Chapter1
Fundamental law of geometrical optics
第一章 几何光学的基本定律
基本概念
§1.1
Basic concept
§1.2
The law of refraction
§1.3
Optical fiber 光纤
§1.4
A perfect optical system 完善光学系统
§1.5
Fermat’s principle 费马原理
衍射定律
<Foreword (introduction)>
﹡The nature of light : the light is a electromagnetic wave, it has both wave and
particle properties.
▲Wave property: may best be described by the phenomenon of diffraction,
interference ,and it has λ-wavelength, ν-frequency.
▲Particles
property: may best be described by interaction of light with
matter :photochemical action, photoelectric action, light
pressure ,it has m-mass, E-energy, P-momentum.
▲Einstein′s
equation
The momentum :
E=hν=hc/λ
h: planck′s constant
ν: frequency of photon
c:
velocity
of
photon
10
c=3*10 cm/s
E=mc 2 =hc/ν
E: energy of photon
m: mass of photon
p=mc=hν/c=h/λ.
The energy is determined only by frequency (for E=hν). Geometrical optics is based
on the particle property of light. Light travel is in <straight
lines>.
Contents: 1 Image size
----Magnification
2 Image position
Newton equation
Gauss
formula
3 Image brightness
Illumination equation
4 Image quality
Aberration
§1.1 basic concept
Ⅰ. The nature of light
Light is a wave motion traveling through space or other transparent medium.
▲
Light sources: principal sources of light are the sun, tungsten lamp, laser .
▲
Detectors: eyes, thermal, detectors, photocells, photographic emulsions.
▲
Characters of light :
Frequency ν: lying between 7.5*10 14 for violet light and 4*10 14 for deep
red light.
Wavelength λ: dividing the velocity by the frequency. λ=v/ν
λ: 0.4μm～0.75μm(micrometer)
Velocity v=3*10 10 cm/sec in empty space
▲
In glass or other transparent media: v, λ reduced by a factor n known as
refractive index.
Wave number: number of wave in a centimeter.from25000 for violet light, to
13.000 for red light.
Ⅱ. Rays and Waves
The path along which light travels are known as rays in a homogeneous
medium, they are straight lines.
﹡The location and brightness of an image can be determined by ray
method.
﹡ The fine structure and distribution of light with an image are
determined by wave method .
﹡The study of ray phenomenon is known as geometrical optics.
The study of light wave is known as physical optics.
Ⅲ. Real and Visual images
﹡Real image :formed outside the system, and can be allowed to fall on
a receiving surface (screen or film)
﹡virtual image: formed inside the system and can be observed by
looking into the end system. Such as: (eyepiece telescope microscope)
﹡Real object : lies outside the system.
Virtual object: formed by other front lens.
B
A′
A
B′
Real object
Real image
image（a）
A
A′
A′:virtual image
image（b）
A
A′
Virtual image
Real image
image（c）
A virtual image point
A virtual object point
﹡Object space of a lens system is the space containing the object.
Image space of a lens system is the space containing the image.
﹡if the object is virtual, we must postulate that the object and image space overlap
each other to infinity in both direction..
§1.2
The law of refraction
A plane wave front (in parallel beam of light) is incident upon a plane
refracting surface. The bottom of the wave touches the surface at A ,the upper end of
the wave B will be distant from the surface by a. when B move to c, the A moves to
BC=a, AD=b, a/b is equal to the ratio of the two refractive indices n′/n.
C
B
I′
n′
I
n
A
D
*
a vt c / n n'



b v ' t c / n' n
sinI
a / AC
a
n'

 
sin I ' b / BC
b
n
(1) nsinI=n′sinI′
well-known law of refraction.
(2) I ,I′ ,N are in the same plane.
*
*Because of the symmetry of the law of refraction (n′sinI′=nsinI), light can travel in
either direction along a ray.
* The law of reflection
For the formula of reflection we can set n′=-n,
N
n
glass
n′
I
-I′
surface
I′
air
n′
So (1) I=-I′ (2) I, I′, N are in the same plane.
(The reason is n=c/v, n′=c/v′, the direction of v′ is opposite to the direction of v)
*Total internal reflection
If a ray in a denser medium of refractive index n is refracted into a medium of
lower refractive index n′.
n>n′, the angle of refraction
sinI=n′/n , sinI′=n′/n
△
If
sinI C =n′/n
,then I′=90° refracted ray will lie along the surface
I C is called critical angle
△
If I> I C , there will be no refracted ray the whole of light will be reflected back to
the denser medium.
This is known as total reflection.
* Interpolation of refractive index in the catalog of the schott optical glass company,
six-term formula is used.
n 2  A0  A12  A2 / 2  A3 / 4  A4 / 6  A5 / 8
All coefficients A 0 ~ A 5 are given explicitly.
§1.3 Optical fiber
Based on total reflection .
Ⅰ. Condition:① light in a optically dense medium approach the boundary of a
lens dense medium.
② incident angle is larger than critical angle.
Ⅱ. Bundles of tiny rods or fibers of clear glass or plastic in orderly array,
used to transmit light images.
﹡ fiber are usually coated with a thin transparent layer of glass or material of
lower refractive index. Total reflection will still take place between the two.
﹡ producing of coated fibers:
① One method is to insert a thick high-refractive index glass rod in tubing of
lower index.
② In a special furnace, the two are drawn down to 1/1000 in diameter.
③ A bundle of such fibers can then be fused together to from a solid mass,
and drawn down a second time .The diameter of the fiber is about 2μm,
resolving power of the fiber is 250 lines per millimeter.
﹡ The application of fiber optics:
▲
The field of medicine: cystoscope, endoscope, enables the surgeon to
observe and operate by remote control.
▲
Optical communication.
Ⅲ. Behavior of rays in straight fiber
In principle:
① A meridional ray transmitted by a fiber will remain in the same azimuthal
plane throughout it passage ,emerging at the same angle α(or-α),depending on the
number of reflection.
② A bundle of rays of small diameter entering at angle α will fill an annulus of
cone of half –angle.
③ A convergent cone is obliquely incident on the fiber ,emergent light in the
form of a hollow cone, divergent angle θ equal to the angle of convergence θ of
incident cone.
Advantages:
① Low loss-transmission.
② High-information carrying.
③ Small size and weight.
④ Immunity to electro-magnetic interference.
⑤ Unparalleled signal security.
Numerical aperture:
NA=n 0 sinI 1 ≦ n12  n22
n 1 :core index .n 2 :cladding index
§1,4 A perfect optical system
A perfect optical system is one which :
Ⅰ ▲ All the light rays from an object point pass through an optical system and from
a single image point .
A′
y
-y
A
Or Ⅱ ▲ The emerging wave front is then spherical (if incident wave front is
spherical).
A′
A
Or Ⅲ▲ Optical path is the sum of products nD along each ray.
k
OP=  nD
1
c
nD= *v*t=ct
v
B
B′
A′
A
C
Ⅱand Ⅲ are called Malus′s law.
C′
§1.5 Fermat’s principle
Ⅰ.The path taken by light ray in going from one point to another through any
set of is(such as to render its optical path (op)) equal, in the first approximation
to other paths closely adjacent to the actual/one.
Optical path can be a maximum, minimum or stationary.
n2
n1
n3
B
A
l2
l3
l1
Ⅱ.Consider a ray of light that pass through a point θ ,after reflection from a
plane , pass though a second point θ″.
θ
N
n
θ′′
n′
I′
I
H
G
M
A
B
C
X
θ′
① To find the real path, we drop a perpendicular to GH, and extend it an equal
distance on the other side to θ′.
② The straight line θ′ θ″ is drawn in.
③ From its intersection B, the line θB is drawn.
④ The real path is θBθ″.
OPL
A
Consider now adjacent path to point like A, C
close to B .Since a straight line is the shortest
path between two points.
so path: θAθ″>θBθ″
θCθ″>θBθ″
so real path θBθ″ is minimum.
C
B
θ
Ⅲ.Consider refraction.:
θ
n
N
d
I
h
I
X
d′
P-X
X
h′
I′
n′
θ′
P
Op  nd  n' d'  n[h 2  (p - x) 2 ] 1/2 n' [h' 2  x 2 ]1/2
n/2
d{op}
n' /2
 2
(-2p  2x) 
* 2x  0
2 1/ 2
dx
[h  ( p  x) ]
[h' 2  x 2 ]1/2
nsinI= n′sinI′
Ⅴ.1 ellipsoid
A
B
c
b
a
θ
F
N
OPL
A
B
C
F′
θ
All rays emanating from a point F is reflected and come to together at the other
focus F′. Optical paths are equal ——stationary.
2. If other reflecting surface like a and c are tangent to the ellipsoid at the point
B,NB is normal to all three surface, and FBF′ is a real path for all three.
(a) Give a maximum light path for the real path to and from reflector a (a such as
sphere).
(c) Give a minimum light path for the real path to and from reflector c (c such as
plane mirror).
OP
OP
B
OP
B
B
θ
(a)
θ
(b)
θ
(c)
1. How comes rainbow?
Rainbow results the combination of refraction, reflection and dispersion from
raintrops, entering ray is refracted and dispersed twice.
2. Why is the sky blue?
Molecules in the atmosphere act as scattering center, Rayleigh scattering has a
1/λ 4 dependence.
3. Why is the sunset red?
Primary mechanism is for the same reason that the sky is blue.