## Theoretical analysis of subwavelength high contrast grating reflectors |

Optics Express, Vol. 18, Issue 16, pp. 16973-16988 (2010)

http://dx.doi.org/10.1364/OE.18.016973

Acrobat PDF (1566 KB)

### 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

## 1. Introduction

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]

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]

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]

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

*predict and design*this extraordinary and unexpected feature of HCGs. A formulation that combines rigor with simplicity remains yet to be presented.

*fully analytic*solution of HCGs, focusing on the high reflectivity phenomenon, without using either coarse rules of thumb or heavy mathematical formalisms, and provide a design algorithm for broadband highly-reflective HCGs. Special attention is paid to the multi-mode nature of such gratings, and to the very quick convergence of their modal representation.

## 2. Theoretical analysis of the grating reflectivity

*n*

_{bar}, which is significantly higher than the refractive index of the surrounding medium (hence the terminology “

*high contrast grating*”). The typical refractive index of the grating bars is

*n*

_{bar}

*=2.8~3.5*, and the outside medium is assumed to be air (

*n*

_{air}

*=1*), even though other low index media, such as silicon dioxide, produce comparable effects [1

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]

*Λ*,

*a*,

*s*and

*t*in Fig. 1a correspond respectively to the period of the grating, the air-gap size, the width of the grating bars and the grating thickness. The grating period is sub-wavelength (

_{g}*Λ<λ*), but remains larger than

*λ/n*

_{bar}. The grating periodicity direction is

*x*. The incident plane wave propagation direction, as indicated by the red arrow, is

*z*. For simplicity, the grating is assumed to be infinite in

*y*direction and infinitely periodic in

*x*direction. We consider two polarizations of incidence: (i) transverse magnetic (TM), in which the electric field is in

*x*direction; (ii) transverse electric (TE), in which the electric field is in

*y*direction. Figure 1a describes both polarizations, the black arrows indicating the directions of the electric field in each case. Figure 1b shows an example of a broadband high reflectivity spectrum exhibited by such HCG.

### 2.1 TM-polarized incidence

*z=-t*(HCG input plane) and

_{g}*z=0*(HCG output plane), as shown in Fig. 2 . We will keep track of the lateral (

*x*,

*y*) field components only, since the longitudinal (

*z*) field component can be easily derived from the lateral components. Since the HCG is infinite in

*y*, the solution is two-dimensional (

*∂/∂y=0*). In region I,

*z<-t*, there are incoming and reflected waves. In region II,

_{g}*-t*, the solutions are modes of a periodic array of slab waveguides, whereby the propagation direction is

_{g}<z<0*z*. In region III,

*z>0*, there exist only the transmitted waves. We first solve for the modes in Region II, and then match the boundary conditions with Regions I and III to solve for reflectivity and transmission spectra.

*Region II*are given in Eq. (1a). They comprise of forward (

*+z*) and backward (

*-z*) propagating components, the coefficients of which are

*a*and

_{m}*a*respectively,

^{ρ}_{m}*m=1,2,…*being the number of the mode and

*β*being the longitudinal (

*z*) wavenumber.

*h*

^{in}

*and*

_{y,m}*e*

^{in}

*in Eq. (1a) indicate the lateral (*

_{x,m}*x*) magnetic and electric field profiles of each mode, where the index “in” stands for: inside HCG.

**a**and

**a**, and the reflection matrix

^{ρ}**ρ**, which relates between these two vectors:It is also instructive at this point to define, based on Eq. (1a), the HCG propagation matrix

**φ**, which is a diagonal matrix containing the accumulated phases of each HCG mode:The lateral field profiles,

*h*

^{in}

*and*

_{y,m}*e*

^{in}

*, in Eq. (1a), are given by:*

_{x,m}*k*;

_{0}=2π/λ*k*and

_{a}*k*are the lateral (

_{s}*x*) wavenumbers inside air-gaps and grating bars, respectively (see Fig. 2). In addition, the solution must be periodic with respect to Λ, and therefore:The nomenclature for Eqs. (1a)–(1e) is shown in Fig. 2:

*β*in Eq. (1a) is given by:Or alternatively:

*z*. The dispersion relation between the lateral wavenumbers

*k*and

_{a}*k*therefore describes such array:

_{s}*x=0*and

*x=a*are matched for all field components. The characteristic Eq. (2) is presented in Fig. 3 for various grating duty cycles (DC), defined as

*DC=s/Λ*. The lower and upper curve-sets in Fig. 3 correspond to the first two solutions of Eq. (2), i.e. the first two modes, while the dashed lines in Fig. 3 are constant wavelength contours, presented in Eq. (1g). The intersections between the dashed lines and the curves, as marked by the black circles, indicate the modes at the specific wavelength. Figure 3 also shows the mode cutoff limit (

*β=0*), which according to Eq. (1f) is given by

*k*

_{s}=n_{bar}

*k*. Above the mode cutoff line, the HCG modes are propagating in

_{a}*z*(real

*β*) and below the mode cutoff line the modes are evanescent in

*z*(imaginary

*β*). Figure 3 shows that in HCG

*k*is always real, while

_{s}*k*can be either imaginary or real, depending on the wavelength. The lowest mode has only imaginary

_{a}*k*values, and therefore its

_{a}*β*has the largest value. Hence, we refer to the lowest mode as the fundamental, or the first, mode. This mode is also the only one not to have cutoff at large wavelengths. In fact, at large wavelengths (

*λ>>Λ*) the first mode resembles a plane wave (

*k*,

_{s}~0*k*). This is because at large wavelengths the exact grating corrugation profile loses effect, and the grating behavior approaches that of a uniform layer with an effective refractive index.

_{a}~0*Region I*are given by Eq. (3a). They comprise of an incident plane wave and multiple reflected modes (propagating in

*–z*direction), the coefficients of which are

*r*whereby

_{n},*n=0,1,2,…*is the number of the reflected mode.

*γ*is the longitudinal (

*-z*) wavenumber, and

*h*

^{out}

*and*

_{y,n}*e*

^{out}

*in Eq. (3a) indicate the lateral (*

_{x,n}*x*) magnetic and electric field profiles, where the index “out” stands for: outside HCG.

*δ*in Eq. (3a) is known as the Kronecker delta function. We are now in the position to define the HCG reflectivity matrix

_{n.0}**R**, which relates between the incident wave coefficient,

*ρ*, and the coefficients of the reflected modes

_{n,0}*r*:

_{n}*x*) field profiles,

*h*

^{out}

*and*

_{y,n}*e*

^{out}

*, in region I (obviously region III will have the same lateral field profiles): Equations (3c) shows that each air-gap center (*

_{x,n}*x=a/2*) is a symmetry plane for all modes in region I. However, each grating bar center (e.g.

*x=*(

*a+Λ*)

*/2*) is a symmetry plane as well. This is because each grating bar center is located half-period away from the adjacent air-gap centers. This is of course also true for the region-II lateral profiles

*h*

^{in}

*and*

_{y,m}*e*

^{in}

*described in Eq. (1d). In addition, the fact that the plane wave incidence is surface normal means that the solution above has no preferred direction among +*

_{x,m}*x*and

*-x*, and therefore the modes in Eqs. (1), (3) have a standing wave (cosine) lateral profile. The lateral symmetry in Eqs. (1), (3) is even (cosine) rather than odd (sine), because the incident plane wave (Eq. (3a)) has a laterally constant profile, and thus it can only excite laterally-even modes.

**τ**for the modes in region III, as well as the HCG transmission matrix

**T**, which relates between the incident wave coefficient,

*ρ*, and the coefficients of the transmitted modes

_{n,0}*τ*:The longitudinal (

_{n}*z*) wavenumber γ in Eqs. (3a) and (3d) is given by:As evident from Eq. (3f), in subwavelength gratings (Λ<λ) only the 0th diffraction order is propagating (

*γ*is real), while the first, second and higher orders are all evanescent (γ

_{0}_{1}, γ

_{2}etc. are imaginary).

*z=0*). We start from the magnetic field

*H*:

_{y}*τ*in terms of the overlaps between the lateral (

_{n}*x*) magnetic field profiles

*h*

^{in}

*and*

_{y,m}*h*

^{out}

*:*

_{y,n}*h*

^{in}

*onto the orthogonal set*

_{y,m}*h*

^{out}

*. The expression (*

_{y,n}*2-ρ*) in Eqs. (4b) accounts for the general fact in Fourier theory - that 0th self-overlap is always twice larger than all the subsequent (cosine) self-overlaps. By repeating the steps in Eqs. (4a), (4b), this time for the electric field

_{n,0}*E*, we can now express the same transmitted coefficients

_{x}*τ*in terms of the overlaps between the lateral

_{n}*electric*field profiles:

**H**and

**E**(both are unit-less), for the magnetic and electric field profiles respectively, based on Eqs. (4b) and (4c), and then rewriting Eqs. (4b) and (4c) in matrix-vector format:

**ρ**-matrix in Eq. (1b):

**a**we can now derive the reflection matrix

**ρ**as a function of the overlap matrixes

**E**and

**H**:

**ρ**in Eq. (6) is typically non-diagonal, which means that the HCG modes in Region II couple into each other during the reflection. This does not contradict the orthogonality of the modes in region II, since the reflection involves interaction with the external modes of region III, which are not orthogonal to the modes in region II.

*z=0*) we now repeat the steps in Eqs. (4)–(6) in order to match the boundary conditions at the HCG

*input*plane (

*z=-t*). For simplicity, we this time omit the details shown in Eqs. (4)–(6) and jump straight to the final equation:

_{g}**E**,

**H**,

**ρ**and

**φ**:

**Z**, which is the equivalent of the corresponding scalar in regular transmission line theory. Using Eq. (7b), the HCG reflectivity matrix

_{in}**R**can finally be calculated, resulting in an equation very common in transmission line theory:

**R**, we now calculate the HCG

*transmission*matrix

**T**. We first derive the coefficient vector

**a**in terms of the matrixes

**E**,

**H**,

**ρ**and

**φ**, using steps similar to Eqs. (4)–(6):

**τ**, from which the HCG transmission matrix

**T**naturally emerges:

*τ*). Had there been more than one transmitted and reflected diffraction orders (i.e. when

_{0}=0*Λ>λ*), obtaining high reflectivity (|

*r*|

_{0}*~1*) through cancellation of multiple orders (

*τ*etc…) would be very difficult, which is why ordinary diffraction gratings are typically

_{0}=0, τ_{1}=0, r_{1}=0,*not*associated with ultra-high reflectivity phenomena.

### 2.2 TE-polarized incidence

*ε*in Faraday’s equation explains why

_{r}*1*, and the minus sign in Faraday’s equation explains the minus sign in the last two entries of Table 1. The inversion of the wavenumber-ratios at the last two entries of Table 1 is also a common difference between TM and TE modes in waveguides.

### 2.3 Solution convergence

*2.1-2.2*converges. Figure 4a shows a convergence example for the TM polarization. For comparison, Fig. 4a also presents the HCG reflectivity calculated using Rigorous Coupled Wave Analysis (RCWA) [9

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]

*double-mode*nature of the HCG high reflectivity phenomenon. The fact that taking only two modes into account is a good approximation is a major advantage of the solution method described in sections

*2.1-2.2*. Such fast convergence is unlike the RCWA solution method, which is based on lateral (

*x*) Fourier expansion of the permittivity

*ε*, and thus requires a significantly larger number of modes in cases of a rectangular profile of refractive index, as considered here. Figure 4a also presents the difference (i.e. the error) between the RCWA solution and the solution described above, showing that when the reflectivity is high, the error between a double-mode solution and the RCWA solution is negligible

_{r}*t*

_{g}(both normalized by

*Λ*). Figure 4b shows that almost all high reflectivity configurations are concentrated between the cutoffs of the second and the third modes – a region where only the first two modes are propagating (

*β*and

_{1}*β*are real) and the third mode is below cutoff (i.e.

_{2}*β*is imaginary). High reflectivity in a triple-mode region is shown to be possible but rare, since the wavelengths of the triple-mode region are too close to the subwavelength limit.

_{3}*Broadband*high reflectivity (i.e. broad red contour sections, such as the one indicated by the red arrow) are non-existent in the triple-mode region. Notably, there are also no high-reflectivity contours in either the single-mode region or below the subwavelength limit.

*z*with the intensity ~25 times lower than the second mode.

## 3. The mech0anism of 100% reflectivity

*double mode*effect, the nature of which is examined in this section.

*τ*. According to Eq. (5a),

_{0}=0**τ**=

**E**(

**a**+

**a**), and therefore we can summarize the 100% reflectivity condition as follows:By using Eqs. (5a) and (3c),

^{ρ}*τ*can be rewritten as follows:

_{0}*τ*) when the lateral average of the first mode and that of the second mode cancel each other at the HCG output plane (

_{0}=0*z=0*). We refer to such cancellation as “destructive interference”. We use quote-marks due to the unusual

*lateral-average*interpretation of interference. The lateral average emerges from the overlap integral ∫

*e*

^{in}

_{x,m}e^{out}

*dx in Eq. (11), as a consequence of*

_{x,0}*e*

^{out}

*being constant with respect to*

_{x,0}*x*. More generally, if we represent the overall field profile (from all grating modes combined) at the output plane

*z=0*in terms of a Fourier series as a function of

^{-}*x*(Λ being the Fourier series periodicity), the lateral average will merely have the mathematical meaning of zeroth Fourier coefficient (“dc”-coefficient). It is easy to show that in general,

*n*th Fourier coefficient in such expansion would correspond directly to the transmission coefficient

*τ*of

_{n}*±n*th transmissive diffraction orders. However, since in subwavelength gratings only the zeroth order carries power along

*z*(as mentioned above), it is only the “dc” Fourier coefficient that we need to suppress, as shown in Eq. (12). Moreover, since the intensities of the first two modes are comparable, as shown in Figs. 5e, 5f, achieving destructive interference between them is fairly straightforward, by merely adjusting the optical path phases of the modes which are determined by HCG thickness (

*t*).

_{g}*a*)

_{1}+a^{ρ}_{1}*Λ*

^{−1}∫e^{in}

*| and |(*

_{x,1}dx*a*)

_{2}+a^{ρ}_{2}*Λ*

^{−1}∫e^{in}

*| respectively (see Eq. (12), are plotted along with their phase difference*

_{x,2}dx*Δϕ=*phase[(

*a*)

_{1}+a^{ρ}_{1}*Λ*

^{−1}∫e^{in}

*]-phase[(*

_{x,1}dx*a*)

_{2}+a^{ρ}_{2}*Λ*

^{−1}∫e^{in}

*]. At the points of 100% reflectivity the modes are at anti-phase (*

_{x,2}dx*Δϕ=π*) with equal intensities, which means that perfect cancellation occurs (Eq. (12). If two such 100% reflectivity points are located at close spectral vicinity, a broad-band of high reflectivity is achieved, as shown in Fig. 6a (top). Figure 6b illustrates the non-traditional “dc”-component interpretation for destructive interference: The lateral field profile (black curve, right plot), given by

*(a*

_{1}+a^{ρ}_{1})E_{y,1}^{II}(

*x,z=0*)

^{-}*+(a*

_{2}+a^{ρ}_{2})E_{y,2}^{II}(

*x,z=0*), is plotted as a function of

^{-}*x*for the case of perfect cancellation, showing that the field profile is zero only in terms of dc-component, but non zero otherwise. The individual field profiles of the first and the second modes,

*(a*

_{1}+a^{ρ}_{1})E_{y,1}^{II}(

*x,z=0*) and

^{-}*(a*

_{2}+a^{ρ}_{2})E_{y,2}^{II}(

*x,z=0*) respectively, are also plotted in Fig. 6b (blue and red curves, left plot). The “dc” components of the first two modes are shown to cancel each other. Had the grating not been subwavelength, this cancellation would no longer be enough, since in order to cancel higher diffraction orders, higher Fourier components (as opposed to only “dc”) would have to be zero as well.

^{-}## 4. Broadband high reflectivity mirror design

*a*,

*s*,

*Λ*and

*t*with respect to the wavelength

_{g}*λ*. Such scalability is intuitively obvious and was reported in [1

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]

*t*,

_{g}/Λ*λ/Λ*and

*DC=s/Λ*, and then the normalized dimensions can be scaled according to the desired wavelength.

- (i) The collection of solutions to Eq. (12), which are the HCG 100% reflectivity contours, are plotted on aFig. 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.
*t*vs._{g}/Λ*λ/Λ*plot. This is repeated for different grating duty cycles (*DC=s/Λ*), as shown in Fig. 7a . Figure 7a shows that these 100% reflectivity curves typically have an s-shape with large sections having very small slopes. Along these sections, reflectivity is 100% across a wide range of wavelengths for nearly the same value of*t*._{g}

- (ii) The next step is choosing a curve section that yields the smallest slope. Such flat region would correspond to broadband high reflectivity, as described above. The arrows in Fig. 7a show examples of two choices corresponding to two different duty cycles. Among these choices, the values of
*DC=s/Λ*,*t*and_{g}/Λ*λ/Λ*yielding the optimal broadest spectrum are selected. - (iii) Finally, having selected the optimal normalized HCG dimensions, the grating period
*Λ*is found by scaling the chosen ratio λ/Λ to fit the wavelength of interest. Having found the period*Λ*, the dimensions*s*and*t*are found using the normalized values_{g}*DC=s/Λ*and*t*from the previous step._{g}/Λ

*t*, is fine-tuned, the bandwidth can be increased through a trade-off with a dip in the reflectivity spectrum. This allows maximizing the bandwidth, given the specific requirement on a minimal reflectivity that can be tolerated by a particular application. For example, if the minimal tolerable reflectivity is 99%, as shown in Fig. 7b, the bandwidth can be increased by ~35% in comparison to the initial spectrally-flat design, which is an output of the first design stage, described in Fig. 7a.

_{g}*broadband*reflectivity is a result of high index contrast between the grating bars and the surrounding medium. Hence, the larger the contrast is, the wider the bandwidth. Figure 8 demonstrates this fact by plotting the same 100% reflectivity contours as in Fig. 7a for a larger grating index,

*n*

_{bar}

*=4*, and for 4 different duty cycles

*s/Λ*. The s-curve slopes in Fig. 8 are smaller and the reflectivity spectra are significantly broader. In addition, the fact that Figs. 7a, 7b and 8 use a normalized scale λ/Λ means that broadband design also automatically results in large fabrication tolerance on

*Λ*. Large fabrication tolerance in high contrast gratings has been experimentally demonstrated by our group in the context of vertical cavity surface emitting lasers [13

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]

## 5. Discussion – advantages and disadvantages of other solution approaches

*z*, which is also the propagation direction of the incident plane wave. We have shown that our approach facilitates a highly efficient convergence – only two modes are required.

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]

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

*±x*. The main advantages of this approach are (i) fast convergence, requiring only 3 leaky modes (ii) harnessing a widely used terminology, such as guided mode resonances. The main disadvantages of this approach are (i) its highly qualitative nature, which lacks mathematical description, making a detailed rigorous physical analysis difficult and (ii) the fact that qualitative agreement with the GMR mode-set is only presented for TM

_{0-3}modes and only for very low index-contrast gratings.

## 6. Conclusion

*double-mode destructive interference*phenomenon. We presented a quickly-converging matrix transmission line formulation for the HCG reflectivity and discussed a graphic design algorithm for broadband HCG mirrors. In the context of broadband design, the high refractive index contrast proves beneficial in terms of both bandwidth and fabrication tolerance.3

## Acknowledgement

## References and links

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. |

2. | C. J. Chang-Hasnain, Y. Zhou, M. C. Y. Huang, and C. Chase, “High-Contrast Grating VCSELs,” IEEE J. Sel. Top. Quantum Electron. |

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 |

4. | M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “A nanoelectromechanical tunable laser,” Nat. Photonics |

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. |

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. |

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 |

9. | M. G. Moharam and T. K. Gaylord, “Rigorous coupled wave analysis of planar grating diffraction,” J. Opt. Soc. Am. |

10. | S. T. Peng, “Rigorous formulation of scattering and guidance by dielectric grating waveguides: general case of oblique incidence,” J. Opt. Soc. Am. A |

11. | L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. |

12. | P. C. Magnusson, G. C. Alexander, V. K. Tripathi, and A. Weisshaar, |

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. |

14. | R. Magnusson and M. Shokooh-Saremi, “Physical basis for wideband resonant reflectors,” Opt. Express |

**OCIS Codes**

(050.2770) Diffraction and gratings : Gratings

(260.2110) Physical optics : Electromagnetic optics

(050.6624) Diffraction and gratings : Subwavelength structures

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: May 24, 2010

Revised Manuscript: June 23, 2010

Manuscript Accepted: July 12, 2010

Published: July 26, 2010

**Citation**

Vadim Karagodsky, Forrest G. Sedgwick, and Connie J. Chang-Hasnain, "Theoretical analysis of subwavelength high contrast grating reflectors," Opt. Express **18**, 16973-16988 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-16-16973

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### References

- 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]
- 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 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]
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