## Periodic and quasi-periodic non-diffracting wave fields generated by superposition of multiple Bessel beams

Optics Express, Vol. 15, Issue 25, pp. 16748-16753 (2007)

http://dx.doi.org/10.1364/OE.15.016748

Acrobat PDF (188 KB)

### Abstract

We discuss a computer generated hologram whose transmittance is defined in terms of the Jacobi-Anger identity. If the hologram is implemented with a continuous phase spatial light modulator it generates integer-order non-diffracting Bessel beams, with a common asymptotic radial frequency, at separated propagation axes. On the other hand, when the hologram is implemented with a low-resolution pixelated phase modulator, it is possible to generate multiple Bessel beams with a common propagation axis. We employ this superposition of multiple Bessel beams to generate non-diffracting periodic and quasi-periodic wave fields.

© 2007 Optical Society of America

## 1. Introduction

1. J. Durnin, “Exact solutions for non-diffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A **4**, 651–654 (1986). [CrossRef]

6. R. P. Macdonal, S. A. Boothroyd, T. Okamoto, J. Chrostowoski, and B. A. Syrett, “Interboard optical data distribution by Bessel beam shadowing,” Opt. Commun. **122**, 169–177 (1996). [CrossRef]

7. J. H. McLeod, “The axicon: a new type of optical element,” J. Opt. Soc. Am. **44**,592–597 (1954). [CrossRef]

8. J. Durnin, J. J. Miceli Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. **58**, 1499–1501 (1987). [CrossRef] [PubMed]

9. S. H. Tao, X-C Yuan, and B. S. Ahluwalia, “The generation of an array of nondiffracting beams by a single composite computer generated hologram,” J. Opt. A: Pure Appl. Opt. **7**, 40–46 (2005). [CrossRef]

12. V. Arrizón, U. Ruiz, R. Carrada, and L. A. González, “Pixelated phase computer holograms for the accurate encoding of scalar complex fields,” J. Opt. Soc. Am. A **24**, 3500–3507(2007). [CrossRef]

13. Z. Bouchal and J. Courtial, “The connection of singular and nondiffracting optics” J. Opt. A: Pure Appl. Opt. **6**, S184–8, 2004. [CrossRef]

14. R. Grunwald, U. Griebner, F. Tschirschwitz, E. T. J. Nibbering, T. Elsaesser, V. Kebbel, H-J Hartmann, and W. Jüptner, “Generation of femtosecond Bessel beams with microaxicon arrays” Opt. Lett. **25**, 981–983, 2000. [CrossRef]

15. V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, “Light fields decomposition in angular harmonics by means of diffractive optics,” J. Mod. Opt. **45**, 1495–1506 (1998). [CrossRef]

9. S. H. Tao, X-C Yuan, and B. S. Ahluwalia, “The generation of an array of nondiffracting beams by a single composite computer generated hologram,” J. Opt. A: Pure Appl. Opt. **7**, 40–46 (2005). [CrossRef]

## 2. Hologram defined in terms of the Jacobi-Anger identity

*q*-

*th*order non-diffracting Bessel beam can be expressed as

*J*denotes the Bessel function of integer order

_{q}*q*, (

*r*,

*θ*) represent the radial and azimuthal polar coordinates, respectively, and

*ρ*

_{0}is the asymptotic radial frequency of the beam. The explicit dependence of the Bessel function

*b*(

_{q}*x*,

*y*) on the rectangular coordinates (

*x*,

*y*), is obtained by means of the relations

*r*=(

*x*

^{2}+

*y*

^{2})

^{1/2}and

*θ*=

*arctan*(

*y*/

*x*). According to the Jacobi-Anger identity [16] the sum of all the non-diffracting Bessel beams is given by

*y*=

*r*sin(

*θ*), this function is identified as the tilted plane wave exp[

*i2πρ*]. Therefore, the Bessel beams can not be physically isolated (by spatial frequency filtering) from the Fourier spectrum of this phase modulation, which is given by an off-axis Dirac delta.

_{0}y*π*(

*u*+

_{0}x*v*), with spatial frequencies (

_{0}y*u*,

_{0}*v*), to the coordinate

_{0}*θ*of the phase function in Eq. (2). Performing this modification, the hologram transmittance is transformed into

*b*(

_{q}*x,y*) is enabled by the linear phase modulation

*exp*[

*i2π*(

*qu*+

_{0}x*qv*)]. Indeed, the Fourier spectrum of the hologram

_{0}y*h*(

*x,y*) is given by

*B*(

_{q}*u,v*) is the Fourier transform of the Bessel beam

*b*(

_{q}*x,y*), and (

*u,v*) denote the spatial frequency coordinates, respectively associated to the spatial variables (

*x,y*). Thus, the function in Eq. (3) represents a phase CGH that encodes the complete set of non-diffracting Bessel beams. The encoded beams occupy different regions at the hologram Fourier spectrum domain. Specifically, the spectrum of the q-th order beam is centered at the spatial frequencies (

*qu*,

_{0}*qv*). The Bessel beams encoded by the CGH can be spatially separated if at least one of the carrier frequencies (

_{0}*u*,

_{0}*v*) is greater than 2ρ

_{0}_{0}.

*h*(

*x,y*) is implemented with a low-resolution pixelated SLM, its Fourier spectrum is formed by laterally shifted replicas of the function

*H*(

*u,v*). If the CGH transmittance implemented with the pixelated SLM is under-sampled, these spectrum replicas show some degree of overlapping. We prove below that an extreme overlapping of the replicas of the spectrum function

*H*(

*u,v*), may be applied for the generation of multiple Bessel beams at a common propagation axis.

*b*. Under such conditions, the Fourier spectrum of the pixelated CGH is given by

*u*=1/

*δx*is the SLM bandwidth,

*H*(

*u,v*) is the Fourier spectrum of the continuous CGH, given by Eq. (5), and

*E*(

*u,v*)=

*b*

^{2}

*sinc*(

*bu*)

*sinc*(

*bv*) is the Fourier transform of the square pixel window. In the expression for

*E*(

*u,v*) we employed the definition

*sinc*(

*α*)≡(

*πα*)

^{-1}

*sin*(

*πα*). A strong under-sampling of the CGH transmittance propitiates a significant overlapping among the different terms

*H*(

*u-n*Δ

*u,v-m*Δ

*u*) in the CGH spectrum. We can take advantage of this overlapping to obtain multiple Bessel beams propagating at a common axis. In particular, these beams are generated if the CGH carrier frequencies are given as

*u*

_{0}=

*v*

_{0}=Δ

*u*/

*Q*, where

*Q*is a positive integer (greater than 1). In this case it is straightforward to prove that the beams spectra

*B*

_{p}_{+jQ}(

*u,v*), which are specified by two fixed integer indices

*p*and

*Q*and a variable index

*j*, appear centered at the spatial frequency coordinates (

*pu*,

_{0}*pu*). The superposition of these spectra terms is expressed as

_{0}*S*(

_{pQ}*u,v*) can be spatially isolated, with an appropriate pupil, from other field contributions that are present at the CGH spectrum. Performing this spatial filtering and neglecting the influence of the factor

*E*(

*u,v*), one obtains the non-diffracting wave field

*S*(

_{pQ}*u,v*). The field

*s*(

_{pQ}*x,y*) generated with the above procedure is formed by the infinite set of non-diffracting Bessel beams of orders

*p*+

*jQ*. This set, which is characterized by the fixed integer indices

*p*and

*Q*, includes the beams of orders

*p*,

*p*±

*Q*,

*p*±

*2Q*, and so on. Different sets of Bessel beams

*s*(

_{pQ}*x,y*), corresponding to different values of the index

*p*, are generated by the CGH with carrier frequencies

*u*

_{0}=

*v*

_{0}=Δ

*u*/

*Q*.

## 3. Computational implementation of under-sampled holograms

*u*

_{0}=

*v*

_{0}=Δ

*u*/

*Q*, with

*Q*=6, to encode Bessel beams of radial frequency

*ρ*

_{0}=

*u*

_{0}/

*4*. In addition the CGH limiting aperture is assumed to be a circle of diameter

*D*=

*6*/

*ρ*. For this combination of parameters the diameter of the CGH support is equivalent to 144 pixels of the phase SLM. The phase distribution of the computed CGH, depicted in Fig. 1(a), shows a periodicity that is originated in a severe under-sampling of the CGH transmittance. The normalized intensity of the CGH Fourier spectrum is partially shown in Fig. 1(b). In this figure the separated arrays of spots correspond to 3 Fourier spectra

_{0}*S*(

_{p6}*u,v*) with

*p*=0, 1, 2 (from top-left to bottom-right). It is remarkable that each Fourier spectrum function

*S*(

_{p6}*u,v*) is formed by 6 symmetrically arranged spots.

*S*(

_{pQ}*u,v*) is formed by

*Q*symmetric spots, placed at the corners of a regular polygon of

*Q*sides. This fact is consequence of a general significant result, namely that the infinite series of Bessel beams in Eq. (8) is equivalent to the sum of

*Q*plane waves. This equivalence is explicitly given by

*2π*/

*Q*. A prove of this relation is obtained by developing and reducing the exponential at the right side of Eq. (9), by means of the Jacobi-Anger identity. The obtained formula represents an interesting generalization of the Jacobi-Anger identity. Indeed, it is straightforward to show that Eq. (9) reduces to Eq. (2) assuming

*Q*=

*1*and

*p*=

*0*. It must be noted that the plane wave corresponding to the index

*m*in Eq. (9) is modulated by a phase shift

*exp*(

*ipm*Δ). This phase shift provides a topological charge of order

*p*to the array of spots in the spectrum function

*S*(

_{pQ}*u,v*).

*s*(

_{06}*x,y*) and

*s*(

_{16}*x,y*) are obtained performing the Fourier transformation of the hexagonal spectra spots arrays

*S*(

_{06}*u,v*) and

*S*(

_{16}*u,v*), respectively. Each one of these spectra functions is isolated by spatial filtering in the CGH Fourier domain. The computed intensities and phases of the wave fields

*s*(

_{06}*x,y*) and

*s*(

_{16}*x,y*) are shown in Fig. 2. The field

*s*(

_{06}*x,y*), which corresponds to the sum of 6 plane waves without phase shifts [Eq. (9)], presents the binary phase of a real function [Fig. 2(b)]. On the other hand, the phase shifts for the 6 plane waves that form the field

*s*(

_{16}*x,y*) increases monotonically with the plane wave index

*m*. In this case a remarkable result is the field intensity distribution with the form of a honeycomb [Fig. 2(c)]. In addition, each cell of this array is centered at a perfect vortex of topological charge 1 [Fig. 2(d)], originated in the topological charge of the array of spots in the spectrum function

*S*(

_{16}*u,v*). For the displayed phase of the field

*s*(

_{16}*x,y*) we have omitted the linear phase factor

*exp*[

*i2πu*(

_{0}*x*+

*y*)].

*s*(

_{pQ}*x,y*), with indices

*Q*=

*2, 3, 4*, and 6. For other values of

*Q*, the field

*s*(

_{pQ}*x,y*) presents a quasi-periodic structure.

## 4. Experimental implementation of holograms

*π*radians is obtained (with approximately 70 unequal steps) illuminating the SLM with a He-Ne laser beam (633 nm). We employed this SLM to implement the CGH described in the first numerical simulation presented in section 3, and experimentally generated the non-diffracting field

*s*(

_{16}*x,y*). The phase distribution of this CGH was shown in Fig. 1(a). The experimentally recorded intensities of the Fourier spectrum

*S*(

_{16}*u,v*), and its corresponding periodic field

*s*(

_{16}*x,y*), are shown in Fig. 3. We also implemented other CGHs for generation of quasi-periodic nondiffracting fields

*s*(

_{1Q}*x,y*) with

*Q*=5 and

*Q*=12. The parameters that we employed for the second CGH [that generates

*s*(

_{15}*x,y*)] are

*u*

_{0}=

*v*

_{0}=Δ

*u*/5,

*ρ*

_{0}=

*u*

_{0}/

*4*, and

*D*=

*12*/

*ρ*(

_{0}*D*is the CGH pupil diameter, equivalent to 240 SLM pixels). The case with

*Q*=12 was implemented with parameters

*u*

_{0}=

*v*

_{0}=Δ

*u*/

*12*,

*ρ*=

_{0}*u*

_{0}/

*3*, and

*D*=

*14*/

*ρ*. The intensities of these quasi-periodic fields, obtained experimentally, are shown in Fig. 4.

_{0}## 5. Concluding remarks

*s*(

_{pQ}*x,y*), equivalent to the Q-plane waves specified above, generates special types of non-diffracting fields. We have established, in numerical simulations, that the non-diffracting fields

*s*(

_{pQ}*x,y*) are transversally periodic (when

*Q*=

*2, 3, 4*, and 6) or quasi-periodic (for other values of

*Q*). This kind of fields can be used to create dynamic photonic lattices in nonlinear media [17

17. Z. Chen, H. Martin, A. Bezryadina, D. Neshev, Y. S. Kivshar, and D. N. Christodoulides, “Experiments on Gaussian beams and vortices in optically induced photonic lattices,” J. Opt. Soc. Am. B **22**, 1395–1405 (2005). [CrossRef]

18. A. S. Desyatnikov, Y. S. Kivshar, V. S. Shchesnovich, S. B. Cavalcanti, and J. M. Hickmann, “Resonant Zener tunneling in two-dimensional periodic photonic lattices,” Opt. Lett. **32**, 325–327 (2007). [CrossRef] [PubMed]

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

*s*(

_{16}*x,y*), and the quasi-periodic fields

*s*(

_{1Q}*x,y*) with

*Q*=5, 12. The experimentally generated fields (Figs. 3 and 4) reproduced with remarkable high fidelity the features of the numerically generated fields. This high fidelity is illustrated in the case of the field

*s*(

_{16}*x,y*) by comparison of Fig. 2(c) with Fig. 3(b). The high fidelity of the generated fields is propitiated by a unique feature of the CGH, namely that the high order diffraction field contributions transmitted by this hologram, form part of the encoded fields. This is not the case of conventional CGHs, for which the high order diffraction contributions represent sources of noise.

## Acknowledgment

## References and links

1. | J. Durnin, “Exact solutions for non-diffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A |

2. | S. Chavez-Cerda. “A new approach to Bessel beams,” J. Mod. Opt. , |

3. | J. Lu and J. F. Greenleaf, “Diffraction-limited beams and their applications for ultrasonic imaging and tissue characterization,” Proc. SPIE |

4. | R. M. Herman and T. A. Wiggins, “Productions and uses of diffraction less beams,” J. Opt. Soc. Am. A |

5. | V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature |

6. | R. P. Macdonal, S. A. Boothroyd, T. Okamoto, J. Chrostowoski, and B. A. Syrett, “Interboard optical data distribution by Bessel beam shadowing,” Opt. Commun. |

7. | J. H. McLeod, “The axicon: a new type of optical element,” J. Opt. Soc. Am. |

8. | J. Durnin, J. J. Miceli Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. |

9. | S. H. Tao, X-C Yuan, and B. S. Ahluwalia, “The generation of an array of nondiffracting beams by a single composite computer generated hologram,” J. Opt. A: Pure Appl. Opt. |

10. | A. Vasara, J. Turunen, and A. T. Friberg, “Realization of general nondiffracting beams with computergenerated holograms,” J. Opt. Soc. Am. A |

11. | V. Arrizón, “Optimum on-axis computer-generated hologram encoded into low-resolution phasemodulation devices,” Opt. Lett. |

12. | V. Arrizón, U. Ruiz, R. Carrada, and L. A. González, “Pixelated phase computer holograms for the accurate encoding of scalar complex fields,” J. Opt. Soc. Am. A |

13. | Z. Bouchal and J. Courtial, “The connection of singular and nondiffracting optics” J. Opt. A: Pure Appl. Opt. |

14. | R. Grunwald, U. Griebner, F. Tschirschwitz, E. T. J. Nibbering, T. Elsaesser, V. Kebbel, H-J Hartmann, and W. Jüptner, “Generation of femtosecond Bessel beams with microaxicon arrays” Opt. Lett. |

15. | V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, “Light fields decomposition in angular harmonics by means of diffractive optics,” J. Mod. Opt. |

16. | G. N. Watson, |

17. | Z. Chen, H. Martin, A. Bezryadina, D. Neshev, Y. S. Kivshar, and D. N. Christodoulides, “Experiments on Gaussian beams and vortices in optically induced photonic lattices,” J. Opt. Soc. Am. B |

18. | A. S. Desyatnikov, Y. S. Kivshar, V. S. Shchesnovich, S. B. Cavalcanti, and J. M. Hickmann, “Resonant Zener tunneling in two-dimensional periodic photonic lattices,” Opt. Lett. |

19. | G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express |

**OCIS Codes**

(070.0070) Fourier optics and signal processing : Fourier optics and signal processing

(090.1760) Holography : Computer holography

(230.6120) Optical devices : Spatial light modulators

(350.4855) Other areas of optics : Optical tweezers or optical manipulation

(080.4865) Geometric optics : Optical vortices

**ToC Category:**

Holography

**History**

Original Manuscript: October 9, 2007

Revised Manuscript: November 5, 2007

Manuscript Accepted: November 6, 2007

Published: December 3, 2007

**Virtual Issues**

Vol. 3, Iss. 1 *Virtual Journal for Biomedical Optics*

**Citation**

Victor Arrizón, Sabino Chavez-Cerda, Ulises Ruiz, and Rosibel Carrada, "Periodic and quasi-periodic non-diffracting wave fields generated by superposition of multiple Bessel beams," Opt. Express **15**, 16748-16753 (2007)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-25-16748

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### References

- J. Durnin, " Exact solutions for nondiffracting beams. I. The scalar theory," J. Opt. Soc. Am. A 4, 651-654 (1986). [CrossRef]
- S. Chavez-Cerda. "A new approach to Bessel beams," J. Mod. Opt., 46, 923-930 (1999).
- J. Lu and J. F. Greenleaf, "Diffraction-limited beams and their applications for ultrasonic imaging and tissue characterization," Proc. SPIE 1733, 92-119 (1992). [CrossRef]
- R. M. Herman and T. A. Wiggins, "Productions and uses of diffraction less beams," J. Opt. Soc. Am. A 8, 932- 942 (1999). [CrossRef]
- V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, K. Dholakia, "Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam," Nature 419, 145-147 (2002). [CrossRef] [PubMed]
- R. P. Macdonal, S. A. Boothroyd, T. Okamoto, J. Chrostowoski, B. A. Syrett, "Interboard optical data distribution by Bessel beam shadowing," Opt. Commun. 122, 169-177 (1996). [CrossRef]
- J. H. McLeod, "The axicon: a new type of optical element," J. Opt. Soc. Am. 44,592-597 (1954). [CrossRef]
- J. Durnin, J. J. Miceli, Jr., J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987). [CrossRef] [PubMed]
- S. H. Tao, X-C Yuan, B. S. Ahluwalia, "The generation of an array of nondiffracting beams by a single composite computer generated hologram," J. Opt. A: Pure Appl. Opt. 7,40-46 (2005). [CrossRef]
- A. Vasara, J. Turunen, A. T. Friberg, "Realization of general nondiffracting beams with computer-generated holograms," J. Opt. Soc. Am. A 6, 1748-1754 (1989). [CrossRef] [PubMed]
- V. Arrizón, "Optimum on-axis computer-generated hologram encoded into low-resolution phase-modulation devices," Opt. Lett. 28, 2521-253 (2003). [CrossRef] [PubMed]
- V. Arrizón, U. Ruiz, R. Carrada, and L. A. González, "Pixelated phase computer holograms for the accurate encoding of scalar complex fields," J. Opt. Soc. Am. A 24, 3500-3507 (2007). [CrossRef]
- Z. Bouchal, J. Courtial, "The connection of singular and nondiffracting optics" J. Opt. A: Pure Appl. Opt. 6, S184-8, 2004. [CrossRef]
- R. Grunwald, U. Griebner, F. Tschirschwitz, E. T. J. Nibbering, T. Elsaesser, V. Kebbel, H-J Hartmann, W. Jüptner, "Generation of femtosecond Bessel beams with microaxicon arrays" Opt. Lett. 25,981-983, 2000. [CrossRef]
- V. V. Kotlyar, S. N. Khonina, V. A. Soifer, "Light fields decomposition in angular harmonics by means of diffractive optics," J. Mod. Opt. 45, 1495-1506 (1998). [CrossRef]
- G. N. Watson, A treatise on the theory of Bessel functions (Cambridge University Press, 1922), p. 22.
- Z. Chen, H. Martin, A. Bezryadina, D. Neshev, Y. S. Kivshar, D. N. Christodoulides, "Experiments on Gaussian beams and vortices in optically induced photonic lattices," J. Opt. Soc. Am. B 22, 1395-1405 (2005). [CrossRef]
- A. S. Desyatnikov, Y. S. Kivshar, V. S. Shchesnovich, S. B. Cavalcanti, J. M. Hickmann, "Resonant Zener tunneling in two-dimensional periodic photonic lattices," Opt. Lett. 32, 325-327 (2007). [CrossRef] [PubMed]
- G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas'ko, S. Barnett, and S. Franke-Arnold, "Free-space information transfer using light beams carrying orbital angular momentum," Opt. Express 12, 5448-5456 (2004). [CrossRef] [PubMed]

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