Transcript Optics - Tensors for Tots

Optics
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The electromagnetic spectrum
Visible light make up only a small part of the entire spectrum of
electromagnetic waves. Unlike sound waves and mechanical
waves, they don’ t require a material medium for their
transmission, they can propagate through a vacuum.
Electromagnetic waves consist of time-varying electric and
magnetic fields that propagate perpendicular to each other and to
the direction of the propagation of the wave. All electromagnetic
waves travel at a fixed speed through a vacuum
regardless of their frequency.
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Dispersion and Refraction
Prisms display the phenomena called dispersion by
separating white light into components of different
wavelength (different colors). The different colors refract at
different angles, splitting white light into a rainbow. When
light passes through a prism, it is refracted twice, when it
enters the prism and when it leaves. Refraction is the change
in direction of light as it leaves one medium and enters a
different medium. Every medium has a characteristic value
called the index of refraction which is a value used to give
the ratio of the speed an electromagnetic wave in vacuum to
the speed of the electromagnetic wave in a particular
medium.
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Refraction
Refraction is the slowing of light by matter and it follows
Snell’s Law. It is the bending of any wave, such as a light or
sound wave, when it passes from one medium into another of
different optical density
Snell's law states that the ratio of the sines of the angles of
incidence and refraction is equivalent to the ratio of
velocities in the two media or the ratio of the index of
refraction for the two media.
or
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Snell’s Law
Normal
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Problem
If a light beam makes an angle of 40º with the surface,
it makes an angle of 50º with the normal; this is the angle
of incidence. We can find the angle of refraction by using
Snell’s Law.
Solution
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Magnification
v is the distance from the lens to the image. u is the distance
from the lens to the object. a is the height of the image, b is
the height of the object.
Thin Lens Equation
u is the distance (measured along the axis) from the image to
the center of the lens
The image is real, inverted, and behind the lens if u is
positive. The image is virtual, upright, and in front of the lens
if u is negative.
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The approximation for thin lenses
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The thin-lens diagam is composed of three rays:
Ray number 1.) A ray from the top of the object
parallel to the centerline and perpendicular to the
lends. It then travels through the focal point on the
other side of the lens.
Ray number 3.) A ray through the center of the lens
which won't be bent.
Ray number 2.) A ray through the focal ponit on the
near side of the lens.
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Problem
An object of height 14 cm is placed 46 cm in front of a converging
lens with a focal length of 26 cm. Where is the image? Is it real or
virtual? Is it upright or inverted? What’s the height of the image?
With u = 46 cm and f = 26 cm, the lens equation gives us
Because v is positive. the image is real. Real images are inverted. The
magnification is
The height of the image is given by
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Huygen’s Principle
Every point of a wave front can be considered the origin of a
new circular or spherical wave, the so-called elemetary
wave. The new position of the wave front results from the
superposition of all of the elementary waves. The Huygens
wave front is a series of concentric circles originating from
the points on a wave front.
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Diffraction in a Single Slit
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If a wave front goes through a single slit, it breaks down
into individual circular waves called elementary waves. As we
know through Huygen’s Principle, the single slit acts a point source
for a new circular or spherical wave. This phenomena is called
diffraction.
The difference between refraction and diffraction
In refraction, the wave is being bent because the speed of
propagation in the medium it is moving into is different than the
medium it is moving out of. For instance, light moving from air
to water at an angle is bent away from the axis that runs
perpendicularly to the surface because light travels slower in
water than in air.
In diffraction, the wave is being bent (often times through very
small slits) when it hits somthing, causing a diffraction pattern to
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form.
Double Slit Interference
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Double Slit Interference
source
screen
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If the crests and troughs of the two waves arrive at the screen at
the same time, then the waves interfere constructively and a
bright band appears on the screen. If the crest of one wave arrives
at the same time as the trough of the other wave, then
the waves interfere destructively and cancel each other out. This
area of the screen will be dark. If the distance to the screen D is
much larger than the distance between the stripes, the conditions
for bright or dark bands are
bright bands
dark bands
where is the wavelength of the light, d is the separation
between the two slits and n is the order of maximum observed
(central maximum is n = 0)
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Problem
When sodium yellow light (¸ = 589 nm) is used in a
double slit experiment, the first order maximum is
0.035 cm from the central maximum. When another
light source of unknown wavelength is used. The first
order maximum occurs at 0.03 cm from the center.
What is this wavelength?
and
,
thus
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