## Analysis of low F-number dual micro-axilens array with binary structures by rigorous electromagnetic theory |

Optics Express, Vol. 19, Issue 11, pp. 10959-10966 (2011)

http://dx.doi.org/10.1364/OE.19.010959

Acrobat PDF (1130 KB)

### Abstract

We investigate a two-dimensional low F-number dual micro-axilens array with binary structures based on a rigorous electromagnetic theory. The focal characteristics of a binary dual micro-axilens array (BDMA), including axial performances (focal depth and focal shift) and transverse performances (focal spot size and diffraction efficiency), have been analyzed in detail for different F-numbers, different incident polarization (TE and TM) waves, and different distances between micro-axilens. Numerical results reveal that the interference effect of a BDMA is not very evident, which is useful for building a BDMA with a high fill factor, and the focal characteristics of a BDMA are sensitive to the polarization of an incident wave. The comparative results have also shown that the diffraction efficiency of a BDMA will increase and the focal spot size of a BDMS will decrease when the F-number increases, for both TE polarization and TM polarization, respectively. It is expected that this investigation will provide useful insight into the design of micro-optical elements with high integration.

© 2011 OSA

## 1. Introduction

1. K. L. Wlodarczyk, E. Mendez, H. J. Baker, R. McBride, and D. R. Hall, “Laser smoothing of binary gratings and multilevel etched structures in fused silica,” Appl. Opt. **49**(11), 1997–2005 (2010). [CrossRef] [PubMed]

2. R. Stevens and T. Miyashita, “Review of standards for microlenses and microlens arrays,” Imaging Sci. J. **58**(4), 202–212 (2010). [CrossRef]

3. N. Davidson, A. A. Friesem, and E. Hasman, “Holographic axilens: high resolution and long focal depth,” Opt. Lett. **16**(7), 523–525 (1991). [CrossRef] [PubMed]

8. G. Druart, J. Taboury, N. Guérineau, R. Haïdar, H. Sauer, A. Kattnig, and J. Primot, “Demonstration of image-zooming capability for diffractive axicons,” Opt. Lett. **33**(4), 366–368 (2008). [CrossRef] [PubMed]

9. J. N. Mait, D. W. Prather, and M. S. Mirotznik, “Design of binary subwavelength diffractive lenses by use of zeroth-order effective-medium theory,” J. Opt. Soc. Am. A **16**(5), 1157–1167 (1999). [CrossRef]

11. D. W. Prather, J. N. Mait, M. S. Mirotznik, and J. P. Collins, “Vector-based synthesis of finite aperiodic subwavelength diffractive optical elements,” J. Opt. Soc. Am. A **15**(6), 1599–1607 (1998). [CrossRef]

10. D. Feng, P. Ou, L. S. Feng, S. L. Hu, and C. X. Zhang, “Binary sub-wavelength diffractive lenses with long focal depth and high transverse resolution,” Opt. Express **16**(25), 20968–20973 (2008). [CrossRef] [PubMed]

12. D. Feng, Y. B. Yan, G. F. Jin, and S. S. Fan, “Beam focusing characteristics of diffractive lenses with binary subwavelength structures,” Opt. Commun. **239**(4-6), 345–352 (2004). [CrossRef]

13. A. Tripathi, T. V. Chokshi, and N. Chronis, “A high numerical aperture, polymer-based, planar microlens array,” Opt. Express **17**(22), 19908–19918 (2009). [CrossRef] [PubMed]

14. D. Wu, S. Z. Wu, L. G. Niu, Q. D. Chen, R. Wang, J. F. Song, H. H. Fang, and H. B. Sun, “High numerical aperture microlens arrays of close packing,” Appl. Phys. Lett. **97**(3), 031109 (2010). [CrossRef]

## 2. Theoretical formulae and analysis model

*x, y*(space components), and

*t*(time component); and

15. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antenn. Propag. **14**(3), 302–307 (1966). [CrossRef]

*l*is the distance between the two binary micro-axilenses and D is the diameter of the individual lens. We assume that a uniform plane wave is incident from the lower surface of the BDMA in the uniform dielectric medium, which fills the lower half space as the substrate; and, as the incident wave suffers from wavefront modulation at the surface of the BDMA with the same material as the substrate, it results in the focusing of the incident light in the free space. To determine the light field distribution, we apply the FDTD method to calculate the whole field transportation process.

6. B. Z. Dong, J. Liu, B. Y. Gu, G. Z. Yang, and J. Wang, “Rigorous electromagnetic analysis of a microcylindrical axilens with long focal depth and high transverse resolution,” J. Opt. Soc. Am. A **18**(7), 1465–1470 (2001). [CrossRef]

10. D. Feng, P. Ou, L. S. Feng, S. L. Hu, and C. X. Zhang, “Binary sub-wavelength diffractive lenses with long focal depth and high transverse resolution,” Opt. Express **16**(25), 20968–20973 (2008). [CrossRef] [PubMed]

*m*is the number of zones; and D and

*f*are the diameter and the geometrical focal length of a lens, respectively. “±” represents the two micro-axilens placed symmetrically at the two sides of the

*y*axis.

17. M. W. Farn, “Binary gratings with increased efficiency,” Appl. Opt. **31**(22), 4453–4458 (1992). [CrossRef] [PubMed]

## 3. Analysis and results

*n*

_{1}and

*n*

_{2}are taken as 1.5 for glass material and 1.0 for free space (air), respectively. The material of lenses and substrates is the same (glass), and the thickness of substrates is assumed as infinite. The size of the aperture of the micro-axilens D is equal to 10 μm. A plane wave with wavelength

*λ*= 1 μm in free space is normally incident on the BDMA along the

*y*axis with TE or TM polarization, as shown in Fig. 1.

*x*axis), and we define focal shift as a value that denotes the difference between the position of the maximum irradiance

*y*

_{max}along the optical axis and the geometrical focal length

*f*as defined in Eq. (2). It can be written as

*y*

_{max}, not the geometry focal plane, due to focal shift). The diffraction efficiency

*η*, which expresses the focusing ability of a BDMA, is defined by the ratio of intensity in the main lobes on the real focal plane to the incident intensity.

*l*= 0 μm) and they have the same parameters and are symmetrically placed on the two sides of the

*y*axis. The total field distribution is calculated by the FDTD method as shown in Fig. 2 , which can give a global review of focusing characteristics for the BDMA with different incidence polarization waves (here (a) for TE polarization and (b) for TM polarization, respectively). It is clearly seen that two focused spots appear in the focal region, and the interference effect is not obvious between each binary micro-axilens, even for zero space between them, but the field distribution is different with TE polarization and TM polarization.

*y*axis; (b) lateral intensity distribution at the real focal plane (

*y*

_{max}= 2.49 μm for TE polarization and

*y*

_{max}= 2.40 μm for TM polarization, respectively). It is clear that there are certain distinctions for different polarization waves. To denote these distinctions more clearly, analysis results are listed as follows: the focal depth of the BDMA is 1.06 μm for TE polarization and 1.27 μm for TM polarization, respectively. The geometrical focal length is 3 μm, but the real focal length is shorter than the geometrical one, and there is focal shift, obviously. The focal shift Δf is 0.51 μm for the TE wave and 0.60 μm for the TM wave, respectively. The focal spot size and diffraction efficiency are 1.17 μm and 54.2% for the TE wave and 1.38 μm and 46.1% for the TM wave, respectively. From these results, we can basically see that the BDMA can get two good focused regions clearly, and meanwhile the focusing characteristics of the BDMA are sensitive to the polarization of incidence waves.

18. M. S. Lee, P. Lalanne, J. C. Rodier, P. Chavel, E. Cambril, and Y. Chen, “Imaging with blazed-binary diffractive elements,” J. Opt. A, Pure Appl. Opt. **4**(5), S119–S124 (2002). [CrossRef]

20. P. Lalanne, “Waveguiding in blazed-binary diffractive elements,” J. Opt. Soc. Am. A **16**(10), 2517–2520 (1999). [CrossRef]

*l*= 0.0 to 10 μm for different polarization waves, and all other parameters are the same as in the previous case except for the spacing. The numerical results are summarized in Table 1 , including real focal length, focal shift Δf, focal spot size, and diffraction efficiency, respectively.

*l*, as shown in Figs. 4(a)–(d) , which are for focal shift, focal depth, focal spot size, and diffraction efficiency, respectively. It is clear that the focusing characteristics of the BDMA are obviously sensitive to the polarization of incidence waves. However, the interference effect is not very obvious between two focused beams, and the diffraction efficiency of the BDMA is 54.2% and 55.6% for distance

*l*= 0 μm and

*l*= 10 μm for TE polarization waves, respectively. In Figs. 4(a)–(d), we can also see that the parameters are slightly oscillating when the distance is increased. Compared with the diameter of a lens, the maximum distance is not so long and is only equal to the diameter; the interaction of two lenses clearly exists due to the vector diffraction nature of binary subwavelength structures, such as multiple electromagnetic scattering and micro-waveguides. For a global view, the total field distribution of a BDMA with space between = 10 μm is shown in Fig. 5 ––(a) for TE polarization and (b) for TM polarization, respectively.

*l*(0, 2, and 4 μm). The analysis results, such as focal depth, focal shift, focal spot size, and diffraction efficiency, have been summarized in Table 2 for different incidence polarization waves. As shown in Table 2, differences of BDMA focusing characteristics between different incidence polarizations are very clear. Besides, it is evident that focal depth, focal spot size, and diffraction efficiency increase as F-numbers increase, and the focal shift represents some oscillation. Compared with the former case (F-number = 0.3), we can see diffraction efficiency is low for a BDMA with a lower F-number due to BDMA special profile structures. With the same diameter of a BDMA, when the F-number is lower, the profile structures of the BDMA lens are more complex, and there are more micro-waveguides with different feature sizes on the profile and more electromagnetic shadowing effects [21

21. O. Sandfuchs, R. Brunner, D. Pätz, S. Sinzinger, and J. Ruoff, “Rigorous analysis of shadowing effects in blazed transmission gratings,” Opt. Lett. **31**(24), 3638–3640 (2006). [CrossRef] [PubMed]

## 4. Conclusion

## Acknowledgments

## References and links

1. | K. L. Wlodarczyk, E. Mendez, H. J. Baker, R. McBride, and D. R. Hall, “Laser smoothing of binary gratings and multilevel etched structures in fused silica,” Appl. Opt. |

2. | R. Stevens and T. Miyashita, “Review of standards for microlenses and microlens arrays,” Imaging Sci. J. |

3. | N. Davidson, A. A. Friesem, and E. Hasman, “Holographic axilens: high resolution and long focal depth,” Opt. Lett. |

4. | J. Sochacki, S. Bará, Z. Jaroszewicz, and A. Kołodziejczyk, “Phase retardation of the uniform-intensity axilens,” Opt. Lett. |

5. | J. Sochacki, A. Kołodziejczyk, Z. Jaroszewicz, and S. Bará, “Nonparaxial design of generalized axicons,” Appl. Opt. |

6. | B. Z. Dong, J. Liu, B. Y. Gu, G. Z. Yang, and J. Wang, “Rigorous electromagnetic analysis of a microcylindrical axilens with long focal depth and high transverse resolution,” J. Opt. Soc. Am. A |

7. | J. S. Ye, B. Z. Dong, B. Y. Gu, G. Z. Yang, and S. T. Liu, “Analysis of a closed-boundary axilens with long focal depth and high transverse resolution based on rigorous electromagnetic theory,” J. Opt. Soc. Am. A |

8. | G. Druart, J. Taboury, N. Guérineau, R. Haïdar, H. Sauer, A. Kattnig, and J. Primot, “Demonstration of image-zooming capability for diffractive axicons,” Opt. Lett. |

9. | J. N. Mait, D. W. Prather, and M. S. Mirotznik, “Design of binary subwavelength diffractive lenses by use of zeroth-order effective-medium theory,” J. Opt. Soc. Am. A |

10. | D. Feng, P. Ou, L. S. Feng, S. L. Hu, and C. X. Zhang, “Binary sub-wavelength diffractive lenses with long focal depth and high transverse resolution,” Opt. Express |

11. | D. W. Prather, J. N. Mait, M. S. Mirotznik, and J. P. Collins, “Vector-based synthesis of finite aperiodic subwavelength diffractive optical elements,” J. Opt. Soc. Am. A |

12. | D. Feng, Y. B. Yan, G. F. Jin, and S. S. Fan, “Beam focusing characteristics of diffractive lenses with binary subwavelength structures,” Opt. Commun. |

13. | A. Tripathi, T. V. Chokshi, and N. Chronis, “A high numerical aperture, polymer-based, planar microlens array,” Opt. Express |

14. | D. Wu, S. Z. Wu, L. G. Niu, Q. D. Chen, R. Wang, J. F. Song, H. H. Fang, and H. B. Sun, “High numerical aperture microlens arrays of close packing,” Appl. Phys. Lett. |

15. | K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antenn. Propag. |

16. | A. Taflove, |

17. | M. W. Farn, “Binary gratings with increased efficiency,” Appl. Opt. |

18. | M. S. Lee, P. Lalanne, J. C. Rodier, P. Chavel, E. Cambril, and Y. Chen, “Imaging with blazed-binary diffractive elements,” J. Opt. A, Pure Appl. Opt. |

19. | P. Lalanne, S. Astilean, P. Chavel, E. Cambril, and H. Launois, “Design and fabrication of blazed binary diffractive elements with sampling periods smaller than the structural cutoff,” J. Opt. Soc. Am. A |

20. | P. Lalanne, “Waveguiding in blazed-binary diffractive elements,” J. Opt. Soc. Am. A |

21. | O. Sandfuchs, R. Brunner, D. Pätz, S. Sinzinger, and J. Ruoff, “Rigorous analysis of shadowing effects in blazed transmission gratings,” Opt. Lett. |

**OCIS Codes**

(050.1970) Diffraction and gratings : Diffractive optics

(220.2560) Optical design and fabrication : Propagating methods

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: February 17, 2011

Revised Manuscript: April 14, 2011

Manuscript Accepted: April 25, 2011

Published: May 20, 2011

**Citation**

Di Feng, Li-Shuang Feng, and Chun-Xi Zhang, "Analysis of low F-number dual micro-axilens array with binary structures by rigorous electromagnetic theory," Opt. Express **19**, 10959-10966 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-11-10959

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### References

- K. L. Wlodarczyk, E. Mendez, H. J. Baker, R. McBride, and D. R. Hall, “Laser smoothing of binary gratings and multilevel etched structures in fused silica,” Appl. Opt. 49(11), 1997–2005 (2010). [CrossRef] [PubMed]
- R. Stevens and T. Miyashita, “Review of standards for microlenses and microlens arrays,” Imaging Sci. J. 58(4), 202–212 (2010). [CrossRef]
- N. Davidson, A. A. Friesem, and E. Hasman, “Holographic axilens: high resolution and long focal depth,” Opt. Lett. 16(7), 523–525 (1991). [CrossRef] [PubMed]
- J. Sochacki, S. Bará, Z. Jaroszewicz, and A. Kołodziejczyk, “Phase retardation of the uniform-intensity axilens,” Opt. Lett. 17(1), 7–9 (1992). [CrossRef] [PubMed]
- J. Sochacki, A. Kołodziejczyk, Z. Jaroszewicz, and S. Bará, “Nonparaxial design of generalized axicons,” Appl. Opt. 31(25), 5326–5330 (1992). [CrossRef] [PubMed]
- B. Z. Dong, J. Liu, B. Y. Gu, G. Z. Yang, and J. Wang, “Rigorous electromagnetic analysis of a microcylindrical axilens with long focal depth and high transverse resolution,” J. Opt. Soc. Am. A 18(7), 1465–1470 (2001). [CrossRef]
- J. S. Ye, B. Z. Dong, B. Y. Gu, G. Z. Yang, and S. T. Liu, “Analysis of a closed-boundary axilens with long focal depth and high transverse resolution based on rigorous electromagnetic theory,” J. Opt. Soc. Am. A 19(10), 2030–2035 (2002). [CrossRef]
- G. Druart, J. Taboury, N. Guérineau, R. Haïdar, H. Sauer, A. Kattnig, and J. Primot, “Demonstration of image-zooming capability for diffractive axicons,” Opt. Lett. 33(4), 366–368 (2008). [CrossRef] [PubMed]
- J. N. Mait, D. W. Prather, and M. S. Mirotznik, “Design of binary subwavelength diffractive lenses by use of zeroth-order effective-medium theory,” J. Opt. Soc. Am. A 16(5), 1157–1167 (1999). [CrossRef]
- D. Feng, P. Ou, L. S. Feng, S. L. Hu, and C. X. Zhang, “Binary sub-wavelength diffractive lenses with long focal depth and high transverse resolution,” Opt. Express 16(25), 20968–20973 (2008). [CrossRef] [PubMed]
- D. W. Prather, J. N. Mait, M. S. Mirotznik, and J. P. Collins, “Vector-based synthesis of finite aperiodic subwavelength diffractive optical elements,” J. Opt. Soc. Am. A 15(6), 1599–1607 (1998). [CrossRef]
- D. Feng, Y. B. Yan, G. F. Jin, and S. S. Fan, “Beam focusing characteristics of diffractive lenses with binary subwavelength structures,” Opt. Commun. 239(4-6), 345–352 (2004). [CrossRef]
- A. Tripathi, T. V. Chokshi, and N. Chronis, “A high numerical aperture, polymer-based, planar microlens array,” Opt. Express 17(22), 19908–19918 (2009). [CrossRef] [PubMed]
- D. Wu, S. Z. Wu, L. G. Niu, Q. D. Chen, R. Wang, J. F. Song, H. H. Fang, and H. B. Sun, “High numerical aperture microlens arrays of close packing,” Appl. Phys. Lett. 97(3), 031109 (2010). [CrossRef]
- K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antenn. Propag. 14(3), 302–307 (1966). [CrossRef]
- A. Taflove, Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech House, 1995).
- M. W. Farn, “Binary gratings with increased efficiency,” Appl. Opt. 31(22), 4453–4458 (1992). [CrossRef] [PubMed]
- M. S. Lee, P. Lalanne, J. C. Rodier, P. Chavel, E. Cambril, and Y. Chen, “Imaging with blazed-binary diffractive elements,” J. Opt. A, Pure Appl. Opt. 4(5), S119–S124 (2002). [CrossRef]
- P. Lalanne, S. Astilean, P. Chavel, E. Cambril, and H. Launois, “Design and fabrication of blazed binary diffractive elements with sampling periods smaller than the structural cutoff,” J. Opt. Soc. Am. A 16(5), 1143–1156 (1999). [CrossRef]
- P. Lalanne, “Waveguiding in blazed-binary diffractive elements,” J. Opt. Soc. Am. A 16(10), 2517–2520 (1999). [CrossRef]
- O. Sandfuchs, R. Brunner, D. Pätz, S. Sinzinger, and J. Ruoff, “Rigorous analysis of shadowing effects in blazed transmission gratings,” Opt. Lett. 31(24), 3638–3640 (2006). [CrossRef] [PubMed]

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