Abstract

We report the development of microscopic size gradient index vortex masks using the modified stack-and-draw technique. The vortex mask has a form of flat surface all-glass plate. Its functionality is determined by an internal nanostructure composed of two types of soft glass nanorods. The generation of optical vortices with charges 1 and 2 is demonstrated.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Optical vortices, i.e. optical beams with donut like intensity profile and helical (or screw-like) phase dislocation and associated charge and angular momentum [1] are subject of intense research into the so-called singular optics [2, 3]. The azimuthal phase variation of the vortex determines its charge, such that charge m corresponds to total phase change of 2πm. Generation and control of optical vortices is of paramount interest because of their application in many areas such as, for instance, optical trapping and manipulation [4, 5], laser micromachining [6, 7], plasmonics [8] or the Stimulated Emission-Depletion microscopy [9].

A number of techniques have been proposed for generation of optical vortices. One of the most direct relies on transmission of optical beam through helical phase mask which imposes screw-like phase dislocation in the beam which transforms it into a vortex [10, 11]. Another approach involves application of computer generated holograms [12, 13]. The diffraction of Gaussian beam from such holograms results in appearance of optical vortices in subsequent diffraction orders. The necessary phase modulation in vortex generation is commonly achieved by using oriented liquid crystals as in spatial light modulators [14]. High quality vortices with large charge (angular momentum) were realized by using spirally structured mirrors [15].

The above techniques are commonly used in fabrication of standard size vortex masks. However, they cannot be directly implemented in fabrication of micron-size beam structuring elements which are of interest for application in fiber and integrated optical devices. Recently a couple of approaches have been used to address this issue. In particular, optical laser material processing via nano-ripple formation [16], two-photon polymerization [17] or direct 3D printing [18, 19] have been successfully employed to create micrometer size vortex masks.

In this work we present for the first time generation of optical vortices by using gradient index phase masks fabricated by a modified stack-and-draw procedure similar to that employed in fabrication of microstructured fibers. Our approach is unique as it leads to flat surfaces, therefore the same component can be used in any medium as air or liquid. Since the mask is developed in glass it is resistant for high power densities. Moreover vortex masks of the diameter of tens of micrometers can be easily integrated into standard single or multimode fibers for potential application in fiber lasers, nonlinear microscopy and plasmonics.

2. Mask design and fabrication

The generation of optical vortex requires that the wave acquires an azimuthal phase change of Δφ = 2πm with m = 1,2.. being integer. In the transmission geometry such a phase modulation can be achieved by helical variation of the thickness or refractive index of the medium. Here we are concerned with the latter approach. The required azimuthal phase variation requires continuous refractive index change (Δn) such that the acquired total phase shift δφ = (2π0nd = 2πm, where λ0 denotes the wavelength in vacuum and d is the thickness of the medium. Figure 1(a) illustrates the azimuthal index distribution. Here the index (and consequently the phase) changes linearly with azimuthal angle. The vortex created by such mask will have ideal donut intensity profile and is called generic. In contrast nongeneric vortices arises from nonuniform azimuthal phase gradient and may have noncircular intensity profile. The problem is that the continuous index modulation cannot be realized in solid media.

 

Fig. 1 Refractive index distribution in gradient index (a) and nanostructured (b) vortex all-glass mask structure. Black and yellow regions in (b) correspond to low and high refractive index glass, respectively.

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To overcome this obstacle we propose to use, the so called, effective medium approach in which the medium consists of mixed nanosized regions of two materials with different refractive indices. In this case the effective refractive index of a nanostructure is determined by a spatial average of the individual refractive indices of the constituent materials. The averaging gives sufficiently accurate results while allowing rapid optimization of highly complex structures. The basic Maxwell-Garnet theory, which gives the effective permittivity εe of a composite structure of mixture of two materials, is expressed as

εe=εmεi(1+2δ)εm(2δ2)εm(2δ)+εi(1δ),
where εi and εm are the relative permittivities of the inclusion and host materials, respectively while δ is the fraction of inclusion material in the composite structure [20]. Effective medium approach is valid if the size of the individual rods is much smaller than the wavelength of the propagating light. In this case light cannot “see” individual rods as such, but rather an effective averaged medium. We have experimentally verified that this condition is valid for all glass nanostructured components if their transverse dimensions are narrower than 1/3 of the operating wavelength.

In our work we used nanorods made of two glasses with high and low refractive indices that correspond to nanosized regions with permittivities εm and εi, respectively. Following the Maxwell Garnet formula and the effective medium theory [20], the appropriate distribution of sub-wavelength nanorods, while discrete, will mimic continuous refractive index distribution. In fact we have already demonstrated effectiveness of this approach in our earlier works fabricating flat gradient lenses and lens arrays [21, 22]. In Fig. 1(b) we show the discrete refractive index distribution which mimics the continuous structure from Fig. 1(a).

For fabrication of nanostructured vortex mask we used a modified stack-and-draw method commonly employed in photonic crystal fiber manufacturing as shown in Fig. 2 [23].The initial preform is assembled manually (see Figs. 2(a) and 2(b)) from high and low refractive index glass rods of 0.5 mm in diameter, according to the pattern determined by the equation Eq. (1) with conjunction with numerical optimization algorithm, the so called simulated annealing [24]. A pair of thermally matched glasses with similar thermal coefficients is selected to ensure uniform drawing process and lack of internal tensions after the process is accomplished. We used nanorods made of two types of borosilicate glasses labeled NC21 and NC32. Their refractive indices at λ = 0.89 μm are: n = 1.518 and n = 1.543, respectively. The dispersive characteristics of both glasses are shown in Fig. 3(a). Difference of refractive indices is almost constant in the near infrared range between 800 and 1600 nm and it is equal to Δn ∝ 0.025 as shown in Fig. 3(b). Both glasses have high transmission in visible and near infrared range between 400 nm and 2100 nm and their expansion coefficient difference Δα = 0.4 × 10−7 K−1 is sufficiently small for joint thermal processing. The glasses have also relatively high laser induced damage threshold ≈1 J/cm2 in femtosecond regime, typical for borosilicate glass [25]. Therefore components based on this type of glasses are well suited for high power applications as laser material processing.

 

Fig. 2 Schematic of stack-and-draw process for vortex mask fabrication: (a) development of individual rods made of two types of thermally matched glasses, (b) assembly of initial preform according to calculated pattern, (c) drawing initial preform at fiber drawing tower, (d) assembly of intermediate preform, (e) drawing intermediate preform, (f) slicing the final fiber into individual vortex masks.

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Fig. 3 (a) Material dispersion of borosilicate glasses NC21 and NC32. (b) the mask thickness vs. beam wavelength for generation of the vortex with unit charge.

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The hexagonal lattice in the preform is placed centrally in capillary made of low index glass NC21. The empty space between inner diameter of the capillary and nanostructured area is filled also with the low index NC21 rods. The preform has an overall diameter of 55 mm and contains 101 glass rods in diagonal. After initial assembly, the structure is drawn-down in a fibre-drawing tower to a rod with the diameter of 2 mm as shown in Fig. 2(c). During this process the glasses are kept at temperature between the curvature and sphere points [23]. The structured rod is then placed into a capillary to create intermediate preforms (see Fig. 2(d)). After the drawing process we end up with a fiber 125 μm in diameter, with the vortex structure positioned at the center having 19 μm along the longer diagonal (16 μm along shorter diagonal). This final diameter corresponds to individual nanorod features of 190 nm making a fabricated component suitable for use under visible and near-IR illumination (Fig. 2(e)). The fiber is subsequently cut into slices, ground and polished such that the thickness of the final mask corresponds to desired vortex charge at a given wavelength (Fig. 2(f)). Fabrication of vortex with any charge is then straightforward task, since the mask is cut from a tens of meters long fiber.

Fabricated vortex mask is depicted in Fig. 4. In general the nanostructurization method allows one to change locally refractive index varying the refractive index of individual rods since assembly of preform allows one to control position of individual glass rods. Further processing does not affect the position of individual rods in the pattern. In the case of vortex mask considered here, the geometrical lateral resolution is 190 nm. Even though the final structure is affected by presence of diffusion between neighboring rods, Fig. 4(c) clearly shows that the features of the mask are way below the 1 micron, and hence the mask fulfills the effective medium condition.

 

Fig. 4 Fabricated vortex mask. (a) Scanning Electron Microscope (SEM) image of the end of the whole drawn fiber. Bright (dark) regions represent high (low) index glass. (b) SEM image of the vortex mask structure. (c) Magnified part of vortex structure with visible individual nanorods.

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3. Generation of optical vortices

The fabricated masks were optically tested by using both continuous wave (CW) as well as short pulse laser beams. While the former’s wavelength was fixed at λ0 = 532 nm, the latter (originating from Coherent Chameleon Ultra II femtosecond laser) could be continuously varied from 680 nm to 1080 nm. In the experiment the mask was glued to an edge of the 130 μm microscope slide. The light beam was collimated and then focused with microscope objective lens × 10 (NA = 0.25) into a 12 μm diameter spot at the center of the mask. The transmitted light was then collimated with microscope objective × 20 and imaged onto the diffusive screen located at 60 cm distance from the mask. The resulting donut-like light intensity distribution was recorded with CCD camera. To determine the charge of the vortex we used two different techniques.

In the experiment with CW light beam, the vortex beam was superposed with the reference beam. This resulted in either fork or spiral intensity pattern for oblique or collinear interference, respectively. Interferometric diagnostic is rather inconvenient for ultrashort pulses. In this case we employed the known effect of vortex deformation by astigmatic transformation [26]. The transformation which can be achieved by transmitting the vortex beam through cylindrical lens, results in vortex breakup and appearance of dark and bright extended regions in the focal region [27]. This is the number of dark regions which indicates the charge of the vortex. For instance, for charge 1 or 2 the focal pattern consists of one or two dark regions separated by bright regions, respectively. In our experiments we introduced the astigmatism by slightly tilting the imaging lens.

The results of our experiments are depicted in Figs. 5-8. In Fig. 5 we show results of illumination of the vortex mask by femtosecond beam at λ = 890 nm. Here the mask is 36 μm thick. The graph in Fig. 5(a) depicts observed light intensity distribution of free propagating vortex beam. Its donut-like profile indicates singularity in the center. This is indeed confirmed by the astigmatic transformation of this beam induced by an angular misalignment of the imaging lens. It results in splitting of the vortex into two distinct bright elongated regions indicating the presence of charge m = 1. This agrees with the calculations from Fig. 2(b) which predict charge one vortex at this wavelength and mask thickness. In the graphs Fig. 5(b) we show results of numerical simulation of vortex generation by using exact discrete representation of the vortex mask as depicted in Fig. 1(b). These simulations employed in-house implementation of fully vectorial beam propagation method with perfectly matched layer boundary conditions. A plane-wave was used as input source. The spatial grid resolution was 0.1 μm. Total area of simulation was 600 μm × 600 μm × 600 μm.

 

Fig. 5 (a) Top row: Light intensity distribution and the orthogonal intensity profiles of the vortex formed after illuminating the vortex mask with femtosecond laser beam at λ0 = 890 nm. Bottom graph depicts the vortex intensity distribution after astigmatic transformation. Single dark stripe confirms charge m = 1 vortex. (b) numerically calculated far field (top row) and phase (bottom row) of the beam generated by vortex mask shown in Fig. 1(b) 500 μm distance behind the mask. The mask length is 36 μm. (c) numerically calculated far filed intensity (top) and phase (bottom) of the 10.2 micron long mask formed using high contrast glasses with refractive indices of n1 = 1.518 (NC21) and n2 = 1.605 (F2 lead-silicate glass).

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Fig. 6 Horizontal and vertical profiles of the optical vortex generated with λ = 890 nm CW beam. Measured intensity contrast in the vortex > 20:1.

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Fig. 7 (a) Light intensity distribution of the vortex formed after illuminating the nanostructured vortex mask with focused CW 532 nm beam. The donut like intensity pattern indicates presence of phase singularity-(b,c) interferometric diagnostic of generated optical vortex by using tilted and collinear (c) reference beam. The presence of the fork having two more black line at the bottom than at the top of the fork structure and a double spiral pattern indicates charge of 2. (d) the light intensity of the vortex beam after astigmatic transformation by tilted lens. The two dark regions confirm charge 2.

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Fig. 8 Horizontal and vertical profiles of the optical vortex generated with the CW 532 nm beam. Measured intensity contrast in the vortex >70:1.

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Numerically determined light intensity distribution 500 micrometers behind the mask is shown in top row of Fig. 5(b). The beam is clearly singular as confirmed by its helical phase structure (bottom row). However, it is also evident that the light intensity is azimuthally nonuniform. This nonuniformity reflects the influence of relatively long beam propagation through the mask and proves that formation of vortex in bulk gradient index medium with low refractive index contrast is different than in case of use a phase only component. In order to minimize the deleterious propagation effect the mask should be shorter which can be achieved by using different types of glasses with higher refractive index difference. This is demonstrated in Fig. 5(c) which shows far field intensity (top) and phase structure (bottom) of the vortex beam generated from structured mask using higher refractive index contrast. In this case the design involved the NC21 glass (n = 1.518) and commercial high-index lead-silicate F2 glass (n = 1.605). It is evident that the quality of the generated vortex beam is greatly improved.

The losses of the vortex mask are limited to Fresnel reflections only. Material loss of glass are negligible since thickness of the component is of the order of tens of micrometers, while attenuation of the bulk glasses is below 5 dB/m. Scattering is not observed since internal structure of vortex is much smaller than the wavelength. For completeness, In Fig. 6 we depict horizontal and vertical intensity profiles across the vortex beam.

In Fig. 7 we depict experiments with our vortex mask but conducted by using CW 532 nm beam. The first graph depicts again the characteristic donut-like intensity distribution. Compared to Fig. 5(a) the central dark region is now wider which points towards higher charge. Indeed, the corresponding interference patterns involving oblique [Fig. 6(b)] and collinear [Fig. 6(c)] reference wave show appearance of two extra fringes and two spirals indicating charge 2 vortex. This charge is also confirmed by the astigmatic transformation of the vortex beam [Fig. 6(d)] which results in appearance of two dark stripes in the focal plane. These observations agree with our calculations [Fig. 2(b)] which show that at λ0 = 532 nm formation of a single charge vortex requires 20.7 μm thick mask. Since our fabricated mask is roughly twice longer it generates charge 2 vortex. Figure 8 shows transverse intensity profiles across the vortex. It is clear that the intensity minimum is now wider than for the vortex at 890 nm. This is a direct consequence of higher charge which leads to broadening of the dark area of the vortex beam [2]. Moreover, the appearance of local intensity maximum in the y-cross-section indicates initial stage of the splitting of the charge 2 vortex into two constituent charge one vortices. Such effect is typically caused by the asymmetry of optical system [10].

4. Conclusions

In conclusion, we used a stack-and-draw technique to fabricate transmission flat phase mask for generation of optical vortices. The technique relies on stacking two types of high and low index glass rods and subsequent drawing of an optical fiber. Because of sub-wavelength dimension of low and high index regions, the light beam propagates in medium with average refractive index. In this way the final mask structure exhibits an azimuthal index gradient which produces azimuthal phase modulation of the transmitted beam resulting in its conversion into an optical vortex. We believe that these flat surface vortex masks of the diameter of tens of micrometers can be easily integrated into standard single and multimode fibers for potential application in fiber lasers, nonlinear microscopy and plasmonics. Flat surface and gradient index nature of vortex mask ensure preservation of charge in any external medium. A glass material used for vortex mask development has a high laser induced damage threshold, therefore the components can be used in applications that requires high power as laser ablation.

Funding

Qatar National Research Fund (NPRP8-246-1-060); Foundation for Polish Science Team Programme from the funds of European Regional Development Fund under Smart Growth Operational Programme (TEAM TECH/2016-1/1).

References and links

1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [PubMed]  

2. G. J. Gbur, Singular Optics (CRC, 2016).

3. D. L. Andrews, ed., Structured Light and Its Applications (Elsevier, 2008).

4. K. T. Gahagan and G. A. Swartzlander Jr., “Optical vortex trapping of particles,” Opt. Lett. 21(11), 827–829 (1996). [PubMed]  

5. V. G. Shvedov, A. V. Rode, Y. V. Izdebskaya, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Giant optical manipulation,” Phys. Rev. Lett. 105(11), 118103 (2010). [PubMed]  

6. C. Hnatovsky, V. G. Shvedov, W. Krolikowski, and A. V. Rode, “Materials processing with a tightly focused femtosecond laser vortex pulse,” Opt. Lett. 35(20), 3417–3419 (2010). [PubMed]  

7. K. Toyoda, F. Takahashi, S. Takizawa, Y. Tokizane, K. Miyamoto, R. Morita, and T. Omatsu, “Transfer of Light Helicity to Nanostructures,” Phys. Rev. Lett. 110(14), 143603 (2013). [PubMed]  

8. Z. J. Hu, P. S. Tan, S. W. Zhu, and X. C. Yuan, “Structured light for focusing surface plasmon polaritons,” Opt. Express 18(10), 10864–10870 (2010). [PubMed]  

9. S. W. Hell and J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission: Stimulated-emission-depletion fluorescence microscopy,” Opt. Lett. 19(11), 780–782 (1994). [PubMed]  

10. V. V. Kotlyar, A. A. Almazov, S. N. Khonina, V. A. Soifer, H. Elfstrom, and J. Turunen, “Generation of phase singularity through diffracting a plane or Gaussian beam by a spiral phase plate,” J. Opt. Soc. Am. A 22(5), 849–861 (2005). [PubMed]  

11. M. Massari, G. Ruffato, M. Gintoli, F. Ricci, and F. Romanato, “Fabrication and characterization of high-quality spiral phase plates for optical applications,” Appl. Opt. 54, 4077–4083 (2015).

12. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221 (1992). [PubMed]  

13. Z. S. Sacks, D. Rozas, and G. A. Swartzlander, “Holographic formation of optical-vortex filaments,” J. Opt. Soc. Am. B 15, 2226–2234 (1998).

14. D. Ganic, X. Gan, M. Gu, M. Hain, S. Somalingam, S. Stankovic, and T. Tschudi, “Generation of doughnut laser beams by use of a liquid-crystal cell with a conversion efficiency near 100,” Opt. Lett. 27(15), 1351–1353 (2002). [PubMed]  

15. G. Campbell, B. Hage, B. Buchler, and P. K. Lam, “Generation of high-order optical vortices using directly machined spiral phase mirrors,” Appl. Opt. 51(7), 873–876 (2012). [PubMed]  

16. E. Brasselet, A. Royon, and L. Canionic, “Dense arrays of microscopic optical vortex generators from femtosecond direct laser writing of radial birefringence in glass,” Appl. Phys. Lett. 100, 181901 (2012).

17. A. Zukauskas, M. Malinauskas, and E. Brasselet, “Monolithic generators of pseudo-nondiffracting optical vortex beams at the microscale,” Appl. Phys. Lett. 103, 181122 (2013).

18. K. Weber, F. Hütt, S. Thiele, T. Gissibl, A. Herkommer, and H. Giessen, “Single mode fiber based delivery of OAM light by 3D direct laser writing,” Opt. Express 25(17), 19672–19679 (2017). [PubMed]  

19. S. Lightman, G. Hurvitz, R. Gvishi, and A. Arie, “Miniature wide-spectrum mode sorter for vortex beams produced by 3D laser printing,” Optica 4, 605–610 (2017).

20. A. Sihvola, Electromagnetic Mixing Formulas and Applications (The Institution of Electrical Engineers, 1999).

21. F. Hudelist, R. Buczynski, A. J. Waddie, and M. R. Taghizadeh, “Design and fabrication of nano-structured gradient index microlenses,” Opt. Express 17(5), 3255–3263 (2009). [PubMed]  

22. R. Buczyński, M. Klimczak, T. Stefaniuk, R. Kasztelanic, B. Siwicki, G. Stępniewski, J. Cimek, D. Pysz, and R. Stępień, “Optical fibers with gradient index nanostructured core,” Opt. Express 23(20), 25588–25596 (2015). [PubMed]  

23. D. Pysz, I. Kujawa, R. Stepien, M. Klimczak, A. Filipkowski, M. Franczyk, L. Kociszewski, J. Buzniak, K. Harasny, and R. Buczynski, “Stack and draw fabrication of soft glass microstructured fiber optics,” Bull. Pol. Acad. Sci. Tech. Sci. 62(4), 667–683 (2014).

24. S. Kirkpatrick, C. D. Gelatt Jr, and M. P. Vecchi, “Optimization by Simulated Annealing,” Science 220(4598), 671–680 (1983). [PubMed]  

25. A. Ben-Yakar and R. L. Byer, “Femtosecond laser ablation properties of borosilicate glass,” J. Appl. Phys. 96(9), 5316–5323 (2004).

26. V. Denisenko, V. Shvedov, A. S. Desyatnikov, D. N. Neshev, W. Krolikowski, A. Volyar, M. Soskin, and Y. S. Kivshar, “Determination of topological charges of polychromatic optical vortices,” Opt. Express 17(26), 23374–23379 (2009). [PubMed]  

27. V. G. Shvedov, C. Hnatovsky, W. Krolikowski, and A. V. Rode, “Efficient beam converter for the generation of high-power femtosecond vortices,” Opt. Lett. 35(15), 2660–2662 (2010). [PubMed]  

References

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  1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
    [PubMed]
  2. G. J. Gbur, Singular Optics (CRC, 2016).
  3. D. L. Andrews, ed., Structured Light and Its Applications (Elsevier, 2008).
  4. K. T. Gahagan and G. A. Swartzlander., “Optical vortex trapping of particles,” Opt. Lett. 21(11), 827–829 (1996).
    [PubMed]
  5. V. G. Shvedov, A. V. Rode, Y. V. Izdebskaya, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Giant optical manipulation,” Phys. Rev. Lett. 105(11), 118103 (2010).
    [PubMed]
  6. C. Hnatovsky, V. G. Shvedov, W. Krolikowski, and A. V. Rode, “Materials processing with a tightly focused femtosecond laser vortex pulse,” Opt. Lett. 35(20), 3417–3419 (2010).
    [PubMed]
  7. K. Toyoda, F. Takahashi, S. Takizawa, Y. Tokizane, K. Miyamoto, R. Morita, and T. Omatsu, “Transfer of Light Helicity to Nanostructures,” Phys. Rev. Lett. 110(14), 143603 (2013).
    [PubMed]
  8. Z. J. Hu, P. S. Tan, S. W. Zhu, and X. C. Yuan, “Structured light for focusing surface plasmon polaritons,” Opt. Express 18(10), 10864–10870 (2010).
    [PubMed]
  9. S. W. Hell and J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission: Stimulated-emission-depletion fluorescence microscopy,” Opt. Lett. 19(11), 780–782 (1994).
    [PubMed]
  10. V. V. Kotlyar, A. A. Almazov, S. N. Khonina, V. A. Soifer, H. Elfstrom, and J. Turunen, “Generation of phase singularity through diffracting a plane or Gaussian beam by a spiral phase plate,” J. Opt. Soc. Am. A 22(5), 849–861 (2005).
    [PubMed]
  11. M. Massari, G. Ruffato, M. Gintoli, F. Ricci, and F. Romanato, “Fabrication and characterization of high-quality spiral phase plates for optical applications,” Appl. Opt. 54, 4077–4083 (2015).
  12. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221 (1992).
    [PubMed]
  13. Z. S. Sacks, D. Rozas, and G. A. Swartzlander, “Holographic formation of optical-vortex filaments,” J. Opt. Soc. Am. B 15, 2226–2234 (1998).
  14. D. Ganic, X. Gan, M. Gu, M. Hain, S. Somalingam, S. Stankovic, and T. Tschudi, “Generation of doughnut laser beams by use of a liquid-crystal cell with a conversion efficiency near 100,” Opt. Lett. 27(15), 1351–1353 (2002).
    [PubMed]
  15. G. Campbell, B. Hage, B. Buchler, and P. K. Lam, “Generation of high-order optical vortices using directly machined spiral phase mirrors,” Appl. Opt. 51(7), 873–876 (2012).
    [PubMed]
  16. E. Brasselet, A. Royon, and L. Canionic, “Dense arrays of microscopic optical vortex generators from femtosecond direct laser writing of radial birefringence in glass,” Appl. Phys. Lett. 100, 181901 (2012).
  17. A. Zukauskas, M. Malinauskas, and E. Brasselet, “Monolithic generators of pseudo-nondiffracting optical vortex beams at the microscale,” Appl. Phys. Lett. 103, 181122 (2013).
  18. K. Weber, F. Hütt, S. Thiele, T. Gissibl, A. Herkommer, and H. Giessen, “Single mode fiber based delivery of OAM light by 3D direct laser writing,” Opt. Express 25(17), 19672–19679 (2017).
    [PubMed]
  19. S. Lightman, G. Hurvitz, R. Gvishi, and A. Arie, “Miniature wide-spectrum mode sorter for vortex beams produced by 3D laser printing,” Optica 4, 605–610 (2017).
  20. A. Sihvola, Electromagnetic Mixing Formulas and Applications (The Institution of Electrical Engineers, 1999).
  21. F. Hudelist, R. Buczynski, A. J. Waddie, and M. R. Taghizadeh, “Design and fabrication of nano-structured gradient index microlenses,” Opt. Express 17(5), 3255–3263 (2009).
    [PubMed]
  22. R. Buczyński, M. Klimczak, T. Stefaniuk, R. Kasztelanic, B. Siwicki, G. Stępniewski, J. Cimek, D. Pysz, and R. Stępień, “Optical fibers with gradient index nanostructured core,” Opt. Express 23(20), 25588–25596 (2015).
    [PubMed]
  23. D. Pysz, I. Kujawa, R. Stepien, M. Klimczak, A. Filipkowski, M. Franczyk, L. Kociszewski, J. Buzniak, K. Harasny, and R. Buczynski, “Stack and draw fabrication of soft glass microstructured fiber optics,” Bull. Pol. Acad. Sci. Tech. Sci. 62(4), 667–683 (2014).
  24. S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by Simulated Annealing,” Science 220(4598), 671–680 (1983).
    [PubMed]
  25. A. Ben-Yakar and R. L. Byer, “Femtosecond laser ablation properties of borosilicate glass,” J. Appl. Phys. 96(9), 5316–5323 (2004).
  26. V. Denisenko, V. Shvedov, A. S. Desyatnikov, D. N. Neshev, W. Krolikowski, A. Volyar, M. Soskin, and Y. S. Kivshar, “Determination of topological charges of polychromatic optical vortices,” Opt. Express 17(26), 23374–23379 (2009).
    [PubMed]
  27. V. G. Shvedov, C. Hnatovsky, W. Krolikowski, and A. V. Rode, “Efficient beam converter for the generation of high-power femtosecond vortices,” Opt. Lett. 35(15), 2660–2662 (2010).
    [PubMed]

2017 (2)

2015 (2)

2014 (1)

D. Pysz, I. Kujawa, R. Stepien, M. Klimczak, A. Filipkowski, M. Franczyk, L. Kociszewski, J. Buzniak, K. Harasny, and R. Buczynski, “Stack and draw fabrication of soft glass microstructured fiber optics,” Bull. Pol. Acad. Sci. Tech. Sci. 62(4), 667–683 (2014).

2013 (2)

K. Toyoda, F. Takahashi, S. Takizawa, Y. Tokizane, K. Miyamoto, R. Morita, and T. Omatsu, “Transfer of Light Helicity to Nanostructures,” Phys. Rev. Lett. 110(14), 143603 (2013).
[PubMed]

A. Zukauskas, M. Malinauskas, and E. Brasselet, “Monolithic generators of pseudo-nondiffracting optical vortex beams at the microscale,” Appl. Phys. Lett. 103, 181122 (2013).

2012 (2)

G. Campbell, B. Hage, B. Buchler, and P. K. Lam, “Generation of high-order optical vortices using directly machined spiral phase mirrors,” Appl. Opt. 51(7), 873–876 (2012).
[PubMed]

E. Brasselet, A. Royon, and L. Canionic, “Dense arrays of microscopic optical vortex generators from femtosecond direct laser writing of radial birefringence in glass,” Appl. Phys. Lett. 100, 181901 (2012).

2010 (4)

2009 (2)

2005 (1)

2004 (1)

A. Ben-Yakar and R. L. Byer, “Femtosecond laser ablation properties of borosilicate glass,” J. Appl. Phys. 96(9), 5316–5323 (2004).

2002 (1)

1998 (1)

1996 (1)

1994 (1)

1992 (2)

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221 (1992).
[PubMed]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[PubMed]

1983 (1)

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by Simulated Annealing,” Science 220(4598), 671–680 (1983).
[PubMed]

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[PubMed]

Almazov, A. A.

Arie, A.

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[PubMed]

Ben-Yakar, A.

A. Ben-Yakar and R. L. Byer, “Femtosecond laser ablation properties of borosilicate glass,” J. Appl. Phys. 96(9), 5316–5323 (2004).

Brasselet, E.

A. Zukauskas, M. Malinauskas, and E. Brasselet, “Monolithic generators of pseudo-nondiffracting optical vortex beams at the microscale,” Appl. Phys. Lett. 103, 181122 (2013).

E. Brasselet, A. Royon, and L. Canionic, “Dense arrays of microscopic optical vortex generators from femtosecond direct laser writing of radial birefringence in glass,” Appl. Phys. Lett. 100, 181901 (2012).

Buchler, B.

Buczynski, R.

Buzniak, J.

D. Pysz, I. Kujawa, R. Stepien, M. Klimczak, A. Filipkowski, M. Franczyk, L. Kociszewski, J. Buzniak, K. Harasny, and R. Buczynski, “Stack and draw fabrication of soft glass microstructured fiber optics,” Bull. Pol. Acad. Sci. Tech. Sci. 62(4), 667–683 (2014).

Byer, R. L.

A. Ben-Yakar and R. L. Byer, “Femtosecond laser ablation properties of borosilicate glass,” J. Appl. Phys. 96(9), 5316–5323 (2004).

Campbell, G.

Canionic, L.

E. Brasselet, A. Royon, and L. Canionic, “Dense arrays of microscopic optical vortex generators from femtosecond direct laser writing of radial birefringence in glass,” Appl. Phys. Lett. 100, 181901 (2012).

Cimek, J.

Denisenko, V.

Desyatnikov, A. S.

V. G. Shvedov, A. V. Rode, Y. V. Izdebskaya, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Giant optical manipulation,” Phys. Rev. Lett. 105(11), 118103 (2010).
[PubMed]

V. Denisenko, V. Shvedov, A. S. Desyatnikov, D. N. Neshev, W. Krolikowski, A. Volyar, M. Soskin, and Y. S. Kivshar, “Determination of topological charges of polychromatic optical vortices,” Opt. Express 17(26), 23374–23379 (2009).
[PubMed]

Elfstrom, H.

Filipkowski, A.

D. Pysz, I. Kujawa, R. Stepien, M. Klimczak, A. Filipkowski, M. Franczyk, L. Kociszewski, J. Buzniak, K. Harasny, and R. Buczynski, “Stack and draw fabrication of soft glass microstructured fiber optics,” Bull. Pol. Acad. Sci. Tech. Sci. 62(4), 667–683 (2014).

Franczyk, M.

D. Pysz, I. Kujawa, R. Stepien, M. Klimczak, A. Filipkowski, M. Franczyk, L. Kociszewski, J. Buzniak, K. Harasny, and R. Buczynski, “Stack and draw fabrication of soft glass microstructured fiber optics,” Bull. Pol. Acad. Sci. Tech. Sci. 62(4), 667–683 (2014).

Gahagan, K. T.

Gan, X.

Ganic, D.

Gelatt, C. D.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by Simulated Annealing,” Science 220(4598), 671–680 (1983).
[PubMed]

Giessen, H.

Gintoli, M.

Gissibl, T.

Gu, M.

Gvishi, R.

Hage, B.

Hain, M.

Harasny, K.

D. Pysz, I. Kujawa, R. Stepien, M. Klimczak, A. Filipkowski, M. Franczyk, L. Kociszewski, J. Buzniak, K. Harasny, and R. Buczynski, “Stack and draw fabrication of soft glass microstructured fiber optics,” Bull. Pol. Acad. Sci. Tech. Sci. 62(4), 667–683 (2014).

Heckenberg, N. R.

Hell, S. W.

Herkommer, A.

Hnatovsky, C.

Hu, Z. J.

Hudelist, F.

Hurvitz, G.

Hütt, F.

Izdebskaya, Y. V.

V. G. Shvedov, A. V. Rode, Y. V. Izdebskaya, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Giant optical manipulation,” Phys. Rev. Lett. 105(11), 118103 (2010).
[PubMed]

Kasztelanic, R.

Khonina, S. N.

Kirkpatrick, S.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by Simulated Annealing,” Science 220(4598), 671–680 (1983).
[PubMed]

Kivshar, Y. S.

V. G. Shvedov, A. V. Rode, Y. V. Izdebskaya, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Giant optical manipulation,” Phys. Rev. Lett. 105(11), 118103 (2010).
[PubMed]

V. Denisenko, V. Shvedov, A. S. Desyatnikov, D. N. Neshev, W. Krolikowski, A. Volyar, M. Soskin, and Y. S. Kivshar, “Determination of topological charges of polychromatic optical vortices,” Opt. Express 17(26), 23374–23379 (2009).
[PubMed]

Klimczak, M.

R. Buczyński, M. Klimczak, T. Stefaniuk, R. Kasztelanic, B. Siwicki, G. Stępniewski, J. Cimek, D. Pysz, and R. Stępień, “Optical fibers with gradient index nanostructured core,” Opt. Express 23(20), 25588–25596 (2015).
[PubMed]

D. Pysz, I. Kujawa, R. Stepien, M. Klimczak, A. Filipkowski, M. Franczyk, L. Kociszewski, J. Buzniak, K. Harasny, and R. Buczynski, “Stack and draw fabrication of soft glass microstructured fiber optics,” Bull. Pol. Acad. Sci. Tech. Sci. 62(4), 667–683 (2014).

Kociszewski, L.

D. Pysz, I. Kujawa, R. Stepien, M. Klimczak, A. Filipkowski, M. Franczyk, L. Kociszewski, J. Buzniak, K. Harasny, and R. Buczynski, “Stack and draw fabrication of soft glass microstructured fiber optics,” Bull. Pol. Acad. Sci. Tech. Sci. 62(4), 667–683 (2014).

Kotlyar, V. V.

Krolikowski, W.

Kujawa, I.

D. Pysz, I. Kujawa, R. Stepien, M. Klimczak, A. Filipkowski, M. Franczyk, L. Kociszewski, J. Buzniak, K. Harasny, and R. Buczynski, “Stack and draw fabrication of soft glass microstructured fiber optics,” Bull. Pol. Acad. Sci. Tech. Sci. 62(4), 667–683 (2014).

Lam, P. K.

Lightman, S.

Malinauskas, M.

A. Zukauskas, M. Malinauskas, and E. Brasselet, “Monolithic generators of pseudo-nondiffracting optical vortex beams at the microscale,” Appl. Phys. Lett. 103, 181122 (2013).

Massari, M.

McDuff, R.

Miyamoto, K.

K. Toyoda, F. Takahashi, S. Takizawa, Y. Tokizane, K. Miyamoto, R. Morita, and T. Omatsu, “Transfer of Light Helicity to Nanostructures,” Phys. Rev. Lett. 110(14), 143603 (2013).
[PubMed]

Morita, R.

K. Toyoda, F. Takahashi, S. Takizawa, Y. Tokizane, K. Miyamoto, R. Morita, and T. Omatsu, “Transfer of Light Helicity to Nanostructures,” Phys. Rev. Lett. 110(14), 143603 (2013).
[PubMed]

Neshev, D. N.

Omatsu, T.

K. Toyoda, F. Takahashi, S. Takizawa, Y. Tokizane, K. Miyamoto, R. Morita, and T. Omatsu, “Transfer of Light Helicity to Nanostructures,” Phys. Rev. Lett. 110(14), 143603 (2013).
[PubMed]

Pysz, D.

R. Buczyński, M. Klimczak, T. Stefaniuk, R. Kasztelanic, B. Siwicki, G. Stępniewski, J. Cimek, D. Pysz, and R. Stępień, “Optical fibers with gradient index nanostructured core,” Opt. Express 23(20), 25588–25596 (2015).
[PubMed]

D. Pysz, I. Kujawa, R. Stepien, M. Klimczak, A. Filipkowski, M. Franczyk, L. Kociszewski, J. Buzniak, K. Harasny, and R. Buczynski, “Stack and draw fabrication of soft glass microstructured fiber optics,” Bull. Pol. Acad. Sci. Tech. Sci. 62(4), 667–683 (2014).

Ricci, F.

Rode, A. V.

Romanato, F.

Royon, A.

E. Brasselet, A. Royon, and L. Canionic, “Dense arrays of microscopic optical vortex generators from femtosecond direct laser writing of radial birefringence in glass,” Appl. Phys. Lett. 100, 181901 (2012).

Rozas, D.

Ruffato, G.

Sacks, Z. S.

Shvedov, V.

Shvedov, V. G.

Siwicki, B.

Smith, C. P.

Soifer, V. A.

Somalingam, S.

Soskin, M.

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[PubMed]

Stankovic, S.

Stefaniuk, T.

Stepien, R.

R. Buczyński, M. Klimczak, T. Stefaniuk, R. Kasztelanic, B. Siwicki, G. Stępniewski, J. Cimek, D. Pysz, and R. Stępień, “Optical fibers with gradient index nanostructured core,” Opt. Express 23(20), 25588–25596 (2015).
[PubMed]

D. Pysz, I. Kujawa, R. Stepien, M. Klimczak, A. Filipkowski, M. Franczyk, L. Kociszewski, J. Buzniak, K. Harasny, and R. Buczynski, “Stack and draw fabrication of soft glass microstructured fiber optics,” Bull. Pol. Acad. Sci. Tech. Sci. 62(4), 667–683 (2014).

Stepniewski, G.

Swartzlander, G. A.

Taghizadeh, M. R.

Takahashi, F.

K. Toyoda, F. Takahashi, S. Takizawa, Y. Tokizane, K. Miyamoto, R. Morita, and T. Omatsu, “Transfer of Light Helicity to Nanostructures,” Phys. Rev. Lett. 110(14), 143603 (2013).
[PubMed]

Takizawa, S.

K. Toyoda, F. Takahashi, S. Takizawa, Y. Tokizane, K. Miyamoto, R. Morita, and T. Omatsu, “Transfer of Light Helicity to Nanostructures,” Phys. Rev. Lett. 110(14), 143603 (2013).
[PubMed]

Tan, P. S.

Thiele, S.

Tokizane, Y.

K. Toyoda, F. Takahashi, S. Takizawa, Y. Tokizane, K. Miyamoto, R. Morita, and T. Omatsu, “Transfer of Light Helicity to Nanostructures,” Phys. Rev. Lett. 110(14), 143603 (2013).
[PubMed]

Toyoda, K.

K. Toyoda, F. Takahashi, S. Takizawa, Y. Tokizane, K. Miyamoto, R. Morita, and T. Omatsu, “Transfer of Light Helicity to Nanostructures,” Phys. Rev. Lett. 110(14), 143603 (2013).
[PubMed]

Tschudi, T.

Turunen, J.

Vecchi, M. P.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by Simulated Annealing,” Science 220(4598), 671–680 (1983).
[PubMed]

Volyar, A.

Waddie, A. J.

Weber, K.

White, A. G.

Wichmann, J.

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[PubMed]

Yuan, X. C.

Zhu, S. W.

Zukauskas, A.

A. Zukauskas, M. Malinauskas, and E. Brasselet, “Monolithic generators of pseudo-nondiffracting optical vortex beams at the microscale,” Appl. Phys. Lett. 103, 181122 (2013).

Appl. Opt. (2)

Appl. Phys. Lett. (2)

E. Brasselet, A. Royon, and L. Canionic, “Dense arrays of microscopic optical vortex generators from femtosecond direct laser writing of radial birefringence in glass,” Appl. Phys. Lett. 100, 181901 (2012).

A. Zukauskas, M. Malinauskas, and E. Brasselet, “Monolithic generators of pseudo-nondiffracting optical vortex beams at the microscale,” Appl. Phys. Lett. 103, 181122 (2013).

Bull. Pol. Acad. Sci. Tech. Sci. (1)

D. Pysz, I. Kujawa, R. Stepien, M. Klimczak, A. Filipkowski, M. Franczyk, L. Kociszewski, J. Buzniak, K. Harasny, and R. Buczynski, “Stack and draw fabrication of soft glass microstructured fiber optics,” Bull. Pol. Acad. Sci. Tech. Sci. 62(4), 667–683 (2014).

J. Appl. Phys. (1)

A. Ben-Yakar and R. L. Byer, “Femtosecond laser ablation properties of borosilicate glass,” J. Appl. Phys. 96(9), 5316–5323 (2004).

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

Opt. Express (5)

Opt. Lett. (6)

Optica (1)

Phys. Rev. A (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[PubMed]

Phys. Rev. Lett. (2)

V. G. Shvedov, A. V. Rode, Y. V. Izdebskaya, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Giant optical manipulation,” Phys. Rev. Lett. 105(11), 118103 (2010).
[PubMed]

K. Toyoda, F. Takahashi, S. Takizawa, Y. Tokizane, K. Miyamoto, R. Morita, and T. Omatsu, “Transfer of Light Helicity to Nanostructures,” Phys. Rev. Lett. 110(14), 143603 (2013).
[PubMed]

Science (1)

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by Simulated Annealing,” Science 220(4598), 671–680 (1983).
[PubMed]

Other (3)

G. J. Gbur, Singular Optics (CRC, 2016).

D. L. Andrews, ed., Structured Light and Its Applications (Elsevier, 2008).

A. Sihvola, Electromagnetic Mixing Formulas and Applications (The Institution of Electrical Engineers, 1999).

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Figures (8)

Fig. 1
Fig. 1 Refractive index distribution in gradient index (a) and nanostructured (b) vortex all-glass mask structure. Black and yellow regions in (b) correspond to low and high refractive index glass, respectively.
Fig. 2
Fig. 2 Schematic of stack-and-draw process for vortex mask fabrication: (a) development of individual rods made of two types of thermally matched glasses, (b) assembly of initial preform according to calculated pattern, (c) drawing initial preform at fiber drawing tower, (d) assembly of intermediate preform, (e) drawing intermediate preform, (f) slicing the final fiber into individual vortex masks.
Fig. 3
Fig. 3 (a) Material dispersion of borosilicate glasses NC21 and NC32. (b) the mask thickness vs. beam wavelength for generation of the vortex with unit charge.
Fig. 4
Fig. 4 Fabricated vortex mask. (a) Scanning Electron Microscope (SEM) image of the end of the whole drawn fiber. Bright (dark) regions represent high (low) index glass. (b) SEM image of the vortex mask structure. (c) Magnified part of vortex structure with visible individual nanorods.
Fig. 5
Fig. 5 (a) Top row: Light intensity distribution and the orthogonal intensity profiles of the vortex formed after illuminating the vortex mask with femtosecond laser beam at λ0 = 890 nm. Bottom graph depicts the vortex intensity distribution after astigmatic transformation. Single dark stripe confirms charge m = 1 vortex. (b) numerically calculated far field (top row) and phase (bottom row) of the beam generated by vortex mask shown in Fig. 1(b) 500 μm distance behind the mask. The mask length is 36 μm. (c) numerically calculated far filed intensity (top) and phase (bottom) of the 10.2 micron long mask formed using high contrast glasses with refractive indices of n1 = 1.518 (NC21) and n2 = 1.605 (F2 lead-silicate glass).
Fig. 6
Fig. 6 Horizontal and vertical profiles of the optical vortex generated with λ = 890 nm CW beam. Measured intensity contrast in the vortex > 20:1.
Fig. 7
Fig. 7 (a) Light intensity distribution of the vortex formed after illuminating the nanostructured vortex mask with focused CW 532 nm beam. The donut like intensity pattern indicates presence of phase singularity-(b,c) interferometric diagnostic of generated optical vortex by using tilted and collinear (c) reference beam. The presence of the fork having two more black line at the bottom than at the top of the fork structure and a double spiral pattern indicates charge of 2. (d) the light intensity of the vortex beam after astigmatic transformation by tilted lens. The two dark regions confirm charge 2.
Fig. 8
Fig. 8 Horizontal and vertical profiles of the optical vortex generated with the CW 532 nm beam. Measured intensity contrast in the vortex >70:1.

Equations (1)

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ε e = ε m ε i (1+2δ) ε m (2δ2) ε m (2δ)+ ε i (1δ) ,

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