Dynamic generation of Ince-Gaussian modes with a

Dynamic generation of Ince-Gaussian modes with a digital micromirror device Yu-Xuan Ren, Zhao-Xiang Fang, Lei Gong, Kun Huang, Yue Chen, and Rong-De Lu Citation: Journal of Applied Physics 117, 133106 (2015); doi: 10.1063/1.4915478 View online: http://dx.doi.org/10.1063/1.4915478 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/117/13?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Generation of cylindrically polarized vector vortex beams with digital micromirror device J. Appl. Phys. 116, 183105 (2014); 10.1063/1.4901574 Generation of two-color continuous variable quantum entanglement at 0.8 and 1.5   μ m Appl. Phys. Lett. 97, 031107 (2010); 10.1063/1.3467045 Parallel microparticle manipulation using an imaging fiber-bundle-based optical tweezer array and a digital micromirror device Appl. Phys. Lett. 89, 194101 (2006); 10.1063/1.2364888 Nonlinear AlGaAs waveguide for the generation of counterpropagating twin photons in the telecom range J. Appl. Phys. 98, 063103 (2005); 10.1063/1.2058197 Generation and Applications of Single Photon States and Entangled Photon States AIP Conf. Proc. 709, 348 (2004); 10.1063/1.1764028

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JOURNAL OF APPLIED PHYSICS 117, 133106 (2015)

Dynamic generation of Ince-Gaussian modes with a digital micromirror device Yu-Xuan Ren,1,a) Zhao-Xiang Fang,2 Lei Gong,3 Kun Huang,4 Yue Chen,2 and Rong-De Lu2,b)

1 National Center for Protein Sciences Shanghai, Institute of Biochemistry and Cell Biology, Shanghai Institutes for Biological Sciences, Shanghai 200031, People’s Republic of China 2 Department of Modern Physics, University of Science and Technology of China, Hefei 230026, People’s Republic of China 3 Department of Optics and Optical Engineering, University of Science and Technology of China, Hefei 230026, People’s Republic of China 4 Department of Electrical and Computer Engineering, National University of Singapore, 4 Engineering Drive 3, Singapore 117576, Republic of Singapore

(Received 20 December 2014; accepted 1 March 2015; published online 1 April 2015) Ince-Gaussian (IG) beam with elliptical profile, as a connection between Hermite-Gaussian (HG) and Laguerre-Gaussian (LG) beams, has showed unique advantages in some applications such as quantum entanglement and optical micromanipulation. However, its dynamic generation with high switching frequency is still challenging. Here, we experimentally reported the quick generation of Ince-Gaussian beam by using a digital micro-mirror device (DMD), which has the highest switching frequency of 5.2 kHz in principle. The configurable properties of DMD allow us to observe the quasi-smooth variation from LG (with ellipticity e ¼ 0) to IG and HG ðe ¼ 1Þ beam. This approach might pave a path to high-speed quantum communication in terms of IG beam. Additionally, the characterized axial plane intensity distribution exhibits a 3D mould potentially C 2015 AIP Publishing LLC. being employed for optical micromanipulation. V [http://dx.doi.org/10.1063/1.4915478]

I. INTRODUCTION

Most of the commercial lasers provide a beam with a transversal Gaussian distribution, since the Gaussian modes are stable modes in most of the laser resonators. Besides, various spatial modes could be solutions in some specifically designed laser resonators, and those modes obey the scalar Helmholtz equation1 ðr2 þ k2 ÞE ¼ 0. Some well known ones are the Hermite-Gaussian modes (HGMs) in Cartesian coordinates and Laguerre-Gaussian modes (LGMs) in cylindrical coordinates. These two modes form two complete sets of orthogonal basis. Arbitrary spatial modes in Cartesian (cylindrical) coordinates could be decomposed by series of HGMs (LGMs) and vice versa. Extensive studies were conducted on the creation, physical property, and applications of either the fundamental modes or the superpositions.2–5 Ince-Gaussian modes (IGMs)6,7 constitute another complete set of transversal eigenmodes for paraxial Helmholtz equation. Mathematically, IGMs could be interpreted in elliptic coordinates. The IGMs were first generated through breaking the symmetry of the cavity of diode-pumped solid state (DPSS) laser and the pumping beam.8,9 Through symmetry breaking, several gain medium have been testified to work for the IGMs, e.g., Nd:YVO4 crystal8 and Nd:GdVO4 crystal.4 The drawback of cavity-based mode generation technique is that the mode order is fixed when the laser cavity is tuned for the specific order. Fine adjustment of the laser cavity is required for switching among various modes. This a)

Electronic mail: [email protected] Electronic mail: [email protected]

b)

0021-8979/2015/117(13)/133106/7/$30.00

lacks flexibility and obstructs various applications requiring the switching among higher order modes. A fast and easyswitching approach is needed to dynamically alter the modes. The digital micromirror device (DMD) is an array of 1024  768 micromirrors.10 Each square mirror has a side length of 13.68 lm and could be switched to two angular states along the main diagonal line. The DMD functions as an amplitude spatial light modulator (SLM) in various recent experimental reports.11–13 Due to the high refreshing rate and fill factor, the DMD has been employed to shape the LaguerreGaussian modes with off-axis interference hologram.11,14 With error diffusion and the division of output and input laser modes, the DMD is able to shape the super-Gaussian modes.15–17 The non-diffracting and self-accelerating beams were also realized through the DMD.18,19 The flexibility and fast switching rate enable the application in dynamical wavefront shaping in scattering biological tissue.20 The DMD has also shown good performance in micro-lithography and optical coherence measurement.21,22 The DMD could be incorporated in high-resolution optical imaging system for the dynamic, parallel, and remote control of neuronal activities.23,24 Compared with the previous cavity based generation technique, this method does not require fine alignment of the optics when switching among higher-order modes.9 Similar to HGM and LGM beams, the IGM modes have arisen a wide spreading interests,25,26 e.g., the high dimensional quantum entanglement, 3-dimensional optical micromanipulation, and construction of complex structures. The IGM modes will attract more and more diverse applications

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in interdisciplinary topics. In this paper, different from intracavity generation, we conducted an experiment using the DMD to shape the IGMs.10 The creation process was verified through measurements of transverse profile as well as longitudinal propagating profiles. The flexibility of the DMD allows the switching among Ince-Gaussian modes with various modal numbers and ellipticities. II. THEORY

The paraxial wave equation (PWE), r2? W þ 2ik@z W ¼ 0, is an approximate form of the Helmholtz equation for traveling waves along the z direction. In elliptic coordinates, the solution for this PWE could be constructed in a modulated version of the Gaussian beam,7,27 IGðrÞ ¼ EðnÞNðgÞ expðiZðzÞÞWG ðrÞ;

IGep;m ðn; g; z; eÞ ¼

(1)

where E, N, Z are real functions of the space coordinates, WG ðrÞ is the complex amplitude of a Gaussian beam in Cartesian coordinates, and IG represents Ince-Gaussian mode. The elliptic coordinates ðn; gÞ are related with the Cartesian coordinates,8 x ¼ f ðzÞcoshðnÞ cosðgÞ; y ¼ f ðzÞ sinhðnÞ sinðgÞ, where n 2 ½0; þ1Þ and g 2 ½0; 2pÞ are the radial and angular elliptic coordinates, and xðzÞ is the beam radius at position z. f ðzÞ ¼ f0 xðzÞ=x0 defines the semifocal separation and it diverges the same manner as the width of a Gaussian beam. In elliptic coordinate, the curves with constant n are confocal ellipses, and curves with constant g are confocal hyperbolas. By inserting the trial solution in Eq. (1) and separation of the variables, the functions EðnÞ and NðgÞ satisfy a unified Ince equation.7,27,28 The Ince equation is a special case of the Hill equation,28,29 d2 X dX þ ða  pe cosð2gÞÞX ¼ 0: þ e sinð2gÞ 2 dg dg

determined by recurrence relations.28 For simplicity, the solutions could be expressed in even and odd Ince polynomials m Cm p ðg; eÞ and Sp ðg; eÞ, respectively, where the elliptic coordinates ðn; gÞ are employed. The order p, degree m, and parameter e jointly determine the transverse profile of the beam. e is the ellipticity parameter. For even Ince polynomials, 0  m  p, while for odd Ince polynomials, 1  m  p, the indices m and p have the same parity, e.g., ð1Þmp ¼ 1. We define another parity parameter d, d ¼ 0 represents even InceGaussian beam; while for odd Ince-Gaussian beam, d ¼ 1. The even and odd Ince-Gaussian modes6,7,30 with mode numbers p, m, and ellipticity parameter e could be, respectively, written as,

IGop;m ðn; g; z; eÞ ¼

Solutions for Eq. (2) are in the form of trigonometric series with unknown coefficients. The unknown coefficients are

(3) Sx0 m Sp ðin; eÞSm g; e Þ pð x ðzÞ ( ) r2  ð p þ 1ÞWGP ðzÞ ;  exp ik x ðzÞ (4)

where WGP ðzÞ ¼ arctanðz=zR Þ is the Guoy phase shift, C and px2

S are normalization constants, zR ¼ k 0 is the Rayleigh length of the beam with waist radius x0. The normalization ensures the same root-mean square value as the trigonometric functions,29 i.e., ðp ðp 2 2 ½Cm ðg; eÞ dg ¼ ½Sm (5) p p ðg; eÞ dg ¼ p: p

(2)

Cx0 m C ðin; eÞCm p ðg; eÞ x ðzÞ p ( ) r2  ð p þ 1ÞWGP ðzÞ ;  exp ik x ðzÞ

p

The transverse shape of the Ince-Gaussian mode is jointly controlled by the mode order and the ellipticity. The ellipticity e, waist radius x0, and the semifocal separation f0 are related by e ¼ 2f02 =x20 . Several transversal intensity

FIG. 1. Transverse beam profile of representative Ince-Gaussian beams.7,8,30 The mode indices and the parities are, respectively, (a) IGe5;1 , (b) IGe6;4 , (c) IGe7;5 , (d) IGo6;4 , (e) IGo7;5 , and (f) IGo8;6 .

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shapes of the IGM IGp;m are shown in Fig. 1, (a)–(c) are for IGe5;1 ; IGe6;4 , and IGe7;5 , respectively; (d)–(f) are for IGo6;4 ; IGo7;5 ; IGo8;6 in sequence. The ellipticity adopts e ¼ 2 throughout Fig. 1. There is a nodal line along x-axis for odd IGMs. The ellipticity parameter e controls the overall transverse shape of the IGM. When e ¼ 1 and e ¼ 0, the IGMs will be reduced to the Hermite-Gaussian and LaguerreGaussian modes, respectively. This implies that the IGMs provide a transition mode set for the basis in Cartesian and cylindrical coordinates. III. EXPERIMENTAL REALIZATION

The amplitude DMD (DLP7000, XGA, Texas Instrument) consists of 786 432 highly reflective, digitally switchable micromirrors.31,32 So many independently switchable micromirrors provide numerous freedom to shape the wavefront. The side length of each square micromirror is 13.68 lm. The micromirror is coated with aluminum to provide high reflectivity of light. Each micromirror can be switched to either þ12 or 12 with respect to the surface normal direction of the DMD.11,21 These two states, respectively, represent “On state” and “Off state.” On state micromirror directs the micro portion of light to the downstream optical path, while the off state micromirror reflects the micro beam onto a light beam absorber. Consequently, the DMD acts as an amplitude type of SLM. The advantages of the DMD are the high fill factor and switching rate. The fill factor of the DMD used in our experiment is 92%, as a comparison, the fill factor of commercial

(a)

0

π/2

π 3π/2

(b)

0 2π/9 4π/9

(c)

liquid crystal SLM is far less than 90%. As a reference, the commercial DLP Discovery 4000 XGA DMDs can transfer data to XGA chips at 25.6 Gb/s, which corresponds to a frame rate of 52 550 frames/s.10 Additionally, such device works in a broadband of spectrum range and is insensitive to incident polarization.11,33,34 Similar to LC-SLM, a hologram has to be numerically calculated and sent to the DMD through a computer (Fig. 3(a)). Various algorithms have been utilized to perform the calculation, e.g., the division of desired fields by input modes and the error diffusion technique,14,15,17 the binary Lee hologram.36–38 We take the advantage of a superpixel based method to encode both the amplitude and phase of the light field35,39,40 using a binary hologram. In the superpixel method, the DMD pixels are divided into square groups of n  n micromirrors. Figures 2(a)–2(c) demonstrate the superpixel composed of 2  2, 3  3, and 4  4 DMD pixels. Each micromirror in the superpixel has a phase difference with respect to the pixel at the top-left corner. As shown in Figs. 2(a)–2(c), the phase differences are written on the squares representing the DMD pixels. By selectively turning on some of the pixels, e.g., the shadowed square pixels in Figs. 2(a)–2(c), a complex field will be synthesized. Black dots represent contributions from the off state DMD pixels, while dark purple dots denote the on state pixels. In Figs. 2(d)–2(f), the dark green dots, which are produced through vector sum of all the dark purple dots, demonstrate the synthesized complex field in the complex number plane.

0 π/8 π/4 3π/8

6π/9 8π/910π/9

π/2 5π/83π/4 7π/8

12π/9 14π/916π/9

π 9π/8 5π/4 11π/8

Re(E)

Im(E)

(g)

Im(E)

Re(E)

Re(E)

(f)

Im(E)

(e)

(h)

Re(E)

(i)

Re(E)

Im(E)

off pixel on pixel complex field

Im(E)

(d)

Im(E)

3π/213π/8 7π/4 15π/8

FIG. 2. Demonstration of superpixels composed of (a) 2  2, (b) 3  3, and (c) 4  4 DMD pixels. The contribution points for each DMD pixel are uniformly scattered in a unit circle (d)–(f). Tiny phase difference separates the contribution points in the complex number plane.35 Suppose some of the DMD pixels are turned on as marked by the shadowed squares in (a)–(c), the active pixels (purple dots) will contribute to the total field complex amplitude (dark green). Through combination of several pixels in the super pixel, various complex field values could be produced (g)–(i).

Re(E)

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In this manner, the phase and the amplitude of the target field can be simultaneously modulated. Therefore, we achieve the complex field modulation. For n  n square pixels, the synthesized complex field will increase dramatically when increasing n. The number of possible fields is 9 for n ¼ 2 as shown in Fig. 2(g). The values increase to 343 and 6561 for n ¼ 3 and n ¼ 4, respectively (Figs. 2(h) and 2(i)). These numbers does not include the degenerate counts. The phase modulation resolution for n ¼ 3 case could be estimated to be 2p=343 ¼ 0:018rad, which corresponds to 1:05 . This resolution is sufficient for the wavefront shaping. As a comparison, many commercial LC-SLM could provide 8-bit modulation, and the phase resolution is 2p=28 ¼ 0:025rad ð1:41 Þ. In the following experiment, we grouped 4  4 pixels into a superpixel, and the resolution reaches 2p=6561 ¼ 9:6  104 rad ð0:05 Þ. A lookup table was created to relate the target complex field and the combinations of on-state micromirrors in a superpixel. The lookup table expedites the hologram calculation. The advantage of the DMD modulation lies in the modulation speed as well as the polarization insensitivity. To perform the shaping experiment, we designed an optical setup which combines the amplitude DMD and the superpixel method. A single mode 633 nm Helion-Neon laser (HJ-1, Nanjing Laser Instrument Company) with nominal output of 1.5 mW was adopted as the coherent light source.17 The coherent laser beam was expanded and collimated through a Keplerian telescope to slightly overfill the aperture of the DMD (Fig. 3(a)). The focal lengths of Lenses L1 and L2 are 25 mm and 200 mm, respectively. Lens L3 (focal length 250 mm) acts as a Fourier lens which transforms the modulated light to the spatial spectrum space on the back focal plane of the lens L3. A pinhole spatial filter was placed on such plane to select light in the first diffraction order.

J. Appl. Phys. 117, 133106 (2015)

The pinhole was located at ðx; yÞ ¼ ða; naÞ, where a ¼  nkf2 d, k is the wavelength of the light, f is the focal length of the first lens, and d is the distance between adjacent micromirrors. This position ensures that the target phase responses of neighbouring pixels inside a superpixel are 2p n2 out of phase in the x-direction and 2p out of phase along y-direction.35 n Specifically, this value corresponds to p8 and p2 out of phase in the x and y directions for pinhole position (x, y) ¼ (0.73, 2.90) mm, where the wavelength and focal length of L3 employed k ¼ 633 nm and f3 ¼ 250 mm, respectively. A fourth Lens L4 (focal length 100 mm) collects the modulated light and projects the light pattern onto a complementary metal oxide semiconductor (CMOS) camera. The CMOS camera (Guangzhou Weiscope, WUS300, pixel resolution 2048  1536, pixel size 3.2 lm) placed on the back focal plane of L4 recorded the transversal beam profile and saved the collected images on a local computer. The combination of two linear polarizers was adopted to adjust the laser power and prevent the camera from being saturated. The control of the DMD was realized through a dual head graphics card. In the static shaping experiment, no feedback control is required, thus this setup could be employed to do open-loop control, provided the holograms are precalculated off-line. As a demonstration, Fig. 3(b) presents the binary hologram produced for the even IGM IGe7;3 (Fig. 3(c)). The square region of the hologram is enlarged for clear visualization. With such hologram projected onto the DMD, we were able to shape the even IGM IGe7;3 mode. Other modes could be achievable with the corresponding holograms. IV. RESULTS AND DISCUSSIONS

In order to verify the feasibility of the proposed method, we produced corresponding binary holograms with superpixel

FIG. 3. (a) Experimental setup for the generation of Ince-Gaussian modes.11,17,18 (b) The binary hologram produced with superpixel method35 for IGe7;3 , the enlarged image provides a closeup view of the region indicated by the white square. (c) The intensity profile for IGe7;3 .

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FIG. 4. Experimentally generated transversal beam profiles of InceGaussian beams. The mode orders for each image are, respectively, (a)IGe5;1 , (b) IGe6;4 , (c) IGe7;5 , (d) IGo6;4 , (e) IGo7;5 , and (f) IGo8;6 .

method for those modes illustrated in Fig. 1. Binary holograms projected onto the DMD modulate the collimated beam. The modulated beam passes a spatial filter and is subsequently collected by the CMOS camera. Fig. 4 shows the experimental results corresponding to the theoretical predictions in Fig. 1. The first row illustrates the even IGMs with modal indices (p,m) ¼ (5, 1), (6, 4), and (7, 5) in sequence, and the second row demonstrated the odd IGMs with (p,m) ¼ (6, 4), (7, 5), and (8, 6), respectively. The shapes demonstrated in Fig. 4 are in good agreement with theoretical predictions shown in Fig. 1. To point out that the created beams are propagating InceGaussian modes. The propagating IGM mode provides a 3D spatial mould suited to the optical manipulation of complex structures.26,41 For verification, we take the IGe6;4 mode as an example to evaluate the propagation behavior. Analysis of the transverse image in Fig. 4(b) shows that the waist radius of the produced IGe6;4 mode is x0 ¼ 0.8 mm. 2Therefore, the px Rayleigh length of the beam is zR ¼ k 0 ¼ 3 m. We employed another lens with focal length (f5 ¼ 100 mm) to converge the beam and measured the propagation behavior near the focus. The optical layout of the propagation measurement is shown in Fig. 5(a). The transverse beam profile (Fig. 5(b)) near the focus of the lens L5 exhibits the similar pattern as shown in Fig. 4(b) except for the reduced size. Further analysis of the transverse pattern near the focus indicates that the beam waist radius is x00 ¼ 0:08mm. Accordingly, the Rayleigh range is reduced to 02 z0R ¼ pxk0 ¼ 30mm. A series of cross-sectional images preceding the front focus were acquired to form an image stack. During the measurement, the increment adopts 1 mm, which is sufficient for the image reconstruction resolution. The image stack was compressed into a short movie as shown in supplementary multimedia file 1. We further reconstructed the x–z intensity distribution of the propagating InceGaussian mode as shown in Fig. 5(c) (Multimedia view). Another stack of images near the rear focus of L5 were collected to reconstruct the focused light field. The image stack and the intensity distribution along the x–z plane are, respectively, shown in supplementary multimedia file 2 and Fig. 5(d) (Multimedia view). The IGM mode was first

converged to a focal point then diverged rapidly with expanding beam size. The propagation measurements demonstrate that the produced Ince-Gaussian beams are propagating beams instead of just imaging the DMD surface in a certain propagation plane. The focus depth could be evaluated from the x–z intensity distribution to be 60 mm, which is consistent with twice the Rayleigh length 2z0R . The ellipticity adopted in Figs. 1 and 4 is e ¼ 2, which determines the overall transverse shape of the produced mode. To verify this phenomenon, a series of IGMs with various e were experimentally created with the DMD-based setup for wavefront shaping. Fig. 6 (Multimedia view)shows the intensity profiles for the even (first row, IGe7;5 ) and odd

FIG. 5. Propagating field measurements for even IGM IGe6;4 with ellipticity e ¼ 0:8. (a) Additional optics was employed to perform the measurement. The lens L5 converges the IGM into its rear focus. The CMOS camera was mounted onto a guide rail for measurements of z-stack images. (b) Transverse profile for the focused IGe6;4 mode. (c) Propagating beam profile before focusing measured at the position marked by a dashed square in (a). (d) Reconstructed x–z intensity distribution near the rear focus of the lens L5. The propagating IGM exhibits a narrow beam waist radius of about 80 lm at the focal plane. The image stacks for extracting the longitudinal field distribution in (c) and (d) are provided in the supplementary media 1 and media 2, respectively (Multimedia view) [URL: http://dx.doi.org/10.1063/ 1.4915478.1] [URL: http://dx.doi.org/10.1063/1.4915478.2].

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FIG. 6. The IGMs with different ellipticity. The first row demonstrates the results for even IGM IGe7;5 , and the second row shows the odd IGM IGo8;6 . The ellipticities are (a) and (d) e ¼ 0:8, (b) and (e) e ¼ 10, and (c) and (f) e ¼ 1000. A multimedia file 3 shows the continuous alteration of the beam profile for even IGM IGe7;5 by increasing the ellipticity (Multimedia view) [URL: http://dx.doi.org/10.1063/ 1.4915478.3].

(second row, IGo8;6 ) IGMs with (a) and (d) e ¼ 0:8, (b) and (e) e ¼ 10, and (c) and (f) e ¼ 1000, respectively. The sectional images shown in Figs. 6(a) and 6(d) (Multimedia l¼6 view) resemble on the LGl¼5 p¼1 and LGp¼1 modes, respectively, while those shown in Figs. 6(c) and 6(f) (Multimedia n¼2 n¼3 and HGm¼5 modes in view) are similar to the HGm¼5 sequence. Continuous alteration of the ellipticity was achieved with the setup. To further display the smooth variation of IGMs from LGs to HGs, we made a video that consisted of a series of experimental intensity profiles for the even IGM IGe7;5 with its ellipticity increasing from e ¼ 0 to e ¼ 100. Dynamical alteration of the transverse mode is achieved as shown in the supplementary media file 3. With varying ellipticity, the IGMs will approach the Laguerre-Gaussian modes when e ¼ 0 or the HermiteGaussian modes when e ¼ 1. The Laguerre-Gaussian and Hermite Gaussian beams are two complete sets of basis.

Therefore, the experimental beam profiles with various ellipticities verified that the produced IGMs are transition modes between the two complete set of basis. Additionally, IGMs provide another basis for conversion among Cartesian, cylindrical, and elliptic coordinates.3 This implies that any transverse profile is able to be decomposed into series of IGMs. The laser modes with orbital angular momentum (OAM) are of particular interest. In elliptic coordinates, such kind of beams with OAM could be constituted by superposition of the IGMs. Further, we constructed the helical Ince-Gaussian modes (HIGM)30 through the superposition of even and odd IGMs, e o HIG6 p;m ðn; g; eÞ ¼ IGp;m ðn; g; eÞ6iIGp;m ðn; g; eÞ;

(6)

whose phase rotates elliptically around the long axis of the ellipse.30 The rotation direction is defined by the sign in Eq. (6).

FIG. 7. Even (a), (d), (g), odd (b), (e), (h), and helical (c), (f), (i) IGMs with p ¼ 10 and m ¼ 6. The ellipticities in each row are (a)–(c) e ¼ 2, (d)–(f) e ¼ 10, and (g)–(i) e ¼ 100, respectively.

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Fig. 7 shows the transverse magnitudes of the IG10;6 ðn; g; eÞ at the beam waist. The ellipticities in each row are e ¼ 2; 10; 100, respectively. Each column corresponds to the even, odd, and helical IG10;6 in sequence. As the ellipticity e increases, the intensity profile elongates along the horizontal axis.25 For helical modes, the central vortex splits into a number of vortices equal to m. As e increases, new vortices will be produced in the outer rings of the pattern as shown in Figs. 7(c), 7(f), and 7(i). All the vortices in the outer rings have a topological charge of 1.25 V. CONCLUSION

In conclusion, we experimentally shaped the IGM with the digital micro-mirror device. The experimental results for IGM are in good agreement with the theoretical predictions. The IGM provides a complete set of eigenmodes in elliptic coordinates. The ellipticity determines the overall shape of the transversal profile. Through varying the ellipticity, this set of eigenmodes transits the HGMs in Cartesian coordinates to the LGMs in cylindrical coordinates. Smooth variation of the transverse mode was achieved through fine adjustment of the ellipticity. Propagation measurements involving the 3D structure of the mode confirm that the produced modes constitute propagating IGM. The creation of helical IGMs demonstrates that the proposed method could shape the light field to possess orbital angular momentum. This provides a means to study the local angular momentum states in elliptic coordinates and even the HG beams in Cartesian coordinates. We anticipate that the proposed method is not limited to Ince-Gaussian beams, but can be applied to beams with other shapes, such as self-accelerating Mathieu beams and Webber beams. The beam shaping with DMD provides possibilities for trapping cold atom clusters with various spatial optical landscapes. Additionally, the control of the ellipticity in IGMs varies the energy landscape of the optical potential. The helical IGMs will be potentially applied to optomechanics, atom cooling, and quantum mechanics. ACKNOWLEDGMENTS

This work was sponsored by the National Natural Science Foundation of China (Grant No. 60974038) and the Project of Provincial Teaching Research in Anhui Institutions of Higher Education (Grant No. 2012jyxm006). K.H. acknowledges the support by National Research Foundation of Singapore under its Competitive Research Program (No. R-263-000-A86-281). 1

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