geometrical optics chapter 24. optics the study of light is called optics some highlights in the...
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Geometrical Optics
Chapter 24
OpticsThe study of light is called opticsSome highlights in the history of optics
Study of optics dates to at least third century BCEyeglasses invented around 1300Microscopes and telescopes invented around 1600
Applications depend on the ability of lenses and mirrors to focus light
Light is an electromagnetic wave and its wave nature needs to be accounted for
Geometrical OpticsApplies to the regime where light travels in straight-
line pathsEffects involving wave interference are not importantDescribes cases in which the wavelength of the light
is much smaller than the size of the objects in the light’s pathWavelength of visible light is less than 1µm
Describes many everyday applicationsIncluding the behavior of mirrors and lenses
RaysRays indicate the path and
direction of propagation of the light wave
In A, the waves pass through a large opening and, to a very good approximation, follow straight lines that pass through the opening
In B, the opening is about the same size or smaller than the wavelength of the light and needs wave optics to explain
Section 24.1
Wave Fronts
Wave front surfaces are determined by the crests and troughs of the wave
They are always perpendicular to the associated raysThe shape of a wave front depends on how the wave is
generated and the distance from the source
Section 24.1
Two Properties of LightThe motion of light along a light ray is reversible
If light can travel in one direction along a ray that connects point A to point B, light can also propagate in the reverse direction, from B to A
The perpendicular distance between two wave fronts is proportional to the speed of lightBecause of the way wave fronts are related to crest
and troughs of a wave
Section 24.1
Ray TracingLight from an object is used by your eye to form an
image of the objectWhen your eye combines the rays to form an image,
your brain extrapolates the rays back to their originThe method of following the individual rays as they
travel from an object to some other point is called ray tracing
Ray tracing involves the use of geometry
Section 24.1
Ray Tracing, cont.The figure shows a few
rays from the objectThere are an infinite
number of actual raysThe light waves
associated with all the rays contribute to the image formed by your eye
In most ray diagrams, we draw just a few rays from the top and bottom of the image
Section 24.1
Image FormationTwo problems must be considered to understand
how images are formedWhat happens to light rays when they reflect from a
surface such as a mirror or a piece of glassWhat happens to light rays when they pass across a
surface from one material to another such as when they pass from air into a piece of glass
You must also distinguish between a flat surface and a curved surface
Section 24.1
Reflection from a Plane MirrorLight rays travel in straight lines until they strike
somethingThe rays may be reflectedRays may be reflected from a plane mirror
A flat surface that reflects all or nearly all the light that strikes it
If the light is a plane wave, all the rays are parallel and strike a surface at many different points
Section 24.2
Reflection from a Plane Mirror, cont.
Characterize the reflection by a single rayThe normal (vertical dashed line in fig. B) is
perpendicular to the mirrorThe direction of the incoming and outgoing rays are
measured relative to the normal
Section 24.2
Law Of Reflection - DefinitionsThe incoming ray is called the incident ray
The angle it makes with the normal is called the angle of incidence, θi
The outgoing ray is called the reflected rayThe angle it makes with the normal is called the angle
of reflection, θr
The Law of Reflection says θi = θr
Reflection from a perfectly flat mirror is called specular reflection
Section 24.2
Diffuse ReflectionIf the reflecting surface is
rough, the reflections from each individual piece of the surface must be analyzed
An incident plane wave will give rise to many reflected rays propagating outward in many different directions
This is called diffuse reflection
Section 24.2
Image Formation – Plane MirrorAn image formed by a
plane mirror is shownTwo representative rays
are shown coming from the objectThere is an infinite
number of rays emanating from each point on the object
The rays that reflected from the mirror and reached your eyes form the image
Section 24.2
Image Formation, cont.To your eye, the location of the image is the point
from which these rays appear to emanateThis point can be found by ray tracingEach ray obeys the law of reflectionApplying geometry will allow the location of the
image to be found
Section 24.2
Image Formation, finalCharacteristics of the image
The distance from the object to the mirror is the same as the distance from the image to the mirror
The size (height) of the image, hi, is the same as the size (height) of the object, ho
The image is virtual The image point is located behind the mirror The light does not actually pass through the image
The same analysis can be applied to multiple mirrors
Section 24.2
RefractionWhen a light ray strikes
a transparent material, some of the light is reflected and some is refractedThe reflected ray obeys
the Law of ReflectionThe refracted ray
passes into the materialThe incident angle is
now denoted as θ1
Section 24.3
Angle of RefractionThe direction of the refracted ray is measured by
using θ2 (refer to fig. 24.10)
The value of this angle depends on the incident angle and the speed of light in the material
The speed of light in a vacuum is 3 x 108 m/sWhen the light travels through a material substance,
its interactions with the atoms of the material slows down the wave
Section 24.3
Snell’s Law
The change in the speed of light from the vacuum to the material changes the direction of the wave
From the geometry of the waves in the material,
1 2sin sincv
Section 24.3
v = λ f
f constantv , λ change
Snell’s Law, cont.
Snell’s Law, cont.The ratio c/v is called the index of refraction and is
denoted by nn = c / vn is unitless
Then, sin θ1 = n sin θ2 This assumes the wave is incident in a vacuum
A more general statement can be applied to any two materials with indices of refraction n1 and n2
n1 sin θ1 = n2 sin θ2
This relationship is called Snell’s Law
Section 24.3
Speeds and n’s for Various Materials
Section 24.3
Applying Snell’s LawRefraction is also reversibleSnell’s Law applies whether light begins in the
material with the larger or smaller index of refractionPossible angles of refraction are always between 0°
and 90°The side with the larger index of refraction has the
smaller angle
Direction of Refracted RayLight is refracted toward
the normal when moving into the substance with the larger index of refraction
Light is refracted away from the normal when moving into the substance with the smaller index of refraction
Section 24.3
Total Internal ReflectionWhen light is incident
from the side with a higher index of refraction, it is bent away from the normal
As the incident angle gets larger, the refracted angle also increases
Eventually, θ2 will reach 90°
Section 24.3
Total Internal Reflection, cont.The angle of incidence for which the angle of
refraction is 90° is called the critical angleIf the angle of incidence is increased beyond the
critical angle, Snell’s Law has no solution for θ2 Physically, there is no refracted ray
This behavior is called total internal reflectionThis is only possible when the light is incident from the
side with the larger index of refraction
Section 24.3
Critical AngleFrom Snell’s Law, with θ2 = 90°, θ1 = θcrit
When the angle of incidence is equal to or greater than the critical angle, light is reflected completely at the interface
1 2
1
sincrit
nn
Section 24.3
Fiber OpticsTotal internal reflection is
used in fiber opticsOptical fibers are
composed of specially made glass and used to carry telecommunication signalsThese signals are sent as
light wavesThey are directed along
the fiber using internal reflection
Section 24.3
DispersionWhen light travels in a
material, the speed depends on the color of the light
This dependence of wave speed on color is called dispersion
Since the index of refraction is slightly different for each color, the angle of refraction will be different for each color
Section 24.3
Dispersion and Prisms
Dispersion is used by a prism to separate a beam of light into its component colors
There are two refractions with the prismThe red and blue show the extremes of the incident
beamsSection 24.3
Curved MirrorsA curved mirror can produce an image of an object
that is magnifiedThe image can be larger or smaller than the object
Magnified images are used in many applicationsTelescopesCar’s review mirrorMany others
Section 24.4
Ray Tracing – Curved MirrorA spherical mirror in one
in which the surface of the mirror forms a section of a spherical shell
The radius, R, of the sphere is the radius of curvature of the mirror
The mirror’s principal axis is the line that extends from the center of curvature, C, to the center of the mirror
Section 24.4
Concave Spherical MirrorProperties of concave
spherical mirrorsIncoming rays that are
close to and parallel to the principal axis reflect through a single point F F is the focal point It is located a distance ƒ,
the focal length, from the mirror
Rays that originate at the focal point reflect from the mirror parallel to the principal axis From reversibility of light
Section 24.4
Image From a Concave Mirror -- Examples
Section 24.4
Image from Concave Mirror – Ray Diagram
Trace rays emanating from the top of the objectThe rays all intersect at a single point
This is the top of the imageA similar result would be found from rays from other
parts of the object
Section 24.4
Drawing A Ray Diagram
Three rays are particularly easy to drawThere are an infinite number of actual rays
The focal rayFrom the tip of the object through the focal pointReflects parallel to the principal axis
Section 24.4
Drawing A Ray Diagram, cont.The parallel ray
From the tip of the object parallel to the principal axisReflects through the focal point
The central rayFrom the tip of the object through the center of
curvature of the mirrorReflects back on itself
The three rays intersect at the tip of the image
Section 24.4
Properties of an ImageMagnification is the ratio of the height of the image,
hi, to the height of the object, ho
By convention, the image height of an inverted image is negativeTherefore, the magnification is also negative
Images smaller than the object are said to be reduced
Section 24.4
i
o
hm
h
Real vs. Virtual ImagesIf the rays that form the image all pass through a
point on the image, the image is called a real imageReal images and virtual images differ
Light rays only appear to emanate from a virtual image, they do not actually pass through the image For a real image, the light rays do actually pass through the
image
An object and its real image are both on the same side the mirror A virtual image is located behind the mirror while the object
is in frontSection 24.4
Concave Mirror and Virtual Images
Use ray tracing to find the image when the object is close to the mirrorCloser than the focal point
Use the same three raysThe rays do not intersect at any point in the front of the
mirror
Section 24.4
Virtual ImagesExtrapolate the rays back behind the mirrorThey intersect at a single image pointThe rays appear to emanate from the image point
behind the mirrorThe image is virtual because light does not actually
pass through any point on the imageThe object and its image are on different sides of the
mirrorThe image is upright and enlarged
Section 24.4
Rules for Ray Tracing – Mirrors Construct a figure showing the mirror and its
principal axisThe figure should also show the focal point and the
center of curvatureDraw the object at the appropriate point
One end of the object will often lie on the principal axisDraw three rays that emanate from the tip of the
objectThe focal ray passes through the focal point and
reflects parallel to the principal axis
Section 24.4
Rules, cont.Three rays, cont.
The parallel ray is parallel to the principal axis and reflects through the focal point
The central ray passes through the center of curvature of the mirror and reflects back through the tip of the object
The point where the three rays intersect is the image pointThis point may be in front of the mirror giving a real imageThis point may be in back of the mirror giving a virtual
image Found by extrapolation of the rays behind the mirror
Section 24.4
Rules, finalThis ray-tracing procedure can be repeated for any
desired point on the objectThis allows you to find other points on the imageIt is usually sufficient to consider just the tip of the
imageOther points may be used if needed
Section 24.4
Ray Tracing – Convex Spherical MirrorsA mirror that curves away
from the object is called a convex mirror
The center of curvature and the focal point lie behind the mirror
After striking the convex surface, the reflected rays diverge from the mirror axis
The parallel rays converge on an image point behind the mirrorThis is the focal point, F
Section 24.4
Ray Tracing – Convex Mirrors, cont.The same three rays are
used as were used for concave mirrors
The focal ray is directed toward the focal point but is reflected at the mirror’s surface, so doesn’t go through F
The three rays extrapolate to a point behind the mirrorProduces virtual image
Section 24.4
Mirror Equation
Geometry can be used to find the characteristics of the image quantitatively
The distance from the object to the mirror is so The distance from the image to the mirror is si The given rays produce similar triangles
Section 24.4
Mirror Equation and Focal LengthFrom the similar triangles,
For an object at (approximately) infinity, 1/so = 0But an “infinite” object will produce parallel raysParallel rays all intersect at the focal pointTherefore, the focal length can be found from the
radius of curvature of the mirror
1 1 2
o is s R
ƒ2R
Section 24.4
Mirror Equation and MagnificationThe mirror equation can be written in terms of the
focal length
The magnification can also be found from the similar triangles shown in fig. 24.30
1 1 1ƒo is s
Section 24.4
i i
o o
h sm
h s
Sign ConventionsAll diagrams with mirrors should be drawn with the
light ray incident on the mirror from the leftThe object distance is positive when the object is to
the left of the mirror and negative if the object is to the right (behind) of the mirror
The image distance is positive when the image is to the left of the mirror and negative if the image is to the right (behind) of the mirrorThe image distance is positive for real images and
negative for virtual images
Section 24.4
Sign Conventions, cont.The focal length is positive for a concave mirror and
negative for a convex mirrorFor a concave mirror, ƒ = R / 2For a convex mirror, ƒ = - R / 2
The object and image heights are positive if the object/image is upright and negative if it is inverted
Section 24.4
Sign Convention, Summary
Section 24.4
LensesA lens uses refraction to form an imageTypical lenses are composed of glass or plasticThe refraction of the light rays as they pass from the
air into the lens and then back into the air causes the rays to be redirectedAlthough refraction occurs at both surfaces of the lens,
for simplicity the rays are drawn to the center of the lens
Section 24.5
Lenses, Focal PointParts B and C show the
simplification of the single deflection of the rays
Parallel rays close to the principal axis intersect at the focal pointThis is true for incident
rays from either side of the lens
The focal points are at equal distances on the two sides of the lens
Section 24.5
Spherical LensesThe simplest lenses
have spherical surfacesThe radii of curvature of
the lenses are called R1 and R2
The radii are not necessarily equal
Section 24.5
Types of Lenses
Converging lensesAll the incoming rays parallel to the principal axis intersect
at the focal point on the opposite sideDiverging lenses
All the incoming rays parallel to the principal axis intersect at the focal point on the same side as the incident rays
Section 24.5
Focal Point – Diverging LensThe parallel incident
rays from the left are refracted away from the axis
The rays on the right appear to emanate from a point F on the left side of the lens
This point F is one of the focal points of the lens
Section 24.5
Image from a Converging LensAn infinite number of
rays emanate from the object
For simplicity, choose three rays that are easy to draw
Start at the tip of the object
Section 24.5
Rays for a Converging LensThe parallel ray is initially
parallel to the principal axisRefracts and passes
through the focal point on the right (FR)
The focal ray passes through the focal point on the left (FL)Refracts and goes parallel
to the principal axis on the right
The center ray passes through the center of the lens, C
Section 24.5
Rays, cont.If the lens is very thin, the center ray is not deflected
by the lensThese three rays come together at the tip of the
image on the right of the lensIn this case, the image is invertedThe image is real
The rays pass through the image
Section 24.5
Rules for Ray Tracing – Lenses Construct a figure showing the lens and its principal
axisThe figure should also show the focal points on both
sides of the lensDraw the object at the appropriate point
One end of the object will often lie on the principal axisDraw three rays that emanate from the tip of the
objectThe parallel ray is initially parallel to the principal axis
and after refraction passes through one of the focal points
Section 24.5
Rules, cont.Three rays, cont.
The focal ray is directed at the other focal point and after refraction the ray is parallel to the principal axis
The central ray passes through the center of the lens and is not deflected
The point where the three rays or their extrapolation intersect is the image pointIf the rays actually pass through the lens, the image is
realIf the rays do not pass through the lens, the image is
virtual
Section 24.5
Rules, finalReal image
When a lens forms a real image, the object and image are on opposite sides of the lens
Virtual imageWhen a lens forms a virtual image, the object and
image are on the same side of the lensAll other rays that pass through the lens will also
pass through the image
Section 24.5
Ray Tracing – Diverging LensFollow the rules for ray
tracing for lensesSince the refracted rays
do not intersect on the right side of the lens, extrapolate the rays back to the left side of the lens
The extrapolations do intersect
The point of intersection is the image point at the tip of the image
Section 24.5
Sign Conventions – Lenses, Diagram
Sign Conventions for LensesAssume light travels through the lens from left to
rightThe object will always be located to the left of the lens
The object distance is positive when the object is to the left of the lensAccording to the first convention, the object distance
will always be positiveThe image distance is positive when the image is to
the right of the lens and negative if the image is to the left of the lens
Section 24.5
Sign Conventions, cont.The focal length is positive for a converging lens and
negative for a diverging lensThe object height is positive if the object extends
above the axis and is negative if the object extends below
The image height is positive if the image is extends above the axis and is negative if the image extends below
Section 24.5
Thin-Lens EquationGeometry can be used
to find a mathematical relation for locating the image produced by a converging lens
The shaded triangles are pairs of similar triangles
Section 24.5
Thin-Lens Equation and MagnificationThe thin-lens equation is found from an analysis of
the similar triangles
The magnification can also be found from the similar triangles shown
These results are identical to the results found for mirrors
1 1 1ƒo is s
Section 24.5
i i
o o
h sm
h s