Abstract

We present an experimental demonstration of a subwavelength diffraction grating performing first-order differentiation of the transverse profile of an incident optical beam with respect to a spatial variable. The experimental results are in a good agreement with the presented analytical model suggesting that the differentiation is performed in transmission at oblique incidence and is associated with the guided-mode resonance of the grating. According to this model, the transfer function of the grating in the vicinity of the resonance is close to the transfer function of an exact differentiator. We confirm this by estimating the transfer function of the fabricated structure on the basis of the measured profiles of the incident and transmitted beams. The considered structure may find application in the design of new photonic devices for beam shaping, optical information processing, and analog optical computing.

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

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References

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    [Crossref] [PubMed]
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2018 (3)

2017 (3)

2016 (2)

Y. Hwang and T. J. Davis, “Optical metasurfaces for subwavelength difference operations,” Appl. Phys. Lett. 109, 181101 (2016).
[Crossref]

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

2015 (3)

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

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

D. A. Bykov and L. L. Doskolovich, “ω−kx Fano line shape in photonic crystal slabs,” Phys. Rev. A 92, 013845 (2015).
[Crossref]

2014 (4)

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

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343, 160–163 (2014).
[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, 1278–1281 (2014).

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] [PubMed]

1996 (1)

1995 (1)

Abdollahramezani, S.

Alù, A.

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

Bezus, E. A.

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]

Bykov, D. A.

Castaldi, G.

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

Davis, T. J.

Doskolovich, L. L.

Engheta, N.

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

Fan, S.

C. Guo, M. Xiao, M. Minkov, Y. Shi, and S. Fan, “Photonic crystal slab Laplace operator for image differentiation,” Optica 5, 251–256 (2018).
[Crossref]

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]

Galdi, V.

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

Gaylord, T. K.

Golovastikov, N. V.

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

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics(McGraw-Hill, 1968).

Grann, E. B.

Guo, C.

Hwang, Y.

Khavasi, A.

Li, L.

Lin, J.

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]

Minkov, M.

Moharam, M. G.

Monticone, F.

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

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]

Pommet, D. A.

Pors, A.

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

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, 601–604 (2015).

Shi, Y.

Silva, A.

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

Soifer, V. A.

Xiao, M.

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.

Yuan, X.-C.

Zangeneh-Nejad, F.

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]

Appl. Phys. Lett. (1)

Y. Hwang and T. J. Davis, “Optical metasurfaces for subwavelength difference operations,” Appl. Phys. Lett. 109, 181101 (2016).
[Crossref]

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

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] [PubMed]

Opt. Commun. (1)

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

Opt. Express (3)

Opt. Lett. (4)

Optica (1)

Phys. Rev. A (1)

D. A. Bykov and L. L. Doskolovich, “ω−kx Fano line shape in photonic crystal slabs,” Phys. Rev. A 92, 013845 (2015).
[Crossref]

Quant. Electron. (1)

N. V. Golovastikov, D. A. Bykov, and L. L. Doskolovich, “Resonant diffraction gratings for spatial differentiation of optical beams,” Quant. Electron. 44, 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, 160–163 (2014).
[Crossref] [PubMed]

Other (2)

J. W. Goodman, Introduction to Fourier Optics(McGraw-Hill, 1968).

A. Saba, M. R. Tavakol, P. Karimi-Khoozani, and A. Khavasi, “Two dimensional edge detection by guided mode resonant metasurface,” https://arxiv.org/abs/1711.01606 .

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Figures (5)

Fig. 1
Fig. 1 Diffraction of an optical beam on a resonant diffraction structure consisting of a grating on top of a slab waveguide layer deposited on a substrate. The transverse profiles of the incident (Pinc), transmitted (Ptr), and reflected beams schematically illustrating the spatial differentiation performed in transmission are shown.
Fig. 2
Fig. 2 Scanning electron microscopy image of the diffraction grating fabricated from ERP-40 electron resist on top of a TiO2 layer.
Fig. 3
Fig. 3 Optical setup of the experiment: (a) light source, (b) 20× microobjective, (c) 40 µm pinhole, (d) and (e) lenses, (f) polarizer, (g) lens with a focal length of 120 mm, (h) sample grating, (i) 8× microobjective, (j) CCD matrix.
Fig. 4
Fig. 4 (a) Measured profile of the incident Gaussian beam (σ = 9.52 µm), (b) measured profile of the transmitted beam, (c) analytically calculated derivative of the incident beam, (d) profile of the transmitted beam calculated taking into account the imperfections of the fabricated structure.
Fig. 5
Fig. 5 (a) Cross sections of the transmitted beam profile: exact derivative (dashed black curve), model (solid blue curve), experiment (red curve with dot markers); (b) intensity (squared absolute value) and phase (argument) of the transfer functions of an exact differentiator (dashed curves) and of the estimated transfer function of the fabricated structure (solid curves with dot markers).

Equations (15)

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H ( k x , k y ) T ( k x cos θ + k 0 2 k x 2 k y 2 sin θ , k y ) ,
T ( k x , k y ) = t v g 2 k x 2 ( ω ω zt η zt k y 2 ) ( ω ω p2 η p2 k y 2 ) v g 2 k x 2 ( ω ω p1 η p1 k y 2 ) ( ω ω p2 η p2 k y 2 ) .
T ( k x , k y ) α ( k x k x 0 ) + β k y 2 .
T ( k x , k y ) α ( k x k x 0 ) .
H ( k x , k y ) i k x .
H ( k x ) a + i k x ,
x = x cos θ + z sin θ , z = x sin θ + z cos θ .
( k x , k y , k z ) = ( k x cos θ + k z sin θ , k y , k z cos θ k x sin θ ) .
Ψ TM = 1 k 0 k z [ k 0 k z 0 k 0 k x k x k y k x 2 + k z 2 k y k z ] e i k x x + i k y y + i k z z , Ψ TM = 1 k 0 k z [ k x k y ( k x 2 + k z 2 ) k y k z k 0 k z 0 k 0 k x ] e i k x x + i k y y + i k z z ,
G ( k x , k y ) = 1 ( 2 π ) 2 P inc ( x , y ) e i k x x i k y y d x d y .
Ψ inc ( x , y , z ) = G ( k x , k y ) k z k z Ψ TM d k x d k y .
T same ( k x , k y ) Ψ TM + T cross ( k x , k y ) Ψ TE ,
Ψ tr ( x , y , z ) = G ( k x , k y ) k z k z [ T same ( k x , k y ) Ψ TM + T cross ( k x , k y ) Ψ TE ] d k x d k y .
Ψ tr ( x , y , z ) = G ( k x , k y ) k z k z T same ( k x , k y ) Ψ TM d k x d k y .
P tr ( x , y ) = G ( k x , k y ) T same ( k x , k y ) e i k x x + i k y y d k x d k y .

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