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Propagation of Light The Electromagnetic Spectrum Maxwell’s description of the EM Wave Intensity and the Poynting Vector

Reflection and Refraction of Light Total Internal Reflection Polarization and Mallus’s Law

Today’s information age is based almost entirely on the physics of electromagnetic waves. The connection between electric and magnetic fields to produce light is own of the greatest achievements produced by physics, and electromagnetic waves are at the core of many fields in science and engineering.

In this chapter we introduce fundamental concepts and explore the properties of electromagnetic waves.

Chapter The Nature of Light

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Light Rays •  Rectilinear Propagation •  Supported particle nature of light •  Intensity I(x) = Io (W/m2)

Point Sources •  Inverse square law •  Intensity I(r) = Io 1/4πr2

Propagation of Light

I = I0

14πr 2

I = I0(W / m2 )

The wavelength/frequency range in which electromagnetic (EM) waves (light) are visible is only a tiny fraction of the entire electromagnetic spectrum

Electromagnetic Spectrum -Newton’s Rainbow (1671)

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Cell phone

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Maxwell (1865) formulates a theory of light as an Electric + Magnetic wave traveling with the speed c = 3x108 m/s.

+ + + - - -

+ + + - - -

•  We say the wave Is polarized along Electric field direction.

Poynting Vector!S = 1

µ0

!E ×!B energy flux

I(W / m2 ) = |!S |=

E0B0

µ0

=E0

2

µ0c

•  The Poynting vector describes the energy flux of the EM or energy transfer. It has units of energy flux (W/m2) or Power/area.

Plane wave

Mathematical Description of Traveling EM Waves

Electric Field: E = Em sin kx −ωt( )Magnetic Field: B = Bm sin kx −ωt( )

Wave Speed: c = 1µ0ε0

AngularFrequency:ω = 2πT

Wave Number : k = 2πλ

υc = c = ωk= λ

T!F = q

!E + q

!υ ×!B → E = cB

All EM waves travel a c in vacuum

EM Wave T

5 E = cB I = 1

µ0c|!E |2

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An LC oscillator causes currents to flow sinusoidally, which in turn produces oscillating electric and magnetic fields, which then propagate through space as EM waves

Oscillation Frequency:

ω = 1

LC

E ~ λ2πε0r

B ~µ0i2πr

i

The Traveling Electromagnetic (EM) Wave, Qualitatively

λ+++

Plane wave

Circular wave pattern from a line source

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Q1: A laser beam delivers 50 mW/mm2 to a target. Find the value of the electric and magnetic field amplitudes of the light wave.

Q2: An observer is 1.8m from a small light source (candle) whose power output is 250W. Calculate the value of the values of the E and B fields at the position of the observer.

I = P / A = 50×10−3W / 10−6 m2 I = 0.05 W / m2

I = |!S |= (1/ µ0 )}E × B | = (1/ µ0 ) E B

Using E = cB − > I = (1/ µ0 ) E B → E2 = µ0c I

E = (4π x 10−7 )(3x108 ) 0.05( ) Erms = 18.8 V / m

B = E / c = 18.8 / 3x108( ) Brms =6.3×10−8 T

I = I0 / 4πr 2 intensity/area of spherical wavefront at r = 1.8m

I = 250W / 4πr 2 = 250 / 4π (1.8m)2 I = 6.2 W / m2

E2 = µ0c I = 2315 V 2 / m2

Erms = 48.1V / m

B = E / c Brms = 1.6×10−7T

In a vacuum light will travel in a straight line. When light encounters a medium it will either reflect or refract (transmit with bend). The velocity of light in a medium of index of refraction n is

Reflection and Refraction

What happens when a narrow beam of light encounters a glass surface?

Reflection: θ1 = θ1 '

Refraction: n1 sinθ1 = n2 sinθ2

Law of Reflection

Snell’s Law

TransmissionCoefficientT=1 - Rn is the index of refraction

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Reflection Coefficient

R =n2 − n1

n2 + n1

2

υn = c / n

n1

n2

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The refractive index can be seen as the factor by which the speed and the wavelength of the radiation are reduced with respect to their vacuum values: the speed of light in a medium is v = c/n, and similarly the wavelength in that medium is λ = λ0/n, where λ0 is the wavelength of that light in vacuum.

υc = f λ0 → f

λ0

n=

f λ0

nυc =

cn

n1<n2 n2

λ =

λ0

nf = f0

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For light going from media n1 to n2

(a) n2 = n1 θ12= θ1

(b) n2 > n1 θ2 < θ1, light bent towards normal

(c) n2 < n1 θ2>θ1, light bent away from normal

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How do metals and other solid materials reflect light ?

In the case of metals the free electrons in the surface layers deflect the light elastically with, θ1=θ2, (specular reflection). Other materials will absorb certain colors and reflect others (diffuse reflection), giving the material its color.

Specular reflection Diffuse reflection

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Indexes of refraction of Some Materials

Chromatic Dispersion

lens

Spectral Analysis

Chromatic Aberration

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Q3: A fisherman observes a fish at viewing angle ϕ =120. He believes the fish to be 1m below the surface of the water. Determine xo and then determine the true depth yo of the fish?

tanφ = 1m / x0

x0 = 1m / tan(120 ) = 4.7m

θ1 = 900 −120 = 780

n1sin(θ1) = n2sin(θ2 )

(1.00)sin(780 ) = (1.33)sin(θ2 )0.978 = 1.33sin(θ2 )θ2 = sin−1(0.978 / 1.33)

= 0.826rdn = 47.30

tan(θ2 ) = x0 / y0 = 1.08

y0 = 4.7m / 1.08 = 4.4m

water

θ1

θ2

ϕ=12o xo

1m

yo

Q4: What percent of light is reflectedAnd transmitted (at normal incidence) at an air water interface?

R = |nw − nair

nw + nair

|2= | 1.33−11.33+1

|2

R = 0.02(2%) T = 98%

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For light that travels from a medium with a larger index of refraction to a medium with a smaller medium of refraction n1>n1 and θ2>θ1, as θ1 increases, θ2 will reach 90o (the largest possible angle for refraction) before θ1 does.

Total Internal Reflection

n1 sinθc = n2 sin90° = n2

n1

n2

Critical Angle: θc = sin−1 n2

n1

When θ2> θc no light is refracted (Snell’s Law does not have a solution!) so no light is transmitted Total Internal Reflection Total internal reflection can be used, for

example, to guide/contain light along an optical fiber

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Fiber Optics are used to transmit signals with low noise and low signal loss.

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Total internal reflection right angle prism.

Retro reflectors Retro reflective tape Corner reflector

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Refraction by lenses Spherical lenses made of glass, n=1.52, can focus or defocus light by the laws of refraction. Rays approaching a converging lense from ∞ will converge to a focal point.

∞

∞

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http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/lenseq.html

The polarization of light describes how the electric field in the EM wave oscillates. This wave is vertically or linearly polarized.

Polarization of Light

Unpolarized or randomly polarized light has its instantaneous polarization direction vary randomly with time.

One can produce different polarizations such as circularly or elliptically polarized light by mixing two linearly polarized beams.

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Polarizing Sheet

Only electric field component along polarizing direction of polarizing sheet is passed (transmitted), the perpendicular component is blocked (absorbed). I = ½ Io 50% transmission of unpolarized light

I0

I=½ I0

Unpolarized light

Incident light

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Transmission Polarized Light through a polarizer

iSince only the component of the incident electric field E parallel to the polarizing axis is transmitted Etransmitted = Ey = E cosθ

iMalus's Law states that the intensity of plane-polarized light that passes through an analyzer varies as the square of the cosine of the angle between the plane of the polarizer and the transmission axes of the analyzer.

I ~| Ey |2 I = I0 cos2θ

i For unpolarized light inpinging on a polarizer I = I0 cos2θ dθ

0

2π

∫ = 12

I0

I = 1

2I0

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Q5: Three polarizing filters are oriented with their axis at 30o with respect to each other. θ1=0o, θ2=30o, θ3=60o. Determine the percent transmission of an unpolarized light source through the three filters.

I = (I0

2)cos2(Δθ12 )cos2(Δθ23)

= (I0

2)cos2(300 )cos2(300 )

= (I0

2)(0.866)2(0.866)2 = 0.28I0

28%

0o

30o

60o 12

I0

12

I0 cos2(300 )

12

I0 cos2(300 )cos2(300 )

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Polarization by Reflection

In which direction does light reflecting off a lake tend to be polarized?

Upon reflection from a surface only a fraction vertical component of the E-field is reflected, resulting in a net horizontal polarization of the reflected light. The angle of perfect polarization is called Brewster’s Angle. Polaroid sun glasses with a vertical polarization axis will cancel the reflected light (glare) sharpening the image.

Brewster's Angleperfect polarization

θB = tan−1(nmedia

nair

)

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STOP HERE

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EM waves have linear momentum as well as energy�light can exert pressure Radiation Pressure

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incidentS!

pΔ

incidentS!

reflectedS!

pΔ

Total absorption: UpcΔΔ =

Total reflection Back along path:

2 UpcΔΔ =

pFt

Δ=Δ

power energy/timearea area

I

U tA

= =

Δ Δ=

(total absorption)

2 (total reflection back along path)

Radiation Pressurer

U IA tIAFcIAFcFpA

Δ = Δ

=

=

=

2 r

Ipc

=

rIpc

=

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The index of refraction n encountered by light in any medium except vacuum depends on the wavelength of the light. So if light consisting of different wavelengths enters a material, the different wavelengths will be refracted differently � chromatic dispersion

Chromatic Dispersion

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Fig. 33-19 Fig. 33-20 n2blue>n2red

Chromatic dispersion can be good (e.g., used to analyze wavelength composition of light) or bad (e.g., chromatic aberration in lenses)

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Rainbows

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Fig. 33-22

Sunlight consists of all visible colors and water is dispersive, so when sunlight is refracted as it enters water droplets, is reflected off the back surface, and again is refracted as it exits the water drops, the range of angles for the exiting ray will depend on the color of the ray. Since blue is refracted more strongly than red, only droplets that are closer the the rainbow center (A) will refract/reflect blue light to the observer (O). Droplets at larger angles will still refract/reflect red light to the observer.

What happens for rays that reflect twice off the back surfaces of the droplets?