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Announcements
-^
Exam 1 is one week from tonight:–
Thursday, 4/17– Open book, notes– 2 hours allowed (should need only 1 hour)– Paper provided– Covers Ch. 1, 2, Sections 3.1-3.2, Ch. 18 and Sections 20.1-
(Elementary optics, geometric optics (ray tracing), spherical aberrations)–
Papers will be returned ~ 1 week later
-^
I will be away 4/16-23 (and out of email contact!)–
See me before class Tuesday 4/15 if you have questions about exam– Class will be held as usual Tuesday 4/22 (substitute TBD)
-^
Lab session tonight: Last names M-Z, in B260 PAB–
Cardinal points, lens defects, laser speckle
Exact (non-paraxial) matricesWant to trace any ray in
meridional (z-y) plane:
Drop tan
θ ∼ θ
assumption (but keep axial symmetry assumption)
:
C = Center of curvature , R = radius^ α, α
’ = Slopes of incident and refracted rays
(relative to optic axis)
θ,^
θ’ = Angles of incidence and refraction
(relative to normal at P)
’ sin
’ = n 1
sin 1
y 1
1
where (see
Nussbaum and Phillips for derivation)
K^1
= (n
’ cos 1
’ - n 1
cos 1
) / R 1
1
c.f.
paraxial case (cos
k
= (n 1
’ - n 1
) / R 1
1
Now refracting power depends upon angle of ray
’ using Snell’s Law: n 1
’ sin 1
’ = n 1
sin 1
1
Horizontal =parallel to axis
α^1
n^1
n^1
R^1 y = y 1
θ^1
α^1
θ^1
’ C (
)^
2
2
2
2
2
2
2
1
1
1
1
1
1
2
2
1
1
1
1
1
1
1
1
1
1
'^
sin
sin
' cos
'^
sin
'^
'^
/^
'^
sin
n^
n^
n^
n
n^
n^
n^
n^
n^
K^
R
−^
−^
θ^
=^
−^
θ^
=^
−^
θ^
Translation in the exact case•
Curvature of surfaces is significant nowCan’t take translation distance T=OV as when cos
α
T = PW or PD for positive or negative front curvature
Note that WB=BD (
⊥^
radius BC bisects chord), so
PW = PB - WBPD = PB + WB,
where
(Note: same factor appears in K
C
R^1
θ^1
Q
B
y
O
S
P
D
positivecurvaturecase
negativecurvaturecase
W V
V’
R^1
OC=C
1
(^
(^2) )
1
(^21)
2
(^21)
sin
sin cos
cos
C
y
R
WB
OC
y
BC
QC
BQ
BC
BC
R
WB
So
2
1
(^21) 1
1
sin
cos (
)
sgn(
sin
cos
α^
C y R R y C
T^
α
= α
α
=
sin ' sin
sin
'^
1 1
T y y^
⎞⎟⎟ ⎠
⎛⎜⎜ ⎝ =^
1 1 0
T
T^1
and PB = OQ - BS = C
cos 1
α
α
Thus
translation:
So
Exact ray tracing •
C^1
= dist from object plane to center of curvature =
s + R
for front surface, = t+R 1
for back surface 2
-^
Cannot just multiply matrices together:–
elements depend on ray parameters:– Must proceed iteratively
for each ray:
•^
Translate s to front surface (T
OBJ
•^
Refract (R
•^
Translate t to back surface (T
•^
Refract (R
) to get output ray 2
•^
...repeat for next ray
R^1
R^2
t
s
=^
T
T
2
1
(^21) 1
1
sin
cos (
)
sgn(
sin
cos
α^
C y R R y C
T^
1
1
1
sin
cos
sin
R
C
y^
sin
y
2
2
2
2
1
1
1
1
1
1
1
'^
sin
sin
n^
n^
n
K^
R
−^
θ^
−^
−^
(^
)^
GLASS^ AIR
AIR
n n
n
K
2
R
(^
)^
AIR GLASS
GLASS
n n
n
K
1
1
R
FYI: Software I’ve prepared See class website, …/labs/progs/545programs.html (also gives links to free ray-tracing software available online) 1. Lenscalc.xls
:^ Calculate single-lens object-image distances (for PC only)
Calculates any one of
l, l' and
f', given the other two. A ray-tracing diagram is drawn.
2. Raytrace1.xls: Calculate thin-lens, paraxial system matrices and cardinal points
(PCs C&C computers) Calculates ray transport matrices for a thin lens (PC spreadsheet version: up to 2 lenses; linux
version, up to 20). When the system is fully specified, cardinal points are calculated from thecomponents of the system matrix. A ray-tracing diagram is drawn.
3. Lensmatrix: Calculate general paraxial system matrices and cardinal points
(Physics PCs and C&C computers) This program calculates ray transport matrices for a generalized thick lens system; i.e., a series
of media of arbitrary thickness, with the radius of curvature specified at each interface(spreadsheet verion: up to 3 interfaces). When the system is fully specified, cardinal pointsare calculated from the components of the system matrix. A ray-tracing diagram is drawn.
4. Raytrace2: Thick lens, meridional ray tracing (C&C only) This program performs ray tracing through a generalized thick lens system; i.e., a series of
media of arbitrary thickness, with the radius of curvature specified at each interface.(Maximum 12 interfaces). Versions are available both on C&C computers and on the AM018PCs. See class website, …/labs/progs/raytrace2g77readme.txt
Aberrations
•^
Aberrations = deviations from ideal lens behavior–
focal point shifted depending upon ray parameters
•^
a = wave aberration–
“can be shown” that
a ~ y
4
•^
b^ x
= longitudinal aberration
-^
b^ y
= transverse aberration
Opticalsystem
planewaves(and rays)
a
b^ x
b^ y
ideal (spherical wave)
y real
x^
z
y θ^ r O=optic axis
P
O
These are the five
3rd order or
Seidel aberrations,
relaxing the sin
θ^
~^
θ^ assumption by one more term:
Ray through point P on lens may have:a ~ r
4
spherical aberration
(^3) r cos
θ
coma (note asymmetry)
(^2) r cos
θ
astigmatism
(^2) r
curvature of field
r cos
θ
distortion
... ! 5 ! 3
sin
5 3
−
−
θ θ θ θ
paraxial
3rd order
Spherical aberration•
Envelope of ray paths for
α
= 0, all x, is the
Caustic Surface
–^
Neck of caustic =
Circle of Least Confusion
-^
= position at which image of distant pinhole is smallest
•^
LSA depends upon y (how marginal ray is)
-^
For a given f’ (PF location) LSA depends on
shape of lens:
–^
Define shape (Coddington factor)
caustic surface
PF
MF
x
| LSA |
y
1 2
1 2
R
R
R
R
S^
Plano-Convex
R^1
= ∞
R^2
= - r
S = -1.
Symmetrical Double-Convex
R^1
= +r
R^2
= - r
S = 0
Convex-Plano
R^1
= +r
R^2
= ∞
S = +1.
S= - 1 S= 0 S= +
Meniscus lenses can have S be + or –dependingupon which way they are oriented
Spherical aberration: LSA
- Example discussed briefly in text (sect. 20-3): thin lens, n=1.5, f’=10 cm
- Can make this lens with any combination of surfaces satisfying– Different combinations of R’s give different shape factor S– LSA depends upon S– “Can be shown” that minimum LSA occurs for
⎛^ ⎜⎜⎝
2 1
R
R
n
f
s s
s s
n n
S^
2
1.2^1 0.80.60.40.2^0 -
-^
0
1
2
shape S
LSA
R^1
R^2
S
-3.
33
-5.
-0.
10
0
5
1
10
2
Minimum LSA(nearly plano-convex)
