Abstract

A general axially symmetric first-order optical system is realized using free propagation and thin convex lenses of fixed focal length. It is shown that not more than five convex lenses of fixed focal length are required to realize the most general first-order optical system, with the required number of lenses depending on the situation. The free propagation distances are evaluated explicitly in each situation. The optimality of the decomposition obtained in each situation is brought out. Decompositions for some familiar subgroups of SL2(R) are also worked out. Convex or concave lenses of arbitrary focal length are realized using three or two convex lenses of fixed focal length, respectively. It is further shown that three convex lenses of arbitrary focal length are sufficient to realize the most general first-order optical system.

© 2014 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2013

D. J. Ming and F. H. Yi, “New decomposition of the Fresnel operator corresponding to the optical transformation in ABCD-systems,” Chin. Phys. B 22, 060302 (2013).
[CrossRef]

2008

2007

2006

M. J. Bastiaans and T. Alieva, “Synthesis of an arbitrary ABCD system with fixed lens positions,” Opt. Lett 31, 2414–2416 (2006).
[CrossRef]

2000

1999

N. M. Atakishiyev, L. E. Vicent, and K. B. Wolf, “Continuous vs. discrete fractional Fourier transforms,” J. Comput. Appl. Math. 107, 73–95 (1999).
[CrossRef]

1995

J. Shamir and N. Cohen, “Root and power transformations in optics,” J. Opt. Soc. Am. A 12, 2415–2423 (1995).
[CrossRef]

Arvind, B. Dutta, N. Mukunda, and R. Simon, “The real symplectic groups in quantum mechanics and optics,” Pramana 45, 471–497 (1995).
[CrossRef]

1990

N. Mukunda, “Role of symmetry and group structure in optics,” Current Sci. 59, 1135–1151 (1990).

1989

R. Simon, N. Mukunda, and E. C. G. Sudarshan, “Hamilton’s theory of turns generalized to Sp(2,R),” Phys. Rev. Lett. 62, 1331–1334 (1989).
[CrossRef]

R. Simon, N. Mukunda, and E. C. G. Sudarshan, “The theory of screws: a new geometric representation for the group SU(1,1),” J. Math. Phys. 30, 1000–1006 (1989).
[CrossRef]

1985

E. C. G. Sudarshan, N. Mukunda, and R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[CrossRef]

1984

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

1983

H. H. Arsenault and B. Macukow, “Factorization of the transfer matrix for symmetrical optical systems,” J. Opt. Soc. Am. 73, 1350–1359 (1983).
[CrossRef]

S. Cornbleet, “Geometrical optics reviewed: a new light on an old subject,” Proc. IEEE 71, 471–502 (1983).
[CrossRef]

1982

1981

1980

M. Nazarathy and J. Shamir, “Fourier optics described by operator algebra,” J. Opt. Soc. Am. 70, 150–159 (1980).
[CrossRef]

H. H. Arsenault, “Generalization of the principal plane concept in matrix optics,” Am. J. Phys. 48, 397–399 (1980).
[CrossRef]

1974

K. B. Wolf, “Canonical transforms. I. Complex linear transforms,” J. Math. Phys. 15, 1295–1301 (1974).
[CrossRef]

1966

Alieva, T.

Arsenault, H. H.

H. H. Arsenault and B. Macukow, “Factorization of the transfer matrix for symmetrical optical systems,” J. Opt. Soc. Am. 73, 1350–1359 (1983).
[CrossRef]

H. H. Arsenault, “Generalization of the principal plane concept in matrix optics,” Am. J. Phys. 48, 397–399 (1980).
[CrossRef]

Arvind,

Arvind, B. Dutta, N. Mukunda, and R. Simon, “The real symplectic groups in quantum mechanics and optics,” Pramana 45, 471–497 (1995).
[CrossRef]

Atakishiyev, N. M.

N. M. Atakishiyev, L. E. Vicent, and K. B. Wolf, “Continuous vs. discrete fractional Fourier transforms,” J. Comput. Appl. Math. 107, 73–95 (1999).
[CrossRef]

Bastiaans, M. J.

Brenner, K. H.

Casperson, L. W.

Cohen, N.

Cornbleet, S.

S. Cornbleet, “Geometrical optics reviewed: a new light on an old subject,” Proc. IEEE 71, 471–502 (1983).
[CrossRef]

Dutta, B.

Arvind, B. Dutta, N. Mukunda, and R. Simon, “The real symplectic groups in quantum mechanics and optics,” Pramana 45, 471–497 (1995).
[CrossRef]

Horn, R. A.

R. A. Horn and C. R. Johnson, “Canonical forms for similarity and triangular factorizations,” in Matrix Analysis (Cambridge University, 2013), pp. 163–224.

Johnson, C. R.

R. A. Horn and C. R. Johnson, “Canonical forms for similarity and triangular factorizations,” in Matrix Analysis (Cambridge University, 2013), pp. 163–224.

Kogelnik, H.

Li, T.

Liu, X.

Macukow, B.

Ming, D. J.

D. J. Ming and F. H. Yi, “New decomposition of the Fresnel operator corresponding to the optical transformation in ABCD-systems,” Chin. Phys. B 22, 060302 (2013).
[CrossRef]

Mukunda, N.

Arvind, B. Dutta, N. Mukunda, and R. Simon, “The real symplectic groups in quantum mechanics and optics,” Pramana 45, 471–497 (1995).
[CrossRef]

N. Mukunda, “Role of symmetry and group structure in optics,” Current Sci. 59, 1135–1151 (1990).

R. Simon, N. Mukunda, and E. C. G. Sudarshan, “Hamilton’s theory of turns generalized to Sp(2,R),” Phys. Rev. Lett. 62, 1331–1334 (1989).
[CrossRef]

R. Simon, N. Mukunda, and E. C. G. Sudarshan, “The theory of screws: a new geometric representation for the group SU(1,1),” J. Math. Phys. 30, 1000–1006 (1989).
[CrossRef]

E. C. G. Sudarshan, N. Mukunda, and R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[CrossRef]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

Nazarathy, M.

Shamir, J.

Simon, R.

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

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

Arvind, B. Dutta, N. Mukunda, and R. Simon, “The real symplectic groups in quantum mechanics and optics,” Pramana 45, 471–497 (1995).
[CrossRef]

R. Simon, N. Mukunda, and E. C. G. Sudarshan, “The theory of screws: a new geometric representation for the group SU(1,1),” J. Math. Phys. 30, 1000–1006 (1989).
[CrossRef]

R. Simon, N. Mukunda, and E. C. G. Sudarshan, “Hamilton’s theory of turns generalized to Sp(2,R),” Phys. Rev. Lett. 62, 1331–1334 (1989).
[CrossRef]

E. C. G. Sudarshan, N. Mukunda, and R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[CrossRef]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

Stavroudis, O. N.

O. N. Stavroudis, “The lens group,” in The Optics of Rays, Wavefronts, and Caustics, H. S. W. Massey and K. A. Brueckner, eds. (Academic, 1972), pp. 281–297.

Sudarshan, E. C. G.

R. Simon, N. Mukunda, and E. C. G. Sudarshan, “Hamilton’s theory of turns generalized to Sp(2,R),” Phys. Rev. Lett. 62, 1331–1334 (1989).
[CrossRef]

R. Simon, N. Mukunda, and E. C. G. Sudarshan, “The theory of screws: a new geometric representation for the group SU(1,1),” J. Math. Phys. 30, 1000–1006 (1989).
[CrossRef]

E. C. G. Sudarshan, N. Mukunda, and R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[CrossRef]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

Vicent, L. E.

N. M. Atakishiyev, L. E. Vicent, and K. B. Wolf, “Continuous vs. discrete fractional Fourier transforms,” J. Comput. Appl. Math. 107, 73–95 (1999).
[CrossRef]

Wolf, K. B.

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

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

N. M. Atakishiyev, L. E. Vicent, and K. B. Wolf, “Continuous vs. discrete fractional Fourier transforms,” J. Comput. Appl. Math. 107, 73–95 (1999).
[CrossRef]

K. B. Wolf, “Canonical transforms. I. Complex linear transforms,” J. Math. Phys. 15, 1295–1301 (1974).
[CrossRef]

K. B. Wolf, “Canonical transformations,” in Geometric Optics on Phase Space (Springer, 2004), pp. 25–46.

K. B. Wolf, “Construction and properties of canonical transforms,” in Integral Transforms in Science and Engineering, A. Miele, ed. (Springer, 1979), pp. 381–416.

Yariv, A.

A. Yariv and P. Yeh, “Rays and optical beams,” in Photonics, A. S. Sedra, ed. (Oxford, 2007), pp. 66–109.

Yeh, P.

A. Yariv and P. Yeh, “Rays and optical beams,” in Photonics, A. S. Sedra, ed. (Oxford, 2007), pp. 66–109.

Yi, F. H.

D. J. Ming and F. H. Yi, “New decomposition of the Fresnel operator corresponding to the optical transformation in ABCD-systems,” Chin. Phys. B 22, 060302 (2013).
[CrossRef]

Am. J. Phys.

H. H. Arsenault, “Generalization of the principal plane concept in matrix optics,” Am. J. Phys. 48, 397–399 (1980).
[CrossRef]

Appl. Opt.

Chin. Phys. B

D. J. Ming and F. H. Yi, “New decomposition of the Fresnel operator corresponding to the optical transformation in ABCD-systems,” Chin. Phys. B 22, 060302 (2013).
[CrossRef]

Current Sci.

N. Mukunda, “Role of symmetry and group structure in optics,” Current Sci. 59, 1135–1151 (1990).

J. Comput. Appl. Math.

N. M. Atakishiyev, L. E. Vicent, and K. B. Wolf, “Continuous vs. discrete fractional Fourier transforms,” J. Comput. Appl. Math. 107, 73–95 (1999).
[CrossRef]

J. Math. Phys.

R. Simon, N. Mukunda, and E. C. G. Sudarshan, “The theory of screws: a new geometric representation for the group SU(1,1),” J. Math. Phys. 30, 1000–1006 (1989).
[CrossRef]

K. B. Wolf, “Canonical transforms. I. Complex linear transforms,” J. Math. Phys. 15, 1295–1301 (1974).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Acta

E. C. G. Sudarshan, N. Mukunda, and R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[CrossRef]

Opt. Lett

M. J. Bastiaans and T. Alieva, “Synthesis of an arbitrary ABCD system with fixed lens positions,” Opt. Lett 31, 2414–2416 (2006).
[CrossRef]

Phys. Rev. A

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

Phys. Rev. Lett.

R. Simon, N. Mukunda, and E. C. G. Sudarshan, “Hamilton’s theory of turns generalized to Sp(2,R),” Phys. Rev. Lett. 62, 1331–1334 (1989).
[CrossRef]

Pramana

Arvind, B. Dutta, N. Mukunda, and R. Simon, “The real symplectic groups in quantum mechanics and optics,” Pramana 45, 471–497 (1995).
[CrossRef]

Proc. IEEE

S. Cornbleet, “Geometrical optics reviewed: a new light on an old subject,” Proc. IEEE 71, 471–502 (1983).
[CrossRef]

Other

R. A. Horn and C. R. Johnson, “Canonical forms for similarity and triangular factorizations,” in Matrix Analysis (Cambridge University, 2013), pp. 163–224.

O. N. Stavroudis, “The lens group,” in The Optics of Rays, Wavefronts, and Caustics, H. S. W. Massey and K. A. Brueckner, eds. (Academic, 1972), pp. 281–297.

K. B. Wolf, “Construction and properties of canonical transforms,” in Integral Transforms in Science and Engineering, A. Miele, ed. (Springer, 1979), pp. 381–416.

K. B. Wolf, “Canonical transformations,” in Geometric Optics on Phase Space (Springer, 2004), pp. 25–46.

A. Yariv and P. Yeh, “Rays and optical beams,” in Photonics, A. S. Sedra, ed. (Oxford, 2007), pp. 66–109.

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Figures (1)

Fig. 1.
Fig. 1.

Curves (p), (q), (r), and (s) for the choice m=5 in the d5k plane. For pairs (d5,k) below curve (p), d2 is positive. For pairs strictly below curve (q) but above (r), both d2 and d1 are positive. This is indicated by the shaded region in the above plot. The curve (s) is below the line k=2, and the curve (r) is above the line k=2. For d3 to be positive, the pairs (d5,k) need to be below curve (s) and thus strictly below curve (r) or be strictly above curve (q). This region in the d5k plane clearly does not intersect with the shaded region in which both d1 and d2 are positive. Thus it is impossible to simultaneously have d1 and d3 be positive.

Tables (3)

Tables Icon

Table 1. Allowed Signatures for a, b, c, and d so that Matrix M Describes a First-Order Optical Systema

Tables Icon

Table 2. Summary of the Decompositions for Ray Transfer Matrices with Nonsingular Entriesa

Tables Icon

Table 3. Summary of Decompositions for Ray Transfer Matrices with Any One or Two of Its Entries To Be Singulara

Equations (83)

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

M=[abcd],
F(d)[1d01],
L(f)[101/f1].
[abcd]=F(d3)L(1)F(d2)L(1)F(d1),
[acd3bad1dd3+cd1d3cdcd1]=[1d2d22+d21d2].
d2=c+2,
d1=d2+d1c=c+d+1c,
d3=d2+a1c=c+a+1c.
[abcd]=F(c+a+1c)L(1)F(c+2)L(1)F(c+d+1c).
X=[0110]=F(1)L(1)F(1).
[abcd]=F(a+1c)XF(c)XF(d+1c).
Y=[0110]=F(2)L(1)F(3)L(1)F(2).
[1001]=F(1)L(1)F(2)L(1)F(1).
Z=[1001]=F(2)L(1)F(2)L(1).
XF(d)X=L(1/d)=YF(d)Y,
XF(d)Y=L(1/d)=YF(d)X.
[1D01],withD>0.
[1D01]=F(d2)L(1)F(d1)=[1d2d1+d2d1d211d1].
[abcd]=F(d4)L(1)F(d3)L(1)F(d2)L(1)F(d1).
F(d3)L(1)F(d4)[abcd]=L(1)F(d2)L(1)F(d1).
1d2=a(1d3)+c(d3d4d3d4),
d1+d2d1d2=b(1d3)+d(d3d4d3d4),
2+d2=a+c(1d4),
12d1d2+d1d2=b+d(1d4).
d2=2+a+cck.
d1=1+a+b+c+d(c+d)ka+cck.
d3=2a+c+12cka+cck.
d1+k=3D.
[abcd]=F(d5)L(1)F(d4)L(1)F(d3)L(1)F(d2)L(1)F(d1).
[abcd]=[acd5bdd5a+c(1d5)b+d(1d5)]
=F(d4)L(1)F(d3)L(1)F(d2)L(1)F(d1).
d1+d5=2D.
[αβγδ]A(α,β,γ,δ)=F(1+αγ)XF(γ)XF(1+δγ).
[αβγδ]A(α,β,γ,δ)=F(1+αγ)XF(γ)XF(1+δγ).
[αβγδ]=XA(γ,δ,α,β)=XF(1+γα)XF(α)XF(1+βα).
[αβγδ]=A(β,α,δ,γ)X=F(1+βδ)XF(δ)XF(1+γδ)X.
[αβγδ]=A(β,α,δ,γ)X=F(1+βδ)XF(δ)XF(1+γδ)X.
[αβγδ]=ZA(α,β,γ,δ)=F(1+αγ)YF(γ)XF(1+δγ).
[αβγδ]=ZA(α,β,γ,δ)=F(1+αγ)YF(γ)XF(1+δγ).
[αβγδ]=YA(γ,δ,α,β)=YF(1+γα)XF(α)XF(1+βα).
[αβγδ]=A(β,α,δ,γ)Y=F(1+βδ)XF(δ)XF(1+γδ)Y.
[αβγδ]=YA(γ,δ,α,β)=YF(1+γα)XF(α)XF(1+βα).
[αβγδ]=XA(δ,γ,β,α)X=XF(1+δβ)XF(β)XF(1+αβ)X.
[αβγδ]=XA(δ,γ,β,α)Y=XF(1+δβ)XF(β)XF(1+αβ)Y.
[αβγδ]=L(α1+γ)F(α)XF(1+βα),
[αβγδ]=F(1+βδ)XF(δ)L(δ1+γ),
[αβγδ]=L(α1+γ)F(α)XF(1+βα).
[αβγδ]=L(β1+δ)F(β)L(β1+α).
[αβγδ]=XF(1+δβ)XF(β)L(β1+α).
[α±βγδ]=F(1+αγ)L(1γ)F(1+δγ).
S(x)=[x001x],
R(θ)=[cosθsinθsinθcosθ],
R(θ,x)=[cosθx2sinθx2sinθcosθ].
S(μ)=[coshμsinhμsinhμcoshμ].
S(μ)=F(1+eμcoshμ)L(1)F(1+coshμ)XF(1+sinhμcoshμ)X,
S(μ)=F(1eμsinhμ)L(1)F(2+sinhμ)L(1)F(1eμsinhμ).
F(D)=[1D01]=XF(2D)XF(D)XF(2D)Y.
[101f1]=YF(1f)X,
[101f1]=F(2f)XF(1f)XF(2f).
c=1,d1=1d,andd2=1a.
d2=2+αγ+γk,
d1=1+αβγ+δ(γ+δ)kαγ+γk,
d3=2αγ+1+2γkαγ+γk.
β+γαδ>1,
δγ>0,
αγ>0,
[abcd]=[α+γd5βδd5αγ+γd5β+δδd5].
d2=(2+2αγ+2γd5)(αγ+γd5)k,
d1=12α+2β+γδ+2(δγ)d5[β+γαδ+(δγ)d5]k(αγ+γd5)k(2αγ+2γd5),
d3=2[(αγ)+γd5]k(1+3αγ+3γd5)(αγ+γd5)k(2αγ+2γd5).
k2+2+γαγ+γd5.
k2+δγ1β+γαδ+(δγ)d5
k>2+γ(αγ)+γd5,
k2+δγ1β+γαδ+(δγ)d5
k<2+γ(αγ)+γd5.
δγ>1,
αγ>1,
k32+γ+1/2(αγ)+γd5
k>2+γ(αγ)+γd5,
k32+γ+1/2(αγ)+γd5
k<2+γ(αγ)+γd5.
α3γ>1,
1+a+b+c+d<0,c+d>1,anda+3c>1,

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