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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 11 — May. 23, 2011
  • pp: 10293–10303
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Generation of optical vortex array with transformation of standing-wave Laguerre-Gaussian mode

Y. C. Lin, T. H. Lu, K. F. Huang, and Y. F. Chen  »View Author Affiliations


Optics Express, Vol. 19, Issue 11, pp. 10293-10303 (2011)
http://dx.doi.org/10.1364/OE.19.010293


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Abstract

We develop a novel method of creating optical vortex array by the conversion of a standing-wave Laguerre-Gaussian (LG) mode. Theoretically, by employing the transformational relation, the standing-wave LG mode is verified to be transformed from a pair of crisscrossed Hermite-Gaussian (HG) modes, embedded with optical vortex array, consists of a TEMn,m mode and a TEMm,n mode. Due to close correspondence between the transformational relation and the mode conversion of astigmatic lenses, we successfully generate the optical vortex array by transforming a standing-wave LG mode into the crisscrossed HG modes via a π/2 cylindrical lens mode converter. The investigation may provide useful insight in the study of the vortex light beam and its further applications.

© 2011 OSA

1. Introduction

In recent decades, optical vortices (OVs) characterized by their distinct features [1

1. K. T. Gahagan and G. A. Swartzlander Jr., “Optical vortex trapping of particles,” Opt. Lett. 21(11), 827–829 (1996). [CrossRef] [PubMed]

] have gained increasing importance in the study of singular optics [2

2. M. S. Soskin and M. V. Vasnetsov, “Singular Optics,” Prog. Opt. 42, 219–276 (2001). [CrossRef]

], light and matter interaction [1

1. K. T. Gahagan and G. A. Swartzlander Jr., “Optical vortex trapping of particles,” Opt. Lett. 21(11), 827–829 (1996). [CrossRef] [PubMed]

,3

3. N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22(1), 52–54 (1997). [CrossRef] [PubMed]

,4

4. H. Adachi, S. Akahoshi, and K. Miyakawa, “Orbital motion of spherical microparticles trapped in diffraction patterns of circularly polarized light,” Phys. Rev. A 75(6), 063409 (2007). [CrossRef]

], and quantum optics [5

5. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001). [CrossRef] [PubMed]

]. Since an OV is defined as the phase singularity with vanishing intensity of a helical-phased light beam, the azimuthally circulated phase term implies the orbital angular momentum (OAM) carried by the light beam [6

6. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef] [PubMed]

]. For practical interest, the characteristics associated with the OAM inspire great applications on optical tweezers [1

1. K. T. Gahagan and G. A. Swartzlander Jr., “Optical vortex trapping of particles,” Opt. Lett. 21(11), 827–829 (1996). [CrossRef] [PubMed]

,3

3. N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22(1), 52–54 (1997). [CrossRef] [PubMed]

,4

4. H. Adachi, S. Akahoshi, and K. Miyakawa, “Orbital motion of spherical microparticles trapped in diffraction patterns of circularly polarized light,” Phys. Rev. A 75(6), 063409 (2007). [CrossRef]

], optical testing [7

7. P. Senthilkumaran, “Optical phase singularities in detection of laser beam collimation,” Appl. Opt. 42(31), 6314–6320 (2003). [CrossRef] [PubMed]

], image processing [8

8. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004). [CrossRef] [PubMed]

,9

9. K. Crabtree, J. A. Davis, and I. Moreno, “Optical processing with vortex-producing lenses,” Appl. Opt. 43(6), 1360–1367 (2004). [CrossRef] [PubMed]

], quantum entanglement [5

5. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001). [CrossRef] [PubMed]

], and nonlinear optics [10

10. K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, “Second-harmonic generation and the orbital angular momentum of light,” Phys. Rev. A 54(5), R3742–R3745 (1996). [CrossRef] [PubMed]

].

Several approaches have been adopted to generate a single OV, such as mode conversions by astigmatic lenses [10

10. K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, “Second-harmonic generation and the orbital angular momentum of light,” Phys. Rev. A 54(5), R3742–R3745 (1996). [CrossRef] [PubMed]

,11

11. M. W. Beijersbergen, L. Allen, H. Vanderveen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96(1-3), 123–132 (1993). [CrossRef]

], spiral phase plates [12

12. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112(5-6), 321–327 (1994). [CrossRef]

], computer generated holograms [13

13. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221–233 (1992). [CrossRef] [PubMed]

], and optical wedges [14

14. Y. Izdebskaya, V. Shvedov, and A. Volyar, “Generation of higher-order optical vortices by a dielectric wedge,” Opt. Lett. 30(18), 2472–2474 (2005). [CrossRef] [PubMed]

]. Since Hermite-Gaussian (HG) modes can be emitted by most laser cavities and are well-known eigenstates for the 2D quantum harmonic oscillator [15

15. G. Nienhuis and L. Allen, “Paraxial wave optics and harmonic oscillators,” Phys. Rev. A 48(1), 656–665 (1993). [CrossRef] [PubMed]

], via the mode conversion, a HG mode has been widely used to create a single OV [11

11. M. W. Beijersbergen, L. Allen, H. Vanderveen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96(1-3), 123–132 (1993). [CrossRef]

13

13. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221–233 (1992). [CrossRef] [PubMed]

] hold by a traveling-wave Laguerre Gaussian (LG) mode features azimuthally phased termexp(ilϕ), where l is known as the topological charge of the vortex. The transformational relation has also been confirmed theoretically by using a Fresnel integral [16

16. E. Abramochkin and V. Volostnikov, “Beam transformation and nontransformed beams,” Opt. Commun. 83(1-2), 123–135 (1991). [CrossRef]

] or quantum operators connected the two complete sets of HG and LG states [17

17. Y. F. Chen, Y. C. Lin, K. F. Huang, and T. H. Lu, “Spatial transformation of coherent optical waves with orbital morphologies,” Phys. Rev. A 82(4), 043801 (2010). [CrossRef]

].

In this work, a novel method is carried out to produce the optical vortex array by the transformation of a standing-wave LG mode (the so-called “flower-like” [25

25. G. Grynberg, A. Maître, and A. Petrossian, “Flowerlike patterns generated by a laser beam transmitted through a rubidium cell with single feedback mirror,” Phys. Rev. Lett. 72(15), 2379–2382 (1994). [CrossRef] [PubMed]

] LG mode). Generation of the flower-like LG modes has been provided experimentally by utilizing a large-aperture CO2 laser [26

26. C. Green, G. B. Mindlin, E. J. D’Angelo, H. G. Solari, and J. R. Tredicce, “Spontaneous symmetry breaking in a laser: The experimental side,” Phys. Rev. Lett. 65(25), 3124–3127 (1990). [CrossRef] [PubMed]

], a solid-state laser cavity compounded of nonlinear medium [25

25. G. Grynberg, A. Maître, and A. Petrossian, “Flowerlike patterns generated by a laser beam transmitted through a rubidium cell with single feedback mirror,” Phys. Rev. Lett. 72(15), 2379–2382 (1994). [CrossRef] [PubMed]

,27

27. Y. F. Chen, T. H. Lu, and K. F. Huang, “Hyperboloid structures formed by polarization singularities in coherent vector fields with longitudinal-transverse coupling,” Phys. Rev. Lett. 97(23), 233903 (2006). [CrossRef]

,28

28. M. P. Thirugnanasambandam, Yu. Senatsky, and K. Ueda, “Generation of very-high order Laguerre-Gaussian modes in Yb:YAG ceramic laser,” Laser Phys. Lett. 7(9), 637–643 (2010). [CrossRef]

], and a vertical-cavity surface-emitting semiconductor laser [29

29. S. F. Pereira, M. B. Willemsen, M. P. van Exter, and J. P. Woerdman, “Pinning of daisy modes in optically pumped vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 73(16), 2239 (1998). [CrossRef]

]. Unlike a traveling-wave LG mode, a flower-like LG mode, formed by coherent superposition of a pair of traveling-wave ones that carry the same topological charges while counter rotational wave fronts, possesses no OV and has been concerned especially in the study of pattern formation [25

25. G. Grynberg, A. Maître, and A. Petrossian, “Flowerlike patterns generated by a laser beam transmitted through a rubidium cell with single feedback mirror,” Phys. Rev. Lett. 72(15), 2379–2382 (1994). [CrossRef] [PubMed]

,27

27. Y. F. Chen, T. H. Lu, and K. F. Huang, “Hyperboloid structures formed by polarization singularities in coherent vector fields with longitudinal-transverse coupling,” Phys. Rev. Lett. 97(23), 233903 (2006). [CrossRef]

29

29. S. F. Pereira, M. B. Willemsen, M. P. van Exter, and J. P. Woerdman, “Pinning of daisy modes in optically pumped vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 73(16), 2239 (1998). [CrossRef]

]. Therefore, it is fascinating and practical to raise the issue for the creation of optical vortex array by the use of the flower-like LG modes. To illustrate the feasibility, we verify firstly that a flower-like LG mode can be transformed from a set of “crisscrossed” HG modes theoretically. The optical vortex array is shown to be located at the cross section of the crisscrossed HG modes established by coherent superposition of a TEMn,m mode and a TEMm,n mode with well-defined relative phase α, where designate the transverse indices in x-y directions. Since the transformational relation has been confirmed to show excellent analogy to the mode conversion of a π/2-cylindrical-lens mode converter (π/2-CLMC) [17

17. Y. F. Chen, Y. C. Lin, K. F. Huang, and T. H. Lu, “Spatial transformation of coherent optical waves with orbital morphologies,” Phys. Rev. A 82(4), 043801 (2010). [CrossRef]

], the investigation enables us to generate the optical vortex array experimentally by transforming the accessible flower-like LG modes through the π/2-CLMC. More importantly, adjustability of the relative phase α is qualitatively displayed in this paper by rotating the mode converter at various angles. In all, we expect the creation and exploration of the vortex light beams in our work may inspirit a more thorough study in the vortex structure and its further applications.

2. Theoretical analysis

HG modes and LG modes are complete orthonormal sets that each can well describe any amplitude distribution by an appropriate complex superposition. Besides, they are eigenmodes that can be emitted by most laser resonators. Due to comprehensive studies and accessibility of the eigenmodes, it can be understood that it is useful to create and investigate the optical vortex array by concerning those well-known eigenmodes. As a result, it may be necessary to provide firstly a brief overview of the eigenmodes and their transformational relationship.

The profile of a HG mode in terms of the Cartesian coordinates (x,y,z) with transverse indices (n,m) under paraxial approximation of the wave equation is given by [15

15. G. Nienhuis and L. Allen, “Paraxial wave optics and harmonic oscillators,” Phys. Rev. A 48(1), 656–665 (1993). [CrossRef] [PubMed]

]
ψn,m(HG)(x,y,z)=ψn,m(HG)(x,y,z)ei(n+m+1)θG(z) eiξ(x,y,z),
(1)
where
ψn,m(HG)(x,y,z)=Cn,m(HG)w(z)   e(x2+y2w(z)2)Hn(2w(z)x)Hm(2w(z)y),
(2)
with ξ(x,y,z)=kz   [1+(x2+y2)/2(z2+zR2)], Cn,m(HG)=(2n+m1πn!m!)1/2, w(z)=w01+(z/zR)2, w0 is the beam radius at the waist, and zR=πw02/λ is the Rayleigh range. Hn() is the Hermite polynomial of order n, k is the wave number, and θG(z)=tan1(z/zR) is the Gouy phase. Likewise, the wavefunction of a LG mode characterized by its azimuthal and radial symmetry associated with a helical phase front exp(ilϕ) can be written in terms of the cylindrical coordinates (r,ϕ,z) with radial index p and azimuthal index l as [15

15. G. Nienhuis and L. Allen, “Paraxial wave optics and harmonic oscillators,” Phys. Rev. A 48(1), 656–665 (1993). [CrossRef] [PubMed]

]
ψp,l(LG)(r,ϕ,z)=ψp,l(LG)(r,ϕ,z)ei(2p+l+1)θG(z)eiξ(r,ϕ,z),
(3)
where
ψp,l(LG)(r,ϕ,z)=Cp,l(LG)w(z)(1)p(2rw(z))lLpl(2r2w(z)2)er2w(z)2   eilϕ,
(4)
where Lpl() is a Laguerre polynomial of azimuthal index l and radial index p,ξ(r,ϕ,z)=kz   [1+r2/2(z2+zR2)], and with condition p,l0. Note that the azimuthal indices with different sign convention (landl) denote the equal and opposite topological charges ±l of the LG modes which implies the OAM possessed by the light beam.

Most important of all, the conversion of the Gaussian modes, which has been demonstrated and verified in detail by [16

16. E. Abramochkin and V. Volostnikov, “Beam transformation and nontransformed beams,” Opt. Commun. 83(1-2), 123–135 (1991). [CrossRef]

,17

17. Y. F. Chen, Y. C. Lin, K. F. Huang, and T. H. Lu, “Spatial transformation of coherent optical waves with orbital morphologies,” Phys. Rev. A 82(4), 043801 (2010). [CrossRef]

], can be expressed in the following with a left-right-double arrow “⇔” signifying the transformational relation
ψp,   l(LG)(r,ϕ,z)   ψn,m(HG)(x,y,z),
(5-a)
ψp,l(LG)(r,ϕ,z)   ψm,   n(HG)(x,y,z),
(5-b)
where p=min(n,m), l=mn, and the relation 2p+l=n+m is hold for the conservation of transverse order under transformation. Explicitly, LG modes of opposite topological charges can be given by the transformation of a HG mode and its replica rotated at 90 degrees as shown in Eq. (5). To clarify the results, simplicity is added by using the Poincaré-sphere resemblance [30

30. M. J. Padgett and J. Courtial, “Poincaré-sphere equivalent for light beams containing orbital angular momentum,” Opt. Lett. 24(7), 430–432 (1999). [CrossRef]

] since the transformation of the LG and HG modes can be well mapped on the Poincaré-sphere that has been verified to be an effective tool in dealing with the conversion of polarization states [31

31. M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

].

To make these formal considerations more meaningful, return to our concern of the optical vortex array. Our goal is to create a network of OVs by coherent superposition of two crisscrossed HG modes of the same order n+m with a well-defined relative phaseα. Hence, the wavefunction of the superposed state composed of the HG modes can straightforwardly be written as
Ψn,   m(x,y,z,α)=ψn,m(HG)(x,y,z)+eiαψm,n(HG)(x,y,z),
(6)
whereα indicates the relative phase between the crisscrossed HG modes. An illustration for and α=π/2 is shown in Fig. 1
Fig. 1 Theoretical demonstration of Eq. (1) for the superposition of ψ0,11(HG) and ψ11,   0(HG) modes with relative phase α=π/2 .
to make Eq. (6) more clearer.

It is worth to mention that coherent superposition is the central concept of many quantum experiments. The phase singularities can only be arisen by coherent rather than incoherent superposition of the Gaussian modes. In [5

5. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001). [CrossRef] [PubMed]

], Alipasha Vaziri et. al. have emphasized the importance of preparing such superposed states for applications in quantum physics including quantum entanglement and quantum information. Though the significance of the relative phase for coherent superposition has also been appreciated in their work, a qualitative analysis and accessible experimental setup could not be seen in the research. As a result, in this article, we focus our attention on the study of the relative phase with the crisscrossed HG modes by a compact experimental configuration in order to find out its physical meaning and pivotal role in the formation of the OVs.

To determine the OVs, it should be noted that they are defined as the phase singularities where the real and imaginary components of the scalar field Ψn,   m(x,y,z,α) are both zero and possess the characteristic of zero intensity in the vortex core [32

32. G. A. Swartzlander Jr., “Dark-soliton prototype devices: analysis by using direct-scattering theory,” Opt. Lett. 17, 789–791 (1992).

,33

33. Y. F. Chen and Y. P. Lan, “Dynamics of the Laguerre Gaussian TEM*0,1 mode in a solid-state laser,” Phys. Rev. A 63(6), 063807 (2001). [CrossRef]

]. In other words, the resulting vortices are dark points in intersects of the nodal lines corresponding to the respective HG modes of the state Ψn,   m(x,y,z,α) with relative phase α apart from an integral multiple ofπ. An illustration of |Ψ0,   11(x,y,z,α)|2 is depicted in Fig. 2
Fig. 2 Theoretical results of Ψ0,   11(x,y,z,α) of various relative phases.
with various relative phaseα ranging from 0 to2π and it can be seen the intensity distribution in the cross-section is altered accordingly. Despite this, Fig. 2 also reveals the nature that |Ψn,   m(x,y,z,α)|2 is repeated within every period of 2π phase shift.

For comprehensive demonstration of the vortex structure, Fig. 3
Fig. 3 (a) Theoretical results of Ψ0,   11(x,y,z,π/2). (b) Phase distribution of (a). (c) Enlarged figure of the box region in (b). (d) linear momentum density p for the box region in (b) of Ψ0,   11(x,y,z,π/2).
depicts the phase distribution of the stateΨ0,   11(x,y,z,π/2). Though there are all 11×11 singularities which situate at the crisscrossed positions of the nodal lines as depicted in Fig. 3(b), the available OVs for practical use of particle trapping with stronger intensity distribution around [1

1. K. T. Gahagan and G. A. Swartzlander Jr., “Optical vortex trapping of particles,” Opt. Lett. 21(11), 827–829 (1996). [CrossRef] [PubMed]

] are estimated at in total within the cross section. The enlarged figure of the box region in Fig. 3(b) is shown in Fig. 3(c) where the black and red dots mark the vortex positions of counter rotational phase fronts. It is conspicuously illustrated in Fig. 3(d) by plotting the transverse linear momentum density pof a linear polarized light beam [6

6. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef] [PubMed]

]
p=iωε02(Ψ*ΨΨΨ*),
(7)
where p=(px,py), =(/x,   /y), ω relates to the frequency of the light beam and ε0 represents the permittivity of free space. Note that the vector field p has been normalized to p/|p| for observing the detailed structures. Since the helical wave fronts signify the OAM carried by the light beam, the swirls in Fig. 3(d) shows the evidence that the superposed stateΨ0,   11(x,y,z,π/2) certainly form an OAM state associated with the vortex array. From quantum perspective, preparing such superposed OAM states has become an important issue concerning quantum entanglement and quantum information [5

5. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001). [CrossRef] [PubMed]

]. Thus, it is crucial to arrange feasible experimental techniques for creating and investigating the superposed states.

3. Experimental results

The experimental configuration can be separated mainly into three parts according to particular purposes as depicted in Fig. 5
Fig. 5 Experimental setup utilized to transform the flower-like LG modes into the crisscrossed HG modes with the cylindrical lenses.
. The microchip solid state laser cavity presented in Fig. 5(a) is utilized to generate the flower-like LG mode discussed on the above as an input mode to be transformed via theπ/2-CLMC. The second part at the external cavity is consisted of a non-spherical lens and theπ/2-CLMC to convert the emitted LG mode into the crisscrossed HG modes as shown in Fig. 5(b). The last part in Fig. 5(c) corresponds apparently to the detection scheme. Furthermore, detailed experimental arrangements are provided in the following.

It can be seen that the laser resonator is composed of a gain medium and a spherical mirror. The laser medium is an a-cut 2.0-at. % Nd: YVO4 crystal with a length of 2 mm and the cross section 3×3 mm2. One side of the Nd: YVO4 crystal is coated for partial reflection at 1064 nm and the other is for antireflection at 1064 nm. The radius of curvature of the cavity mirror is R = 25 cm and its reflectivity is 97% at 1064 nm. The pump source is an 808 nm fiber-coupled laser diode with pump core of 100 μm in radius, a numerical aperture of 0.16, and a maximum output power of 1 W. A focusing lens with focal length of 20 mm and 90% coupling efficiency is used to reimage the pump beam into the laser crystal. To produce a high-order flower-like LG mode, which are processed from the astigmatism and imperfection dominated by the cylindrical symmetric laser cavity, the valid key point is using a doughnut pump profile and defocusing the standard fiber-coupled diode [32

32. G. A. Swartzlander Jr., “Dark-soliton prototype devices: analysis by using direct-scattering theory,” Opt. Lett. 17, 789–791 (1992).

,33

33. Y. F. Chen and Y. P. Lan, “Dynamics of the Laguerre Gaussian TEM*0,1 mode in a solid-state laser,” Phys. Rev. A 63(6), 063807 (2001). [CrossRef]

]. The pump spot radius is controlled to be around 50-200μm. The effective cavity length is set in the range of 1-1.5cm. A non-spherical lens with focal length f=40 mm mounted on a translation stage is exploited to provide the mode matching condition by collimating the input light beam in the midway between the following cylindrical lenses. The flower-like LG modes generated by the laser cavity are converted into the crisscrossed HG modes by passing through a rotatable -CLMC comprised of two identical cylindrical lenses with focal lengthf=25mm, separated by2f. To observe the far-field pattern via a CCD camera, the output beam is directly projected to a paper screen.

What needs to be emphasized especially is the controllable relative phaseαbetween the two crisscrossed HG modes, which can be qualitatively altered by rotating the π/2-CLMC at various angle. Since the relative phase has been confirmed to be the decisive factor that contributes to the formation of the phase singularities according to the above investigation, the adjustability of the relative phase in the experiment appears to be absolutely crucial to the production of the OAM state. For practical consideration, the investigation reveals the possibility of particle manipulation in two-transverse-dimension for the developing applications as it can be informed from Fig. 6(b) that the resulting vortices with fixed relative positions are simultaneously rotated with the CLMC by an angleθ. Moreover, since the Gaussian beams satisfy the property of bilinear transformation [34

34. A. E. Siegman, Lasers (University Science, 1986).

], which indicates that the profiles are preserved under propagation in free space through the Fourier transformation, the resulting vortex array can maintain its spatial distribution while being focused. Namely, it enables us to quantitatively determine the features of the optical vortex array that focused into the optical traps.

In addition to constructing the optical vortex array that embedded in the superposed stateΨn,   m(x,y,z,α) with vanishing transverse index n illustrated on the above, we are now considering more complicated vortex structures determined by increasing transverse index p with multi-ring LG modes. As an illustration, Fig. 8(b)
Fig. 8 Theoretical analysis: (a) LG modes with non-vanishing radial indexn. (b) The resulting modes converted from the LG modes. (c) Phase distribution corresponding to (b).
demonstrates the theoretical results of superposed statesΨ1,10(x,y,z,π/2),Ψ2,10(x,y,z,π/2), and Ψ3,10(x,y,z,π/2)with non-vanishing transverse index p corresponding to the flower-like LG modes ofΦ1,9(r,ϕ,z,π/2),Φ2,8(r,ϕ,z,π/2), andΦ3,7(r,ϕ,z,π/2) as shown in Fig. 8(a). To make it clear, the associated phase distribution is demonstrated in Fig. 8(c) which explicitly shows the variation of the position of the singularities defined by the points of intersection. Since several methods have been adopted to generate the multi-ring LG modes [25

25. G. Grynberg, A. Maître, and A. Petrossian, “Flowerlike patterns generated by a laser beam transmitted through a rubidium cell with single feedback mirror,” Phys. Rev. Lett. 72(15), 2379–2382 (1994). [CrossRef] [PubMed]

29

29. S. F. Pereira, M. B. Willemsen, M. P. van Exter, and J. P. Woerdman, “Pinning of daisy modes in optically pumped vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 73(16), 2239 (1998). [CrossRef]

], our investigation may provoke further application for the creation of the exotic vortex structures by transforming the available higher-order LG modes through the mode converter.

4. Conclusion

Acknowledgments

The authors acknowledge the National Science Council of Taiwan (NSCT) for their financial support of this research under contract NSC96-2112-M-009-027-MY3.

References and links

1.

K. T. Gahagan and G. A. Swartzlander Jr., “Optical vortex trapping of particles,” Opt. Lett. 21(11), 827–829 (1996). [CrossRef] [PubMed]

2.

M. S. Soskin and M. V. Vasnetsov, “Singular Optics,” Prog. Opt. 42, 219–276 (2001). [CrossRef]

3.

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22(1), 52–54 (1997). [CrossRef] [PubMed]

4.

H. Adachi, S. Akahoshi, and K. Miyakawa, “Orbital motion of spherical microparticles trapped in diffraction patterns of circularly polarized light,” Phys. Rev. A 75(6), 063409 (2007). [CrossRef]

5.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001). [CrossRef] [PubMed]

6.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef] [PubMed]

7.

P. Senthilkumaran, “Optical phase singularities in detection of laser beam collimation,” Appl. Opt. 42(31), 6314–6320 (2003). [CrossRef] [PubMed]

8.

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004). [CrossRef] [PubMed]

9.

K. Crabtree, J. A. Davis, and I. Moreno, “Optical processing with vortex-producing lenses,” Appl. Opt. 43(6), 1360–1367 (2004). [CrossRef] [PubMed]

10.

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, “Second-harmonic generation and the orbital angular momentum of light,” Phys. Rev. A 54(5), R3742–R3745 (1996). [CrossRef] [PubMed]

11.

M. W. Beijersbergen, L. Allen, H. Vanderveen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96(1-3), 123–132 (1993). [CrossRef]

12.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112(5-6), 321–327 (1994). [CrossRef]

13.

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221–233 (1992). [CrossRef] [PubMed]

14.

Y. Izdebskaya, V. Shvedov, and A. Volyar, “Generation of higher-order optical vortices by a dielectric wedge,” Opt. Lett. 30(18), 2472–2474 (2005). [CrossRef] [PubMed]

15.

G. Nienhuis and L. Allen, “Paraxial wave optics and harmonic oscillators,” Phys. Rev. A 48(1), 656–665 (1993). [CrossRef] [PubMed]

16.

E. Abramochkin and V. Volostnikov, “Beam transformation and nontransformed beams,” Opt. Commun. 83(1-2), 123–135 (1991). [CrossRef]

17.

Y. F. Chen, Y. C. Lin, K. F. Huang, and T. H. Lu, “Spatial transformation of coherent optical waves with orbital morphologies,” Phys. Rev. A 82(4), 043801 (2010). [CrossRef]

18.

S. Vyas and P. Senthilkumaran, “Interferometric optical vortex array generator,” Appl. Opt. 46(15), 2893–2898 (2007). [CrossRef] [PubMed]

19.

K. O’Holleran, M. J. Padgett, and M. R. Dennis, “Topology of optical vortex lines formed by the interference of three, four, and five plane waves,” Opt. Express 14(7), 3039–3044 (2006). [CrossRef] [PubMed]

20.

K. Otsuka and S. C. Chu, “Generation of vortex array beams from a thin-slice solid-state laser with shaped wide-aperture laser-diode pumping,” Opt. Lett. 34(1), 10–12 (2009). [CrossRef]

21.

J. Masajada, “Small-angle rotations measurement using optical vortex interferometer,” Opt. Commun. 239(4-6), 373–381 (2004). [CrossRef]

22.

M. D. Levenson, T. J. Ebihara, G. Dai, Y. Morikawa, N. Hayashi, and S. M. Tan, “Optical vortex mask via levels,” J. Microlithogr. Microfabr. Microsyst. 3(2), 293–304 (2004). [CrossRef]

23.

K. Ladavac and D. G. Grier, “Microoptomechanical pumps assembled and driven by holographic optical vortex arrays,” Opt. Express 12(6), 1144–1149 (2004). [CrossRef] [PubMed]

24.

G. H. Kim, J. H. Jeon, Y. C. Noh, K. H. Ko, H. J. Moon, J. H. Lee, and J. S. Chang, “An array of phase singularities in a self-defocusing medium,” Opt. Commun. 147(1-3), 131–137 (1998). [CrossRef]

25.

G. Grynberg, A. Maître, and A. Petrossian, “Flowerlike patterns generated by a laser beam transmitted through a rubidium cell with single feedback mirror,” Phys. Rev. Lett. 72(15), 2379–2382 (1994). [CrossRef] [PubMed]

26.

C. Green, G. B. Mindlin, E. J. D’Angelo, H. G. Solari, and J. R. Tredicce, “Spontaneous symmetry breaking in a laser: The experimental side,” Phys. Rev. Lett. 65(25), 3124–3127 (1990). [CrossRef] [PubMed]

27.

Y. F. Chen, T. H. Lu, and K. F. Huang, “Hyperboloid structures formed by polarization singularities in coherent vector fields with longitudinal-transverse coupling,” Phys. Rev. Lett. 97(23), 233903 (2006). [CrossRef]

28.

M. P. Thirugnanasambandam, Yu. Senatsky, and K. Ueda, “Generation of very-high order Laguerre-Gaussian modes in Yb:YAG ceramic laser,” Laser Phys. Lett. 7(9), 637–643 (2010). [CrossRef]

29.

S. F. Pereira, M. B. Willemsen, M. P. van Exter, and J. P. Woerdman, “Pinning of daisy modes in optically pumped vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 73(16), 2239 (1998). [CrossRef]

30.

M. J. Padgett and J. Courtial, “Poincaré-sphere equivalent for light beams containing orbital angular momentum,” Opt. Lett. 24(7), 430–432 (1999). [CrossRef]

31.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

32.

G. A. Swartzlander Jr., “Dark-soliton prototype devices: analysis by using direct-scattering theory,” Opt. Lett. 17, 789–791 (1992).

33.

Y. F. Chen and Y. P. Lan, “Dynamics of the Laguerre Gaussian TEM*0,1 mode in a solid-state laser,” Phys. Rev. A 63(6), 063807 (2001). [CrossRef]

34.

A. E. Siegman, Lasers (University Science, 1986).

OCIS Codes
(030.4070) Coherence and statistical optics : Modes
(140.3480) Lasers and laser optics : Lasers, diode-pumped

ToC Category:
Physical Optics

History
Original Manuscript: March 16, 2011
Revised Manuscript: April 29, 2011
Manuscript Accepted: May 8, 2011
Published: May 10, 2011

Citation
Y. C. Lin, T. H. Lu, K. F. Huang, and Y. F. Chen, "Generation of optical vortex array with transformation of standing-wave Laguerre-Gaussian mode," Opt. Express 19, 10293-10303 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-11-10293


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References

  1. K. T. Gahagan and G. A. Swartzlander., “Optical vortex trapping of particles,” Opt. Lett. 21(11), 827–829 (1996). [CrossRef] [PubMed]
  2. M. S. Soskin and M. V. Vasnetsov, “Singular Optics,” Prog. Opt. 42, 219–276 (2001). [CrossRef]
  3. N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22(1), 52–54 (1997). [CrossRef] [PubMed]
  4. H. Adachi, S. Akahoshi, and K. Miyakawa, “Orbital motion of spherical microparticles trapped in diffraction patterns of circularly polarized light,” Phys. Rev. A 75(6), 063409 (2007). [CrossRef]
  5. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001). [CrossRef] [PubMed]
  6. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef] [PubMed]
  7. P. Senthilkumaran, “Optical phase singularities in detection of laser beam collimation,” Appl. Opt. 42(31), 6314–6320 (2003). [CrossRef] [PubMed]
  8. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004). [CrossRef] [PubMed]
  9. K. Crabtree, J. A. Davis, and I. Moreno, “Optical processing with vortex-producing lenses,” Appl. Opt. 43(6), 1360–1367 (2004). [CrossRef] [PubMed]
  10. K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, “Second-harmonic generation and the orbital angular momentum of light,” Phys. Rev. A 54(5), R3742–R3745 (1996). [CrossRef] [PubMed]
  11. M. W. Beijersbergen, L. Allen, H. Vanderveen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96(1-3), 123–132 (1993). [CrossRef]
  12. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112(5-6), 321–327 (1994). [CrossRef]
  13. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221–233 (1992). [CrossRef] [PubMed]
  14. Y. Izdebskaya, V. Shvedov, and A. Volyar, “Generation of higher-order optical vortices by a dielectric wedge,” Opt. Lett. 30(18), 2472–2474 (2005). [CrossRef] [PubMed]
  15. G. Nienhuis and L. Allen, “Paraxial wave optics and harmonic oscillators,” Phys. Rev. A 48(1), 656–665 (1993). [CrossRef] [PubMed]
  16. E. Abramochkin and V. Volostnikov, “Beam transformation and nontransformed beams,” Opt. Commun. 83(1-2), 123–135 (1991). [CrossRef]
  17. Y. F. Chen, Y. C. Lin, K. F. Huang, and T. H. Lu, “Spatial transformation of coherent optical waves with orbital morphologies,” Phys. Rev. A 82(4), 043801 (2010). [CrossRef]
  18. S. Vyas and P. Senthilkumaran, “Interferometric optical vortex array generator,” Appl. Opt. 46(15), 2893–2898 (2007). [CrossRef] [PubMed]
  19. K. O’Holleran, M. J. Padgett, and M. R. Dennis, “Topology of optical vortex lines formed by the interference of three, four, and five plane waves,” Opt. Express 14(7), 3039–3044 (2006). [CrossRef] [PubMed]
  20. K. Otsuka and S. C. Chu, “Generation of vortex array beams from a thin-slice solid-state laser with shaped wide-aperture laser-diode pumping,” Opt. Lett. 34(1), 10–12 (2009). [CrossRef]
  21. J. Masajada, “Small-angle rotations measurement using optical vortex interferometer,” Opt. Commun. 239(4-6), 373–381 (2004). [CrossRef]
  22. M. D. Levenson, T. J. Ebihara, G. Dai, Y. Morikawa, N. Hayashi, and S. M. Tan, “Optical vortex mask via levels,” J. Microlithogr. Microfabr. Microsyst. 3(2), 293–304 (2004). [CrossRef]
  23. K. Ladavac and D. G. Grier, “Microoptomechanical pumps assembled and driven by holographic optical vortex arrays,” Opt. Express 12(6), 1144–1149 (2004). [CrossRef] [PubMed]
  24. G. H. Kim, J. H. Jeon, Y. C. Noh, K. H. Ko, H. J. Moon, J. H. Lee, and J. S. Chang, “An array of phase singularities in a self-defocusing medium,” Opt. Commun. 147(1-3), 131–137 (1998). [CrossRef]
  25. G. Grynberg, A. Maître, and A. Petrossian, “Flowerlike patterns generated by a laser beam transmitted through a rubidium cell with single feedback mirror,” Phys. Rev. Lett. 72(15), 2379–2382 (1994). [CrossRef] [PubMed]
  26. C. Green, G. B. Mindlin, E. J. D’Angelo, H. G. Solari, and J. R. Tredicce, “Spontaneous symmetry breaking in a laser: The experimental side,” Phys. Rev. Lett. 65(25), 3124–3127 (1990). [CrossRef] [PubMed]
  27. Y. F. Chen, T. H. Lu, and K. F. Huang, “Hyperboloid structures formed by polarization singularities in coherent vector fields with longitudinal-transverse coupling,” Phys. Rev. Lett. 97(23), 233903 (2006). [CrossRef]
  28. M. P. Thirugnanasambandam, Yu. Senatsky, and K. Ueda, “Generation of very-high order Laguerre-Gaussian modes in Yb:YAG ceramic laser,” Laser Phys. Lett. 7(9), 637–643 (2010). [CrossRef]
  29. S. F. Pereira, M. B. Willemsen, M. P. van Exter, and J. P. Woerdman, “Pinning of daisy modes in optically pumped vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 73(16), 2239 (1998). [CrossRef]
  30. M. J. Padgett and J. Courtial, “Poincaré-sphere equivalent for light beams containing orbital angular momentum,” Opt. Lett. 24(7), 430–432 (1999). [CrossRef]
  31. M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).
  32. G. A. Swartzlander., “Dark-soliton prototype devices: analysis by using direct-scattering theory,” Opt. Lett. 17, 789–791 (1992).
  33. Y. F. Chen and Y. P. Lan, “Dynamics of the Laguerre Gaussian TEM*0,1 mode in a solid-state laser,” Phys. Rev. A 63(6), 063807 (2001). [CrossRef]
  34. A. E. Siegman, Lasers (University Science, 1986).

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