研究者業績

宮本 克彦

ミヤモト カツヒコ  (Katsuhiko Miyamoto)

基本情報

所属
千葉大学 大学院工学研究院 教授
学位
博士(工学)

研究者番号
20375158
ORCID ID
 https://orcid.org/0000-0002-2611-882X
J-GLOBAL ID
200901053531392102
Researcher ID
E-6713-2015
researchmap会員ID
5000103464

外部リンク

研究キーワード

 2

受賞

 6

論文

 201
  • Keigo Masuda, Ryo Shinozaki, Ami Shiraishi, Mitsuki Ichijo, Keisaku Yamane, Katsuhiko Miyamoto, Takashige Omatsu
    JOURNAL OF NANOPHOTONICS 14(1) 2020年1月  
    We report on the first demonstration of picosecond optical vortex-induced chiral surface relief in an azo-polymer film due to two-photon absorption isomerization. The chiral surface relief exhibits an extremely narrow defocusing tolerance without undesired outer rings due to the Airy pattern of highly focused light. Such chiral surface relief reflects a z-polarized electric field with an azimuthal helical phase caused by spin-orbital angular momentum coupling. (C) The Authors. Published by SPIE under a Creative Commons Attribution 4.0 Unported License.
  • Ablimit Ablez, Kohei Toyoda, Katsuhiko Miyamoto, Ryuji Morita, Takashige Omatsu
    OPTICAL MANIPULATION AND STRUCTURED MATERIALS CONFERENCE 2020 11522 2020年  
    We investigate the structures on a silicon (111) substrate produced by illumination of optical vortex. The fabricated structures on silicon (111) exhibit polycrystalline properties associated with rather complicated 7x7 constructed surface.
  • Ami Shiraishi, Keigo Masuda, Keisaku Yamane, Kohei Toyoda, Katsuhiko Miyamoto, Takashige Omatsu
    Optics InfoBase Conference Papers Part F183-CLEO-SI 2020 2020年  
    We demonstrate the formation of chiral surface structure of azo-polymers via optical vortex induced two-photon-absorption (TPA). TPA-induced isomerization allows us to create the three-dimensional chiral structures with high longitudinal and transverse spatial resolution beyond the diffraction limit.
  • Haruki Kawaguchi, Kei Umesato, Keisaku Yamane, Katsuhiko Miyamoto, Takashige Omatsu
    Proceedings of SPIE - The International Society for Optical Engineering 11522 2020年  
    We discover an entirely novel phenomenon, so-called the formation of curved "spin-jet", in which an irradiated fractional optical vortex provides a donor film non-axisymmetric torque to form a "spin-jet" with a curved trajectory. This phenomenon allows the development of a novel pattering technology to scan the ejected donor dots without any mechanical systems.
  • Ryosuke Nakamura, Haruki Kawaguchi, Muneaki Iwata, Akihiro Kaneko, Ryo Nagura, Satoyuki Kawano, Kohei Toyoda, Katsuhiko Miyamoto, Takashige Omatsu
    Optics Express 2019年12月23日  査読有り
  • Roukuya Mamuti, Shunsuke Goto, Katsuhiko Miyamoto, Takashige Omatsu
    Optics Express 2019年12月9日  査読有り
  • Katsuhiko Miyamoto, Kazuki Sano, Takahiro Miyakawa, Hiromasa Niinomi, Kohei Toyoda, Adam Vallés, Takashige Omatsu
    Optics Express 27(22) 31840-31840 2019年10月28日  査読有り筆頭著者責任著者
  • 川口 晴生, 有田 佳彦, 李 俊亨, 松尾 怜門, 宮本 克彦, Kishan Dholakia, 尾松 孝茂
    応用物理学会学術講演会講演予稿集 2019.2 563-563 2019年9月4日  
  • Yuanyuan Ma, Jung-Chen Tung, Yung-Fu Chen, Adam Valle-Mari, Kohei Toyoda, Katsuhiko Miyamoto, Takashige Omatsu
    2019 Conference on Lasers and Electro-Optics Europe and European Quantum Electronics Conference, CLEO/Europe-EQEC 2019 2019年6月1日  
    Optical vortices [1,2] carry an annular spatial profile and an orbital angular momentum of (where ℓ is the topological charge) per photon owing to the phase singularity. In particular, optical vortices enable the creation of nanoscale cliiral structures in various materials, such as azo-polymers [3,4], silicon and metal. Such chiral nanostructures formation strongly desires compact visible vortex sources and their selective handedness (the sign of the topological charge) control.
  • Roukuya Mamuti, Shigeki Nishida, Katsuhiko Miamoto, Takashige Omatsu
    2019 Conference on Lasers and Electro-Optics Europe and European Quantum Electronics Conference, CLEO/Europe-EQEC 2019 2019年6月1日  
    Optical vortices with a ring-shaped spatial form and an orbital angular momentum (OAM) of Ιφ [1], arising from their helical wavefronts characterized by an azimuthal phase, exp(iΙφ) (where Ι is an integer termed the topological charge), have been widely exploiting a variety of research fields. These include optical trapping and manipulation [2], fluorescent microscopes with a high spatial resolution beyond the diffraction limit [3] and materials processing [4,5]. The aforementioned applications strongly desire wavelength-tunability and OAM-versatility to optical vortex sources [6].
  • Keigo Masuda, Yoshinori Kinezuka, Mitsuki Ichijo, Ryo Shinozaki, Keisaku Yamane, Kouhei Toyoda, Katsuhiko Miyamoto, Takashige Omatsu
    2019 Conference on Lasers and Electro-Optics, CLEO 2019 - Proceedings 2019年5月  
    We have demonstrated two-photon-absorption induced chiral surface relief formation in an azo-polymer film by the illumination of picosecond 1-μm optical vortex pulses. The chiral surface relief formation required at least several times the response time of trans-cis isomerization of the azo-polymers. The resulting chiral surface relief exhibited a diameter of 3-μm, corresponding to 0.7 times the diffraction limit. We also observed spin-orbital angular momentum coupling in the surface relief formation, even via two-photon-absorption.
  • Junhyung Lee, Yoshihiko Arita, Reimon Matsuo, Haruki Kawaguchi, Katsuhiko Miyamoto, Kishan Dholakia, Takashige Omatsu
    Optics InfoBase Conference Papers 2019 2019年  
    An optical vortex possesses an orbital angular momentum (OAM) and a ring-shaped spatial profile due to its spiral wavefront, characterized by a topological charge ℓ [1]. The OAM of an optical vortex can twist the irradiated materials, such as metal, silicon, azo-polymer films [2,3], to form helical structured materials. Thus, the optical vortex has been intensely studied in laser materials processing.
  • Roukuya Mamuti, Shunsuke Goto, Katsuhiko Miyamoto, Takashige Omatsu
    Optics InfoBase Conference Papers 2019 2019年  
  • Yuanyuan Ma, Adam Vallés, Katsuhiko Miyamoto, Takashige Omatsu
    Optics InfoBase Conference Papers 2019 2019年  
  • Katsuhiko Miyamoto, Takashige Omatsu
    Optics InfoBase Conference Papers 2019 2019年  
  • Yuanyuan Ma, Jung-Chen Tung, Katsuhiko Miyamoto, Takashige Omatsu
    Optics InfoBase Conference Papers 2019 2019年  
    We have demonstrated the direct generation of visible structured light beams, scalar and vector vortex modes with a wavelength of 640nm from a diode-pumped Pr3+:YLF laser by employing an off-axis pumping technique. A maximum output of ~100 mW was achieved at a pump power of 1.2 W. Such red structured light beams will be potentially applied to super-resolution fluorescent microscopes and micro-fabrications.
  • Roukuya Mamuti, Shigeki Nishida, Katsuhiko Miyamoto, Takashige Omatsu
    Proceedings of SPIE - The International Society for Optical Engineering 11141 2019年  
    We demonstrate a tunable vortex laser with versatile orbital angular momentum (OAM) states based on a singly resonant optical parametric oscillator (OPO) formed of a non-critical phase-matching LiB3O5 (NCPM-LBO) crystal. The selective generation of a signal (idler) output with three OAMs, including an up-converted (negative) OAM, is achieved simply by appropriate shortening (or extending) of the cavity. The compact cavity configuration also allows for the generation of the signal (idler) output with various OAMs by simply tuning the signal wavelength. The vortex output is tuned within the wavelength region of 0.74 μm to 1.86 μm with maximum pulse energy of 1.99 mJ from pump energy of 8.5 mJ.
  • Yuanyuan Ma, Jung-Chen Tung, Yung-Fu Chen, Katsuhiko Miyamoto, Takashige Omatsu
    Proceedings of SPIE - The International Society for Optical Engineering 11141 2019年  
    We successfully demonstrated the visible vortex mode generation from a diode-pumped Pr3+:YLF laser by employing an off-axis pumping technique. A maximum output of ∼100 mW was obtained at a pump power of 1.2 W. Such visible structured light beams will be potentially applied to super-resolution fluorescent microscope and micro-fabrications.
  • Junhyung Lee, Yoshihiko Arita, Reimon Matsuo, Haruki Kawaguchi, Katsuhiko Miyamoto, Kishan Dholakia, Takashige Omatsu
    Proceedings of SPIE - The International Society for Optical Engineering 11141 2019年  
    We successfully created helical microfibers by irradiating picosecond optical vortex pulses with a wavelength of 532 nm to ultraviolet curing resin via a two-photon-photopolymerization process. Permanent refractive index change of resin caused by laser triggers the self-focusing of light and forms the Light-induced self-written waveguides. In this process, the orbital angular momentum of the light is transferred to make a helical microfiber. The resulting helical microfibers exhibited a length of ∼300 μm. Also, we can control the direction of twists by changing the sign of the topological charge of optical vortex.
  • Nakamura R, Iwata M, Kaneko A, Toyoda K, Miyamoto K, Omatsu T
    2019 Conference on Lasers and Electro-Optics, CLEO 2019 - Proceedings 2019年  査読有り
  • Masuda K, Yoshizawa T, Akiyama T, Okada-Shudo Y, Murata T, Toyoda K, Miyamoto K, Omatsu T
    Proceedings of SPIE - The International Society for Optical Engineering 11141 2019年  査読有り
  • Keigo Masuda, Yoshinori Kinezuka, Mitsuki Ichijo, Ryo Shinozaki, Keisaku Yamane, Kouhei Toyoda, Katsuhiko Miyamoto, Takashige Omatsu
    Optics InfoBase Conference Papers Part F129-CLEO_SI 2019 2019年  
    We have demonstrated two-photon-absorption induced chiral surface relief formation in an azo-polymer film by the illumination of picosecond 1-μm optical vortex pulses. The chiral surface relief formation required at least several times the response time of trans-cis isomerization of the azo-polymers. The resulting chiral surface relief exhibited a diameter of 3-μm, corresponding to 0.7 times the diffraction limit. We also observed spin-orbital angular momentum coupling in the surface relief formation, even via two-photon-absorption.
  • Keigo Masuda, Mitsuki Ichijo, Ryo Shinozaki, Keisaku Yamane, Katsuhiko Miyamoto, Takashige Omatsu
    Proceedings of SPIE - The International Society for Optical Engineering 11141 2019年  
    We demonstrate chiral mass-transport of azo-polymers by illumination of tightly focused 1-μm picosecond optical vortex pulses with a pulse width of 8-ps through two-photon absorption. The chiral surface relief formation requires picosecond pulses with a relatively long pulse duration. In fact, it is also worth noting that it is difficult to create such chiral surface relief by employing 2-ps optical vortex pulses.
  • Keigo Masuda, Mitsuki Ichijo, Ryo Sinozaki, Keisaku Yamane, Katsuhiko Miyamoto, Takashige Omatsu
    Optics InfoBase Conference Papers Part F146-JSAP 2019 2019年  
  • Ma, Y., Vallés, A., Tung, J.-C., Chen, Y.-F., Miyamoto, K., Omatsu, T.
    Optics Express 27(13) 2019年  査読有り
  • Niinomi, H., Sugiyama, T., Uda, S., Tagawa, M., Ujihara, T., Miyamoto, K., Omatsu, T.
    Crystal Growth and Design 19(2) 2019年  査読有り
  • Bisson, J.-F., Miyamoto, K., Omatsu, T.
    Japanese Journal of Applied Physics 58(3) 2019年  査読有り
  • Omatsu, T., Miyamoto, K., Toyoda, K., Morita, R., Arita, Y., Dholakia, K.
    Advanced Optical Materials 7(14) 2019年  査読有り
  • Roukuya Manuti, Shungo Araki, Shigeki Nishida, Katsuhiko Miyamoto, Takashige Omatsu
    Applied Optics 57(34) 10004-10008 2018年12月  査読有り
  • Taximaiti Yusufu, Suijian Niu, Paerhatijiang Tuersun, Yusufu Tulake, Katsuhiko Miyamoto, Takashige Omatsu
    Japanese Journal of Applied Physics 57(12) 122701 2018年12月1日  査読有り
  • Keigo Masuda, Ryo Shinozaki, Yoshinori Kinezuka, Junhyung Lee, Seigo Ohno, Shun Hashiyada, Hiromi Okamoto, Daisuke Sakai, Kenji Harada, Katsuhiko Miyamoto, Takashige Omatsu
    Optics Express 26(17) 22197-22207 2018年8月  査読有り
  • Hiromasa Niinomi, Teruki Sugiyama, Miho Tagawa, Shunta Harada, Toru Ujihara, Satoshi Uda, Katsuhiko Miyamoto, Takashige Omatsu
    Cryst. Growth Des. 18(8) 4230-4239 2018年8月  査読有り
    We have found that large chiral symmetry breaking in chiral crystallization can be achieved by irradiating a several milliwatts focused laser to a plasmonic nanolattice immersed in a stagnant NaClO3 saturated aqueous solution. Several hundreds of chiral crystals with the same handedness showed up in the solution after the laser irradiation in contrast to spontaneous crystallization. In situ microscopic observation for the early stage of the crystallization in the vicinity of the focal spot revealed that microbubble generation followed by large supersaturation increase, in which supersaturation reaches 360%, promotes several numbers of crystal nucleation in the vicinity of the bubble as "mother" crystal. The generation of the microbubble induced Marangoni convection, the velocity of which reaches several hundreds of micrometers per second, crushing the first appearing chiral crystal into pieces by microfluidic shear. Namely, secondary nucleation caused by microfluidic shear amplified the number of "daughter" crystals with the same handedness. This spatiotemporally controllable micromixing experiment realized by laser irradiation gives us not only a novel route bridging a light and chiral symmetry breaking but also the novel method to observe the early stage dynamics of the secondary nucleation, which was hard to observe by conventional observation technique, in real time.
  • Katsuhiko Miyamoto, Kazuki Sano, Tomohito Yamasaki, Takahiro Miyakawa, Takashige Omatsu
    2018 Conference on Lasers and Electro-Optics Pacific Rim, CLEO-PR 2018 2018年7月2日  
    We have developed a widely tunable terahertz (THz) vortex source with an average power of 3 μW and a frequency tuning range of 2-6 THz. The handedness of the THz vortex output in this system is also controlled.
  • Shungo Araki, Kana Ando, Katsuhiko Miyamoto, Takashige Omatsu
    Applied Optics 57(4) 620-624 2018年2月1日  査読有り
    We demonstrate the generation of an ultra-widely tunable mid-infrared (6–18 μm) optical vortex output with a moderate pulse energy from a AgGaSe2 difference frequency generator pumped by an optical vortex parametric oscillator. The handedness of the vortex output can be controlled/selected by swapping the lasing frequencies of the signal and idler outputs and rotating the AgGaSe2 crystal by 90 deg.
  • Roukuya Mamuti, Shigeki Nishida, Katsuhiko Miamoto, Takashige Omatsu
    Optics InfoBase Conference Papers 2018 2018年  
    Helical light beams, i.e., optical vortices, carry an orbital angular momentum (OAM), characterized by an azimuthal phase term, ℓΦ, where ℓ is a topological charge and Φ is the azimuthal angle [1]. Optical vortices have been attracting much attention in a variety of research fields, for instance, material processing for chiral structured materials [2], and microscopy with high spatial resolution [3]. The above-mentioned applications strongly desire controllable OAM states versatility to optical vortex sources.
  • Keigo Masuda, Yoshinori Kinezuka, Mitsuki Ichijo, Ryo Sinozaki, Katsuhiko Miyamoto, Takashige Omatsu
    Optics InfoBase Conference Papers 2018 2018年  
    Azo-polymer exhibits mass-transport through cis-trans photo-isomerization by irradiation of visible light, so as to establish a surface relief [1]. To date, we have discovered that azo-polymer film is deformed to shape a spiral surface relief ('OAM-induced chiral surface relief') on micro-scale by illumination of a cw green optical vortex beam, in which orbital angular momentum (OAM) of the vortex beam twists the azo-polymer [2-3]. In recent years, surface relief formation in the azo-polymer film via two-photon absorption process by illumination of tightly focused femto-/pico- second pulses has been reported to create structures on a na-noscale beyond the diffraction limit [4]. In this presentation, we report on, for the first time, OAM-induced chiral surface relief formation in azo-polymer film via two-photon absorption (TPA) by illumination of picosecond 1µm optical vortex pulses.
  • Yuanyuan Ma, Jung-Chen Tung, Katsuhiko Miyamoto, Yung-Fu Chen, Takashige Omatsu
    Optics InfoBase Conference Papers 2018 2018年  
    Optical bottle beams1,2) carry a 3D dark core, i.e. 'zero intensity' region, surrounded by bright region, and they have widely received much attention in a variety of fields, such as optical tweezers for atom trapping and particle guiding, optical imaging and fluorescence microscopes with high 3D spatial resolution.
  • Mitsuki Ichijo, Keigo Masuda, Yoshinori Kinezuka, Ryo Shinozaki, Yuuki Kuramoto, Katsuhiko Miyamoto, Takashige Omatsu
    Optics InfoBase Conference Papers 2018 2018年  
    We discovered that 1-µm picosecond optical vortex pulses direct azo-polymers towards clockwise or counter-clockwise direction to form chiral surface relief, mirroring a helical wavefront of the optical vortex, through two-photon absorption. The vortex pulse width required for the chiral surface relief formation is addressed.
  • Yuanyuan Ma, Jung-Chen Tung, Katsuhiko Miyamoto, Takashige Omatsu
    Optics InfoBase Conference Papers 2018 2018年  
    We demonstrate the direct generation of a green bottle beam with high potential barrier from an intra-cavity frequency-doubled Nd:YVO4 laser with a nearly hemispherical cavity configuration. Also, numerically analysis of the generated bottle beam is conducted by coherent superposition of a series of frequency-locked Laguerre-Gaussian modes.
  • JunHyung Lee, Yoshihiko Arita, Reimon Matsuo, Haruki Kawaguchi, Katsuhiko Miyamoto, Kishan Dholakia, Takashige Omatsu
    Optics InfoBase Conference Papers 2018 2018年  
    We report on the first demonstration of self-writing helical photo-polymerized fibers, formed by self-trapping effects and orbital angular momentum transfer effects in photo-cure resin, by picosecond optical vortex illumination via two photon absorption. Two photon absorption enables us to extend significantly the helical fiber with a sub-milli-meter length. Such helical fiber will be potentially applied as a waveguide of optical vortex mode.
  • Bong Joo Kang, Katsuhiko Miyamoto, Won Tae Kim, Kwang Jun Ahn, Takashige Omatsu, Fabian Rotermund
    Optics InfoBase Conference Papers 2018 2018年  
    High-power THz vortex beam was generated based on optical rectification in nonlinear optical crystal and mode conversion by employing a polymeric spiral phase plate. Subsequent application of THz vortex beam for nonlinear spectroscopy in graphene shows interesting enhanced nonlinear characteristics.
  • Roukuya Mamuti, Shungo Araki, Katsuhiko Miyamoto, Takashige Omatsu
    Optics InfoBase Conference Papers 2018 2018年  
    We report on a tunable 2 µm optical vortex laser source formed of a periodically poled stoichiometric lithium tantalite (PPSLT) optical parametric oscillator with a singly-resonant cavity configuration. A milli-joule level vortex output was generated at a low lasing threshold (1.2 mJ), with its wavelength being tuned within a range of 1.85-2.03 µm.
  • Taximaiti Yusufu, Ranziguli Fulati, Niu Sujian, Katsuhiko Miyamoto, Takashige Omatsu
    Optics InfoBase Conference Papers 2018 2018年  
    We generate the first millijoule-level tunable mid-infrared (3.362-3.677µm) optical vortex output with a topological charge l of ±1 by a 1 µm vortex pumped quasi-phase matching MgO-doped periodically poled lithium niobate (MgO:PPLN) optical parametric oscillator.
  • Jean-François Bisson, Katsuhiko Miyamoto, Takashige Omatsu
    Optics InfoBase Conference Papers 2018 2018年  
    We present a device that can modulate the orbital angular momentum (AOM) of a laser beam The proposed design can be scaled up to several GHz using existing commercial devices, while the output power can also be scaled to values determined by state-of-the-art high-power lasers. This device will find applications in laser processing and for the characterization of metamaterials.
  • Yuri Nakamura, Ryosuke Nakamura, Chisa Koriyama, Katsuhiko Miyamoto, Tsukasa Torimoto, Yasuyuki Tsuboi, Takashige Omatsu
    Proceedings of SPIE - The International Society for Optical Engineering 10712 2018年  
    We present the new structured materials, i.e. string-shaped Au nano-structures, formed by employing the optical vortex ablation processing on an Au thin film. We also address the filament formation of Au particles by deposition of optical vortex pulse. Such structured Au particles have the potential to pave the pathway towards advanced chemical reaction.
  • Lee, J., Arita, Y., Toyoshima, S., Miyamoto, K., Panagiotopoulos, P., Wright, E.M., Dholakia, K., Omatsu, T.
    ACS Photonics 5(10) 2018年  査読有り
  • Tung, J.C., Ma, Y.Y., Miyamoto, K., Chen, Y.F., Omatsu, T.
    Scientific Reports 8(1) 2018年  査読有り
  • Hiromasa Niinomi, Teruki Sugiyama, Katsuhiko Miyamoto, Takashige Omatsu
    Crystal Growth and Design 18(2) 734-741 2018年  査読有り
    We reversibly controlled phase conversion between a microdroplet of a NaClO3 unsaturated aqueous solution and a metastable single crystal, which is usually a short-lived phase in spontaneous crystallization, simply by irradiating a tightly focused visible continuous-wave (CW) laser to the microdroplet. The laser irradiation allowed the metastable crystal to generate and stably grow without a polymorphic transformation. This successful metastable phase control is attributed to the combination of the advantage of optical trapping-induced nucleation that nucleation takes place from unsaturated mother solution and the advantage of microdroplet method, which suppresses additional nucleation leading to the transformation. In situ observation shows the crystal dissolves when the laser irradiation is stopped, whereas the laser irradiation stabilizes the crystal even if the size of the crystal becomes larger than that of focal spot. These observations indicate that a change in the relative magnitudes of chemical potentials between solution/crystalline phases. This change is possibly promoted via crystal growth by trapping of crystalline clusters in optical potential well formed on a crystal surfaces originating from "light propagation" through the crystal. Our results shed a light not only on polymorph control but also on a method to prepare a longer-lived achiral precursor for analysis on achiral-chiral transition by "freezing" a kinetic pathway of chiral crystallization.
  • Takashige Omatsu, Katsuhiko Miyamoto, Andrew J. Lee
    JOURNAL OF OPTICS 19(12) 123002 2017年12月  査読有り
    The unique properties of optical vortex beams, in particular their spiral wavefront, have resulted in the emergence of a wide range of unique applications for this type of laser output. These applications include optical tweezing, free space optical communications, microfabrication, environmental optics, and astrophysics. However, much like the laser in its infancy, the adaptation of this type of laser output requires a diversity of wavelengths. We report on recent progress on development of optical vortex laser sources and in particular, focus on their wavelength extension, where nonlinear optical processes have been used to generate vortex laser beams with wavelengths which span the ultraviolet to infrared. We show that nonlinear optical conversion can be used to not only diversify the output wavelength of these sources, but can be used to uniquely engineer the wavefront and spatial properties of the laser output.
  • Yuta Sasaki, Koki Yamaguchi, Jun Shibakawa, Katsuhiko Miyamoto, Takashige Omatsu
    APPLIED OPTICS 56(29) 8075-8080 2017年10月  査読有り
    Picosecond pulsed frequency-doubled optical vortices were generated using a pair of beta-BaB2O4 crystals with their c axes inverted. This arrangement produced high-quality ultraviolet vortex output with low spatial separation of the phase singularity at a conversion efficiency of similar to 40%. We also discuss the theoretical spatial form and beam propagation of the ultraviolet vortex output. (C) 2017 Optical Society of America

MISC

 73

書籍等出版物

 1

講演・口頭発表等

 94

所属学協会

 4

共同研究・競争的資金等の研究課題

 16

社会貢献活動

 4