Abstract

We propose an optimized method for the inverse design of guided-mode resonance (GMR) filters using one- and two-dimensional (1D and 2D) grating structures. This work for 2D state is based on developing the effective permittivity of 1D grating structures along three orthogonal axes to predict the physical dimensions of the structure, for the first time to our knowledge. Also, we compare three optimization methods to reach the optimized conditions based on the characteristics of multilayer structures. Both the transfer matrix method and rigorous coupled-wave analysis are used to simulate and show the reflection and transmission of the proposed 2D GMR filters. The results show that insensitivity to polarization, the best accuracy in resonance location design, and a high quality factor can be achieved for both the rectangular and cylindrical structures as the ideal 2D GMR filters. Also, the effect of each layer thickness on the resonance location and the full width at half-maximum is illustrated. Finally, we investigate three different reasons for decreasing the FWHM of the output reflection of the GMR filters.

© 2020 Optical Society of America

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    [Crossref]

2019 (2)

Y. Khorrami and D. Fathi, “Broadband thermophotovoltaic emitter using magnetic polaritons based on optimized one- and two-dimensional multilayer structures,” J. Opt. Soc. Am. B 36, 662–666 (2019).
[Crossref]

M. H. Tahersima, K. Kojima, T. Koike-Akino, D. Jha, B. Wang, C. Lin, and K. Parsons, “Deep neural network inverse design of integrated photonic power splitters,” Sci. Rep. 9, 1368 (2019).
[Crossref]

2018 (3)

S. Molesky, Z. Lin, A. Y. Piggott, W. Jin, J. Vucković, and A. W. Rodriguez, “Inverse design in nanophotonics,” Nat. Photonics 12, 659–670 (2018).
[Crossref]

H. S. Bark, G. J. Kim, and T.-I. Jeon, “Tunable terahertz guided-mode resonance filter with a variable grating period,” Opt. Express 26, 29353–29362 (2018).
[Crossref]

H. S. Bark, G. J. Kim, and T.-I. Jeon, “Transmission characteristics of all-dielectric guided-mode resonance filter in the THz region,” Sci. Rep. 8, 13570 (2018).
[Crossref]

2017 (1)

P. K. Sahoo, S. Sarkar, and J. Joseph, “High sensitivity guided-mode resonance optical sensor employing phase detection,” Sci. Rep. 7, 7607 (2017).
[Crossref]

2016 (2)

K. Marák, “Characterization of the inverse problem in critical dimension measurement of diffraction gratings,” Period. Polytech. Electr. Eng. Comput. Sci. 60, 178–186 (2016).
[Crossref]

Z. Lin, X. Liang, M. Lončar, S. G. Johnson, and A. W. Rodriguez, “Cavity-enhanced second-harmonic generation via nonlinear-overlap optimization,” Optica 3, 233–238 (2016).
[Crossref]

2015 (1)

2014 (3)

Y. Deng, Z. Liu, C. Song, J. Wu, Y. Liu, and Y. Wu, “Topology optimization-based computational design methodology for surface plasmon polaritons,” Plasmonics 10, 569–583 (2014).
[Crossref]

K. Han and C.-H. Chang, “Numerical modeling of sub-wavelength anti-reflective structures for solar module applications,” Nanomaterials 4, 87–128 (2014).
[Crossref]

D. B. Mazulquim, K. J. Lee, J. W. Yoon, L. V. Muniz, B.-H. V. Borges, L. G. Neto, and R. Magnusson, “Efficient band-pass color filters enabled by resonant modes and plasmons near the Rayleigh anomaly,” Opt. Express 22, 30843–30851 (2014).
[Crossref]

2013 (1)

A. B. Khanikaev, C. Wu, and G. Shvets, “Fano-resonant metamaterials and their applications,” Nanophotonics 2, 247–264 (2013).
[Crossref]

2012 (1)

J. H. Barton, R. C. Rumpf, R. W. Smith, C. Kozikowski, and P. Zellner, “All-dielectric frequency selective surfaces with few number of periods,” Prog. Electromagn. Res. B 41, 269–283 (2012).
[Crossref]

2011 (2)

2010 (3)

2008 (1)

J. Hao and L. Zhou, “Electromagnetic wave scatterings by anisotropic metamaterials: generalized 4 × 4 transfer-matrix method,” Phys. Rev. B 77, 094201 (2008).
[Crossref]

2007 (1)

A. A. Mehta, R. C. Rumpf, Z. A. Roth, and E. G. Johnson, “Guided mode resonance filter as a spectrally selective feedback element in a double-cladding optical fiber laser,” IEEE Photon. Tech. Lett. 19, 2030–2032 (2007).
[Crossref]

2006 (2)

Y. Weng, W. Xu, Y. Wu, K. Zhou, and B. Guo, “2D shape deformation using nonlinear least squares optimization,” Visual Comput. 22, 653–660 (2006).
[Crossref]

S. Boonruang, A. Greenwell, and M. G. Moharam, “Multiline two-dimensional guided-mode resonant filters,” Appl. Opt. 45, 5740–5747 (2006).
[Crossref]

2005 (1)

T. Kobayashi, Y. Kanamori, and K. Hane, “Surface laser emission from solid polymer dye in a guided mode resonant grating filter structure,” Appl. Phys Lett. 87, 151106 (2005).
[Crossref]

2003 (2)

P. S. Priambodo, T. A. Maldonado, and R. Magnusson, “Fabrication and characterization of high-quality waveguide-mode resonant optical filters,” Appl. Phys. Lett. 83, 3248–3250 (2003).
[Crossref]

S. T. Thurman and G. M. Morris, “Controlling the spectral response in guided-mode resonance filter design,” Appl. Opt. 42, 3225–3233 (2003).
[Crossref]

2002 (1)

2000 (1)

1998 (1)

J. A. Cox, R. A. Morgan, R. Wilke, and C. Ford, “Guided-mode grating resonant filter for VCSEL applications,” Proc. SPIE 3291, 70–76 (1998).
[Crossref]

1997 (2)

1995 (1)

1994 (2)

E. B. Grann, M. G. Moharam, and D. A. Pommet, “Artificial uniaxial and biaxial dielectrics with use of two-dimensional subwavelength binary gratings,” J. Opt. Soc. Am. A 11, 2695–2703 (1994).
[Crossref]

R. Magnusson, S. S. Wang, T. D. Black, and A. Sohn, “Resonance properties of dielectric waveguide gratings: Theory and experiments at 4–18  GHz,” IEEE Trans. Antennas Propag. 42, 567–569 (1994).
[Crossref]

1993 (1)

1992 (1)

R. Magnusson and S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992).
[Crossref]

1907 (1)

L. Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).
[Crossref]

Akozbek, N.

Andkjær, J.

J. Andkjær and O. Sigmund, “Topology optimized low-contrast all-dielectric optical cloak,” Appl. Phys. Lett. 98, 021112 (2011).
[Crossref]

Bark, H. S.

H. S. Bark, G. J. Kim, and T.-I. Jeon, “Tunable terahertz guided-mode resonance filter with a variable grating period,” Opt. Express 26, 29353–29362 (2018).
[Crossref]

H. S. Bark, G. J. Kim, and T.-I. Jeon, “Transmission characteristics of all-dielectric guided-mode resonance filter in the THz region,” Sci. Rep. 8, 13570 (2018).
[Crossref]

Barton, J. H.

J. H. Barton, R. C. Rumpf, R. W. Smith, C. Kozikowski, and P. Zellner, “All-dielectric frequency selective surfaces with few number of periods,” Prog. Electromagn. Res. B 41, 269–283 (2012).
[Crossref]

Bergner, B. C.

Bhargava, R.

Bianco, G. V.

Black, T. D.

R. Magnusson, S. S. Wang, T. D. Black, and A. Sohn, “Resonance properties of dielectric waveguide gratings: Theory and experiments at 4–18  GHz,” IEEE Trans. Antennas Propag. 42, 567–569 (1994).
[Crossref]

Boonruang, S.

Borges, B.-H. V.

Bruno, G.

Ceglia, D. D.

Chang, C.-H.

K. Han and C.-H. Chang, “Numerical modeling of sub-wavelength anti-reflective structures for solar module applications,” Nanomaterials 4, 87–128 (2014).
[Crossref]

Chang, J.-Y.

Cox, J. A.

J. A. Cox, R. A. Morgan, R. Wilke, and C. Ford, “Guided-mode grating resonant filter for VCSEL applications,” Proc. SPIE 3291, 70–76 (1998).
[Crossref]

Cunningham, B. T.

D’Orazio, A.

De Vittorio, M.

Deng, Y.

Y. Deng, Z. Liu, C. Song, J. Wu, Y. Liu, and Y. Wu, “Topology optimization-based computational design methodology for surface plasmon polaritons,” Plasmonics 10, 569–583 (2014).
[Crossref]

Dunn, S. C.

Fathi, D.

Ford, C.

J. A. Cox, R. A. Morgan, R. Wilke, and C. Ford, “Guided-mode grating resonant filter for VCSEL applications,” Proc. SPIE 3291, 70–76 (1998).
[Crossref]

Gaylord, T. K.

Germer, T. A.

Grande, M.

Grann, E. B.

Greenwell, A.

Guo, B.

Y. Weng, W. Xu, Y. Wu, K. Zhou, and B. Guo, “2D shape deformation using nonlinear least squares optimization,” Visual Comput. 22, 653–660 (2006).
[Crossref]

Guo, H.

Han, K.

K. Han and C.-H. Chang, “Numerical modeling of sub-wavelength anti-reflective structures for solar module applications,” Nanomaterials 4, 87–128 (2014).
[Crossref]

Hane, K.

T. Kobayashi, Y. Kanamori, and K. Hane, “Surface laser emission from solid polymer dye in a guided mode resonant grating filter structure,” Appl. Phys Lett. 87, 151106 (2005).
[Crossref]

Hao, J.

J. Hao and L. Zhou, “Electromagnetic wave scatterings by anisotropic metamaterials: generalized 4 × 4 transfer-matrix method,” Phys. Rev. B 77, 094201 (2008).
[Crossref]

Hsu, C.-L.

Iwata, K.

Jacob, D. K.

Jeon, T.-I.

H. S. Bark, G. J. Kim, and T.-I. Jeon, “Tunable terahertz guided-mode resonance filter with a variable grating period,” Opt. Express 26, 29353–29362 (2018).
[Crossref]

H. S. Bark, G. J. Kim, and T.-I. Jeon, “Transmission characteristics of all-dielectric guided-mode resonance filter in the THz region,” Sci. Rep. 8, 13570 (2018).
[Crossref]

Jha, D.

M. H. Tahersima, K. Kojima, T. Koike-Akino, D. Jha, B. Wang, C. Lin, and K. Parsons, “Deep neural network inverse design of integrated photonic power splitters,” Sci. Rep. 9, 1368 (2019).
[Crossref]

Jin, W.

S. Molesky, Z. Lin, A. Y. Piggott, W. Jin, J. Vucković, and A. W. Rodriguez, “Inverse design in nanophotonics,” Nat. Photonics 12, 659–670 (2018).
[Crossref]

Johnson, E. G.

A. A. Mehta, R. C. Rumpf, Z. A. Roth, and E. G. Johnson, “Guided mode resonance filter as a spectrally selective feedback element in a double-cladding optical fiber laser,” IEEE Photon. Tech. Lett. 19, 2030–2032 (2007).
[Crossref]

Johnson, S. G.

Joseph, J.

P. K. Sahoo, S. Sarkar, and J. Joseph, “High sensitivity guided-mode resonance optical sensor employing phase detection,” Sci. Rep. 7, 7607 (2017).
[Crossref]

Kanamori, Y.

T. Kobayashi, Y. Kanamori, and K. Hane, “Surface laser emission from solid polymer dye in a guided mode resonant grating filter structure,” Appl. Phys Lett. 87, 151106 (2005).
[Crossref]

Khanikaev, A. B.

A. B. Khanikaev, C. Wu, and G. Shvets, “Fano-resonant metamaterials and their applications,” Nanophotonics 2, 247–264 (2013).
[Crossref]

Khorrami, Y.

Kikuta, H.

Kim, G. J.

H. S. Bark, G. J. Kim, and T.-I. Jeon, “Transmission characteristics of all-dielectric guided-mode resonance filter in the THz region,” Sci. Rep. 8, 13570 (2018).
[Crossref]

H. S. Bark, G. J. Kim, and T.-I. Jeon, “Tunable terahertz guided-mode resonance filter with a variable grating period,” Opt. Express 26, 29353–29362 (2018).
[Crossref]

Kobayashi, T.

T. Kobayashi, Y. Kanamori, and K. Hane, “Surface laser emission from solid polymer dye in a guided mode resonant grating filter structure,” Appl. Phys Lett. 87, 151106 (2005).
[Crossref]

Koike-Akino, T.

M. H. Tahersima, K. Kojima, T. Koike-Akino, D. Jha, B. Wang, C. Lin, and K. Parsons, “Deep neural network inverse design of integrated photonic power splitters,” Sci. Rep. 9, 1368 (2019).
[Crossref]

Kojima, K.

M. H. Tahersima, K. Kojima, T. Koike-Akino, D. Jha, B. Wang, C. Lin, and K. Parsons, “Deep neural network inverse design of integrated photonic power splitters,” Sci. Rep. 9, 1368 (2019).
[Crossref]

Kozikowski, C.

J. H. Barton, R. C. Rumpf, R. W. Smith, C. Kozikowski, and P. Zellner, “All-dielectric frequency selective surfaces with few number of periods,” Prog. Electromagn. Res. B 41, 269–283 (2012).
[Crossref]

Lai, Z.

Lalanne, P.

Lee, C.-C.

Lee, K. J.

Li, L. F.

Liang, X.

Lin, C.

M. H. Tahersima, K. Kojima, T. Koike-Akino, D. Jha, B. Wang, C. Lin, and K. Parsons, “Deep neural network inverse design of integrated photonic power splitters,” Sci. Rep. 9, 1368 (2019).
[Crossref]

Lin, Z.

S. Molesky, Z. Lin, A. Y. Piggott, W. Jin, J. Vucković, and A. W. Rodriguez, “Inverse design in nanophotonics,” Nat. Photonics 12, 659–670 (2018).
[Crossref]

Z. Lin, X. Liang, M. Lončar, S. G. Johnson, and A. W. Rodriguez, “Cavity-enhanced second-harmonic generation via nonlinear-overlap optimization,” Optica 3, 233–238 (2016).
[Crossref]

Liu, J.-N.

Liu, W.

Liu, Y.

Y. Deng, Z. Liu, C. Song, J. Wu, Y. Liu, and Y. Wu, “Topology optimization-based computational design methodology for surface plasmon polaritons,” Plasmonics 10, 569–583 (2014).
[Crossref]

W. Liu, Z. Lai, H. Guo, and Y. Liu, “Guided-mode resonance filters with shallow grating,” Opt. Lett. 35, 865–867 (2010).
[Crossref]

Liu, Z.

Y. Deng, Z. Liu, C. Song, J. Wu, Y. Liu, and Y. Wu, “Topology optimization-based computational design methodology for surface plasmon polaritons,” Plasmonics 10, 569–583 (2014).
[Crossref]

Loncar, M.

Magnusson, R.

D. B. Mazulquim, K. J. Lee, J. W. Yoon, L. V. Muniz, B.-H. V. Borges, L. G. Neto, and R. Magnusson, “Efficient band-pass color filters enabled by resonant modes and plasmons near the Rayleigh anomaly,” Opt. Express 22, 30843–30851 (2014).
[Crossref]

P. S. Priambodo, T. A. Maldonado, and R. Magnusson, “Fabrication and characterization of high-quality waveguide-mode resonant optical filters,” Appl. Phys. Lett. 83, 3248–3250 (2003).
[Crossref]

R. Magnusson, S. S. Wang, T. D. Black, and A. Sohn, “Resonance properties of dielectric waveguide gratings: Theory and experiments at 4–18  GHz,” IEEE Trans. Antennas Propag. 42, 567–569 (1994).
[Crossref]

S. S. Wang and R. Magnusson, “Theory and applications of guided-mode resonance filters,” Appl. Opt. 32, 2606–2613 (1993).
[Crossref]

R. Magnusson and S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992).
[Crossref]

Maldonado, T. A.

P. S. Priambodo, T. A. Maldonado, and R. Magnusson, “Fabrication and characterization of high-quality waveguide-mode resonant optical filters,” Appl. Phys. Lett. 83, 3248–3250 (2003).
[Crossref]

Marák, K.

K. Marák, “Characterization of the inverse problem in critical dimension measurement of diffraction gratings,” Period. Polytech. Electr. Eng. Comput. Sci. 60, 178–186 (2016).
[Crossref]

Mazulquim, D. B.

Mehta, A. A.

A. A. Mehta, R. C. Rumpf, Z. A. Roth, and E. G. Johnson, “Guided mode resonance filter as a spectrally selective feedback element in a double-cladding optical fiber laser,” IEEE Photon. Tech. Lett. 19, 2030–2032 (2007).
[Crossref]

Mizutani, A.

Moharam, M. G.

Molesky, S.

S. Molesky, Z. Lin, A. Y. Piggott, W. Jin, J. Vucković, and A. W. Rodriguez, “Inverse design in nanophotonics,” Nat. Photonics 12, 659–670 (2018).
[Crossref]

Morgan, R. A.

J. A. Cox, R. A. Morgan, R. Wilke, and C. Ford, “Guided-mode grating resonant filter for VCSEL applications,” Proc. SPIE 3291, 70–76 (1998).
[Crossref]

Morris, G. M.

Muniz, L. V.

Neto, L. G.

Parsons, K.

M. H. Tahersima, K. Kojima, T. Koike-Akino, D. Jha, B. Wang, C. Lin, and K. Parsons, “Deep neural network inverse design of integrated photonic power splitters,” Sci. Rep. 9, 1368 (2019).
[Crossref]

Petruzzelli, V.

Piggott, A. Y.

S. Molesky, Z. Lin, A. Y. Piggott, W. Jin, J. Vucković, and A. W. Rodriguez, “Inverse design in nanophotonics,” Nat. Photonics 12, 659–670 (2018).
[Crossref]

Pommet, D. A.

Priambodo, P. S.

P. S. Priambodo, T. A. Maldonado, and R. Magnusson, “Fabrication and characterization of high-quality waveguide-mode resonant optical filters,” Appl. Phys. Lett. 83, 3248–3250 (2003).
[Crossref]

Rayleigh, L.

L. Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).
[Crossref]

Rodriguez, A. W.

S. Molesky, Z. Lin, A. Y. Piggott, W. Jin, J. Vucković, and A. W. Rodriguez, “Inverse design in nanophotonics,” Nat. Photonics 12, 659–670 (2018).
[Crossref]

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[Crossref]

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P. K. Sahoo, S. Sarkar, and J. Joseph, “High sensitivity guided-mode resonance optical sensor employing phase detection,” Sci. Rep. 7, 7607 (2017).
[Crossref]

M. H. Tahersima, K. Kojima, T. Koike-Akino, D. Jha, B. Wang, C. Lin, and K. Parsons, “Deep neural network inverse design of integrated photonic power splitters,” Sci. Rep. 9, 1368 (2019).
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Figures (11)

Fig. 1.
Fig. 1. Allowed resonance regions of the GMR filter, with various values of the angle of incidence versus the normalized wavelength with the grating period. The first three allowed modes correspond to $m = \pm 1$, $ \pm {2}$, and $ \pm {3}$. Also, the structure is a 1D grating with ${n_1} = {1}$ and ${n_2} = {1.5}$.
Fig. 2.
Fig. 2. (a) Preliminary homogenous two-layered schematic and (b) the desired infinitely periodic 2D GMR filter that we are looking for the design process.
Fig. 3.
Fig. 3. Convergence diagram versus the iteration number for the Fsolve method: (a) the function value and (b) the objectives variations. The final (minimum) value of the error function has been depicted as the current function value on the figure, and the optimized values of objectives have been shown in the inset table.
Fig. 4.
Fig. 4. Convergence diagram versus the iteration number for the LSO method: (a) the function value and (b) the objectives variations. The final (minimum) value of the error function has been depicted as the current function value on the figure, and the optimized values of objectives have been shown in the inset table.
Fig. 5.
Fig. 5. Convergence diagram versus the generation number for the GA. The best (final) and mean values of the fitness function have been depicted on the figure.
Fig. 6.
Fig. 6. Reflection and transmission spectrums versus the wavelength, for the structure of Fig. 2(a), using the TMM. The inset table shows the optimized parameters extracted from the LSO.
Fig. 7.
Fig. 7. Allowed region for ${\varepsilon _{{\rm eff}}}$ versus the filling fractions along the $x$ and $y$ directions, based on Eqs. (10)–(13).
Fig. 8.
Fig. 8. Flowchart demonstrating the process of inverse design for the 1D and 2D GMR filters presented in this paper.
Fig. 9.
Fig. 9. 1D complex grating as a GMR filter. The filling fractions are ${f_1}$, ${f_2}$, and ${f_3}$ along the $x$ axis, and the grating period is shown by $\Lambda$.
Fig. 10.
Fig. 10. Reflection and transmission spectrums of the designed 2D GMR filter for both polarizations. The right insets show the schematic of rectangular and cylindrical GMR filters, respectively. The left insets depict the normalized magnetic field amplitude distribution of both schematics at the resonance wavelength.
Fig. 11.
Fig. 11. Reflection spectrum of the designed 2D rectangular GMR filter for various thicknesses of (a) the grating layer and (b) the waveguide layer. The remaining parameters are the same as in Fig. 10.

Equations (15)

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n r e f / t r n sin θ r e f / t r n = n i n c sin θ i n c m λ 0 Λ sin ϕ ,
n 1 | n 1 sin θ i n c m λ 0 Λ sin ϕ | < n 2 ,
λ c Λ = n 1 + n 2 2 | m | .
ε E K = ε 2 f + ε 1 ( 1 f ) ,
ε E K = ( f ε 2 + 1 f ε 1 ) 1 ,
ε E K y = ε 2 f y + ε 1 ( 1 f y ) .
ε e f f , z E K = ε E K y f x + ε 1 ( 1 f x ) ,
ε e f f , z E K = ε 2 f x f y + ε 1 ( 1 f x f y ) .
ε e f f , z E K = ( f x f y ε 2 + ( 1 f x f y ) ε 1 ) 1 .
ε e f f , x h i g h = ( f x f y ε 2 + ( 1 f y ) ε 1 + ( 1 f x ) ε 1 ) 1 ,
ε e f f , x l o w = ( 1 f x ) ε 1 + f x ( f y ε 2 + ( 1 f y ) ε 1 ) 1 ,
ε e f f , y h i g h = ( f y f x ε 2 + ( 1 f x ) ε 1 + ( 1 f y ) ε 1 ) 1 ,
ε e f f , y l o w = ( 1 f y ) ε 1 + f y ( f x ε 2 + ( 1 f x ) ε 1 ) 1 .
ε e f f = ε 2 f 1 + ε 1 f 2 + ε 2 f 3 + ε 1 ( 1 f 1 f 2 f 3 ) ,
f = f 1 + f 3 ,

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