## Tunable Bessel light modes: engineering the axial propagation

Optics Express, Vol. 17, Issue 18, pp. 15558-15570 (2009)

http://dx.doi.org/10.1364/OE.17.015558

Acrobat PDF (1529 KB)

### Abstract

Due to their immunity to diffraction, Bessel light modes potentially offer advantages in various applications. However, they do exhibit significant intensity variations along their axial propagation length which hampers their applicability. In this paper we present a technique to generate Bessel beams with a tunable axial intensity within the accessible range of spatial frequencies. The beam may be engineered to have a constant intensity along its propagation length. Finally, we demonstrate how one can form a Bessel beam with a varying propagation constant along its axial extent which results in a tunable scaling of its lateral cross-section.

© 2009 Optical Society of America

## 1. Introduction

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

2. J. Durnin, “Exact solutions for nondifracting beams. I. The scalar theory,” J. Opt. Soc. Am. A **4**, 651–641 (1987).
[CrossRef]

3. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. **99**, 213901 (2007).
[CrossRef]

4. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nature Photonics **2**, 675–678 (2008).
[CrossRef]

5. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. **29**, 44–46 (2004).
[CrossRef] [PubMed]

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

8. V. Karásek, T. Čižmár, O. Brzobohatýmánek, V. Garcés-Chávez, and K. Dholakia, “Long-range one-dimensional longitudinal optical binding,” Phys. Rev. Lett. **101**, 143601 (2008).
[CrossRef] [PubMed]

9. X. Tsampoula, V. Garcés-Chávez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett. **91**, 053902 (2007).
[CrossRef]

10. K.-S. Lee and J. P. Rolland, “Bessel beam spectral-domain high-resolution optical coherence tomography with micro-optic axicon providing extended focusing range,” Opt. Lett. **33**, 1696–1698 (2008).
[CrossRef] [PubMed]

11. P. Johannisson, D. Anderson, M. Lisak, and M. Marklund, “Nonlinear Bessel beams,” Opt. Commun. **222**, 107–115 (2003).
[CrossRef]

12. P. Dufour, M. Piché, Y. D. Koninck, and N. McCarthy, “Two-photon excitation fluorescence microscopy with a high depth of field using an axicon,” Appl. Opt. **45**, 9246–9252 (2006).
[CrossRef] [PubMed]

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

13. Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. **151**, 207–211 (1998).
[CrossRef]

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

15. T. Čižmár, V. Kollárová, X. Tsampoula, F. Gunn-Moore, W. Sibbett, Z. Bouchal, and K. Dholakia, “Generation of multiple Bessel beams for a biophotonics workstation,” Opt. Express **16**, 14024–14035 (2008).
[CrossRef]

## 2. Azimuthally symmetric fields

*U*(

*x*,

*y*) defined in one lateral plane can be used to decompose any optical field into its plane wave components. Each plane wave component is defined by a wave-vector

**k**=[

*k*,

_{x}*k*,

_{y}*k*] in the orthogonal system of coordinates [

_{z}*x*,

*y*,

*z*] and if we assign a zero value to the axial position of the selected plane we can express this decomposition as:

*J*

_{0}is the zero-order Bessel function. The inverse transform to (2) is given by:

*k*is the axial wave-vector component:

_{z}*k*〉. Each of these Bessel beam components then has a specific spatial frequency along the optical axis which contribute to the resulting on-axis field which as we know is a consequence of the interference of the individual modes. Using the inverse transform (3) we obtain for the on-axis field the following:

*k*∊〈0,

_{z}*k*〉.

## 3. Axial intensity of frequently used quasi Bessel beams

## 3.1. Quasi-Bessel beam generated by an annular aperture

*a*and

*b*are given by:

*R*is the radius of the annular aperture,

*d*is the annulus thickness,

*f*is the focal length of the lens as shown in Fig. 1.

## 3.2. Axicon-generated Bessel beam

16. V. Jarutis, R. Paškauskas, and A. Stabinis, “Focusing of Laguerre-Gaussian beams by axicon,” Opt. Commun. **184**, 105–112 (2000).
[CrossRef]

*w*is the half-width of the Gaussian beam and

*α*is the angle under which the light is refracted by the axicon. This, however, is the case where an optimal axicon is used. In reality axicons suffer a number of imperfections, with the oblate tip being the most important one. Such an imperfection causes a strong on-axis oscillations that destroys the overall smooth profile of the propagating Bessel beam [14]. An example of such a field is shown in Fig. 3.

## 4. Shaping of the axial intensity profile of a quasi-Bessel beam

*k*=

_{r}*δ*(

*k*

_{r0}). Let us define a quasi-Bessel beam with an arbitary axial envelope

*A*(

*z*) defined on the optical axis as:

*k*∊〈0,

_{z}*k*〉. This limits the possibility of the axial beam shaping and so we have to circumvent these spectral values in the optical field design.

## 4.1. Bessel beam with an uniform axial intensity

*r*illumination of the axicon in their theoretical study [17

17. M. Honkanen and J. Turunen, “Tandem systems for efficient generation of uniform-axial-intensity Bessel fields,” Opt. Commun. **154**, 368–375 (1998).
[CrossRef]

*A*(

*z*) from the relation (8) be a step function defined by:

*k*

_{r0}. In reality we cannot even use the whole range of possible spatial frequencies

*k*∊〈0,

_{r}*k*〉 because of the limited aperture size of the optics. Equation (4) however reveals which on-axis intensity profile one can reach using the spectrum with such a limited range of the spatial frequencies. Numerical evaluation of (4) for the spectrum presented in the Fig. 4 and for the range of the shown frequencies is seen in Fig. 5. Here we see some on-axis oscillations that are present because of the hard-aperturing of the spatial spectrum. To avoid this we can introduce a Gaussian amplitude envelope around the spatial spectrum centered at the value of

*k*

_{r0}. The resulting on-axis intensity is then shown in Fig. 6. These types of fields have their spectrum concentrated in the vicinity of a ring with the radius of

*k*

_{r0}and have their on-axis intensity uniform over a limited range. From this perspective this would represent the best achievable approximation to the ideal Bessel light mode. It is important to say that the selection of the Gaussian envelope width directly influences the magnitude of the persisting oscillations. The narrower envelope used the smaller are the residual oscillations. The price for this is the reduction of the uniform intensity extent because of rounding the sharp corners of the selected step function (9). This way one has to trade-off these two influences for a particular application. The magnitude of the persisting oscillations gets always smaller towards the central part of the Bessel beam region (see Fig. 5). In our case presented in the Fig. 6 the magnitude of the oscillations in the central region covering 50% of the beam axial extent the ratio between the oscillation and the mean intensity does not exceed the level of 10

^{-7}. If this region is extended to 80% of the beam existence the ratio increases to the value of 10

^{-4}and for 90% we still get less than 10

^{-2}. We believe that these values are more than satisfactory for the majority of applications but it is worth remarking that this evaluation was done for the spatial spectrum enveloped by the Gaussian function and that a different enveloping function might bring a future improvement.

## 4.2. Bessel beams with shaped on-axis intensity and the propagation constant

*A*(

*z*) to yield an arbitrary shape. In several cases the spatial spectrum can be found in an analytical form. For example, this includes the generation of a Bessel beam with either a uniformly increasing or decreasing on-axis intensity or even the axicon-generated Bessel beam with the on-axis intensity described by the approximative formula (6). For other on-axis axial intensity profiles one can find the spatial spectrum by numerical evaluation of the formula (8). For brevity we do not describe all of the possible cases but concentrate upon the example of a Bessel mode with an uniformly growing axial intensity. We present the relevant spatial spectrum in the Fig. 7 and the resulting intensity after the spectrum enveloped by the Gaussian function as in the previous sub-section (Fig. 8).

*A*(

*z*) and

*k*

_{z0}(

*z*) and calculate the corresponding spatial spectrum evaluating the equation (8). Several examples of this will be shown in the next section together with their experimentally realized counterparts.

## 5. Experimental realization of the novel quasi-Bessel beam types

## 5.1. The hologram design

18. M. Pasienski and B. DeMarco, “A high-accuracy algorithm for designing arbitrary holographic atom traps,” Opt. Express **16**, 2176–2190 (2008).
[CrossRef] [PubMed]

18. M. Pasienski and B. DeMarco, “A high-accuracy algorithm for designing arbitrary holographic atom traps,” Opt. Express **16**, 2176–2190 (2008).
[CrossRef] [PubMed]

## 5.2. Experimental procedure

*f*=400mm) placed at the distance of one focal length from the SLM. In the front focal plane we obtained the pseudo-Bessel beam that was separated from the unused light by a pinhole. The size of the pinhole was set so it corresponded to the restricted area in the hologram generating algorithm. This beam was then demagnified by a telescope consisting of

*f*=250mm and

*f*=50mm lenses to reduce the quasi-Bessel beam to the operating range of our motorized actuator (Newport CMA-12CC). The resulting field was imaged onto a CCD camera chip (Basler piA640-210gm) by a micro-scope objective (20x NA=0.4). The imaging system of the camera and the objective was placed on a positioning stage controlled by the motorized actuator. The quasi-Bessel beam intensity was then captured at 1200 axial positions along the axial distance of 10mm. The experimental results are summarized in the following section.

## 6. Experimental results

## 7. Conclusions

17. M. Honkanen and J. Turunen, “Tandem systems for efficient generation of uniform-axial-intensity Bessel fields,” Opt. Commun. **154**, 368–375 (1998).
[CrossRef]

15. T. Čižmár, V. Kollárová, X. Tsampoula, F. Gunn-Moore, W. Sibbett, Z. Bouchal, and K. Dholakia, “Generation of multiple Bessel beams for a biophotonics workstation,” Opt. Express **16**, 14024–14035 (2008).
[CrossRef]

20. D. Palima, C. A. Alonzo, P. J. Rodrigo, and J. Glückstad, “Generalized phase contrast matched to Gaussian illumination,” Opt. Express **15**, 11971–11977 (2007).
[CrossRef] [PubMed]

## Acknowledgments

## References and links

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

2. | J. Durnin, “Exact solutions for nondifracting beams. I. The scalar theory,” J. Opt. Soc. Am. A |

3. | G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. |

4. | J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nature Photonics |

5. | M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. |

6. | T. Čižmár, V. Garcés-Chávez, K. Dholakia, and P. Zemánek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. |

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

8. | V. Karásek, T. Čižmár, O. Brzobohatýmánek, V. Garcés-Chávez, and K. Dholakia, “Long-range one-dimensional longitudinal optical binding,” Phys. Rev. Lett. |

9. | X. Tsampoula, V. Garcés-Chávez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett. |

10. | K.-S. Lee and J. P. Rolland, “Bessel beam spectral-domain high-resolution optical coherence tomography with micro-optic axicon providing extended focusing range,” Opt. Lett. |

11. | P. Johannisson, D. Anderson, M. Lisak, and M. Marklund, “Nonlinear Bessel beams,” Opt. Commun. |

12. | P. Dufour, M. Piché, Y. D. Koninck, and N. McCarthy, “Two-photon excitation fluorescence microscopy with a high depth of field using an axicon,” Appl. Opt. |

13. | Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. |

14. | O. Brzobohatß ižmár and P. Zemánek, “High quality quasi-Bessel beam generated by round-tip axicon,” Opt. Express |

15. | T. Čižmár, V. Kollárová, X. Tsampoula, F. Gunn-Moore, W. Sibbett, Z. Bouchal, and K. Dholakia, “Generation of multiple Bessel beams for a biophotonics workstation,” Opt. Express |

16. | V. Jarutis, R. Paškauskas, and A. Stabinis, “Focusing of Laguerre-Gaussian beams by axicon,” Opt. Commun. |

17. | M. Honkanen and J. Turunen, “Tandem systems for efficient generation of uniform-axial-intensity Bessel fields,” Opt. Commun. |

18. | M. Pasienski and B. DeMarco, “A high-accuracy algorithm for designing arbitrary holographic atom traps,” Opt. Express |

19. | R. Gerchberg and W. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik |

20. | D. Palima, C. A. Alonzo, P. J. Rodrigo, and J. Glückstad, “Generalized phase contrast matched to Gaussian illumination,” Opt. Express |

**OCIS Codes**

(140.3300) Lasers and laser optics : Laser beam shaping

(090.1995) Holography : Digital holography

(070.6120) Fourier optics and signal processing : Spatial light modulators

**ToC Category:**

Physical Optics

**History**

Original Manuscript: July 7, 2009

Revised Manuscript: August 17, 2009

Manuscript Accepted: August 17, 2009

Published: August 18, 2009

**Virtual Issues**

Vol. 4, Iss. 10 *Virtual Journal for Biomedical Optics*

**Citation**

Tomáš Čižmár and Kishan Dholakia, "Tunable Bessel light modes: engineering
the axial propagation," Opt. Express **17**, 15558-15570 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-18-15558

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### References

- J. Durnin, J. J. Miceli, and J. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987). [CrossRef] [PubMed]
- J. Durnin, "Exact solutions for nondifracting beams. I. The scalar theory," J. Opt. Soc. Am. A 4, 651-641 (1987). [CrossRef]
- G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Observation of Accelerating Airy Beams," Phys. Rev. Lett. 99, 213901 (2007). [CrossRef]
- J. Baumgartl, M. Mazilu, and K. Dholakia, "Optically mediated particle clearing using Airy wavepackets," Nature Photonics 2, 675-678 (2008). [CrossRef]
- M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, "Parabolic nondiffracting optical wave fields," Opt. Lett. 29, 44-46 (2004). [CrossRef] [PubMed]
- T. Čižmár, V. Garcés-Chávez, K. Dholakia, and P. Zemánek, "Optical conveyor belt for delivery of submicron objects," Appl. Phys. Lett. 86, 174,101:1-3 (2005).
- V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, "Simultaneous micromanipulation in multiple planes using a self- reconstructing light beam," Nature 419, 145-147 (2002). [CrossRef] [PubMed]
- V. Karásek T. Čižmár, O. Brzobohatý, P. Zemánek, V. Garcés-Chávez, and K . Dholakia, "Long-range onedimensional longitudinal optical binding," Phys. Rev. Lett. 101, 143601 (2008). [CrossRef] [PubMed]
- X. Tsampoula, V. Garcés-Chávez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, "Femtosecond cellular transfection using a nondiffracting light beam," Appl. Phys. Lett. 91, 053902 (2007). [CrossRef]
- K.-S. Lee and J. P. Rolland, "Bessel beam spectral-domain high-resolution optical coherence tomography with micro-optic axicon providing extended focusing range," Opt. Lett. 33, 1696-1698 (2008). [CrossRef] [PubMed]
- P. Johannisson, D. Anderson, M. Lisak, and M. Marklund, "Nonlinear Bessel beams," Opt. Commun. 222, 107-115 (2003). [CrossRef]
- P. Dufour, M. Piché, Y. D. Koninck, and N. McCarthy, "Two-photon excitation fluorescence microscopy with a high depth of field using an axicon," Appl. Opt. 45, 9246-9252 (2006). [CrossRef] [PubMed]
- Z. Bouchal, J. Wagner, and M. Chlup, "Self-reconstruction of a distorted nondiffracting beam," Opt. Commun. 151, 207-211 (1998). [CrossRef]
- O. Brzobohatý, T. Čižmár, and P. Zemánek, "High quality quasi-Bessel beam generated by round-tip axicon," Opt. Express 16, 12688-12700 (2008).
- T. Čižmár, V. Kollárová, X. Tsampoula, F. Gunn-Moore, W. Sibbett, Z. Bouchal, and K. Dholakia, "Generation of multiple Bessel beams for a biophotonics workstation," Opt. Express 16, 14024-14035 (2008). [CrossRef]
- V. Jarutis, R. Paškauskas, and A. Stabinis, "Focusing of Laguerre-Gaussian beams by axicon," Opt. Commun. 184, 105-112 (2000). [CrossRef]
- M. Honkanen and J. Turunen, "Tandem systems for efficient generation of uniform-axial-intensity Bessel fields," Opt. Commun. 154, 368 - 375 (1998). [CrossRef]
- M. Pasienski and B. DeMarco, "A high-accuracy algorithm for designing arbitrary holographic atom traps," Opt. Express 16, 2176-2190 (2008). [CrossRef] [PubMed]
- R. Gerchberg and W. Saxton, "A practical algorithm for the determination of the phase from image and diffraction plane pictures," Optik 35, 237-246 (1972).
- D. Palima, C. A. Alonzo, P. J. Rodrigo, and J. Glückstad, "Generalized phase contrast matched to Gaussian illumination," Opt. Express 15, 11971-11977 (2007). [CrossRef] [PubMed]

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