1. Introduction

Two-dimensional (2D) photonic-crystal (PC) lasers [1–17] have recently attracted considerable attention for their potential as high-quality and functional semiconductor lasers. This is because they utilize a 2D PC as a resonator to realize 2D broad-area single-mode oscillation by Bragg reflection and thus a very narrow (<1°) divergence angle [9–11, 16]. In addition, by appropriately designing the PC structure [8, 10, 12, 13, 15, 16], their polarization [3, 7, 10, 12], beam pattern [10, 12, 16], and even beam direction [14, 17] can be controlled. The resonant characteristics of 2D PC lasers are determined by the photonic band structure [18, 19], namely, the dispersion relation between the wavevector and the frequency of light inside the PC. Generally, lasing oscillation occurs at any one of the band edges at highly symmetric points; therefore, the band structure is normally evaluated in highly symmetrical directions.

Here, we report that a very weak dual-streak pattern (around the strong main peak) is occasionally (depending on the PC design) observed and is not along these special directions. This newly discovered phenomenon cannot be explained by the normal evaluation of the band structure. In this study, we evaluated the omnidirectional band structures both experimentally and theoretically and clarified that non-lasing bands may contribute to the generation of this unusual beam pattern. In Section 2, we explain the evaluation methods together with the device structure, procedures for beam-pattern and omnidirectional band-structure measurement, and the calculation method. In Section 3, we show the experimental results and discuss their significance. The observed unusual beam pattern, the experimentally measured omnidirectional band structure, and the calculated structure are discussed. Finally, concluding remarks are presented in Section 4.

2. Evaluation method

2.1 Device structure

As shown in Fig. 1(a) , the device structure comprised a 2D PC layer placed near the active layer between two cladding layers. The PC was patterned by electron-beam lithography using a resist, and the pattern was transferred by dry etching on the PC layer. The PC area was square shape and had an area of 400 × 400 μm. As seen in the scanning electron microscope image in Fig. 1(b), the pattern comprised a triangular lattice with a lattice constant of 336 nm and a filling factor, i.e., the area ratio of the trapezoid shape, of 20%. The bottom p-electrode had an area of 200 × 200 μm. Since the distance between the active layer and the p-electrode was small (approximately 2 μm), the injected current covered an area almost equal to that of the p-electrode. Light was generated over this area and output via a window in the top n-electrode.

 

Fig. 1 Device structure. (a) Layout of the 2D PC laser. (b) Scanning electron microscope image of the fabricated PC layer before burial. The Γ-J and the Γ-X directions are defined as indicated by the black arrows.

Download Full Size | PDF

2.2 Measurement of beam pattern

The beam pattern was observed by a beam profiler (A3267-15, Hamamatsu Photonics) through a neutral density (ND) filter in order to extract the unusual beam pattern around the very strong central peak.

2.3 Measurement of omnidirectional band structure

The band structure was evaluated by measuring the angular distribution of spontaneous emission spectra below the lasing threshold [18] using the system shown in Fig. 2 . In this setup, an optical fiber is mounted on a rotating arm with a collimation lens. Emission from the device in a specific direction is measured by a multi-channel spectrometer through the optical fiber. We set the spherical polar coordinates around the device as shown in Fig. 2. The polar angle θ and the azimuthal angle ϕ are transformed into the normalized wavevector (k_x, k_y) on the reciprocal lattice space as follows:

 

Fig. 2 Schematic diagram of measurement apparatus for omnidirectional band structure.

Download Full Size | PDF

The measurement results for the omnidirectional band structures were plotted on 2D planes by cutting the 3D arrays (k_x, k_y, f) vertically or horizontally. When a vertical cross section of the omnidirectional band structure is plotted for a highly symmetric direction, such as the Γ-X or Γ-J direction, it becomes a normal (or conventional) band structure. An important point here is that we simultaneously extract the horizontal cross sections (or equal frequency bands) in order to clarify the origin of the unusual lasing patterns.

2.4 Calculation of omnidirectional band structure

In the real space and reciprocal lattice space of the triangular photonic crystal (Figs. 3(a) and 3(b), respectively), a₁ and a₂ are the primitive translation vectors and b₁ and b₂ are the primitive reciprocal lattice vectors. There are six highly symmetric directions, i.e., Γ–X, Γ–X′, Γ–X″, Γ–J, Γ–J′, and Γ–J″ directions. It should be noted that because of the asymmetric nature of the trapezoidal holes, it is necessary to distinguish these six directions, although three Γ–X or Γ–J directions are equivalent in the case of the rotationally symmetric holes.

 

Fig. 3 (a) Real space of the triangular PC. (b) Reciprocal lattice space of the triangular PC.

Download Full Size | PDF

We use the 2D plain-wave expansion method to calculate the band structure. Since we used an active layer in which the transverse electric (TE) mode is likely to oscillate, the equation for the magnetic wave in the surface normal direction (H_z) is used as follows [20]:

3. Results and discussion

3.1 Beam pattern

Figures 4(a) and 4(b) show the lasing beam patterns through the ND filters implemented with optical densities (ODs) of eight and five, respectively. These measurements were made under continuous wave (CW) operation (100 mA) at room temperature. In Fig. 4(b), we can clearly observe an unusual dual-streak pattern around the central main peak. It should be noted that the intensity of the unusual pattern is 10³–10⁵ times lower than that of the central peak.

 

Fig. 4 Lasing beam patterns under CW operation (100 mA). (a) With ND filter (OD = 8); (b) with ND filter (OD = 5).

Download Full Size | PDF

3.2 Band structure

We carried out the evaluation of the band structure to clarify the origin of the unusual lasing beam patterns in two-step approach. First, the experimental and theoretical band structures are compared. Next, the relationship between the unusual lasing beam pattern and the band structure are discussed.

We measured the omnidirectional band structures, as shown in Fig. 5 , using the method described in Section 2.3. As we mentioned in section 2.4, due to the PC is rotational asymmetric, we measured the omnidirectional band structure for all horizontal directions. The lasing threshold was CW 95 mA. We measured the omnidirectional band structure under CW 93 mA (less than the lasing threshold) and the lasing spectrum under CW 100 mA (above the lasing threshold). There are six bands labeled A, B, C, D, E and F from bottom to top. The dashed box indicates the equi-frequency plane of the lasing frequency. From the vertical cross section and lasing spectrum shown in Figs. 6(a) and 6(b), respectively, we can see that lasing occurs at the edge of the lowest band (labeled Band A in Fig. 6(a)). Moreover, the horizontal cross section at the lasing frequency (0.3433 c/a; Fig. 7 ) shows a pattern similar to that observed in Fig. 4(b).

 

Fig. 5 Measured omnidirectional band structure under CW operation (93 mA).

Download Full Size | PDF

 

Fig. 6 (a) Vertical cross section of measured omnidirectional band structure. (b) Lasing spectrum.

Download Full Size | PDF

 

Fig. 7 Horizontal cross section of measured omnidirectional band structure at lasing frequency (0.3433 c/a) under CW operation (93 mA).

Download Full Size | PDF

Then, we calculated the omnidirectional band structure for six bands (bands A, B, C, D, E, and F) around the second-order Γ point, as shown in Fig. 8 . Band A, which is the main band related to lasing oscillation, has the shape of a six-sided pyramid, as shown in Fig. 8(a), and lasing oscillation occurs at the top. Bands B and C has six ridges that are convex upward (it has components along directions other than the specific directions Γ–X and Γ–J for the lasing frequency). In addition, we calculated the plots for bands A, B, and C at the frequency at the top of band A (0.3412 c/a), as shown in Fig. 9 . The difference from the measured value should come from estimation error of the refractive index. Figure 9 can be directly compared with Fig. 7 and shows good correspondence with it. Therefore, we found that Fig. 7 corresponds to the plots of bands A, B, and C at the lasing frequency, where the omnidirectional band structures of bands B and C are shown in Figs. 8(b) and 8(c), respectively.

 

Fig. 8 Calculated omnidirectional band structures. (a) Band A, (b) band B, (c) band C, (d) band D, (e) band E, (f) band F.

Download Full Size | PDF

 

Fig. 9 Horizontal cross section of calculated omnidirectional band structure at frequency corresponding to the edge of band A. Bands A, B, and C are plotted with blue, green, and red dots, respectively.

Download Full Size | PDF

These results indicate that the non-lasing bands (bands B and C) may diffract the laser beam to generate the dual-streak pattern. Here, we note that the wavenumber of band C is significantly distant from that of the Γ point of band A, and so, ideally, the laser light cannot be scattered by band C. Therefore, we suggest that some scattering mechanisms might be present to broaden the wavenumber. For example, if some imperfections are made in the PC structure during fabrication processes (for example, electron-beam lithography, dry etching, and so on), they works as scattering centers. The wavenumber of the lasing light scattered at the scattering centers will be broaden.

From these results, it is expected that noiseless high-quality 2D PC lasers are always obtained (so far, dual-streak noise pattern is occasionally observed) by designing the PC structure in which there is no other band structures near the lasing point on horizontal cross section of the lasing frequency (a detailed discussion and experimental results will be given elsewhere).

4. Summary

We investigated the effects of a non-lasing band in 2D PC lasers. An occasionally observed very weak dual-streak pattern around the strong main peak has the same frequency as the main peak and might be caused by the non-lasing band, even though the wavenumber is significantly different. This result indicates a new direction of a designing method of 2D PC lasers to avoid weak dual-streak pattern. Therefore, this paper will contribute to stable developing of high-power, high-quality surface emitting lasers. In addition, we believe that the method of omnidirectional measurement of the band structure will also be useful for developing various high-functionality PC lasers, including beam-steering lasers that contain dual periods [14, 17] as it provide not only along special directions but also comprehensive information about the band structure.

Appendix

Fourier expansion of trapezoid

As shown in Eq. (5), to calculate the band structure by the 2D plain-wave expansion method, the Fourier expansion of the reciprocal of the dielectric constant should be obtained. We derive the expression for a complicated trapezoid shape by dividing the shape into a rectangle and a right-angled triangle, as shown in Fig. 10 .

 

Fig. 10 Conceptual drawing of calculation procedure.

Download Full Size | PDF

The primitive translation vectors a₁ and a₂ in Fig. 3(a) are expressed as follows:

(6)$$a_1=\left(a,0\right),a_2=\left(a\cos 60°,a\sin 60°\right),$$

where a is the lattice constant. Then, the primitive reciprocal lattice vectors b₁ and b₂ in Fig. 3(b) are expressed as follows:

(7)$$b_1=\frac{2\pi }{a\sin 60°}\left(\sin 60°,-\cos 60°\right),b_2=\frac{2\pi }{a\sin 60°}(0,1).$$

Therefore, the reciprocal lattice vector G in Eqs. (4) and (5) is expressed as follows:

(8)$$G=m{b}_1+n{b}_2,$$

where m and n are integers. The Fourier coefficient κ(G) is given as follows:

(9)$$\kappa \left(G\right)=\frac{1}{a^2\sin 60°}{\displaystyle {\int }_{Shape}\frac{1}{\Delta \epsilon }\exp \left(iG\cdot r\right)}dxdy,$$

where a²sin60° corresponds to the area of a unit cell, and Δε is the difference in the dielectric constant between the inside and outside of the shape.

First, we calculate the Fourier coefficient κ₁(G) for the rectangular shape shown in Fig. 11(a) . To avoid dividing by zero, the results for four cases should be considered separately as follows:

 

Fig. 11 (a) Rectangle, (b) shifted rectangle, (c) right-angled triangle.

Download Full Size | PDF

(10)$$\begin{array}{l}{\kappa }_1\left(G\right)=\\ \frac{4hl}{a^2\sin 60°\Delta \epsilon }:m=n=0,\\ \frac{2l}{n\pi a\Delta \epsilon }\sin \left(\frac{2\pi nh}{a\sin 60°}\right):m=0,n\ne 0,\\ \frac{2h}{m\pi a\sin 60°\Delta \epsilon }\sin \left(\frac{2\pi ml}{a}\right):m\ne 0,m=2n,\\ \frac{1}{m{\pi }^2(n-m\cos 60°)\Delta \epsilon }\sin \left(\frac{2\pi ml}{a}\right)\sin \left(\frac{2\pi \left(n-m\cos 60°\right)h}{a\sin 60°}\right):m\ne 0,m\ne 2n.\end{array}$$

Then, the Fourier coefficient κ₂(G) for the shifted rectangle shown in Fig. 11(b) can be simply calculated as follows:

(11)$${\kappa }_2\left(G\right)={\kappa }_1\left(G\right)\exp \left(i\frac{2m\pi l}{a}\right).$$

Next, we calculate the Fourier coefficient κ₃(G) for the right-angled triangular shape shown in Fig. 11(c) as follows:

(12)$$\begin{array}{l}{\kappa }_3\left(G\right)=\\ \frac{2h^2}{a^2\sin 60°\tan \phi \Delta \epsilon }:m=n=0,\\ i\frac{1}{2n\pi a\tan \phi \Delta \epsilon }\left\{\frac{a\sin 60°}{n\pi }\sin \left(\frac{2n\pi h}{a\sin 60°}\right)+2h\exp \left(-i\frac{2n\pi h}{a\sin 60°}\right)\right\}:m=0,n\ne 0,\\ i\frac{1}{2m\pi a\sin 60°\Delta \epsilon }\left\{\frac{a\tan \phi }{m\pi }\exp \left(i\frac{2m\pi h}{a\tan \phi }\right)\sin \left(\frac{2m\pi h}{a\tan \phi }\right)+2h\right\}:m\ne 0,m=2n,\\ -i\frac{1}{2m\pi a\sin 60°\Delta \epsilon }\left\{2h\exp \left(i\frac{2m\pi h}{a\tan \phi }\right)-\frac{a\sin 60°}{\pi \left(n-m\cos 60°\right)}\sin \left(\frac{2\pi h(n-m\cos 60°)}{a\sin 60°}\right)\right\}:m\ne 0,m=n,\\ -i\frac{1}{2m{\pi }^2\Delta \epsilon }\left[\begin{array}{l}\frac{\exp \left(i\frac{2m\pi h}{a\tan \phi }\right)}{n-m\cos 60°-m\frac{\sin 60°}{\tan \phi }}\sin \left\{\frac{2\pi h}{a\sin 60°}\left(n-m\cos 60°-m\frac{\sin 60°}{\tan \phi }\right)\right\}\\ -\frac{1}{n-m\cos 60°}\sin \left\{\frac{2\pi h}{a\sin 60°}\left(n-m\cos 60°\right)\right\}\end{array}\right]:m\ne 0,n\ne 0,m\ne 2n.\end{array}$$

Finally, the Fourier coefficient κ₄(G) for the trapezoid is simply given by the sum of those for the shifted rectangle and the right-angled triangle as follows:

(13)$${\kappa }_4\left(G\right)={\kappa }_2\left(G\right)+{\kappa }_3\left(G\right).$$

Acknowledgments

The authors express their thanks to A. Higuchi, S. Furuta, K. Shibata, and K. Masuda for carrying out laser processing.

References and links

1. M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett. 75(3), 316–318 (1999). [CrossRef]  

2. M. Meier, A. Mekis, A. Dodabalapur, A. Timko, R. E. Slusher, J. D. Joannopoulos, and O. Nalamasu, “Laser action from two-dimensional distributed feedback in photonic crystals,” Appl. Phys. Lett. 74(1), 7–9 (1999). [CrossRef]  

3. S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science 293(5532), 1123–1125 (2001). [CrossRef]   [PubMed]  

4. M. Notomi, H. Suzuki, and T. Tamamura, “Directional lasing oscillation of two-dimensional organic photonic crystal lasers at several photonic band gaps,” Appl. Phys. Lett. 78(10), 1325–1327 (2001). [CrossRef]  

5. M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, “Multidirectionally distributed feedback photonic crystal lasers,” Phys. Rev. B 65(19), 195306 (2002). [CrossRef]  

6. H.-Y. Ryu, S.-H. Kwon, Y.-J. Lee, Y.-H. Lee, and J.-S. Kim, “Very-low-threshold photonic band-edge lasers from free-standing triangular photonic crystal slabs,” Appl. Phys. Lett. 80(19), 3476–3478 (2002). [CrossRef]  

7. M. Yokoyama and S. Noda, “Polarization mode control of two-dimensional photonic crystal laser having a square lattice structure,” IEEE J. Quantum Electron. 39(9), 1074–1080 (2003). [CrossRef]  

8. I. Vurgaftman and J. R. Meyer, “Design optimization for high-brightness surface-emitting photonic-crystal distributed-feedback lasers,” IEEE J. Quantum Electron. 39(6), 689–700 (2003). [CrossRef]  

9. D. Ohnishi, T. Okano, M. Imada, and S. Noda, “Room temperature continuous wave operation of a surface-emitting two-dimensional photonic crystal diode laser,” Opt. Express 12(8), 1562–1568 (2004). [CrossRef]   [PubMed]  

10. E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, “Photonics: lasers producing tailored beams,” Nature 441(7096), 946 (2006). [CrossRef]   [PubMed]  

11. H. Matsubara, S. Yoshimoto, H. Saito, Y. Jianglin, Y. Tanaka, and S. Noda, “GaN photonic-crystal surface-emitting laser at blue-violet wavelengths,” Science 319(5862), 445–447 (2008), doi:. [CrossRef]   [PubMed]  

12. E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, “Linearly polarized single-lobed beam in a surface-emitting photonic-crystal laser,” Appl. Phys. Express 1, 062002 (2008). [CrossRef]  

13. Y. Kurosaka, K. Sakai, E. Miyai, and S. Noda, “Controlling vertical optical confinement in two-dimensional surface-emitting photonic-crystal lasers by shape of air holes,” Opt. Express 16(22), 18485–18494 (2008). [CrossRef]   [PubMed]  

14. Y. Kurosaka, S. Iwahashi, Y. Liang, K. Sakai, E. Miyai, W. Kunishi, D. Ohnishi, and S. Noda, “On-chip beam-steering photonic-crystal lasers,” Nat. Photonics 4(7), 447–450 (2010). [CrossRef]  

15. S. Iwahashi, K. Sakai, Y. Kurosaka, and S. Noda, “Air-hole design in a vertical direction for high-power two-dimensional photonic-crystal surface-emitting lasers,” J. Opt. Soc. Am. B 27(6), 1204–1207 (2010). [CrossRef]  

16. S. Iwahashi, Y. Kurosaka, K. Sakai, K. Kitamura, N. Takayama, and S. Noda, “Higher-order vector beams produced by photonic-crystal lasers,” Opt. Express 19(13), 11963–11968 (2011). [CrossRef]   [PubMed]  

17. S. Iwahashi, T. Nobuoka, Y. Kurosaka, and S. Noda, “Comprehensive investigation of composite photonic-crystal cavities emitting arbitrary-angled laser beams,” in Proceedings of IPC2011 (IEEE, 2011), paper WP1.

18. K. Sakai, E. Miyai, T. Sakaguchi, D. Ohnishi, T. Okano, and S. Noda, “Lasing band-edge identification for a surface-emitting photonic crystal laser,” IEEE J. Sel. Areas Comm. 23(7), 1335–1340 (2005). [CrossRef]  

19. Y. Kurosaka, S. Iwahashi, K. Sakai, E. Miyai, W. Kunishi, D. Ohnishi, and S. Noda, “Band structure observation of 2D photonic crystal with various V-shaped air-hole arrangements,” IEICE Electron. Express 6(13), 966–971 (2009). [CrossRef]  

20. K. Sakoda, “Eigenmodes of photonic crystals,” in Optical Properties of Photonic Crystals, K. Sakoda, ed. (Springer-Verlag, Heidelberg, 2005).