Abstract

Spatial differentiation is important in image-processing applications such as image sharpening and edge-based segmentation. In these applications, of particular importance is the Laplacian, the simplest isotropic derivative operator in two dimensions. Spatial differentiation can be implemented electronically. However, in applications requiring real-time and high-throughput image differentiation, conventional digital computations become challenging. Optical analog computing may overcome this challenge by offering high-throughput low-energy-consumption operations using compact devices. However, previous works on spatial differentiation with nanophotonic structures are restricted to either one-dimensional differentiation or reflection mode, whereas operating in the transmission mode is important because it is directly compatible with standard image processing/recognition systems. Here, we show that the Laplacian can be implemented in the transmission mode by a photonic crystal slab device. We theoretically derive the criteria for realizing the Laplacian using the guided resonances in a photonic crystal slab. Guided by these criteria, we show that the Laplacian can be implemented using a carefully designed photonic crystal slab with a non-trivial isotropic band structure near the Γ point. Our work points to new opportunities in optical analog computing as provided by nanophotonic structures.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
OSA Recommended Articles
Extraordinary refraction and dispersion in two-dimensional photonic-crystal slabs

Wounjhang Park and Christopher J. Summers
Opt. Lett. 27(16) 1397-1399 (2002)

Optical characterization of photonic crystal slabs using orthogonally oriented polarization filters

Yousef Nazirizadeh, Jürgen G. Müller, Ulf Geyer, Detlef Schelle, Ernst-Bernhard Kley, Andreas Tünnermann, Uli Lemmer, and Martina Gerken
Opt. Express 16(10) 7153-7160 (2008)

Confined waveguide modes in slot photonic crystal slab

Tao Liu and Roberto Panepucci
Opt. Express 15(7) 4304-4309 (2007)

References

  • View by:
  • |
  • |
  • |

  1. R. Gonzales and R. Woods, Digital Image Processing, 3rd ed. (Prentice Hall, 2008).
  2. R. Markham, S. Frey, and G. Hills, “Methods for the enhancement of image detail and accentuation of structure in electron microscopy,” Virology 20, 88–102 (1963).
    [Crossref]
  3. M. D. Abràmoff, P. J. Magalhães, and S. J. Ram, “Image processing with ImageJ,” Biophoton. Int. 11, 36–42 (2004).
  4. T. Brosnan and D. W. Sun, “Improving quality inspection of food products by computer vision—a review,” J. Food Eng. 61, 3–16 (2004).
    [Crossref]
  5. N. Dalal and B. Triggs, “Histograms of oriented gradients for human detection,” in IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR) (IEEE, 2005), pp. 886–893.
  6. A. Rosenfeld and A. C. Kak, Digital Picture Processing (Academic, 1982).
  7. D. L. Pham, C. Xu, and J. L. Prince, “Current methods in medical image segmentation,” Annu. Rev. Biomed. Eng. 2, 315–337 (2000).
    [Crossref]
  8. R. J. Holyer and S. H. Peckinpaugh, “Edge detection applied to satellite imagery of the oceans,” IEEE Trans. Geosci. Remote Sens. 27, 46–56 (1989).
    [Crossref]
  9. D. R. Solli and B. Jalali, “Analog optical computing,” Nat. Photonics 9, 704–706 (2015).
    [Crossref]
  10. D. Görlitz and F. Lanzl, “A holographic spatial filter for direction independent differentiation,” Jpn. J. Appl. Phys. 14, 223–228 (1975).
    [Crossref]
  11. J. Eu, C. Liu, and A. Lohmann, “Spatial filters for differentiation,” Opt. Commun. 9, 168–171 (1973).
    [Crossref]
  12. E. R. Reinhardt and W. H. Bloss, “Optical differential operation processor,” Opt. Eng. 17, 69–72 (1978).
    [Crossref]
  13. R. Sirohi and V. R. Mohan, “Differentiation by spatial filtering,” Opt. Acta Int. J. Opt. 24, 1105–1113 (1977).
    [Crossref]
  14. H. Kasprzak, “Differentiation of a noninteger order and its optical implementation,” Appl. Opt. 21, 3287–3291 (1982).
    [Crossref]
  15. S. K. Yao and S. H. Lee, “Spatial differentiation and integration by coherent optical-correlation method,” J. Opt. Soc. Am. 61, 474–477 (1971).
    [Crossref]
  16. K. S. Nesteruk, I. P. Nikolaev, and A. V. Larichev, “Image differentiation with the aid of a phase knife,” Opt. Spectrosc. 91, 295–299 (2001).
    [Crossref]
  17. C. Warde and J. Thackara, “Operating modes of the microchannel spatial light modulator,” Opt. Eng. 22, 695–703 (1983).
    [Crossref]
  18. J. A. Davis, W. V. Brandt, D. M. Cottrell, and R. M. Bunch, “Spatial image differentiation using programmable binary optical elements,” Appl. Opt. 30, 4610–4614 (1991).
    [Crossref]
  19. J. Lancis, T. Szoplik, E. Tajahuerce, V. Climent, and M. Fernández-Alonso, “Fractional derivative Fourier plane filter for phase-change visualization,” Appl. Opt. 36, 7461–7464 (1997).
    [Crossref]
  20. T. Zhu, Y. Zhou, Y. Lou, H. Ye, M. Qiu, Z. Ruan, and S. Fan, “Plasmonic computing of spatial differentiation,” Nat. Commun. 8, 15391 (2017).
    [Crossref]
  21. N. V. Golovastikov, D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “Spatiotemporal optical pulse transformation by a resonant diffraction grating,” J. Exp. Theor. Phys. 121, 785–792 (2015).
    [Crossref]
  22. N. V. Golovastikov, D. A. Bykov, and L. L. Doskolovich, “Resonant diffraction gratings for spatial differentiation of optical beams,” Quantum Electron. 44, 984–988 (2014).
    [Crossref]
  23. A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alu, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343, 160–163 (2014).
    [Crossref]
  24. A. Youssefi, F. Zangeneh-Nejad, S. Abdollahramezani, and A. Khavasi, “Analog computing by Brewster effect,” Opt. Lett. 41, 3467–3470 (2016).
    [Crossref]
  25. S. AbdollahRamezani, K. Arik, A. Khavasi, and Z. Kavehvash, “Analog computing using graphene-based metalines,” Opt. Lett. 40, 5239–5242 (2015).
    [Crossref]
  26. A. Pors, M. G. Nielsen, and S. I. Bozhevolnyi, “Analog computing using reflective plasmonic metasurfaces,” Nano Lett. 15, 791–797 (2015).
    [Crossref]
  27. C.-S. Guo, Q.-Y. Yue, G.-X. Wei, L.-L. Lu, and S.-J. Yue, “Laplacian differential reconstruction of in-line holograms recorded at two different distances,” Opt. Lett. 33, 1945–1947 (2008).
    [Crossref]
  28. J. P. Ryle, D. Li, and J. T. Sheridan, “Dual wavelength digital holographic Laplacian reconstruction,” Opt. Lett. 35, 3018–3020 (2010).
    [Crossref]
  29. D. A. Bykov, L. L. Doskolovich, E. A. Bezus, and V. A. Soifer, “Optical computation of the Laplace operator using phase-shifted Bragg grating,” Opt. Express 22, 25084–25092 (2014).
    [Crossref]
  30. E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
    [Crossref]
  31. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).
  32. W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38, 1–74 (2014).
    [Crossref]
  33. R. N. Bracewell, The Fourier Transform and Its Applications, 3rd ed. (McGraw-Hill, 1999).
  34. S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112 (2002).
    [Crossref]
  35. K. X. Wang, Z. Yu, S. Sandhu, and S. Fan, “Fundamental bounds on decay rates in asymmetric single-mode optical resonators,” Opt. Lett. 38, 100–102 (2013).
    [Crossref]
  36. T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,” Phys. Rev. B 63, 125107 (2001).
    [Crossref]
  37. L. C. Andreani and D. Gerace, “Photonic-crystal slabs with a triangular lattice of triangular holes investigated using a guided-mode expansion method,” Phys. Rev. B 73, 235114 (2006).
    [Crossref]
  38. M. Minkov and V. Savona, “Automated optimization of photonic crystal slab cavities,” Sci. Rep. 4, 5124 (2015).
    [Crossref]
  39. K. Sakoda, Optical Properties of Photonic Crystals, Vol. 80 of Springer Series in Optical Sciences (Springer, 2005).
  40. S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
    [Crossref]

2017 (1)

T. Zhu, Y. Zhou, Y. Lou, H. Ye, M. Qiu, Z. Ruan, and S. Fan, “Plasmonic computing of spatial differentiation,” Nat. Commun. 8, 15391 (2017).
[Crossref]

2016 (1)

2015 (5)

S. AbdollahRamezani, K. Arik, A. Khavasi, and Z. Kavehvash, “Analog computing using graphene-based metalines,” Opt. Lett. 40, 5239–5242 (2015).
[Crossref]

M. Minkov and V. Savona, “Automated optimization of photonic crystal slab cavities,” Sci. Rep. 4, 5124 (2015).
[Crossref]

N. V. Golovastikov, D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “Spatiotemporal optical pulse transformation by a resonant diffraction grating,” J. Exp. Theor. Phys. 121, 785–792 (2015).
[Crossref]

A. Pors, M. G. Nielsen, and S. I. Bozhevolnyi, “Analog computing using reflective plasmonic metasurfaces,” Nano Lett. 15, 791–797 (2015).
[Crossref]

D. R. Solli and B. Jalali, “Analog optical computing,” Nat. Photonics 9, 704–706 (2015).
[Crossref]

2014 (4)

N. V. Golovastikov, D. A. Bykov, and L. L. Doskolovich, “Resonant diffraction gratings for spatial differentiation of optical beams,” Quantum Electron. 44, 984–988 (2014).
[Crossref]

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alu, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343, 160–163 (2014).
[Crossref]

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38, 1–74 (2014).
[Crossref]

D. A. Bykov, L. L. Doskolovich, E. A. Bezus, and V. A. Soifer, “Optical computation of the Laplace operator using phase-shifted Bragg grating,” Opt. Express 22, 25084–25092 (2014).
[Crossref]

2013 (1)

2010 (1)

2008 (1)

2006 (1)

L. C. Andreani and D. Gerace, “Photonic-crystal slabs with a triangular lattice of triangular holes investigated using a guided-mode expansion method,” Phys. Rev. B 73, 235114 (2006).
[Crossref]

2004 (2)

M. D. Abràmoff, P. J. Magalhães, and S. J. Ram, “Image processing with ImageJ,” Biophoton. Int. 11, 36–42 (2004).

T. Brosnan and D. W. Sun, “Improving quality inspection of food products by computer vision—a review,” J. Food Eng. 61, 3–16 (2004).
[Crossref]

2002 (2)

S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112 (2002).
[Crossref]

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

2001 (2)

T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,” Phys. Rev. B 63, 125107 (2001).
[Crossref]

K. S. Nesteruk, I. P. Nikolaev, and A. V. Larichev, “Image differentiation with the aid of a phase knife,” Opt. Spectrosc. 91, 295–299 (2001).
[Crossref]

2000 (2)

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
[Crossref]

D. L. Pham, C. Xu, and J. L. Prince, “Current methods in medical image segmentation,” Annu. Rev. Biomed. Eng. 2, 315–337 (2000).
[Crossref]

1997 (1)

1991 (1)

1989 (1)

R. J. Holyer and S. H. Peckinpaugh, “Edge detection applied to satellite imagery of the oceans,” IEEE Trans. Geosci. Remote Sens. 27, 46–56 (1989).
[Crossref]

1983 (1)

C. Warde and J. Thackara, “Operating modes of the microchannel spatial light modulator,” Opt. Eng. 22, 695–703 (1983).
[Crossref]

1982 (1)

1978 (1)

E. R. Reinhardt and W. H. Bloss, “Optical differential operation processor,” Opt. Eng. 17, 69–72 (1978).
[Crossref]

1977 (1)

R. Sirohi and V. R. Mohan, “Differentiation by spatial filtering,” Opt. Acta Int. J. Opt. 24, 1105–1113 (1977).
[Crossref]

1975 (1)

D. Görlitz and F. Lanzl, “A holographic spatial filter for direction independent differentiation,” Jpn. J. Appl. Phys. 14, 223–228 (1975).
[Crossref]

1973 (1)

J. Eu, C. Liu, and A. Lohmann, “Spatial filters for differentiation,” Opt. Commun. 9, 168–171 (1973).
[Crossref]

1971 (1)

1963 (1)

R. Markham, S. Frey, and G. Hills, “Methods for the enhancement of image detail and accentuation of structure in electron microscopy,” Virology 20, 88–102 (1963).
[Crossref]

Abdollahramezani, S.

Abràmoff, M. D.

M. D. Abràmoff, P. J. Magalhães, and S. J. Ram, “Image processing with ImageJ,” Biophoton. Int. 11, 36–42 (2004).

Alleman, A.

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
[Crossref]

Alu, A.

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alu, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343, 160–163 (2014).
[Crossref]

Andreani, L. C.

L. C. Andreani and D. Gerace, “Photonic-crystal slabs with a triangular lattice of triangular holes investigated using a guided-mode expansion method,” Phys. Rev. B 73, 235114 (2006).
[Crossref]

Arik, K.

Bezus, E. A.

Bloss, W. H.

E. R. Reinhardt and W. H. Bloss, “Optical differential operation processor,” Opt. Eng. 17, 69–72 (1978).
[Crossref]

Bozhevolnyi, S. I.

A. Pors, M. G. Nielsen, and S. I. Bozhevolnyi, “Analog computing using reflective plasmonic metasurfaces,” Nano Lett. 15, 791–797 (2015).
[Crossref]

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications, 3rd ed. (McGraw-Hill, 1999).

Brandt, W. V.

Brosnan, T.

T. Brosnan and D. W. Sun, “Improving quality inspection of food products by computer vision—a review,” J. Food Eng. 61, 3–16 (2004).
[Crossref]

Bunch, R. M.

Bykov, D. A.

N. V. Golovastikov, D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “Spatiotemporal optical pulse transformation by a resonant diffraction grating,” J. Exp. Theor. Phys. 121, 785–792 (2015).
[Crossref]

N. V. Golovastikov, D. A. Bykov, and L. L. Doskolovich, “Resonant diffraction gratings for spatial differentiation of optical beams,” Quantum Electron. 44, 984–988 (2014).
[Crossref]

D. A. Bykov, L. L. Doskolovich, E. A. Bezus, and V. A. Soifer, “Optical computation of the Laplace operator using phase-shifted Bragg grating,” Opt. Express 22, 25084–25092 (2014).
[Crossref]

Castaldi, G.

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alu, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343, 160–163 (2014).
[Crossref]

Chadha, A.

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38, 1–74 (2014).
[Crossref]

Chow, E.

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
[Crossref]

Chuwongin, S.

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38, 1–74 (2014).
[Crossref]

Climent, V.

Cottrell, D. M.

Dalal, N.

N. Dalal and B. Triggs, “Histograms of oriented gradients for human detection,” in IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR) (IEEE, 2005), pp. 886–893.

Davis, J. A.

Doskolovich, L. L.

N. V. Golovastikov, D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “Spatiotemporal optical pulse transformation by a resonant diffraction grating,” J. Exp. Theor. Phys. 121, 785–792 (2015).
[Crossref]

N. V. Golovastikov, D. A. Bykov, and L. L. Doskolovich, “Resonant diffraction gratings for spatial differentiation of optical beams,” Quantum Electron. 44, 984–988 (2014).
[Crossref]

D. A. Bykov, L. L. Doskolovich, E. A. Bezus, and V. A. Soifer, “Optical computation of the Laplace operator using phase-shifted Bragg grating,” Opt. Express 22, 25084–25092 (2014).
[Crossref]

Engheta, N.

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alu, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343, 160–163 (2014).
[Crossref]

Eu, J.

J. Eu, C. Liu, and A. Lohmann, “Spatial filters for differentiation,” Opt. Commun. 9, 168–171 (1973).
[Crossref]

Fan, S.

T. Zhu, Y. Zhou, Y. Lou, H. Ye, M. Qiu, Z. Ruan, and S. Fan, “Plasmonic computing of spatial differentiation,” Nat. Commun. 8, 15391 (2017).
[Crossref]

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38, 1–74 (2014).
[Crossref]

K. X. Wang, Z. Yu, S. Sandhu, and S. Fan, “Fundamental bounds on decay rates in asymmetric single-mode optical resonators,” Opt. Lett. 38, 100–102 (2013).
[Crossref]

S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112 (2002).
[Crossref]

Fernández-Alonso, M.

Frey, S.

R. Markham, S. Frey, and G. Hills, “Methods for the enhancement of image detail and accentuation of structure in electron microscopy,” Virology 20, 88–102 (1963).
[Crossref]

Galdi, V.

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alu, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343, 160–163 (2014).
[Crossref]

Gerace, D.

L. C. Andreani and D. Gerace, “Photonic-crystal slabs with a triangular lattice of triangular holes investigated using a guided-mode expansion method,” Phys. Rev. B 73, 235114 (2006).
[Crossref]

Gippius, N. A.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

Golovastikov, N. V.

N. V. Golovastikov, D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “Spatiotemporal optical pulse transformation by a resonant diffraction grating,” J. Exp. Theor. Phys. 121, 785–792 (2015).
[Crossref]

N. V. Golovastikov, D. A. Bykov, and L. L. Doskolovich, “Resonant diffraction gratings for spatial differentiation of optical beams,” Quantum Electron. 44, 984–988 (2014).
[Crossref]

Gonzales, R.

R. Gonzales and R. Woods, Digital Image Processing, 3rd ed. (Prentice Hall, 2008).

Görlitz, D.

D. Görlitz and F. Lanzl, “A holographic spatial filter for direction independent differentiation,” Jpn. J. Appl. Phys. 14, 223–228 (1975).
[Crossref]

Guo, C.-S.

Hills, G.

R. Markham, S. Frey, and G. Hills, “Methods for the enhancement of image detail and accentuation of structure in electron microscopy,” Virology 20, 88–102 (1963).
[Crossref]

Holyer, R. J.

R. J. Holyer and S. H. Peckinpaugh, “Edge detection applied to satellite imagery of the oceans,” IEEE Trans. Geosci. Remote Sens. 27, 46–56 (1989).
[Crossref]

Hou, H.

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
[Crossref]

Ishihara, T.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

Jalali, B.

D. R. Solli and B. Jalali, “Analog optical computing,” Nat. Photonics 9, 704–706 (2015).
[Crossref]

Joannopoulos, J. D.

S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112 (2002).
[Crossref]

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
[Crossref]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).

Johnson, S. G.

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
[Crossref]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).

Kak, A. C.

A. Rosenfeld and A. C. Kak, Digital Picture Processing (Academic, 1982).

Kasprzak, H.

Kavehvash, Z.

Khavasi, A.

Lancis, J.

Lanzl, F.

D. Görlitz and F. Lanzl, “A holographic spatial filter for direction independent differentiation,” Jpn. J. Appl. Phys. 14, 223–228 (1975).
[Crossref]

Larichev, A. V.

K. S. Nesteruk, I. P. Nikolaev, and A. V. Larichev, “Image differentiation with the aid of a phase knife,” Opt. Spectrosc. 91, 295–299 (2001).
[Crossref]

Lee, S. H.

Li, D.

Lin, S. Y.

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
[Crossref]

Liu, C.

J. Eu, C. Liu, and A. Lohmann, “Spatial filters for differentiation,” Opt. Commun. 9, 168–171 (1973).
[Crossref]

Liu, V.

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38, 1–74 (2014).
[Crossref]

Lohmann, A.

J. Eu, C. Liu, and A. Lohmann, “Spatial filters for differentiation,” Opt. Commun. 9, 168–171 (1973).
[Crossref]

Lou, Y.

T. Zhu, Y. Zhou, Y. Lou, H. Ye, M. Qiu, Z. Ruan, and S. Fan, “Plasmonic computing of spatial differentiation,” Nat. Commun. 8, 15391 (2017).
[Crossref]

Lu, L.-L.

Ma, Z.

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38, 1–74 (2014).
[Crossref]

Magalhães, P. J.

M. D. Abràmoff, P. J. Magalhães, and S. J. Ram, “Image processing with ImageJ,” Biophoton. Int. 11, 36–42 (2004).

Markham, R.

R. Markham, S. Frey, and G. Hills, “Methods for the enhancement of image detail and accentuation of structure in electron microscopy,” Virology 20, 88–102 (1963).
[Crossref]

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).

Minkov, M.

M. Minkov and V. Savona, “Automated optimization of photonic crystal slab cavities,” Sci. Rep. 4, 5124 (2015).
[Crossref]

Mohan, V. R.

R. Sirohi and V. R. Mohan, “Differentiation by spatial filtering,” Opt. Acta Int. J. Opt. 24, 1105–1113 (1977).
[Crossref]

Monticone, F.

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alu, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343, 160–163 (2014).
[Crossref]

Muljarov, E. A.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

Nesteruk, K. S.

K. S. Nesteruk, I. P. Nikolaev, and A. V. Larichev, “Image differentiation with the aid of a phase knife,” Opt. Spectrosc. 91, 295–299 (2001).
[Crossref]

Nielsen, M. G.

A. Pors, M. G. Nielsen, and S. I. Bozhevolnyi, “Analog computing using reflective plasmonic metasurfaces,” Nano Lett. 15, 791–797 (2015).
[Crossref]

Nikolaev, I. P.

K. S. Nesteruk, I. P. Nikolaev, and A. V. Larichev, “Image differentiation with the aid of a phase knife,” Opt. Spectrosc. 91, 295–299 (2001).
[Crossref]

Ochiai, T.

T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,” Phys. Rev. B 63, 125107 (2001).
[Crossref]

Peckinpaugh, S. H.

R. J. Holyer and S. H. Peckinpaugh, “Edge detection applied to satellite imagery of the oceans,” IEEE Trans. Geosci. Remote Sens. 27, 46–56 (1989).
[Crossref]

Pham, D. L.

D. L. Pham, C. Xu, and J. L. Prince, “Current methods in medical image segmentation,” Annu. Rev. Biomed. Eng. 2, 315–337 (2000).
[Crossref]

Pors, A.

A. Pors, M. G. Nielsen, and S. I. Bozhevolnyi, “Analog computing using reflective plasmonic metasurfaces,” Nano Lett. 15, 791–797 (2015).
[Crossref]

Prince, J. L.

D. L. Pham, C. Xu, and J. L. Prince, “Current methods in medical image segmentation,” Annu. Rev. Biomed. Eng. 2, 315–337 (2000).
[Crossref]

Qiu, M.

T. Zhu, Y. Zhou, Y. Lou, H. Ye, M. Qiu, Z. Ruan, and S. Fan, “Plasmonic computing of spatial differentiation,” Nat. Commun. 8, 15391 (2017).
[Crossref]

Ram, S. J.

M. D. Abràmoff, P. J. Magalhães, and S. J. Ram, “Image processing with ImageJ,” Biophoton. Int. 11, 36–42 (2004).

Reinhardt, E. R.

E. R. Reinhardt and W. H. Bloss, “Optical differential operation processor,” Opt. Eng. 17, 69–72 (1978).
[Crossref]

Rosenfeld, A.

A. Rosenfeld and A. C. Kak, Digital Picture Processing (Academic, 1982).

Ruan, Z.

T. Zhu, Y. Zhou, Y. Lou, H. Ye, M. Qiu, Z. Ruan, and S. Fan, “Plasmonic computing of spatial differentiation,” Nat. Commun. 8, 15391 (2017).
[Crossref]

Ryle, J. P.

Sakoda, K.

T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,” Phys. Rev. B 63, 125107 (2001).
[Crossref]

K. Sakoda, Optical Properties of Photonic Crystals, Vol. 80 of Springer Series in Optical Sciences (Springer, 2005).

Sandhu, S.

Savona, V.

M. Minkov and V. Savona, “Automated optimization of photonic crystal slab cavities,” Sci. Rep. 4, 5124 (2015).
[Crossref]

Seo, J.-H.

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38, 1–74 (2014).
[Crossref]

Sheridan, J. T.

Shuai, Y.-C.

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38, 1–74 (2014).
[Crossref]

Silva, A.

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alu, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343, 160–163 (2014).
[Crossref]

Sirohi, R.

R. Sirohi and V. R. Mohan, “Differentiation by spatial filtering,” Opt. Acta Int. J. Opt. 24, 1105–1113 (1977).
[Crossref]

Soifer, V. A.

N. V. Golovastikov, D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “Spatiotemporal optical pulse transformation by a resonant diffraction grating,” J. Exp. Theor. Phys. 121, 785–792 (2015).
[Crossref]

D. A. Bykov, L. L. Doskolovich, E. A. Bezus, and V. A. Soifer, “Optical computation of the Laplace operator using phase-shifted Bragg grating,” Opt. Express 22, 25084–25092 (2014).
[Crossref]

Solli, D. R.

D. R. Solli and B. Jalali, “Analog optical computing,” Nat. Photonics 9, 704–706 (2015).
[Crossref]

Sun, D. W.

T. Brosnan and D. W. Sun, “Improving quality inspection of food products by computer vision—a review,” J. Food Eng. 61, 3–16 (2004).
[Crossref]

Szoplik, T.

Tajahuerce, E.

Thackara, J.

C. Warde and J. Thackara, “Operating modes of the microchannel spatial light modulator,” Opt. Eng. 22, 695–703 (1983).
[Crossref]

Tikhodeev, S. G.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

Triggs, B.

N. Dalal and B. Triggs, “Histograms of oriented gradients for human detection,” in IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR) (IEEE, 2005), pp. 886–893.

Vawter, G. A.

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
[Crossref]

Villeneuve, P. R.

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
[Crossref]

Wang, K. X.

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38, 1–74 (2014).
[Crossref]

K. X. Wang, Z. Yu, S. Sandhu, and S. Fan, “Fundamental bounds on decay rates in asymmetric single-mode optical resonators,” Opt. Lett. 38, 100–102 (2013).
[Crossref]

Warde, C.

C. Warde and J. Thackara, “Operating modes of the microchannel spatial light modulator,” Opt. Eng. 22, 695–703 (1983).
[Crossref]

Wei, G.-X.

Wendt, J. R.

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
[Crossref]

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).

Woods, R.

R. Gonzales and R. Woods, Digital Image Processing, 3rd ed. (Prentice Hall, 2008).

Xu, C.

D. L. Pham, C. Xu, and J. L. Prince, “Current methods in medical image segmentation,” Annu. Rev. Biomed. Eng. 2, 315–337 (2000).
[Crossref]

Yablonskii, A. L.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

Yang, H.

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38, 1–74 (2014).
[Crossref]

Yao, S. K.

Ye, H.

T. Zhu, Y. Zhou, Y. Lou, H. Ye, M. Qiu, Z. Ruan, and S. Fan, “Plasmonic computing of spatial differentiation,” Nat. Commun. 8, 15391 (2017).
[Crossref]

Youssefi, A.

Yu, Z.

Yue, Q.-Y.

Yue, S.-J.

Zangeneh-Nejad, F.

Zhao, D.

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38, 1–74 (2014).
[Crossref]

Zhou, W.

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38, 1–74 (2014).
[Crossref]

Zhou, Y.

T. Zhu, Y. Zhou, Y. Lou, H. Ye, M. Qiu, Z. Ruan, and S. Fan, “Plasmonic computing of spatial differentiation,” Nat. Commun. 8, 15391 (2017).
[Crossref]

Zhu, T.

T. Zhu, Y. Zhou, Y. Lou, H. Ye, M. Qiu, Z. Ruan, and S. Fan, “Plasmonic computing of spatial differentiation,” Nat. Commun. 8, 15391 (2017).
[Crossref]

Zubrzycki, W.

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
[Crossref]

Annu. Rev. Biomed. Eng. (1)

D. L. Pham, C. Xu, and J. L. Prince, “Current methods in medical image segmentation,” Annu. Rev. Biomed. Eng. 2, 315–337 (2000).
[Crossref]

Appl. Opt. (3)

Biophoton. Int. (1)

M. D. Abràmoff, P. J. Magalhães, and S. J. Ram, “Image processing with ImageJ,” Biophoton. Int. 11, 36–42 (2004).

IEEE Trans. Geosci. Remote Sens. (1)

R. J. Holyer and S. H. Peckinpaugh, “Edge detection applied to satellite imagery of the oceans,” IEEE Trans. Geosci. Remote Sens. 27, 46–56 (1989).
[Crossref]

J. Exp. Theor. Phys. (1)

N. V. Golovastikov, D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “Spatiotemporal optical pulse transformation by a resonant diffraction grating,” J. Exp. Theor. Phys. 121, 785–792 (2015).
[Crossref]

J. Food Eng. (1)

T. Brosnan and D. W. Sun, “Improving quality inspection of food products by computer vision—a review,” J. Food Eng. 61, 3–16 (2004).
[Crossref]

J. Opt. Soc. Am. (1)

Jpn. J. Appl. Phys. (1)

D. Görlitz and F. Lanzl, “A holographic spatial filter for direction independent differentiation,” Jpn. J. Appl. Phys. 14, 223–228 (1975).
[Crossref]

Nano Lett. (1)

A. Pors, M. G. Nielsen, and S. I. Bozhevolnyi, “Analog computing using reflective plasmonic metasurfaces,” Nano Lett. 15, 791–797 (2015).
[Crossref]

Nat. Commun. (1)

T. Zhu, Y. Zhou, Y. Lou, H. Ye, M. Qiu, Z. Ruan, and S. Fan, “Plasmonic computing of spatial differentiation,” Nat. Commun. 8, 15391 (2017).
[Crossref]

Nat. Photonics (1)

D. R. Solli and B. Jalali, “Analog optical computing,” Nat. Photonics 9, 704–706 (2015).
[Crossref]

Nature (1)

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983–986 (2000).
[Crossref]

Opt. Acta Int. J. Opt. (1)

R. Sirohi and V. R. Mohan, “Differentiation by spatial filtering,” Opt. Acta Int. J. Opt. 24, 1105–1113 (1977).
[Crossref]

Opt. Commun. (1)

J. Eu, C. Liu, and A. Lohmann, “Spatial filters for differentiation,” Opt. Commun. 9, 168–171 (1973).
[Crossref]

Opt. Eng. (2)

E. R. Reinhardt and W. H. Bloss, “Optical differential operation processor,” Opt. Eng. 17, 69–72 (1978).
[Crossref]

C. Warde and J. Thackara, “Operating modes of the microchannel spatial light modulator,” Opt. Eng. 22, 695–703 (1983).
[Crossref]

Opt. Express (1)

Opt. Lett. (5)

Opt. Spectrosc. (1)

K. S. Nesteruk, I. P. Nikolaev, and A. V. Larichev, “Image differentiation with the aid of a phase knife,” Opt. Spectrosc. 91, 295–299 (2001).
[Crossref]

Phys. Rev. B (4)

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112 (2002).
[Crossref]

T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,” Phys. Rev. B 63, 125107 (2001).
[Crossref]

L. C. Andreani and D. Gerace, “Photonic-crystal slabs with a triangular lattice of triangular holes investigated using a guided-mode expansion method,” Phys. Rev. B 73, 235114 (2006).
[Crossref]

Prog. Quantum Electron. (1)

W. Zhou, D. Zhao, Y.-C. Shuai, H. Yang, S. Chuwongin, A. Chadha, J.-H. Seo, K. X. Wang, V. Liu, Z. Ma, and S. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quantum Electron. 38, 1–74 (2014).
[Crossref]

Quantum Electron. (1)

N. V. Golovastikov, D. A. Bykov, and L. L. Doskolovich, “Resonant diffraction gratings for spatial differentiation of optical beams,” Quantum Electron. 44, 984–988 (2014).
[Crossref]

Sci. Rep. (1)

M. Minkov and V. Savona, “Automated optimization of photonic crystal slab cavities,” Sci. Rep. 4, 5124 (2015).
[Crossref]

Science (1)

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alu, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343, 160–163 (2014).
[Crossref]

Virology (1)

R. Markham, S. Frey, and G. Hills, “Methods for the enhancement of image detail and accentuation of structure in electron microscopy,” Virology 20, 88–102 (1963).
[Crossref]

Other (6)

R. Gonzales and R. Woods, Digital Image Processing, 3rd ed. (Prentice Hall, 2008).

N. Dalal and B. Triggs, “Histograms of oriented gradients for human detection,” in IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR) (IEEE, 2005), pp. 886–893.

A. Rosenfeld and A. C. Kak, Digital Picture Processing (Academic, 1982).

K. Sakoda, Optical Properties of Photonic Crystals, Vol. 80 of Springer Series in Optical Sciences (Springer, 2005).

R. N. Bracewell, The Fourier Transform and Its Applications, 3rd ed. (McGraw-Hill, 1999).

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).

Supplementary Material (1)

NameDescription
» Supplement 1       Supplemental document

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1.
Fig. 1. (a) Geometry of the photonic crystal slab differentiator, which consists of a photonic crystal slab separated from a uniform dielectric slab by an air gap. For ε=12, the geometry parameters are: d=0.55a, r=0.111a, ds=0.07a, dg=0.21a, where a is the lattice constant. The plane above and below shows the input and output field profiles, respectively. The transmitted field profile (e.g., ring) through the device is the Laplacian of the incident field profile (e.g., disk) illuminated by a normally incident unpolarized light with frequency ω0=0.47656×2πc/a. (b) The coordinate system. (c) The Brillouin zone of the system.
Fig. 2.
Fig. 2. Band structure of the photonic crystal slab with the dielectric constant ε=12, thickness d=0.55a, and radius r=0.111a of the holes. (a)–(c) Band structure near ω0=0.38749×2πc/a as a typical example of conventional anisotropic bands that are doubly degenerate at Γ. (a) Band dispersion diagram along ΓX and ΓM. (b) Constant frequency contours of the lower band with respect to (kx,ky). (c) Constant frequency contours of the upper band. (d)–(f) Band structure at ω0=0.47656×2πc/a, which is nearly isotropic. (d) Band dispersions along ΓX and ΓM. (e) Constant frequency contours of the lower band. (f) Constant frequency contours of the upper band.
Fig. 3.
Fig. 3. Transmittances of the two-slab structure as illustrated in Fig. 1. The photonic crystal slab has the same structural parameters as those in Fig. 2. The thickness of the uniform dielectric slab is ds=0.07a, and the air gap width is dg=2.1a. (a), (b) Transmittance near ω0=0.38749×2πc/a as a function of |k| and frequency (ωω0) along a general direction ϕ=14° for (a) S light |ts|=|tss|2+|tps|2 and (b) P light |tp|=|tpp|2+|tsp|2. Both bands couple to S and P light. (c), (d) Transmittance near ω0=0.47656×2πc/a along ϕ=14° for (c) S light |ts|=|tss| and (d) P light |tp|=|tpp|. S light couples with only the upper B band while P with the lower A band, showing the single-band excitation effect.
Fig. 4.
Fig. 4. Contour plots of transmittance |t| as a function of kx and ky at frequency ω0=0.47656×2πc/a for (a) S light |tss|, (b) P light |tpp|, and (c) unpolarized light |tu|, which are all isotropic near Γ. (d) |tu| as a function of |k| along the ϕ=14° direction, and the quadratic fitting tu=αu|k|2 where αu is the only fitting parameter.
Fig. 5.
Fig. 5. (a) Incident Stanford emblem with a size of 2610a×1729a. (b) Calculated transmitted image with unpolarized light, which clearly shows the edges with different orientations. (c) Incident slot patterns with length 500a and widths 100, 50, 30, and 20a. (d) Calculated transmitted images with unpolarized light, which show that the spatial resolution of our design is around 30a.

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

t(kx,ky)(tss(kx,ky)tsp(kx,ky)tps(kx,ky)tpp(kx,ky))=(α(kx2+ky2)00α(kx2+ky2)),
t(kx,ky)=(αs(kx2+ky2)00αp(kx2+ky2)),
t(ω,k)=td+fγ(k)i[ωω(k)]+γ(k),
f=td=1,
t(ω,k)=1γ(k)i[ωω(k)]+γ(k).
ω=ω0.
t(ω0,k)=0+tω(k)|Γδω(k)+tγ(k)|Γδγ(k),
δω(k)=ω(k)ω0,δγ(k)=γ(k)γ0,
tω(k)|Γ=iγ0,tγ(k)|Γ=0,
t(ω0,k)=iγ0δω(k).
H^(k)=(ω0iγ0+a|k|2)I^+b(kx2ky2)σ^z+ckxkyσ^x,
ω±(k)iγ±(k)=ω0iγ0+a|k|2±b2|k|4+(c24b2)kx2ky2.
ω±(k)iγ±(k)=ω0iγ0+(a±b)|k|2.
R^(ϕ)H^(k)R^1(ϕ)=H^(R(ϕ)k),
|R(ϕ)k,A=R^(ϕ)|k,A,|R(ϕ)k,B=R^(ϕ)|k,B.
|R(ϕ)k,S=R^(ϕ)|k,S,|R(ϕ)k,P=R^(ϕ)|k,P.
kx,S|kx,A=0,kx,P|kx,B=0.
k,S|k,A=kx,S|R^1(ϕ)R^(ϕ)|kx,A=0,k,P|k,B=kx,P|R^1(ϕ)R^(ϕ)|kx,B=0.
|tu|=|tss|2+|tpp|22.

Metrics