Abstract

A study of light and charged-particle optical systems with inversion, reflection, rotation, translation, and/or glide symmetries is presented. The constraints imposed by the various symmetries on the first-order properties of a lens are investigated. In particular, the mathematical structures of the deflection vectors and the transfer matrices are described for various symmetrical systems. In the course of studying the translation and the glide symmetries, a simple technique for characterizing a general system of N identical components in series (or cascade) is also developed, based on the linear algebra theory of factoring matrices into Jordan canonical forms. Applications of these results are presented in a follow-up paper [ J. Opt. Soc. Am. A 12, 2760 ( 1995)].

© 1995 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Szilagyi, Electron and Ion Optics (Plenum, New York, 1988).
    [CrossRef]
  2. S. Bhagavantam, T. Venkatarayudu, Theory of Groups and Its Application to Physical Problems (Academic, New York, 1969).
  3. O. Scherzer, “Über einige Fehler von Elektronenlinsen,” Z. Phys. 101, 593–603 (1936).
    [CrossRef]
  4. M. Szilagyi, “Electron optical synthesis and optimization,” Proc. IEEE 73, 412–418 (1985).
    [CrossRef]
  5. J. Szép, M. Szilagyi, “A novel approach to the synthesis of electrostatic lenses with minimized aberrations,” IEEE Trans. Electron Devices 35, 1181–1183 (1988).
    [CrossRef]
  6. M. Szilagyi, J. Szép, “Optimum design of electrostatic lenses,” J. Vac. Sci. Technol. B 6, 953–957 (1988).
    [CrossRef]
  7. J. C. Burfoot, “Correction of electrostatic lenses by departure from rotational symmetry,” Proc. Phys. Soc. B 66, 775–792 (1953).
    [CrossRef]
  8. A. Septier, Advances in Optical and Electron Microscopy, R. Barer, V. E. Cosslett, eds. (Academic, New York, 1966), Vol. 1, pp. 204–274.
  9. P. W. Hawkes, Quadrupoles in Electron Lens Design (Academic, New York, 1970).
  10. P. H. Mui, M. Szilagyi, “Synthesis of monopole-and-quadrupole focusing columns,” J. Vac. Sci. Technol. B 12, 3036–3045 (1994).
    [CrossRef]
  11. M. Szilagyi, P. H. Mui, “Synthesis of focusing-and-deflection columns,” J. Vac. Sci. Technol. B 13, 375–382 (1995).
    [CrossRef]
  12. Y. Li, “Application of group theory to electron optics,” Adv. Electron. Electron Phys. 85, 231–258 (1993).
    [CrossRef]
  13. D. C. Carey, The Optics of Charged Particle Beams (Harwood, New York, 1987).
  14. S. Humphries, Principles of Charged Particle Acceleration (Wiley, New York, 1986).
  15. M. Szilagyi, P. H. Mui, “Symmetries in geometrical optics: applications,” J. Opt. Soc. Am. 12, 2760–2766 (1995).
    [CrossRef]
  16. Results in this section are not generally valid for systems with magnetic components.
  17. Results for cases (1), (2), and (4) in this section are not generally valid for systems with magnetic components.
  18. P. W. Milonni, J. H. Eberly, Lasers (Wiley-Interscience, New York, 1988).
  19. C.-T. Chen, Linear System Theory and Design (Holt, Rinehart & Winston, New York, 1970).

1995 (2)

M. Szilagyi, P. H. Mui, “Synthesis of focusing-and-deflection columns,” J. Vac. Sci. Technol. B 13, 375–382 (1995).
[CrossRef]

M. Szilagyi, P. H. Mui, “Symmetries in geometrical optics: applications,” J. Opt. Soc. Am. 12, 2760–2766 (1995).
[CrossRef]

1994 (1)

P. H. Mui, M. Szilagyi, “Synthesis of monopole-and-quadrupole focusing columns,” J. Vac. Sci. Technol. B 12, 3036–3045 (1994).
[CrossRef]

1993 (1)

Y. Li, “Application of group theory to electron optics,” Adv. Electron. Electron Phys. 85, 231–258 (1993).
[CrossRef]

1988 (2)

J. Szép, M. Szilagyi, “A novel approach to the synthesis of electrostatic lenses with minimized aberrations,” IEEE Trans. Electron Devices 35, 1181–1183 (1988).
[CrossRef]

M. Szilagyi, J. Szép, “Optimum design of electrostatic lenses,” J. Vac. Sci. Technol. B 6, 953–957 (1988).
[CrossRef]

1985 (1)

M. Szilagyi, “Electron optical synthesis and optimization,” Proc. IEEE 73, 412–418 (1985).
[CrossRef]

1953 (1)

J. C. Burfoot, “Correction of electrostatic lenses by departure from rotational symmetry,” Proc. Phys. Soc. B 66, 775–792 (1953).
[CrossRef]

1936 (1)

O. Scherzer, “Über einige Fehler von Elektronenlinsen,” Z. Phys. 101, 593–603 (1936).
[CrossRef]

Bhagavantam, S.

S. Bhagavantam, T. Venkatarayudu, Theory of Groups and Its Application to Physical Problems (Academic, New York, 1969).

Burfoot, J. C.

J. C. Burfoot, “Correction of electrostatic lenses by departure from rotational symmetry,” Proc. Phys. Soc. B 66, 775–792 (1953).
[CrossRef]

Carey, D. C.

D. C. Carey, The Optics of Charged Particle Beams (Harwood, New York, 1987).

Chen, C.-T.

C.-T. Chen, Linear System Theory and Design (Holt, Rinehart & Winston, New York, 1970).

Eberly, J. H.

P. W. Milonni, J. H. Eberly, Lasers (Wiley-Interscience, New York, 1988).

Hawkes, P. W.

P. W. Hawkes, Quadrupoles in Electron Lens Design (Academic, New York, 1970).

Humphries, S.

S. Humphries, Principles of Charged Particle Acceleration (Wiley, New York, 1986).

Li, Y.

Y. Li, “Application of group theory to electron optics,” Adv. Electron. Electron Phys. 85, 231–258 (1993).
[CrossRef]

Milonni, P. W.

P. W. Milonni, J. H. Eberly, Lasers (Wiley-Interscience, New York, 1988).

Mui, P. H.

M. Szilagyi, P. H. Mui, “Synthesis of focusing-and-deflection columns,” J. Vac. Sci. Technol. B 13, 375–382 (1995).
[CrossRef]

M. Szilagyi, P. H. Mui, “Symmetries in geometrical optics: applications,” J. Opt. Soc. Am. 12, 2760–2766 (1995).
[CrossRef]

P. H. Mui, M. Szilagyi, “Synthesis of monopole-and-quadrupole focusing columns,” J. Vac. Sci. Technol. B 12, 3036–3045 (1994).
[CrossRef]

Scherzer, O.

O. Scherzer, “Über einige Fehler von Elektronenlinsen,” Z. Phys. 101, 593–603 (1936).
[CrossRef]

Septier, A.

A. Septier, Advances in Optical and Electron Microscopy, R. Barer, V. E. Cosslett, eds. (Academic, New York, 1966), Vol. 1, pp. 204–274.

Szép, J.

M. Szilagyi, J. Szép, “Optimum design of electrostatic lenses,” J. Vac. Sci. Technol. B 6, 953–957 (1988).
[CrossRef]

J. Szép, M. Szilagyi, “A novel approach to the synthesis of electrostatic lenses with minimized aberrations,” IEEE Trans. Electron Devices 35, 1181–1183 (1988).
[CrossRef]

Szilagyi, M.

M. Szilagyi, P. H. Mui, “Symmetries in geometrical optics: applications,” J. Opt. Soc. Am. 12, 2760–2766 (1995).
[CrossRef]

M. Szilagyi, P. H. Mui, “Synthesis of focusing-and-deflection columns,” J. Vac. Sci. Technol. B 13, 375–382 (1995).
[CrossRef]

P. H. Mui, M. Szilagyi, “Synthesis of monopole-and-quadrupole focusing columns,” J. Vac. Sci. Technol. B 12, 3036–3045 (1994).
[CrossRef]

M. Szilagyi, J. Szép, “Optimum design of electrostatic lenses,” J. Vac. Sci. Technol. B 6, 953–957 (1988).
[CrossRef]

J. Szép, M. Szilagyi, “A novel approach to the synthesis of electrostatic lenses with minimized aberrations,” IEEE Trans. Electron Devices 35, 1181–1183 (1988).
[CrossRef]

M. Szilagyi, “Electron optical synthesis and optimization,” Proc. IEEE 73, 412–418 (1985).
[CrossRef]

M. Szilagyi, Electron and Ion Optics (Plenum, New York, 1988).
[CrossRef]

Venkatarayudu, T.

S. Bhagavantam, T. Venkatarayudu, Theory of Groups and Its Application to Physical Problems (Academic, New York, 1969).

Adv. Electron. Electron Phys. (1)

Y. Li, “Application of group theory to electron optics,” Adv. Electron. Electron Phys. 85, 231–258 (1993).
[CrossRef]

IEEE Trans. Electron Devices (1)

J. Szép, M. Szilagyi, “A novel approach to the synthesis of electrostatic lenses with minimized aberrations,” IEEE Trans. Electron Devices 35, 1181–1183 (1988).
[CrossRef]

J. Opt. Soc. Am. (1)

M. Szilagyi, P. H. Mui, “Symmetries in geometrical optics: applications,” J. Opt. Soc. Am. 12, 2760–2766 (1995).
[CrossRef]

J. Vac. Sci. Technol. B (3)

P. H. Mui, M. Szilagyi, “Synthesis of monopole-and-quadrupole focusing columns,” J. Vac. Sci. Technol. B 12, 3036–3045 (1994).
[CrossRef]

M. Szilagyi, P. H. Mui, “Synthesis of focusing-and-deflection columns,” J. Vac. Sci. Technol. B 13, 375–382 (1995).
[CrossRef]

M. Szilagyi, J. Szép, “Optimum design of electrostatic lenses,” J. Vac. Sci. Technol. B 6, 953–957 (1988).
[CrossRef]

Proc. IEEE (1)

M. Szilagyi, “Electron optical synthesis and optimization,” Proc. IEEE 73, 412–418 (1985).
[CrossRef]

Proc. Phys. Soc. B (1)

J. C. Burfoot, “Correction of electrostatic lenses by departure from rotational symmetry,” Proc. Phys. Soc. B 66, 775–792 (1953).
[CrossRef]

Z. Phys. (1)

O. Scherzer, “Über einige Fehler von Elektronenlinsen,” Z. Phys. 101, 593–603 (1936).
[CrossRef]

Other (10)

M. Szilagyi, Electron and Ion Optics (Plenum, New York, 1988).
[CrossRef]

S. Bhagavantam, T. Venkatarayudu, Theory of Groups and Its Application to Physical Problems (Academic, New York, 1969).

A. Septier, Advances in Optical and Electron Microscopy, R. Barer, V. E. Cosslett, eds. (Academic, New York, 1966), Vol. 1, pp. 204–274.

P. W. Hawkes, Quadrupoles in Electron Lens Design (Academic, New York, 1970).

D. C. Carey, The Optics of Charged Particle Beams (Harwood, New York, 1987).

S. Humphries, Principles of Charged Particle Acceleration (Wiley, New York, 1986).

Results in this section are not generally valid for systems with magnetic components.

Results for cases (1), (2), and (4) in this section are not generally valid for systems with magnetic components.

P. W. Milonni, J. H. Eberly, Lasers (Wiley-Interscience, New York, 1988).

C.-T. Chen, Linear System Theory and Design (Holt, Rinehart & Winston, New York, 1970).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (1)

Fig. 1
Fig. 1

Illustration of the retracing of the original ray through the reverse system. Note that the directional angle of the retracing ray on the input/output side is just the negative of that of the original ray on the output/ input side. Retraceability generally does not hold in magnetic systems.

Equations (44)

Equations on this page are rendered with MathJax. Learn more.

P 1 P 2 n d s = P 1 P 2 n ( x , y , z ) ( 1 + x 2 + y 2 ) 1 / 2 d z .
n ( x , y , z ) = p + q A . ê υ
[ x i x i y i y i ] = [ h 11 h 12 h 13 h 14 h 21 h 22 h 23 h 24 h 31 h 32 h 33 h 34 h 41 h 42 h 43 h 44 ] [ x o x o y o y o ] + [ x c x c y c y c ]
X _ i = H X _ o + X _ c ,
X _ i = H X _ o + X _ c
X _ o = H 1 X _ i H 1 X _ c .
T rev X _ o = H T rev X _ i + X _ c ,
T rev [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] .
X _ o = T rev H T rev X _ i + T rev X _ c .
H = T rev H 1 T rev , X _ c = T rev H 1 X _ c .
X _ i = T rev H 1 T rev X _ o T rev H 1 X _ c .
( I ) X _ i = T rev H 1 T rev ( I ) X _ o T rev H 1 X _ c ,
( T rev H 1 T rev H ) X _ i = 0 _ , ( I T rev H 1 ) X _ i = 0 _ .
( I T rev H 1 ) X _ c = 0 _
( I HT rev ) X _ c = 0 _ ; T rev H 1 T rev H = 0
H 1 = T rev HT rev .
T ref x [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] , T ref y [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] .
T ref x X _ i = T rev H 1 T rev T ref x X _ o T rev H 1 X _ c .
( T ref x T rev H 1 T rev T ref x H ) X _ i = ( I + T ref x T rev H 1 ) X _ c ,
T ref rev H 1 T ref rev H = 0 ,
T ref rev T ref x T rev = T rev T ref x = [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] .
( I + T ref rev H 1 ) X _ c = ( I + H T ref rev ) X _ c = 0 _ .
[ h 11 h 12 h 13 h 14 h 21 h 22 h 23 h 24 h 31 h 32 h 33 h 34 h 41 h 42 h 43 h 44 ] = [ h 11 h 12 h 13 h 14 h 21 h 22 h 23 h 24 h 31 h 32 h 33 h 34 h 41 h 42 h 43 h 44 ] .
( I + HT ref rev ) = [ 1 + h 11 h 12 h 13 h 14 h 21 1 h 22 h 23 h 24 h 31 h 32 1 h 33 h 34 h 41 h 42 h 43 1 + h 44 ] .
[ h 11 h 12 h 13 h 14 h 21 h 22 h 23 h 24 h 31 h 32 h 33 h 34 h 41 h 42 h 43 h 44 ] = [ h 11 h 12 0 0 h 21 h 22 0 0 0 0 h 33 h 34 0 0 h 43 h 44 ] .
T rot n = [ cos θ 0 sin θ 0 0 cos θ 0 sin θ sin θ 0 cos θ 0 0 sin θ 0 cos θ ] ,
T rot n X _ f = HT rot n X _ i + X _ c ( n = 0 , 1 , . . . N 1 ) .
( T rot nT HT rot n H ) X _ i = ( I T rot nT ) X _ c ( n = 0 , 1 , . . . N 1 ) ,
[ h 11 h 12 h 13 h 14 h 21 h 22 h 23 h 24 h 31 = h 13 h 32 = h 14 h 33 = h 11 h 34 = h 12 h 41 = h 23 h 42 = h 24 h 43 = h 21 h 44 = h 22 ]
X _ f = H N X _ i + ( i = 0 N 1 H i ) X _ c .
[ A B C D ] N = 1 sin φ × [ A sin N φ sin ( N 1 ) φ B sin N φ C sin N φ D sin N φ sin ( N 1 ) φ ] ,
H N = Q 1 G N Q , i = 0 N 1 H i = Q 1 ( i = 0 N 1 G i ) Q ,
X _ f = Q 1 G N Q X _ i + Q 1 ( i = 0 N 1 G i ) Q X _ c .
[ λ 1 0 0 0 0 λ 2 0 0 0 0 λ 3 0 0 0 0 λ 4 ] , [ λ 1 1 0 0 0 λ 1 0 0 0 0 λ 2 0 0 0 0 λ 3 ] , [ λ 1 1 0 0 0 λ 1 0 0 0 0 λ 2 1 0 0 0 λ 2 ] , [ λ 1 1 0 0 0 λ 1 1 0 0 0 λ 1 0 0 0 0 λ 2 ] , [ λ 1 1 0 0 0 λ 1 1 0 0 0 λ 1 1 0 0 0 λ 1 ] .
G i = [ λ 1 i 0 0 0 0 λ 2 i 0 0 0 0 λ 3 i 0 0 0 0 λ 4 i ] , G i = [ λ 1 i i λ 1 i 1 0 0 0 λ 1 i 0 0 0 0 λ 2 i 0 0 0 0 λ 3 i ] , G i = [ λ 1 i i λ 1 i 1 0 0 0 λ 1 i 0 0 0 0 λ 2 i i λ 2 i 1 0 0 0 λ 2 i ] , G i = [ λ 1 i i λ 1 i 1 i ( i 1 ) 2 λ 1 i 2 0 0 λ 1 i i λ 1 i 1 0 0 0 λ 1 i 0 0 0 0 λ 2 i ] , G i = [ λ 1 i i λ 1 i 1 i ( i 1 ) 2 λ 1 i 2 i ( i 2 ) ( i 1 ) 6 λ 1 i 3 0 λ 1 i i λ 1 i 1 i ( i 1 ) 2 i λ 1 i 2 0 0 λ 1 i i λ 1 i 1 0 0 0 λ 1 i ] .
Γ ( x ) x N 1 x 1 = i = 0 N 1 x i ( for finite N ) .
Γ I ( x ) = x N 1 [ ( N 1 ) x N ] + 1 ( x 1 ) 2 ,
Γ II ( x ) = ( N 2 ) x N 1 [ ( N 1 ) x 2 N ] + N ( N 1 ) x N 2 2 ( x 1 ) 3 ,
Γ III ( x ) = 1 ( x 1 ) 4 { ( N 2 ) ( N 1 ) x N 3 [ ( N 3 ) x 3 N ] 3 N ( N 3 ) x N 2 [ ( N 2 ) x ( N 1 ) ] + 6 } .
i = 0 N 1 G i = [ Γ ( λ 1 ) 0 0 0 0 Γ ( λ 2 ) 0 0 0 0 Γ ( λ 3 ) 0 0 0 0 Γ ( λ 4 ) ] , i = 0 N 1 G i = [ Γ ( λ 1 ) Γ I ( λ 1 ) 0 0 0 Γ ( λ 1 ) 0 0 0 0 Γ ( λ 2 ) 0 0 0 0 Γ ( λ 3 ) ] , i = 0 N 1 G i = [ Γ ( λ 1 ) Γ ( λ 1 ) 0 0 0 Γ ( λ 1 ) 0 0 0 0 Γ ( λ 2 ) Γ ( λ 2 ) 0 0 0 Γ ( λ 2 ) ] , i = 0 N 1 G i = [ Γ ( λ 1 ) Γ I ( λ 1 ) Γ II ( λ 1 ) 2 0 0 Γ ( λ 1 ) Γ I ( λ 1 ) 0 0 0 Γ ( λ 1 ) 0 0 0 0 Γ ( λ 2 ) ] , i = 0 N 1 G i = [ Γ ( λ 1 ) Γ I ( λ 1 ) Γ II ( λ 1 ) 2 Γ III ( λ 1 ) 6 0 Γ ( λ 1 ) Γ I ( λ 1 ) Γ II ( λ 1 ) 2 0 0 Γ ( λ 1 ) Γ I ( λ 1 ) 0 0 0 Γ ( λ 1 ) ] .
X _ 1 = H X _ 0 + X _ c .
X _ 2 = T 1 ( HT X _ 1 + X _ c ) .
X _ 2 = T 1 ( HT ) H X _ 0 + T 1 ( I + HT ) X _ c .
X _ N = T 1 N ( HT ) N 1 H X _ 0 + T 1 N [ i = 0 N 1 ( HT ) i ] X _ c .

Metrics