Rotating beams in isotropic optical system

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 Optical Society of America OCIS codes: (070.2590) ABCD transforms; (070.2575) Fractional Fourier transforms; (070.3185) Invariant optical fields; (070.2580) Paraxial wave optics. References and links 1. E. Abramochkin, and V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102(3-4), 336–350 (1993). 2. E. Abramochkin, and V. Volostnikov, “Spiral-type beams: optical and quantum aspects,” Opt. Commun. 125(46), 302–323 (1996). 3. E. Abramochkin, and V. Volostnikov, “Spiral light beams,” Phys. Usp. 47(12), 1177–1203 (2004). 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). 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). 6. S. A. Collins, Jr., “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60(9), 1168–1177 (1970). 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). 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). 9. G. F. Calvo, “Wigner representation and geometric transformations of optical orbital angular momentum spatial modes,” Opt. Lett. 30(10), 1207–1209 (2005). 10. T. Alieva, and M. J. Bastiaans, “Orthonormal mode sets for the two-dimensional fractional Fourier transformation,” Opt. Lett. 32(10), 1226–1228 (2007). 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). 13. T. Alieva, and M. J. Bastiaans, “Mode mapping in paraxial lossless optics,” Opt. Lett. 30(12), 1461–1463 (2005). 14. A. Wünsche, “General Hermite and Laguerre two-dimensional polynomials,” J. Phys. Math. Gen. 33(17), 1603– 1629 (2000). 15. E. Abramochkin, and V. Volostnikov, “Generalized Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6(5), S157– S161 (2004). 16. T. Alieva, and A. Barbé, “Self-fractional Fourier images,” J. Mod. Opt. 46, 83–99 (1999).


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][2][3][4][5].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], represented by operator T R , whose kernel is parameterized by the real symplectic ray transformation matrix T, which relates the position ( , ) t x y = r and direction ( , ) t x y p p = p of an incoming ray to those of the outgoing ray .
Here we use dimensionless variables.Note that the variables ( , ) r p form, in paraxial approximation, optical phase space.Using the modified Iwasawa decomposition [7], the properly normalized ray transformation matrix T can be written as a product of three matrices, 1 , where the first matrix L T with the symmetric matrix respectively.In particular, the FrFT is responsible for the rotation in ( , ) x x p and ( , ) y y p planes at angles x γ and y γ , respectively.The signal rotator produces the rotation in ( , ) x y and ( , ) x y p p planes.The gyrator operator describes cross ( , ) x y p and ( , ) y x p rotations.Note that a cascade of two rotators defined by 1 U and 2 U corresponds to a phase-space rotator described by the matrix 2 1 U U .The experimental setups capable to perform these phase-space rotators were discussed in Ref [8].

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, ( , ) , is inherently different.Indeed, its unitary matrix ( , ) commutes with any unitary matrix.The det ( , ) exp( 2) while the determinant of the matrix describing other phase-space rotators, indicated as ar U which can be expressed as a product of ( , ) 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].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, ar T R , 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, ( , ) ) and the anisotropic phasespace rotator ar T R will be identical Below the operator for the canonical transform T R associated with orthosymplectic matrix T will be denoted by U R .

Design of rotating beams
Let us consider the way how to generate the beams which satisfy Eq. ( 2).Note that any ar U may be factored as , where 0 U is also unitary matrix [12].It has been shown [13,14] that for any unitary matrix 0 U with elements jk U ( , 1, 2 j k = ) there exists a complete orthonormal set of Gaussian modes { } 0 , ( ), , 0,1, which are eigenfunctions for the transform 0 ( , )

R . The corresponding eigenvalues are given by exp[ ( ) ]
i m n γ − − .Moreover, these modes are also eigenfunctions for the symmetric FrFT, defined by ( , ) Hermite-Laguerre-Gaussian modes [15]: where ( , ) ( cos , sin ) can also be represented as a linear combination of modes further call as anisotropic rotators, describe the movements on the orbital Poincaré sphere[9,10].Any IOS, which may consist of centered spherical lenses, mirrors and free space intervals, is described by the ray transformation matrix with scalar a particular, for free space propagation (Fresnel diffraction) 1 a d = = , 0 c = and the angle φ is limited: 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 (HG) and Laguerre-Gaussian (LG) modes respectively: #118426 -$15.00USD Received 11 Oct 2009; revised 19 Jan 2010; accepted 28 Jan 2010; published 4 Feb 2010 (C) 2010 OSA ( )

U
eigenfunctions for the FrFT at angles (γ,−γ) and therefore at any pair of angles x γ and y γ .LG modes are eigenfunctions for the signal rotator 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 0 with the same of the indices, m n − , is an eigenfunction for the transform associated with 0 ( , ) ar γ U U 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

U
during symmetric FrFT at angle ϕ , we can rewrite φ as ( , ) [ (1 ) (1 ) 1] complex mn c and the mode indices m and n, which satisfy the relation