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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 24 — Nov. 24, 2008
  • pp: 19934–19949
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Vortex array laser beam generation from a Dove prism-embedded unbalanced Mach-Zehnder interferometer

Shu-Chun Chu, Chao-Shun Yang, and Kenju Otsuka  »View Author Affiliations


Optics Express, Vol. 16, Issue 24, pp. 19934-19949 (2008)
http://dx.doi.org/10.1364/OE.16.019934


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Abstract

This paper proposes a new scheme for generating vortex laser beams from a laser. The proposed system consists of a Dove prism embedded in an unbalanced Mach-Zehnder interferometer configuration. This configuration allows controlled construction of p×p vortex array beams from Ince-Gaussian modes, IGe p, p modes. An incident IGe p, p laser beam of variety order p can easily be generated from an end-pumped solid-state laser system with an off-axis pumping mechanism. This study simulates this type of vortex array laser beam generation, analytically derives the vortex positions of the resulting vortex array laser beams, and discusses beam propagation effects. The resulting vortex array laser beam can be applied to optical tweezers and atom traps in the form of two-dimensional arrays, or used to study the transfer of angular momentum to micro particles or atoms (Bose-Einstein condensate).

© 2008 Optical Society of America

1. Introduction

Optical vortices possess several special properties, including carrying optical angular momentum (OAM) and exhibiting zero intensity. As a result, vortex laser beams are widely used as optical tweezers [1

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

, 2

2. D. W. Zhang, X. -. Yuan, E. Santamato, A. Sasso, B. Piccirillo, and A. Vella, “Optical angular momentum transfer to transparent isotropic particles using laser beam carrying zero average angular momentum,” Opt. Express10, 871–878 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-17-871.

] and in the study of the transfer of angular momentum to micro particles or atoms [3

3. L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912–914 (2001). [CrossRef] [PubMed]

6

6. J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun. 198, 21–27 (2001). [CrossRef]

]. Researchers use several different approaches to study the generation of vortex array laser beams including three plane waves interference [7

7. A. Dreischuh, S. Chervenkov, D. Neshev, G. G. Paulus, and H. Walther, “Generation of lattice structures of optical vortices,” J. Opt. Soc. Am. B 19, 550–556 (2002) [CrossRef]

], propagation in a saturable nonlinear medium [8

8. P. Kurzynowski and M. Borwińska, “Generation of vortex-type markers in a one-wave setup,” Appl. Opt. 46, 676–679 (2007) [CrossRef] [PubMed]

], or interfering beams passing through birefringent elements [9

9. K. J. Moh, X. -. Yuan, W. C. Cheong, L. S. Zhang, J. Lin, B. P. S. Ahluwalia, and H. Wang, “High-power efficient multiple optical vortices in a single beam generated by a kinoform-type spiral phase plate,” Appl. Opt. 45, 1153–1161 (2006) [CrossRef] [PubMed]

]. The promising applications of vortex array laser beams to laser tweezers and other applications, such as atom guiding, raise the important issue of propagation dynamics: Does a vortex array laser beam repeat its vortex array pattern during propagation and focusing? These questions are important for laser trapping and other applications. To our best knowledge, there are two primary approaches to producing stable vortex array laser beams that preserve the lateral functional structure during propagation. The first approach is to pass a TEM00 laser beam through well-designed 2-dimensional phase plates [10

10. G. A. Turnball, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre-Gaussian modes at millimetre-wave frequencies by use of a spiral phase plate,” Opt. Commun. 127, 183–188 (1996). [CrossRef]

] or co-propagate a vortex beam with a phase singularity possessing multiple (p) charges, resulting in separated p single-charge vortices [10

10. G. A. Turnball, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre-Gaussian modes at millimetre-wave frequencies by use of a spiral phase plate,” Opt. Commun. 127, 183–188 (1996). [CrossRef]

]. These charges can be generated using phase plates [11

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

] or astigmatic mode converters for higher-order Hermite-Gaussian modes [12

12. M. Brambilla, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” Phys. Rev. A 43, 5090–5113 (1991). [CrossRef] [PubMed]

]. A second approach might be to employ vortex array beams called “optical vortices crystals,” which are generated spontaneously from wide-aperture lasers due to the intrinsic optical nonlinearity of lasers [13

13. J. Scheuer and M. Orenstein, “Optical vortices crystals: Spontaneous generation in nonlinear semiconductor microcavities,” Science 285, 230–233 (1999). [CrossRef] [PubMed]

, 14

14. M. A. Bandres and Julio C. Gutiérrez-Vega, “Ince-Gaussian beams,” Opt. Lett. 29144–146 (2004) [CrossRef] [PubMed]

].

M. A. Bandres et al. proposed a new complete family of transverse modes in 2004, the Ince-Gaussian modes (IGMs), which differs from the well-known Hermite-Gaussian modes (HGMs) and Laguerre-Gaussian modes (LGMs). Ince-Gaussian modes constitute the continuous transition modes between HGMs and LGMs [15

15. M. A. Bandres and J. C. Gutiérrez-Vega, “Ince-Gaussian modes of the paraxial wave equation and stable resonators,” J. Opt. Soc. Am. A 21, 873–880 (2004). [CrossRef]

, 16

16. M. A. Bandres, “Elegant Ince-Gaussian beams,” Opt. Lett. 29, 1724, (2004). [CrossRef] [PubMed]

], and several authors have recently explored the characteristics of this novel family of IGMs [17

17. J. C. Gutiérrez-Vega and M. A. Bandres, “Ince-Gaussian beam in quadratic index medium,” J. Opt. Soc. Am. A , 22, 306–309, (2005). [CrossRef]

20

20. E. L. Wooten, R. L. Stone, E. W. Miles, and E. M. Bradley, “Rapidly tunable narrow band wavelength filter using LiNbO3 unbalanced Mach-Zehnder interferometers,” J. Lightwave Technol. 14, 2530–2536 (1996). [CrossRef]

]. This paper proposes a new method of vortex array beam generation based on IGMs that involves a Dove prism embedded unbalanced Mach-Zehnder interferometer. This interferometer can convert even Ince-Gaussian modes, the IGep, p modes, into vortex array laser beams consisting of ap×p number of well-aligned vortices. This study simulates vortex array laser beam formation based on IGep, p laser beams, which can be generated from solid-state lasers (SSLs) with off-axis laser-diode pumping, and a method to solve the vortex positions of the vortex array laser beams. The resulting vortex array laser beam from the proposed interferometer maintains its beam profile during both propagation and focusing. Thus, the proposed vortex array laser beams hold great promise for application in optical tweezers and atom traps in the form of two-dimensional arrays, as well as the study of the transfer of angular momentum to micro particles or atoms (Bose-Einstein condensate).

This paper is organized as follows. Section 2 briefly reviews the basic formalism of the Ince-Gaussian modes used in this paper, the IGep, p modes. Section 3 describes the basic idea of a Dove prism-embedded unbalanced Mach-Zehnder interferometer, including a variable phase retarder. Section 4 numerically demonstrates the generation of vortex array laser beam from an end-pumped SSL. Section 5 presents a method to solve the cross-section vortex positions of the resulting vortex array beam. Section 6 discusses the effects of ellipticity parameter, phase retardation, and power difference for two IGMs combined into vortex array patterns. Section 6 also discusses the propagation dynamics of the resulting vortex array laser beam and practical applications of the proposed system. Section 7 summarizes the numerical and analytical results of this study.

2. Basic formalism of Ince-Gaussian modes

The Ince-Gaussian modes propagating along the z axis of an elliptic coordinate system r=(ξ, η, z), with mode numbers p and m and ellipticity ε, are given by [16

16. M. A. Bandres, “Elegant Ince-Gaussian beams,” Opt. Lett. 29, 1724, (2004). [CrossRef] [PubMed]

]

IGp,me(r,ε)=C[w0w(z)]Cpm(iξ,ε)Cpm(η,ε)exp[r2w2(z)]
×expi[kz+{kr22R(z)}(p+1)ψz(z)],
(1)
IGp,mo(r,ε)=S[w0w(z)]Spm(iξ,ε)Spm(η,ε)exp[r2w2(z)]
×expi[kz+{kr22R(z)}(p+1)ψz(z)],
(2)

Fig. 1. Some analytical patterns of Ince-Gaussian modes.

3. Dove prism embedded unbalanced Mach-Zehnder interferometer

Although more than one specific interferometer configuration is possible, this study demonstrates superposed generation of vortex array laser beam with a typical interferometer configuration for simplicity. Figure 2(a) shows the proposed configuration, which is similar to the unbalanced Mach-Zehnder interferometer [21

21. S.-C. Chu, “Generation of multiple vortex beams with specified vortex number from lasers with controlled Ince-Gaussian modes,” Jap. J. Appl. Phys. 475297–5303 (2008) [CrossRef]

], but with a Dove prism embedded in one arm. As Fig. 2(b) shows, the embedded Dove prism rotates about the optical axis, the z-axis, at 45 degrees. Passing a nearly-collimated IGep, p laser beam through the interferometer produces vortex array laser beams with p×p vortices. The vortex array is generated by the precise control of the relative phase shift between the IGep, p mode and its rotated replica, namely the [IGep, p]T mode, with a variable phase retarder (e.g., commercial nano-stage with a precision motorized actuator with <100-nm resolution and/or liquid crystal with high phase shift adjustment precision inserted into one arm). An incident IGep, p laser beam of variety order p can be easily generated from an end-pumped SSL with an off-axis pumping mechanism [22

22. W. J. Smith, Modern Optical Engineering (McGraw-Hill, 2000) pp. 105–107

].

Fig. 2. (a) Schematic diagram of the interferometer configuration. (b) Dove prism setup in the proposed interferometer. The field polarization states at all stations are drawn in green.

Figure 2 shows that the incident linear polarized IGep, p laser beam (TE wave) splits into two sub-beams after passing through the beam splitter (BS). Mirrors M1 and M2 reflect one sub-beam, and the other sub-beam passes through the rotated Dove prism. Note that the Dove prism has a very interesting effect on the orientation of incident beams. If the Dove prism rotates to angle θ, the beam passing through the Dove prism rotates to angle 2θ [23

23. M. Endo, M. Kawakami, K. Nanri, S. Takeda, T. Fujioka, and M. Endo, “Numerical simulation of an optical resonator for generation of a doughnut-like laser beam”, Opt. Express12, 1959-1965 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1959. [CrossRef] [PubMed]

]. Thus, after passing through the Dove prism rotated at 45 degrees, the sub-beam rotates about the optical axis at 90 degrees. The two sub-beams are recombined into one beam after passing through a polarizing beam splitter (PBS), which eliminates minor total internal reflectioninduced TM field created by the Dove prism. For a co-axial beam passing through the Dove prism, the distance between the axis of the incident IGep, p mode to the bottom of the Dove prism, h, can be calculated by h=[tan-1δ/(1+tan-1δ)] ×(L/2), where L is the base length of the Dove prism and angle δ is the ray deviation due to refraction at the front surface of the Dove prism. Note that mirrors M1 and M2 are set such that the optical lengths along both paths are nearly the same, i.e., h×[(nD/sinδ)-(1/tanδ)] ~l. (nD : refractive index of a Dove prism). This setup helps prevent any mismatch of the wave-front curvatures of the two modes combined at the output port PBS. The phase difference between the two sub-beams at the output port PBS, Δϕ, is set as π/2 using a variable phase retarder (e.g., electrically-adjustable nano-stage/phase-adjustable liquid crystals). Note that because the total internal reflectance (TIR) in the Dove prism causes a phase delay ϕD in to one sub-beam, the variable phase retarder only needs to add a short optical path length (OPL), less than λ/4, to the other sub-beam, where the TIRintroduced phase delay on TE wave can be easily calculated from Jones Matrices calculation.

The superposed field from the two sub-beams creates a vortex array laser beam with p × p vortices embedded, are observed by CCD images. The effect of the π/2 phase delay between the two sub-beams introduces a coefficient i between the two superposed sub-beams. Thus, the resulting vortex array laser beam UVL is

UVL=IGp,pe+i×[IGp,pe]T,
(3)

where the notation []T denotes a transpose operation, i.e., the field in the square bracket rotates by 90 degrees.

4. Simulation of vortex array laser beam generation

The vortex array laser beams generated by the proposed interference mechanism consist of incident IGep, p laser beams that can be generated directly from a end-pumped SSL [22

22. W. J. Smith, Modern Optical Engineering (McGraw-Hill, 2000) pp. 105–107

]. This paper uses simulation codes based on Endo’s simulation method [24

24. U. T. Schwarz, M. A. Bandres, and J. C. Gutiérrez-Vega, “Observation of Ince-Gaussian modes in stable resonators,” Opt. Lett. 291870–1872 (2004) [CrossRef] [PubMed]

, 25

25. T. Ohtomo, K. Kamikariya, K. Otsuka, and S. Chu, “Single-frequency Ince-Gaussian mode operations of laser-diode-pumped microchip solid-state lasers,” Opt. Express 15, 10705–10717 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-17-10705. [CrossRef] [PubMed]

], which simulates a single-wavelength, single/multi-mode oscillation in unstable/stable laser cavities [24

24. U. T. Schwarz, M. A. Bandres, and J. C. Gutiérrez-Vega, “Observation of Ince-Gaussian modes in stable resonators,” Opt. Lett. 291870–1872 (2004) [CrossRef] [PubMed]

, 25

25. T. Ohtomo, K. Kamikariya, K. Otsuka, and S. Chu, “Single-frequency Ince-Gaussian mode operations of laser-diode-pumped microchip solid-state lasers,” Opt. Express 15, 10705–10717 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-17-10705. [CrossRef] [PubMed]

]. Another study recently used this code to demonstrate the forced single IGep, p mode operation using off-axis tight focus pumping [22

22. W. J. Smith, Modern Optical Engineering (McGraw-Hill, 2000) pp. 105–107

]. Figure 3 shows the simulation model of the typical half-symmetric laser resonator which this study uses to generate IGep, p laser beam. This model is similar to laser systems in real experiments [26

26. M. Mansuripur, Classical Optics and its Applications (Cambridge University Press, 2001), p. 200.

, 27

27. S.-C. Chu, T. Ohtomo, and K. Otsuka, “Generation of donutlike vortex beam with tunable orbital angular momentum from lasers with controlled Hermite-Gaussian modes,” Appl. Opt. 47, 2583–2591 (2008) [CrossRef] [PubMed]

]. The cavity in this simulation is formed by one planar mirror and a concave mirror with a curvature radius of R2=100 mm at a distance of Lc=10 mm from the planar mirror. The planar mirror is actually the high-reflection coated surface of a laser crystal. This study assumes the refractive index of the crystal to be the index of Nd:GdVO4, n=2. The laser beam wavelength is set as the lasing wavelength of laser crystal Nd:GdVO4, 1064 nm. The end-side pumping beam size d is set to be half the waist spot size of the fundamental modes, HG0, 0 mode, of the laser cavity at the position of laser crystal. The effective gain region in this simulation is the area of the pumping beam on the laser crystal. Figure 3 shows that simply shifting the lateral off axis position r of the pumping beam focus to the location of the most outside lobe of the IGep, p mode, i.e., changing the effective gain region distribution in the laser cavity, which turns the initial spontaneous random field into the IGep, p mode after it propagates back and forth in the cavity several times.

Fig. 3. Diagram of the simulation model of a half-symmetric cavity

Figure 4 shows the simulation results. Figure 4(a) shows the simulated lasing IGep, p mode distributions from the laser resonator using an off-axis pumping mechanism [22

22. W. J. Smith, Modern Optical Engineering (McGraw-Hill, 2000) pp. 105–107

]. Figures 4(b) and 4(c) show the amplitude and phase distributions of the corresponding vortex array laser beams, respectively, which are obtained by superposing two sub-beams. One sub-beam is the lasing IGep, p mode and the other sub-beam is the rotated IGep, p mode with a π/2 phase delay introduced by a variable phase retarder, e.g., an electrically-driven liquid crystal plate. Figure 4(c) also shows that the resulting vortex array laser beams contain several vortices, which are the result of phase singularities mixing with the wavefront curvature. Figure 4(d) shows the interferogram of these vortex array laser beams with a tilted plane wave, which further indicates the cross-section positions and the order of vortices (i.e., topological charge) in the resulting vortex array laser beams [28

28. Z. Chen, M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Self-Trapping of Dark Incoherent Light Beams,” Science 280, 889–892 (1998) [CrossRef] [PubMed]

]. In Fig. 4(d), the characteristic forks are all one fringe split into two, i.e., these vortices are all first order vortices. The red spots plotted in Fig. 4 indicate the positions of vortices, which are actually the dark spots appearing in Fig. 4(b).

Fig. 4. (a) Amplitude distributions of selected IGep, p modes from p=2 to 4. (b) (c) Amplitude and phase distributions of vortex array laser beams created by superposing the IGep, p mode and its rotated replica with a π/2 phase delay. (d) The Interferogram (calculation of interference fringes of the vortex array laser beam with a tilted plane wave). The widow widths of all interferograms are half of Fig. 4(a), (b) and (c).

Fig. 5. (a) Amplitude distributions of forced single IGep, p mode oscillations in a simulated endpumped solid-state laser system (from p=5 to 10). (b) Amplitude distributions of the vortex array laser beams generated by superposing the IGep, p mode and its rotated replica with a π/2 phase delay.

5. Cross-section vortex positions of the vortex array laser beam

This section presents a method of solving the cross-section vortex positions of the vortex array laser beams generated by the proposed interferometer. The results provided in this section are useful for further applications of the resultant vortex array laser beams. A well--known property of optical vortices is the phase distribution of a vortex under multiple of 2π phase shift in one circle around the phase singularity. For the amplitude distribution to be a single valued it must go to zero at the center, i.e., the dark spots of the resulting vortex array laser beams [28

28. Z. Chen, M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Self-Trapping of Dark Incoherent Light Beams,” Science 280, 889–892 (1998) [CrossRef] [PubMed]

]. It is therefore necessary to determine the cross-section zero intensity position of the vortex array laser beam distribution, UVL. Equation 3 shows that the field distribution of UVL is actually the IGep, p mode with an additional rotated IGep, p mode which is multiplied by a constant i.

One possible approach to finding the dark spot positions of UVL, i.e., the vortex positions, is to find the points of zero amplitude in both the IGep, p field distribution and the rotated IGep, p field distribution, [IGep, p]T. Figure 5(a) shows that the amplitude of the parabolic nodal lines in the IGep, p mode is zero. The amplitude of these parabolic nodal lines still remains at zero even when rotating the IGep, p mode. Thus, the positions of the dark spots in the resulting vortex array laser beam are actually the crisscrossed positions of the parabolic nodal lines of the IGep, p mode and the rotated IGep, p mode. The following discussion details the method used to find the dark spots of UVL.

To determine the cross-section vortex positions of resultant vortex array laser beams, i.e., the crisscrossed position of the parabolic nodal lines of the IGep, p mode and the rotated IGep, p mode, first find the nodal lines of an IGep, p mode. Any point on the nodal line of an IGep, p mode should have zero intensity. Equation (1) gives that

Cpp(iξ,ε)Cpp(η,ε)=0,
(4)

Any parabolic nodal line is described in elliptical-cylindrical coordinate by two specific angles η and -η and the parameter ξ under a value of [0, ∞). Since the mode distribution of the IGep, p mode should not be a zero solution, the Cpp(, ε) cannot be zero. Thus, the specific angle η of parabolic nodal lines could be solved from the equation

Cpp(η,ε)=0.
(5)

Eq. (5) is divided into two cases, depending on whether the value of order p is even or odd. According to the appendix of Ref. [16

16. M. A. Bandres, “Elegant Ince-Gaussian beams,” Opt. Lett. 29, 1724, (2004). [CrossRef] [PubMed]

], while p is even, Eq. (5) will be

C2n2n(η,ε)=r=0nArcos2rη,p=2nandnint,
{(p2+1)εA1=aA0,(p2+2)εA2=pεA0(4a)A1,(p2+r+2)εAr+2=[a4(r+1)2]Ar+1+(rp2)εAr.
(6)

While p is odd, Eq. (6) will be

C2n+12n+1(η,ε)=r=0nArcos(2r+1)η,p=2n+1andnint,
{(p+3)ε2A1=[aε2(p+1)1]A0,(p+2r+3)ε2Ar+1=[a(2r+1)2]Ar+(2rp1)ε2Ar1.
(7)

The symbol a in Eq. (6) and (7) is the largest eigenvalue of the ordinary equation [16

16. M. A. Bandres, “Elegant Ince-Gaussian beams,” Opt. Lett. 29, 1724, (2004). [CrossRef] [PubMed]

]

d2Ndη2+εsin2ηdNdη+(apεcos2η)N=0,
(8)

which was generated from the derivation of Ince-Gaussian modes, the solution of the paraxial wave equation. Solving Eq. (6) or Eq. (7) (depending on whether the value of order p is even or odd) produces 2p specific angles η of parabolic nodal lines, denoted by {±η1, ±η2, …, and ±ηp}. The equations of IGep, p mode parabolic nodal lines can be described by equations

x2f(z)2cos2ηy2f(z)2sin2η=1,(η=±η1,±η2,or±ηp).
(9)

The equations of the parabolic nodal lines of the rotated IGep, p mode is simply Eq. (9) that exchanges the position of variables x and y, i.e.,

y2f(z)2cos2ηx2f(z)2sin2η=1,(η=±η1,±η2,or±ηp).
(10)

The cross-section vortex positions of the resulting vortex array laser beams are simply the (x, y) solution of the combined Eq. (9) and (10).

Figure 6 plots the parabolic nodal lines overlapping the amplitude and phase distribution of IGep, p mode-superposed vortex array laser beams using derived Eq. (9) and (10). To include the two cases discussed above, Fig. 6 shows vortex array laser beams converted from two kinds of IGep, p modes, IGe4, 4 and IGe5, 5 mode. Since the IGep, p mode has p nodal lines, the exact dark points appearing in the cross-section of the resulting vortex array beams should be p×p, which can be indentified from the phase distribution of vortex array laser beams (Fig. 6(b)). However, because the beam intensity near the crisscrossed position of the marginal nodal lines is much weaker than the center of the vortex array laser beam, these vortices are hard to be observed in the beam intensity distributions and are therefore difficult to manipulate further. These marginal vortices are indicated by red spots in Fig. 6.

Fig. 6. The (a) amplitude and (b) phase distribution of the resulting vortex array laser beam with the nodal lines of the IGep, p mode and rotated IGep, p mode plotted in blue. Red spots indicate marginal vortices in these field distributions.

6. Discussion of vortex array laser beam properties

6.1 Effect of ellipticity parameter on vortex array pattern

This section discusses some properties of the resulting vortex array laser beam. Figure 7(a) illustrates simulated excited single IGep, p mode oscillation from an end-pumped SSL with different azimuthal focus pumping beam shapes. The red circles in Fig. 7(a) plot the effective gain region used in simulation of the single IGep, p laser beam generation from end-pumped SSL. The ellipticity parameter ε of the selected IGep, p laser beams is estimated by comparing the analytical IGep, p mode patterns and are shown in Fig. 7(a). Figure 7(b) shows the corresponding vortex array laser beam generated by superposing the selected IGep, p mode and its rotated replica [IGep, p]T with a π/2 phase delay. As Ref. [22

22. W. J. Smith, Modern Optical Engineering (McGraw-Hill, 2000) pp. 105–107

] demonstrates, the elliptical parameter of incident IGep, p laser beams from an end-pumped SSL can be easily changed by simply controlling the azimuthal pumping beam shapes on the laser crystal in the laser cavity. In addition, the discussion in Section 5 above shows that the cross-section vortex positions of the resulting vortex array laser beams are the crisscross positions of the parabolic nodal lines of the IGep, p mode and the rotated IGep, p mode. Since the IGM patterns are different in terms of ellipticity parameter ε, changing the ellipticity parameter ε of the incident IGep, p laser beam also changes the curvature of its parabolic nodal line and the resulting vortex positions. It is a property of IGMs that while increasing the ellipticity parameter ε, the IGM pattern will approach a HGM pattern and the nodal lines of IGMs approach straight lines. As Fig. 7 shows, increasing the ellipticity parameter ε of the incident IGep, p laser beam decreases the curvature of the IGep, p mode parabolic nodal lines, and the resulting vortices toward aligning in a square array.

Fig. 7. (a) Simulated excited single IGep, p mode oscillations from an end-pumped solid-state laser system with different azimuthal pumping beam shapes (plotted by red circles). (b) The vortex array laser beams generated by superposing the selected IGep, p mode and its rotated replica, [IGep, p]T with a π/2 phase delay.

Figure 8(a) provides example vortex arrays calculated in the limit of infinite ellipticity parameter ε for an incident IGep, p laser beam, i.e. HGp, 0 mode. HGp, 0 mode is proved to be achieved experimentally by overlapping the focus pumping beam on the outer most lobe of HGp, 0 mode distribution at the laser crystal in end-pumped SSL system [29

29. K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, and H. Masuhara, “Optical trapping of a metal particle and a water droplet by a scanning laser beam,” Appl. Phys. Lett. 60, 807-809 (1992). [CrossRef]

]. In this case, the nodal lines of the incident IGep, p laser beam become straight lines, and the spatial arrangement of the resulting vortices form an exactly square array. However, a comparison of Fig. 8(a) and 8(b) shows that because the vertical length of the center lobes of the HGp, 0 mode are much shorter than the IGep, p mode, only the central lobes of the HGp, 0 mode can create visible vortices after passing through the proposed interferometer. The IGep, p mode has an advantage in that its beam energy is distributed more uniformly in longer center lobes with a larger mode area. Thus, when passing through the proposed interferometer, the IGep, p laser beam generates more useful visible dark spots than the HGp, 0 laser beam. Figure 8(c) shows a tilted IGep, p laser beam whose principle axis forms an angle of 45 degrees with the (x, y) axes along with the corresponding resultant vortex array formed by the proposed interferometer. A tilted IGep, p laser beam can be easily generated in an end-pumped SSL by simply shifting the pumping beam focus at the outer-most lobe of the tilted IGep, p mode [22

22. W. J. Smith, Modern Optical Engineering (McGraw-Hill, 2000) pp. 105–107

]. Figure 8(c) shows that the vortices in a vortex array laser beam formed from a tilted incident IGep, p laser beam are also tilted by 45 degrees against the (x, y) axes. This result implies the possibility of controlling the rotation of vortex arrays for the future manipulation of particles in two dimensions.

Fig. 8. Three kinds of incident laser beams and its corresponding resultant vortex laser beams from Dove prism embedded Mach-Zehnder interferometer. Three kinds of incident laser beams are (a) HG10, 0 mode, (b) IGe10, 10 mode and (c) tilted IGe10, 10 mode.

6.2. Effect of modal difference on combined pattern

This section discusses the effect of the relative phase difference Δϕ and the power difference between the two sub-beams, i.e., the IGep, p mode and its rotated replica [IGep, p]T, on laser beams generated by the proposed interferometer. Figure 9 shows the intensity and phase distributions of laser beams generated by superposing two sub-beams, the IGe4, 4 modes and rotated IGe4, 4 modes, with a phase difference Δϕ ranging from π/8 to 2π in 16 steps. This figure also shows their interferogram with a tilted plane wave, where the characteristic forks in interferograms indicate both the position and the topological charge of each vortex [28

28. Z. Chen, M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Self-Trapping of Dark Incoherent Light Beams,” Science 280, 889–892 (1998) [CrossRef] [PubMed]

]. In this figure, red spots indicate the vortices embedded in the resulting laser beam. These patterns are repeated in every period of 2π phase retardation. As Fig. 9 shows, laser beams produced by the proposed interferometer vary along with the relative phase difference Δϕ between the two sub-beams.

It should be noted that all superposed fields with phase difference Δϕ except an integral multiple of π are found to possess smooth phase rotations of 2π around dark spots, which are actually situated at the crisscrossed positions of the nodal lines of the IGep, p modes and their rotated replicas. Such “flexible” IGep, p mode based formations of vortex array beams including a large number of vortices in a wide region of the phase difference Δϕ, as compared with other methods, is considered to arise from the fact that the dark spots in the resulting vortex array laser beam are formed at the crisscrossed positions of the zero-amplitude “parabolic” nodal lines of the IGep, p mode and those of the rotated IGep, p mode whose “parabolic” nodal lines still remains at zero even when rotating the IGep, p mode as shown in section 5.

In general laser trapping, particles whose refractive index is higher than surrounding medium are confined within the focal region of laser beam. At this situation, general TEM0, 0 laser beam and the dark beams (such as donut beams) can both apply to trap such particles [30

30. P. H. Jones, E. Stride, and N. Saffari, “Trapping and manipulation of microscopic bubbles with a scanning optical tweezer,” Appl. Phys. Lett. 89, 081113 (2006). [CrossRef]

]. Dark beams can further apply to trap some special particles, such as low refractive-index particles, metal fragments, or strongly absorbing particles, because the net force repels these types of particles from the region of stronger light intensity [31, 32]. These kinds of particles thus trapped around the weakest point of the beam, where all the resultant force from surrounding field is balanced there. Therefore, for applying proposed vortex array beams in the actual dark-beam trap of particles, the dark spots (i.e., phase singularities) must be surrounded by “stronger fields” in both cases. How wreaker of the surrounding stronger fields is acceptable in experiments depends on the property of particles to trap. While, in the present case, well-defined surrounding fields are visible only when the phase difference Δϕ is set around the ideal value of ±π/8 as red boxes indicated in Fig. 9. Therefore, the allowed error in the phase difference for dark-beam traps is estimated to be roughly ±π/8 for generating vortex array beams with enough surrounding field intensities. IGep, p modes of other order p have also been checked.

Fig. 9. Intensity distribution, phase distributions and the Interferogram (calculation of interference fringes of the vortex array laser beam with a tilted plane wave) of laser beams superposed by two sub-beams, i.e., the IGe4, 4 modes and its rotated replica [IGe4, 4]T, with a phase difference Δϕ of values ranging from π/8 to 2π in 16 steps.

Refer to scheme diagram Fig. 2, the figure shows that the straight path contains more interfaces than the mirror path, which arise in the power difference between the two sub-beams, i.e., the IGep, p mode and its rotated replica [IGep, p]T. Figure 10 shows the intensity and phase distributions of laser beams generated by superposing two sub-beams, the IGe4, 4 mode and rotated IGe4, 4 mode, with different power ratios τ=P2/P1 from 0.5 to 1 in 6 steps. This figure also shows the interferogram with a tilted plane wave, where P1 and P2 denote the power of the IGe4, 4 mode and its rotated replica [IGe4, 4]T, respectively. Figure 10 shows that the power disparity between the two sub-beams does not influence the resulting vortex position in the beam cross-section, but only causes a slight dissymmetric intensity distribution, as depicted by blue boxes. When the Dove prism in the interferometer is uncoated, the transmission efficiency form straight path is about 86%. An anti-reflection-coated Dove prism further increases transmission efficiency from the straight path to 95%. Both power efficiencies of the passing TE wave can easily be calculated from Jones Matrices calculation. Simply adding an attenuator to the mirror path decreases this unbalance-intensity effect. Considering the application of the present vortex array laser beam to real situations, such as laser tweezers and atom traps, the required laser power varies case by case. For example, the laser power for trapping bio-samples is about several milli-watts. Since the selected IGep, p laser beam from an end-pumped SSL is about several hundred milli-watts [27

27. S.-C. Chu, T. Ohtomo, and K. Otsuka, “Generation of donutlike vortex beam with tunable orbital angular momentum from lasers with controlled Hermite-Gaussian modes,” Appl. Opt. 47, 2583–2591 (2008) [CrossRef] [PubMed]

] and the efficiency in converting IGep, p modes to vortex array laser beams is greater than 86%, the resulting vortex array laser beams are powerful enough for the bio-sample trapping applications.

Fig. 10. Intensity distributions, phase distributions and the interferograms (calculations of interference fringes of the vortex array laser beam with a tilted plane wave) of laser beams superposed by two sub-beams, i.e., the IGe4, 4 mode and its rotated replica [IGe4, 4]T, with a power ratio τ for two sub-beams ranging from 0.5 to 1 in 6 steps.

6.3. Propagation effect

To apply vortex array laser beams to optical tweezers and other applications, such as atom guiding and trapping, the propagation effect and the focusing property of vortex beams must be identified. Figure 11 shows the intensity distributions, phase distributions, and interferograms of vortex array laser beams at different situations. In this figures, a well-collimated IGep, p laser beam of 1 mm width value passes through the interferometer. The window is 12 times as wide as the TEM00 mode spot size. A CCD first observes and records the resulting vortex array laser beam in Fig. 2, which is 1 m away from the waist of the passing IGep, p laser beam. Figure 11(a) shows the patterns of the vortex array laser beam after propagating another 4 m behind CCD. Figure 11(b) is the focused pattern of the vortex array beam observed at the back focal plane of a lens with a focal length of 1.5 m. Other beam waist spot sizes and focal lengths can result in similar patterns.

Fig. 11. Intensity distribution, phase distributions and the interferogram (calculation of interference fringes of the vortex array laser beam with a tilted plane wave). (a) After another 4 m of propagation behind CCD. (b) After passing through a focal lens and observing at the back focal plane of a lens with a focal length of 1.5 m.

Figure 11(a) shows that while propagating, the intensity of the resulting vortex array laser beam still maintains its array pattern. In short, the wavefront of the vortex array beam mixes with the spherical wavefront as the distance from the beam waist increases. However, the special vortex phase distribution of each optical vortex is still embedded in the spherical wavefront, as the red spots indicate. Figure 11(b) shows that the resulting vortex array laser beam maintains its vortex array pattern while being focused. Actually, the far-field pattern of the resulting vortex array laser beam that was not show here is actually the same as the pattern shown in Fig. 11(b). This reflects the well-known property of Gaussian beams: the Fourier transform of a Gaussian near-field pattern, i.e., the far-field pattern, exhibits a Gaussian distribution and any Gaussian beam can repeat itself while focusing. In the present case, the focused pattern is simply a coherent superposition of two focused Gaussian beams, the IGep, p mode and its rotated replica [IGep, p]T with a constant π/2 phase shift, thus resulting in the same vortex array pattern.

The effective OPL difference between two sub-beams can actually be much greater than λ/4. When OPL difference between two sub-beams was ΔL=δL+λ/4=100λ+λ/4, simulated resulting vortex array laser beam patterns cannot be distinguished from patterns shown in Fig. 11. This is because the profile of one sub-beam changes very little when propagates 100λ+λ/4 optical path length. This suggests that when an effective constant of π/2 phase shift is ensured, the superposing of the IGep, p mode and its rotated replica [IGep, p]T will produce the same vortex array pattern even if δL≫λ. An important question then arises: How great can the OPL difference between two sub-beams be while still allowing for the formation of stable vortex array beams? Refer to Eq. (1), in which all items are slowly variation functions of propagation distance z, except for the phase function, ϕ(z)=expi[kz+{kr2/2R(z)}-(p+1)ψz(z)]. Name each item by three phase functions, ϕ1=kz, ϕ2=kr2/2R(z), and ϕ3=-ψz(z), and the phase function becomes ϕ(z)=expi[ϕ1(z) 2(z)+(p+1)ϕ3(z)]. The resulting phase change while propagating an IGep, p mode or its rotated replica [IGep, p]T can be estimated by (dϕi/dz)Δz, where (dϕi/dz) are

{dϕ1(z)dz=kdϕ2(z)dz=kr2(1zR2z2)2(z+zR2z)2dϕ3(z)dz=1(1+zzR2)zR.
(11)

Fig. 12. (a) Phase errors resulting from inserting the λ/4 optical path length difference between two sub-beams in the interferometer. (b) Phase errors from function (a) ϕ2(z), (b) ϕ3(z), and (c) the sum of two phase function, ϕ2(z) and ϕ3(z). The passing IGep, p laser beam waist sizes are indicated above each sub-figures.

Figure 12 shows that all phase errors arising at all position z are much less than 1 degree. The primary phase error arises from ϕ2(z). Note that though (dϕ2/dz) is a function of radial distance r, when position z approaches to infinity, the limit value of (dϕ2/dz) approaches to a small constant value -1/zR, where the radial distance r in (dϕ2/dz) is estimated by the beam spot size w(z). (For the IGep, p mode, rmax is about p×w(z) and the phase error maximum is multiplied value in Fig. 12 by p2.) A IGep, p laser beam passing through the interferometer produces phase changes in the entire beam cross-section are all very small, and will not significantly change the resulting vortex array pattern. For an IGep, p laser beam with a waist spot size of 10 µm, for instance, the observed pattern at postion z=10 m away from the laser beam waist still preserves the desiredvortex array pattern. As the IGep, p laser beam becomes more collimated, the resulting phase changes from the unwanted ϕ2(z) and ϕ3(z) phase terms become less significant. In addition, from Fig. 12, we can also estimate how great the OPL difference between the two sub-beams can be while still producing the same vortex array pattern. For an IGep, p laser beam with a waist spot size of 100µm, the OPL difference between two sub-beams is 100λ+λ/4, and the maximum phase change from ϕ2(z) in the beam transverse cross-section is only about 100/(1/4)×p2×0.0005=3.2 degrees. This suggests that a IGep, p laser beam with a wavelength of 1.064 µm passing through the interferometer produces a pattern which retains a vortex array pattern after propagation even if the OPL difference between the two sub-beams is on the order of 1 mm.

7. Conclusion

In summary, this paper proposes a Dove prism-embedded Mach-Zehnder interferometer capable of converting an incident collimated even Ince-Gaussian mode, the IGep, p mode, into a vortex array laser beam consisting of p×p optical vortices. The incident IGep, p laser beams can easily be generated from an end-pumped solid-state laser. The embedded vortex number of the resulting vortex array laser beam can be increased by simply increasing the order p of the incident IGep, p laser beam; that is, simply increasing the off-axis position of the pumping beam in an end-pumped solid-state laser system. The resulting robust vortex array laser beam, which maintains vortex array profile during both propagation and focusing, is applicable to optical tweezers and atom traps in the form of two-dimensional arrays, and can be used to study the transfer of angular momentum to micro particles or atoms (Bose-Einstein condensate).

Acknowledgment

This work was supported in part by a grant from the National Science Council of Taiwan, R.O.C., under contract no. NSC 96-2112-M-006-019-MY3

References and links

1.

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

2.

D. W. Zhang, X. -. Yuan, E. Santamato, A. Sasso, B. Piccirillo, and A. Vella, “Optical angular momentum transfer to transparent isotropic particles using laser beam carrying zero average angular momentum,” Opt. Express10, 871–878 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-17-871.

3.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912–914 (2001). [CrossRef] [PubMed]

4.

Y. Song, D. Milam, and W. T. Hill, “Long, narrow all-light atom guide,” Opt. Lett. 24, 1805–1807 (1999). [CrossRef]

5.

X. Xu, K. Kim, W. Jhe, and N. Kwon, “Efficient optical guiding of trapped cold atoms by a hollow laser beam,” Phy. Rev. A 63, 3401 (2001). [CrossRef]

6.

J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun. 198, 21–27 (2001). [CrossRef]

7.

A. Dreischuh, S. Chervenkov, D. Neshev, G. G. Paulus, and H. Walther, “Generation of lattice structures of optical vortices,” J. Opt. Soc. Am. B 19, 550–556 (2002) [CrossRef]

8.

P. Kurzynowski and M. Borwińska, “Generation of vortex-type markers in a one-wave setup,” Appl. Opt. 46, 676–679 (2007) [CrossRef] [PubMed]

9.

K. J. Moh, X. -. Yuan, W. C. Cheong, L. S. Zhang, J. Lin, B. P. S. Ahluwalia, and H. Wang, “High-power efficient multiple optical vortices in a single beam generated by a kinoform-type spiral phase plate,” Appl. Opt. 45, 1153–1161 (2006) [CrossRef] [PubMed]

10.

G. A. Turnball, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre-Gaussian modes at millimetre-wave frequencies by use of a spiral phase plate,” Opt. Commun. 127, 183–188 (1996). [CrossRef]

11.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993). [CrossRef]

12.

M. Brambilla, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” Phys. Rev. A 43, 5090–5113 (1991). [CrossRef] [PubMed]

13.

J. Scheuer and M. Orenstein, “Optical vortices crystals: Spontaneous generation in nonlinear semiconductor microcavities,” Science 285, 230–233 (1999). [CrossRef] [PubMed]

14.

M. A. Bandres and Julio C. Gutiérrez-Vega, “Ince-Gaussian beams,” Opt. Lett. 29144–146 (2004) [CrossRef] [PubMed]

15.

M. A. Bandres and J. C. Gutiérrez-Vega, “Ince-Gaussian modes of the paraxial wave equation and stable resonators,” J. Opt. Soc. Am. A 21, 873–880 (2004). [CrossRef]

16.

M. A. Bandres, “Elegant Ince-Gaussian beams,” Opt. Lett. 29, 1724, (2004). [CrossRef] [PubMed]

17.

J. C. Gutiérrez-Vega and M. A. Bandres, “Ince-Gaussian beam in quadratic index medium,” J. Opt. Soc. Am. A , 22, 306–309, (2005). [CrossRef]

18.

M. A. Bandres and J. C. Gutiérrez-Vega, “Ince-Gaussian series representation of the two-dimensional fractional Fourier transform,” Opt. Lett. 30, 540–542 (2005). [CrossRef] [PubMed]

19.

T. Xu and S. Wang, “Propagation of Ince-Gaussian beams in a thermal lens medium,” Opt. Commun. 265, 1–5 (2006). [CrossRef]

20.

E. L. Wooten, R. L. Stone, E. W. Miles, and E. M. Bradley, “Rapidly tunable narrow band wavelength filter using LiNbO3 unbalanced Mach-Zehnder interferometers,” J. Lightwave Technol. 14, 2530–2536 (1996). [CrossRef]

21.

S.-C. Chu, “Generation of multiple vortex beams with specified vortex number from lasers with controlled Ince-Gaussian modes,” Jap. J. Appl. Phys. 475297–5303 (2008) [CrossRef]

22.

W. J. Smith, Modern Optical Engineering (McGraw-Hill, 2000) pp. 105–107

23.

M. Endo, M. Kawakami, K. Nanri, S. Takeda, T. Fujioka, and M. Endo, “Numerical simulation of an optical resonator for generation of a doughnut-like laser beam”, Opt. Express12, 1959-1965 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1959. [CrossRef] [PubMed]

24.

U. T. Schwarz, M. A. Bandres, and J. C. Gutiérrez-Vega, “Observation of Ince-Gaussian modes in stable resonators,” Opt. Lett. 291870–1872 (2004) [CrossRef] [PubMed]

25.

T. Ohtomo, K. Kamikariya, K. Otsuka, and S. Chu, “Single-frequency Ince-Gaussian mode operations of laser-diode-pumped microchip solid-state lasers,” Opt. Express 15, 10705–10717 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-17-10705. [CrossRef] [PubMed]

26.

M. Mansuripur, Classical Optics and its Applications (Cambridge University Press, 2001), p. 200.

27.

S.-C. Chu, T. Ohtomo, and K. Otsuka, “Generation of donutlike vortex beam with tunable orbital angular momentum from lasers with controlled Hermite-Gaussian modes,” Appl. Opt. 47, 2583–2591 (2008) [CrossRef] [PubMed]

28.

Z. Chen, M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Self-Trapping of Dark Incoherent Light Beams,” Science 280, 889–892 (1998) [CrossRef] [PubMed]

29.

K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, and H. Masuhara, “Optical trapping of a metal particle and a water droplet by a scanning laser beam,” Appl. Phys. Lett. 60, 807-809 (1992). [CrossRef]

30.

P. H. Jones, E. Stride, and N. Saffari, “Trapping and manipulation of microscopic bubbles with a scanning optical tweezer,” Appl. Phys. Lett. 89, 081113 (2006). [CrossRef]

OCIS Codes
(120.4820) Instrumentation, measurement, and metrology : Optical systems
(140.3480) Lasers and laser optics : Lasers, diode-pumped
(140.7010) Lasers and laser optics : Laser trapping

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: September 15, 2008
Revised Manuscript: November 11, 2008
Manuscript Accepted: November 12, 2008
Published: November 19, 2008

Virtual Issues
Vol. 4, Iss. 1 Virtual Journal for Biomedical Optics

Citation
Shu-Chun Chu, Chao-Shun Yang, and Kenju Otsuka, "Vortex array laser beam generation from a Dove prism-embedded unbalanced Mach-Zehnder interferometer," Opt. Express 16, 19934-19949 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-24-19934


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References

  1. K. T. Gahagan and G. A. Swartzlander, Jr, "Optical vortex trapping of particles," Opt. Lett. 21, 827-829 (1996). [CrossRef] [PubMed]
  2. D. W. Zhang and X.  -. Yuan, "Optical doughnut for optical tweezers," Opt. Lett. 28, 740-742 (2003)
  3. E. Santamato, A. Sasso, B. Piccirillo and A. Vella, "Optical angular momentum transfer to transparent isotropic particles using laser beam carrying zero average angular momentum," Opt. Express 10, 871-878 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-17-871. [CrossRef] [PubMed]
  4. L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, "Controlled rotation of optically trapped microscopic particles," Science 292, 912-914 (2001). [CrossRef]
  5. Y. Song, D. Milam, and W. T. Hill, "Long, narrow all-light atom guide," Opt. Lett. 24, 1805-1807 (1999). [CrossRef]
  6. X. Xu, K. Kim, W. Jhe, and N. Kwon, "Efficient optical guiding of trapped cold atoms by a hollow laser beam," Phy. Rev. A 63, 3401 (2001). [CrossRef]
  7. J. Masajada and B. Dubik, "Optical vortex generation by three plane wave interference," Opt. Commun. 198, 21-27 (2001). [CrossRef]
  8. A. Dreischuh, S. Chervenkov, D. Neshev, G. G. Paulus, and H. Walther, "Generation of lattice structures of optical vortices," J. Opt. Soc. Am. B 19, 550-556 (2002) [CrossRef] [PubMed]
  9. P. Kurzynowski and M. Borwińska, "Generation of vortex-type markers in a one-wave setup," Appl. Opt. 46, 676-679 (2007) [CrossRef] [PubMed]
  10. K. J. Moh, X. -. Yuan, W. C. Cheong, L. S. Zhang, J. Lin, B. P. S. Ahluwalia, and H. Wang, "High-power efficient multiple optical vortices in a single beam generated by a kinoform-type spiral phase plate," Appl. Opt. 45, 1153-1161 (2006) [CrossRef]
  11. G. A. Turnball, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, "The generation of free-space Laguerre-Gaussian modes at millimetre-wave frequencies by use of a spiral phase plate," Opt. Commun. 127, 183-188 (1996). [CrossRef]
  12. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123-132 (1993). [CrossRef] [PubMed]
  13. M. Brambilla, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, "Transverse laser patterns. I. Phase singularity crystals," Phys. Rev. A 43, 5090-5113 (1991). [CrossRef] [PubMed]
  14. J. Scheuer, M. Orenstein, "Optical vortices crystals: Spontaneous generation in nonlinear semiconductor microcavities," Science 285, 230-233 (1999). [CrossRef] [PubMed]
  15. M. A. Bandres and JulioC. Gutiérrez-Vega, "Ince-Gaussian beams," Opt. Lett. 29, 144-146 (2004) [CrossRef]
  16. M. A. Bandres and J. C. Gutiérrez-Vega, "Ince-Gaussian modes of the paraxial wave equation and stable resonators," J. Opt. Soc. Am. A 21, 873-880 (2004). [CrossRef] [PubMed]
  17. M. A. Bandres, "Elegant Ince-Gaussian beams," Opt. Lett. 29, 1724, (2004). [CrossRef]
  18. J. C. Gutiérrez-Vega and M. A. Bandres, "Ince-Gaussian beam in quadratic index medium," J. Opt. Soc. Am. A 22, 306-309, (2005). [CrossRef] [PubMed]
  19. M. A. Bandres and J. C. Gutiérrez-Vega, "Ince-Gaussian series representation of the two-dimensional fractional Fourier transform," Opt. Lett. 30, 540-542 (2005). [CrossRef]
  20. T. Xu and S. Wang, "Propagation of Ince-Gaussian beams in a thermal lens medium," Opt. Commun. 265, 1-5 (2006). [CrossRef]
  21. E. L. Wooten, R. L. Stone, E. W. Miles, and E. M. Bradley, "Rapidly tunable narrow band wavelength filter using LiNbO3 unbalanced Mach-Zehnder interferometers," J. Lightwave Technol. 14, 2530-2536 (1996). [CrossRef]
  22. S.-C. Chu, "Generation of multiple vortex beams with specified vortex number from lasers with controlled Ince-Gaussian modes," Jpn. J. Appl. Phys. 47, 5297-5303 (2008)
  23. W. J. Smith, Modern Optical Engineering (McGraw-Hill, 2000) pp. 105-107 [CrossRef] [PubMed]
  24. M. Endo, M. Kawakami, K. Nanri, S. Takeda and T. Fujioka, "Two-dimensional Simulation of an Unstable Resonator with a Stable Core," Appl. Opt. 38, 3298-3307 (1999). [CrossRef] [PubMed]
  25. M. Endo, "Numerical simulation of an optical resonator for generation of a doughnut-like laser beam," Opt. Express 12, 1959-1965 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1959. [CrossRef] [PubMed]
  26. U. T. Schwarz, M. A. Bandres and J. C. Gutiérrez-Vega, "Observation of Ince-Gaussian modes in stable resonators," Opt. Lett. 29, 1870-1872 (2004)
  27. T. Ohtomo, K. Kamikariya, K. Otsuka, and S. Chu, "Single-frequency Ince-Gaussian mode operations of laser-diode-pumped microchip solid-state lasers," Opt. Express 15, 10705-10717 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-17-10705. [CrossRef] [PubMed]
  28. M. Mansuripur, Classical Optics and its Applications (Cambridge University Press, 2001), p. 200. [CrossRef] [PubMed]
  29. S.-C. Chu, T. Ohtomo, and K. Otsuka, "Generation of donutlike vortex beam with tunable orbital angular momentum from lasers with controlled Hermite-Gaussian modes," Appl. Opt. 47, 2583-2591 (2008) [CrossRef]
  30. Z. Chen, M. Mitchell, M. Segev, T. H. Coskun, D. N. Christodoulides, "Self-Trapping of Dark Incoherent Light Beams," Science 280, 889-892 (1998) [CrossRef]
  31. K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, and H. Masuhara, "Optical trapping of a metal particle and a water droplet by a scanning laser beam," Appl. Phys. Lett. 60, 807-809 (1992).
  32. P. H. Jones, E. Stride and N. Saffari, "Trapping and manipulation of microscopic bubbles with a scanning optical tweezer," Appl. Phys. Lett. 89, 081113 (2006).

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