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Chapter 24: Geometric optics
M. C. Escher
What will we learn in this chapter?
Contents:
Reflection at a plane surface
Reflection at a spherical surface
Mirrors
Refraction at a spherical surface
Thin lenses
Graphical methods for lenses
Reflection at a plane surface
Several rays of light diverge from the
object point P and are reflected at the
mirror.
The rays diverging from P are reflected
as if their direction had come from the
image point P’.
We say that the mirror forms
an image of point P’.
The rays do not come from P’, but
their directions are the same as
though they had come from that point.
Refraction at a plane surface
A similar behavior can be found
at a refracting surface.
Rays coming from P are refracted
at the interface.
When the angles of incidence are
small it seems as if the rays are
coming from the image point P’.
When na > nb ( na < nb ) P’
seems closer (farther) to the
surface than P.
plane mirror
Geometric (optics) construction
The ray PV is normal
to the mirror and thus
retraces its path.
The distance s = PV is
called object distance, the
distance s’ = VP’ image
distance.
The ray PB makes an
angle θ with PV and
strikes the mirror
at an angle of incidence θ and is reflected at the same angle. |s| = |s! |
When both rays are extended backwards, they intersect at P’.
Both triangles PVB and P’VB are congruent, thus PV = VP’.
This can be repeated for all rays originating from P (real image). The
direction of all outgoing rays are the same as though they had
originated at P’ (virtual image).
Sign rules
Sign rules for object and image distances:
Object distance: !When the object is on the same side of the reflecting
! ! ! ! ! or refracting surface as the incoming light, the object
! ! ! ! ! distance s is positive, otherwise negative.
Image distance: ! When the image is on the same side of the reflecting
! ! ! ! ! or refracting surface as the outgoing light, the image
! ! ! ! ! distance s’ is positive, otherwise negative.
Note:
These rules seem unnecessarily complex at this point, but will be
very handy later for more complex situations.
For a plane mirror s = −s! .
Sign rules illustrated
In both cases the object distance s
is positive because the object point
P is on the incoming side of the
reflecting/refracting surface.
The image distance is not on
the same side as the outgoing
ray, hence s’ < 0.
The object distance is on
the same side as the incoming
ray, i.e., s > 0.
The image distance is on the
other side of the outgoing
ray, i.e., s’ < 0.
Objects with finite size || mirror
The object is represented by
the arrow PQ and has height y.
The image is represented by
the arrow P’Q’ and has height y’.
Because the triangles PQV and
P’Q’V have equal angles it follows
that object and image have the
same size, i.e., y = y’.
object
image
Lateral magnification: For an object of height y and image y’, the lateral
magnification m is
y!
m=
y
For a plane mirror, m = 1.
Objects with finite size || mirror
contd.
Conventions:
Objects are represented by arrows in diagrams.
When the image arrow points in the same direction, we call the
image erect, otherwise it is inverted.
Note: the image formed by a plane mirror is always erect.
A negative value of the lateral magnification m
means the image is inverted.
3D objects on plane mirrors:
The size of 3D objects are
the same, but they are
reversed even though the
image is erect.
Multiple reflecting/refracting surfaces
The image of P in
mirror 1 (P1’)
serves as “object”
for the image in
mirror 2 (P3’).
Therefore more
complex setups
can be treated.
Note: Later these
concepts will be
used for
successive curved
refraction in a
lens.
Reflection at a spherical surface
’
vertex
Setup:
Radius of curvature R at the
center of curvature C.
Concave side faces incident light.
P is an object point.
For now assume PV > R.
Point V is called the vertex
of the mirror.
The line PVC is called the
optic axis.
The ray PB is reflected at the mirror with an angle θ and passes the
point P’.
Reflection at a spherical surface contd.
One can show that all rays
emitted from P and reflected
in the mirror pass point P’.
This means P’ is a real image
point.
’
vertex
Using plane geometry one can
show that
α + β = 2φ
Reflection at a spherical surface contd.
Sign rule for the radius of curvature: When the center of curvature C is
on the same side as the outgoing (reflected) light, the radius of
curvature R is positive; otherwise negative.
For a convex [concave] surface R is negative [positive].
R>0
R<0
Calculation of image distance s’ (spherical)
tan β =
tan φ =
s!
h
−δ
h
R−δ
vertex
Goal: ! compute the image
! ! distance s’.
Using trigonometry one can
show that:
h
tan α =
s−δ
Because these equations cannot be solved, we can use a smallangle approximation for tan(x) and neglect the distance δ :
1
h
h
h
tan x = x + x3 + . . .
α=
β= !
φ=
3
s
s
R
Calculation of image distance s’ contd.
Using α + β = 2φ we obtain an
expression for the image distance.
Image distance for a spherical mirror:
1
1
2
+ ! =
s s
R
Note:
This is an approximation valid only if
the angle α is small (paraxial approximation).
As the angle α increases, the point P’ moves closer to the
spherical mirror and there is no precise point image any longer.
This is called spherical aberration.
If R = ! (plane mirror) we obtain s = –s’ as before.
Focal point (spherical mirror)
When P is very far from the mirror (s = !) the incoming rays can be
treated as parallel. It follows:
R
s! =
2
The distance FV is called focal length:
f=
R
2
It follows:
1
1
1
+ ! =
s s
f
Note: the focal length will be of
importance when we study optical
instruments.
V
Focal point (spherical mirror) contd.
When the source of the rays in a
spherical mirror lies at the focal point,
they are reflected parallel to the
optical axis.
This construction is generally used
for flashlights.
Focal point of a concave spherical mirror:
1. ! Any incoming ray parallel to the optical axis is reflected trough the
! focal point.
2.! Any incoming ray that passes trough the focal point is reflected
! parallel to the optical axis.
For spherical mirrors this is only true for paraxial rays, for parabolic
mirrors this is exactly true.
Lateral magnification of a spherical mirror
What happens when we have an object of finite size PQ = y (arrow)?
One can show for the lateral magnification:
s!
m=−
s
A negative sign means the image is inverted w.r.t. the object.
Convex mirrors
Now the incident light faces a convex spherical mirror.
Because the center of the curvature is on the other side of the
incoming rays, R < 0.
One can show that all rays along PB are reflected and, when
projected backward intersect the optical axis at P’, provided that the
angle α is small.
P’ is thus the image
point of P.
Note: s > 0, s’ < 0.
One can show:
1
1
1
+ ! =
s s
f
Determination of the size of an image
Perform a similar construction as before:
As with a concave spherical mirror we obtain
s!
s
The derived relations are valid for both convex and concave
spherical mirrors!
m=−
Convex mirrors contd.
When R < 0, incoming parallel rays
are reflected as if they would come
from the focal point F behind at a
distance f the mirror.
In this case F is called a virtual focal
point.
Applications of spherical mirrors:
Wide-angle view
mirrors at
intersections.
Solar reactors.
…
Graphical methods for mirrors
Idea:
Graphical methods to determine the size an position of reflected
images in mirrors by drawing a selection of principal rays.
Strategy:
Identify known quantities such as s, s’, R and f and determine the
unknowns.
Pay attention to the signs of the objects!
Use a ruler and draw the lines carefully.
If the lines do not converge, extend the optical axis further.
If necessary, color code the rays.
Graphical methods for mirrors (concave)
Graphical methods for mirrors (convex)
Refraction at a spherical surface
Goal: ! Study refraction at a spherical interface between two optical
! ! materials with different indices of refraction na and nb.
Setup:
P is the object point, P’ the image.
We use the same rules as for the spherical mirror.
Refraction at a spherical surface contd.
When the angle α is small, we can do a similar calculation than
before.
Use the law of refraction na sin θa = nb sin θb and the paraxial
approximation to obtain na θa = nb θb .
Result:
na
nb
nb − na
+ ! =
s
s
R
Note:
Since the equation does not contain the angle α it is valid for all
paraxial rays from P.
The derived equations can be applied to both convex and concave
interfaces. Use the sign rules consistently!
Magnification at a spherical surface contd.
Study the triangles PQV and P’Q’V. It follows:
y
−y !
tan θa =
tan θb = !
use also na sin θa = nb sin θb
s
s
and the approximation that for small angles tan(x) ≈ sin(x) .
We obtain na y/s = −nb y ! /s!
y!
na s!
m=
=−
y
nb s
Special case: plane surface
For a plane surface R = !. In this case
na
nb
m=1
+ ! =0
s
s
It follows that
s! = −
nb
s
na
If nb > na then objects are closer than
they appear.
Example: ! 2m deep pool. How deep does it
! ! ! seem?
s! = −
nb
1.00
s=−
2m = −1.5m
na
1.33
Thin lenses
Definition: ! A thin lens is made from two spherical surfaces so close
! ! ! ! that their distance is negligible.
optic axis
second focal point
Characteristics:
The center of the lens defines the
optic axis.
When parallel beams pass the lens
they converge at the second focal
point F2.
Similarly, one can invert the picture:
focal length
Rays coming from F2 (F1) exit
parallel to the optic axis.
F1 and F2 are called the first and second focal point, respectively.
Regardless of the lens curvature, the focal length f is the same on
both sides.
Determining an image trough a thin lens
Let s and s’ be the object and image distances, respectively, and y and
y’ their heights. Study rays QOQ’ and QAQ’.
PQO and PQ’O’ are
similar triangles, it
follows:
y!
s!
=−
y
s
equate
The negative sign is because the image is below the optic axis.
The triangles OAF2 and P’Q’F2 are similar, it follows: y !
s! − f
=−
y
f
Determining an image contd.
We obtain for a thin lens:
1
1
1
+ ! =
s s
f
s!
m=−
s
Note:
The negative sign tells us that when both s and s’ are positive, the
image is inverted and y and y’ have opposite signs.
The equations are the same as for spherical mirrors. So are the
sign rules.
Unlike mirrors, the image is not
reversed, but inverted: a left
hand is a left hand, but the
fingers point in opposite
directions.
Converging vs diverging lenses
Converging lens:
Studied so far.
Diverging lens:
The focal length is negative. The focal points are virtual and
reversed w.r.t. the focal points of a converging lens.
The same equations as before apply (!)
General definitions
Converging lens: !a lens that is thicker at the center than at the
! ! ! ! ! edges (f > 0).
Samples:
meniscus
planoconvex
double convex
Diverging lens:! a lens that is thinner at the center than at the
! ! ! ! ! edges (f < 0).
Samples:
meniscus
planoconcave
double concave
Thin-lens equation
Goal: ! derive a relationship between the focal length f, the index of
! ! refraction n and the radii of curvature R1 and R2 of an
! ! arbitrary lens.
Setup:
Study the general problem of two spherical interfaces separating
three materials with indices of refraction na, nb, nc.
Assume the distance t between the interfaces is small enough that
it can be neglected.
The idea is to daisy-chain the different interfaces and compute
relations between the distances.
Thin-lens equation contd.
The object and image distances are shown above.
Note that for example s2 = –s’1 depending from which surface it is
seen from.
Use the single-surface equation twice to obtain:
na
nb
nb − na
+ ! =
s1
s1
R1
nb
nc
nc − nb
+ ! =
s2
s2
R2
Thin-lens equation contd.
In general, we study a lens in air/vacuum, i.e., na = nc = 1 and nb = n.
We obtain by using s2 = –s’1:
1
n
n−1
n
1
1−n
+ ! =
− ! + ! =
s1
s1
R1
s1
s2
R2
Add both equations:
!
"
1
1
1
1
+ ! = (n − 1)
−
s1
s2
R1
R2
Since we treat the lens as a single unit, we set s1 = s and s’2 = s…
Lensmaker equation (thin-lens equation):
1
1
1
+ ! = = (n − 1)
s s
f
!
1
1
−
R1
R2
"
Thin lens equation sign conventions
Previous sign conventions still work.
Example:
s, s’ and R1 are positive, R2 is negative.
Graphical methods for lenses
Converging lens:
Graphical methods for lenses contd.
Diverging lens:
Graphical methods for lenses contd.