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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 4 — Feb. 15, 2010
  • pp: 3568–3573
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Rotating beams in isotropic optical system

Tatiana Alieva, Eugeny Abramochkin, Ana Asenjo-Garcia, and Evgeniya Razueva  »View Author Affiliations


Optics Express, Vol. 18, Issue 4, pp. 3568-3573 (2010)
http://dx.doi.org/10.1364/OE.18.003568


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Abstract

Based on the ray transformation matrix formalism, we propose a simple method for generation of paraxial beams performing anisotropic rotation in the phase space during their propagation through isotropic optical systems. The widely discussed spiral beams are the particular case of these beams. The propagation of these beams through the symmetric fractional Fourier transformer is demonstrated by numerical simulations.

© 2010 OSA

1. Introduction

The analysis and synthesis of spiral paraxial beams, whose transversal intensity distribution rotates around the axis without changing its form (except for some scaling) during its propagation in free space, have been treated in many publications [1

1. E. Abramochkin and V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102(3-4), 336–350 (1993). [CrossRef]

5

5. A. Bekshaev and M. Soskin, “Rotational transformations and transverse energy flow in paraxial light beams: linear azimuthons,” Opt. Lett. 31(14), 2199–2201 (2006). [CrossRef] [PubMed]

]. Here we propose an alternative approach for the description of spiral beams based on the ray transformation matrix formalism which is suitable for the analysis of their propagation through any isotropic paraxial optical system (IOS). Moreover, this method is also valid for the design of beams which perform other types of phase-space rotations during their propagation through an IOS.

Beam propagation through a lossless paraxial optical system is described by the canonical integral transformation [6

6. S. A. Collins Jr., “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60(9), 1168–1177 (1970). [CrossRef]

], represented by operator RT, whose kernel is parameterized by the real symplectic ray transformation matrix T, which relates the position r=(x,y)t and direction p=(px,py)t of an incoming ray to those of the outgoing ray

(ropo)=[ABCD](ripi)=T(ripi).

2. Isotropic and anisotropic phase-space rotators

All the transforms associated with orthogonal ray transformation matrix produce rotation in phase space. Nevertheless, one of them, corresponding to the symmetric FrFT, RTf(φ,φ), is inherently different. Indeed, its unitary matrix Uf(φ,φ) is scalar, Uf(φ,φ)=exp(iφ)I, and commutes with any unitary matrix. The detUf(φ,φ)=exp(2iφ) while the determinant of the matrix describing other phase-space rotators, indicated as Uar which can be expressed as a product of Uf(γ,γ), Ur(θ) and Ug(α)equals one: detUar=1. The last transformsRTar, which we will further call as anisotropic rotators, describe the movements on the orbital Poincaré sphere [9

9. G. F. Calvo, “Wigner representation and geometric transformations of optical orbital angular momentum spatial modes,” Opt. Lett. 30(10), 1207–1209 (2005). [CrossRef] [PubMed]

,10

10. T. Alieva and M. J. Bastiaans, “Orthonormal mode sets for the two-dimensional fractional Fourier transformation,” Opt. Lett. 32(10), 1226–1228 (2007). [CrossRef] [PubMed]

].

Any IOS, which may consist of centered spherical lenses, mirrors and free space intervals, is described by the ray transformation matrix with scalar A=aI, B=bI, C=cI and D=dI. Therefore the matrices G=gI, S=sI, and U=Uf(φ,φ)=exp(iφ)I are also scalar ones, where g=(ac+bd)/(a2+b2), s=(a2+b2)1/2, and φ=γx=γy=arg(a+ib). In particular, for free space propagation (Fresnel diffraction) a=d=1, c=0 and the angle φ is limited: φ[0,π/2]. We observe that in the case of IOS, the orthogonal matrix in the decomposition (1) corresponds to the symmetric FrFT. Since the lens and scaler transformations don't change the form of the beam intensity, then the intensity distribution at the output plane of the IOS is described by the symmetric FrFT power spectrum with a proper scaling |RTIOS[fi(ri)](ro)|2=|RTf(φ,φ)[fi(ri)](ro/s)|2. Correspondingly, the construction of the spiral beams during the propagation through the IOS reduces to the generation of such beams for the symmetric fractional Fourier transformer. We recall that beam propagation through optical fiber with a quadratic refractive index profile corresponds to the symmetric FrFT of its complex field amplitude at angles φ defined by the propagation distance z and the refractive index gradient g: φ = gz [11

11. H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2001).

]. Note that in this case φ can cover the interval of several periods of 2π. Other fractional Fourier transformers can be constructed using one or two spherical lenses.

The realization of other phase-space rotators, RTar, requires the application of asymmetric optical elements such as cylindrical lenses or mirrors. Nevertheless, it is possible, as for the spiral beams, to design beams Ψ(r) for which the evolution of their intensity distribution during the propagation through the IOS (symmetric FrFT, RTf(φ,φ)) and the anisotropic phase-space rotator RTar will be identical
|RTar[Ψ(ri)](ro)|2=|RTf(φ,φ)[Ψ(ri)](ro)|2.
(2)
Below the operator for the canonical transform RT associated with orthosymplectic matrix T will be denoted by RU.

3. Design of rotating beams

Let us consider the way how to generate the beams which satisfy Eq. (2). Note that any Uar may be factored as Uar(γ,U0)=U0Uf(γ,γ)U01, where U0 is also unitary matrix [12

12. M. J. Bastiaans and T. Alieva, “First-order optical systems with unimodular eigenvalues,” J. Opt. Soc. Am. A 23(8), 1875–1883 (2006). [CrossRef]

]. It has been shown [13

13. T. Alieva and M. J. Bastiaans, “Mode mapping in paraxial lossless optics,” Opt. Lett. 30(12), 1461–1463 (2005). [CrossRef] [PubMed]

,14

14. A. Wünsche, “General Hermite and Laguerre two-dimensional polynomials,” J. Phys. Math. Gen. 33(17), 1603–1629 (2000). [CrossRef]

] that for any unitary matrix U0 with elements Ujk (j,k=1,2) there exists a complete orthonormal set of Gaussian modes {Hm,nU0(r)m,n=0,1,},
Hm,nU0(r)=(1)m+nexp(x2+y2)2m+n1/2(πm!n!)1/2(U11x+U12y)m(U21x+U22y)nexp(2x22y2),
(3)
which are eigenfunctions for the transform RUar(γ,U0). The corresponding eigenvalues are given by exp[i(mn)γ]. Moreover, these modes are also eigenfunctions for the symmetric FrFT, defined by Uf(φ,φ), with eigenvalues exp[i(m+n+1)φ]. For U0=Ug(α) the formula (3) reduces to Hermite-Laguerre-Gaussian modes [15

15. E. Abramochkin and V. Volostnikov, “Generalized Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6(5), S157–S161 (2004). [CrossRef]

]: Hm,nUg(α)(r)=(π2m+n1m!n!)1/2inGm,n(r|α). In particular, Hm,nUg(0)(r) and Hm,nUg(±π/4)(r)are Hermite-Gaussian (HG) and Laguerre-Gaussian (LG) modes respectively:
Hm,nUg(0)(r)=Hm,nI(r)=in(π2m+n1m!n!)1/2exp(x2y2)Hm(2x)Hn(2y),Hm,nUg(±π/4)(r)=(±i)n(1)min(π/2)1/2min!max!exp(r2±i(mn)ψ)(2r)|mn|Lmin|mn|(2r2),
(4)
where r=(x,y)t=(rcosψ,rsinψ)t, min=min(m,n), max=max(m,n).

HG modes are eigenfunctions for the FrFT at angles (γ,−γ) and therefore at any pair of angles γx andγy. LG modes are eigenfunctions for the signal rotator Ur(±θ)=Ug(±π/4)Uf(θ,θ)Ug(π/4) with eigenvalue exp[±i(mn)θ]. Note that mode Hm,nU0(r) satisfies Eq. (2) for Uar(γ,U0).

A linear combination of the orthosymplectic modes Hm,nU0(r) of the same order m+n and U0 is also an eigenfunction for the symmetric FrFT for all possible angles φ. It means that all the modes in this decomposition accumulate the same Gouy phase during the propagation through an IOS or the corresponding symmetric fractional Fourier transformer. Similarly, a linear superposition of the modes Hm,nU0(r) with the same difference of the indices, mn, is an eigenfunction for the transform associated with Uar(γ,U0) for any γ.

Analogously, a beam Ψ(r) which undergoes the same transformation during propagation through the symmetric fractional Fourier transformer, except for a constant phase factor, as during the propagation through the optical system described by the one parametric unitary matrix Uar(γ,U0),
RUf(φ,φ)[Ψ(r)]=exp(iϕ)RUar(γ,U0)[Ψ(r)],
(5)
can also be represented as a linear combination of modes Hm,nU0(r). Using the relation Uar1(γ,U0)=Uar(γ,U0) the last equation can be rewritten in the following form

RUar(γ,U0)RUf(φ,φ)[Ψ(r)]=exp(iϕ)Ψ(r).

We observe that Ψ(r) has to be an eigenfunction of the transform described by the unitary matrix Uar(γ,U0)Uf(φ,φ). As we have mentioned above the modes Hm,nU0(r) are eigenfunctions for the symmetric FrFT, RUf(φ,φ), and for the phase-space rotator RUar(γ,U0) with eigenvalues exp[i(m+n+1)φ] and exp[i(mn)γ] respectively and therefore
RUar(γ,U0)RUf(φ,φ)[Hm,nU0(r)]=exp(iϕ)Hm,nU0(r),
(6)
where ϕ=(mn)γ(m+n+1)φ. Representing γ as γ=vφ, where v indicates the velocity of the phase-space rotation associated with Uar(γ,U0) during symmetric FrFT at angle φ, we can rewrite ϕ as ϕ(v,φ)=[m(1v)+n(1+v)+1]φ. Then a linear combination of these modes
ΨU0(r,v)=m,ncmnHm,nU0(r)
(7)
with the arbitrary complex cmn and the mode indices m and n, which satisfy the relation
m(1v)+n(1+v)=const,
(8)
is also an eigenfunction of the canonical integral transform RUar(vφ,U0)RUf(φ,φ):

RUar(vφ,U0)RUf(φ,φ)[ΨU0(r,v)]=exp[iϕ(v,φ)]ΨU0(r,v).
(9)

If v is irrational we have a trivial case where only one mode Hm,nU0(r) satisfies relation (8). Moreover, it is easy to see that for v=1 the beam anisotropically rotating in phase space during symmetric FrFT is given by
ΨnU0(r,1)=m=0cmnHm,nU0(r),
(10)
where n is fixed. Similar expression for fixed m and arbitrary n can be obtained for v=1. Since any field amplitude f(r) can be represented as a linear superposition of the orthonormal modes Hm,nU0(r), it also can be written as a sum of the beams rotating in the phase space with the same velocity v=1 as
f(r)=m=0n=0fmnHm,nU0(r)=n=0ΨnU0(r,1),
(11)
where ΨnU0(r,1) is given by Eq. (10) and fmn=cmn.

Note that the sign of the parameter v indicates the direction of the rotation, while its absolute value corresponds to the number of complete loops which beam makes during the 2π-interval of φ. It can be bigger or less than one if a beam has certain symmetry in the phase space. The velocity of rotation is defined by the indices of any pair of modes Hm1,n1U0(r) and Hm2,n2U0(r) in the composition (7) as a ratio of the differences of the eigenvalues of these modes for symmetric FrFT and anisotropic phase-space rotator
m1(1v)+n1(1+v)=m2(1v)+n2(1+v)v=(m2+n2)(m1+n1)(m2n2)(m1n1)=kl
with nonzero integers k and l. Note that a linear combination of any two modes Hm,nU0(r) with fixed U0 and such indices that m1+n1m2+n2 and m1n1m2n2 always forms an anisotropically rotating beam. Since phase-space rotators are periodic with period 2π (the symmetric FrFT except for the phase): Uf(φ+2π,φ+2π)=Uf(φ,φ) and Uar(γ+2π,U0)=Uar(γ,U0) and Uf(2π,2π)=Uar(2π,U0)=I, we observe from Eq. (9) that ΨnU0(r,v)=ΨnU0(r,k/l) is an eigenfuction of the symmetric FrFT at angles 2π/v=2πl/k. Taking into account the periodicity of the rotators [16

16. T. Alieva and A. Barbé, “Self-fractional Fourier images,” J. Mod. Opt. 46, 83–99 (1999).

] we can conclude that ΨnU0(r,k/l) is an eigenfunction for the operator RUf(2π/k,2π/k). Similarly, one can prove that ΨnU0(r,k/l) is an eigenfunction for the operator RUar(2π/l,U0).

As an example we consider a family of rotating beams with the rotation velocity v=1, which can be expressed in the integral form (used for numerical simulations) or according with Eq. (10) by series

Ψ2Ug(α)(r,1)=n=0(3n+2)!(3n)!33nH3n+2,2Ug(α)(r).
(12)

The gyrator matrix U0=Ug(α) defines anisotropic phase-space rotator Uar(γ,Ug(α)). For α=π/4 it reduces to the signal rotator, H3n+2,2Ug(π/4)(r) are LG modes, and Ψ2Ug(π/4)(r,1) is a spiral beam shown in Fig. 1a
Fig. 1 Intensities (top row) and phases (bottom row) of the beam Ψ2Ug(α)(r,1), given by Eq. (12), for various values of the angle α: (a) α=π/4 (a spiral beam), (b) α=π/8, (c) α=0 (combination of HG modes). (Media 1) Intensity and phase evolution of the beam |RUf(φ,φ)[Ψ2Ug(α)(r,1)]|2for φ[0,2π/3].
. For α=π/8 the beam rotating in other phase-space planes is obtained (see Fig. 1b). If α=0 then H3n+2,2Ug(0)(r)are HG modes and the beam Ψ2Ug(0)(r,1), displayed in Fig. 1c, performs the anisotropic rotations in (x,px) and (y,py) planes during the propagation. In the video connected to the figure the evolution of the intensity distribution of these beams during their propagation through the symmetric fractional Fourier transformer, which can be an optical fiber, is shown. Note that Ψ2Ug(α)(r,1) are eigenfunctions for the symmetric FrFT for angle φ0=2π/3 and we display only a part of the loop corresponding to the intensity evolution in the interval φ[0,2π/3] (observe that the bottom circle of the spiral beam is at the left side in the end of the video). For the spiral beam we observe the simple rotation of the intensity distribution at the transversal plane (x,y), while for other ones the intensity distribution is changing since the anisotropic phase-space rotation is performed in another plane.

3. Conclusion

We have shown that beams performing anisotropic rotation in phase space during their propagation in isotropic systems may be represented as a linear combination of the Gaussian modes expressed by Eq. (3). While the parameter U0 defines the plane of the phase-space rotation, the indices of the modes participated in the beam synthesis determine the velocity of the rotation v. These beams can be generated using spatial light modulators. The application of these beams for light-matter interaction is under investigation. We only mention that spiral beams have been found useful for optical trapping.

Acknowledgments

T. Alieva thanks Spanish Ministry of Education and Science (project TEC2008-04105/TEC).

References and links

1.

E. Abramochkin and V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102(3-4), 336–350 (1993). [CrossRef]

2.

E. Abramochkin and V. Volostnikov, “Spiral-type beams: optical and quantum aspects,” Opt. Commun. 125(4-6), 302–323 (1996). [CrossRef]

3.

E. Abramochkin and V. Volostnikov, “Spiral light beams,” Phys. Usp. 47(12), 1177–1203 (2004). [CrossRef]

4.

A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Centrifugal transformation of the transverse structure of freely propagating paraxial light beams,” Opt. Lett. 31(6), 694–696 (2006). [CrossRef] [PubMed]

5.

A. Bekshaev and M. Soskin, “Rotational transformations and transverse energy flow in paraxial light beams: linear azimuthons,” Opt. Lett. 31(14), 2199–2201 (2006). [CrossRef] [PubMed]

6.

S. A. Collins Jr., “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60(9), 1168–1177 (1970). [CrossRef]

7.

R. Simon and K. B. Wolf, “Structure of the set of paraxial optical systems,” J. Opt. Soc. Am. A 17(2), 342–355 (2000). [CrossRef]

8.

J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Optical system design for orthosymplectic transformations in phase space,” J. Opt. Soc. Am. A 23(10), 2494–2500 (2006). [CrossRef]

9.

G. F. Calvo, “Wigner representation and geometric transformations of optical orbital angular momentum spatial modes,” Opt. Lett. 30(10), 1207–1209 (2005). [CrossRef] [PubMed]

10.

T. Alieva and M. J. Bastiaans, “Orthonormal mode sets for the two-dimensional fractional Fourier transformation,” Opt. Lett. 32(10), 1226–1228 (2007). [CrossRef] [PubMed]

11.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2001).

12.

M. J. Bastiaans and T. Alieva, “First-order optical systems with unimodular eigenvalues,” J. Opt. Soc. Am. A 23(8), 1875–1883 (2006). [CrossRef]

13.

T. Alieva and M. J. Bastiaans, “Mode mapping in paraxial lossless optics,” Opt. Lett. 30(12), 1461–1463 (2005). [CrossRef] [PubMed]

14.

A. Wünsche, “General Hermite and Laguerre two-dimensional polynomials,” J. Phys. Math. Gen. 33(17), 1603–1629 (2000). [CrossRef]

15.

E. Abramochkin and V. Volostnikov, “Generalized Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6(5), S157–S161 (2004). [CrossRef]

16.

T. Alieva and A. Barbé, “Self-fractional Fourier images,” J. Mod. Opt. 46, 83–99 (1999).

OCIS Codes
(070.2580) Fourier optics and signal processing : Paraxial wave optics
(070.2590) Fourier optics and signal processing : ABCD transforms
(070.2575) Fourier optics and signal processing : Fractional Fourier transforms
(070.3185) Fourier optics and signal processing : Invariant optical fields

ToC Category:
Fourier Optics and Signal Processing

History
Original Manuscript: October 11, 2009
Revised Manuscript: January 19, 2010
Manuscript Accepted: January 28, 2010
Published: February 4, 2010

Citation
Tatiana Alieva, Eugeny Abramochkin, Ana Asenjo-Garcia, and Evgeniya Razueva, "Rotating beams in isotropic optical system," Opt. Express 18, 3568-3573 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-4-3568


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References

  1. E. Abramochkin and V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102(3-4), 336–350 (1993). [CrossRef]
  2. E. Abramochkin and V. Volostnikov, “Spiral-type beams: optical and quantum aspects,” Opt. Commun. 125(4-6), 302–323 (1996). [CrossRef]
  3. E. Abramochkin and V. Volostnikov, “Spiral light beams,” Phys. Usp. 47(12), 1177–1203 (2004). [CrossRef]
  4. A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Centrifugal transformation of the transverse structure of freely propagating paraxial light beams,” Opt. Lett. 31(6), 694–696 (2006). [CrossRef] [PubMed]
  5. A. Bekshaev and M. Soskin, “Rotational transformations and transverse energy flow in paraxial light beams: linear azimuthons,” Opt. Lett. 31(14), 2199–2201 (2006). [CrossRef] [PubMed]
  6. S. A. Collins., “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60(9), 1168–1177 (1970). [CrossRef]
  7. R. Simon and K. B. Wolf, “Structure of the set of paraxial optical systems,” J. Opt. Soc. Am. A 17(2), 342–355 (2000). [CrossRef]
  8. J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Optical system design for orthosymplectic transformations in phase space,” J. Opt. Soc. Am. A 23(10), 2494–2500 (2006). [CrossRef]
  9. G. F. Calvo, “Wigner representation and geometric transformations of optical orbital angular momentum spatial modes,” Opt. Lett. 30(10), 1207–1209 (2005). [CrossRef] [PubMed]
  10. T. Alieva and M. J. Bastiaans, “Orthonormal mode sets for the two-dimensional fractional Fourier transformation,” Opt. Lett. 32(10), 1226–1228 (2007). [CrossRef] [PubMed]
  11. H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2001).
  12. M. J. Bastiaans and T. Alieva, “First-order optical systems with unimodular eigenvalues,” J. Opt. Soc. Am. A 23(8), 1875–1883 (2006). [CrossRef]
  13. T. Alieva and M. J. Bastiaans, “Mode mapping in paraxial lossless optics,” Opt. Lett. 30(12), 1461–1463 (2005). [CrossRef] [PubMed]
  14. A. Wünsche, “General Hermite and Laguerre two-dimensional polynomials,” J. Phys. Math. Gen. 33(17), 1603–1629 (2000). [CrossRef]
  15. E. Abramochkin and V. Volostnikov, “Generalized Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6(5), S157–S161 (2004). [CrossRef]
  16. T. Alieva and A. Barbé, “Self-fractional Fourier images,” J. Mod. Opt. 46, 83–99 (1999).

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