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Chapter 35
The Nature of Light and the
Laws of Geometric Optics
예제 35.1 피조의 톱니바퀴를 이용한 빛의 속력 측정
360개의 톱니를 가진 피조의 톱니바퀴가 27.5회/초로 회전하고 있다. 그림 35.2에서
톱니의 홈 A를 통과한 빛이 반사되어 돌아올 때, 톱니 B에 의하여 막히게 된다.
거울까지의 거리가 7500 m일 때, 빛의 속력을 구하라.
풀이
톱니바퀴의 홈은 360개.
빛이 거울까지 왕복 운동하는 동안 톱니바퀴는 1/720회 회전.
t 



(1 / 720)rev
 5.05 10 5 s
27.5rev / s
c
2d
2(7500m)
8


2
.
97

10
m/s
5
t 5.05  10 s
35.3 기하 광학에서의 광선 근사
(The Ray Approximation in Geometric Optics)
기하 광학(geometric optics) : 빛은 균일한 매질을 지날 때에는 직선 방향으로
고정된 방향으로 진행하고, 다른 매질과의 경계면을 만나거나, 시간적으로
또는 공간에 따라 매질의 광학적 성질이 불균일할 때에는 진행 방향을
바꾼다고 가정.
광선 근사(ray approximation) : 광선이란 평면파의 경우 파면에 수직인 직선.
광선 근사에서, 매질을 통하여 이동하는 파동은 광선의 방향을 따라
직선으로 진행한다.
The Ray Approximation in
Geometric Optics



Geometric optics involves the study of the
propagation of light
It uses the assumption that light travels in a
straight-line path in a uniform medium and
changes its direction when it meets the
surface of a different medium or if the optical
properties of the medium are nonuniform
The ray approximation is used to represent
beams of light
Ray Approximation


The rays are straight
lines perpendicular to
the wave fronts
With the ray
approximation, we
assume that a wave
moving through a
medium travels in a
straight line in the
direction of its rays
Ray Approximation, cont.

If a wave meets a
barrier, we will assume
that λ<<d



d is the diameter of the
opening
This approximation is
good for the study of
mirrors, lenses, prisms,
etc.
Other effects occur for
openings of other sizes
Law of Reflection

The normal is a line
perpendicular to the surface



It is at the point where the
incident ray strikes the
surface
The incident ray makes an
angle of θ1 with the normal
The reflected ray makes an
angle of θ1’ with the normal
Law of Reflection, cont.


The angle of reflection is equal to the angle of
incidence
θ1’= θ1



This relationship is called the Law of Reflection
The incident ray, the reflected ray and the
normal are all in the same plane
Because this situation happens often, an
analysis model, wave under reflection, is
identified
Retroreflection




Assume the angle between two mirrors is 90o
The reflected beam returns to the source
parallel to its original path
This phenomenon is called retroreflection
Applications include



Measuring the distance to the Moon
Automobile taillights
Traffic signs
예제 35.2 이중으로 반사된 광선
그림과 같이 두 개의 거울이 서로 120°의 각을 이루고 놓여있다. 거울 M1에 65°의
입사각으로 들어온 광선이 거울 M2 로부터 반사될 때의 방향을 구하라.
풀이
첫 번째 반사 광선이 수평면과 이루는 각도
  90  65  25
  180  25  120  35
 M  90  35  55
2
 M   M  55
1
2
그림에서 입사 광선과 반사 광선을 거울 뒤로 연장하면 두 광선은 60°의 각으로
교차할 것이며, 광선의 방향 변화는 모두 120°가 된다. 이는 곧 거울의 사이각이다.
광선의 방향 변화는 항상 거울의 사이각과 일치하지는 않으며, 위의 경우는
특별한 경우이다. 사이각이 90도인 경우 방향 변화는 180도가 되어 반사된 빛은
원래의 방향으로 돌아가게 된다.
Refraction of Light


When a ray of light traveling through a
transparent medium encounters a boundary
leading into another transparent medium, part
of the energy is reflected and part enters the
second medium
The ray that enters the second medium is
bent at the boundary

This bending of the ray is called refraction
Refraction, 2


The incident ray, the reflected ray, the
refracted ray, and the normal all lie on the
same plane
The angle of refraction depends upon the
material and the angle of incidence

sin θ2 v 2

sin θ1 v1
v1 is the speed of the light in the first medium and
v2 is its speed in the second
Refraction of Light, 3

The path of the light
through the refracting
surface is reversible
 For example, a ray
that travels from A to
B
 If the ray originated
at B, it would follow
the line AB to reach
point A
Following the Reflected and
Refracted Rays





Ray  is the incident
ray
Ray  is the reflected
ray
Ray  is refracted into
the lucite
Ray  is internally
reflected in the lucite
Ray  is refracted as it
enters the air from the
lucite
Light in a Medium




The light enters from the left
The light may encounter an
electron
The electron may absorb
the light, oscillate, and
reradiate the light
The absorption and
radiation cause the average
speed of the light moving
through the material to
c
decrease
1
v
The Index of Refraction


The speed of light in any material is less than
its speed in vacuum
The index of refraction, n, of a medium can
be defined as
speed of light in a vacuum c
n

speed of light in a medium v
Index of Refraction, cont.

For a vacuum, n = 1



We assume n = 1 for air also
For other media, n > 1
n is a dimensionless number greater than
unity

n is not necessarily an integer
Some Indices of Refraction
Frequency Between Media

As light travels from
one medium to another,
its frequency does not
change


Both the wave speed and
the wavelength do
change
The wavefronts do not
pile up, nor are created
or destroyed at the
boundary, so ƒ must stay
the same
공기
ƒ1 = ƒ2
물
Index of Refraction Extended


The frequency stays the same as the wave
travels from one medium to the other
v = ƒλ


ƒ1 = ƒ2 but v1  v2 so λ1  λ2
The ratio of the indices of refraction of the
two media can be expressed as various
ratios
c
λ1 v1
n1 n2



λ2 v 2 c
n1
n2
More About Index of
Refraction


The previous relationship can be simplified to
compare wavelengths and indices: λ1n1 =
λ2n2
In air, n1 = 1 and the index of refraction of the
material can be defined in terms of the
wavelengths
λ
n
λn
 λ in vacuum 


λ
in
a
medium


Snell’s Law – Example



Light is refracted into a
crown glass slab
θ1 = 30.0o, θ2 = ?
n1 = 1.00 and n2 = 1.52



From Table 35.1
air
n2  1
θ2 = sin-1(n1 / n2) sin θ1 =
19.2o
The ray bends toward the
normal, as expected
water
n1  1.33
d ' 허상
d 실제 깊이
예제 35.3 유리의 굴절각
파장이 589nm인 빛이 공기 중에서 투명하고 평평한 크라운 유리로 법선과 이루는
입사각 30.0°인 상태로 입사한다. (A) 굴절각, (B) 유리에서 이 빛의 속력 및 (C) 빛의
파장을 구하라.
풀이
(A) 굴절각  2 ; sin  2 
n1
sin 1
n2
이므로
 1.00 
sin  2  
 sin 30.0  0.329
 1.52 
 2  sin 1 (0.329)  19.2
(B) 속력 v2 ,
파장 2 ;
c
3.00 108 m / s
v2 

 1.97 108 m / s
n2
1.52
2 
1
n2

589nm
 388nm
1.52
예제 35.4 평행판을 투과하는 빛
빛이 매질 1로부터 매질 2, 즉 굴절률이 n2인 두꺼운 평행판을 투과한다. 투과된 빛이
입사한 빛과 평행함을 보여라.
풀이
(1)
sin  2 
n1
sin 1
n2
( 2)
sin  3 
n2
sin  2
n1
sin 3 
n2
n1
 n1

 sin 1   sin 1
 n2

만약 평행판의 두께가 두 배로 되면 두 광로 사이의 거리 d도 두 배로 되는가?
a
t
cos  2
d 
and
t
sin( 1   2 )
cos  2
d  a sin   a sin( 1  2 )
광로 사이의 거리 d가 평행판 두께에 비례한다.
Observing the Rainbow



If a raindrop high in the sky is observed, the red ray is
seen
A drop lower in the sky would direct violet light to the
observer
The other colors of the spectra lie in between the red and
the violet
Total Internal Reflection

A phenomenon called total internal
reflection can occur when light is directed
from a medium having a given index of
refraction toward one having a lower index of
refraction
Possible Beam Directions


Possible directions of
the beam are indicated
by rays numbered 1
through 5
The refracted rays are
bent away from the
normal since n1 > n2
n1 sin 1  n2 sin 2
n2
n2
sin 1 
sin  2 
n1
n1
Critical Angle

There is a particular
angle of incidence that
will result in an angle of
refraction of 90°

This angle of incidence is
called the critical angle,
θC
n2
sin θC 
(for n1  n2 )
n1
n1 sin 1  n2 sin 2
n2
n2
sin 1 
sin  2 
n1
n1
예제 35.6 물고기의 눈에 보이는 광경
물의 굴절율이 1.33이라면, 물과 공기의 경계면에서 임계각은 얼마인가?
풀이
sin  C 
sin  C 
n2
sin  2
n1
이므로
1
sin 90  0.752
1.33
공기
 C  48.8
물 속에서 밖을 볼 때,
1  48.8 이면
물 속에서 밖을 볼 수 없다.
물
Fiber Optics



An application of
internal reflection
Plastic or glass rods
are used to “pipe” light
from one place to
another
Applications include


Medical examination of
internal organs
Telecommunications
Fiber Optics, cont.


A flexible light pipe is
called an optical fiber
A bundle of parallel
fibers (shown) can be
used to construct an
optical transmission line
Construction of an Optical
Fiber

The transparent core is
surrounded by cladding



The cladding has a lower n
than the core
This allows the light in the
core to experience total
internal reflection
The combination is
surrounded by the
jacket