In this paper we present an extensive theoretical and numerical analysis of monolithic high-index contrast grating, facilitating simple manufacture of compact mirrors for very broad spectrum of vertical-cavity surface-emitting lasers (VCSELs) emitting from ultraviolet to mid-infrared. We provide the theoretical background explaining the phenomenon of high reflectance in monolithic subwavelength gratings. In addition, by using a three-dimensional, fully vectorial optical model, verified by comparison with the experiment, we investigate the optimal parameters of high-index contrast grating enabling more than 99.99% reflectance in the diversity of photonic materials and in the broad range of wavelengths.
© 2015 Optical Society of America
Vertical-cavity surface-emitting lasers (VCSELs) have a number of unique features that distinguish them from conventional semiconductor lasers  and made them widely used in optical interconnects: low power consumption, on wafer testing, inherent single-longitudinal-mode operation, low-divergence nonastigmatic circular output beams, suitability for integration into two-dimensional laser arrays. However, there are still some challenges like the need of increase of the modulation speed and down-sizing. They require new approaches mostly in reduction of the dimensions of the distributed Bragg reflectors (DBRs) which provide optical feedback in VCSELs.
High-index contrast gratings (HCGs) can be very appealing answer to this issue. Having a thickness less than one third of the wavelength, HCGs ensure above 99.99% reflection for surface-normal incident light  and additionally offer local phase and polarization control of the light unavailable with DBRs. The physical principles of their operation can be explained on the basis of destructive interference between two periodic waveguide modes propagating in the grating  or as a Fano resonance between the modes guided in the grating and the incident wave . Basing on this understanding, in order to provide the high reflection, the HCG layers—consisting of high refractive index stripes—must be surrounded by low index media. Practically, such low/high index stack can be fabricated in several ways: as a HCG freely suspended in the air , embedded in silicon-dioxide  or placed on the top of the cladding layer of low refractive index which typically is aluminum-oxide  or silicon-dioxide . In each case, low refractive index layers are electrical insulators of poor thermal conductivities. Arsenide-based HCG VCSELs can be grown in single process since low index material beneath the HCG stripes can be lattice matched aluminum-oxide layer. The weakness of such construction is their mechanical instability in the case of thick layers of aluminum-oxide . In phosphide and nitride-based technology the lack of lattice-matched low refractive index materials makes fabrication of integrated HCGs impossible.
To overcome those limitations the work of S. Goemanwe et al.  proposes alternative approach. The authors demonstrated a monolithic grating mirror (named GIRO grating) whose reflectance is as high as 85% for polarized 1550 nm wavelength. Based on this work a monolithic GaN subwavelength grating was demonstrated by J. Lee et al.  which reflectance is close to unity for polarized 450 nm wavelength.
In this paper we investigate monolithic high-index contrast grating (MHCG) which can be a very attractive alternative for DBRs as well as HCGs. MHCGs bring unprecedented simplification of the VCSELs designs and offer the freedom in the material choice, opening new prospects for the realizations of VCSELs emitting at wavelength bands hardly achievable in the past. In the case of phosphide-based and nitride-based VCSELs, the fabrication of monolithically integrated DBRs is very challenging. The trade-off between high refractive index contrast and low lattice mismatch of binary stacked structures is practically impossible. As the consequence, DBRs consisting of 90 layers and more are required to achieve a reflectance larger than 99.8% [12, 13]. Growth of such large number of layers is very time consuming process which must be carefully controlled with respect to the layers thickness and their mol fractions. Currently experimentally realized nitride- and phosphide-based VCSELs require sophisticated technology involving implementation of dielectric mirrors [14, 15] or bonding semiconductor DBRs to the cavity . For those reasons MHCGs can be appealing solution for those VCSELs. Although arsenide-based VCSELs are significantly more efficient compared to phosphide and nitride counterparts and their technology is mature, there is still a room for an improvement which is particularly driven by their integration with electronic circuits . The thickness of the DBRs is more than 90% of the total thickness of the epitaxial VCSEL structure. Replacement of both DBRs by 20-40 times thinner MHCGs reduces the thickness of the VCSEL more than 10 times which scales down the epitaxial growth costs and facilitates radical miniaturization of the VCSELs. Further consequence of the VCSEL downsizing is a shortening of the light round trip which enhances the bit rate of the operating VCSELs. The freedom in the MHCG material choice facilitates electrically conductive mirrors and eliminates the mechanical strains in the structure. MHCG opens a fascinating prospective of possible fabrication of very thin mirrors made of wide variety of materials monolithically integrated with the VCSEL cavity.
In this paper we demonstrate the phenomena explaining nearly 100% reflection mechanism of MHCG. We verify our numerical model by comparison with the experiment and finally we perform the exhaustive numerical analysis of the geometrical parameters of MHCG assuring very high reflectance. Our analysis is dominantly oriented on the application of MHCG in VCSELs, which motivates the choice of four photonic materials: GaAs, InP, GaN and Si taken under consideration as potential VCSEL mirrors. Nonetheless the final conclusions are of much more general meaning and can be applied to MHCGs realized in numerous possible materials. Although the presented analysis is limited to very highly reflecting MHCGs, they offer also engineering of the light transmission and phase which can be used in selective mirrors/transmitters , light steering  etc.
2. Monolithic grating structure
A schematic of a MHCG is shown in Fig. 1. In the theoretical and numerical analysis the dimensions of the mirror are infinite in the x-y plane. Thickness of the substrate and the air beneath the mirror are assumed to be infinite as well. All MHCG parameters (L – period of the grating, h - height of the stripe, F - fill factor) will be varied to design MHCGs of high reflectance.
The spectral dependence of the refractive indices and absorptions of the four considered photonic materials, which will be used in the calculations are illustrated in Fig. 2.
3. Numerical model and its verification
To determine reflectance of a MHCG mirror realized in the four chosen photonic materials in broad range of geometrical MHCG parameters we applied the plane-wave reflection transformation method . We simulate a single period of a MHCG combined with periodic boundary conditions. In the calculations from 20 up to 30 plane-waves for single MHCG period were used to reach convergence depending on the MHCG configuration. The assumption of infinite dimensions of MHCG (see section 2) and the plane wave reflections may at first seem an unrealistic structure for an MHCG interacting with laser light. However, the reflectivity of a finite size beam is very close to that assumed in the model, since the fundamental-mode Gaussian profile (the plane-wave component perpendicular to the grating) is dominant in both cases. In the reflectance calculations all the wave vectors are considered. Particularly those which also have non-zero transversal components. However, the power reflectance is calculated for the zero diffraction order only, since only this beam component takes part in the process of stimulated emission in the VCSELs.
In order to verify the theoretical model of reflectance we fabricated MHCG directly on a 2 inch GaAs wafer, 500 μm thick. The size of the patterned area was 1 cm2. Critical dimensions of the grating allowed us to apply standard optical lithography (contact printing with chromium mask). On the top of the wafer the photoresist layer was span-off. Exposed photoresist formed a stripe pattern. In the next step GaAs wafer was etched using Inductively Coupled Plasma Reactive Ion Etching (ICP RIE) with chlorine- argon BCl3/Ar (~1.5:1) plasma. Finally the photoresist was removed using proper solvent solutions. The parameters of the MHCG are given in Table 1. which assure nearly 100% reflection according to the numerical analysis presented in the section 6. The MHCG is designed for the wavelength of 4.7 μm. Although there is no VCSELs emitting the light of such wavelength the demonstration of MHCG reflectance can be performed for arbitrary wavelengths since Maxwell's equations can be linearly scaled. MCHG requires less fabrication effort at longer wavelengths however light detection becomes more challenging. Hence the wavelength of 4.7 μm is consensual in our case.
In the experiment we achieved the MHCG parameters deviated from the assumed ones: h = 1.1 μm, L = 4.0 μm, F = 0.48 for which the calculated reflectance is on the level of 80%. Roughness of the stripes and fluctuations of the period contributes to further reduction of the reflectance, however shift of the reflectance spectrum is expected to be negligible . Hence verification of the numerical model with experiment is expected to rely on a qualitative comparison. Figure 3(a) shows a scanning electron microscope (SEM) image of a fabricated stand-alone MHCG mirror.
We used Fourier transform infrared (FTIR) spectrometer (Vertex 80 from Bruker) to measure the reflectance of the grating. The sample was illuminated by the polarized probing beam from the globar. The electric field of the electromagnetic field was polarized parallel to MHCG stripes. The beam was collimated by parabolic mirror and directed under the angle of 12° with respect to normal direction of MHCG [Fig. 3(b)], which is the smallest angle achievable in used spectrometer. Such deviation from the normal direction reduces the reflectance of the mirror which according to the calculations is expected to be on the level of 60%. Figure 4 illustrates the comparison of the experimental and the calculated spectra. In the calculations we assume the incident beam propagating through air/GaAs interface and through GaAs only. To take into account additional reflection from the Air/GaAs interface we assumed that coherence length of the light emitted by the globar is shorter than the doubled sample’s thickness (500 μm). In such a case there are no Fabry-Perot oscillations in the calculated spectrum. In the measured spectrum Fabry-Perot oscillations are visible, however their amplitude is small especially in the wavelength region of the interest below 6 μm. Additionally, fabrication imperfections of the MHCG, particularly: deviation of the period over the sample and roughness of the side walls reduce the measured reflectance. However qualitative comparison concerning the appearance of the maxima in the reflectance spectrum proves that our model reflects correctly optical phenomena taking part in MHCG. In the deep subwavelength regime (long wavelengths) MHCG behaves as quasi-uniform layer hence experimental and calculated spectra reach close value of the reflectance. The dashed line in Fig. 4 illustrates the reflectance of the beam which propagates through GaAs only, which closely relates to the light propagating in VCSEL with one MHCG mirror. In the experiment the MHCG imperfections can be reduced using electron beam lithography (EBL) in the preparation of the mask. The above comparison of experimental characterization and calculations serves to confirm the validity of the numerical model which is used to investigate the designs of monolithic subwavelength grating.
4. Principle of operation
We consider a plane-wave which is perpendicularly incident from the bottom side of the grating (Fig. 1) with the free-space wavelength not much longer than the grating period (near-subwavelength regime). In the uniform high-index region (R1) this wave is accompanied by the propagating and evanescent reflected waves of different orders. Depending on the grating period and the layer refractive index, there are two or three reflected propagating diffraction orders and the infinite number of evanescent waves. The second layer (R2)—the grating itself—supports exactly two forward- and backward-propagating modes in the near-subwavelength regime . Finally, in the top air layer (R3) one transmitted propagating mode can exist only, however, it is accompanied by the infinite number of evanescent modes as in the first layer.
The modes are mutually orthogonal in all three layers, however they may couple to each other at the layer boundaries. Hence, the single 0-th order incident wave can excite reflected and transmitted modes of all orders: both evanescent and propagating. The evanescent modes play important role in this coupling as they are necessary to ensure fulfillment of electromagnetic boundary conditions at grating edges (one may consider them as Fourier basis in which the grating modes are expanded). However, only the propagating modes are the ones transferring the energy. Hence, it is sufficient to suppress the zero-order transferred mode in R3 by the means of negative interference to obtain 100% reflectance.
In case of conventional index-confined grating the reflected wave has only zero-order propagating mode, so naturally 100% of the energy is reflected into this mode. In case of the monolithic HCG there are more than one reflected propagating mode. However, in the presented case all modes except the zero-order one vanish at the wavelength corresponding to 100% reflectance. Figure 5 illustrates that despite the fact that the 100% reflectivity peak wavelength is shorter than second-order mode cut-off, at this particular wavelength all modes except the zero-order one have reflectance of 0% for carefully chosen parameters of MHCG.
Consider a non-monolithic grating that has a cladding layer (region R1 in Fig. 1) with refractive index nB different from the refractive index of the stripes (region R2). Depending on this cladding refractive index, the cut-off wavelength of the first-order reflected mode is changing: the higher its value (nB), the longer the wavelengths (λ) up to which the first order mode is a propagating one. For the wavelengths longer than the cut-off, we have a situation ofa confined grating, where only zero-order reflected mode can transfer the energy. At this regime 100% reflectivity peaks can appear as already well theoretically explained . Figure 6 shows the wavelengths of two such peaks for TM polarization as a function of nB. These two peaks show qualitatively different behaviors: the lower-wavelength one (red line) exists only above the cut-off (black line) and its wavelength strongly depends on the value of nB. Most of the hitherto reported high-reflectivity peaks [3,5]—both broadband and narrow-band—are of this nature. However, the long-wavelength peak (blue line) behaves differently: it is nearly independent on nB and this mere fact is the clue to MHCG's nearly 100% reflection. Below the cut-off, the cancelation of the transmitted zero-order mode ensures that all the energy is back-reflected into zero-order reflected mode (nearly 100% reflection), as no other mode can transfer reflected energy. In other words, the reflection coefficient for the zero-order mode is 1. In case this coefficient is nearly independent on nB in both its value and wavelength (which can be observed in Fig. 6), the higher-order reflected modes—even when they become physically possible—cannot transfer any energy due to the total energy conservation. Hence, by increasing the nB up to the value of the grating refractive index (so there is no index contrast between the cladding and the grating bars), we can convert the 100% reflective confined HCG into a 100% reflective MHCG.
For TE polarization one can find similar mode independent on the cladding refractive index, although for different grating parameters. However, the physics of such mode is exactly the same as described above, hence, it is possible to make an MHCG operating on TE mode as well. Particular examples of such modes are presented in section 5.
5. Numerical analysis of monolithic grating properties
In order to find optimal MHCG parameters, we have investigated its properties varying height of the stripes (h), their period (L), fill factor (F), and assuming four different refractive indices of MHCG, which correspond to the refractive indices of GaAs at 980 nm (n = 3.53), Si at 1300 nm (n = 3.5), InP at 1550 nm (n = 3.16), and GaN at 420 nm (n = 2.49). We havecomputed MHCGs reflectivities for both TE (electric field parallel to stripes) and TM (electric field perpendicular to stripes) polarizations of the perpendicularly incident wave. The results, presented in Fig. 7, allow to identify several regions of high reflectance (RHRs) that we define by total reflectance greater than 99%, which is of great importance for VCSEL applications.
As known from the section 4, the regions of high reflectance appear only for the MHCG periods (L) shorter than the wavelength. In particular, the RHRs for TE polarization [Fig. 7(a)] are localized within the range 1.0L < λ < 1.5L and for TM polarization [Fig. 7(b)] they can extend up to wavelength of 2.5L. However, if refractive index decreases, the long-wavelength limit of RHRs gets reduced to 1.0L for both polarizations. Different behavior can be observed when analyzing reflectance dependence on the etching depth (h): RHRs are forming periodic (dark red) islands in the reflectance maps. Their close inspection reveals that the largest reflectance of the MHCG is usually found within the RHR island for the most shallow etching depths and the size of the RHR island reduces while refractive index decreases. The parameter that has the lowest impact on the reflectance is the fill-factor (F). For high-index materials there are high-reflectivity regions in a wide range of fill-factors,however decreasing the refractive index results in more strict constraint on this parameter: large RHRs are present only in the proximity of F = 0.5.
To analyze high reflectance of MHCG more precisely we have performed an exhaustive multidimensional optimization of the geometrical parameters of MHCG (varying F, L, and h) for both TE and TM polarizations in order to determine the maximum possible grating reflectance (i.e. the maximum reflectance was the criterion of optimal MHCG parameters choice). Figure 8 shows the results. It illustrates the dependence of the TE and the TM reflectances on MHCG period (L), fill factor (F) and etching depth (h) at given wavelengths for four abovementioned materials (GaAs, Si, InP, and GaN). The maps of the TE reflectance as the function of L and F show relatively broad range of MHCG parameters for which the reflectance is larger than 99%. In case of GaAs, such high reflectance can be maintained with the following deviations from optimal parameters: ΔF = 0.14, ΔL/L = 0.10, and Δh/h = 0.09. To achieve the reflectance larger than 95% the constrains in MHCG fabrication precision are less strict and the following deviations from optimal values are acceptable: ΔF = 0.44, ΔL/L = 0.25, Δh/h = 0.18. The analogous tolerances in the case of GaN, which is the material of the smallest refractive index considered here, are ΔF = 0.14, ΔL/L = 0.06, Δh/h = 0.09 for 99% reflectance and ΔF = 0.44, ΔL/L = 0.12, Δh/h = 0.18 for 95% reflectance. Hence, we can see that the required relative precision of the MHCG period (ΔL/L) gets stricter with the decrease of the refractive index, however the necessary precisions of the fill factor (ΔF) and the etching depth (Δh/h) are independent on the refractive index. A careful analysis of Fig. 7 reveals that there is a trade-off between the large error tolerance and the high reflectance. The RHR containing the maximum of the reflectance is not necessarily the same RHR that can tolerate larger manufacturing errors. From the practical point of view the etching depths (h) are of great importance. In this view TE configuration, which is nearly twice shallower than TM configuration, is expected to be more feasible in realization.
The optimized MHCGs, assuring maximal reflectance, show similar electromagnetic field profiles in all material systems (Fig. 9). In the TE configuration, the light is confined dominantly within the top part of the stripes and there is exactly one light intensity lobe per stripe. In the TM configuration the highest light intensity appears in the air between the stripes as rapidly oscillating function of the position. The only exception is the GaN MHCG, which has the lowest reflectivity of all the analyzed cases (Fig. 8) and Fig. 9 shows the explanation of this fact: there is a very strong light leakage out from the GaN stripes.
Such light distributions in the TE and TM configurations intuitively suggest similarities to waveguiding and anti-waveguiding mechanisms observed in laser arrays . In those devices the waveguiding focuses the light within the regions of high refractive index by means of total internal reflection. Anti-waveguiding mechanism, relying on resonant reflections, confines the light within the region of lower refractive index. The weakening of waveguiding phenomenon caused by the reduction of the refractive index contrast manifests in equalization of the mode amplitudes in the periodic regions of lower and higher refractive indices. Weakening of anti-waveguiding mechanism results in leakage of strongly oscillating waves to the region of higher refractive index. Adequate similarities to waveguiding and anti-waveguiding mechanisms can be observed in the TE and TM configurations of MHCG, respectively. In both cases the highest field intensities are located in the stripes layer: the maximal intensity in this layer is 4 times larger than the light intensity in the bulk material below the mirror in TE configuration and nearly 10 times larger in the TM configuration. Additionally the peak of the light intensity is located at the edges of the stripes in TM configuration. Such light distribution in TM configuration may cause susceptibility of the grating reflectance to the stripe imperfections and the surface absorption, which favors TE configuration as practical realization of MHCG.
We have also performed the spectral analysis of the maximum grating reflectance. By means of multidimensional optimization we have determined the MHCG parameters (L, F, h) providing the maximal reflectance for given wavelengths (λ) and calculated the spectral band (Δλ) in which the reflectance is greater than 99%.
Figure 10 illustrates the normalized spectral RHR (Δλ/λ) as the function of the wavelength for considered photonic materials (Si, GaAs, InP and GaN). We have assumed the material dispersion and absorption as it is shown in Fig. 2. Comparing Fig. 2(b) with Fig. 10(a) (which corresponds to the TE configuration) one can conclude that the main factor assuring Δλ/λ > 0, for a particular refractive index, is the material absorption. The inter-band absorption limits high reflectance for short wavelengths and subband absorption do the same for long wavelengths. Consequently, the very high reflectance can be achieved only in the following bands: λ > 980 nm for GaAs and InP, λ > 1000 nm for Si, and 400 nm < λ < 5 μm for GaN. The subband absorption deteriorates the reflectance of GaAs and InP for λ < 2 μm and λ < 5 μm, respectively, however, it does not drops below 99%. In the wavelength bands free of the absorption, Δλ/λ depends mostly on the value of the refractive index and reaches: 0.066 for Si, 0.064 for GaAs, 0.061 for InP, and 0.04 for GaN. These spectral ranges are significantly smaller than the ones of previously reported HCGs, however, they are comparable to spectral high reflectivity ranges of DBRs (Table 2).
Two and threefold broader spectral RHR (Δλ/λ) of conventional HCGs with respect to MHCG is achieved due to proper choice of HCG parameters assuring two spectrally overlapping Fano resonances . In MHCG only single Fano resonance exists, however spectral RHR of MHCG is comparable to spectral RHR of typical arsenide-based DBR which proves the functionality of monolithic subwavelength grating.
Figure 10(b) illustrates normalized spectral RHR as the function of the refractive index for TM configuration of MHCG. Δλ/λ is 30% lower in the maximum and of narrower spectral band with respect to the TE configuration. GaN MHCG do not reach 99% level of the reflectance in the first RHR island [see Fig. 7(b)] and it is not considered here, since necessary aspect ratio a/h is far from feasibility in the present stage of etching technology. GaAs and InP MHCGs reveal strong susceptibility to the absorption and reduce rapidly with increasing absorption.
The geometrical parameters of the TE configuration of MHCGs providing maximal reflectance are collected in Fig. 11. The increase of the incident wavelength contributes to the decrease of the refractive indices of all photonic materials considered here [Fig. 2(a)]. Since the optical field is mostly confined in the stripes of the MHCG the optimal MHCG parameters depends on the dispersion of photonic materials. The decrease of the refractive indices imposes the increase of the normalized dimensions of the construction parameters (L/λ, h/λ) and contributes to the increase of the fill factor (F).
Finally, Fig. 12 illustrates the dependence of the maximal TE reflectance as the function of the refractive index, neglecting the absorption of the particular materials. It can be seen that nearly 100% reflection can occur for all the refractive indices larger than 1.75, which opens the possibility to fabricate the MHCG with most of the materials used in the modern photonics. Normalized spectral RHR reaches its maximum (Δλ/λ = 0.0667) for the refractive index n = 3.45 which corresponds to the refractive index of GaAs at λ = 1.15 μm.
In this paper we have presented the results of an extensive theoretical, numerical and experimental analysis of the monolithic high-refractive-index contrast grating (MHCG). Theoretical analysis revealed the possibility of nearly 100% reflection in the monolithic HCG configuration which relied on elimination of all higher diffraction orders except the 0th-order one. To numerically investigate the MHCGs we have applied three dimensional, fully vectorial, plane-wave reflection transformation method, which was validated by the comparison with experimental results obtained for a GaAs-based MHCG designed for 4.7 μm wavelength. Good agreement between experiment and numerical model confirmed adequacy of the method to the analysis of MHCG. Based on the numerical analysis we have proved that MHCGs realized in GaAs, Si, InP and GaN can provide reflectances larger than 99% in broad range of the wavelengths. In the case of considered photonic materials the absorption was the only factor limiting the spectrum of the very high reflectance. TE configuration of the MHCG assured high reflectance in broader spectral band we have concluded that the fabrication of TE MHCG configuration can be better controlled with respect to TM configuration.
Our analysis showed that MHCG is an appealing candidate to replace distributed Bragg reflectors (DBRs) in VCSELs. MHCG can be scaled with the wavelength and fabricated in the vast variety of the photonic materials of refractive index larger than 1.75 without the need of the combination of low and high refractive index materials. The freedom of use of various materials allows to provide more efficient current injection and better heat flow through the mirror, in contrary to the conventional HCGs. It has a great application potential in passive and active optoelectronic devices, but mostly it has great prospects in application to monolithically integrated phosphide and nitride-based VCSELs that lack monolithically integrated materials of high refractive index contrast. MHCG can simplify the construction of VCSELs, reducing their epitaxial design to monolithic wafer with carrier confinement and active region inside and etched stripes on both surfaces in post processing.
In future perspective, MHCGs can produce three dimensional confinement  by lateral Bragg reflections  which makes MHCG vertical-cavity lasers interesting candidates as micro-cavities as single photon source .
This work is jointly supported by the Polish National Centre of Research and Development and by Singapore A*STAR (grant no. 122 070 3063, ‘A Novel Photonic Crystal Surface Emitting Laser Incorporating a High-Index-Contrast Grating’). It is also partially supported by grants funded by Ministry of Education, Singapore. (MOE2011-T2-2-147). M. Gebski acknowledges the support by the Polish National Science Centre within the project PRELUDIUM 2013/11/N/ST7/02826. A. Szerling acknowledges the support of the Polish National Center for Research and Development within the project LIDER/34/70/L-3/11/NCBR/2012. M. Motyka acknowledges the support by the Polish National Science Centre within the project DEC-2011/03/D/ST3/02640.
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