Abstract

In this paper, the relation between gain and resolution of an ideal analog optical differentiator in two different cases and their fundamental limits are investigated. Based on this relation, a figure of merit for comparison of the designed differentiators in recent papers is proposed. The differentiators are optimized using this figure of merit, and they are compared with each other to determine the best one. Also, a new differentiator is presented based on the dielectric slab waveguide in which the trade-off between its gain and resolution is easily controllable, and its best operating point is determined.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]

2019 (1)

T. Zhu, Y. Lou, Y. Zhou, J. Zhang, J. Huang, Y. Li, H. Luo, S. Wen, S. Zhu, Q. Gong, M. Qiu, and Z. Ruan, “Generalized spatial differentiation from the spin hall effect of light and its application in image processing of edge detection,” Phys. Rev. Appl. 11(3), 034043 (2019).
[Crossref]

2018 (2)

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

A. Saba, M. R. Tavakol, P. Karimi-Khoozani, and A. Khavasi, “Two-dimensional edge detection by guided mode resonant metasurface,” IEEE Photonics Technol. Lett. 30(9), 853–856 (2018).
[Crossref]

2017 (2)

T. Zhu, Y. Zhou, Y. Lou, H. Ye, M. Qiu, Z. Ruan, and S. Fan, “Plasmonic computing of spatial differentiation,” Nat. Commun. 8(1), 15391 (2017).
[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]

2016 (1)

2015 (3)

2014 (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]

2013 (2)

2012 (1)

J. Ren, Y. Li, Y. Lin, Y. Qin, R. Wu, J. Yang, Y. F. Xiao, H. Yang, and Q. Gong, “Spin hall effect of light reflected from a magnetic thin film,” Appl. Phys. Lett. 101(17), 171103 (2012).
[Crossref]

2009 (1)

2008 (2)

1986 (1)

J. Canny, “A computational approach to edge detection,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-8(6), 679–698 (1986).
[Crossref]

Abdollahramezani, S.

Ahn, T. J.

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]

Ayotte, N.

Azaña, J.

Canny, J.

J. Canny, “A computational approach to edge detection,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-8(6), 679–698 (1986).
[Crossref]

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]

Chen, J.

T. Yang, J. Dong, L. Lu, L. Zhou, A. Zheng, X. Zhang, and J. Chen, “All-optical differential equation solver with constant-coefficient tunable based on a single microring resonator,” Sci. Rep. 4(1), 5581 (2015).
[Crossref]

Dong, J.

Dong, J. J.

Doucet, S.

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]

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(1), 15391 (2017).
[Crossref]

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]

Gao, D. S.

Gong, Q.

T. Zhu, Y. Lou, Y. Zhou, J. Zhang, J. Huang, Y. Li, H. Luo, S. Wen, S. Zhu, Q. Gong, M. Qiu, and Z. Ruan, “Generalized spatial differentiation from the spin hall effect of light and its application in image processing of edge detection,” Phys. Rev. Appl. 11(3), 034043 (2019).
[Crossref]

J. Ren, Y. Li, Y. Lin, Y. Qin, R. Wu, J. Yang, Y. F. Xiao, H. Yang, and Q. Gong, “Spin hall effect of light reflected from a magnetic thin film,” Appl. Phys. Lett. 101(17), 171103 (2012).
[Crossref]

Hu, S.

Huang, J.

T. Zhu, Y. Lou, Y. Zhou, J. Zhang, J. Huang, Y. Li, H. Luo, S. Wen, S. Zhu, Q. Gong, M. Qiu, and Z. Ruan, “Generalized spatial differentiation from the spin hall effect of light and its application in image processing of edge detection,” Phys. Rev. Appl. 11(3), 034043 (2019).
[Crossref]

Huang, T. L.

Janner, D.

Karimi, P.

A. P. M. Q. Vafa, P. Karimi, and A. Khavasi, “Recent advances in spatial analog optical computing,” in 2018 Fifth International Conference on Millimeter-Wave and Terahertz Technologies (MMWaTT), (IEEE, 2018), pp. 6–11.

Karimi-Khoozani, P.

A. Saba, M. R. Tavakol, P. Karimi-Khoozani, and A. Khavasi, “Two-dimensional edge detection by guided mode resonant metasurface,” IEEE Photonics Technol. Lett. 30(9), 853–856 (2018).
[Crossref]

Khavasi, A.

A. Saba, M. R. Tavakol, P. Karimi-Khoozani, and A. Khavasi, “Two-dimensional edge detection by guided mode resonant metasurface,” IEEE Photonics Technol. Lett. 30(9), 853–856 (2018).
[Crossref]

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

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

A. P. M. Q. Vafa, P. Karimi, and A. Khavasi, “Recent advances in spatial analog optical computing,” in 2018 Fifth International Conference on Millimeter-Wave and Terahertz Technologies (MMWaTT), (IEEE, 2018), pp. 6–11.

LaRochelle, S.

Lei, L.

Li, M.

Li, Y.

T. Zhu, Y. Lou, Y. Zhou, J. Zhang, J. Huang, Y. Li, H. Luo, S. Wen, S. Zhu, Q. Gong, M. Qiu, and Z. Ruan, “Generalized spatial differentiation from the spin hall effect of light and its application in image processing of edge detection,” Phys. Rev. Appl. 11(3), 034043 (2019).
[Crossref]

J. Ren, Y. Li, Y. Lin, Y. Qin, R. Wu, J. Yang, Y. F. Xiao, H. Yang, and Q. Gong, “Spin hall effect of light reflected from a magnetic thin film,” Appl. Phys. Lett. 101(17), 171103 (2012).
[Crossref]

Lin, Y.

J. Ren, Y. Li, Y. Lin, Y. Qin, R. Wu, J. Yang, Y. F. Xiao, H. Yang, and Q. Gong, “Spin hall effect of light reflected from a magnetic thin film,” Appl. Phys. Lett. 101(17), 171103 (2012).
[Crossref]

Liu, F.

Lou, Y.

T. Zhu, Y. Lou, Y. Zhou, J. Zhang, J. Huang, Y. Li, H. Luo, S. Wen, S. Zhu, Q. Gong, M. Qiu, and Z. Ruan, “Generalized spatial differentiation from the spin hall effect of light and its application in image processing of edge detection,” Phys. Rev. Appl. 11(3), 034043 (2019).
[Crossref]

T. Zhu, Y. Zhou, Y. Lou, H. Ye, M. Qiu, Z. Ruan, and S. Fan, “Plasmonic computing of spatial differentiation,” Nat. Commun. 8(1), 15391 (2017).
[Crossref]

Lu, L.

T. Yang, J. Dong, L. Lu, L. Zhou, A. Zheng, X. Zhang, and J. Chen, “All-optical differential equation solver with constant-coefficient tunable based on a single microring resonator,” Sci. Rep. 4(1), 5581 (2015).
[Crossref]

Luo, H.

T. Zhu, Y. Lou, Y. Zhou, J. Zhang, J. Huang, Y. Li, H. Luo, S. Wen, S. Zhu, Q. Gong, M. Qiu, and Z. Ruan, “Generalized spatial differentiation from the spin hall effect of light and its application in image processing of edge detection,” Phys. Rev. Appl. 11(3), 034043 (2019).
[Crossref]

Maier, S. A.

S. A. Maier, Plasmonics: fundamentals and applications (Springer Science & Business Media, 2007).

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]

Park, Y.

Pruneri, V.

Q. Vafa, A. P. M.

A. P. M. Q. Vafa, P. Karimi, and A. Khavasi, “Recent advances in spatial analog optical computing,” in 2018 Fifth International Conference on Millimeter-Wave and Terahertz Technologies (MMWaTT), (IEEE, 2018), pp. 6–11.

Qiang, L.

Qin, Y.

J. Ren, Y. Li, Y. Lin, Y. Qin, R. Wu, J. Yang, Y. F. Xiao, H. Yang, and Q. Gong, “Spin hall effect of light reflected from a magnetic thin film,” Appl. Phys. Lett. 101(17), 171103 (2012).
[Crossref]

Qiu, M.

T. Zhu, Y. Lou, Y. Zhou, J. Zhang, J. Huang, Y. Li, H. Luo, S. Wen, S. Zhu, Q. Gong, M. Qiu, and Z. Ruan, “Generalized spatial differentiation from the spin hall effect of light and its application in image processing of edge detection,” Phys. Rev. Appl. 11(3), 034043 (2019).
[Crossref]

T. Zhu, Y. Zhou, Y. Lou, H. Ye, M. Qiu, Z. Ruan, and S. Fan, “Plasmonic computing of spatial differentiation,” Nat. Commun. 8(1), 15391 (2017).
[Crossref]

F. Liu, T. Wang, L. Qiang, T. Ye, Z. Zhang, M. Qiu, and Y. Su, “Compact optical temporal differentiator based on silicon microring resonator,” Opt. Express 16(20), 15880–15886 (2008).
[Crossref]

Rejaei, B.

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

Ren, J.

J. Ren, Y. Li, Y. Lin, Y. Qin, R. Wu, J. Yang, Y. F. Xiao, H. Yang, and Q. Gong, “Spin hall effect of light reflected from a magnetic thin film,” Appl. Phys. Lett. 101(17), 171103 (2012).
[Crossref]

Ruan, Z.

T. Zhu, Y. Lou, Y. Zhou, J. Zhang, J. Huang, Y. Li, H. Luo, S. Wen, S. Zhu, Q. Gong, M. Qiu, and Z. Ruan, “Generalized spatial differentiation from the spin hall effect of light and its application in image processing of edge detection,” Phys. Rev. Appl. 11(3), 034043 (2019).
[Crossref]

T. Zhu, Y. Zhou, Y. Lou, H. Ye, M. Qiu, Z. Ruan, and S. Fan, “Plasmonic computing of spatial differentiation,” Nat. Commun. 8(1), 15391 (2017).
[Crossref]

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]

Saba, A.

A. Saba, M. R. Tavakol, P. Karimi-Khoozani, and A. Khavasi, “Two-dimensional edge detection by guided mode resonant metasurface,” IEEE Photonics Technol. Lett. 30(9), 853–856 (2018).
[Crossref]

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]

Slavík, R.

Stax, O.

O. Stax and P. P. Urone, College Physics (OpenStax College, Rice University, 2019).

Su, Y.

Tan, S.

Tavakol, M. R.

A. Saba, M. R. Tavakol, P. Karimi-Khoozani, and A. Khavasi, “Two-dimensional edge detection by guided mode resonant metasurface,” IEEE Photonics Technol. Lett. 30(9), 853–856 (2018).
[Crossref]

Urone, P. P.

O. Stax and P. P. Urone, College Physics (OpenStax College, Rice University, 2019).

Wang, T.

Wen, S.

T. Zhu, Y. Lou, Y. Zhou, J. Zhang, J. Huang, Y. Li, H. Luo, S. Wen, S. Zhu, Q. Gong, M. Qiu, and Z. Ruan, “Generalized spatial differentiation from the spin hall effect of light and its application in image processing of edge detection,” Phys. Rev. Appl. 11(3), 034043 (2019).
[Crossref]

Wu, R.

J. Ren, Y. Li, Y. Lin, Y. Qin, R. Wu, J. Yang, Y. F. Xiao, H. Yang, and Q. Gong, “Spin hall effect of light reflected from a magnetic thin film,” Appl. Phys. Lett. 101(17), 171103 (2012).
[Crossref]

Wu, Z.

Xiang, L.

Xiao, Y. F.

J. Ren, Y. Li, Y. Lin, Y. Qin, R. Wu, J. Yang, Y. F. Xiao, H. Yang, and Q. Gong, “Spin hall effect of light reflected from a magnetic thin film,” Appl. Phys. Lett. 101(17), 171103 (2012).
[Crossref]

Yang, H.

J. Ren, Y. Li, Y. Lin, Y. Qin, R. Wu, J. Yang, Y. F. Xiao, H. Yang, and Q. Gong, “Spin hall effect of light reflected from a magnetic thin film,” Appl. Phys. Lett. 101(17), 171103 (2012).
[Crossref]

Yang, J.

J. Ren, Y. Li, Y. Lin, Y. Qin, R. Wu, J. Yang, Y. F. Xiao, H. Yang, and Q. Gong, “Spin hall effect of light reflected from a magnetic thin film,” Appl. Phys. Lett. 101(17), 171103 (2012).
[Crossref]

Yang, T.

T. Yang, J. Dong, L. Lu, L. Zhou, A. Zheng, X. Zhang, and J. Chen, “All-optical differential equation solver with constant-coefficient tunable based on a single microring resonator,” Sci. Rep. 4(1), 5581 (2015).
[Crossref]

Yao, J.

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(1), 15391 (2017).
[Crossref]

Ye, T.

Youssefi, A.

Yu, Y.

Zangeneh-Nejad, F.

Zhang, J.

T. Zhu, Y. Lou, Y. Zhou, J. Zhang, J. Huang, Y. Li, H. Luo, S. Wen, S. Zhu, Q. Gong, M. Qiu, and Z. Ruan, “Generalized spatial differentiation from the spin hall effect of light and its application in image processing of edge detection,” Phys. Rev. Appl. 11(3), 034043 (2019).
[Crossref]

Zhang, Q.

Zhang, X.

Zhang, X. L.

Zhang, Z.

Zheng, A.

T. Yang, J. Dong, L. Lu, L. Zhou, A. Zheng, X. Zhang, and J. Chen, “All-optical differential equation solver with constant-coefficient tunable based on a single microring resonator,” Sci. Rep. 4(1), 5581 (2015).
[Crossref]

Zheng, A. L.

Zhou, L.

T. Yang, J. Dong, L. Lu, L. Zhou, A. Zheng, X. Zhang, and J. Chen, “All-optical differential equation solver with constant-coefficient tunable based on a single microring resonator,” Sci. Rep. 4(1), 5581 (2015).
[Crossref]

Zhou, Y.

T. Zhu, Y. Lou, Y. Zhou, J. Zhang, J. Huang, Y. Li, H. Luo, S. Wen, S. Zhu, Q. Gong, M. Qiu, and Z. Ruan, “Generalized spatial differentiation from the spin hall effect of light and its application in image processing of edge detection,” Phys. Rev. Appl. 11(3), 034043 (2019).
[Crossref]

T. Zhu, Y. Zhou, Y. Lou, H. Ye, M. Qiu, Z. Ruan, and S. Fan, “Plasmonic computing of spatial differentiation,” Nat. Commun. 8(1), 15391 (2017).
[Crossref]

Zhu, S.

T. Zhu, Y. Lou, Y. Zhou, J. Zhang, J. Huang, Y. Li, H. Luo, S. Wen, S. Zhu, Q. Gong, M. Qiu, and Z. Ruan, “Generalized spatial differentiation from the spin hall effect of light and its application in image processing of edge detection,” Phys. Rev. Appl. 11(3), 034043 (2019).
[Crossref]

Zhu, T.

T. Zhu, Y. Lou, Y. Zhou, J. Zhang, J. Huang, Y. Li, H. Luo, S. Wen, S. Zhu, Q. Gong, M. Qiu, and Z. Ruan, “Generalized spatial differentiation from the spin hall effect of light and its application in image processing of edge detection,” Phys. Rev. Appl. 11(3), 034043 (2019).
[Crossref]

T. Zhu, Y. Zhou, Y. Lou, H. Ye, M. Qiu, Z. Ruan, and S. Fan, “Plasmonic computing of spatial differentiation,” Nat. Commun. 8(1), 15391 (2017).
[Crossref]

Zou, J.

Appl. Phys. Lett. (1)

J. Ren, Y. Li, Y. Lin, Y. Qin, R. Wu, J. Yang, Y. F. Xiao, H. Yang, and Q. Gong, “Spin hall effect of light reflected from a magnetic thin film,” Appl. Phys. Lett. 101(17), 171103 (2012).
[Crossref]

IEEE Photonics Technol. Lett. (1)

A. Saba, M. R. Tavakol, P. Karimi-Khoozani, and A. Khavasi, “Two-dimensional edge detection by guided mode resonant metasurface,” IEEE Photonics Technol. Lett. 30(9), 853–856 (2018).
[Crossref]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

J. Canny, “A computational approach to edge detection,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-8(6), 679–698 (1986).
[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(1), 15391 (2017).
[Crossref]

Opt. Commun. (1)

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

Opt. Express (4)

Opt. Lett. (5)

Phys. Rev. Appl. (1)

T. Zhu, Y. Lou, Y. Zhou, J. Zhang, J. Huang, Y. Li, H. Luo, S. Wen, S. Zhu, Q. Gong, M. Qiu, and Z. Ruan, “Generalized spatial differentiation from the spin hall effect of light and its application in image processing of edge detection,” Phys. Rev. Appl. 11(3), 034043 (2019).
[Crossref]

Sci. Rep. (1)

T. Yang, J. Dong, L. Lu, L. Zhou, A. Zheng, X. Zhang, and J. Chen, “All-optical differential equation solver with constant-coefficient tunable based on a single microring resonator,” Sci. Rep. 4(1), 5581 (2015).
[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]

Other (3)

A. P. M. Q. Vafa, P. Karimi, and A. Khavasi, “Recent advances in spatial analog optical computing,” in 2018 Fifth International Conference on Millimeter-Wave and Terahertz Technologies (MMWaTT), (IEEE, 2018), pp. 6–11.

O. Stax and P. P. Urone, College Physics (OpenStax College, Rice University, 2019).

S. A. Maier, Plasmonics: fundamentals and applications (Springer Science & Business Media, 2007).

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Figures (17)

Fig. 1.
Fig. 1. Systematic demonstration of an ideal edge detector performance with $H_1(k_x)=jG'k_x/k_0$ . This system can detect edges of its input with zero resolution.
Fig. 2.
Fig. 2. Systematic demonstration of an ideal limited-bandwidth edge detector performance with $H_1(k_x)=jG'k_x/k_0$ and $H_2(k_x)=\sqcap (k_x/(2\Delta {k}))$ . This system can detect edges of its input with the resolution of $\pi /(\lambda _0\Delta {k})$ . The maximum allowable value for $\Delta {k}$ is $k_0$ .
Fig. 3.
Fig. 3. The ideal frequency response of the passive differentiator with limited-bandwidth: a) Magnitude b) Phase.
Fig. 4.
Fig. 4. The ideal frequency response of an typical differentiator in optics: a) Magnitude b) Phase.
Fig. 5.
Fig. 5. Comparison of the relationship between resolution and gain in two different investigated cases. In case $1$ , the transfer function of the edge detector system is the one plotted in Fig. 3 and in case $2$ , the transfer function of the edge detector system is the one plotted in Fig. 4.
Fig. 6.
Fig. 6. Green’s function of the differentiator based on Brewster effect for various refractive indices.
Fig. 7.
Fig. 7. The resolution versus gain of the Brewster differentiator for different refractive indices (n=2,3,…,10) of second medium. The first medium is assumed air. The incident angle for each refractive index is the Brewster angle $\theta _B=\tan ^{-1}(n)$ .
Fig. 8.
Fig. 8. FOM versus different refractive indices (n=2,3,…,10) for the second medium in Brewster differentiator to determine the optimum operating point for Brewster differentiator used in edge detector. The first medium is assumed air. The incident angle for each refrective index is the Brewster angle $\theta _B=\tan ^{-1}(n)$ .
Fig. 9.
Fig. 9. The structure of the proposed differentiator using a dielectric slab waveguide that has the feature of tunability of the gain and resolution.
Fig. 10.
Fig. 10. Green’s function of the proposed tunable differentiator for different values of parameter $d$ (the distance between the dielectric slab and the prism). a) TE polarization by incident angle of $67^\circ$ and b) TM polarization by incident angle of $59.7^\circ$ .
Fig. 11.
Fig. 11. The resolution versus gain of the proposed tunable differentiator for different values of parameter $d$ (the distance between the dielectric slab and the prism) for both TE and TM polarization with $\theta _i=67^\circ$ and $\theta _i=59.7^\circ$ , respectively.
Fig. 12.
Fig. 12. FOM of the proposed tunable differentiator for different values of parameter $d$ (the distance between the dielectric slab and the prism) for both TE and TM polarization with $\theta _i=67^\circ$ and $\theta _i=59.7^\circ$ , respectively.
Fig. 13.
Fig. 13. Green’s function of the half-wavelength slab differentiator for different refractive indices and incident angles. a) TE polarization and b) TM polarization.
Fig. 14.
Fig. 14. FOM of the half-wavelength slab differentiator for TE polarization for different values of $n_2$ and $\theta _1$ .
Fig. 15.
Fig. 15. The spatial spectral transfer function of plasmonic differentiator for different metals.
Fig. 16.
Fig. 16. The spatial spectral transfer function of differentiator based on photonic spin hall effect for different materials as the second medium and different angles of incidence. The first medium is assumed to be air.
Fig. 17.
Fig. 17. The FOM of differentiator based on photonic spin Hall effect for different refractive indices and incident angles for x-polarized incident wave.

Tables (1)

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Table 1. FOM for selected structures that can be used for edge detection

Equations (17)

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q 1 ( x ) = δ ( x x 0 ) δ ( x + x 0 )
Δ k = k 0 G
q 2 ( x ) = q 1 ( x ) F 1 ( H 2 ( k x ) ) = 2 Δ k [ s i n c ( Δ k π ( x x 0 ) ) s i n c ( Δ k π ( x + x 0 ) ) ]
Δ x = 1 λ 0 π Δ k = 1 λ 0 π G k 0 = 0.5 G Δ x 0.5 G 1 = 0.5
E x i ( x , z ) = P ( k x ) e j k x x e j k z z d k x
P ( k x ) = 1 2 π p ( x ) e j k x x d x
P ( k x ) = sin k x x 0 π k x = x 0 π s i n c ( x 0 π k x )
q ( x ) = k 0 k 0 Q ( k x ) e j k x x e j k z z d k x
Δ x = 0.345 G + 0.155 Δ x 0.5 G 1 = 0.345
F O M = | Δ x 0.5 G 1 |
θ B = tan 1 ( n )
G = n 2 1 2 n 3
L = π k 0 n 2 cos θ 2
H ( k x ) e j ϕ B j k x
H ( k y ) = j ( r s + r p ) 4 ( e j k y δ e j k y δ )
H ( k y ) δ ( r s + r p ) 2 k y
| k y | k 0 1 2 cot θ 1

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