Abstract

Geometric optics can be completely derived from Fermat’s principle, as classical mechanics can be obtained by the application of the Hamilton principle. In Lagrangian optics, for optical systems with rotational symmetry, is known the invariant L3, the Lagrange optical invariant. For systems built only with spherical lenses, we demonstrate there are two other optical invariants, L1 and L2, analogous to L3. A proof based on Snell’s law, the Weierstrass–Erdman jump condition, and the expression of the ray between two optical surfaces in the Hamiltonian formalism is reported. The presence of a conserved vector, L, allows us to write the equation of an emerging ray without any approximation.

© 2011 Optical Society of America

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References

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  1. R. Feynman, R. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison-Wesley, 1963).
  2. A. Marasco and A. Romano, Il Nuovo Cimento B 121, 91(2006).
  3. M. S. Mahoney, The Mathematical Career of Pierre de Fermat, 1601-1665, 2nd ed. (Princeton U. Press, 1994).
  4. A. Romano, Geometric Optics (Springer, 2010).
    [CrossRef]
  5. V. Lakshminarayanan, A. Ghatak, and K. Thyagarajan, Lagrangian Optics (Springer, 2001).
    [CrossRef]
  6. D. J. Gross, Proc. Natl. Acad. Sci. USA 93, 14256 (1996).
    [CrossRef] [PubMed]
  7. F. Corrente, “Questioni di Ottica Geometrica,” Master’sthesis (Università degli studi di Napoli Federico II, 2005).
  8. L. P. Lebedev and M. J. Cloud, The Calculus of Variations and Functional Analysis: With Optimal Control and Applications in Mechanics (World Scientific, 2003).
    [CrossRef] [PubMed]
  9. R. K. Luneburg, Mathematical Theory of Optics (U. of California Press, 1964).
  10. H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. Press, 1970).

2006 (1)

A. Marasco and A. Romano, Il Nuovo Cimento B 121, 91(2006).

1996 (1)

D. J. Gross, Proc. Natl. Acad. Sci. USA 93, 14256 (1996).
[CrossRef] [PubMed]

Buchdahl, H. A.

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. Press, 1970).

Cloud, M. J.

L. P. Lebedev and M. J. Cloud, The Calculus of Variations and Functional Analysis: With Optimal Control and Applications in Mechanics (World Scientific, 2003).
[CrossRef] [PubMed]

Corrente, F.

F. Corrente, “Questioni di Ottica Geometrica,” Master’sthesis (Università degli studi di Napoli Federico II, 2005).

Feynman, R.

R. Feynman, R. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison-Wesley, 1963).

Ghatak, A.

V. Lakshminarayanan, A. Ghatak, and K. Thyagarajan, Lagrangian Optics (Springer, 2001).
[CrossRef]

Gross, D. J.

D. J. Gross, Proc. Natl. Acad. Sci. USA 93, 14256 (1996).
[CrossRef] [PubMed]

Lakshminarayanan, V.

V. Lakshminarayanan, A. Ghatak, and K. Thyagarajan, Lagrangian Optics (Springer, 2001).
[CrossRef]

Lebedev, L. P.

L. P. Lebedev and M. J. Cloud, The Calculus of Variations and Functional Analysis: With Optimal Control and Applications in Mechanics (World Scientific, 2003).
[CrossRef] [PubMed]

Leighton, R.

R. Feynman, R. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison-Wesley, 1963).

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (U. of California Press, 1964).

Mahoney, M. S.

M. S. Mahoney, The Mathematical Career of Pierre de Fermat, 1601-1665, 2nd ed. (Princeton U. Press, 1994).

Marasco, A.

A. Marasco and A. Romano, Il Nuovo Cimento B 121, 91(2006).

Romano, A.

A. Marasco and A. Romano, Il Nuovo Cimento B 121, 91(2006).

A. Romano, Geometric Optics (Springer, 2010).
[CrossRef]

Sands, M.

R. Feynman, R. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison-Wesley, 1963).

Thyagarajan, K.

V. Lakshminarayanan, A. Ghatak, and K. Thyagarajan, Lagrangian Optics (Springer, 2001).
[CrossRef]

Il Nuovo Cimento B (1)

A. Marasco and A. Romano, Il Nuovo Cimento B 121, 91(2006).

Proc. Natl. Acad. Sci. USA (1)

D. J. Gross, Proc. Natl. Acad. Sci. USA 93, 14256 (1996).
[CrossRef] [PubMed]

Other (8)

F. Corrente, “Questioni di Ottica Geometrica,” Master’sthesis (Università degli studi di Napoli Federico II, 2005).

L. P. Lebedev and M. J. Cloud, The Calculus of Variations and Functional Analysis: With Optimal Control and Applications in Mechanics (World Scientific, 2003).
[CrossRef] [PubMed]

R. K. Luneburg, Mathematical Theory of Optics (U. of California Press, 1964).

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. Press, 1970).

M. S. Mahoney, The Mathematical Career of Pierre de Fermat, 1601-1665, 2nd ed. (Princeton U. Press, 1994).

A. Romano, Geometric Optics (Springer, 2010).
[CrossRef]

V. Lakshminarayanan, A. Ghatak, and K. Thyagarajan, Lagrangian Optics (Springer, 2001).
[CrossRef]

R. Feynman, R. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison-Wesley, 1963).

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Equations (28)

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OPL γ L ( x , x ˙ , t ) d t = γ N ( x ) x ˙ 1 2 + x ˙ 2 2 + x ˙ 3 2 d t ,
L ( x α , x ˙ α , x 3 ) = N ( x ) 1 + x ˙ 1 2 + x ˙ 2 2 , α = 1 , 2 ,
L 3 p 1 x 2 p 2 x 1 ,
p i = L x ˙ i = N x ˙ i 1 + x ˙ 1 2 + x ˙ 2 2 ,
[ [ p ] ] = λ n ,
λ = N cos i N cos i = N 1 sin 2 i N cos i .
N sin i = N sin i .
x α ( x 3 ) = p α p 3 ( x 3 x 3 , 0 ) + x α , 0 , α = 1 , 2 ,
x α ( x 3 ) = p α p 3 ( x 3 x 3 , 0 ) + x α , 0 , α = 1 , 2 .
x α , 0 = p α p 3 ( x 3 , 0 x 3 , 0 ) + x α , 0 , α = 1 , 2 ,
n i = 1 | F | F x i = 1 | F | F ρ 2 ρ 2 x i = 2 x i , 0 1 | F | F ρ 2 .
L 3 = L 3 .
x 1 2 + x 2 2 + x 3 2 = R 2 ,
x i ( x 3 , 0 ) = x i , 0 , p i ( x 3 , 0 ) = p i , 0 ,
x α ( x 3 ) = p α + λ n α p 3 + λ n 3 ( x 3 x 3 , int ) + p α p 3 ( x 3 , int x 0 ) + x α , 0
n = 1 R ( x 1 ( t int ) , x 2 ( t int ) , x 3 , int ) ) .
L 1 = L 1 , L 2 = L 2 ,
N ( ρ ) = 2 ρ 2 , 0 ρ 1 .
cos i = p · x ( t int ) N R = N 2 R 2 L 2 N R .
λ = N 2 R 2 L 2 N 2 R 2 L 2 R .
L 1 = x 3 , 0 p 2 , L 2 = x 3 , 0 p 1 x 1 , 0 p 3 , L 3 = x 1 , 0 p 2 , x 1 , 0 2 + x 3 , 0 2 = R D 2 .
f ( x 1 , x 2 , x 3 , p 1 , p 2 ) = 0.
p 1 = L 1 L 2 + N 2 R 2 L 2 L 3 R D L 1 2 + L 3 2 , p 2 = L 1 2 + L 3 2 R D , x 1 , 0 = R D L 3 L 1 2 + L 3 2 , x 3 , 0 = R D L 1 L 1 2 + L 3 2 .
x 1 ( x 3 ) = p 1 p 3 ( x 3 x 3 , 0 ) + x 0 , x 2 ( x 3 ) = p 2 p 3 ( x 3 x 3 , 0 ) .
2 ρ ( x 3 , D ) D
p i = ± N ( x i ξ i ) ( x 1 ξ 1 ) 2 + ( x 2 ξ 2 ) 2 + ( x 3 ξ 3 ) 2 .
x i ( x 3 ) = x i 1 + x i 3 + x i 5 + , i = 1 , 2 ,
ρ ( x 3 ) 2 = ( x 1 2 + x 2 2 ) + ( x 1 1 x 1 3 + x 2 1 x 2 3 ) + ( x 1 1 x 1 5 + x 2 1 x 2 5 + x 1 3 + x 2 3 ) + .

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