Chapter 4. The Properties of Light - 성균관대학교...

Hecht by YHLEE;100510; 4-1

Chapter 4. The Properties of Light 4.1 Introduction Scattering → Transmission, reflection, and refraction (microscopic) (macroscopic) 4.2 Rayleigh Scattering Scattering of sunlight

Sunlight in the air → Ground-state vibration of → Re-emission of light. nitrogen, oxygen, etc.

Higher freq. of light → Larger amplitude of ground-state vibration. Stronger scattering.

The intensity of the scattered light ~ ν4 Blue scatters more strongly than red (Blue sky)

Rayleigh scattering : Scattering from particles < λ /15 A. Scattering and Interference

Rare medium (separation ≥ λ ).

→ Optical path difference to P >> λ . → Intensites are added at P The dense medium (separation ≤ λ ).

→ Electric fields are added at P. → Less lateral scattering due to interference.


Forward Propagation The same optical path length to P

→ Constructive interference in forwar direction.

B. The Transmission of Light Through Dense Media Little scatterings in the lateral or the backward directions

A fixed phase difference among wavelets in the lateral direction. → Sumed to zero More dense, uniform and ordered medium → More complete lateral destructive interference → Forward propagation without diminish Example Glass, plastic : amorphous solids → Lateral scattering Quartz, mica : crystals → Smaller lateral scattering


C. Transmission and the Index of Refraction A primary wave in a dielectric. → Ground-state vibrations of atoms → Spherical wavelets → Interference of wavelets to form secondary wave. The primary + The secondary wave ⇒ The transmitted wave ↑ ↑ ↑ Same speed of c The phase velocity =c , <c, >c. Refractive index change. • Primary wave → Electron oscillator → Secondary wave ↑ ↑ 0 ~ π phase shift 90o phase lag, natural result Lorentz model (3.5) For ω ω<< o : The secondary lags the primary by 90o For ω ω≈ o , at resonance : 180o out of phase. Reduced refracted wave (absorption) For ω ω>> o : 270o phase lag

Dashed : reduced damping

• Accumulated phase lag or lead → Speed change of the wave.


4.3 Reflection A beam of light in a dense medium → Scattering mostly in the forward direction A beam of light across an interface → Some backward scattering. Reflection The change of n over a distance > λ → Little reflection The change of n over a distance < λ /4 → Abrupt interface Internal and External Reflection Unpaired atomic oscillators → Reflection Indep. of glass thickness

Beam I : External reflection (n ni t< )

Beam II : Internal reflection (n ni t> ), 180o phase shift Huygens’s Principle Every point on a primary wavefront behaves as a point source of spherical secondary wavelet. The secondary wavelets propagate with the same speed and frequency with the primary wave. The wave at a later time is the superposition of these wavelets. Rays A ray is a line drawn in the direction of light propagation. In most cases, ray is straight and perpendicular to the wavefront A plane wave is represented by a single ray. A. The Law of Reflection A plane wave into a flat medium ( λ >> atomic spacing) → Spherical wavelets from the atoms.

→ Constructive interference only in one direction.


• Derivation of the law At t=0, the wavefront is AB At t= t1 , the wavefront is CD Note v t BD ADi i1 = = sinθ ,

v t AC ADr r1 = = sinθ

→ sin sinθ θi

i

r

rv v=

Since v vi r=

→ θ θi r= : Law of reflection (Part I) 4.4 Refraction The incident rays are bent at an interface → Refraction A. The Law of Refraction At t=0 the wavefront is AB At t t= Δ the wavefront is ED v t BD ADi iΔ = = sinθ

v t AE ADt tΔ = = sinθ

→ sin sinθ θi

i

t

tv v=

Since v cni

i= , v c

ntt

=

→ n ni i t tsin sinθ θ= : Law of refraction, Snell’s law A weak electric field

→ A linear response of the atom → A simple harmonic vibration of the atom

→ The frequencies of the incident, reflected and refracted waves are equal. 4.5 Fermat’s Principle Hero proposed the principle of shortest path → θ θi r= S, P and B are in the plane of incidence

Fermat proposed the principle of least time → Light takes the path that takes the least time


• Reflection by Fermat’s principle The time from S to P

( )222 2

i t i t

b a xSO OP h xtv v v v

+ −+= + ⇒ +

→ sin sinθ θi

i

t

tv v= : Snell’s law

↑ dt dx/ = 0 • Optical Path Length The transit time from S to P

t sv c

n si

ii

m

i ii

m

= ⇒= =∑ ∑

1 1

1

↑ Optical path length (OPL) In an inhomogeneous medium

( )P

SOPL n s ds= ∫

• Modern Fermat’s Principle

The optical path length of the actual light path is stationary with respect to variations of the path

↑ dfdx

= 0

↑ ↑ Not allowed in the principle of least time

Rays slightly deviate from the stationary path → The same OPL → Constructive interference • Stationary paths in an ellipsoidal mirror

• Fermat and Mirages [Fig. 4.31-33] Bending of rays due to Fermat’s principle


kr

ki

θr

θir

k i

θi

θtk t

4.6 The Electromagnetic Approach A. Waves at an Interface An incident plane wave ( )cosi oi i iE E k r t= • − ω

The reflected and transmitted waves ( )cosr or r r rE E k r t= • − ω + ε

( )cost ot t t tE E k r t= • − ω + ε

, ,i r tε ε ε are constant phases

The boundary conditions ( ) ( ) ( )tangential tangential tangentiali r tE E E+ =

↑ ↑ ↑ u En i× u En r× u En t× This relation should be satisfied regardless of r and t → ω ω ωi r t= =

k r k r k ri r r t t• = • + = • +ε ε (1) • From the first two of (1) ( )i r rk k r− • = ε : r is on the interface plane

→ ( ) ( )k k r ri r o− • − = 0 : ro is a point on the interface plane

→ ( ) //k k ui r n− : un is the surface normal → ki , kr and un form a plane (Plane of incidence)

k ki i r rsin sinθ θ= → θ θi r=

↑ k ki r=

From the first and last of (1) ( )i t tk k r− • = ε

→ ( ) ( )k k r ri t o− • − = 0

→ ( ) i tk k− ⊥ The interface plane

→ ki , kt and un form the plane of incidence

k ki i t tsin sinθ θ= → n ni i t tsin sinθ θ=

↑ k n c= ω /


1 0.2

0.8

B. The Fresnel Eqs. Case 1. E ⊥ The plane of incidence The relation among E H k, , and

( ) ˆ//E H k× , ( )ˆ //k E H×

At the interface E E Eoi or ot+ = (1)

( ) ( ) ( )tangential tangential tangentialoi or otH H H+ =

↑ ↑ ↑ −H xoi icosθ H xor rcosθ −H xot tcosθ Since H E v= /μ

( )1 1cos cosoi or i ot ti i t t

E E Ev v

− θ = θμ μ

(2)

From (1) and (2) with μ μ μ μi r t o= = = , v c n= / Amplitude reflection coefficient

cos coscos cos

or t t i i

oi t t i i

E n n rE n n ⊥

⊥

⎛ ⎞ θ − θ= − ≡⎜ ⎟ θ + θ⎝ ⎠

Amplitude transmission coefficient

2 cos

cos cosot i i

oi t t i i

E n tE n n ⊥

⊥

⎛ ⎞ θ= ≡⎜ ⎟ θ + θ⎝ ⎠

The physical meaning of π phase shift in the reflected wave when n nt i> .


Case 2. E // The plane of incidence

E tangential should be continuous across the interface

→ ( ) ( ) ( )tangential tangential tangentialoi or otE E E+ = (3)

↑ ↑ ↑ E xoi icosθ , −E xor rcosθ , E xot tcosθ , : E is such that B points outward H tangential should be continuous across the interface

→ ( ) ( ) ( )tangential tangential tangentialoi or otH H H+ = (4)

↑ ↑ ↑

1μi i

oivE z 1

μr rorv

E z 1μt t

otvE z

From (3) and (4) with θ θi r= , v vi r= , μ μ μ μi r t o= = = , v c n= / Amplitude reflection coefficient

////

cos coscos cos

or t i i t

oi t i i t

E n n rE n n

⎛ ⎞ θ − θ= ≡⎜ ⎟ θ + θ⎝ ⎠

Amplitude transmission coefficient

////

2 coscos cos

ot i i

oi t i i t

E n tE n n

⎛ ⎞ θ= ≡⎜ ⎟ θ + θ⎝ ⎠

]

• Applying Snell’s law assuming θi ≠ 0 , Fresnel Eqs. become ↑ n ni i t tsin sinθ θ=

( )( )

sinsin

t i

t ir⊥

θ − θ=

θ + θ ( )( )//

tantan

t i

t ir

θ − θ= −

θ + θ

( )2sin cossin

t i

t it⊥

θ θ=

θ + θ ( ) ( )//

2sin cossin cos

t i

t i t it θ θ

=θ + θ θ − θ


C. Interpretation of the Fresnel Eqs. Amplitude Coefficients At normal incidence, θi = 0

t i

t i

n nr rn n⊥

−= =

+

• The external reflection ( ), t i i tn n> θ > θ → r⊥ < 0 .

// 0r = when ( ) 90ot iθ + θ = : Brewster angle, Polarization angle of i pθ = θ .

• The internal reflection ( ), i t t in n> θ > θ

→ 1r⊥ = when 90otθ = : Critical angle of i cθ = θ in sini i tn nθ =

// 0r = when ( ) 90ot iθ + θ = : Brewster angle of 'i pθ = θ .

( 90op p′θ + θ = )

n nt i> , nt = 15. n n ni t i> =, . 15 Stronger reflection at glacing angle Reflectance and Transmittance The power per unit area : S = ×b e , poynting vector In phasor form : ( )*1

2S E H= ×

The intensity ( )2/W m : Irradiance

→ 212 o r o

cI S En

= = ε ε : Average energy per unit time per unit area


The cross sectional area of the incident beam = A icosθ “ “ reflected beam = A rcosθ “ “ transmitted beam = A tcosθ The reflectance

R I AI A

II

EE

rr r

i i

r

i

or

oi≡ ⇒ ⇒ = =

Reflected powerIncident power

coscos

θθ

22

The transmittance

⎛ ⎞θ θ θ

≡ ⇒ ⇒ = ⎜ ⎟θ θ θ⎝ ⎠

22cos cos cosTransmitted power

Incident power cos cos cost t ot t t t t

i i oi i i i i

I A E n nT t

I A E n n

• Energy conservation I A I A I Ai i r r t tcos cos cosθ θ θ= +

→ n E n E n Ei oi i i or i t ot t2 2 2cos cos cosθ θ θ= +

→ 2 2

cos1cos

or t t ot

oi i i oi

E n EE n E

⎛ ⎞ ⎛ ⎞ ⎛ ⎞θ= +⎜ ⎟ ⎜ ⎟ ⎜ ⎟θ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

↑ ↑ R T


4.7 Total Internal reflection The Snell’s law for n ni t>

sin sinθ θit

it

nn

= : θ θi t<

At the critical angle, θt = 90o

sin θct

i

nn

=

For θ θi c>

→ All the incoming energy is reflected back into the incident medium

Total Internal Reflection Internal reflection and TIR: Transition from (a) to (e) without discontinuity. (Reflection increases while transmission decreases)

TIR in prisms The critical angle at air-glass interface : 42o

TIR in terms of scattering

A surface wave when θt

o= 90


A. The Evanescent Wave Using Snell’s law we rewrite Fresnel Eq. as

( )

( )

2 2

2 2

/ sin coscos coscos cos / sin cos

t i i it t i i

t t i i t i i i

n nn nr

n n n n⊥

− θ − θθ − θ= − ⇒ −

θ + θ − θ + θ

( ) ( )

( ) ( )

2 22

// 2 22

/ sin / coscos coscos cos / sin / cos

t i i t i it i i t

t i i t t i i t i i

n n n nn nrn n n n n n

− θ − θθ − θ= − ⇒

θ + θ − θ + θ

r r⊥ , // become complex when θ θi c>

→ r r r r R⊥ ⊥ = = =*// //

* 1

The transmitted wave: ( )ti k r tt otE E e • −ω= where t tx tyk k x k y= +

k k k nntx t t t

i

ti= ⇒sin sinθ θ

2

2cos 1 sin ity t t t i

t

nk k k

n⎛ ⎞

= θ ⇒ ± − θ⎜ ⎟⎝ ⎠

⎡ ⎤⎛ ⎞⎢ ⎥⇒ ± θ −⎜ ⎟⎢ ⎥⎝ ⎠⎢ ⎥⎣ ⎦

22 sin 1i

t it

ni kn

↑ ↑ ↑ = β Snell’s law θ θi c>

The transmitted wave : sini

t it

nik x y i tn

t otE E eθ β ω−

=∓

, Evanescent wave → It advances in x-direction but exponential decay along y-axis → Constant phase (yz-plane) ⊥ Constant amplitude (xz-plane), Inhomogeneous wave No net energy flow across the interface. • Frustrated Total Internal Reflection (FTIR) Dense medium → Rare medium → Dense medium (Energy transfer) ↑ ↑ TIR Evanescent wave [Fig. 4.55] FTIR [Fig. 4.56] Beamsplitter using FTIR Low-index space controls the transmittance


4.8 Optical Properties of Metals Free electrons in metals → J E= σ ↑ ↑ ↑ Conductivity Unbound Current density A perfect conductor : σ = ∞ → Electrons follow the electric field exactly (No restoring force, no natural freq., no absorption, only reemission) In real metals : σ ≠ ∞ Collision of electrons with lattice or imperfections → Energy loss by heat Waves in a metal The Maxwell’s eqs. in metals

∇ × = −E Bt

∂∂

, ∇ × = +H Et

Eε∂∂

σ

→ ∂

∂

∂

∂

∂

∂με

∂

∂μσ

∂∂

2

2

2

2

2

2

2

2E

xE

yE

zE

tEt

+ + = + ( )2 2o o on i E⇒ −ω μ ε − ωμ σ 2 2

o oo

n i E⎛ ⎞σ

⇒ −ω μ ε +⎜ ⎟ωε⎝ ⎠

↑ ↑ Damping ( )22

c R In n in≡ = + The plane wave solution

I Rn y i n y i tik r i t c c

o oE E e E eω ω ωω − + −• −= ⇒

↑ ˆ o o ck n y= ω μ ε The irradiance

( ) ( )0 yI y I e−α= , 2 2= =In fcωα π μσ : attenuation coefficient

For y =1α

the irradiance drops by a factor of e −1 : skin depth, δ

Example Skin depth of Copper For UV ( )100o nmλ ≈ δ = 06. nm

For IR ( )10,000o nmλ ≈ δ = 6nm Little penetration → High reflection of light Metals reflect almost all the incident light (85%~95%) regardless of wavelengths → Colorless (Silvery gray)


The Dispersion Equation Vibration of a bound electron due to the electric field

( ) ( )2 2/

o

q mx t E ti

=ω − ω − γω

: q > 0 , x measures from - to +

No restoring force in metals : ωo = 0

→ ( )x t is always 180o out of phase with ( )E t → The reradiated wave cancels the incoming wave • The Dispersion Relation Neglect bound charges and neglect γ assuming high frequency

( )22

221 1 p

o

Nqnm

ω⎛ ⎞ω ≈ − = − ⎜ ⎟ωε ω ⎝ ⎠

: ω p = plasma frequency

For ω ω< p , n becomes complex. Exponential decay of the wave

for ω ω> p , n becomes real. Small absorption. The conductor becomes transparent

Ionosphere : Distribution of free electrons n < 1 and real for ω ω> p

Reflection from a metal At normal incidence on a metal

( )( )

* 2 2

2 2

11 11 1 1

R Ic c

c c R I

n nn nRn n n n

− +⎛ ⎞ ⎛ ⎞− −= ⇒⎜ ⎟ ⎜ ⎟+ + + +⎝ ⎠ ⎝ ⎠

: c R In n in= +

If nI = 0 → Dielectric material If nI > 0 → R becomes larger If n nI R>> → nc purely imaginary, R=1

Reflectance from an absorbing medium nI and R depend on ω [p130] Visor of space suit Thin gold coating → 70% reflection (Reduction of IR transmission still transmitting VIS)


4.9 The Interaction of Light and Matter Reflection of all visible frequency → White color 70%~80% reflection → Shiny gray of metal Thomas Young : Colors can be generated by mixing three beams of light well separated in frequency Three primary colors combine to produce white light : No unique set The common primary colors : R, G, B

• Two complementary colors combine to produce white color

M G WC R WY B W

+ =+ =+ =

,,

• A saturated color contains no white light (deep and intense) An example of an unsaturated color ( ) ( )M Y R B R G W R+ = + + + = + : Pink • The characteristic color comes from selective absorption

Example: (1) Yellow stained glass White light → Resonance in blue → Yellow is seen at the opposite side ↑ ↑ Red + Green Strong absorption in blue

(2) H O2 has resonance in IR and red → No red at ~30m underwater (3) Blue ink looks blue in either reflection or transmission Dried blue ink on a glass slide looks red. → Very strong absorption of red. Strong absorber is a strong reflector due to large nI . Resonance of materials Most atoms and molecules → Resonances in UV and IR Pigment molecules. → Resonances in VIS Organic dye molecules → Resonance in VIS • Subtractive coloration Blue light → Yellow filter → Black at the other side ↑ It removes blue

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