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Transcript
R&D of Exploratory Electro-Optic
Materials and Applications
(前瞻光電材料與應用之開發)
Introduction &
Optics (EM) and Electronics Fundamentals
Professor: Chungpin Liao (Hovering)
廖重賓 (飛翔)
[email protected]
[email protected]
Coherent Control Lab. (http://coherent.nfu.edu.tw)
RDExploreEO -- Hovering
1
Syllabus (課程表)
 Language for course material: English mostly
 Objective: Exploratory research capability buildup
need your curiosity, imagination,
innovative literature mining,
research efforts and notes keeping
 Presentation: Lecture notes PowerPoint presentation
 Evaluation:
Active attendance (30%), Hand-in’s (50%), Final Report (2
RDExploreEO -- Hovering
2
Main issues confronting us and efforts of CCL:
• Electro-Optics is actually Electro and Optics separately
at present. (SPR)
• Magnetics remains an area of very low frequency,
far from light frequencies. (r)
• Conductivity remains quantity of relatively low frequency
far from light frequencies.
• New possibilities in optics emerge in the nanotechnology
era. (dipole engineering: FreqPush, AsymScat, newBrew
• Applications based on interactions between lights and
bio-systems / bio-materials are thriving. (Kirlian, SEA,
light-protein, LLLT, GdIONP magnetic fluid hypertherm
• High efficiency solar energy harvester is in need.
(BlackbodySolar, chl-a organic cell)
RDExploreEO -- Hovering
3
 Coverage: (only as leads for more unexpected)
1-5
6, 7
8
9
10
11
12
13
14
15
16
Fundamentals of photonics and optics
Radiating dipoles and refractive index
Fresnel equations -- EM form & traditional Brewster angle
Fresnel equations -- scattering form & new Brewster angle
Double layer caused asymmetric refraction & strobe light
(PVDF, LiNbO3)
Large refractive index & Clausius-Mossotti equation
 small-area spiral inductor, etc.
Frequency Push
Visible light-induced ITO surface plasmon resonance
(incl. Landau damping)
Strange mirror
Open-air Weibel instability amplifier
Possible relation between Light-interference & protein releas
And, occasionally, some invited speakers …
RDExploreEO -- Hovering
4
 As a matter of taste: an informal, relaxed but thoughtful environment
 Come with ease, vigor, confidence and lots of laughter
 Play a scholar, bring in whatever piles of reference materials you like
 Have your coffee and donuts (only) ready
 Bring in your favorite pencil (or fountain pen), eraser, and notebook
 Read and survey as deep and wide as you can (until I stop you)
 Use web and INSPEC etc. a lot
 Ask questions on many open issues
 Use whatever background, knowledge, technique, personal network,
guts, instinct and intuition you possess
 Come upfront to share your points with others, even start a new issue
 Feel free to carry arguments into debates (but still be kind to others)
 Form study/research groups with people you like
(experience the importance of cooperation and discussion)
 You can even form a special research project with me on topics of
your dream, and if the results are good, have it published
 You want, you get it! (A living corpse simply can not.)
RDExploreEO -- Hovering
5
To boggle your mind, there remain piles of questions ….
Is light a particle or wave? What’s their difference in terms of applications?
A light beam can heat up a chunk of wood or a piece of metal. Can it be
solely explained from energy flux perspective? What about charge?
Why can light slow down in media? What’s changed? Wavelength
or frequency? Why? Does it have to be so always?
Is the color sequence of a rainbow fixed? What has caused it so?
An electron is always anti-magnetic (diamagnetic), and every substance is
made of electrons, how come there are paramagnetic and even
ferromagnetic materials? What about at light frequencies?
(An electron can certainly catch up with the light frequency since we
already see abundant light-responsive dielectrics resulting from
electronic responses to lights.)
RDExploreEO -- Hovering
6
If everything in this universe is a mere cluster of dipoles (electric and
magnetic) in the eyes of lights, why cannot we have new, revolutionary
optical materials and devices?
Further, why cannot we manipulate lights in biological ways such that
they become favorable to human beings, creatures, plants, foods,
energy production, and environmental protection?
If physical forms of dipoles can be identified within the realm of EM waves,
why cannot those relevant to earthquakes be exploited, recognized, and
controlled?
Can we only aim to treat, even destroy something surrounded by other
same kind or different types of things optically or acoustically (隔山打牛)?
For example, unconventional treatment on cancer cells.
Can we realistically make Harry Potter’s invisible cape?
RDExploreEO -- Hovering
7
Is it theoretically possible to have a reflector for all frequencies of lights?
Is there realistic handheld left-hand materials? (i.e., excluding SPR)
Is there a light highway system within each human body or each plant?
If so, is it why a plant can react to human touch immediately even though
it possesses no nerve system?
If so, can light medicine (or, even quantum medicine) be realized?
The vast knowledge of organic chemistry is largely a know-how under
room temperature and heat, not one under specifically tailored spectra.
What a strange new world would the latter bring us?
Can there be microwave creatures? How to make them, in principle?
RDExploreEO -- Hovering
8
Optics (EM) and Electronics Fundamentals
 Photonics in general
 Optics treatment overview






Lightwave, phase velocity and phasor
Maxwell’s equations
Plane waves
Photons, dipole scattering model
Index of refraction, phase lag or lead
Dispersion (色散)
 A brief review checklist (Homework 1)
 The EM approach at an interface: Fresnel equations
RDExploreEO -- Hovering
9
Photonics (光電) in general
 Origin of light + electronics:
• Optics is an old subject involving the generation, propagation and
detection of light.
• Then, 3 major developments in the past 30 years:
 invention of laser
 fabrication of low-loss optic fibers
 birth of semiconductor optical devices
• As a result, new disciplines emerged:
 electro-optics
 optoelectronics
 quantum electronics
 quantum optics
 lightwave technology
 photonics
RDExploreEO -- Hovering
10
 Photonics (光電) area categorization:
About optical devices in which electrical
effect plays a role (lasers,
electro-optic modulators, and switches)
Electro-optics
Optoelectronics
Quantum electronics
Quantum optics
Lightwave technology
Referring to devices and systems that are
essentially electronic in nature but involve
light (LED, LCD, array photodetectors)
For devices and systems that rely
principally on the interaction of light with
matter (lasers and nonlinear optical devices
used for optical amplification and
wave mixing)
About studies of quantum and coherent
properties of light
Describing devices and systems used in optical
communication and optical signal processing
RDExploreEO -- Hovering
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Electronics
Photonics
Controlling electric charge flow in free space
and in matter
Controlling photons in free space and in matter
We’ll treat applications within either of these two domains, and those
spanning both domains.
RDExploreEO -- Hovering
12
Optics treatment overview:
 Domain of validity for each theoretical model in optics (and general radiation):
Quantum optics
(QED, QFT)
Electromagnetic (EM)
optics (Maxwell’s equ’s)
Scalar (純量) wave optics
(Physical optics)
(scalar wave equation)
Ray optics
(geometric optics)
RDExploreEO -- Hovering
13
 Quantum optics
(quantum electrodynamics)
provides an
explanation of virtually all
optical phenomena. Good for:
Understanding of physics,
semiconductor lasers,
detectors, quantum information,
new materials, new inventions
Quantum optics
Electromagnetic (EM)
optics
Wave optics
Ray optics
 The electromagnetic theory of light (EM optics) provides the most complete treatment
of light within the confines of classical optics. Good for:
Polarization, scattering, energy transfer, nonlinear optics, wave guide, plasma
 Wave optics (physical optics) is a scalar approximation of EM optics. Good for:
Interference, diffraction, Fourier optics
 Ray optics (geometric optics) is the limit of wave optics when the wavelength is
very short. Good for:
Rectilinear ray tracing, lens & mirrors design, image formation
RDExploreEO -- Hovering
14
 The above categorization of application domains for each theoretical model is only
rough and pertains to applications already known.
 New thoughts and new inventions may urge us to cross the domain for more
appropriate theoretical tools.
For example, in describing reflection and refraction, concepts from ray optics to
quantum optics will be employed for different applications.
 We need to cover all levels of the theoretical models in order to thrive in the
now rapidly changing worlds of optics and photonics.
 For example: A complete hierarchical understanding of the theory of optics is
absolutely favorable concerning the major developments which rejuvenated the
old topics of optics and photonics:
lasers, low-loss optic fibers, semiconductor optic materials/devices, solar system,
photochemstry, and bio-photonics.
 Not to mention the capability you need to face up to the largely unknown future.
RDExploreEO -- Hovering
15
Lightwave, phase velocity and phasor (相位向量)
 In general wave propagations, although the energy-carrying disturbance
advances through the medium, the individual participating atoms remain
in the vicinity of their equilibrium positions; the disturbance advances,
not the material medium.
– a crucial feature to allow waves to propagate at high speeds.
disturbance  in moving:
(must be a function of x and t)
Let the fixed wave shape (x, t) = f(x, t)
= f(x, 0) = f(x):
 then, f(x’) = f(x) too.
 So, f(x, t) =?
x'  x   t
But x’ = x – vt (or,  (x – vt) relativistically),
a propagating wave is described by:
 = f(x – vt).
the most general form of the 1-D wavefunction
RDExploreEO -- Hovering
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For the moment, limit to waves of constant shape:
is a bell-shaped propagating wave
Equivalently,
RDExploreEO -- Hovering
17
The differential wave equation (linear perturbation (擾動))
RDExploreEO -- Hovering
18
Harmonic waves (諧波)
Why study harmonic waves (or, sinusoidal waves)?
(originated from simple harmonic motions in all disciplines (學門))
Any wave shape can be synthesized (合成) by a superposition (疊加)
of harmonic waves.
RDExploreEO -- Hovering
19
RDExploreEO -- Hovering
20
Phase and phase velocity (相速度)
RDExploreEO -- Hovering
21
Any point on a harmonic wave having a fixed magnitude moves such that the
phase is constant in time. In fact, this is true for all waves, periodic or not.
RDExploreEO -- Hovering
22
Superposition principle (疊加原則)
Most fundamental assumption: linear (weak) disturbance of medium
The resulting (linear) disturbance at each point is the region of overlap (重疊) is the
algebraic sum of the individual constituent (成份)waves at that location.
In-phase: constructive interference
0 (建設性干涉  增強)
Out-of-phase: destructive interference
180 (破壞性干涉 抵銷)
RDExploreEO -- Hovering
23
The complex (複變數) representation
Representing harmonic waves by sine (正弦) and cosine (餘弦) functions
are awkward for our purpose, since trigonometric (三角函數) manipulation
is unattractive.
Complex exponentials (複變指數) ei are used extensively in classical and
quantum mechanics, and of course, optics.
The key reason: a convention (常規) (never forget and then fool yourself):
RDExploreEO -- Hovering
24
RDExploreEO -- Hovering
25
Phasors (相位向量) and the addition of waves
RDExploreEO -- Hovering
26
Plane waves (平面波)
 
 

i ( k r t )
 the simplest of 3-D waves: plane wave: (r , t )  A cos(k  r  t )  Ae
 Why studying plane waves (i.e., sinusoidal waves, or harmonic waves)?
• Any 3-D wave can be expressed as a combination of plane waves.
• Physically, sinusoidal waves can be generated relatively simply by using some
form of harmonic oscillators (諧波振盪器).
• By using optical devices, we can readily produce light resembling plane waves.
(We thus have geometric (ray) optics as an ordinary thing.)
• A plane waves is a good local approximation to a large spherical wave.
RDExploreEO -- Hovering
27
Wavefront (波前): at any given time, the surfaces joining all points of equal phase.
Note that the wavefunction will have a constant value over the wavefront only if the
Amplitude A has a fixed value at every point on the wavefront. In general, however,
A is a function of position.
 


 i ( kr t )
( r , t )  A( r ) cos( k  r  t )  A( r ) e
RDExploreEO -- Hovering
28
3-D differential wave equation
RDExploreEO -- Hovering
29
 Light is most certainly electromagnetic in nature macroscopically, after the work of
J. Clerk Maxwell and followers.
That is, light is an EM wave phenomenon described by the same theoretical principles
that govern all forms of electromagnetic radiation.
 Classical electrodynamics gives the picture of continuous transfer of energy by way
of electromagnetic waves. The EM wave propagates in the form of two mutually
coupled vector waves, an electric-field (E) wave and a magnetic-field (M) wave.
 Nevertheless, it is possible to describe many optical phenomena using a scalar wave
theory (or, physical optics), in which light is described by
a single scalar wavefunction .
 When light waves propagate through and around objects whose dimensions are
much greater than the wavelength, the wave nature of light is not readily discerned
(e.g., what a lens would see the light), so that its behavior can be adequately
described by rays obeying a set of geometrical rules.
Ray optics is thus called the geometric optics, and is the simplest
approximation of light, concerning mainly location and direction of light rays.
RDExploreEO -- Hovering
30
 The more modern view: quantum electrodynamics (QED, 量子電動力學) describes
electromagnetic interactions (交互作用) and the transfer of energy in terms of
massless elementary “particles” known as photons. (Though the quantum nature of
radiant energy is not always apparent, nor is it always of practical concern in Optics.)
 The photon has zero mass, and therefore exceedingly large numbers of low-energy
photons can be envisioned as present in a beam of light. Within this model, dense
streams of photons act on the average to produce well-defined (i.e., via Maxwell’s
equations) classical fields.
RDExploreEO -- Hovering
31
 Further, the quantum-mechanical treatment associates a wave equation with a
particle, be it a photon (光子), electron, proton (質子).
 In the case of material particle, the wave aspects are introduced by way of the
field equation: Schrodinger equation (薛丁格方程式).
 For light photon, we have a representation of its wave nature in the
form of the classical electromagnetic field equations of Maxwell.
 With these as a starting point, one can construct a QM theory of photons and their
interaction with charges.  QED (量子電動力學)
 QED explanation of the dual nature of light has been evidenced:
• Wave nature: light propagate thru media in a wave-like fashion
(Though in QED, the EM radiant energy is created and destroyed in quanta
or photons and not continuously as a classical wave.)
• Particle nature: streams of photons give rise to emission and absorption in
media (QED: E = h)
RDExploreEO -- Hovering
32
 Q: Since photons propagate at the speed of light in vacuum, how come light gets
slowed down in media other than vacuum? Is QED self-consistent at all?
 A: Photons does propagate at the speed of light in the void (vacuo) within the media,
and interaction of light with atomic dipoles (i.e., scattering) of the medium
does add phase to the propagating light (equivalent to slowing it down).
 Does it make sense at all?
 From another perspective, since v = f, a slowing down in v indicates the decrease of
the frequency f or the wavelength  of a photon? Why?
RDExploreEO -- Hovering
33
The real picture:
Induced dipoles
Stream of photons
  v
Photons disappear temporarily and are added phases.
 Since the energy transported by a large number of photons is, on the average,
equivalent to the energy transferred by a corresponding classical EM wave,
so after all, the EM theory is a fairly useful model for many purposes.
 We would therefore, spend time to know the classical EM wave theory better
in the following.
RDExploreEO -- Hovering
34
Maxwell’s equations: Basic laws of electromagnetic theory
 First of all, a very fundamental question, (have you ever asked yourself?)
Q: what is the nature of all force-at-a-distance (超距力)? Take
electrostatic force (靜電力) as an example.
 QED explanation: each charge (電荷) emits (and absorbs) a stream of
undetected particles (short life) (virtual photons (虛光子)). The exchange of
these “particles” among the charges is regarded as the mode of interaction.
(An analogy: throwing a rope to somebody to pull him over)
 Classical EM approach: imagine instead that each charge is surrounded by
something called an electric field. We then need only suppose that each charge
interacts directly with the electric field in which it is immersed, and experience


a force: F  qE . (cf. in QED, such charge and field are the same thing,
and so when calculating a charge distribution and the associated field, we have only
to do it once for all, never one for charge, another for field.)
 A moving charge experiences an extra field, B field, and subsequently a force:


 
F  qE  qv  B . However, there is interdependence between E and B….
 Maxwell’s equations
RDExploreEO -- Hovering
35
Before getting into the details of Maxwell’s equations, let’s familiarize ourselves
With  (gradient),  (divergence),  (curl) (right-hand rule),
and 2 (Laplacian) first (differential geometry 微分幾何):
RDExploreEO -- Hovering
36
Faraday’s induction (磁感電) law
Q: Do you keep
 Michael Faraday jotted “convert magnetism into electricity” a notebook for
wild new ideas?
on his notebook in 1822,and started working on it in 1831.
How do you
keep a reading habit
 He found that change of a magnetic flux M is the
and store
driving “force” (called emf: electromotive “force”)
information
to create an electric field or current:
effectively?
 
 M  B A  BA  BA cos   B  ds
A

 
 M
B 


 ds  emf  E  dl
A
C
t
t

 
 E  dl 
C
A

 
(  E )  ds


B
  E  
t
RDExploreEO -- Hovering
37


B
For    E  
t
the electric field arises not from the presence of electric
charge. With no charges to act as sources or sinks, the
field lines close on themselves.
RDExploreEO -- Hovering
38
Gauss’s law – electric & magnetic
 Sources of E field: electric charge & time rate of change of B field
 Source of B field: moving charge & time rate of change of E field
(no magnetic charge (or, magnetic monopole))
 In the absence of B(t), E field comes from charges q’s (source)
 E field behaves like classical incompressible flow, namely,
E 
 
E  ds  0 when there is no source or sink within the closed surface A
A
ds
E
i.e., in flux of E = out flux of E
A
RDExploreEO -- Hovering
39
 Consider a closed spherical surface A, which confines charge q:
E 
A
E 
 
E  ds  E
1
4 0
E 
q
0
A
ds  E  4r 2
q
r2
(Coulomb’s law: an experimental observation)
Vacuum electric permittivity 0 = 8.8542 x 10-12 C2/N-m2
(真空介電常數)
However, since the closed surface A does not have to be spherical, but can be any
shape, therefore:
E 

q
1

E  ds 

A
But, 
A
We have:
0
 
E  ds 
V
0


(  E ) dV
q
1
0
V dV
(體積分)
流出量


E 
0
RDExploreEO -- Hovering
40
 In a sense, the permittivity of a material  ( = KE 0 ) is a measure of the degree
to which the material is permeated (穿透) by the electric field in which it is
immersed (浸泡).
 KE ( =  / 0) is dielectric coefficient (or, relative permittivity), and is related to
the speed of light within the medium. (Hence is related to the medium’s index of
refraction)
 Similarly, since there is no magnetic monopole (i.e., magnetic charge), therefore:
M 
 
B  ds 
A

  B  0
V

(  B) dV  0 
V 0 dV
 Therefore a B field must be in the shape of a closed loop.
RDExploreEO -- Hovering
41
Ampere’s circuital law (電感磁)
C
 A “work rule” found out long time ago:
 
B  dl   0
i
 If the current has a nonuniform cross section, then using the notion of current
 
 
density J (A/m2), we have:
B  dl  
J  ds (*)
C
0
A
 0 = 4 x 10-7 N-s2/C2 is the permeability of free space.
 In general,  = KM 0, KM being the dimensionless
relative permeability (相對導磁係數)
 However, equation (*) does not tell the whole story.
RDExploreEO -- Hovering
42
• While charging a capacitor:
Ampere’s law is not particular about
the open surface used, provided it’s
bounded by the curve C. It can be
A1 or A2 (encompassing one capacitor
plate) to get the same answer.
C
 
B  dl   0
• If the flat A1 is used, then a net
current i flows thru it, and according
to equ. (*), there is a B field along C.
 
J  ds
A
A2
(*)
i=0
• If A2 is used as the open surface, no
net current flows thru it, and
according to (*), B field must now be
zero, even if nothing physical has been
changed. Something wrong!
• Moving charges are not the only
source of B field. E/t while charging
a capacitor will contribute too.
RDExploreEO -- Hovering
43
• From Gauss law, E field within the
capacitor is: E = Q/(A).
• During charging the capacitor, charges
on the plate changes, and so does the
E field, giving rise to an effective E
current density JD
(“displacement current density”), i.e.,

E
i

 JD
t
A
A
z
Substance filled
• The complete Ampere’s law is thus:
C
 
B  dl  


E  
 J  
  ds
A
t 


• Differential form:
On each xy plane, B(z) varies
according to
Jtotal = Jconductor + JD
(owing to Maxwell)



E
  B  J  
t
RDExploreEO -- Hovering
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Maxwell’s equations – the most complete classical description of optics (or, EM)
Integral form

1

E

d
s


0
A

V
dV
 
B  ds  0
A
Gauss-B

 
B 
 
 ds   E  dl
A t
C
C
 
B  dl  
Gauss-E


E  
J 
  ds
A

t



Faraday
Ampere
Differential form


E 
0

 B  0


B
 E  
t
Magnetic
induction
Displacement
current



E
  B  J   
t
Q: Can you write down the scalar component equations of the differential forms?
Q: The Maxwell’s equations look linear, where would nonlinearities come from?
 E and B are coupled
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Electromagnetic (EM) waves (電磁波)
 Starting from Maxwell’s equations, kaleidoscopes of phenomena can be explained
(in the macroscopic picture), and tremendous amount of new researches and
practical applications are still actively going on today.
Why? New physics comes in thru general anisotropic, nonlinear ,  tensors (張量) of
materials and situations (system configuration, parameters of interest, etc.)
can vary very widely.
 So, development of an intuitive appreciation of Maxwell’s equations thru EM waves
(a result of interplay between E and B) is now due.
 Symmetry: a time-varying E field generates a B field, which is everywhere
perpendicular to E; a time-varying B field generates a E field, which is everywhere
perpendicular to B. (電磁相生)
I.e., EM waves are generated under the cooperation of the magnetic induction and
the displacement current.
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 EM wave generation from charge acceleration:
• A charge originally at rest has its E field extending from it in all directions
to the infinity.
• At the instant the charge begins to move, the E field is altered in the vicinity of
the charge, and this alteration propagates out into space at some finite speed.
The time-varying E field induces a perpendicular time-varying B field.
• If the charge is then brought to a constant speed, E/t is constant, and thus
B is constant. E, B fields are attached to the charge.
• If now the charge is again accelerated, E/t is no longer constant, and
B is now time-varying. (Ampere’s law)
• The time-varying B field generates a new time-varying E field. (Faraday’s law)
• The process continues with E and B couple in the form of a pulse (脈衝).
• The pulse “detaches” itself from the charge and moves out from one point to the
next thru space, giving rise to a propagating EM wave radiation
(see Radiation later).
E(t)
B(t)
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 Bound together as a single entity, the time-varying electric and magnetic fields
regenerate each other in an endless cycle. (e.g., light from galaxy)
 The wave aspect of E and B can be seen from (e.g., in free space):

2


E
 2 E   0 0
t 2

2


B
 2 B   0 0
t 2
  0,  conductor  0,  r  1, r  1
Cf. In general,  
1

 The two wave equations are equivalent to 6 scalar equations. In Cartesian
coordinate (直角座標), each can be written in the form (with  = Ex, Ey, Ez, Bx,
By, Bz): 2
2
2
2
      1     1  3 108 m / s  c
 2  2  2 2
2
0 0
x
y
z
 t
That is, (scalar) wave theory, or physical optics (物理光學).
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Purely transverse (TEM) wave in vacuum (free space)
 Consider a plane wave propagating in vacuum along x
where E field is constant over each of an infinite set
of planes perpendicular to the x axis
So, E = E(x,t), and Ex = 0
 recall that E is independent of y, z, E = 0 means
Ex/x = 0. So, Ex can be a constant or zero.

E y
B
Let E  ˆjE y ( x, t ) Then we have:
 z
x
t
i.e., E (t) is generated by B (t)
y
(Faraday’s law)
z
 Thus, in free space, the plane EM wave is transverse, that is, TEM.
 If we let Ey(x,t) = E0y cos[ (t - x/c) +  ], then using Faraday’s law, we
arrive at Ey = c Bz.
 Apparently, the wave propagates along the direction of
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 
EB
49
Right-hand rule
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Let’s look at the plane wave more closely:
▪ The plane wave exists at a given time, when all the surfaces upon which a
disturbance has constant phase form a set of planes, each generally
perpendicular to the propagation direction.
▪ The mathematical expression for a plane, which is perpendicular to a given
vector k and which passes thru some point (x0, y0, z0) can be derived as follows.
  

By setting (r  r0 )  k  0
k
we force the vector (r-r0) to sweep out a plane
z (x, y, z)
perpendicular to k, and its end point (x, y, z)
takes on all allowed values.
 
With k = (kx, ky, kz), we have:
 r  r0
r
 (x0, y0, z0)
r0
x

r  ( x, y , z )
kx x  k y y  kz z  a
y
where a  k x x0  k y y0  k z z0
 
k r  a
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 a plane
51
▪ The most concise form of
the equation of a plane
perpendicular to k is then just:
z
 
k r  a
▪ The plane is the locus of all points whose
projection onto the k-direction (rk in the
figure blow) is a constant.
(We have chosen our axes that way so that
k vector is originated at (0, 0, 0).)
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 
r  r0
 (x0, y0, z0)
r0
y

k
x
z
 
,
k  r  ai s
▪ At any fixed point in space where r is constant,
the phase is constant, and so too, is (r),
in short, the planes are motionless.
(x, y, z)

r
▪ We now construct a set of planes over which
(r) varies in space sinusoidally, i.e.,
modulation
variable





 (r )  A sin( k  r ) or A cos( k  r )


ik  r
Or ,  (r )  Ae

k

r r

k
y
x
52
▪ To get things moving, (r) must be made
to vary in time, i.e.,
 

i ( k r t )
 (r , t )  Ae
(*)
▪ As this disturbance (r) travels along in the k direction, we can assign a phase
( = kr t) corresponding to it at each point in space and time.
At any given time, the surfaces joining all points of equal phase are known as
wavefronts, or wave surfaces.
▪ The phase velocity of a plane wave given by equation (*) is equivalent to the
propagation velocity of the wavefront. (A = constant)
▪ The disturbance on a wavefront is constant, so that after a time dt, if the front
moves along k a distance drk, we must have: (r, t )  (rk , t )  (rk  drk , t  dt )

or, in exponential form:
i ( kr t )
i ( kr  kdr t dt )
i ( k r t )
Ae
 Ae
k
 Ae
k
k
k drk   dt
Therefore,
and the magnitude of the wave velocity (phase velocity here), drk/dt, is:
drk

   v
dt
k
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Energy and momentum
Poynting vector
 Since we are now treating the classical field as if real and continuous
(strictly speaking, wrong; but useful), we want to further find out the energy stored
in the E and B fields, as a tool.
 It is known that E field energy density uE = (0/2)E2 and B field energy
density uB = (1/20)B2. Since c = (00) -½, we have uE = uB. This means the energy
streaming thru space in the form of an EM wave is shared equally between the
constituent electric and magnetic fields.
 Since u = uE + uB = 0E2 = (1/0)B2 = (0/0)½ E B = [(1/0) EB] / c, it was convenient
for a guy, named Poynting, to define an energy flux quantity, Poynting S (= uc):
 1  
 
2
S
E  B  c 0 E  B
e.g.
0
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About photons
 Light is absorbed and emitted in tiny discrete bursts, in particles of electromagnetic
stuff, known as photons, and has been confirmed and well-established.
 However, the question of whether or not light is “really” a stream of photons is far
from settled.
 Ordinarily, a light beam delivers so many minute energy quanta that its inherent
granular nature is totally hidden and a continuous phenomenon is observed
macroscopically.
cf. individual gas molecules in a wind give rise to a “continuous” pressure
 In 1900, Max Planck attacked the blackbody radiation problem and
incidentally discovered the quantization of light energy,
violating classical wave theory. E = nh
 J. J. Thomson discovered in 1903 that when a high-energy EM wave (X ray) was shone
onto a gas, only certain atoms in random positions got ionized. The light beam has
“hot spots” within itself, rather than having its energy continuously distributed
continuously over the wavefront.
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Radiation
 Origins:
• linearly accelerated charge
• synchrotron (同步輻射) (circularly accelerated charge)
• dipole radiation
• atom line radiation (de-excitation of excited atom)
Linearly accelerated charge:
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Linearly accelerating charges
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Note the kink in E field, which
gives rise to an EM pulse propagating
away from the accelerating charge
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Linearly accelerating
charges
Synchrotron radiation
(同步輻射)
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Electric dipole radiation
Origins:
The simplest and most important EM wave
o atoms
producing mechanism
o molecules
o nuclei
o any charge distribution
o defects
+q
-q
• Both light and UV radiation arise primarily from the rearrangement of the
outermost, or weakly bound, electrons.
• The rate of energy emission from a material system, although a QM process,
can be envisioned in terms of the classical oscillating electric dipole.
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The higher the frequency, the stronger the dipole radiation.
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The emission of light from atoms
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Light in bulk matter – material aspect
Optical concern : the response of dielectric ( i.e., nonconducting )
materials to EM Fields transparent dielectric : lens, prizm, air, plates,
films, water.
Phase speed of light in a homogeneous,
isotropic dielectric (介電質) :
Absolute index of refraction :
   r   0
,
n
   r  0
c



  
1
 
0   0
 n   r r
Since normally dielectrics of interest are nonmagnetic,
r  1
n  n()   r
Maxwell’s relation
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Dispersion (色散) :
In short, n = n (ω)
∵
material
hν
atom : scatters light
1
1
2
ω<ω0 (or hν< hν0= Ei→j )
absorb hν then re-emit photons (redirect light)
only
Non-resonant scattering (ground state)
2
Matching Ei→j:
excitation of atomic e-’s (quantum jump)
absorption
dense medium
collision
Thermalization
(randomization)
Dissipative absorption
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An r rule, the closer the frequency of the incident beam is to an atomic
resonance, the more strongly will the interaction occur, and in dense
materials, the more energy will be dissipatively (耗散) absorbed.
Selective absorption
Your color of skin, hair, eyes, clothing, leaves, fruits, etc.
Why? (math)
an atom
Dipole moment per unit volume :
P = (ε-ε0)E
e-
e+
e
e-
-
For isotropic homogeneous and linear media :
FE=qeE(t)=q0E0cosωt
( driving force on an e- )
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  F  ma
an atom
e-
e-
 qe E0 cost  kx  me
me
e
-
e-
k
dt 2
=
+
2
d x
x
2
me =  0
ω0 = natural freq. (resonant)
of the bound e- dipole
Medium=  (dipole)i
i
( 0 i )
Now, let x(t) = x0cosωt
( 0   )
2
2
E0 cost
≡
(a guess)
 x(t ) 
qe me
E(t )


:
x
(
t
)
//
E
(
t
)
in-phase


0

   0 : x(t ) //  E(t ) 180° out-of-phase
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2
 P  q0  x  N 
qe NE / me
(0   )
2
 P(t )  (   0)E(t )
2
( e ' s / m 3 )
2
&
P
qe NE / me
For rarefied gas
2
qe N / m e
P( t )
   0 
 0 
2
E(t )
(0   2 )
(0   2 )
2
Now, since n  K E 
2

0
∴We arrive at dispersion relation :
2
q
N
1 
2
e 
n ( )  1 
 2

2

 0 me   0  
In general, n ( )  1 
2
N qe2
 0 me

j
fj
 oj 2   2  i r j 
Works for rarefied gas (稀薄)
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cf. In dense media :
n2  1
N qe2
fj

 2
2
n  2 3 0 me j  oj   2  i r j 
2
2
if  oj    i r j  ( little dissipation )
→
n2  1
N qe2
fj


n 2  2 3 0 me j  oj 2   2
&
n  IR
Colorless, transparent media have their ω0's outside the
visible range. ( ω02 >> ω2 )
Visible range
So, n ≈ const.
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A brief review checklist (Homework 1.1):
1.
折射率(Index of refraction)這名詞來自當年牛頓所作的白光經三綾
鏡的色散(dispersion)實驗,並不是一個很恰當的名詞。它其實是代
表一光學材料對某頻率的光波的反應程度。
折射率的值離開1越遠代表該頻率的光波與該物質的交互作用越強
(亦即愈不像真空,因真空之折射率為1)。儘管如此,折射率的巨觀
物理定義為何(即:與光速的關係)?
2.
所有巨觀(macroscopic)的光學現象,包括折射、反射、繞射等,都
是微觀散射(microscopic scattering)的集總結果,而以折射率代表。
請描述這當中基態散射(ground-state scattering; non-resonant
scattering)的真正物理過程。
(hint: 電偶極振盪子(electric dipole oscillators))
3.
請描繪出(作圖)材料中的電偶極及磁偶極如何被外來的光波感應產
生?它們的定義各為何?
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4.
光子的速度一律為c(= 3 x 108 m/s)不論其波長或頻率為何。
但是為什麼光線在玻璃中的速度卻小於c?請解釋之(亦即:折
射率(Index of refraction)的微觀物理原因為何?)。
5.
請寫下光速與真空介電(permittivity)及導磁(permeability)
係數之關係,James Clerk Maxwell當年即是由此懷疑光波是一
種電磁波。
6.
折射率與相對介電及導磁係數之關係為何?
7.
請將可見光(以七色表)的能量(或頻率)由大到小列出。
再將一般玻璃透鏡對這七色可見光的折射率由大到小排列出。
(用 “>” 符號明確表示)
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8.
當一可見光(具一頻率範圍)射入一透明(對可見光)石英中時,
我們通常都“知道” 該光波在該石英中的速度變慢,波長變短,但
頻率不變。請問,為什麼是頻率不變而不是波長不變?這當中的基
本假設是什麼?
倘若該可見光是一高功率(100 MW)脈衝(pulsed)雷射束,請
問結果還是一樣嗎?為什麼?
9.
當一可見光(具一頻率範圍)射入一透明(對可見光)材料中時,
我們發現該材料中之感應振盪子(induced oscillators)對入射光作
基態散射(ground-state scattering)的程度(含振幅、相位、能量)
隨著頻率可觀的增大(事實上與4成正比,因為振盪子即為
dipoles)。由此,我們知道該材料極可能在UV區至少有一個共振
點(resonance)。
請問,我們是如何知道的?請由相位變化()入手,說明相位遲
延(phase lag)或領先(phase lead)與相速度(phase velocity)的
關係,進而說明與折射率(n)的關係,並繪出相關n-的函數圖。
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10. 當一材料的能隙(energy bandgap; Eg = Ec -Ev)大於入射光波的能
量(E = h)時,該材料對該光波為透明(transparent);否則為
不透明。然而,對高能X-ray(如:100 keV)而言,其頻率因為遠
遠超過大部分材料中的電子們所能夠跟上的頻率,故大部分材料對
強X-ray而言皆似真空般,以致折射率n 都近乎1,亦即透明。在這
個情形裡,顯然h >>> Eg,似與前述能隙說法不合。試問要如何
正確解決這 “矛盾”?
11. 請問光波對材料表面施加壓力(即:光壓)的機制為何?是透過該
材料上的什麼東西以致之?
(hint:用電磁波的場的觀念;可分導電與非導電材料)
12. 在一均勻的透明(對可見光)石英中,試說明為何入射的可見光雷
射束能保持直進,而非側邊散射(lateral scattering)或後散射
(backward propagation)?
(hint: check out E. Hecht’s Optics (any edition) for an intuitive
understanding)
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The EM Approach at Boundaries: Fresnel equations
– a complete description within the framework of classical physics
Waves at an interface
▪ Let the incident wave be a plane wave:
(Any form of light at  can be represented by
two orthogonal linear polarized waves.)
▪ Without making any assumption, the
reflected and transmitted waves can be
written as:
 


Ei  E0i cos( ki  r   t )
 


Er  E0 r cos( k r  r   t   r )
 


Et  E0t cos( kt  r   t   t )
▪ Boundary conditions (BC’s): (def.: Q = Qside-1 – Qside-2) E|| = 0
The component of the E-field that is tangent to the interface must be
continuous cross it.
Let Ûn denote the unit vector normal to the interface, we then have (tangent
to the interface y = b):



not true when
Uˆ n  Ei  Uˆ n  Er  Uˆ n  Et
across a
double layer
(No absorption, no delayed emission, only pure linear and isotropic.)
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▪ This relationship must hold at any instant of time and at any point on the
interface (y = b). Hence, Ei, Er, and Et must have the same functional



ˆ
ˆ
ˆ
dependence on the variables r and t.
U n  Ei  U n  Er  U n  Et
 
 
(ki  r  i t ) | y b  (k r  r  r t   r ) | y b
 
▪ That is,
 (kt  r  t t   t ) | y b
▪ However, this has to be true for all
values of time, thus the coefficients of t
must be equal: i = r = t.
(linear media)
▪ Consequently, we have:
 
 
( k i  r ) | y b  ( k r  r   r ) | y b
 
 ( kt  r   t ) | y b
(*)
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 
▪ From equation (*), we obtain: [(k  k )  r] |  
i
r
y b
r
 an equation of plane!
i.e., the endpoint of r sweeps out a plane
(the interface) perpendicular to (ki – kr).
But r terminates on
the interface at
y = b.
 
Or, (ki – kr) has no projection on the interface, i.e., (ki  kr ) || Uˆ n
 
ˆ
Therefore, U n  (ki  kr )  0
and so,


ˆ
ˆ
Or , U n  ki  sin i  U n  kr  sin  r
ki sin i  kr sin  r
Note that since the incident and the reflected waves are in the same medium
so ki = kr.
Hence, we have the Law of Reflection: i = r.
▪ Further, since (ki – kr) // Ûn, all ki, kr, Ûn are all in the same plane, i.e., the
plane-of-incidence.
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 Similarly, from equation (*), we obtain:
  
[(ki  kt )  r ] | y b   t
But r terminates on
the interface at y = b.
So, again, an equation of plane
(i.e., interface) perpendicular to (ki –kt).
 
ˆ
U n  (ki  kt )  0
 Thus, ki, kr, kt, and Ûn are all coplanar. And (ki –kt) // Ûn.


ˆ
ˆ
Or , U n  ki  sin i  U n  kt  sin t
 Therefore,
and so:
ki sin i  kt sin t
 Since i = t = , multiplying both sides with c/ gives us:
ni sin i  nr sin  r
This is just the familiar Snell’s law.
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The Fresnel equations
▪ We have arrived at laws of reflection and refraction from the phases of Ei(r,t),
Er(r,t), and Et(r,t) at the boundary.
▪ There is still an interdependence shared by the amplitudes of these 3 field
vectors.
Case 1: E perpendicular to the plane-of-incidence (POI) (s-wave)
Since E  POI, so B // POI, due to the fact:
 
  
 
 
Br , t 
  E r , t   
 k  E k ,     Bk ,  
t
Fourier-Laplace transform
 
  
 
  E r , t   0  k  E k ,   0



ˆ
ˆ
ˆ
From U n  Ei  U n  Er  U n  Et
We have:



E 0 i  E0 r  E0 t
B  E / v v   / k
only good for source-free
Since all 3 vectors are parallel and
defined to be out of the POI, we have:
E0i  E0 r  E0t
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▪ Due to the boundary conditions derived
from the Maxwell’s equations:
no surface current
E|| = H|| = 0
and Bn = Dn = 0 
together with
B H
(linear media)
we have the continuity of the
tangential component of B /  at interface:
x̂ :

Bi
i
cos  i 
Br
r
cos  r  
Bt
t
cos  t
Since Bi  Ei / vi Br  Er / v r Bt  Et / vt
And i = r, ki = kr, vi = vr
(in the same medium) and i = r,
We have:
1
Ei  Er  cos i  1 Et cos t
i v i
t v t
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With
1
Ei  Er  cos i  1 Et cos t
i v i
t v t
We further eliminate equal phases, and multiply with c, the amplitude matching
at interface becomes:
ni
i
E0i  E0 r  cos  i

nt
t
E0t cos  t
Then, together with E0i  E0 r  E0t
We arrive at (: E being perpendicular to POI):
 E0 r

 E0 i
 E0 t

 E0 i

 


 

ni
i
cos  i  t cos  t
nt
nt
cos


i
i
 t cos  t
ni
2 nii cos  i
nt
cos


i
i
 t cos  t
ni
* For any linear, isotropic, and
homogeneous materials.
(&)
Fresnel equations
For i  t  0 (i.e., non-magnetic):
 E0 r
r  
 E0i

n cos  i  nt cos  t
  i
  ni cos  i  nt cos  t
 E0t
t   
 E0i

2ni cos  i
 
  ni cos  i  nt cos  t
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Case 2: E parallel to the plane-of-incidence (p-wave)
Similarly, continuity of the tangential
components of E at interface gives:
E0i cosi  E0 r cos r  E0t cost
While continuity of the tangential
components of B /  yields:
no surface current
1
1
1
E0i 
E0 r 
E0t
i v i
r v r
t v t
Using i = r, i = r, we can combine
these 2 equations to obtain:
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 E0 r
r//  
 E0 i

 
 //
 E0 t
t //  
 E0 i

 
 //
nt
cos  i  nii cos  t
ni
cos  t 
t
i
nt
t
cos  i
2 i cos  i
ni
ni
i
cos  t 
nt
t
Fresnel equations
* For any linear, isotropic, and
homogeneous materials.
cos  i
For i  t  0 (i.e., non-magnetic):
 E0 r
r//  
 E0i

nt cos  i  ni cos  t
 
 // ni cos  t  nt cos  i
 E0t
t //  
 E0i

2ni cos  i
 
 // ni cos  t  nt cos  i
//: E being parallel to POI
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All Fresnel equations (going with the figures!)
 E0 r
r  
 E0i

n cos  i  nt cos  t
  i
  ni cos  i  nt cos  t
 E0 r
r//  
 E0i

nt cos  i  ni cos  t
 
 // ni cos  t  nt cos  i
 E0t
t   
 E0i

2ni cos  i
 
  ni cos  i  nt cos  t
 E0t
t //  
 E0i

2ni cos  i
 
 // ni cos  t  nt cos  i
Snell’s law:
sin      sin  cos   sin  cos 
ni sin i  nt sin t
i.e., 
   
   
sin  
cos   cos   2 sin 
 
sin


sin


2
cos
sin
2
cos     cos  cos   sin  sin 
2 sin  cos   sin 2
sin   sin   2 sin  2  cos
r  
sin  i   t 
sin  i   t 
2 sin  t cos  i
t 
sin  i   t 
sin  i
sin  i
2
 
2
 
 
2
2
積化和差

和差化積
r// 
tan  i   t 
tan  i   t 
t // 
2 sin  t cos  i
sin  i   t  cos i   t 
Homework 1.2
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Interpretation of Fresnel equations
Amplitude coefficients
▪ For i  0 (nearly normal incidence)
 r//  r  nt  ni
(cos i  cos t  1)
n n
t
i
The equality of the reflection coefficients arises
because the plane-of-incidence (POI) is no
longer specified when t  0.
E 
n cos  i  nt cos  t
r   0 r   i
 E0i   ni cos  i  nt cos  t
E 
2ni cos  i
t    0t  
 E0i   ni cos  i  nt cos  t
E 
n cos  i  ni cos  t
r//   0 r   t
 E0i  // ni cos  t  nt cos  i
E 
2ni cos  i
t //   0t  
 E0i  // ni cos  t  nt cos  i
▪ When nt > ni, from Snell’s law we have i > t,
and r < 0 (i.e., E is reflected with direction flipped)
for all values of i.
▪ In contrast (still nt > ni), r// starts out positive at
i = 0 and decreases gradually until it equals 0 when
(i + t) = 90.
This particular angle of i is always denoted p and
referred to as polarization angle or Brewster angle.
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r  
t 
sin  i   t 
sin  i   t 
2 sin  t cos  i
sin  i   t 
r// 
tan  i   t 
tan  i   t 
t // 
2 sin  t cos  i
sin  i   t  cos i   t 
83
nt > ni: external reflection
nt < ni: internal reflection
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Picture of the Brewster angle for p-waves:
When i = p, we have
r + t = 90.
Reflected light is caused by
the SHG motion of dipoles in
the material.
When Et  kr,
There is no reflection
Induced dipole field:
far field: E // kr
near field: Er
Idea: use of Er (e.g., lift)
in optic tweezer
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