## Shaping of light beams along curves in three dimensions |

Optics Express, Vol. 21, Issue 18, pp. 20544-20555 (2013)

http://dx.doi.org/10.1364/OE.21.020544

Acrobat PDF (5012 KB)

### Abstract

We present a method for efficient and versatile generation of beams whose intensity and phase are prescribed along arbitrary 3D curves. It comprises a non-iterative beam shaping technique that does not require solving inversion problems of light propagation. The generated beams have diffraction-limited focusing with high intensity and controlled phase gradients useful for applications such as laser micro-machining and optical trapping. Its performance and feasibility are experimentally demonstrated on several examples including multiple trapping of micron-sized particles.

© 2013 Optical Society of America

## 1. Introduction

1. M. A. Seldowitz, J. P. Allebach, and D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. **26**, 2788–2798 (1987). [CrossRef] [PubMed]

3. G. Whyte and J. Courtial, “Experimental demonstration of holographic three-dimensional light shaping using a Gerchberg–Saxton algorithm,” New J.
Phys. **7**, 117 (2005). [CrossRef]

4. T. D. Gerke and R. Piestun, “Aperiodic volume optics,” Nat. Photonics **4**, 188–193 (2010). [CrossRef]

5. Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. **100**, 013602 (2008). [CrossRef] [PubMed]

6. M. Woerdemann, C. Alpmann, M. Esseling, and C. Denz, “Advanced optical trapping by complex beam shaping,” Laser Photonics Rev. 1–16 (2012). [CrossRef]

5. Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. **100**, 013602 (2008). [CrossRef] [PubMed]

7. M. Padgett and L. Allen, “Light with a twist in its tail,” Contemp. Phys. **41**, 275–285 (2000). [CrossRef]

9. A. Jesacher, S. Fürhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte, “Holographic optical tweezers for object manipulations at an air-liquid surface,” Opt. Express **14**, 6342–6352 (2006). [CrossRef] [PubMed]

9. A. Jesacher, S. Fürhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte, “Holographic optical tweezers for object manipulations at an air-liquid surface,” Opt. Express **14**, 6342–6352 (2006). [CrossRef] [PubMed]

*true*3D vortex trap generated by focusing a

*helical*Bessel beam was experimentally demonstrated in [10

10. Y. Roichman and D. G. Grier, “Three-dimensional holographic ring traps,” Proc. SPIE **6483**, 64830F (2007). [CrossRef]

*tilted*with respect to the focal plane, see [11

11. E. R. Shanblatt and D. G. Grier, “Extended and knotted optical traps in three dimensions,” Opt. Express **19**, 5833–5838 (2011). [CrossRef] [PubMed]

11. E. R. Shanblatt and D. G. Grier, “Extended and knotted optical traps in three dimensions,” Opt. Express **19**, 5833–5838 (2011). [CrossRef] [PubMed]

11. E. R. Shanblatt and D. G. Grier, “Extended and knotted optical traps in three dimensions,” Opt. Express **19**, 5833–5838 (2011). [CrossRef] [PubMed]

*solenoid*beam that exhibits a fixed spiral shape around the optical axis [12

12. S.-H. Lee, Y. Roichman, and D. G. Grier, “Optical solenoid beams,” Opt. Express **18**, 6988–6993 (2010). [CrossRef] [PubMed]

*tractor*beam. It was achieved by imposing helical phases to a collinear superposition of Bessel beams.

13. E. G. Abramochkin and V. G. Volostnikov, “Spiral light beams,” Physics-Uspekhi **47**, 1177–1203 (2004). [CrossRef]

## 2. Principle of the technique

### 2.1. HIG beams shaped along 2D curves

*F*(

*x*,

*y*,

*l*) reported in [13

13. E. G. Abramochkin and V. G. Volostnikov, “Spiral light beams,” Physics-Uspekhi **47**, 1177–1203 (2004). [CrossRef]

*xy*–plane. The beam shape is preserved during propagation along the axial direction

*l*except for scaling and rotation. Specifically, the complex field amplitude of a beam with Gaussian intensity profile along the curve is given, in the plane

*l*= 0, by the expression: where

**r**= (

*x*,

*y*),

**c**

_{2}(

*t*) = (

*x*

_{0}(

*t*),

*y*

_{0}(

*t*)) is the curve in the

*xy*–plane with

*t*∈ [0,

*T*],

*w*

_{0}is a constant,

**c′**

_{2}(

*t*) = d

**c**

_{2}/d

*t*, and is a term that shapes the phase of the beam along the curve [13

13. E. G. Abramochkin and V. G. Volostnikov, “Spiral light beams,” Physics-Uspekhi **47**, 1177–1203 (2004). [CrossRef]

*σ*= 1, however, further we will use

*σ*as a free parameter for controlling the phase gradient along the curve. Below a shorter notation

*F*(

**r**|

**c**

_{2}) is used in cases when the curve

**c**

_{2}does not need a detailed description.

*light drawing tool*in the way that the function

*r*

^{2}=

*x*

^{2}+

*y*

^{2}, works as a

*pencil tip*moved along the prescribed curve. These Gaussian beam terms are coherently assembled according to the trajectory

**c**

_{2}(

*t*), where Φ(

**r**,

*t*) guarantees the phase matching in the superposition, Eq. (1). Note that beams described by a ring curve

*x*

_{0}(

*t*) + i

*y*

_{0}(

*t*) =

*R*

_{0}exp (i

*t*),

*t*∈ [0, 2

*π*], coincide with helical Laguerre-Gaussian modes of topological charge

**47**, 1177–1203 (2004). [CrossRef]

*m*, is obtained by complex conjugation:

*F*

^{*}(

**r**|

**c**

_{2}).

*λ*the light wavelength in the medium. Nevertheless, the Gaussian generating term does not provide high axial intensity gradient under the focusing and has to be replaced with a more appropriate one. In particular, a plane wave can be used as generating term because it is transformed into an extremely focused spot yielding both high transversal and axial intensity gradients. Therefore, the expression Eq. (1) turns into in order to generate a focusing beam with high intensity gradients prescribed along the curve. We underline that the plane waves

**r**,

*t*), see Eq. (3), control the position of each focused spot. While the term

*σ*can be any real number that allows varying the phase gradient if needed. In practice, the beam is truncated by the circular aperture of the focusing lens, which is described by the circle function circ (2

*r/D*) with diameter

*D*. In this case the beam is circ (2

*r/D*)

*G*(

**r**|

**c**

_{2}) and thus the

*pencil tip*in the focal plane is a tiny spot of size

*δ*= 2.44

*λ*f/

*D*corresponding to the central maximum of the well-known Airy disc.

*F*(

**r**|

**c**

_{2}) and

*G*(

**r**|

**c**

_{2}), see expressions Eq. (1) and Eq. (4) respectively. In this example we consider the ring curve

*x*

_{0}(

*t*) + i

*y*

_{0}(

*t*) =

*R*

_{0}exp (i

*t*) for the generation of a vortex beam, with topological charge

*m*= 30 and radius

*R*= 0.56 mm in the focal plane, by using both methods. The setup required for the beam generation is sketched in Fig. 1(a). A phase-only hologram that encodes the amplitude and phase of the beam is addressed into the spatial light modulator (SLM), see Methods for further details (section 5.1). In our case, the hologram is illuminated by a collimated laser beam (

*λ*= 532 nm) and the resulting beam is then focused by a lens (FL) with focal length f = 10 cm. As observed in Fig. 1(b)–(d), the beam

*F*(

**r**|

**c**

_{2}) has in the focal plane a Gaussian intensity profile [Fig. 1(b), red color line] whereas the beam

*G*(

**r**|

**c**

_{2}) has tighter intensity profile with high transversal intensity gradient [Fig. 1(b), blue color line]. The propagation of

*G*(

**r**|

**c**

_{2}) in the focusing region, displayed in Fig. 1(d) for the

*yz*–plane, confirms the generation of high axial intensity gradient as well. In contrast, the Gaussian counterpart shown in Fig. 1(c) has low axial intensity gradient. We underline that in the case of the ring curve we recover both the Gaussian-vortex [Fig. 1(c)] and the ring-vortex [Fig. 1(d)] previously studied in [10

10. Y. Roichman and D. G. Grier, “Three-dimensional holographic ring traps,” Proc. SPIE **6483**, 64830F (2007). [CrossRef]

*G*(

**r**|

**c**

_{2}) provides additional degrees of freedom. For instance, the topological charge

*m*is controlled by the parameter

*σ*without altering the radius

*R*of the ring-vortex. In this case,

*G*(

**r**|

**c**

_{2}) is a helical Bessel beam:

*J*(

_{m}*kr*) exp (i

*mθ*), where

*J*(

_{m}*kr*) is the

*m*–th order Bessel function of first kind with

*m*=

*kR*

_{0}

*σ*, and

*θ*is the azimuthal angle, see Appendix. The opposite topological charge is obtained by changing the sign of

*σ*. Although the radius of the ring-vortex does not depend on the topological charge, the position

*ρ*of the first maximum of

_{m}*J*(

_{m}*kr*) does it. Therefore, for a certain value of

*m*the Bessel beam is significantly truncated by the lens aperture (i.e. circ (2

*r*/

*D*) pupil function). Note that this beam truncation degrades both the shape and

*light efficiency*of the generated light curve. To avoid this effect and preserve the high axial intensity gradient, as the one observed in Fig. 1(d), the constraint

*D*≫ 2

*ρ*has to be fulfilled. The loss of axial intensity gradient caused by increasing the topological charge was pointed out in [10

_{m}10. Y. Roichman and D. G. Grier, “Three-dimensional holographic ring traps,” Proc. SPIE **6483**, 64830F (2007). [CrossRef]

*light efficiency*of arbitrary light curves needs further research out of the scope of this work.

*G*(

**r**|

**c**

_{2}) are displayed in Fig. 1(e). As observed, these beams show a Bessel-like structure and are focused yielding the 2D light curves displayed in the first row of Fig. 2. The experimental results shown in the second row of Fig. 2 are in good agreement with the theoretical ones (first row) and clearly exhibit a HIG profile along the target curve. It is worth to mention that the exact generation of the hologram is not possible in practice and thus the generated light curve is slightly

*blurred*with respect to the theoretical one. In our case the phase-only hologram is addressed into a programmable reflective LCoS-SLM (Holoeye PLUTO, 8-bit gray-level, pixel pitch of 8

*μ*m and 1920 × 1080 pixels) calibrated for a 2

*π*phase shift at the wavelength

*λ*= 532 nm and corrected from static aberrations as reported in [14

14. J. A. Rodrigo, T. Alieva, A. Cámara, O. Martínez-Matos, P. Cheben, and M. L. Calvo, “Characterization of holographically generated beams via phase retrieval based on Wigner distribution projections,” Opt.
Express **19**, 6064–6077 (2011). [CrossRef] [PubMed]

*μ*m), see Methods (section 5.3), and stored as a video ( Media 1).

### 2.2. HIG beams shaped along 3D curves

**r**,

*t*) with the quadratic phase function: according to the prescribed curve

**c**

_{3}(

*t*) = (

*x*

_{0}(

*t*),

*y*

_{0}(

*t*),

*z*

_{0}(

*t*)), where

*z*

_{0}(

*t*) is a

*defocusing*distance defined along the curve

**c**

_{2}(

*t*) and f is the focal length of the focusing lens. Finally, these new beam terms are coherently assembled using the expression which is a generalization of Eq. (4) for the 3D curve. We recall that the

*pencil tip*associated with the expression Eq. (6) takes the form of a highly focused spot of diameter

*δ*= 2.44

*λ*f

*/D*in the focal region of a lens with aperture diameter

*D*. Note that for

*z*

_{0}(

*t*) = 0 the beam given by Eq. (6) is projected into the focal plane following the 2D curve

**c**

_{2}(

*t*).

*z*= 0) are shown in the third row of Fig. 3, whereas the second and fourth row correspond to a plane before and after the focal one, respectively. For each kind of beam, the intensity distribution in the overall focusing region was also measured and stored in ( Media 2). To illustrate the 3D shape of the light curve, the images stored in ( Media 2) are used as 2D slices to perform the volumetric beam reconstruction displayed in Fig. 4. These results (Fig. 3 and Fig. 4) demonstrate that the HIG profile is preserved according to the 3D curve. Note that in the case of the Archimedean spiral [Fig. 3 (c) and Fig. 4(c)] the light propagating near the curve’s center does not converge and diverge in the same way as the rest of the light curve. In particular, this beam present a vertical flame-like structure in the spiral’s center, where the curvature is higher, as observed in the volumetric reconstruction displayed in Fig. 4(c). Moreover, the limited resolution of the hologram impedes encoding the entire beam

*H*(

**r**|

**c**

_{3}) required to obtain high axial intensity gradient in this region of the curve. This effect is also observed in the 2D Archimedean spiral beam previously studied, see Fig. 2(b) and ( Media 1).

*H*(

**r**|

**c**

_{3}) is focused into two symmetric loops (or foils) appearing on top of each other and intersecting in

*z*= 0, see Fig. 3(b) and Fig. 4(b). We underline that both loops present a well-defined intensity profile without cross-talk in spite of the same light structure is generated at different propagation distances. This fact is also verified in the case of the trefoil-knotted curve, Fig. 3(d). In this case, the light propagating from a focused segment of the knot at

*z*< 0 overlaps with the next segment focused for

*z*> 0, see Fig. 3(d).

## 3. Optical traps generated by beams shaped in 3D

*H*(

**r**|

**c**

_{3}) is encoded into the SLM as a phase hologram. The generated beam is projected and expanded to fill the back-aperture of the microscope objective (Olympus UPLSAPO, 1.4 NA, 100×, f = 1.8 mm,

*D*= 5 mm, oil immersion), which highly focuses the beam on the sample enclosed into a chamber made by attaching a glass coverslip (thickness 0.17 mm) to a standard microscope slide with double-sided tape. Our trapping setup is an inverted microscope developed by us comprising the same SLM used in the previous experiments, see Methods (section 5.4) for further details. The input laser beam (Laser Quantum, Ventus,

*λ*= 532 nm) was collimated and expanded to illuminate the SLM display. The measured light power at the exit of the microscope objective is about 10 mW.

*μ*m diameter polystyrene sphere, Spherotech Lot. AD01, diluted in distilled water) trapped by the following beams: a ring-vortex of radius

*R*= 10

*μ*m and topological charge

*m*= 30, contained in the focal plane [Fig. 5(a), Media 3] and a 14° tilted one with respect to it [Fig. 5(b), Media 4]; the Archimedean spiral extended along the optical axis [Fig. 5(c), Media 5]; and the Viviani’s curve [Fig. 5(d), Media 6]. The plane

*z*= 0 of the beam trap, see Figs. 5(a)–5(c), is 10

*μ*m above the chamber bottom (glass coverslip). We underline that the trapping region obtained using oil immersion objectives is limited in deep due to the presence of spherical aberrations caused by the mismatch in the refractive indices of the immersion and specimen media, as reported elsewhere [15

15. S. N. S. Reihani and L. B. Oddershede, “Optimizing immersion media refractive index improves optical trapping by compensating spherical aberrations,” Opt.
Lett. **32**, 1998–2000 (2007). [CrossRef] [PubMed]

*μ*m. To deal with this condition, in the case of Viviani’s light curve (which is wider extended in the axial direction than the other considered 3D curves) the plane

*z*= 0 is set at 5

*μ*m in deep. Thus one loop

*touches*the bottom surface of the chamber.

*z*> 0 to

*z*< 0, see Fig. 5(b) and ( Media 4). It is due to the phase gradient forces exerted over the particle along the curve. These results are in good agreement with the previously reported ones in [11

**19**, 5833–5838 (2011). [CrossRef] [PubMed]

*σ*from 1 to −1, yielding clockwise and anticlockwise rotation, respectively. In the case of the Archimedean spiral the movement of the particles clearly reveals a spiral shape Fig. 5(c). For

*σ*= −1 they perform a downstream motion towards the tail of the spiral, whereas for

*σ*= +1 the particles move retrograde upstream along the spiral towards its center as observed in Fig. 5(c), see ( Media 5).

*xy*–plane is also drawn (white dashed circle) in Fig. 5(d). Specifically, the particle moves in clockwise direction along the curve according to the phase gradient. This movement starts in the position 1 in the bottom loop as depicted in Fig. 5(d), where the trapping is affected by the bottom surface of the chamber. Nevertheless, the phase gradient forces are strong enough to drive the particle to positions 2 and 3. When the particle arrives position 3, near to the loop intersection point, it keeps moving along the second loop. This loop is far enough from the glass surface yielding efficient trapping, and thus the particle motion is stable along the loop (see Media 6). The trajectory of the particle projected in the

*xy*–plane, between position 3 and 6, coincides with the curve projection (white dashed circle), as expected.

## 4. Discussion

16. W. T. M. Irvine and D. Bouwmeester, “Linked and knotted beams of light,” Nat. Physics **4**, 716–720 (2008). [CrossRef]

*H*(

**r**|

**c**

_{3}), Eq. (6), its properties are preserved under focusing in the nonparaxial regime yielding the target 3D light curve. Indeed, the high intensity gradients together with the phase gradients defined along the light curve permit its application as an optical trap. It has been experimentally demonstrated that micron-sized particles confined in this type of trap perform forward and backward motion to the light source according to the phase gradient. Nevertheless, further research is required for the generation of beams with a tunable phase gradient defined along the curve. This can be achieved by setting the parameter

*σ*in Eq. (3) as a function of the curve coordinates:

*σ*=

*σ*(

**c**

_{3}(

*t*)).

*H*(

**r**|

**c**

_{3}) can be understood as a Bessel-like beam extended in different geometries. For instance,

*H*(

**r**|

**c**

_{3}) is a helical Bessel beam for the case of a ring curve. It is well-known that the Bessel beam shows self-reconstructing behavior and preserve its properties in the nonparaxial regime, which is useful in optical tweezers [17

17. V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nat. **419**, 145–147 (2002). [CrossRef]

18. F. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics **4**, 780–785 (2010). [CrossRef]

## 5. Methods

### 5.1. Holographic technique used for beam generation

*A*(

**r**) =

*a*(

**r**)exp [

*iϕ*(

**r**)], where

*a*(

**r**) and

*ϕ*(

**r**) are respectively the amplitude and phase distributions, can be encoded as a phase computer generated hologram (CGH) with the transmittance function The encoding phase

*ψ*[

*a*(

**r**),

*ϕ*(

**r**)] can be found representing Ψ(

**r**) as a Fourier expansion in the domain of

*ϕ*(

**r**). For instance, in [19

19. J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, “Encoding amplitude information onto phase-only filters,” Appl. Opt. **38**, 5004–5013 (1999). [CrossRef]

*A*(

**r**) is recovered from the first-order term of such Fourier expansion by using

*ψ*[

*a*(

**r**),

*ϕ*(

**r**)] =

*g*[

*a*(

**r**)]

*ϕ*(

**r**), if the condition sinc [1 −

*g*[

*a*(

**r**)]] =

*a*(

**r**) is fulfilled for every value of the normalized amplitude

*a*(

**r**) ∈ [0, 1], where sinc (

*t*) = sin (

*πt*)/

*πt*. To isolate the encoded field

*A*(

**r**) from other terms of the Fourier expansion, a fixed carrier phase

*φ*= 2

_{c}*π*(

*u*

_{0}

*x*+

*v*

_{0}

*y*) is added to the latter modulation function:

*ψ*[

*a*(

**r**),

*ϕ*(

**r**) +

*φ*] mod 2

_{c}*π*. The spatial filtering, that is performed using the 4-f telescopic system comprising the trapping setup, permits selecting the corresponding term according to the value of spatial frequencies (

*u*

_{0},

*v*

_{0}). Further technical details are reported in [14

14. J. A. Rodrigo, T. Alieva, A. Cámara, O. Martínez-Matos, P. Cheben, and M. L. Calvo, “Characterization of holographically generated beams via phase retrieval based on Wigner distribution projections,” Opt.
Express **19**, 6064–6077 (2011). [CrossRef] [PubMed]

### 5.2. Closed and open parametric curves

**c**

_{3}(

*t*) = (

*x*

_{0}(

*t*),

*y*

_{0}(

*t*),

*z*

_{0}(

*t*)), where

*t*∈ [0,

*T*], used for generation of the beams

*H*(

**r**|

**c**

_{3}). For all cases

*T*= 2

*π*except the Archimedean spiral, where

*T*= 1. The tilted ring with scaling factor

*s*≠ 1 yields a tilted ellipse. Note that the 2D curves,

**c**

_{2}(

*t*), presented in Fig. 2 are obtained using

*z*

_{0}(

*t*) = 0.

### 5.3. Measurement of the beam propagation and its volumetric reconstruction

*H*(

**r**|

**c**

_{3}) exp [i

*πz*(

*x*

^{2}+

*y*

^{2})/

*λ*f

^{2}] as the hologram given by Eq. (7), for each propagation distance

*z*. The latter approach avoids misalignment in the measurement due to the beam propagation. In our case 251 slices were measured to obtain the volumetric reconstruction.

### 5.4. Optical trapping setup

*λ*= 532 nm) is 6× expanded to illuminate the CGH. The resulting beam,

*H*(

**r**|

**c**

_{3}), is projected to fill the back-aperture of the microscope objective (MO, Super apochromatic, Olympus, 1.4 NA, 100×), using a 1.5× Keplerian telescope comprising the relay lenses RL1 (focal length of 10 cm) and RL2 (focal length of 15 cm). A LED light source illuminates the sample, which is imaged on the CMOS color camera (Thorlabs, DCC1240C) using the tube lens (TL, focal length of 15 cm) in combination with a dichroic-mirror filter (DF) that prevents backscattered laser light from saturating the camera. A 10× Nikon objective is used as the condenser.

## Appendix

*x*

_{0}(

*t*) =

*R*

_{0}cos

*t*and

*y*

_{0}(

*t*) =

*R*

_{0}sin

*t*, the beam

*G*(

**r**|

**c**

_{2}) given by Eq. (4) corresponds to a helical Bessel one, as discussed in section 2.1. Let us consider polar coordinates

*x*=

*r*cos

*θ*and

*y*=

*r*sin

*θ*. It is easy to find that the terms comprising Φ (

**r**,

*t*) are:

## Acknowledgments

*Ministerio de Economía y Competitividad*is acknowledged for the project TEC2011-23629.

## References

1. | M. A. Seldowitz, J. P. Allebach, and D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. |

2. | V. A. Soifer, ed., |

3. | G. Whyte and J. Courtial, “Experimental demonstration of holographic three-dimensional light shaping using a Gerchberg–Saxton algorithm,” New J.
Phys. |

4. | T. D. Gerke and R. Piestun, “Aperiodic volume optics,” Nat. Photonics |

5. | Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. |

6. | M. Woerdemann, C. Alpmann, M. Esseling, and C. Denz, “Advanced optical trapping by complex beam shaping,” Laser Photonics Rev. 1–16 (2012). [CrossRef] |

7. | M. Padgett and L. Allen, “Light with a twist in its tail,” Contemp. Phys. |

8. | K. Ladavac and D. Grier, “Microoptomechanical pumps assembled and driven by holographic optical vortex arrays,” Opt. Express |

9. | A. Jesacher, S. Fürhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte, “Holographic optical tweezers for object manipulations at an air-liquid surface,” Opt. Express |

10. | Y. Roichman and D. G. Grier, “Three-dimensional holographic ring traps,” Proc. SPIE |

11. | E. R. Shanblatt and D. G. Grier, “Extended and knotted optical traps in three dimensions,” Opt. Express |

12. | S.-H. Lee, Y. Roichman, and D. G. Grier, “Optical solenoid beams,” Opt. Express |

13. | E. G. Abramochkin and V. G. Volostnikov, “Spiral light beams,” Physics-Uspekhi |

14. | J. A. Rodrigo, T. Alieva, A. Cámara, O. Martínez-Matos, P. Cheben, and M. L. Calvo, “Characterization of holographically generated beams via phase retrieval based on Wigner distribution projections,” Opt.
Express |

15. | S. N. S. Reihani and L. B. Oddershede, “Optimizing immersion media refractive index improves optical trapping by compensating spherical aberrations,” Opt.
Lett. |

16. | W. T. M. Irvine and D. Bouwmeester, “Linked and knotted beams of light,” Nat. Physics |

17. | V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nat. |

18. | F. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics |

19. | J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, “Encoding amplitude information onto phase-only filters,” Appl. Opt. |

**OCIS Codes**

(090.1760) Holography : Computer holography

(140.3300) Lasers and laser optics : Laser beam shaping

(140.7010) Lasers and laser optics : Laser trapping

(090.1995) Holography : Digital holography

**ToC Category:**

Holography

**History**

Original Manuscript: July 2, 2013

Revised Manuscript: August 8, 2013

Manuscript Accepted: August 8, 2013

Published: August 26, 2013

**Virtual Issues**

Vol. 8, Iss. 10 *Virtual Journal for Biomedical Optics*

**Citation**

José A. Rodrigo, Tatiana Alieva, Eugeny Abramochkin, and Izan Castro, "Shaping of light beams along curves in three dimensions," Opt. Express **21**, 20544-20555 (2013)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-21-18-20544

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### References

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