Abstract

We have introduced a method to find optimized structured media exhibiting large internal electric field amplitudes. The method is based on a genetic algorithm in which a spatial fitness function according to the computed field distribution in the interior of media is defined and maximized. The main feature of our method is that it enables localization of light at a desired layer (or more) within the structure. The enhancements are demonstrated to be up to about 70-fold in |E|2 by use of only seven layers. The results are interesting for nonlinear and sensor applications, which due to compact size and few number of structure layers, are also desirable for fabrication purposes.

© 2013 Optical Society of America

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References

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  1. R. W. Boyd, “Material slow light and structural slow light: similarities and differences for nonlinear optics,” J. Opt. Soc. Am. B 28, A38–A44 (2011).
    [CrossRef]
  2. A. Figotin and I. Vitebskiy, “Slow wave phenomena in photonic crystals,” Laser Photon. Rev. 5, 201–213 (2010).
    [CrossRef]
  3. M. A. Muriel and A. Carballar, “Internal field distributions in fiber Bragg gratings,” IEEE Photon. Technol. Lett. 9, 955–957 (1997).
    [CrossRef]
  4. A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron., QE-9919–933 (1973).
    [CrossRef]
  5. U. Bandelow and U. Leonhardt, “Light propagation in one-dimensional lossless dielectrica: transfer matrix method and coupled mode theory,” Opt. Commun. 101, 92–99(1993).
    [CrossRef]
  6. R. Feced, M. N. Zervas, and M. A. Muriel, “Efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
    [CrossRef]
  7. H. Alisafaee and M. Ghanaatshoar, “Optimization of all-garnet magneto-optical magnetic field sensors with genetic algorithm,” Appl. Opt. 51, 5144–5148 (2012).
    [CrossRef]
  8. C. Kontis, M. R. Mueller, C. Kuechenmeister, K. T. Kallis, and J. Knoch, “Optimizing the identification of mono- and bilayer graphene on multilayer substrates,” Appl. Opt. 51, 385–389 (2012).
    [CrossRef]
  9. J. F. Wu, Y. Y. Chen, and T. S. Wang, “Flat field concave holographic grating with broad spectral region and moderately high resolution,” Appl. Opt. 51, 509–514 (2012).
    [CrossRef]
  10. L. Jiang, W. Jia, G. Zheng, and X. Li, “Design and fabrication of rod-type two-dimensional photonic crystal slabs with large high-order bandgaps in near-infrared wavelengths,” Opt. Lett. 37, 1424–1426 (2012).
    [CrossRef]
  11. T. Takenaka and T. Moriyama, “Inverse scattering approach based on the field equivalence principle: inversion without a priori information on incident fields,” Opt. Lett. 37, 3432–3434 (2012).
    [CrossRef]

2012 (5)

H. Alisafaee and M. Ghanaatshoar, “Optimization of all-garnet magneto-optical magnetic field sensors with genetic algorithm,” Appl. Opt. 51, 5144–5148 (2012).
[CrossRef]

C. Kontis, M. R. Mueller, C. Kuechenmeister, K. T. Kallis, and J. Knoch, “Optimizing the identification of mono- and bilayer graphene on multilayer substrates,” Appl. Opt. 51, 385–389 (2012).
[CrossRef]

J. F. Wu, Y. Y. Chen, and T. S. Wang, “Flat field concave holographic grating with broad spectral region and moderately high resolution,” Appl. Opt. 51, 509–514 (2012).
[CrossRef]

L. Jiang, W. Jia, G. Zheng, and X. Li, “Design and fabrication of rod-type two-dimensional photonic crystal slabs with large high-order bandgaps in near-infrared wavelengths,” Opt. Lett. 37, 1424–1426 (2012).
[CrossRef]

T. Takenaka and T. Moriyama, “Inverse scattering approach based on the field equivalence principle: inversion without a priori information on incident fields,” Opt. Lett. 37, 3432–3434 (2012).
[CrossRef]

2011 (1)

R. W. Boyd, “Material slow light and structural slow light: similarities and differences for nonlinear optics,” J. Opt. Soc. Am. B 28, A38–A44 (2011).
[CrossRef]

2010 (1)

A. Figotin and I. Vitebskiy, “Slow wave phenomena in photonic crystals,” Laser Photon. Rev. 5, 201–213 (2010).
[CrossRef]

1999 (1)

R. Feced, M. N. Zervas, and M. A. Muriel, “Efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

1997 (1)

M. A. Muriel and A. Carballar, “Internal field distributions in fiber Bragg gratings,” IEEE Photon. Technol. Lett. 9, 955–957 (1997).
[CrossRef]

1993 (1)

U. Bandelow and U. Leonhardt, “Light propagation in one-dimensional lossless dielectrica: transfer matrix method and coupled mode theory,” Opt. Commun. 101, 92–99(1993).
[CrossRef]

1973 (1)

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron., QE-9919–933 (1973).
[CrossRef]

Alisafaee, H.

H. Alisafaee and M. Ghanaatshoar, “Optimization of all-garnet magneto-optical magnetic field sensors with genetic algorithm,” Appl. Opt. 51, 5144–5148 (2012).
[CrossRef]

Bandelow, U.

U. Bandelow and U. Leonhardt, “Light propagation in one-dimensional lossless dielectrica: transfer matrix method and coupled mode theory,” Opt. Commun. 101, 92–99(1993).
[CrossRef]

Boyd, R. W.

R. W. Boyd, “Material slow light and structural slow light: similarities and differences for nonlinear optics,” J. Opt. Soc. Am. B 28, A38–A44 (2011).
[CrossRef]

Carballar, A.

M. A. Muriel and A. Carballar, “Internal field distributions in fiber Bragg gratings,” IEEE Photon. Technol. Lett. 9, 955–957 (1997).
[CrossRef]

Chen, Y. Y.

J. F. Wu, Y. Y. Chen, and T. S. Wang, “Flat field concave holographic grating with broad spectral region and moderately high resolution,” Appl. Opt. 51, 509–514 (2012).
[CrossRef]

Feced, R.

R. Feced, M. N. Zervas, and M. A. Muriel, “Efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

Figotin, A.

A. Figotin and I. Vitebskiy, “Slow wave phenomena in photonic crystals,” Laser Photon. Rev. 5, 201–213 (2010).
[CrossRef]

Ghanaatshoar, M.

H. Alisafaee and M. Ghanaatshoar, “Optimization of all-garnet magneto-optical magnetic field sensors with genetic algorithm,” Appl. Opt. 51, 5144–5148 (2012).
[CrossRef]

Jia, W.

L. Jiang, W. Jia, G. Zheng, and X. Li, “Design and fabrication of rod-type two-dimensional photonic crystal slabs with large high-order bandgaps in near-infrared wavelengths,” Opt. Lett. 37, 1424–1426 (2012).
[CrossRef]

Jiang, L.

L. Jiang, W. Jia, G. Zheng, and X. Li, “Design and fabrication of rod-type two-dimensional photonic crystal slabs with large high-order bandgaps in near-infrared wavelengths,” Opt. Lett. 37, 1424–1426 (2012).
[CrossRef]

Kallis, K. T.

C. Kontis, M. R. Mueller, C. Kuechenmeister, K. T. Kallis, and J. Knoch, “Optimizing the identification of mono- and bilayer graphene on multilayer substrates,” Appl. Opt. 51, 385–389 (2012).
[CrossRef]

Knoch, J.

C. Kontis, M. R. Mueller, C. Kuechenmeister, K. T. Kallis, and J. Knoch, “Optimizing the identification of mono- and bilayer graphene on multilayer substrates,” Appl. Opt. 51, 385–389 (2012).
[CrossRef]

Kontis, C.

C. Kontis, M. R. Mueller, C. Kuechenmeister, K. T. Kallis, and J. Knoch, “Optimizing the identification of mono- and bilayer graphene on multilayer substrates,” Appl. Opt. 51, 385–389 (2012).
[CrossRef]

Kuechenmeister, C.

C. Kontis, M. R. Mueller, C. Kuechenmeister, K. T. Kallis, and J. Knoch, “Optimizing the identification of mono- and bilayer graphene on multilayer substrates,” Appl. Opt. 51, 385–389 (2012).
[CrossRef]

Leonhardt, U.

U. Bandelow and U. Leonhardt, “Light propagation in one-dimensional lossless dielectrica: transfer matrix method and coupled mode theory,” Opt. Commun. 101, 92–99(1993).
[CrossRef]

Li, X.

L. Jiang, W. Jia, G. Zheng, and X. Li, “Design and fabrication of rod-type two-dimensional photonic crystal slabs with large high-order bandgaps in near-infrared wavelengths,” Opt. Lett. 37, 1424–1426 (2012).
[CrossRef]

Moriyama, T.

T. Takenaka and T. Moriyama, “Inverse scattering approach based on the field equivalence principle: inversion without a priori information on incident fields,” Opt. Lett. 37, 3432–3434 (2012).
[CrossRef]

Mueller, M. R.

C. Kontis, M. R. Mueller, C. Kuechenmeister, K. T. Kallis, and J. Knoch, “Optimizing the identification of mono- and bilayer graphene on multilayer substrates,” Appl. Opt. 51, 385–389 (2012).
[CrossRef]

Muriel, M. A.

R. Feced, M. N. Zervas, and M. A. Muriel, “Efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

M. A. Muriel and A. Carballar, “Internal field distributions in fiber Bragg gratings,” IEEE Photon. Technol. Lett. 9, 955–957 (1997).
[CrossRef]

Takenaka, T.

T. Takenaka and T. Moriyama, “Inverse scattering approach based on the field equivalence principle: inversion without a priori information on incident fields,” Opt. Lett. 37, 3432–3434 (2012).
[CrossRef]

Vitebskiy, I.

A. Figotin and I. Vitebskiy, “Slow wave phenomena in photonic crystals,” Laser Photon. Rev. 5, 201–213 (2010).
[CrossRef]

Wang, T. S.

J. F. Wu, Y. Y. Chen, and T. S. Wang, “Flat field concave holographic grating with broad spectral region and moderately high resolution,” Appl. Opt. 51, 509–514 (2012).
[CrossRef]

Wu, J. F.

J. F. Wu, Y. Y. Chen, and T. S. Wang, “Flat field concave holographic grating with broad spectral region and moderately high resolution,” Appl. Opt. 51, 509–514 (2012).
[CrossRef]

Yariv, A.

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron., QE-9919–933 (1973).
[CrossRef]

Zervas, M. N.

R. Feced, M. N. Zervas, and M. A. Muriel, “Efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

Zheng, G.

L. Jiang, W. Jia, G. Zheng, and X. Li, “Design and fabrication of rod-type two-dimensional photonic crystal slabs with large high-order bandgaps in near-infrared wavelengths,” Opt. Lett. 37, 1424–1426 (2012).
[CrossRef]

Appl. Opt. (3)

H. Alisafaee and M. Ghanaatshoar, “Optimization of all-garnet magneto-optical magnetic field sensors with genetic algorithm,” Appl. Opt. 51, 5144–5148 (2012).
[CrossRef]

C. Kontis, M. R. Mueller, C. Kuechenmeister, K. T. Kallis, and J. Knoch, “Optimizing the identification of mono- and bilayer graphene on multilayer substrates,” Appl. Opt. 51, 385–389 (2012).
[CrossRef]

J. F. Wu, Y. Y. Chen, and T. S. Wang, “Flat field concave holographic grating with broad spectral region and moderately high resolution,” Appl. Opt. 51, 509–514 (2012).
[CrossRef]

IEEE J. Quantum Electron. (2)

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron., QE-9919–933 (1973).
[CrossRef]

R. Feced, M. N. Zervas, and M. A. Muriel, “Efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

M. A. Muriel and A. Carballar, “Internal field distributions in fiber Bragg gratings,” IEEE Photon. Technol. Lett. 9, 955–957 (1997).
[CrossRef]

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

R. W. Boyd, “Material slow light and structural slow light: similarities and differences for nonlinear optics,” J. Opt. Soc. Am. B 28, A38–A44 (2011).
[CrossRef]

Laser Photon. Rev. (1)

A. Figotin and I. Vitebskiy, “Slow wave phenomena in photonic crystals,” Laser Photon. Rev. 5, 201–213 (2010).
[CrossRef]

Opt. Commun. (1)

U. Bandelow and U. Leonhardt, “Light propagation in one-dimensional lossless dielectrica: transfer matrix method and coupled mode theory,” Opt. Commun. 101, 92–99(1993).
[CrossRef]

Opt. Lett. (2)

L. Jiang, W. Jia, G. Zheng, and X. Li, “Design and fabrication of rod-type two-dimensional photonic crystal slabs with large high-order bandgaps in near-infrared wavelengths,” Opt. Lett. 37, 1424–1426 (2012).
[CrossRef]

T. Takenaka and T. Moriyama, “Inverse scattering approach based on the field equivalence principle: inversion without a priori information on incident fields,” Opt. Lett. 37, 3432–3434 (2012).
[CrossRef]

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Figures (6)

Fig. 1.
Fig. 1.

Illustration of zones for which electric field distribution will be computed and considered as a separate parameter in GA.

Fig. 2.
Fig. 2.

Plots of real (blue) and imaginary (red) parts of electric field distribution in Bragg structures with N=20 (a) δn=0.2 and (b) δn=1.2. (c) Transmission spectra of Bragg structure (a) as blue curve and (b) as red curve. The insets show field intensity.

Fig. 3.
Fig. 3.

Plots of real (blue) and imaginary (red) parts of electric field distribution of structures in Table 1. The insets show field intensity. Dimensionless wavelength is 0.15 (in units of 2πc/ω), and δn=3, and a normal incidence is considered in the all cases. Location of the first interface is at 1 (2πc/ω).

Fig. 4.
Fig. 4.

Contour plots of field distribution of structures in Table 1 as a function of wavelength. Horizontal black lines show interfaces of structure. The first interface is located at 2 (2πc/ω).

Fig. 5.
Fig. 5.

Plots of peak intensity change (in units of decibel) as the thickness value of each layer changes. The plots correspond to insets of Fig. 3.

Fig. 6.
Fig. 6.

Plots of peak intensity change (in units of decibel) as the thickness value of each structure’s layer changes by 1%, 2%, or 3% due to any possible fabrication error. The layers are considered quarterwave of optical thickness, and the symmetry of structure is close to Fig. 3(c). These data should be compared to Fig. 5(c). Insets show the worst case (with 3% error) for a traditional design (solid curve) and that of Fig. 3(c) (dashed curve).

Tables (1)

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Table 1. Thickness Values Found for Optimized Structures (Values in Units of 2πc/ω×104)

Equations (3)

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F=(F2F1)+(F2F3).
F=(F2F1+F2),
F=F22F12+F22.

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