Abstract

A new method, employing Lie algebraic tools, is presented for characterizing optical systems and computing aberrations. It represents the action of each separate element of a compound optical system, including all departures from Gaussian optics, by a certain operator. These operators can then be concatenated, following well-defined rules, to obtain a resultant operator that characterizes the entire system. New insight into the origin and possible correction of aberrations is provided. With some effort, it should be possible to produce, by manual calculations, explicit formulas for the third-, fourth-, and fifth-order aberrations of a general optical system including systems without axial symmetry. With the aid of symbolic manipulation computer programs, it should be possible to compute routinely explicit formulas for aberrations of seventh, eighth, and ninth order, and probably beyond.

© 1982 Optical Society of America

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References

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  1. W. R. Hamilton, Trans. R. Irish Acad. 15, 69 (1828); Trans. R. Irish Acad. 16, 1 (1830); Trans. R. Irish Acad. 16, 93 (1831); Trans. R. Irish Acad. 17, 1 (1837). Reprinted in The Mathematical Papers of Sir W. R. Hamilton, Vol. I, Geometrical Optics, A. W. Conway and J. L. Synge, eds. (Cambridge U. Press, Cambridge, 1931).
  2. See, for example, M. Born and E. Wolf, Principles of Optics, 2nd ed. (Macmillan, New York, 1964), pp. 134, 211, 221.
  3. Matrix methods are described, for example, by K. Halbach, Am. J. Phys.,  32, 90 (1964); M. Klein, Optics (Wiley, New York, 1970), p. 84; W. Brouwer, Matrix Methods in Optical Instrument Design (Benjamin, New York, 1964).
    [Crossref]
  4. H. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968); An Introduction to Hamiltonian Optics (Cambridge U. Press, Cambridge, 1970), p. 36. See also the following papers of H. Buchdahl: J. Opt. Soc. Am. 62, 1314 (1972); Optik 37, 571 (1973); Optik 40, 460 (1974); Optik 46, 287 (1976); Optik 46, 393 (1976); Optik 48, 53 (1977).
    [Crossref]
  5. R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1953), Vol. 1, pp. 164–190.
  6. A. Ghatak and K. Thyagarajan, Contemporary Optics (Plenum, New York, 1978), p. 15.
  7. There does not seem to be any particularly transparent derivation of this condition in the literature. However, it may be inferred, for example, from H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, Reading, Mass., 1980), pp. 394–396; E. Salatan and A. Cromer, Theoretical Mechanics (Wiley, New York, 1971), p. 222.
  8. H. Weyl, The Classical Groups (Princeton U. Press, Princeton, N.J., 1946), p. 165.
  9. O. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972), p. 245. This book contains an extensive discussion of optics from a group theoretical perspective. The reader is also referred to the paper of M. Herzberger, Trans. Am. Math. Soc. 53, 218 (1943), for an early discussion of the consequences of what is essentially the symplectic condition.
    [Crossref]
  10. A. Dragt and J. Finn, J. Math. Phys. 17, 2215 (1976).
    [Crossref]
  11. It is not necessarily the case that every symplectic map corresponds to a physically realizable optical system.
  12. A. Ghatak and K. Thyagarajan, Contemporary Optics (Plenum, New York, 1978), p. 52.
  13. The program accosv is a product of Scientific Calculations, Inc.

1976 (1)

A. Dragt and J. Finn, J. Math. Phys. 17, 2215 (1976).
[Crossref]

1964 (1)

Matrix methods are described, for example, by K. Halbach, Am. J. Phys.,  32, 90 (1964); M. Klein, Optics (Wiley, New York, 1970), p. 84; W. Brouwer, Matrix Methods in Optical Instrument Design (Benjamin, New York, 1964).
[Crossref]

1828 (1)

W. R. Hamilton, Trans. R. Irish Acad. 15, 69 (1828); Trans. R. Irish Acad. 16, 1 (1830); Trans. R. Irish Acad. 16, 93 (1831); Trans. R. Irish Acad. 17, 1 (1837). Reprinted in The Mathematical Papers of Sir W. R. Hamilton, Vol. I, Geometrical Optics, A. W. Conway and J. L. Synge, eds. (Cambridge U. Press, Cambridge, 1931).

Born, M.

See, for example, M. Born and E. Wolf, Principles of Optics, 2nd ed. (Macmillan, New York, 1964), pp. 134, 211, 221.

Buchdahl, H.

H. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968); An Introduction to Hamiltonian Optics (Cambridge U. Press, Cambridge, 1970), p. 36. See also the following papers of H. Buchdahl: J. Opt. Soc. Am. 62, 1314 (1972); Optik 37, 571 (1973); Optik 40, 460 (1974); Optik 46, 287 (1976); Optik 46, 393 (1976); Optik 48, 53 (1977).
[Crossref]

Courant, R.

R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1953), Vol. 1, pp. 164–190.

Dragt, A.

A. Dragt and J. Finn, J. Math. Phys. 17, 2215 (1976).
[Crossref]

Finn, J.

A. Dragt and J. Finn, J. Math. Phys. 17, 2215 (1976).
[Crossref]

Ghatak, A.

A. Ghatak and K. Thyagarajan, Contemporary Optics (Plenum, New York, 1978), p. 52.

A. Ghatak and K. Thyagarajan, Contemporary Optics (Plenum, New York, 1978), p. 15.

Goldstein, H.

There does not seem to be any particularly transparent derivation of this condition in the literature. However, it may be inferred, for example, from H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, Reading, Mass., 1980), pp. 394–396; E. Salatan and A. Cromer, Theoretical Mechanics (Wiley, New York, 1971), p. 222.

Halbach, K.

Matrix methods are described, for example, by K. Halbach, Am. J. Phys.,  32, 90 (1964); M. Klein, Optics (Wiley, New York, 1970), p. 84; W. Brouwer, Matrix Methods in Optical Instrument Design (Benjamin, New York, 1964).
[Crossref]

Hamilton, W. R.

W. R. Hamilton, Trans. R. Irish Acad. 15, 69 (1828); Trans. R. Irish Acad. 16, 1 (1830); Trans. R. Irish Acad. 16, 93 (1831); Trans. R. Irish Acad. 17, 1 (1837). Reprinted in The Mathematical Papers of Sir W. R. Hamilton, Vol. I, Geometrical Optics, A. W. Conway and J. L. Synge, eds. (Cambridge U. Press, Cambridge, 1931).

Hilbert, D.

R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1953), Vol. 1, pp. 164–190.

Stavroudis, O.

O. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972), p. 245. This book contains an extensive discussion of optics from a group theoretical perspective. The reader is also referred to the paper of M. Herzberger, Trans. Am. Math. Soc. 53, 218 (1943), for an early discussion of the consequences of what is essentially the symplectic condition.
[Crossref]

Thyagarajan, K.

A. Ghatak and K. Thyagarajan, Contemporary Optics (Plenum, New York, 1978), p. 52.

A. Ghatak and K. Thyagarajan, Contemporary Optics (Plenum, New York, 1978), p. 15.

Weyl, H.

H. Weyl, The Classical Groups (Princeton U. Press, Princeton, N.J., 1946), p. 165.

Wolf, E.

See, for example, M. Born and E. Wolf, Principles of Optics, 2nd ed. (Macmillan, New York, 1964), pp. 134, 211, 221.

Am. J. Phys. (1)

Matrix methods are described, for example, by K. Halbach, Am. J. Phys.,  32, 90 (1964); M. Klein, Optics (Wiley, New York, 1970), p. 84; W. Brouwer, Matrix Methods in Optical Instrument Design (Benjamin, New York, 1964).
[Crossref]

J. Math. Phys. (1)

A. Dragt and J. Finn, J. Math. Phys. 17, 2215 (1976).
[Crossref]

Trans. R. Irish Acad. (1)

W. R. Hamilton, Trans. R. Irish Acad. 15, 69 (1828); Trans. R. Irish Acad. 16, 1 (1830); Trans. R. Irish Acad. 16, 93 (1831); Trans. R. Irish Acad. 17, 1 (1837). Reprinted in The Mathematical Papers of Sir W. R. Hamilton, Vol. I, Geometrical Optics, A. W. Conway and J. L. Synge, eds. (Cambridge U. Press, Cambridge, 1931).

Other (10)

See, for example, M. Born and E. Wolf, Principles of Optics, 2nd ed. (Macmillan, New York, 1964), pp. 134, 211, 221.

It is not necessarily the case that every symplectic map corresponds to a physically realizable optical system.

A. Ghatak and K. Thyagarajan, Contemporary Optics (Plenum, New York, 1978), p. 52.

The program accosv is a product of Scientific Calculations, Inc.

H. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968); An Introduction to Hamiltonian Optics (Cambridge U. Press, Cambridge, 1970), p. 36. See also the following papers of H. Buchdahl: J. Opt. Soc. Am. 62, 1314 (1972); Optik 37, 571 (1973); Optik 40, 460 (1974); Optik 46, 287 (1976); Optik 46, 393 (1976); Optik 48, 53 (1977).
[Crossref]

R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1953), Vol. 1, pp. 164–190.

A. Ghatak and K. Thyagarajan, Contemporary Optics (Plenum, New York, 1978), p. 15.

There does not seem to be any particularly transparent derivation of this condition in the literature. However, it may be inferred, for example, from H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, Reading, Mass., 1980), pp. 394–396; E. Salatan and A. Cromer, Theoretical Mechanics (Wiley, New York, 1971), p. 222.

H. Weyl, The Classical Groups (Princeton U. Press, Princeton, N.J., 1946), p. 165.

O. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972), p. 245. This book contains an extensive discussion of optics from a group theoretical perspective. The reader is also referred to the paper of M. Herzberger, Trans. Am. Math. Soc. 53, 218 (1943), for an early discussion of the consequences of what is essentially the symplectic condition.
[Crossref]

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Figures (2)

Fig. 1
Fig. 1

Optical system consisting of an optical device preceded and followed by simple transit. A ray originates at Pi with location ri and direction ŝi and terminates at Pf with location rf and direction ŝf.

Fig. 2
Fig. 2

Lens with planar entrance and exit faces. It is composed of two media having indices of refraction n1 and n2, respectively, that are separated by a curved interface.

Tables (1)

Tables Icon

Table 1 Table of Aberrations

Equations (70)

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n i ŝ i = - V / r i ,
n f ŝ f = V / r f .
d s = [ ( d z ) 2 + ( d x ) 2 + ( d y ) 2 ] 1 / 2 = [ 1 + ( x ) 2 + ( y ) 2 ] 1 / 2 d z .
A = z i z f n ( x , y , z ) [ 1 + ( x ) 2 + ( y ) 2 ] 1 / 2 d z .
d / d z ( L / x ) - L / x = 0 , d / d z ( L / y ) - L / y = 0 ,
L = n ( x , y , z ) [ 1 + ( x ) 2 + ( y ) 2 ] 1 / 2 .
p x = L / x ,             p y = L / y ,
p x = n ( r ) x / [ 1 + ( x ) 2 + ( y ) 2 ] 1 / 2 , p y = n ( r ) y / [ 1 + ( x ) 2 + ( y ) 2 ] 1 / 2 .
H = - [ n 2 ( r ) - p x 2 - p y 2 ] 1 / 2 .
w = ( w 1 , w 2 , w 3 , w 4 ) = ( q x , q y , p x , p y ) .
w f = M w i .
M α β = W α f / w β i .
J = ( 0 I - I 0 ) .
M ˜ J M = J .
[ f , g ] = Σ ( f / q i ) ( g / p i ) - ( f / p i ) ( g / q i ) .
: f : g = [ f , g ] .
: f : 2 g = [ f , [ f , g ] ] .
: f : 0 g = g .
exp ( : f : ) = n = 0 : f : n / n ! .
exp ( : f : ) g = g + [ f , g ] + [ f , [ f , g ] ] / 2 ! + .
M = exp ( : f 2 : ) exp ( : f 3 : ) exp ( : f 4 : ) .
M = exp ( : f 2 : ) ,
f 2 = - { l / ( 2 n ) } ( p i ) 2 .
: f 2 : p α i = - { l / ( 2 n ) } [ ( p i ) 2 , p α i ] = 0 ,
: f 2 : q α i = - { l / ( 2 n ) } [ ( p i ) 2 , q α i ] = ( l / n ) p α i ,
: f 2 : 2 q α i = ( l / n ) : f 2 : p α i = 0 , etc .
p α f = M p α i = exp ( : f 2 : ) p α i = p α i , q α f = M q α i = exp ( : f 2 : ) q α i = q α i + ( l / n ) p α i .
M = exp { - l / ( 2 n ) : ( p i ) 2 : } Transit by distance l through medium of refractive index n in the Gaussian approximation .
f 2 = { ( n 2 - n 1 ) / ( 2 r ) } ( q i ) 2 .
: f 2 : q α i = { ( n 2 - n 1 ) / ( 2 r ) } [ ( q i ) 2 , q α i ] = 0 ,
: f 2 : p α i = { ( n 2 - n 1 ) / ( 2 r ) } [ ( q i ) 2 , p α i ] = { ( n 2 - n 1 ) / r } q α i ,
: f 2 : 2 p α i = { ( n 2 - n 1 ) / r } : f 2 : q α i = 0 , etc .
q α f = M q α i = exp ( : f 2 : ) q α i = q α i , p α f = M p α i = exp ( : f 2 : ) p α i = p α i + { ( n 2 - n 1 ) / r } q α i .
M = exp { ( n 2 - n 1 ) / ( 2 r ) : ( q i ) 2 : } Refraction in the Gaussian approximation by a spherical surface of radius r separating media having refractive indices n 1 and n 2 .
f 2 = - σ ( p i · q i ) ,
q f = ( e σ ) q i , p f = ( e - σ ) p i .
M = exp ( : f 2 : ) exp ( : f 4 : ) exp ( : f 6 : ) .
M = exp ( : f 2 : ) exp ( : f 4 : ) .
w ¯ = exp ( : f 4 : ) w = w + [ f 4 , w ] + [ f 4 , [ f 4 , w ] ] / 2 ! .
f 4 = A ( p 2 ) 2 + B p 2 ( p · q ) + C ( p · q ) 2 + D p 2 q 2 + E ( p · q ) q 2 + F ( q 2 ) 2 .
q ¯ α = q α + [ A ( p 2 ) 2 , q α ] .
Δ q α = q ¯ α - q α = - A ( p 2 ) 2 / p α = - 4 A p 2 p α .
M = exp { l : ( n 2 - p 2 ) 1 / 2 : } = exp { ( - l ) / ( 2 n ) : p 2 : } × exp { ( - l ) / ( 8 n 3 ) : ( p 2 ) 2 : } .
z = - α ( x 2 + y 2 ) + β ( x 2 + y 2 ) 2 + γ ( x 2 + y 2 ) 3 + .
z = - r + ( r 2 - x 2 - y 2 ) 1 / 2 .
α = 1 / ( 2 r ) , β = - α 3 = - 1 / ( 8 r 3 ) .
M = exp { ( - t ) / ( 2 n 1 ) : p 2 : } exp { ( - t ) / ( 8 n 1 3 ) : ( p 2 ) 2 : } × exp { α ( n 2 - n 1 ) : q 2 : } exp ( : f 4 : ) .
A = 0 ,             B = 0 ,             C = 0 , D = α ( n 2 - n 1 ) / ( 2 n 1 n 2 ) , E = 2 α 2 ( n 1 - n 2 ) / n 1 , F = α 3 ( n 1 - n 2 ) { n 1 [ ( β / α 3 ) + 2 ] - 2 n 2 } / n 1 .
M = exp { ( - d 1 / 2 ) : p 2 : } exp { ( - d 1 / 8 ) : ( p 2 ) 2 : } × exp { ( - t ) / ( 2 n ) : p 2 : } exp { ( - t ) / ( 8 n 3 ) : ( p 2 ) 2 : } × exp { α ( 1 - n ) : q 2 : } exp ( : f 4 : ) × exp { ( - d 2 / 2 ) : p 2 : } exp { ( - d 2 / 8 ) : ( p 2 ) 2 : } .
1 / ( d 1 + t / n ) + 1 / d 2 = 1 / f .
1 / f = ( n - 1 ) / r = 2 α ( n - 1 ) .
M = M G exp ( : f 4 * : ) .
M G = exp { ( - d 1 / 2 ) : p 2 : } × exp { ( - t ) / ( 2 n ) : p 2 : } exp { α ( 1 - n ) : q 2 : } × exp { ( - d 2 / 2 ) : p 2 : } ,
( 1 d 2 0 1 ) ( 1 0 - 1 / f 1 ) ( 1 t / n 0 1 ) ( 1 d 1 0 1 ) .
( m 0 - 1 / f 1 / m ) ,
m = - d 2 / ( d 1 + t / n ) .
exp ( : f 2 : ) w i = j M i j f w j , etc .
exp ( : f 2 : ) exp ( : g 2 : ) w i = exp ( : f 2 : ) j M i j g w j = j M i j g exp ( : f 2 : ) w j = j M i j g k M j k f w k = k ( M g M f ) i k w k .
A * = - m 4 ( d 1 + t n - 3 ) / 8 + ( d 2 2 D - d 2 3 E + d 2 4 F ) - d 2 / 8 , B * = - m 3 ( d 1 + t n - 3 ) / ( 2 f ) + ( - 2 d 2 D + 3 d 2 2 E - 4 d 2 3 F ) , C * = - m 2 ( d 1 + t n - 3 ) / ( 2 f 2 ) + ( - 2 d 2 E + 4 d 2 2 F ) , D * = - m 2 ( d 1 + t n - 3 ) / ( 4 f 2 ) + ( D - d 2 E + 2 d 2 2 F ) , E * = - m ( d 1 + t n - 3 ) / ( 2 f 3 ) + ( E - 4 d 2 F ) , F * = - ( d 1 + t n - 3 ) / ( 8 f 4 ) + F .
2 D * - C * = 2 D = α ( n 2 - n 1 ) / ( n 1 n 2 )
M = exp { d : ( n 2 - p 2 ) 1 / 2 : } .
n 2 = n 0 2 [ 1 - α ( x 2 + y 2 ) + β ( x 2 + y 2 ) 2 + γ ( x 2 + y 2 ) 3 + ] .
M = exp ( : f 2 : ) exp ( : f 4 : ) .
f 2 = - d { p 2 / ( 2 n 0 ) + n 0 α q 2 / 2 } .
f 4 = - ( f 2 ) 2 / ( 2 n 0 d ) + d h 4 .
h 4 = ( n 0 β / 2 ) 0 1 d λ { [ q cos ( λ d α ) + [ p / ( n 0 α ) ] sin ( λ d α ) ] 2 } 2 .
d 1 = 19.0 , t = 1.5 , n = 1.5 , r = 5.0 , d 2 = 20.0
A d = β + 1 / ( 8 r 3 ) ,
A d = 1.1450 × 10 - 3 .
A d = 1.1449653 × 10 - 3 .