FT2224 Applied Optics

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FT222/4 Applied Optics

• Module 2
• Exact ray trace.
• Aberrations, analysis and optimisation

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(A)spheric lenses
Ideal lens has elliptical profile Hyperbolic lens
- elliptical surface.
Most optics are Spherical.
Why?
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Paraxial Region
y
For a circle shifted R from the origin y2
(x-R) 2 R2 Solving for x x R2 /- (R2 -y2
)1/2 Using binomial expansion and taking the -
term Check
R
x
f
But the equation for a parabola is y2 4fx
In the paraxial region circle and parabola are
the same
R 2f
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Aberrations
Chromatic Aberrations Refractive index n
f(l) ? Focal length varies with wavelength
(Cauchy or Sellmeier formula)
n
l
Achromat. Lens combination showing no chromatic
distortion for two designated wavelengths
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Monochromatic Aberrations In paraxial region
Sinq ? q More correctly
Sinq ? q is referred to as the first-order
approximation.
If we include second term - third-order
approximation.
Primary (or Seidal) aberrations are the
difference between paraxial, first-order
approximations and more correct third-order
approximations. There are 5 .
Aside An exact ray trace will use higher-order
terms.
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Spherical Aberration SA
Dependence of focal length on distance of ray
from axis.
T.SA
SLc
Paraxial
Paraxial focus F
L.SA
Longitudinal SA Distance between the axial
intersection of a ray and the paraxial focus
Transverse (or Lateral) SA Distance above the
axis that a ray crosses a plane at the paraxial
focus
Circle of Least Confusion SLC Position at which
image will have the least blur
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Coma
Coma is a varying lateral angular
magnification. For an object point off-axis its
image, as formed by different zones from the
axis, are displaced from the axis by different
amounts.
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Astigmatism
Variation in the sagittal and meridional
focus. For any off-axis object the beam becomes
elliptical and there are separate sagittal and
meridional line foci with an intermediate circle
of least confusion.
Circle of Least Confusion
FS
FT
Note NOT due to any circular asymmetry
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Petzval Field Curvature
In a lens having none of the previous
aberrations, the image of a plane object lie on a
curved image plane.
In the presence of astigmatism tangential and
sagittal curvature will differ.
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Distortion
When all other aberrations are removed, there can
be a varying radial magnification.
Pincushion
Barrel
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Seidel Terms (aside)
An object point emits spherical wavefronts. The
image wavefront is perturbed from spherical
aberration. Wavefront aberration W f (aperture
co-ords, x,y field co-ords, x,h).
Rotationally symmetric systems, only
need x2y2, xx yh, x2 h2 Rotationally
symmetric gt consider one axis gt let x 0 The
polynomial is therefore
The two 1st degree terms, a1 and a2, are defocus
lateral shift. The five 2nd degree terms are
the Seidel (Primary) Aberrations in
order Spherical Aberration, Coma, Astigmatism,
Petzval Curvature, Distortion
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Analysing Aberrations Image Analysis
Primary Aberrations

• Chromatic aberration

• Field aberrations

• Longitudinal and transverse ray aberrations

Image evaluation based on detailed analysis of
rays

• Spot diagrams

• Modulation transfer function MTF

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Chromatic aberration
Plot variation of focal length with wavelength
Biconvex
Achromat
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Field Aberrations Graph
This graphs shows astigmatism, distortion and
lateral colour with field size. Vertical axis
- field size. Horizontal axis -
aberration. The data is obtained by tracing 10
chief rays at equal angle increments up to the
specified field size. Astigmatism plot shows
the Tangential (solid) and Sagittal fields
(dotted). Distortion is the difference between
the real and the paraxial image height.
Lateral Colour is the difference in image
height for the chief rays traced at the short
and long wavelengths.
Biconvex lens
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Longitudinal and Transverse Ray Aberrations
Lateral and longitudinal displacement of a ray
with respect to the Chief Ray of that ray fan.
Can be resolved into orthogonal components.
Chief ray
Long
TRA
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Spot diagrams
0
Spot diagrams plot the transverse ray aberrations
of concentric rings of rays, for on and off axis
file positions simultaneously. Each horizontal
set of 5 boxes plots the ray aberrations at
different defocuses. The central box in a row
will represent the paraxial image surface, with
the left hand boxes showing defocus toward the
lens.
Off-axis
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Modulation transfer function MTF
Variation in contrast with detail size at a given
position.

Low
High
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Optimisation
First order Layout a first pass Given design
constraints e.g. magnification, pupil, etc,
Minimise S l component powers l or Minimise S
l component powers.diameters l Avoiding very
low f/ - divide fat lenses This approach
should help to minimise aberration
5 variables 3 component powers and two spaces
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Lens Bending
Recall 1/f n2 - n1 (1/ R1 - 1/ R2) There
are an infinite number of refractive index and
radii combinations with same focal length. Can
be adjusted to reduce Spherical Aberration and
Coma for a given object position.
Exercise investigate single lenses with
Winlens Note effect of changing object distance
Handbook of Optics, Vol II, Chap 1 (OSA)
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Moving the Aperture
Coma, Astigmatism, and Distortion are described
with respect to the Chief Ray. Chief Ray defined
by position of Aperture Aberrations aperture
stop position and diameter
E.g. A thin lens system will have no distortion
if the aperture is at the optical centre
Exercise Investigate for a balanced pair of
plano-convex lenses - monitor field aberrations.
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Chromatic Aberration
Simple Solution - Use a metal mirror Almost no
dispersion and little variation in
reflectivity. Especially useful in UV and IR
regions
Achromat Combine two lenses of differing
refractive index to cancel out shift in focal
length for two wavelengths
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Achromats and Abbe numbers
Hecht Optics, Chap 6
In air 1/f n2 - n1 (1/ R1 - 1/ R2) (ni - 1)
ri
Two thin lenses touching 1/f 1/f1 1/f2
Consider two wavelengths, Red R, and Blue B For
an achromat we want 1/fR 1/fB
Taking an intermediate wavelength Yellow Y,
Blue lt Yellow lt Red
Where V1, the Abbe number, indicates the
dispersion of the material
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Achromat contd
Manufacturers usually quote a standard Abbe Number
Where d 5875 nm F 4861 nm C 6562 nm
So to design an achromat at wavelength d
Giving
and
Thus to avoid short f, and thus strongly curved
surfaces (V1d - V2d) should be maximised,
typically gt20.
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Cost
Factors affecting cost
Diameter
Material
Number of components
Stock v custom
Number of moving parts
Position sensitivity a manufacture
Number of copies
http//www.thorlabs.com/
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Thorlabs UV Fused Silica PLANO-CONVEX Lens (Price
2000)
Diam fmm Price e
f75
f50.8
f25.4
f6 12
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Merit function
Automated design is based on varying the system
variables, e.g. lens curvatures, aperture, lens
thickness, refractive indices, etc.
Some variables may be restricted, e.g. 1.54 lt n lt
1.62. These are known as boundary conditions.
Automated design will seek to optimise a single
number known as a Merit Function (or error
function). This is a sum of squares of quantities
called operands that characterise the system such
as aberration coefficients, exact ray
displacements, etc. that must be minimised.
This is done given certain constraints, which
must be satisfied exactly, e.g., paraxial focus,
or exit pupil.
There is usually no exact solution - task is to
choose a good solution