March 15, 2014 / Vol. 39, No. 6 / OPTICS LETTERS
1585
Controllable and enhanced photonic jet generated by fiber combined with spheroid Lu Han,1,* Yiping Han,1 Jiajie Wang,1 Gerard Gouesbet,2 and Gerard Gréhan2 1
School of physics and optoelectronic engineering, Xidian University, Xi’an, shannxi 710071, China Laboratoire d’Electromagnétisme des Systèmes Particulaires (LESP), Unité Mixte de Recherche (UMR) 6614 du Centre National de la Recherche Scientifique (CNRS), Complexe de Recherche Interprofessionnel en Aérothermochimie (CORIA), Université de Rouen, and Institut National des Sciences Appliquées (INSA) de Rouen BP12, Avenue de l’Université, Technopôle du Madrillet, Saint-Etienne-du Rouvray 76801, France *Corresponding author: [email protected] 2
Received January 9, 2014; revised February 7, 2014; accepted February 11, 2014; posted February 12, 2014 (Doc. ID 204472); published March 12, 2014 Dielectric microparticles are used as simple and low-cost means to achieve strong intensity confinement below the standard diffraction limit. Here we report the demonstration of enhanced light intensity in the vicinity of optical fiber combined with dielectric spheroidal particles. Specific attention is paid to the study of the influences of the spheroid’s ellipticity (ratio of horizontal length to vertical length) as well as the refractive index on the intensity enhancement and focal distance. It reveals that simply varying the ellipticity, it is possible to obtain localized photon fluxes having different characteristics. This could yield a new superenhanced intensity device with excellent optical properties and low manufacturing cost for using visible light in many areas of biology, material sciences, chemistry, medicine, and tissue engineering. © 2014 Optical Society of America OCIS codes: (220.4000) Microstructure fabrication; (260.3160) Interference; (290.4020) Mie theory; (350.3950) Micro-optics. http://dx.doi.org/10.1364/OL.39.001585
Optical properties of dielectric microparticles have attracted a lot of attention during last decade due to their ability to squeeze light and enhance laser intensity on the shadow side. The localization does not just exist in very close vicinity to the surface of the particle but stretches beyond. The high-intensity optical flux with rather narrow transverse dimensions and very low divergence angle is referred to as a “photonic jet” [1–5]. By properly selecting the optical properties of particulate material and the particle geometry, it is possible to obtain the photonic jet that can maintain a subwavelength FWHM transverse beam width along a path. The interest in the related optical properties has prompted important papers reporting advances in high-resolution (nanometer scale) optical sensors [6,7], ultrahigh density optical data storage devices [8,9], subdiffraction-resolution optical virtual imaging [10,11], and ultra-directional optical antennas [12]. Due to the distinctive features of the photonic jet, the feasibilities of using the photonic jet to design tools for precision cell surgery and tumor detection [13,14], and optical tweezers [15,16], as well as to achieve enhanced Raman scattering [17–19] were reported. Given the current trend of miniaturization, it is natural to consider the possibility of photonic jet formation and the maximum value of the near-field optical intensity enhancement. Most of previous work has focused on photonic jets from spheres or cylinders; the use of a spheroid, however, offers many advantages. The size and shape of a spheroid can be easily controlled to high precision in the fabrication process. Mendes et al. [20] studied the light focusing properties of dielectric spheroid with mesoscopic sizes under plane wave illumination, and investigated the influences of spheroid ellipticity and material on the photonic jet’s properties. The ability of a spheroidal particle to generate confined light under focused Gaussian beam illumination has not been reported in the previous literature to the best of our knowledge. 0146-9592/14/061585-04$15.00/0
In this Letter, we theoretically demonstrate that the fiber end face combined with a dielectric spheroidal particle can form enhanced light intensity, within the framework of generalized Lorenz–Mie theory (GLMT). GLMT can serve as a rigorous theory in describing the interaction between the incident beam and spheroid. The intensity enhancement and focal distance of the photonic jet depend strongly on the spheroids’ ellipticity, as well as refractive index. The dielectric spheroids offer attractive solutions to design an enhanced intensity device with excellent optical properties and compactness at minimal costs, and other microsystems, allowing the integration of huge optical components. Such enhanced intensity can be used for sensing, single molecule detection, surface-enhanced Raman scattering, medicine, biology, and other applications. A schematic diagram of the spheroid on the fiber end face is shown in Fig. 1(a). A microwell corresponding to the spheroid on the fiber distal face can be produced by wet-chemical etching. This technique is based on the fiber configuration and some microfabrication processes. Neglecting the divergence in the microvoid formed between fiber and particle, the optical beam that emerges from the fiber can be assumed to have the Gaussianbeam waist w0 at O in the xyz coordinate system in the schematic. The center of the spheroid is also located at the point O. The ellipticity of the spheroid is defined as the ratio of the horizontal length a to its vertical length b, with the geometric configuration described in Fig. 1(b), similar definitions were used by Han et al. [21]. Prolate spheroid with a∕b > 1 is elongated along a line, whereas oblate spheroid with a∕b < 1 is contracted. The case of a∕b 1 reduces to a sphere. Through this Letter, the propagation axis of the beam always coincides with the revolution axis z of the spheroidal particle, and the time-dependent part of the electromagnetic fields is exp−iωt. © 2014 Optical Society of America
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OPTICS LETTERS / Vol. 39, No. 6 / March 15, 2014
Hw
∞ k2 X r1 in γ n Mr1 o1n cII ; ξ; η; ϕ − iδn Ne1n cII ; ξ; η; ϕ: ωμ2 n1
(6) In the above equations, the field strength E 0 has been set equal to unity. Also, c is equal to kf , with k as the wavenumber and f as the semi-interfocal distance of the spheroid. The subscripts I, II on c refer to the outer or inner region of the particle, respectively. Gn cI and F n cI are the expansion coefficients of the incident beam, i.e., the beam shape coefficients [23,25,26]. As the incident field is known, the unknown coefficients of the scattered and internal electromagnetic fields can be determined by using the well-known boundary conditions of continuity of the tangential electromagnetic field over the spheroidal surface. In the external near-surface field, the intensities of the scattered field may be expressed as [22–24] Fig. 1. (a) Schematic diagram for the optical fiber distal face etched with spheroid. (b) Geometry of the spheroid under study.
For a prolate spheroidal particle, the expansion of the incident, scattered, and internal fields can be expanded in terms of the spheroidal vector wave functions [22–24]: (for a TE mode) incident field, Ei
∞ X n1
Hi −
r1 in Gn cI Me1n cI ; ξ; η; ϕ
r1 iF n cI No1n cI ; ξ; η; ϕ;
(1)
(2)
scattered field, Es
∞ X n1
Hs
r3 r3 in βn Me1n cI ; ξ; η; ϕ iαn No1n cI ; ξ; η; ϕ; (3)
∞ k1 X r3 r3 in αn Mo1n cI ; ξ; η; ϕ − iβn Ne1n cI ; ξ; η; ϕ; ωμ1 n1
(4) internal field, Ew
∞ X n1
r1 r1 in δn Me1n cII ; ξ; η; ϕ iγ n No1n cII ; ξ; η; ϕ;
(5)
(7)
while the intensities in the internal electromagnetic field can be written as w w w w w I w E w E w E w η Eη Eξ Eξ Eϕ Eϕ ;
(8)
in which the superscript is the complex conjugate. It is important to note that the results for the oblate spheroid can be evaluated in analogy with the method used above for the prolate spheroid by the transformations [23,27] cI → −icI ;
∞ k1 X r1 in Gn cI Mo1n cI ; ξ; η; ϕ ωμ1 n1
r1 cI ; ξ; η; ϕ; − iF n cI Ne1n
s s s s I s E s E s E sη E s η Eξ Eξ Eϕ Eϕ ;
cII → −icII ;
ξ → iξ:
(9)
The influence of the spheroids’ ellipticity on the photonic jet formation is shown in Fig. 2. The GeSiO2 -FSiO2 core-clad optical fiber has a core diameter of 4 μm and cladding diameter of 10 μm. The fundamental mode of the optical fiber at 632.8 nm wavelength is launched inside the fiber, and the optical beam that emerges from the fiber has a Gaussian-beam waist w0 3λ at the point O. The vertical length b of the silica spheroid (refractive index nII 1.43) is a constant equal to the fiber core diameter, while the horizontal length a is a variable. The surrounding medium is taken to be air (refractive index nI 1.0). The total field outside the spheroid is considered as the sum of the incident and scattered fields. Here, the intensity distributions over the x–z plane are shown. Gradations of the intensity are given by the color-bar, and the particle borders are represented by the white lines in the figures. Strong focusing occurs in the vicinity, mainly due to the focusing effect of the curvature surface of the spheroidal particle. It is evident that the intensity enhancement of the photonic jet depends significantly on the spheroids’ ellipticity. From the comparison of Figs. 2(a)–2(e), we observe that, with each increase of the ellipticity a∕b, the maximum value of the optical intensity enhancement increases, the point of maximum intensity shifts close to the center of the spheroid along the z direction, and the photonic jet significantly shortens in the z direction. Finally, the focus
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Fig. 2. Spatial distributions of the photonic jet relative intensity formed in the vicinity of silica spheroids with different ellipticities: (a) a∕b 0.7634, (b) a∕b 0.8929, (c) a∕b 1, (d) a∕b 1.09, and (e) a∕b 1.18. The refractive index of the silica spheroid is nII 1.43.
tends shifting inside the particle [see Fig. 2(e) a∕b 1.18]. The intensity maximum outside the spheroid is multiplied by more than 26 [see Fig. 2(d) a∕b 1.09]. Different refractive index for the spheroid also will affect the photonic jet. Figure 3 demonstrates the influence of the spheroids’ ellipticity on the main characteristics of the photonic jet for three refractive indices of the spheroid nII 1.43, 1.5, 1.59, respectively. The parameters of the incident beam and surrounding medium are the same as used in Fig. 2. The vertical length b is a constant equal to 4 μm, while the horizontal length a is a variable. Obviously, as the ellipticity a∕b increases, the maximum intensity enhancement increases with superimposed oscillations, and the point of maximum intensity shifts close to the center of the spheroid. Assuming initial value of the refractive index as nII 1.43, we see that the increasing of the refractive index leads to increase of the maximum intensity enhancement and to reduction of distance between the point of maximum intensity and the spheroid’s center. These results offer good evidence that the oblate spheroid (a∕b < 1) can generate lower intensity, but longer focal distance than the corresponding sphere (a∕b 1). On the other hand, the prolate spheroids (a∕b > 1) demonstrate shorter longitudinal waists, but higher intensity than the various realizations of the sphere (a∕b 1). It may be advantageous to choose spheroidal particle on the fiber end face to obtain the photonic jet with desired properties in application. For example, in technologies of direct-write nanopatterning, the preferred photonic jet from the prolate spheroid has maximal intensity, and in live probe, the oblate spheroid could be selected to generate a maximally extended photonic jet.
Fig. 3. Evolution of (a) the maximum intensity enhancement and (b) the point of the maximum intensity as the spheroid’s ellipticity increases for three refractive indices of the spheroid (nII 1.43, 1.5, 1.59).
To obtain an enhanced intensity spot in the far field by utilizing the excellent properties of the photonic jet as depicted in Fig. 2(d), we design a fiber bundle containing a central element and surrounding a hexagonal ring of
Fig. 4. (a) Schematic of the optical fiber bundle with spheroids focus on the photoresist plane. (b) The two-dimensional intensity distribution in the photoresist plane, calculated by our three-dimensional GLMT code.
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elements, as illustrated in Fig. 4(a). The light intensity of the initial incident beam on every spheroid is uniform, so the light intensities of the induced photonic jet fields are the same correspondingly. The photonic jets could be focused by a lens, which is placed at maximum intensity surface in the transverse plane. The centrosymmetry arrangement results in coherent enhancement to form the focus point and no phase difference at point along the propagation axis. Figure 4(b) shows the intensity distribution of the focal spot on the photoresist. We see the intensity maximum is about 270 times larger than the incident intensity. The results demonstrate obviously that the structure we designed is efficient to form super-enhanced intensity in the far field due to constructive interference of the photonic nanojet fields. In conclusion, we have analyzed the enhancement of localized photonic jets generated at the shadow-side surface of optical fiber, which combines with dielectric spheroidal particles, by virtue of the GLMT. Numerical results concerning the influences of spheroids’ ellipticity, as well as the refractive index on the intensity enhancement and focal distance of the photonic jet are presented. The prolate spheroids demonstrate higher intensity, but shorter longitudinal waists than the various realizations of the oblate ones. It may be advantageous to choose a jet-producing particle to obtain the photonic jet with desired properties in application. A bundle of fibers with spheroids can generate a superenhanced intensity spot in the far field by utilizing the excellent properties of the photonic jet. The proposed novel fiber with spheroid can be used for superenhanced intensity and other microsystems allowing the integration of huge optical components at minimal costs. Well, let us wait that these calculations come in reality. References 1. Z. Chen, A. Taflove, and V. Backman, Opt. Express 12, 1214 (2004). 2. A. Itagi and W. Challener, J. Opt. Soc. Am. A 22, 2847 (2005). 3. S. Lecler, Y. Takakura, and P. Meyrueis, Opt. Lett. 30, 2641 (2005). 4. A. Devilez, B. Stout, N. Bonod, and E. Popov, Opt. Express 16, 14200 (2008).
5. P. Ferrand, J. Wenger, A. Devilez, M. Pianta, B. Stout, N. Bonod, E. Popov, and H. Rigneault, Opt. Express 16, 6930 (2008). 6. X. Li, Z. G. Chen, A. Taflove, and V. Backman, Opt. Express 13, 526 (2005). 7. P. Ghenuche, H. Rigneault, and J. Wenger, Appl. Opt. 51, 8637 (2012). 8. S. C. Kong, A. Sahakian, A. Taflove, and V. Backman, Opt. Express 16, 13713 (2008). 9. S. C. Kong, A. V. Sahakian, A. Heifetz, A. Taflove, and V. Backman, Appl. Phys. Lett. 92, 211102 (2008). 10. Z. Wang, W. Guo, L. Li, B. Luk’yanchuk, A. Khan, Z. Liu, Z. Chen, and M. Hong, Nat. Commun. 2, 218 (2011). 11. Y. Duan, G. Barbastathis, and B. Zhang, Opt. Lett. 38, 2988 (2013). 12. A. Devilez, B. Stout, and N. Bonod, ACS Nano 4, 3390 (2010). 13. H. Seidfaraji, M. Hasan, and J. J. Simpson, International Conference on Electromagnetics in Advanced Applications (ICEAA) (2012), p. 949. 14. V. N. Astratov, A. Darafsheh, M. D. Kerr, K. W. Allen, N. M. Fried, A. N. Antoszyk, and H. S. Ying, SPIE Newsroom 12, 32 (2010). 15. X. Cui, D. Erni, and C. Hafner, Opt. Express 16, 13560 (2008). 16. V. Yannopapas, Opt. Commun. 285, 2952 (2012). 17. K. J. Yi, H. Wang, Y. F. Lu, and Z. Y. Yang, J. Appl. Phys. 101, 063528 (2007). 18. C. L. Du, J. Kasim, Y. M. You, D. N. Shi, and Z. X. Shen, J. Raman Spectrosc. 42, 145 (2011). 19. J. Kasim, T. Yu, Y. M. You, J. P. Liu, A. See, L. J. Li, and Z. X. Shen, Opt. Express 16, 7976 (2008). 20. M. J. Mendes, I. T. Galicia, A. M. Vega, and A. L. López, Opt. Express 19, 16207 (2011). 21. Y. Han, L. Méès, K. Ren, G. Gouesbet, S. Wu, and G. Gréhan, Opt. Commun. 210, 1 (2002). 22. G. Gouesbet and G. Gréhan, Generalized Lorenz-Mie Theories (Springer, 2011). 23. Y. Han and Z. Wu, Appl. Opt. 40, 2501 (2001). 24. L. Han, Y. Han, J. Wang, and G. Gouesbet, J. Quant. Spectrosc. Radiat. Transfer 126, 38 (2013). 25. F. Xu, K. Ren, G. Gouesbet, G. Gréhan, and X. Cai, J. Opt. Soc. Am. A 24, 119 (2007). 26. G. Gouesbet, F. Xu, and Y. Han, J. Quant. Spectrosc. Radiat. Transfer 112, 1 (2011). 27. C. Flammer, Spheroidal Wave Functions (Stanford University, 1957).
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Controllable and enhanced photonic jet generated by fiber combined with spheroid Lu Han,1,* Yiping Han,1 Jiajie Wang,1 Gerard Gouesbet,2 and Gerard Gréhan2 1
School of physics and optoelectronic engineering, Xidian University, Xi’an, shannxi 710071, China Laboratoire d’Electromagnétisme des Systèmes Particulaires (LESP), Unité Mixte de Recherche (UMR) 6614 du Centre National de la Recherche Scientifique (CNRS), Complexe de Recherche Interprofessionnel en Aérothermochimie (CORIA), Université de Rouen, and Institut National des Sciences Appliquées (INSA) de Rouen BP12, Avenue de l’Université, Technopôle du Madrillet, Saint-Etienne-du Rouvray 76801, France *Corresponding author: [email protected] 2
Received January 9, 2014; revised February 7, 2014; accepted February 11, 2014; posted February 12, 2014 (Doc. ID 204472); published March 12, 2014 Dielectric microparticles are used as simple and low-cost means to achieve strong intensity confinement below the standard diffraction limit. Here we report the demonstration of enhanced light intensity in the vicinity of optical fiber combined with dielectric spheroidal particles. Specific attention is paid to the study of the influences of the spheroid’s ellipticity (ratio of horizontal length to vertical length) as well as the refractive index on the intensity enhancement and focal distance. It reveals that simply varying the ellipticity, it is possible to obtain localized photon fluxes having different characteristics. This could yield a new superenhanced intensity device with excellent optical properties and low manufacturing cost for using visible light in many areas of biology, material sciences, chemistry, medicine, and tissue engineering. © 2014 Optical Society of America OCIS codes: (220.4000) Microstructure fabrication; (260.3160) Interference; (290.4020) Mie theory; (350.3950) Micro-optics. http://dx.doi.org/10.1364/OL.39.001585
Optical properties of dielectric microparticles have attracted a lot of attention during last decade due to their ability to squeeze light and enhance laser intensity on the shadow side. The localization does not just exist in very close vicinity to the surface of the particle but stretches beyond. The high-intensity optical flux with rather narrow transverse dimensions and very low divergence angle is referred to as a “photonic jet” [1–5]. By properly selecting the optical properties of particulate material and the particle geometry, it is possible to obtain the photonic jet that can maintain a subwavelength FWHM transverse beam width along a path. The interest in the related optical properties has prompted important papers reporting advances in high-resolution (nanometer scale) optical sensors [6,7], ultrahigh density optical data storage devices [8,9], subdiffraction-resolution optical virtual imaging [10,11], and ultra-directional optical antennas [12]. Due to the distinctive features of the photonic jet, the feasibilities of using the photonic jet to design tools for precision cell surgery and tumor detection [13,14], and optical tweezers [15,16], as well as to achieve enhanced Raman scattering [17–19] were reported. Given the current trend of miniaturization, it is natural to consider the possibility of photonic jet formation and the maximum value of the near-field optical intensity enhancement. Most of previous work has focused on photonic jets from spheres or cylinders; the use of a spheroid, however, offers many advantages. The size and shape of a spheroid can be easily controlled to high precision in the fabrication process. Mendes et al. [20] studied the light focusing properties of dielectric spheroid with mesoscopic sizes under plane wave illumination, and investigated the influences of spheroid ellipticity and material on the photonic jet’s properties. The ability of a spheroidal particle to generate confined light under focused Gaussian beam illumination has not been reported in the previous literature to the best of our knowledge. 0146-9592/14/061585-04$15.00/0
In this Letter, we theoretically demonstrate that the fiber end face combined with a dielectric spheroidal particle can form enhanced light intensity, within the framework of generalized Lorenz–Mie theory (GLMT). GLMT can serve as a rigorous theory in describing the interaction between the incident beam and spheroid. The intensity enhancement and focal distance of the photonic jet depend strongly on the spheroids’ ellipticity, as well as refractive index. The dielectric spheroids offer attractive solutions to design an enhanced intensity device with excellent optical properties and compactness at minimal costs, and other microsystems, allowing the integration of huge optical components. Such enhanced intensity can be used for sensing, single molecule detection, surface-enhanced Raman scattering, medicine, biology, and other applications. A schematic diagram of the spheroid on the fiber end face is shown in Fig. 1(a). A microwell corresponding to the spheroid on the fiber distal face can be produced by wet-chemical etching. This technique is based on the fiber configuration and some microfabrication processes. Neglecting the divergence in the microvoid formed between fiber and particle, the optical beam that emerges from the fiber can be assumed to have the Gaussianbeam waist w0 at O in the xyz coordinate system in the schematic. The center of the spheroid is also located at the point O. The ellipticity of the spheroid is defined as the ratio of the horizontal length a to its vertical length b, with the geometric configuration described in Fig. 1(b), similar definitions were used by Han et al. [21]. Prolate spheroid with a∕b > 1 is elongated along a line, whereas oblate spheroid with a∕b < 1 is contracted. The case of a∕b 1 reduces to a sphere. Through this Letter, the propagation axis of the beam always coincides with the revolution axis z of the spheroidal particle, and the time-dependent part of the electromagnetic fields is exp−iωt. © 2014 Optical Society of America
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OPTICS LETTERS / Vol. 39, No. 6 / March 15, 2014
Hw
∞ k2 X r1 in γ n Mr1 o1n cII ; ξ; η; ϕ − iδn Ne1n cII ; ξ; η; ϕ: ωμ2 n1
(6) In the above equations, the field strength E 0 has been set equal to unity. Also, c is equal to kf , with k as the wavenumber and f as the semi-interfocal distance of the spheroid. The subscripts I, II on c refer to the outer or inner region of the particle, respectively. Gn cI and F n cI are the expansion coefficients of the incident beam, i.e., the beam shape coefficients [23,25,26]. As the incident field is known, the unknown coefficients of the scattered and internal electromagnetic fields can be determined by using the well-known boundary conditions of continuity of the tangential electromagnetic field over the spheroidal surface. In the external near-surface field, the intensities of the scattered field may be expressed as [22–24] Fig. 1. (a) Schematic diagram for the optical fiber distal face etched with spheroid. (b) Geometry of the spheroid under study.
For a prolate spheroidal particle, the expansion of the incident, scattered, and internal fields can be expanded in terms of the spheroidal vector wave functions [22–24]: (for a TE mode) incident field, Ei
∞ X n1
Hi −
r1 in Gn cI Me1n cI ; ξ; η; ϕ
r1 iF n cI No1n cI ; ξ; η; ϕ;
(1)
(2)
scattered field, Es
∞ X n1
Hs
r3 r3 in βn Me1n cI ; ξ; η; ϕ iαn No1n cI ; ξ; η; ϕ; (3)
∞ k1 X r3 r3 in αn Mo1n cI ; ξ; η; ϕ − iβn Ne1n cI ; ξ; η; ϕ; ωμ1 n1
(4) internal field, Ew
∞ X n1
r1 r1 in δn Me1n cII ; ξ; η; ϕ iγ n No1n cII ; ξ; η; ϕ;
(5)
(7)
while the intensities in the internal electromagnetic field can be written as w w w w w I w E w E w E w η Eη Eξ Eξ Eϕ Eϕ ;
(8)
in which the superscript is the complex conjugate. It is important to note that the results for the oblate spheroid can be evaluated in analogy with the method used above for the prolate spheroid by the transformations [23,27] cI → −icI ;
∞ k1 X r1 in Gn cI Mo1n cI ; ξ; η; ϕ ωμ1 n1
r1 cI ; ξ; η; ϕ; − iF n cI Ne1n
s s s s I s E s E s E sη E s η Eξ Eξ Eϕ Eϕ ;
cII → −icII ;
ξ → iξ:
(9)
The influence of the spheroids’ ellipticity on the photonic jet formation is shown in Fig. 2. The GeSiO2 -FSiO2 core-clad optical fiber has a core diameter of 4 μm and cladding diameter of 10 μm. The fundamental mode of the optical fiber at 632.8 nm wavelength is launched inside the fiber, and the optical beam that emerges from the fiber has a Gaussian-beam waist w0 3λ at the point O. The vertical length b of the silica spheroid (refractive index nII 1.43) is a constant equal to the fiber core diameter, while the horizontal length a is a variable. The surrounding medium is taken to be air (refractive index nI 1.0). The total field outside the spheroid is considered as the sum of the incident and scattered fields. Here, the intensity distributions over the x–z plane are shown. Gradations of the intensity are given by the color-bar, and the particle borders are represented by the white lines in the figures. Strong focusing occurs in the vicinity, mainly due to the focusing effect of the curvature surface of the spheroidal particle. It is evident that the intensity enhancement of the photonic jet depends significantly on the spheroids’ ellipticity. From the comparison of Figs. 2(a)–2(e), we observe that, with each increase of the ellipticity a∕b, the maximum value of the optical intensity enhancement increases, the point of maximum intensity shifts close to the center of the spheroid along the z direction, and the photonic jet significantly shortens in the z direction. Finally, the focus
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Fig. 2. Spatial distributions of the photonic jet relative intensity formed in the vicinity of silica spheroids with different ellipticities: (a) a∕b 0.7634, (b) a∕b 0.8929, (c) a∕b 1, (d) a∕b 1.09, and (e) a∕b 1.18. The refractive index of the silica spheroid is nII 1.43.
tends shifting inside the particle [see Fig. 2(e) a∕b 1.18]. The intensity maximum outside the spheroid is multiplied by more than 26 [see Fig. 2(d) a∕b 1.09]. Different refractive index for the spheroid also will affect the photonic jet. Figure 3 demonstrates the influence of the spheroids’ ellipticity on the main characteristics of the photonic jet for three refractive indices of the spheroid nII 1.43, 1.5, 1.59, respectively. The parameters of the incident beam and surrounding medium are the same as used in Fig. 2. The vertical length b is a constant equal to 4 μm, while the horizontal length a is a variable. Obviously, as the ellipticity a∕b increases, the maximum intensity enhancement increases with superimposed oscillations, and the point of maximum intensity shifts close to the center of the spheroid. Assuming initial value of the refractive index as nII 1.43, we see that the increasing of the refractive index leads to increase of the maximum intensity enhancement and to reduction of distance between the point of maximum intensity and the spheroid’s center. These results offer good evidence that the oblate spheroid (a∕b < 1) can generate lower intensity, but longer focal distance than the corresponding sphere (a∕b 1). On the other hand, the prolate spheroids (a∕b > 1) demonstrate shorter longitudinal waists, but higher intensity than the various realizations of the sphere (a∕b 1). It may be advantageous to choose spheroidal particle on the fiber end face to obtain the photonic jet with desired properties in application. For example, in technologies of direct-write nanopatterning, the preferred photonic jet from the prolate spheroid has maximal intensity, and in live probe, the oblate spheroid could be selected to generate a maximally extended photonic jet.
Fig. 3. Evolution of (a) the maximum intensity enhancement and (b) the point of the maximum intensity as the spheroid’s ellipticity increases for three refractive indices of the spheroid (nII 1.43, 1.5, 1.59).
To obtain an enhanced intensity spot in the far field by utilizing the excellent properties of the photonic jet as depicted in Fig. 2(d), we design a fiber bundle containing a central element and surrounding a hexagonal ring of
Fig. 4. (a) Schematic of the optical fiber bundle with spheroids focus on the photoresist plane. (b) The two-dimensional intensity distribution in the photoresist plane, calculated by our three-dimensional GLMT code.
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elements, as illustrated in Fig. 4(a). The light intensity of the initial incident beam on every spheroid is uniform, so the light intensities of the induced photonic jet fields are the same correspondingly. The photonic jets could be focused by a lens, which is placed at maximum intensity surface in the transverse plane. The centrosymmetry arrangement results in coherent enhancement to form the focus point and no phase difference at point along the propagation axis. Figure 4(b) shows the intensity distribution of the focal spot on the photoresist. We see the intensity maximum is about 270 times larger than the incident intensity. The results demonstrate obviously that the structure we designed is efficient to form super-enhanced intensity in the far field due to constructive interference of the photonic nanojet fields. In conclusion, we have analyzed the enhancement of localized photonic jets generated at the shadow-side surface of optical fiber, which combines with dielectric spheroidal particles, by virtue of the GLMT. Numerical results concerning the influences of spheroids’ ellipticity, as well as the refractive index on the intensity enhancement and focal distance of the photonic jet are presented. The prolate spheroids demonstrate higher intensity, but shorter longitudinal waists than the various realizations of the oblate ones. It may be advantageous to choose a jet-producing particle to obtain the photonic jet with desired properties in application. A bundle of fibers with spheroids can generate a superenhanced intensity spot in the far field by utilizing the excellent properties of the photonic jet. The proposed novel fiber with spheroid can be used for superenhanced intensity and other microsystems allowing the integration of huge optical components at minimal costs. Well, let us wait that these calculations come in reality. References 1. Z. Chen, A. Taflove, and V. Backman, Opt. Express 12, 1214 (2004). 2. A. Itagi and W. Challener, J. Opt. Soc. Am. A 22, 2847 (2005). 3. S. Lecler, Y. Takakura, and P. Meyrueis, Opt. Lett. 30, 2641 (2005). 4. A. Devilez, B. Stout, N. Bonod, and E. Popov, Opt. Express 16, 14200 (2008).
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