Upload
140557
View
243
Download
0
Embed Size (px)
Citation preview
7/27/2019 Opti GeometricalgeoOptics I
1/31
Geometrical Optics
Geometrical light rays
Ray matrices and ray vectors
Matrices for various optical
components
The Lens Makers Formula
Imaging and the Lens Law
Mapping angle to position
Prof. Rick Trebino, Georgia Tech
www.frog.gatech.edu
7/27/2019 Opti GeometricalgeoOptics I
2/31
Geometrical optics (ray optics) is the
simplest version of optics.
Ray
optics
Its the roughestapproximation of
optical reality.
Itll handle
reflection andrefraction, but
not interference
or diffraction.
However, when it
does work, it will
handle complex
problems much
more easily than,
say, wave optics.
7/27/2019 Opti GeometricalgeoOptics I
3/31
How will geometrical optics go wrong?
It neglects the phase.
As a result, the ray pictureimplies that a lens can focus a
beam to a point with zero
diameterand so obtain infinite
intensity and infinitely good
spatial resolution.
Not true. The smallest possible
focal spot is actually about the
wavelength, l. Same for the
best spatial resolution of an
image. This is fundamentallydue to the wave nature of
light, which is not included in
geometrical optics.
But it will be easy to use.
> ~l
~0
Ray optics
Reality
7/27/2019 Opti GeometricalgeoOptics I
4/31
Ray Optics
We'll define light rays as directions in space, corresponding,
roughly, to k-vectors of light waves.
Each optical system will have an optic axis, and all light rays
will be assumed to propagate at small angles to it. This is called
the Paraxial Approximation.
Optic axis
Light ray
Optical system
7/27/2019 Opti GeometricalgeoOptics I
5/31
The Optic Axis
A mirror deflects the optic axis into a new direction with theangle of reflection equal to the angle of incidence.
This ring laser has an optic axis that scans out a rectangle.
Optic axis A ray propagatingthrough this system
We define all rays relative to the relevant optic axis.
7/27/2019 Opti GeometricalgeoOptics I
6/31
7/27/2019 Opti GeometricalgeoOptics I
7/31
Ray
Vectors
At every position,z, along the optic axis, a light ray can be definedby two co-ordinates:
its position,x
its slope, q
Optic axis
x
q
These parameters define aray vector,
which will change with distance,z, as
the ray propagates through optics.
x
q
zz0
xin, qin
xout, qout
Optic
axisz0
0
7/27/2019 Opti GeometricalgeoOptics I
8/31
Ray Matrices
An optical elements effect on a ray is found by multiplying the rayvector by the elements ray matrix.
Ray vector
before lens
after lens before lens
after lens before lens
x xA B
C Dq q
Lens 2 x 2 ray matrix
For many optical components,we can define 2 x 2 ray matrices.
Distance 2 x 2 ray matrix
Ray matrix
for lens
Ray vector
after lens
xin, qin
xout, qout
We can do thesame for theother lenses andthe distances.
7/27/2019 Opti GeometricalgeoOptics I
9/31
Ray Matrices
as Derivatives
We can write
these equationsin matrix form.
out in
out inD
B x
C
x A
q q
out
in
q
q
out
in
x
x
out
inx
q
out
in
x
q
Angular
magnification
Spatial
magnification
out i
outout
i n
i
n i
n nx x
xx
xq
q
out in i
outut
nin i
o
nx
xq
q
q
Since the displacements,xin andxout,
and angles, qin and qout, are all
assumed to be small, we can think in
terms of partial derivatives.
7/27/2019 Opti GeometricalgeoOptics I
10/31
For cascaded elements, we simply multiply
together all the individual ray matrices.
3 2 1 3 2 1
out in in
out in in
x x xO O O O O O
q q q
Notice that the order looks opposite to what it should be,
but it makes sense when you think about it.
O1 O3O2in
in
x
q
out
out
x
q
Component #1 Component #2 Component #3
7/27/2019 Opti GeometricalgeoOptics I
11/31
Ray Matrix for Free Space or a Medium
Ifxin and qin are the position and slope upon entering, letxoutand qout
be the position and slope after propagating an arbitrary distance,z.
out in in
out in
x x zq
q q
xin, qin
0
xout, qout
z
10 1
out in
out in
x xz
q q
Rewriting these expressions
in matrix notation:
1=
0 1space
zO
7/27/2019 Opti GeometricalgeoOptics I
12/31
Ray Matrix for
an Interface
At the interface:
qout= [n1 / n2]qin
qin
n1
qout
n2
xin xout
1 2
1 0
0 /interfaceO
n n
which, for small angles, becomes: n1q
in = n
2q
out
Snell's Law says: n1sin(qin) = n2sin(qout)
Now calculate qout:
xout=xin
7/27/2019 Opti GeometricalgeoOptics I
13/31
7/27/2019 Opti GeometricalgeoOptics I
14/31
A thin lens is just two curved interfaces.
1 2 1 2
1 0
( / 1) / /curvedinterface
On n R n n
Well neglect the glass in between (its a
really thin lens!), and well take n1 = 1.
2 1 2 1
1 0 1 0( 1) / [(1/ ) 1] / 1/
thin lens curved curved interface interface
O O On R n n R n
2 1 2 1
1 0 1 0
( 1) / [(1/ ) 1] / (1/ ) ( 1) / (1 ) / 1n R n n R n n n R n R
2 1
1 0
( 1)(1/ 1/ ) 1n R R
1 0
1/ 1f
This can be written:
1 21/ ( 1)(1/ 1/ )f n R R The Lens-Makers Formulawhere:
n1= 1
R1 R2
n2 = n 1
n1= 1
7/27/2019 Opti GeometricalgeoOptics I
15/31
Ray Matrix for a Lens
The quantity,f, is the focal lengthof the lens. Its the single most
important parameter of a lens. It can be positive or negative.
1 0=
-1/ 1lensO
f
Iff > 0, the lens deflects
rays toward the axis.
f > 0
Iff < 0, the lens deflects
rays away from the axis.
1 21/ ( 1)(1/ 1/ )f n R R
R1 > 0
R2 < 0f < 0
R1 < 0
R2 > 0
Its easy to extend the Lens Makers Formula to real lenses of
greater thickness.
Sign convention:R > 0 if the sphere
center is to the
right (z> 0), and
R < 0 if the sphere
center is to the
left (z< 0).
7/27/2019 Opti GeometricalgeoOptics I
16/31
Types of Lenses
Lens nomenclature
Which type of lens to use (and how to orient it) depends on the
aberrations and application.
7/27/2019 Opti GeometricalgeoOptics I
17/31
A lens focuses parallel rays to a point
one focal length away.
01 1 0 0
/0 1 1/ 1 0 1/ 1 0
out in in
out in
x f x f x
x ff fq
f
f
At the focal plane, all rays
converge to thezaxis (xout
= 0)
independent of input position.
Parallel rays at a different angle
focus at a differentxout.
A lens followed by propagation by one focal length:
Assume all input
rays have qin = 0
For all rays,
xout
= 0!
Looking from right to left, rays diverging from a point are made parallel.
7/27/2019 Opti GeometricalgeoOptics I
18/31
f
f
f
f
And it maps input position
to angle:
Lenses can simultan-
eously map angle to
position and position
to angle.
From input to output, use:
1) A distance f
2) A lens of focal lengthf3) Another distancef
1 1 0 1
0 1 1/ 1 0 1
1 1
0 1 1/ 1
0
/1/ 0
out in
out in
in
in
in in
in in
x xf f
f
xf f
f
x ff
x ff
q q
q
q
q
out inx q
out inxq
f
So this arrangement maps
input angle to position:
independent of
input position
independent of
input angle
7/27/2019 Opti GeometricalgeoOptics I
19/31
Spectrometers
f
f
Entrance
slit
Diffraction
grating
ff
Camera
To best distinguish different wave-
lengths, a slit confines the beam to
the optic axis. A lens collimates the
beam, and a diffraction gratingdisperses the colors. A second
lens focuses the beam to a
point that depends on its
beam input angle (i.e.,
the wavelength).
q ll0 There aremany
types ofspectrom-
eters. But
most are
based on
this principle.
7/27/2019 Opti GeometricalgeoOptics I
20/31
Lenses and Phase Delay
Equal phasedelays
Focus
ff
Ordinarily phase isnt considered in geometrical optics, but its
worth computing the phase delay vs.x andy for a lens.
It turns out that all paths through a lens to its focus have the same
phase delay, and hence yield constructive interference there!
7/27/2019 Opti GeometricalgeoOptics I
21/31
Lenses and
Phase Delay
( , ) ( 1) ( , )lens x y n k x y
2 2 2
1( , ) ( 1) ( )lens x y n k R x y d
neglecting phase delays
independent ofx andy.2 2
1( , ) ( 1)( / 2 )( )lens x y n k R x y
2 22 2 2 2 2 2
1 1 1 1
1
1 ( ) /2
x yR x y R x y R R
R
2 2 2
1( , )x y R x y d
( , )x yd
First consider variation (the
x andy dependence) in the
path through the lens.
But:
Extra phase
delay due to
the glass
7/27/2019 Opti GeometricalgeoOptics I
22/31
7/27/2019 Opti GeometricalgeoOptics I
23/31
Ray Matrix for a Curved Mirror
Like a lens, a curved mirror will focus a beam. Its focal length isR/2.
Note that a flat mirror hasR = and hence an identity ray matrix.
1 ( )2 /
out s in s s
in inx R
q q q q q q q
Consider a mirror with radius of curvature,R, with its optic axis
perpendicular to the mirror:
qin
qout
xin = xout
R
z
q1
q1
qs
1 /in s s inx Rq q q q
1 0
= 2 / 1mirrorO R
7/27/2019 Opti GeometricalgeoOptics I
24/31
Two flat mirrors, the flat-flatlaser cavity, is difficult to align
and maintain aligned.
Two concave curved mirrors,
the usually stable laser cavity,
is generally easy to align and
maintain aligned.
Two convex mirrors, the
unstable laser cavity, is
impossible to align!
Mirror curvatures matter in lasers.Laser Cavities
7/27/2019 Opti GeometricalgeoOptics I
25/31
But an unstable cavity (orunstable resonator) can be useful!
In fact, it produces a large beam, useful for high-power lasers, which
must have large beams.
The mirror curvatures
determine the beam size,
which, for a stable resonator,
is small (100 mm to 1 mm).
An unstable resonator can
have a very large beam. But
the gain must be high. And
the beam has a hole in it.
Unstable Resonators
7/27/2019 Opti GeometricalgeoOptics I
26/31
Consecutive Lenses
2 1 1 2
1 0 1 0 1 0= =
1/ 1 1/ 1 1/ 1/ 1tot
O
f f f f
f1 f2
Consider two lenses right next to
each other (with no space inbetween).
1 21/ = 1/ +1/totf f f
So two consecutive lenses act as one whose focal length iscomputed by the resistive sum.
As a result, we define a measure ofinverse lens focal length, the
diopter:
1 diopter = 1 m-1
7/27/2019 Opti GeometricalgeoOptics I
27/31
A system images an object when B= 0.
WhenB = 0, all rays from a
point xin arrive at a point xout,
independent of angle.
xout= A xin WhenB = 0, A is the magnification.
0out in in
out in in in
x x A xAC x DC Dq q q
Lens
ImageObject
do di
f
7/27/2019 Opti GeometricalgeoOptics I
28/31
/
1/ 1/ 1/
0
o i o i
o i o i
B d d d d f
d d d d f
if
1 1 0 1
0 1 1/ 1 0 1
11
1/ 1 /0 1
1 / /
1/ 1 /
i o
oi
o
i o i o i
o
d dO
f
dd
f d f
d f d d d d f
f d f
The Lens Law
From the object to the
image, we have:
1) A distance do2) A lens of focal lengthf
3) A distance di
1 1 1
o id d f
This is the Lens Law.
Lens
ImageObject
do di
f
7/27/2019 Opti GeometricalgeoOptics I
29/31
Imaging
Magnification
1 1 1
o id d f
1 11 / 1i i
o i
A d f dd d
i
o
dM
d
If the imaging condition,
is satisfied, then:
1 / 0
1/ 1 /
i
o
d fO
f d f
1 11 / 1o o
o i
D d f dd d
1/o
i
dM
d
0
1/ 1/
MO
f M
So:
Lens
ImageObject
do di
f
Angular
magnification
7/27/2019 Opti GeometricalgeoOptics I
30/31
Negative-flenses have virtual images, and positive-flenses do
also if the object is less than one focal length away.
Object
f > 0
Virtual
image
Virtual
Images
f < 0
Virtual
image
Simply looking at a flat mirror yields a virtual image.
A virtual image occurs when the outgoing rays
from a point on the object never actually intersect
at a point but can be traced backwards to one.
Object infinitely
far away
7/27/2019 Opti GeometricalgeoOptics I
31/31
The F-number, f / #, of a lens is the ratio of its focal length and its
diameter.
f / # = f / d
f
f
d1 f
f
d2
f / # = 1 f / # = 2
Large f-number lenses collect more light but are harder to engineer
F-Number