## 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, 2] and in the study of the transfer of angular momentum to micro particles or atoms [3–6]. Researchers use several different approaches to study the generation of vortex array laser beams including three plane waves interference [7], propagation in a saturable nonlinear medium [8], or interfering beams passing through birefringent elements [9]. 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 TEM_{00} laser beam through well-designed 2-dimensional phase plates [10] or co-propagate a vortex beam with a phase singularity possessing multiple (*p*) charges, resulting in separated *p* single-charge vortices [10]. These charges can be generated using phase plates [11] or astigmatic mode converters for higher-order Hermite-Gaussian modes [12]. 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, 14].

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, 16], and several authors have recently explored the characteristics of this novel family of IGMs [17–20]. 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 *IG ^{e}p, p* modes, into vortex array laser beams consisting of a

*p*×

*p*number of well-aligned vortices. This study simulates vortex array laser beam formation based on

*IG*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).

^{e}p, pThis paper is organized as follows. Section 2 briefly reviews the basic formalism of the Ince-Gaussian modes used in this paper, the *IG ^{e}p, 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]

$$\times \mathrm{exp}\phantom{\rule{.5em}{0ex}}i\left[\mathrm{kz}+\{k{r}^{2}\u20442R\left(z\right)\}-\left(p+1\right){\psi}_{z}\left(z\right)\right],$$

$$\times \mathrm{exp}\phantom{\rule{.3em}{0ex}}i\left[\mathrm{kz}+\{k{r}^{2}\u20442R\left(z\right)\}-\left(p+1\right){\psi}_{z}\left(z\right)\right],$$

where the elliptic coordinate is defined in a transverse z plane as x=*f*(z) cos ξ cos *η*, y=*f*(z) sin ξ sin η, and ξ∈[0, ∞], *η* ∈[0, 2π]. *f(z)* is the semifocal separation of IGMs defined as the Gaussian beam width, i.e., *f*(*z*)=*f _{0}w*(z)

*/w*, where

_{0}*f*and

_{0}*w*are the semifocal separation and beam width at the z=0 plane, respectively.

_{0}*w*(

*z*)=

*w*(1+

_{0}*z*/

^{2}*z*)

_{R}^{2}^{1/2}describes the beam width,

*z*=

_{R}*kw*is the Rayleigh length, and

_{0}^{2}/2*k*is the wave number of the beam. The terms

*C*and

*S*are normalization constants, and subscripts

*e*and

*o*refer to even and odd IGMs, respectively.

*C*(. , ε) and

_{p}^{m}*S*(. , ε) are the even and odd Ince polynomials [16] of order

_{p}^{m}*p*, degree

*m*, and ellipticity parameter ε, respectively. In Eq. (1) and (2),

*r*is the radial distance from the central axis of the cavity,

*R*(

*z*)=

*z*+

*z*is the radius of curvature of the phase front, and

_{R}^{2}/z*ψ*(

_{z}*z*)=arctan(

*z/z*). The parameters of ellipticity

_{R}*ε*, waist

*w*and the semifocal seperation

_{0}*f*are not independent, but related by ε=2

_{0}*f*. IGMs patterns can be recognized by 2 rules: degree

^{2}_{0}/w^{2}_{0}*m*corresponds to the number of hyperbolic nodal lines and (

*p-m*)/2 is the number of elliptic nodal lines. Figure 1 plots some analytical patterns of the

*IG*modes obtained from Eq. (1) and (2). Figure 1(a) shows the typical structure of the

_{p, m}*IG*modes used in this paper, i.e.,

^{e}_{p, p}*IG*modes have only parabolic nodal lines, and no elliptical nodal lines.

_{p, p}## 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], 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 *IG ^{e}_{p, 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

*IG*mode and its rotated replica, namely the [

^{e}_{p, p}*IG*]

^{e}_{p, 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

*IG*laser beam of variety order

^{e}_{p, p}*p*can be easily generated from an end-pumped SSL with an off-axis pumping mechanism [22].

Figure 2 shows that the incident linear polarized *IG ^{e}_{p, p}* laser beam (TE wave) splits into two sub-beams after passing through the beam splitter (BS). Mirrors M

_{1}and M

_{2}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 2

*θ*[23]. 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

*IG*mode to the bottom of the Dove prism,

^{e}_{p, p}*h*, can be calculated by

*h*=[tan

^{-1}δ/(1+tan

^{-1}δ)] ×(

*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 M

_{1}and M

_{2}are set such that the optical lengths along both paths are nearly the same, i.e.,

*h*×[(

*n*/sin

_{D}*δ*)-(1/tanδ)] ~

^{l}. (

*n*: 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, Δ

_{D}*ϕ*, 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

*ϕ*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.

_{D}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 *U _{VL}* is

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 *IG ^{e}_{p, p}* laser beams that can be generated directly from a end-pumped SSL [22]. This paper uses simulation codes based on Endo’s simulation method [24, 25], which simulates a single-wavelength, single/multi-mode oscillation in unstable/stable laser cavities [24, 25]. Another study recently used this code to demonstrate the forced single

*IG*mode operation using off-axis tight focus pumping [22]. Figure 3 shows the simulation model of the typical half-symmetric laser resonator which this study uses to generate

^{e}_{p, p}*IG*laser beam. This model is similar to laser systems in real experiments [26, 27]. The cavity in this simulation is formed by one planar mirror and a concave mirror with a curvature radius of

^{e}_{p, p}*R*=100

_{2}*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:GdVO

_{4},

*n*=2. The laser beam wavelength is set as the lasing wavelength of laser crystal Nd:GdVO

_{4}, 1064 nm. The end-side pumping beam size

*d*is set to be half the waist spot size of the fundamental modes,

*HG*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

_{0, 0}*r*of the pumping beam focus to the location of the most outside lobe of the

*IG*mode, i.e., changing the effective gain region distribution in the laser cavity, which turns the initial spontaneous random field into the

^{e}_{p, p}*IG*mode after it propagates back and forth in the cavity several times.

^{e}_{p, p}Figure 4 shows the simulation results. Figure 4(a) shows the simulated lasing *IG ^{e}_{p, p}* mode distributions from the laser resonator using an off-axis pumping mechanism [22]. 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

*IG*mode and the other sub-beam is the rotated

^{e}_{p, p}*IG*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]. 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).

^{e}_{p, p}The vortex number of the resulting vortex array laser beams increase when using an incident *IG ^{e}_{p, p}* mode with a higher-order

*p*. It is easy to force a single

*IG*mode oscillation in end-pumped SSL by simply controlling the off-axis distance between the pumping beam focus

^{e}_{p, p}*r*and the optical axis of the laser cavity. Figures 5(a) and 5(b) show excited

*IG*laser beams with increased order

^{e}_{p, p}*p*from the off-axis pumping mechanism and the corresponding vortex array laser beams. These laser beams contain many dark spots, which are aligned in an array trend. These dark spots actually indicate the positions of many first order vortices. The vortex array laser beam with embedded vortex array can be implemented in a new class of laser tweezers or the study of the transfer of angular momentum to micro particles or atoms (Bose-Einstein condensate) without having to rely on the previous methods of using sophisticated phase plates [10] or the collinear propagation of plane wave and vortex beams with a phase singularity possessing multiple charges [12].

## 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]. It is therefore necessary to determine the cross-section zero intensity position of the vortex array laser beam distribution, *U _{VL}*. Equation 3 shows that the field distribution of

*U*is actually the

_{VL}*IG*mode with an additional rotated

^{e}_{p, p}*IG*mode which is multiplied by a constant

^{e}_{p, p}*i*.

One possible approach to finding the dark spot positions of *U _{VL}*, i.e., the vortex positions, is to find the points of zero amplitude in both the

*IG*field distribution and the rotated

^{e}_{p, p}*IG*field distribution, [

^{e}_{p, p}*IG*]

^{e}_{p, p}^{T}. Figure 5(a) shows that the amplitude of the parabolic nodal lines in the

*IG*mode is zero. The amplitude of these parabolic nodal lines still remains at zero even when rotating the

^{e}_{p, p}*IG*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

^{e}_{p, p}*IG*mode and the rotated

^{e}_{p, p}*IG*mode. The following discussion details the method used to find the dark spots of

^{e}_{p, p}*U*.

_{VL}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 *IG ^{e}_{p, p}* mode and the rotated

*IG*mode, first find the nodal lines of an

^{e}_{p, p}*IG*mode. Any point on the nodal line of an

^{e}_{p, p}*IG*mode should have zero intensity. Equation (1) gives that

^{e}_{p, p}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 *IG ^{e}_{p, p}* mode should not be a zero solution, the

*C*(

_{p}^{p}*iξ*, ε) cannot be zero. Thus, the specific angle η of parabolic nodal lines could be solved from the equation

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

$$\{\begin{array}{c}\left(p\u20442+1\right)\epsilon {A}_{1}=a{A}_{0},\\ \left(p\u20442+2\right)\epsilon {A}_{2}=-p\epsilon {A}_{0}-\left(4-a\right){A}_{1},\\ \left(p\u20442+r+2\right)\epsilon {A}_{r+2}=\left[a-4{\left(r+1\right)}^{2}\right]{A}_{r+1}+(r-p\u20442)\epsilon {A}_{r}.\end{array}$$

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

$$\{\begin{array}{c}\left(p+3\right)\frac{\epsilon}{2}{A}_{1}=\left[a-\frac{\epsilon}{2}\left(p+1\right)-1\right]{A}_{0},\\ \left(p+2r+3\right)\frac{\epsilon}{2}{A}_{r+1}=\left[a-{\left(2r+1\right)}^{2}\right]{A}_{r}+\left(2r-p-1\right)\frac{\epsilon}{2}{A}_{r-1}.\end{array}$$

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

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 2*p* specific angles *η* of parabolic nodal lines, denoted by {±*η _{1}, ±η_{2}*, …, and ±

*η*}. The equations of

_{p}*IG*mode parabolic nodal lines can be described by equations

^{e}_{p, p}The equations of the parabolic nodal lines of the rotated *IG ^{e}_{p, p}* mode is simply Eq. (9) that exchanges the position of variables

*x*and

*y*, i.e.,

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 *IG ^{e}_{p, 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

*IG*modes,

^{e}_{p, p}*IG*and

^{e}_{4, 4}*IG*mode. Since the

^{e}_{5, 5}*IG*mode has

^{e}_{p, p}*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.

## 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 *IG ^{e}_{p, 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

*IG*laser beam generation from end-pumped SSL. The ellipticity parameter ε of the selected

^{e}_{p, p}*IG*laser beams is estimated by comparing the analytical

^{e}_{p, p}*IG*mode patterns and are shown in Fig. 7(a). Figure 7(b) shows the corresponding vortex array laser beam generated by superposing the selected

^{e}_{p, p}*IG*mode and its rotated replica [

^{e}_{p, p}*IG*]

^{e}_{p, p}^{T}with a π/2 phase delay. As Ref. [22] demonstrates, the elliptical parameter of incident

*IG*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

^{e}_{p, p}*IG*mode and the rotated

^{e}_{p, p}*IG*mode. Since the IGM patterns are different in terms of ellipticity parameter

^{e}_{p, p}*ε*, changing the ellipticity parameter

*ε*of the incident

*IG*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

^{e}_{p, p}*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

*IG*laser beam decreases the curvature of the

^{e}_{p, p}*IG*mode parabolic nodal lines, and the resulting vortices toward aligning in a square array.

^{e}_{p, p}Figure 8(a) provides example vortex arrays calculated in the limit of infinite ellipticity parameter *ε* for an incident *IG ^{e}_{p, p}* laser beam, i.e.

*HG*mode.

_{p, 0}*HG*mode is proved to be achieved experimentally by overlapping the focus pumping beam on the outer most lobe of

_{p, 0}*HG*mode distribution at the laser crystal in end-pumped SSL system [29]. In this case, the nodal lines of the incident

_{p, 0}*IG*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

^{e}_{p, p}*HG*mode are much shorter than the

_{p, 0}*IG*mode, only the central lobes of the

^{e}_{p, p}*HG*mode can create visible vortices after passing through the proposed interferometer. The

_{p, 0}*IG*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

^{e}_{p, p}*IG*laser beam generates more useful visible dark spots than the

^{e}_{p, p}*HG*laser beam. Figure 8(c) shows a tilted

_{p, 0}*IG*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

^{e}_{p, p}*IG*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

^{e}_{p, p}*IG*mode [22]. Figure 8(c) shows that the vortices in a vortex array laser beam formed from a tilted incident

^{e}_{p, p}*IG*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.

^{e}_{p, p}#### 6.2. Effect of modal difference on combined pattern

This section discusses the effect of the relative phase difference Δ*ϕ* and the power difference between the two sub-beams, i.e., the *IG ^{e}_{p, p}* mode and its rotated replica [

*IG*]

^{e}_{p, 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

*IG*modes and rotated

^{e}_{4, 4}*IG*modes, with a phase difference Δ

^{e}_{4, 4}*ϕ*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]. 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 Δ

*ϕ*between the two sub-beams.

It should be noted that all superposed fields with phase difference Δ*ϕ* 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 *IG ^{e}_{p, p}* modes and their rotated replicas. Such “flexible”

*IG*mode based formations of vortex array beams including a large number of vortices in a wide region of the phase difference Δ

^{e}_{p, p}*ϕ*, 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

*IG*mode and those of the rotated

^{e}_{p, p}*IG*mode whose “parabolic” nodal lines still remains at zero even when rotating the

^{e}_{p, p}*IG*mode as shown in section 5.

^{e}_{p, p}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 TEM_{0, 0} laser beam and the dark beams (such as donut beams) can both apply to trap such particles [30]. 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 Δ*ϕ* 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. *IG ^{e}_{p, p}* modes of other order

*p*have also been checked.

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 *IG ^{e}_{p, p}* mode and its rotated replica [

*IG*]

^{e}_{p, p}^{T}. Figure 10 shows the intensity and phase distributions of laser beams generated by superposing two sub-beams, the

*IG*mode and rotated

^{e}_{4, 4}*IG*mode, with different power ratios τ=

^{e}_{4, 4}*P*from 0.5 to 1 in 6 steps. This figure also shows the interferogram with a tilted plane wave, where

_{2}/P_{1}*P*and

_{1}*P*denote the power of the

_{2}*IG*mode and its rotated replica [

^{e}_{4, 4}*IG*]

^{e}_{4, 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

*IG*laser beam from an end-pumped SSL is about several hundred milli-watts [27] and the efficiency in converting

^{e}_{p, p}*IG*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.

^{e}_{p, p}#### 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 *IG ^{e}_{p, p}* laser beam of 1 mm width value passes through the interferometer. The window is 12 times as wide as the TEM

_{00}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

*IG*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.

^{e}_{p, p}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 *IG ^{e}_{p, p}* mode and its rotated replica [

*IG*]

^{e}_{p, 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 *IG ^{e}_{p, p}* mode and its rotated replica [

*IG*]

^{e}_{p, 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)*=exp

*i[kz+{kr*-(p+1)

^{2}/2R(z)}*ψz(z)*]. Name each item by three phase functions,

*ϕ*=kz,

_{1}*ϕ*=

_{2}*kr*), and

^{2}/2R(z*ϕ*=-

_{3}*ψ*, and the phase function becomes

_{z}(z)*ϕ(z)*=exp

*i*[

*ϕ*

_{1}(z)*+ϕ*]. The resulting phase change while propagating an

_{2}(z)+(p+1)ϕ_{3}(z)*IG*mode or its rotated replica [

^{e}_{p, p}*IG*]

^{e}_{p, p}^{T}can be estimated by (d

*ϕ*/dz)Δz, where (d

_{i}*ϕ*/dz) are

_{i}The phase change rate from *ϕ _{1}*(z) is a constant

*k*, and the desired π/2 phase delays are introduced by adding the OPL difference Δz=λ/4 between the two sub-beams. The phase change rates of

*ϕ*(z) and

_{2}*ϕ*(

_{3}*z*) are related to the beam diverging rates, i.e.,

*z*or

_{R}*w*. Note that

_{0}*ϕ*(z) is a function of radial distance

_{2}*r*. Figures 12(a), (b), and (c) plot the resulting phase error caused by

*ϕ*(z),

_{2}*ϕ*(

_{3}*z*) and sum of two phase errors, respectively, for three different waist sizes of

*w*=10µm, 100µm and 1mm, assuming Δz=λ/4. Here, the radial distance

_{0}*r*in (d

*ϕ*/dz) is estimated by the beam spot size

_{2}*w*(

*z*).

Figure 12 shows that all phase errors arising at all position z are much less than 1 degree. The primary phase error arises from *ϕ _{2}*(z). Note that though (d

*ϕ*/dz) is a function of radial distance

_{2}*r*, when position z approaches to infinity, the limit value of (d

*ϕ*/dz) approaches to a small constant value -1/

_{2}*z*, where the radial distance

_{R}*r*in (d

*ϕ*/dz) is estimated by the beam spot size

_{2}*w*(z). (For the I

*G*mode,

^{e}_{p, p}*r*is about

_{max}*p×w(z)*and the phase error maximum is multiplied value in Fig. 12 by

*p*.) A

^{2}*IG*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

^{e}_{p, p}*IG*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

^{e}_{p, p}*IG*laser beam becomes more collimated, the resulting phase changes from the unwanted

^{e}_{p, p}*ϕ*(z) and

_{2}*ϕ*(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

_{3}*IG*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

^{e}_{p, p}*ϕ*(z) in the beam transverse cross-section is only about 100/(1/4)×

_{2}*p*×0.0005=3.2 degrees. This suggests that a

^{2}*IG*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.

^{e}_{p, p}## 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 *IG ^{e}_{p, p}* mode, into a vortex array laser beam consisting of

*p*×

*p*optical vortices. The incident

*IG*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

^{e}_{p, p}*p*of the incident

*IG*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).

^{e}_{p, p}## 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

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