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Electrostriction-induced third-order nonlinear optical susceptibility in metamaterials

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Abstract

In this paper, a theoretical method is presented to calculate third-order nonlinear optical susceptibility resulting from electrostriction (TNSRE) in metamaterials. The presented model is based on homogenization and effective medium theory. We have demonstrated that electrostriction is the dominant mechanism in the enhancement and suppression of the TNSRE. To provide a comprehensive framework that includes different types of metamaterials, we examined all known effective elastic models and effective optical models separately. The numerical results showed that TNSRE has a little dependency on the frequency which decreases at higher frequencies.

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Acknowledgements

This work was supported by NSAF, China No. U1830123, the National Natural Science Foundation of China (No. 61627802), and the High-Level Educational Innovation Team Introduction Plan of Jiangsu, China.

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Authors and Affiliations

  1. School of Electronic and Optical Engineering, Nanjing University of Science and Technology, Nanjing, 210094, China

    Omid Khakpour, Bo Yang & Li Li

  2. Laser Fusion Research Center, China Academy of Engineering Physics (CAEP), Mianyang, 621900, China

    Guo Chao & Lin Honghuan

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  1. Omid Khakpour
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  2. Bo Yang
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  3. Guo Chao
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  4. Lin Honghuan
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Correspondence to Li Li.

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Appendices

Appendix A

Here, we review different analytical methods to obtain the effective elastic properties of metamaterials [18, 21, 22].

  1. (1)

    Paul equation:

    $${E}_{\mathrm{eff}}=\frac{{E}_{\mathrm{m}}^{2}+\left({E}_{\mathrm{m}}{E}_{\mathrm{i}}-{E}_{\mathrm{m}}^{2}\right){f}^{2/3}}{{E}_{\mathrm{m}}+\left({E}_{\mathrm{i}}-{E}_{\mathrm{m}}\right){f}^{2/3}\left(1-{f}^{1/3}\right)}$$

    where \({E}_{\mathrm{eff}}\) is effective the elastic modulus,\({E}_{m}\) is the elastic modulus of the matrix, and \({E}_{\mathrm{i}}\) is the elastic modulus of the inclusion.

  2. (2)

    Wu equation:

    $$\frac{1}{{E}_{\mathrm{eff}}}=\left[\frac{1}{{E}_{\mathrm{m}}}-\frac{{\left(\frac{1}{{E}_{\mathrm{m}}}-\frac{1}{{E}_{\mathrm{i}}}\right)}^{2}}{\lambda \left(\frac{1}{{E}_{\mathrm{m}}}+\frac{\left(1-f\right)}{f{E}_{\mathrm{m}}}\right)}\right](1-f)+\frac{f}{{E}_{\mathrm{i}}}$$

    where \(\lambda\) is a parameter that has been found experimentally near to 1 for \({E}_{m}\ll {E}_{f}\) [22].

  3. (3)

    Voigt equation:

    $${E}_{\mathrm{eff}}=f{E}_{\mathrm{m}}+(1-f){E}_{\mathrm{i}}$$
  4. (4)

    Reuss equation:

    $$\frac{1}{{E}_{\mathrm{eff}}}=\frac{(1-f)}{{E}_{\mathrm{m}}}+\frac{f}{{E}_{\mathrm{i}}}$$
  5. (5)

    Einstein equation:

    $${E}_{\mathrm{eff}}={E}_{\mathrm{m}}\left(1+2.5 f\right)$$
  6. (6)

    Guth and Smallwood Equation:

    $${E}_{\mathrm{eff}}={E}_{\mathrm{m}}\left(1+2.5f+14.1{f}^{2}\right)$$
  7. (7)

    Euler and Van Dyck equation:

    $${E}_{\mathrm{eff}}={E}_{\mathrm{m}}\left(1+\frac{kf}{1-sf}\right)$$

    where \(k\) and \(s\) take the values of 1.25 and 1.20, respectively.

  8. (8)

    Narkis equation:

    $${E}_{\mathrm{eff}}=\frac{{E}_{\mathrm{m}}}{\psi (1-{f}^{1/3})}$$

    where \(1.4\le \psi \le 1.7\)

  9. (9)

    Mooney equation:

    $${E}_{\mathrm{eff}}={E}_{\mathrm{m}}\mathrm{exp}\frac{1+2.5f}{1-5f}$$
  10. (10)

    Counto model:

    $$\frac{1}{{E}_{\mathrm{eff}}}=\frac{1-{f}^{1/2}}{{E}_{\mathrm{m}}}+\frac{1}{\left[\left(1-{f}^{1/2}\right)/{f}^{1/2}\right]{E}_{\mathrm{m}}+{E}_{\mathrm{i}}}$$
  11. (11)

    Takahashi equation:

    $$\frac{{E}_{\mathrm{eff}}}{{E}_{\mathrm{m}}}=1+\left(1-{\nu }_{\mathrm{m}}\right)f\frac{{E}_{i}\left(1-{2\nu }_{\mathrm{m}}\right)-{E}_{\mathrm{m}}\left(1-{\nu }_{\mathrm{i}}\right)+10\left(1+{\nu }_{\mathrm{m}}\right){E}_{\mathrm{i}}\left(1+{\nu }_{\mathrm{m}}\right)-{E}_{\mathrm{m}}(1+{\nu }_{\mathrm{m}})}{{E}_{\mathrm{i}}\left(1+{\nu }_{\mathrm{m}}\right)+{2E}_{\mathrm{m}}\left(1-{2\nu }_{\mathrm{i}}\right)+2{E}_{\mathrm{i}}\left(4-{5\nu }_{\mathrm{m}}\right)\left(1+{\nu }_{\mathrm{m}}\right)+{E}_{\mathrm{m}}(7-5{\nu }_{\mathrm{m}})(1+{\nu }_{\mathrm{i}})}$$

    where \({\nu }_{\mathrm{m}}\) and \({\nu }_{\mathrm{i}}\) are Poisson ratios of the matrix and inclusion, respectively.

  12. (12)

    Kerner equations:

    $$\frac{{E}_{\mathrm{eff}}}{{E}_{\mathrm{m}}}=\frac{\frac{f{\mu }_{i}}{\left(7-{5\nu }_{\mathrm{m}}\right){\mu }_{\mathrm{m}}+(8-{10\nu }_{\mathrm{m}}){\mu }_{\mathrm{i}}}+\frac{f}{15(1-{\nu }_{\mathrm{m}})}}{\frac{f{\mu }_{m}}{\left(7-{5\nu }_{\mathrm{m}}\right){\mu }_{\mathrm{m}}+(8-{10\nu }_{\mathrm{m}}){\mu }_{\mathrm{i}}}+\frac{(1-f)}{15(1-{\nu }_{\mathrm{m}})}}$$
  13. (13)

    Hashin and Shtrikman bounds (upper and lower bounds):

    $${{E}_{\mathrm{eff}}}_{H-S}^{\mathrm{upper}}=\frac{9\left({K}_{m}+\frac{f}{\frac{1}{{K}_{i}-{K}_{m}}+\frac{3(1-f)}{{3K}_{m}+4{\mu }_{\mathrm{m}}}}\right)\left({\mu }_{\mathrm{m}}+\frac{f}{\frac{1}{{\mu }_{\mathrm{i}}-{\mu }_{\mathrm{m}}}+\frac{6({K}_{m}+2{\mu }_{m})(1-f)}{5(3{K}_{m}+4{\mu }_{m}){\mu }_{\mathrm{m}}}}\right)}{3\left({K}_{m}+\frac{f}{\frac{1}{{K}_{i}-{K}_{m}}+\frac{3(1-f)}{{3K}_{m}+4{\mu }_{\mathrm{m}}}}\right)+\left({\mu }_{\mathrm{m}}+\frac{f}{\frac{1}{{\mu }_{\mathrm{i}}-{\mu }_{\mathrm{m}}}+\frac{6({K}_{m}+2{\mu }_{\mathrm{m}})(1-f)}{5(3{K}_{m}+4{\mu }_{\mathrm{m}}){\mu }_{\mathrm{m}}}}\right)}$$
    $${{E}_{\mathrm{eff}}}_{H-S}^{\mathrm{lower}}=\frac{9\left({K}_{i}+\frac{(1-f)}{\frac{1}{{K}_{m}-{K}_{i}}+\frac{3f}{3{K}_{i}+4{\mu }_{\mathrm{i}}}}\right)\left({\mu }_{\mathrm{i}}+\frac{(1-f)}{\frac{1}{{\mu }_{\mathrm{m}}-{\mu }_{\mathrm{i}}}+\frac{6({K}_{i}+2{\mu }_{\mathrm{i}})f}{5(3{K}_{i}+4{\mu }_{\mathrm{i}}){\mu }_{\mathrm{i}}}}\right)}{3\left({K}_{f}+\frac{(1-f)}{\frac{1}{{K}_{m}-{K}_{i}}+\frac{3f}{3{K}_{i}+4{\mu }_{\mathrm{i}}}}\right)+\left({\mu }_{\mathrm{i}}+\frac{(1-f)}{\frac{1}{{\mu }_{\mathrm{m}}-{\mu }_{\mathrm{i}}}+\frac{6({K}_{i}+2{\mu }_{\mathrm{i}})f}{5(3{K}_{i}+4{\mu }_{\mathrm{i}}){\mu }_{\mathrm{i}}}}\right)}$$

It is necessary to mention that Eq. (29) is also exact for a metamaterial consisting of spheres of different sizes embedded in a matrix [19].

Appendix B

In this section, we derived \(\frac{\partial \varepsilon }{\partial \omega }\) and \(\frac{{\partial }^{2}\varepsilon }{\partial \rho \partial \omega }\) for some materials.

For \({\mathrm{SiO}}_{2}\):

$$\frac{\partial \varepsilon }{\partial \omega }=(-\frac{2\lambda {A}_{1}{\left({B}_{1}\right)}^{2}}{{\left({\lambda }^{2}-{\left({B}_{1}\right)}^{2}\right)}^{2}}-\frac{2\lambda {C}_{1}{\left({D}_{1}\right)}^{2}}{{\left({\lambda }^{2}-{\left({D}_{1}\right)}^{2}\right)}^{2}}-\frac{2\lambda {E}_{1}{\left({F}_{1}\right)}^{2}}{{\left({\lambda }^{2}-{\left({F}_{1}\right)}^{2}\right)}^{2}})(-\frac{{\lambda }^{2}}{c})$$
$$\frac{{\partial }^{2}\varepsilon }{\partial \rho \partial \omega }=(\frac{2\left({{P}_{11}}_{1}+{{P}_{12}}_{1}\right)}{3\rho })(\frac{{A}_{1}{\lambda }^{2}}{{\lambda }^{2}-{\left({B}_{1}\right)}^{2}}+\frac{{C}_{1}{\lambda }^{2}}{{\lambda }^{2}-{\left({D}_{1}\right)}^{2}}+\frac{{E}_{1}{\lambda }^{2}}{{\lambda }^{2}-{\left({F}_{1}\right)}^{2}}+1)(-\frac{2\lambda {A}_{1}{\left({B}_{1}\right)}^{2}}{{\left({\lambda }^{2}-{\left({B}_{1}\right)}^{2}\right)}^{2}}-\frac{2\lambda {C}_{1}{\left({D}_{1}\right)}^{2}}{{\left({\lambda }^{2}-{\left({D}_{1}\right)}^{2}\right)}^{2}}-\frac{2\lambda {E}_{1}{\left({F}_{1}\right)}^{2}}{{\left({\lambda }^{2}-{\left({F}_{1}\right)}^{2}\right)}^{2}})(-\frac{{\lambda }^{2}}{c})$$

\({A}_{1}=0.6961663;{B}_{1}=0.0684043;{C}_{1}=0.4079426;{D}_{1}=0.1162414;{E}_{1}=0.8974794;{F}_{1}=9.896161;{{P}_{11}}_{1}=0.12;{{P}_{12}}_{1}=0.27\)

For \({\mathrm{As}}_{2}{\mathrm{S}}_{3}\):

$$\frac{\partial \varepsilon }{\partial \omega }=(-\frac{2{A}_{2}{B}_{2}\lambda }{{\left({\lambda }^{2}-{B}_{2}\right)}^{2}}-\frac{2{C}_{2}{D}_{2}\lambda }{{\left({\lambda }^{2}-{D}_{2}\right)}^{2}}-\frac{2{E}_{2}{F}_{2}\lambda }{{\left({\lambda }^{2}-{F}_{2}\right)}^{2}}-\frac{2{G}_{2}{H}_{2}\lambda }{{\left({\lambda }^{2}-{H}_{2}\right)}^{2}}-\frac{2{K}_{2}{L}_{2}\lambda }{{\left({\lambda }^{2}-{L}_{2}\right)}^{2}})\left(-\frac{{\lambda }^{2}}{c}\right)$$
$$\frac{{\partial }^{2}\varepsilon }{\partial \rho \partial \omega }=(\frac{2\left({{P}_{11}}_{2}+{{P}_{12}}_{2}\right)}{3\rho })(\frac{{A}_{2}{\lambda }^{2}}{{\lambda }^{2}-{B}_{2}}+\frac{{C}_{2}{\lambda }^{2}}{{\lambda }^{2}-{D}_{2}}+\frac{{E}_{2}{\lambda }^{2}}{{\lambda }^{2}-{F}_{2}}+\frac{{G}_{2}{\lambda }^{2}}{{\lambda }^{2}-{H}_{2}}+\frac{{K}_{2}{\lambda }^{2}}{{\lambda }^{2}-{L}_{2}}+1)(-\frac{2{A}_{2}{B}_{2}\lambda }{{\left({\lambda }^{2}-{B}_{2}\right)}^{2}}-\frac{2{C}_{2}{D}_{2}\lambda }{{\left({\lambda }^{2}-{D}_{2}\right)}^{2}}-\frac{2{E}_{2}{F}_{2}\lambda }{{\left({\lambda }^{2}-{F}_{2}\right)}^{2}}-\frac{2{G}_{2}{H}_{2}\lambda }{{\left({\lambda }^{2}-{H}_{2}\right)}^{2}}-\frac{2{K}_{2}{L}_{2}\lambda }{{\left({\lambda }^{2}-{L}_{2}\right)}^{2}})\left(-\frac{{\lambda }^{2}}{c}\right)$$

\({A}_{2}=1.8983678;{B}_{2}=0.0225;{C}_{2}=1.9222979;{D}_{2}=0.0625;{E}_{2}=0.8765134;{F}_{2}=0.1225;{G}_{2}=0.1188704;{H}_{2}=0.2025; {K}_{2}=0.9569903;{L}_{2}=750;{{P}_{11}}_{2}=0.25;{{P}_{12}}_{2}=0.24\)

For \(\mathrm{Si}\):

$$\frac{\partial \varepsilon }{\partial \omega }=2\left({A}_{3}+\frac{{B}_{3} {H}^{2}}{{\lambda }^{2}}+\frac{{C}_{3} {H}^{4}}{{\lambda }^{4}}\right)\left(-\frac{2{B}_{3} {H}^{2}}{{\lambda }^{3}}-\frac{4{C}_{3} {H}^{4}}{{\lambda }^{5}}\right)\left(-\frac{{\lambda }^{2}}{c}\right)$$
$$\frac{{\partial }^{2}\varepsilon }{\partial \rho \partial \omega }=2\left(\frac{2\left({{P}_{11}}_{3}+{{P}_{12}}_{3}\right)}{3\rho }\right){({A}_{3}+\frac{{B}_{3} {H}^{2}}{{\lambda }^{2}}+\frac{{C}_{3} {H}^{4}}{{\lambda }^{4}})}^{2}\left({A}_{3}+\frac{{B}_{3} {H}^{2}}{{\lambda }^{2}}+\frac{{C}_{3} {H}^{4}}{{\lambda }^{4}}\right)\left(-\frac{2{B}_{3} {H}^{2}}{{\lambda }^{3}}-\frac{4{C}_{3} {H}^{4}}{{\lambda }^{5}}\right)\left(-\frac{{\lambda }^{2}}{c}\right)$$

\({A}_{3}= 3.4164 ;{ B}_{3}=0.085818; {C}_{3}=0.010149; H=1.2398419;{{p}_{11}}_{3}=-0.094;{{p}_{12}}_{3}=0.017\)

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Khakpour, O., Yang, B., Chao, G. et al. Electrostriction-induced third-order nonlinear optical susceptibility in metamaterials. Appl. Phys. A 127, 376 (2021). https://doi.org/10.1007/s00339-021-04450-8

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