## Rigorous electromagnetic analysis of two dimensional micro-axicon by boundary integral equations

Optics Express, Vol. 17, Issue 3, pp. 1466-1471 (2009)

http://dx.doi.org/10.1364/OE.17.001466

Acrobat PDF (298 KB)

### Abstract

The focal performance of the micro-axicon and the Fresnel axicon (fraxicon) are investigated, for the first time, by the rigorous electromagnetic theory and boundary element method. The micro-axicon with different angle of apex and the fraxicon with various period and angle of apex are investigated. The dark segments of the fraxicon are explored numerically. Rigorous results of focal performance of the micro-axicon and the fraxicon are different from the results given by the approximation of geometrical optics and the scalar diffraction theory. The scattering effects are dominant in the fraxicon with small size of feature. It is expected that our study can provides very useful information in analyzing the axicon in optical trapping systems.

© 2009 Optical Society of America

## 1. Introduction

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

2. J. H. McLeod, “Axicons and their uses,” J. Opt. Soc. Am. **50**, 166–169 (1960). [CrossRef]

3. I. Golub and R. Tremblay, “Light focusing and guiding by an axicon-pair-generated tubular light beam,” J. Opt. Soc. Am. B **7**, 1264–1267 (1990). [CrossRef]

5. J. Fan, E. Parra, and H. M. Milchberg, “Resonant self-trapping and absorption of intense Bessel beams,” Phys. Rev. Lett. **84**, 3085–3088 (2000). [CrossRef] [PubMed]

6. D. Mcgloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. **46**, 15–28 (2005). [CrossRef]

12. P.-A. Bélanger and M. Rioux, “Ring pattern of a lens-axicon doublet illuminated by a Gaussian beam,” Appl. Opt. **17**, 1080–1088 (1978). [CrossRef] [PubMed]

13. Z. Ding, H. Ren, Y. Zhao, J. S. Nelson, and Z. Chen, “High-resolution optical coherence tomography over a large depth range with an axicon lens,” Opt. Lett. **27**, 243–245 (2002). [CrossRef]

14. D. J. Fischer, C. J. Harkrider, and D. T. Moore, “Design and manufacture of a gradient-index axicon,” Appl. Opt. **39**, 2687–2694 (2000). [CrossRef]

17. J. A. Monsoriu, C. J. Zapata-Rodŕguez, and W. D. Furlan, “Fractal axicons,” Opt. Commun. **263**, 1–5 (2006). [CrossRef]

20. L. Lin, S. Lee, K. Pister, and M.C. Wu, “Three-dimensional micro-Fresnel optical elements fabricated bymicroma-chining technique,” Electron. Lett. **30**, 448–449 (1994). [CrossRef]

18. A. Burvall, P. Martinsson, and A. Friberg, “Communication modes applied to axicons,” Opt. Express **12**377–383 (2004). [CrossRef] [PubMed]

19. C.J. Zapata-Rodríguez and F.E. Hernńndez, “Focal squeeze in axicons,” Opt. Commun. **254**, 3–9 (2005). [CrossRef]

23. K. Yashiro and S. Ohkawa, “Boundary element method for electromagnetic scattering from cylinders,” IEEE Trans. Antennas Propag. **AP-33**, 383–389 (1985). [CrossRef]

24. K. Hirayama, E. N. Glytsis, T. K. Gaylord, and D. W. Wilson, “Rigorous electromagnetic analysis of diffractive cylindrical lenses,” J. Opt. Soc. Am. A **13**, 2219–2231 (1996). [CrossRef]

## 2. Boundary integral equations

*S*

_{1}and region

*S*

_{2}, by axicon’s boundary Γ. The region

*S*

_{1}is occupied by the 2D axicon with refractive index

*n*

_{1}and the region

*S*

_{2}usually is the air with the refractive index

*n*

_{2}=1.0. The 2D axicon is infinite along

*z*-axis. A TE parallel beam is assumed incident along opposite direction of

*y*-axis. After propagating through the axicon, the beams get interfered in region

*S*

_{2}. To calculate the optical field distributions in region

*S*

_{2}, the Green’s theorem and Green function are applied to Maxwell’s equations. By incorporating with Sommerfeld radiation condition, we can obtain the boundary integral equation about the boundary scattering fields and their normal derivatives. The boundary integral equations are discretized by using of the BEM. Through matrix inversion and Gaussian elimination method, the scattering fields and their normal derivatives can be determined. Therefore, the transmitted optical field in region

*S*

_{2}is computed numerically by [24

24. K. Hirayama, E. N. Glytsis, T. K. Gaylord, and D. W. Wilson, “Rigorous electromagnetic analysis of diffractive cylindrical lenses,” J. Opt. Soc. Am. A **13**, 2219–2231 (1996). [CrossRef]

**r**and

**r**′

_{Γ}represent any point in region

*S*

_{2}and on micro-axicon boundary Γ, respectively.

**E**

_{Γ}and

**n**̂ is normal to the boundary Γ with orientation to region

*S*

_{1}, as shown in Fig. 1(a).

*G*

_{2}is the 2D Green function in the region

*S*

_{2}and is described as a zeroth-order Hankel function of the second kind, i.e.,

*G*

_{2}(

**r**,

**r**′

_{Γ}) = (-

*j*/4)

*H*

_{0}

^{(2)}(

*k*

_{2}|

**r**-

**r**′

_{Γ}|).

*k*

_{2}= 2

*πn*

_{2}/

*λ*is the wave number in the region

*S*

_{2}, and

*λ*is the incident wavelength in free space.

## 3. Transmission characteristics of 2D micro-axicons

*n*

_{1}), the size of the aperture (

*D*= 2

*R*) and the angle of the apex (

*θ*), where

*R*is radial coordinate. The incident wavelength is

*λ*= 1.0

*μ*m. The micro-axicon and the fraxicon are assumed made of AlAs crystal and the refractive index is

*n*

_{1}= 2.857 at

*λ*= 1.0

*μ*m [25]. According to the principles of geometrical optics, the finite interference zone is given by

*y*

_{max}=

*R*tan

*γ*-

*R*tan(

*θ*/2), where 2

*γ*=

*θ*+2

*β*-

*π*, and

*β*= arcsin[

*n*

_{1}cos(

*θ*/2)], respectively. For a large apex angle,

*y*

_{max}≈

*R*tan

*γ*. However,

*y*

_{max}may fail to describe the focal performance when the scattering effect on the boundary of the micro-axicon is more obviously enhanced as the decreasing in the size of axicon. For the fraxicon, which is shown in Fig. 1(b), the dark segments caused by introducing the Fresnel zone will appear, and the diffraction and scattering effects are more obviously if the feature size is approximately equal to the incident wavelength

*λ*. When the collimated beam is propagating from the region

*S*

_{1}to the region

*S*

_{2}, the total reflection is occurred and the corresponding critical angle of apex is given as 2arccos(1/

*n*

_{1}) ≈ 139°. In our work, the diameter of the 2D micro-axicons are 50.0

*μ*m and the angle of apex are chose as 130°, 140°, 150°, 160°, and 170°, respectively.

*θ*= 170°. The gray-level representation of the electric-field distribution in the region

*S*

_{2}is drawn in Fig. 2(a), where the bright (dark) regions mark the locations with high (low) electric field intensities. The interference occurs in the region from

*y*= 0 to -148.52

*μ*m. It is obviously seen that the interference fringes are parallel to the

*y*-axis and occur in a rhombus-like region surrounded by the red lines. In this region, seven strong interference fringes appear in the marked region. In the large propagating range, the fringes keep parallel each other and the results can be seen in Fig. 2(b). The solid curve represents the lateral intensity distribution at

*y*= -100.0

*μ*m, while the red dashed curve for

*y*= -40.0

*μ*m. Along the propagation, the positions of the innermost five interference fringes are almost unchangeable and the distances between interference fringes are approximately constant. In fact, owing to the diffraction and scattering effects, blurry interference also appears in

*y*=-148.52

*μ*m to -160

*μ*m, as shown in Fig. 2(a). Meanwhile, as shown in Fig. 2(c), the intensity distribution along

*y*-axis is fluctuated. The dotted-dashed line is located at 80.0% maximum of the intensity. Obviously there are two ranges for the intensity larger than 80.0% maximum of the intensity. At

*y*= -74.69

*μ*m, an intensity well is formed, which disagrees with the result given by the scalar diffractive theory [6

6. D. Mcgloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. **46**, 15–28 (2005). [CrossRef]

*θ*= 160° and 150° are illustrated in Fig. 3(a) and (b), respectively. As the decreasing in the angle of apex, the interference fringes are located in a rhombus-like region and its shape is decayed. The theoretical ranges are

*y*

_{max}= 65.25 and 32.27

*μ*m for

*θ*= 160° and 150°, respectively. In fact, as shown in Fig. 3(c), the ranges are reduced owing to the scattering effect and the intensity distribution is fluctuated in a range that is smaller than the range predicted by geometrical optics. When the angle of apex is smaller than the critical angle 2arccos(1/

*n*

_{1}), weak power is transmitted into the region

*S*

_{2}.

*T*= 5.0 and 2.5

*μ*m, respectively. Under the approximation of geometrical optics, the interference fringes along

*y*-axis are not continuous: it consists of several dark segments which displays in Fig. 1(b). Furthermore, the number of the dark segments is equal to

*N*- 1, where

*N*=

*R*/

*T*is the number of Fresnel zones. For the fraxicon with

*T*= 5.0 and 2.5

*μ*m, the number of dark segments are 4 and 9, respectively. However, the numerical result departs from the results in the geometric optical approximation. In Fig. 4(a), serval peaks appear on the intensity distributions on the observation plane

*y*= -100.0

*μ*m. It indicates that the interference caused by different Fresnel zones occurs in the region

*S*

_{2}. The position of interference has changed for the fraxicons. The interference occur in more positions for the smaller

*T*. The much more obvious differences are easily seen from the axial intensity distributions. As shown in Fig. 4(b), for the fraxicon with

*T*= 5.0

*μ*m, only one obvious dark segment is emerged in the curve a at

*y*= -40

*μ*m nearby. For the fraxicon with

*T*= 2.5

*μ*m, five dark segments are observed in the curve b. It is obviously that the actual number of dark segments is much less than the predicted number.

*θ*= 170°, which indicate there are two dark segments exist. With the further decrease of the apex angle, the positions of intensity wells will get increased, for instance, there are three dark segments for

*θ*= 160°. For the cases of

*θ*= 150° and 140°, number of dark segments are four. It is obviously that the number of dark segments is different for different angle of apex and the number is not monotonically changed with the angle of apex. According to the feature of fraxicon, the maximal thickness of each zone is increasing while the angle of apex is decreasing. That indicates the feature size of the fraxicon is increasing. Therefore the scattering of the fraxicon with small angle of apex is relatively weak compared with the fraxicon with large angle of apex. It is obviously that the focal performance of fraxicon is governed by scattering effect. The scattering effect is more evident as the decreasing in period of the fraxicon.

## 4. Conclusion

## Acknowledgments

## References and links

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

2. | J. H. McLeod, “Axicons and their uses,” J. Opt. Soc. Am. |

3. | I. Golub and R. Tremblay, “Light focusing and guiding by an axicon-pair-generated tubular light beam,” J. Opt. Soc. Am. B |

4. | Y. Qian and Y. Z. Wang, “Theoretical analysis of a collimated hollow-laser-beam generated by a singel axicon using diffraction integral,” Chin. Opt. Lett. |

5. | J. Fan, E. Parra, and H. M. Milchberg, “Resonant self-trapping and absorption of intense Bessel beams,” Phys. Rev. Lett. |

6. | D. Mcgloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. |

7. | V. V. Kotlyar, A. A. Kovalev, V. A. Soifer, C. S. Tuvey, and J. A. Davis, “Sidelobe contrast reduction for optical vortex beams using a helical axicon,” Opt. Lett. |

8. | O. Brzobohatý, T. Čižmár, and P. Zemánek, “High quality quasi-Bessel beam generated by round-tip axicon,” Opt. Express |

9. | M. Florjanczyk and R. Tremblay, “Guiding of atoms in a travelling-wave laser trap formed by the axicon,” Opt. Commun. |

10. | R. Arimoto, C. Saloma, T. Tanaka, and S. Kawata, “Imaging properties of axicon in a scanning optical system,” Appl. Opt. |

11. | T. Tanaka and S. Yamamoto, “Comparison of aberration between axicon and lens,” Opt. Commun. |

12. | P.-A. Bélanger and M. Rioux, “Ring pattern of a lens-axicon doublet illuminated by a Gaussian beam,” Appl. Opt. |

13. | Z. Ding, H. Ren, Y. Zhao, J. S. Nelson, and Z. Chen, “High-resolution optical coherence tomography over a large depth range with an axicon lens,” Opt. Lett. |

14. | D. J. Fischer, C. J. Harkrider, and D. T. Moore, “Design and manufacture of a gradient-index axicon,” Appl. Opt. |

15. | J. Monsoriu, G. Saavedra, and W. Furlan, “Fractal zone plates with variable lacunarity,” Opt. Express |

16. | I. Golub, “Fresnel axicon,” Opt. Lett. |

17. | J. A. Monsoriu, C. J. Zapata-Rodŕguez, and W. D. Furlan, “Fractal axicons,” Opt. Commun. |

18. | A. Burvall, P. Martinsson, and A. Friberg, “Communication modes applied to axicons,” Opt. Express |

19. | C.J. Zapata-Rodríguez and F.E. Hernńndez, “Focal squeeze in axicons,” Opt. Commun. |

20. | L. Lin, S. Lee, K. Pister, and M.C. Wu, “Three-dimensional micro-Fresnel optical elements fabricated bymicroma-chining technique,” Electron. Lett. |

21. | W. C. Cheong, B. P. S. Ahluwalia, X.-C. Yuan, L.-S. Zhang, H. B. Niu, and X. Peng, “Fabrication of efficient microaxicon by direct electron-beam lithography for long nondiffracting distance of Bessel beams for optical manipulation,” Appl. Phys. Lett. |

22. | B. P. S. Ahluwalia, W. C. Cheong, X.-C. Yuan, L.-S. Zhang, S.-H. Tao, J. Bu, and H. Wang, “Design and fabrication of a double-axicon for generation of tailorable self-imaged three-dimensional intensity voids,” Opt. Lett. |

23. | K. Yashiro and S. Ohkawa, “Boundary element method for electromagnetic scattering from cylinders,” IEEE Trans. Antennas Propag. |

24. | K. Hirayama, E. N. Glytsis, T. K. Gaylord, and D. W. Wilson, “Rigorous electromagnetic analysis of diffractive cylindrical lenses,” J. Opt. Soc. Am. A |

25. | M. Bass, E. W. Van Stryland, D. R. Williams, and W. L. Wolfe, “Handbooks of Optics, Vol. 2: Devices, Measurements and Properties,” (McGraw-Hill, New York, 1995). |

**OCIS Codes**

(050.1970) Diffraction and gratings : Diffractive optics

(230.3990) Optical devices : Micro-optical devices

(260.1960) Physical optics : Diffraction theory

**ToC Category:**

Physical Optics

**History**

Original Manuscript: October 30, 2008

Revised Manuscript: December 16, 2008

Manuscript Accepted: December 19, 2008

Published: January 26, 2009

**Virtual Issues**

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

**Citation**

Jie Lin, Jiubin Tan, Jian Liu, and Shutian Liu, "Rigorous electromagnetic analysis of two dimensional micro-axicon by boundary integral equations," Opt. Express **17**, 1466-1471 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-3-1466

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### References

- J. H. McLeod, "The axicon: a new type of optical element," J. Opt. Soc. Am. 44, 592-597 (1954). [CrossRef]
- J. H. McLeod, "Axicons and their uses," J. Opt. Soc. Am. 50, 166-169 (1960). [CrossRef]
- I. Golub and R. Tremblay, "Light focusing and guiding by an axicon-pair-generated tubular light beam," J. Opt. Soc. Am. B 7, 1264-1267 (1990). [CrossRef]
- Y. Qian and Y. Z. Wang, "Theoretical analysis of a collimated hollow-laser-beam generated by a singel axicon using diffraction integral," Chin. Opt. Lett. 2, 232-234 (2004).
- J. Fan, E. Parra, and H. M. Milchberg, "Resonant self-trapping and absorption of intense Bessel beams," Phys. Rev. Lett. 84, 3085-3088 (2000). [CrossRef] [PubMed]
- D. Mcgloin and K. Dholakia, "Bessel beams: diffraction in a new light," Contemp. Phys. 46, 15-28 (2005). [CrossRef]
- V. V. Kotlyar, A. A. Kovalev, V. A. Soifer, C. S. Tuvey, and J. A. Davis, "Sidelobe contrast reduction for optical vortex beams using a helical axicon," Opt. Lett. 32, 921-923 (2007). [CrossRef] [PubMed]
- O. Brzobohatý, T .Čižmár, and P. Zemánek, "High quality quasi-Bessel beam generated by round-tip axicon," Opt. Express 16, 12688-12700 (2008). [PubMed]
- M. Florjanczyk and R. Tremblay, "Guiding of atoms in a travelling-wave laser trap formed by the axicon," Opt. Commun. 74, 448-450 (1989). [CrossRef]
- R. Arimoto, C. Saloma, T. Tanaka, and S. Kawata, "Imaging properties of axicon in a scanning optical system," Appl. Opt. 31, 6653-6657 (1992). [CrossRef] [PubMed]
- T. Tanaka and S. Yamamoto, "Comparison of aberration between axicon and lens," Opt. Commun. 184, 113-118 (2000). [CrossRef]
- P.-A. Bélanger and M. Rioux, "Ring pattern of a lens-axicon doublet illuminated by a Gaussian beam," Appl. Opt. 17, 1080-1088 (1978). [CrossRef] [PubMed]
- Z. Ding, H. Ren, Y. Zhao, J. S. Nelson, and Z. Chen, "High-resolution optical coherence tomography over a large depth range with an axicon lens," Opt. Lett. 27, 243-245 (2002). [CrossRef]
- D. J. Fischer, C. J. Harkrider, and D. T. Moore, "Design and manufacture of a gradient-index axicon," Appl. Opt. 39, 2687-2694 (2000). [CrossRef]
- J. Monsoriu, G. Saavedra, and W. Furlan, "Fractal zone plates with variable lacunarity," Opt. Express 12, 4227-4234 (2004). [CrossRef] [PubMed]
- I. Golub, "Fresnel axicon," Opt. Lett. 31, 1890-1892 (2006). [CrossRef] [PubMed]
- J. A. Monsoriu, C. J. Zapata-Rodrıguez, and W. D. Furlan, "Fractal axicons," Opt. Commun. 263, 1-5 (2006). [CrossRef]
- A. Burvall, P. Martinsson, and A. Friberg, "Communication modes applied to axicons," Opt. Express 12377-383 (2004). [CrossRef] [PubMed]
- C. J. Zapata-Rodríguez, and F. E. Hernńndez, "Focal squeeze in axicons," Opt. Commun. 254, 3-9 (2005). [CrossRef]
- L. Lin, S. Lee, K. Pister, and M. C. Wu, "Three-dimensional micro-Fresnel optical elements fabricated by micromachining technique," Electron. Lett. 30, 448-449 (1994). [CrossRef]
- W. C. Cheong, B. P. S. Ahluwalia, X.-C. Yuan, L.-S. Zhang, H. B. Niu, and X. Peng, "Fabrication of efficient microaxicon by direct electron-beam lithography for long nondiffracting distance of Bessel beams for optical manipulation," Appl. Phys. Lett. 87, 024104-1-3 (2005). [CrossRef]
- B. P. S. Ahluwalia, W. C. Cheong, X.-C. Yuan, L.-S. Zhang, S.-H. Tao, J. Bu, and H. Wang, "Design and fabrication of a double-axicon for generation of tailorable self-imaged three-dimensional intensity voids," Opt. Lett. 31, 987-989 (2006). [CrossRef] [PubMed]
- K. Yashiro and S. Ohkawa, "Boundary element method for electromagnetic scattering from cylinders," IEEE Trans. Antennas Propag. AP-33, 383-389 (1985). [CrossRef]
- K. Hirayama, E. N. Glytsis, T. K. Gaylord, and D. W. Wilson, "Rigorous electromagnetic analysis of diffractive cylindrical lenses," J. Opt. Soc. Am. A 13, 2219-2231 (1996). [CrossRef]
- M. Bass, E. W. Van Stryland, D. R. Williams, and W. L. Wolfe, "Handbooks of Optics, Vol. 2: Devices, Measurements and Properties," (McGraw-Hill, New York, 1995).

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