Abstract

The quest for less costly and more compact high-energy particle accelerators makes research on alternative acceleration mechanisms an important enterprise. From the multitude of suggested concepts, the photonic accelerator design by B. M. Cowan [Phys. Rev. ST Accel. Beams 11, 011301 (2008)] stands out by its distinct potential of creating an accelerator on a chip [Proposal E-163, SLAC (2001)]. Herein, electrons are accelerated by the axial electric field of a strongly confined optical mode of an air waveguide within a silicon-based three-dimensional photonic band-gap material. Using a combination of direct laser writing and silicon double inversion, we here present the first experimental realization of this complex structure. Optical spectroscopy provides unambiguous evidence for the existence of an accelerating waveguide mode with axial polarization.

© 2012 OSA

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  1. S. John, “Strong Localization of Photons in Certain Disordered Dielectric Superlattices,” Phys. Rev. Lett. 58(23), 2486–2489 (1987).
    [CrossRef] [PubMed]
  2. E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. 58(20), 2059–2062 (1987).
    [CrossRef] [PubMed]
  3. Z.-Y. Li and K. M. Ho, “Waveguides in three-dimensional layer-by-layer photonic crystals,” J. Opt. Soc. Am. B 20(5), 801–809 (2003).
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    [CrossRef]
  5. R. H. Siemann, “Energy efficiency of laser driven, structure based accelerators,” Phys. Rev. ST Accel. Beams  7(6), 061303 (2004).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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2011

2010

2009

C. McGuinness, E. R. Colby, and R. L. Byer, “Accelerating electrons with lasers and photonic crystals,” J. Mod. Opt. 56(18-19), 2142–2147 (2009).
[CrossRef]

2008

S. Rinne, F. García-Santamaría, and P. V. Braun, “Embedded cavities and waveguides in three-dimensional silicon photonic crystals,” Nat. Photonics 2(1), 52–56 (2008).
[CrossRef]

B. M. Cowan, “Three-dimensional dielectric photonic crystal structures for laser-driven acceleration,” Phys. Rev. ST Accel. Beams  11(1), 011301 (2008).
[CrossRef]

C. Sears, E. Colby, R. Ischebeck, C. McGuinness, J. Nelson, R. Noble, R. Siemann, J. Spencer, D. Walz, T. Plettner, and R. Byer, “Production and characterization of attosecond electron bunch trains,” Phys. Rev. ST Accel. Beams  11(6), 061301 (2008).
[CrossRef]

C. M. S. Sears, E. Colby, R. J. England, R. Ischebeck, C. McGuinness, J. Nelson, R. Noble, R. H. Siemann, J. Spencer, D. Walz, T. Plettner, and R. L. Byer, “Phase stable net acceleration of electrons from a two-stage optical accelerator,” Phys. Rev. ST Accel. Beams  11(10), 101301 (2008).
[CrossRef]

2006

N. Tétreault, G. von Freymann, M. Deubel, M. Hermatschweiler, F. Pérez-Willard, S. John, M. Wegener, and G. A. Ozin, “New Route to Three-Dimensional Photonic Bandgap Materials: Silicon Double Inversion of Polymer Templates,” Adv. Mater. 18(4), 457–460 (2006).
[CrossRef]

2005

M. Mero, J. Liu, W. Rudolph, D. Ristau, and K. Starke, “Scaling laws of femtosecond laser pulse induced breakdown in oxide films,” Phys. Rev. B 71(11), 115109 (2005).
[CrossRef]

2004

M. Deubel, G. von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, “Direct laser writing of three-dimensional photonic-crystal templates for telecommunications,” Nat. Mater. 3(7), 444–447 (2004).
[CrossRef] [PubMed]

R. H. Siemann, “Energy efficiency of laser driven, structure based accelerators,” Phys. Rev. ST Accel. Beams  7(6), 061303 (2004).
[CrossRef]

2003

2001

1994

V. Lujala, J. Skarp, M. Tammenmaa, and T. Suntola, “Atomic layer epitaxy growth of doped zinc oxide thin films from organometals,” Appl. Surf. Sci. 82–83, 34–40 (1994).
[CrossRef]

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: New layer-by-layer periodic structures,” Solid State Commun. 89(5), 413–416 (1994).
[CrossRef]

1987

S. John, “Strong Localization of Photons in Certain Disordered Dielectric Superlattices,” Phys. Rev. Lett. 58(23), 2486–2489 (1987).
[CrossRef] [PubMed]

E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. 58(20), 2059–2062 (1987).
[CrossRef] [PubMed]

Biswas, R.

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: New layer-by-layer periodic structures,” Solid State Commun. 89(5), 413–416 (1994).
[CrossRef]

Braun, P. V.

S. Rinne, F. García-Santamaría, and P. V. Braun, “Embedded cavities and waveguides in three-dimensional silicon photonic crystals,” Nat. Photonics 2(1), 52–56 (2008).
[CrossRef]

Busch, K.

Byer, R.

C. Sears, E. Colby, R. Ischebeck, C. McGuinness, J. Nelson, R. Noble, R. Siemann, J. Spencer, D. Walz, T. Plettner, and R. Byer, “Production and characterization of attosecond electron bunch trains,” Phys. Rev. ST Accel. Beams  11(6), 061301 (2008).
[CrossRef]

Byer, R. L.

C. McGuinness, E. R. Colby, and R. L. Byer, “Accelerating electrons with lasers and photonic crystals,” J. Mod. Opt. 56(18-19), 2142–2147 (2009).
[CrossRef]

C. M. S. Sears, E. Colby, R. J. England, R. Ischebeck, C. McGuinness, J. Nelson, R. Noble, R. H. Siemann, J. Spencer, D. Walz, T. Plettner, and R. L. Byer, “Phase stable net acceleration of electrons from a two-stage optical accelerator,” Phys. Rev. ST Accel. Beams  11(10), 101301 (2008).
[CrossRef]

Chan, C. T.

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: New layer-by-layer periodic structures,” Solid State Commun. 89(5), 413–416 (1994).
[CrossRef]

Colby, E.

C. M. S. Sears, E. Colby, R. J. England, R. Ischebeck, C. McGuinness, J. Nelson, R. Noble, R. H. Siemann, J. Spencer, D. Walz, T. Plettner, and R. L. Byer, “Phase stable net acceleration of electrons from a two-stage optical accelerator,” Phys. Rev. ST Accel. Beams  11(10), 101301 (2008).
[CrossRef]

C. Sears, E. Colby, R. Ischebeck, C. McGuinness, J. Nelson, R. Noble, R. Siemann, J. Spencer, D. Walz, T. Plettner, and R. Byer, “Production and characterization of attosecond electron bunch trains,” Phys. Rev. ST Accel. Beams  11(6), 061301 (2008).
[CrossRef]

Colby, E. R.

C. McGuinness, E. R. Colby, and R. L. Byer, “Accelerating electrons with lasers and photonic crystals,” J. Mod. Opt. 56(18-19), 2142–2147 (2009).
[CrossRef]

Cowan, B. M.

B. M. Cowan, “Three-dimensional dielectric photonic crystal structures for laser-driven acceleration,” Phys. Rev. ST Accel. Beams  11(1), 011301 (2008).
[CrossRef]

Deubel, M.

N. Tétreault, G. von Freymann, M. Deubel, M. Hermatschweiler, F. Pérez-Willard, S. John, M. Wegener, and G. A. Ozin, “New Route to Three-Dimensional Photonic Bandgap Materials: Silicon Double Inversion of Polymer Templates,” Adv. Mater. 18(4), 457–460 (2006).
[CrossRef]

M. Deubel, G. von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, “Direct laser writing of three-dimensional photonic-crystal templates for telecommunications,” Nat. Mater. 3(7), 444–447 (2004).
[CrossRef] [PubMed]

England, R. J.

C. M. S. Sears, E. Colby, R. J. England, R. Ischebeck, C. McGuinness, J. Nelson, R. Noble, R. H. Siemann, J. Spencer, D. Walz, T. Plettner, and R. L. Byer, “Phase stable net acceleration of electrons from a two-stage optical accelerator,” Phys. Rev. ST Accel. Beams  11(10), 101301 (2008).
[CrossRef]

Essig, S.

Frölich, A.

García-Santamaría, F.

S. Rinne, F. García-Santamaría, and P. V. Braun, “Embedded cavities and waveguides in three-dimensional silicon photonic crystals,” Nat. Photonics 2(1), 52–56 (2008).
[CrossRef]

Hermatschweiler, M.

N. Tétreault, G. von Freymann, M. Deubel, M. Hermatschweiler, F. Pérez-Willard, S. John, M. Wegener, and G. A. Ozin, “New Route to Three-Dimensional Photonic Bandgap Materials: Silicon Double Inversion of Polymer Templates,” Adv. Mater. 18(4), 457–460 (2006).
[CrossRef]

Ho, K. M.

Z.-Y. Li and K. M. Ho, “Waveguides in three-dimensional layer-by-layer photonic crystals,” J. Opt. Soc. Am. B 20(5), 801–809 (2003).
[CrossRef]

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: New layer-by-layer periodic structures,” Solid State Commun. 89(5), 413–416 (1994).
[CrossRef]

Ischebeck, R.

C. M. S. Sears, E. Colby, R. J. England, R. Ischebeck, C. McGuinness, J. Nelson, R. Noble, R. H. Siemann, J. Spencer, D. Walz, T. Plettner, and R. L. Byer, “Phase stable net acceleration of electrons from a two-stage optical accelerator,” Phys. Rev. ST Accel. Beams  11(10), 101301 (2008).
[CrossRef]

C. Sears, E. Colby, R. Ischebeck, C. McGuinness, J. Nelson, R. Noble, R. Siemann, J. Spencer, D. Walz, T. Plettner, and R. Byer, “Production and characterization of attosecond electron bunch trains,” Phys. Rev. ST Accel. Beams  11(6), 061301 (2008).
[CrossRef]

Ishizaki, K.

Joannopoulos, J. D.

John, S.

N. Tétreault, G. von Freymann, M. Deubel, M. Hermatschweiler, F. Pérez-Willard, S. John, M. Wegener, and G. A. Ozin, “New Route to Three-Dimensional Photonic Bandgap Materials: Silicon Double Inversion of Polymer Templates,” Adv. Mater. 18(4), 457–460 (2006).
[CrossRef]

S. John, “Strong Localization of Photons in Certain Disordered Dielectric Superlattices,” Phys. Rev. Lett. 58(23), 2486–2489 (1987).
[CrossRef] [PubMed]

Johnson, S. G.

Kawashima, S.

Li, Z.-Y.

Lin, X. E.

X. E. Lin, “Photonic band gap fiber accelerator,” Phys. Rev. ST Accel. Beams  4(5), 051301 (2001).
[CrossRef]

Liu, J.

M. Mero, J. Liu, W. Rudolph, D. Ristau, and K. Starke, “Scaling laws of femtosecond laser pulse induced breakdown in oxide films,” Phys. Rev. B 71(11), 115109 (2005).
[CrossRef]

Lujala, V.

V. Lujala, J. Skarp, M. Tammenmaa, and T. Suntola, “Atomic layer epitaxy growth of doped zinc oxide thin films from organometals,” Appl. Surf. Sci. 82–83, 34–40 (1994).
[CrossRef]

McGuinness, C.

C. McGuinness, E. R. Colby, and R. L. Byer, “Accelerating electrons with lasers and photonic crystals,” J. Mod. Opt. 56(18-19), 2142–2147 (2009).
[CrossRef]

C. M. S. Sears, E. Colby, R. J. England, R. Ischebeck, C. McGuinness, J. Nelson, R. Noble, R. H. Siemann, J. Spencer, D. Walz, T. Plettner, and R. L. Byer, “Phase stable net acceleration of electrons from a two-stage optical accelerator,” Phys. Rev. ST Accel. Beams  11(10), 101301 (2008).
[CrossRef]

C. Sears, E. Colby, R. Ischebeck, C. McGuinness, J. Nelson, R. Noble, R. Siemann, J. Spencer, D. Walz, T. Plettner, and R. Byer, “Production and characterization of attosecond electron bunch trains,” Phys. Rev. ST Accel. Beams  11(6), 061301 (2008).
[CrossRef]

Mero, M.

M. Mero, J. Liu, W. Rudolph, D. Ristau, and K. Starke, “Scaling laws of femtosecond laser pulse induced breakdown in oxide films,” Phys. Rev. B 71(11), 115109 (2005).
[CrossRef]

Nelson, J.

C. M. S. Sears, E. Colby, R. J. England, R. Ischebeck, C. McGuinness, J. Nelson, R. Noble, R. H. Siemann, J. Spencer, D. Walz, T. Plettner, and R. L. Byer, “Phase stable net acceleration of electrons from a two-stage optical accelerator,” Phys. Rev. ST Accel. Beams  11(10), 101301 (2008).
[CrossRef]

C. Sears, E. Colby, R. Ischebeck, C. McGuinness, J. Nelson, R. Noble, R. Siemann, J. Spencer, D. Walz, T. Plettner, and R. Byer, “Production and characterization of attosecond electron bunch trains,” Phys. Rev. ST Accel. Beams  11(6), 061301 (2008).
[CrossRef]

Noble, R.

C. Sears, E. Colby, R. Ischebeck, C. McGuinness, J. Nelson, R. Noble, R. Siemann, J. Spencer, D. Walz, T. Plettner, and R. Byer, “Production and characterization of attosecond electron bunch trains,” Phys. Rev. ST Accel. Beams  11(6), 061301 (2008).
[CrossRef]

C. M. S. Sears, E. Colby, R. J. England, R. Ischebeck, C. McGuinness, J. Nelson, R. Noble, R. H. Siemann, J. Spencer, D. Walz, T. Plettner, and R. L. Byer, “Phase stable net acceleration of electrons from a two-stage optical accelerator,” Phys. Rev. ST Accel. Beams  11(10), 101301 (2008).
[CrossRef]

Noda, S.

Ozin, G. A.

N. Tétreault, G. von Freymann, M. Deubel, M. Hermatschweiler, F. Pérez-Willard, S. John, M. Wegener, and G. A. Ozin, “New Route to Three-Dimensional Photonic Bandgap Materials: Silicon Double Inversion of Polymer Templates,” Adv. Mater. 18(4), 457–460 (2006).
[CrossRef]

Pereira, S.

M. Deubel, G. von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, “Direct laser writing of three-dimensional photonic-crystal templates for telecommunications,” Nat. Mater. 3(7), 444–447 (2004).
[CrossRef] [PubMed]

Pérez-Willard, F.

N. Tétreault, G. von Freymann, M. Deubel, M. Hermatschweiler, F. Pérez-Willard, S. John, M. Wegener, and G. A. Ozin, “New Route to Three-Dimensional Photonic Bandgap Materials: Silicon Double Inversion of Polymer Templates,” Adv. Mater. 18(4), 457–460 (2006).
[CrossRef]

Plettner, T.

C. M. S. Sears, E. Colby, R. J. England, R. Ischebeck, C. McGuinness, J. Nelson, R. Noble, R. H. Siemann, J. Spencer, D. Walz, T. Plettner, and R. L. Byer, “Phase stable net acceleration of electrons from a two-stage optical accelerator,” Phys. Rev. ST Accel. Beams  11(10), 101301 (2008).
[CrossRef]

C. Sears, E. Colby, R. Ischebeck, C. McGuinness, J. Nelson, R. Noble, R. Siemann, J. Spencer, D. Walz, T. Plettner, and R. Byer, “Production and characterization of attosecond electron bunch trains,” Phys. Rev. ST Accel. Beams  11(6), 061301 (2008).
[CrossRef]

Rinne, S.

S. Rinne, F. García-Santamaría, and P. V. Braun, “Embedded cavities and waveguides in three-dimensional silicon photonic crystals,” Nat. Photonics 2(1), 52–56 (2008).
[CrossRef]

Ristau, D.

M. Mero, J. Liu, W. Rudolph, D. Ristau, and K. Starke, “Scaling laws of femtosecond laser pulse induced breakdown in oxide films,” Phys. Rev. B 71(11), 115109 (2005).
[CrossRef]

Rudolph, W.

M. Mero, J. Liu, W. Rudolph, D. Ristau, and K. Starke, “Scaling laws of femtosecond laser pulse induced breakdown in oxide films,” Phys. Rev. B 71(11), 115109 (2005).
[CrossRef]

Sears, C.

C. Sears, E. Colby, R. Ischebeck, C. McGuinness, J. Nelson, R. Noble, R. Siemann, J. Spencer, D. Walz, T. Plettner, and R. Byer, “Production and characterization of attosecond electron bunch trains,” Phys. Rev. ST Accel. Beams  11(6), 061301 (2008).
[CrossRef]

Sears, C. M. S.

C. M. S. Sears, E. Colby, R. J. England, R. Ischebeck, C. McGuinness, J. Nelson, R. Noble, R. H. Siemann, J. Spencer, D. Walz, T. Plettner, and R. L. Byer, “Phase stable net acceleration of electrons from a two-stage optical accelerator,” Phys. Rev. ST Accel. Beams  11(10), 101301 (2008).
[CrossRef]

Siemann, R.

C. Sears, E. Colby, R. Ischebeck, C. McGuinness, J. Nelson, R. Noble, R. Siemann, J. Spencer, D. Walz, T. Plettner, and R. Byer, “Production and characterization of attosecond electron bunch trains,” Phys. Rev. ST Accel. Beams  11(6), 061301 (2008).
[CrossRef]

Siemann, R. H.

C. M. S. Sears, E. Colby, R. J. England, R. Ischebeck, C. McGuinness, J. Nelson, R. Noble, R. H. Siemann, J. Spencer, D. Walz, T. Plettner, and R. L. Byer, “Phase stable net acceleration of electrons from a two-stage optical accelerator,” Phys. Rev. ST Accel. Beams  11(10), 101301 (2008).
[CrossRef]

R. H. Siemann, “Energy efficiency of laser driven, structure based accelerators,” Phys. Rev. ST Accel. Beams  7(6), 061303 (2004).
[CrossRef]

Sigalas, M.

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: New layer-by-layer periodic structures,” Solid State Commun. 89(5), 413–416 (1994).
[CrossRef]

Skarp, J.

V. Lujala, J. Skarp, M. Tammenmaa, and T. Suntola, “Atomic layer epitaxy growth of doped zinc oxide thin films from organometals,” Appl. Surf. Sci. 82–83, 34–40 (1994).
[CrossRef]

Soukoulis, C. M.

M. Deubel, G. von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, “Direct laser writing of three-dimensional photonic-crystal templates for telecommunications,” Nat. Mater. 3(7), 444–447 (2004).
[CrossRef] [PubMed]

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: New layer-by-layer periodic structures,” Solid State Commun. 89(5), 413–416 (1994).
[CrossRef]

Spencer, J.

C. M. S. Sears, E. Colby, R. J. England, R. Ischebeck, C. McGuinness, J. Nelson, R. Noble, R. H. Siemann, J. Spencer, D. Walz, T. Plettner, and R. L. Byer, “Phase stable net acceleration of electrons from a two-stage optical accelerator,” Phys. Rev. ST Accel. Beams  11(10), 101301 (2008).
[CrossRef]

C. Sears, E. Colby, R. Ischebeck, C. McGuinness, J. Nelson, R. Noble, R. Siemann, J. Spencer, D. Walz, T. Plettner, and R. Byer, “Production and characterization of attosecond electron bunch trains,” Phys. Rev. ST Accel. Beams  11(6), 061301 (2008).
[CrossRef]

Starke, K.

M. Mero, J. Liu, W. Rudolph, D. Ristau, and K. Starke, “Scaling laws of femtosecond laser pulse induced breakdown in oxide films,” Phys. Rev. B 71(11), 115109 (2005).
[CrossRef]

Staude, I.

Suntola, T.

V. Lujala, J. Skarp, M. Tammenmaa, and T. Suntola, “Atomic layer epitaxy growth of doped zinc oxide thin films from organometals,” Appl. Surf. Sci. 82–83, 34–40 (1994).
[CrossRef]

Tammenmaa, M.

V. Lujala, J. Skarp, M. Tammenmaa, and T. Suntola, “Atomic layer epitaxy growth of doped zinc oxide thin films from organometals,” Appl. Surf. Sci. 82–83, 34–40 (1994).
[CrossRef]

Tétreault, N.

N. Tétreault, G. von Freymann, M. Deubel, M. Hermatschweiler, F. Pérez-Willard, S. John, M. Wegener, and G. A. Ozin, “New Route to Three-Dimensional Photonic Bandgap Materials: Silicon Double Inversion of Polymer Templates,” Adv. Mater. 18(4), 457–460 (2006).
[CrossRef]

Thiel, M.

von Freymann, G.

I. Staude, G. von Freymann, S. Essig, K. Busch, and M. Wegener, “Waveguides in three-dimensional photonic-band-gap materials by direct laser writing and silicon double inversion,” Opt. Lett. 36(1), 67–69 (2011).
[CrossRef] [PubMed]

I. Staude, M. Thiel, S. Essig, C. Wolff, K. Busch, G. von Freymann, and M. Wegener, “Fabrication and characterization of silicon woodpile photonic crystals with a complete bandgap at telecom wavelengths,” Opt. Lett. 35(7), 1094–1096 (2010).
[CrossRef] [PubMed]

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Figures (3)

Fig. 1
Fig. 1

Accelerating mechanism. (a) Design sketch of the waveguide geometry inside the woodpile based silicon structure [6]. (b) z-component of the electric field for the accelerating mode as seen by a speed-of-light particle. (c) Schematic of a train of electron bunches situated at the peaks of the accelerating phase. (d) Snap shots of the accelerating field and the particle bunch train at two different time steps (not to scale), if short laser pulses with vg < ve ~c are used to excite the accelerating mode. (e) Concept of a staged laser accelerator. Each cavity is powered by an external laser coupled into the cavity. A single laser pulse is split several times to power a series of cavities. For a 1 ps long laser pulse, and a group velocity of 0.26 × c, each cavity length is 105 µm. Image: Courtesy of Greg Stewart, SLAC.

Fig. 2
Fig. 2

Pilot sample of a photonic accelerator segment. (a) Top-view electron micrograph of a fabricated silicon structure. The waveguide position is marked by the dashed white lines. The two sampling areas for optical transmittance measurements in the waveguide region and in the reference area are highlighted by the white shaded circles. The colored arrows aim at connecting the sampling areas and the indicated polarizations of the incident electric fields to the measured spectra shown in Fig. 3(a) and (b). On the right hand side of the sample we have performed focused ion beam (FIB) milling in order to reveal the waveguide buried inside. (b) Oblique-view electron micrograph of the same sample looking at the FIB cross section with the waveguide opening at its centre. (c) and (d) Close-ups of the waveguide opening for two different characteristic z-positions of the FIB-cut separated from each other by approximately half a rod distance.

Fig. 3
Fig. 3

Optical characterization of fabricated samples and comparison with theory. (a) Linear-optical transmittance spectra of the sample shown in Fig. 2, collected both in the waveguide region (wg) and in the undisturbed reference area (ref). In these measurements we have used linearly polarized light with the electric field oriented either parallel (||) or perpendicular (⊥) to the waveguide. The position of the woodpile photonic stop band in spectroscopy direction calculated for experimental structure parameters is shaded in light gray, the calculated complete three-dimensional photonic band gap in dark gray. (b) Close-up of the measured data on a linear scale, clearly revealing the waveguide signature for longitudinally polarized incident light. (c) Numerical simulations: Dispersion diagram for a structure with parameters modeled after that depicted in Fig. 2. Black solid lines are MPB results. HFSS has been used for refined recalculation of the dispersion of the speed-of-light mode and of the mode existing at experimentally excitable k-values, showing excellent agreement with the MPB results. Also included in this panel are peak positions from the measurements shown in Fig. 3(d) (blue circles). (d) Angle-resolved transmittance spectra for a nominally identical sample.

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