Abstract

Nanojet has been emerging as an interesting topic in variety photonics applications. In this paper, inspired by the properties of generalized Luneburg lens (GLLs), a two-dimensional photonic nanojet system has been developed, which focal distance can be tuned by engineering the refractive index profile of GLLs. Simulation and analysis results show that the maximum light intensity, transverse and longitudinal dimensions of the photonic nanojet are dependent on the focal distance of the GLLs, thereby, by simply varying the focal distance, it is possible to obtain localized photon fluxes with different power characteristics and spatial dimensions. This can be of interest for many promising applications, such as high-resolution optical detection, optical manipulation, technology of direct-write nano-patterning and nano-lithography.

© 2015 Optical Society of America

1. Introduction

A photonic nanojet (PNJ) is defined as an extremely narrow local light field that propagates over a distance longer than λ while maintaining a sub-wavelength full width at half-maximum (FWHM). PNJ phenomenon was originally discovered through numerical calculations by Chen et al. in 2004 [1]. It is usually associated with the focusing of the incident plane wave or Gaussian beam by relatively large lossless dielectric microspheres [2], microspheroids [3], microcylinders [4, 5], microdisks [6], microcuboids [7, 8], or microcones [9], where PNJs spurt directly from the rear surface of the structures in the transmission mode or the front surface of the structures in the reflection mode. Then this field has drawn increasing interest due to potential applications in detecting [10] and manipulating [11] nanoscale objects, maskless subwavelength nanopatterning and nanolithography [12, 13], low-loss coupled resonator optical waveguide [14, 15], and ultrahigh-density optical data storage [16, 17].

In all the mentioned applications, nanojets are expected to extend as far as possible in the forward direction with relatively small FWHM. For microsphere with uniform refractive index, simple ray optics predicts that the PNJ phenomenon in the presence of air would not occur unless the sphere refractive index n is less than 2 [18], which restricts the maximum value of n. And the smaller the refractive index is, the longer the depth of focus is. Nonetheless, the lowest available refractive index from conventional optical materials in the visible range is around 1.37 [19], which decides the largest distance a nanojet can reach is only several wavelengths. Meanwhile, in [20], Shen et al. have shown that the FWHM of PNJ will increase with the refractive index decreasing. Consequently, in order to obtain a long nanojet with small FWHM, the refractive index distribution of microsphere has to be engineered so that the energy power flow near the focal point is essentially parallel to achieve small angular deviation [20]. Recently, the multilayer microspheres with graded refractive index were proposed, and exhibited elongated nanojets [20–22].

The Luneburg lens (LLs), which is rotationally symmetric graded refractive index lens, is able to manipulate the ray paths so that, for example, an omnidirectional source can be transformed into a directive beam [23]. This lens has been implemented in parallel-plate and three-dimensional systems in order to create directive antennas for microwave applications. By engineering the refractive index distribution, the generalized Luneburg lens (GLLs), of which focal point can be tuned either inside or at a distance from the surface of the lens, can be formed. Multiple focal points of GLLs also can be generated by tailoring the refractive index distribution [24]. Especially, when the focal length of GLLs is large, caustic will occur near the focal point, which predicts a parallel power flow near the focal point. Furthermore, double-focus GLLs permits arbitrary positions of the two focal points, which can localize the power flow between the two focal spots. As aforementioned, the parallel power flow implies a long nanojet. Therefore, we anticipate the model of GLLs in microscale can form a tunable PNJ by tailoring the focal length.

In this letter, we demonstrate the formation of the PNJ based on GLLs in microscale. Through simple trajectory of the ray optics, we qualitatively find a possibly feasible method to project the PNJs by changing the focal distance of double-focus and single-focus GLLs. The numerical simulation results quantitatively confirm the prediction of ray optics. Although the model for the GLLs has been widely studied in the theory and experiment [24–26] and introduced into the area of PNJ by Geints et al. in 2011 [27, 28], to the best of our knowledge, double-focus and single-focus GLLs, which could form tunable PNJs by tailoring the focus distance, have not reported till now.

The paper is organized as follows. Section 2 gives a brief introduction on the theoretical model for the GLLs. Section 3 includes a discussion on the influence of the focal distance of the GLLs on the intensity, transverse and longitudinal dimensions of the PNJs. Section 4 serves as a conclusion.

2. Model of the GLLs

First, we show the refractive index distribution of single-focus GLLs. The refractive index contrast, which is relative to the background medium n0, can be expressed as [25]:

n(r)=n0[1+f2(rR)2]1/2/f
Where r is the radial coordinate, R is the radius of GLLs, n(r) denotes the refractive index distribution of GLLs, and f is the focal length normalized radii of GLLs. Equation (1) reduces to the refractive index distribution of a classical LLs when f = 1. First, we present the ray tracing results to show focusing properties of GLLs. As we all know, the ray optics is only suitable for πd/λ≫1. A typical diameter d of GLLs for nanojets is about 2 μm and visible light is used (λ = 0.4–0.6 μm), which has been out of the range of ray optics. Even though ray optics cannot quantitatively describe the situation, it can provide a useful qualitative rule of thumb which GLLs parameters might lead to PNJs with the desired properties [5]. Figures 1(a)-1(d) display the results of ray trajectories and corresponding refractive index distributions for different focal distances of single-focus GLLs. Figures 1(e)-1(h) display the results of ray trajectories and corresponding refractive index distributions for double-focus GLLs.

 figure: Fig. 1

Fig. 1 The refractive index distributions of single-focus GLLs with various normalized focal length for (a) f = 1.0, (b) f = 1.25, (c) f = 1.5, (d) f = 1.75. The refractive index distributions of double-focus GLLs for (e) f2 = 1.7, (f) f2 = 2.2, (g) f2 = 2.7, (h) f2 = 3.2 when the first focal length f1 is set as 1.2. Insets in each subfigures show the ray tracing results.

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The double-focus GLLs has a more complex form, which can be described as below [24]:

n(ρ)={exp[ω1(ρ,f2)],Paρ1exp[ω1(ρ,f2)+ω2(ρ,f1,Pa)+ω2(ρ,f2,Pa)],0ρPa
ω1(ρ,f2)=1πρ1arcsin(k/f2)dk(k2ρ2)1/2
ω2(ρ,i,Pa)=1πρPaarcsin(k/i)dk(k2ρ2)1/2
Where n(ρ) is the refractive index, f1 and f2 are the focal length normalized radii of the lens, i and k represent two variables. The argument ρ can be defined as ρ(r) = n(r)r, which is an invertible function of the radial position r (both normalized to unity at the edge of the lens). Pa is a parameter to denote the position of the interface between inner region and outer shell of the lens. When Pa = n(a)a≤1, n(a) denotes the value of the refractive index corresponding to radius a (0<a≤1), and a denotes the radius of the inner region of the lens, for which all the rays entering this region will be perfectly focused at the nearer focal points as shown in Figs. 1(e)-1(h).

Then, we calculate the transmitted ray path inside the GLLs by means of ray optics, these profiles are graphed in Fig. 1 for f = 1.0, 1.25, 1.5, and 1.75 for single-focus GLLs, and Pa = 0.5, f2 = 1.7, 2.2, 2.7, and 3.2 when the first focal length f1 is set as 1.2 for double-focus GLLs. According to the refractive index distributions in Fig. 1, we can find that the refractive index value decreases with the focal length increasing for the double-focus and single-focus GLLs. From the results of ray trajectories in Fig. 1, especially the red rectangular area in Figs. 1(d) and 1(h), we could intuitively see the light focuses on a certain range rather than only a spot in Fig. 1(a) so that the length of PNJ may be rather long. At the same time, the light is concentrated in horizontal two different points in Fig. 1(h) and some rays can be observed between the two spots, thus it is greatly conducive to adjust the length of PNJ by the change of the distance between two focal points.

3. Numerical results and discussion

As aforementioned, the GLLs in microscale is expected to generate a PNJ. To verify this situation, numerical simulation is performed by means of finite-difference time-domain (FDTD) method [29]. The size of the computational cell in our numerical experiment is selected as 0.01 μm and perfectly matched layers (PMLs) are chosen as the boundary conditions to minimize non-physical reflections.

The PNJ can be characterized with three main parameters as follows: the maximum light intensity enhancement Imax, the longitudinal waist L defined as the distance from the edge of the lens on the shadow side (x/λ = 1 in the intensity profile plots) to the spatial point where the intensity drops to twice of that of the incident light similar to the definition in [20, 21], and the FWHM of the central diffraction lobe. Figures 2(a)-2(d) depict the simulated envelope of the electric field intensity of single-focus GLLs for the various normalized focal length f = 1.0, 1.25, 1.5, and 1.75, respectively. Transverse magnetic (TM) incident light with 400 nm wavelength is shinned on a GLLs with 2 μm diameter. The incident light propagates along x-axis. It can be concluded that the electric hot spot for f = 1.0 is extended to a line or a jet with decreased peak intensity and broadened FWHM as the normalized focal length of GLLs increases.

 figure: Fig. 2

Fig. 2 Intensity distributions of single-focus GLLs with various normalized focal length for (a) f = 1.0, (b) f = 1.25, (c) f = 1.5, and (d) f = 1.75 for a 400 nm incident wavelength.

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To clarify the variation of L and FWHM with different f, Figs. 3(a) and 3(b) show the on-axis intensity of PNJ parallel to the x-axis passing through the center of the GLLs and the FWHM at the maximum intensity point, respectively. The inset in Fig. 3(a) depicts the variation of L with f increasing. It’s clearly that the L increase from 1.5λ to 13.77λ with large focal length, however, the FWHM increase from 0.4075λ to 0.975λ in Fig. 3(b). Thus, by simply varying the focal distance f, it is possible to obtain localized photon fluxes with different power characteristics and spatial dimensions.

 figure: Fig. 3

Fig. 3 Intensity distributions of single-focus GLLs (a) along the x axis passing through center of GLLs, and (b) along the y axis at maximum intensity spots. The insets of the right upper corner each figure correspond to the concrete value of L and FWHM with different normalized focal length f.

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Furthermore, we also numerically explore the focusing properties of double-focus GLLs in microscale by means of FDTD method. Figure 4 shows the electric field intensity envelopes of double-focus GLLs with different focal lengths. The simulated parameters are same as Fig. 3, except the varied focal lengths. To explore the effect of double-focus GLLs on PNJ, one focal length has to be set as fixed and then the other focal length is varied to observe the properties of PNJ. First, we scan f2 with a fixed f1. Figures 4(a)-4(d) show the intensity distribution of GLLs with f2 = 1.7, 2.2, 2.7, and 3.2, respectively. The first focal length is set as fixed value (f1 = 1.2). Similar to the single-focus GLLs, the increasing f2 leads to elongated PNJ, accompany with decreased peak intensity and broadened FWHM. Figures 4(e) and 4(f) show the corresponding L and FWHM for Figs. 4(a)-4(d). According to Figs. 4(e) and 4(f), the L of PNJ extend from 6.10λ to 12.10λ while the FWHM increase from 0.6625λ to 0.97λ. Certainly, the longitudinal waist L of PNJ and the corresponding FWHM are mutual contradiction, a trade-off has to be decided according to specific application.

 figure: Fig. 4

Fig. 4 Intensity distributions of double-focus GLLs with various normalized focal length for (a) f2 = 1.7, (b) f2 = 2.2, (c) f2 = 2.7, and (d) f2 = 3.2 when the first focal length f1 is set as 1.2 for a 400 nm incident wavelength. Intensity distributions of double-focus GLLs (e) along the x axis passing through center of GLLs, and (f) along the y axis at maximum intensity spots. The insets of the right upper corner each figure correspond to the concrete value of L and FWHM with different normalized focal length f2.

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Then, we study the influence of different first focal length f1 with a settled f2 (f2 = 2.7) for double-focal GLLs in Fig. 5. Figures 5(a)-5(c) show the intensity distribution of GLLs with f1 = 1.0, 1.2, 1.5, respectively. We could see that the focal region will gradually leave off the rear surface of the GLLs when the first focal length f1 increases. This will benefit to extend the optical detectable extent to far field region. At the same time, the changed tendency of the longitudinal waist L and the corresponding FWHM of PNJ in Fig. 5(d) is consistent to Fig. 4.

 figure: Fig. 5

Fig. 5 Intensity distributions of double-focus GLLs with various normalized focal length for (a) f1 = 1.0, (b) f1 = 1.2, (c) f1 = 1.5 when the second focal length f2 is set as 2.7 for a 400 nm incident wavelength. The corresponding L and FWHM of double-focus GLLs (d).

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Besides, we also study the longitudinal waist L and the corresponding FWHM of different first focal length f1 when the second focal length is set as fixed values (f2 = 1.7, 2.2, 2.7, 3.2) for double-focal GLLs in Fig. 6. According to Figs. 6(a) and 6(b), for a given f2, the L increases with large f1 accompany with increasing FWHM as well. The increasing FWHM will weaken the localization of light energy on axis, the nanojet, thus, cannot be formed strictly. This tendency is similar with that of the single-focus GLLs.

 figure: Fig. 6

Fig. 6 The longitudinal waist L and the corresponding FWHM of double-focus GLLs with various normalized first focal length f1 when the second focal length f2 is fixed as 1.7, 2.2, 2.7, 3.2 respectively for a 400 nm incident wavelength.

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For both single-focus and double-focus GLLs, the PNJ can be tuned by varying the focal length f. The L of PNJ will be elongated with longer focal length of GLLs at the price of weaker intensity and broadened FWHM. This situation can also be understood with ray tracing method, which has been carried out in Fig. 1 in Section 2. For the single-focus GLLs, Fig. 1(a) shows the ray trajectories are converged rapidly near the focal point, which also unavoidably leads to a fast divergence and thus a shorter L. However, the FWHM formed focal point would be smaller than diffraction limit (about 0.4λ). Figure 1(d) shows that the ray trajectories experience a much slower convergence and the divergence, and the longer PNJ is thus produced. Furthermore, Figs. 1(e)-1(h) display the two focal spots are located on the axis of GLLs, some rays can be observed between the two points. Therefore, by tailoring f1 or f2 with fixed another, the length of PNJ can be tuned.

Although we can qualitatively analysis and predict the formation dynamics of PNJ by ray optics, it should be emphasized that the behavior of the energy power flow (time-averaged Poynting vector) cannot be correctly inferred from the trajectory of the ray optics, as the dimensions of the layers are of the same orders of magnitude as the wavelength. Fullwave treatment is required to calculate numerically correct behavior. Figures 7(a) and 7(b) show the energy power flow in the 2-D electromagnetic field corresponding to Figs. 1(a) and 1(d). Figures 7(c) and 7(d) are arrow depictions of the energy power flow corresponding to Figs. 1(e) and 1(h). Comparing to Figs. 7(a) and 7(b), the energy power flow within the single-focus GLLs (f = 1.75) is directed to its shadow side more smoothly in space, and with smaller transverse components than the single-focus GLLs (f = 1.0). Likely, the mechanism and tendency of Figs. 7(c) and 7(d) are same to Figs. 7(a) and 7(b).

 figure: Fig. 7

Fig. 7 Arrow-plot of the energy power flow in the x-y plane. Single-focus GLLs for (a) f = 1.0 and (b) f = 1.75. Double-focus GLLs for (c) f2 = 1.7 and (d) f2 = 3.2. For double-focus GLLs, the first focal length f1 is set as 1.2.

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The manufacturing of GLLs in microscale, which is not considered in this paper, is a separate technical issue. However, plasmonic LLs has been experimently studied by Zentgraf et al. [26]. Besides, Kong et al. [21] have already mentioned that fabrication of such microlens should also be possible by utilizing nanoshell techniques [30]. The controlled synthesis of materials with refractive indices in the range 1.05 to 1.28 has been reported [31] by varying the composition and porosity of silica glass. Higher refractive indices from 1.4 to 1.9 could be obtained from various available glasses [32].

4. Conclusion

In this work, the PNJs generated by GLLs are presented. Numerically results show that the special attention has been devoted to the influence of the focal distance on the main characteristics (the maximum light intensity enhancement, the longitudinal waist, and the FWHM) of the PNJs. We have clearly shown that it is possible to control the PNJ parameters by altering the focal distance of the GLLs. At last, the fabrication of such GLLs is discussed as well. Varieties of applications could benefit from the PNJs emerging from the GLLs, especially ultra-microscopy, optical tweezers, and optical data storage.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China (Grant No. 61177061, 11204208 and 61405088); the Open Research Fund of the State Key Laboratory of Transient Optics and Photonics (Chinese Academy of Sciences); Key project of Nature Science Foundation of Tianjin No. 14JCZDJC31400.

References and links

1. Z. Chen, A. Taflove, and V. Backman, “Photonic nanojet enhancement of backscattering of light by nanoparticles: a potential novel visible-light ultramicroscopy technique,” Opt. Express 12(7), 1214–1220 (2004). [CrossRef]   [PubMed]  

2. S. Lecler, Y. Takakura, and P. Meyrueis, “Properties of a three-dimensional photonic jet,” Opt. Lett. 30(19), 2641–2643 (2005). [CrossRef]   [PubMed]  

3. L. Han, Y. Han, G. Gouesbet, J. Wang, and G. Gréhan, “Photonic jet generated by spheroidal particle with Gaussian-beam illumination,” J. Opt. Soc. Am. B 31(7), 1476–1483 (2014). [CrossRef]  

4. A. V. Itagi and W. A. Challener, “Optics of photonic nanojets,” J. Opt. Soc. Am. A 22(12), 2847–2858 (2005). [CrossRef]   [PubMed]  

5. G. Gu, R. Zhou, Z. Chen, H. Xu, G. Cai, Z. Cai, and M. Hong, “Super-long photonic nanojet generated from liquid-filled hollow microcylinder,” Opt. Lett. 40(4), 625–628 (2015). [CrossRef]   [PubMed]  

6. D. Ju, H. Pei, Y. Jiang, and X. Sun, “Controllable and enhanced nanojet effects excited by surface plasmon polariton,” Appl. Phys. Lett. 102(17), 171109 (2013). [CrossRef]  

7. V. V. Kotlyar, S. S. Stafeev, and A. Feldman, “Photonic nanojets generated using square-profile microsteps,” Appl. Opt. 53(24), 5322–5329 (2014). [CrossRef]   [PubMed]  

8. I. V. Minin, O. V. Minin, V. Pacheco-Peña, and M. Beruete, “Localized photonic jets from flat, three-dimensional dielectric cuboids in the reflection mode,” Opt. Lett. 40(10), 2329–2332 (2015). [CrossRef]   [PubMed]  

9. Y. E. Geints, A. A. Zemlyanov, and E. K. Panina, “Microaxicon-generated photonic nanojets,” J. Opt. Soc. Am. B 32(8), 1570–1574 (2015). [CrossRef]  

10. X. Li, Z. Chen, A. Taflove, and V. Backman, “Optical analysis of nanoparticles via enhanced backscattering facilitated by 3-D photonic nanojets,” Opt. Express 13(2), 526–533 (2005). [CrossRef]   [PubMed]  

11. X. Cui, D. Erni, and C. Hafner, “Optical forces on metallic nanoparticles induced by a photonic nanojet,” Opt. Express 16(18), 13560–13568 (2008). [CrossRef]   [PubMed]  

12. E. Mcleod and C. B. Arnold, “Subwavelength direct-write nanopatterning using optically trapped microspheres,” Nat. Nanotechnol. 3(7), 413–417 (2008). [CrossRef]   [PubMed]  

13. W. Wu, A. Katsnelson, O. G. Memis, and H. Mohseni, “A deep sub-wavelength process for the formation of highly uniform arrays of nanoholes and nanopillars,” Nanotechnology 18(48), 485302 (2007). [CrossRef]  

14. A. M. Kapitonov and V. N. Astratov, “Observation of nanojet-induced modes with small propagation losses in chains of coupled spherical cavities,” Opt. Lett. 32(4), 409–411 (2007). [CrossRef]   [PubMed]  

15. S. Yang and V. N. Astratov, “Photonic nanojet-induced modes in chains of size-disordered microspheres with an attenuation of only 0.08 dB per sphere,” Appl. Phys. Lett. 92(26), 261111 (2008). [CrossRef]  

16. S.-C. Kong, A. V. Sahakian, A. Heifetz, A. Taflove, and V. Backman, “Robust detection of deeply subwavelength pits in simulated optical data-storage disks using photonic jets,” Appl. Phys. Lett. 92(21), 211102 (2008). [CrossRef]  

17. S.-C. Kong, A. Sahakian, A. Taflove, and V. Backman, “Photonic nanojet-enabled optical data storage,” Opt. Express 16(18), 13713–13719 (2008). [CrossRef]   [PubMed]  

18. A. Heifetz, S.-C. Kong, A. V. Sahakian, A. Taflove, and V. Backman, “Photonic Nanojets,” J. Comput. Theor. Nanosci. 6(9), 1979–1992 (2009). [CrossRef]   [PubMed]  

19. E. F. Schubert, J. K. Kim, and J.-Q. Xi, “Low-refractive-index materials: A new class of optical thin-film materials,” Phys. Status Solidi B 244(8), 3002–3008 (2007). [CrossRef]  

20. Y. Shen, L. V. Wang, and J.-T. Shen, “Ultralong photonic nanojet formed by a two-layer dielectric microsphere,” Opt. Lett. 39(14), 4120–4123 (2014). [CrossRef]   [PubMed]  

21. S.-C. Kong, A. Taflove, and V. Backman, “Quasi one-dimensional light beam generated by a graded-index microsphere,” Opt. Express 17(5), 3722–3731 (2009). [CrossRef]   [PubMed]  

22. C. M. Ruiz and J. J. Simpson, “Detection of embedded ultra-subwavelength-thin dielectric features using elongated photonic nanojets,” Opt. Express 18(16), 16805–16812 (2010). [CrossRef]   [PubMed]  

23. R. C. Mitchell-Thomas, O. Quevedo-Teruel, T. M. McManus, S. A. R. Horsley, and Y. Hao, “Lenses on curved surfaces,” Opt. Lett. 39(12), 3551–3554 (2014). [CrossRef]   [PubMed]  

24. J. Sochacki, “Multiple-foci Luneberg lenses,” Appl. Opt. 23(23), 4444–4449 (1984). [CrossRef]   [PubMed]  

25. J. A. Lock, “Scattering of an electromagnetic plane wave by a Luneburg lens. I. Ray theory,” J. Opt. Soc. Am. A 25(12), 2971–2979 (2008). [CrossRef]   [PubMed]  

26. T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnol. 6(3), 151–155 (2011). [CrossRef]   [PubMed]  

27. Y. E. Geints, A. A. Zemlyanov, and E. K. Panina, “Photonic nanojet effect in multilayer micrometre-sized spherical particles,” Quantum Electron. 41(6), 520–525 (2011). [CrossRef]  

28. Y. E. Geints, A. A. Zemlyanov, and E. K. Panina, “Photonic nanojet calculations in layered radially inhomogennous micrometer-sized spherical particles,” J. Opt. Soc. Am. B 28(8), 1825–1830 (2011). [CrossRef]  

29. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).

30. E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003). [CrossRef]   [PubMed]  

31. J. F. Poco and L. W. Hrubesh, “Method of producing optical quality glass having a selected refractive index,” U.S. Patent 6, 158, 244, (2008).

32. O. V. Mazurin, M. V. Streltsina, and T. P. Shvaiko-Shavaikovskaya, Handbook of Glass Data (Elsevier, 1993).

References

  • View by:

  1. Z. Chen, A. Taflove, and V. Backman, “Photonic nanojet enhancement of backscattering of light by nanoparticles: a potential novel visible-light ultramicroscopy technique,” Opt. Express 12(7), 1214–1220 (2004).
    [Crossref] [PubMed]
  2. S. Lecler, Y. Takakura, and P. Meyrueis, “Properties of a three-dimensional photonic jet,” Opt. Lett. 30(19), 2641–2643 (2005).
    [Crossref] [PubMed]
  3. L. Han, Y. Han, G. Gouesbet, J. Wang, and G. Gréhan, “Photonic jet generated by spheroidal particle with Gaussian-beam illumination,” J. Opt. Soc. Am. B 31(7), 1476–1483 (2014).
    [Crossref]
  4. A. V. Itagi and W. A. Challener, “Optics of photonic nanojets,” J. Opt. Soc. Am. A 22(12), 2847–2858 (2005).
    [Crossref] [PubMed]
  5. G. Gu, R. Zhou, Z. Chen, H. Xu, G. Cai, Z. Cai, and M. Hong, “Super-long photonic nanojet generated from liquid-filled hollow microcylinder,” Opt. Lett. 40(4), 625–628 (2015).
    [Crossref] [PubMed]
  6. D. Ju, H. Pei, Y. Jiang, and X. Sun, “Controllable and enhanced nanojet effects excited by surface plasmon polariton,” Appl. Phys. Lett. 102(17), 171109 (2013).
    [Crossref]
  7. V. V. Kotlyar, S. S. Stafeev, and A. Feldman, “Photonic nanojets generated using square-profile microsteps,” Appl. Opt. 53(24), 5322–5329 (2014).
    [Crossref] [PubMed]
  8. I. V. Minin, O. V. Minin, V. Pacheco-Peña, and M. Beruete, “Localized photonic jets from flat, three-dimensional dielectric cuboids in the reflection mode,” Opt. Lett. 40(10), 2329–2332 (2015).
    [Crossref] [PubMed]
  9. Y. E. Geints, A. A. Zemlyanov, and E. K. Panina, “Microaxicon-generated photonic nanojets,” J. Opt. Soc. Am. B 32(8), 1570–1574 (2015).
    [Crossref]
  10. X. Li, Z. Chen, A. Taflove, and V. Backman, “Optical analysis of nanoparticles via enhanced backscattering facilitated by 3-D photonic nanojets,” Opt. Express 13(2), 526–533 (2005).
    [Crossref] [PubMed]
  11. X. Cui, D. Erni, and C. Hafner, “Optical forces on metallic nanoparticles induced by a photonic nanojet,” Opt. Express 16(18), 13560–13568 (2008).
    [Crossref] [PubMed]
  12. E. Mcleod and C. B. Arnold, “Subwavelength direct-write nanopatterning using optically trapped microspheres,” Nat. Nanotechnol. 3(7), 413–417 (2008).
    [Crossref] [PubMed]
  13. W. Wu, A. Katsnelson, O. G. Memis, and H. Mohseni, “A deep sub-wavelength process for the formation of highly uniform arrays of nanoholes and nanopillars,” Nanotechnology 18(48), 485302 (2007).
    [Crossref]
  14. A. M. Kapitonov and V. N. Astratov, “Observation of nanojet-induced modes with small propagation losses in chains of coupled spherical cavities,” Opt. Lett. 32(4), 409–411 (2007).
    [Crossref] [PubMed]
  15. S. Yang and V. N. Astratov, “Photonic nanojet-induced modes in chains of size-disordered microspheres with an attenuation of only 0.08 dB per sphere,” Appl. Phys. Lett. 92(26), 261111 (2008).
    [Crossref]
  16. S.-C. Kong, A. V. Sahakian, A. Heifetz, A. Taflove, and V. Backman, “Robust detection of deeply subwavelength pits in simulated optical data-storage disks using photonic jets,” Appl. Phys. Lett. 92(21), 211102 (2008).
    [Crossref]
  17. S.-C. Kong, A. Sahakian, A. Taflove, and V. Backman, “Photonic nanojet-enabled optical data storage,” Opt. Express 16(18), 13713–13719 (2008).
    [Crossref] [PubMed]
  18. A. Heifetz, S.-C. Kong, A. V. Sahakian, A. Taflove, and V. Backman, “Photonic Nanojets,” J. Comput. Theor. Nanosci. 6(9), 1979–1992 (2009).
    [Crossref] [PubMed]
  19. E. F. Schubert, J. K. Kim, and J.-Q. Xi, “Low-refractive-index materials: A new class of optical thin-film materials,” Phys. Status Solidi B 244(8), 3002–3008 (2007).
    [Crossref]
  20. Y. Shen, L. V. Wang, and J.-T. Shen, “Ultralong photonic nanojet formed by a two-layer dielectric microsphere,” Opt. Lett. 39(14), 4120–4123 (2014).
    [Crossref] [PubMed]
  21. S.-C. Kong, A. Taflove, and V. Backman, “Quasi one-dimensional light beam generated by a graded-index microsphere,” Opt. Express 17(5), 3722–3731 (2009).
    [Crossref] [PubMed]
  22. C. M. Ruiz and J. J. Simpson, “Detection of embedded ultra-subwavelength-thin dielectric features using elongated photonic nanojets,” Opt. Express 18(16), 16805–16812 (2010).
    [Crossref] [PubMed]
  23. R. C. Mitchell-Thomas, O. Quevedo-Teruel, T. M. McManus, S. A. R. Horsley, and Y. Hao, “Lenses on curved surfaces,” Opt. Lett. 39(12), 3551–3554 (2014).
    [Crossref] [PubMed]
  24. J. Sochacki, “Multiple-foci Luneberg lenses,” Appl. Opt. 23(23), 4444–4449 (1984).
    [Crossref] [PubMed]
  25. J. A. Lock, “Scattering of an electromagnetic plane wave by a Luneburg lens. I. Ray theory,” J. Opt. Soc. Am. A 25(12), 2971–2979 (2008).
    [Crossref] [PubMed]
  26. T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnol. 6(3), 151–155 (2011).
    [Crossref] [PubMed]
  27. Y. E. Geints, A. A. Zemlyanov, and E. K. Panina, “Photonic nanojet effect in multilayer micrometre-sized spherical particles,” Quantum Electron. 41(6), 520–525 (2011).
    [Crossref]
  28. Y. E. Geints, A. A. Zemlyanov, and E. K. Panina, “Photonic nanojet calculations in layered radially inhomogennous micrometer-sized spherical particles,” J. Opt. Soc. Am. B 28(8), 1825–1830 (2011).
    [Crossref]
  29. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).
  30. E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003).
    [Crossref] [PubMed]
  31. J. F. Poco and L. W. Hrubesh, “Method of producing optical quality glass having a selected refractive index,” U.S. Patent 6, 158, 244, (2008).
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2015 (3)

2014 (4)

2013 (1)

D. Ju, H. Pei, Y. Jiang, and X. Sun, “Controllable and enhanced nanojet effects excited by surface plasmon polariton,” Appl. Phys. Lett. 102(17), 171109 (2013).
[Crossref]

2011 (3)

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnol. 6(3), 151–155 (2011).
[Crossref] [PubMed]

Y. E. Geints, A. A. Zemlyanov, and E. K. Panina, “Photonic nanojet effect in multilayer micrometre-sized spherical particles,” Quantum Electron. 41(6), 520–525 (2011).
[Crossref]

Y. E. Geints, A. A. Zemlyanov, and E. K. Panina, “Photonic nanojet calculations in layered radially inhomogennous micrometer-sized spherical particles,” J. Opt. Soc. Am. B 28(8), 1825–1830 (2011).
[Crossref]

2010 (1)

2009 (2)

S.-C. Kong, A. Taflove, and V. Backman, “Quasi one-dimensional light beam generated by a graded-index microsphere,” Opt. Express 17(5), 3722–3731 (2009).
[Crossref] [PubMed]

A. Heifetz, S.-C. Kong, A. V. Sahakian, A. Taflove, and V. Backman, “Photonic Nanojets,” J. Comput. Theor. Nanosci. 6(9), 1979–1992 (2009).
[Crossref] [PubMed]

2008 (6)

S. Yang and V. N. Astratov, “Photonic nanojet-induced modes in chains of size-disordered microspheres with an attenuation of only 0.08 dB per sphere,” Appl. Phys. Lett. 92(26), 261111 (2008).
[Crossref]

S.-C. Kong, A. V. Sahakian, A. Heifetz, A. Taflove, and V. Backman, “Robust detection of deeply subwavelength pits in simulated optical data-storage disks using photonic jets,” Appl. Phys. Lett. 92(21), 211102 (2008).
[Crossref]

S.-C. Kong, A. Sahakian, A. Taflove, and V. Backman, “Photonic nanojet-enabled optical data storage,” Opt. Express 16(18), 13713–13719 (2008).
[Crossref] [PubMed]

X. Cui, D. Erni, and C. Hafner, “Optical forces on metallic nanoparticles induced by a photonic nanojet,” Opt. Express 16(18), 13560–13568 (2008).
[Crossref] [PubMed]

E. Mcleod and C. B. Arnold, “Subwavelength direct-write nanopatterning using optically trapped microspheres,” Nat. Nanotechnol. 3(7), 413–417 (2008).
[Crossref] [PubMed]

J. A. Lock, “Scattering of an electromagnetic plane wave by a Luneburg lens. I. Ray theory,” J. Opt. Soc. Am. A 25(12), 2971–2979 (2008).
[Crossref] [PubMed]

2007 (3)

W. Wu, A. Katsnelson, O. G. Memis, and H. Mohseni, “A deep sub-wavelength process for the formation of highly uniform arrays of nanoholes and nanopillars,” Nanotechnology 18(48), 485302 (2007).
[Crossref]

A. M. Kapitonov and V. N. Astratov, “Observation of nanojet-induced modes with small propagation losses in chains of coupled spherical cavities,” Opt. Lett. 32(4), 409–411 (2007).
[Crossref] [PubMed]

E. F. Schubert, J. K. Kim, and J.-Q. Xi, “Low-refractive-index materials: A new class of optical thin-film materials,” Phys. Status Solidi B 244(8), 3002–3008 (2007).
[Crossref]

2005 (3)

2004 (1)

2003 (1)

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003).
[Crossref] [PubMed]

1984 (1)

Arnold, C. B.

E. Mcleod and C. B. Arnold, “Subwavelength direct-write nanopatterning using optically trapped microspheres,” Nat. Nanotechnol. 3(7), 413–417 (2008).
[Crossref] [PubMed]

Astratov, V. N.

S. Yang and V. N. Astratov, “Photonic nanojet-induced modes in chains of size-disordered microspheres with an attenuation of only 0.08 dB per sphere,” Appl. Phys. Lett. 92(26), 261111 (2008).
[Crossref]

A. M. Kapitonov and V. N. Astratov, “Observation of nanojet-induced modes with small propagation losses in chains of coupled spherical cavities,” Opt. Lett. 32(4), 409–411 (2007).
[Crossref] [PubMed]

Backman, V.

Beruete, M.

Cai, G.

Cai, Z.

Challener, W. A.

Chen, Z.

Cui, X.

Erni, D.

Feldman, A.

Geints, Y. E.

Gouesbet, G.

Gréhan, G.

Gu, G.

Hafner, C.

Halas, N. J.

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003).
[Crossref] [PubMed]

Han, L.

Han, Y.

Hao, Y.

Heifetz, A.

A. Heifetz, S.-C. Kong, A. V. Sahakian, A. Taflove, and V. Backman, “Photonic Nanojets,” J. Comput. Theor. Nanosci. 6(9), 1979–1992 (2009).
[Crossref] [PubMed]

S.-C. Kong, A. V. Sahakian, A. Heifetz, A. Taflove, and V. Backman, “Robust detection of deeply subwavelength pits in simulated optical data-storage disks using photonic jets,” Appl. Phys. Lett. 92(21), 211102 (2008).
[Crossref]

Hong, M.

Horsley, S. A. R.

Itagi, A. V.

Jiang, Y.

D. Ju, H. Pei, Y. Jiang, and X. Sun, “Controllable and enhanced nanojet effects excited by surface plasmon polariton,” Appl. Phys. Lett. 102(17), 171109 (2013).
[Crossref]

Ju, D.

D. Ju, H. Pei, Y. Jiang, and X. Sun, “Controllable and enhanced nanojet effects excited by surface plasmon polariton,” Appl. Phys. Lett. 102(17), 171109 (2013).
[Crossref]

Kapitonov, A. M.

Katsnelson, A.

W. Wu, A. Katsnelson, O. G. Memis, and H. Mohseni, “A deep sub-wavelength process for the formation of highly uniform arrays of nanoholes and nanopillars,” Nanotechnology 18(48), 485302 (2007).
[Crossref]

Kim, J. K.

E. F. Schubert, J. K. Kim, and J.-Q. Xi, “Low-refractive-index materials: A new class of optical thin-film materials,” Phys. Status Solidi B 244(8), 3002–3008 (2007).
[Crossref]

Kong, S.-C.

S.-C. Kong, A. Taflove, and V. Backman, “Quasi one-dimensional light beam generated by a graded-index microsphere,” Opt. Express 17(5), 3722–3731 (2009).
[Crossref] [PubMed]

A. Heifetz, S.-C. Kong, A. V. Sahakian, A. Taflove, and V. Backman, “Photonic Nanojets,” J. Comput. Theor. Nanosci. 6(9), 1979–1992 (2009).
[Crossref] [PubMed]

S.-C. Kong, A. Sahakian, A. Taflove, and V. Backman, “Photonic nanojet-enabled optical data storage,” Opt. Express 16(18), 13713–13719 (2008).
[Crossref] [PubMed]

S.-C. Kong, A. V. Sahakian, A. Heifetz, A. Taflove, and V. Backman, “Robust detection of deeply subwavelength pits in simulated optical data-storage disks using photonic jets,” Appl. Phys. Lett. 92(21), 211102 (2008).
[Crossref]

Kotlyar, V. V.

Lecler, S.

Li, X.

Liu, Y.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnol. 6(3), 151–155 (2011).
[Crossref] [PubMed]

Lock, J. A.

Mcleod, E.

E. Mcleod and C. B. Arnold, “Subwavelength direct-write nanopatterning using optically trapped microspheres,” Nat. Nanotechnol. 3(7), 413–417 (2008).
[Crossref] [PubMed]

McManus, T. M.

Memis, O. G.

W. Wu, A. Katsnelson, O. G. Memis, and H. Mohseni, “A deep sub-wavelength process for the formation of highly uniform arrays of nanoholes and nanopillars,” Nanotechnology 18(48), 485302 (2007).
[Crossref]

Meyrueis, P.

Mikkelsen, M. H.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnol. 6(3), 151–155 (2011).
[Crossref] [PubMed]

Minin, I. V.

Minin, O. V.

Mitchell-Thomas, R. C.

Mohseni, H.

W. Wu, A. Katsnelson, O. G. Memis, and H. Mohseni, “A deep sub-wavelength process for the formation of highly uniform arrays of nanoholes and nanopillars,” Nanotechnology 18(48), 485302 (2007).
[Crossref]

Nordlander, P.

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003).
[Crossref] [PubMed]

Pacheco-Peña, V.

Panina, E. K.

Pei, H.

D. Ju, H. Pei, Y. Jiang, and X. Sun, “Controllable and enhanced nanojet effects excited by surface plasmon polariton,” Appl. Phys. Lett. 102(17), 171109 (2013).
[Crossref]

Prodan, E.

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003).
[Crossref] [PubMed]

Quevedo-Teruel, O.

Radloff, C.

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003).
[Crossref] [PubMed]

Ruiz, C. M.

Sahakian, A.

Sahakian, A. V.

A. Heifetz, S.-C. Kong, A. V. Sahakian, A. Taflove, and V. Backman, “Photonic Nanojets,” J. Comput. Theor. Nanosci. 6(9), 1979–1992 (2009).
[Crossref] [PubMed]

S.-C. Kong, A. V. Sahakian, A. Heifetz, A. Taflove, and V. Backman, “Robust detection of deeply subwavelength pits in simulated optical data-storage disks using photonic jets,” Appl. Phys. Lett. 92(21), 211102 (2008).
[Crossref]

Schubert, E. F.

E. F. Schubert, J. K. Kim, and J.-Q. Xi, “Low-refractive-index materials: A new class of optical thin-film materials,” Phys. Status Solidi B 244(8), 3002–3008 (2007).
[Crossref]

Shen, J.-T.

Shen, Y.

Simpson, J. J.

Sochacki, J.

Stafeev, S. S.

Sun, X.

D. Ju, H. Pei, Y. Jiang, and X. Sun, “Controllable and enhanced nanojet effects excited by surface plasmon polariton,” Appl. Phys. Lett. 102(17), 171109 (2013).
[Crossref]

Taflove, A.

Takakura, Y.

Valentine, J.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnol. 6(3), 151–155 (2011).
[Crossref] [PubMed]

Wang, J.

Wang, L. V.

Wu, W.

W. Wu, A. Katsnelson, O. G. Memis, and H. Mohseni, “A deep sub-wavelength process for the formation of highly uniform arrays of nanoholes and nanopillars,” Nanotechnology 18(48), 485302 (2007).
[Crossref]

Xi, J.-Q.

E. F. Schubert, J. K. Kim, and J.-Q. Xi, “Low-refractive-index materials: A new class of optical thin-film materials,” Phys. Status Solidi B 244(8), 3002–3008 (2007).
[Crossref]

Xu, H.

Yang, S.

S. Yang and V. N. Astratov, “Photonic nanojet-induced modes in chains of size-disordered microspheres with an attenuation of only 0.08 dB per sphere,” Appl. Phys. Lett. 92(26), 261111 (2008).
[Crossref]

Zemlyanov, A. A.

Zentgraf, T.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnol. 6(3), 151–155 (2011).
[Crossref] [PubMed]

Zhang, X.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnol. 6(3), 151–155 (2011).
[Crossref] [PubMed]

Zhou, R.

Appl. Opt. (2)

Appl. Phys. Lett. (3)

D. Ju, H. Pei, Y. Jiang, and X. Sun, “Controllable and enhanced nanojet effects excited by surface plasmon polariton,” Appl. Phys. Lett. 102(17), 171109 (2013).
[Crossref]

S. Yang and V. N. Astratov, “Photonic nanojet-induced modes in chains of size-disordered microspheres with an attenuation of only 0.08 dB per sphere,” Appl. Phys. Lett. 92(26), 261111 (2008).
[Crossref]

S.-C. Kong, A. V. Sahakian, A. Heifetz, A. Taflove, and V. Backman, “Robust detection of deeply subwavelength pits in simulated optical data-storage disks using photonic jets,” Appl. Phys. Lett. 92(21), 211102 (2008).
[Crossref]

J. Comput. Theor. Nanosci. (1)

A. Heifetz, S.-C. Kong, A. V. Sahakian, A. Taflove, and V. Backman, “Photonic Nanojets,” J. Comput. Theor. Nanosci. 6(9), 1979–1992 (2009).
[Crossref] [PubMed]

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (3)

Nanotechnology (1)

W. Wu, A. Katsnelson, O. G. Memis, and H. Mohseni, “A deep sub-wavelength process for the formation of highly uniform arrays of nanoholes and nanopillars,” Nanotechnology 18(48), 485302 (2007).
[Crossref]

Nat. Nanotechnol. (2)

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnol. 6(3), 151–155 (2011).
[Crossref] [PubMed]

E. Mcleod and C. B. Arnold, “Subwavelength direct-write nanopatterning using optically trapped microspheres,” Nat. Nanotechnol. 3(7), 413–417 (2008).
[Crossref] [PubMed]

Opt. Express (6)

Opt. Lett. (6)

Phys. Status Solidi B (1)

E. F. Schubert, J. K. Kim, and J.-Q. Xi, “Low-refractive-index materials: A new class of optical thin-film materials,” Phys. Status Solidi B 244(8), 3002–3008 (2007).
[Crossref]

Quantum Electron. (1)

Y. E. Geints, A. A. Zemlyanov, and E. K. Panina, “Photonic nanojet effect in multilayer micrometre-sized spherical particles,” Quantum Electron. 41(6), 520–525 (2011).
[Crossref]

Science (1)

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003).
[Crossref] [PubMed]

Other (3)

J. F. Poco and L. W. Hrubesh, “Method of producing optical quality glass having a selected refractive index,” U.S. Patent 6, 158, 244, (2008).

O. V. Mazurin, M. V. Streltsina, and T. P. Shvaiko-Shavaikovskaya, Handbook of Glass Data (Elsevier, 1993).

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).

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Figures (7)

Fig. 1
Fig. 1 The refractive index distributions of single-focus GLLs with various normalized focal length for (a) f = 1.0, (b) f = 1.25, (c) f = 1.5, (d) f = 1.75. The refractive index distributions of double-focus GLLs for (e) f2 = 1.7, (f) f2 = 2.2, (g) f2 = 2.7, (h) f2 = 3.2 when the first focal length f1 is set as 1.2. Insets in each subfigures show the ray tracing results.
Fig. 2
Fig. 2 Intensity distributions of single-focus GLLs with various normalized focal length for (a) f = 1.0, (b) f = 1.25, (c) f = 1.5, and (d) f = 1.75 for a 400 nm incident wavelength.
Fig. 3
Fig. 3 Intensity distributions of single-focus GLLs (a) along the x axis passing through center of GLLs, and (b) along the y axis at maximum intensity spots. The insets of the right upper corner each figure correspond to the concrete value of L and FWHM with different normalized focal length f.
Fig. 4
Fig. 4 Intensity distributions of double-focus GLLs with various normalized focal length for (a) f2 = 1.7, (b) f2 = 2.2, (c) f2 = 2.7, and (d) f2 = 3.2 when the first focal length f1 is set as 1.2 for a 400 nm incident wavelength. Intensity distributions of double-focus GLLs (e) along the x axis passing through center of GLLs, and (f) along the y axis at maximum intensity spots. The insets of the right upper corner each figure correspond to the concrete value of L and FWHM with different normalized focal length f2.
Fig. 5
Fig. 5 Intensity distributions of double-focus GLLs with various normalized focal length for (a) f1 = 1.0, (b) f1 = 1.2, (c) f1 = 1.5 when the second focal length f2 is set as 2.7 for a 400 nm incident wavelength. The corresponding L and FWHM of double-focus GLLs (d).
Fig. 6
Fig. 6 The longitudinal waist L and the corresponding FWHM of double-focus GLLs with various normalized first focal length f1 when the second focal length f2 is fixed as 1.7, 2.2, 2.7, 3.2 respectively for a 400 nm incident wavelength.
Fig. 7
Fig. 7 Arrow-plot of the energy power flow in the x-y plane. Single-focus GLLs for (a) f = 1.0 and (b) f = 1.75. Double-focus GLLs for (c) f2 = 1.7 and (d) f2 = 3.2. For double-focus GLLs, the first focal length f1 is set as 1.2.

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

n(r)= n 0 [ 1+ f 2 ( r R ) 2 ] 1/2 /f
n( ρ )={ exp[ ω 1 ( ρ, f 2 ) ], P a ρ1 exp[ ω 1 ( ρ, f 2 )+ ω 2 ( ρ, f 1 , P a )+ ω 2 ( ρ, f 2 , P a ) ],0ρ P a
ω 1 (ρ, f 2 )= 1 π ρ 1 arcsin( k/ f 2 )dk ( k 2 ρ 2 ) 1/2
ω 2 ( ρ,i, P a )= 1 π ρ P a arcsin( k/i )dk ( k 2 ρ 2 ) 1/2

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