Abstract

In the paraxial regime of three-dimensional optics, two evolution Hamiltonians are equivalent when one can be transformed to the other modulo scale by similarity through an optical system. To determine the equivalence sets of paraxial optical Hamiltonians one requires the orbit analysis of the algebra sp(4, ℜ) of 4×4 real Hamiltonian matrices. Our strategy uses instead the isomorphic algebra so(3, 2) of 5×5 matrices with metric (+1, +1, +1, -1, -1) to find four orbit regions (strata), six isolated orbits at their boundaries, and six degenerate orbits at their common point. We thus resolve the degeneracies of the eigenvalue classification.

© 2002 Optical Society of America

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References

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  1. H. M. Ozaktas, Z. Zalevsky, M. Alper Kutay, The Fractional Fourier Transform (Wiley, Chichester, UK, 2001).
  2. S. Steinberg, “Lie series, Lie transformations, and their applications,” in Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986), Chap. 3, pp. 45–102.
    [CrossRef]
  3. A. J. Dragt, E. Forest, K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics,” in Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986) Chap. 4, pp. 105–158.
  4. R. Simon, K. B. Wolf, “Structure of the set of paraxial optical systems,” J. Opt. Soc. Am. A 17, 342–355 (2000).
    [CrossRef]
  5. R. Simon, K. B. Wolf, “Fractional Fourier transforms in two dimensions,” J. Opt. Soc. Am. A 17, 2368–2381 (2000).
    [CrossRef]
  6. R. Gilmore, Lie Groups, Lie Algebras, and Some of Their Applications (Wiley, New York, 1974).
  7. A. J. Dragt, “Elementary and advanced Lie algebraic methods with applications to accelerator design, electron microscopes, and light optics,” Nucl. Instrum. Methods Phys. Res. A 258, 339–354 (1987).
    [CrossRef]
  8. P. A. M. Dirac, “A remarkable representation of the 3+2 de Sitter group,” J. Math. Phys. 4, 901–909 (1963).
    [CrossRef]
  9. C. Fronsdal, “Elementary particles in a curved space,” Rev. Mod. Phys. 37, 201–223 (1965).
    [CrossRef]
  10. T. O. Philips, E. P. Wigner, “De Sitter space and positive energy,” in Group Theory and Its Applications, E. M. Loebl, ed. (Academic, New York, 1968), pp. 631–676.
  11. A. J. Laub, K. Meyer, “Canonical forms for symplectic and Hamiltonian matrices,” Celest. Mech. 9, 213–238 (1974).
    [CrossRef]
  12. J. Patera, R. T. Sharp, P. Winternitz, H. Zassenhaus, “Continuous subgroups of the fundamental groups of physics. III. The de Sitter groups,” J. Math. Phys. 18, 2259–2288 (1977).
    [CrossRef]
  13. M. Moshinsky, P. Winternitz, “Quadratic Hamiltonians in phase space and their eigenstates,” J. Math. Phys. 21, 1667–1682 (1980).
    [CrossRef]
  14. J. Patera, P. Winternitz, H. Zassenhaus, “Maximal abelian subalgebras of real and complex symplectic Lie algebras,” J. Math. Phys. 24, 1973–1985 (1983).
    [CrossRef]
  15. R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
    [CrossRef] [PubMed]
  16. R. Simon, N. Mukunda, “Optical phase space, Wigner representation, and invariant quality parameters,” J. Opt. Soc. Am. A 17, 342–355 (2000).
    [CrossRef]
  17. M. Kauderer, Symplectic Matrices, First Order Systems and Special Relativity (World Scientific, Singapore, 1994).
  18. K. B. Wolf, “Dynamical groups for the point rotor and the hydrogen atom,” Nuovo Cimento Suppl. 5, 1041–1050 (1967).
  19. K. B. Wolf, “The group-theoretical treatment of aberrating systems. III. The classification of asymmetric aberrations,” J. Math. Phys. 28, 2498–2507 (1987).
    [CrossRef]

2000

1987

K. B. Wolf, “The group-theoretical treatment of aberrating systems. III. The classification of asymmetric aberrations,” J. Math. Phys. 28, 2498–2507 (1987).
[CrossRef]

A. J. Dragt, “Elementary and advanced Lie algebraic methods with applications to accelerator design, electron microscopes, and light optics,” Nucl. Instrum. Methods Phys. Res. A 258, 339–354 (1987).
[CrossRef]

1985

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[CrossRef] [PubMed]

1983

J. Patera, P. Winternitz, H. Zassenhaus, “Maximal abelian subalgebras of real and complex symplectic Lie algebras,” J. Math. Phys. 24, 1973–1985 (1983).
[CrossRef]

1980

M. Moshinsky, P. Winternitz, “Quadratic Hamiltonians in phase space and their eigenstates,” J. Math. Phys. 21, 1667–1682 (1980).
[CrossRef]

1977

J. Patera, R. T. Sharp, P. Winternitz, H. Zassenhaus, “Continuous subgroups of the fundamental groups of physics. III. The de Sitter groups,” J. Math. Phys. 18, 2259–2288 (1977).
[CrossRef]

1974

A. J. Laub, K. Meyer, “Canonical forms for symplectic and Hamiltonian matrices,” Celest. Mech. 9, 213–238 (1974).
[CrossRef]

1967

K. B. Wolf, “Dynamical groups for the point rotor and the hydrogen atom,” Nuovo Cimento Suppl. 5, 1041–1050 (1967).

1965

C. Fronsdal, “Elementary particles in a curved space,” Rev. Mod. Phys. 37, 201–223 (1965).
[CrossRef]

1963

P. A. M. Dirac, “A remarkable representation of the 3+2 de Sitter group,” J. Math. Phys. 4, 901–909 (1963).
[CrossRef]

Alper Kutay, M.

H. M. Ozaktas, Z. Zalevsky, M. Alper Kutay, The Fractional Fourier Transform (Wiley, Chichester, UK, 2001).

Dirac, P. A. M.

P. A. M. Dirac, “A remarkable representation of the 3+2 de Sitter group,” J. Math. Phys. 4, 901–909 (1963).
[CrossRef]

Dragt, A. J.

A. J. Dragt, “Elementary and advanced Lie algebraic methods with applications to accelerator design, electron microscopes, and light optics,” Nucl. Instrum. Methods Phys. Res. A 258, 339–354 (1987).
[CrossRef]

A. J. Dragt, E. Forest, K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics,” in Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986) Chap. 4, pp. 105–158.

Forest, E.

A. J. Dragt, E. Forest, K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics,” in Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986) Chap. 4, pp. 105–158.

Fronsdal, C.

C. Fronsdal, “Elementary particles in a curved space,” Rev. Mod. Phys. 37, 201–223 (1965).
[CrossRef]

Gilmore, R.

R. Gilmore, Lie Groups, Lie Algebras, and Some of Their Applications (Wiley, New York, 1974).

Kauderer, M.

M. Kauderer, Symplectic Matrices, First Order Systems and Special Relativity (World Scientific, Singapore, 1994).

Laub, A. J.

A. J. Laub, K. Meyer, “Canonical forms for symplectic and Hamiltonian matrices,” Celest. Mech. 9, 213–238 (1974).
[CrossRef]

Meyer, K.

A. J. Laub, K. Meyer, “Canonical forms for symplectic and Hamiltonian matrices,” Celest. Mech. 9, 213–238 (1974).
[CrossRef]

Moshinsky, M.

M. Moshinsky, P. Winternitz, “Quadratic Hamiltonians in phase space and their eigenstates,” J. Math. Phys. 21, 1667–1682 (1980).
[CrossRef]

Mukunda, N.

R. Simon, N. Mukunda, “Optical phase space, Wigner representation, and invariant quality parameters,” J. Opt. Soc. Am. A 17, 342–355 (2000).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[CrossRef] [PubMed]

Ozaktas, H. M.

H. M. Ozaktas, Z. Zalevsky, M. Alper Kutay, The Fractional Fourier Transform (Wiley, Chichester, UK, 2001).

Patera, J.

J. Patera, P. Winternitz, H. Zassenhaus, “Maximal abelian subalgebras of real and complex symplectic Lie algebras,” J. Math. Phys. 24, 1973–1985 (1983).
[CrossRef]

J. Patera, R. T. Sharp, P. Winternitz, H. Zassenhaus, “Continuous subgroups of the fundamental groups of physics. III. The de Sitter groups,” J. Math. Phys. 18, 2259–2288 (1977).
[CrossRef]

Philips, T. O.

T. O. Philips, E. P. Wigner, “De Sitter space and positive energy,” in Group Theory and Its Applications, E. M. Loebl, ed. (Academic, New York, 1968), pp. 631–676.

Sharp, R. T.

J. Patera, R. T. Sharp, P. Winternitz, H. Zassenhaus, “Continuous subgroups of the fundamental groups of physics. III. The de Sitter groups,” J. Math. Phys. 18, 2259–2288 (1977).
[CrossRef]

Simon, R.

Steinberg, S.

S. Steinberg, “Lie series, Lie transformations, and their applications,” in Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986), Chap. 3, pp. 45–102.
[CrossRef]

Sudarshan, E. C. G.

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[CrossRef] [PubMed]

Wigner, E. P.

T. O. Philips, E. P. Wigner, “De Sitter space and positive energy,” in Group Theory and Its Applications, E. M. Loebl, ed. (Academic, New York, 1968), pp. 631–676.

Winternitz, P.

J. Patera, P. Winternitz, H. Zassenhaus, “Maximal abelian subalgebras of real and complex symplectic Lie algebras,” J. Math. Phys. 24, 1973–1985 (1983).
[CrossRef]

M. Moshinsky, P. Winternitz, “Quadratic Hamiltonians in phase space and their eigenstates,” J. Math. Phys. 21, 1667–1682 (1980).
[CrossRef]

J. Patera, R. T. Sharp, P. Winternitz, H. Zassenhaus, “Continuous subgroups of the fundamental groups of physics. III. The de Sitter groups,” J. Math. Phys. 18, 2259–2288 (1977).
[CrossRef]

Wolf, K. B.

R. Simon, K. B. Wolf, “Fractional Fourier transforms in two dimensions,” J. Opt. Soc. Am. A 17, 2368–2381 (2000).
[CrossRef]

R. Simon, K. B. Wolf, “Structure of the set of paraxial optical systems,” J. Opt. Soc. Am. A 17, 342–355 (2000).
[CrossRef]

K. B. Wolf, “The group-theoretical treatment of aberrating systems. III. The classification of asymmetric aberrations,” J. Math. Phys. 28, 2498–2507 (1987).
[CrossRef]

K. B. Wolf, “Dynamical groups for the point rotor and the hydrogen atom,” Nuovo Cimento Suppl. 5, 1041–1050 (1967).

A. J. Dragt, E. Forest, K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics,” in Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986) Chap. 4, pp. 105–158.

Zalevsky, Z.

H. M. Ozaktas, Z. Zalevsky, M. Alper Kutay, The Fractional Fourier Transform (Wiley, Chichester, UK, 2001).

Zassenhaus, H.

J. Patera, P. Winternitz, H. Zassenhaus, “Maximal abelian subalgebras of real and complex symplectic Lie algebras,” J. Math. Phys. 24, 1973–1985 (1983).
[CrossRef]

J. Patera, R. T. Sharp, P. Winternitz, H. Zassenhaus, “Continuous subgroups of the fundamental groups of physics. III. The de Sitter groups,” J. Math. Phys. 18, 2259–2288 (1977).
[CrossRef]

Celest. Mech.

A. J. Laub, K. Meyer, “Canonical forms for symplectic and Hamiltonian matrices,” Celest. Mech. 9, 213–238 (1974).
[CrossRef]

J. Math. Phys.

J. Patera, R. T. Sharp, P. Winternitz, H. Zassenhaus, “Continuous subgroups of the fundamental groups of physics. III. The de Sitter groups,” J. Math. Phys. 18, 2259–2288 (1977).
[CrossRef]

M. Moshinsky, P. Winternitz, “Quadratic Hamiltonians in phase space and their eigenstates,” J. Math. Phys. 21, 1667–1682 (1980).
[CrossRef]

J. Patera, P. Winternitz, H. Zassenhaus, “Maximal abelian subalgebras of real and complex symplectic Lie algebras,” J. Math. Phys. 24, 1973–1985 (1983).
[CrossRef]

P. A. M. Dirac, “A remarkable representation of the 3+2 de Sitter group,” J. Math. Phys. 4, 901–909 (1963).
[CrossRef]

K. B. Wolf, “The group-theoretical treatment of aberrating systems. III. The classification of asymmetric aberrations,” J. Math. Phys. 28, 2498–2507 (1987).
[CrossRef]

J. Opt. Soc. Am. A

Nucl. Instrum. Methods Phys. Res. A

A. J. Dragt, “Elementary and advanced Lie algebraic methods with applications to accelerator design, electron microscopes, and light optics,” Nucl. Instrum. Methods Phys. Res. A 258, 339–354 (1987).
[CrossRef]

Nuovo Cimento Suppl.

K. B. Wolf, “Dynamical groups for the point rotor and the hydrogen atom,” Nuovo Cimento Suppl. 5, 1041–1050 (1967).

Phys. Rev. A

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[CrossRef] [PubMed]

Rev. Mod. Phys.

C. Fronsdal, “Elementary particles in a curved space,” Rev. Mod. Phys. 37, 201–223 (1965).
[CrossRef]

Other

T. O. Philips, E. P. Wigner, “De Sitter space and positive energy,” in Group Theory and Its Applications, E. M. Loebl, ed. (Academic, New York, 1968), pp. 631–676.

R. Gilmore, Lie Groups, Lie Algebras, and Some of Their Applications (Wiley, New York, 1974).

H. M. Ozaktas, Z. Zalevsky, M. Alper Kutay, The Fractional Fourier Transform (Wiley, Chichester, UK, 2001).

S. Steinberg, “Lie series, Lie transformations, and their applications,” in Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986), Chap. 3, pp. 45–102.
[CrossRef]

A. J. Dragt, E. Forest, K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics,” in Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986) Chap. 4, pp. 105–158.

M. Kauderer, Symplectic Matrices, First Order Systems and Special Relativity (World Scientific, Singapore, 1994).

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Figures (1)

Fig. 1
Fig. 1

Eigenvalue plane Γ–Δ of three-dimensional paraxial Hamiltonians. The Hamiltonian orbits are parabolas Δ =14α2Γ2, 0αR, that we project on a circle, and degenerate points at the origin. There are four strata (HHΘ, HRΘ, RRΘ, and RMΘ). On their boundaries we find six isolated orbits (HF±, R–F, FM±, and M), and six orbits coexist at the origin (FF0,±, X, F–I and F–X).

Equations (55)

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H(m; p, q)=12 cp2-aqp-12 bq2m=abc-a.
m=αMmM-1,
m=0-110,0110,or 0010.
H:Harmonic,H0-110=12 p2+12 q2,
Oscillatingtrajectories,Δ>0,λ=±i.
R:Repulsive,H0110=12 p2-12 q2,
Hyperbolictrajectories,Δ<0,λ=±1.
F:Freehomogeneous,H0010=12 p2,
Straighttrajectories,Δ=0,λ=0(double).
-12 r1,312 (r2,3-r1,2)12 (r2,3+r1,2)12 r1,30r1,2r1,3-r1,20r2,3r1,3r2,30.
M Ω MT=Ω,Ω=0-110,
m Ω=-Ω mT,m=abc-a,b=bT,c=cT,
orbit(m)={αMmM-1|MSp(4, R),0αR}.
H(m;p,q)=12cxpx2+12cypy2+cpxpy-12bxqx2-12byqy2-bqxqy-axqxpx-ayqypy-axyqxpy-ayxqypxm=[axaxybxbayxaybbycxc-ax-ayxccy-axy-ay].
H([m1, m2]; p, q)={H(m1; p, q), H(m2; p, q)}.
12[r1,5¯+r3,4¯r1,2+r2,5¯-r1,3-r1,4¯-r3,5¯+r4¯,5¯r2,3+r2,4¯-r1,2+r2,5¯-r1,5¯+r3,4¯r2,3+r2,4¯r1,3+r14¯-r3,5¯+r4¯,5¯r1,3-r1,4¯-r3,5¯-r4¯,5¯-r2,3+r2,4¯-r1,5¯-r3,4¯r1,2-r2,5¯-r2,3+r2,4¯-r1,3+r1,4¯-r3,5¯-r4¯,5¯-r1,2-r2,5¯r1,5¯-r3,4¯]
[0r1,2r1,3r1,4¯r1,5¯-r1,20r2,3r2,4¯r2,5¯-r1,3-r2,30r3,4¯r3,5¯r1,4¯r2,4¯r3,4¯0r4¯,5¯r1,5¯r2,5¯r3,5¯-r4¯,5¯0],
H(r)=1m<n3 rm,njm,n+m=1,2,3n¯=4¯,5¯ rm,n¯jm,n¯+r4¯,5¯j4¯,5¯.
j1,2=12 (qxpy-qypx),12-angularmomentum;
j1,3=14 (-px2+py2-qx2+qy2),
counter-harmonic 12 (-Hx+Hy),
j2,3=12 (pxpy+qxqy),cross-harmonic,
j4¯,5¯=14 (px2+py2+qx2+qy2),
isotropicharmonic 12 (Hx+Hy).
j1,4¯=14 (px2-py2-qx2+qy2),counter-repulsive 12 (Rx-Ry),
j2,4¯=12 (-pxpy+qxqy),-cross-repulsive,
j3,5¯=14 (|p|2-|q|2),isotropicrepulsive 12 (Rx+Ry).
j2,5¯=12 (pxqx-pyqy),counter-imager 12 (Ix-Iy),
j2,5¯=12 (pxqy+pyqx),cross-imager,
j3,4¯=12pq,isotropicimager 12 (Ix+Iy).
j^m,n=vmn-vnm,j^m,n¯=vmn¯+vn¯m,
j^m¯,n¯=-vm¯n¯+vn¯m¯.
[jα,β, jγ,δ]=+gα,δ jβ,γ+gβ,γ jα,δ+gγ,α jδ,β+gδ,β jγ,α,
T(α, β)(τ)T(γ, δ)(τ):rα,γrα,δrβ,γrβ,δ
=T(α, β)(τ)rα,γrα,δrβ,γrβ,δT(γ, δ)(τ)T,
T(α, β)(τ)=cos τ-sin τsin τcos ττS1whenα, β{1, 2, 3}orα,β{4¯, 5¯}; cosh τ-sinh τ-sinh τcosh ττRwhenα{1, 2, 3}βorβ{4¯, 5¯}α.
GΘ(p, q)=Gx(px, qx)cos Θ+Gy(py, qy)sin Θ,
0Θ<π,
SeparableHamiltoniansRangeNotation13H-HΘstratum:HxcosΘ+HysinΘ,-14π<Θ14π,3-1<λ1,H-RΘstratum:HxcosΘ+RysinΘ,0<Θ<12π,20<λ<,R-RΘstratum:RxcosΘ+RysinΘ,0Θ14π,10λ1,H-Fσorbits:Hx+σFy,σ{-1,+1},5σ=±,R-Forbit:Rx+Fy,4,F-Fσorbits:Fx+σFy,σ{-1,0+1},6σ=0,±.
LorentzianHamiltoniansRangeNotation13R-MΘstratum:RcosΘ+MsinΘ,0<Θ<12π,9λ>0,Morbit:M,7,Xorbit:qxpy,-.
tα±=jα,3 ± jα,4¯,α{1, 2, 5¯},
t±=(t1±, t2±, t5¯±)T,
t-=(-12 (px2-py2),pxpy,12 (px2+py2))T,
t+=t-(p  q).
TE(2,-)(τ) : j1,2=j1,2-τt1-,
TE(2,-)(τ) : t1+=t1++τj1,2-12 τ2t1-,
mm+τe1+,e1-e1--τm-12 τ2e1+.
TE(τ) : (cj+et)=cj+[e+E(τ×c)]t.
EuclideanHamiltoniansNotation13F-M±orbits:F±M=12p2±12(qxpy-qypx),8ε=±,F-Iorbit:12(px2-py2+pxqy+pyqx)[orpxpy+12(pxqx-pyqy)],10,F-Xorbit:12px2+pyqx,11.
det(m-λ1)=λ4+Γλ2+Δ=0,
Γ=axbxcx-ax+aybycy-ay+2axybc-ayx,
Δ=det m=Δnoncross+Δcross,
Δnoncross=axbxcx-axaybycy-ay-2axayaxyayx+axy2ayx2+axy2bycx+ayx2bxcy,
Δcross=b2c2-b2cxcy-c2bxby+2bc(axay+axyayx)-2b(axayzcy+ayaxycx)-2c(axaxyby+ayayxbx).
λ=±  -12 Γ±(12 Γ)2-Δ1/2.

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