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We established the angular conditions that maintain the quasi-phase matching conditions for enhanced second-harmonic generation. To do that, we investigated the equifrequency surfaces of the resonant Bloch modes of a two-dimensional periodic, hole-array photonic crystal etched into a GaN/sapphire epitaxial structure. The equifrequency surfaces exhibit remarkable shapes, in contrast to the simpler surfaces of a one-dimensional structure. The observed anisotropy agrees well with the surfaces calculated by a scattering matrix method. The equifrequency surfaces at fundamental and second-harmonic frequencies provide the values of polar and azimuthal angles that maintain quasi-phase matching conditions for enhanced second-harmonic generation over an extended tuning range. The predicted values for quasi phase-matching conditions show that frequency tuning for the two-dimensional case covers an about two times larger fractional bandwidth relative to the one-dimensional case.
©2004 Optical Society of America
1. Introduction
Photonic crystals (PhCs) are periodic dielectric structures that interact strongly with light [1–3]. In recent work, the non-linear properties of PhCs have received increasing attention [4]. The photonic band structure of PhCs can exhibit multiple photonic bandgaps, large dispersion and strong anisotropy in the allowed bands. By exploiting these properties together with nonlinearity, a number of next generation optoelectronic devices such as second-harmonic (SH) generators can be envisaged. In this context, GaN-based PhCs will be attractive structures for optical wavelength conversion from ultraviolet to infrared. Although the second-order coefficient of GaN [5–10] is smaller than that of materials such as LiNbO3 or GaAs [11] (Table 1), GaN possesses three advantages: i) a wide useful spectral transmission range, from the electronic band gap at 365 nm [12] to the single phonon Reststrahlen absorption band at 13.5 µm [13]; ii) a mechanical ruggedness; and iii) a high optical damage threshold. It is in part the combination of transparency, ruggedness and strong enough non-linearity that makes GaN (and related materials) interesting. One- or two-dimensional PhC structures can significantly enhance second-harmonic generation (SHG) by providing quasi-phase matching (QPM) conditions in a chosen propagation environment [14–18]. In the case of an external beam incident on the surface of the PhC from the top half-space [19, 20], the QPM conditions are fulfilled when both the fundamental and the second harmonic (SH) fields are strongly localized via resonant Bloch modes of the periodic structure [21–24]. In this configuration, theoretical predictions [22] indicate that it is possible to achieve nearly six orders of magnitude enhancement in the SH signal. However these calculations have only been performed for the situation where the incident fundamental beam is propagating along one of the major symmetry axes of the PhC; the QPM conditions are then satisfied only at one or more specific optical frequencies. The practical application of SHG using QPM in PhC structures will benefit strongly if it becomes possible to tune the optical frequency while retaining the enhanced SHG behavior.
In recent work [25], we have shown that even one-dimensional structures etched in epitaxial GaN films can provide both the transmission/reflection resonant behavior and the strong anisotropy required to satisfy the QPM conditions, leading to a giant enhancement in the SHG. Although the structure used in these experiments has certainly not been optimized, we have already demonstrated experimentally enhancement factors as large as 5000, producing short wavelength light over an approximately 3% bandwidth range around 400 nm. The successful use of one-dimensional structures naturally raises the question of what further possible enhancements in the SHG behavior could be obtained by using two-dimensional PhC structures – and what significant differences will be exhibited between one-dimensional and two-dimensional structures from the point-of-view of SHG.
Appropriate issues for comparison of the nonlinear properties of one- and two-dimensional PhCs, with SHG and more general {χ(2)} processes in mind, are i) the sensitivity of the QPM conditions to small changes in the optical frequency or the direction of the incident wave-vector, and ii) the optical frequency tuning range that can be obtained by deliberate modification of the wave-vector. To address these issues properly it is necessary to establish methods for calculating and measuring the detailed band-structure of candidate two-dimensional photonic crystal structures in the linear regime, taking account of their azimuthal direction, over a range of optical frequencies in which there is a requirement for the exploitation of SHG. In the present work, we shall describe the linear characterization of an already fabricated two-dimensional GaN PhC structure, showing that it potentially can provide the desired SHG enhancement in a region of the optical spectrum that is of immediate importance for practical applications. The spectral regions of specific interest in the present work are a fundamental frequency range (approaching a wavelength of 1 µm) in the near infra-red where efficient high-power semiconductor laser technology is well-established- and the corresponding SH spectral region, which is in the blue-green part of the spectrum.
This paper is devoted to both construction of the equifrequency surfaces (EFS) [26–33] of the resonant Bloch modes of the PhC and determination of angular geometries that maintain QPM conditions. The EFS are obtained by cutting through the dispersion surfaces at constant frequency.
In Section 2 of the paper, we present the parameters of the two-dimensional PhC fabricated in a GaN/sapphire film. In Section 3, we have studied, experimentally as well as numerically, the linear dispersion properties of the resonant modes that characterise the two-dimensional GaN PhC for various polar and azimuthal directions. The theory based on a rigorous scattering matrix method [34] has provided a good description of the photonic bands and of the EFS and has enabled us to construct the EFS for the frequencies unavailable experimentally. In Section 4, we have showed that by adjusting both the angle of incidence and the azimuthal angle continuously, the frequency of the enhanced SHG can be tuned over a large range. It is shown, in this context, that the numerical modelling of the linear optical properties can be of great help in determining the experimental configurations that satisfy the QPM conditions. An important aspect of the present paper is the comparison to be made between the properties of the one-dimensional and two-dimensional PhC structures from the point of view of QPM for SHG. Section 5 concludes this paper.
Table 1. Second-order non linear coefficients dij for several dielectric materials.
2. Experimental set-up
Our photonic structure consists of a 260-nm thick epitaxial GaN layer on a sapphire substrate (refractive index nGaN=2.35 and nsapphire=1.78). The GaN layer is patterned with a triangular lattice of circular holes. The lattice constant a is 500 nm. The PhC was fabricated by high-resolution electron beam lithography and reactive ion etching techniques. The hole depth and the average air-filling factor f are estimated to be 185±5 nm and 0.21±0.02, respectively, using AFM measurements. Etch profiles tended to be overcut; this tendency is, in general, a characteristic of GaN; and f therefore varies with depth.
Transmission and reflection spectra were measured systematically for collimated white light incident on the surface of the PhC over a range of both angles θ (0°≤θ≤54°) and φ (0°≤φ≤30°), where θ is the angle of incidence relative to the surface normal, and φ denotes the azimuthal angle between the plane of incidence and the Γ-K direction [35] (Fig. 1). For certain θ values, depending both on the frequency and on φ, light can couple to the resonant Bloch modes of the PhC. Details of the variable-angles spectrometry used here are described in Ref. 25.
Fig. 1. Schematic diagram illustrating the sample, coordinate system, and experimental geometry used for angular resolved reflection and transmission experiments to investigate the resonant Bloch modes dispersion.
3. Results
Figures 2(a)-(d) show experimental transmission spectra for s-and p-polarized light along the Γ-K (φ=0°) and Γ-M (φ=30°) directions. Along these directions, the modes have either purely s- or purely p-polarization [36]; in the following text, the modes will be labeled according to their symmetry (s or p) along Γ-K, as shown in Figs. 2(a)-(d). The label number associated with each resonance (1s, 2s,…) was chosen according to its order of appearance in the spectra from higher to lower wavelengths.
Figures 3(a) and 3(b) show experimental reflection spectra of the PhC for s-s and p-p polarization at θ=25°, when the sample is rotated about its surface normal at different specific φ values (in the range 0°≤φ≤30°). The labels s-s, p-p refers to the incident and detected polarizations in that order. We have measured the reflectivity spectra over the entire first Brillouin zone and have been able to follow the change of polarization of the resonances corresponding to the photonic modes. As Figs. 3(a) and 3(b) show, the resonance 1s displays prominent feature in the s-s reflectivity spectra and does not appear in p-p; 1s is purely s-polarized along Γ-K as it can be established by the symmetry arguments. For a direction slightly different from Γ-K and s-polarized incident light 1s disappears, whereas 1p emerges. This 1p resonance becomes purely s-polarized along the Γ-M. These results provide clear evidence of polarization mixing behavior for propagation along general directions.
Fig. 2. Experimental transmission spectra for the two-dimensional GaN PhC for various angles of incidence, with s and p polarized light along the Γ-K and Γ-M lattice directions. The angle of incidence is varied from 0° to 39° with a step of 2° then 3°. (a) Γ-K, s-polarized incident light; (b) Γ-K, p-polarized incident light; (c) Γ-M, s-polarized incident light; (d) Γ-M, p-polarized incident light. The marked features in the higher angles spectra correspond to resonant Bloch modes. Along the symmetry directions the modes are excited by either s- or p-polarized light. For propagation directions away from the symmetry directions, these two polarizations are mixed and modes radiate an elliptically polarized field. For convenience in their labeling, the modes are referred to as s or p by continuity according to their polarization at φ=0°.
Fig. 3. Reflection spectra at θ=25°, φ varies from 0° (Γ-K) to 30° (Γ-M). (a) s-s refers to incident-wave s-polarization and reflected-wave analyzer also s-polarization. (b) p-p refers to incident-wave p-polarization and reflected-wave also p-polarization.
Figure 4 shows reflectivity spectra obtained with the specific propagation direction φ=16°, which is close to exactly half-way between the high-symmetry Γ-K and Γ-M directions. The labels p-s corresponds to the situation of an incident-wave p-polarization and a reflected-wave s-polarization. Put another way, the resonance 2s along a Γ-K direction appears to be elliptically polarized at φ=16°, with a strength in s comparable to the strength in p.
Fig. 4. Reflection spectra at θ=25° and φ=16° ; s-p refers to the incident and detected polarizations.
The positions in frequency of the resonances (in units of ωa/2πc) and the in-plane wave-vector locations are displayed in Fig. 5, for the two high symmetry directions Γ-K and Γ-M. For the modes that appear in color, we have plotted the EFS.
Fig. 5. Photonic band structure of the two-dimensional GaN PhC for s-polarized light as determined from the experimental transmission spectra (solid dots) and from theoretical calculation (solid lines and open dots) along the high symmetry directions. The resonant Bloch modes 1p, 2s and 6s are respectively in red, in green, and in violet. The two large circles indicate the points involved in the QPM condition satisfied for fundamental field at 0.48 and SH field at 0.96. The solid lines show the edges of the cone of incident light.
The EFS of the three modes 1p, 2s, and 6s exhibit, for s-polarized incident light, a variety of remarkable shapes in contrast to the circular or ellipsoidal surfaces obtained for classical optical materials. For the mode 1p, the most notable feature in the range 0.455≤ωa/2πc≤0.50 [Fig. 6(a)] is their almost perfect hexagonal shapes, reflecting the symmetry of the PhC. The EFS shrink as the frequency increases, according to the curvature of the lowest dispersion band shown in red in Fig. 5. The EFS shape becomes rounded when the in-plane wave-vectors approach the Γ-symmetry point (ωa/2πc≤0.52). This effect can be explain by the fact that, in a PhC, the band mixing is more pronounced around the symmetry points, close to the band-edge [30]. Along the Γ-K direction the EFS are not complete, because the resonance 1s is predominant and resonance 1p has disappeared. For the mode 2s [Fig. 6(b)], the greater the EFS size, the more noticeable is the distortion. Note that the almost hexagonal shape at 0.52 is rotated through 30° with respect to the similarly shaped surfaces in Fig. 6(a), i.e., with peak magnitudes along Γ-M directions rather than Γ-K directions. For the mode 6s, all of the EFS have a quasi-circular shape centered on Γ-point [Fig. 6(c)]. The corresponding dispersion band, in violet in Fig. 5, possesses positive curvature and is separated from the higher surfaces by a small band gap.
We have calculated reflection spectra using the scattering matrix method at constant values of in-plane wave-vector, along Γ-K and Γ-M [34]. The calculation contains all the knowledge related to the etched photonic structure, such as refractive, layer thickness (260 nm) and etch depth (180 nm). The frequency dispersion of the refractive indices of GaN and sapphire are taken into account by calculating each band and EFS with appropriate values of refractive indices taken at an average frequency, and we emphasized that the filling factor f was the sole adjustable parameter. The value of f was then varied around the value estimated from AFM measurements, for the calculations of the photonic band structure in the vicinity of the Γ-point. For f=0.22, the theoretical photonic band structure agrees well with the experimental results as shown in Fig. 5 for s-polarization. Small discrepancies could be related to the experimental uncertainty in the angles θ and φ, and to the precision of the energy position estimation for the Fano-shaped experimental resonances. The calculated EFS show very close agreement with the measured ones, even if the discrepancies increase for the portion of the Brillouin zone where the bands are less dispersive [Figs. 6(a)-6(c)]. In this case, the determination of the wave-vector corresponding to a given frequency is less accurate. The validity of the calculations using the scattering matrix method is experimentally confirmed. Therefore the theoretical approach that allows a significant reduction in experimental efforts becomes a fundamental tool for the construction of the complete photonic band structure, in the overall 2k-space. Optical measurements along the high symmetry directions only, or even just at the Γ-point, thus can be used to check and to confirm the parameters of the etched PhC, before calculations.
Fig. 6. EFS for different normalized frequencies as determined from the experimental transmission spectra (dots) and from calculation (solid), for the three resonant Bloch modes, and for s-polarized incident light; (a) 1p in red, (b) 2s in green, and (c) 6s in violet.
4. EFS for obtaining the angular configurations that maintain QPM conditions
Using the experimental EFS, we now illustrate the possibilities for successfully achieving QPM conditions for SHG in a large domain of frequencies for the GaN PhC described above. In the nonlinear conversion involving resonant modes, the enhancement in the magnitude of SH occurs when both the fundamental and SH fields are resonant with the photonic modes. This resonant to resonant condition or QPM condition arises when k ∥(2ω)=2k ∥(ω)+G, where k ∥(ω) and k ∥(2ω) are the in-plane wave-vector of the fundamental field and SH field, respectively; 𝙶=ni G i+nj G j is one of the reciprocal lattice vectors, G i and G j are the reciprocal lattice vectors for the triangular lattice and ni, nj are integers.
To exemplify the evolution of the QPM conditions with frequency, the EFS at ω and at 2ω are plotted in Figs. 7(a)-(d) for four fundamental wavelengths at 0.98 µm, 1.00 µm, 1.02 µm, and 1.04 µm, respectively. The red experimental EFS come from the mode 1p at frequency ω, whereas the blue EFS are extracted from the mode 10s, at 2ω, and plotted at half their in-plane wave-vectors. In Figs. 7(b), 7(c) and 7(d), black circles denote points of coincidence for the two EFS where the QPM conditions are obtained, i.e. where both of the fields, the fundamental at ω and second harmonic at 2ω are coupling into the resonant Bloch modes. So, it illustrates the possibility to adjust the frequency position of the modes and to adjust the values of θ and φ, until the QPM conditions are fulfilled to obtain an optimal SH conversion efficiency. The direction of the wave-vectors involved in the QPM conditions vary very rapidly with frequency, as illustrated in Figs. 7(b)-(d). Figures 7(a) and 7(d) give the upper and lower wavelengths of the tuning range involving the 1p and 10s resonant modes.
Fig. 7. Experimental EFS used to predict angular conditions for enhanced SHG (red curve for fundamental frequency, blue curve for SH frequency). Black circles denote the achievement of QPM conditions. (a) EFS at ω from 1p at 0.51 and EFS at 2ω extracted from 10s at 1.02 plotted at half its in-plane wave-vectors. (b) Same but for fundamental frequency at 0.50 and SH frequency at 1.00. (c) Same but for fundamental frequency at 0.49 and SH frequency at 0.98. (d) The particular configuration where the coincidence point is along the Γ-M direction; fundamental frequency at 0.48 and SH frequency at 0.96.
Notice also, for Fig. 7(b) and Fig. 7(c), that there are twelve propagation directions that are in principle completely equivalent. The case of Fig. 7(d) corresponds to the particular configuration for which the fundamental beam is incident and the SH beam is generated along the Γ-M direction (φ=30°). The circles in Fig. 5 show the points of the band structure involved in the particular doubly-resonant SHG process of Fig. 7(d). The degeneracy of the situation in Fig. 7(d) means that there are now only six equivalent propagation directions along which the resonance and QPM conditions are satisfied. The enhanced SHG frequency can be tuned by fine control of θ and φ within the range of coincidence of two EFS at ω and 2ω such that the QPM conditions are maintained. In the case of the GaN PhC used in our experiments, the tuning range involving the 1p and 10s Bloch modes extended from 0.96 (λ=0.52 µm) to 1.02 (λ=0.49 µm). Additional tuning ranges are attainable using higher-lying modes.
As indicated in the introduction, we have already demonstrated giant enhancement in the generation of SH radiation through the use of a one-dimensional epitaxial GaN-sapphire PhC structure [25]. The experimental SHG behavior was shown to conform closely to the QPM conditions specified by both the measured and calculated band structures of the PhC at the single specified fundamental frequency corresponding to a wavelength value of 791 nm. Figures 8(a)-(c) show extensions of the previously reported results for the one-dimensional structure to cover a range of frequencies (and corresponding wavelengths). In particular, the results of Figs. 8(a) and 8(c) show the upper and lower wavelengths, respectively, at which the QPM condition is satisfied. The fractional bandwidth corresponding to this wavelength range is approximately 3% and obtaining it requires both a total rotation of the propagation direction (φ) through 90 degrees and a substantial progressive change in the angle of incidence (θ) to maintain the QPM conditions. Figures 8 (a)-(c) show that the EFS for the one-dimensional PhC with comparable parameters to the two-dimensional PhC of the present paper are elliptical in appearance for the particular bands, labeled as 2p and 4s, used.
Fig. 8. Calculated EFS of the one-dimensional GaN PhC (a=500 nm) involved in enhanced SHG (red full squares for fundamental frequency, blue open circles for SH frequency). Black circles denote the achievement of QPM conditions. (a) EFS at ω from the mode 2p of the one-dimensional GaN PhC at frequency 0.629 (in units of ωa/2πc); the mode 2p is elliptically polarized for propagation along general directions. EFS at 2ω extracted from the mode4s at 1.258 (in units of ωa/2πc) plotted at half its in-plane wave-vectors. (b) Same but for fundamental frequency at 0.637 and SH frequency at 1.274. (c) Same but for fundamental frequency at 0.645 and SH frequency at 1.290.
There is clearly a difference between the behaviors observed and predicted for the one-dimensional PhC structure and those for the two-dimensional PhC structure. Fairly obviously, the six-fold rotational symmetry of the two-dimensional structure means that the full tuning range possible for QPM of SHG must lie within a 30 degree sector, whereas the two-fold rotational symmetry of the one-dimensional structure implies a three times greater (90 degrees) variation. But, in association with the more complex frequency surfaces, the frequency tuning for the two-dimensional case covers an approximately factor of two large fractional bandwidth – and this larger bandwidth is obtained over a factor of three smaller angular tuning range. The corresponding angular tuning ‘velocity’ (rate of change of matching frequency with angle) is therefore much greater (approximately six times) for the two-dimensional case than for the one-dimensional case. This comparison clearly has a strongly anecdotal character, given the large parameter space within which the comparison is being carried out. But we believe that the relative behavior observed here will continue in more comprehensive future studies. A further potentially important observation for practical applications of PhC based SHG is that the amount of tuning of the angle of incidence, θ, required to maintain QPM conditions, as the frequency (wavelength) of the fundamental is varied, is much smaller for the two-dimensional case than for the one-dimensional case.
5. Conclusions
The experimental and theoretical linear optical properties, photonic band structure and EFS of a two-dimensional GaN PhC have been studied to probe possible QPM conditions for enhanced SHG. A numerical modelling of the linear optical properties has been performed by using the scattering matrix approach. Our calculations show excellent agreement with experimental results. For future investigations, the modeling approach of the linear properties can avoid the task of tedious angular resolved experiments, especially in the case of PhCs with etched area smaller than 50×50 µm2.
The polar and azimuthal angular geometry to achieve enhancement of the SHG caused by QPM conditions can be directly obtained by the construction of the experimental or theoretical EFS at ω and at 2ω. We have demonstrated that the EFS plot is a powerful tool for obtaining the angular configurations that maintain QPM conditions for enhanced SHG with extended tuning range.
The comparisons that we have been able to make between the angular tuning characteristics of one-dimensional and two-dimensional structures for SHG functionality are strictly limited. Nevertheless these comparisons suggest that two-dimensional periodicity has potential advantages in terms, in particular, of greater tunability. This work was restricted to the characterization of the linear properties of the PhC to find a suitable range of frequencies that might be exploited in SHG. However, in order to accurately design and analyze non linear optical processes in terms of figures-of-merit, it is necessary to develop appropriate rigorous modelling tools. A Green’s function approach has been already proposed and tested [37]. An extension of the scattering approach to the treatment of the non linear polarization field should also be appropriate to simulate the SHG in PhCs.
These results are promising for the practical realization of non linear frequency conversion devices based on epitaxial GaN technology.
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