Chapter 35 Interference © 2016 Pearson Education Inc.

Learning Goals for Chapter 35 What happens when two waves combine (interfere) in space. Understand interference pattern formed by interference of two coherent light waves. Calculate intensity at locations of interference pattern. Understand how interference occurs when light reflects from two surfaces of a thin film. Interferometry, & how interference enables measurement of extremely small distances. © 2016 Pearson Education Inc.

Introduction Why soap bubbles show vibrant color patterns, even though soapy water is colorless! What causes the multicolored reflections from CDs & DVDs? Look at optical effects that depend on wave nature of light. © 2016 Pearson Education Inc.

Principle of superposition Interference occurs in any situation where two or more waves overlap in space. Total wave at any point at any instant of time is governed by principle of superposition: When two or more waves overlap, resultant displacement at any point & at any instant is found by adding instantaneous displacements that would be produced by individual waves if each were present alone. © 2016 Pearson Education Inc.

Wave fronts from a disturbance Interference effects most easily seen when combine sinusoidal waves with a single frequency & wavelength. © 2016 Pearson Education Inc.

Constructive and destructive interference Two identical sources of monochromatic waves, S1 and S2. Two sources permanently in phase vibrating in unison. Constructive interference occurs at point a equidistant from both sources. © 2016 Pearson Education Inc.

Conditions for Constructive Interference Distance S2 to point b is exactly two (2) wavelengths greater than distance S1 to b. Both waves arrive in phase Waves “reinforce” each other. © 2016 Pearson Education Inc.

Conditions for Destructive Interference Distance S1 to point c is half-integral # of wavelengths greater than distance S2 to c. Both waves cancel or partly cancel each other. © 2016 Pearson Education Inc.

Constructive and Destructive interference Start with 2 identical sources of monochromatic waves, S1 & S2, in phase. Red curves show all positions where constructive interference occurs (antinodal curves). Nodal curves would show where destructive interference occurs. © 2016 Pearson Education Inc.

Constructive and destructive interference Concepts of constructive & destructive interference apply to water waves as well as to light & sound waves. © 2016 Pearson Education Inc.

Young’s Double-Slit Experiment If light is a wave, interference effects will be seen, where one part of a wave front can interact with another part. One way to study this is to do a double-slit experiment: Figure 34-5. (a) Young’s double-slit experiment. (b) If light consists of particles, we would expect to see two bright lines on the screen behind the slits. (c) In fact, many lines are observed. The slits and their separation need to be very thin.

Young’s Double-Slit Experiment If light is a wave, interference effects will be seen, where one part of a wave front can interact with another part. One way to study this is to do a double-slit experiment: Figure 34-5. (a) Young’s double-slit experiment. (b) If light consists of particles, we would expect to see two bright lines on the screen behind the slits. (c) In fact, many lines are observed. The slits and their separation need to be very thin.

Young’s Double-Slit Experiment If light is a wave, interference effects will be seen, where one part of a wave front can interact with another part. One way to study this is to do a double-slit experiment: Figure 34-5. (a) Young’s double-slit experiment. (b) If light consists of particles, we would expect to see two bright lines on the screen behind the slits. (c) In fact, many lines are observed. The slits and their separation need to be very thin.

Interference – Young’s Double-Slit Experiment If light is a wave, there should be an interference pattern. Figure 34-6. If light is a wave, light passing through one of two slits should interfere with light passing through the other slit.

Two-source interference of light Young’s Experiment (1801) Interference of waves from slits S1 & S2 produces a pattern on the screen. (Check out weblink!) © 2016 Pearson Education Inc.

Two-source interference of light Thomas Young “The last man who knew everything” Dr. of Medicine (haemodynamics) Wave Theory of Light Young’s Modulus (Stress/Strain) Founder of physiological optics Music (Young temperament) Translator of Rosetta Stone… © 2016 Pearson Education Inc.

Young’s Double-Slit Experiment Interference occurs because each point on screen is NOT same distance from both slits. Depending on the path length difference, wave can interfere constructively (bright spot) or destructively (dark spot). Figure 34-7. How the wave theory explains the pattern of lines seen in the double-slit experiment. (a) At the center of the screen the waves from each slit travel the same distance and are in phase. (b) At this angle θ, the lower wave travels an extra distance of one whole wavelength, and the waves are in phase; note from the shaded triangle that the path difference equals d sin θ. (c) For this angle θ, the lower wave travels an extra distance equal to one-half wavelength, so the two waves arrive at the screen fully out of phase. (d) A more detailed diagram showing the geometry for parts (b) and (c).

Young’s Double-Slit Experiment Between the maxima and the minima, the interference varies smoothly. Figure 34-9. (a) Interference fringes produced by a double-slit experiment and detected by photographic film placed on the viewing screen. The arrow marks the central fringe. (b) Graph of the intensity of light in the interference pattern. Also shown are values of m for Eq. 34–2a (constructive interference) and Eq. 34–2b (destructive interference).

Young’s Double-Slit Experiment Interference pattern lines. (a) Will there be an infinite number of points on the viewing screen where constructive and destructive interference occur, or only a finite number of points? (b) Are neighboring points of constructive interference uniformly spaced, or is the spacing between neighboring points of constructive interference not uniform? Solution. a. Due to the fact that sin θ cannot be greater than 1, the maximum value of m is equal to the (truncated) value of d/λ. b. The spacing increases with θ, although for small θ it is close to being uniform.

Constructive Interference from two slits Constructive interference (reinforcement) occurs at points where path difference is integral number of wavelengths path difference d = mλ Bright regions on screen occur at angles θ for which © 2016 Pearson Education Inc.

Destructive Interference from two slits Destructive interference (cancellation) occurs, forming dark regions on screen, at points for which path difference is a half-integral number of wavelengths. © 2016 Pearson Education Inc.

Line spacing for double-slit interference. A screen containing two slits 0.100 mm apart is 1.20 m from the viewing screen. Light of wavelength λ = 500 nm falls on th slits from a distant source. Approximately how far apart will adjacent bright interference fringes be on the screen? Figure 34-10. Examples 34–2 and 34–3. For small angles θ (give θ in radians), the interference fringes occur at distance x = θl above the center fringe (m = 0); θ1 and x1 are for the first-order fringe (m = 1), θ2 and x2 are for m = 2. Solution: Using the geometry in the figure, x ≈ lθ for small θ, so the spacing is 6.0 mm.

Example: Changing wavelength (a) What happens to interference pattern in previous example if incident light (500 nm) is replaced by light of wavelength 700 nm? (b) What happens if wavelength stays at 500 nm but slits are moved farther apart? Check out http://www.walter-fendt.de/html5/phen/doubleslit_en.htm Solution: a. As the wavelength increases, the fringes move farther apart. b. Increasing the slit spacing causes the fringes to move closer together.

Young’s Double-Slit Experiment Since position of maxima (except central one) depends on wavelength, first- and higher-order fringes contain a spectrum of colors. Figure 34-11. First-order fringes are a full spectrum, like a rainbow.

Example : Wavelengths from double-slit interference. White light passes through two slits 0.50 mm apart, and an interference pattern is observed on a screen 2.5 m away. First-order fringe resembles a rainbow with violet and red light at opposite ends. The violet light is about 2.0 mm and the red 3.5 mm from the center of the central white fringe. Estimate wavelengths for the violet and red light. Solution: Using the geometry of the previous examples and the small-angle approximation, the wavelength for the first-order fringe is given by dx/l = 400 nm for the violet light and 700 nm for the red light.

Two-source interference of light If distance R to screen is much greater than distance d between slits…. Use approximate geometry (c). © 2016 Pearson Education Inc.

Electric field in interference patterns Find intensity at any point in two-source interference pattern, by combining two sinusoidally varying fields. If two sources are in phase, then waves arriving at point P differ in phase by amount ϕ proportional to difference in path lengths © 2016 Pearson Education Inc.

Electric field in interference patterns Assuming amplitudes of two waves are both approximately equal to E at point P, combined amplitude is: © 2016 Pearson Education Inc.

Phasor diagram for superposition To add two sinusoidal functions with a phase difference, use phasor representation for simple harmonic motion & voltages & currents in AC circuits © 2016 Pearson Education Inc.

Phasor diagram for superposition Each sinusoidal function is represented by rotating vector (phasor) Projection on horizontal axis at any instant represents instantaneous value of sinusoidal function. © 2016 Pearson Education Inc.

Intensity in Double-Slit Interference Pattern The two waves can be added using phasors, to take the phase difference into account: Figure 34-11. Phasor diagram for double-slit interference pattern.

Intensity in interference patterns Intensity at any point in two-source interference pattern: I0 is maximum intensity = 2 e0cE2 = four times larger than intensity from each individual source. Averaged over all phase differences, I = I0/2, just twice intensity of each source. Total energy isn’t changed – just redistributed in space. © 2016 Pearson Education Inc.

Intensity in Double-Slit Interference Pattern The electric fields at the point P from the two slits are given by . Figure 34-12. Determining the intensity in a double-slit interference pattern. Not to scale: in fact l >> d and the two rays become essentially parallel. where

Intensity in Double-Slit Interference Pattern The time-averaged intensity is proportional to the square of the field:

Intensity in Double-Slit Interference Pattern Intensity as a function of angle. Figure 34-14. Intensity I as a function of phase difference δ and position on screen y (assuming y << l).

Intensity in interference patterns Phase difference is: Must use wavelength in medium! l = l0 / n © 2016 Pearson Education Inc.

Interference in thin films © 2016 Pearson Education Inc.

Phase shifts during reflection © 2016 Pearson Education Inc.

Phase shifts during reflection © 2016 Pearson Education Inc.

Interference in thin films: No relative shift For light of normal incidence on thin film wavelength λ in film, where neither (or both) reflected waves have a half-cycle phase shift: © 2016 Pearson Education Inc.

Interference in thin films: Half-wave shift For light of normal incidence on a thin film wavelength λ in film, when only one of reflected waves has a half-cycle phase shift: © 2016 Pearson Education Inc.

Nonreflective coatings © 2016 Pearson Education Inc.

Sketch of a Michelson Interferometer Laser Beam Splitter End Mirror Screen Viewing

Michelson interferometer © 2016 Pearson Education Inc.

LIGO

LIGO sites LIGO (Washington) LIGO (Louisiana) (4km and 2km) (4km) Funded by the National Science Foundation; operated by Caltech and MIT; the research focus for more than 670 LIGO Scientific Collaboration members worldwide.

The LIGO Observatories LIGO Hanford Observatory (LHO) H1 : 4 km arms H2 : 2 km arms 10 ms LIGO Livingston Observatory (LLO) L1 : 4 km arms Adapted from “The Blue Marble: Land Surface, Ocean Color and Sea Ice” at visibleearth.nasa.gov NASA Goddard Space Flight Center Image by Reto Stöckli (land surface, shallow water, clouds). Enhancements by Robert Simmon (ocean color, compositing, 3D globes, animation). Data and technical support: MODIS Land Group; MODIS Science Data Support Team; MODIS Atmosphere Group; MODIS Ocean Group Additional data: USGS EROS Data Center (topography); USGS Terrestrial Remote Sensing Flagstaff Field Center (Antarctica); Defense Meteorological Satellite Program (city lights).

Sensing the Effect of a Gravitational Wave Gravitational wave changes arm lengths and amount of light in signal Change in arm length is 10-18 meters, or about 2/10,000,000,000,000,000 inches Laser signal

Gravitational Wave Detection Suspended Interferometers Suspended mirrors in “free-fall” Fabry-Perot cavity 4km g.w. output port power recycling mirror LIGO design length sensitivity: 10-18m

Core Optics Suspension and Control Optics suspended as simple pendulums Shadow sensors & voice-coil actuators provide damping and control forces Mirror is balanced on 30 micron diameter wire to 1/100th degree of arc

Seismic Isolation – Springs and Masses damped spring cross section

How Small is 10-18 Meter? One meter, about 40 inches Human hair, about 100 microns Wavelength of light, about 1 micron Atomic diameter, 10-10 meter Nuclear diameter, 10-15 meter LIGO sensitivity, 10-18 meter

Gravitational Waves Transverse distortions of spacetime due to motion of massive astronomical bodies. Expected sources: Inspiraling neutron stars/black holes (Asymmetric) supernovae Rotating pulsars Cosmic gravitational-wave background Expected properties: Quadrupole polarization Propagating at speed of light Strains of ΔL/L = 10-21 or less

Binary: a typical GW source Strength of GWs For binary system shown in the figure, strain h=2δl/l becomes h~R1R2/Dd For neutron stars of 1.4M⊙out at Virgo cluster (15Mpc): h~10-21 Crazy but within current reach! Binary: a typical GW source

Hulse-Taylor Binary Pulsar · 17 / sec ~ 8 hr PSR 1913 + 16, measured in 1975 System should lose energy through gravitational radiation Stars get closer together Orbital period gets shorter

Why Are We Looking? “Chirp Signal” We can use weak-field gravitational waves to study strong-field general relativity.

Fabry-Perot Michelson Interferometer Uses light interference to measure path length difference between two arms Each arm is a Fabry-Perot cavity, effectively increasing arm length Geometry ideally suited for quadrupole radiation

Fabry-Perot Michelson Interferometer Each arm is Fabry-Perot cavity, effectively increasing arm length FP cavities increase path length by ~280 times, making equivalent length 1120 km long!

Latest Detection: Neutron Star Collision