Coma aberration

Lens Design OPTI 517

Prof. Jose Sasian Coma

0.25 wave 1.0 wave 2.0 waves 4.0 waves Spot diagram

   2  2 W H, ,  W200 H W020 W111H cos  4  3  2 2 2 W040 W131H cos W222 H cos  2 2 3  4 W220 H W311H cos W400 H   ...

Prof. Jose Sasian Coma though focus

Prof. Jose Sasian Cases of zero coma

1 u  WAAy131    2 n

•At y=0, surface is at an image •A=0, On axis beam concentric with center of curvature •A-bar=0, Off-axis beam concentric, chief ray goes through the center of curvature •Aplanatic points

Prof. Jose Sasian Cases of zero coma

Prof. Jose Sasian Aplanatic-concentric

Prof. Jose Sasian Coma as a variation of magnification with aperture I

1 u WH131 ,   AAyH    2 n

Prof. Jose Sasian Coma as a variation of magnification with aperture II

S S’

m=s’/s

Prof. Jose Sasian Sine condition Coma aberration can be considered as a variation of magnification with respect to the aperture. If the paraxial magnification is equal to the real ray marginal magnification, then an optical system would be free of coma. Spherical aberration can be considered as a variation of the focal length with the aperture.

u sinU U U’  u' sinU '

Prof. Jose Sasian Sine condition

On-axis L Sine condition

YY’ L' O’ h YY’ O’ UU’L' L O h’

PP’ PP’

Optical path length between y and y’ is Optical path length between O and O’ is Laxis and does not depend on Y or Y’ Loff-axis = Laxis + L’ - L

L = L + h’ n’ sin(U’) - h n sin(U) Lhsin( U ) L ''sin(') hU off-axis axis u sinU hn' 'sin( U ') hn sin( U )  u' sinU ' That is: OPD has no linear phase errors as a function Prof.of field Jose of Sasian view! Imaging a grating

m sin(U )  d d sin(U )  m  d'sin(U ')

Prof. Jose Sasian Contribution from an aspheric surface 1 y W  8Ay4  n 1312 y 4

y-bar chief ray

y marginal ray

Prof. Jose Sasian   Aberrations and symmetry 2  2 W H, ,  W200 H W020 W111H cos  4  3  2 2 2 W040 W131H cos W222 H cos  2 2 3  4 W220 H W311H cos W400 H   ... •Coma is an odd aberration with respect to the stop •Natural stop position to cancel coma by symmetry

Prof. Jose Sasian Structural coefficients: Thin lens (stop at lens) n  2 A  1 nn 1 2 S  y 4 3 AX 2  BXY  CY 2  D I 4

1 22 4n 1 S Ж y  EX FY B  II 2 n n 1 2 SIII  Ж  3n  2 C  1 n 2 c1  c2 r2  r1 SIV  Ж  X   n c1  c2 r2  r1 n 2 D  2 SV  0 1 m u'u n 1 Y   1 1 m u'u n 1 C  y2 E  L  nn 1   nc  (n 1)(c1  cx ) CT  0 2n 1 F  n

Prof. Jose Sasian Coma vs Bending

Coma

 II  EX  FY

Shape factor X

Prof. Jose Sasian Principal surface

In an aplanat working at m=0 the equivalent refracting surface is a hemisphere

Prof. Jose Sasian Cassegrain’s principal surface

Since the equivalent refracting surface in a Cassegrain telescope is a paraboloid then the coma of that Cassegrain is the same of a paraboloid mirror with the same focal length.

Prof. Jose Sasian Aplanat doublets

Prof. Jose Sasian Kingslake’s cemented aplanat

Chromatic correction Spherical aberration correction Coma correction Still cemented

Prof. Jose Sasian Control of coma in the presence of an aspheric mirror near a pupil

In the presence of a strong aspheric surface near the stop or pupil, coma aberration can be corrected by moving the surface

1 y W  8Ay4  n 1312 y 4

Prof. Jose Sasian Coma correction by natural stop position Original Natural y stop position stop position S  S II y I

Optical system

Object plane Image plane

In the presence of spherical aberration there is a stop position for which coma is zero. At that stop position spherical aberration might be corrected. Then the system becomes aplanatic and the stop

Prof. Jose Sasian can be shifted back to its original position. The aplanatic member(s) in a family of solutions

Doublet Ritchey-Chretien

Prof. Jose Sasian Camera Schmidt

Aspheric plate at mirror center of curvature A-bar=0 Stop aperture at aspheric plate Note symmetry about mirror CC No spherical aberration No coma No astigmatism. Anastigmatic over a wide field of view! Prof. Jose Sasian Satisfies Conrady’s D-d sum Notes

• Need to make aplanatic zero-field systems (that are fast). The alignment becomes easier. • Lenses for lasers diodes/optical fibers • Microscope objectives

Prof. Jose Sasian Summary

• Coma aberration • Coma as an odd aberration • Sine condition • Natural stop position • Aplanatic doublets • Zero-field systems

Prof. Jose Sasian