qft01 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。

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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））. 請描繪出（作圖）材料 ...

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。

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1

Professor: Chungpin Liao (Hovering)

廖重賓 （飛翔）

cpliao@alum.mit.edu

cpliaq@nfu.edu.tw

Coherent Control Lab. (http://coherent.nfu.edu.tw)

Introduction &

Optics (EM) and Electronics Fundamentals

R&D of Exploratory Electro-Optic Materials and Applications

（前瞻光電材料與應用之開發）

2

• 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 (20%)
Syllabus (課程表)

3

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, newBrewster)

• Applications based on interactions between lights and

bio-systems / bio-materials are thriving. (Kirlian, SEA,
light-protein, LLLT, GdIONP magnetic fluid hyperthermia)

• High efficiency solar energy harvester is in need.

(BlackbodySolar, chl-a organic cell)

4

• Coverage: (only as leads for more unexpected)

1-5 Fundamentals of photonics and optics
6, 7 Radiating dipoles and refractive index
8 Fresnel equations -- EM form & traditional Brewster angle
9 Fresnel equations -- scattering form & new Brewster angle
10 Double layer caused asymmetric refraction & strobe light
(PVDF, LiNbO₃)
11 Large refractive index & Clausius-Mossotti equation
 small-area spiral inductor, etc.
12 Frequency Push
13 Visible light-induced ITO surface plasmon resonance
(incl. Landau damping)
14 Strange mirror
15 Open-air Weibel instability amplifier
16 Possible relation between Light-interference & protein release
And, occasionally, some invited speakers …

5

• 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.)

6

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.)

7

• 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?

8

• 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?

9

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

10

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

11

• Photonics (光電) area categorization:

About optical devices in which electrical
effect plays a role (lasers,
electro-optic modulators, and switches)
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
Electro-optics
Optoelectronics
Quantum electronics
Quantum optics
Lightwave technology
Photonics

12

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.

13

• 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)
Optics treatment overview:

14

Quantum optics
Electromagnetic (EM)
optics
Wave optics
Ray optics

• 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

• 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

15

• 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.

16

• 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)
the most general form of the 1-D wavefunction
Lightwave, phase velocity and phasor (相位向量)
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) =?

But x’ = x – vt (or,  (x – vt) relativistically),
a propagating wave is described by:
 = f(x – vt).

17

is a bell-shaped propagating wave
Equivalently,
For the moment, limit to waves of constant shape:

18

The differential wave equation (linear perturbation (擾動))

19

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.

20

21

Phase and phase velocity (相速度)

22

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.

23

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)

24

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) e^i 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):

25

26

Phasors (相位向量) and the addition of waves

27

Plane waves (平面波)

• the simplest of 3-D waves: plane wave:

• 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.

28

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.

29

3-D differential wave equation

30

• 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.

31

• 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.

32

• 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)

• 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)

33

• 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?

34

• 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.
  v
Stream of photons
Induced dipoles
Photons disappear temporarily and are added phases.
The real picture:

35

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: . (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:

. However, there is interdependence between E and B….
 Maxwell’s equations

36

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):

37

Faraday’s induction (磁感電) law

• Michael Faraday jotted “convert magnetism into electricity”

on his notebook in 1822,and started working on it in 1831.

• He found that change of a magnetic flux _M is the

driving “force” (called emf: electromotive “force”)
to create an electric field or current:
Q: Do you keep
a notebook for
wild new ideas?
How do you
keep a reading habit
and store
information
effectively?

38

For

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.

39

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,

ds

A

E

when there is no source or sink within the closed surface A
i.e., in flux of E = out flux of E

40

• Consider a closed spherical surface A, which confines charge q:

(Coulomb’s law: an experimental observation)
However, since the closed surface A does not have to be spherical, but can be any
shape, therefore:
Vacuum electric permittivity ₀ = 8.8542 x 10^-12 C²/N-m²
(真空介電常數)
But,
We have:
（體積分）
流出量

41

• Similarly, since there is no magnetic monopole (i.e., magnetic charge), therefore:

• Therefore a B field must be in the shape of a closed loop.

• In a sense, the permittivity of a material  ( = K_E 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).

• K_E ( =  / 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)

42

Ampere’s circuital law (電感磁)

• A “work rule” found out long time ago:

• If the current has a nonuniform cross section, then using the notion of current

density J (A/m²), we have:

• ₀ = 4 x 10^-7 N-s²/C² is the permeability of free space.

• In general,  = K_M ₀, K_M being the dimensionless

relative permeability (相對導磁係數)

• However, equation (*) does not tell the whole story.

(*)

43

(*)

• 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.

• 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.

• 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.

i = 0

A₂

44

• 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 J_D
(“displacement current density”), i.e.,

• The complete Ampere’s law is thus:

• Differential form:

A

Substance filled

z

On each xy plane, B(z) varies
according to
J_total = J_conductor + J_D
(owing to Maxwell)

45

Maxwell’s equations – the most complete classical description of optics (or, EM)

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

Integral form

Differential form

Gauss-E

Gauss-B

Faraday

Ampere

Magnetic
induction
Displacement
current

46

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.

47

• 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)

48

• 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):

• 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  = E_x, E_y, E_z, B_x,
B_y, B_z):
That is, (scalar) wave theory, or physical optics (物理光學).
Cf. In general,

49

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 E_x = 0

• recall that E is independent of y, z, E = 0 means

E_x/x = 0. So, E_x can be a constant or zero.

• Thus, in free space, the plane EM wave is transverse, that is, TEM.

• If we let E_y(x,t) = E_0y cos[ (t - x/c) +  ], then using Faraday’s law, we

arrive at E_y = c B_z.

• Apparently, the wave propagates along the direction of

Let
Then we have:
i.e., E_y(t) is generated by B_z(t)
(Faraday’s law)

50

Right-hand rule

51

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
we force the vector (r-r₀) to sweep out a plane
perpendicular to k, and its end point (x, y, z)
takes on all allowed values.
With k = (k_x, k_y, k_z), we have:
 a plane
(x, y, z)
(x₀, y₀, z₀)
x
y
z

52

• The most concise form of

the equation of a plane
perpendicular to k is then just:

• The plane is the locus of all points whose

projection onto the k-direction (r_k in the
figure blow) is a constant.
(We have chosen our axes that way so that
k vector is originated at (0, 0, 0).)

• We now construct a set of planes over which

(r) varies in space sinusoidally, i.e.,

• 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

variable
modulation
(x, y, z)
(x₀, y₀, z₀)
x
y
z

53

• To get things moving, (r) must be made

to vary in time, i.e.,

• 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 dr_k, we must have:
or, in exponential form:
Therefore,
and the magnitude of the wave velocity (phase velocity here), dr_k/dt, is:

54

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 u_E = (₀/2)E² and B field energy

density u_B = (1/2₀)B². Since c = (₀₀) ^-½, we have u_E = u_B. 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 = u_E + u_B = 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):
e.g.

55

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.

56

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:

57

Linearly accelerating charges

58

Note the kink in E field, which
gives rise to an EM pulse propagating
away from the accelerating charge

59

Synchrotron radiation
(同步輻射)

Linearly accelerating

charges

60

Electric dipole radiation

+q
-q
Origins:

• atoms
• molecules
• nuclei
• any charge distribution
• defects

The simplest and most important EM wave
producing mechanism

• 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.

61

The higher the frequency, the stronger the dipole radiation.

62

The emission of light from atoms

63

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) :
Since normally dielectrics of interest are nonmagnetic,
Absolute index of refraction :
Maxwell’s relation
,
Light in bulk matter – material aspect

64

In short, n = n (ω)

1

ω<ω₀ (or hν< hν₀= E_i→j )
material
atom : scatters light
∵
hν

1

2

absorb hν then re-emit photons (redirect light)
only
Non-resonant scattering (ground state)

2

Matching E_i→j:
excitation of atomic e^-’s (quantum jump) absorption
dense medium
collision
Thermalization
(randomization)
Dissipative absorption
Dispersion (色散) :

65

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)

+

e^-
e^-
e^-
e^-
an atom
Dipole moment per unit volume :
P = (ε-ε₀)E

For isotropic homogeneous and linear media :

F_E=q_eE(t)=q₀E₀cosωt

( driving force on an e^- )

66

+

e^-
e^-
e^-
e^-
an atom
ω₀ = natural freq. (resonant)
of the bound e^- dipole
≡
=
Medium=
Now, let x(t) = x₀cosωt
(a guess)
=
in-phase
180° out-of-phase

67

For rarefied gas
&
Now, since
∴We arrive at dispersion relation :
In general,
Works for rarefied gas (稀薄)

68

Colorless, transparent media have their ω₀'s outside the
visible range. ( ω_0² >> ω² )
cf. In dense media :
( little dissipation )
if
&
→
Visible range
So, n ≈ const.

69

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）？

• 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））

• 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？

A brief review checklist (Homework 1.1):

70

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？）。

4. 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。

4. 折射率與相對介電及導磁係數之關係為何？

4. 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）

71

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？

8. 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。

72

10. 當一材料的能隙（energy bandgap; E_g = E_c -E_v）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 >>> E_g，似與前述能隙說法不合。試問要如何正確解決這 “矛盾”？

10. 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）

10. 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)

73

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:

• Boundary conditions (BC’s): (def.: Q = Q_side-1 – Q_side-2) E_|| = 0

The component of the E-field that is tangent to the interface must be
continuous cross it.
Let Û_n denote the unit vector normal to the interface, we then have (tangent
to the interface y = b):

(No absorption, no delayed emission, only pure linear and isotropic.)
not true when
across a
double layer

74

• This relationship must hold at any instant of time and at any point on the

interface (y = b). Hence, E_i, E_r, and E_t must have the same functional
dependence on the variables r and t.

• That is,

• 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:

(*)

75

• From equation (*), we obtain:

 an equation of plane!
i.e., the endpoint of r sweeps out a plane
(the interface) perpendicular to (k_i – k_r).
Or, (k_i – k_r) has no projection on the interface, i.e.,

Therefore,
and so,
Note that since the incident and the reflected waves are in the same medium
so k_i = k_r.
Hence, we have the Law of Reflection: _i = _r.

• Further, since (k_i – k_r) // Û_n, all k_i, k_r, Û_n are all in the same plane, i.e., the

plane-of-incidence.
But r terminates on
the interface at
y = b.

76

• Similarly, from equation (*), we obtain:

So, again, an equation of plane
(i.e., interface) perpendicular to (k_i –k_t).

• Thus, k_i, k_r, k_t, and Û_n are all coplanar. And (k_i –k_t) // Û_n.

• Therefore,

and so:

• Since _i = _t = , multiplying both sides with c/ gives us:

This is just the familiar Snell’s law.
But r terminates on
the interface at y = b.

77

The Fresnel equations

• We have arrived at laws of reflection and refraction from the phases of E_i(r,t),

E_r(r,t), and E_t(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:
Fourier-Laplace transform
From
We have:
Since all 3 vectors are parallel and
defined to be out of the POI, we have:
(&)
only good for source-free

78

• Due to the boundary conditions derived

from the Maxwell’s equations:
E_|| = H_|| = 0
and B_n = D_n = 0
together with
(linear media)
we have the continuity of the
tangential component of B /  at interface:
Since
And _i = _r, k_i = k_r, v_i = v_r
(in the same medium) and _i = _r,
We have:
no surface current

79

With
We further eliminate equal phases, and multiply with c, the amplitude matching
at interface becomes:
Then, together with
(&)
We arrive at (: E being perpendicular to POI):
* For any linear, isotropic, and
homogeneous materials.
For _i  _t  ₀ (i.e., non-magnetic):
Fresnel equations

80

Case 2: E parallel to the plane-of-incidence (p-wave)
Similarly, continuity of the tangential
components of E at interface gives:
While continuity of the tangential
components of B /  yields:
Using _i = _r, _i = _r, we can combine
these 2 equations to obtain:
no surface current

81

* For any linear, isotropic, and
homogeneous materials.
For _i  _t  ₀ (i.e., non-magnetic):
//: E being parallel to POI
Fresnel equations

82

All Fresnel equations (going with the figures!)
Snell’s law:
積化和差

和差化積
Homework 1.2

83

Interpretation of Fresnel equations
Amplitude coefficients

• For _i  0 (nearly normal incidence)


The equality of the reflection coefficients arises
because the plane-of-incidence (POI) is no
longer specified when _t  0.

• When n_t > n_i, 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 n_t > n_i), 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.
(cos _i  cos _t  1)

84

n_t > n_i: external reflection
n_t < n_i: internal reflection

85

Picture of the Brewster angle for p-waves:
When E_t  k_r,
There is no reflection
When _i = _p, we have
_r + _t = 90.
Reflected light is caused by
the SHG motion of dipoles in
the material.
Induced dipole field:
far field: E_ // k_r
near field: E_r
Idea: use of E_r (e.g., lift)
in optic tweezer