chapter 1_introduction

1 Introduction and Ray Optics
Optics is the study of light and its interaction with matter.
Light is visible electromagnetic radiation, which transports energy and momentum
(linear and angular) from source to detector.
Photonics includes the generation, transmission, modulation, amplification,
frequency conversion and detection of light.
,Methods of studying light, in historical order are:
-ray optics
-wave optics
-electromagnetic optics
-photon optics (E & M fields are wavefunctions of photons).
In this course, we will focus on electromagnetic optics.
Maxwell’s equations give accurate descriptions of most optical phenomena.
However, for pedagogical reasons, we begin, with ray optics.
1.1 Ray Optics
Ray optics is the simplest theory of light. Rays travel in optical media according
to a set of geometrical rules; hence ray optics is also called geometrical optics.
Ray optics is an approximate theory, but describes accurately a variety of
phenomena.
Ray optics is concerned with the locations and directions of light rays, which
carry photons and light energy (They also carry momentum, but the direction of
the momentum may be different from the ray direction). It is useful in describing
image formation, the guiding of light, and energy transport.
Optical systems are often centered around an axis, called the optical axis.
If rays are nearly parallel to such an axis, they are called paraxial rays. The
study of paraxial rays is called paraxial optics.
1.2 Postulates of Ray Optics
1. Light travels in the form of rays (can think of rays as photon currents).
Rays are emitted by light sources, and can be observed by light detectors.
2. An optical medium (through which rays propagate) is characterized
by a real scalar quantity n
1, called the refractive index. The speed of light
in vacuum is c = 3 × 108m/s.The speed of light in a medium is v = c/n; this
is the definition of the refractive index. The time taken by light to cover a
distance d is t = nd/c; it is proportional to nd, which is called the optical path
length.
1
3. In an inhomogeneous medium, the refractive index n(r) varies with
position; hence the optical path length OPL between two points A and B is
OPL =

B
A
n(r)ds
where ds is an element of length along the path. The time t taken by light to
go from A to B is t = OPL/c.
4. Fermat
sPrinciple Light rays between the points A and B follow a
path such that the time of travel, relative to neighboring paths, is an extremum
(minimum). This means that the variation in the travel time, or, equivalently,
in the optical path lenght, is zero. That is,


B
A
n(r)ds = 0 (1)
Usually, the extremum is a minimum; then light rays travel along the path of
least time. If there are many paths with the minimum time, then light rays
travel along all of these simultaneously.
Why should Fermat’s principle work? Were is the physics? (Fermat’s
principle is the main principle of quantum electrodynamics! It is a consequence
of Huygen’s principle: waves with extremal paths contribute the most due to
constructive interference.)
1.2.1 Propagation in a homogeneous medium
In a homogeneous medium, the refractive index n is the same everywhere, so
is the speed of light v. Therefore the optical path length of least time is the
shortest one - that is, a straight line.
Proof: Suppose the path taken by light is along the curve described by y(x).
The optical path length is
OPL =

B
A
nds = n

B
A
ds (2)
We want to minimize the OPL. Suppose y0(x) is the shortest path. Then we
must have that any other path is longer. Consider the path y(x) = y0(x)+(x) ,
2
where (x) is small. We note that
ds =

dx2 + dy2 = dx

1 + (
dy
dx
)2 = dx

1 + y
2(x) (3)
Since y
(x) = y


0(x) + 
(x), we can write
ds = dx

1 + y
2
0 + 2y

o
+ 
2 (4)
If 
is also small, we have, approximately
ds = dx

1 + y

0
2

1 +
2y

o

1 + y

0
2  dx

1 + y

0
2 + dx

y

0 
1 + y

0
2
(5)
and the optical path length which we varied, OPLv is
OPLv = n

B
A
(

1 + y

0
2 +

y

0 
1 + y

0
2
)dx (6)
Now we said that y0(x) is the shortest path, that is, the shortest opical path
length OPLS
OPLS = n

B
A
(

1 + y

0
2)dx (7)
is a minimum. This means that OPLv cannot be less than OPLS.
If we could choose 
freely, then we could argue that the coefficient of 

must be equal to zero, otherwise, by choosing 
appopriately, we chould make
OPLv less than OPLS, in violation of our original assumption. However, we
cannot choose 
freely, since we have the constraint that (x) = 0 at the end
points. (This is because all paths must go through the end points, and since
y0(x) certainly does, 
(x) must vanish there.)
We therefore recall that

B
A
udv = uv|B
A ?

B
A
vdu (8)
and write

B
A
y

0 
1 + y

0
2

dx =
y

0 
1 + y

0
2
|B
A ?

B
A

d
dx
(
y

0 
1 + y

0
2
) (9)
Since  vanishes at the end points, the first term on the rhs is zero, and we have
A
OPLv = n

B
A
(

1 + y

0
2 ? n

B
A

d
dx
(
y

0 
1 + y

0
2
) (10)
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