Optics Basics
A Physics MOSAIC
MIT Haystack Observatory RET 2010
Background image from NASA
Representing Light
As we have already seen, light (and all electromagnetic waves)
consists of electric and magnetic fields oscillating perpendicular to
each other and to the direction of wave propagation.
While a source of electromagnetic radiation will (generally) send
waves out in all directions from the source, it is often convenient to
represent these waves as rays, traveling in straight lines and
represented by arrows.
Some sources, like light
bulbs and stars, send
light rays out in all
directions.
Image by SKMay
Thomasbrightbill, from flickr, Creative Commons
Other sources, like lasers, send
out light rays in (primarily) one
direction.
Inverse Square Law
• Like sound, the gravitational force, and the electrostatic force, light (and
other electromagnetic waves) follow an inverse square law for intensity.
• As the distance from a source of light increases, the intensity detected
decreases with the square of the distance.
• This should make some sense, since the light from the source is being
spread over larger areas (proportional to distance2) as the distance
increases.
Image from NASA
Which Object is Brightest?
Image found at Nasa.gov, photo credit: Dave Jurasevich, Mt. Wilson Observatory
Reflection for Light
• What does it mean to be reflective?
– As with other types of waves, light reflects off boundaries it
encounters at changes in media.
– Materials absorb some of the light that encounters their
surface, transmit some of the light, and reflect some of the
light.
– If most of the light is reflected rather than absorbed, we think
of it as a reflective surface.
– This depends on the chemical properties of the material.
• Glass transmits most of the light through the medium,
but some is reflected. Why can you see yourself in your
windows at night but see outside during the day?
Fermat’s Principle of Least Time
• Fermat’s Principle of Least Time states that of all
the possible paths for light to take between two
points, it will take the one that requires the least
amount of time.
• For single medium situations, this corresponds to
straight paths.
• The path of least time for a ray to reflect off a
mirror results in equal angles of incidence and
reflection. This is called the Law of Reflection.
Law of Reflection
• Law of Reflection: The angle of incidence of a ray equals
the angle of reflection.
• Both angles should be measured from the normal, or line
perpendicular to the surface.
• The law of reflection applies to all reflection, regardless of
the surface or the type of wave being reflected.
Image from Wikipedia, Drawn by Johan Arvelius 2005-09-26, Creative Commons
Diffuse and Specular Reflection
• If the surface is smooth, the reflection is specular.
That is, the light rays all reflect in the same direction.
• If the surface is not smooth, the reflection is diffuse.
That is, the light rays reflect in different directions
Images from http://twistedphysics.typepad.com/cocktail_party_physics/optics/
Smooth?
• The “smooth”-ness of a surface depends on the size of
the irregularities on the surface compared to the
wavelength of the wave being reflected.
• This means that what constitutes smooth for radio
waves is different than for visible light or x-rays.
(Remember the EM Spectrum)
• In order to be considered smooth, any bumps on the
surface must be smaller than the size of the wavelength
being reflected. For telescopes, microscopes, and other
high precision devices, even smaller irregularities are
tolerable. Typically, astronomers seek 1/20 l smoothness
for their instruments.
Smooth to What?
James Clerk Maxwell
Telescope
Water on Calm Day
Satellite TV Dish
Image from National Research Council of Canada
Two Properties of Being Shiny
• In conclusion, in order to be “shiny” for a
particular electromagnetic wave, a surface must
be both
– Reflective: it must neither absorb nor transmit the
majority of the light incident upon it.
– Smooth: with bumps and irregularities on the same
size scale of the wavelength of the radiation being
reflected
MOSAIC and reflection
• The MOSAIC system reflects radio waves at a
frequency of 11.072 GHz that come from ozone
in Earth’s mesosphere onto a feed at the focal
point of the dish.
• Because it reflects radio waves, it does not look
shiny in visible light, but does look shiny to the
11.072 GHz radiation we are interested in.
Basics of Concave Mirrors
• By making a shiny surface in
an appropriate curved
shape, you can create a
mirror where parallel rays
approaching the mirror
reflect to a single point (the
focal point).
• This is still an example of
specular reflection, and
each individual ray reflects
according to the law of
reflection.
qi
qr
qi
qr
qi
qr
From Wikipedia, Image by AndrewBuck, Creative Commons
Properties of Concave Mirrors
• Focal Length (f)
– Distance between the mirror and the focal point (F).
– Depends on radius curvature of mirror (r): for spherical mirrors, f = r/2
Image from Telescopes from the Ground Up, Amazing Space,
http://amazing-space.stsci.edu/resources/explorations/groundup/,
Image in public domain
• Light Gathering Power
– Ability of the mirror to gather light. Especially important for telescopes.
– Depends on area of telescope. Proportional to (diameter)2
• Resolution
– Ability of the mirror to differentiate small features.
– Depends on wavelength being observed and diameter of telescope.
• Improves proportional to diameter of telescope.
• Improves indirectly with wavelength.
Arecibo Radio Telescope
The 300 m telescope at Arecibo,
Puerto Rico is designed to observe
and transmit radio waves to and
from space. As we have seen, radio
waves have a much longer
wavelength than the visible
spectrum. How is that reflected in
the size of this telescope? Its
smoothness?
Photo by SKMay
Interferometry
Another solution to the problem of poor resolution from radio waves
involves mathematically combining the signals from multiple
telescopes spread over a large area. This is called interferometry.
Combining the signals in this way creates an “effective diameter” for
the purposes of telescope resolution equal to the distance between
the dishes.
The Very Large Array (VLA) in Socorro,
NM consists of 27 identical telescopes
in a Y-shaped configuration to allow
for interferometry of the signals from
all the dishes.
The dishes are on tracks which allow
them to be placed at different
distances for different observational
needs.
Image from NRAO
Two Types of Curved Mirrors
• Curved mirrors are either
concave or convex in shape.
• As we have seen, concave
mirrors tend to converge
light rays (and thus are used
in telescopes).
• Convex mirrors tend to
diverge light rays.
• Both types of curved
mirrors can produce images,
and both have practical
applications.
Images from Telescopes from the Ground Up, Amazing Space, http://amazing-space.stsci.edu/resources/explorations/groundup/. Image in public domain
Special Rays for Concave Mirrors
• Using what we know about the properties of concave mirrors, we can
develop some shortcuts for tracing rays that encounter the surface.
4
1
3
principal axis
•
C
•
F
2
• Special Rays for Concave Mirrors
1.
2.
3.
4.
A ray approaching the mirror parallel to the principal axis will reflect through
the focal point.
A ray approaching the mirror along a path through the focal point will reflect
parallel to the principal axis.
A ray approaching the mirror along a path through the center of curvature
(for a spherical mirror) will reflect back along its incoming path. (The angle of
incidence is zero.)
A ray encountering the mirror at the center will reflect with an angle of
reflection equal to the angle of incidence. (The normal is the principal axis.)
Special Rays for Convex Mirrors
• Instead of focusing parallel rays to a single focal point, a convex mirror disperses
incoming parallel rays, as if they are coming from a single focal point, located
behind the mirror.
• Using this, we can develop the same sort of shortcuts for rays that we did with
concave mirrors.
4
1
•
F
principal axis
• Special Rays for Convex Mirrors 3
1.
2.
3.
4.
•
C
2
A ray approaching the mirror parallel to the principal axis will reflect as if it coming from the
focal point.
A ray approaching the mirror along a path directed towards the focal point will reflect
parallel to the principal axis.
A ray approaching the mirror along a path directed towards the center of curvature (for a
spherical mirror) will reflect back along its incoming path. (The angle of incidence is zero.)
A ray encountering the mirror at the center will reflect with an angle of reflection equal to
the angle of incidence. (The normal is the principal axis.)
Image Formation from Curved Mirrors
• Helpful facts about image formation
– Images that form from the actual convergence of
light rays are called real images. These are formed in
front of mirrors.
– Images that form from the apparent convergence of
light rays are called virtual images. These are formed
behind mirrors, as in the flat mirror you may have in
your bedroom or bathroom.
– Your brain always interprets a light ray coming into
your optical system as having traveled in a straight
path.
Ray Diagram: Concave Mirrors
• When the object is located closer to the mirror
than the focal point, a virtual, upright, larger
image is produced. The image is located behind
the mirror.
Images from Wikipedia, user Cronholm144, Creative Commons
do < f
The image is located using two
special rays.
Light leaves the object in all
directions, but if we choose to
sketch the path of rays following
our special ray paths, we can
easily find the point of
convergence, which, in this case,
is behind the mirror.
Ray Diagrams: Concave Mirrors
do = f
No image is formed when the
object is placed at the focal point.
All light rays reflect off the mirror
parallel to each other, so there is
no convergence.
f < do < 2f
A real, inverted, larger image is
formed when the object is placed
between the focal point (F) and
the center of curvature (C or 2F).
The image is beyond 2F (or C).
Images from Wikipedia, user Cronholm144, Creative Commons
Ray Diagrams: Concave Mirrors
do = 2f
A real, inverted, same-size
image is formed when the
object is placed at the center of
curvature (C or 2F). The image is
also located at 2F (or C).
do > 2f
A real, inverted, smaller image is
formed when the object is placed
beyond the center of curvature (C
or 2F). The image is located
between the F and 2F (or C).
Images from Wikipedia, user Cronholm144, Creative Commons
Ray Diagrams: Flat Mirrors
• No matter there the object is placed near a flat
mirror, a virtual, upright, and same-size object will
be produced. The image is located behind the
mirror at the same distance as the object.
From Wikipedia, user Fffred, Public Domain
Ray Diagrams: Convex Mirrors
• No matter where the object is located relative to the
surface of the convex mirror, a virtual, upright,
smaller image is produced. The image is located
behind the mirror.
Images from Wikipedia, user Cronholm144, Creative Commons
Applications of Mirrors
The Bean, Chicago
Image by SKMay
James Webb Space Telescope Side View Mirror
Image from NASA
Image by SKMay
Magnifying Mirror
Rearview Mirror
Security Mirror
Image by steve loya, Flickr, Creative Commons
Image by SKMay
Image by Leo Reynolds, Flickr, Creative Commons
Summary:
Types of Images and Sizes
• Depending on the location of the object
relative to the focal point, concave mirrors can
produce real images that are larger, the same
size, or smaller than the object or virtual
images that are larger than the object.
• Convex mirrors always produce virtual images
that are smaller than the object.
• Flat mirrors always produce virtual images
that are the same size as the object.
Mirror Equation
• The relationship between the position of an object, image,
and focal length of the mirror is given by the mirror
equation: 1
1 1
f

d0

di
• This equation is completely consistent with the images
produced using ray diagrams on the earlier slides.
• In order to take advantage of the equation, we must be
very careful with signs.
– Focal length (f): positive for converging mirrors (concave),
negative for diverging mirrors (convex)
– Object distance (do): positive for real objects
– Image distance (di): positive for real images (in front of mirror),
negative for virtual images (behind mirror)
Magnification
• The amount that an image is enlarged or reduced as
compared to the object is expressed in its
magnification (m). m  hi   d i
ho
do
• Magnification is defined as the ratio of the height of
the image (hi) to the height of the object (ho). It is
also equivalent to the opposite of the ratio of the
image distance to the object distance.
• The height of the image (hi) will be positive when
the image is upright (and therefore virtual) and
negative when the image is inverted (and therefore
real).
Spherical Aberration and Parabolas
• Spherical reflectors introduce an aberration (error) in focal point.
– This is called spherical aberration, and consists of the light rays not coming
to a perfect focus at a single point.
• This aberration does not exist when a parabola is used.
– Because of the properties of a parabola, light rays that come into the mirror
parallel to the center line reflect exactly to a single focal point.
Image from Telescopes from the Ground Up, Amazing Space, http://amazing-space.stsci.edu/resources/explorations/groundup/. Image in public domain
Applications of Parabolic Optics
From Wikipedia, Creative Commons, User Matěj Baťha
The world's larges solar energy dish
at the Ben-Gurion National Solar
Energy Center in Sde Boker, Israel.
From Wikipedia, Creative Commons,
user David Shankbone.
From Wikipedia, Creative Commons, User Duk
Parabolic Hot Dog Cooker, from
nycg46, found on Flickr
Offset Parabola
• Design:
– Uses only part of a
parabola, allowing the
focal point to be below the
dish doing the receiving.
• Advantages
– The receiver (located at
the focal point) does not
need to block any of the
signal
• Note that the dish appears
to be pointing in a
different direction than it
actually is.
Image of Green Bank Radio Telescope (Green Bank, WV) from NRAO / AUI / NSF, from
RET 2009
MOSAIC’s Offset Parabola
Like the Green Bank Radio Telescope,
MOSAIC (and all small television satellite
dishes) are offset parabolas. While this
dish appears to be pointed towards the
ground, it is actually pointing 8˚ above
the horizon.
Images by SKMay
Refraction
• Refraction Basics
– Recall that the speed of a wave depends on the properties of
the medium it is traveling through.
– For light, the speed of the wave depends on the optical density
of the medium.
– Light only travels at c (3.0 x 108 m/s), the speed of light in a
vacuum, when it is in a vacuum. (Go figure!)
• Index of Refraction
– We can quantify the effect of different media on the speed of
light with the index of refraction (n).
– Greater n, slower speeds.
c
n
v
Some Indices of Refraction
What is the speed of light in water?
Material
Index of
Refraction
Vacuum
1 (exactly)
Air
1.000277
Water
1.333
Crown Glass
1.52
Flint Glass
1.66
Ice
1.309
Diamond
2.417
Cubic Zirconia
2.20
Human Cornea
1.373
c
n
v
1.33 
v
3.00  108 m
s
v
3.00 108 m
1.33
Still pretty fast!
s  2.26 108 m
s
Refraction and
Fermat’s Principle of Least Time
• Refraction is the bending of a wave due to a
change in the speed of the wave in different
media.
• This can be thought of a consequence of
Fermat’s Principle.
– Light travels more slowly in optically dense media, so
it spends less time in them.
– Light travels more quickly in media that are less
optically dense, thus spending more time.
Example: Fermat’s Principle for
Lifeguards
sand
Lifeguard
water
flailing
swimmer
The lifeguard will spend more time
running along the beach to get to the
flailing swimmer than in the water,
because he is a faster runner than he is a
swimmer.
How would this ideal path change if the
lifeguard were a seal?
I’m a much better swimmer
than runner!
Image by SKMay
Example: Fermat’s Principle for Light
air
laser pointer
qi
The light from the laser will
take a path that spends
more time in air than water,
since it travels faster in air
than it does in water.
water
qr
target
Note that this results in a
smaller angle of refraction
than the angle of incidence,
since both are measured
from the normal.
Light Rays in Media
Note that
air
glass
qi
• Light bends towards the normal
when entering a slower medium
(higher n)
– qi > qr when ni < nr
qr
qi
• Light bends away from the
normal when entering a faster
medium (lower n)
– qi < qr when ni > nr
The amount of bending will depend
on how much slower or faster the
new medium is.
Note also
• The light is partially reflected
(following the law of reflection)
at each boundary.
Snell’s Law
• Willebrord Snellius (the disputed funniest name in all of physics)
recorded the quantitative law governing refraction of light in 1621.
• The law was previously discovered in 984 by Ibn Sahl. Ptolemy
also collected the data in table form, but didn’t know about
trigonometry.
• Still, there’s a nice song about Snell and his (dubious)
accomplishment.
• Snell’s Law states n1 sin q1  n2 sin q 2
• As was true for the Law of Reflection, all angles must be measured
from the normal.
Image from Wikipedia, user Oleg Alexandrov,
Public Domain
Snell’s Law and
Total Internal Reflection
• When a ray of light encounters a medium with a lower
optical density than the one it is currently in (lower n,
higher speed), there are three possibilities for its
behavior.
– For low angles of incidence, the ray will refract, bending
away from the normal.
– At some critical angle, the angle of refraction will be 90˚.
– For angles of incidence greater than the critical angle, the
ray will reflect perfectly, staying within the incident
material. This is called total internal reflection.
• When a ray of light encounters a medium with higher
optical density, there will always be refraction. There is
no critical angle in this case.
Critical Angles and TIR
angle of incidence less
than critical angle
qi < qc
Air
n = 1.00
qr
angle of incidence equal
to critical angle
q i = qc
Air
n = 1.00
qr
angle of incidence greater
than critical angle
qi > qc
Air
n = 1.00
qi qrl
qi qrl
qi qrl
Water
n = 1.33
Water
n = 1.33
For the partially
reflected ray:
For the partially
reflected ray:
For the totally reflected
ray:
For the refracted ray:
For the refracted ray:
There is no refracted
ray!
qi = qrl.
ni sin qi = nr sin qr
Water
n = 1.33
qi = qrl.
ni sin qc = nr sin 90˚
sin qc = nr / ni
qi = qrl.
Total Internal Reflection
Wikipedia, user Hustvedt, Creative Commons
Wikipedia, user Mina Zinkova, Creative Commons
Wikipedia, Public Domain
Convex Lens Basics
• By cleverly changing the shape of a refractive
medium, you can produce a lens.
• Parallel rays approaching the lens converge to a single
point.
• Each individual light ray bends according to Snell’s
Law.
Images from Telescopes from the Ground Up, Amazing Space, http://amazing-space.stsci.edu/resources/explorations/groundup/. Image in public domain
Properties of a Convex Lens
• Focal Length
– Depends on n
• The greater the change in the speed of light, the greater the bending, and therefore,
the smaller the focal length.
– Depends on curvature
• The smaller the radius, the greater the bending, and therefore, the smaller the focal
length.
Image by SKMay
• Light Gathering Power: as with concave mirrors, proportional to
area.
• Resolution: as with concave mirrors, improves with diameter and
gets worse as the wavelength of the observed radiation increases.
Two Types of Lenses
• By cleverly changing the
shape of the medium, you
can produce a lens.
• Just as with mirrors, lenses
can be either concave or
convex in shape.
• While Snell’s Law governs
the interaction of each light
ray with the lens, we can
develop some shortcuts by
considering special rays, as
we did with mirrors.
Images from Telescopes from the Ground Up, Amazing Space, http://amazing-space.stsci.edu/resources/explorations/groundup/. Image in public domain
Special Rays for Convex Lenses
• Using what we know about the properties of convex lenses, we
can develop some shortcuts for tracing rays that encounter the
surface.
2
1
3
principal axis
•
2F
•
F
•
F
•
2F
• Special Rays for Convex Lenses
1. A ray approaching the lens parallel to the principal axis will refract
through the focal point on the opposite of the lens.
2. A ray approaching the lens along a path through the focal point will
refract parallel to the principal axis on the opposite side of the lens.
3. A ray passing through the optical center of the lens will be
undeflected.
Special Rays for Concave Lenses
• Using what we know about the properties of concave lenses,
we can develop some shortcuts for tracing rays that encounter
the surface.
3
1
principal axis
•
F
•
2F
•
F
•
2F
2
• Special Rays for Convex Lenses
1. A ray approaching the lens parallel to the principal axis will refract as if
it is coming from the focal point on the side of the lens it comes from.
2. A ray approaching the lens along a path toward the focal point on the
opposite site of the lens will refract parallel to the principal axis.
3. A ray passing through the optical center of the lens will be
undeflected.
Image Formation from Lenses
• Just as we saw with image formation from
mirrors, we can note the following helpful facts:
– Images that form from the actual convergence of
light rays are called real images. These are formed
on the opposite side of a lens as the object.
– Images that form from the apparent convergence of
light rays are called virtual images. These are formed
on the same side of the lens as the object.
– Your brain always interprets a light ray coming into
your optical system as having traveled in a straight
path.
Ray Diagrams: Convex Lens
• Just as with mirrors, we can identify the nature and location of an
image formed by a lens by tracing a few special rays leaving the
object as they refract through the lens and ultimately converge.
A
A
do > 2f
The image produced is real, inverted, and smaller than the object. The image is
located between F and 2F.
do = 2f
A
A
The image produced is real, inverted, and the same size as the object. The image
is located at 2F.
Ray Diagrams: Convex Lens
2f > do > f
A
A
The image produced is real, inverted, and larger than the object. The image is
located beyond 2f.
A
do = f
There is no image produced. All refracted rays emerge from the lens parallel to
each other.
Ray Diagrams:
Convex and Concave Lens
A
A
do < f
The image produced is virtual, upright, and larger than the object. The image is
located on the same side of the lens as the object.
A
A
do
The image produced is virtual, upright, and smaller than the object. The image
is located on the same side of the lens as the object.
Applications of Lenses
Image by SKMay
Image from chrisjohnbecket, from Flickr, Creative Commons
Images by SKMay
Image by Yerkes Observatory
Summary:
Types of Images and Sizes
• Depending on the location of the object relative
to the focal point, convex lenses can produce real
images that are larger, the same size, or smaller
than the object or virtual images that are larger
than the object.
• Concave lenses always produce virtual images
that are smaller than the object.
Lens Equation
• The relationship between the position of an object, image,
and focal length of the lens is given by the lens equation,
which is identical to the mirror equation: 1
1 1
f

d0

di
• This equation is completely consistent with the images
produced using ray diagrams on the earlier slides.
• In order to take advantage of the equation, we must be
very careful with signs.
– Focal lengths (f): positive for converging lenses (convex),
negative for diverging lenses (concave)
– Object distance (do): positive for real objects
– Image distance (di): positive for real images (opposite side of
lens as object), negative for virtual images (same side of lens
as object)
Magnification
• Just as with mirrors, the amount that an image is
enlarged or reduced as compared to the object is
expressed in its magnification (m). m  hi   d i
ho
do
• Magnification is defined as the ratio of the height of
the image (hi) to the height of the object (ho). It is
also equivalent to the opposite of the ratio of the
image distance to the object distance.
• The height of the image (hi) will be positive when
the image is upright (and therefore virtual) and
negative when the image is inverted (and therefore
real).
Dispersion
• The index of refraction is not constant for all
frequencies. This is called dispersion.
• Because n (the index of refraction) is different for
different wavelengths and frequencies, different
wavelengths and frequencies will bend different
amounts in the same medium.
• Effects
– Prisms
– Rainbows
– Chromatic Aberration
– Atmospheric Effect on radio waves from space
Image by Marlene May
Image of Prism from Telescopes from the Ground Up, Amazing Space, http://amazing-space.stsci.edu/resources/explorations/groundup/. Image in public domain
Chromatic Aberration
• Because different wavelengths have slightly different n,
they will bend different amounts in a lens, and therefore
have different focal lengths.
• This can be (partially) corrected with an achromatic lens,
which introduces a second lens of a different n to bring
two wavelengths to the same focal point.
Image from Wikipedia, user DrBob, Creative Commons
Image from Wikipedia, user DrBob, Creative Commons
Dispersion of Radio Waves from Space
• Earth’s Ionosphere acts as a dispersive medium
for radio signals from space.
• As the ionosphere changes (due to space weather
such as solar flares), the dispersive properties
change.
• GPS signals become unpredictable during high
variability in solar activity due to this.
• Atmospheric scientists can use the dispersive
properties of the ionosphere to study its
variations.
Reflecting vs. Refracting Telescopes
• Most large telescopes are reflectors rather than
refractors. Why?
– Only one surface to make perfectly shaped, which
makes it less expensive and less difficult to produce.
– No sagging (glass is viscous fluid) in mirrors; mirrors
can be supported from the bottom
– No chromatic aberration in mirrors, and a parabolic
shape eliminates spherical aberration
– More compact designs are possible; with secondary
reflectors, the telescope can be smaller than the
focal length.
Another Optical Effect
• Gravitational Lensing: gravity from massive
objects (usually galaxies) causes light from
beyond the massive object to bend.
Images from NASA