Electromagnetic Waves Chapter 34, sections 4-9 Energy and pressure Polarization Reflection and Refraction

Consider these equations in a vacuum: no charges or currents Maxwell’s Equations in a Vacuum

Plane Electromagnetic Waves x EyEy BzBz E(x, t) = E P sin (kx-  t) B(x, t) = B P sin (kx-  t) ˆ z ˆ j c Solved by: Works for any wavelength =2  /k as long as Since  =2  f and k=2 , this means f=c. is inversely proportional to f.

The Electromagnetic Spectrum Radio waves  -wave infra -red  -rays x-rays ultra -violet

Energy in Electromagnetic Waves Electric and magnetic fields contain energy, the potential energy stored in the field: u E = (1/2)  0 E 2 electric field energy density u B = (1/2  0 ) B 2 magnetic field energy density The energy is put into the oscillating fields by the sources that generate them. This energy then propagates to locations far away, at the velocity of light.

B E Energy in Electromagnetic Waves area A dx c propagation direction

B E Energy in Electromagnetic Waves area A dx Energy per unit volume in an EM wave: u = u E + u B c propagation direction

B E Energy in Electromagnetic Waves area A dx Energy per unit volume in an EM wave: u = u E + u B Thus the energy dU in a box of area A and length dx is c propagation direction

B E Energy in Electromagnetic Waves area A dx Energy per unit volume in an EM wave: u = u E + u B Thus the energy dU in a box of area A and length dx is Let the length dx equal cdt. Then all of this energy flows through the front face in time dt. Thus energy flows at the rate c propagation direction

Energy in Electromagnetic Waves area A dx c propagation direction B E Rate of energy flow:

Energy in Electromagnetic Waves area A dx c propagation direction We define the intensity S as the rate of energy flow per unit area: B E Rate of energy flow:

Energy in Electromagnetic Waves area A dx c propagation direction We define the intensity S, as the rate of energy flow per unit area: Rearranging by substituting E=cB and B=E/c, we get B E Rate of energy flow:

The Poynting Vector area A dx B E propagation direction In general we write: S = (1/  0 ) E x B S is a vector that points in the direction of propagation of the wave and represents the rate of energy flow per unit area. We call this the “Poynting vector”. Units of S are Jm -2 s -1, or Watts/m 2.

The Poynting Vector For a plane EM wave the intensity is

The Poynting Vector For a plane EM wave the intensity is Because the fields depend on position and time, so does the intensity:

The Poynting Vector For a plane EM wave the intensity is Because the fields depend on position and time, so does the intensity: If you sit at a certain position S will change in time. The average is Sometimes the notation S is used for S avg. _

Poynting vector for spherical waves Source r A point source of light, or any EM radiation, spreads out as a spherical wave: Source Power, P, flowing through sphere is same for any radius.

Example: An observer is 1.8 m from a point light source whose average power P= 250 W. Calculate the rms fields in the position of the observer. Intensity of light at a distance r is S= P / 4  r 2

Wave Momentum and Radiation Pressure It is somewhat surprising to discover that EM radiation possesses momentum as well as energy. The momentum and energy of a wave are related by p = U / c.

Wave Momentum and Radiation Pressure It is somewhat surprising to discover that EM radiation possesses momentum as well as energy. The momentum and energy of a wave are related by p = U / c. If light carries momentum then it follows that a beam of light falling on an object exerts a pressure: Force = dp/dt = (dU/dt)/c Pressure (radiation) = Force / unit area P = (dU/dt) / (A c) = S / c Radiation Pressure 

Example: Serious proposals have been made to “sail” spacecraft to the outer solar system using the pressure of sunlight. How much sail area must a 1000 kg spacecraft have if its acceleration is to be 1 m/s 2 at the Earth’s orbit? Make the sail reflective. Can ignore gravity. Need F=ma=(1000kg)(1 m/s 2 )=1000 N This comes from pressure: F=PA, so A=F/P. Here P is the radiation pressure of sunlight: Sun’s power = 4 x W, so S=power/(4  r 2 ) gives S = (4 x W) / (4  (1.5x10 11 m) 2 )= 1.4kW/m 2. Thus the pressure due to this light, reflected, is: P = 2S/c = 2(1400W/m 2 ) / 3x10 8 m/s = 9.4x10 -6 N/m 2 Hence A=1000N / 9.4x10 -6 N/m 2 =1.0x10 8 m 2 = 100 km 2

Polarizatio n The direction of polarization of a wave is the direction of the electric field. Most light is randomly polarized, which means it contains a mixture of waves of different polarizations. x EyEy BzBz Polarization direction

 Polarizatio n A polarizer lets through light of only one polarization: E0E0 E E = E 0 cos  hence S = S 0 cos 2  Malus’s Law If the initial beam has bits with random polarizations, then S = S 0 (cos 2  avg = S 0 /2: half gets through. Transmitted light has its E in the direction of the polarizer’s transmission axis.

OPTICS

Geometrical Optics Optics is the study of the behavior of light (not necessarily visible light). This behavior can be described by Maxwell’s equations. However, when the objects with which light interacts are larger that its wavelength, the light travels in straight lines called rays, and its wave nature can be ignored. This is the realm of geometrical optics. The wave properties of light show up in phenomena such as interference and diffraction.

Geometrical Optics Light can be described using geometrical optics, as long as the objects with which it interacts are much larger than the wavelength of the light. This can be described using geometrical optics This requires the use of full wave optics (Maxwell’s equations)

Reflection and Transmission Some materials reflect light. For example, metals reflect light because an incident oscillating light beam causes the metal’s nearly free electrons to oscillate, setting up another (reflected) electromagnetic wave. Opaque materials absorb light (by, say, moving electrons into higher atomic orbitals). Transparent materials are usually insulators whose electrons are bound to atoms, and which would require more energy to move to higher orbitals than in materials which are opaque.

11  1 = angle of incidence Geometrical Optics Surface Normal to surface Incident ray Angles are measured with respect to the normal to the surface

Reflection The Law of Reflection: Light reflected from a surface stays in the plane formed by the incident ray and the surface normal; and the angle of reflection equals the angle of incidence (measured to the normal) 11 ’1’1  1 =  ’ 1 This is called “specular” reflection

Refraction More generally, when light passes from one transparent medium to another, part is reflected and part is transmitted. The reflected ray obeys  1 =  ’ 1. 11 ’1’1 22 Medium 1 Medium 2

Refraction 11 ’1’1 22 Medium 1 Medium 2 More generally, when light passes from one transparent medium to another, part is reflected and part is transmitted. The reflected ray obeys  1 =  ’ 1. The transmitted ray obeys Snell’s Law of Refraction: It stays in the plane, and the angles are related by n 1 sin  1 = n 2 sin  2 Here n is the “index of refraction” of a medium.

Refraction n  index of refraction n i = c / v i v i = velocity of light in medium i Incident ray 11 ’1’1 22 Medium 1 Medium 2 Reflected ray Refracted ray  1 = angle of incidence  ’ 1 = angle of reflection  1 = angle of refraction Law of Reflection  1 =  ’ 1 Law of Refraction n 1 sin  1 = n 2 sin  2

Refraction The little shaded triangles have the same hypotenuse: so 1 /sin  1 = 2 /sin  2, or v 1 /sin  1 =v 2 /sin  2 1 =v 1 T 2 =v 2 T 1 2 11 22 11 22 Define the index of refraction: n=c/v. Then Snell’s law is: n 1 sin  1 = n 2 sin  2 The period T doesn’t change, but the speed of light can be different. in different materials. Then the wavelengths 1 and 2 are unequal. This also gives rise to refraction.

Example: air-water interface If you shine a light at an incident angle of 40 o onto the surface of a pool 2m deep, where does the beam hit the bottom? air water 40  2m Air: n=1.00 Water: n=1.33 (1.00)sin40 = (1.33)sin  sin  =sin40/1.33 so  =28.9 o Then d/2=tan28.9 o which gives d=1.1 m. d

Example: air-water interface If you shine a light at an incident angle of 40 o onto the surface of a pool 2m deep, where does the beam hit the bottom? air water 40  2m Air: n=1.00 Water: n=1.33 (1.00)sin40 = (1.33)sin  sin  =sin40/1.33 so  =28.9 o Then d/2=tan28.9 o which gives d=1.1 m. d

Example: air-water interface If you shine a light at an incident angle of 40 o onto the surface of a pool 2m deep, where does the beam hit the bottom? air water 40  2m Air: n=1.00 Water: n=1.33 (1.00) sin(40) = (1.33) sin  Sin  = sin(40)/1.33 so  = 28.9 o Then d/2 = tan(28.9 o ) which gives  d=1.1 m. d Turn this around: if you shine a light from the bottom at this position it will look like it’s coming from further right.

Air-water interface air water 11 22 Air: n 1 = 1.00 Water: n 2 = 1.33 When the light travels from air to water (n 1 < n 2 ) the ray is bent towards the normal. When the light travels from water to air (n 2 > n 1 ) the ray is bent away from the normal. n 1 sin  1 = n 2 sin  2 n 1 /n 2 = sin  2 / sin  1 This is valid for any pair of materials with n 1 < n 2

Total Internal Reflection Suppose the light goes from medium 1 to 2 and that n 2 <n 1 (for example, from water to air). Snell’s law gives sin  2 = (n 1 / n 2 ) sin  1. Since sin  2 <= 1 there must be a maximum value of  1. At angles bigger than this “critical angle”, the beam is totally reflected. The critical angle is when  2 =  /2, which gives  c =sin -1 (n 2 /n 1 ).

cc 22 22 11 11 11 n2n2 n1n1 Some light is refracted and some is reflected Total internal reflection: no light is refracted Total Internal Reflection n 2 sin  = n 1 sin  1... sin  1 = sin  c = n 2 / n 1 n 1 > n 2

Example: Fiber Optics An optical fiber consists of a core with index n 1 surrounded by a cladding with index n 2, with n 1 >n 2. Light can be confined by total internal reflection, even if the fiber is bent and twisted. Exercise: For n 1 = 1.7 and n 2 = 1.6 find the minimum angle of incidence for guiding in the fiber. Answer: sin  C = n 2 / n 1   C = sin -1 (n 2 / n 1 ) = sin -1 (1.6/1.7) = 70 o. (Need to graze at < 20 o )

Dispersion The index of refraction depends on frequency or wavelength: n = n( ) Typically many optical materials, (glass, quartz) have decreasing n with increasing wavelength in the visible region of spectrum Dispersion by a prism: 700 nm 400 nm nm n

Example: dispersion at a right angle prism Find the angle between outgoing red ( r = 700nm) and violet ( v = 400nm) light [ n 400 =1.538, n 700 = 1.516,  1 = 40° ]. 11 red violet 22 Red: sin(40°) = 1 sin  400   400 = sin -1 ( ) = 81.34° Violet: sin(40°) = 1 sin  700   700 = sin -1 ( ) = 77.02°   = 4.32°  angular dispersion of the beam n 1 sin  1 = n 2 sin  2 n 2 = 1 (air)

Reflection and Transmission at Normal Incidence Geometrical optics can’t tell how much is reflected and how much transmitted at an interface. This can be derived from Maxwell’s equations. These are described in terms of the reflection and transmission coefficients R and T, which are, respectively, the fraction of incident intensity reflected and transmitted. For the case of normal incidence, one finds: Notice that when n 1 =n 2 (so that there is not really any interface), R=0 and T=1. I RI TI

Reflection and Transmission at Oblique Incidence In this case R and T depend on the angle of incidence in a complicated way – and on the polarization of the incident beam. We relate polarization to the plane of the three rays. E parallel reflected incident transmitted E perpendicular n1n1 n2n2

Angle of incidence R (%) Reflection and Transmission at Oblique Incidence perp parallel Light with the perpendicular polarization is reflected more strongly than light with the parallel polarization. Hence if unpolarized light is incident on a surface, the reflected beam will be partially polarized. Notice that at grazing incidence everything is reflected.

Angle of incidence R (%) Polarizing angle, or “Brewster’s angle” Brewster’s angle of incidence is the angle at which light polarized in the plane is not reflected but transmitted 100% All the reflected light has perpendicular polarization. Reflection and Transmission at Oblique Incidence pp perp parallel