- Email: [email protected]
Photonic nanojet shaping of dielectric non-spherical microparticles
Physica E 64 (2014) 23–28
Contents lists available at ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Photonic nanojet shaping of dielectric non-spherical microparticles Cheng-Yang Liu 1 Department of Mechanical and Electro-Mechanical Engineering, Tamkang University, New Taipei City, Taiwan
H I G H L I G H T S
G R A P H I C A L
We present the photonic nanojet shaping of dielectric non-spherical microparticles. The dimensions of photonic nanojet depending on the cutting thickness are studied. The results provide a new way to distinguish nanoscale specimens.
We present the photonic nanojet shaping effect in the dielectric non-spherical microparticles.
art ic l e i nf o
a b s t r a c t
Article history: Received 8 May 2014 Received in revised form 27 June 2014 Accepted 30 June 2014 Available online 5 July 2014
The photonic nanojet shaping effect in the dielectric non-spherical microparticles is reported. The specific spatial electromagnetic field is studied by using finite-difference time-domain calculation which constitutes the so-called photonic nanojet. The dielectric non-spherical microparticle is truncated by the cutting thickness. The latitudinal and longitudinal dimensions of the photonic nanojet and its peak intensity depending on the variation of cutting thickness are numerically researched. The shape dependence of the photonic nanojet in the non-spherical microparticles has been investigated by quality criterion. The practical results are drawn concerning the possible procedures to gain the control over the properties of photonic nanojet in the non-spherical microparticles. The shaping mechanism has a significant impact on the use of photonic nanojet to distinguish nanoscale specimens. & 2014 Elsevier B.V. All rights reserved.
Keywords: Photonic nanojet Microparticle Subwavelength
A B S T R A C T
1. Introduction The question of overcoming the diffraction limit in the spatial resolution of optical microscopy has become the highlight of the study in progressive optics [1–3]. The important aspect of this question is the study of the distribution of the optical field near a surface of slightly absorbing microspheres from the viewpoint of the possibility of subwavelength focusing of the incident optical beam. The term, a photonic nanojet, is generally used recently that is characterized by a particular spatial localized and high-intensity region in the near-field scattering of a lightwave at dielectric microspheres [4–9]. This specific electromagnetic field has a subdiffraction transverse dimension and expands conserving its shape to the distances of several wavelengths. This phenomenon of
E-mail address: [email protected] Address: No. 151, Ying-chuan Road, Tamsui District, New Taipei City, Taiwan. Tel: 886-2-26215656 # 2061, Fax: 886-2-26209745 1
http://dx.doi.org/10.1016/j.physe.2014.06.026 1386-9477/& 2014 Elsevier B.V. All rights reserved.
photonic nanojet attracts a particular interest and provides wide application for designing the optical nano-scale sensors with high resolution [10–12]. The other possible practical applications of a photonic nanojet include the enhanced Raman scattering of the silicon photon modes [13], the two-photon fluorescence enhancement [14], the technology of nano-photonic lithography [15], an optical tool for nano-object manipulation [16], and the optical data storage with ultra-high resolution [17]. The review paper devoted to theoretical and experimental investigations of the photonic nanojet can be found in the Ref. [18]. The principles of functioning of the mentioned applications are based on the point that photonic nanojet can provide the high intensity of the optical field in a localized space near a microsphere. When nano-objects fall within the region of photonic nanojet, the photonic nanojet interacts with these objects and results in specific changes of optical properties. Hence, the issue moves to the characteristic manipulation of photonic nanojet. The utilization of photonic nanojet has been limited by the short effective length of photonic nanojet. The short effective length
24
C.-Y. Liu / Physica E 64 (2014) 23–28
only permits photonic nanojet to measure the near surface features. The photonic nanojets created by a composite inhomogeneous sphere consisting of several concentric shells with different materials are reported [19–24]. The properties of photonic nanojet can be changed significantly and it becomes possible to elongate or enhance the photonic nanojet abnormally. The researchers examined the photonic nanojets from the dielectric microparticles consisting of two and even more concentric layers illuminated by an optical source. The radial variation of the refractive index under certain conditions allowed the photonic nanojet to be elongated up to 20 μm. However, the cost for this elongation of photonic nanojet is the transverse size widening and the reduction of the peak intensity. Moreover, the fabrication of the layered inhomogeneous microparticles is very difficult and expensive. Therefore, we are interested in the photonic nanojet shaping for far-field projection by a simple and cheap way. In this aim, a dielectric non-spherical microparticle for the photonic nanojet shaping is proposed by the author. In this paper, we theoretically study the shaping effect of photonic nanojets generated at the shadow side surfaces of nonspherical microparticles illuminated by a plane wave. The dielectric non-spherical microparticle is truncated by the cutting thickness. The latitudinal and longitudinal dimensions of the photonic nanojet and its peak intensity depending on the variation of cutting thickness are numerically researched. The numerical model of photonic nanojet for the dielectric non-spherical microparticle is presented in Section 2. The shaping effect of photonic nanojet is presented in Section 3. We summarize the upshot and discuss the potential employments in Section 4.
Fig. 1. Schematic diagram of a non-spherical microparticle for photonic nanojet.
photonic nanojet can be less than the classical diffraction limit. The third, the nano-scale targets perturb the far-filed backscattered power of the illuminated microsphere by inserting within a photonic nanojet. The perturbation behavior leads to the possibility of detecting a nano-scale target through its perturbation of the backscattering from the microsphere. Later, Section 3 will demonstrate that the key parameters of photonic nanojet formed in the vicinity of homogeneous dielectric non-spherical microparticle under exposure of laser radiation are studied theoretically.
3. Photonic nanojet shaping 2. Numerical model The main object of this paper is to consider the formation of the photonic nanojet in a wide class of non-spherical microparticles having different cutting thicknesses. Considering that the finitedifference time-domain (FDTD) method [25] based on vector electromagnetic wave theory can accurately illustrate the propagation of lightwave in the dielectric media, the excitation and depletion fields of photonic nanojet is simulated by using the FDTD calculation in the MATLABs code. During the operation of simulation, a three-dimensional computational region of 12 μm 8 μm 8 μm is separated into a uniform mesh. The mesh dimension is set to be 10 nm with the perfectly matched layers [26] after the convergence verification performed by the decrease of the mesh dimension. We can acquire lightwave propagation depending on time by using discrete time and lattices according to the Yee algorithm. The mesh dimension and time sampling are selected to ensure enough accuracy, high calculation speed, and numerical stability of the algorithm. Fig. 1 shows the schematic diagram of a non-spherical microparticle for photonic nanojet. The diameter of microparticle is d and the cutting thickness is t. The refractive indices of the microparticle and the surrounding medium (air) are nd ¼1.6 and ns ¼1. As a light source of the field is set a plane monochromatic wave with the unit amplitude (1 V/m) generated at the left boundary of the computational region. The focal length from the surface of the microparticle to the point of maximum intensity of photonic nanojet is f. The decay length is the radial distance from the point of maximum intensity at which the electromagnetic field decays to 1/e of the maximum intensity. The effective length, decay length, full-width half-maximum, and maximum intensity of the photonic nanojet are L, g, FWHM, and I, respectively. The narrow, high intensity photonic nanojet that emerges from a dielectric microsphere has been found to present several key properties. First, photonic nanojet is capable of maintaining a subwavelength FWHM. Second, the transverse width of
Many studies have shown that the shape and intensity of photonic nanojet depend significantly on the dimension and optical materials of a generating microsphere. Due to the simple structure and low cost, the photonic nanojet shaping based on non-spherical microparticle has attracted our attention. The numerical FDTD code permits the simulation of the electromagnetic fields throughout entire computational space when an incident lightwave interacts in a non-spherical microparticle. Fig. 2 to Fig. 4 show the normalized electric field intensity distribution of a non-spherical microparticle at different diameters and cutting thicknesses. A light source with wavelength 633 nm is incident from the left and impinges on the non-spherical microparticle. The 633 nm wavelength light source is chosen as an example. The key parameters of the photonic nanojets are scalable in respect of the incident wavelength [27]. The influence of the cutting thickness of the non-spherical microparticle on the photonic nanojet is illustrated in Fig. 2 to Fig. 4. It can be seen from these figures that the shape of field intensity distribution changes as the cutting thickness increases. For the microspheres (Fig. 2(a), Fig. 3 (a), and Fig. 4(a)), the photonic nanojet acts like a water droplet. As the cutting thickness increases, the region of the maximal intensity shifts beyond a water droplet that becomes a photonic flux. In the situation with larger microparticle (Fig. 4(d)), the configuration of intensity distribution becomes more complex. The additional areas of the high intensity are formed rather far from the microparticle surface. Thus, simply varying the cutting thickness, it is possible to achieve localized photonic fluxes having different spatial dimensions and intensity properties. In order to study the characteristics of photonic nanojet, it is necessary primarily to define its spatial length and width. Fig. 5 and Fig. 6 show the normalized intensity distribution of photonic nanojets for nonspherical microparticles along propagation axis (x axis) and transversal axis (y axis) at different diameters and cutting thicknesses. The longitudinal profile (Fig. 5) is acquired as a section of the two-dimensional intensity distribution by the straight line
C.-Y. Liu / Physica E 64 (2014) 23–28
Fig. 2. Normalized electric field intensity distribution of a 3 μm non-spherical microparticle at (a) t ¼ 0 (sphere), (b) t¼ 0.25d ¼0.75 μm, (c) t ¼0.5d ¼1.5 μm, and (d) t ¼0.75d ¼2.25 μm. The 633 nm lightwave propagates from left to right.
25
Fig. 3. Normalized electric field intensity distribution of a 5 μm non-spherical microparticle at (a) t ¼0 (sphere), (b) t¼ 0.25d¼ 1.25 μm, (c) t ¼0.5d ¼2.5 μm, and (d) t¼ 0.75d ¼3.75 μm. The 633 nm lightwave propagates from left to right.
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C.-Y. Liu / Physica E 64 (2014) 23–28
Fig. 5. Normalized intensity distribution of photonic nanojets for non-spherical microparticles along propagation axis (x axis) at (a) d ¼3 μm, (b) d¼ 5 μm, and (c) d¼ 8 μm.
Fig. 4. Normalized electric field intensity distribution of an 8 μm non-spherical microparticle at (a) t¼ 0 (sphere), (b) t¼ 0.25d ¼2 μm, (c) t ¼0.5d ¼ 4 μm, and (d) t¼ 0.75d ¼6 μm. The 633 nm lightwave propagates from left to right.
located at y¼ 0. The transverse profile (Fig. 6) is acquired as a section by the straight line parallel to axis y and passing through the peak of the maximum intensity. The all intensity profiles of
photonic nanojet for non-spherical microparticles are normalized to the intensity profiles for the microsphere in the same diameter. At small cutting thickness, the photonic nanojet adhered to the surface of microparticle and emerges from it in the form of the exponentially decaying trail. However, the lengths of photonic nanojets for 3 μm, 5 μm, and 8 μm non-spherical microparticles at cutting thickness t ¼0.75d are 3.38 μm, 5.05 μm, and 5.31 μm. The non-spherical microparticles at cutting thickness t ¼0.75d demonstrate the elongated photonic nanojets with the extent ranging from 5λ to 8λ. Depending on the cutting thickness of microparticle, we observe that not only the dimensions and intensity of photonic nanojet change but the separation of the photonic nanojet from the microparticle surface changes also. In Fig. 6, it is significant
C.-Y. Liu / Physica E 64 (2014) 23–28
27
Fig. 7. Focal length as a function of the cutting thickness for non-spherical microparticles.
Fig. 8. Decay length as a function of the cutting thickness for non-spherical microparticles.
Fig. 6. Normalized intensity distribution of photonic nanojets for non-spherical microparticles along transversal axis (y axis) at (a) d ¼ 3 μm, (b) d¼ 5 μm, and (c) d ¼8 μm.
that we have small transverse width of photonic nanojet among all considered situations. The length and location of the photonic nanojet depend on the cutting thickness, but the FWHM of photonic nanojet is almost the same at the different cutting thicknesses. Fig. 7 shows the focal length as a function of the cutting thickness for non-spherical microparticles. It should be noted that all the three studied features demonstrate the nearly linear growth with the increase in cutting thickness. As a result, the focal length and the location of the photonic nanojets can be controlled by cutting thickness. Fig. 8 shows the decay length as a function of the cutting thickness for non-spherical microparticles. The decay length increases as cutting thickness increases. The long decay length can be used to overcome the diffraction limit at the
Fig. 9. Full-width half-maximum of photonic nanojet as a function of the cutting thickness for non-spherical microparticles.
focusing of lightwave by a non-spherical microparticle. Fig. 9 shows the full-width half-maximum of photonic nanojet as a function of the cutting thickness for non-spherical microparticles. The morphological type of non-spherical microparticles optimally combines the high spatial localization of the photonic nanojet with sufficiently high intensity. The photonic nanojet is formed rather far from the surface of the non-spherical microparticles at large cutting thickness, and a decrease in cutting thickness the coordinate of the intensity maximum reaches the microparticle rim. The optimal balance between these key parameters of photonic nanojet is realized in the non-spherical microparticles
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thickness. We believe that this novel technique has a large analysis range and will be helpful in the study of nano-optics and nanobiotechnology. The observed optical emission offers the potential use of the non-spherical microparticle as a high resolution optical lens. The photonic nanojet may allow the coupling of light beam from the photonic molecule into other photonic micro-devices such as optical waveguides or coaxial cables. The optical systems would be not only with sub-wavelength spatial precision, but also with high optical power in comparison with traditional coupling techniques.
Acknowledgments
Fig. 10. Normalized quality criterion of photonic nanojet as a function of the cutting thickness for non-spherical microparticles.
This work is funded by National Science Council in Taiwan under Grant number NSC 102–2221-E-032-009.
References with cutting thickness. Combining these key parameters of photonic nanojet, we propose to describe the photonic nanojets by a quality criterion as Q¼(L I)/FWHM. Fig. 10 shows the normalized quality criterion of photonic nanojet as a function of the cutting thickness for non-spherical microparticles. The quality criterion Q increases as cutting thickness t increases. The quality criterion accounts for the useful value of photonic nanojet to the solution of practical issues. When the quality criterion is high, the effective length of photonic nanojet is long, its FWHM is small, and the peak intensity is high. The quality criterion possesses the high values at larger cutting thickness. These results offer excellent evidence that the elongated photonic nanojet provides the action of a nearly one-dimensional light beam over a significant range of spectral frequencies. The generation of photonic nanojet is based on a nonspherical microparticle which provides highly confined electromagnetic fields to efficiently detect nanoscale targets. 4. Conclusion We have numerically investigated the photonic nanojet shaping generated at the shadow side surface of a non-spherical microparticle. The basic key characteristics of photonic nanojets formed in the vicinity of dielectric non-spherical microparticles with different cutting thicknesses are studied by using FDTD simulation. It makes possible to modulate the shape and location of photonic nanojet. The latitudinal and longitudinal profiles of a photonic nanojet depending on the cutting thickness of non-spherical microparticles are numerically demonstrated. We can gain command of the key parameters for photonic nanojet through the variation of the cutting
[1] H.C. van de Hulst, Light Scattering by Small Particles, Dover Publications, New York, 1981. [2] E. Hecht, Optics, Addison Wesley, San Francisco, 2002. [3] L. Novotny, B. Hecht, Principles of Nano-Optics, Cambridge University Press, New York, 2006. [4] Z. Chen, A. Taflove, V. Backman, Opt. Express 12 (2004) 1214. [5] A.V. Itagi, W.A. Challener, J. Opt. Soc. Am. A 22 (2005) 2847. [6] C. Li, G.W. Kattawar, P. Zhai, Opt. Express 13 (2005) 4554. [7] P. Ferrand, J. Wenger, A. Devilez, M. Pianta, B. Stout, N. Bonod, E. Popov, H. Rigneault, Opt. Express 16 (2008) 6930. [8] A. Devilez, B. Stout, N. Bonod, E. Popov, Opt. Express 16 (2008) 14200. [9] M. Kim, T. Scharf, S. Mühlig, C. Rockstuhl, H.P. Herzig, Opt. Express 19 (2011) 10206. [10] X. Li, Z. Chen, A. Taflove, V. Backman, Opt. Express 13 (2005) 526. [11] Z. Wang, W. Guo, L. Li, B. Lukyanchuk, A. Khan, Z. Liu, Z. Chen, M. Hong, Nat. Commun. 2 (2011) 218. [12] A. Darafsheh, G.F. Walsh, L. Dal Negro, V.N. Astratov, Appl. Phys. Lett. 101 (2012) 141128. [13] K.J. Yi, Y.F. Lu, Z.Y. Yang, J. Appl. Phys. 101 (2007) 063528. [14] S. Lecler, S. Haacke, N. Lecong, O. Crégut, J. Rehspringer, C. Hirlimann, Opt. Express 15 (2007) 4935. [15] W. Wu, A. Katsnelson, O. Memis, H. Mohseni, Nanotechnology 18 (2007) 485302. [16] X. Cui, D. Erni, C. Hafner, Opt. Express 16 (2008) 13560. [17] S. Kong, A. Sahakian, A. Taflove, V. Backman, Opt. Express 16 (2008) 13713. [18] A. Heifetz, S. Kong, A. Sahakian, A. Taflove, V. Backman, J. Comput. Theor. Nanosci. 6 (2009) 1979. [19] S. Kong, A. Taflove, V. Backman, Opt. Express 17 (2009) 3722. [20] Y. Geints, E. Panina, A. Zemlyanov, Opt. Commun. 283 (2010) 4775. [21] Y. Geints, A. Zemlyanov, E. Panina, J. Opt. Soc. Am. B 28 (2011) 1825. [22] C.-Y. Liu, Phys. Lett. A 376 (2012) 1856. [23] C.-Y. Liu, Prog. Electromagn. Res. Lett. 37 (2013) 153. [24] C.-Y. Liu, J. Optoelectron. Adv. Mater. 15 (2013) 150. [25] A. Taflove, S.C. Hagness, Computational Electrodynamics: The Finite Difference Time Domain Method, Artech House, Boston, 1998. [26] J. Berenger, J. Comput. Phys. 114 (1994) 185. [27] C.-Y. Liu, Y. Wang, Physica E 61 (2014) 141.
Contents lists available at ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Photonic nanojet shaping of dielectric non-spherical microparticles Cheng-Yang Liu 1 Department of Mechanical and Electro-Mechanical Engineering, Tamkang University, New Taipei City, Taiwan
H I G H L I G H T S
G R A P H I C A L
We present the photonic nanojet shaping of dielectric non-spherical microparticles. The dimensions of photonic nanojet depending on the cutting thickness are studied. The results provide a new way to distinguish nanoscale specimens.
We present the photonic nanojet shaping effect in the dielectric non-spherical microparticles.
art ic l e i nf o
a b s t r a c t
Article history: Received 8 May 2014 Received in revised form 27 June 2014 Accepted 30 June 2014 Available online 5 July 2014
The photonic nanojet shaping effect in the dielectric non-spherical microparticles is reported. The specific spatial electromagnetic field is studied by using finite-difference time-domain calculation which constitutes the so-called photonic nanojet. The dielectric non-spherical microparticle is truncated by the cutting thickness. The latitudinal and longitudinal dimensions of the photonic nanojet and its peak intensity depending on the variation of cutting thickness are numerically researched. The shape dependence of the photonic nanojet in the non-spherical microparticles has been investigated by quality criterion. The practical results are drawn concerning the possible procedures to gain the control over the properties of photonic nanojet in the non-spherical microparticles. The shaping mechanism has a significant impact on the use of photonic nanojet to distinguish nanoscale specimens. & 2014 Elsevier B.V. All rights reserved.
Keywords: Photonic nanojet Microparticle Subwavelength
A B S T R A C T
1. Introduction The question of overcoming the diffraction limit in the spatial resolution of optical microscopy has become the highlight of the study in progressive optics [1–3]. The important aspect of this question is the study of the distribution of the optical field near a surface of slightly absorbing microspheres from the viewpoint of the possibility of subwavelength focusing of the incident optical beam. The term, a photonic nanojet, is generally used recently that is characterized by a particular spatial localized and high-intensity region in the near-field scattering of a lightwave at dielectric microspheres [4–9]. This specific electromagnetic field has a subdiffraction transverse dimension and expands conserving its shape to the distances of several wavelengths. This phenomenon of
E-mail address: [email protected] Address: No. 151, Ying-chuan Road, Tamsui District, New Taipei City, Taiwan. Tel: 886-2-26215656 # 2061, Fax: 886-2-26209745 1
http://dx.doi.org/10.1016/j.physe.2014.06.026 1386-9477/& 2014 Elsevier B.V. All rights reserved.
photonic nanojet attracts a particular interest and provides wide application for designing the optical nano-scale sensors with high resolution [10–12]. The other possible practical applications of a photonic nanojet include the enhanced Raman scattering of the silicon photon modes [13], the two-photon fluorescence enhancement [14], the technology of nano-photonic lithography [15], an optical tool for nano-object manipulation [16], and the optical data storage with ultra-high resolution [17]. The review paper devoted to theoretical and experimental investigations of the photonic nanojet can be found in the Ref. [18]. The principles of functioning of the mentioned applications are based on the point that photonic nanojet can provide the high intensity of the optical field in a localized space near a microsphere. When nano-objects fall within the region of photonic nanojet, the photonic nanojet interacts with these objects and results in specific changes of optical properties. Hence, the issue moves to the characteristic manipulation of photonic nanojet. The utilization of photonic nanojet has been limited by the short effective length of photonic nanojet. The short effective length
24
C.-Y. Liu / Physica E 64 (2014) 23–28
only permits photonic nanojet to measure the near surface features. The photonic nanojets created by a composite inhomogeneous sphere consisting of several concentric shells with different materials are reported [19–24]. The properties of photonic nanojet can be changed significantly and it becomes possible to elongate or enhance the photonic nanojet abnormally. The researchers examined the photonic nanojets from the dielectric microparticles consisting of two and even more concentric layers illuminated by an optical source. The radial variation of the refractive index under certain conditions allowed the photonic nanojet to be elongated up to 20 μm. However, the cost for this elongation of photonic nanojet is the transverse size widening and the reduction of the peak intensity. Moreover, the fabrication of the layered inhomogeneous microparticles is very difficult and expensive. Therefore, we are interested in the photonic nanojet shaping for far-field projection by a simple and cheap way. In this aim, a dielectric non-spherical microparticle for the photonic nanojet shaping is proposed by the author. In this paper, we theoretically study the shaping effect of photonic nanojets generated at the shadow side surfaces of nonspherical microparticles illuminated by a plane wave. The dielectric non-spherical microparticle is truncated by the cutting thickness. The latitudinal and longitudinal dimensions of the photonic nanojet and its peak intensity depending on the variation of cutting thickness are numerically researched. The numerical model of photonic nanojet for the dielectric non-spherical microparticle is presented in Section 2. The shaping effect of photonic nanojet is presented in Section 3. We summarize the upshot and discuss the potential employments in Section 4.
Fig. 1. Schematic diagram of a non-spherical microparticle for photonic nanojet.
photonic nanojet can be less than the classical diffraction limit. The third, the nano-scale targets perturb the far-filed backscattered power of the illuminated microsphere by inserting within a photonic nanojet. The perturbation behavior leads to the possibility of detecting a nano-scale target through its perturbation of the backscattering from the microsphere. Later, Section 3 will demonstrate that the key parameters of photonic nanojet formed in the vicinity of homogeneous dielectric non-spherical microparticle under exposure of laser radiation are studied theoretically.
3. Photonic nanojet shaping 2. Numerical model The main object of this paper is to consider the formation of the photonic nanojet in a wide class of non-spherical microparticles having different cutting thicknesses. Considering that the finitedifference time-domain (FDTD) method [25] based on vector electromagnetic wave theory can accurately illustrate the propagation of lightwave in the dielectric media, the excitation and depletion fields of photonic nanojet is simulated by using the FDTD calculation in the MATLABs code. During the operation of simulation, a three-dimensional computational region of 12 μm 8 μm 8 μm is separated into a uniform mesh. The mesh dimension is set to be 10 nm with the perfectly matched layers [26] after the convergence verification performed by the decrease of the mesh dimension. We can acquire lightwave propagation depending on time by using discrete time and lattices according to the Yee algorithm. The mesh dimension and time sampling are selected to ensure enough accuracy, high calculation speed, and numerical stability of the algorithm. Fig. 1 shows the schematic diagram of a non-spherical microparticle for photonic nanojet. The diameter of microparticle is d and the cutting thickness is t. The refractive indices of the microparticle and the surrounding medium (air) are nd ¼1.6 and ns ¼1. As a light source of the field is set a plane monochromatic wave with the unit amplitude (1 V/m) generated at the left boundary of the computational region. The focal length from the surface of the microparticle to the point of maximum intensity of photonic nanojet is f. The decay length is the radial distance from the point of maximum intensity at which the electromagnetic field decays to 1/e of the maximum intensity. The effective length, decay length, full-width half-maximum, and maximum intensity of the photonic nanojet are L, g, FWHM, and I, respectively. The narrow, high intensity photonic nanojet that emerges from a dielectric microsphere has been found to present several key properties. First, photonic nanojet is capable of maintaining a subwavelength FWHM. Second, the transverse width of
Many studies have shown that the shape and intensity of photonic nanojet depend significantly on the dimension and optical materials of a generating microsphere. Due to the simple structure and low cost, the photonic nanojet shaping based on non-spherical microparticle has attracted our attention. The numerical FDTD code permits the simulation of the electromagnetic fields throughout entire computational space when an incident lightwave interacts in a non-spherical microparticle. Fig. 2 to Fig. 4 show the normalized electric field intensity distribution of a non-spherical microparticle at different diameters and cutting thicknesses. A light source with wavelength 633 nm is incident from the left and impinges on the non-spherical microparticle. The 633 nm wavelength light source is chosen as an example. The key parameters of the photonic nanojets are scalable in respect of the incident wavelength [27]. The influence of the cutting thickness of the non-spherical microparticle on the photonic nanojet is illustrated in Fig. 2 to Fig. 4. It can be seen from these figures that the shape of field intensity distribution changes as the cutting thickness increases. For the microspheres (Fig. 2(a), Fig. 3 (a), and Fig. 4(a)), the photonic nanojet acts like a water droplet. As the cutting thickness increases, the region of the maximal intensity shifts beyond a water droplet that becomes a photonic flux. In the situation with larger microparticle (Fig. 4(d)), the configuration of intensity distribution becomes more complex. The additional areas of the high intensity are formed rather far from the microparticle surface. Thus, simply varying the cutting thickness, it is possible to achieve localized photonic fluxes having different spatial dimensions and intensity properties. In order to study the characteristics of photonic nanojet, it is necessary primarily to define its spatial length and width. Fig. 5 and Fig. 6 show the normalized intensity distribution of photonic nanojets for nonspherical microparticles along propagation axis (x axis) and transversal axis (y axis) at different diameters and cutting thicknesses. The longitudinal profile (Fig. 5) is acquired as a section of the two-dimensional intensity distribution by the straight line
C.-Y. Liu / Physica E 64 (2014) 23–28
Fig. 2. Normalized electric field intensity distribution of a 3 μm non-spherical microparticle at (a) t ¼ 0 (sphere), (b) t¼ 0.25d ¼0.75 μm, (c) t ¼0.5d ¼1.5 μm, and (d) t ¼0.75d ¼2.25 μm. The 633 nm lightwave propagates from left to right.
25
Fig. 3. Normalized electric field intensity distribution of a 5 μm non-spherical microparticle at (a) t ¼0 (sphere), (b) t¼ 0.25d¼ 1.25 μm, (c) t ¼0.5d ¼2.5 μm, and (d) t¼ 0.75d ¼3.75 μm. The 633 nm lightwave propagates from left to right.
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Fig. 5. Normalized intensity distribution of photonic nanojets for non-spherical microparticles along propagation axis (x axis) at (a) d ¼3 μm, (b) d¼ 5 μm, and (c) d¼ 8 μm.
Fig. 4. Normalized electric field intensity distribution of an 8 μm non-spherical microparticle at (a) t¼ 0 (sphere), (b) t¼ 0.25d ¼2 μm, (c) t ¼0.5d ¼ 4 μm, and (d) t¼ 0.75d ¼6 μm. The 633 nm lightwave propagates from left to right.
located at y¼ 0. The transverse profile (Fig. 6) is acquired as a section by the straight line parallel to axis y and passing through the peak of the maximum intensity. The all intensity profiles of
photonic nanojet for non-spherical microparticles are normalized to the intensity profiles for the microsphere in the same diameter. At small cutting thickness, the photonic nanojet adhered to the surface of microparticle and emerges from it in the form of the exponentially decaying trail. However, the lengths of photonic nanojets for 3 μm, 5 μm, and 8 μm non-spherical microparticles at cutting thickness t ¼0.75d are 3.38 μm, 5.05 μm, and 5.31 μm. The non-spherical microparticles at cutting thickness t ¼0.75d demonstrate the elongated photonic nanojets with the extent ranging from 5λ to 8λ. Depending on the cutting thickness of microparticle, we observe that not only the dimensions and intensity of photonic nanojet change but the separation of the photonic nanojet from the microparticle surface changes also. In Fig. 6, it is significant
C.-Y. Liu / Physica E 64 (2014) 23–28
27
Fig. 7. Focal length as a function of the cutting thickness for non-spherical microparticles.
Fig. 8. Decay length as a function of the cutting thickness for non-spherical microparticles.
Fig. 6. Normalized intensity distribution of photonic nanojets for non-spherical microparticles along transversal axis (y axis) at (a) d ¼ 3 μm, (b) d¼ 5 μm, and (c) d ¼8 μm.
that we have small transverse width of photonic nanojet among all considered situations. The length and location of the photonic nanojet depend on the cutting thickness, but the FWHM of photonic nanojet is almost the same at the different cutting thicknesses. Fig. 7 shows the focal length as a function of the cutting thickness for non-spherical microparticles. It should be noted that all the three studied features demonstrate the nearly linear growth with the increase in cutting thickness. As a result, the focal length and the location of the photonic nanojets can be controlled by cutting thickness. Fig. 8 shows the decay length as a function of the cutting thickness for non-spherical microparticles. The decay length increases as cutting thickness increases. The long decay length can be used to overcome the diffraction limit at the
Fig. 9. Full-width half-maximum of photonic nanojet as a function of the cutting thickness for non-spherical microparticles.
focusing of lightwave by a non-spherical microparticle. Fig. 9 shows the full-width half-maximum of photonic nanojet as a function of the cutting thickness for non-spherical microparticles. The morphological type of non-spherical microparticles optimally combines the high spatial localization of the photonic nanojet with sufficiently high intensity. The photonic nanojet is formed rather far from the surface of the non-spherical microparticles at large cutting thickness, and a decrease in cutting thickness the coordinate of the intensity maximum reaches the microparticle rim. The optimal balance between these key parameters of photonic nanojet is realized in the non-spherical microparticles
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thickness. We believe that this novel technique has a large analysis range and will be helpful in the study of nano-optics and nanobiotechnology. The observed optical emission offers the potential use of the non-spherical microparticle as a high resolution optical lens. The photonic nanojet may allow the coupling of light beam from the photonic molecule into other photonic micro-devices such as optical waveguides or coaxial cables. The optical systems would be not only with sub-wavelength spatial precision, but also with high optical power in comparison with traditional coupling techniques.
Acknowledgments
Fig. 10. Normalized quality criterion of photonic nanojet as a function of the cutting thickness for non-spherical microparticles.
This work is funded by National Science Council in Taiwan under Grant number NSC 102–2221-E-032-009.
References with cutting thickness. Combining these key parameters of photonic nanojet, we propose to describe the photonic nanojets by a quality criterion as Q¼(L I)/FWHM. Fig. 10 shows the normalized quality criterion of photonic nanojet as a function of the cutting thickness for non-spherical microparticles. The quality criterion Q increases as cutting thickness t increases. The quality criterion accounts for the useful value of photonic nanojet to the solution of practical issues. When the quality criterion is high, the effective length of photonic nanojet is long, its FWHM is small, and the peak intensity is high. The quality criterion possesses the high values at larger cutting thickness. These results offer excellent evidence that the elongated photonic nanojet provides the action of a nearly one-dimensional light beam over a significant range of spectral frequencies. The generation of photonic nanojet is based on a nonspherical microparticle which provides highly confined electromagnetic fields to efficiently detect nanoscale targets. 4. Conclusion We have numerically investigated the photonic nanojet shaping generated at the shadow side surface of a non-spherical microparticle. The basic key characteristics of photonic nanojets formed in the vicinity of dielectric non-spherical microparticles with different cutting thicknesses are studied by using FDTD simulation. It makes possible to modulate the shape and location of photonic nanojet. The latitudinal and longitudinal profiles of a photonic nanojet depending on the cutting thickness of non-spherical microparticles are numerically demonstrated. We can gain command of the key parameters for photonic nanojet through the variation of the cutting
[1] H.C. van de Hulst, Light Scattering by Small Particles, Dover Publications, New York, 1981. [2] E. Hecht, Optics, Addison Wesley, San Francisco, 2002. [3] L. Novotny, B. Hecht, Principles of Nano-Optics, Cambridge University Press, New York, 2006. [4] Z. Chen, A. Taflove, V. Backman, Opt. Express 12 (2004) 1214. [5] A.V. Itagi, W.A. Challener, J. Opt. Soc. Am. A 22 (2005) 2847. [6] C. Li, G.W. Kattawar, P. Zhai, Opt. Express 13 (2005) 4554. [7] P. Ferrand, J. Wenger, A. Devilez, M. Pianta, B. Stout, N. Bonod, E. Popov, H. Rigneault, Opt. Express 16 (2008) 6930. [8] A. Devilez, B. Stout, N. Bonod, E. Popov, Opt. Express 16 (2008) 14200. [9] M. Kim, T. Scharf, S. Mühlig, C. Rockstuhl, H.P. Herzig, Opt. Express 19 (2011) 10206. [10] X. Li, Z. Chen, A. Taflove, V. Backman, Opt. Express 13 (2005) 526. [11] Z. Wang, W. Guo, L. Li, B. Lukyanchuk, A. Khan, Z. Liu, Z. Chen, M. Hong, Nat. Commun. 2 (2011) 218. [12] A. Darafsheh, G.F. Walsh, L. Dal Negro, V.N. Astratov, Appl. Phys. Lett. 101 (2012) 141128. [13] K.J. Yi, Y.F. Lu, Z.Y. Yang, J. Appl. Phys. 101 (2007) 063528. [14] S. Lecler, S. Haacke, N. Lecong, O. Crégut, J. Rehspringer, C. Hirlimann, Opt. Express 15 (2007) 4935. [15] W. Wu, A. Katsnelson, O. Memis, H. Mohseni, Nanotechnology 18 (2007) 485302. [16] X. Cui, D. Erni, C. Hafner, Opt. Express 16 (2008) 13560. [17] S. Kong, A. Sahakian, A. Taflove, V. Backman, Opt. Express 16 (2008) 13713. [18] A. Heifetz, S. Kong, A. Sahakian, A. Taflove, V. Backman, J. Comput. Theor. Nanosci. 6 (2009) 1979. [19] S. Kong, A. Taflove, V. Backman, Opt. Express 17 (2009) 3722. [20] Y. Geints, E. Panina, A. Zemlyanov, Opt. Commun. 283 (2010) 4775. [21] Y. Geints, A. Zemlyanov, E. Panina, J. Opt. Soc. Am. B 28 (2011) 1825. [22] C.-Y. Liu, Phys. Lett. A 376 (2012) 1856. [23] C.-Y. Liu, Prog. Electromagn. Res. Lett. 37 (2013) 153. [24] C.-Y. Liu, J. Optoelectron. Adv. Mater. 15 (2013) 150. [25] A. Taflove, S.C. Hagness, Computational Electrodynamics: The Finite Difference Time Domain Method, Artech House, Boston, 1998. [26] J. Berenger, J. Comput. Phys. 114 (1994) 185. [27] C.-Y. Liu, Y. Wang, Physica E 61 (2014) 141.