Abstract

A simple analytic analysis of the ultra-high reflectivity feature of subwavelength dielectric gratings is developed. The phenomenon of ultra high reflectivity is explained to be a destructive interference effect between the two grating modes. Based on this phenomenon, a design algorithm for broadband grating mirrors is suggested.

© 2010 OSA

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References

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  1. C. F. R. Mateus, M. C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang-Hasnain, “Ultra-broadband mirror using low index cladded subwavelength grating,” IEEE Photon. Technol. Lett. 16(2), 518–520 (2004).
    [CrossRef]
  2. C. J. Chang-Hasnain, Y. Zhou, M. C. Y. Huang, and C. Chase, “High-Contrast Grating VCSELs,” IEEE J. Sel. Top. Quantum Electron. 15, 869 (2009).
    [CrossRef]
  3. M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “A surface-emitting laser incorporating a high index-contrast subwavelength grating,” Nat. Photonics 1(2), 119–122 (2007).
    [CrossRef]
  4. M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “A nanoelectromechanical tunable laser,” Nat. Photonics 2(3), 180–184 (2008).
    [CrossRef]
  5. M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “Polarization mode control in high contrast subwavelength grating VCSEL”, Conference on Lasers and Electro-Optics (2008).
  6. P. Debernardi, J. M. Ostermann, M. Feneberg, C. Jalics, and R. Michalzik, “Reliable polarization control of VCSELs through monolithically integrated surface gratings: a comparative theoretical and experimental study,” IEEE J. Sel. Top. Quantum Electron. 11(1), 107–116 (2005).
    [CrossRef]
  7. J. M. Ostermann, P. Debernardi, C. Jalics, and R. Michalzik, “Shallow surface gratings for high-power VCSELs with one preferred polarization for all modes,” IEEE Photon. Technol. Lett. 17(8), 1593–1595 (2005).
    [CrossRef]
  8. Y. Zhou, M. Moewe, J. Kern, M. C. Y. Huang, and C. J. Chang-Hasnain, “Surface-normal emission of a high-Q resonator using a subwavelength high-contrast grating,” Opt. Express 16(22), 17282–17287 (2008).
    [CrossRef] [PubMed]
  9. M. G. Moharam and T. K. Gaylord, “Rigorous coupled wave analysis of planar grating diffraction,” J. Opt. Soc. Am. 71(7), 811 (1981).
    [CrossRef]
  10. S. T. Peng, “Rigorous formulation of scattering and guidance by dielectric grating waveguides: general case of oblique incidence,” J. Opt. Soc. Am. A 6(12), 1869 (1989).
    [CrossRef]
  11. L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40(4), 553–573 (1993).
    [CrossRef]
  12. P. C. Magnusson, G. C. Alexander, V. K. Tripathi, and A. Weisshaar, Transmission lines and wave propagation, 4th edition (CRC Press, 2001).
  13. Y. Zhou, M. C. Y. Huang, and C. J. Chang-Hasnain, “Large fabrication tolerance for VCSELs using high contrast grating,” IEEE Photon. Technol. Lett. 20(6), 434–436 (2008).
    [CrossRef]
  14. R. Magnusson and M. Shokooh-Saremi, “Physical basis for wideband resonant reflectors,” Opt. Express 16(5), 3456–3462 (2008).
    [CrossRef] [PubMed]

2009

C. J. Chang-Hasnain, Y. Zhou, M. C. Y. Huang, and C. Chase, “High-Contrast Grating VCSELs,” IEEE J. Sel. Top. Quantum Electron. 15, 869 (2009).
[CrossRef]

2008

Y. Zhou, M. C. Y. Huang, and C. J. Chang-Hasnain, “Large fabrication tolerance for VCSELs using high contrast grating,” IEEE Photon. Technol. Lett. 20(6), 434–436 (2008).
[CrossRef]

M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “A nanoelectromechanical tunable laser,” Nat. Photonics 2(3), 180–184 (2008).
[CrossRef]

R. Magnusson and M. Shokooh-Saremi, “Physical basis for wideband resonant reflectors,” Opt. Express 16(5), 3456–3462 (2008).
[CrossRef] [PubMed]

Y. Zhou, M. Moewe, J. Kern, M. C. Y. Huang, and C. J. Chang-Hasnain, “Surface-normal emission of a high-Q resonator using a subwavelength high-contrast grating,” Opt. Express 16(22), 17282–17287 (2008).
[CrossRef] [PubMed]

2007

M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “A surface-emitting laser incorporating a high index-contrast subwavelength grating,” Nat. Photonics 1(2), 119–122 (2007).
[CrossRef]

2005

P. Debernardi, J. M. Ostermann, M. Feneberg, C. Jalics, and R. Michalzik, “Reliable polarization control of VCSELs through monolithically integrated surface gratings: a comparative theoretical and experimental study,” IEEE J. Sel. Top. Quantum Electron. 11(1), 107–116 (2005).
[CrossRef]

J. M. Ostermann, P. Debernardi, C. Jalics, and R. Michalzik, “Shallow surface gratings for high-power VCSELs with one preferred polarization for all modes,” IEEE Photon. Technol. Lett. 17(8), 1593–1595 (2005).
[CrossRef]

2004

C. F. R. Mateus, M. C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang-Hasnain, “Ultra-broadband mirror using low index cladded subwavelength grating,” IEEE Photon. Technol. Lett. 16(2), 518–520 (2004).
[CrossRef]

1993

L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40(4), 553–573 (1993).
[CrossRef]

1989

1981

Chang-Hasnain, C. J.

C. J. Chang-Hasnain, Y. Zhou, M. C. Y. Huang, and C. Chase, “High-Contrast Grating VCSELs,” IEEE J. Sel. Top. Quantum Electron. 15, 869 (2009).
[CrossRef]

M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “A nanoelectromechanical tunable laser,” Nat. Photonics 2(3), 180–184 (2008).
[CrossRef]

Y. Zhou, M. C. Y. Huang, and C. J. Chang-Hasnain, “Large fabrication tolerance for VCSELs using high contrast grating,” IEEE Photon. Technol. Lett. 20(6), 434–436 (2008).
[CrossRef]

Y. Zhou, M. Moewe, J. Kern, M. C. Y. Huang, and C. J. Chang-Hasnain, “Surface-normal emission of a high-Q resonator using a subwavelength high-contrast grating,” Opt. Express 16(22), 17282–17287 (2008).
[CrossRef] [PubMed]

M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “A surface-emitting laser incorporating a high index-contrast subwavelength grating,” Nat. Photonics 1(2), 119–122 (2007).
[CrossRef]

C. F. R. Mateus, M. C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang-Hasnain, “Ultra-broadband mirror using low index cladded subwavelength grating,” IEEE Photon. Technol. Lett. 16(2), 518–520 (2004).
[CrossRef]

Chase, C.

C. J. Chang-Hasnain, Y. Zhou, M. C. Y. Huang, and C. Chase, “High-Contrast Grating VCSELs,” IEEE J. Sel. Top. Quantum Electron. 15, 869 (2009).
[CrossRef]

Debernardi, P.

P. Debernardi, J. M. Ostermann, M. Feneberg, C. Jalics, and R. Michalzik, “Reliable polarization control of VCSELs through monolithically integrated surface gratings: a comparative theoretical and experimental study,” IEEE J. Sel. Top. Quantum Electron. 11(1), 107–116 (2005).
[CrossRef]

J. M. Ostermann, P. Debernardi, C. Jalics, and R. Michalzik, “Shallow surface gratings for high-power VCSELs with one preferred polarization for all modes,” IEEE Photon. Technol. Lett. 17(8), 1593–1595 (2005).
[CrossRef]

Deng, Y.

C. F. R. Mateus, M. C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang-Hasnain, “Ultra-broadband mirror using low index cladded subwavelength grating,” IEEE Photon. Technol. Lett. 16(2), 518–520 (2004).
[CrossRef]

Feneberg, M.

P. Debernardi, J. M. Ostermann, M. Feneberg, C. Jalics, and R. Michalzik, “Reliable polarization control of VCSELs through monolithically integrated surface gratings: a comparative theoretical and experimental study,” IEEE J. Sel. Top. Quantum Electron. 11(1), 107–116 (2005).
[CrossRef]

Gaylord, T. K.

Huang, M. C. Y.

C. J. Chang-Hasnain, Y. Zhou, M. C. Y. Huang, and C. Chase, “High-Contrast Grating VCSELs,” IEEE J. Sel. Top. Quantum Electron. 15, 869 (2009).
[CrossRef]

M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “A nanoelectromechanical tunable laser,” Nat. Photonics 2(3), 180–184 (2008).
[CrossRef]

Y. Zhou, M. C. Y. Huang, and C. J. Chang-Hasnain, “Large fabrication tolerance for VCSELs using high contrast grating,” IEEE Photon. Technol. Lett. 20(6), 434–436 (2008).
[CrossRef]

Y. Zhou, M. Moewe, J. Kern, M. C. Y. Huang, and C. J. Chang-Hasnain, “Surface-normal emission of a high-Q resonator using a subwavelength high-contrast grating,” Opt. Express 16(22), 17282–17287 (2008).
[CrossRef] [PubMed]

M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “A surface-emitting laser incorporating a high index-contrast subwavelength grating,” Nat. Photonics 1(2), 119–122 (2007).
[CrossRef]

C. F. R. Mateus, M. C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang-Hasnain, “Ultra-broadband mirror using low index cladded subwavelength grating,” IEEE Photon. Technol. Lett. 16(2), 518–520 (2004).
[CrossRef]

Jalics, C.

P. Debernardi, J. M. Ostermann, M. Feneberg, C. Jalics, and R. Michalzik, “Reliable polarization control of VCSELs through monolithically integrated surface gratings: a comparative theoretical and experimental study,” IEEE J. Sel. Top. Quantum Electron. 11(1), 107–116 (2005).
[CrossRef]

J. M. Ostermann, P. Debernardi, C. Jalics, and R. Michalzik, “Shallow surface gratings for high-power VCSELs with one preferred polarization for all modes,” IEEE Photon. Technol. Lett. 17(8), 1593–1595 (2005).
[CrossRef]

Kern, J.

Li, L.

L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40(4), 553–573 (1993).
[CrossRef]

Magnusson, R.

Mateus, C. F. R.

C. F. R. Mateus, M. C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang-Hasnain, “Ultra-broadband mirror using low index cladded subwavelength grating,” IEEE Photon. Technol. Lett. 16(2), 518–520 (2004).
[CrossRef]

Michalzik, R.

P. Debernardi, J. M. Ostermann, M. Feneberg, C. Jalics, and R. Michalzik, “Reliable polarization control of VCSELs through monolithically integrated surface gratings: a comparative theoretical and experimental study,” IEEE J. Sel. Top. Quantum Electron. 11(1), 107–116 (2005).
[CrossRef]

J. M. Ostermann, P. Debernardi, C. Jalics, and R. Michalzik, “Shallow surface gratings for high-power VCSELs with one preferred polarization for all modes,” IEEE Photon. Technol. Lett. 17(8), 1593–1595 (2005).
[CrossRef]

Moewe, M.

Moharam, M. G.

Neureuther, A. R.

C. F. R. Mateus, M. C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang-Hasnain, “Ultra-broadband mirror using low index cladded subwavelength grating,” IEEE Photon. Technol. Lett. 16(2), 518–520 (2004).
[CrossRef]

Ostermann, J. M.

J. M. Ostermann, P. Debernardi, C. Jalics, and R. Michalzik, “Shallow surface gratings for high-power VCSELs with one preferred polarization for all modes,” IEEE Photon. Technol. Lett. 17(8), 1593–1595 (2005).
[CrossRef]

P. Debernardi, J. M. Ostermann, M. Feneberg, C. Jalics, and R. Michalzik, “Reliable polarization control of VCSELs through monolithically integrated surface gratings: a comparative theoretical and experimental study,” IEEE J. Sel. Top. Quantum Electron. 11(1), 107–116 (2005).
[CrossRef]

Peng, S. T.

Shokooh-Saremi, M.

Zhou, Y.

C. J. Chang-Hasnain, Y. Zhou, M. C. Y. Huang, and C. Chase, “High-Contrast Grating VCSELs,” IEEE J. Sel. Top. Quantum Electron. 15, 869 (2009).
[CrossRef]

M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “A nanoelectromechanical tunable laser,” Nat. Photonics 2(3), 180–184 (2008).
[CrossRef]

Y. Zhou, M. Moewe, J. Kern, M. C. Y. Huang, and C. J. Chang-Hasnain, “Surface-normal emission of a high-Q resonator using a subwavelength high-contrast grating,” Opt. Express 16(22), 17282–17287 (2008).
[CrossRef] [PubMed]

Y. Zhou, M. C. Y. Huang, and C. J. Chang-Hasnain, “Large fabrication tolerance for VCSELs using high contrast grating,” IEEE Photon. Technol. Lett. 20(6), 434–436 (2008).
[CrossRef]

M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “A surface-emitting laser incorporating a high index-contrast subwavelength grating,” Nat. Photonics 1(2), 119–122 (2007).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

C. J. Chang-Hasnain, Y. Zhou, M. C. Y. Huang, and C. Chase, “High-Contrast Grating VCSELs,” IEEE J. Sel. Top. Quantum Electron. 15, 869 (2009).
[CrossRef]

P. Debernardi, J. M. Ostermann, M. Feneberg, C. Jalics, and R. Michalzik, “Reliable polarization control of VCSELs through monolithically integrated surface gratings: a comparative theoretical and experimental study,” IEEE J. Sel. Top. Quantum Electron. 11(1), 107–116 (2005).
[CrossRef]

IEEE Photon. Technol. Lett.

J. M. Ostermann, P. Debernardi, C. Jalics, and R. Michalzik, “Shallow surface gratings for high-power VCSELs with one preferred polarization for all modes,” IEEE Photon. Technol. Lett. 17(8), 1593–1595 (2005).
[CrossRef]

Y. Zhou, M. C. Y. Huang, and C. J. Chang-Hasnain, “Large fabrication tolerance for VCSELs using high contrast grating,” IEEE Photon. Technol. Lett. 20(6), 434–436 (2008).
[CrossRef]

C. F. R. Mateus, M. C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang-Hasnain, “Ultra-broadband mirror using low index cladded subwavelength grating,” IEEE Photon. Technol. Lett. 16(2), 518–520 (2004).
[CrossRef]

J. Mod. Opt.

L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40(4), 553–573 (1993).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Nat. Photonics

M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “A surface-emitting laser incorporating a high index-contrast subwavelength grating,” Nat. Photonics 1(2), 119–122 (2007).
[CrossRef]

M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “A nanoelectromechanical tunable laser,” Nat. Photonics 2(3), 180–184 (2008).
[CrossRef]

Opt. Express

Other

M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “Polarization mode control in high contrast subwavelength grating VCSEL”, Conference on Lasers and Electro-Optics (2008).

P. C. Magnusson, G. C. Alexander, V. K. Tripathi, and A. Weisshaar, Transmission lines and wave propagation, 4th edition (CRC Press, 2001).

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Figures (8)

Fig. 1
Fig. 1

(a) High Contrast Grating (HCG) schematics. The red arrow indicates the direction of wave incidence. The black arrows correspond to the E-field direction in both TE and TM polarizations of incidence. The grating comprises of rectangular dielectric bars having a refractive index in the semiconductor range, surrounded by a low index medium (air or oxide). The high refractive index contrast between the grating bars and the surrounding medium is beneficial for the reflectivity bandwidth and the fabrication tolerance of the grating. The nomenclature for the HCG dimensions is as follows: Λ is the grating period, a is the air-gap width, s is the bar width and tg is the grating thickness. HCG has subwavelength periodicity (Λ<λ). (b) Example of the broadband high reflectivity spectrum. The HCG parameters for this plot are: n bar =3.21, Λ=0.779μm, s/Λ=0.77, tg=0.508μm and TM polarization of incidence.

Fig. 2
Fig. 2

Nomenclature for Eqs. (1a)(1e): The HCG input plane is z=-tg and the output plane is z=0. ka and ks are the x-wavenumbers in the air-gaps and in the grating bars respectively. The z-wavenumber β is the same in both the air-gaps and the bars. The x-wavenumber (green arrow) outside the grating (Regions I and III) is determined by the grating periodicity: 2πn/Λ.

Fig. 3
Fig. 3

Graphic representation of dispersion relations for a HCG with a refractive index n bar =3.48, assuming TM polarization of incidence. Lower and upper curves are the first two solutions of Eq. (2), i.e. the first two HCG modes. Dashed lines are the constant wavelength contours, given by Eq. (1g). Black circles indicate the intersections between the dashed lines and the curves, thus describing the modes at a specific wavelength. The mode cutoff (β=0) is given by ks=n bar ka , according to Eq. (1f). Above the cutoff β is real and below the cutoff β is imaginary.

Fig. 4
Fig. 4

(a) Convergence of the analytical solution in sections 2.1-2.2 towards the RCWA [9] simulation result as a function of the number of modes taken into consideration. The double-mode solution is in very good agreement with RCWA, especially when the reflectivity is high. The HCG parameters for this plot are: n bar =3.214, tg/Λ=0.627, s/Λ=0.62 and TM polarization of incidence. (b) >99% reflectivity contour as a function of wavelength λ and thickness tg. Almost all high-reflectivity configurations are shown to reside within the double-mode region, i.e. between the cutoffs of the second and third modes.

Fig. 5
Fig. 5

(a-d) Convergence of the intensity profiles as a function of the number of modes taken into consideration, corresponding to the case of ~100% reflectivity. The grating bars are marked by the white dashed squares. When three or more modes are used, the boundary condition matching is almost perfect, but two modes are already a good approximation. The high reflectivity is clearly demonstrated by the standing wave profile created above the grating by the incident and reflected plane waves. (e-h) Contour plots of each mode separately inside the HCG. The grating bars are marked by the white dashed squares. The contours take into account both the forward ( +z) and the backward (-z) propagating components of each mode, including their coefficients. The HCG parameters for this plot are: n bar =3.48, λ/Λ=1.509, tg/Λ=0.2, s/Λ=0.4 and TE polarization of incidence.

Fig. 6
Fig. 6

(a) Double-mode solution exhibiting perfect cancellation at the HCG output plane (z=0- ) leading to 100% reflectivity. At the wavelengths of 100% reflectivity both modes have the same magnitude of the “dc” lateral Fourier component (|(a1+aρ 1 )Λ−1∫e in x,1dx|=|(a2+aρ 2 )Λ−1∫e in x,2dx|), but opposite phases: ΔΦ=phase[(a1+aρ 1 )Λ−1∫e in x,1dx]-phase[(a2+aρ 2 )Λ−1∫e in x,2dx]=π. This means that the overall “dc” Fourier component is zero (Eq. (12), which leads to cancellation of the 0th transmissive diffraction order. When two perfect-cancellation points are located in close spectral vicinity of each other, a broad band of high-reflectivity is achieved. HCG parameters for this plot are: n bar =3.214, s/Λ=0.62, tg/Λ=0.627 and TM polarization of incidence. (b) Double-mode solution for the overall field profile at the HCG output plane (z=0- ) in the case of perfect cancellation (black curve, right plot). The cancellation is shown to be only in terms of the “dc” Fourier component. The higher Fourier components do not need to be zero, since subwavelength gratings have no diffraction orders other than 0th. The left plot shows the decomposition of the overall field profile into the first two modes, whereby the dc-components of these two modes cancel each other. HCG parameters for this plot are: n bar =3.48, s/Λ=0.4, tg/Λ=0.2, λ/Λ=1.563 and TE polarization of incidence.

Fig. 7
Fig. 7

(a) First stage of broadband reflectivity design: The broadband spectra are found along flat sections in 100% reflectivity contours. The 100% reflectivity contours are the solutions of Eq. (12) for different duty cycles s/Λ, marked by different colors. These contours are shown to form s-type shapes (only the lowest s-shapes are plotted). The optimal dimensions leading to the broadest spectra (indicated by arrows) are chosen. (b) Second stage of broadband reflectivity design: The bandwidth can be further increased through a tradeoff with a spectral dip, by fine-tuning the grating thickness. For example, if the minimal tolerable reflectivity for a particular application is 99%, the bandwidth can be increased by ~35% in comparison to a spectrally-flat configuration. Both figures (a) and (b) use normalized units, since HCG solution is scalable.

Fig. 8
Fig. 8

Same design as in Fig. 7a repeated for a larger refractive index: n bar =4 and four different duty cycles s/Λ. The increased refractive index contrast is shown to have a beneficial effect on the achievable high-reflectivity bandwidths.

Tables (1)

Tables Icon

Table 1 Differences between TM and TE polarizations of incidence. Check-marks indicate the changes required for each equation. Equations not listed in this table require no changes

Equations (29)

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

H y II ( x , t g z 0 ) = m = 1 h y , m in ( x ) [ a m exp ( j β m z ) a m ρ exp ( + j β m z ) ] E x II ( x , t g z 0 ) = m = 1 e x , m in ( x ) [ a m exp ( j β m z ) + a m ρ exp ( + j β m z ) ]
Coefficient c vectors: a = Δ ( a 1 a 2 ... ) T ; a ρ = Δ ( a 1 ρ a 2 ρ ... ) T Reflection matrix ρ at the output plane ( z = 0 ): a ρ ρ a
φ n . m = exp ( j β m t g ) for n = m and 0 for n m
h y , m in ( 0 < x < a ) inside air gaps = cos ( k s , m s / 2 ) cos [ k a , m ( x a / 2 ) ] ; e x , m in = ( β m / k 0 ) η h y . m h y , m in ( a < x < Λ ) inside HCG bars = cos ( k a , m a / 2 ) cos { k s , m [ x ( a + Λ ) / 2 ] } ; e x , m in = ( β m / k 0 ) n bar 2 η h y . m
h y , m i n ( x < 0 o r x > Λ ) = h y , m i n ( x modulo Λ ) e x , m i n ( x < 0 o r x > Λ ) = e x , m i n ( x modulo Λ )
β m 2 = ( 2 π / λ ) 2 k a , m 2 = ( 2 π n bar / λ ) 2 k s , m 2
k s , m 2 k a , m 2 = ( 2 π / λ ) 2 ( n bar 2 1 )
n bar 2 k s , m tan ( k s , m s / 2 ) = k a , m tan ( k a , m a / 2 )
H y I ( x , z t g ) = exp [ j ( 2 π / λ ) ( z + t g ) ] incident plane wave n = 0 r n h y , n out ( x ) exp [ + j γ n ( z + t g ) ] reflected = n = 0 ( δ n , 0 r n ) h y , n out ( x ) exp [ + j γ n ( z + t g ) ] E x I ( x , z t g ) = η exp [ j ( 2 π / λ ) ( z + t g ) ] incident plane wave + n = 0 r n e x , n out ( x ) exp [ + j γ n ( z + t g ) ] reflected = = n = 0 ( δ n , 0 + r n ) e x , n out ( x ) exp [ + j γ n ( z + t g ) ] δ n , 0 = { 1 , n = 0 0 , otherwise
HCG Reflectivity Matrix R : ( r 0 r 1 r 2 ... ) T R ( 1 0 0 ... ) T δ n , 0
h y , n out = cos [ ( 2 n π / Λ ) ( x a / 2 ) ] ; e x , n out = ( γ n / k 0 ) η h y , n out
H y III ( x , z 0 ) = n = 0 τ n h y , n out ( x ) exp ( j γ n z ) E x III ( x , z 0 ) = n = 0 τ n e x , n out ( x ) exp ( j γ n z )
Transmitted coefficient vector: τ = Δ ( τ 0 τ 1 ... ) T HCG Transmission Matrix T : ( τ 0 τ 1 τ 2 ... ) T T ( 1 0 0 ... ) T δ n , 0
γ n 2 = ( 2 π / λ ) 2 ( 2 n π / Λ ) 2
n = 0 τ n h y , n out ( x ) = H y III ( x , z = 0 ) , eq . 3d = m = 1 h y , m in ( x ) ( a m a m ρ ) = H y II ( x , z = 0 ) , eq . 1a
τ n Λ 1 0 Λ [ h y , n out ( x ) ] 2 d x = ( 2 δ n , 0 ) 1 , see eq . 3c = m = 1 ( a m a m ρ ) Λ 1 0 Λ h y , m in ( x ) h y , n out ( x ) d x
τ n Λ 1 0 Λ [ e x , n out ( x ) ] 2 d x = ( η γ n / k 0 ) 2 ( 2 δ n , 0 ) 1 , see eq . 3c = m = 1 ( a m + a m ρ ) Λ 1 0 Λ e x , m in ( x ) e x , n out ( x ) d x
H n , m = Λ 1 ( 2 δ n , 0 ) 0 Λ h y , m in ( x ) h y , n out ( x ) d x , see eq . 4b E n , m = Λ 1 ( 2 δ n , 0 ) ( η γ n / k 0 ) 2 0 Λ e x , m in ( x ) e x , n out ( x ) d x , see eq . 4c Eqs . 4b,c in matrix-vector format: τ = H ( a a ρ ) = E ( a + a ρ )
τ = H ( I ρ ) a = E ( I + ρ ) a
H ( I ρ ) = E ( I + ρ ) ρ = ( I + H 1 E ) 1 ( I H 1 E )
H y I ( x , z = t g ) = H y II ( x , z = t g ) and E x I ( x , z = t g ) = E x II ( x , z = t g ) same steps as in eqs . 4-6 ( I R ) 1 H ( φ 1 ρ φ ) = ( I + R ) 1 E ( φ 1 + ρ φ )
E ( I + φ ρ φ ) ( I φ ρ φ ) 1 H 1 = ( I + R ) ( I R ) 1 = Δ Z i n
R = ( Z i n + I ) 1 ( Z i n I ) , where Z i n = E ( I + φ ρ φ ) ( I φ ρ φ ) 1 H 1
a = 2 φ [ ( Z i n 1 + I ) E ( I + φ ρ φ ) ] 1 ( 1 0 0 ... ) T δ n , 0
τ = eq . 5b E ( I + ρ ) a = eq . 8a 2 E ( I + ρ ) φ [ ( Z i n 1 + I ) E ( I + φ ρ φ ) ] 1 = Δ T , HCG Transmission Matrix ( 1 0 0 ... ) T
HCG Reflectivity = | R 00 | 2 HCG Transmission = | T 00 | 2 | R 00 | 2 + | T 00 | 2 1 when Λ < λ
| R 00 | = 100 % τ 0 = [ E ( a + a ρ ) ] 0 = m E 0 m ( a m + a m ρ ) = 0
τ 0 = m E 0 m ( a m + a m ρ ) = eq . 5a ( η γ 0 / k 0 ) 2 m ( a m + a m ρ ) Λ 1 0 Λ e x , m in ( x ) e x , 0 out ( x ) d x = eq . 3c ( η γ 0 / k 0 ) 1 m ( a m + a m ρ ) Λ 1 0 Λ e x , m in ( x ) d x = 0
| R 00 | = 100 % ( a 1 + a 1 ρ ) Λ 1 0 Λ e x , 1 in ( x ) d x lateral average of the first mode (forward + backward) + ( a 2 + a 2 ρ ) Λ 1 0 Λ e x , 2 in ( x ) d x lateral average of the second mode (forward + backward) = 0 "destructive interference" (cancellation) between the first and the second modes at z = 0

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