- Email: [email protected]
Photonic jet shaping of mesoscale dielectric spherical particles: Resonant and non-resonant jet formation
Journal of Quantitative Spectroscopy & Radiative Transfer 126 (2013) 44–49
Contents lists available at SciVerse ScienceDirect
Journal of Quantitative Spectroscopy & Radiative Transfer journal homepage: www.elsevier.com/locate/jqsrt
Photonic jet shaping of mesoscale dielectric spherical particles: Resonant and non-resonant jet formation Yuri E. Geints n, Ekaterina K. Panina, Alexander A. Zemlyanov Zuev Institute of Atmospheric Optics SB RAS, 1 Zuev Square, Tomsk 634021, Russia
a r t i c l e in f o
abstract
Article history: Received 29 May 2012 Received in revised form 19 July 2012 Accepted 20 July 2012 Available online 28 July 2012
A descriptive analysis of the morphological shapes of photonic jets (PJs) these being a specific spatially localized and high-intensity area formed near micron-sized transparent spherical dielectric particles illuminated by a visible laser radiation, is presented. The PJ shape characterization is based on the numerical calculations of the near-field distribution according to the Mie theory and accounts for jet dimensions and shape complexity. The special situation of the morphology-dependent resonances excitation in the internal field of the particle is also considered. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Mie scattering Photonic jet Dielectric particles Near-field distribution
1. Introduction The transparent dielectric sphere exposed to a light wave tends to focus it and that leads to a marked increase of laser intensity in the near-field region of forward scattering. This high-intensity spatial area possesses the shape of a jet-like photonic flux emerging from the particle shadow surface and is commonly referred to as a ‘‘photonic jet’’ (PJ) [1]. Its main features are strong transverse spatial localization (up to sub-diffraction scale), enormous elongation and high intensity that opens promising prospects for the practical application of this phenomenon in various fields of science and technology [2–5]. It should be clear that although the PJ is nothing else but the area of external focus of an optical wave created by a spherical particle, this wave structure has very specific spatial parameters (i.e., the relations between the length and width of the focal waist), which are untypical for the well-known thin convex lenses. The
n
Corresponding author. E-mail address: [email protected] (Y.E. Geints).
0022-4073/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jqsrt.2012.07.023
focal region formed by a micron-sized particle is abnormally extended along the direction of laser incidence and takes the shape of a pronounced jet. Thus, because a microparticle can be treated as a mesoscopic scatterer for visible radiation the formation of PJ from such a particle is strongly affected by the diffraction effects accompanying the scattering and, by the interference of the waves transmitted through and refracted by a particle. The study of PJ basic characteristics (transverse dimension, length and peak intensity) was the subject of numerous theoretical [6–8] and experimental works [9,10]. Most of these papers deal either with only single morphological type of photonic jets or with the individual jet parameters. High variability of PJ shapes with the variation of particle size and its optical structure, and the absence of adequate criteria for comprehensive estimation of PJ parameters make proper choice of jet-producing particle to obtain the PJ with desired properties difficult. Actually, different applications require different types of PJ having specific characteristics. For example, in optical nano-sensing [11] or in laser surgery [2] maximally narrow and extended photonic jets are needed. On the other side, if dielectric microspheres are used for the contact perforation of cell membranes [12], the capability
Y.E. Geints et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 126 (2013) 44–49
of a particle to focus the incident optical field comes to the forefront, i.e., the PJs having the maximal intensity are preferred, while their length has no significant importance. Therefore, it would be instructive to catalog the main jet shapes, and to partition them into several morphological groups or classes, within which the particular jet parameters could be further optimized. The purpose of this paper is to give such a qualitative morphological analysis based on the numerical calculations of jet shapes for different structural composition of spherical microparticles. Besides, we give special attention to the case of morphology-dependent resonances (MDR) excitation in the particle internal field that can lead to jet parameter modifications. 2. PJ structural types As a key parameter for our analysis we used the existence of the pronounced gap between the shadow surface of the microsphere and PJ intensity maximum. Visually, the presence or the absence of this space looks like either the flare-type (Fl) jet or the dagger-type (Dg) jet, respectively. Thus, we introduce two morphological types of jets, Dg and Fl, which will be further named as jet classes. These two classes are supplemented by detailed attributes in accordance with the following: (I) the type of jet latitudinal width variation, namely: a. inhomogeneous PJ, which is characterized by the presence of a pronounced waist (or waists) along the jet; b. homogeneous PJ, i.e., with the monotonic variation of the jet width; (II) the type of longitudinal intensity distribution along the jet: a. discontinuous PJs, having secondary intensity maxima in the direction of light incidence; b. continuous PJ, which do not have these maxima; (III) the type of transverse intensity distribution: a. branched PJ, having marked side lobes in the intensity distribution of the field at the sphere surface; b. simple PJ, i.e., without marked side lobes. For simplicity, we will use the alphanumeric designation of introduced PJ classes. For example, the Dg: Ia–IIb–IIIa class indicates a dagger-type inhomogeneous continuous jet with pronounced side lobes. Table 1 summarizes the proposed jet classification. The proposed PJ classification is made on the basis of numerical calculations and analysis of the spatial form of photonic jets formed in the vicinity of airborne nonabsorbing spherical dielectric particles with radii a0 ¼1– 5 mm, and refractive indexes in the range na ¼1–2 being irradiated by a monochromatic linearly polarized plane light wave with l ¼0.532 mm. This is dictated by the fact that, on the one hand, for smaller particle sizes the pronounced photonic flux does not form yet, and on the other hand, for larger particles the near-field scattering region of optical radiation (the region of jet formation) is blurred and, therefore, cannot be referred to as a PJ [13].
45
Table 1 Classification of the photonic jet shapes. PJ Class (Dg) dagger-type
(Fl) flare-type Attributes
(I) Jet width z-variation (II) Longitudinal profile a Inhomogeneous b Homogeneous
Discontinuous Continuous
(III) Jet complexity
Branched Simple
The specific range of particle sizes and the specific laser radiation wavelength that we have chosen do not reduce a generality of our PJ types analysis because in the Lorenz–Mie theory [14] the relevant quantity is not the particle size itself but the dimensionless ratio of particle radius a0 to the light wavelength l called the size parameter: xa ¼ 2pa0/l. Thus, when considering Mie scattering in a different wavelengths range one has to change the particle radius a0 as well. Most practical PJ applications require a photon flux having submicron dimensions and therefore the subject of our study was micron-sized particles, which produce PJs of desired transverse widths. Our choice of the specific range of particles refractive index na is motivated by the considerations of practical capabilities of microsphere production. Below we present a more detailed discussion of issues, related to the peculiarities of the formation of photonic jets from the most common structural types of dielectric spherical microparticles, as well as illustrate the PJ classification by numerical near-field calculations. 3. Nonresonant excitation of spherical particles 3.1. Homogeneous particles It is known that the character of light intensity distribution near homogeneous nonabsorbing spherical particles differs as the particle size increases. According to the calculations [13], for spherical particles with the radius a0 r3 mm, the half-width of photonic flux R remains smaller than the diffraction-limited value determined as the focal waist radius upon convex lens focusing. Such particles will be referred to here as small scatterers. For small particles, the spatial area of the external optical field maximum, i.e., the area of photonic jet, is characterized by smooth variations along and across the jet and has an arrow-like shape. We name this type of jets as a dagger-type PJ. As an example of a typical PJ, Fig. 1 shows the false color 2D-distributions of the relative optical field intensity B(x,z)¼I(x,z)I0 (I0 is the incident intensity) in the vicinity of a quartz sphere with the radius a0 ¼ 3 mm. The intensity in this and in the following 2D-graphs is normalized to its maximal value Bmax, which is indicated in each figure by the corresponding number in the upper left corner. For the convenience we added a bar showing the 1 mm scale.
46
Y.E. Geints et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 126 (2013) 44–49
Fig. 1. PJ formation in the vicinity of a homogeneous sphere with a0 ¼ 3 mm and na ¼ 1.5 (PJ class Dg: Ib–IIb–IIIb). The light wave is incident from the left. The dashed circle shows the particle rim, and the bar shows the 1 mm scale. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
Hereafter we follow the situation when the incident plane wave is linearly polarized along the x-axis and propagates in the positive z-direction. Therefore, we choose the x–z plane as a plane where the photonic jets will be analyzed and classified. According to the Mieformulae, the state of incident light polarization does not significantly affect scattered intensity spatial distribution, but affects only its magnitude. Thus, the perpendicular y–z plane will not be considered. As the particle radius increases, the PJ intensity maximum shifts farther from the particle boundary. The PJ configuration becomes more complicated: the jet extends along the radiation incidence (in absolute units), a pronounced waist appears in the transverse direction, and the secondary maxima are formed in the longitudinal direction. The Lorenz–Mie theory states that the key parameter in light scattering is not the refractive index of a particle na itself, but its ratio to the refractive index of the ambient medium (nm), nr ¼na/nm, often referred to as the optical contrast. The particles with low nr values form photonic jets of the classical flare-like type, which is clearly separated from the particle surface. This type of jets is characterized by the morphological class Fl:Ib–IIb–IIIb (homogeneous continuous jet without marked side lobes). With increasing parameter nr, the center of the external focal waist first approaches the particle surface and then shifts inside the particle. This is caused by an increase in the optical strength of the lens, whose role is to play as a spherical particle for the incident optical wave. The samples of intensity distributions corresponding to the situations of higher particle optical contrast demonstrate photonic jets of the different morphological class, namely the dagger-type PJ. Thus, the comparative analysis of PJ spatial shapes in the case of non-resonant excitation of the optical field of homogeneous non-absorbing spherical particles indicates the possibility of realization of both PJ classes.
transverse dimensions and the peak intensity of photonic jets from dielectric layered microspheres of different sizes. The calculations were made for a particle consisting of a core with the radius a0 and four concentric layers of equal thickness h with the radii as, s¼ 0,y,N. The refractive indices of the core, n0, and the outermost shell, nN, are fixed: n0 ¼1.5, nN ¼1.1. To quantitatively describe the layer-bylayer refractive index variation, the following functional g dependence is introduced: ns =n0 ¼ ðnN =n0 Þðs=NÞ , where s (integer) is the layer number, and the parameter g (g40) defines the type of grading. Following [15], according to the radial dependence of interlayer optical contrast, gs ¼ns/ns þ 1, we will consider three principal types of particle refractive index grading: ‘‘concave’’, ‘‘convex’’, and ‘‘neutral grading’’. The particles of the first ‘‘concave’’ structural type (type A, g o1) have decreasing refractive index contrast between neighboring shells. Particles of the ‘‘convex’’ structural type (type B, g 41) are characterized by increasing optical contrast from the core to the peripheral shell. And, finally, the particles of the third structural type (type C, g¼1) have a constant interlayer optical contrast. For radially inhomogeneous multilayer spherical particles the spatial shape and the intensity of PJ are mainly determined by the type of the optical contrast grading of neighboring shells. Particles of C-type can form high-intensity extended and simultaneously spatially localized ‘‘dagger’’ type jets of the class Dg:Ib–IIb–IIIa. The multilayer spheres of other structural types (A and B) lead to different PJ morphological classes. Thus, for A-type particles we have the class Fl: Ib–IIb–IIIb, while the PJs emerging from B-type spheres can be classified as Dg: Ia–IIa–IIIa jets. As shown by the numerical calculations [15], the particles with the sublinear n-grading (A-type), with gradual decrease in refractive index between neighboring shells (see Fig. 2a,b), produce optimal photon fluxes in the
3.2. Radially-inhomogeneous multilayered particles In Ref. [15], the numerical calculations within the Mie theory extended to the case of light scattering at multilayered spheres were used to study the longitudinal and
Fig. 2. (a) Refractive index-grading scheme of the particle; and (b) Photonic jet from seven-layered spheres of sub-linear structural type (A-type) with a0 ¼3 mm (PJ class Fl: Ib–IIb–IIIb).
Y.E. Geints et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 126 (2013) 44–49
sense of the relation between PJ intensity, its length and transverse dimension. With an increase of the outer diameter of multilayered particles of the type A, the characteristic length L and width R of the photonic jet do not change significantly and depend only on the value of structural parameter g. On the contrary, photonic jets produced by B-type particles exhibit the strong dependence on the size of parent sphere and vary only slightly on the parameter g. When the particle radius grows, the intensity of the jet, its length and width increase, and, what is more important, the PJ longitudinal structure becomes more complex. In addition to the main intensity maximum near the particle surface, the secondary field bunch (‘‘light bullet’’) appears in the radial direction at a distance of about a wavelength (see Fig. 3). This circumstance allows a significant increase in the total area of the photonic jet, which may be of principal importance for various practical applications.
47
optical energy is accumulated by the mode. Although in general the spatial shape of a PJ at optical resonance changes only insignificantly, an increase of the near-surface intensity can affect markedly the class of the PJ. The ‘‘sticking’’ of the photonic jet to the microparticle surface can change its class from the flare-type jet, in the absence of resonance, to dagger-type jet at a resonance. Our calculations show that this PJ type changing is characteristic only for high-quality WGMs with typical values, Q4103. The excitation of resonances with low Q-factors does not lead to PJ type change. Moreover, we have found that at a resonance, the length and the mean halfwidth of a photonic jet remain nearly the same as in the nonresonant case (see Fig. 4a–c). The effect of transverse PJ squeezing investigated in [6] is prominent only close to the particle surface at distances of about a quarter-wavelength, and after this, the PJ widens to its ‘‘normal’’ size.
4. Resonant PJ excitation from homogeneous spherical particles One of the striking phenomena in light scattering is the excitation of morphological resonances in a transparent spherical particle. This means that by properly choosing the wavelength of incident light and the particle radius it is possible to selectively excite certain spatial configurations (eigenmodes) of electromagnetic field inside a dielectric particle. At resonance, the maximal intensity of the internal optical field becomes much higher than that of off-resonance, which could also influence the external optical field, and namely, the PJ region. As known, the electromagnetic field inside a spherical particle can be represented as an expansion in two families of eigenmodes: TE-waves and TM-waves. The interference of these two mode families results in a specific pattern of the scattering field. At resonance, only one certain type of eigenmodes is sharply amplified, and therefore the spatial configuration of the internal optical field follows the spatial profile of the excited resonant mode. The resonant optical field inside the particle is concentrated in a narrow annular layer near the particle rim forming the so-called whispering-gallery mode (WGM) [16]. The WGMs of different types also have different quality factors Q (Q-factors) showing how efficiently the
Fig. 3. Example of discontinuous PJ with pronounced side lobes from five-layered spherical particle with a0 ¼ 4 mm, g ¼2 (B-type particle, PJ class Dg: Ia–IIa–IIIa).
Fig. 4. (a) PJ of the class Fl: Ib–IIb–IIIa produced upon resonant water droplet excitation (TM 241 , Q ¼4 104 ); (b) PJ of the class Dg: Ib–IIb–IIIa (TM 138 , Q ¼2 107); and (c) non-resonant PJ of the class Fl: Ib–IIb–IIIa.
48
Y.E. Geints et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 126 (2013) 44–49
the ratio of WGM width to laser radiation linewidth and decreases with this ratio decrease. Practically, this can considerably limit the influence of field resonances on PJ parameters.
5. Conclusion
Fig. 5. Radial intensity profile of the near-field scattering of spherical particle, and PJ half-width (the insert) in the y–z plane on- and offresonance for TM 140 and TM 240 modes. The dashed line marks the particle boundary.
Following the proposed classification, we can state that at resonant WGM excitation for the data presented in Fig. 4a,b the homogeneous continuous PJs are formed with weak lateral lobes, representing the morphological classes Fl: Ib–IIb–IIIa and Dg: Ib–IIb–IIIa, respectively. This qualitative analysis is also confirmed by our largescale numerical calculations performed for several dozens of resonances with different polarizations and orders excited in the particle with the radii ranging from 1 to 5 mm. For example, as shown in [17], the resonant excitation of the TM 140 mode in the particle with a0 ¼2.5790068 mm leads to an increase of the optical intensity in the near-field region up to the values B 1200. After that, the intensity decreases quickly, and at a distance of approximately 300 nm from the particle surface, i.e., in the PJ area, it becomes even lower than that in the off-resonance case (particle radius, a0 ¼ 2.88 mm). The near-field intensity at the TM 240 resonance (a0 ¼2.8748 mm) demonstrates a similar behavior; however, its increase on the particle surface is about twofold smaller (B 600) and progressive decay is slower too. In the PJ region, the leaking resonant mode interferes with the residual off-resonance modes. Therefore, just near the particle surface the influence of excited WGM on PJ properties is dominant, thus leading to some narrowing of the PJ compared to the nonresonant case. This is clear from Fig. 5, which shows the PJ’s R parameter as a function of the longitudinal coordinate (the insert). Indeed, when the internal field is in resonance, the width of the PJ becomes nearly halved with respect to its normal off-resonant value and equals only about to 200 nm. This ultrathin jet exists at a distance of several hundreds of nanometers from the particle surface; after that it widens up to its normal value. In this on-resonance jet formation a certain rule holds true, which claims that jets produced by the resonance with highest Q-factor possess the smallest close-to particle width. Here it is important to point out that the morphologydependent resonances can be spectrally relatively narrow; the spectral width of a WGM is inversely proportional to its Q-factor. Then, the efficiency of WGM excitation in a spherical particle is strongly affected by
In conclusion, the specific highly-localized spatial region of near-field light scattering of a plane optical wave on transparent micron-sized particle called a photonic jet is considered. Based on the Lorenz–Mie formulation we have calculated the spatial structure of the optical field in the vicinity of homogeneous and layered spherical particles with the size parameters xo60 and optical contrast g o2. Both resonant and non-resonant internal field excitations of a particle are examined. The presented examples of the photonic jet shapes indicate a high variability of PJ spatial forms. Qualitative systematization of these shapes allowed us to propose an approximate PJ classification with the selection of their basic types. The proposed classification is based on the presence or absence of spatial separation between the photon flux and parent microsphere leading to ‘‘dagger’’, or ‘‘flare’’ shapes of the jet. Each morphological PJ type is associated with the indicative characteristics such as the formation of secondary intensity maxima and the presence of marked side lobes, as well as the type of jet width variations along the distance from the particle. The positive aspect of such PJ classification is the facilitating of selection of desired morphological jets types that match the specific requirements of an experiment or practical applications without having to perform numerical calculations.
Acknowledgments We acknowledge the financial support by the Division of Physical Sciences of the Russian Academy of Sciences. References [1] Chen Z, Taflove A, Backman V. Photonic nanojet enhancement of backscattering of light by nanoparticles: a potential novel visiblelight ultramicroscopy technique. Opt Express 2004;12(7):1214–20. [2] Astratov VN, Darafsheh A, Kerr MD, Allen KW, Fried NM, Antoszyk AN, et al. Photonic nanojets for laser surgery. SPIE Newsroom 2010. http://dx.doi.org/10.1117/2.1201002.002578. [3] Cui X, Erni D, Hafner C. Optical forces on metallic nanoparticles induced by a photonic nanojet. Opt Express 2008;16:13560–8. [4] Kong SC, Sahakian AV, Heifetz A, Taflove A, Backman V. Robust detection of deeply subwavelength pits in simulated optical datastorage disks using photonic jets. Appl Phys Lett 2008;92: 211102–3. [5] Utzinger U, Richards-Kortum RR. Fiber optic probes for biomedical optical spectroscopy. J Biomed Opt 2003;8:121–47. [6] Heifetz A, Simpson JJ, Kong SC, Taflove A, Backman V. Subdiffraction optical resolution of a gold nanosphere located within the nanojet of a Mie-resonant dielectric microsphere. Opt Express 2007;15(25): 17334–42. [7] Devilez A, Bonod N, Stout B, Gerard D, Wenger J, Rigneault H, et al. Three-dimensional subwavelength confinement of light with dielectric microspheres. Opt Express 2009;17(4):2089–94. [8] Itagi AV, Challener WA. Optics of photonic nanojets. J Opt Soc Am A 2005;22(12):2847–58.
Y.E. Geints et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 126 (2013) 44–49
[9] Lecler S, Takakura Y, Meyrueis P. Properties of a three-dimensional photonic jet. Opt Lett 2005;30:2641–3. [10] Heifetz A, Huang K, Sahakian AV, Li X, Taflove A, Backman V. Experimental confirmation of backscattering enhancement induced by a photonic jet. Appl Phys Lett 2006;89(22) 221118-3. [11] Li X, Chen Z, Taflove A, Backman V. Optical analysis of nanoparticles via enhanced backscattering facilitated by 3-D photonic nanojets. Opt Express 2005;13(22):526–33. [12] Wu W, Katsnelson A, Memis OG, Mohseni HA. A deep subwavelength process for the formation of highly uniform arrays of nanoholes and nanopillars. Nanotechnology 2007;8:485302. [13] Geints YuE, Panina EK, Zemlyanov AA. Control over parameters of photonic nanojets of dielectric microspheres. Opt Commun 2010;283:4775–81.
49
[14] Boren CF, Hafmen PR. In: Absorption and scattering of light by small particles. New York: Wiley; 1983. [15] Geints YuE, Panina EK, Zemlyanov AA. A photonic nanojet calculations in layered radially-inhomogeneous micrometer-sized spherical particles. J Opt Soc Am B 2011;28:1825–30. [16] Barber PW, Hill SC. In: Light Scattering by Particles. Singapore: World Scientific; 1990. [17] Geints YuE, Zemlyanov AA, Panina EK. Photonic jets from resonantly excited transparent dielectric microspheres. J Opt Soc Am B 2012;29(4):758–62.
Contents lists available at SciVerse ScienceDirect
Journal of Quantitative Spectroscopy & Radiative Transfer journal homepage: www.elsevier.com/locate/jqsrt
Photonic jet shaping of mesoscale dielectric spherical particles: Resonant and non-resonant jet formation Yuri E. Geints n, Ekaterina K. Panina, Alexander A. Zemlyanov Zuev Institute of Atmospheric Optics SB RAS, 1 Zuev Square, Tomsk 634021, Russia
a r t i c l e in f o
abstract
Article history: Received 29 May 2012 Received in revised form 19 July 2012 Accepted 20 July 2012 Available online 28 July 2012
A descriptive analysis of the morphological shapes of photonic jets (PJs) these being a specific spatially localized and high-intensity area formed near micron-sized transparent spherical dielectric particles illuminated by a visible laser radiation, is presented. The PJ shape characterization is based on the numerical calculations of the near-field distribution according to the Mie theory and accounts for jet dimensions and shape complexity. The special situation of the morphology-dependent resonances excitation in the internal field of the particle is also considered. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Mie scattering Photonic jet Dielectric particles Near-field distribution
1. Introduction The transparent dielectric sphere exposed to a light wave tends to focus it and that leads to a marked increase of laser intensity in the near-field region of forward scattering. This high-intensity spatial area possesses the shape of a jet-like photonic flux emerging from the particle shadow surface and is commonly referred to as a ‘‘photonic jet’’ (PJ) [1]. Its main features are strong transverse spatial localization (up to sub-diffraction scale), enormous elongation and high intensity that opens promising prospects for the practical application of this phenomenon in various fields of science and technology [2–5]. It should be clear that although the PJ is nothing else but the area of external focus of an optical wave created by a spherical particle, this wave structure has very specific spatial parameters (i.e., the relations between the length and width of the focal waist), which are untypical for the well-known thin convex lenses. The
n
Corresponding author. E-mail address: [email protected] (Y.E. Geints).
0022-4073/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jqsrt.2012.07.023
focal region formed by a micron-sized particle is abnormally extended along the direction of laser incidence and takes the shape of a pronounced jet. Thus, because a microparticle can be treated as a mesoscopic scatterer for visible radiation the formation of PJ from such a particle is strongly affected by the diffraction effects accompanying the scattering and, by the interference of the waves transmitted through and refracted by a particle. The study of PJ basic characteristics (transverse dimension, length and peak intensity) was the subject of numerous theoretical [6–8] and experimental works [9,10]. Most of these papers deal either with only single morphological type of photonic jets or with the individual jet parameters. High variability of PJ shapes with the variation of particle size and its optical structure, and the absence of adequate criteria for comprehensive estimation of PJ parameters make proper choice of jet-producing particle to obtain the PJ with desired properties difficult. Actually, different applications require different types of PJ having specific characteristics. For example, in optical nano-sensing [11] or in laser surgery [2] maximally narrow and extended photonic jets are needed. On the other side, if dielectric microspheres are used for the contact perforation of cell membranes [12], the capability
Y.E. Geints et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 126 (2013) 44–49
of a particle to focus the incident optical field comes to the forefront, i.e., the PJs having the maximal intensity are preferred, while their length has no significant importance. Therefore, it would be instructive to catalog the main jet shapes, and to partition them into several morphological groups or classes, within which the particular jet parameters could be further optimized. The purpose of this paper is to give such a qualitative morphological analysis based on the numerical calculations of jet shapes for different structural composition of spherical microparticles. Besides, we give special attention to the case of morphology-dependent resonances (MDR) excitation in the particle internal field that can lead to jet parameter modifications. 2. PJ structural types As a key parameter for our analysis we used the existence of the pronounced gap between the shadow surface of the microsphere and PJ intensity maximum. Visually, the presence or the absence of this space looks like either the flare-type (Fl) jet or the dagger-type (Dg) jet, respectively. Thus, we introduce two morphological types of jets, Dg and Fl, which will be further named as jet classes. These two classes are supplemented by detailed attributes in accordance with the following: (I) the type of jet latitudinal width variation, namely: a. inhomogeneous PJ, which is characterized by the presence of a pronounced waist (or waists) along the jet; b. homogeneous PJ, i.e., with the monotonic variation of the jet width; (II) the type of longitudinal intensity distribution along the jet: a. discontinuous PJs, having secondary intensity maxima in the direction of light incidence; b. continuous PJ, which do not have these maxima; (III) the type of transverse intensity distribution: a. branched PJ, having marked side lobes in the intensity distribution of the field at the sphere surface; b. simple PJ, i.e., without marked side lobes. For simplicity, we will use the alphanumeric designation of introduced PJ classes. For example, the Dg: Ia–IIb–IIIa class indicates a dagger-type inhomogeneous continuous jet with pronounced side lobes. Table 1 summarizes the proposed jet classification. The proposed PJ classification is made on the basis of numerical calculations and analysis of the spatial form of photonic jets formed in the vicinity of airborne nonabsorbing spherical dielectric particles with radii a0 ¼1– 5 mm, and refractive indexes in the range na ¼1–2 being irradiated by a monochromatic linearly polarized plane light wave with l ¼0.532 mm. This is dictated by the fact that, on the one hand, for smaller particle sizes the pronounced photonic flux does not form yet, and on the other hand, for larger particles the near-field scattering region of optical radiation (the region of jet formation) is blurred and, therefore, cannot be referred to as a PJ [13].
45
Table 1 Classification of the photonic jet shapes. PJ Class (Dg) dagger-type
(Fl) flare-type Attributes
(I) Jet width z-variation (II) Longitudinal profile a Inhomogeneous b Homogeneous
Discontinuous Continuous
(III) Jet complexity
Branched Simple
The specific range of particle sizes and the specific laser radiation wavelength that we have chosen do not reduce a generality of our PJ types analysis because in the Lorenz–Mie theory [14] the relevant quantity is not the particle size itself but the dimensionless ratio of particle radius a0 to the light wavelength l called the size parameter: xa ¼ 2pa0/l. Thus, when considering Mie scattering in a different wavelengths range one has to change the particle radius a0 as well. Most practical PJ applications require a photon flux having submicron dimensions and therefore the subject of our study was micron-sized particles, which produce PJs of desired transverse widths. Our choice of the specific range of particles refractive index na is motivated by the considerations of practical capabilities of microsphere production. Below we present a more detailed discussion of issues, related to the peculiarities of the formation of photonic jets from the most common structural types of dielectric spherical microparticles, as well as illustrate the PJ classification by numerical near-field calculations. 3. Nonresonant excitation of spherical particles 3.1. Homogeneous particles It is known that the character of light intensity distribution near homogeneous nonabsorbing spherical particles differs as the particle size increases. According to the calculations [13], for spherical particles with the radius a0 r3 mm, the half-width of photonic flux R remains smaller than the diffraction-limited value determined as the focal waist radius upon convex lens focusing. Such particles will be referred to here as small scatterers. For small particles, the spatial area of the external optical field maximum, i.e., the area of photonic jet, is characterized by smooth variations along and across the jet and has an arrow-like shape. We name this type of jets as a dagger-type PJ. As an example of a typical PJ, Fig. 1 shows the false color 2D-distributions of the relative optical field intensity B(x,z)¼I(x,z)I0 (I0 is the incident intensity) in the vicinity of a quartz sphere with the radius a0 ¼ 3 mm. The intensity in this and in the following 2D-graphs is normalized to its maximal value Bmax, which is indicated in each figure by the corresponding number in the upper left corner. For the convenience we added a bar showing the 1 mm scale.
46
Y.E. Geints et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 126 (2013) 44–49
Fig. 1. PJ formation in the vicinity of a homogeneous sphere with a0 ¼ 3 mm and na ¼ 1.5 (PJ class Dg: Ib–IIb–IIIb). The light wave is incident from the left. The dashed circle shows the particle rim, and the bar shows the 1 mm scale. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
Hereafter we follow the situation when the incident plane wave is linearly polarized along the x-axis and propagates in the positive z-direction. Therefore, we choose the x–z plane as a plane where the photonic jets will be analyzed and classified. According to the Mieformulae, the state of incident light polarization does not significantly affect scattered intensity spatial distribution, but affects only its magnitude. Thus, the perpendicular y–z plane will not be considered. As the particle radius increases, the PJ intensity maximum shifts farther from the particle boundary. The PJ configuration becomes more complicated: the jet extends along the radiation incidence (in absolute units), a pronounced waist appears in the transverse direction, and the secondary maxima are formed in the longitudinal direction. The Lorenz–Mie theory states that the key parameter in light scattering is not the refractive index of a particle na itself, but its ratio to the refractive index of the ambient medium (nm), nr ¼na/nm, often referred to as the optical contrast. The particles with low nr values form photonic jets of the classical flare-like type, which is clearly separated from the particle surface. This type of jets is characterized by the morphological class Fl:Ib–IIb–IIIb (homogeneous continuous jet without marked side lobes). With increasing parameter nr, the center of the external focal waist first approaches the particle surface and then shifts inside the particle. This is caused by an increase in the optical strength of the lens, whose role is to play as a spherical particle for the incident optical wave. The samples of intensity distributions corresponding to the situations of higher particle optical contrast demonstrate photonic jets of the different morphological class, namely the dagger-type PJ. Thus, the comparative analysis of PJ spatial shapes in the case of non-resonant excitation of the optical field of homogeneous non-absorbing spherical particles indicates the possibility of realization of both PJ classes.
transverse dimensions and the peak intensity of photonic jets from dielectric layered microspheres of different sizes. The calculations were made for a particle consisting of a core with the radius a0 and four concentric layers of equal thickness h with the radii as, s¼ 0,y,N. The refractive indices of the core, n0, and the outermost shell, nN, are fixed: n0 ¼1.5, nN ¼1.1. To quantitatively describe the layer-bylayer refractive index variation, the following functional g dependence is introduced: ns =n0 ¼ ðnN =n0 Þðs=NÞ , where s (integer) is the layer number, and the parameter g (g40) defines the type of grading. Following [15], according to the radial dependence of interlayer optical contrast, gs ¼ns/ns þ 1, we will consider three principal types of particle refractive index grading: ‘‘concave’’, ‘‘convex’’, and ‘‘neutral grading’’. The particles of the first ‘‘concave’’ structural type (type A, g o1) have decreasing refractive index contrast between neighboring shells. Particles of the ‘‘convex’’ structural type (type B, g 41) are characterized by increasing optical contrast from the core to the peripheral shell. And, finally, the particles of the third structural type (type C, g¼1) have a constant interlayer optical contrast. For radially inhomogeneous multilayer spherical particles the spatial shape and the intensity of PJ are mainly determined by the type of the optical contrast grading of neighboring shells. Particles of C-type can form high-intensity extended and simultaneously spatially localized ‘‘dagger’’ type jets of the class Dg:Ib–IIb–IIIa. The multilayer spheres of other structural types (A and B) lead to different PJ morphological classes. Thus, for A-type particles we have the class Fl: Ib–IIb–IIIb, while the PJs emerging from B-type spheres can be classified as Dg: Ia–IIa–IIIa jets. As shown by the numerical calculations [15], the particles with the sublinear n-grading (A-type), with gradual decrease in refractive index between neighboring shells (see Fig. 2a,b), produce optimal photon fluxes in the
3.2. Radially-inhomogeneous multilayered particles In Ref. [15], the numerical calculations within the Mie theory extended to the case of light scattering at multilayered spheres were used to study the longitudinal and
Fig. 2. (a) Refractive index-grading scheme of the particle; and (b) Photonic jet from seven-layered spheres of sub-linear structural type (A-type) with a0 ¼3 mm (PJ class Fl: Ib–IIb–IIIb).
Y.E. Geints et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 126 (2013) 44–49
sense of the relation between PJ intensity, its length and transverse dimension. With an increase of the outer diameter of multilayered particles of the type A, the characteristic length L and width R of the photonic jet do not change significantly and depend only on the value of structural parameter g. On the contrary, photonic jets produced by B-type particles exhibit the strong dependence on the size of parent sphere and vary only slightly on the parameter g. When the particle radius grows, the intensity of the jet, its length and width increase, and, what is more important, the PJ longitudinal structure becomes more complex. In addition to the main intensity maximum near the particle surface, the secondary field bunch (‘‘light bullet’’) appears in the radial direction at a distance of about a wavelength (see Fig. 3). This circumstance allows a significant increase in the total area of the photonic jet, which may be of principal importance for various practical applications.
47
optical energy is accumulated by the mode. Although in general the spatial shape of a PJ at optical resonance changes only insignificantly, an increase of the near-surface intensity can affect markedly the class of the PJ. The ‘‘sticking’’ of the photonic jet to the microparticle surface can change its class from the flare-type jet, in the absence of resonance, to dagger-type jet at a resonance. Our calculations show that this PJ type changing is characteristic only for high-quality WGMs with typical values, Q4103. The excitation of resonances with low Q-factors does not lead to PJ type change. Moreover, we have found that at a resonance, the length and the mean halfwidth of a photonic jet remain nearly the same as in the nonresonant case (see Fig. 4a–c). The effect of transverse PJ squeezing investigated in [6] is prominent only close to the particle surface at distances of about a quarter-wavelength, and after this, the PJ widens to its ‘‘normal’’ size.
4. Resonant PJ excitation from homogeneous spherical particles One of the striking phenomena in light scattering is the excitation of morphological resonances in a transparent spherical particle. This means that by properly choosing the wavelength of incident light and the particle radius it is possible to selectively excite certain spatial configurations (eigenmodes) of electromagnetic field inside a dielectric particle. At resonance, the maximal intensity of the internal optical field becomes much higher than that of off-resonance, which could also influence the external optical field, and namely, the PJ region. As known, the electromagnetic field inside a spherical particle can be represented as an expansion in two families of eigenmodes: TE-waves and TM-waves. The interference of these two mode families results in a specific pattern of the scattering field. At resonance, only one certain type of eigenmodes is sharply amplified, and therefore the spatial configuration of the internal optical field follows the spatial profile of the excited resonant mode. The resonant optical field inside the particle is concentrated in a narrow annular layer near the particle rim forming the so-called whispering-gallery mode (WGM) [16]. The WGMs of different types also have different quality factors Q (Q-factors) showing how efficiently the
Fig. 3. Example of discontinuous PJ with pronounced side lobes from five-layered spherical particle with a0 ¼ 4 mm, g ¼2 (B-type particle, PJ class Dg: Ia–IIa–IIIa).
Fig. 4. (a) PJ of the class Fl: Ib–IIb–IIIa produced upon resonant water droplet excitation (TM 241 , Q ¼4 104 ); (b) PJ of the class Dg: Ib–IIb–IIIa (TM 138 , Q ¼2 107); and (c) non-resonant PJ of the class Fl: Ib–IIb–IIIa.
48
Y.E. Geints et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 126 (2013) 44–49
the ratio of WGM width to laser radiation linewidth and decreases with this ratio decrease. Practically, this can considerably limit the influence of field resonances on PJ parameters.
5. Conclusion
Fig. 5. Radial intensity profile of the near-field scattering of spherical particle, and PJ half-width (the insert) in the y–z plane on- and offresonance for TM 140 and TM 240 modes. The dashed line marks the particle boundary.
Following the proposed classification, we can state that at resonant WGM excitation for the data presented in Fig. 4a,b the homogeneous continuous PJs are formed with weak lateral lobes, representing the morphological classes Fl: Ib–IIb–IIIa and Dg: Ib–IIb–IIIa, respectively. This qualitative analysis is also confirmed by our largescale numerical calculations performed for several dozens of resonances with different polarizations and orders excited in the particle with the radii ranging from 1 to 5 mm. For example, as shown in [17], the resonant excitation of the TM 140 mode in the particle with a0 ¼2.5790068 mm leads to an increase of the optical intensity in the near-field region up to the values B 1200. After that, the intensity decreases quickly, and at a distance of approximately 300 nm from the particle surface, i.e., in the PJ area, it becomes even lower than that in the off-resonance case (particle radius, a0 ¼ 2.88 mm). The near-field intensity at the TM 240 resonance (a0 ¼2.8748 mm) demonstrates a similar behavior; however, its increase on the particle surface is about twofold smaller (B 600) and progressive decay is slower too. In the PJ region, the leaking resonant mode interferes with the residual off-resonance modes. Therefore, just near the particle surface the influence of excited WGM on PJ properties is dominant, thus leading to some narrowing of the PJ compared to the nonresonant case. This is clear from Fig. 5, which shows the PJ’s R parameter as a function of the longitudinal coordinate (the insert). Indeed, when the internal field is in resonance, the width of the PJ becomes nearly halved with respect to its normal off-resonant value and equals only about to 200 nm. This ultrathin jet exists at a distance of several hundreds of nanometers from the particle surface; after that it widens up to its normal value. In this on-resonance jet formation a certain rule holds true, which claims that jets produced by the resonance with highest Q-factor possess the smallest close-to particle width. Here it is important to point out that the morphologydependent resonances can be spectrally relatively narrow; the spectral width of a WGM is inversely proportional to its Q-factor. Then, the efficiency of WGM excitation in a spherical particle is strongly affected by
In conclusion, the specific highly-localized spatial region of near-field light scattering of a plane optical wave on transparent micron-sized particle called a photonic jet is considered. Based on the Lorenz–Mie formulation we have calculated the spatial structure of the optical field in the vicinity of homogeneous and layered spherical particles with the size parameters xo60 and optical contrast g o2. Both resonant and non-resonant internal field excitations of a particle are examined. The presented examples of the photonic jet shapes indicate a high variability of PJ spatial forms. Qualitative systematization of these shapes allowed us to propose an approximate PJ classification with the selection of their basic types. The proposed classification is based on the presence or absence of spatial separation between the photon flux and parent microsphere leading to ‘‘dagger’’, or ‘‘flare’’ shapes of the jet. Each morphological PJ type is associated with the indicative characteristics such as the formation of secondary intensity maxima and the presence of marked side lobes, as well as the type of jet width variations along the distance from the particle. The positive aspect of such PJ classification is the facilitating of selection of desired morphological jets types that match the specific requirements of an experiment or practical applications without having to perform numerical calculations.
Acknowledgments We acknowledge the financial support by the Division of Physical Sciences of the Russian Academy of Sciences. References [1] Chen Z, Taflove A, Backman V. Photonic nanojet enhancement of backscattering of light by nanoparticles: a potential novel visiblelight ultramicroscopy technique. Opt Express 2004;12(7):1214–20. [2] Astratov VN, Darafsheh A, Kerr MD, Allen KW, Fried NM, Antoszyk AN, et al. Photonic nanojets for laser surgery. SPIE Newsroom 2010. http://dx.doi.org/10.1117/2.1201002.002578. [3] Cui X, Erni D, Hafner C. Optical forces on metallic nanoparticles induced by a photonic nanojet. Opt Express 2008;16:13560–8. [4] Kong SC, Sahakian AV, Heifetz A, Taflove A, Backman V. Robust detection of deeply subwavelength pits in simulated optical datastorage disks using photonic jets. Appl Phys Lett 2008;92: 211102–3. [5] Utzinger U, Richards-Kortum RR. Fiber optic probes for biomedical optical spectroscopy. J Biomed Opt 2003;8:121–47. [6] Heifetz A, Simpson JJ, Kong SC, Taflove A, Backman V. Subdiffraction optical resolution of a gold nanosphere located within the nanojet of a Mie-resonant dielectric microsphere. Opt Express 2007;15(25): 17334–42. [7] Devilez A, Bonod N, Stout B, Gerard D, Wenger J, Rigneault H, et al. Three-dimensional subwavelength confinement of light with dielectric microspheres. Opt Express 2009;17(4):2089–94. [8] Itagi AV, Challener WA. Optics of photonic nanojets. J Opt Soc Am A 2005;22(12):2847–58.
Y.E. Geints et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 126 (2013) 44–49
[9] Lecler S, Takakura Y, Meyrueis P. Properties of a three-dimensional photonic jet. Opt Lett 2005;30:2641–3. [10] Heifetz A, Huang K, Sahakian AV, Li X, Taflove A, Backman V. Experimental confirmation of backscattering enhancement induced by a photonic jet. Appl Phys Lett 2006;89(22) 221118-3. [11] Li X, Chen Z, Taflove A, Backman V. Optical analysis of nanoparticles via enhanced backscattering facilitated by 3-D photonic nanojets. Opt Express 2005;13(22):526–33. [12] Wu W, Katsnelson A, Memis OG, Mohseni HA. A deep subwavelength process for the formation of highly uniform arrays of nanoholes and nanopillars. Nanotechnology 2007;8:485302. [13] Geints YuE, Panina EK, Zemlyanov AA. Control over parameters of photonic nanojets of dielectric microspheres. Opt Commun 2010;283:4775–81.
49
[14] Boren CF, Hafmen PR. In: Absorption and scattering of light by small particles. New York: Wiley; 1983. [15] Geints YuE, Panina EK, Zemlyanov AA. A photonic nanojet calculations in layered radially-inhomogeneous micrometer-sized spherical particles. J Opt Soc Am B 2011;28:1825–30. [16] Barber PW, Hill SC. In: Light Scattering by Particles. Singapore: World Scientific; 1990. [17] Geints YuE, Zemlyanov AA, Panina EK. Photonic jets from resonantly excited transparent dielectric microspheres. J Opt Soc Am B 2012;29(4):758–62.