Wave Interference and Diffraction

Diffraction of Light Diffraction is the ability of light waves to bend around obstacles placed in their path. Ocean Beach Fuzzy Shadow Light rays Water waves easily bend around obstacles, but light waves also bend, as evidenced by the lack of a sharp shadow on the wall.

A new set of waves is observed emerging from the gap to the wall. Water Waves A wave generator sends periodic water waves into a barrier with a small gap, as shown below. A new set of waves is observed emerging from the gap to the wall.

Figure 2 on p. 459- both waves encounter same obstacle (blue object) Figure 2A Figure 2B Short wavelengths Less diffraction (red arrows) Longer wavelengths Greater amount of diffraction (red arrows) Figure 3 on p. 460- all 3 diagrams have the same slit opening. The wavelength increases from a-c, so does the amount of diffraction Figure 4 on p. 460- Fig 4a and b have a wave with the same wavelength, however slit opening decreases. As slit width decreases, diffraction increases, as wavelength is kept constant.

Interference of Water Waves Interference occurs when 2 waves in the same medium interact.

The Superposition Principle The resultant displacement of two simul-taneous waves (blue and green) is the algebraic sum of the two displacements. The resultant wave is shown in yellow. Constructive Interference Destructive Interference The superposition of two light waves results in light and dark fringes on a screen.

Constructive Interference- two interfering waves have displacement in the same direction. The waves must be in phase and combine together. Destructive Interference- two interfering waves have displacement in opposite directions when they superimpose. The amplitude of the superimposed wave is smaller then the interacting waves.

Constructive Interference Destructive Interference

3 conditions for Interference to occur: Short distance apart, so they can interact in same medium. Waves come together at a common point. Waves must have same frequencies and fixed positions.

2 Point Source Interference Continuous interference from two sources. If the source of waves produces circular waves, then the waves will meet within the medium to produce a pattern. The pattern is characterized by a collection of nodes and antinodes referred to as antinodal lines and nodal lines. Figure 8 on p. 463

Two Point Interference Pattern When we have two identical point sources that are side by side, in phase, and have identical frequencies, we can analyse the interference pattern that is produced to learn more about the waves. ripple tank simulation of two point source pattern: http://www.falstad.com/ripple/

Two Point Interference Pattern Antinodes Nodes Sources

Two Point Interference Pattern Thin line = trough In the diagram, crests are represented by thick lines and troughs are represented by thin lines. We get constructive interference then whenever a thick line meets thick line, or when a thin line meets a thin line. This constructive interference causes antinodes, shown by the red dots. Thick line = crest

Two Point Interference Pattern Destructive interference occurs whenever a thick line meets a thin line. These points form nodes, which are represented by a blue dot. The nodes and antinodes appear to ‘stand still’ which makes this a standing wave pattern.

Two Point Interference Pattern The number of nodal lines increases when you do any of the following: Increase frequency of the sources Decrease the wavelength of the waves Increase separation between the sources

https://www.youtube.com/watch?v=1FwM1oF5e6o -

Young’s Experiment In 1801 Thomas Young was able to offer some very strong evidence to support the wave model of light. He placed a screen that had two slits cut into it in front of a monochromatic light. The results of Young's Double Slit Experiment should be very different if light is a wave or a particle.

Light as a Particle

Light as a Wave

In Young’s experiment, light from a monochromatic source falls on two slits, setting up an interference pattern similar to that with water waves. Light source S1 S2

Young’s Interference Pattern Constructive Bright fringe Destructive Dark fringe Constructive Bright fringe

Interference of the waves determines the intensity of the light. The alternating areas of dark and light fringes are created by alternating areas of constructive and destructive interference.

Conditions for Bright Fringes Bright fringes occur when the difference in path is an integral multiple of one wave length l. l l l p1 p2 p3 p4

What does this mean? Waves must be in phase Path length needs to be a whole number wavelength difference for constructive interference to occur. FORMULA for bright fringes, also known as MAXIMA (ON BOARD)

Conditions for Dark Fringes Dark fringes occur when the difference in path is an odd multiple of one-half of a wave length l/2. l l p1 n = odd n = 1,3,5 … p2 p3 p3

What does this mean? Waves must be out of phase Path length needs to be a half number wavelength difference for destructive interference to occur. FORMULA for dark fringes, also known as MINIMA (ON BOARD)

https://www.youtube.com/watch?v=Iuv6hY6zsd0

The Diffraction Grating A diffraction grating consists of thousands of parallel slits etched on glass so that brighter and sharper patterns can be observed than with Young’s experiment. Equation is similar. d sin q q d d sin q = nl n = 1, 2, 3, …

The Grating Equation 3l The grating equation: 2l l 1st order d = slit width (spacing) l = wavelength of light q = angular deviation n = order of fringe 2nd order 2l 4l 6l

To find slit separation, we take reciprocal of 300 lines/mm: Example 2: Light (600 nm) strikes a grating ruled with 300 lines/mm. What is the angular deviation of the 2nd order bright fringe? To find slit separation, we take reciprocal of 300 lines/mm: 300 lines/mm n = 2 Lines/mm  mm/line

Angular deviation of second order fringe is: Example (Cont.) 2: A grating is ruled with 300 lines/mm. What is the angular deviation of the 2nd order bright fringe? 300 lines/mm n = 2 l = 600 nm Angular deviation of second order fringe is: q2 = 21.10

A compact disk acts as a diffraction grating A compact disk acts as a diffraction grating. The colors and intensity of the reflected light depend on the orientation of the disc relative to the eye.

Interference From Single Slit When monochromatic light strikes a single slit, diffraction from the edges produces an interference pattern as illustrated. Pattern Exaggerated Relative intensity The interference results from the fact that not all paths of light travel the same distance some arrive out of phase.

When looking at single slit diffraction we must assume: Slit width is narrow enough for diffraction to occur. (NOT too small as thought it acts like a point source) Light is monochromatic with a λ. Light source is far enough from the slit to ensure all rays hitting the slit are parallel.

The pattern observed from single slit diffraction is called Fraunhofer diffraction. (fig.2 p. 512) This pattern shows a bright central fringe called the central maximum and is flanked by less bright fringes called secondary maxima and dark fringes called minima. Huygen’s Principle helps to explain and predict the diffraction patterns. All points on a wave front can be thought of as new sources of spherical waves called wavelets. (diagram on board)

Single Slit Interference Pattern Each point inside slit acts as a source. a/2 a 1 2 4 3 5 For rays 1 and 3 and for 2 and 4: First dark fringe: For every ray there is another ray that differs by this path and therefore interferes destructively.

Single Slit Interference Pattern 1 2 4 3 5 First dark fringe: Other dark fringes occur for integral multiples of this fraction l/a.

Example 3: Monochromatic light shines on a single slit of width 0 Example 3: Monochromatic light shines on a single slit of width 0.45 mm. On a screen 1.5 m away, the first dark fringe is displaced 2 mm from the central maximum. What is the wavelength of the light? q x = 1.5 m y a = 0.35 mm l = ? l = 600 nm

Diffraction for a Circular Opening Circular diffraction D The diffraction of light passing through a circular opening produces circular interference fringes that often blur images. For optical instruments, the problem increases with larger diameters D.

Resolution of Images Consider light through a pinhole. As two objects get closer the interference fringes overlap, making it difficult to distinguish separate images. d1 Clear image of each object d2 Separate images barely seen

Resolution Limit Images are just resolved when central maximum of one pattern coincides with first dark fringe of the other pattern. d2 Resolution limit Separate images Resolution Limit

Resolving Power of Instruments The resolving power of an instrument is a measure of its ability to produce well-defined separate images. Limiting angle D q For small angles, sin q  q, and the limiting angle of resolution for a circular opening is: Limiting angle of resolution:

Resolution and Distance q so p D Limiting angle qo Limiting Angle of Resolution:

Example 4: The tail lights (l = 632 nm) of an auto are 1 Example 4: The tail lights (l = 632 nm) of an auto are 1.2 m apart and the pupil of the eye is around 2 mm in diameter. How far away can the tail lights be resolved as separate images? q so p Eye D Tail lights p = 3.11 km

Summary Young’s Experiment: Monochromatic light falls on two slits, producing interference fringes on a screen. x y d sin q s1 s2 d q p1 p2 Bright fringes: Dark fringes:

Summary (Cont.) The grating equation: d = slit width (spacing) l = wavelength of light q = angular deviation n = order of fringe

Interference from a single slit of width a: Summary (Cont.) Interference from a single slit of width a: Pattern Exaggerated Relative Intensity

Summary (cont.) The resolving power of instruments. p D so q so p D Limiting angle qo Limiting Angle of Resolution:

CONCLUSION: Chapter 37 Interference and Diffraction