Abstract

We introduce the paraxial group, the group of symmetries of the paraxial-wave equation and its action on paraxial beams. The transformations, elements of the group, are used to obtain closed-form expressions for the propagation of any paraxial beam through misaligned ABCD optical systems. We prove that any paraxial beam is form-invariant under these transformations.

© 2008 Optical Society of America

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References

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  1. M. Guizar-Sicairos and J. C. Gutiérrez-Vega, Opt. Lett. 31, 2912 (2006).
    [CrossRef] [PubMed]
  2. M. A. Bandres and J. C. Gutiérrez-Vega, Opt. Lett. 32, 3459 (2007).
    [CrossRef] [PubMed]
  3. M. A. Bandres and J. C. Gutiérrez-Vega, Opt. Express 15, 16719 (2007).
    [CrossRef] [PubMed]
  4. M. A. Bandres and J. C. Gutiérrez-Vega, Opt. Lett. 33, 177 (2008).
    [CrossRef] [PubMed]
  5. M. A. Bandres, Opt. Lett. 33, 1678 (2008).
    [CrossRef] [PubMed]
  6. M. A. Bandres and J. C. Gutiérrez-Vega, “Elliptical beams” (submitted to Opt. Express).
  7. W. Miller, Symmetry and Separation of Variables (Cambridge U. Press, 1984).
  8. K. B. Wolf, Geometric Optics on Phase Space (Springer, 2004).
  9. W. Shaomin and L. Ronchi, in Progress in Optics, Vol. XXV, E.Wolf, ed. (Elsevier, 1988), pp. 281-348.

2008 (2)

2007 (2)

2006 (1)

Bandres, M. A.

Guizar-Sicairos, M.

Gutiérrez-Vega, J. C.

Miller, W.

W. Miller, Symmetry and Separation of Variables (Cambridge U. Press, 1984).

Ronchi, L.

W. Shaomin and L. Ronchi, in Progress in Optics, Vol. XXV, E.Wolf, ed. (Elsevier, 1988), pp. 281-348.

Shaomin, W.

W. Shaomin and L. Ronchi, in Progress in Optics, Vol. XXV, E.Wolf, ed. (Elsevier, 1988), pp. 281-348.

Wolf, K. B.

K. B. Wolf, Geometric Optics on Phase Space (Springer, 2004).

Opt. Express (1)

Opt. Lett. (4)

Other (4)

M. A. Bandres and J. C. Gutiérrez-Vega, “Elliptical beams” (submitted to Opt. Express).

W. Miller, Symmetry and Separation of Variables (Cambridge U. Press, 1984).

K. B. Wolf, Geometric Optics on Phase Space (Springer, 2004).

W. Shaomin and L. Ronchi, in Progress in Optics, Vol. XXV, E.Wolf, ed. (Elsevier, 1988), pp. 281-348.

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Figures (1)

Fig. 1
Fig. 1

(a) Misaligned optical system. (b) and (c) show the propagation of a HzG beam through the optical system for Δ y = 3 mm and Δ ϕ x = 2 ° . Dashed lines indicate the position of the lenses.

Equations (26)

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( 2 + i 2 k z ) Ψ ( r , z ) = 0 ,
M A B C D = M I M P = [ A 0 C 1 A ] [ 1 B A 0 1 ] ,
I C ( M I ) Ψ ( r , z 0 ) = 1 A exp ( i k C r 2 2 A ) Ψ ( r A , z 0 ) ,
I C ( M A B C D ) Ψ ( r , 0 ) = 1 A exp ( i k C r 2 2 A ) Ψ ( r A , B A ) ,
P ( M A B C D ) Ψ ( r , z ) = 1 A + C z exp [ i k C r 2 2 ( A + C z ) ] × Ψ ( r A + C z , B + D z A + C z ) ,
T ( u , v , ρ ) Ψ ( r , z ) = exp [ i ρ + i 2 ( 2 r z k u ) u ] × Ψ ( r z u k + v , z ) .
T ( u , v , ρ ) T ( u , v , ρ ) = T ( u + u , v + v , ρ + ρ + u v ) .
R ( θ ) Ψ ( r , z ) = Ψ ( Θ r , z ) ,
Θ ( θ ) = [ cos θ sin θ sin θ cos θ ] .
R ( θ ) T ( u , v , ρ ) R ( θ ) = T ( Θ u , Θ v , ρ ) ,
P ( M 1 ) T ( u , v , ρ ) P ( M ) = T ( u , v , ρ ) ,
u = A u + k C v ,
v = D v + B u k ,
ρ = ρ + ( u v u v ) 2 .
G = T [ S L ( 2 , C ) × S O ( 2 ) ] ,
G ( M , θ , u , v , ρ ) = P ( M ) R ( θ ) T ( u , v , ρ ) .
G ( g ) Ψ ( r , z ) = 1 A + C z exp [ i k C r 2 2 ( A + C z ) ] × exp { i ρ + i 2 ( A + C z ) [ 2 Θ r ( B + D z ) u k ] u } × Ψ ( Θ r ( B + D z ) u k A + C z + v , B + D z A + C z ) ,
g f = { M M , θ + θ , u f , v f , ρ f } ,
u f = u + Θ ( θ ) ( A u + k C v ) ,
v f = v + Θ ( θ ) ( D v + B u k ) ,
ρ f = ρ + ρ + 1 2 [ ( u f + u ) ( v f v ) u v ] .
G [ I , 0 , k ε , ε L ε , k ε ( ε + L ε ) ] × G ( M A B C D , 0 , 0 , 0 , 0 ) G ( I , 0 , k ε , ε , 0 ) Ψ ( r , z ) ,
G { M A B C D , 0 , k ( e + A g ) B , g , k ε ( ε + L ε ) 2 + [ k ( e + A g ) B k ε ] ( g ε ) 2 } Ψ ( r , z ) ,
e = ( α ε + β ε ) ,
g = ( B γ D α ) ε + ( B δ D β ) ε ,
[ α β γ δ ] = [ 1 A L B C 1 D ] .

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