Abstract

Plasmonic hybridized transverse magnetic like (TM-like) cavity modes in multi-layered metal-dielectric circular nanoantenna are systematically studied. The main purpose is to explore the symmetry features of the vertical modal profile and its impact on the in-plane interference of gap plasmonic waves that are responsible to the resonant mode. It is found that only vertically in-phase modes are excitable when illuminated by a plane wave under normal incidence and more could be selectively excited using a dipole source, within the wavelength range from 430 nm-1250 nm. More specifically, the excitation of localized cavity modes is shown to be highly sensitive to the dipole position which determines symmetry matching and the degree of field overlap between the dipole source and the cavity mode pattern. Furthermore, we show that the resonance frequencies can be approximately predicted by the dispersion relations of plasmonic wave in the corresponding two-dimensional multilayered structure. Our results would be helpful for the design of photonic nanoantennas with alternative metal and dielectric medium.

© 2015 Optical Society of America

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    [Crossref]
  27. B. M. Ross and L. P. Lee, “Comparison of near- and far-field measures for plasmon resonance of metallic nanoparticles,” Opt. Lett. 34(7), 896–898 (2009).
    [Crossref] [PubMed]
  28. J. Zuloaga and P. Nordlander, “On the energy shift between near-field and far-field peak intensities in localized plasmon systems,” Nano Lett. 11(3), 1280–1283 (2011).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
  35. M. A. Kats, D. Woolf, R. Blanchard, N. Yu, and F. Capasso, “Spoof plasmon analogue of metal-insulator-metal waveguides,” Opt. Express 19(16), 14860–14870 (2011).
    [Crossref] [PubMed]

2015 (1)

Q. Zhang, J. J. Xiao, X. M. Zhang, D. Z. Han, and L. Gao, “Core-shell-structured dielectric−metal circular nanodisk antenna: gap plasmon assisted magnetic toroid-like cavity modes,” ACS Photonics 2(1), 60–65 (2015).
[Crossref]

2014 (10)

C. Guclu, T. S. Luk, G. T. Wang, and F. Capolino, “Radiative emission enhancement using nano-antennas made of hyperbolic metamaterial resonators,” Appl. Phys. Lett. 105(12), 123101 (2014).
[Crossref]

H. Zhang, H. V. Demir, and A. O. Govorov, “Plasmonic metamaterials and nanocomposites with the narrow transparency window effect in broad extinction spectra,” ACS Photonics 1(9), 822–832 (2014).
[Crossref]

A. Mohtashami, T. Coenen, A. Antoncecchi, A. Polman, and A. F. Koenderink, “Nanoscale execitation mapping of plasmonic patch antennas,” ACS Photonics 1(11), 1134–1143 (2014).
[Crossref]

F. Lou, M. Yan, L. Thylen, M. Qiu, and L. Wosinski, “Whispering gallery mode nanodisk resonator based on layered metal-dielectric waveguide,” Opt. Express 22(7), 8490–8502 (2014).
[Crossref] [PubMed]

X. M. Zhang, J. J. Xiao, and Q. Zhang, “Interaction between single nano-emitter and plasmonic disk–ring nanostructure with multiple Fano resonances,” J. Opt. Soc. Am. B 31(9), 2193–2200 (2014).
[Crossref]

N. Zhang, Y. J. Liu, J. Yang, X. Su, J. Deng, C. C. Chum, M. Hong, and J. Teng, “High sensitivity molecule detection by plasmonic nanoantennas with selective binding at electromagnetic hotspots,” Nanoscale 6(3), 1416–1422 (2014).
[Crossref] [PubMed]

T. Coenen, F. Bernal Arango, A. Femius Koenderink, and A. Polman, “Directional emission from a single plasmonic scatterer,” Nat. Commun. 5, 3250 (2014).
[Crossref] [PubMed]

F. Minkowski, F. Wang, A. Chakrabarty, and Q. H. Wei, “Resonant cavity modes of circular plasmonic patch nanoantennas,” Appl. Phys. Lett. 104(2), 021111 (2014).
[Crossref]

S. R. K. Rodriguez, F. Bernal Arango, T. P. Steinbusch, M. A. Verschuuren, A. F. Koenderink, and J. Gómez Rivas, “Breaking the symmetry of forward-backward light emission with localized and collective magnetoelectric resonances in arrays of pyramid-shaped aluminum nanoparticles,” Phys. Rev. Lett. 113(24), 247401 (2014).
[Crossref] [PubMed]

H. L. Ning, N. A. Krueger, X. Sheng, H. Keum, C. Zhang, K. D. Choquette, X. L. Li, S. Kim, J. A. Rogers, and P. Braun, “Transfer-printing of tunable porous silicon microcavities with embedded emitters,” ACS Photonics 1(11), 1144–1150 (2014).
[Crossref]

2013 (6)

W. D. Newman, C. L. Cortes, and Z. Jacob, “Enhanced and directional single-photon emission in hyperbolic metamaterials,” J. Opt. Soc. Am. B 30(4), 766–775 (2013).

X. Ren, W. E. I. Sha, and W. C. H. Choy, “Tuning optical responses of metallic dipole nanoantenna using graphene,” Opt. Express 21(26), 31824–31829 (2013).
[Crossref] [PubMed]

M. Frederiksen, V. E. Bochenkov, R. Ogaki, and D. S. Sutherland, “Onset of bonding plasmon hybridization preceded by gap modes in dielectric splitting of metal disks,” Nano Lett. 13(12), 6033–6039 (2013).
[Crossref] [PubMed]

G. Lu, J. Liu, T. Zhang, H. Shen, P. Perriat, M. Martini, O. Tillement, Y. Gu, Y. He, Y. Wang, and Q. Gong, “Enhancing molecule fluorescence with asymmetrical plasmonic antennas,” Nanoscale 5(14), 6545–6551 (2013).
[Crossref] [PubMed]

T. Zhan, X. Shi, Y. Dai, X. Liu, and J. Zi, “Transfer matrix method for optics in graphene layers,” J. Phys. Condens. Matter 25(21), 215301 (2013).
[Crossref] [PubMed]

D. Aronovich and G. Bartal, “Nonlinear hyperlens,” Opt. Lett. 38(4), 413–415 (2013).
[Crossref] [PubMed]

2012 (3)

Y. C. Chang, S. M. Wang, H. C. Chung, C. B. Tseng, and S. H. Chang, “Observation of absorption-dominated bonding dark plasmon mode from metal-insulator-metal nanodisk arrays fabricated by nanospherical-lens lithography,” ACS Nano 6(4), 3390–3396 (2012).
[Crossref] [PubMed]

R. Filter, J. Qi, C. Rockstuhl, and F. Lederer, “Circular optical nanoantennas: an analytical theory,” Phys. Rev. B 85(12), 125429 (2012).
[Crossref]

A. Chakrabarty, F. Wang, F. Minkowski, K. Sun, and Q. H. Wei, “Cavity modes and their excitations in elliptical plasmonic patch nanoantennas,” Opt. Express 20(11), 11615–11624 (2012).
[PubMed]

2011 (4)

N. Liu, M. L. Tang, M. Hentschel, H. Giessen, and A. P. Alivisatos, “Nanoantenna-enhanced gas sensing in a single tailored nanofocus,” Nat. Mater. 10(8), 631–636 (2011).
[Crossref] [PubMed]

J. Zuloaga and P. Nordlander, “On the energy shift between near-field and far-field peak intensities in localized plasmon systems,” Nano Lett. 11(3), 1280–1283 (2011).
[Crossref] [PubMed]

M. A. Kats, D. Woolf, R. Blanchard, N. Yu, and F. Capasso, “Spoof plasmon analogue of metal-insulator-metal waveguides,” Opt. Express 19(16), 14860–14870 (2011).
[Crossref] [PubMed]

O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, “Dipole radiation near hyperbolic metamaterials: applicability of effective-medium approximation,” Opt. Lett. 36(13), 2530–2532 (2011).
[Crossref] [PubMed]

2010 (1)

2009 (4)

M. Liu, T. W. Lee, S. K. Gray, P. Guyot-Sionnest, and M. Pelton, “Excitation of dark plasmons in metal nanoparticles by a localized emitter,” Phys. Rev. Lett. 102(10), 107401 (2009).
[Crossref] [PubMed]

B. Min, E. Ostby, V. Sorger, E. Ulin-Avila, L. Yang, X. Zhang, and K. Vahala, “High-Q surface-plasmon-polariton whispering-gallery microcavity,” Nature 457(7228), 455–458 (2009).
[Crossref] [PubMed]

B. M. Ross and L. P. Lee, “Comparison of near- and far-field measures for plasmon resonance of metallic nanoparticles,” Opt. Lett. 34(7), 896–898 (2009).
[Crossref] [PubMed]

Z. Y. Fang, F. Lin, S. Huang, W. T. Song, and X. Zhu, “Focusing surface plasmon polariton trapping of colloidal particles,” Appl. Phys. Lett. 94(6), 063306 (2009).
[Crossref]

2008 (1)

F. Neubrech, A. Pucci, T. W. Cornelius, S. Karim, A. García-Etxarri, and J. Aizpurua, “Resonant plasmonic and vibrational coupling in a tailored nanoantenna for infrared detection,” Phys. Rev. Lett. 101(15), 157403 (2008).
[Crossref] [PubMed]

2007 (2)

X. Wei, X. Luo, X. Dong, and C. Du, “Localized surface plasmon nanolithography with ultrahigh resolution,” Opt. Express 15(21), 14177–14183 (2007).
[Crossref] [PubMed]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[Crossref] [PubMed]

2006 (1)

E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311(5758), 189–193 (2006).
[Crossref] [PubMed]

1998 (1)

Aizpurua, J.

F. Neubrech, A. Pucci, T. W. Cornelius, S. Karim, A. García-Etxarri, and J. Aizpurua, “Resonant plasmonic and vibrational coupling in a tailored nanoantenna for infrared detection,” Phys. Rev. Lett. 101(15), 157403 (2008).
[Crossref] [PubMed]

Alivisatos, A. P.

N. Liu, M. L. Tang, M. Hentschel, H. Giessen, and A. P. Alivisatos, “Nanoantenna-enhanced gas sensing in a single tailored nanofocus,” Nat. Mater. 10(8), 631–636 (2011).
[Crossref] [PubMed]

Andersen, D. R.

Antoncecchi, A.

A. Mohtashami, T. Coenen, A. Antoncecchi, A. Polman, and A. F. Koenderink, “Nanoscale execitation mapping of plasmonic patch antennas,” ACS Photonics 1(11), 1134–1143 (2014).
[Crossref]

Aronovich, D.

Bartal, G.

Bernal Arango, F.

S. R. K. Rodriguez, F. Bernal Arango, T. P. Steinbusch, M. A. Verschuuren, A. F. Koenderink, and J. Gómez Rivas, “Breaking the symmetry of forward-backward light emission with localized and collective magnetoelectric resonances in arrays of pyramid-shaped aluminum nanoparticles,” Phys. Rev. Lett. 113(24), 247401 (2014).
[Crossref] [PubMed]

T. Coenen, F. Bernal Arango, A. Femius Koenderink, and A. Polman, “Directional emission from a single plasmonic scatterer,” Nat. Commun. 5, 3250 (2014).
[Crossref] [PubMed]

Blanchard, R.

Bochenkov, V. E.

M. Frederiksen, V. E. Bochenkov, R. Ogaki, and D. S. Sutherland, “Onset of bonding plasmon hybridization preceded by gap modes in dielectric splitting of metal disks,” Nano Lett. 13(12), 6033–6039 (2013).
[Crossref] [PubMed]

Braun, P.

H. L. Ning, N. A. Krueger, X. Sheng, H. Keum, C. Zhang, K. D. Choquette, X. L. Li, S. Kim, J. A. Rogers, and P. Braun, “Transfer-printing of tunable porous silicon microcavities with embedded emitters,” ACS Photonics 1(11), 1144–1150 (2014).
[Crossref]

Capasso, F.

Capolino, F.

C. Guclu, T. S. Luk, G. T. Wang, and F. Capolino, “Radiative emission enhancement using nano-antennas made of hyperbolic metamaterial resonators,” Appl. Phys. Lett. 105(12), 123101 (2014).
[Crossref]

Chakrabarty, A.

F. Minkowski, F. Wang, A. Chakrabarty, and Q. H. Wei, “Resonant cavity modes of circular plasmonic patch nanoantennas,” Appl. Phys. Lett. 104(2), 021111 (2014).
[Crossref]

A. Chakrabarty, F. Wang, F. Minkowski, K. Sun, and Q. H. Wei, “Cavity modes and their excitations in elliptical plasmonic patch nanoantennas,” Opt. Express 20(11), 11615–11624 (2012).
[PubMed]

Chang, S. H.

Y. C. Chang, S. M. Wang, H. C. Chung, C. B. Tseng, and S. H. Chang, “Observation of absorption-dominated bonding dark plasmon mode from metal-insulator-metal nanodisk arrays fabricated by nanospherical-lens lithography,” ACS Nano 6(4), 3390–3396 (2012).
[Crossref] [PubMed]

Chang, Y. C.

Y. C. Chang, S. M. Wang, H. C. Chung, C. B. Tseng, and S. H. Chang, “Observation of absorption-dominated bonding dark plasmon mode from metal-insulator-metal nanodisk arrays fabricated by nanospherical-lens lithography,” ACS Nano 6(4), 3390–3396 (2012).
[Crossref] [PubMed]

Choquette, K. D.

H. L. Ning, N. A. Krueger, X. Sheng, H. Keum, C. Zhang, K. D. Choquette, X. L. Li, S. Kim, J. A. Rogers, and P. Braun, “Transfer-printing of tunable porous silicon microcavities with embedded emitters,” ACS Photonics 1(11), 1144–1150 (2014).
[Crossref]

Choy, W. C. H.

Chum, C. C.

N. Zhang, Y. J. Liu, J. Yang, X. Su, J. Deng, C. C. Chum, M. Hong, and J. Teng, “High sensitivity molecule detection by plasmonic nanoantennas with selective binding at electromagnetic hotspots,” Nanoscale 6(3), 1416–1422 (2014).
[Crossref] [PubMed]

Chung, H. C.

Y. C. Chang, S. M. Wang, H. C. Chung, C. B. Tseng, and S. H. Chang, “Observation of absorption-dominated bonding dark plasmon mode from metal-insulator-metal nanodisk arrays fabricated by nanospherical-lens lithography,” ACS Nano 6(4), 3390–3396 (2012).
[Crossref] [PubMed]

Coenen, T.

A. Mohtashami, T. Coenen, A. Antoncecchi, A. Polman, and A. F. Koenderink, “Nanoscale execitation mapping of plasmonic patch antennas,” ACS Photonics 1(11), 1134–1143 (2014).
[Crossref]

T. Coenen, F. Bernal Arango, A. Femius Koenderink, and A. Polman, “Directional emission from a single plasmonic scatterer,” Nat. Commun. 5, 3250 (2014).
[Crossref] [PubMed]

Cornelius, T. W.

F. Neubrech, A. Pucci, T. W. Cornelius, S. Karim, A. García-Etxarri, and J. Aizpurua, “Resonant plasmonic and vibrational coupling in a tailored nanoantenna for infrared detection,” Phys. Rev. Lett. 101(15), 157403 (2008).
[Crossref] [PubMed]

Cortes, C. L.

Dai, Y.

T. Zhan, X. Shi, Y. Dai, X. Liu, and J. Zi, “Transfer matrix method for optics in graphene layers,” J. Phys. Condens. Matter 25(21), 215301 (2013).
[Crossref] [PubMed]

Demir, H. V.

H. Zhang, H. V. Demir, and A. O. Govorov, “Plasmonic metamaterials and nanocomposites with the narrow transparency window effect in broad extinction spectra,” ACS Photonics 1(9), 822–832 (2014).
[Crossref]

Deng, J.

N. Zhang, Y. J. Liu, J. Yang, X. Su, J. Deng, C. C. Chum, M. Hong, and J. Teng, “High sensitivity molecule detection by plasmonic nanoantennas with selective binding at electromagnetic hotspots,” Nanoscale 6(3), 1416–1422 (2014).
[Crossref] [PubMed]

Djurišic, A. B.

Dong, X.

Du, C.

Elazar, J. M.

Fang, Z. Y.

Z. Y. Fang, F. Lin, S. Huang, W. T. Song, and X. Zhu, “Focusing surface plasmon polariton trapping of colloidal particles,” Appl. Phys. Lett. 94(6), 063306 (2009).
[Crossref]

Femius Koenderink, A.

T. Coenen, F. Bernal Arango, A. Femius Koenderink, and A. Polman, “Directional emission from a single plasmonic scatterer,” Nat. Commun. 5, 3250 (2014).
[Crossref] [PubMed]

Filter, R.

R. Filter, J. Qi, C. Rockstuhl, and F. Lederer, “Circular optical nanoantennas: an analytical theory,” Phys. Rev. B 85(12), 125429 (2012).
[Crossref]

Frederiksen, M.

M. Frederiksen, V. E. Bochenkov, R. Ogaki, and D. S. Sutherland, “Onset of bonding plasmon hybridization preceded by gap modes in dielectric splitting of metal disks,” Nano Lett. 13(12), 6033–6039 (2013).
[Crossref] [PubMed]

Gao, L.

Q. Zhang, J. J. Xiao, X. M. Zhang, D. Z. Han, and L. Gao, “Core-shell-structured dielectric−metal circular nanodisk antenna: gap plasmon assisted magnetic toroid-like cavity modes,” ACS Photonics 2(1), 60–65 (2015).
[Crossref]

García-Etxarri, A.

F. Neubrech, A. Pucci, T. W. Cornelius, S. Karim, A. García-Etxarri, and J. Aizpurua, “Resonant plasmonic and vibrational coupling in a tailored nanoantenna for infrared detection,” Phys. Rev. Lett. 101(15), 157403 (2008).
[Crossref] [PubMed]

Giessen, H.

N. Liu, M. L. Tang, M. Hentschel, H. Giessen, and A. P. Alivisatos, “Nanoantenna-enhanced gas sensing in a single tailored nanofocus,” Nat. Mater. 10(8), 631–636 (2011).
[Crossref] [PubMed]

Gómez Rivas, J.

S. R. K. Rodriguez, F. Bernal Arango, T. P. Steinbusch, M. A. Verschuuren, A. F. Koenderink, and J. Gómez Rivas, “Breaking the symmetry of forward-backward light emission with localized and collective magnetoelectric resonances in arrays of pyramid-shaped aluminum nanoparticles,” Phys. Rev. Lett. 113(24), 247401 (2014).
[Crossref] [PubMed]

Gong, Q.

G. Lu, J. Liu, T. Zhang, H. Shen, P. Perriat, M. Martini, O. Tillement, Y. Gu, Y. He, Y. Wang, and Q. Gong, “Enhancing molecule fluorescence with asymmetrical plasmonic antennas,” Nanoscale 5(14), 6545–6551 (2013).
[Crossref] [PubMed]

Govorov, A. O.

H. Zhang, H. V. Demir, and A. O. Govorov, “Plasmonic metamaterials and nanocomposites with the narrow transparency window effect in broad extinction spectra,” ACS Photonics 1(9), 822–832 (2014).
[Crossref]

Gray, S. K.

M. Liu, T. W. Lee, S. K. Gray, P. Guyot-Sionnest, and M. Pelton, “Excitation of dark plasmons in metal nanoparticles by a localized emitter,” Phys. Rev. Lett. 102(10), 107401 (2009).
[Crossref] [PubMed]

Gu, Y.

G. Lu, J. Liu, T. Zhang, H. Shen, P. Perriat, M. Martini, O. Tillement, Y. Gu, Y. He, Y. Wang, and Q. Gong, “Enhancing molecule fluorescence with asymmetrical plasmonic antennas,” Nanoscale 5(14), 6545–6551 (2013).
[Crossref] [PubMed]

Guclu, C.

C. Guclu, T. S. Luk, G. T. Wang, and F. Capolino, “Radiative emission enhancement using nano-antennas made of hyperbolic metamaterial resonators,” Appl. Phys. Lett. 105(12), 123101 (2014).
[Crossref]

Guyot-Sionnest, P.

M. Liu, T. W. Lee, S. K. Gray, P. Guyot-Sionnest, and M. Pelton, “Excitation of dark plasmons in metal nanoparticles by a localized emitter,” Phys. Rev. Lett. 102(10), 107401 (2009).
[Crossref] [PubMed]

Han, D. Z.

Q. Zhang, J. J. Xiao, X. M. Zhang, D. Z. Han, and L. Gao, “Core-shell-structured dielectric−metal circular nanodisk antenna: gap plasmon assisted magnetic toroid-like cavity modes,” ACS Photonics 2(1), 60–65 (2015).
[Crossref]

He, Y.

G. Lu, J. Liu, T. Zhang, H. Shen, P. Perriat, M. Martini, O. Tillement, Y. Gu, Y. He, Y. Wang, and Q. Gong, “Enhancing molecule fluorescence with asymmetrical plasmonic antennas,” Nanoscale 5(14), 6545–6551 (2013).
[Crossref] [PubMed]

Hentschel, M.

N. Liu, M. L. Tang, M. Hentschel, H. Giessen, and A. P. Alivisatos, “Nanoantenna-enhanced gas sensing in a single tailored nanofocus,” Nat. Mater. 10(8), 631–636 (2011).
[Crossref] [PubMed]

Hong, M.

N. Zhang, Y. J. Liu, J. Yang, X. Su, J. Deng, C. C. Chum, M. Hong, and J. Teng, “High sensitivity molecule detection by plasmonic nanoantennas with selective binding at electromagnetic hotspots,” Nanoscale 6(3), 1416–1422 (2014).
[Crossref] [PubMed]

Huang, S.

Z. Y. Fang, F. Lin, S. Huang, W. T. Song, and X. Zhu, “Focusing surface plasmon polariton trapping of colloidal particles,” Appl. Phys. Lett. 94(6), 063306 (2009).
[Crossref]

Jacob, Z.

Karim, S.

F. Neubrech, A. Pucci, T. W. Cornelius, S. Karim, A. García-Etxarri, and J. Aizpurua, “Resonant plasmonic and vibrational coupling in a tailored nanoantenna for infrared detection,” Phys. Rev. Lett. 101(15), 157403 (2008).
[Crossref] [PubMed]

Kats, M. A.

Keum, H.

H. L. Ning, N. A. Krueger, X. Sheng, H. Keum, C. Zhang, K. D. Choquette, X. L. Li, S. Kim, J. A. Rogers, and P. Braun, “Transfer-printing of tunable porous silicon microcavities with embedded emitters,” ACS Photonics 1(11), 1144–1150 (2014).
[Crossref]

Kidwai, O.

Kim, S.

H. L. Ning, N. A. Krueger, X. Sheng, H. Keum, C. Zhang, K. D. Choquette, X. L. Li, S. Kim, J. A. Rogers, and P. Braun, “Transfer-printing of tunable porous silicon microcavities with embedded emitters,” ACS Photonics 1(11), 1144–1150 (2014).
[Crossref]

Koenderink, A. F.

S. R. K. Rodriguez, F. Bernal Arango, T. P. Steinbusch, M. A. Verschuuren, A. F. Koenderink, and J. Gómez Rivas, “Breaking the symmetry of forward-backward light emission with localized and collective magnetoelectric resonances in arrays of pyramid-shaped aluminum nanoparticles,” Phys. Rev. Lett. 113(24), 247401 (2014).
[Crossref] [PubMed]

A. Mohtashami, T. Coenen, A. Antoncecchi, A. Polman, and A. F. Koenderink, “Nanoscale execitation mapping of plasmonic patch antennas,” ACS Photonics 1(11), 1134–1143 (2014).
[Crossref]

Krueger, N. A.

H. L. Ning, N. A. Krueger, X. Sheng, H. Keum, C. Zhang, K. D. Choquette, X. L. Li, S. Kim, J. A. Rogers, and P. Braun, “Transfer-printing of tunable porous silicon microcavities with embedded emitters,” ACS Photonics 1(11), 1144–1150 (2014).
[Crossref]

Lederer, F.

R. Filter, J. Qi, C. Rockstuhl, and F. Lederer, “Circular optical nanoantennas: an analytical theory,” Phys. Rev. B 85(12), 125429 (2012).
[Crossref]

Lee, H.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[Crossref] [PubMed]

Lee, L. P.

Lee, T. W.

M. Liu, T. W. Lee, S. K. Gray, P. Guyot-Sionnest, and M. Pelton, “Excitation of dark plasmons in metal nanoparticles by a localized emitter,” Phys. Rev. Lett. 102(10), 107401 (2009).
[Crossref] [PubMed]

Li, X. L.

H. L. Ning, N. A. Krueger, X. Sheng, H. Keum, C. Zhang, K. D. Choquette, X. L. Li, S. Kim, J. A. Rogers, and P. Braun, “Transfer-printing of tunable porous silicon microcavities with embedded emitters,” ACS Photonics 1(11), 1144–1150 (2014).
[Crossref]

Lin, F.

Z. Y. Fang, F. Lin, S. Huang, W. T. Song, and X. Zhu, “Focusing surface plasmon polariton trapping of colloidal particles,” Appl. Phys. Lett. 94(6), 063306 (2009).
[Crossref]

Liu, J.

G. Lu, J. Liu, T. Zhang, H. Shen, P. Perriat, M. Martini, O. Tillement, Y. Gu, Y. He, Y. Wang, and Q. Gong, “Enhancing molecule fluorescence with asymmetrical plasmonic antennas,” Nanoscale 5(14), 6545–6551 (2013).
[Crossref] [PubMed]

Liu, M.

M. Liu, T. W. Lee, S. K. Gray, P. Guyot-Sionnest, and M. Pelton, “Excitation of dark plasmons in metal nanoparticles by a localized emitter,” Phys. Rev. Lett. 102(10), 107401 (2009).
[Crossref] [PubMed]

Liu, N.

N. Liu, M. L. Tang, M. Hentschel, H. Giessen, and A. P. Alivisatos, “Nanoantenna-enhanced gas sensing in a single tailored nanofocus,” Nat. Mater. 10(8), 631–636 (2011).
[Crossref] [PubMed]

Liu, X.

T. Zhan, X. Shi, Y. Dai, X. Liu, and J. Zi, “Transfer matrix method for optics in graphene layers,” J. Phys. Condens. Matter 25(21), 215301 (2013).
[Crossref] [PubMed]

Liu, Y. J.

N. Zhang, Y. J. Liu, J. Yang, X. Su, J. Deng, C. C. Chum, M. Hong, and J. Teng, “High sensitivity molecule detection by plasmonic nanoantennas with selective binding at electromagnetic hotspots,” Nanoscale 6(3), 1416–1422 (2014).
[Crossref] [PubMed]

Liu, Z.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[Crossref] [PubMed]

Lou, F.

Lu, G.

G. Lu, J. Liu, T. Zhang, H. Shen, P. Perriat, M. Martini, O. Tillement, Y. Gu, Y. He, Y. Wang, and Q. Gong, “Enhancing molecule fluorescence with asymmetrical plasmonic antennas,” Nanoscale 5(14), 6545–6551 (2013).
[Crossref] [PubMed]

Luk, T. S.

C. Guclu, T. S. Luk, G. T. Wang, and F. Capolino, “Radiative emission enhancement using nano-antennas made of hyperbolic metamaterial resonators,” Appl. Phys. Lett. 105(12), 123101 (2014).
[Crossref]

Luo, X.

Majewski, M. L.

Martini, M.

G. Lu, J. Liu, T. Zhang, H. Shen, P. Perriat, M. Martini, O. Tillement, Y. Gu, Y. He, Y. Wang, and Q. Gong, “Enhancing molecule fluorescence with asymmetrical plasmonic antennas,” Nanoscale 5(14), 6545–6551 (2013).
[Crossref] [PubMed]

Min, B.

B. Min, E. Ostby, V. Sorger, E. Ulin-Avila, L. Yang, X. Zhang, and K. Vahala, “High-Q surface-plasmon-polariton whispering-gallery microcavity,” Nature 457(7228), 455–458 (2009).
[Crossref] [PubMed]

Minkowski, F.

F. Minkowski, F. Wang, A. Chakrabarty, and Q. H. Wei, “Resonant cavity modes of circular plasmonic patch nanoantennas,” Appl. Phys. Lett. 104(2), 021111 (2014).
[Crossref]

A. Chakrabarty, F. Wang, F. Minkowski, K. Sun, and Q. H. Wei, “Cavity modes and their excitations in elliptical plasmonic patch nanoantennas,” Opt. Express 20(11), 11615–11624 (2012).
[PubMed]

Mohtashami, A.

A. Mohtashami, T. Coenen, A. Antoncecchi, A. Polman, and A. F. Koenderink, “Nanoscale execitation mapping of plasmonic patch antennas,” ACS Photonics 1(11), 1134–1143 (2014).
[Crossref]

Neubrech, F.

F. Neubrech, A. Pucci, T. W. Cornelius, S. Karim, A. García-Etxarri, and J. Aizpurua, “Resonant plasmonic and vibrational coupling in a tailored nanoantenna for infrared detection,” Phys. Rev. Lett. 101(15), 157403 (2008).
[Crossref] [PubMed]

Newman, W. D.

Ning, H. L.

H. L. Ning, N. A. Krueger, X. Sheng, H. Keum, C. Zhang, K. D. Choquette, X. L. Li, S. Kim, J. A. Rogers, and P. Braun, “Transfer-printing of tunable porous silicon microcavities with embedded emitters,” ACS Photonics 1(11), 1144–1150 (2014).
[Crossref]

Nordlander, P.

J. Zuloaga and P. Nordlander, “On the energy shift between near-field and far-field peak intensities in localized plasmon systems,” Nano Lett. 11(3), 1280–1283 (2011).
[Crossref] [PubMed]

Ogaki, R.

M. Frederiksen, V. E. Bochenkov, R. Ogaki, and D. S. Sutherland, “Onset of bonding plasmon hybridization preceded by gap modes in dielectric splitting of metal disks,” Nano Lett. 13(12), 6033–6039 (2013).
[Crossref] [PubMed]

Ostby, E.

B. Min, E. Ostby, V. Sorger, E. Ulin-Avila, L. Yang, X. Zhang, and K. Vahala, “High-Q surface-plasmon-polariton whispering-gallery microcavity,” Nature 457(7228), 455–458 (2009).
[Crossref] [PubMed]

Ozbay, E.

E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311(5758), 189–193 (2006).
[Crossref] [PubMed]

Pelton, M.

M. Liu, T. W. Lee, S. K. Gray, P. Guyot-Sionnest, and M. Pelton, “Excitation of dark plasmons in metal nanoparticles by a localized emitter,” Phys. Rev. Lett. 102(10), 107401 (2009).
[Crossref] [PubMed]

Perriat, P.

G. Lu, J. Liu, T. Zhang, H. Shen, P. Perriat, M. Martini, O. Tillement, Y. Gu, Y. He, Y. Wang, and Q. Gong, “Enhancing molecule fluorescence with asymmetrical plasmonic antennas,” Nanoscale 5(14), 6545–6551 (2013).
[Crossref] [PubMed]

Polman, A.

A. Mohtashami, T. Coenen, A. Antoncecchi, A. Polman, and A. F. Koenderink, “Nanoscale execitation mapping of plasmonic patch antennas,” ACS Photonics 1(11), 1134–1143 (2014).
[Crossref]

T. Coenen, F. Bernal Arango, A. Femius Koenderink, and A. Polman, “Directional emission from a single plasmonic scatterer,” Nat. Commun. 5, 3250 (2014).
[Crossref] [PubMed]

Pucci, A.

F. Neubrech, A. Pucci, T. W. Cornelius, S. Karim, A. García-Etxarri, and J. Aizpurua, “Resonant plasmonic and vibrational coupling in a tailored nanoantenna for infrared detection,” Phys. Rev. Lett. 101(15), 157403 (2008).
[Crossref] [PubMed]

Qi, J.

R. Filter, J. Qi, C. Rockstuhl, and F. Lederer, “Circular optical nanoantennas: an analytical theory,” Phys. Rev. B 85(12), 125429 (2012).
[Crossref]

Qiu, M.

Rakic, A. D.

Ren, X.

Rockstuhl, C.

R. Filter, J. Qi, C. Rockstuhl, and F. Lederer, “Circular optical nanoantennas: an analytical theory,” Phys. Rev. B 85(12), 125429 (2012).
[Crossref]

Rodriguez, S. R. K.

S. R. K. Rodriguez, F. Bernal Arango, T. P. Steinbusch, M. A. Verschuuren, A. F. Koenderink, and J. Gómez Rivas, “Breaking the symmetry of forward-backward light emission with localized and collective magnetoelectric resonances in arrays of pyramid-shaped aluminum nanoparticles,” Phys. Rev. Lett. 113(24), 247401 (2014).
[Crossref] [PubMed]

Rogers, J. A.

H. L. Ning, N. A. Krueger, X. Sheng, H. Keum, C. Zhang, K. D. Choquette, X. L. Li, S. Kim, J. A. Rogers, and P. Braun, “Transfer-printing of tunable porous silicon microcavities with embedded emitters,” ACS Photonics 1(11), 1144–1150 (2014).
[Crossref]

Ross, B. M.

Sha, W. E. I.

Shen, H.

G. Lu, J. Liu, T. Zhang, H. Shen, P. Perriat, M. Martini, O. Tillement, Y. Gu, Y. He, Y. Wang, and Q. Gong, “Enhancing molecule fluorescence with asymmetrical plasmonic antennas,” Nanoscale 5(14), 6545–6551 (2013).
[Crossref] [PubMed]

Sheng, X.

H. L. Ning, N. A. Krueger, X. Sheng, H. Keum, C. Zhang, K. D. Choquette, X. L. Li, S. Kim, J. A. Rogers, and P. Braun, “Transfer-printing of tunable porous silicon microcavities with embedded emitters,” ACS Photonics 1(11), 1144–1150 (2014).
[Crossref]

Shi, X.

T. Zhan, X. Shi, Y. Dai, X. Liu, and J. Zi, “Transfer matrix method for optics in graphene layers,” J. Phys. Condens. Matter 25(21), 215301 (2013).
[Crossref] [PubMed]

Sipe, J. E.

Song, W. T.

Z. Y. Fang, F. Lin, S. Huang, W. T. Song, and X. Zhu, “Focusing surface plasmon polariton trapping of colloidal particles,” Appl. Phys. Lett. 94(6), 063306 (2009).
[Crossref]

Sorger, V.

B. Min, E. Ostby, V. Sorger, E. Ulin-Avila, L. Yang, X. Zhang, and K. Vahala, “High-Q surface-plasmon-polariton whispering-gallery microcavity,” Nature 457(7228), 455–458 (2009).
[Crossref] [PubMed]

Steinbusch, T. P.

S. R. K. Rodriguez, F. Bernal Arango, T. P. Steinbusch, M. A. Verschuuren, A. F. Koenderink, and J. Gómez Rivas, “Breaking the symmetry of forward-backward light emission with localized and collective magnetoelectric resonances in arrays of pyramid-shaped aluminum nanoparticles,” Phys. Rev. Lett. 113(24), 247401 (2014).
[Crossref] [PubMed]

Su, X.

N. Zhang, Y. J. Liu, J. Yang, X. Su, J. Deng, C. C. Chum, M. Hong, and J. Teng, “High sensitivity molecule detection by plasmonic nanoantennas with selective binding at electromagnetic hotspots,” Nanoscale 6(3), 1416–1422 (2014).
[Crossref] [PubMed]

Sun, C.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[Crossref] [PubMed]

Sun, K.

Sutherland, D. S.

M. Frederiksen, V. E. Bochenkov, R. Ogaki, and D. S. Sutherland, “Onset of bonding plasmon hybridization preceded by gap modes in dielectric splitting of metal disks,” Nano Lett. 13(12), 6033–6039 (2013).
[Crossref] [PubMed]

Tang, M. L.

N. Liu, M. L. Tang, M. Hentschel, H. Giessen, and A. P. Alivisatos, “Nanoantenna-enhanced gas sensing in a single tailored nanofocus,” Nat. Mater. 10(8), 631–636 (2011).
[Crossref] [PubMed]

Teng, J.

N. Zhang, Y. J. Liu, J. Yang, X. Su, J. Deng, C. C. Chum, M. Hong, and J. Teng, “High sensitivity molecule detection by plasmonic nanoantennas with selective binding at electromagnetic hotspots,” Nanoscale 6(3), 1416–1422 (2014).
[Crossref] [PubMed]

Thylen, L.

Tillement, O.

G. Lu, J. Liu, T. Zhang, H. Shen, P. Perriat, M. Martini, O. Tillement, Y. Gu, Y. He, Y. Wang, and Q. Gong, “Enhancing molecule fluorescence with asymmetrical plasmonic antennas,” Nanoscale 5(14), 6545–6551 (2013).
[Crossref] [PubMed]

Tseng, C. B.

Y. C. Chang, S. M. Wang, H. C. Chung, C. B. Tseng, and S. H. Chang, “Observation of absorption-dominated bonding dark plasmon mode from metal-insulator-metal nanodisk arrays fabricated by nanospherical-lens lithography,” ACS Nano 6(4), 3390–3396 (2012).
[Crossref] [PubMed]

Ulin-Avila, E.

B. Min, E. Ostby, V. Sorger, E. Ulin-Avila, L. Yang, X. Zhang, and K. Vahala, “High-Q surface-plasmon-polariton whispering-gallery microcavity,” Nature 457(7228), 455–458 (2009).
[Crossref] [PubMed]

Vahala, K.

B. Min, E. Ostby, V. Sorger, E. Ulin-Avila, L. Yang, X. Zhang, and K. Vahala, “High-Q surface-plasmon-polariton whispering-gallery microcavity,” Nature 457(7228), 455–458 (2009).
[Crossref] [PubMed]

Verschuuren, M. A.

S. R. K. Rodriguez, F. Bernal Arango, T. P. Steinbusch, M. A. Verschuuren, A. F. Koenderink, and J. Gómez Rivas, “Breaking the symmetry of forward-backward light emission with localized and collective magnetoelectric resonances in arrays of pyramid-shaped aluminum nanoparticles,” Phys. Rev. Lett. 113(24), 247401 (2014).
[Crossref] [PubMed]

Wang, F.

F. Minkowski, F. Wang, A. Chakrabarty, and Q. H. Wei, “Resonant cavity modes of circular plasmonic patch nanoantennas,” Appl. Phys. Lett. 104(2), 021111 (2014).
[Crossref]

A. Chakrabarty, F. Wang, F. Minkowski, K. Sun, and Q. H. Wei, “Cavity modes and their excitations in elliptical plasmonic patch nanoantennas,” Opt. Express 20(11), 11615–11624 (2012).
[PubMed]

Wang, G. T.

C. Guclu, T. S. Luk, G. T. Wang, and F. Capolino, “Radiative emission enhancement using nano-antennas made of hyperbolic metamaterial resonators,” Appl. Phys. Lett. 105(12), 123101 (2014).
[Crossref]

Wang, S. M.

Y. C. Chang, S. M. Wang, H. C. Chung, C. B. Tseng, and S. H. Chang, “Observation of absorption-dominated bonding dark plasmon mode from metal-insulator-metal nanodisk arrays fabricated by nanospherical-lens lithography,” ACS Nano 6(4), 3390–3396 (2012).
[Crossref] [PubMed]

Wang, Y.

G. Lu, J. Liu, T. Zhang, H. Shen, P. Perriat, M. Martini, O. Tillement, Y. Gu, Y. He, Y. Wang, and Q. Gong, “Enhancing molecule fluorescence with asymmetrical plasmonic antennas,” Nanoscale 5(14), 6545–6551 (2013).
[Crossref] [PubMed]

Wei, Q. H.

F. Minkowski, F. Wang, A. Chakrabarty, and Q. H. Wei, “Resonant cavity modes of circular plasmonic patch nanoantennas,” Appl. Phys. Lett. 104(2), 021111 (2014).
[Crossref]

A. Chakrabarty, F. Wang, F. Minkowski, K. Sun, and Q. H. Wei, “Cavity modes and their excitations in elliptical plasmonic patch nanoantennas,” Opt. Express 20(11), 11615–11624 (2012).
[PubMed]

Wei, X.

Woolf, D.

Wosinski, L.

Xiao, J. J.

Q. Zhang, J. J. Xiao, X. M. Zhang, D. Z. Han, and L. Gao, “Core-shell-structured dielectric−metal circular nanodisk antenna: gap plasmon assisted magnetic toroid-like cavity modes,” ACS Photonics 2(1), 60–65 (2015).
[Crossref]

X. M. Zhang, J. J. Xiao, and Q. Zhang, “Interaction between single nano-emitter and plasmonic disk–ring nanostructure with multiple Fano resonances,” J. Opt. Soc. Am. B 31(9), 2193–2200 (2014).
[Crossref]

Xiong, Y.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[Crossref] [PubMed]

Yan, M.

Yang, J.

N. Zhang, Y. J. Liu, J. Yang, X. Su, J. Deng, C. C. Chum, M. Hong, and J. Teng, “High sensitivity molecule detection by plasmonic nanoantennas with selective binding at electromagnetic hotspots,” Nanoscale 6(3), 1416–1422 (2014).
[Crossref] [PubMed]

Yang, L.

B. Min, E. Ostby, V. Sorger, E. Ulin-Avila, L. Yang, X. Zhang, and K. Vahala, “High-Q surface-plasmon-polariton whispering-gallery microcavity,” Nature 457(7228), 455–458 (2009).
[Crossref] [PubMed]

Yu, N.

Zhan, T.

T. Zhan, X. Shi, Y. Dai, X. Liu, and J. Zi, “Transfer matrix method for optics in graphene layers,” J. Phys. Condens. Matter 25(21), 215301 (2013).
[Crossref] [PubMed]

Zhang, C.

H. L. Ning, N. A. Krueger, X. Sheng, H. Keum, C. Zhang, K. D. Choquette, X. L. Li, S. Kim, J. A. Rogers, and P. Braun, “Transfer-printing of tunable porous silicon microcavities with embedded emitters,” ACS Photonics 1(11), 1144–1150 (2014).
[Crossref]

Zhang, H.

H. Zhang, H. V. Demir, and A. O. Govorov, “Plasmonic metamaterials and nanocomposites with the narrow transparency window effect in broad extinction spectra,” ACS Photonics 1(9), 822–832 (2014).
[Crossref]

Zhang, N.

N. Zhang, Y. J. Liu, J. Yang, X. Su, J. Deng, C. C. Chum, M. Hong, and J. Teng, “High sensitivity molecule detection by plasmonic nanoantennas with selective binding at electromagnetic hotspots,” Nanoscale 6(3), 1416–1422 (2014).
[Crossref] [PubMed]

Zhang, Q.

Q. Zhang, J. J. Xiao, X. M. Zhang, D. Z. Han, and L. Gao, “Core-shell-structured dielectric−metal circular nanodisk antenna: gap plasmon assisted magnetic toroid-like cavity modes,” ACS Photonics 2(1), 60–65 (2015).
[Crossref]

X. M. Zhang, J. J. Xiao, and Q. Zhang, “Interaction between single nano-emitter and plasmonic disk–ring nanostructure with multiple Fano resonances,” J. Opt. Soc. Am. B 31(9), 2193–2200 (2014).
[Crossref]

Zhang, T.

G. Lu, J. Liu, T. Zhang, H. Shen, P. Perriat, M. Martini, O. Tillement, Y. Gu, Y. He, Y. Wang, and Q. Gong, “Enhancing molecule fluorescence with asymmetrical plasmonic antennas,” Nanoscale 5(14), 6545–6551 (2013).
[Crossref] [PubMed]

Zhang, X.

B. Min, E. Ostby, V. Sorger, E. Ulin-Avila, L. Yang, X. Zhang, and K. Vahala, “High-Q surface-plasmon-polariton whispering-gallery microcavity,” Nature 457(7228), 455–458 (2009).
[Crossref] [PubMed]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[Crossref] [PubMed]

Zhang, X. M.

Q. Zhang, J. J. Xiao, X. M. Zhang, D. Z. Han, and L. Gao, “Core-shell-structured dielectric−metal circular nanodisk antenna: gap plasmon assisted magnetic toroid-like cavity modes,” ACS Photonics 2(1), 60–65 (2015).
[Crossref]

X. M. Zhang, J. J. Xiao, and Q. Zhang, “Interaction between single nano-emitter and plasmonic disk–ring nanostructure with multiple Fano resonances,” J. Opt. Soc. Am. B 31(9), 2193–2200 (2014).
[Crossref]

Zhu, X.

Z. Y. Fang, F. Lin, S. Huang, W. T. Song, and X. Zhu, “Focusing surface plasmon polariton trapping of colloidal particles,” Appl. Phys. Lett. 94(6), 063306 (2009).
[Crossref]

Zhukovsky, S. V.

Zi, J.

T. Zhan, X. Shi, Y. Dai, X. Liu, and J. Zi, “Transfer matrix method for optics in graphene layers,” J. Phys. Condens. Matter 25(21), 215301 (2013).
[Crossref] [PubMed]

Zuloaga, J.

J. Zuloaga and P. Nordlander, “On the energy shift between near-field and far-field peak intensities in localized plasmon systems,” Nano Lett. 11(3), 1280–1283 (2011).
[Crossref] [PubMed]

ACS Nano (1)

Y. C. Chang, S. M. Wang, H. C. Chung, C. B. Tseng, and S. H. Chang, “Observation of absorption-dominated bonding dark plasmon mode from metal-insulator-metal nanodisk arrays fabricated by nanospherical-lens lithography,” ACS Nano 6(4), 3390–3396 (2012).
[Crossref] [PubMed]

ACS Photonics (4)

Q. Zhang, J. J. Xiao, X. M. Zhang, D. Z. Han, and L. Gao, “Core-shell-structured dielectric−metal circular nanodisk antenna: gap plasmon assisted magnetic toroid-like cavity modes,” ACS Photonics 2(1), 60–65 (2015).
[Crossref]

H. Zhang, H. V. Demir, and A. O. Govorov, “Plasmonic metamaterials and nanocomposites with the narrow transparency window effect in broad extinction spectra,” ACS Photonics 1(9), 822–832 (2014).
[Crossref]

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Figures (6)

Fig. 1
Fig. 1 Geometry of the multilayered metal-dielectric circular antenna.

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Fig. 2
Fig. 2 Optical cross sections of the antenna with plane wave excitation. (a) Insets show the resonant electric fields E z in the xz plane and the xy plane across the top dielectric layer for the two peaks for two different incident plane wave configurations. (b) Inserts show E x in the xz plane and the xy plane across the top dielectric layer for the resonances.

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Fig. 3
Fig. 3 Non-radiative decay rate spectra for electric dipole at P 1 , P 2 , P 3 , and P 4 .

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Fig. 4
Fig. 4 Selective mode excitation by a dipole source. Shown in the table are xz plane and xy plane electric field when dipole source is positioned at P 1 , P 2 , P 3 , and P 4 . The resonant wavelength and mode labeling are shown for each column.

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Fig. 5
Fig. 5 Cavity resonant frequencies (symbols) versus χ ' nm /(R+d) for different radius of the nanoantenna. The 2D theoretically calculated dispersion (black lines) of hybridized SPP are also shown.

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Fig. 6
Fig. 6 In-plane field ratio | E // / E tot | 2 measured along the z axis for the three different resonance modes excited by a dipole source at P 1 . The symbols “M” and “D” represent the metal and dielectric layer, respectively.

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Equations (5)

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∂ ∂ρ (ρ ∂ E z ∂ρ )+ 1 ρ 2 ∂ 2 E z ∂ ϕ 2 + ∂ 2 E z ∂ z 2 + k 2 i E z =0
E z (ρ,ϕ,z)=a(z)[ H m (1) ( k gsp ρ)+ r m H m (2) ( k gsp ρ)] e imϕ
k gsp R eff =χ ' nm
D ij = 1 2 [ 1+ η ij    1− η i j 1− η i j    1+ η ij ]
ϕ j =[ e i k z,j Δz    0 0        e i k z,j Δz ]

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