Introduction to aberrations OPTI 518 Lecture 13

Prof. Jose Sasian OPTI 518 Topics

• Aspheric surfaces • Stop shifting • Field curve concept

Prof. Jose Sasian OPTI 518 Aspheric Surfaces

• Meaning not spherical • Conic surfaces: Sphere, prolate ellipsoid, hyperboloid, paraboloid, oblate ellipsoid or spheroid • Cartesian Ovals • Polynomial surfaces • Infinite possibilities for an aspheric surface • Ray tracing for quadric surfaces uses closed formulas; for other surfaces iterative algorithms are used

Prof. Jose Sasian OPTI 518 Aspheric surfaces

The concept of the sag of a surface

2 cS 4 6 8 10 SZ  4 6 8 10SASASASA  ...  )1(11 ScK 22

Sxy222 K   2

K is the conic constant K=0, sphere K=-1, parabola C is 1/r where r is the radius of curvature; K is the K<-1, hyperola conic constant (the eccentricity squared); -10, oblate ellipsoid

Prof. Jose Sasian OPTI 518 Conic surfaces focal properties

• Focal points for Ellipsoid case mirrors Hyperboloid case Oblate ellipsoid • Focal points of lenses

K  n2

Hecht-Zajac Optics Prof. Jose Sasian OPTI 518 Refraction at a spherical surface Aspheric surface description

2 cS 4 6 8 10 SZ  4 6 8 10SASASASA  ...  )1(11 ScK 22

11 22 ZxyKxyAxy221  22 22 aspheric 28rr 4

Prof. Jose Sasian OPTI 518 Cartesian Ovals

ln  l'n' Cte .

Prof. Jose Sasian OPTI 518 Aspheric cap

Aspheric surface

The aspheric surface can be thought of as comprising a base sphere and an aspheric cap

Cap Spherical base surface

Prof. Jose Sasian OPTI 518 Aspheric surface contributions

Prof. Jose Sasian OPTI 518 2 1 y Wa220   4 y

Prof. Jose Sasian OPTI 518 Stop at aspheric surface

  4 2 WHcap  ,   Ayn4  

Prof. Jose Sasian OPTI 518 Upon stop shifting

  4 2 WHcap  ,  Ayn4  shift  shift 

Prof. Jose Sasian OPTI 518 Aspheric cap contributions

New stop

y y

Old stop

Prof. Jose Sasian OPTI 518 Stop shifting

  4 2 WHcap  ,  Ayn4  shift  shift 

   shift SH

yy y 0 y S  new old new yyy

Prof. Jose Sasian OPTI 518   2 Expansion of shift shift 

2   2      2SH  S H  H  shift shift    2   2SH  S H  H   2    2    44SH   S2  H      2 24S234 HH  S HH H S HH

Prof. Jose Sasian OPTI 518 Aberration function upon stop shifting for an aspheric cap

  4 2 WHcap ,  Ayn4  shift  shift   442   Ay44 n    4 Ay  n S H   

2  42 42  42Ay44 n S H  Ay n S H  H  

  2 43 44 4Ay44 n S H  H H  Ay n S H  H

112   12 aaSHaSH    2  82 2 11    1   2 aS23 H H aS H H H  aS4  H  H 42 8

4 aAyn8 4 

Prof. Jose Sasian OPTI 518 Aspheric contributions

Prof. Jose Sasian OPTI 518 Aberration function of base sphere and aspheric cap

   WH ,,,  Wspherical  H Wcap H

Case of an aspherical surface with the stop at the surface:

2221 24u   WAyAyn040      4    8 n

222 Sag Sphere A4  x y

Prof. Jose Sasian OPTI 518 Stop shifting

Prof. Jose Sasian OPTI 518 Stop shifting

Prof. Jose Sasian OPTI 518 Stop shifting

• How do the aberration coefficients change as we shift the stop? • Used to ease calculations • Most importantly, the formulas give insights • The concept of object shifting

Prof. Jose Sasian OPTI 518 Stop shifting parameter

Prof. Jose Sasian OPTI 518 Stop shifting formulas

Prof. Jose Sasian OPTI 518 Derivation for A

Ж Ay11 Ay

Ж A22y Ay

0 A12AyAyy  12 

yy y A AAAS21   yy AAAS  Prof. Jose Sasian OPTI 518 Derivation of new Seidel sums upon stop shifting

AASA 

2 u SAyI   n  SSII And similarly for: uu  SAAyAASAyII       nn   uu2  SSIV IV AAy   S A y   nn   23 SSSSSVV IVIII 33  SSSS III   SSSSII II I

2 2 uu  SAyASAyIII       nn 

222u A 2 AAS  S A y  n  2 SSIII III2 SSSS II  I

Prof. Jose Sasian OPTI 518 Field focus curve concept

2  w

Provides limit for a given wavefront variance. Prof. Jose Sasian OPTI 518 Field focus curve concept

Prof. Jose Sasian OPTI 518 Field focus curve concept

•Must preserve DOF

Prof. Jose Sasian OPTI 518 Summary

• Variety in aspheric surfaces • Aspheric surface contributions to fourth- order aberrations • Stop shifting and how aberrations change • The concept of field focus curve

Prof. Jose Sasian OPTI 518