chapter 4. the properties of light - skkulab.icc.skku.ac.kr/~yeonlee/optics/hecht_4.pdf · 2012....
TRANSCRIPT
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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.
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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
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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. 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 : 270
o phase lag
Dashed : reduced damping
• Accumulated phase lag or lead → Speed change of the wave.
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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.
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• 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 θii
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 θii
t
tv v=
Since v cni i
= , v cnt t
=
→ 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
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• 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 θii
t
tv v= : Snell’s law
↑ dt dx/ = 0 • Optical Path Length The transit time from S to P
t sv c
n siii
m
i ii
m
= ⇒= =∑ ∑
1 1
1
↑ Optical path length (OPL) In an inhomogeneous medium
( )PS
OPL 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
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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 /
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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> .
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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
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Hecht by YHLEE;100510; 4-10
C. Interpretation of the Fresnel Eqs. Amplitude Coefficients At normal incidence, θi = 0
t it 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 90
otθ = : 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 : ( )*12S E H= × The intensity ( )2/W m : Irradiance → 2
12 o r o
cI S En
= = ε ε : Average energy per unit time per unit area
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Hecht by YHLEE;100510; 4-11
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 ri 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
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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 = 90
o
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
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Hecht by YHLEE;100510; 4-14
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 it
nk k k
n⎛ ⎞
= θ ⇒ ± − θ⎜ ⎟⎝ ⎠
⎡ ⎤⎛ ⎞⎢ ⎥⇒ ± θ −⎜ ⎟⎢ ⎥⎝ ⎠⎢ ⎥⎣ ⎦
22 sin 1it i
t
ni kn
↑ ↑ ↑ = β Snell’s law θ θi c>
The transmitted wave : sinit i
t
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
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Hecht by YHLEE;100510; 4-15
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 2o oo
n i E⎛ ⎞σ
⇒ −ω μ ε +⎜ ⎟ωε⎝ ⎠
↑ ↑ Damping ( )22c 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)
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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)
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Hecht by YHLEE;100510; 4-17
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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