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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 18 — Aug. 27, 2012
  • pp: 19799–19805
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A hollow beam supercontinuum generation by the supermode superposition in a GeO2 doped triangular-core photonic crystal fiber

Xin-ben Zhang, Xian Zhu, Xiang Chen, Hai-qing Li, Jing-gang Peng, Neng-li Dai, and Jin-yan Li  »View Author Affiliations


Optics Express, Vol. 20, Issue 18, pp. 19799-19805 (2012)
http://dx.doi.org/10.1364/OE.20.019799


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Abstract

A GeO2 doped triangular-core photonic-crystal fiber (PCF) is designed and fabricated to allow the generation of a hollow beam through a nonlinear-optical transformation by femtosecond pulses at 1040 nm from a high power Yb-doped PCF laser oscillator. The hollow beam supercontinuum is obtained at far field by adjusting incident light polarization to excite the high order supermode, behaving as a mode convertor. The supercontinuum ranging from 540 to 1540 nm is achieved with an average power of 1.04 W.

© 2012 OSA

1. Introduction

The generation of hollow optical potentials has found extensive applications in both life and physical sciences. Hollow beam is a light field whose intensity along the central axis is the minimum or vanishes in the ideal case along the beam axis. Various proposals have been carried out to generate this type beam. Such as axicon [1

1. I. Manek, Y. B. Ovchinnikov, and R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147(1-3), 67–70 (1998). [CrossRef]

3

3. S. K. Tiwari, S. R. Mishra, and S. P. Ram, “Generation of a variable-diameter collimated hollow laser beam using metal axicon mirrors,” Opt. Eng. 50(1), 014001 (2011). [CrossRef]

], hollow optical systems [4

4. S. Marksteiner, C. M. Savage, P. Zoller, and S. L. Rolston, “Coherent atomic waveguides from hollow optical fibers: Quantized atomic motion,” Phys. Rev. A 50(3), 2680–2690 (1994). [CrossRef] [PubMed]

9

9. H. R. Li and J. P. Yin, “Generation of a vectorial elliptic hollow beam by an elliptic hollow fiber,” Opt. Lett. 36(4), 457–459 (2011). [CrossRef] [PubMed]

], multimode light guides [10

10. C. L. Zhao, Y. J. Cai, F. Wang, X. H. Lu, and Y. Z. Wang, “Generation of a high-quality partially coherent dark hollow beam with a multimode fiber,” Opt. Lett. 33(12), 1389–1391 (2008). [CrossRef] [PubMed]

, 11

11. G. Schweiger, R. Nett, B. Özel, and T. Weigel, “Generation of hollow beams by spiral rays in multimode light guides,” Opt. Express 18(5), 4510–4517 (2010). [CrossRef] [PubMed]

], waveguide with metal rods [12

12. C. C. Yan, D. H. Zhang, D. D. Li, H. J. Bian, Z. J. Xu, and Y. K. Wang, “Metal nanorod-based metamaterials for beam splitting and a subdiffraction-limited dark hollow light cone,” J. Opt. 13(8), 085102 (2011). [CrossRef]

], and mode convertors [13

13. A. Witkowska, S. G. Leon-Saval, A. Pham, and T. A. Birks, “All-fiber LP11 mode convertors,” Opt. Lett. 33(4), 306–308 (2008). [CrossRef] [PubMed]

, 14

14. N. Bokor and N. Davidson, “Generation of a hollow dark spherical spot by 4pi focusing of a radially polarized Laguerre-Gaussian beam,” Opt. Lett. 31(2), 149–151 (2006). [CrossRef] [PubMed]

]. An impressive performances have been obtained by using multi-mode holey fiber [15

15. M. L. Hu, C. Y. Wang, Y. J. Song, Y. F. Li, L. Chai, E. E. Serebryannikov, and A. M. Zheltikov, “A hollow beam from a holey fiber,” Opt. Express 14(9), 4128–4134 (2006). [CrossRef] [PubMed]

], suggesting attractive fiber optic solutions. Recently it was shown that the optical vortex beam can be generated by using the coherent-superposition of multi-beams in a radially symmetric configuration [16

16. L. G. Wang, L. Q. Wang, and S. Y. Zhu, “Formation of optical vortices using coherent laser beam arrays,” Opt. Commun. 282(6), 1088–1094 (2009). [CrossRef]

18

18. G. Q. Zhou, “Propagation of a radial phased-locked Lorentz beam array in turbulent atmosphere,” Opt. Express 19(24), 24699–24711 (2011). [CrossRef] [PubMed]

]. However, it is impossible to keep the initial phase of the laser arrays stable in practical engineering excepting that the phase noise is corrected [19

19. P. Zhou, X. L. Wang, Y. X. Ma, H. T. Ma, X. J. Xu, and Z. J. Liu, “Generation of a hollow beam by active phasing of a laser array using a stochastic parallel gradient descent algorithm,” J. Opt. 12(1), 015401 (2010). [CrossRef]

, 20

20. Y. Zheng, X. Wang, F. Shen, and X. Li, “Generation of dark hollow beam via coherent combination based on adaptive optics,” Opt. Express 18(26), 26946–26958 (2010). [CrossRef] [PubMed]

]. The difficulty of this correction inspires us to explore the realization of laser beam array in the multi-core PCF.

Meanwhile, some biophotonics special applications need combining the hollow beam with various spectroscopic techniques, which can simultaneously manipulate and characterize a single trapped cell by gathering spectral information [21

21. P. Li, K. B. Shi, and Z. W. Liu, “Manipulation and spectroscopy of a single particle by use of white-light optical tweezers,” Opt. Lett. 30(2), 156–158 (2005). [CrossRef] [PubMed]

, 22

22. K. B. Shi, P. Li, and Z. W. Liu, “Broadband coherent anti-Stokes Raman scattering spectroscopy in supercontinuum optical trap,” Appl. Phys. Lett. 90(14), 141116 (2007). [CrossRef]

]. Taking advantage of the design freedom of photonic crystal fibers, we demonstrate a GeO2 doped triangular-core (three core) enhanced nonlinear PCF. With the femtosecond laser pulse is coupled into the triangular-core PCF, supercontinuum can be generated and a hollow beam can be produced in far-field region through the supermode superposition. Unlike a narrowband laser, supercontinuum hollow beam has an additional degree of freedom. Through mapping of different wavelengths to different positions by diffractive–dispersive optical elements, it is possible to create a chromatic optical trap that can bring new capability and versatility to manipulation.

2. Fiber fabrication and properties

PCFs are widely investigated both theoretically and experimentally due to their flexible management of dispersion and nonlinearity [23

23. J. C. Knight, T. A. Birks, P. S. J. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. 21(19), 1547–1549 (1996). [CrossRef] [PubMed]

, 24

24. S. G. Yang, Y. J. Zhang, X. Z. Peng, Y. Lu, S. H. Xie, J. Li, W. Chen, Z. Jiang, J. Peng, and H. Li, “Theoretical study and experimental fabrication of high negative dispersion photonic crystal fiber with large area mode field,” Opt. Express 14(7), 3015–3023 (2006). [CrossRef] [PubMed]

]. The multi-core PCFs we designed is formed by three missing central neighboring air holes and made by the standard stack-and-draw process. A scanning electron microscope (SEM) image of the GeO2-doped core PCF is displayed in Fig. 1
Fig. 1 Scanning electron micrograph of PCF end face, and white circles depict GeO2 regions.
. The fiber cladding lattice has a pitch (hole spacing, Λ) of 3.7 μm and a hole diameter to pitch ratio (d/Λ) of 0.81. Generally, GeO2 is always doped into the fiber core as a means of additional method to enhance nonlinearity and engineer the dispersion. Here we show that the GeO2 dopant can obviously influence the field distribution of PCF, resulting in a novel characteristic of PCF for mode control. The GeO2 doped regions (depicted by the white circles in Fig. 1) are oval with a dimension of 1.5 μm × 1 μm and have a relative refractive index difference (Δ) of 0.35%. This three-fold symmetry structure promises special polarization properties and far field distributions.

More detailed supermode field profiles of the PCF are calculated in Fig. 3
Fig. 3 Calculated near (top) and far field (bottom) distribution of the in-phase ((a), (d)) and out-phase ((b), (c), (e), (f)) mode, where (e) is the far field of out-phase supermode (b) and (f) is superposition of out-phase supermode (b) and (c) .
. The near field distribution of the in-phase supermode (Fig. 3(a)) features a strong coupling strength between the adjacent cores, and the three fields nearly coupled together due to in-phase condition. The phase of each core is identical, which corresponds with the situation of in-phase coupling A1 = [1,1,1] , and Gaussian-shaped supermode is obtained in far-filed [Fig. 3(d) ]. Furthermore, there are two out-phase supermodes which are depicted in Fig. 3(b) and Fig. 3(c), the field phases of the cores are consistent with the eigenvectors of the coupled equation A2 = [-1, 1, 0], A3 = [-1, 0, 1] respectively. One far field profile of them is also given in Fig. 3(e), what is noteworthy is that the intensity in center almost vanishes. Generally, hollow-beam modes arise as a superposition of two or more modes [15

15. M. L. Hu, C. Y. Wang, Y. J. Song, Y. F. Li, L. Chai, E. E. Serebryannikov, and A. M. Zheltikov, “A hollow beam from a holey fiber,” Opt. Express 14(9), 4128–4134 (2006). [CrossRef] [PubMed]

, 27

27. S. Konorov, E. Serebryannikov, A. Zheltikov, P. Zhou, A. Tarasevitch, and D. von der Linde, “Mode-controlled colors from microstructure fibers,” Opt. Express 12(5), 730–735 (2004). [CrossRef] [PubMed]

].When propagating in the fiber, the effective refractive index of these involved modes should be close enough for co-excitation and have comparable intensities. For the PCF considered here, both of the out-phase supermodes will contributed to the hollow-beam modes, and the effective refractive index difference of the two modes is around 8 × 10−5 at wavelength of 600 nm. The normalized superposition of their far filed is displayed Fig. 3(f), apparently a good-shape hollow-beam.

3. Experiments and discussion

The fabricated PCF was pumped by a femtosecond Yb-doped PCF laser oscillator working in a dissipative mode-locking regime [28

28. C. Xie, M. L. Hu, D. P. Zhang, C. L. Gu, Y. J. Song, L. Chai, and C. Y. Wang, “Generation of 25-fs High Energy Pulses by SPM-Induced Spectral Broadening in a Photonic Crystal Fiber Laser System,” IEEE Photon. Technol. Lett. 24(7), 551–553 (2012). [CrossRef]

]. Nonlinear polarization rotation technique assisted with a spectral-filtering effect enables the generation of 10 W of average power at a 49 MHz repetition rate at a central wavelength of 1038 nm. A transmission grating compressor dechirped the output pulses down to 95 fs. An optical isolator is placed at the output of the oscillator to prevent light feedback into laser system. The pump power is controlled by the combination of a half-wave plate and a polarizer and the polarization of the input pulses is controlled with another half-wave plate placed in front of the microscope objective. Laser radiation in our experiments was coupled into the fiber with a coupling efficiency up to 65%. It is worth mentioning that the input beam is aligned exactly along the axis of the PCF without tilting in the whole experiment. The output beam was coupled into a high-resolution spectrometer (ANDO 6315A) to record spectrum.

As mentioned in section 2, the triangular-core PCF was slightly birefringent, so laser polarization was aligned to parallel to one of the principal axis using the half-wave plate and then laser field propagated in the fundamental in-phase supermode within the PCF. The visible laser field nearly uniformly distributed in the three fiber cores (Fig. 4(a)
Fig. 4 Results of experiment for first order mode. (a) the near field profile of the PCF output, (b) the far field profile of the PCF output and (c) the spectral intensity of the PCF output as a function of the wavelength and the average power of the input field.
, recorded at pump power of 1.04 W). The discrepancy from Fig. 3(d) should be caused by measurement. A typical six-fold symmetry pattern called in-phase supermode (Fig. 4(b)) was obtained in far-field recorded by a CCD camera. In addition, there was a annular violet outside the pattern which was not expected from our design and was probably due to a Cherenkov radiation generation in a higher mode [30

30. A. V. Gorbach, D. V. Skryabin, J. M. Stone, and J. C. Knight, “Four-wave mixing of solitons with radiation and quasi-nondispersive wave packets at the short-wavelength edge of a supercontinuum,” Opt. Express 14(21), 9854–9863 (2006). [CrossRef] [PubMed]

]. Figure 5(c)
Fig. 5 Experiment results of the out-phase supermode superposition. (a) the far field profile of the PCF when one mode’s output is visible, (b) the far field profile of the superposed mode and (c) the spectral intensity of the PCF output with a input power of 1.04 W.
shows the spectrum evolving with the input power up to 1.04 W. A supercontinuum is obtained in the output of fiber thanks to the small effective mode area (Fig. 2(b)) and high peak power. The spectrum is characteristic of red-shifted soliton and trapping dispersive waves, lining with group-velocity matching. A relative high power on short-wavelength side of the pump (occupying about 56% of the total energy at pump power of 1.04 W) is due to that the pump pulses are close to the ZDW, allowing more spectral component of the solitons to cross ZDW and transform to dispersive wave.

4. Conclusion

In conclusion, we experimentally demonstrated a hollow beam supercontinuum generation in a GeO2 doped triangular-core PCF. The GeO2-doping assists in structuring the guiding region: Each doped region can act as an independent waveguide and its phase can be controlled by the incident polarization. The hollow beam is a superposition of out-phase supermode but can be easily excited with a suitable incidence polarization. In this case, the PCF behaves as a simple mode convertors along with nonlinear-optical transformation. In addition, an in-phase supermode was obtained when the laser was polarized parallel to one of the main axis of the PCF. The high pump peak power gives rise to supercontinuum generations both in hollow mode and in-phase supermode, and the spectra can be controlled by input power due to the inherent feature of nonlinear broadening. The output wideband free-space hollow beam supplies a flexible laser source in both life and physical sciences.

Acknowledgments

We are grateful to Minglie Hu, professor of Ultrafast Laser Lab of Tianjin University, for enlightening guidance and modification on the manuscript. We also wish to thank Lili Huang, Xiaohui Fang and Dapeng Zhang for kindly providing the femtosecond laser and their help.

References and links

1.

I. Manek, Y. B. Ovchinnikov, and R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147(1-3), 67–70 (1998). [CrossRef]

2.

S. R. Mishra, S. K. Tiwari, S. P. Ram, and S. C. Mehendale, “Generation of hollow conic beams using a metal axicon mirror,” Opt. Eng. 46(8), 084002 (2007). [CrossRef]

3.

S. K. Tiwari, S. R. Mishra, and S. P. Ram, “Generation of a variable-diameter collimated hollow laser beam using metal axicon mirrors,” Opt. Eng. 50(1), 014001 (2011). [CrossRef]

4.

S. Marksteiner, C. M. Savage, P. Zoller, and S. L. Rolston, “Coherent atomic waveguides from hollow optical fibers: Quantized atomic motion,” Phys. Rev. A 50(3), 2680–2690 (1994). [CrossRef] [PubMed]

5.

J. P. Yin, H. R. Noh, K. I. Lee, K. H. Kim, Y. Z. Wang, and W. Jhe, “Generation of a dark hollow beam by a small hollow fiber,” Opt. Commun. 138(4-6), 287–292 (1997). [CrossRef]

6.

Y. I. Shin, K. Kim, J. A. Kim, H. R. Noh, W. Jhe, K. Oh, and U. C. Paek, “Diffraction-limited dark laser spot produced by a hollow optical fiber,” Opt. Lett. 26(3), 119–121 (2001). [CrossRef] [PubMed]

7.

H. R. Noh and W. Jhe, “Atom optics with hollow optical systems,” Phys. Rep. 372(3), 269–317 (2002). [CrossRef]

8.

T. G. Euser, M. A. Schmidt, N. Y. Joly, C. Gabriel, C. Marquardt, L. Y. Zang, M. Förtsch, P. Banzer, A. Brenn, D. Elser, M. Scharrer, G. Leuchs, and P. S. J. Russell, “Birefringence and dispersion of cylindrically polarized modes in nanobore photonic crystal fiber,” J. Opt. Soc. Am. B 28(1), 193–198 (2011). [CrossRef]

9.

H. R. Li and J. P. Yin, “Generation of a vectorial elliptic hollow beam by an elliptic hollow fiber,” Opt. Lett. 36(4), 457–459 (2011). [CrossRef] [PubMed]

10.

C. L. Zhao, Y. J. Cai, F. Wang, X. H. Lu, and Y. Z. Wang, “Generation of a high-quality partially coherent dark hollow beam with a multimode fiber,” Opt. Lett. 33(12), 1389–1391 (2008). [CrossRef] [PubMed]

11.

G. Schweiger, R. Nett, B. Özel, and T. Weigel, “Generation of hollow beams by spiral rays in multimode light guides,” Opt. Express 18(5), 4510–4517 (2010). [CrossRef] [PubMed]

12.

C. C. Yan, D. H. Zhang, D. D. Li, H. J. Bian, Z. J. Xu, and Y. K. Wang, “Metal nanorod-based metamaterials for beam splitting and a subdiffraction-limited dark hollow light cone,” J. Opt. 13(8), 085102 (2011). [CrossRef]

13.

A. Witkowska, S. G. Leon-Saval, A. Pham, and T. A. Birks, “All-fiber LP11 mode convertors,” Opt. Lett. 33(4), 306–308 (2008). [CrossRef] [PubMed]

14.

N. Bokor and N. Davidson, “Generation of a hollow dark spherical spot by 4pi focusing of a radially polarized Laguerre-Gaussian beam,” Opt. Lett. 31(2), 149–151 (2006). [CrossRef] [PubMed]

15.

M. L. Hu, C. Y. Wang, Y. J. Song, Y. F. Li, L. Chai, E. E. Serebryannikov, and A. M. Zheltikov, “A hollow beam from a holey fiber,” Opt. Express 14(9), 4128–4134 (2006). [CrossRef] [PubMed]

16.

L. G. Wang, L. Q. Wang, and S. Y. Zhu, “Formation of optical vortices using coherent laser beam arrays,” Opt. Commun. 282(6), 1088–1094 (2009). [CrossRef]

17.

L. G. Wang and W. W. Zheng, “The effect of atmospheric turbulence on the propagation properties of optical vortices formed by using coherent laser beam arrays,” J. Opt. A, Pure Appl. Opt. 11(6), 065703 (2009). [CrossRef]

18.

G. Q. Zhou, “Propagation of a radial phased-locked Lorentz beam array in turbulent atmosphere,” Opt. Express 19(24), 24699–24711 (2011). [CrossRef] [PubMed]

19.

P. Zhou, X. L. Wang, Y. X. Ma, H. T. Ma, X. J. Xu, and Z. J. Liu, “Generation of a hollow beam by active phasing of a laser array using a stochastic parallel gradient descent algorithm,” J. Opt. 12(1), 015401 (2010). [CrossRef]

20.

Y. Zheng, X. Wang, F. Shen, and X. Li, “Generation of dark hollow beam via coherent combination based on adaptive optics,” Opt. Express 18(26), 26946–26958 (2010). [CrossRef] [PubMed]

21.

P. Li, K. B. Shi, and Z. W. Liu, “Manipulation and spectroscopy of a single particle by use of white-light optical tweezers,” Opt. Lett. 30(2), 156–158 (2005). [CrossRef] [PubMed]

22.

K. B. Shi, P. Li, and Z. W. Liu, “Broadband coherent anti-Stokes Raman scattering spectroscopy in supercontinuum optical trap,” Appl. Phys. Lett. 90(14), 141116 (2007). [CrossRef]

23.

J. C. Knight, T. A. Birks, P. S. J. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. 21(19), 1547–1549 (1996). [CrossRef] [PubMed]

24.

S. G. Yang, Y. J. Zhang, X. Z. Peng, Y. Lu, S. H. Xie, J. Li, W. Chen, Z. Jiang, J. Peng, and H. Li, “Theoretical study and experimental fabrication of high negative dispersion photonic crystal fiber with large area mode field,” Opt. Express 14(7), 3015–3023 (2006). [CrossRef] [PubMed]

25.

X. H. Fang, M. L. Hu, Y. F. L. Chai, and C. Y. Wang, “Spatially Flat In-Phase Supermode in Multicore Hybrid Photonic Crystal Fiber,” J. Lightwave Technol. 29(22), 3428–3432 (2011). [CrossRef]

26.

M. Digonnet and H. J. Shaw, “Wavelength multiplexing in single-mode fiber couplers,” Appl. Opt. 22(3), 484–491 (1983). [CrossRef] [PubMed]

27.

S. Konorov, E. Serebryannikov, A. Zheltikov, P. Zhou, A. Tarasevitch, and D. von der Linde, “Mode-controlled colors from microstructure fibers,” Opt. Express 12(5), 730–735 (2004). [CrossRef] [PubMed]

28.

C. Xie, M. L. Hu, D. P. Zhang, C. L. Gu, Y. J. Song, L. Chai, and C. Y. Wang, “Generation of 25-fs High Energy Pulses by SPM-Induced Spectral Broadening in a Photonic Crystal Fiber Laser System,” IEEE Photon. Technol. Lett. 24(7), 551–553 (2012). [CrossRef]

29.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78(4), 1135–1184 (2006). [CrossRef]

30.

A. V. Gorbach, D. V. Skryabin, J. M. Stone, and J. C. Knight, “Four-wave mixing of solitons with radiation and quasi-nondispersive wave packets at the short-wavelength edge of a supercontinuum,” Opt. Express 14(21), 9854–9863 (2006). [CrossRef] [PubMed]

31.

N. Nishizawa and T. Goto, “Pulse trapping by ultrashort soliton pulses in optical fibers across zero-dispersion wavelength,” Opt. Lett. 27(3), 152–154 (2002). [CrossRef] [PubMed]

32.

A. Kudlinski, G. Bouwmans, O. Vanvincq, Y. Quiquempois, A. Le Rouge, L. Bigot, G. Mélin, and A. Mussot, “White-light cw-pumped supercontinuum generation in highly GeO2-doped-core photonic crystal fibers,” Opt. Lett. 34(23), 3631–3633 (2009). [CrossRef] [PubMed]

33.

G. Genty, M. Lehtonen, and H. Ludvigsen, “Effect of cross-phase modulation on supercontinuum generated in microstructured fibers with sub-30 fs pulses,” Opt. Express 12(19), 4614–4624 (2004). [CrossRef] [PubMed]

34.

M. H. Frosz, P. M. Moselund, P. D. Rasmussen, C. L. Thomsen, and O. Bang, “Increasing the blue-shift of a supercontinuum by modifying the fiber glass composition,” Opt. Express 16(25), 21076–21086 (2008). [CrossRef] [PubMed]

OCIS Codes
(190.4370) Nonlinear optics : Nonlinear optics, fibers
(190.7110) Nonlinear optics : Ultrafast nonlinear optics
(060.5295) Fiber optics and optical communications : Photonic crystal fibers
(320.6629) Ultrafast optics : Supercontinuum generation

ToC Category:
Ultrafast Optics

History
Original Manuscript: June 4, 2012
Revised Manuscript: July 26, 2012
Manuscript Accepted: July 27, 2012
Published: August 14, 2012

Citation
Xin-ben Zhang, Xian Zhu, Xiang Chen, Hai-qing Li, Jing-gang Peng, Neng-li Dai, and Jin-yan Li, "A hollow beam supercontinuum generation by the supermode superposition in a GeO2 doped triangular-core photonic crystal fiber," Opt. Express 20, 19799-19805 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-18-19799


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References

  1. I. Manek, Y. B. Ovchinnikov, R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147(1-3), 67–70 (1998). [CrossRef]
  2. S. R. Mishra, S. K. Tiwari, S. P. Ram, S. C. Mehendale, “Generation of hollow conic beams using a metal axicon mirror,” Opt. Eng. 46(8), 084002 (2007). [CrossRef]
  3. S. K. Tiwari, S. R. Mishra, S. P. Ram, “Generation of a variable-diameter collimated hollow laser beam using metal axicon mirrors,” Opt. Eng. 50(1), 014001 (2011). [CrossRef]
  4. S. Marksteiner, C. M. Savage, P. Zoller, S. L. Rolston, “Coherent atomic waveguides from hollow optical fibers: Quantized atomic motion,” Phys. Rev. A 50(3), 2680–2690 (1994). [CrossRef] [PubMed]
  5. J. P. Yin, H. R. Noh, K. I. Lee, K. H. Kim, Y. Z. Wang, W. Jhe, “Generation of a dark hollow beam by a small hollow fiber,” Opt. Commun. 138(4-6), 287–292 (1997). [CrossRef]
  6. Y. I. Shin, K. Kim, J. A. Kim, H. R. Noh, W. Jhe, K. Oh, U. C. Paek, “Diffraction-limited dark laser spot produced by a hollow optical fiber,” Opt. Lett. 26(3), 119–121 (2001). [CrossRef] [PubMed]
  7. H. R. Noh, W. Jhe, “Atom optics with hollow optical systems,” Phys. Rep. 372(3), 269–317 (2002). [CrossRef]
  8. T. G. Euser, M. A. Schmidt, N. Y. Joly, C. Gabriel, C. Marquardt, L. Y. Zang, M. Förtsch, P. Banzer, A. Brenn, D. Elser, M. Scharrer, G. Leuchs, P. S. J. Russell, “Birefringence and dispersion of cylindrically polarized modes in nanobore photonic crystal fiber,” J. Opt. Soc. Am. B 28(1), 193–198 (2011). [CrossRef]
  9. H. R. Li, J. P. Yin, “Generation of a vectorial elliptic hollow beam by an elliptic hollow fiber,” Opt. Lett. 36(4), 457–459 (2011). [CrossRef] [PubMed]
  10. C. L. Zhao, Y. J. Cai, F. Wang, X. H. Lu, Y. Z. Wang, “Generation of a high-quality partially coherent dark hollow beam with a multimode fiber,” Opt. Lett. 33(12), 1389–1391 (2008). [CrossRef] [PubMed]
  11. G. Schweiger, R. Nett, B. Özel, T. Weigel, “Generation of hollow beams by spiral rays in multimode light guides,” Opt. Express 18(5), 4510–4517 (2010). [CrossRef] [PubMed]
  12. C. C. Yan, D. H. Zhang, D. D. Li, H. J. Bian, Z. J. Xu, Y. K. Wang, “Metal nanorod-based metamaterials for beam splitting and a subdiffraction-limited dark hollow light cone,” J. Opt. 13(8), 085102 (2011). [CrossRef]
  13. A. Witkowska, S. G. Leon-Saval, A. Pham, T. A. Birks, “All-fiber LP11 mode convertors,” Opt. Lett. 33(4), 306–308 (2008). [CrossRef] [PubMed]
  14. N. Bokor, N. Davidson, “Generation of a hollow dark spherical spot by 4pi focusing of a radially polarized Laguerre-Gaussian beam,” Opt. Lett. 31(2), 149–151 (2006). [CrossRef] [PubMed]
  15. M. L. Hu, C. Y. Wang, Y. J. Song, Y. F. Li, L. Chai, E. E. Serebryannikov, A. M. Zheltikov, “A hollow beam from a holey fiber,” Opt. Express 14(9), 4128–4134 (2006). [CrossRef] [PubMed]
  16. L. G. Wang, L. Q. Wang, S. Y. Zhu, “Formation of optical vortices using coherent laser beam arrays,” Opt. Commun. 282(6), 1088–1094 (2009). [CrossRef]
  17. L. G. Wang, W. W. Zheng, “The effect of atmospheric turbulence on the propagation properties of optical vortices formed by using coherent laser beam arrays,” J. Opt. A, Pure Appl. Opt. 11(6), 065703 (2009). [CrossRef]
  18. G. Q. Zhou, “Propagation of a radial phased-locked Lorentz beam array in turbulent atmosphere,” Opt. Express 19(24), 24699–24711 (2011). [CrossRef] [PubMed]
  19. P. Zhou, X. L. Wang, Y. X. Ma, H. T. Ma, X. J. Xu, Z. J. Liu, “Generation of a hollow beam by active phasing of a laser array using a stochastic parallel gradient descent algorithm,” J. Opt. 12(1), 015401 (2010). [CrossRef]
  20. Y. Zheng, X. Wang, F. Shen, X. Li, “Generation of dark hollow beam via coherent combination based on adaptive optics,” Opt. Express 18(26), 26946–26958 (2010). [CrossRef] [PubMed]
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