geometric optics 3

INTRODUCTION
It has been shown that electromagnetic waves in matter
travel at a speed that is reduced, and can be expressed
in terms of the refractive index n:
? =
?
?
where Maxwell’s equations were used to predict that
the refractive index ? is given by:
? = ??? ??
For non-ferromagnetic materials the relative permeability
is very close to unity and then ? = ??? where
?? is the dielectric constant. The plane EM wave in
matter has the relation between the perpendicular coupled
electromagnetic fields ? = ?? and travels in the
??E × ??B direction.
In the absence of any obstructions or media effects,
electromagnetic radiation travels in straight lines. The
fact that light travels in straight lines, and casts a
sharp shadow, led Newton to believe that light involved
particle motion not wave motion. Solutions of
the wave equation also predict electromagnetic waves
traveling in a straight line. For example, we saw that
a plane wave of the form ??E = ?0bi
sin(?? ? ??) travels
in the ? direction. However, last lecture it was shown
that waves behave differently when obstructions are
encountered. The difference between particle and wave
motion is that, for obstructions comparable in size to
the wavelength, waves are diffracted outwards at the
edges. The angle of the first minimum for diffraction
at a slit of width d is given by sin ? = ?
? . Diffraction
effects become negligible when ? ?? ?, which is
the realm of geometric optics. For many applications
of wave propagation, such as optics, diffraction effects
can be ignored, to first order, allowing use of geometrical
optics, the topic of today’s lecture.
Although the propagation of waves can be predicted
exactly using mathematical solutions of the wave
equation, with known boundary conditions, you may
Figure 1 Huygens construction for propagation to the
right of (?) a plane wave, and (?) an outgoing spherical
wave.
find it helpful to consider the Huygens construct of
secondary wavelets to explain wave motion. The Huygens
construct assumes that each point on a primary
wavefront serves as the source of spherical secondary
wavelets that advance with the speed and frequency of
the primary wavefront. Figure 1 illustrates the Huygens
construct for the propagation of plane and spherical
waves. The secondary wavefront is the locus of
the secondary wavelets. Note that the wavelets do
spread outwards into the shadow at the edge of the
plane wave. Calculation of the secondary wavefront requires
including the relative phases and amplitudes of
the wavelets involved. The Huygens construct is useful
in understanding reflection, refraction and diffraction.
The trajectory taken by an electromagnetic wave also
is predicted by Fermat’s principle which states that
the wave-front follows a path that minimizes the transit
time.
REFLECTION AND REFRACTION AT
BOUNDARIES
The reflection and transmission of electromagnetic waves
incident normally at the boundary between two media
has been discussed and the reflectivity and transmission
calculated. When waves are incident upon a
boundary at non-normal incidence, the reflected and
transmitted waves are emitted at definite non-normal
angles, and the reflectivity and transmission differ from
the case of normal incidence discussed earlier. The
angles of incidence, reflection and refraction are most
conveniently measured with respect to the normal to
the boundary surface. The plane of incidence is the
surface defined by the incident ray and the normal to
the surface.
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Figure 2 Wave with velocity v1in medium 1 incident
upon a boundary with medium 2 in which the wave has
velocity v2? The angle of incidence to the normal to the
surface is ??? the angle of reflection to the normal is ??,
and the angle of refraction relative to the normal is ?? ?
Reflection
Reflected and incident waves are in the same material
and therefore have the same velocity. Therefore, using
the Huygens construct shown in figure 2, it can be seen
that the reflected wave is in the plane of incidence and
the angle of reflection equals the angle of incidence.
?? = ??
Refraction
The transmitted wave travels in a different medium,
with different speed compared to the incident wave.
Using the Huygens construct, one can see that in a
given time ??? the incident and transmitted waves
travel the distances ???? and ?? ?? respectively. Since
the two wavefronts have a common hypotenuse, then
we have the relation that gives the angle of refraction
of the transmitted wave:
sin ??
sin ??
=
?? ?
???
that is:
sin ??
??
=
sin ??
??
These two relations for the angles of reflection and
refraction are true for any type of travelling wave. For
electromagnetic waves, the velocity can be expressed
in terms of the refractive index, that is ? = ?
? ? Using
this, leads to Snell’s Law that gives the refracted wave
is in the plane of incidence at an angle of refraction
given by:
?? sin ?? = ?? sin ?? (Snell’s Law)
As an example, consider light in air incident upon
glass of refractive index ? = 1?5 as shown in figure
Figure 3 Refraction of light by air-glass interface.
Figure 4 Total internal reflection when n1 ? ?2?
3. For normal incidence ?? = ?? = ?? = 0? For an
incident angle ?? = 45?, the angle of reflection is 45?
and the angle of refraction is given by:
sin ?? =
1
1?5
sin 45?
which gives ?? = 28?13?? At the maximum angle of
incidence ?? = 90?, then Snell’s law gives ?? = 41?81??
Critical angle
When electromagnetic radiation strikes a boundary going
from higher to lower refractive index, the transmitted
wave is bent away from the normal at a larger angle
than the angle of incidence as shown in figure 4. For
incident angles at the critical angle, ??? the angle of
refraction is 90?. That is,
sin ?? =
?2
?1
(Critical angle)
For angles of incidence ?? ? ?? there is no solution to
Snell’s law since there is no angle for which the sine
exceeds unity. For angles of incidence greater than
the critical angle, the incident wave is totally internally
reflected, there is no transmitted wave. Thus
the surface behaves as a perfect mirror. Underwater
swimmers are familiar with this phenomenon, they
can see upwards through only an angular region of
the water surface within ? = 48?8? of the normal to
can be totally internally reflected twice to exactly reverse
the direction of travel using a right-angle prism.
Four such prisms are used in binoculars to correct for
inversion of the image by the lens. An important technical
application of total internal reflection is the optical
fiber illustrated in figure 6. This has applications
in medicine as well as for long-distance transmission of
signals.
Polarization
Since electromagnetic waves are transverse waves it
can be shown that when the angle of reflection plus angle
of refraction sum to 90?, the reflected wave is plane
polarized as illustrated in figure 7. The reflected wave
is fully plane polarized at the Brewster angle given by:
tan ????????? =
?2
?1
Polaroid sunglasses can be used to attenuate the polarized
reflected light from a surface such as a window.
REFLECTION AT SINGLE SURFACE
Geometrical optics for reflective systems, that is when
? ?? ?, is dominated by the statement that the reflected
wave is in the plane of incidence and the angle
of reflection equals the angle of incidence. Let us consider
simple reflective systems.
Figure 7 Polarization of reflected light.
Figure 8 Image formed by reflection from a plane
mirror.
Plane mirror
As shown in figure 8, radiation reflected from the
mirror appears to emanate from a virtual image that
is the same distance behind the mirror as is the object
in front of the mirror. An important feature of the
mirror image is that depth inversion results in a righthanded
system,bi
×bj
= bk
having a left-handed mirror
image wherebi
×bj
= ?bk? This is illustrated in figure
9. The behaviour of physics under transformation
of a right-handed system into a left-handed mirrorimage
system is an important fundamental symmetry
of nature.
Concave mirrors
Consider the concave spherical surface of radius ? shown
in figure 10? Let the object be a distance ? and image
a distance ? from the mirror. From elementary
Figure 9 Mirror image of a right-handthe surface. For gl