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Optica Publishing Group

Flexible and broadband graphene polarizer based on surface silicon-core microfiber

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Abstract

A graphene-coated polarizer based on the surface silicon-core microfiber has been proposed and numerically analyzed. The numerical discussion shows that our proposed polarizer serves alternatively as TM-pass or TE-pass polarizers depending on the change of the radius of the silicon core. Their polarization extinction ratios can be adjusted by the structure parameters of the ellipticity, the major and minor diameter of the microfiber, and the thickness of the graphene film. When the length of the graphene film is 1.5 mm, extinction ratios of ~30 and ~32 dB can be respectively achieved at the wavelength of 1.55 μm for TM-pass and TE-pass polarizers. The operation wavelength range of the TM- and TE-pass polarizer could cover ~400 and ~1000 nm, respectively. Our proposed polarizer provides a flexible way to manipulate the polarization of the light in the fiber and highly integrated optical fiber system.

© 2017 Optical Society of America

1. Introduction

Optical polarizers are important elements for many optical systems such as fiber-optic networks, optical fiber sensors, biological sensors, optical signal process systems, and so on [1–4], because good line-polarization state lights could provide high signal-to-noise ratio and avoid signal fading. The manipulation of polarization, however, is still a challenge in the integrated optical fiber system. So far, many integratable optical polarizers [5–9] have been proposed with various kind of technology. The common technology is based on the polarization mode selectivity of the surface plasmon resonances (SPRs) [5,6], since the SPRs can only be excited by a transverse electromagnetic field perpendicular to a very thin metal surface with phase matching. However, the working spectral ranges of these optical polarizers are usually limited by their structures [6]. In order to enlarge the working spectral ranges, an in-fiber linear polarizer based on the tilted grating has been performed for TM polarization with 40dB polarization extinction ratio over 50 nm wavelength range [7]. However, its working spectral ranges cannot meet the demand in the optical fiber communication system yet. In 2011, a broadband fiber polarizer was achieved by combing the 3 mm graphene sheet with the side-polished fiber for large working spectral ranges from 800 to 1650 nm [8]. This side-polished approach, however, greatly reduced the strength and durability of the fiber, which would impose limits on the possible applications and service life. Subsequently, J. T. Kim et al., fabricated a graphene polarizer by using the planar-lightwave-circuit (PLC) and graphene [9], where the insertion losses of the TM and TE polarization light for air cladding are 4.14 and 2.18 dB/μm, and TM and TE losses for polymer cladding are 5.4 and 9.4 dB/μm, respectively. Nevertheless, such polarizer cannot integrate with the fiber system.

Recently, silica-cladding silicon core optical fibers [10,11] have attracted much interest due to their infrared transparency [11], strong evanescent field, high optical confining factor, and strong optical nonlinearities, which offer great potential for many applications in sensing, biomedicine, high-speed, all-optical signal processing, supercontinuum spectrum generation [12–14]and so on. Especially, the tightly strong evanescent field of surface-core fiber [15] could enable it to combine with graphene sheet for a novel polarizer with easy-integrated, durable, and broadband performances. In this paper, a flexible and broadband graphene polarizer based on the surface silicon-core microfiber has been proposed, in which the surface silicon-core locates tangent to the cladding of the microfiber. The optical broadband polarization properties of such polarizer are investigated in detail. The impacts of the structure parameters such as ellipticity, major and minor diameter of the microfiber, the size of the silicon core, and the thickness of the graphene film on its performances are discussed.

2. Theoretical analysis and design

The schematic cross section of our proposed surface silicon-core microfiber (SSCM) polarizer with surface-covered graphene film is shown in Fig. 1. A monolayer or few-layer graphene film covers upon the side of the microfiber, where the cladding of the microfiber and the graphene film locate tangent to the surface silicon-core, as shown in Fig. 1. A tight and strong evanescent field can be achieved by its small surface silicon core, which could increase the interaction between the evanescent light and the graphene film. The asymmetric dielectric structure model consists of air, graphene, silicon core, and silica cladding layers. In Fig. 1, a and b is respectively the major and minor diameter of the microfiber, and ellipticity is defined as e = b/a. The complex effective refractive index neff of such waveguide can be obtained by solving Maxwell’s equations [8]:

×H=jωεrε0E+σE
×E=jωμ0H
By matching the boundary conditions with the Maxwell’s equation, the dispersion relation of the graphene-microfiber waveguide could be achieved [8]:
atan(Ca)+atan(Cb)+mπ=γ2d
where m is the order of the mode, and d is the diameter of the silicon core. In Eq. (3) for TM polarization modes,Ca=(γ1ε2ε1γ2)(1+σγ1ωε0ε1)-1,Cb=γ3ε2ε3γ2; and for TE polarization mode Ca=(γ1+ωμ0σγ1),Cb=γ3γ2, where γ1=k0neff2-n12,γ2=k0n22-neff2and γ3=k0neff2-ε3. where, εj is the relative permittivity of region j. n1 is the refractive index of the silica cladding and n2 is the refractive index of the silicon core.

 figure: Fig. 1

Fig. 1 The schematic cross section of the polarizer, where r is the radius of the silicon core. a and b is respectively the major and minor diameter of the microfiber. n1 = 1.4378, n2 = 3.4784.

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The complex relative permittivity of the graphene ε3 is derived by [16]:

ε3=1Im(σ)ωε0dg+iRe(σ)ωε0dg
where dg is the thickness of graphene film, σ is the complex conductivity of graphene and can be expressed by Kubo formula, including interband and intraband contributions [17]:
σ(ω,μc,Γ,T)=je2(ωjτ-1)π[1(ωjτ-1)20ε(fd(ε)εfd(ε)ε)dε0fd(ε)fd(ε)(ωj2Γ)24(ε/)2dε]
where e is the charge of an electron,=h/2πis the reduced Planck’s constant, fd(ε)=(e(εμc)/kBT+1)1is the Fermi-Dirac distribution, and kB is Boltzmann’s constant. Thus, the complex effective refractive index neff of the waveguide can be obtained by combining Eq. (3),(4) and (5). The birefringence can be calculated by the real part of the effective refractive index of TE-polarized and TM-polarized modes, that isB=Re(nTE)Re(nTM). The attenuation constant α is related with image of the effective refractive index of such waveguide, which can be expressed as:

α=Im(neff)k0

The extinction ratio (ER) of such graphene-microfiber polarizer can be calculated by multiplying the difference in the attenuation constant between the TE-polarized and TM-polarized modes by the propagation distance:

E=(αTEαTM)Lg
where Lg is the graphene film length, and αTE and αTM are the losses of TE and TM modes.

3. Polarizer properties discussion

In order to simplify the experiment and optimize the design process, it is necessary to carry out simulations of the proposed structures prior to an experimental investigation. Figure 2(a) and 2(c) show the birefringence varies with the change of the silicon core radius for different ellipticity e and different cladding size. With the change of the ellipticity from 0.5 to 1, the radius of the silicon core of ~0.152 μm related with the maximal birefringence Bmax remains unchanged, and the value of the Bmax slightly decreases. It can be seen that the ellipticity of the microfiber and the major and minor diameters of the microfiber has nothing to do with the silicon-core radius of such polarizer with the maximal birefringence, but it could affect the value of Bmax. The extinction ratio of two polarized modes is calculated by Eq. (7) with the graphene film length of 1.5 mm, which is varied with the silicon core under different ellipticity, the major and minor diameter in Fig. 2(b) and 2(d). It can be seen that, when the radius of the silicon core is low to ~0.147 μm, the fundamental TE mode losses is larger than that of fundamental TM mode. However, when the radius of the silicon core is larger than ~0.147 μm, the fundamental TM mode suffers greater loss than the fundamental TE mode. At the silicon core radius of 0.132 μm, a TM-pass polarizer has been obtained with the highest extinction ratio of ~30 dB, while a good TE-pass polarizer with 30 dB ER has been obtained at the silicon core radius of 0.17 μm. The extinction ratio of two orthogonal polarizations is a consequence of the attenuations difference between two orthogonal polarizations of the fundamental mode (TM- polarized and TE-polarized modes). So the TM-pass mode with the lowest loss in the SSCM is generated at the silicon core radius of 0.132 μm, while the TE-pass mode with the lowest loss in the SSCM is realized at the silicon core radius of 0.17 μm. The extinction ratios of ~30 and ~32 dB can be achieved at the wavelength of 1.55 μm for TM and TE-pass polarizer, respectively. Furthermore, the ellipticity, the major diameter, and the minor diameter of the microfiber cannot affect the polarization status of such polarizer. From Fig. 2(b) and 2(d), it is concluded that the polarization status of such polarizer is just determined by the size of the silicon core. Thus, the above phenomenon confirms that changing the silicon core of the SSCM provides a potential way to manipulate the polarization property.

 figure: Fig. 2

Fig. 2 (a) Birefringence as a function of the radius of the silicon core with different ellipticity. (b) Extinction ratio versus the radius of the silicon core with different ellipticity. (c) Birefringence as a function of the radius of the silicon core with different major and minor diameter. (d) Extinction ratio versus the radius of the silicon core with different major and minor diameter.

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In order to explore deeply the reason why the polarization of the SSCM is mainly dependent on the radius of the silicon core, the electric field distributions for TM-pass and TE-pass modes at the chemical potential of 0 meV has been presented, as shown in Fig. 3. Under the silicon core radius of 0.132 μm, the electric field distributions for TE and TM modes at e = 0.5, λ = 1.55 μm are shown in Fig. 3(a) and 3(b), respectively. It can be seen that TE polarization direction is along the graphene film, and the TM polarization direction is perpendicular to the graphene film surface. Figure 3(c) is the intensity of the electric field of the core center along the x-axis at y = 1 μm, in which the Cartesian coordinate origin is at the center of the microfiber cladding. Compared to the TM mode, a large of electric field energy in TE modes is out of the silicon core to come into the graphene film by absorption. Thus, the SSCM with the silicon core radius of 0.132 μm presents as a TM-pass polarizer. Figure 3(d) and 3(e) are respectively the electric field distributions for TE and TM modes at r = 0.17 μm, e = 0.5, λ = 1.55 μm. Compared the electric field distribution of the SSCM at r = 0.132 μm (Fig. 3 (a) and 3(b)) with that of r = 0.17 μm (Fig. 3(d) and 3(e)), the larger silicon core could constraint more energy in the core, which could result in the change of the attenuations difference of two orthogonal polarizations modes. Figure 3(f) is the intensity of the electric field of the core center in the SSCM at r = 0.17 μm along the y-axis at x = 0 μm, where the Cartesian coordinate origin is also at the center of the microfiber cladding. There is more energy of the TM mode coupled into the graphene film than that of the TE mode, thus achieving a TE-pass polarizer at r = 0.17 μm.

 figure: Fig. 3

Fig. 3 The electric field distribution on the cross section of the SSCM polarizer with surface-covered graphene film for TE and TM mode light at the radius of the silicon core of 0.132 μm ((a) and (b)) and 0.17 μm ((d) and (e)). (c) The amplitudes of the electric field as a function of x coordinate axis along y = 1μm at the silicon core radius of 0.132μm, a = 4 μm and b = 2 μm. (f) The intensity of the electric field as a function of y coordinate axis along x = 0 at the silicon core radius of 0.17 μm, a = 4 μm and b = 2 μm.

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To further discuss the impact of the ellipticity, major and minor diameter on TM-pass and TE-pass polarizer, the birefringence and propagation losses of the SSCM as a function of the ellipticity, major and minor diameter has been analyzed in Fig. 4, under keeping r = 0.132 μm and r = 0.17 μm. With the increase of the ellipticity, it is found in Fig. 4(a) and 4(c) that the birefringence at r = 0.132 μm and r = 0.17 μm all rapidly decrease, while the difference of the TE and TM modes increases at r = 0.132 μm while it decreases at r = 0.17 μm. Thus, the ellipticity should be set at 0.2 to obtain a high-ER TE-pass polarizer while it should be set at 1 to obtain a high-ER TM-pass polarizer. When the major and minor diameters increase, the birefringence at r = 0.132 μm and r = 0.17 μm increases from Fig. 4(b) and 4(d). The differences of the TE and TM modes for two polarizers at r = 0.132 μm and r = 0.17 μm both increase with the increase of the major and minor diameters. It can be concluded that the large-sized cladding in the SSCM contributes to improve the ER of the polarizer.

 figure: Fig. 4

Fig. 4 The effect of the ellipticity, major and minor diameter on the birefringence and propagation loss for TE and TM modes. (a) and (b) for r = 0.132 μm, (c) and (d) for r = 0.17 μm.

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The impact of the graphene thickness on the birefringence and propagation loss is shown in Fig. 5. When the graphene layer is varied from 1 to 7, the propagation losses of the TM and TE modes increases, and on the contrary birefringence decreases. From Fig. 5, it can be observed that, in order to obtain the same high ER on polarization, the length of monolayer graphene film should be longer along the propagation direction than that of the multi-layer graphene. To be concluded, increasing the graphene layer is helpful for decreasing the length of the graphene film.

 figure: Fig. 5

Fig. 5 The effect of the graphene thickness on the birefringence and propagation loss for TE and TM modes. (a) r = 0.132 μm and (b) r = 0.17 μm.

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The large optical absorption bandwidth is an important property for the polarizer. For analyzing the bandwidth of such SSCM polarizer, the propagation losses of the TM and TE modes have been calculated with the variation of the wavelength, as shown in Fig. 6. When the silicon core radius is 0.132 μm, the losses between the TE and TM modes dramatically increase and the difference of two polarized modes obviously decreases by varying the wavelength from 1400 to 1800 nm. On the contrary, in Fig. 6(b), by increasing the wavelength from 800 to 1750 nm, the losses of the TE and TM modes at r = 0.17 μm remarkably increase and the difference of two polarized modes also increases. For the TM-pass polarizer at r = 0.132 μm, the polarization bandwidth is about 400 nm, while the TE polarization mode wave propagation bandwidth is close to ~1000 nm when the silicon core radius is set at 0.17 μm.

 figure: Fig. 6

Fig. 6 Propagation losses of TE and TM modes against the wavelength (a) r = 0.132 μm and (b) r = 0.17 μm.

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4. Summary

We proposed a graphene-coated SSCM polarizer and numerically analyzed the birefringence and losses of two polarization modes in detail. The birefringence of the SSCM has no impact on the light polarization. By adjusting the radius of the silicon core from 0.132 to 0.17 μm, we can achieve TM-pass and TE-pass polarizers. The extinction ratios of the TM-pass polarizer can be increased by enlarging the ellipticity, while the extinction ratios of the TE-pass polarizer can be increased by reducing the ellipticity. The extinction ratios of the TM-pass and TE-pass polarizer can be improved by increasing the size of the cladding in the SSCM. Multi-layer the graphene is helpful for decreasing the used length of graphene film covered on the SSCM. The extinction ratios of ~30 and ~32 dB can be achieved at the wavelength of 1550 nm for TM- and TE-pass polarizers respectively, when the length of the graphene film is 1.5 mm. The operation wavelength range of the TM-pass and TE-pass polarizer could cover ~400 and ~1000 nm, respectively. Such graphene-coated SSCM polarizer can be manipulated by the physical changes on the surface silicon core without destroying the fiber structure, and therefore this polarized component has advantages of simple fabrication, easy-integration, durability, broadband and low cost, and also provide a flexible way to manipulate the polarization of the light in fiber and integrated optical fiber system.

Funding

National Natural Science Foundation of China (NSFC) (No.61675046, 61205132); a Doctoral Program of Higher Education of China (20120071120023).

References and links

1. H. Sunnerud, M. Karlsson, C. Xie, and P. A. Andrekson, “Polarization-mode dispersion in high-speed fiber-optic transmission systems,” J. Lightwave Technol. 20(12), 2204–2219 (2002). [CrossRef]  

2. D. C. Cullen, R. G. Brown, and C. R. Lowe, “Detection of immuno-complex formation via surface plasmon resonance on gold-coated diffraction gratings,” Biosensors 3(4), 211–225 (1987). [CrossRef]   [PubMed]  

3. M. Piliarik, J. Homola, Z. Manikova, and J. Ctyroky, “Surface plasmon resonance sensor based on a single-mode polarization-maintaining optical fiber,” Sens. Actuators B Chem. 90(1-3), 236–242 (2003). [CrossRef]  

4. H. C. Su and L. A. Wang, “A highly efficient polarized superfluorescent fiber source for fiber-optic gyroscope applications,” IEEE Photonics Technol. Lett. 15(10), 1357–1359 (2003). [CrossRef]  

5. T. K. Ng, M. Z. M. Khan, A. Al-Jabr, and B. S. Ooi, “Analysis of CMOS compatible Cu-based TM-pass optical polarizer,” IEEE Photonics Technol. Lett. 24(9), 724–726 (2012). [CrossRef]  

6. Y. Ma, G. Farrel, Y. Semenova, B. Li, J. Yuan, B. Li, J. Yuan, X. Sang, B. Yan, C. Yu, and Q. Wu, “Optical microfiber-loaded surface plasmonic TE-pass polarizer,” Opt. Laser Technol. 78, 110–114 (2015).

7. H. Esmaeilzadeh, E. Arzi, M. Mozafari, and A. Hassani, “A broadband optical fiber based inline polarizer for telecom wavelength range,” Sens. Actuators A Phys. 185, 59–65 (2012). [CrossRef]  

8. Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics 5(7), 411–415 (2011). [CrossRef]  

9. J. T. Kim and C. G. Choi, “Graphene-based polymer waveguide polarizer,” Opt. Express 20(4), 3556–3562 (2012). [CrossRef]   [PubMed]  

10. P. Mehta, N. Healy, N. F. Baril, P. J. A. Sazio, J. V. Badding, and A. C. Peacock, “Nonlinear transmission properties of hydrogenated amorphous silicon core optical fibers,” Opt. Express 18(16), 16826–16831 (2010). [CrossRef]   [PubMed]  

11. S. Morris, T. Hawkins, P. Foy, J. Hudson, L. Zhu, R. Stolen, R. Rice, and J. Ballato, “On loss in silicon core optical fiber,” Opt. Mater. Express 2(11), 1511–1519 (2012). [CrossRef]  

12. D. Graham-Rowe, “Fibres get functional,” Nat. Photonics 5(2), 66–67 (2011). [CrossRef]  

13. B. Jalali, V. Raghunathan, D. Dimitrophoulos, and O. Boyraz, “Raman-based silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12(3), 412–421 (2006). [CrossRef]  

14. J. Ballato, T. Hawkins, P. Foy, B. Yazgan-Kokuoz, C. Mcmillen, L. Burka, S. Morris, R. Stolen, and R. Rice, “Advancements in semiconductor core optical fiber,” Opt. Fiber Technol. 16(6), 399–408 (2010). [CrossRef]  

15. L. Yang, T. Hu, R. Hao, C. Qiu, C. Xu, H. Yu, Y. Xu, X. Jiang, Y. Li, and J. Yang, “Low-chirp high-extinction-ratio modulator based on graphene-silicon waveguide,” Opt. Lett. 38(14), 2512–2515 (2013). [CrossRef]   [PubMed]  

16. G. W. Hanson, “Quasi-transverse electromagnetic modes supported by a graphene parallel-plate waveguide,” J. Appl. Phys. 104(8), 084314 (2008). [CrossRef]  

17. C. Guan, L. Yuan, F. Tian, and Q. Dai, “Characteristics of near-surface-core optical fibers,” J. Lightwave Technol. 29(19), 3004–3008 (2011). [CrossRef]  

References

  • View by:

  1. H. Sunnerud, M. Karlsson, C. Xie, and P. A. Andrekson, “Polarization-mode dispersion in high-speed fiber-optic transmission systems,” J. Lightwave Technol. 20(12), 2204–2219 (2002).
    [Crossref]
  2. D. C. Cullen, R. G. Brown, and C. R. Lowe, “Detection of immuno-complex formation via surface plasmon resonance on gold-coated diffraction gratings,” Biosensors 3(4), 211–225 (1987).
    [Crossref] [PubMed]
  3. M. Piliarik, J. Homola, Z. Manikova, and J. Ctyroky, “Surface plasmon resonance sensor based on a single-mode polarization-maintaining optical fiber,” Sens. Actuators B Chem. 90(1-3), 236–242 (2003).
    [Crossref]
  4. H. C. Su and L. A. Wang, “A highly efficient polarized superfluorescent fiber source for fiber-optic gyroscope applications,” IEEE Photonics Technol. Lett. 15(10), 1357–1359 (2003).
    [Crossref]
  5. T. K. Ng, M. Z. M. Khan, A. Al-Jabr, and B. S. Ooi, “Analysis of CMOS compatible Cu-based TM-pass optical polarizer,” IEEE Photonics Technol. Lett. 24(9), 724–726 (2012).
    [Crossref]
  6. Y. Ma, G. Farrel, Y. Semenova, B. Li, J. Yuan, B. Li, J. Yuan, X. Sang, B. Yan, C. Yu, and Q. Wu, “Optical microfiber-loaded surface plasmonic TE-pass polarizer,” Opt. Laser Technol. 78, 110–114 (2015).
  7. H. Esmaeilzadeh, E. Arzi, M. Mozafari, and A. Hassani, “A broadband optical fiber based inline polarizer for telecom wavelength range,” Sens. Actuators A Phys. 185, 59–65 (2012).
    [Crossref]
  8. Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics 5(7), 411–415 (2011).
    [Crossref]
  9. J. T. Kim and C. G. Choi, “Graphene-based polymer waveguide polarizer,” Opt. Express 20(4), 3556–3562 (2012).
    [Crossref] [PubMed]
  10. P. Mehta, N. Healy, N. F. Baril, P. J. A. Sazio, J. V. Badding, and A. C. Peacock, “Nonlinear transmission properties of hydrogenated amorphous silicon core optical fibers,” Opt. Express 18(16), 16826–16831 (2010).
    [Crossref] [PubMed]
  11. S. Morris, T. Hawkins, P. Foy, J. Hudson, L. Zhu, R. Stolen, R. Rice, and J. Ballato, “On loss in silicon core optical fiber,” Opt. Mater. Express 2(11), 1511–1519 (2012).
    [Crossref]
  12. D. Graham-Rowe, “Fibres get functional,” Nat. Photonics 5(2), 66–67 (2011).
    [Crossref]
  13. B. Jalali, V. Raghunathan, D. Dimitrophoulos, and O. Boyraz, “Raman-based silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12(3), 412–421 (2006).
    [Crossref]
  14. J. Ballato, T. Hawkins, P. Foy, B. Yazgan-Kokuoz, C. Mcmillen, L. Burka, S. Morris, R. Stolen, and R. Rice, “Advancements in semiconductor core optical fiber,” Opt. Fiber Technol. 16(6), 399–408 (2010).
    [Crossref]
  15. L. Yang, T. Hu, R. Hao, C. Qiu, C. Xu, H. Yu, Y. Xu, X. Jiang, Y. Li, and J. Yang, “Low-chirp high-extinction-ratio modulator based on graphene-silicon waveguide,” Opt. Lett. 38(14), 2512–2515 (2013).
    [Crossref] [PubMed]
  16. G. W. Hanson, “Quasi-transverse electromagnetic modes supported by a graphene parallel-plate waveguide,” J. Appl. Phys. 104(8), 084314 (2008).
    [Crossref]
  17. C. Guan, L. Yuan, F. Tian, and Q. Dai, “Characteristics of near-surface-core optical fibers,” J. Lightwave Technol. 29(19), 3004–3008 (2011).
    [Crossref]

2015 (1)

Y. Ma, G. Farrel, Y. Semenova, B. Li, J. Yuan, B. Li, J. Yuan, X. Sang, B. Yan, C. Yu, and Q. Wu, “Optical microfiber-loaded surface plasmonic TE-pass polarizer,” Opt. Laser Technol. 78, 110–114 (2015).

2013 (1)

2012 (4)

S. Morris, T. Hawkins, P. Foy, J. Hudson, L. Zhu, R. Stolen, R. Rice, and J. Ballato, “On loss in silicon core optical fiber,” Opt. Mater. Express 2(11), 1511–1519 (2012).
[Crossref]

T. K. Ng, M. Z. M. Khan, A. Al-Jabr, and B. S. Ooi, “Analysis of CMOS compatible Cu-based TM-pass optical polarizer,” IEEE Photonics Technol. Lett. 24(9), 724–726 (2012).
[Crossref]

H. Esmaeilzadeh, E. Arzi, M. Mozafari, and A. Hassani, “A broadband optical fiber based inline polarizer for telecom wavelength range,” Sens. Actuators A Phys. 185, 59–65 (2012).
[Crossref]

J. T. Kim and C. G. Choi, “Graphene-based polymer waveguide polarizer,” Opt. Express 20(4), 3556–3562 (2012).
[Crossref] [PubMed]

2011 (3)

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics 5(7), 411–415 (2011).
[Crossref]

D. Graham-Rowe, “Fibres get functional,” Nat. Photonics 5(2), 66–67 (2011).
[Crossref]

C. Guan, L. Yuan, F. Tian, and Q. Dai, “Characteristics of near-surface-core optical fibers,” J. Lightwave Technol. 29(19), 3004–3008 (2011).
[Crossref]

2010 (2)

J. Ballato, T. Hawkins, P. Foy, B. Yazgan-Kokuoz, C. Mcmillen, L. Burka, S. Morris, R. Stolen, and R. Rice, “Advancements in semiconductor core optical fiber,” Opt. Fiber Technol. 16(6), 399–408 (2010).
[Crossref]

P. Mehta, N. Healy, N. F. Baril, P. J. A. Sazio, J. V. Badding, and A. C. Peacock, “Nonlinear transmission properties of hydrogenated amorphous silicon core optical fibers,” Opt. Express 18(16), 16826–16831 (2010).
[Crossref] [PubMed]

2008 (1)

G. W. Hanson, “Quasi-transverse electromagnetic modes supported by a graphene parallel-plate waveguide,” J. Appl. Phys. 104(8), 084314 (2008).
[Crossref]

2006 (1)

B. Jalali, V. Raghunathan, D. Dimitrophoulos, and O. Boyraz, “Raman-based silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12(3), 412–421 (2006).
[Crossref]

2003 (2)

M. Piliarik, J. Homola, Z. Manikova, and J. Ctyroky, “Surface plasmon resonance sensor based on a single-mode polarization-maintaining optical fiber,” Sens. Actuators B Chem. 90(1-3), 236–242 (2003).
[Crossref]

H. C. Su and L. A. Wang, “A highly efficient polarized superfluorescent fiber source for fiber-optic gyroscope applications,” IEEE Photonics Technol. Lett. 15(10), 1357–1359 (2003).
[Crossref]

2002 (1)

1987 (1)

D. C. Cullen, R. G. Brown, and C. R. Lowe, “Detection of immuno-complex formation via surface plasmon resonance on gold-coated diffraction gratings,” Biosensors 3(4), 211–225 (1987).
[Crossref] [PubMed]

Al-Jabr, A.

T. K. Ng, M. Z. M. Khan, A. Al-Jabr, and B. S. Ooi, “Analysis of CMOS compatible Cu-based TM-pass optical polarizer,” IEEE Photonics Technol. Lett. 24(9), 724–726 (2012).
[Crossref]

Andrekson, P. A.

Arzi, E.

H. Esmaeilzadeh, E. Arzi, M. Mozafari, and A. Hassani, “A broadband optical fiber based inline polarizer for telecom wavelength range,” Sens. Actuators A Phys. 185, 59–65 (2012).
[Crossref]

Badding, J. V.

Ballato, J.

S. Morris, T. Hawkins, P. Foy, J. Hudson, L. Zhu, R. Stolen, R. Rice, and J. Ballato, “On loss in silicon core optical fiber,” Opt. Mater. Express 2(11), 1511–1519 (2012).
[Crossref]

J. Ballato, T. Hawkins, P. Foy, B. Yazgan-Kokuoz, C. Mcmillen, L. Burka, S. Morris, R. Stolen, and R. Rice, “Advancements in semiconductor core optical fiber,” Opt. Fiber Technol. 16(6), 399–408 (2010).
[Crossref]

Bao, Q.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics 5(7), 411–415 (2011).
[Crossref]

Baril, N. F.

Boyraz, O.

B. Jalali, V. Raghunathan, D. Dimitrophoulos, and O. Boyraz, “Raman-based silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12(3), 412–421 (2006).
[Crossref]

Brown, R. G.

D. C. Cullen, R. G. Brown, and C. R. Lowe, “Detection of immuno-complex formation via surface plasmon resonance on gold-coated diffraction gratings,” Biosensors 3(4), 211–225 (1987).
[Crossref] [PubMed]

Burka, L.

J. Ballato, T. Hawkins, P. Foy, B. Yazgan-Kokuoz, C. Mcmillen, L. Burka, S. Morris, R. Stolen, and R. Rice, “Advancements in semiconductor core optical fiber,” Opt. Fiber Technol. 16(6), 399–408 (2010).
[Crossref]

Choi, C. G.

Ctyroky, J.

M. Piliarik, J. Homola, Z. Manikova, and J. Ctyroky, “Surface plasmon resonance sensor based on a single-mode polarization-maintaining optical fiber,” Sens. Actuators B Chem. 90(1-3), 236–242 (2003).
[Crossref]

Cullen, D. C.

D. C. Cullen, R. G. Brown, and C. R. Lowe, “Detection of immuno-complex formation via surface plasmon resonance on gold-coated diffraction gratings,” Biosensors 3(4), 211–225 (1987).
[Crossref] [PubMed]

Dai, Q.

Dimitrophoulos, D.

B. Jalali, V. Raghunathan, D. Dimitrophoulos, and O. Boyraz, “Raman-based silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12(3), 412–421 (2006).
[Crossref]

Esmaeilzadeh, H.

H. Esmaeilzadeh, E. Arzi, M. Mozafari, and A. Hassani, “A broadband optical fiber based inline polarizer for telecom wavelength range,” Sens. Actuators A Phys. 185, 59–65 (2012).
[Crossref]

Farrel, G.

Y. Ma, G. Farrel, Y. Semenova, B. Li, J. Yuan, B. Li, J. Yuan, X. Sang, B. Yan, C. Yu, and Q. Wu, “Optical microfiber-loaded surface plasmonic TE-pass polarizer,” Opt. Laser Technol. 78, 110–114 (2015).

Foy, P.

S. Morris, T. Hawkins, P. Foy, J. Hudson, L. Zhu, R. Stolen, R. Rice, and J. Ballato, “On loss in silicon core optical fiber,” Opt. Mater. Express 2(11), 1511–1519 (2012).
[Crossref]

J. Ballato, T. Hawkins, P. Foy, B. Yazgan-Kokuoz, C. Mcmillen, L. Burka, S. Morris, R. Stolen, and R. Rice, “Advancements in semiconductor core optical fiber,” Opt. Fiber Technol. 16(6), 399–408 (2010).
[Crossref]

Graham-Rowe, D.

D. Graham-Rowe, “Fibres get functional,” Nat. Photonics 5(2), 66–67 (2011).
[Crossref]

Guan, C.

Hanson, G. W.

G. W. Hanson, “Quasi-transverse electromagnetic modes supported by a graphene parallel-plate waveguide,” J. Appl. Phys. 104(8), 084314 (2008).
[Crossref]

Hao, R.

Hassani, A.

H. Esmaeilzadeh, E. Arzi, M. Mozafari, and A. Hassani, “A broadband optical fiber based inline polarizer for telecom wavelength range,” Sens. Actuators A Phys. 185, 59–65 (2012).
[Crossref]

Hawkins, T.

S. Morris, T. Hawkins, P. Foy, J. Hudson, L. Zhu, R. Stolen, R. Rice, and J. Ballato, “On loss in silicon core optical fiber,” Opt. Mater. Express 2(11), 1511–1519 (2012).
[Crossref]

J. Ballato, T. Hawkins, P. Foy, B. Yazgan-Kokuoz, C. Mcmillen, L. Burka, S. Morris, R. Stolen, and R. Rice, “Advancements in semiconductor core optical fiber,” Opt. Fiber Technol. 16(6), 399–408 (2010).
[Crossref]

Healy, N.

Homola, J.

M. Piliarik, J. Homola, Z. Manikova, and J. Ctyroky, “Surface plasmon resonance sensor based on a single-mode polarization-maintaining optical fiber,” Sens. Actuators B Chem. 90(1-3), 236–242 (2003).
[Crossref]

Hu, T.

Hudson, J.

Jalali, B.

B. Jalali, V. Raghunathan, D. Dimitrophoulos, and O. Boyraz, “Raman-based silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12(3), 412–421 (2006).
[Crossref]

Jiang, X.

Karlsson, M.

Khan, M. Z. M.

T. K. Ng, M. Z. M. Khan, A. Al-Jabr, and B. S. Ooi, “Analysis of CMOS compatible Cu-based TM-pass optical polarizer,” IEEE Photonics Technol. Lett. 24(9), 724–726 (2012).
[Crossref]

Kim, J. T.

Li, B.

Y. Ma, G. Farrel, Y. Semenova, B. Li, J. Yuan, B. Li, J. Yuan, X. Sang, B. Yan, C. Yu, and Q. Wu, “Optical microfiber-loaded surface plasmonic TE-pass polarizer,” Opt. Laser Technol. 78, 110–114 (2015).

Y. Ma, G. Farrel, Y. Semenova, B. Li, J. Yuan, B. Li, J. Yuan, X. Sang, B. Yan, C. Yu, and Q. Wu, “Optical microfiber-loaded surface plasmonic TE-pass polarizer,” Opt. Laser Technol. 78, 110–114 (2015).

Li, Y.

Lim, C. H. Y. X.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics 5(7), 411–415 (2011).
[Crossref]

Loh, K. P.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics 5(7), 411–415 (2011).
[Crossref]

Lowe, C. R.

D. C. Cullen, R. G. Brown, and C. R. Lowe, “Detection of immuno-complex formation via surface plasmon resonance on gold-coated diffraction gratings,” Biosensors 3(4), 211–225 (1987).
[Crossref] [PubMed]

Ma, Y.

Y. Ma, G. Farrel, Y. Semenova, B. Li, J. Yuan, B. Li, J. Yuan, X. Sang, B. Yan, C. Yu, and Q. Wu, “Optical microfiber-loaded surface plasmonic TE-pass polarizer,” Opt. Laser Technol. 78, 110–114 (2015).

Manikova, Z.

M. Piliarik, J. Homola, Z. Manikova, and J. Ctyroky, “Surface plasmon resonance sensor based on a single-mode polarization-maintaining optical fiber,” Sens. Actuators B Chem. 90(1-3), 236–242 (2003).
[Crossref]

Mcmillen, C.

J. Ballato, T. Hawkins, P. Foy, B. Yazgan-Kokuoz, C. Mcmillen, L. Burka, S. Morris, R. Stolen, and R. Rice, “Advancements in semiconductor core optical fiber,” Opt. Fiber Technol. 16(6), 399–408 (2010).
[Crossref]

Mehta, P.

Morris, S.

S. Morris, T. Hawkins, P. Foy, J. Hudson, L. Zhu, R. Stolen, R. Rice, and J. Ballato, “On loss in silicon core optical fiber,” Opt. Mater. Express 2(11), 1511–1519 (2012).
[Crossref]

J. Ballato, T. Hawkins, P. Foy, B. Yazgan-Kokuoz, C. Mcmillen, L. Burka, S. Morris, R. Stolen, and R. Rice, “Advancements in semiconductor core optical fiber,” Opt. Fiber Technol. 16(6), 399–408 (2010).
[Crossref]

Mozafari, M.

H. Esmaeilzadeh, E. Arzi, M. Mozafari, and A. Hassani, “A broadband optical fiber based inline polarizer for telecom wavelength range,” Sens. Actuators A Phys. 185, 59–65 (2012).
[Crossref]

Ng, T. K.

T. K. Ng, M. Z. M. Khan, A. Al-Jabr, and B. S. Ooi, “Analysis of CMOS compatible Cu-based TM-pass optical polarizer,” IEEE Photonics Technol. Lett. 24(9), 724–726 (2012).
[Crossref]

Ni, Z.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics 5(7), 411–415 (2011).
[Crossref]

Ooi, B. S.

T. K. Ng, M. Z. M. Khan, A. Al-Jabr, and B. S. Ooi, “Analysis of CMOS compatible Cu-based TM-pass optical polarizer,” IEEE Photonics Technol. Lett. 24(9), 724–726 (2012).
[Crossref]

Peacock, A. C.

Piliarik, M.

M. Piliarik, J. Homola, Z. Manikova, and J. Ctyroky, “Surface plasmon resonance sensor based on a single-mode polarization-maintaining optical fiber,” Sens. Actuators B Chem. 90(1-3), 236–242 (2003).
[Crossref]

Qiu, C.

Raghunathan, V.

B. Jalali, V. Raghunathan, D. Dimitrophoulos, and O. Boyraz, “Raman-based silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12(3), 412–421 (2006).
[Crossref]

Rice, R.

S. Morris, T. Hawkins, P. Foy, J. Hudson, L. Zhu, R. Stolen, R. Rice, and J. Ballato, “On loss in silicon core optical fiber,” Opt. Mater. Express 2(11), 1511–1519 (2012).
[Crossref]

J. Ballato, T. Hawkins, P. Foy, B. Yazgan-Kokuoz, C. Mcmillen, L. Burka, S. Morris, R. Stolen, and R. Rice, “Advancements in semiconductor core optical fiber,” Opt. Fiber Technol. 16(6), 399–408 (2010).
[Crossref]

Sang, X.

Y. Ma, G. Farrel, Y. Semenova, B. Li, J. Yuan, B. Li, J. Yuan, X. Sang, B. Yan, C. Yu, and Q. Wu, “Optical microfiber-loaded surface plasmonic TE-pass polarizer,” Opt. Laser Technol. 78, 110–114 (2015).

Sazio, P. J. A.

Semenova, Y.

Y. Ma, G. Farrel, Y. Semenova, B. Li, J. Yuan, B. Li, J. Yuan, X. Sang, B. Yan, C. Yu, and Q. Wu, “Optical microfiber-loaded surface plasmonic TE-pass polarizer,” Opt. Laser Technol. 78, 110–114 (2015).

Stolen, R.

S. Morris, T. Hawkins, P. Foy, J. Hudson, L. Zhu, R. Stolen, R. Rice, and J. Ballato, “On loss in silicon core optical fiber,” Opt. Mater. Express 2(11), 1511–1519 (2012).
[Crossref]

J. Ballato, T. Hawkins, P. Foy, B. Yazgan-Kokuoz, C. Mcmillen, L. Burka, S. Morris, R. Stolen, and R. Rice, “Advancements in semiconductor core optical fiber,” Opt. Fiber Technol. 16(6), 399–408 (2010).
[Crossref]

Su, H. C.

H. C. Su and L. A. Wang, “A highly efficient polarized superfluorescent fiber source for fiber-optic gyroscope applications,” IEEE Photonics Technol. Lett. 15(10), 1357–1359 (2003).
[Crossref]

Sunnerud, H.

Tang, D. Y.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics 5(7), 411–415 (2011).
[Crossref]

Tian, F.

Wang, B.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics 5(7), 411–415 (2011).
[Crossref]

Wang, L. A.

H. C. Su and L. A. Wang, “A highly efficient polarized superfluorescent fiber source for fiber-optic gyroscope applications,” IEEE Photonics Technol. Lett. 15(10), 1357–1359 (2003).
[Crossref]

Wang, Y.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics 5(7), 411–415 (2011).
[Crossref]

Wu, Q.

Y. Ma, G. Farrel, Y. Semenova, B. Li, J. Yuan, B. Li, J. Yuan, X. Sang, B. Yan, C. Yu, and Q. Wu, “Optical microfiber-loaded surface plasmonic TE-pass polarizer,” Opt. Laser Technol. 78, 110–114 (2015).

Xie, C.

Xu, C.

Xu, Y.

Yan, B.

Y. Ma, G. Farrel, Y. Semenova, B. Li, J. Yuan, B. Li, J. Yuan, X. Sang, B. Yan, C. Yu, and Q. Wu, “Optical microfiber-loaded surface plasmonic TE-pass polarizer,” Opt. Laser Technol. 78, 110–114 (2015).

Yang, J.

Yang, L.

Yazgan-Kokuoz, B.

J. Ballato, T. Hawkins, P. Foy, B. Yazgan-Kokuoz, C. Mcmillen, L. Burka, S. Morris, R. Stolen, and R. Rice, “Advancements in semiconductor core optical fiber,” Opt. Fiber Technol. 16(6), 399–408 (2010).
[Crossref]

Yu, C.

Y. Ma, G. Farrel, Y. Semenova, B. Li, J. Yuan, B. Li, J. Yuan, X. Sang, B. Yan, C. Yu, and Q. Wu, “Optical microfiber-loaded surface plasmonic TE-pass polarizer,” Opt. Laser Technol. 78, 110–114 (2015).

Yu, H.

Yuan, J.

Y. Ma, G. Farrel, Y. Semenova, B. Li, J. Yuan, B. Li, J. Yuan, X. Sang, B. Yan, C. Yu, and Q. Wu, “Optical microfiber-loaded surface plasmonic TE-pass polarizer,” Opt. Laser Technol. 78, 110–114 (2015).

Y. Ma, G. Farrel, Y. Semenova, B. Li, J. Yuan, B. Li, J. Yuan, X. Sang, B. Yan, C. Yu, and Q. Wu, “Optical microfiber-loaded surface plasmonic TE-pass polarizer,” Opt. Laser Technol. 78, 110–114 (2015).

Yuan, L.

Zhang, H.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics 5(7), 411–415 (2011).
[Crossref]

Zhu, L.

Biosensors (1)

D. C. Cullen, R. G. Brown, and C. R. Lowe, “Detection of immuno-complex formation via surface plasmon resonance on gold-coated diffraction gratings,” Biosensors 3(4), 211–225 (1987).
[Crossref] [PubMed]

IEEE J. Sel. Top. Quantum Electron. (1)

B. Jalali, V. Raghunathan, D. Dimitrophoulos, and O. Boyraz, “Raman-based silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12(3), 412–421 (2006).
[Crossref]

IEEE Photonics Technol. Lett. (2)

H. C. Su and L. A. Wang, “A highly efficient polarized superfluorescent fiber source for fiber-optic gyroscope applications,” IEEE Photonics Technol. Lett. 15(10), 1357–1359 (2003).
[Crossref]

T. K. Ng, M. Z. M. Khan, A. Al-Jabr, and B. S. Ooi, “Analysis of CMOS compatible Cu-based TM-pass optical polarizer,” IEEE Photonics Technol. Lett. 24(9), 724–726 (2012).
[Crossref]

J. Appl. Phys. (1)

G. W. Hanson, “Quasi-transverse electromagnetic modes supported by a graphene parallel-plate waveguide,” J. Appl. Phys. 104(8), 084314 (2008).
[Crossref]

J. Lightwave Technol. (2)

Nat. Photonics (2)

D. Graham-Rowe, “Fibres get functional,” Nat. Photonics 5(2), 66–67 (2011).
[Crossref]

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics 5(7), 411–415 (2011).
[Crossref]

Opt. Express (2)

Opt. Fiber Technol. (1)

J. Ballato, T. Hawkins, P. Foy, B. Yazgan-Kokuoz, C. Mcmillen, L. Burka, S. Morris, R. Stolen, and R. Rice, “Advancements in semiconductor core optical fiber,” Opt. Fiber Technol. 16(6), 399–408 (2010).
[Crossref]

Opt. Laser Technol. (1)

Y. Ma, G. Farrel, Y. Semenova, B. Li, J. Yuan, B. Li, J. Yuan, X. Sang, B. Yan, C. Yu, and Q. Wu, “Optical microfiber-loaded surface plasmonic TE-pass polarizer,” Opt. Laser Technol. 78, 110–114 (2015).

Opt. Lett. (1)

Opt. Mater. Express (1)

Sens. Actuators A Phys. (1)

H. Esmaeilzadeh, E. Arzi, M. Mozafari, and A. Hassani, “A broadband optical fiber based inline polarizer for telecom wavelength range,” Sens. Actuators A Phys. 185, 59–65 (2012).
[Crossref]

Sens. Actuators B Chem. (1)

M. Piliarik, J. Homola, Z. Manikova, and J. Ctyroky, “Surface plasmon resonance sensor based on a single-mode polarization-maintaining optical fiber,” Sens. Actuators B Chem. 90(1-3), 236–242 (2003).
[Crossref]

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Figures (6)

Fig. 1
Fig. 1 The schematic cross section of the polarizer, where r is the radius of the silicon core. a and b is respectively the major and minor diameter of the microfiber. n1 = 1.4378, n2 = 3.4784.
Fig. 2
Fig. 2 (a) Birefringence as a function of the radius of the silicon core with different ellipticity. (b) Extinction ratio versus the radius of the silicon core with different ellipticity. (c) Birefringence as a function of the radius of the silicon core with different major and minor diameter. (d) Extinction ratio versus the radius of the silicon core with different major and minor diameter.
Fig. 3
Fig. 3 The electric field distribution on the cross section of the SSCM polarizer with surface-covered graphene film for TE and TM mode light at the radius of the silicon core of 0.132 μm ((a) and (b)) and 0.17 μm ((d) and (e)). (c) The amplitudes of the electric field as a function of x coordinate axis along y = 1μm at the silicon core radius of 0.132μm, a = 4 μm and b = 2 μm. (f) The intensity of the electric field as a function of y coordinate axis along x = 0 at the silicon core radius of 0.17 μm, a = 4 μm and b = 2 μm.
Fig. 4
Fig. 4 The effect of the ellipticity, major and minor diameter on the birefringence and propagation loss for TE and TM modes. (a) and (b) for r = 0.132 μm, (c) and (d) for r = 0.17 μm.
Fig. 5
Fig. 5 The effect of the graphene thickness on the birefringence and propagation loss for TE and TM modes. (a) r = 0.132 μm and (b) r = 0.17 μm.
Fig. 6
Fig. 6 Propagation losses of TE and TM modes against the wavelength (a) r = 0.132 μm and (b) r = 0.17 μm.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

× H = j ω ε r ε 0 E + σ E
× E = j ω μ 0 H
a tan ( C a ) + a tan ( C b ) + m π = γ 2 d
ε 3 = 1 Im ( σ ) ω ε 0 d g + i Re ( σ ) ω ε 0 d g
σ ( ω , μ c , Γ , T ) = j e 2 ( ω j τ -1 ) π [ 1 ( ω j τ -1 ) 2 0 ε ( f d ( ε ) ε f d ( ε ) ε ) d ε 0 f d ( ε ) f d ( ε ) ( ω j 2 Γ ) 2 4 ( ε / ) 2 d ε ]
α = Im ( n e f f ) k 0
E = ( α T E α T M ) L g

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