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Aberrations
Aberrations
Aberrations are distortions that occur in images, usually due to imperfections in lenses, some unavoidable, some avoidable.
They include:
Chromatic aberration
Spherical aberration
Astigmatism
Coma
Curvature of field
distortion (Pincushion and Barrel )
Most aberrations can’t be modeled with ray matrices. Designers beat them with lenses of multiple elements, that is, several lenses in a row. Some zoom lenses can have as many as a dozen or more elements.
Aberrations
Aberrations - deviations from Gaussian optics.Chromatic aberrations - n depends on wavelengthMonochromatic aberrations - rays deviate from Gaussian optics
• spherical aberrations• coma• astigmatism• field curvature• distortion
Paraxial approximation: sin
Third order theory:!3
sin3
Departures from the first order theory observed in the third order leave to the following primary aberrations:
Taylor series: ...!7!5!3
sin753
“The five Seidel aberrations”
Philipp Ludwig Seidel(1821-1896)
Third order aberrations
• sin terms in Snell’s law can be expanded in power series
n sin = n’ sin ’
n ( - 3/3! + 5/5! + …) = n’ ( ’ - ’3/3! + ’5/5! +
…)
• Paraxial ray approximation: keep only terms (first order optics;
rays propagate nearly along optical axis)
• Third order aberrations: result from adding 3 terms
– Spherical aberration, coma, astigmatism, .....
Spherical Aberration
.
Spherical Aberration in Mirrors
For all rays to converge to a point a distance f away from a curved mirror requires a paraboloidal surface.
But this only works for rays with in = 0.
Spherical surface Paraboloidal surface
Hubble Space Telescope suffered from Spherical Aberration
• In a Cassegrain telescope, the hyperboloid of the primary mirror must match the specs of the secondary mirror. For HST they didn’t match.
Hubble telescope
COSTAR - corrective optics space telescope axial replacement module
Spherical Aberration in Lenses
So we use spherical surfaces, which work better for a wider range of input angles.
Nevertheless, off-axis rays see a different focal length, so lenses have spherical aberration, too.
Spherical aberrations
Paraxial approximation (First order):
R
nn
s
n
s
n
io
1221
Third order:
2
2
2
121221 11
2
11
2 iiooio sRs
n
Rss
nh
R
nn
s
n
s
n
deviation from the first-order theory
Paraxial approximation: sin
Third order theory:!3
sin3
Spherical Aberration in Lenses
So we use spherical surfaces, which work better for a wider range of input angles.
Nevertheless, off-axis rays see a different focal length, so lenses have spherical aberration, too.
Spherical aberrations
LC - circle of least confusion, smallest image blur
L.SA = longitudinal spherical aberrations image of an on-axis object is longitudinally stretched
T.SA = Transverse (lateral) spherical aberrations image of an on-axis object is blurred in image plane
Lens arrangement and S.A
Spherical aberration depends on lens arrangement:
.
Different ways to illustrate optical aberrations
Side view of a fan of rays
(No aberrations)
“Spot diagram”: Image at different focus positions
Shows “spots” where rays would strike a detector
1 2 3 4 5
1 2 3 4 5
Spherical aberration
Through-focus spot diagram for spherical aberration
Rays from a spherically aberrated wavefront focus
at different planes
• How to fixed spherical aberration
Coddington Shape factor
2 1
2 1
R Rq
R R
q lens1 = [(+R+(-R)]/[R-(-R)] =0
q len2=[(-R+(+R)]/[(-R)-(+R)]=0
q = ?
R1 = Front surface radius of curvature
R2 = Back surface radius of curvature
Coddington Shape factor curves
Coddigton Position factor
• .
oi
oiP
o i
Example:
o =5 cm
i=10
P=[10+(-5)]/[10-(-5) =1/3
Minimum S.A
• .
oi
oiP
Pn
n
2
)1(2 2
12
12
RR
RR
From “introduction to Geometrical and Physical optics” p 86-94;
Minimum S.A happens when:
The radius of lens with minimum of S.A
• From lens maker equation:
1
)1(2
1
)1(2
)11
)(1(1
2
1
12
12
21
nfR
nfR
RR
RR
RRn
f
Spherical aberration can
be also minimized
using additional lenses
The additional lenses cancel the spherical aberration of the first.
Refractive Gradients
• Spherical aberration can be also minimized using gradient index lens .
no – SA: (gradient lens)
SA: normal lens
Theory of refractive Gradient
• .L
L`
n
n`yn
dLy
n
n
Ln
y
n
1
.1
R
L = R
The angle is a function of the gradient index n and the length of path traversed.
Theory of refractive Gradient
dLy
n
n
Ln
y
n
1
.1
The intergral is taken over the length L, traversed by the light. If:
1-The L is relatively short
2- The medium is honomgeneous
y
nn
R
LL
Ly
n
nRRL
Ly
n
ny
1
1
Example
Determine
-The angle of deflection of light
-The radius of curvature which the light is bent.
Solution
n=1.4
n=1.6
52.0
5.1
53.102667.0)01.0(05.0
2.0
5.1
1
1
0
y
nn
R
rad
Ly
n
ny
n = (1.4+1.6)/2=1.5
5cm
1cm
• Coma• Coma
causes rays from an off-axis point of light in the object plane to create a trailing "comet-like" blur directed away from the optic axis.
For objects-image pints that are off-axis the
aberration is called coma
• Coma
The lens magnification is depend on the lens diameter and
ComaComa causes rays from an off-axis point of light in the object plane to create a trailing "comet-like" blur directed away from the optic axis. A lens with considerable coma may produce a sharp image in the center of the field, but become increasingly blurred toward the edges. For a single lens, coma can be caused or partially corrected by tilting the lens.
Coma
point off the axis depicted as comet shaped blob
Marc Pollefeys
Coma (comatic aberration)
Reason: principal planes are not flat but curved surfacesFocal length is different for off-axis points/rays
Negative coma: meridional rays focus closer to the principal axis
Coma (comatic aberration)
Vertical coma
Horizontal coma
Removing the of coma by asymmetric lens
• Off axes rays and similar optical path for asymmetric lens
How Coma can be fixed
These distortions are fixed by an “orthoscopic doublet” or a “Zeiss orthometer.”
• Astigmatism
Where does astigmatism come from?
From Ian McLean, UCLA
Different view of astigmatism
AstigmatismWhen the optical system lacks perfect cylindrical symmetry, we say it has astigmatism. A simple cylindrical lens or off-axis curved-mirror reflection will cause this problem.
Cure astigmatism with another cylindrical lens or off-axis curved mirror.
Model astigmatism
by separate x and y analyses.
Astigmatism
http://www.microscopyu.com/tutorials/java/aberrations/astigmatism/
Focal length for rays in Sagittal and Meridional planes differ for off-axis points
Oblique astigmatism
• Occurs when rays of light strike a spherical lens obliquely-
• Line of sight not parallel with principal axis of lens
Astigmatism
Different focal length for inclined rays
Marc Pollefeys
•Curvature of field
Curvature of field
Curvature of field causes a planar object to project a curved (non-planar) image. Rays at a large angle see the lens as having an effectively smaller diameter and an effectively smaller focal length, forming the image of the off axis points closer to the lens.
Field curvature
Focal plane is curved:Petzval field curvature aberration
http://www.microscopyu.com/tutorials/java/aberrations/curvatureoffield/index.html
Negative lens has field plane that curves away from the image plane:Can use a combination of positive and negative lenses to cancel the effect
•distortion
Pincushion and Barrel Distortion
These distortions are fixed by an “orthoscopic doublet” or a “Zeiss orthometer.”
Barrel and pincushion distortion
Barrel
Pincushion
DistortionTransverse magnification MT may be a function of off-axis image distance: distortions
http://www.microscopyu.com/tutorials/java/aberrations/distortion/index.html
Positive (pincushion) distortion
Negative (barrel) distortion
Correcting monochromatic aberrations
• Use combinations of lenses with mutually canceling aberration effects
• Use apertures• Use aspherical elements
Example:
Chromatic Aberration
Because the lens material has a different refractive index for each wavelength, the lens will have a different focal length for each wavelength. Recall the lens-maker’s formula:
1 21/ ( ) ( ( ) 1)(1/ 1/ )f n R R
Here, the refractive index is larger for blue than red, so the focal length is less for blue than red.
You can model spherical aberration using ray matrices, but only one color at a time.
Chromatic aberration can be minimized using additional lenses
In an Achromat, the second lens cancels the dispersion of the first.
Achromats use two different materials, and one has a negative focal length.
Chromatic aberrations
Refraction index n depends on wavelength
21
111
1
RRn
f l
Chromatic aberrations
A.CA: axial chromatic aberration
L.CA: lateral chromatic aberration
21
111
1
RRn
f l
Achromatic Doublets
http://www.microscopyu.com/tutorials/java/aberrations/chromatic/index.html
Combine positive and negative lenses so that red and blue rays focus at the same point
Achromatized for red and blue
For two thin lenses d apart:2121
111
ff
d
fff
21
111
1
RRn
f l
22112211 1111
1 nndnnf
Achromat: fR=fB
22112211
22112211
1111
1111
BBBB
RRRR
nndnn
nndnn
Achromatic Doublets
22112211
22112211
1111
1111
BBBB
RRRR
nndnn
nndnn
Simple case: d = 0
RB
RB
nn
nn
11
22
2
1
Focal length in yellow light
(between red and blue):
11
YY
nf
Y
Y
Y
Y
f
f
n
n
1
2
1
2
2
1
1
1
1
1
111
222
1
2
YRB
YRB
Y
Y
nnn
nnn
f
f
Combine:
Achromatic Doublets
http://www.microscopyu.com/tutorials/java/aberrations/chromatic/index.html
1
1
111
222
1
2
YRB
YRB
Y
Y
nnn
nnn
f
f
12
22
Y
RB
n
nnDispersive powers:
11
11
Y
RB
n
nn
Abbe numbers (dispersive indices, V-numbers):RB
Y
nn
nV
22
21
1
RB
Y
nn
nV
11
12
1
2
1
1
2
V
V
f
f
Y
Y 01122 VfVf YY
Typical BYR colors: B = 486.1327 nm (F-line of hydrogen)Y = 587.5618 nm (D3 line of helium)R = 656.2816 nm (C-line of hydrogen)Table of V numbers - page 270
Achromatic lenses CrownFlint
http://www.microscopyu.com/tutorials/java/aberrations/chromatic/index.html
Achromatic triplet:focus match for 3 wavelengths
Cooke triplet(Denis Taylor, 1893)
Flint
Crown
Achromatic doublet: exampleDesign an achromatic doublet with f = 50 cm Use thin lens approximation.
Solution:21
111
fff01122 VfVf
ff
fff
1
12
01121
1
VfVff
ff
fV
VVf
1
211
f
V
VVf
2
122
Technically: want smaller R, i.e. longest possible f1 and f2
Solution: use two materials with drastically different V
Use figure 6.39 (page 271)
Achromatic doublet: example
V
V1= 63.46
V2= 36.37
Achromatic doublet: exampleDesign an achromatic doublet with f = 50 cm
Solution:f
V
VVf
1
211
f
V
VVf
2
122
m 2134.050.0
46.63
37.3646.631
f m 3724.02 f
21
111
1
RRn
f l
f1 f2 n1=1.51009 (for yellow line!)n2 = 1.62004
Negative lens:
11
11
12
2 Rn
f m 2309.01221 nfR
Positive lens:
m2309.0
111
1
11
1 Rn
f m 2059.01 R
Spherical aberration as “the parent of all other aberrations”
• Coma and astigmatism can be thought of as the aberrations from a de-centered bundle of spherically aberrated rays
• Ray bundle on axis shows spherical aberration only
• Ray bundle slightly de-centered shows coma
• Ray bundle more de-centered shows astigmatism
• All generated from subsets of a larger centered bundle of spherically aberrated rays
– (diagrams follow)
Spherical aberration as the parent of coma
Big bundle of spherically aberrated rays
De-centered subset of rays produces coma
Coma
Through-focus spot diagram for coma
Rays from a comatic wavefront
Note that centroid shifts:
Spherical aberration as the parent of astigmatism
Big bundle of spherically aberrated rays
More-decentered subset of rays produces astigmatism
Astigmatism
Through-focus spot diagram for astigmatism
Side view of rays
Top view of rays
Concept Question
• How do you suppose eyeglasses correct for astigmatism?
Off-axis object is equivalent to having a de-
centered ray bundle
Ray bundle from an off-axis object. How to view this as a de-centered ray bundle?
For any field angle there will be an optical axis, which is to the surface of the optic and || to the incoming ray bundle. The bundle is de-centered wrt this axis.
Spherical surface
New optical axis
•Extra
Zernike Polynomials
• Convenient basis set for expressing wavefront aberrations over a circular pupil
• Zernike polynomials are orthogonal to each other
• A few different ways to normalize – always check definitions!
Piston
Tip-tilt
Astigmatism(3rd order)
Defocus
Trefoil
Coma
Spherical
“Ashtray”
Astigmatism(5th order)
Units: Radians of phase / (D / r0)5/6
Reference: Noll
Tip-tilt is single biggest contributor
Focus, astigmatism, coma also big
High-order terms go on and on….
Seidel polynomials vs. Zernike polynomials
• Seidel polynomials also describe aberrations
• At first glance, Seidel and Zernike aberrations look very similar
• Zernike aberrations are an orthogonal set of functions used to decompose a given wavefront at a given field point into its components
– Zernike modes add to the Seidel aberrations the correct amount of low-order modes to minimize rms wavefront error
• Seidel aberrations are used in optical design to predict the aberrations in a design and how they will vary over the system’s field of view
• The Seidel aberrations have an analytic field-dependence—proportional to some power of field angle
References for Zernike Polynomials
• Books:– Hardy pages 95-96 READ THIS
• Pivotal Paper: Noll, R. J. 1976, “Zernike polynomials and atmospheric turbulence”, JOSA 66, page 207
Review of important points
• Both lenses and mirrors can focus and collimate light• Equations for system focal lengths, magnifications are quite similar for
lenses and for mirrors– But be careful of sign conventions
• Telescopes are combinations of two or more optical elements– Main function: to gather lots of light– Secondary function: magnification
• Aberrations occur both due to your local instrument’s optics and to the atmosphere– Can describe both with Zernike polynomials
Spherical aberrations
Paraxial approximation:
R
nn
s
n
s
n
io
1221
Third order:
2
2
2
121221 11
2
11
2 iiooio sRs
n
Rss
nh
R
nn
s
n
s
n
deviation from the first-order theory
L.SA = longitudinal spherical aberrations image of an on-axis object is longitudinally stretched
T.SA = Transverse (lateral) spherical aberrations image of an on-axis object is blurred in image plane
positive L.SA - marginal rays intersect in front of Fparaxial
Wavefront aberrations
John William Strutt(Lord Rayleigh)
1842-1919
Lord Rayleigh criterion: wavefront aberration of /4 produces noticeably degraded image (light intensity of a point object image drops by ~20%)
Zero SA
For points P and P’ SA is zero
Oil-immersion microscope objective
http://micro.magnet.fsu.edu/primer/java/aberrations/spherical/
Minimizing spherical aberration in a focus
2 1
2 1
R Rq
R R
Plano-convex lenses (with their flat surface facing the focus) are best for minimizing spherical aberration when focusing.
One-to-one imaging works best with a symmetrical lens (q = ∞).
R1 = Front surface radius of curvature
R2 = Back surface radius of curvature
