wave motion - skkulab.icc.skku.ac.kr/~yeonlee/display optics/plane wave.pdf · 2012. 10. 23. · 5....
TRANSCRIPT
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Wave Motion 1. One Dimensional Waves A wave : A self-sustaining disturbance of the medium
Waves in a spring
A longitudinal wave : Medium displacement // Direction of the wave A transverse wave : Medium displacement ⊥ Direction of the wave The disturbance advances, but not the medium (This is why waves can propagate at great speeds) A wave is given by a function of position and time: ψ = f x t,b g The shape of the disturbance at a certain time → The profile of the wave ψ x f x f x, ,0 0b g b g b g= ≡ : Taking a “photograph” at t = 0
A wave on a string
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Propagation of a pulse
Moving reference frame
A new coordinate system, ′S , moving with the wave → ψ = ′f xb g : Indep. of time. The profile is measured along x ' In this case x x vt'= − → ψ x t f x vt,b g b g= − : 1-D wavefunction • At a later time t t+ Δ f x v t t f x vt− + ⇒ −Δb g b g : Same profile
↑ change position x x v t→ + Δ • A wave in -x direction ψ = +f x vtb g
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2cosA xπψλ
⎛ ⎞= ⎜ ⎟⎝ ⎠
cosAψ ϕ=
3 5 7 0 2 3 2 2 2 2 2π π π π πϕ π π π= −
3 5 3 7 0 4 4 2 4 4 2 4
x λ λ λ λ λ π πλ= −
x
The Differential Wave Eq. Start with ψ = f x vt∓b g and x x vt'= ∓
∂ψ∂
∂∂
∂∂
∂∂x
fx
xx
fx
= ='
'', → ∂ ψ
∂
∂
∂
2
2
2
2xf
x=
'
Similarly
∂ψ∂
∂∂
∂∂
∂∂t
fx
xt
v fx
= ='
''
∓ , → ∂ ψ
∂
∂
∂
2
22
2
2tv f
x=
′
Combining the eqs. : ∂ ψ
∂
∂ ψ
∂
2
2 2
2
21
x v t= , 1-D differential wave eq.
Its solution : ψ = − + +C f x vt C g x vt1 2b g b g There are two kinds of 1-D wave(+ and – waves) → 1-D differential wave eq. should be of second-order. 2. Harmonic waves A wave with sine or cosine profile ( ) ( ),0 cosx A kx f xψ = = k : propagation number A : amplitude A traveling wave by replacing x by ( )x vt− ( ) [ ] ( ), cos ( )x t A k x vt f x vtψ = − = − : Periodic in both space and time • The spatial period : Wavelength, λ [m]
ψ λ ψx t x t± =, ,b g b g [ ]cos[ ( )] cos ( ) cos[ ( ) ]k x vt k x vt k x vt kλ λ− ⇒ ± − ⇒ − ±
↑ ↑ position change by ±λ ↑ =2π → k = 2π λ/ phase, ϕ • The relation between ϕ and x
• The temporal period : Frequency, ν τ= 1/ [Hertz] ψ τ ψx t x t, ,± =b g b g [ ] ( ){ }cos[ ( )] cos ( ) cosk x vt k x v t k x vt vτ τ⎡ ⎤− ⇒ − ± ⇒ −⎣ ⎦∓
↑ time change by ±τ , ↑ = 2π /k → τ λ= /v The temporal frequency : ν τ=1/ → v = νλ Angular frequency : ω πν= 2
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3. The Superposition Principle Two waves ψ ψ1 2 and both satisfy the differential wave eq.
2 2
1 12 2 2
1x v tψ ψ∂ ∂
=∂ ∂
, 2 2
2 22 2 2
1x v tψ ψ∂ ∂
=∂ ∂
Add two eqs.
( ) ( )2 2
1 2 1 22 2 2
1x v t
ψ ψ ψ ψ∂ ∂+ = +
∂ ∂
→ ( )1 2ψ ψ+ is also a solution of the diff. wave eq. → Two waves can add algebraically in the overlap region. Outside the overlap region, each propagates unaffected. Superposition of two waves
Waves that are out-of-phase diminish each other → Interference
(a) Two waves are in-phase (b) Two waves are π /3 out-of-phase (c) “ 2 3π / “ (d) “ π “
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4. The Complex Representation The complex number ~z x iy= + : i = −1 ↑ ↑ real, imaginary parts In polar coords. x r y r= =cos , sinθ θ
→ ~ cos sinz r ir rei= + ⇒θ θ θ r : magnitude, modulus, absolute value ↑ θ : phase angle Euler formula
e iiθ θ θ= +cos sin The complex conjugate : ~* *z x iy x iy re i= + = − = −b g b g θ A complex exponential
e e e ez x iy x iy~= =+ → Modulus e ez x~
=
It is periodic e e e ez i z i z~ ~ ~+ = =2 2π π Any complex number
~ Re ~ Im ~ cos sinz z i z r ir= + = +b g b g θ θ : Either real or imaginary part can represent ↑ ↑ a harmonic wave 1
2~ ~*z z+e j , 1
2i z z~ ~*−e j A harmonic wave is usually represented by the real part of ~z
ψ ω εω εx t Ae A kx ti kx t, Re cosb g b gb g= LNMOQP⇒ − +− +
The wave function, in general
ψ ω εx t Aei kx t,b g b g= − +
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5. Plane Waves The surface of a constant phase form a plane. The phase plane is perpendicular to the propagation direction. • A plane wave ψ r Aeik rb g = •
A constant phase requires k r a• = ↑ constant Assume oa k r= • ↑ constant vector → ( )r r ko− • = 0
→ r forms a plane that is perpendicular to k
• The periodic nature of a plane wave ( ) ( )ψ λ ψr k r+ = k : unit vector of k
↑ ↑
( )Aeik r k• +λ
, Aeik r•
→ k k=2πλ
: Propagation vector
Surfaces joining points of equal phase → wavefronts
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• A plane wave with time dependence
( )ψ ω∓r t Aeik r i t, = • The plane wave in Cartesian coord.
ψωx y z t Aei k x k y k z tx y z, , ,b g d i=
+ + ∓ : k k k kx y z= + +2 2 2
The phase velocity (Velocity of the wavefront) ψ ψr rk t t r t+ + ⇒Δ Δ, , e j b g
↑ ↑ eik r rk i t t∓• + +( ) ( )Δ Δω eik r i t∓• ω → k r tΔ = ±ωΔ
→ drdt k
v= ± = ±ω
• Two plane waves with the same wavelength → k k1 2 2= = π λ/
Wave 1 : A e A eik r i t ik z i t1 1
1 1• − −=ω ω ,
Wave 2 : A e A eik r i t ik y z i t2 2
2 2• − + −=ω θ θ ωsin cos b g
In general
ψα β γ ωx y z t Aei k x y z t, , ,b g b g=
+ + ∓ (1)
where k k k k kx y z= = + +2 2 2
→ 2 2 2 1α + β + γ = : Direction cosines ↑ Cosine of the angle subtended by k x and
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6. The Three-Dimensional Differential Wave Equation From (1)
∂ ψ
∂α ψ
2
22 2
xk= − , ∂ ψ
∂β ψ
2
22 2
yk= − , ∂ ψ
∂γ ψ
2
22 2
zk= −
→ ∂ ψ
∂
∂ ψ
∂
∂ ψ
∂ψ
2
2
2
2
2
22
x y zk+ + = −
Similarly
→ ∂ ψ
∂ω ψ
2
22
t= −
Combine the two eqs.
∇ =22
2
21
ψ∂ ψ
∂v t : Three dimensional differential wave eq.
where ∇ = + +22
2
2
2
2
2∂
∂
∂
∂
∂
∂x y z : Laplacian operator
One of the solution is a plane wave
ψω α β γx y z t Ae Ae
i k x k y k z i t ik x y z vtx y z, , ,b g d i b g= =+ + + +∓ ∓
Note that the following is also a solution ψ x y z t C f k r vt C g k r vt, , ,b g e j e j= • − + • +1 2
7. Spherical Waves A spherical wave is spherically symmetrical, independent of θ φ and ( ) ( ) ( ), ,r r rψ ψ θ φ ψ= = , The differential wave eq. in this case
∂
∂ψ
∂
∂ψ
2
2 2
2
21
rr
v trb g b g=
The solution
r r t f r vtψ ,b g b g= ∓ → ψ r tr
f r vt,b g b g=1 ∓
The harmonic spherical wave
ψ r t A er
ik r vt,b g
b g=
∓
The wavefront is obtained from kr = constant → Concentric spheres
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• The spherical wave decreases in amplitude as it propagates.
A part of a spherical wave resembles a part of a plane wave at far distance
8. Cylindrical Waves The cylindrical symmetry requires ψ ψ θ ψr r z rb g b g b g= =, , The differential wave eq.
1 12
2
2r rr
r v t∂∂
∂ψ∂
∂ ψ
∂FHGIKJ =
The solution is given by Bessel functions. As r → ∞
ψ r t Ar
eik r vt,b g b g≈ ∓
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Electromagnetic Theory, Photons, and Light Classical Electrodynamics : Energy transfer by electromagnetic waves Quantum Electrodynamics : Energy transfer by massless particles(photons) 1. Basic Laws of Electromagnetic Theory Electric charges, time varying magnetic fields → Electric field Electric currents, time varying electric fields → Magnetic field The force on a charge q F qE qv B= + × : Lorentz force A. Faraday’s Induction Law A time-varying magnetic flux through a loop → Induced Electromotive Force(emf) or voltage
emf ddt B= − Φ
↑ ↑ E dl
C•z . B dS
A•zz
→ E dl ddt
B dS dBdt
dSC A A
• = − • ⇒ − •z zz zz : Faraday’s law
(Wrong direction of dl in the book)
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B. Gauss’s Law-Electric Electric flux : E A A
D dS D dS⊥Φ = = •∫∫ ∫∫
The total electric flux from a closed surface = The total enclosed charge
A VD dS dVρ• =∫∫ ∫∫∫ ,
where D Eε= , ε ε ε= r o , electric permittivity ↑ ↑ Permittivity of free space Relative permittivity, Dielectric constant
A point charge at the origin → D is constant over a sphere, 24E D rπΦ = . → E qΦ =
Coulomb’s law, 24 o
qErπε
=
C. Gauss’s Law-Magnetic No isolated magnetic charge B dS
A• =zz 0
D. Ampere’s Law A current penetrating a closed loop induces magnetic field around the loop
Using current density 2[ / ]J A m
C A
H dl J dS• = •∫ ∫∫ : Ampere’s law
where B Hμ= with μ μ μ= r o , permeability ↑ ↑ Permeability of free space Relative permeability
• A capacitor Between the capacitor plates
E QA
=ε
→ ε∂∂Et
iA
J D= ≡ ,
↑ ↑ Differentiate both sides ↑ Displacement current density The generalized Ampere’s law
C A
DH dl J dSt
∂∂
⎛ ⎞• = + •⎜ ⎟
⎝ ⎠∫ ∫∫ : Maxwell
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E. Maxwell’s Equations
In free space : , , 0o o Jε ε μ μ ρ= = = =
C A
BE dl dSt
∂∂
• = − •∫ ∫∫ → BEt
∂∂
∇ × = − (1)
C A
DH dl dSt
∂∂
• = •∫ ∫∫ → DHt
∂∂
∇ × = (2)
0AB dS• =∫∫ → 0B∇ • = (3)
0AD dS• =∫∫ → 0D∇ • = (4)
↑ Differential form
2. Electromagnetic Waves Three basic properties
(1) Perpendicularity of the fields A time-varying D generates H that is perpendicular to D “ B “ E “ B → Transverse nature of E and H (2) Interdependence of and E H → Time varying E B and regenerate each other endlessly (3) Symmetry of the equations → The propagation direction will be symmetrical to both and E H Differential wave equation From (1)
∇ × ∇ × = − ∇ ×Et
Be j e j∂∂
Using (2)
∇ ∇ • − ∇ = −E E Eto oe j 22
2ε μ∂
∂ → ∇ =2
2
2E Eto oε μ
∂
∂
↑ =0 • In Cartesian coord. by separation of variables
∂
∂
∂
∂
∂
∂ε μ
∂
∂
2
2
2
2
2
2
2
2Ex
Ey
Ez
Et
x x xo o
x+ + = , : o
Wave equation known long before Maxwell
with 1/ ov ε μ=
Maxwell calculated
v m so o
= =× • ×
≈ ×− −
1 1
8 854 10 4 103 10
12 78
ε μ π./ :
Fizeau's measurement, c=315,300 /Km s
→ Maxwell concluded that light is an electromagnetic wave.
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A. Transverse Waves
A plane wave propagating along x-axis
→ ∂
∂
∂
∂
E x y z ty
E x y z tz
, , , , , ,b g b g= = 0
→ ( ), ( , ) ( , ) ( , )x y zE E x t E x t x E x t y E x t z= ⇒ + +
From divergence eq. (4)
∇ • =E 0 → ∂∂
∂
∂∂∂
Ex
Ey
Ez
x y z+ + = 0 → ∂∂Ex
x = 0 ,
↑ ↑ E constx = (not a traveling wave) = 0, = 0
→ 0xE =
( , ) ( , )y zE E x t y E x t z= + : Transverse wave
• Assume E yE x ty= ,b g From (1)
x y z
x y zE
Bt
y
∂∂
∂∂
∂∂
∂∂
0 0
= − → zEx
xBt
yBt
zBt
y x y z∂
∂∂∂
∂
∂∂∂
= − − − (5)
↑ ↑ = 0, = 0, to match z → Only E By z and exist : Transverse wave
• A plane harmonic wave
( ) [ ], cosy oE x t E kx t= − ω + ε The B- field from (5)
[ ]1 cosyz o
EB dt E kx t
x c∂
= − ⇒ − ω + ε∂∫
→ E cBy z= : E B and are in phase
E B× // Beam propagation direction
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3 Light in Matter In a homogeneous, isotropic dielectric (nonconducting material) ε ε μ μo o→ →, → v = 1/ εμ The absolute index of refraction
n cv o o
r r≡ = ⇒εμε μ
ε μ → n r= ε , Maxwell’s relation
↑ μr ≈ 1, except for ferromagnetic materials (μr = − × −1 22 10 5. for diamond)
The index of refraction depends on frequency
→ Dispersion
A. Dispersion Macroscopic view: A matter responses to the electric or magnetic field via ε or μ Microscopic view: The atom interacts with electric field via electric dipole Applied electric field → Distorted charge distribution → Internal field (External) (Electric dipole moment) The electric polarization P : Dipole moment per unit volume oD E P Eε ε= + ⇒ NaCl,… The eq. of motion
Restoring force of the electron for small x : F kx= − From Newton’s second law
2
22
i to o
d x dxqE e m x m mdtdt
− ω = ω + + γ : ωo k m2 = /
↑ ↑ ↑ ↑ Damping effect (Energy loss during oscillation due to ↑ ↑ Mass X Acceleration interaction between neighboring atoms)
↑ Restoring force Driving force from the incident wave The solution
( ) ( )2 2
/ i to
o
q mx t E ei
− ω⎡ ⎤⎢ ⎥=⎢ ⎥ω − ω − γω⎣ ⎦
(1) Without the driving force (no incident wave) → The oscillator vibrates at the resonance freq. ωo (2) For oω ω : E tb g and x tb g are in phase
(3) For oω ω : x tb g is 180o out of phase with E tb g
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• The electric polarization, The permittivity
( )
2
2 2
/oo
o
q NE mP qx N
i= ⇒
ω − ω − γω,
( )( ) ( )
2
2 2
/o o
o
P t q N mE t i
ε = ε + ⇒ ε +ω − ω − γω
The dispersion relation
2
22 2
11o o
Nqnm i
⎛ ⎞= + ⎜ ⎟⎜ ⎟ε ω − ω − γω⎝ ⎠
> <1 for ω ωo
< >1 for ω ωo For oω << ω : ω can be neglected in the eq. → constant n For oω→ ω : n gradually increases with frequency (normal dispersion) For oω ≈ ω : The damping term becomes dominant (strong absorption)
dndω
< 0 (anomalous dispersion)
Shaded regions: absorption bands Shaded region: visible band (Note rise in UV and fall in IR) A material opaque near the resonance frequency can be transparent at other frequencies.
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The Properties of Light 1. 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 θi
i
r
rv v=
Since v vi r=
→ θ θi r= : Law of reflection (Part I) 2. 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.
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3. 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)
k ki i r rsin sinθ θ= → θ θi r=
From the first and last of (1)
k ki i t tsin sinθ θ= → n ni i t tsin sinθ θ=
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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⎝ ⎠
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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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C. Interpretation of the Fresnel Eqs.
Amplitude Coefficients
At normal incidence, θi = 0
t i
t i
n nr r
n n⊥
−= =
+
The external reflection ( ), t i i tn n> θ > θ
→ r⊥ < 0 .
// 0r = when ( ) 90ot iθ + θ = : Polarization angle of i pθ = θ .
The internal reflection ( ), i t t in n> θ > θ
→ 1r⊥ = when 90otn = : Critical angle of i cθ = θ in sini i tn nθ =
// 0r = when ( ) 90ot iθ + θ = : Polarization 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
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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
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4. 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
5. 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
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• 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
Hecht by YHLEE;090901; 0-25