Abstract

We investigate the optimum emitter position within reflecting parabolic antennas whose size is comparable to the emission wavelength. Using finite-element modeling we calculate the dependence of the amount of power directed into a 20° half-angle cone on the emitter’s position and compare with results obtained using geometrical optics. The spatially varying density of states within the wavelength-scale reflector is mapped and its impact discussed. In addition, it is demonstrated that changing the characteristic size of the reflector within the range from 0.5 to 1.5 times the emission wavelength has a strong bearing on the optimum emitter position, a position that does not in general coincide with the parabola’s focus. We calculate that the optimal antenna size and emitter position allow for the maximum directed power to exceed that obtained in the geometrical optics regime.

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References

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  1. C. Spencer, “Headlamp developments with DMC reflectors including homofocal arrangements,” in SAE Technical Paper (1984), paper 840041.
  2. C. A. Balanis, Antenna Theory, 3rd ed. (Wiley, 2005).
  3. M. A. Lieb and A. J. Meixner, “A high numerical aperture parabolic mirror as imaging device for confocal microscopy,” Opt. Express 8, 458–474 (2001).
    [Crossref]
  4. J. Stadler, C. Stanciu, C. Stupperich, and A. J. Meixner, “Tighter focusing with a parabolic mirror,” Opt. Lett. 33, 681–683 (2008).
    [Crossref]
  5. S. Morozov, M. Gaio, S. A. Maier, and R. Sapienza, “Metal-dielectric parabolic antenna for directing single photons,” Nano Lett. 18, 3060–3065 (2018).
    [Crossref]
  6. D. T. Schoen, T. Coenen, F. J. Garc, M. L. Brongersma, and A. Polman, “The planar parabolic optical antenna,” Nano Lett. 13, 188–193 (2013).
    [Crossref]
  7. N. H. Wan, B. J. Shields, D. Kim, S. Mouradian, B. Lienhard, M. Walsh, H. Bakhru, T. Schröder, and D. Englund, “Efficient extraction of light from a nitrogen-vacancy center in a diamond parabolic reflector,” Nano Lett. 18, 2787–2793 (2018).
    [Crossref]
  8. A. W. Schell, T. Neumer, Q. Shi, J. Kaschke, J. Fischer, M. Wegener, and O. Benson, “Laser-written parabolic micro-antennas for efficient photon collection,” Appl. Phys. Lett. 105, 231117 (2014).
    [Crossref]
  9. J. P. Dowling, “Spontaneous emission in cavities: How much more classical can you get?” Found. Phys. 23, 895–905 (1993).
    [Crossref]
  10. E. Snoeks, A. Lagendijk, and A. Polman, “Measuring and modifying the spontaneous emission rate of erbium near an interface,” Phys. Rev. Lett. 74, 2459–2462 (1995).
    [Crossref]
  11. M. Wubs and W. L. Vos, “Förster resonance energy transfer rate in any dielectric nanophotonic medium with weak dispersion,” New J. Phys. 18, 053037 (2016).
    [Crossref]
  12. R. R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978).
    [Crossref]
  13. R. Sprik, B. A. van Tiggelen, and A. Lagendijk, “Optical emission in periodic dielectrics,” Europhys. Lett. 35, 265–270 (1996).
    [Crossref]
  14. COMSOL AB, COMSOL Multiphysics.
  15. J. Straton and I. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
    [Crossref]
  16. COMSOL, The RF module user guide (2012).
  17. D. K. Cheng, Field and Wave Electromagentics, 2nd ed. (Addison-Wesley, 1991).
  18. R. K. Wangness, Electromagnetic Fields, 2nd ed. (Wiley, 1986).
  19. Z. Živkovi, D. Senic, C. Bodendorf, J. Skrzypczynski, and A. Šaroli, “Radiation pattern and impedance of a quarter wavelength monopole antenna above a finite ground plane,” in 20th International Conference on Software, Telecommunications and Computer Networks (SoftCOM) (2012), pp. 1–5.
  20. K. H. Drexhage, “Influence of a dielectric interface on fluorescence decay time,” J. Lumin. 1-2, 693–701 (1970).
    [Crossref]
  21. S. Haroche, “Cavity quantum electrodynamics,” in Fundamental Systems in Quantum Optics (North-Holland, 1992), Chap. 13, pp. 768–940.
  22. W. L. Barnes, “Fluorescence near interfaces: the role of photonic mode density,” J. Mod. Opt. 45, 661–699 (1998).
    [Crossref]
  23. H.-S. Ee, S.-K. Kim, S.-H. Kwon, and H.-G. Park, “Design of polarization-selective light emitters using one-dimensional metal grating mirror,” Opt. Express 19, 1609–1616 (2011).
    [Crossref]

2018 (2)

S. Morozov, M. Gaio, S. A. Maier, and R. Sapienza, “Metal-dielectric parabolic antenna for directing single photons,” Nano Lett. 18, 3060–3065 (2018).
[Crossref]

N. H. Wan, B. J. Shields, D. Kim, S. Mouradian, B. Lienhard, M. Walsh, H. Bakhru, T. Schröder, and D. Englund, “Efficient extraction of light from a nitrogen-vacancy center in a diamond parabolic reflector,” Nano Lett. 18, 2787–2793 (2018).
[Crossref]

2016 (1)

M. Wubs and W. L. Vos, “Förster resonance energy transfer rate in any dielectric nanophotonic medium with weak dispersion,” New J. Phys. 18, 053037 (2016).
[Crossref]

2014 (1)

A. W. Schell, T. Neumer, Q. Shi, J. Kaschke, J. Fischer, M. Wegener, and O. Benson, “Laser-written parabolic micro-antennas for efficient photon collection,” Appl. Phys. Lett. 105, 231117 (2014).
[Crossref]

2013 (1)

D. T. Schoen, T. Coenen, F. J. Garc, M. L. Brongersma, and A. Polman, “The planar parabolic optical antenna,” Nano Lett. 13, 188–193 (2013).
[Crossref]

2011 (1)

2008 (1)

2001 (1)

1998 (1)

W. L. Barnes, “Fluorescence near interfaces: the role of photonic mode density,” J. Mod. Opt. 45, 661–699 (1998).
[Crossref]

1996 (1)

R. Sprik, B. A. van Tiggelen, and A. Lagendijk, “Optical emission in periodic dielectrics,” Europhys. Lett. 35, 265–270 (1996).
[Crossref]

1995 (1)

E. Snoeks, A. Lagendijk, and A. Polman, “Measuring and modifying the spontaneous emission rate of erbium near an interface,” Phys. Rev. Lett. 74, 2459–2462 (1995).
[Crossref]

1993 (1)

J. P. Dowling, “Spontaneous emission in cavities: How much more classical can you get?” Found. Phys. 23, 895–905 (1993).
[Crossref]

1978 (1)

R. R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978).
[Crossref]

1970 (1)

K. H. Drexhage, “Influence of a dielectric interface on fluorescence decay time,” J. Lumin. 1-2, 693–701 (1970).
[Crossref]

1939 (1)

J. Straton and I. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
[Crossref]

Bakhru, H.

N. H. Wan, B. J. Shields, D. Kim, S. Mouradian, B. Lienhard, M. Walsh, H. Bakhru, T. Schröder, and D. Englund, “Efficient extraction of light from a nitrogen-vacancy center in a diamond parabolic reflector,” Nano Lett. 18, 2787–2793 (2018).
[Crossref]

Balanis, C. A.

C. A. Balanis, Antenna Theory, 3rd ed. (Wiley, 2005).

Barnes, W. L.

W. L. Barnes, “Fluorescence near interfaces: the role of photonic mode density,” J. Mod. Opt. 45, 661–699 (1998).
[Crossref]

Benson, O.

A. W. Schell, T. Neumer, Q. Shi, J. Kaschke, J. Fischer, M. Wegener, and O. Benson, “Laser-written parabolic micro-antennas for efficient photon collection,” Appl. Phys. Lett. 105, 231117 (2014).
[Crossref]

Bodendorf, C.

Z. Živkovi, D. Senic, C. Bodendorf, J. Skrzypczynski, and A. Šaroli, “Radiation pattern and impedance of a quarter wavelength monopole antenna above a finite ground plane,” in 20th International Conference on Software, Telecommunications and Computer Networks (SoftCOM) (2012), pp. 1–5.

Brongersma, M. L.

D. T. Schoen, T. Coenen, F. J. Garc, M. L. Brongersma, and A. Polman, “The planar parabolic optical antenna,” Nano Lett. 13, 188–193 (2013).
[Crossref]

Chance, R. R.

R. R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978).
[Crossref]

Cheng, D. K.

D. K. Cheng, Field and Wave Electromagentics, 2nd ed. (Addison-Wesley, 1991).

Chu, I.

J. Straton and I. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
[Crossref]

Coenen, T.

D. T. Schoen, T. Coenen, F. J. Garc, M. L. Brongersma, and A. Polman, “The planar parabolic optical antenna,” Nano Lett. 13, 188–193 (2013).
[Crossref]

Dowling, J. P.

J. P. Dowling, “Spontaneous emission in cavities: How much more classical can you get?” Found. Phys. 23, 895–905 (1993).
[Crossref]

Drexhage, K. H.

K. H. Drexhage, “Influence of a dielectric interface on fluorescence decay time,” J. Lumin. 1-2, 693–701 (1970).
[Crossref]

Ee, H.-S.

Englund, D.

N. H. Wan, B. J. Shields, D. Kim, S. Mouradian, B. Lienhard, M. Walsh, H. Bakhru, T. Schröder, and D. Englund, “Efficient extraction of light from a nitrogen-vacancy center in a diamond parabolic reflector,” Nano Lett. 18, 2787–2793 (2018).
[Crossref]

Fischer, J.

A. W. Schell, T. Neumer, Q. Shi, J. Kaschke, J. Fischer, M. Wegener, and O. Benson, “Laser-written parabolic micro-antennas for efficient photon collection,” Appl. Phys. Lett. 105, 231117 (2014).
[Crossref]

Gaio, M.

S. Morozov, M. Gaio, S. A. Maier, and R. Sapienza, “Metal-dielectric parabolic antenna for directing single photons,” Nano Lett. 18, 3060–3065 (2018).
[Crossref]

Garc, F. J.

D. T. Schoen, T. Coenen, F. J. Garc, M. L. Brongersma, and A. Polman, “The planar parabolic optical antenna,” Nano Lett. 13, 188–193 (2013).
[Crossref]

Haroche, S.

S. Haroche, “Cavity quantum electrodynamics,” in Fundamental Systems in Quantum Optics (North-Holland, 1992), Chap. 13, pp. 768–940.

Kaschke, J.

A. W. Schell, T. Neumer, Q. Shi, J. Kaschke, J. Fischer, M. Wegener, and O. Benson, “Laser-written parabolic micro-antennas for efficient photon collection,” Appl. Phys. Lett. 105, 231117 (2014).
[Crossref]

Kim, D.

N. H. Wan, B. J. Shields, D. Kim, S. Mouradian, B. Lienhard, M. Walsh, H. Bakhru, T. Schröder, and D. Englund, “Efficient extraction of light from a nitrogen-vacancy center in a diamond parabolic reflector,” Nano Lett. 18, 2787–2793 (2018).
[Crossref]

Kim, S.-K.

Kwon, S.-H.

Lagendijk, A.

R. Sprik, B. A. van Tiggelen, and A. Lagendijk, “Optical emission in periodic dielectrics,” Europhys. Lett. 35, 265–270 (1996).
[Crossref]

E. Snoeks, A. Lagendijk, and A. Polman, “Measuring and modifying the spontaneous emission rate of erbium near an interface,” Phys. Rev. Lett. 74, 2459–2462 (1995).
[Crossref]

Lieb, M. A.

Lienhard, B.

N. H. Wan, B. J. Shields, D. Kim, S. Mouradian, B. Lienhard, M. Walsh, H. Bakhru, T. Schröder, and D. Englund, “Efficient extraction of light from a nitrogen-vacancy center in a diamond parabolic reflector,” Nano Lett. 18, 2787–2793 (2018).
[Crossref]

Maier, S. A.

S. Morozov, M. Gaio, S. A. Maier, and R. Sapienza, “Metal-dielectric parabolic antenna for directing single photons,” Nano Lett. 18, 3060–3065 (2018).
[Crossref]

Meixner, A. J.

Morozov, S.

S. Morozov, M. Gaio, S. A. Maier, and R. Sapienza, “Metal-dielectric parabolic antenna for directing single photons,” Nano Lett. 18, 3060–3065 (2018).
[Crossref]

Mouradian, S.

N. H. Wan, B. J. Shields, D. Kim, S. Mouradian, B. Lienhard, M. Walsh, H. Bakhru, T. Schröder, and D. Englund, “Efficient extraction of light from a nitrogen-vacancy center in a diamond parabolic reflector,” Nano Lett. 18, 2787–2793 (2018).
[Crossref]

Neumer, T.

A. W. Schell, T. Neumer, Q. Shi, J. Kaschke, J. Fischer, M. Wegener, and O. Benson, “Laser-written parabolic micro-antennas for efficient photon collection,” Appl. Phys. Lett. 105, 231117 (2014).
[Crossref]

Park, H.-G.

Polman, A.

D. T. Schoen, T. Coenen, F. J. Garc, M. L. Brongersma, and A. Polman, “The planar parabolic optical antenna,” Nano Lett. 13, 188–193 (2013).
[Crossref]

E. Snoeks, A. Lagendijk, and A. Polman, “Measuring and modifying the spontaneous emission rate of erbium near an interface,” Phys. Rev. Lett. 74, 2459–2462 (1995).
[Crossref]

Prock, A.

R. R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978).
[Crossref]

Sapienza, R.

S. Morozov, M. Gaio, S. A. Maier, and R. Sapienza, “Metal-dielectric parabolic antenna for directing single photons,” Nano Lett. 18, 3060–3065 (2018).
[Crossref]

Šaroli, A.

Z. Živkovi, D. Senic, C. Bodendorf, J. Skrzypczynski, and A. Šaroli, “Radiation pattern and impedance of a quarter wavelength monopole antenna above a finite ground plane,” in 20th International Conference on Software, Telecommunications and Computer Networks (SoftCOM) (2012), pp. 1–5.

Schell, A. W.

A. W. Schell, T. Neumer, Q. Shi, J. Kaschke, J. Fischer, M. Wegener, and O. Benson, “Laser-written parabolic micro-antennas for efficient photon collection,” Appl. Phys. Lett. 105, 231117 (2014).
[Crossref]

Schoen, D. T.

D. T. Schoen, T. Coenen, F. J. Garc, M. L. Brongersma, and A. Polman, “The planar parabolic optical antenna,” Nano Lett. 13, 188–193 (2013).
[Crossref]

Schröder, T.

N. H. Wan, B. J. Shields, D. Kim, S. Mouradian, B. Lienhard, M. Walsh, H. Bakhru, T. Schröder, and D. Englund, “Efficient extraction of light from a nitrogen-vacancy center in a diamond parabolic reflector,” Nano Lett. 18, 2787–2793 (2018).
[Crossref]

Senic, D.

Z. Živkovi, D. Senic, C. Bodendorf, J. Skrzypczynski, and A. Šaroli, “Radiation pattern and impedance of a quarter wavelength monopole antenna above a finite ground plane,” in 20th International Conference on Software, Telecommunications and Computer Networks (SoftCOM) (2012), pp. 1–5.

Shi, Q.

A. W. Schell, T. Neumer, Q. Shi, J. Kaschke, J. Fischer, M. Wegener, and O. Benson, “Laser-written parabolic micro-antennas for efficient photon collection,” Appl. Phys. Lett. 105, 231117 (2014).
[Crossref]

Shields, B. J.

N. H. Wan, B. J. Shields, D. Kim, S. Mouradian, B. Lienhard, M. Walsh, H. Bakhru, T. Schröder, and D. Englund, “Efficient extraction of light from a nitrogen-vacancy center in a diamond parabolic reflector,” Nano Lett. 18, 2787–2793 (2018).
[Crossref]

Silbey, R.

R. R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978).
[Crossref]

Skrzypczynski, J.

Z. Živkovi, D. Senic, C. Bodendorf, J. Skrzypczynski, and A. Šaroli, “Radiation pattern and impedance of a quarter wavelength monopole antenna above a finite ground plane,” in 20th International Conference on Software, Telecommunications and Computer Networks (SoftCOM) (2012), pp. 1–5.

Snoeks, E.

E. Snoeks, A. Lagendijk, and A. Polman, “Measuring and modifying the spontaneous emission rate of erbium near an interface,” Phys. Rev. Lett. 74, 2459–2462 (1995).
[Crossref]

Spencer, C.

C. Spencer, “Headlamp developments with DMC reflectors including homofocal arrangements,” in SAE Technical Paper (1984), paper 840041.

Sprik, R.

R. Sprik, B. A. van Tiggelen, and A. Lagendijk, “Optical emission in periodic dielectrics,” Europhys. Lett. 35, 265–270 (1996).
[Crossref]

Stadler, J.

Stanciu, C.

Straton, J.

J. Straton and I. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
[Crossref]

Stupperich, C.

van Tiggelen, B. A.

R. Sprik, B. A. van Tiggelen, and A. Lagendijk, “Optical emission in periodic dielectrics,” Europhys. Lett. 35, 265–270 (1996).
[Crossref]

Vos, W. L.

M. Wubs and W. L. Vos, “Förster resonance energy transfer rate in any dielectric nanophotonic medium with weak dispersion,” New J. Phys. 18, 053037 (2016).
[Crossref]

Walsh, M.

N. H. Wan, B. J. Shields, D. Kim, S. Mouradian, B. Lienhard, M. Walsh, H. Bakhru, T. Schröder, and D. Englund, “Efficient extraction of light from a nitrogen-vacancy center in a diamond parabolic reflector,” Nano Lett. 18, 2787–2793 (2018).
[Crossref]

Wan, N. H.

N. H. Wan, B. J. Shields, D. Kim, S. Mouradian, B. Lienhard, M. Walsh, H. Bakhru, T. Schröder, and D. Englund, “Efficient extraction of light from a nitrogen-vacancy center in a diamond parabolic reflector,” Nano Lett. 18, 2787–2793 (2018).
[Crossref]

Wangness, R. K.

R. K. Wangness, Electromagnetic Fields, 2nd ed. (Wiley, 1986).

Wegener, M.

A. W. Schell, T. Neumer, Q. Shi, J. Kaschke, J. Fischer, M. Wegener, and O. Benson, “Laser-written parabolic micro-antennas for efficient photon collection,” Appl. Phys. Lett. 105, 231117 (2014).
[Crossref]

Wubs, M.

M. Wubs and W. L. Vos, “Förster resonance energy transfer rate in any dielectric nanophotonic medium with weak dispersion,” New J. Phys. 18, 053037 (2016).
[Crossref]

Živkovi, Z.

Z. Živkovi, D. Senic, C. Bodendorf, J. Skrzypczynski, and A. Šaroli, “Radiation pattern and impedance of a quarter wavelength monopole antenna above a finite ground plane,” in 20th International Conference on Software, Telecommunications and Computer Networks (SoftCOM) (2012), pp. 1–5.

Adv. Chem. Phys. (1)

R. R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978).
[Crossref]

Appl. Phys. Lett. (1)

A. W. Schell, T. Neumer, Q. Shi, J. Kaschke, J. Fischer, M. Wegener, and O. Benson, “Laser-written parabolic micro-antennas for efficient photon collection,” Appl. Phys. Lett. 105, 231117 (2014).
[Crossref]

Europhys. Lett. (1)

R. Sprik, B. A. van Tiggelen, and A. Lagendijk, “Optical emission in periodic dielectrics,” Europhys. Lett. 35, 265–270 (1996).
[Crossref]

Found. Phys. (1)

J. P. Dowling, “Spontaneous emission in cavities: How much more classical can you get?” Found. Phys. 23, 895–905 (1993).
[Crossref]

J. Lumin. (1)

K. H. Drexhage, “Influence of a dielectric interface on fluorescence decay time,” J. Lumin. 1-2, 693–701 (1970).
[Crossref]

J. Mod. Opt. (1)

W. L. Barnes, “Fluorescence near interfaces: the role of photonic mode density,” J. Mod. Opt. 45, 661–699 (1998).
[Crossref]

Nano Lett. (3)

S. Morozov, M. Gaio, S. A. Maier, and R. Sapienza, “Metal-dielectric parabolic antenna for directing single photons,” Nano Lett. 18, 3060–3065 (2018).
[Crossref]

D. T. Schoen, T. Coenen, F. J. Garc, M. L. Brongersma, and A. Polman, “The planar parabolic optical antenna,” Nano Lett. 13, 188–193 (2013).
[Crossref]

N. H. Wan, B. J. Shields, D. Kim, S. Mouradian, B. Lienhard, M. Walsh, H. Bakhru, T. Schröder, and D. Englund, “Efficient extraction of light from a nitrogen-vacancy center in a diamond parabolic reflector,” Nano Lett. 18, 2787–2793 (2018).
[Crossref]

New J. Phys. (1)

M. Wubs and W. L. Vos, “Förster resonance energy transfer rate in any dielectric nanophotonic medium with weak dispersion,” New J. Phys. 18, 053037 (2016).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. (1)

J. Straton and I. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
[Crossref]

Phys. Rev. Lett. (1)

E. Snoeks, A. Lagendijk, and A. Polman, “Measuring and modifying the spontaneous emission rate of erbium near an interface,” Phys. Rev. Lett. 74, 2459–2462 (1995).
[Crossref]

Other (8)

COMSOL, The RF module user guide (2012).

D. K. Cheng, Field and Wave Electromagentics, 2nd ed. (Addison-Wesley, 1991).

R. K. Wangness, Electromagnetic Fields, 2nd ed. (Wiley, 1986).

Z. Živkovi, D. Senic, C. Bodendorf, J. Skrzypczynski, and A. Šaroli, “Radiation pattern and impedance of a quarter wavelength monopole antenna above a finite ground plane,” in 20th International Conference on Software, Telecommunications and Computer Networks (SoftCOM) (2012), pp. 1–5.

C. Spencer, “Headlamp developments with DMC reflectors including homofocal arrangements,” in SAE Technical Paper (1984), paper 840041.

C. A. Balanis, Antenna Theory, 3rd ed. (Wiley, 2005).

S. Haroche, “Cavity quantum electrodynamics,” in Fundamental Systems in Quantum Optics (North-Holland, 1992), Chap. 13, pp. 768–940.

COMSOL AB, COMSOL Multiphysics.

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Figures (5)

Fig. 1.
Fig. 1. A section of an infinite parabola (red) on Cartesian axes, defined by being equidistant from both the focus (red dot) and the directrix (blue, dashed) at all points. The distance from vertex to the focal point is the focal length f. Emission from the focal point (green lines) is collimated by specular refection from a parabolic surface.
Fig. 2.
Fig. 2. (a) Contour plot of the calculated electric field amplitude in the y=0 plane, surrounding an oscillating electric point dipole within a nanoscale parabolic reflector. The emission wavelength λ=600nm. The dipole is located at the parabola’s focus (on axis at z=0.5λ) and is oscillating in the x^ direction as indicated by the red arrow. The amplitude of the field has been normalized to unity. (b) Corresponding far-field contour plot, in wave-vector (kx, ky) space. The amplitude of the far field has been normalized to unity. The white circles mark emission inclination angles as labeled, with the dashed circle corresponding to the 20° half-angle cone used to measure the performance of the reflector.
Fig. 3.
Fig. 3. Geometrical optics solution for power directed into a 20° half-angle cone for an isotropic emitter at varying positions within a finite parabolic reflector. As the solution is independent of the size of the reflector in the geometrical optics regime, the dimensions are given in terms of the vertex to opening distance y0. Power values are normalized to the total emitted power. The red dot marks the focus of the paraboloid.
Fig. 4.
Fig. 4. Directed power for dipole emitters as a function of their position within a wavelength-scale parabolic reflector (top row), corresponding total radiated power (middle row), and fraction of total radiated power within the target cone (bottom row). The emission wavelength λ=600nm. The dipole orientation is given by the red arrows, where Av. denotes the geometric mean of the three orthogonal dipole orientations (where the dipole oscillation in the y case is obtained from the oscillation in the x solution shown). In (a)–(c) the power directed into a 20° half-angle cone is normalized to the total power emitted in the absence of any reflector, as is the total radiated power (d)–(f). Results for dipoles orientated in the z^ direction and the geometric mean possess rotational symmetry about the z axis, while perpendicular orientations have symmetry about the x=0 and y=0 planes. Red dots indicate the geometrical focal point of the paraboloids.
Fig. 5.
Fig. 5. (a) Variation in the power directed into a 20° half-angle cone with emitter position along the reflector’s central axis and with emission wavelength. The emission wavelength is given in units of the design wavelength λ0=600nm, which determines the (fixed) size of the reflector. The size and shape of the reflector remains as in Fig. 4, with a vertex to opening distance of λ0=600nm. Directed power values are normalized to the total power radiated by an equivalent emitter (at the specified emission wavelength) in free space. The black squares indicate values obtained by simulation, and the connecting black lines are visual guides. The dashed red lines indicate the position of the geometrical focus. The “λ=0” values represent the solution in the small wavelength limit in which the geometrical optics approximation may accurately be applied, and the solution does not depend on wavelength. (b) As in (a), but showing the total radiated power.

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