Interference and Diffraction of EM waves

Interference and Diffraction of EM waves

Maxwell Equations in General Form Differential form Integral Form D v B E H 0 B t D J t L L DdS s s E dl H dl B ds s v J dv 0 t v s B ds D ds t Gauss s Law for E field. Gauss s Law for H field. Nonexistence of monopole Faraday s Law Ampere s Circuit Law

Terms E = electric field intensity [V/m] D = electric field density H = magnetic field intensity, [A/m] B = magnetic field density, [Teslas]

Maxwell s Equations v Additionally the equation of continuity J t D Maxwell added the term to Ampere s Law so that it not only works for static conditions but also for time-varying situations. This added term is called the displacement current density, while J is the conduction current. t

Energy (intensity) of electromagnetic waves The frequency of light is very high, There is no such detector to measure the electric field changes, We are able only to measure the mean value of the square root of the electric field, our eyes can detect only intensity of light, not phase.

Energy (intensity) of electromagnetic waves Poynting vector Intensity of EM wave Light intensity I is a mean velue of square root of the electric field intensity and is defined in W/m² Taking into account the spectral characteristic of human eye, light intensity is defined in candelas, lumens or lux

Two waves interfering with each other If two monochromatic waves described as: will overlap in some plane x=const, then: Responsible for interference

Two waves interfering with each other Responsible for interference For : > 0 constructive interference = 0 < 0 destructive interference

The same phases The oposite phases Constructive interference Destructive interference

The key principle: Huygen s Principle All points in a wavefront serve as point sources of spherical secondary waves. After a time t, the new wavefront will be the tangent to all the resulting spherical waves. Christian Huygens 1629-1695

Huygen s Principle For plane waves entering a single slit, the waves emerging from the slit start spreading out, diffracting

Young s Double Slit Experiment For waves entering two slits, the emerging waves interfere and form an interference (diffraction) pattern Young experiment in 1801: light is wave phenomenon First plane wave through a small slit yields coherent spherical wave Then interposed two slits: interference of two spherical waves on a screen

Interference Phase difference between two waves can change for paths of different lengths Each point on the screen is determined by the path length difference DL of the rays reaching that point Path Length Difference: DLdsin

Interference If DL dsin integer bright fringe Maxima-bright fringes: d sin m for m 0,1, 2, 1 Minima-dark fringes: d sin m for m 0,1,2, 2 m 1 2 2 bright fringe at: sin d m 1 1.5 1 dark fringe at: sin d

When the interference is possible Two sources can produce an interference that is stable over time, if their light has a phase relationship that does not change with time: E(t)=E 0 cos(wt+f). Coherent sources: Phase f must be well defined and constant Sunlight is coherent over a short length and time range Since laser light is produced by cooperative behavior of atoms, it is coherent of long length and time ranges Incoherent sources: f jitters randomly in time, no stable interference occurs

Example of interference Red laser light (=633nm) goes through two slits d=1cm apart, and produces a diffraction pattern on a screen L = 55cm away. How far apart are the fringes near the center? If the fringes are near the center, we can use sin ~, and then m=dsin~d => =m/d is the angle for each maximum (in radians) D= /d =is the angular separation. The distance between the fringes is then Dx=LD=L/d=55cm 633nm/1cm=35 mm For the spacing to be 1mm, we need d~ L/1mm=0.35mm

Example 2 In a double slit experiment, we can measure the wavelength of the light if we know the distances between the slits and the angular separation of the fringes. If the separation between the slits is 0.5mm and the first order maximum of the interference pattern is at an angle of 0.059 o from the center of the pattern, what is the wavelength and color of the light used? d sin=m => =0.5mm sin(0.059 o ) = 5.15 x 10-7 m=515nm ~ green

Intensity in Double-Slit Interference E1 E0 sin wt and E2 E0 sin wt f 2 1 2d I 4I0 cos 2f f sin E 2 E 1 1 2 d Maxima when: 2 f m for m 0,1, 2, f 2m sin d sin m for m 0,1, 2, (maxima) Minima when: f m d sin m for m 0,1, 2, (minima) 1 1 1 2 2 2 I 2I avg 0

Example A double slit experiment has a screen 120cm away from the slits, which are 0.25cm apart. The slits are illuminated with coherent 600nm light. At what distance above the central maximum is the average intensity on the screen 75% of the maximum? I/I 0 =4cos 2 f/2 ; I/I max =cos 2 f/2 =0.75 => f=2cos 1 (0.75) 1/2 =60 o =/3 rad f=(2d/)sin => = sin -1 (/2d)f0.0022 o 40 mrad (small!) y=l48mm

Interferometers Michelson s Mach-Zehnder s Ring

Temporal coherence

White light LED SLED LD Gas laser He-Ne D

Measuring the distance with Michelson s interferometer Optical coherence tomography

Diffraction

Huygen s Principle All points in a wavefront serve as point sources of spherical secondary waves. After a time t, the new wavefront will be the tangent to all the resulting spherical waves. Christian Huygens 1629-1695

Huygen s Principle For plane waves entering a single slit, the waves emerging from the slit start spreading out, diffracting

Young s Double Slit Experiment For waves entering two slits, the emerging waves interfere and form an interference (diffraction) pattern Young experiment in 1801: light is wave phenomenon First plane wave through a small slit yields coherent spherical wave Then interposed two slits: interference of two spherical waves on a screen

Diffraction by a Single Slit: Locating the Minima Path length difference between rays r 1 and r 2 is /2 Two rays out of phase at P 1 resulting in destructive interference Path length difference is distance from starting point of r 2 at center of the slit to point b For D>>a, the path length difference between rays r 1 and r 2 is (a/2) sin

Repeat previous analysis for pairs of rays, each separated by a vertical distance of a/2 at the slit. Setting path length difference to /2 for each pair of rays, we obtain the first dark fringes at: a sin asin 2 2 (first minimum) For second minimum, divide slit into 4 zones of equal widths a/4 (separation between pairs of rays). Destructive interference occurs when the path length difference for each pair is /2. a sin asin 2 4 2 (second minimum) Dividing the slit into increasingly larger even numbers of zones, we can find higher order minima: asin m, for m 1, 2,3 (minima-dark fringes)

Diffraction and the Wave Theory of Light Diffraction pattern from a single narrow slit. Side or secondary maxima Light Central maximum Fresnel Bright Spot. These patterns cannot be explained using geometrical optics! Light Bright spot

When light goes through a narrow slit, it spreads out to form a diffraction pattern. Single Slit Diffraction

Intensity in Single-Slit Diffraction, Quantitatively Here we will show that the intensity at the screen due to a single slit is: sin I I m 2 (36-5) (1) 1 a where f sin (36-6) (2) 2 In Eq. 1, minima occur when: m, for m1, 2,3 If we put this into Eq. 2 we find: a m sin, for m1, 2,3 or asin m, for m 1,2,3 (minima-dark fringes)

Diffraction of a laser through a slit (example) 1.2 cm Light from a helium-neon laser ( = 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? y L (0.012 m) (2.00 m) 1 1 0.0060 rad a 7 (6.3310 m) 3 sin (6.0010 rad) 1 1 4 1.06 10 m 0.106 mm

Width of a Single-Slit Diffraction Pattern -y 1 0 y 1 y 2 y 3 y p w pl ; p 1, 2,3, (positions of dark fringes) a w 2L a (width of diffraction peak from min to min)

1000m 10 m 1m 1m You are doing 137 mph on I-10 and you pass a little old lady doing 55mph when a cop, Located 1km away fires his radar gun, which has a 10 cm opening. Can he tell you from the L.O.L. if the gun Is X-band? What about Laser? X-band: =10cm Laser-band: =1mm w 2L a 0.1m 1000m 2 0.1m 2000m w 2L a 0.000001m 1000m 2 0.1m 0.02m

Angles of the Secondary Maxima The diffraction minima are precisely at the angles where sin q = p l/a and a = pp (so that sin a=0). However, the diffraction maxima are not quite at the angles where sin q = (p+½) l/a and a = (p+½)p (so that sin a =1). 1 l = 633 nm a = 0.2 mm 2 sin Imax I 2 3 p (p+½) /a Max 1 0.00475 0.00453 2 0.00791 0.00778 3 0.01108 0.01099 4 0.01424 0.01417 5 0.01741 0.01735 4 5 q (radians)

Diffraction Gratings A device with N slits (rulings) can be used to manipulate light, such as separate different wavelengths of light that are contained in a single beam. How does a diffraction grating affect monochromatic light? Fig. 36-17 Fig. 36-18 d sin m for m 0,1,2 (maxima-lines) (36-11)

Circular Apertures Single slit of aperture a Hole of diameter D When light passes through a circular aperture instead of a vertical slit, the diffraction pattern is modified by the 2D geometry. The minima occur at about 1.22/D instead of /a. Otherwise the behavior is the same, including the spread of the diffraction pattern with decreasing aperture.

The Rayleigh Criterion The Rayleigh Resolution Criterion says that the minimum separation to separate two objects is to have the diffraction peak of one at the diffraction minimum of the other, i.e., D 1.22 /D. Example: The Hubble Space Telescope has a mirror diameter of 4 m, leading to excellent resolution of close-lying objects. For light with wavelength of 500 nm, the angular resolution of the Hubble is D = 1.53 x 10-7 radians.

Example A spy satellite in a 200km low-earth orbit is imaging the Earth in the visible wavelength of 500nm. How big a diameter telescope does it need to read a newspaper over your shoulder from Outer Space?

Example Solution D 1.22 /D (The smaller the wavelength or the bigger the telescope opening the better the angular resolution.) Letters on a newspaper are about Dx = 10mm apart. Orbit altitude R = 200km & D is telescope diameter. Formula: Dx = RD = R(1.22/D) R D D = R(1.22/Dx) Dx = (200x10 3 m)(1.22x500x10 9 m)/(10x10 3 m) = 12.2m

Holography

Brief history of holography Invented in 1948 by Dennis Gabor to improve the resolution in electron microscopy, before the invention of the laser (this time light sources were not coherent) Leith and Upatnieks (1962) applied laser light to holography and introduced an important off-axis technique (the first holographic picture, laser was necessary to see the picture) The pioneer of holography in Poland prof. Mieczysław Wolfke (professor from Faculty of Physics WUT),

Holos - whole, grapho drawing Holography is a method of producing a three-dimensional (3-D) image of an object. (The three dimensions are height, width, and depth.) Later the object can be reconstructed. The hologram is actually a recording of the difference between two beams of coherent light Can be used as optical store disk, in information processing,

Conventional vs. Holographic picture Conventional: 2-d version of a 3-d scene Photograph lacks depth perception or parallax Film sensitive only to radiant energy Phase relation (i.e. interference) are lost

Conventional photography Light Object Reflected wave Photographic film: The intensity is recorded

Conventional photography Light

Conventional vs. Holographic picture Hologram: Freezes the intricate wavefront of light that carries all the visual information of the scene To view a hologram, the wavefront is reconstructed View what we would have seen if present at the original scene through the window defined by the hologram Provides depth perception and parallax

Conventional vs. Holographic picture Converts phase information into amplitude information (in-phase - maximum amplitude, out-of-phase minimum amplitude) Interfere wavefront of light from a scene with a reference wave The hologram is a complex interference pattern of microscopically spaced fringes

How hologram is made? Need a laser, lenses, mirror, photographic film, and an object The laser light is separated into two beams, reference beam and object beam Reference beam enlarged and aimed at a piece of holographic film

How hologram is made? Object beam directed at subject to be recorded and expanded to illuminate subject, Object beam reflects off of object and meets reference beam at film, Produces interference pattern which is recorded,

How hologram is made? Photographic film. Interference of reference and reflected waves is recorded Reference wave

How hologram is made? Film is developed, Hologram illuminated at same angle as reference beam during original exposure to reveal holographic image,

Types of holograms Transmission hologram: reference and object waves traverse the film from the same side Invented by Benton, Can be reconstructed in normal light Reflection hologram: reference and object waves traverse the emulsion from opposite sides

Applications of holography Credit cards carry monetary value, Supermarket scanners, Optical Computers, Used in aircraft heads-up display, Art, Archival Recording of fragile museum artifacts,

Holography in the future Medical Purposes Gaming Systems Personal Defense Computers Artwork Amusement Park Rides Movie production Holodeck from Star Trek Holodeck Clip Star Wars Chess Game

Summary Interference only for coherent light, i.e., with a phase relationship that is time independent Intensity in double-slit interference: I 4I cos f 2 1 0 2 f 2d Use Huygens Principle to find positions of diffraction minima of a single slit by subdividing the aperture asin m, for m 1, 2,3 sin (minima-dark fringes)

Summary Diffraction of light occurs at openings of the order of the wave length of the light Double slit experiment: Maxima-bright fringes: d sin m for m 0,1, 2, 1 Minima-dark fringes: d sin m for m 0,1,2, Intensity in double-slit interference: 2 I 4I cos f 2 1 0 2 f 2d sin

Summary To predict the interference pattern of a multi-slit system, we must combine interference and diffraction effects. Rayleigh s Criterion for the separability of two points Intensity in single-slit diffraction: sin I I m Double-slit diffraction: 2 1 a (36-5) where f sin (36-6) 2 2 sin I I m cos (double slit) 2