Download Nature of light - Art of Problem Solving

The Nature of Light
Nature
of
light
How does light carry energy from one point to another?
18th Century: 2 “competing” theories.
Newton: Particle (corpuscular) theory.
Why? Rectilinear propagation  no bending around
corners.
Huygens: Wave Theory
Why? Light beams can pass through each other
without influence.
Course divided into two “halves”:
Geometric and wave optics
Geometric Optics: based upon Newton’s idea.
Propagation of light is governed by Fermat’s Principle.
Wave (Physical) Optics: based upon Hygens’ idea.
Propagation of light is governed by Huygens’ Principle
(wave motion).
Geometric (Ray) Optics
Geometric
optics
How to make a light ray?
Deals with the direction of propagation of light “rays”.
Source
Light Ray
Aperture
Aperture
Light emitted by any point on the source can be thought of as
being composed of a “bundle” of rays.
Reflection / refraction
Two important means by which a light ray direction can be
changed:
Reflection:
Reflecting surface
Refraction:
Medium 1
(incident)
Medium 2
(transmitting)
Ray Optics
Possible path of a light ray through an optical system.
Fermat’s Principle
Fermat’s principle
Original formulation: Light travels from point A B along the
path which takes the least time.
Medium 3
Medium 4
Medium 5
v3
v4
v5
Medium 1
v1
Medium 2
v2
Assumption: Light travels at different speeds in different media.
This was postulated by Huygens and Newton and later verified
by experiment.
The propagation speed in vacuum  c
Fermat’s Principle
Fermat’s
principle
2
It's not quite right to call this the principle of least time. In the
figure below, for example, we consider light emitted at the centre
of an elliptical mirror. The four physically possible paths by which
a ray can return to the center consist of two shortest-time paths
and two longest-time paths !!
Fermat’s Principle
Fermat’s principle 3
More precise formulation: Time taken for light to travel along
its true or actual path from point A B is equal “in first order
approximation” to the time taken along other (hypothetical) paths
closely adjacent to the true path.
Optical path length
Homogeneous medium: Light travels from point A B along a
straight line path. The time taken is:
Path length
Speed of light in
the medium
Optical path length (OPL): the OPL is proportional to the time
taken:
Index of Refraction (of a medium)
Index of refraction
For materials we’ll be considering (transparent):
By definition:
Note:
n1
nvacuum = 1
nair  1.000293  nvacuum
Terminology: if medium 2 has refractive index n2 and medium 1
has refractive index n1 > n2 then medium 1 is “optically denser”.
Note: refractive index n is colour dependent (“dispersion”)
Law of Reflection by Fermat’s Principle
Law of reflection: Fermat (1)
Hypothetical light paths from point A B by reflection.
Which is the true path?
Reflecting surface (xz plane)
Law of Reflection by Fermat’s Principle
Law of reflection: Fermat (2)
Set up the problem for solution by Fermat’s Principle:
The “family” of hypothetical light paths from point A B
is determined by the various possible values of x.
Law of Reflection by Fermat’s Principle
Law of reflection: Fermat (3)
Fermat’s Principle: The true path OPL is equal to the
OPL of nearby (adjacent) hypothetical paths (to “first
order” approximation).
Law of Reflection by Fermat’s Principle
Law of reflection: Fermat (4)
Here OPLAB =OPLAB(x):
Law of Reflection by Fermat’s Principle
Law of reflection: Fermat (5)
For a nearby path, the OPL is OPLAB(x+ ) for small  .
When does OPLAB(x+ ) = OPLAB(x) (to first order)?
Recall from elementary calculus that:
Law of Reflection by Fermat’s Principle
Law of reflection: Fermat (6)
Apply the definition of the 1st order derivative to OPLAB(x):
For small  :
“first order approximation”
Law of Reflection by Fermat’s Principle
Law of reflection: Fermat (7)
Thus OPLAB(x+ ) = OPLAB(x) (to first order) when the 1st order
derivative of OPLAB(x) vanishes:
The value of x for which the 1st order derivative of
OPLAB(x) vanishes will determine the true path of the
light ray from A  B.
Law of Reflection by Fermat’s Principle
Law of reflection: Fermat (8)
Solve this equation:
Law of Reflection by Fermat’s Principle
Law of reflection: Fermat (9)
To make the solution more
obvious, introduce the angles
i and r (the angles of
incidence and reflection).
Law of Reflection by Fermat’s Principle
Law of reflection: Fermat (10)
Law of Reflection (partially!)
Fermat’s Principle
Fermat least time
Note: The condition of
“least time” and of “nearly
equal OPLs for nearby
paths” are equivalent.
The 2nd condition is more
meaningful (later!).
Law of Refraction by Fermat’s Principle
Law of Refraction by Fermat’s
Points A and B are separated by an interface between two
principle 1
different media.
Incident
Medium
Interface
Transmitting
Medium
Possible (hypothetical) paths connecting points A and B.
Which is the true path?
Law of Refraction by Fermat’s Principle
Law of Refraction by Fermat’s
Set up the problem for solution by Fermat’s Principle.
principle 2
Law of Refraction by Fermat’s Principle
Law of Refraction by Fermat’s
The OPL for the path A  P  B is thus:
principle 3
Fermat’s Principle is satisfied for the condition:
Solving this equation for x will determine the true path.
Law of Refraction by Fermat’s Principle
Law of Refraction by Fermat’s
Define angles  (the angle of incidence ) and  (the angle of
principle
4
transmittance):
i
t
Law of Refraction by Fermat’s Principle
Law of Refraction by Fermat’s
principle 5
Law of Refraction by Fermat’s Principle
Law of Refraction by Fermat’s
Fermat’s Principle gave:
principle 6
Thus:
Snell’s Law of Refraction (partial)
Summary: Law of Reflection
Summary: Law of Reflection
Normal to reflecting
surface at P.
Incident ray:
Reflected ray
Vectors lie in the same plane,
“plane of incidence”
Summary: Law of Reflection (alt)
Summary: Law of Reflection
Consider a light ray emanating from point O (unit vector r ) and
reflected at point P to arrive at (alt)
Q (unit vector r ). The reflecting
1
2
surface is the xy plane.
Reflected ray
Incident ray:
Reflection at the xy plane gives the transformation of the
incident ray unit vector to the reflected ray unit vector:
Application: Corner Cube (Retro) Reflector
Application: Corner CubeIncident
ray
Reflected
ray
Corner Cube: Three
mutually orthogonal
reflecting surfaces.
Unit vector r1 gives the incident light ray direction and unit vector
r2 gives the reflected ray direction. Reflection at the 3 orthogonal
surfaces transforms the incident ray unit vector to a reflected ray
unit vector pointing in exactly the opposite direction.
Summary: (Snell’s) Law of Refraction
Summary: Law Normal
of Refraction
to the
interface at P.
Transmitted
ray
Vectors lie in the same plane,
“plane of incidence”
Reversibility Principle
Reversibility
Fermat’s Principle gives the light path from A B independent
of the direction in which light travels along this path. In other
words, Fermat’s Principle applied to light travelling from A B
gives the same path when applied to light travelling from B A
Uniform Medium
Uniform Medium
In applying Fermat’s Principle to reflection and refraction, we
assumed straight line propagation in a uniform medium. This
can be proved using Fermat’s Principle (calculus of variations).
Uniform medium
(constant n)
Non-uniform medium
(non-constant n)