Abstract

We show that a very simple structure consisting of a single subwavelength dielectric ridge on the surface of a slab waveguide enables spatial integration and differentiation of the profile of optical beams propagating in the waveguide. The integration and differentiation operations are performed in reflection and in transmission, respectively, at oblique incidence of the beam impinging on the ridge. The implementation of these operations is associated with the resonant excitation of a cross-polarized eigenmode of the ridge. We demonstrate that the quality factor of the resonances strongly varies along the dispersion curves and allows one to achieve the required tradeoff between the integration (or differentiation) quality and the amplitude of the resulting beam. The presented rigorous numerical simulation results confirm high-quality integration and differentiation. The proposed integrated structure may find application in ultrafast all-optical analog computing and signal processing systems.

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

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  1. J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  2. A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343(6167), 160–163 (2014).
    [Crossref] [PubMed]
  3. A. Pors, M. G. Nielsen, and S. I. Bozhevolnyi, “Analog computing using reflective plasmonic metasurfaces,” Nano Lett. 15(1), 791–797 (2015).
    [Crossref]
  4. A. Chizari, S. Abdollahramezani, M. V. Jamali, and J. A. Salehi, “Analog optical computing based on a dielectric meta-reflect array,” Opt. Lett. 41(15), 3451–3454 (2016).
    [Crossref] [PubMed]
  5. H. Babashah, Z. Kavehvash, S. Koohi, and A. Khavasi, “Integration in analog optical computing using metasurfaces revisited: toward ideal optical integration,” J. Opt. Soc. Am. B 34(6), 1270–1279 (2017).
    [Crossref]
  6. L. L. Doskolovich, D. A. Bykov, E. A. Bezus, and V. A. Soifer, “Spatial differentiation of optical beams using phase-shifted Bragg grating,” Opt. Lett. 39(5), 1278–1281 (2014).
    [Crossref] [PubMed]
  7. N. V. Golovastikov, D. A. Bykov, L. L. Doskolovich, and E. A. Bezus, “Spatial optical integrator based on phase-shifted Bragg gratings,” Opt. Commun. 338, 457–460 (2015).
    [Crossref]
  8. 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] [PubMed]
  9. Z. Ruan, “Spatial mode control of surface plasmon polariton excitation with gain medium: from spatial differentiator to integrator,” Opt. Lett. 40(4), 601–604 (2015).
    [Crossref] [PubMed]
  10. F. Zangeneh-Nejad and A. Khavasi, “Spatial integration by a dielectric slab and its planar graphene-based counterpart,” Opt. Lett. 42(10), 1954–1957 (2017).
    [Crossref] [PubMed]
  11. F. Zangeneh-Nejad, A. Khavasi, and B. Rejaei, “Analog optical computing by half-wavelength slabs,” Opt. Commun. 407, 338–343 (2018).
    [Crossref]
  12. A. Youssefi, F. Zangeneh-Nejad, S. Abdollahramezani, and A. Khavasi, “Analog computing by Brewster effect,” Opt. Lett. 41(15), 3467–3470 (2016).
    [Crossref] [PubMed]
  13. N. V. Golovastikov, D. A. Bykov, and L. L. Doskolovich, “Spatiotemporal pulse shaping using resonant diffraction gratings,” Opt. Lett. 40(15), 3492–3495 (2015).
    [Crossref] [PubMed]
  14. N. V. Golovastikov, D. A. Bykov, and L. L. Doskolovich, “Resonant diffraction gratings for spatial differentiation of optical beams,” Quantum Electron. 44(10), 984–988 (2014).
    [Crossref]
  15. D. A. Bykov, L. L. Doskolovich, A. A. Morozov, V. V. Podlipnov, E. A. Bezus, P. Verma, and V. A. Soifer, “First-order optical spatial differentiator based on a guided-mode resonant grating,” Opt. Express 26(8), 10997–11006 (2018).
    [Crossref] [PubMed]
  16. Z. Dong, J. Si, X. Yu, and X. Deng, “Optical spatial differentiator based on subwavelength high-contrast gratings,” Appl. Phys. Lett. 112(18), 181102 (2018).
    [Crossref]
  17. G. Calafiore, A. Koshelev, S. Dhuey, A. Goltsov, P. Sasorov, S. Babin, V. Yankov, S. Cabrini, and C. Peroz, “Holographic planar lightwave circuit for on-chip spectroscopy,” Light. Sci. Appl. 3, e203 (2014).
    [Crossref]
  18. T. Mossberg, “Planar holographic optical processing devices,” Opt. Lett. 26(7), 414–416 (2001).
    [Crossref]
  19. L. L. Doskolovich, E. A. Bezus, N. V. Golovastikov, D. A. Bykov, and V. A. Soifer, “Planar two-groove optical differentiator in a slab waveguide,” Opt. Express 25(19), 22328–22340 (2017).
    [Crossref] [PubMed]
  20. C. L. Zou, J. M. Cui, F. W. Sun, X. Xiong, X. B. Zou, Z. F. Han, and G. C. Guo, “Guiding light through optical bound states in the continuum for ultrahigh-Q microresonators,” Laser Photon. Rev. 9(1), 114–119 (2015).
    [Crossref]
  21. E. A. Bezus, D. A. Bykov, and L. L. Doskolovich, “Bound states in the continuum and high-Q resonances supported by a dielectric ridge on a slab waveguide,” https://arxiv.org/abs/1807.01888 .
  22. E. Silberstein, P. Lalanne, J.-P. Hugonin, and Q. Cao, “Use of grating theories in integrated optics,” J. Opt. Soc. Am. A 18(11), 2865–2875 (2001).
    [Crossref]
  23. J. P. Hugonin and P. Lalanne, “Perfectly matched layers as nonlinear coordinate transforms: a generalized formalization,” J. Opt. Soc. Am. A 22(9), 1844–1849 (2005).
    [Crossref]
  24. M. Hammer, A. Hildebrandt, and J. Förstner, “How planar optical waves can be made to climb dielectric steps,” Opt. Lett. 40(16), 3711–3714 (2015).
    [Crossref] [PubMed]
  25. R. D. Kekatpure, A. C. Hryciw, E. S. Barnard, and M. L. Brongersma, “Solving dielectric and plasmonic waveguide dispersion relations on a pocket calculator,” Opt. Express 17(26), 24112–24129 (2009).
    [Crossref]
  26. G. Lifante, Integrated Photonics: Fundamentals (Wiley, 2003).
    [Crossref]
  27. D. A. Bykov, L. L. Doskolovich, N. V. Golovastikov, and V. A. Soifer, “Time-domain differentiation of optical pulses in reflection and in transmission using the same resonant grating,” J. Opt. 15(10), 105703 (2013).
    [Crossref]
  28. N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, “Optical properties of photonic crystal slabs with an asymmetrical unit cell,” Phys. Rev. B 72(4), 045138 (2005).
    [Crossref]
  29. V. Karagodsky, C. Chase, and C. J. Chang-Hasnain, “Matrix Fabry–Perot resonance mechanism in high-contrast gratings,” Opt. Lett. 36(9), 1704–1706 (2011).
    [Crossref] [PubMed]
  30. F. Zangeneh-Nejad and R. Fleury, “Performing mathematical operations using high-index acoustic metamaterials,” New J. Phys. 20, 073001 (2018).
    [Crossref]
  31. D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “On the ability of resonant diffraction gratings to differentiate a pulsed optical signal,” J. Exp. Theor. Phys. 114(5), 724–730 (2012).
    [Crossref]
  32. L. L. Doskolovich, E. A. Bezus, D. A. Bykov, and V. A. Soifer, “Spatial differentiation of Bloch surface wave beams using an on-chip phase-shifted Bragg grating,” J. Opt. 18(11), 115006 (2016).
    [Crossref]

2018 (4)

F. Zangeneh-Nejad, A. Khavasi, and B. Rejaei, “Analog optical computing by half-wavelength slabs,” Opt. Commun. 407, 338–343 (2018).
[Crossref]

Z. Dong, J. Si, X. Yu, and X. Deng, “Optical spatial differentiator based on subwavelength high-contrast gratings,” Appl. Phys. Lett. 112(18), 181102 (2018).
[Crossref]

F. Zangeneh-Nejad and R. Fleury, “Performing mathematical operations using high-index acoustic metamaterials,” New J. Phys. 20, 073001 (2018).
[Crossref]

D. A. Bykov, L. L. Doskolovich, A. A. Morozov, V. V. Podlipnov, E. A. Bezus, P. Verma, and V. A. Soifer, “First-order optical spatial differentiator based on a guided-mode resonant grating,” Opt. Express 26(8), 10997–11006 (2018).
[Crossref] [PubMed]

2017 (4)

2016 (3)

2015 (6)

Z. Ruan, “Spatial mode control of surface plasmon polariton excitation with gain medium: from spatial differentiator to integrator,” Opt. Lett. 40(4), 601–604 (2015).
[Crossref] [PubMed]

N. V. Golovastikov, D. A. Bykov, and L. L. Doskolovich, “Spatiotemporal pulse shaping using resonant diffraction gratings,” Opt. Lett. 40(15), 3492–3495 (2015).
[Crossref] [PubMed]

M. Hammer, A. Hildebrandt, and J. Förstner, “How planar optical waves can be made to climb dielectric steps,” Opt. Lett. 40(16), 3711–3714 (2015).
[Crossref] [PubMed]

A. Pors, M. G. Nielsen, and S. I. Bozhevolnyi, “Analog computing using reflective plasmonic metasurfaces,” Nano Lett. 15(1), 791–797 (2015).
[Crossref]

N. V. Golovastikov, D. A. Bykov, L. L. Doskolovich, and E. A. Bezus, “Spatial optical integrator based on phase-shifted Bragg gratings,” Opt. Commun. 338, 457–460 (2015).
[Crossref]

C. L. Zou, J. M. Cui, F. W. Sun, X. Xiong, X. B. Zou, Z. F. Han, and G. C. Guo, “Guiding light through optical bound states in the continuum for ultrahigh-Q microresonators,” Laser Photon. Rev. 9(1), 114–119 (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(10), 984–988 (2014).
[Crossref]

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343(6167), 160–163 (2014).
[Crossref] [PubMed]

G. Calafiore, A. Koshelev, S. Dhuey, A. Goltsov, P. Sasorov, S. Babin, V. Yankov, S. Cabrini, and C. Peroz, “Holographic planar lightwave circuit for on-chip spectroscopy,” Light. Sci. Appl. 3, e203 (2014).
[Crossref]

L. L. Doskolovich, D. A. Bykov, E. A. Bezus, and V. A. Soifer, “Spatial differentiation of optical beams using phase-shifted Bragg grating,” Opt. Lett. 39(5), 1278–1281 (2014).
[Crossref] [PubMed]

2013 (1)

D. A. Bykov, L. L. Doskolovich, N. V. Golovastikov, and V. A. Soifer, “Time-domain differentiation of optical pulses in reflection and in transmission using the same resonant grating,” J. Opt. 15(10), 105703 (2013).
[Crossref]

2012 (1)

D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “On the ability of resonant diffraction gratings to differentiate a pulsed optical signal,” J. Exp. Theor. Phys. 114(5), 724–730 (2012).
[Crossref]

2011 (1)

2009 (1)

2005 (2)

J. P. Hugonin and P. Lalanne, “Perfectly matched layers as nonlinear coordinate transforms: a generalized formalization,” J. Opt. Soc. Am. A 22(9), 1844–1849 (2005).
[Crossref]

N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, “Optical properties of photonic crystal slabs with an asymmetrical unit cell,” Phys. Rev. B 72(4), 045138 (2005).
[Crossref]

2001 (2)

Abdollahramezani, S.

Alù, A.

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343(6167), 160–163 (2014).
[Crossref] [PubMed]

Babashah, H.

Babin, S.

G. Calafiore, A. Koshelev, S. Dhuey, A. Goltsov, P. Sasorov, S. Babin, V. Yankov, S. Cabrini, and C. Peroz, “Holographic planar lightwave circuit for on-chip spectroscopy,” Light. Sci. Appl. 3, e203 (2014).
[Crossref]

Barnard, E. S.

Bezus, E. A.

D. A. Bykov, L. L. Doskolovich, A. A. Morozov, V. V. Podlipnov, E. A. Bezus, P. Verma, and V. A. Soifer, “First-order optical spatial differentiator based on a guided-mode resonant grating,” Opt. Express 26(8), 10997–11006 (2018).
[Crossref] [PubMed]

L. L. Doskolovich, E. A. Bezus, N. V. Golovastikov, D. A. Bykov, and V. A. Soifer, “Planar two-groove optical differentiator in a slab waveguide,” Opt. Express 25(19), 22328–22340 (2017).
[Crossref] [PubMed]

L. L. Doskolovich, E. A. Bezus, D. A. Bykov, and V. A. Soifer, “Spatial differentiation of Bloch surface wave beams using an on-chip phase-shifted Bragg grating,” J. Opt. 18(11), 115006 (2016).
[Crossref]

N. V. Golovastikov, D. A. Bykov, L. L. Doskolovich, and E. A. Bezus, “Spatial optical integrator based on phase-shifted Bragg gratings,” Opt. Commun. 338, 457–460 (2015).
[Crossref]

L. L. Doskolovich, D. A. Bykov, E. A. Bezus, and V. A. Soifer, “Spatial differentiation of optical beams using phase-shifted Bragg grating,” Opt. Lett. 39(5), 1278–1281 (2014).
[Crossref] [PubMed]

E. A. Bezus, D. A. Bykov, and L. L. Doskolovich, “Bound states in the continuum and high-Q resonances supported by a dielectric ridge on a slab waveguide,” https://arxiv.org/abs/1807.01888 .

Bozhevolnyi, S. I.

A. Pors, M. G. Nielsen, and S. I. Bozhevolnyi, “Analog computing using reflective plasmonic metasurfaces,” Nano Lett. 15(1), 791–797 (2015).
[Crossref]

Brongersma, M. L.

Bykov, D. A.

D. A. Bykov, L. L. Doskolovich, A. A. Morozov, V. V. Podlipnov, E. A. Bezus, P. Verma, and V. A. Soifer, “First-order optical spatial differentiator based on a guided-mode resonant grating,” Opt. Express 26(8), 10997–11006 (2018).
[Crossref] [PubMed]

L. L. Doskolovich, E. A. Bezus, N. V. Golovastikov, D. A. Bykov, and V. A. Soifer, “Planar two-groove optical differentiator in a slab waveguide,” Opt. Express 25(19), 22328–22340 (2017).
[Crossref] [PubMed]

L. L. Doskolovich, E. A. Bezus, D. A. Bykov, and V. A. Soifer, “Spatial differentiation of Bloch surface wave beams using an on-chip phase-shifted Bragg grating,” J. Opt. 18(11), 115006 (2016).
[Crossref]

N. V. Golovastikov, D. A. Bykov, L. L. Doskolovich, and E. A. Bezus, “Spatial optical integrator based on phase-shifted Bragg gratings,” Opt. Commun. 338, 457–460 (2015).
[Crossref]

N. V. Golovastikov, D. A. Bykov, and L. L. Doskolovich, “Spatiotemporal pulse shaping using resonant diffraction gratings,” Opt. Lett. 40(15), 3492–3495 (2015).
[Crossref] [PubMed]

L. L. Doskolovich, D. A. Bykov, E. A. Bezus, and V. A. Soifer, “Spatial differentiation of optical beams using phase-shifted Bragg grating,” Opt. Lett. 39(5), 1278–1281 (2014).
[Crossref] [PubMed]

N. V. Golovastikov, D. A. Bykov, and L. L. Doskolovich, “Resonant diffraction gratings for spatial differentiation of optical beams,” Quantum Electron. 44(10), 984–988 (2014).
[Crossref]

D. A. Bykov, L. L. Doskolovich, N. V. Golovastikov, and V. A. Soifer, “Time-domain differentiation of optical pulses in reflection and in transmission using the same resonant grating,” J. Opt. 15(10), 105703 (2013).
[Crossref]

D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “On the ability of resonant diffraction gratings to differentiate a pulsed optical signal,” J. Exp. Theor. Phys. 114(5), 724–730 (2012).
[Crossref]

E. A. Bezus, D. A. Bykov, and L. L. Doskolovich, “Bound states in the continuum and high-Q resonances supported by a dielectric ridge on a slab waveguide,” https://arxiv.org/abs/1807.01888 .

Cabrini, S.

G. Calafiore, A. Koshelev, S. Dhuey, A. Goltsov, P. Sasorov, S. Babin, V. Yankov, S. Cabrini, and C. Peroz, “Holographic planar lightwave circuit for on-chip spectroscopy,” Light. Sci. Appl. 3, e203 (2014).
[Crossref]

Calafiore, G.

G. Calafiore, A. Koshelev, S. Dhuey, A. Goltsov, P. Sasorov, S. Babin, V. Yankov, S. Cabrini, and C. Peroz, “Holographic planar lightwave circuit for on-chip spectroscopy,” Light. Sci. Appl. 3, e203 (2014).
[Crossref]

Cao, Q.

Castaldi, G.

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343(6167), 160–163 (2014).
[Crossref] [PubMed]

Chang-Hasnain, C. J.

Chase, C.

Chizari, A.

Cui, J. M.

C. L. Zou, J. M. Cui, F. W. Sun, X. Xiong, X. B. Zou, Z. F. Han, and G. C. Guo, “Guiding light through optical bound states in the continuum for ultrahigh-Q microresonators,” Laser Photon. Rev. 9(1), 114–119 (2015).
[Crossref]

Deng, X.

Z. Dong, J. Si, X. Yu, and X. Deng, “Optical spatial differentiator based on subwavelength high-contrast gratings,” Appl. Phys. Lett. 112(18), 181102 (2018).
[Crossref]

Dhuey, S.

G. Calafiore, A. Koshelev, S. Dhuey, A. Goltsov, P. Sasorov, S. Babin, V. Yankov, S. Cabrini, and C. Peroz, “Holographic planar lightwave circuit for on-chip spectroscopy,” Light. Sci. Appl. 3, e203 (2014).
[Crossref]

Dong, Z.

Z. Dong, J. Si, X. Yu, and X. Deng, “Optical spatial differentiator based on subwavelength high-contrast gratings,” Appl. Phys. Lett. 112(18), 181102 (2018).
[Crossref]

Doskolovich, L. L.

D. A. Bykov, L. L. Doskolovich, A. A. Morozov, V. V. Podlipnov, E. A. Bezus, P. Verma, and V. A. Soifer, “First-order optical spatial differentiator based on a guided-mode resonant grating,” Opt. Express 26(8), 10997–11006 (2018).
[Crossref] [PubMed]

L. L. Doskolovich, E. A. Bezus, N. V. Golovastikov, D. A. Bykov, and V. A. Soifer, “Planar two-groove optical differentiator in a slab waveguide,” Opt. Express 25(19), 22328–22340 (2017).
[Crossref] [PubMed]

L. L. Doskolovich, E. A. Bezus, D. A. Bykov, and V. A. Soifer, “Spatial differentiation of Bloch surface wave beams using an on-chip phase-shifted Bragg grating,” J. Opt. 18(11), 115006 (2016).
[Crossref]

N. V. Golovastikov, D. A. Bykov, L. L. Doskolovich, and E. A. Bezus, “Spatial optical integrator based on phase-shifted Bragg gratings,” Opt. Commun. 338, 457–460 (2015).
[Crossref]

N. V. Golovastikov, D. A. Bykov, and L. L. Doskolovich, “Spatiotemporal pulse shaping using resonant diffraction gratings,” Opt. Lett. 40(15), 3492–3495 (2015).
[Crossref] [PubMed]

L. L. Doskolovich, D. A. Bykov, E. A. Bezus, and V. A. Soifer, “Spatial differentiation of optical beams using phase-shifted Bragg grating,” Opt. Lett. 39(5), 1278–1281 (2014).
[Crossref] [PubMed]

N. V. Golovastikov, D. A. Bykov, and L. L. Doskolovich, “Resonant diffraction gratings for spatial differentiation of optical beams,” Quantum Electron. 44(10), 984–988 (2014).
[Crossref]

D. A. Bykov, L. L. Doskolovich, N. V. Golovastikov, and V. A. Soifer, “Time-domain differentiation of optical pulses in reflection and in transmission using the same resonant grating,” J. Opt. 15(10), 105703 (2013).
[Crossref]

D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “On the ability of resonant diffraction gratings to differentiate a pulsed optical signal,” J. Exp. Theor. Phys. 114(5), 724–730 (2012).
[Crossref]

E. A. Bezus, D. A. Bykov, and L. L. Doskolovich, “Bound states in the continuum and high-Q resonances supported by a dielectric ridge on a slab waveguide,” https://arxiv.org/abs/1807.01888 .

Engheta, N.

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343(6167), 160–163 (2014).
[Crossref] [PubMed]

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] [PubMed]

Fleury, R.

F. Zangeneh-Nejad and R. Fleury, “Performing mathematical operations using high-index acoustic metamaterials,” New J. Phys. 20, 073001 (2018).
[Crossref]

Förstner, J.

Galdi, V.

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343(6167), 160–163 (2014).
[Crossref] [PubMed]

Gippius, N. A.

N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, “Optical properties of photonic crystal slabs with an asymmetrical unit cell,” Phys. Rev. B 72(4), 045138 (2005).
[Crossref]

Golovastikov, N. V.

L. L. Doskolovich, E. A. Bezus, N. V. Golovastikov, D. A. Bykov, and V. A. Soifer, “Planar two-groove optical differentiator in a slab waveguide,” Opt. Express 25(19), 22328–22340 (2017).
[Crossref] [PubMed]

N. V. Golovastikov, D. A. Bykov, and L. L. Doskolovich, “Spatiotemporal pulse shaping using resonant diffraction gratings,” Opt. Lett. 40(15), 3492–3495 (2015).
[Crossref] [PubMed]

N. V. Golovastikov, D. A. Bykov, L. L. Doskolovich, and E. A. Bezus, “Spatial optical integrator based on phase-shifted Bragg gratings,” Opt. Commun. 338, 457–460 (2015).
[Crossref]

N. V. Golovastikov, D. A. Bykov, and L. L. Doskolovich, “Resonant diffraction gratings for spatial differentiation of optical beams,” Quantum Electron. 44(10), 984–988 (2014).
[Crossref]

D. A. Bykov, L. L. Doskolovich, N. V. Golovastikov, and V. A. Soifer, “Time-domain differentiation of optical pulses in reflection and in transmission using the same resonant grating,” J. Opt. 15(10), 105703 (2013).
[Crossref]

Goltsov, A.

G. Calafiore, A. Koshelev, S. Dhuey, A. Goltsov, P. Sasorov, S. Babin, V. Yankov, S. Cabrini, and C. Peroz, “Holographic planar lightwave circuit for on-chip spectroscopy,” Light. Sci. Appl. 3, e203 (2014).
[Crossref]

Goodman, J.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Guo, G. C.

C. L. Zou, J. M. Cui, F. W. Sun, X. Xiong, X. B. Zou, Z. F. Han, and G. C. Guo, “Guiding light through optical bound states in the continuum for ultrahigh-Q microresonators,” Laser Photon. Rev. 9(1), 114–119 (2015).
[Crossref]

Hammer, M.

Han, Z. F.

C. L. Zou, J. M. Cui, F. W. Sun, X. Xiong, X. B. Zou, Z. F. Han, and G. C. Guo, “Guiding light through optical bound states in the continuum for ultrahigh-Q microresonators,” Laser Photon. Rev. 9(1), 114–119 (2015).
[Crossref]

Hildebrandt, A.

Hryciw, A. C.

Hugonin, J. P.

Hugonin, J.-P.

Ishihara, T.

N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, “Optical properties of photonic crystal slabs with an asymmetrical unit cell,” Phys. Rev. B 72(4), 045138 (2005).
[Crossref]

Jamali, M. V.

Karagodsky, V.

Kavehvash, Z.

Kekatpure, R. D.

Khavasi, A.

Koohi, S.

Koshelev, A.

G. Calafiore, A. Koshelev, S. Dhuey, A. Goltsov, P. Sasorov, S. Babin, V. Yankov, S. Cabrini, and C. Peroz, “Holographic planar lightwave circuit for on-chip spectroscopy,” Light. Sci. Appl. 3, e203 (2014).
[Crossref]

Lalanne, P.

Lifante, G.

G. Lifante, Integrated Photonics: Fundamentals (Wiley, 2003).
[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] [PubMed]

Monticone, F.

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343(6167), 160–163 (2014).
[Crossref] [PubMed]

Morozov, A. A.

Mossberg, T.

Nielsen, M. G.

A. Pors, M. G. Nielsen, and S. I. Bozhevolnyi, “Analog computing using reflective plasmonic metasurfaces,” Nano Lett. 15(1), 791–797 (2015).
[Crossref]

Peroz, C.

G. Calafiore, A. Koshelev, S. Dhuey, A. Goltsov, P. Sasorov, S. Babin, V. Yankov, S. Cabrini, and C. Peroz, “Holographic planar lightwave circuit for on-chip spectroscopy,” Light. Sci. Appl. 3, e203 (2014).
[Crossref]

Podlipnov, V. V.

Pors, A.

A. Pors, M. G. Nielsen, and S. I. Bozhevolnyi, “Analog computing using reflective plasmonic metasurfaces,” Nano Lett. 15(1), 791–797 (2015).
[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] [PubMed]

Rejaei, B.

F. Zangeneh-Nejad, A. Khavasi, and B. Rejaei, “Analog optical computing by half-wavelength slabs,” Opt. Commun. 407, 338–343 (2018).
[Crossref]

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] [PubMed]

Z. Ruan, “Spatial mode control of surface plasmon polariton excitation with gain medium: from spatial differentiator to integrator,” Opt. Lett. 40(4), 601–604 (2015).
[Crossref] [PubMed]

Salehi, J. A.

Sasorov, P.

G. Calafiore, A. Koshelev, S. Dhuey, A. Goltsov, P. Sasorov, S. Babin, V. Yankov, S. Cabrini, and C. Peroz, “Holographic planar lightwave circuit for on-chip spectroscopy,” Light. Sci. Appl. 3, e203 (2014).
[Crossref]

Si, J.

Z. Dong, J. Si, X. Yu, and X. Deng, “Optical spatial differentiator based on subwavelength high-contrast gratings,” Appl. Phys. Lett. 112(18), 181102 (2018).
[Crossref]

Silberstein, E.

Silva, A.

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343(6167), 160–163 (2014).
[Crossref] [PubMed]

Soifer, V. A.

D. A. Bykov, L. L. Doskolovich, A. A. Morozov, V. V. Podlipnov, E. A. Bezus, P. Verma, and V. A. Soifer, “First-order optical spatial differentiator based on a guided-mode resonant grating,” Opt. Express 26(8), 10997–11006 (2018).
[Crossref] [PubMed]

L. L. Doskolovich, E. A. Bezus, N. V. Golovastikov, D. A. Bykov, and V. A. Soifer, “Planar two-groove optical differentiator in a slab waveguide,” Opt. Express 25(19), 22328–22340 (2017).
[Crossref] [PubMed]

L. L. Doskolovich, E. A. Bezus, D. A. Bykov, and V. A. Soifer, “Spatial differentiation of Bloch surface wave beams using an on-chip phase-shifted Bragg grating,” J. Opt. 18(11), 115006 (2016).
[Crossref]

L. L. Doskolovich, D. A. Bykov, E. A. Bezus, and V. A. Soifer, “Spatial differentiation of optical beams using phase-shifted Bragg grating,” Opt. Lett. 39(5), 1278–1281 (2014).
[Crossref] [PubMed]

D. A. Bykov, L. L. Doskolovich, N. V. Golovastikov, and V. A. Soifer, “Time-domain differentiation of optical pulses in reflection and in transmission using the same resonant grating,” J. Opt. 15(10), 105703 (2013).
[Crossref]

D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “On the ability of resonant diffraction gratings to differentiate a pulsed optical signal,” J. Exp. Theor. Phys. 114(5), 724–730 (2012).
[Crossref]

Sun, F. W.

C. L. Zou, J. M. Cui, F. W. Sun, X. Xiong, X. B. Zou, Z. F. Han, and G. C. Guo, “Guiding light through optical bound states in the continuum for ultrahigh-Q microresonators,” Laser Photon. Rev. 9(1), 114–119 (2015).
[Crossref]

Tikhodeev, S. G.

N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, “Optical properties of photonic crystal slabs with an asymmetrical unit cell,” Phys. Rev. B 72(4), 045138 (2005).
[Crossref]

Verma, P.

Xiong, X.

C. L. Zou, J. M. Cui, F. W. Sun, X. Xiong, X. B. Zou, Z. F. Han, and G. C. Guo, “Guiding light through optical bound states in the continuum for ultrahigh-Q microresonators,” Laser Photon. Rev. 9(1), 114–119 (2015).
[Crossref]

Yankov, V.

G. Calafiore, A. Koshelev, S. Dhuey, A. Goltsov, P. Sasorov, S. Babin, V. Yankov, S. Cabrini, and C. Peroz, “Holographic planar lightwave circuit for on-chip spectroscopy,” Light. Sci. Appl. 3, e203 (2014).
[Crossref]

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] [PubMed]

Youssefi, A.

Yu, X.

Z. Dong, J. Si, X. Yu, and X. Deng, “Optical spatial differentiator based on subwavelength high-contrast gratings,” Appl. Phys. Lett. 112(18), 181102 (2018).
[Crossref]

Zangeneh-Nejad, F.

F. Zangeneh-Nejad, A. Khavasi, and B. Rejaei, “Analog optical computing by half-wavelength slabs,” Opt. Commun. 407, 338–343 (2018).
[Crossref]

F. Zangeneh-Nejad and R. Fleury, “Performing mathematical operations using high-index acoustic metamaterials,” New J. Phys. 20, 073001 (2018).
[Crossref]

F. Zangeneh-Nejad and A. Khavasi, “Spatial integration by a dielectric slab and its planar graphene-based counterpart,” Opt. Lett. 42(10), 1954–1957 (2017).
[Crossref] [PubMed]

A. Youssefi, F. Zangeneh-Nejad, S. Abdollahramezani, and A. Khavasi, “Analog computing by Brewster effect,” Opt. Lett. 41(15), 3467–3470 (2016).
[Crossref] [PubMed]

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] [PubMed]

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] [PubMed]

Zou, C. L.

C. L. Zou, J. M. Cui, F. W. Sun, X. Xiong, X. B. Zou, Z. F. Han, and G. C. Guo, “Guiding light through optical bound states in the continuum for ultrahigh-Q microresonators,” Laser Photon. Rev. 9(1), 114–119 (2015).
[Crossref]

Zou, X. B.

C. L. Zou, J. M. Cui, F. W. Sun, X. Xiong, X. B. Zou, Z. F. Han, and G. C. Guo, “Guiding light through optical bound states in the continuum for ultrahigh-Q microresonators,” Laser Photon. Rev. 9(1), 114–119 (2015).
[Crossref]

Appl. Phys. Lett. (1)

Z. Dong, J. Si, X. Yu, and X. Deng, “Optical spatial differentiator based on subwavelength high-contrast gratings,” Appl. Phys. Lett. 112(18), 181102 (2018).
[Crossref]

J. Exp. Theor. Phys. (1)

D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “On the ability of resonant diffraction gratings to differentiate a pulsed optical signal,” J. Exp. Theor. Phys. 114(5), 724–730 (2012).
[Crossref]

J. Opt. (2)

L. L. Doskolovich, E. A. Bezus, D. A. Bykov, and V. A. Soifer, “Spatial differentiation of Bloch surface wave beams using an on-chip phase-shifted Bragg grating,” J. Opt. 18(11), 115006 (2016).
[Crossref]

D. A. Bykov, L. L. Doskolovich, N. V. Golovastikov, and V. A. Soifer, “Time-domain differentiation of optical pulses in reflection and in transmission using the same resonant grating,” J. Opt. 15(10), 105703 (2013).
[Crossref]

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (1)

Laser Photon. Rev. (1)

C. L. Zou, J. M. Cui, F. W. Sun, X. Xiong, X. B. Zou, Z. F. Han, and G. C. Guo, “Guiding light through optical bound states in the continuum for ultrahigh-Q microresonators,” Laser Photon. Rev. 9(1), 114–119 (2015).
[Crossref]

Light. Sci. Appl. (1)

G. Calafiore, A. Koshelev, S. Dhuey, A. Goltsov, P. Sasorov, S. Babin, V. Yankov, S. Cabrini, and C. Peroz, “Holographic planar lightwave circuit for on-chip spectroscopy,” Light. Sci. Appl. 3, e203 (2014).
[Crossref]

Nano Lett. (1)

A. Pors, M. G. Nielsen, and S. I. Bozhevolnyi, “Analog computing using reflective plasmonic metasurfaces,” Nano Lett. 15(1), 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] [PubMed]

New J. Phys. (1)

F. Zangeneh-Nejad and R. Fleury, “Performing mathematical operations using high-index acoustic metamaterials,” New J. Phys. 20, 073001 (2018).
[Crossref]

Opt. Commun. (2)

N. V. Golovastikov, D. A. Bykov, L. L. Doskolovich, and E. A. Bezus, “Spatial optical integrator based on phase-shifted Bragg gratings,” Opt. Commun. 338, 457–460 (2015).
[Crossref]

F. Zangeneh-Nejad, A. Khavasi, and B. Rejaei, “Analog optical computing by half-wavelength slabs,” Opt. Commun. 407, 338–343 (2018).
[Crossref]

Opt. Express (3)

Opt. Lett. (9)

M. Hammer, A. Hildebrandt, and J. Förstner, “How planar optical waves can be made to climb dielectric steps,” Opt. Lett. 40(16), 3711–3714 (2015).
[Crossref] [PubMed]

V. Karagodsky, C. Chase, and C. J. Chang-Hasnain, “Matrix Fabry–Perot resonance mechanism in high-contrast gratings,” Opt. Lett. 36(9), 1704–1706 (2011).
[Crossref] [PubMed]

T. Mossberg, “Planar holographic optical processing devices,” Opt. Lett. 26(7), 414–416 (2001).
[Crossref]

A. Youssefi, F. Zangeneh-Nejad, S. Abdollahramezani, and A. Khavasi, “Analog computing by Brewster effect,” Opt. Lett. 41(15), 3467–3470 (2016).
[Crossref] [PubMed]

N. V. Golovastikov, D. A. Bykov, and L. L. Doskolovich, “Spatiotemporal pulse shaping using resonant diffraction gratings,” Opt. Lett. 40(15), 3492–3495 (2015).
[Crossref] [PubMed]

Z. Ruan, “Spatial mode control of surface plasmon polariton excitation with gain medium: from spatial differentiator to integrator,” Opt. Lett. 40(4), 601–604 (2015).
[Crossref] [PubMed]

F. Zangeneh-Nejad and A. Khavasi, “Spatial integration by a dielectric slab and its planar graphene-based counterpart,” Opt. Lett. 42(10), 1954–1957 (2017).
[Crossref] [PubMed]

L. L. Doskolovich, D. A. Bykov, E. A. Bezus, and V. A. Soifer, “Spatial differentiation of optical beams using phase-shifted Bragg grating,” Opt. Lett. 39(5), 1278–1281 (2014).
[Crossref] [PubMed]

A. Chizari, S. Abdollahramezani, M. V. Jamali, and J. A. Salehi, “Analog optical computing based on a dielectric meta-reflect array,” Opt. Lett. 41(15), 3451–3454 (2016).
[Crossref] [PubMed]

Phys. Rev. B (1)

N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, “Optical properties of photonic crystal slabs with an asymmetrical unit cell,” Phys. Rev. B 72(4), 045138 (2005).
[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(10), 984–988 (2014).
[Crossref]

Science (1)

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343(6167), 160–163 (2014).
[Crossref] [PubMed]

Other (3)

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

G. Lifante, Integrated Photonics: Fundamentals (Wiley, 2003).
[Crossref]

E. A. Bezus, D. A. Bykov, and L. L. Doskolovich, “Bound states in the continuum and high-Q resonances supported by a dielectric ridge on a slab waveguide,” https://arxiv.org/abs/1807.01888 .

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Figures (4)

Fig. 1
Fig. 1 Geometry of the problem of diffraction of a waveguide mode on a dielectric step.
Fig. 2
Fig. 2 Reflectance (a) and transmittance (b) of the ridge vs. the ridge length l (horizontal axis) and the angle of incidence θ (vertical axis). Dashed lines show the boundaries of the resonance region and correspond to the cutoff angles of the TM-modes outside the ridge (at hc = 80 nm, upper line) and inside the ridge (hr = 110 nm, lower line). Dotted curves show the dispersion of the cross-polarized modes of the ridge. The points marked with white and black asterisks correspond to the spatial integration and differentiation examples considered below, respectively.
Fig. 3
Fig. 3 Amplitudes (absolute values) and phases (arguments) of the reflection transfer functions HR (ku,inc) calculated at the points (lunity,1, θunity,1) = (0.355 μm, 53.28°) (a), (b) and (lunity,2, θunity,2) = (0.36 μm, 53.37°) (d), (e). Dotted curves in (a) and (d) show the spectrum of the incident beam, dashed curves in (a), (b) and (d), (e) show the amplitudes and phases of the TF approximations calculated using Eq. (7). (c), (f) Absolute values of the calculated profiles of the reflected beams corresponding to the TFs shown in (a), (b) and (d), (e), respectively. Dotted curves in (c) and (f) show the incident beam, dashed curves show the absolute values of the analytically calculated integral.
Fig. 4
Fig. 4 Amplitudes (absolute values) and phases (arguments) of the transmission transfer functions HT (ku,inc) calculated at the points (lzero,1, θzero,1) = (0.2 μm, 48.62°) (a) and (lzero,2, θzero,2) = (0.24 μm, 50.14°) (c). Dotted curves in (a) and (c) show the normalized spectrum of the incident Gaussian beam. (b), (d) Absolute values of the calculated profiles of the transmitted beams corresponding to the TFs shown in (a) and (c), respectively. Dotted curves show the incident Gaussian beam, dashed curves show the absolute value of the analytically calculated derivative of the Gaussian function.

Equations (8)

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l = π m + arg r ( θ ) k 0 n r , TM cos θ ,
Ψ inc , wg ( u inc , v inc ) = exp ( i k 0 n wg , TE v inc ) P inc , wg ( u inc ) = exp ( i k 0 n wg , TE v inc ) G ( k u , inc ) exp ( i k u , inc u inc ) d k u , inc ,
H R ( k u , inc ) = R TE ( k x ( k u , inc ) ) , H T ( k u , inc ) = T TE ( k x ( k u , inc ) ) ,
k x ( k u , inc ) = k 0 n wg , TE sin ( θ inc + θ 0 ) k u , inc cos θ 0 + k x , 0 ,
P refl ( u refl ) = G ( k u , inc ) R ( k x ( k u , inc ) ) exp ( i k u , inc u refl ) d k u , inc , P tr ( u tr ) = G ( k u , inc ) T ( k x ( k u , inc ) ) exp ( i k u , inc u tr ) d k u , inc .
R TE ( k r ) r + b k x k x , p ,
H R ( k u , inc ) = γ k u , inc i γ ,
H T ( k u , inc ) α T k u , inc .