Abstract

A finite-element method for calculating the illumination-dependence of absorption in three-dimensional nanostructures is presented based on the radio frequency module of the Comsol Multiphysics software package (Comsol AB). This method is capable of numerically determining the optical response and near-field distribution of subwavelength periodic structures as a function of illumination orientations specified by polar angle, φ, and azimuthal angle, γ. The method was applied to determine the illumination-angle-dependent absorptance in cavity-based superconducting-nanowire single-photon detector (SNSPD) designs. Niobium-nitride stripes based on dimensions of conventional SNSPDs and integrated with ∼ quarter-wavelength hydrogen-silsesquioxane-filled nano-optical cavity and covered by a thin gold film acting as a reflector were illuminated from below by p-polarized light in this study. The numerical results were compared to results from complementary transfer-matrix-method calculations on composite layers made of analogous film-stacks. This comparison helped to uncover the optical phenomena contributing to the appearance of extrema in the optical response. This paper presents an approach to optimizing the absorptance of different sensing and detecting devices via simultaneous numerical optimization of the polar and azimuthal illumination angles.

© 2011 Optical Society of America

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  1. G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. M. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, “Picosecond superconducting single-photon optical detector,” Appl. Phys. Lett. 79, 705–708 (2001).
    [CrossRef]
  2. K. M. Rosfjord, J. K. W. Yang, E. A. Dauler, A. J. Kerman, V. Anant, B. M. Voronov, G. N. Gol’tsman, and K. K. Berggren, “Nanowire single-photon detector with an integrated optical cavity and anti-reflection coating,” Opt. Express 14, 527–534 (2006).
    [CrossRef] [PubMed]
  3. V. Anant, A. J. Kermann, E. A. Dauler, J. K. W. Yang, K. M. Rosfjord, and K. K. Berggren, “Optical properties of superconducting nanowire single-photon detectors,” Opt. Express 16, 10750–10761 (2008).
    [CrossRef] [PubMed]
  4. M. Born and E. Wolf, Principles of Optics (Pergamon, 1964).
  5. E. F. C. Driessen and M. J. A. de Dood, “The perfect absorber,” Appl. Phys. Lett. 94, 171109 (2009).
    [CrossRef]
  6. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
    [CrossRef]
  7. X. J. Yu and H. S. Kwok, “Optical wire-grid polarizers at oblique angles of incidence,” J. Appl. Phys. 93, 4407–4412 (2003).
    [CrossRef]
  8. E. F. C. Driessen, F. R. Braakman, E. M. Reiger, S. N. Dorenbos, V. Zviller, and M. J. A. de Dood, “Impedance model for the polarization-dependent optical absorption of superconducting single-photon detectors,” Eur. Phys. J. Appl. Phys. 47, 10701(2009).
    [CrossRef]
  9. L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993).
    [CrossRef]
  10. B. Saleh-Teich, Fundamentals of Photonics (Wiley, 2007): The relationship between the intensity and E-field magnitude is: I=|Eo|22η,where the impedance η is defined in terms of free space impedance ηo via η=ηo/ηsapphire, and taking into account that our substrate is a nonmagnetic medium, μ=μo, whereupon η=μo/ε; also, sapphire is essentially non-absorbing at λ=1550 nm; therefore ε=nsapphire, so the amplitude of the E-field was determined as follows: Eo=2Iη=2PA1ηsapphireμoεo.Based on this calculation, the power-flow at the source boundary corresponds to Eo=2.32×106 V/mE-field amplitude. This value for Eo is the reference amplitude necessary to interpret the near-field distribution determined by COMSOL models.
  11. X. Hu, C. W. Holzwarth, D. Masciarelli, E. A. Dauler, and K. K. Berggren, “Efficiently coupling light to superconducting nanowire single-photon detectors,” IEEE Trans. Appl. Supercond. 19, 336–340 (2009).
    [CrossRef]
  12. G. R. Bird and M. Parrish, “The wire grid as a near-infrared polarizer,” J. Opt. Soc. Am. 50, 886–891 (1960).
    [CrossRef]
  13. D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photon. 4, 83–91 (2010).
    [CrossRef]
  14. X. Hu, E. A. Dauler, R. J. Molnar, and K. K. Berggren, “Superconducting nanowire single-photon detectors integrated with optical nano-antennae,” Opt. Express 19, 17–31(2011).
    [CrossRef] [PubMed]

2011 (1)

2010 (1)

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photon. 4, 83–91 (2010).
[CrossRef]

2009 (3)

X. Hu, C. W. Holzwarth, D. Masciarelli, E. A. Dauler, and K. K. Berggren, “Efficiently coupling light to superconducting nanowire single-photon detectors,” IEEE Trans. Appl. Supercond. 19, 336–340 (2009).
[CrossRef]

E. F. C. Driessen and M. J. A. de Dood, “The perfect absorber,” Appl. Phys. Lett. 94, 171109 (2009).
[CrossRef]

E. F. C. Driessen, F. R. Braakman, E. M. Reiger, S. N. Dorenbos, V. Zviller, and M. J. A. de Dood, “Impedance model for the polarization-dependent optical absorption of superconducting single-photon detectors,” Eur. Phys. J. Appl. Phys. 47, 10701(2009).
[CrossRef]

2008 (1)

2006 (1)

2003 (1)

X. J. Yu and H. S. Kwok, “Optical wire-grid polarizers at oblique angles of incidence,” J. Appl. Phys. 93, 4407–4412 (2003).
[CrossRef]

2001 (1)

G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. M. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, “Picosecond superconducting single-photon optical detector,” Appl. Phys. Lett. 79, 705–708 (2001).
[CrossRef]

1993 (1)

L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993).
[CrossRef]

1981 (1)

1960 (1)

Anant, V.

Berggren, K. K.

Bird, G. R.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1964).

Bozhevolnyi, S. I.

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photon. 4, 83–91 (2010).
[CrossRef]

Braakman, F. R.

E. F. C. Driessen, F. R. Braakman, E. M. Reiger, S. N. Dorenbos, V. Zviller, and M. J. A. de Dood, “Impedance model for the polarization-dependent optical absorption of superconducting single-photon detectors,” Eur. Phys. J. Appl. Phys. 47, 10701(2009).
[CrossRef]

Chulkova, G.

G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. M. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, “Picosecond superconducting single-photon optical detector,” Appl. Phys. Lett. 79, 705–708 (2001).
[CrossRef]

Dauler, E. A.

de Dood, M. J. A.

E. F. C. Driessen, F. R. Braakman, E. M. Reiger, S. N. Dorenbos, V. Zviller, and M. J. A. de Dood, “Impedance model for the polarization-dependent optical absorption of superconducting single-photon detectors,” Eur. Phys. J. Appl. Phys. 47, 10701(2009).
[CrossRef]

E. F. C. Driessen and M. J. A. de Dood, “The perfect absorber,” Appl. Phys. Lett. 94, 171109 (2009).
[CrossRef]

Dorenbos, S. N.

E. F. C. Driessen, F. R. Braakman, E. M. Reiger, S. N. Dorenbos, V. Zviller, and M. J. A. de Dood, “Impedance model for the polarization-dependent optical absorption of superconducting single-photon detectors,” Eur. Phys. J. Appl. Phys. 47, 10701(2009).
[CrossRef]

Driessen, E. F. C.

E. F. C. Driessen, F. R. Braakman, E. M. Reiger, S. N. Dorenbos, V. Zviller, and M. J. A. de Dood, “Impedance model for the polarization-dependent optical absorption of superconducting single-photon detectors,” Eur. Phys. J. Appl. Phys. 47, 10701(2009).
[CrossRef]

E. F. C. Driessen and M. J. A. de Dood, “The perfect absorber,” Appl. Phys. Lett. 94, 171109 (2009).
[CrossRef]

Dzardanov, A.

G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. M. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, “Picosecond superconducting single-photon optical detector,” Appl. Phys. Lett. 79, 705–708 (2001).
[CrossRef]

Gaylord, T. K.

Gol’tsman, G. N.

K. M. Rosfjord, J. K. W. Yang, E. A. Dauler, A. J. Kerman, V. Anant, B. M. Voronov, G. N. Gol’tsman, and K. K. Berggren, “Nanowire single-photon detector with an integrated optical cavity and anti-reflection coating,” Opt. Express 14, 527–534 (2006).
[CrossRef] [PubMed]

G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. M. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, “Picosecond superconducting single-photon optical detector,” Appl. Phys. Lett. 79, 705–708 (2001).
[CrossRef]

Gramotnev, D. K.

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photon. 4, 83–91 (2010).
[CrossRef]

Holzwarth, C. W.

X. Hu, C. W. Holzwarth, D. Masciarelli, E. A. Dauler, and K. K. Berggren, “Efficiently coupling light to superconducting nanowire single-photon detectors,” IEEE Trans. Appl. Supercond. 19, 336–340 (2009).
[CrossRef]

Hu, X.

X. Hu, E. A. Dauler, R. J. Molnar, and K. K. Berggren, “Superconducting nanowire single-photon detectors integrated with optical nano-antennae,” Opt. Express 19, 17–31(2011).
[CrossRef] [PubMed]

X. Hu, C. W. Holzwarth, D. Masciarelli, E. A. Dauler, and K. K. Berggren, “Efficiently coupling light to superconducting nanowire single-photon detectors,” IEEE Trans. Appl. Supercond. 19, 336–340 (2009).
[CrossRef]

Kerman, A. J.

Kermann, A. J.

Kwok, H. S.

X. J. Yu and H. S. Kwok, “Optical wire-grid polarizers at oblique angles of incidence,” J. Appl. Phys. 93, 4407–4412 (2003).
[CrossRef]

Li, L.

L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993).
[CrossRef]

Lipatov, A.

G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. M. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, “Picosecond superconducting single-photon optical detector,” Appl. Phys. Lett. 79, 705–708 (2001).
[CrossRef]

Masciarelli, D.

X. Hu, C. W. Holzwarth, D. Masciarelli, E. A. Dauler, and K. K. Berggren, “Efficiently coupling light to superconducting nanowire single-photon detectors,” IEEE Trans. Appl. Supercond. 19, 336–340 (2009).
[CrossRef]

Moharam, M. G.

Molnar, R. J.

Okunev, O.

G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. M. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, “Picosecond superconducting single-photon optical detector,” Appl. Phys. Lett. 79, 705–708 (2001).
[CrossRef]

Parrish, M.

Reiger, E. M.

E. F. C. Driessen, F. R. Braakman, E. M. Reiger, S. N. Dorenbos, V. Zviller, and M. J. A. de Dood, “Impedance model for the polarization-dependent optical absorption of superconducting single-photon detectors,” Eur. Phys. J. Appl. Phys. 47, 10701(2009).
[CrossRef]

Rosfjord, K. M.

Semenov, A.

G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. M. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, “Picosecond superconducting single-photon optical detector,” Appl. Phys. Lett. 79, 705–708 (2001).
[CrossRef]

Smirnov, K.

G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. M. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, “Picosecond superconducting single-photon optical detector,” Appl. Phys. Lett. 79, 705–708 (2001).
[CrossRef]

Sobolewski, R.

G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. M. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, “Picosecond superconducting single-photon optical detector,” Appl. Phys. Lett. 79, 705–708 (2001).
[CrossRef]

Voronov, B. M.

K. M. Rosfjord, J. K. W. Yang, E. A. Dauler, A. J. Kerman, V. Anant, B. M. Voronov, G. N. Gol’tsman, and K. K. Berggren, “Nanowire single-photon detector with an integrated optical cavity and anti-reflection coating,” Opt. Express 14, 527–534 (2006).
[CrossRef] [PubMed]

G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. M. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, “Picosecond superconducting single-photon optical detector,” Appl. Phys. Lett. 79, 705–708 (2001).
[CrossRef]

Williams, C.

G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. M. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, “Picosecond superconducting single-photon optical detector,” Appl. Phys. Lett. 79, 705–708 (2001).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1964).

Yang, J. K. W.

Yu, X. J.

X. J. Yu and H. S. Kwok, “Optical wire-grid polarizers at oblique angles of incidence,” J. Appl. Phys. 93, 4407–4412 (2003).
[CrossRef]

Zviller, V.

E. F. C. Driessen, F. R. Braakman, E. M. Reiger, S. N. Dorenbos, V. Zviller, and M. J. A. de Dood, “Impedance model for the polarization-dependent optical absorption of superconducting single-photon detectors,” Eur. Phys. J. Appl. Phys. 47, 10701(2009).
[CrossRef]

Appl. Phys. Lett. (2)

G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. M. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, “Picosecond superconducting single-photon optical detector,” Appl. Phys. Lett. 79, 705–708 (2001).
[CrossRef]

E. F. C. Driessen and M. J. A. de Dood, “The perfect absorber,” Appl. Phys. Lett. 94, 171109 (2009).
[CrossRef]

Eur. Phys. J. Appl. Phys. (1)

E. F. C. Driessen, F. R. Braakman, E. M. Reiger, S. N. Dorenbos, V. Zviller, and M. J. A. de Dood, “Impedance model for the polarization-dependent optical absorption of superconducting single-photon detectors,” Eur. Phys. J. Appl. Phys. 47, 10701(2009).
[CrossRef]

IEEE Trans. Appl. Supercond. (1)

X. Hu, C. W. Holzwarth, D. Masciarelli, E. A. Dauler, and K. K. Berggren, “Efficiently coupling light to superconducting nanowire single-photon detectors,” IEEE Trans. Appl. Supercond. 19, 336–340 (2009).
[CrossRef]

J. Appl. Phys. (1)

X. J. Yu and H. S. Kwok, “Optical wire-grid polarizers at oblique angles of incidence,” J. Appl. Phys. 93, 4407–4412 (2003).
[CrossRef]

J. Mod. Opt. (1)

L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993).
[CrossRef]

J. Opt. Soc. Am. (2)

Nat. Photon. (1)

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photon. 4, 83–91 (2010).
[CrossRef]

Opt. Express (3)

Other (2)

B. Saleh-Teich, Fundamentals of Photonics (Wiley, 2007): The relationship between the intensity and E-field magnitude is: I=|Eo|22η,where the impedance η is defined in terms of free space impedance ηo via η=ηo/ηsapphire, and taking into account that our substrate is a nonmagnetic medium, μ=μo, whereupon η=μo/ε; also, sapphire is essentially non-absorbing at λ=1550 nm; therefore ε=nsapphire, so the amplitude of the E-field was determined as follows: Eo=2Iη=2PA1ηsapphireμoεo.Based on this calculation, the power-flow at the source boundary corresponds to Eo=2.32×106 V/mE-field amplitude. This value for Eo is the reference amplitude necessary to interpret the near-field distribution determined by COMSOL models.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1964).

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Figures (4)

Fig. 1
Fig. 1

(a) Schematic drawing of structures studied: NbN patterns with 200 nm periodicity, 4 nm thickness, and 50% filling factor, covered by 2 nm NbNO x layer, are arrayed below HSQ-filled nano-optical cavities having 279 nm thickness, and covered by continuous gold film with 60 nm thickness. (b) Coordinate system as defined for the device in (a), indicating how the integrated structures are illuminated in conical-mounting by p- polarized λ = 1550 nm light from the bottom (sapphire substrate) side. The φ polar angle is measured relative to the surface normal, while the azimuthal orientation is specified by the γ angle between the plane of light incidence and the long axis of the NbN stripes. (c) Relative orientation of the plane of incidence with respect to P-( γ = 0 ° ) and S-structures ( γ = 90 ° ).

Fig. 2
Fig. 2

Dual-angle-dependent absorptance of the 200 nm period NbN stripes embedded in an optical cavity. The calculations were performed over the γ = [ 0 90 ° ] and φ = [ 0 85 ° ] intervals, with Δ γ = Δ φ = 5 ° resolution.

Fig. 3
Fig. 3

(a–b) Comparison of the optical responses γ = 90 ° as determined by FEM (lines with symbols) and by TMM (lines). (b) Results of FEM calculations in [ 34.5 ° , 35.5 ° ] interval with Δ φ = 0.005 ° resolution are incorporated into the graph prepared in [ 34 ° , 36 ° ] interval with Δ φ = 0.05 ° resolution, and all of these data are incorporated into (a) graphs originating from computation performed with Δ φ = 1 ° resolution in [ 0 ° , 85 ° ] region. The “absorptance—reflectance—transmittance TMM” refers to the optical response of composites made of NbN - NbNO x - HSQ - Au and HSQ-Au film stacks, according to the 50% fill factor in 200 nm periodic cavity-based structure. There is a local and global minimum on the NbN absorptance at 35 ° and 57 ° , respectively. The origins of these minima are detailed in the text.

Fig. 4
Fig. 4

(a) Schematic drawing showing two unit cells in a plane perpendicular to the NbN stripes. The pattern period is 200 nm . (b) E-field distribution in presence of nanocavities and gold reflector taken at different polar angles at the plane shown in (a). The pictures in the upper row are plotted using the same scale in [ V / m ], while the pictures on bottom are scaled to illustrate better small variations in the field.

Tables (1)

Tables Icon

Table 1 Real ( n 1 ) and Imaginary ( n 2 ) Parts of the Refractive Indices of Materials in the Investigated Optical System

Equations (8)

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H x exp { j [ ( k x x ) + ( k y y ) + ( k z z ) ] } , where     H x = H o cos γ ,
H y exp { j [ ( k x x ) + ( k y y ) + ( k z z ) ] } , where     H y = H o sin γ ,
H z exp { j [ ( k x x ) + ( k y y ) + ( k z z ) ] } , with     H z = 0 ;
k x = k o ( sin φ sin γ ) ,
k y = k o ( sin φ cos γ ) ,
k z = k o cos φ .
I = | E o | 2 2 η ,
E o = 2 I η = 2 P A 1 η sapphire μ o ε o .

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