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ECE 5616 OE System Design
Robert McLeod 187
Ray and wavefront aberrationsGeneral picture and relationship
Entrance pupil Exit pupil
Paraxial focus
Reference sphere
Wavefront
• A spherical wavefront in the exit pupil forms a perfect focus• The difference between the actual wavefront and the reference sphere is the “wavefront aberration”.• Non-spherical wavefronts cause rays to cross the optical axis NOT at the paraxial focus = “ray aberration”
Ray aberration
•Turning ideal imaging systems into real optics–Aberrations
ECE 5616 OE System Design
Robert McLeod 188
Connecting paraxial and finite optics
•Turning ideal imaging systems into real optics–Aberrations
Mouroulis & Macdonald 5.2
yEP
sinu
EP
-h
EP
tanu
Sine condition Tangent condition
Paraxial to non-paraxial conversion
Ray drawings
Sine condition
n0
sin
0
nx
hnhn sinsin
sinsin nn
Mhhxx
Tangent condition
constanttantan
ECE 5616 OE System Design
Robert McLeod 189
Ray and wavefront aberrationsMathematical relationship
Exit Pupil
Paraxial image plane
Reference sphere
Wavefront
= wavefront error
m, pp yxW
m, pp yxy
py
y
p
pp
y
yxW
n
Ry
,
R = Radius of reference sphere
•Turning ideal imaging systems into real optics–Aberrations
= transverse ray aberration
ECE 5616 OE System Design
Robert McLeod 190
Ray aberration polynomialConcept
hy
x
Object
Paraxial image plane
r
Entrance pupil
•Turning ideal imaging systems into real optics–Aberrations
1. Assume a rotationally-symmetric optical system2. Expand the coordinates of the ray intersect with the
paraxial image plane (x,y) in a polynomial of the object ray parameters (h, r, ).
3. Reject all terms which don’t meet symmetry assumption.
ECE 5616 OE System Design
Robert McLeod 191
Ray aberration polynomialFirst and third order terms
3
5
243
22
31
21
243
22
31
1
Cos3
)2Cos2(
Cos
Cos
Sin
2Sin
Sin
Sin
hB
rhBB
hrB
rB
hArAy
rhBB
hrB
rB
rAx
W. Smith, Modern Optical Engineering, p. 58, McGraw-Hill, 1990
•Turning ideal imaging systems into real optics–Aberrations
The x intercept in the paraxial image plane in
terms of the object height (h) , and the ray
coordinates in the pupil (r, ).
The y intercept in the paraxial image plane in terms of the object
height (h) , and the ray coordinates in the
pupil (r, ).
ECE 5616 OE System Design
Robert McLeod 192
Linear termsDefocus and magnification
•Turning ideal imaging systems into real optics–Aberrations
Cos
Sin
1
1
rAy
rAx
r Cos y
y
r Cos
Ray intercept coordinates linearly proportional to pupil
coordinates.
Ray intercept coordinates linearly proportional to pupil
coordinates.
hAy 2 Magnification
ECE 5616 OE System Design
Robert McLeod 193
Spherical aberrationRay aberration definitions & summary
LSA
TSA
CLC
Par
axia
l im
age
plan
e
• B1>0 is positive SA (shown): marginal rays focus more strongly.• Only 3rd order aberration that occurs for on-axis imaging.• Radius of CLC is ¼ TSA (for pure 3rd order SA)• CLC is located ¾ LSA from paraxial focus (for pure 3rd order SA)• LSA TSA / r, thus LSA r2
Cos
Sin3
1
31
rBy
rBx
Equation for circle of radius B1 r3,
thus TSA to aperture size3
but is independent of field
Marginal ray
Paraxial ray
•Turning ideal imaging systems into real optics–Aberrations
ECE 5616 OE System Design
Robert McLeod 194
Spherical aberrationVarious views
Ray fan at paraxial focus
Ray fan at CLC
Transverse ray aberration plot at focus showing expected 3rd order dependence on pupil coordinate
•Turning ideal imaging systems into real optics–Aberrations
Sections through focus
en.wikipedia.org/wiki/Spherical_aberration
SA < 0
SA = 0
SA > 0
ECE 5616 OE System Design
Robert McLeod 195
Spherical aberrationWavefront aberration plots
Reference sphere
Wavefront aberration
r
•Turning ideal imaging systems into real optics–Aberrations
4040WW
-1
-0.5
0
0.5
1-1
-0.5
0
0.5
1
-4
-3
-2
-1
0
-1
-0.5
0
0.5
1x
y
Interferogram
ECE 5616 OE System Design
Robert McLeod 196
Spherical aberration at CLC
-1
-0.5
0
0.5
1-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
-1
-0.5
0
0.5
1
2234
040 WW
Wavefront error in pupilW040 = 1
Interferogram
W
x
y
x
y
•Turning ideal imaging systems into real optics–Aberrations
ECE 5616 OE System Design
Robert McLeod 197
ComaRay aberration definitions & summary
2Cos2
2Sin2
2
22
hrBy
hrBx Equation for double circle of radius B2 r2 h plus focal shift
of twice this amount
hy
x
Object
Paraxial image plane
r
Aperture stop
• B2>0 is positive coma: cone of rays (see next) opens away from optical axis• Linearly dependent on h: important even at small field angles• Defocus does not yield increase in power density (no CLC)
•Turning ideal imaging systems into real optics–Aberrations
ECE 5616 OE System Design
Robert McLeod 198
ComaRay aberration plots
•Turning ideal imaging systems into real optics–Aberrations
Paraxial focus
02/
xhrBy 2
2,03
Ray fan Transverse ray aberration plot at focus
Detail of rays at paraxial focus for fixed r
ECE 5616 OE System Design
Robert McLeod 199
-1
-0.5
0
0.5
1-1
-0.5
0
0.5
1
-2
-1
0
1
2
-1
-0.5
0
0.5
1
Coma2
01313
0131 cos xxWxWW
Wavefront error in pupilW131 = 1
Interferogram
W
x
x
y
y
•Turning ideal imaging systems into real optics–Aberrations
ECE 5616 OE System Design
Robert McLeod 200
AstigmatismRay aberration definitions & summary
•Turning ideal imaging systems into real optics–Aberrations
• Defocus can correct astigmatism in one plane and one field• Dependent on h2: important at large field angles• CLC is located ½ way between meridional and sagital foci and is a disk with diameter equal to half their length.
Cos3
Sin2
3
23
rhBy
rhBx
Defocus dependent on h2 and unequal in x and y
h
r
Sagital focus
Meridional focus
x
y
ECE 5616 OE System Design
Robert McLeod 201
Astigmatism (negative)Ray aberration definitions & summary
•Turning ideal imaging systems into real optics–Aberrations
• Defocus can correct astigmatism in one plane and one field• Dependent on h2: important at large field angles• CLC is located ½ way between meridional and sagital foci and is a disk with diameter equal to half their length.
Cos3
Sin2
3
23
rhBy
rhBx
Defocus dependent on h2 and unequal in x and y
h
r
Sagital focus
Meridional focus
x
y
ECE 5616 OE System Design
Robert McLeod 202
AstigmatismRay & wavefront aberration plots
Sagital plane
Meridional plane
Plane of least confusion
y in focus x defocused
y defocused x in focus
= and opposite defocus
y
x
y
x
y
x
x
y
x
y
x
y
•Turning ideal imaging systems into real optics–Aberrations
ECE 5616 OE System Design
Robert McLeod 203
-1
-0.5
0
0.5
1-1
-0.5
0
0.5
1
-1
-0.75
-0.5
-0.25
0
-1
-0.5
0
0.5
1
Astigmatism22
0222222
0222 cos xxWxWW
Wavefront error in pupilW222 = 1
Interferogram
W
x
x
y
y
•Turning ideal imaging systems into real optics–Aberrations
ECE 5616 OE System Design
Robert McLeod 204
-1
-0.5
0
0.5
1-1
-0.5
0
0.5
1
-0.5
-0.25
0
0.25
0.5
-1
-0.5
0
0.5
1
Astigmatism at CLC
Wavefront error in pupilW222 = 1
Interferogram
W
x
x
y
y
2212
212
02222
21222
0222 cos yxxWxWW
•Turning ideal imaging systems into real optics–Aberrations
ECE 5616 OE System Design
Robert McLeod 205
Field guide to aberrationsAt paraxial focus
Diffraction-limted Defocus Spherical
Spherical+defocus Astigmatism Coma
astron.berkeley.edu/ ~jrg/Aberrations/node5.html
ECE 5616 OE System Design
Robert McLeod 206
Petzval curvatureRay aberration definitions & summary
Cos
Sin2
4
24
rhBy
rhBx
Defocus, equal in x and y, that depends on h2
h
Petzval focal surface
•Turning ideal imaging systems into real optics–Aberrations
ECE 5616 OE System Design
Robert McLeod 207
-1
-0.5
0
0.5
1-1
-0.5
0
0.5
1
-2
-1.5
-1
-0.5
0
-1
-0.5
0
0.5
1
Field curvature22
0220 xWW
Wavefront error in pupilW220 = 1
Interferogram
W
x
x
y
y
•Turning ideal imaging systems into real optics–Aberrations
ECE 5616 OE System Design
Robert McLeod 208
Field curvatureRelationship of astigmatism and Petzval
Sin243 rhBBx
Cos3 243 rhBBy
Paraxial image plane
Petzval image surface
Meridional image surface
Surface of least confusion
Sagital image surface
OA
34 2BB In this plot,
•Turning ideal imaging systems into real optics–Aberrations
Petzval
Astigmatism
ECE 5616 OE System Design
Robert McLeod 209
Amount of field curvaturePetzval radius of single lens
•Turning ideal imaging systems into real optics–Aberrations
h
cos/~00 tt
1~t
1001
coscos1
1cos1~11
~1
tftftft
][2
cos~
422
1
11
Of
t
tt
h’
22
21
2 th
f
cos~1t
Write imaging equation for distances along diagonals.
Field curvature D is on-axis image distance minus off-axis image distance
Plug in expression for and expand in 1~t
Expand equation for circle of radius in transverse coordinateh
So radius of curvature equals focal length.
ECE 5616 OE System Design
Robert McLeod 210
Distortion
35hBy Magnification = B5h2
Image of square object:
B5>0 = “pincusion”
B5=0
B5<0 = “barrel”
•Turning ideal imaging systems into real optics–Aberrations
ECE 5616 OE System Design
Robert McLeod 211
Distortion
-1
-0.5
0
0.5
1-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
xxWxWW 30311
30311 cos
Wavefront error in pupilW311 = 1
Interferogram
W
x
x
y
y
•Turning ideal imaging systems into real optics–Aberrations
ECE 5616 OE System Design
Robert McLeod
Zernike polynomialsand relationship to Seidel
212
m
mRZ m
nm
n sin
cos,
kn
mn
k
kmn kmn
kn
k
knR 2
2
0 2
21
mmnnmn
mn nmnmCddZZ
,,
2
0
1
0
,,,,,
1
sin cos
12 2 2sin2 2cos2
sin23 2 cos23 2 3sin3 3cos3
•Turning ideal imaging systems into real optics–Aberrations
Matlab file exchange zernfun, Darryl Meister “Dispensing Optics” 2010
Field curvature
Astig-matism
Astig-matism
Spherical
24 34
Coma Coma
Distortion
Piston
Distortion
Orthogonality
ECE 5616 OE System Design
Robert McLeod 213
Strehl RatioA measure of coherent image quality
22221ratio Strehl
Strehl ratio is the on-axis intensity in the presence of aberrations relative to the on-axis intensity w/o aberrations.
The relationship between SR and RMS wavefront error () is
Smith, Modern Optical Engineering, Chapter 11
P-V OPD RMS OPD SR Energy in Airy Energy in rings0.0 0.0 1.0 84% 16%/16 0.018 /8 0.036 /4 0.07 /2 0.14 /4 0.21 0.29 *SR does not correlate well with image quality at these low levels.
RMS OPD = P-V OPD/3.5 to P-V OPD/5 depending on the aberration type.
•Turning ideal imaging systems into real optics–Imaging in non-ideal systems
SR also gives fiber coupling efficiency in many cases.
Exact for defocus, close for most.
Strehl ratio
0 1 2 3-1-2-30
1
r/r0
I/I m
ax