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Chapter 5
Geometrical Optics
Phys 322Lecture 12
Geometrical optics (ray optics) is the simplest version of optics.
Rayoptics
Ray Optics
We'll define light rays as directions in space, corresponding, roughly, to k-vectors of light waves.
We won’t worry about the phase.
Each optical system will have an axis, and all light rays will be assumed to propagate at small angles to it. This is called the Paraxial Approximation.
axis
RaysRays are orthogonal trajectories of the wavefronts.
Normal congruence: group of rays for which you can find a surface that is orthogonal to all of them.
Theorem of Malus and Dupin:A group of rays will preserve its normal congruence after any
number of reflections and refractions.
Is geometrical optics the whole story?
No. We neglect the phase.
Also, our ray pictures seem to imply that, if we could just remove all aberrations, we could focus a beam to a point and obtain infinitely good spatial resolution.
Not true. The smallest possible focal spot is the wavelength, . Same for the best spatial resolution of an image. This is due to diffraction, which has not been included in geometrical optics.
>
~0
Focus
Each point on an illuminated or a self-illuminating surface is a source of spherical waves:Rays diverge from that point
Spherical wave can also converge to a point
A point from (to) which a portion of spherical wave diverges (converges) is a focus of the bundle of rays
Stigmatic optical system - perfect image
Reversibility: SPP and S are conjugate points
object space image space
Lens- refractive device that changes the wavefront curvature
The wavefront changed from convex to concave
OPL should be the same for red and blue rays
Qualitatively:Insert a transparent object with n>1 that is thicker in center and thinner at the edges
Aspherical surfacesThe shape of the interface:
constant1 ADnAFn ti
Time to ravel from S to DD’:
ti
ADAFvv
1
Time of travel from S to DD’ must be the same for any point in plane DD’:
or: cADnAFn ti
c1
constant1 ADnnAF
i
t
nti nt/ni > 1 - hyperbola nti nt/ni < 1 - ellipsoid
rays can be reversed
Change spherical wave to plane wave
Aspherical surfaces
Convex lensesConverging lenses
Concave lensDiverging lens
F: Focal points
When a bundle of parallel rays passes through a lens, the point to which they converge (converging lens) or the point from which they appear to diverge (diverging lens) is called focal point
real image
virtual image
object distance
image distance
Vertex
Opticalaxis
Spherical lensUse Fermat’s Principle
OPL=n1lo+ n2li
Law of cosines for SAC and ACP:
2/1222 cos2 RsRRsRl ooo
2/1222 cos2 RsRRsRl iii cos180cos
02
sin2
sin 21
i
i
o
o
lRsRn
lRsRn
dOPLd
0
122
0
1 1lsn
lsn
Rln
ln o
i
i
iFor different P will be different
object distance
image distance
Vertex
Opticalaxis
Spherical lens
0
122
0
1 1lsn
lsn
Rln
ln o
i
i
i
Approximate: for small :
ii
o
slsl
0
sin1cos
Rnn
sn
sn
i
122
0
1
The position of P is independent of the location of A over small area close to optical axis.Paraxial rays: rays that form small angles with respect to optical axisParaxial approximation: consider paraxial rays only
Spherical lens: focal length
Rnn
sn
sn
io
1221
Focal point F0 : si =
Rnn
fn
o
121 0
Rnn
nfo12
1
First focal length:
(object focal length)
Reverse:
Rnn
nfi12
2
Second focal length:
(image focal length)
object(or first)focus
image(or second)focus
R > 0, n2 > n1 f > 0 - converging lens
Spherical lens: focal lengthWhat if R is negative?
Rnn
nfo12
1
an image is virtual:it appears on object side
Rnn
nfi12
1
an object is virtual:it appears to be in the image side
object distance
image distance
Vertex
Opticalaxis
Sign convention
so, fo + left of Vsi, fi + right of VR + if C is right of V
Assume light entering from left:
Rnn
sn
sn
io
1221
Lens classification
Thicker inthe middle
Thinner inthe middle
R1<0 R2<0R1<0 R2=
R1>0 R2<0
(negative)(positive)
Thin lens equationR
nnsn
sn
i
122
0
1
For the first surface:
111 Rnn
sn
sn ml
i
l
o
m
221 Rnn
sn
dsn lm
i
m
i
l
Second surface:
Add two eq-ns and simplify using nm=1 (air) and d0:
Thin-lens equation(Lensmaker’s formula)
21
11111RR
nss l
io
Gaussian lens formula
21
11111RR
nss l
io
Find focal lengths (so, or si)fo = fi f
21
1111RR
nf l
Gaussian lens formula:
fss io
111
This is one of the most widely used equations.All one needs to know about the lens is its focal length.
s0 si
f fxo xi
Newtonian form: 2fxx io
Example
Plano-convex spherical lens
R = 50 mmn = 1.5
What is a focal length of this lens?
Solution
mm 100/1mm 50
1115.11111
21
RRn
f l
mm 100f
Example
Object is placed at 600 mm, 200 mm, 150 mm, 100 mm, 50 mm.Where would be the image?
Solutionfss io
111 so si
600 120200 200150 300100 80 -400
f = 100 mm
fsfss
o
oi
Focal plane
All bundles of parallel rays converge to focal points that lay on one plane: second, or back focal plane
Thin lens + paraxial approximation:All rays that go through center O do not bend
second, or backfocal plane
Fo- lies on first, or front focal plane.
Imaging with a lens
Each point in object plane is a point source of spherical waves and the lens will image them to respective points in the image plane.
Converging lens: principal rays
1) Rays parallel to principal axis pass through focal point Fi.2) Rays through center of lens are not refracted.3) Rays through Fo emerge parallel to principal axis.
Fo
Fi
Object
ImageOptical axis
In this case image is real, inverted and enlargedAssumptions:
• monochromatic light
• thin lens.• rays are all “near” the principal axis
(paraxial).
Since n is function of , in reality each color has different focal point: chromatic aberration. Contrast to mirrors: angle of incidence/reflection not a function of
Principal rays:
Diverging lens: forming image
1) Rays parallel to principal appear to come from focal point Fi.2) Rays through center of lens are not refracted.3) Rays toward Fo emerge parallel to principal axis.
Fi
Fo
Object Image
O.A.
Image is virtual, upright and reduced.
Principal rays:
Assumptions:• paraxial monochromatic rays• thin lens