Abstract

The detailed microscopic characterization of photonic crystal (PC) structures is challenging due to their small sizes. Generally, only the gross macroscopic behavior can be determined. This leaves in question the performance at the basic structure level. The single-incident-angle plane-wave transmittances of one-dimensional photonic crystal (PC) structures are extracted from multiple-incident-angle, focused-beam measurements. In the experimental apparatus, an infrared beam is focused by a reflecting microscope objective to produce an incident beam. This beam can be modeled as multiple, variable-intensity plane waves incident on the PC structure. The transmittance of the structure in response to a multiple- incident-angle composite beam is measured. The composite beam measurement is repeated at various incident angle orientations with respect to the sample normal so that, at each angular orientation, the included set of single-angle plane-wave components is unique. A set of measurements recorded over a range of angular orientations results in an underspecified matrix algebra problem. Regularization techniques can be applied to the problem to extract the single-angle plane-wave response of the structure from the composite measurements. Experimental results show very good agreement between the measured and theoretical single-angle plane-wave transmittances.

© 2009 Optical Society of America

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  1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059-2062(1987).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  6. E. Miyai, M. Okano, M. Mochizuki, and S. Noda, “Analysis of coupling between two-dimensional photonic crystal waveguide and external waveguide,” Appl. Phys. Lett. 81, 3729-3731 (2002).
    [CrossRef]
  7. S. Olivier, C. J. M. Smith, H. Benisty, C. Weisbuch, T. Krauss, R. Houdré, and U. Oesterle, “Cascaded photonic crystal guides and cavities: spectral studies and their impact on integrated optics design,” IEEE J. Quantum Electron. 38, 816-824 (2002).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  12. G. R. Kilby, Infrared Methods Applied to Photonic Crystal Device Development (Georgia Institute of Technology, 2005).
  13. S. Rowson, A. Chelnokov, C. Cuisin, and J.-M. Lourtioz, “Three-dimensional characterization of a two-dimensional photonic bandgap reflector at mid-infrared wavelengths,” IEE Proc. Optelectron. 145, 403-408 (1998).
    [CrossRef]

2004 (1)

T. K. Gaylord and G. R. Kilby, “Optical single-angle plane-wave transmittances/reflectances from Schwarzschild objective variable-angle measurements,” Rev. Sci. Instrum. 75, 317-323 (2004).
[CrossRef]

2002 (3)

E. Miyai, M. Okano, M. Mochizuki, and S. Noda, “Analysis of coupling between two-dimensional photonic crystal waveguide and external waveguide,” Appl. Phys. Lett. 81, 3729-3731 (2002).
[CrossRef]

S. Olivier, C. J. M. Smith, H. Benisty, C. Weisbuch, T. Krauss, R. Houdré, and U. Oesterle, “Cascaded photonic crystal guides and cavities: spectral studies and their impact on integrated optics design,” IEEE J. Quantum Electron. 38, 816-824 (2002).
[CrossRef]

M. Imada, S. Noda, A. Chutinan, M. Mochizuiki, and T. Tanaka, “Channel drop filter using a single defect in a 2-d photonic crystal slab waveguide,” J. Lightwave Technol. 20, 873-878 (2002).
[CrossRef]

1999 (1)

1998 (1)

S. Rowson, A. Chelnokov, C. Cuisin, and J.-M. Lourtioz, “Three-dimensional characterization of a two-dimensional photonic bandgap reflector at mid-infrared wavelengths,” IEE Proc. Optelectron. 145, 403-408 (1998).
[CrossRef]

1996 (1)

A. Mekis, J. Chen, I. Kurland, S. Fan, P. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787-3790(1996).
[CrossRef] [PubMed]

1987 (2)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059-2062(1987).
[CrossRef] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486-2489(1987).
[CrossRef] [PubMed]

Arsenin, V. Y.

A. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Winston and Sons, 1977).

Benisty, H.

S. Olivier, C. J. M. Smith, H. Benisty, C. Weisbuch, T. Krauss, R. Houdré, and U. Oesterle, “Cascaded photonic crystal guides and cavities: spectral studies and their impact on integrated optics design,” IEEE J. Quantum Electron. 38, 816-824 (2002).
[CrossRef]

Chelnokov, A.

S. Rowson, A. Chelnokov, C. Cuisin, and J.-M. Lourtioz, “Three-dimensional characterization of a two-dimensional photonic bandgap reflector at mid-infrared wavelengths,” IEE Proc. Optelectron. 145, 403-408 (1998).
[CrossRef]

Chen, J.

A. Mekis, J. Chen, I. Kurland, S. Fan, P. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787-3790(1996).
[CrossRef] [PubMed]

Chutinan, A.

M. Imada, S. Noda, A. Chutinan, M. Mochizuiki, and T. Tanaka, “Channel drop filter using a single defect in a 2-d photonic crystal slab waveguide,” J. Lightwave Technol. 20, 873-878 (2002).
[CrossRef]

Cuisin, C.

S. Rowson, A. Chelnokov, C. Cuisin, and J.-M. Lourtioz, “Three-dimensional characterization of a two-dimensional photonic bandgap reflector at mid-infrared wavelengths,” IEE Proc. Optelectron. 145, 403-408 (1998).
[CrossRef]

Fan, S.

A. Mekis, J. Chen, I. Kurland, S. Fan, P. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787-3790(1996).
[CrossRef] [PubMed]

Gaylord, T. K.

T. K. Gaylord and G. R. Kilby, “Optical single-angle plane-wave transmittances/reflectances from Schwarzschild objective variable-angle measurements,” Rev. Sci. Instrum. 75, 317-323 (2004).
[CrossRef]

Hansen, P. C.

P. C. Hansen, Rank-Deficient and Discrete Ill Posed Problems Numerical Aspects of Linear Inversion (Society for Industrial and Applied Mathematics, 1998).
[CrossRef]

Houdré, R.

S. Olivier, C. J. M. Smith, H. Benisty, C. Weisbuch, T. Krauss, R. Houdré, and U. Oesterle, “Cascaded photonic crystal guides and cavities: spectral studies and their impact on integrated optics design,” IEEE J. Quantum Electron. 38, 816-824 (2002).
[CrossRef]

Imada, M.

M. Imada, S. Noda, A. Chutinan, M. Mochizuiki, and T. Tanaka, “Channel drop filter using a single defect in a 2-d photonic crystal slab waveguide,” J. Lightwave Technol. 20, 873-878 (2002).
[CrossRef]

Joannopoulos, J. D.

A. Mekis, J. Chen, I. Kurland, S. Fan, P. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787-3790(1996).
[CrossRef] [PubMed]

John, S.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486-2489(1987).
[CrossRef] [PubMed]

Kilby, G. R.

T. K. Gaylord and G. R. Kilby, “Optical single-angle plane-wave transmittances/reflectances from Schwarzschild objective variable-angle measurements,” Rev. Sci. Instrum. 75, 317-323 (2004).
[CrossRef]

G. R. Kilby, Infrared Methods Applied to Photonic Crystal Device Development (Georgia Institute of Technology, 2005).

Krauss, T.

S. Olivier, C. J. M. Smith, H. Benisty, C. Weisbuch, T. Krauss, R. Houdré, and U. Oesterle, “Cascaded photonic crystal guides and cavities: spectral studies and their impact on integrated optics design,” IEEE J. Quantum Electron. 38, 816-824 (2002).
[CrossRef]

Kurland, I.

A. Mekis, J. Chen, I. Kurland, S. Fan, P. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787-3790(1996).
[CrossRef] [PubMed]

Lee, R. K.

Lourtioz, J.-M.

S. Rowson, A. Chelnokov, C. Cuisin, and J.-M. Lourtioz, “Three-dimensional characterization of a two-dimensional photonic bandgap reflector at mid-infrared wavelengths,” IEE Proc. Optelectron. 145, 403-408 (1998).
[CrossRef]

Mekis, A.

A. Mekis, J. Chen, I. Kurland, S. Fan, P. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787-3790(1996).
[CrossRef] [PubMed]

Miyai, E.

E. Miyai, M. Okano, M. Mochizuki, and S. Noda, “Analysis of coupling between two-dimensional photonic crystal waveguide and external waveguide,” Appl. Phys. Lett. 81, 3729-3731 (2002).
[CrossRef]

Mochizuiki, M.

M. Imada, S. Noda, A. Chutinan, M. Mochizuiki, and T. Tanaka, “Channel drop filter using a single defect in a 2-d photonic crystal slab waveguide,” J. Lightwave Technol. 20, 873-878 (2002).
[CrossRef]

Mochizuki, M.

E. Miyai, M. Okano, M. Mochizuki, and S. Noda, “Analysis of coupling between two-dimensional photonic crystal waveguide and external waveguide,” Appl. Phys. Lett. 81, 3729-3731 (2002).
[CrossRef]

Noda, S.

E. Miyai, M. Okano, M. Mochizuki, and S. Noda, “Analysis of coupling between two-dimensional photonic crystal waveguide and external waveguide,” Appl. Phys. Lett. 81, 3729-3731 (2002).
[CrossRef]

M. Imada, S. Noda, A. Chutinan, M. Mochizuiki, and T. Tanaka, “Channel drop filter using a single defect in a 2-d photonic crystal slab waveguide,” J. Lightwave Technol. 20, 873-878 (2002).
[CrossRef]

Oesterle, U.

S. Olivier, C. J. M. Smith, H. Benisty, C. Weisbuch, T. Krauss, R. Houdré, and U. Oesterle, “Cascaded photonic crystal guides and cavities: spectral studies and their impact on integrated optics design,” IEEE J. Quantum Electron. 38, 816-824 (2002).
[CrossRef]

Okano, M.

E. Miyai, M. Okano, M. Mochizuki, and S. Noda, “Analysis of coupling between two-dimensional photonic crystal waveguide and external waveguide,” Appl. Phys. Lett. 81, 3729-3731 (2002).
[CrossRef]

Olivier, S.

S. Olivier, C. J. M. Smith, H. Benisty, C. Weisbuch, T. Krauss, R. Houdré, and U. Oesterle, “Cascaded photonic crystal guides and cavities: spectral studies and their impact on integrated optics design,” IEEE J. Quantum Electron. 38, 816-824 (2002).
[CrossRef]

Rowson, S.

S. Rowson, A. Chelnokov, C. Cuisin, and J.-M. Lourtioz, “Three-dimensional characterization of a two-dimensional photonic bandgap reflector at mid-infrared wavelengths,” IEE Proc. Optelectron. 145, 403-408 (1998).
[CrossRef]

Scherer, A.

Smith, C. J. M.

S. Olivier, C. J. M. Smith, H. Benisty, C. Weisbuch, T. Krauss, R. Houdré, and U. Oesterle, “Cascaded photonic crystal guides and cavities: spectral studies and their impact on integrated optics design,” IEEE J. Quantum Electron. 38, 816-824 (2002).
[CrossRef]

Tanaka, T.

M. Imada, S. Noda, A. Chutinan, M. Mochizuiki, and T. Tanaka, “Channel drop filter using a single defect in a 2-d photonic crystal slab waveguide,” J. Lightwave Technol. 20, 873-878 (2002).
[CrossRef]

Tikhonov, A.

A. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Winston and Sons, 1977).

Villeneuve, P.

A. Mekis, J. Chen, I. Kurland, S. Fan, P. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787-3790(1996).
[CrossRef] [PubMed]

Weisbuch, C.

S. Olivier, C. J. M. Smith, H. Benisty, C. Weisbuch, T. Krauss, R. Houdré, and U. Oesterle, “Cascaded photonic crystal guides and cavities: spectral studies and their impact on integrated optics design,” IEEE J. Quantum Electron. 38, 816-824 (2002).
[CrossRef]

Xu, Y.

Yablonovitch, E.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059-2062(1987).
[CrossRef] [PubMed]

Yariv, A.

Appl. Phys. Lett. (1)

E. Miyai, M. Okano, M. Mochizuki, and S. Noda, “Analysis of coupling between two-dimensional photonic crystal waveguide and external waveguide,” Appl. Phys. Lett. 81, 3729-3731 (2002).
[CrossRef]

IEE Proc. Optelectron. (1)

S. Rowson, A. Chelnokov, C. Cuisin, and J.-M. Lourtioz, “Three-dimensional characterization of a two-dimensional photonic bandgap reflector at mid-infrared wavelengths,” IEE Proc. Optelectron. 145, 403-408 (1998).
[CrossRef]

IEEE J. Quantum Electron. (1)

S. Olivier, C. J. M. Smith, H. Benisty, C. Weisbuch, T. Krauss, R. Houdré, and U. Oesterle, “Cascaded photonic crystal guides and cavities: spectral studies and their impact on integrated optics design,” IEEE J. Quantum Electron. 38, 816-824 (2002).
[CrossRef]

J. Lightwave Technol. (1)

M. Imada, S. Noda, A. Chutinan, M. Mochizuiki, and T. Tanaka, “Channel drop filter using a single defect in a 2-d photonic crystal slab waveguide,” J. Lightwave Technol. 20, 873-878 (2002).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. Lett. (3)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059-2062(1987).
[CrossRef] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486-2489(1987).
[CrossRef] [PubMed]

A. Mekis, J. Chen, I. Kurland, S. Fan, P. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787-3790(1996).
[CrossRef] [PubMed]

Rev. Sci. Instrum. (1)

T. K. Gaylord and G. R. Kilby, “Optical single-angle plane-wave transmittances/reflectances from Schwarzschild objective variable-angle measurements,” Rev. Sci. Instrum. 75, 317-323 (2004).
[CrossRef]

Other (4)

P. C. Hansen, Rank-Deficient and Discrete Ill Posed Problems Numerical Aspects of Linear Inversion (Society for Industrial and Applied Mathematics, 1998).
[CrossRef]

Department of Mathematical Modeling, “Regularization tools: a Matlab package for analysis and solution of discrete ill-posed problems. Version 3.1 for Matlab 6.0,” Technical University of Denmark, Building 305, DK-2800 Lyngby, Denmark, 2001.

A. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Winston and Sons, 1977).

G. R. Kilby, Infrared Methods Applied to Photonic Crystal Device Development (Georgia Institute of Technology, 2005).

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Figures (5)

Fig. 1
Fig. 1

(a) Optical configuration of a Schwarzschild reflecting focusing objective. Light passes through an aperture in the large outer mirror and is reflected by the smaller mirror back to the large mirror. This mirror reflects light to the focal point. (b) Light rays from a Schwarzschild objective are focused onto a photonic crystal sample. The objective axis is at an angle θ S with respect to the normal of the sample. The minimum and maximum ray angles of the objective are given by θ ob , min and θ ob , max , respectively. An example single-angle plane-wave incident at an angle of θ k with respect to the sample normal is shown.

Fig. 2
Fig. 2

As the objective axis is rotated with respect to the sample normal in the half-plane in front of the sample, various sets of incident angle plane waves are selected by the objective. By rotating and measuring the transmittance or reflectance of the composite beam in discrete increments, new angles are added while some are removed in each subsequent measurement. If θ S is rotated to sufficient limits, all angles in the half-plane before the sample can be selected. The increment from one axis angular orientation to the next is given as Δ θ S .

Fig. 3
Fig. 3

Scanning electron microscope image showing one- dimensional photonic crystal structure on an SOI substrate. The one-dimensional PC was etched to a depth of approximately 40 μm . The inset shows the roughness of the PC surface caused by the etch and passivation cycling inherent to the Bosch process.

Fig. 4
Fig. 4

Composite normalized FTIR microspectroscopy transmittance measurements and normalized theoretical transmittance calculations of a PC structure. The measured and theoretical transmittances are shown for objective axis positions of θ S = 20 ° , θ S = 12 ° , θ S = 8 ° , and θ S = 0 ° with respect to the sample normal.

Fig. 5
Fig. 5

FTIR system transmission-based single- angle plane-wave normalized transmittance and theoretical single-angle plane-wave ideal normalized transmittance of a PC structure. The measured and theoretical transmittances are shown for objective axis positions of θ k = 1 ° , θ k = 9 ° , θ k = 21 ° , and θ k = 25 ° .

Equations (2)

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T ob , θ S = [ a 1 a 2 a 3 a 4 ] [ t θ k 1 t θ k 2 t θ k 3 t θ k 4 ] ,
[ T ob , 30 ° T ob , 20 ° T ob , 10 ° T ob , 0 ° T ob , 10 ° T ob , 20 ° T ob , 30 ° ] = [ a 1 a 2 a 3 0 a 4 a 5 a 6 0 0 0 0 0 0 0 a 1 a 2 a 3 0 a 4 a 5 a 6 0 0 0 0 0 0 0 a 1 a 2 a 3 0 a 4 a 5 a 6 0 0 0 0 0 0 0 a 1 a 2 a 3 0 a 4 a 5 a 6 0 0 0 0 0 0 0 a 1 a 2 a 3 0 a 4 a 5 a 6 0 0 0 0 0 0 0 a 1 a 2 a 3 0 a 4 a 5 a 6 0 0 0 0 0 0 0 a 1 a 2 a 3 0 a 4 a 5 a 6 ] [ t 60 ° t 50 ° t 40 ° t 30 ° t 20 ° t 10 ° t 0 ° t 10 ° t 20 ° t 30 ° t 40 ° t 50 ° t 60 ° ] .

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