Astigmatism Field Curvature Distortion

Lens Design OPTI 517

Prof. Jose Sasian Earliest through focus images

T.1. Young, “On the mechanism of the eye,”

Phil2. Trans Royal Soc Lond 1801; 91: 23–88 and plates.

Prof. Jose Sasian Astigmatism through focus

Prof. Jose Sasian Astigmatism

2 2 WH(,) W111 H cos() W 020 W200 H 4 3 22 2 22 3 4 WWHWH040 131 cos( ) 222 cos  ( )  WHWHWH220 311  cos(  ) 400

Prof. Jose Sasian Anastigmatic

• Aplanatic: free from spherical aberration and coma. • Stigmatic ~ pointy • Astigmatism: No pointy • Anastigmatic: No-No pointy = pointy • Anastigmatic: free from spherical aberration, coma, and astigmatism • Aplanatic: coined by John Herschel • Astigmatism: coined by George Airy

Prof. Jose Sasian Cases of zero astigmatism

1 2 u  WAy222    2 n

Prof. Jose Sasian   Field behavior 2 2 2 2 2 W (H, )  W222 H cos ( ) W220 H 

1 2 u  1122u WAy222    WAy220    Ж P 2 n 42n

Prof. Jose Sasian Review of aberrations coefficients 1 WS 040 8 I 1 WS 131 2 II 1 WS 222 2 III 1 WS 220P 4 IV 1 WS 311 2 V 1 WC  020 2 L WC Prof. Jose Sasian 111 T Structural coefficients

Prof. Jose Sasian Seidel sum for 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 Жy  EX FY B  II 2 n n 1 2 SIII  Ж  3n  2 C  n 2 1 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 Cy 2 E  L  nn 1   nc  (n 1)(c1  cx ) CT  0 2n 1 F  n

Prof. Jose Sasian Thin lens astigmatism

2 SIII  Ж 

When the stop is a the thin lens astigmatism is fixed.

Shifting the stop in the presence of spherical aberration or coma Allows changing astigmatism

*2  III III2SS II  I

Prof. Jose Sasian Controlling astigmatism

Prof. Jose Sasian 1) Stop position: singlet lens

Coma and astigmatism are zero! u 0 *2  SSIII III2 SSSS II I n 1 A  0 Prof. Jose Sasian2 2) Canceling/balancing negative and positive astigmatism

Prof. Jose Sasian 3-a) Adding a degree of freedom

• In this case one adds a lens which contributes the opposite amount of astigmatism. • The spherical aberration and coma of the new lens are corrected by the system that has the degrees of freedom for such. • New lens hopefully contributes little coma and spherical aberration.

Prof. Jose Sasian 3-b) Adding a degree of freedom Ritchey-Chretien I 1.7 waves of astigmatism @ f.3.3

At best surface (Sagittal field surface)

Prof. Jose Sasian 3-c) Adding a degree of freedom Ritchey-Chretien II

0.0 waves of astigmatism @ f/1.9 after conic tweak

Prof. Jose Sasian 4) Shells near the image plane (or aspheric plate)

Prof. Jose Sasian Offner unit magnification relay

•Offner relay system: •Three spherical mirrors •Negative unit magnification •No primary aberrations •Ring field concept •Improvement of field with shell

Prof. Jose Sasian However; beware of ghosts

Prof. Jose Sasian Field curvature

1 22u   1 W220 Ж PA    y PC  4 n  n

11 nn' Petzval sum:  nn''kk11 nnr'

1  For a system of thin lenses:   'k n

Prof. Jose Sasian Field curvature interpretation

• Assume same glass and consider sag 22 of Petzval surface at a height y: ynny'  2'  nr 2 • If the Petzval sum is zero then the lens k has constant thickness across the aperture or across the field. n  1 • Compare with the image displacement S  t S caused by a plano parallel plate: n

• The conclusion is that Petzval field curvature arises because the overall lens thickness variation across the aperture (in the general case the index of refraction enters as a weight).

Prof. Jose Sasian Thickness variation in a telecentric lens

Prof. Jose Sasian Four classical ways • 1) A thick meniscus lens can contribute optical power but no field curvature if both surfaces have the same radius. Consider double Gasuss lens. Note the correction for color.

• 2) Separated thin lenses: Bulges and constrictions Consider the Cooke triplet and lenses for microlithography.

• 3) A field flattener: Fully contributes to Petzval but not to spherical, coma, or astigmatism. Also there is little contribution to optical power. Consider Petzval lens with a field flattener.

• 4) New achromat: use to advantage new glass types. 1    'k n Prof. Jose Sasian Four classical ways

Use of a thick meniscus lens Use of a field flattener lens

Prof. Jose Sasian Four classical ways

Creating beam bulges and constrictions

Prof. Jose Sasian Four classical ways: Use of glass

V-number for flint increases V-number for crown decreases

N for crown increases N for flint decreases

f a  a  f b  b  F   a  b 

F=100 mm 1    'k n BK7 BK7-F2 SSKN5-LF5 P=-152 mm P=-139 mm P=-219 mm Prof. Jose Sasian Distortion

2 2 WH(,) W111 H cos() W 020 W200 H 4 3 22 2 22 3 4 WWHWH040 131 cos( ) 222 cos  ( )  WHWHWH220 311  cos(  ) 400

With respect to chief ray, geometrical or physical centroid

W311 H3cos() W511 H5cos()

Hh Distortion  100 h

Prof. Jose Sasian Distortion

Top row, (barrel) distortion:0%, 2.5%, 5% and 10%. Bottom row, (pincushion) distortion 0%, 2.5%, 5% and 10%.

Prof. Jose Sasian 1) By Symmetry about the stop or phantom stop

Distortion is an odd aberration: It can be cancelled by symmetry About the stop

Prof. Jose Sasian 2) Aspheric plate or bending a field flattener

Prof. Jose Sasian Exercise: Galilean telescope

A plano-convex lens objective with a focal length of about 750-1000 mm. A plano-concave lens for the eyepiece (ocular) with a focal length of about 50 mm. The objective lens was stopped down to an aperture of 12.5 to 25 mm. The field of view is about 15 arc-minutes. The instrument's magnifying power is 15-20.

Prof. Jose Sasian