a Fresnel diffraction pattern. Fresnel diffraction patterns

¨ The patterns of a circular aperture and a opaque disk are complements of each other.

¨ Poisson spot at the center of the pattern caused by the constructive interference of the light waves diffracted from the edge of the disk.

Poisson spot

Opaque disk Circular aperture Fresnel diffraction patterns 2

¨ Fresnel Diffraction Patters of a rectangular aperture (left) and a straightedge (right).

Straightedge

Rectangular aperture Demo: 3

¨ Arago's Bright Spot (Poisson Spot) ¤ Demonstration of Poisson spot

¨ Point and Eye of Needle ¤ Demonstration of diffraction due to a needle

¨ Crossed Slits ¤ Two dimensional diffraction pattern

¨ Knife Edge Diffraction “Ai r y d i sk s” o f a ¨ Aperture Diffraction (Airy Disk) binary star viewed by a 2.56 m telescope ¤ Demonstration of diffraction due to circular aperture Rayleigh’s criterion for resolution

¨ When the first minimum of the diffraction pattern of one source falls on the central maximum of the other source, these objects are just resolved by Rayleigh’s criterion for resolution. ¨ The critical angle subtended by the sources just resolved by Rayleigh’s criterion for resolution is

2αc αc Quiz: 4 Example 2

¨ The Hubble Space Telescope has a mirror with a diameter of 2.4 m. For light with wavelength of 500 nm, what is the angular resolution of the Hubble? Example 3

¨ How far from the grains of red sand must you be to position yourself just at the limit of resolving the grains if your pupil diameter is 1.5 mm, the grains are spherical with radius 50.0 µm, and the light from the grains has wavelength 650 nm? If the grains were blue, and the light from them had wavelength 400 nm, would the answer be larger or smaller?