Open Access
Add to BibSonomyAbstract
In this paper, we present a numerical and experimental study of W3-4 photonic crystal (PhC) waveguide lasers fabricated on InP substrate. In such a PhC waveguide, the dispersion curve of the fundamental mode folds in the two-dimensional gap of the triangular lattice. Folding occurs at the Brillouin zone edge as in the case of genuine distributed feedback (DFB) lasers. Single-mode emission is presently observed in both electrical and optical pumping configurations. This behavior is attributed to the different levels of out-of-plane losses experienced by the two DFB mode components. Three-dimensional finite-difference-time-domain calculations are used to finely quantify the quality factors of the waveguide modes. The modal discrimination is shown to be reinforced when lasing occurs far from the conduction band edge. This trend is also predicted for other canonical waveguides in triangular PhCs as for instance W2-3 waveguides.
©2005 Optical Society of America
1. Introduction
Photonic crystals (PhC) waveguide lasers have received particular attention [1–6] as they could possibly replace more traditional distributed feedback (DFB) lasers in future photonic-integrated-circuits. Single-mode emission with a large side-mode suppression ratio (SMSR) has been obtained from different types of PhC waveguide lasers on InP substrate [3–5]. Recently, standard structures such as W3 waveguides simply formed by three rows of missing holes in the ΓK direction of a triangular PhC were operated under optical pumping with the emitted wavelength above the gap [3]. Actually, W3 lasers were shown to behave as second-order DFB lasers, as lasing occurred at the second fold of the fundamental mode (at the Γ point). Single-mode operation can be achieved with these lasers since only one of the two DFB mode components is well confined in the guide, while the other spreads over the whole crystal [6, 7]. A different situation is a priori expected for lasers whose fundamental mode folds at the Brillouin zone edge in-gap. Due to the 2D bandgap, the two DFB components should experience a similar level of confinement in the guide, thus leading to a dual-mode emission.
In this paper, we present a numerical and experimental study of W3-4 PhC waveguide lasers whose dispersion curve of the fundamental mode indeed folds in the two-dimensional gap (at the M point) of the triangular lattice. Lasers are fabricated on InP substrate. Despite the bandgap confinement of the two DFB components, we show that single-mode emission with large SMSR is obtained near 1.5μm in both electrical and optical pumping configurations. A plane wave model is used to calculate the PhC band diagram, while three-dimensional finite-difference-time-domain (FDTD) computations are used to finely quantify the eigenmode losses. The single-mode behavior is correlated with the different levels of out-of-plane losses experienced by the two DFB mode components. The mode discrimination is shown to be reinforced when lasing occurs far from the conduction band edge. The same trend is observed for other standard waveguides oriented in the Γ-M direction of triangular PhCs.
2. Modeling
W3-4 waveguides are oriented in the ΓM direction of the triangular lattice, their width being determined by alternating three and four missing holes. For a standard air filling factor (30%) of the lattice and a 3.21 dielectric refractive index for the guiding layer, the third fold of the fundamental mode’s dispersion curve occurs in gap at the M point. Figure 1 (left) represents the W3-4 waveguide band diagram calculated from a two-dimensional plane wave expansion with the previous parameters. The third fold of the fundamental mode occurs at the normalized frequency a/λ ~ 0.273 inside the gap which extends from a/λ ~ 0.22 to a/λ ~ 0.28 (a being the PhC period). In the same way as genuine DFB lasers, a mini stop band opens around the folding frequency. The group velocities of the two DFB modes then tend to cancel each other, where as the two-dimensional photonic gap confines both of them well into the guide core. These conditions are very favorable for laser action of each DFB component [8]. This situation is different from that previously reported for W3 PhC waveguides oriented in the ΓK direction, where the laser action took place above the gap [3]. It also contrasts with recent results obtained on W1 PhC waveguides oriented in the ΓX direction of a square lattice PhC, where the laser emission occurred in the absence of a gap [9]. In all these previously reported cases, the laser emission coincided with the second fold of the fundamental mode. As for second-order DFB lasers, mode selection was provided by first-order radiation losses [10–13] since there was no gap to stop in-plane radiation. Only one DFB component was confined in the guide while the lossy one spread over the whole crystal [6, 7].
If we omit parasitic reflections at the waveguide ends, mode selection in W3-4 waveguide lasers can only be due to a different level of out-of-plane losses experienced by the two DFB components. A 2D phenomenological approach of this mechanism can be made, where losses are modeled by adding an imaginary dielectric constant ε″ to the air holes [14]. Because there is an arbitrary part in the choice of ε″, this method only allows comparing the relative losses of the two DFB modes. The respective mode lifetimes and frequencies are obtained from first order perturbation calculations by assuming a small value of ε″ (here ε″=0.05 and hence ε holes=1+i0.05):
where εp is the perturbed dielectric pattern, H is the magnetic field previously computed from the plane wave expansion method (Fig. 1 right), and ω is a complex eigenfrequency whose imaginary part is the inverse lifetime of the mode. From such calculations, it appears that the lifetime of the low frequency mode is 75 % longer than the lifetime of the high frequency mode. This actually indicates that the respective fields of the two DFB modes do not overlap the PhC holes in the same proportion.
Fig. 1. Left: band diagram of the W3-4 waveguide in TE polarization calculated from a 2D plane wave model with the supercell method. The thick curve is for the fundamental mode. Right: H field patterns of the modes at normalized wave vector k=0.5 and normalized frequencies a/λ=0.2725 and 0.273, respectively.
However, three-dimensional calculations are obviously needed to precisely quantify the mode lifetimes. The supercell presently used for three-dimensional FDTD calculations is depicted in the inset of Fig. 2. The vertically guiding hetero-structure consists of a 188 nm thick layer of refractive index n = 3.4 embedded in an InP cladding of refractive index n = 3.17. The 1 μm thick upper InP cladding is covered by a 360 nm thick air layer. The PhC hole depth is 4.5 μm and the PhC period is a = 432 nm. The FDTD model is basically the same as the one described in [15], though it is adapted to our geometry: an infinitely long periodic waveguide is simulated by applying Bloch-type boundary conditions in the ΓM direction. Perfectly matched layer (PML) conditions are applied in the other directions. The FDTD space step (dx) is equal to a√3/48 = 15.6 nm, and the time step (dt) is equal to 2.6 × 10-5 ps. For the sake of simplicity, neither gain nor absorption are included in the model. Instead of the Padé approximant technique used in [15], the Prony’s method [16] is used to accurately extract the frequency and lifetime of the guided modes from the time domain simulations. This method consists in writing the time-dependant field as a sum of damped complex exponentials An exp(-t/2τn + iωnt). The frequency ωn/2π , the lifetime τn and the amplitude An of each mode can then be directly found by solving an appropriate set of linear equations.
Figure 2 (left) represents the dispersion diagram of the odd symmetry modes with respect to the XZ plane. In the frequency range of interest, modes with an even symmetry exhibit a shorter lifetime τ (approximately one order of magnitude shorter than odd modes at the Mpoint in the frequency range of interest), and thus are not represented. Figure 2 (right) shows the quality factor Q = ωτ of the odd modes as a function of the normalized frequency. As seen, the mode frequencies calculated from 3D FDTD simulations are in good agreement with those calculated from the 2D plane wave model in TE polarization. This indicates that only TE-like modes propagate in this range of frequencies.
Figure. 2 (right) reveals that near the frequency of the third fold (a/λ = 0.271), the two DFB components exhibit very different evolutions in terms of quality factor [17]. The quality factor of the mode at the lower frequency increases suddenly when approaching the Brillouin zone edge while the one at the higher frequency decreases. As the net result, the lifetime of the low frequency mode is ~ 5 times longer than that of the high frequency mode at the Brillouin zone edge (50 ps versus 9 ps). This confirms the trend previously observed in 2D simulations where a phenomenological approach was used. Hence, out-of-plane diffraction losses clearly discriminate between the two modes. Only the low frequency mode of a W3-4 waveguide should lase as long as facet effects are neglected, i.e., the waveguide is of sufficient length.
Fig. 2. Left: dispersion curves of the odd modes with the longest lifetime τ represented near the Brillouin zone edge around the normalized frequency a/λ=0.27. Calculations are performed with the 3D FDTD model. Right: Quality factor Q=τω of the modes versus the normalized frequency. Inset: Schematic view of the simulated W3-4 waveguide with the active layer in dark gray.
It can be informative to estimate the W3-4 waveguide losses from 3D calculations. This is presently achieved in the frequency region just below the lasing frequency. For this, the normalized group velocity vng = vg/c of each mode is estimated by taking the derivative of the function Ũ (k), where U is a rational fraction that interpolates the computed normalized frequencies U = a/λ. Propagation losses α of the waveguide mode are then deduced from the following formula α = (2πU)/(Q vng a√3), where α is in cm-1 if α is expressed in cm [15, 17]. Assuming that only the fundamental mode propagates, a loss coefficient of ~ 13 dB/cm is calculated at the normalized frequency of 0.25 (i.e., at the normalized wave vector k =0.33 (Fig. 2)). This loss coefficient is close to that directly measured on W3 waveguides oriented in the ΓK direction of a triangular lattice [18]. Actually, for a PhC period of 432 nm, the width of the W3-4 waveguide varies between 1.5 and 1.9 μm, these values are between the widths of the W3 (1.25 μm) and W7 (2.75 μm) waveguides measured in [18].
3. Experimental results
The W3-4 PhC waveguide lasers were fabricated in an InP/InGaAsP/InP laser structure including six compressively strained InGaAsP quantum wells whose emission was centered at 1550 nm. The triangular lattice of holes was defined by electron beam lithography. The lattice period was chosen to be 432 nm and the air filling factor 30% as in the simulations. The holes were etched around 4 μm deep through the entire semiconductor heterostructure [19]. The W3-4 waveguides were 170 μm long (~ 400 PhC periods). They were terminated at one end by a cleaved facet, and at the other end by a PhC mirror with a lattice period of 320 nm chosen to optimize the mirror reflectivity. Finally, metallic contacts were deposited on the whole sample surface. Care was taken to avoid a deep in-filling of air holes with metal. Another set of samples exempt from metallic contacts, but with similar photonic crystal patterns, was also prepared for optical pumping experiments.
The W3-4 waveguide lasers were operated in pulsed regime. The electrical pulses were sufficiently long to reach a quasi-steady state regime where the laser relaxation oscillations were damped. The mode selection mechanisms were thus studied in this regime. Figure 3 shows the output power of the electrically pumped W3-4 laser measured as a function of time. As seen, only three relaxation oscillations occur before the laser output starts to reproduce the electrical pulse shape. The frequency of the relaxation oscillations is about 5 GHz, which should roughly correspond to the laser bandwidth itself.
Fig. 3: Measured output power of the W3-4 laser as a function of time. Inset: measured electrical pulse injected into the laser.
Figure 4 shows the emission spectra of the electrically pumped W3-4 laser measured at room temperature under pulsed electrical injection (pulse duration ~ 20 ns). The spectrum consists of a high-intensity laser mode (single peak at λ =1576 nm, i.e. a/λ = 0.274) accompanied with multiple low-intensity Fabry-Perot modes between 1530 and 1570 nm (i.e., 0.275 ≤ a/λ ≤ 0.283). Two dips appear in the spectrum at 1520 and 1575 nm, respectively. The latter corresponds to the mini-stop band revealed by simulations (Figs. 1 and 2) at the third fold of the fundamental mode. As a major result, it is shown that the lasing mode is the DFB component on the low-frequency side of the mini-stop band, thereby vindicating theoretical predictions from 3D FDTD simulations (Fig. 2, right). It is also worthwhile noticing the large side mode suppression ratio (≥ 40 dB) of the single-mode PhC laser.
The single mode behavior of W3-4 waveguide lasers has also been confirmed in optical pumping experiments. This is illustrated in the inset of Fig. 4, which shows the lasing spectrum of a 300 μm long W3-4 waveguide with a PhC period of 424 nm. The waveguide is pumped by a pulsed YAG laser emitting at 1.06 μm. The pulse duration was fixed to 15 ns and the repetition rate was 10 kHz. Despite the use of somewhat different parameters, it is seen that the optically pumped laser emits on a single frequency near the normalized value of 0.274. Indeed, W3-4 waveguide lasers are intrinsically single mode. Although no dip appears aside the laser line, another dip is clearly resolved around the wavelength of 1500 nm (normalized frequency of 0.283). This dip actually corresponds to the one observed at 1520 nm for electrically pumped lasers (Fig. 4). It is also predicted in numerical simulations: as seen in Fig. 2 right, the quality factor is abruptly decreased by a factor of 5 near a/λ=0.28. The reason is that the fundamental mode couples to a lossy higher-order mode around this frequency (see also Fig. 1 around the coordinates k=0.44 and a/λ=0.284). Hence, waveguide losses are strongly increased at this point. Note that a weak laser emission is observed near a/λ=0.281 under optical pumping while it is not detected under electrical excitation. This weak laser line could be attributed to one of the high-order slow modes that are predicted in the frequency range of interest above the gap (Fig. 1). Such modes extend deeper into the photonic crystal walls than the fundamental mode. Their excitation under optical pumping is likely due to the pump-laser spot size (~10 μm) used in our experiments.
Fig. 4. Emission spectrum of the W3-4 waveguide laser with a lattice period of 432 nm under electrical injection (Pulse duration ~20 ns, repetition rate 20 kHz, peak voltage 8 V). Inset: emission spectrum of the W3-4 waveguide laser with a lattice period of 424 nm under optical pumping (Pulse duration 15 ns, repetition rate 10 kHz)
4. Evolution of the DFB mode quality factor within gap
It is important to verify that the single-mode behavior reported above for W3-4 waveguide lasers is not fortuitous. For this purpose, additional calculations were performed for different values of photonic crystal parameters. The PhC hole diameters were varied in such a way that the third fold of the fundamental mode occurred at different positions with respect to the photonic gap edges.
Figure 5 (left) represents the evolution of the two DFB mode frequencies calculated as a function of the hole radius, r, for k=0.5. Despite the fact that the air filling factor varies from ~ 23 % (r/a=0.25) to ~ 45% (r/a=0.35), the relative variations of the DFB mode frequencies are less than 0.5 %. This contrasts with the strong broadening (× 3) of the photonic gap represented by the white area in Fig. 5 (left). As a consequence, the two DFB components lie above the gap for hole radii smaller than ~0.275 while they get close to mid-gap for hole radii approaching ~0.32.
Figure 5 (right) shows the evolutions of the quality factors calculated versus hole radius for the two DFB modes at the Brillouin zone edge. Calculations were performed with the 3D FDTD model using the same parameters as those of Section II. As seen in Fig. 5, the quality factor of the high frequency mode remains approximately constant whereas the quality factor of the low frequency mode significantly increases with the hole radius. A Q-maximum of 90.000 is reached when the DFB mode frequency is close to mid-gap (r/a~0.32). Hence, the discrimination between the two modes can either be enhanced or suppressed by properly choosing the PhC hole radii. Single mode laser operation is favored for sufficiently large PhC hole radii (like in our experiments), while a standard DFB laser behavior with two lasing modes is expected for small PhC radii around 0.275, i.e. for a laser emission approaching the upper gap edge. The exact origin of the steep rise observed for the Q-factor evolution reported in Fig. 5 is still unclear, but photonic bandgap effects associated to a high index contrast between air holes and the semiconductor are likely to contribute to this result. Let us notice that a similar evolution has been calculated at the Γ-point for a W1 waveguide created in a PhC slab [17].
Fig. 5. Left : Evolutions of the TE gap (white area) and of the normalized DFB laser frequencies at k=0.5 (solid lines) versus the hole radius in normalized units r/a. Right: Evolutions of the Q-factors of the high and low frequency modes at k=0.5 with the hole radius (3D FDTD simulations).
It is also interesting to compare our theoretical predictions to recent results reported on W2-3 waveguide lasers [3]. The latter are actually narrower homologues of the W3-4 ones, their width being determined by alternating two and three missing holes. Taking the same vertical layer structure as for the W3-4 waveguides (Section II) and a PhC hole radius of 0.287 (i.e., a 30% air filling factor), the third fold of the fundamental mode of W2-3 waveguides occurs in the gap near the upper photonic gap edge. The corresponding DFB mode frequencies are a/λ=0.275 and a/λ=0.277, respectively. In this case, 3D FDTD simulations indicate that the quality factor of the low frequency mode is only slightly higher than that of the high frequency mode. As has been observed in the experiments [3], the mode discrimination is less efficient, and lasing can simultaneously occur on the two modes. In contrast, calculations show that an increase of the PhC hole radius, i.e. a fundamental mode fold closer to mid-gap, again tends to enhance the Q-factor of the low frequency DFB mode. This trend thus appears to be general for canonical waveguides in triangular PhCs.
5. Conclusion
We have presented a detailed, theoretical and experimental study of W3-4 waveguide lasers in triangular photonic crystals fabricated on InP substrate. Experiments were carried out separately under electrical current injection and in the optical pumping regime. In both cases, single-mode emission was obtained in the gap with a side mode suppression ratio of ~ 40 dB. Experimental results were found in good agreement with theoretical predictions, which have revealed a strong discrimination between DFB modes associated with out-of-plane diffraction losses. The quality factor of the low frequency DFB mode was shown to increase when the photonic crystal parameters were chosen to operate deeper in the gap. Correspondingly, the discrimination between the two DFB modes was enhanced at the same time. This transition from weak- to strong- mode discrimination has been verified in other canonical waveguide lasers operating in-gap such as W2-3 waveguide lasers. Perhaps the major result of this study is the fact that the quality factor and loss of PhC waveguide modes above the light line can be controlled in the substrate approach by varying the photonic crystal parameters and thus the mode frequency within the gap.
Acknowledgments
This work has been supported by the French RNRT CRISTEL project. The authors would like to thank Juliette Mangeney and Nicolas Zerounian from IEF for many fruitful discussions.
References and Links
1. A. Sugitatsu and S. Noda, “Room temperature operation of 2D photonic crystal slab defect-waveguide laser with optical pump,” Electron. Lett. 39, 213-215 (2003). [CrossRef]
2. K. Inoue, H. Sasaki, K. Ishida, Y. Sugimoto, N. Ikeda, Y. Tanaka, S. Ohkouchi, Y Nakamura, and K. Asakawa, “InAs quantum-dot laser utilizing GaAs photonic-crystal line-defect waveguide,” Opt. Express , 12, 5502-5509 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5502. [CrossRef] [PubMed]
3. X. Checoury, P. Boucaud, J-M. Lourtioz, F. Pommereau, C. Cuisin, E. Derouin, O. Drisse, L. Legouezigou, F. Lelarge, F. Poingt, G.H. Duan, D. Mulin, S. Bonnefont, O. Gauthier-Lafaye, J. Valentin, F. Lozes, and A. Talneau, “Distributed feedback regime of photonic crystal waveguide lasers at 1.5 μm,” Appl. Phys. Lett. , 85, 5502-5504 (2004) [CrossRef]
4. T. D. Happ, M. Kamp, A. Forchel, J. L. Gentner, and L. Goldstein, “Two-dimensional photonic crystal coupled-defect laser diode,” Appl. Phys. Lett. 82, 4-6 (2003). [CrossRef]
5. A. Talneau, L. LeGratiet, J. L. Gentner, A. Berrier, M. Mulot, S. Anand, and S. Olivier “High external efficiency in a monomode full-photonic-crystal laser under continuous wave electrical injection,” Appl. Phys. Lett. , 85, 1913-1915 (2004) [CrossRef]
6. X. Checoury, P. Boucaud, J-M. Lourtioz, F. Pommereau, C. Cuisin, E. Derouin, O. Drisse, L. Legouezigou, O.L. Legouezigou, F. Lelarge, F. Poingt, G.H. Duan, S. Bonnefont, D. Mulin, J. Valentin, O. Gauthier-Lafaye, F. Lozes-Dupuy, and A. Talneau, «Distributed Feedback-like Laser Emission in Photonic Crystal Waveguides on InP Substrate,” IEEE J. Sel. Top. Quantum. Electron., to appear in Sept. 2005.
7. O. Gauthier-Lafaye, D. Mulin, S. Bonnefont, X. Checoury, J-M. Lourtioz, A. Talneau, and F. Lozes-Dupuy, “Highly monomode W1 waveguide square lattice photonic crystal lasers,” IEEE Photonics Technol. Lett. 17, 1587-1589 (2005) [CrossRef]
8. K. Sakoda, Optical Properties of Photonic Crystals Springer-Verlag (Berlin Heidelberg2001)
9. X. Checoury, P. Boucaud, J-M. Lourtioz, O. Gauthier-Lafaye, S. Bonnefont, D. Mulin, J. Valentin, F. Lozes-Dupuy, F. Pommereau, C. Cuisin, E. derouin, O. Drisse, L. Legouezigou, F. Lelarrge, F. Poingt, G-H. Duan, and A. Talneau “1.5 μm room-temperature emission of square-lattice photonic-crystal waveguide lasers with a single defect line,” Appl. Phys. Lett. 86, 151111 (2005) [CrossRef]
10. C. H. Henry, R. F. Kazarinov, R. A. Logan, and R. Yen, “Observation of destructive interference in the radiation loss of second-order distributed feedback lasers,” IEEE J. Quantum Electron. 21, 151–153 (1985) [CrossRef]
11. R. F. Kazarinov and C. H. Henry, “Second-order distributed feedback lasers with mode selection provided by first-order radiation losses,” IEEE J. Quantum Electron. 21, 144–150 (1985) [CrossRef]
12. A. Shams-Zadeh-Amiri, J. Hong, X. Li, and W.-P. Huang, “Second-and higher order resonant gratings with gain or loss-Part I: Green’s Function Analysis,” IEEE J. Quantum Electron. 36, 1421–1430 (2000) [CrossRef]
13. A. Shams-Zadeh-Amiri, J. Hong, X. Li, and W.-P. Huang, “Second-and higher order resonant gratings with gain or loss-Part II: designing complex-coupled DFB lasers with second-order gratings,” IEEE J. Quantum Electron. 36, 1431–1437 (2000) [CrossRef]
14. H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, A. Béraud, D. Cassagne, and C. Jouanin, “Radiation losses of waveguide-based two-dimensional photonic crystals : positive role of the substrate,” Appl. Phys. Lett. 76, 532–534 (2000). [CrossRef]
15. Q. Chen, Y.-Z. Huang, W.-H. Guo, and Li-Juan YuCalculation of propagation loss in photonic crystal waveguides by FDTD technique and Padé approximation,” Opt. Commun. 248, 309–315 (2005). [CrossRef]
16. S. L. Marple Digital Spectral analysis with applications, (Prentice-Hall,New Jersey1987)
17. L. C. Andreani and M. Agio, «Intrinsic diffraction losses in photonic crystal waveguides with line defects» Appl. Phys. Lett. , 82, 2011–2013 (2003). [CrossRef]
18. J. Zimmermann, H. Scherer, M. Kamp, S. Deubert, J.P. Reithmaier, A. Forchel, R. März, and S. Anand,“Photonic crystal waveguides with propagation losses in the 1 dB/mm range,” J. Vac. Sci. Technol. B 22 (6), 3356–3358 (2004). [CrossRef]
19. F. Pommereau, L. Legouezigou, S. Hubert, S. Sainson, J. P. Chandouineau, S. Fabre, G. H. Duan, B. Lombardet, R. Ferrini, and R. Houdre, “Fabrication of low loss two-dimensional InP photonic crystals by inductively coupled plasma etching,” J. Appl. Phys. 95, 2242–2245 (2004). [CrossRef]
