ray and wavefront aberrations - university of colorado...

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


Robert McLeod 188

Connecting paraxial and finite optics


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


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


= transverse ray aberration


Robert McLeod 190

Ray aberration polynomialConcept

hy

x

Object


r

Entrance pupil


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.


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


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, ).


Robert McLeod 192

Linear termsDefocus and magnification


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


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



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


Sections through focus

en.wikipedia.org/wiki/Spherical_aberration

SA < 0

SA = 0

SA > 0


Robert McLeod 195

Spherical aberrationWavefront aberration plots

Reference sphere

Wavefront aberration

r


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


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



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


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)



Robert McLeod 198

ComaRay aberration plots


Paraxial focus

02/

xhrBy 2

2,03

Ray fan Transverse ray aberration plot at focus

Detail of rays at paraxial focus for fixed r


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


Interferogram

W

x

x

y

y



Robert McLeod 200

AstigmatismRay aberration definitions & summary


• 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


Robert McLeod 201

Astigmatism (negative)Ray aberration definitions & summary


• 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


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



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


Interferogram

W

x

x

y

y



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


Interferogram

W

x

x

y

y

2212

212

02222

21222

0222 cos yxxWxWW



Robert McLeod 205

Field guide to aberrationsAt paraxial focus

Diffraction-limted Defocus Spherical

Spherical+defocus Astigmatism Coma

astron.berkeley.edu/ ~jrg/Aberrations/node5.html


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



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


Interferogram

W

x

x

y

y



Robert McLeod 208

Field curvatureRelationship of astigmatism and Petzval

Sin243 rhBBx

Cos3 243 rhBBy


Petzval image surface

Meridional image surface

Surface of least confusion

Sagital image surface

OA

34 2BB In this plot,


Petzval

Astigmatism


Robert McLeod 209

Amount of field curvaturePetzval radius of single lens


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.


Robert McLeod 210

Distortion

35hBy Magnification = B5h2

Image of square object:

B5>0 = “pincusion”

B5=0

B5<0 = “barrel”



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


Interferogram

W

x

x

y

y



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


Matlab file exchange zernfun, Darryl Meister “Dispensing Optics” 2010

Field curvature

Astig-matism

Astig-matism

Spherical

24 34

Coma Coma

Distortion

Piston

Distortion

Orthogonality


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

ray and wavefront aberrations - university of colorado...

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