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Sub-wavelength focusing of cylindrical vector beams (CVBs) has attracted great attention due to the specific physical effects and the applications in many areas. More powerful, flexible and effective ways to modulate the focus transversally and also longitudinally are always being pursued. In this paper, cylindrically symmetric lens composed of negative-index one-dimensional photonic crystal is proposed to make a breakthrough. By revealing the relationship between focal length and the exit surface shape of the lens, a quite simple and effective principle of designing the lens structure is presented to realize specific focus modulation. Plano-concave lenses are parameterized to modulate the focal length and the number of focuses. An axicon constructed by one-dimensional photonic crystal is proposed for the first time to obtain a large depth of focus and an optical needle focal field with almost a theoretical minimum FWHM of 0.362λ is achieved under radially polarized incident light. Because of the almost identical negative refractive index for TE and TM polarization states, all the modulation methods can be applied for any arbitrary polarized CVBs. This work offers a promising methodology for designing negative-index lenses in related application areas.
© 2015 Optical Society of America
1. Introduction
Sub-wavelength focusing of cylindrical vector beams (CVBs) has wide applications in many domains such as optical trapping, sub-wavelength imaging, super-resolution microscopy, laser machining and so on, due to the specific physical effects for the cylindrically symmetric polarization and intensity distribution of CVBs [1–4]. Many researches have been reported about how to modulate the focus [5–9]. Traditional lenses can focus any arbitrary polarized CVBs, and by changing the polarization states of the incident light or introducing other optical elements, the focusing can be modulated transversally and also longitudinally [5,6]. Meanwhile, it is a hot issue to break the diffraction limit and to obtain smaller transversal FWHM [10–12], and meanwhile, to modulate the focus in longitudinal direction effectively.
Indeed, a focus with a smaller transversal FWHM but an ultra-large depth of focus (DOF), such as optical needles obtained by focusing radially polarized beams [13–19], has attracted great interest recently. Such kinds of needle-shaped focus can be achieved by focusing the radial polarization with aplanatic lenses aided by binary filters [15–19], or with a parabolic mirror [13]. Both methods take advantage of the transformation from radial to longitudinal polarization to extremely enlarge the DOF and compress the transversal FWHM of the focal field. With aplanatic lenses aided by binary filters, thanks to the pure longitudinal polarization in the focal field [18,19], almost the smallest transversal FWHM is achieved [20]. However, using aplanatic lenses aided by binary filters cannot achieve 90°-bending of light rays, so that the DOF and the transversal FWHM of optical needles are limited. With a parabolic mirror, the light ray can be bent in 90°, and the proportion of the longitudinal polarization in focal field is bigger than that of other methods, and so as the DOF of focus. Thus, based on a proper mirror size, a DOF of 50000λ has been reported [15]. But the application of parabolic mirror may be limited because the focal field and the incident field locate at the same side of the mirror.
A tiny focus of CVBs, especially of the radially polarized light, can be efficiently achieved with plasmonic lenses [11], while modulation of DOF and other characteristics depends on the design of plasmonic lens structure [12]. However, the application of plasmonic lens in focusing CVBs is still restricted to certain polarization states, and the focal length is limited also. The first restriction lies in the polarization condition of surface plasmon polariton (SPP) excitation in the structure of plasmonic lens, so the excitation condition is usually not satisfied for all states of polarization of CVBs. The limitation of the focal length is due to the fact that SPP is a kind of surface wave, which would further limit the longitudinal focus modulation.
Fortunately, the usage of negative-index lenses is known to be free of diffraction limited resolution, making it a promising avenue for high-resolution imaging and high-precision field manipulation [21]. And once the negative-index is not restricted by the polarization states, the negative-index lenses will be expected to focus specific polarized light beams such as CVBs. We have proposed a kind of concave lens constructed by one-dimensional photonic crystal (1D-PC) with effective negative refraction index to focus CVBs into a sub-wavelength scale [22]. It is found that the negative refraction effect occurs in this 1D-PC for both orthogonal states of polarization. So both polarization components of CVBs can be focused, but have different focal fields. Thus, by varying polarization components of the incident beam, the focal field will be transversally modulated. Before considering the polarization components of the incident CVBs, the structure of the lens should first be designed to achieve specific focusing effect.
This paper thoroughly illuminates the physical mechanism and the design principle of negative-index 1D-PC lens, and reveals the relationship between the focal length and the profile of lens surfaces. By using the design principle, lenses with different structures are constructed to achieve different focus modulation in the longitudinal direction. Cylindrically symmetric plano-concave lenses are designed to realize specified focal lengths and multiple focuses. And an axicon constructed by 1D-PC is proposed for the first time to focus radially polarized light to obtain an optical needle with large DOF and the smallest transversal FWHM.
2. Design principle of negative-index 1D-PC lens
The refractive index and the profile of the lens strictly codetermine the position and the shape of focus. In order to obtain negative index, the 1D-PC structure is designed to locate the operating wavelength in the second passband of the 1D-PC [22]. Figure 1(a) shows the band structure of 1D-PC composed of MgF2 and GaN, with material refractive indices 1.38 and 2.67, respectively. To ensure the second passband covers the visible region, a and b are chosen to be 10 nm and 140 nm, respectively. The band structures are identical for both TE and TM polarization states in condition of normal incidence, so the negative refraction effect occurs for both states of polarization. The value of the refractive index is calculated by analyzing the equi-frequency surfaces (EFSs). The equi-frequency surfaces at λ = 532 nm of both polarization states in the structure of 1D-PC are depicted in Fig. 1(b), with the blue circle indicating equi-frequency surface in air. In order to analyze the negative refraction at the interface of the 1D-PC and air, an isosceles right-angled triangle prism is constructed. When light is incident from the right angle side, it will be negatively refracted at the hypotenuse interface. Under the continuity condition of the wave vector component along the interface marked as a black dashed line in Fig. 1(b), along the gradient of equi-frequency surfaces, the group velocity direction of refracted light is acquired and presented by the red arrow marked with “R”. The negative refraction phenomena for TE and TM polarization states are depicted in Figs. 1(c) and 1(d), respectively. The refracted directions acquired from the simulated negative refraction phenomena through the prism are almost the same for TE and TM polarization states because of the two almost superimposed equi-frequency surfaces. Therefore, for a cylindrically symmetric 1D-PC structure, the negative refractions of radially and azimuthally polarized incident light are expected to be practically identical. Thus, for brevity in this paper, only the case of radially polarized CVB is discussed.
Fig. 1 (a) Band structure at λ = 532 nm; (b) Equi-frequency surfaces (EFSs), the blue circle indicates the EFS in air; (c) and (d) indicate a negative refraction phenomenon of the incident light with TE and TM polarization states, respectively, simulated by finite element method (FEM).
The value of the negative index can be obtained by analyzing the triangle △ABC shown in Fig. 1(b), where αi and αr indicate the incident and refractive angles, respectively. In k space, the light propagates along the gradient direction of equi-frequency surface. When light propagates from left to right with ky = 0 and kx = −0.266 (2π/d), the wave vector of refracted light in air is k = 2π/λ = 0.282 (2π/d). Considering the law of sines in △ABC, the negative index of 1D-PC is
Thus, the refraction angle is exactly αr = arcsin(nsin(αi)) = −41.66°. Once the negative index is known, it is quite convenient to predict refractive angle for negative refraction in this way. There is no need to calculate the entire equi-frequency surface, because only one value of kx is needed when ky = 0. As a result of the negative refraction, CVBs incident from the plane side of a cylindrically symmetric plano-concave lens shown in Fig. 2(a), may finally be focused at the concave side.A comparison between diffractive effect of a periodicity onto wave optics and equivalent-slope upon ray optics is made to simplify the analysis on design principle of the lenses. Diffracted waves from every periodicity with different orientations can be seen as light rays refracted at every equivalent-slope between the two adjacent periods. Thus, to achieve the focusing process, the plano-concave lens should be tailored to refract all the light rays pointing to a single spot. In Fig. 2(b), connecting lines between each two adjacent periods are drawn to mimic a concave plane, with blue and red arrow lines indicating incident and exit light rays, respectively. It is assumed that each single light ray incidents at the center of rm and rm+1, and is negative refracted at the hypotenuse interface of each triangle element. The direction of refracted light ray depends on the slope of each triangle element. When all light rays are refracted across the same point, focusing process exhibits and the focal length is also determined at the same time.
The focal length of the plano-concave lens is quite dependent on the profile of the structure. The relationship between the focal length and the profile of concave plane is derived by tracking each light ray. Here, the focal length represents the distance between the focus and the bottom of the exit surface. Considering an incident light between rm-1 and rm refracted to a pre-set point F in Fig. (2), it is obviously that ∠BAC = αi and ∠EDF = αi + |αr|. The relationship is deduced from the geometry of the triangle and Snell’s laws, as follows,
In these equations, f is the focal length, m is a positive integer indicating the number of each period, n is negative index, and d = a + b is the period of 1D-PC. For a specific focal length, each coordinate of the naked tip of PC can be derived from the recurrence relation between rm and rm-1 by substituting the initial value like r0 = 0. With the help of the method mentioned above, modulation of focus positions, numbers and DOFs can be easily realized by tailoring the shape of exit surface.
3. Focus modulation by using plano-concave lens
Cylindrically symmetric plano-concave lens composed of negative-index 1D-PC can be used to focus CVBs to sub-wavelength scale [22], by tailoring the exit concave surface the focus can be modulated further. It is worth mentioning that the design principle is also definitely suitable for the situation of linearly polarized incidence [23].
To realize a focus modulation in the longitudinal direction, the focal length should be discussed first. Four cylindrically symmetric plano-concave lenses composed of 1D-PC with 30 periods as examples are designed with the above principle to obtain four different specified focal lengths as 4.5 um, 5.5 um, 6.5 um and 7.5 um. The simulated electric intensity distributions by FEM method under radial polarization incidence are depicted in Fig. 3. The focal lengths read from simulation results are 4.45 um, 5.45 um, 6.52 um and 7.50 um, and the transversal FWHMs of the four focuses are 214 nm (0.40λ), 229 nm (0.43λ), 251 nm (0.47λ) and 259 nm (0.49λ), respectively. The transversal FWHM of focus increases with the focal lengths but remains smaller than the diffraction limit. The purer the longitudinal polarization in the focus field is, the more non-diffractive the focus is, which means the more the proportion of longitudinal polarization is, the smaller the focus is. In case of the proposed 1D-PC plano-concave lens, the smaller the focal length is required, the bigger the bending angle of the light ray should be. And the bending angle just determines the conversion ratio of the longitudinal polarization transformed from radial polarization. As a result, the smaller the focal length is, the smaller the transversal FWHMs is.
Fig. 3 Modulation of focal length by plano-concave lens through tailoring the shape of the exit surface. The specified f are (a) 4.5 um; (b) 5.5 um; (c) 6.5 um; (d) 7.5 um, respectively.
Multiple focuses can also be realized by shaping the profile exit surface. The four lenses designed above can only focus light to one specific spot, because almost all the exiting light rays arrow at a single point. But in fact, focusing light to a single point just needs part of the light rays refracted from part of the surface, and the rest part of surface can be tailored to focus light into another spot by using the same principle. For example, if two focal spots with focal length 4.5 um and 7.0 um are expected, the lower 11 periods may be tailored for one focus and the rest upper part for the other. As can be seen from the simulated result shown in Fig. 4(a), the two focal lengths are f1 = 4.5 um and f2 = 6.8 um, respectively. The first focus is located exactly at the expected position, while the other one exhibits a little difference of 0.2 um. Since each focal spot is approximately formed by a number of exit light rays coming from the exit concave surface. If more focal spots are required, more periods are needed. To acquire three focal spots with focal length of 4.5 um, 7.0 um and 9.5 um, a plano-concave lens with 50 periods is designed. The simulated focal lengths are f1 = 4.5 um, f2 = 7.0 um and f3 = 9.2 um, depicted in Fig. 4(b). The difference of the third focal length is 0.3 um compared with the expected value. It can be seen from the result that, a larger focal length leads to a larger difference. In our further work, the underlying mechanism will be explored.
Fig. 4 Focal field of radially polarized beams by plano-concave lens focuses. (a) two focuses; (b) three focuses.
4. Large DOF realized by using axicon
Focuses with large DOFs but also small transversal FWHMs are always being pursued. For example, a non-diffractive optical needle can be observed under the incidence of radially polarized beams, as the combination of a series of longitudinally polarized focal spots [16]. The transformation from radial to longitudinal polarization will extremely compress the transversal FWHM and enlarge the DOF of the focal field. The key point of the polarization transformation is bending light adequately. 1D-PC lens with negative index is expected to make the breakthrough by appropriately tailoring the profile of the exit surface, which makes a 90° bending of propagation direction of light rays. At the same time, a smallest transversal FWHM can also be obtained.
An axicon composed of 1D-PC is designed. The 90° bending of light rays requires the specific profile of the axicon shown in Fig. 5(a). Considering Snell’s law with n = −0.94, the incident angle should be αi = 46.77° to realize αi + |αr| = 90°. So the angle α of the cone equals 43.23°. The focusing of radially polarized beams is illustrated in Fig. 5(b), and the simulated optical needle is shown in Fig. 5(c). The transversal FWHM is 192.6 nm (0.362λ), practically approaching to the theoretically smallest one.
Fig. 5 (a) Profile of 1D-PC axicon at r-z plane; (b) Focusing radially polarized light by axicon. (c) Optical needle focal field, and the inserted graph is the enlarged view of region marked with white dashed rectangle, black arrows indicate the polarization directions.
Compared with the traditional axicon, which can also be used to obtain a large DOF of focus [24–26], there are several differences and also advantages of the axicon constructed by 1D-PC. For the traditional axicons, the convex ones focus light, while the concave ones scatter light. However, due to the negative refraction effect of the 1D-PC, the axicon constructed by 1D-PC can be design to be concave to focus light, and also transform the radially polarized states to longitudinally polarized states. Thus, a non-diffractive optical needle is obtained. It can be seen from Fig. 5(c) that, in the focal domain, electric field is purely longitudinally polarized and the field intensity is quite flat. The DOF of optical needle is codetermined by the size of the lens and the incident field distribution, so it can be easily enlarged.
5. Conclusion
Cylindrically symmetric lens composed of negative-index 1D-PC can be used to realize sub-wavelength focusing and fine focus modulation of CVBs. This paper reveals the relationship between the focal length and the shape of exit surface considering Snell’s law with negative index of 1D-PC. A simple and effective principle for designing the structure of the 1D-PC lens is proposed. By properly tailoring the exit surface profile of the plano-concave lens, specified focal lengths are acquired and multiple focuses are also realized. Furthermore, an axicon constructed by 1D-PC is proposed for the first time to obtain the non-diffractive optical needle with the smallest transversal FWHM. Compared with the traditional modulation method, this novel achievement is due to the perfect transformation from radial to longitudinal polarization. Owing to the negative index, the axicon is concave, and the focal field locates at the other side of the lens.
Although only longitudinal focus modulations for radial polarization situation are discussed in this paper, focus modulation for other states of polarization can be realized in the same way. And the transversal modulation can also be achieved conveniently by changing the polarization components of CVBs, since the negative index of 1D-PC for both TE and TM polarization states are almost identical, which also overcomes the limit of polarization states brought by plasmonic lenses. The lenses composed of negative-index 1D-PC presented in this paper can be easily tailored by the design principle, and effectively modulation the focus of CVBs is realized. This work would provide a promising methodology for designing negative-index lenses in related application areas.
Acknowledgments
This work was supported by National Natural Science Foundation of China (Grant No. 11404170), Natural Science Foundation of Jiangsu Province (Grant No. BK20131383), and Basic Research Program of Jiangsu Education Department (Grant No. 14KJB140010).
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