Andrew M. Weiner, Editor-in-Chief
Yong Liang, Chao Peng, Kenji Ishizaki, Seita Iwahashi, Kyosuke Sakai, Yoshinori Tanaka, Kyoko Kitamura, and Susumu Noda
Yong Liang,* Chao Peng, Kenji Ishizaki, Seita Iwahashi, Kyosuke Sakai, Yoshinori Tanaka, Kyoko Kitamura, and Susumu Noda
Department of Electronic Science and Engineering, Kyoto University, Kyoto-Daigaku-Katsura, Nishikyo-ku, Kyoto 615-8510,
*Corresponding author: firstname.lastname@example.org
Three-dimensional coupled-wave theory is extended to model triangular-lattice photonic-crystal surface-emitting lasers with transverse-electric polarization. A generalized coupled-wave equation is derived to describe the sixfold symmetry of the eigenmodes in a triangular lattice. The extended theory includes the effects of both surface radiation and in-plane losses in a finite-size laser structure. Modal properties of interest including the band structure, radiation constant, threshold gain, field intensity profile, and far-field pattern (FFP) are calculated. The calculated band structure and FFP, as well as the predicted lasing mode, agree well with experimental observations. The effect of air-hole size on mode selection is also studied and confirmed by experiment.
© 2013 Optical Society of America
Yong Liang, Chao Peng, Kyosuke Sakai, Seita Iwahashi, and Susumu Noda
Opt. Express 20(14) 15945-15961 (2012)
Chao Peng, Yong Liang, Kyosuke Sakai, Seita Iwahashi, and Susumu Noda
Opt. Express 19(24) 24672-24686 (2011)
Kyosuke Sakai, Jianglin Yue, and Susumu Noda
Opt. Express 16(9) 6033-6040 (2008)
Zhixin Wang, Yong Liang, Xuefan Yin, Chao Peng, Weiwei Hu, and Jérôme Faist
Opt. Express 25(10) 11997-12007 (2017)
Yi Yang, Chao Peng, and Zhengbin Li
Opt. Express 21(18) 20588-20600 (2013)
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(a) Schematic structure of a photonic-crystal surface-emitting laser device with a triangular lattice. (b) Band structure of triangular-lattice PCs calculated by the 2D plane-wave expansion method for transverse-electric (TE) mode . The red circle indicates the second-order Γ point. The inset shows the high-symmetry points at the corners of the irreducible Brillouin zone (shaded light blue).
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(a) Schematic of a triangular-lattice PC in real space. The blue arrows denote the primitive translation vectors a1 and a2, and a is the lattice constant. (b) Reciprocal lattice space of a triangular-lattice PC. The colored arrows indicate the six basic waves: R1, S1, R2, S2, R3, and S3 at the second-order Γ point, whose wavenumber is equal to
Calculated (a) band structure and (b) radiation constant of the eigenmodes in the vicinity of the second-order Γ point for a triangular-lattice PC-SEL with circular air holes. Six modes near the Γ point are referred to as modes: A, B1, B2, C, D1, and D2, in the order of increasing frequency. Modes B1 and B2, as well as D1 and D2, are doubly degenerate at the Γ point. (c) E-field vector distribution (arrows) and H-field patterns (in color) of the individual band-edge modes. The black circles indicate the air holes. Band-edge modes A and C are known as hexapole and monopole modes, respectively . In the calculations, we use the structural parameters listed in Table 1, the lattice constant a = 341 nm, and the air-hole filling factor f = 0.15.
(a) Normalized threshold gain (αL) as a function of normalized mode frequency deviation (δL). The fundamental band-edge modes (A, B1, B2, and C) and an additional mode W are indicated by arrows. (b) Field intensity envelopes of the modes indicated by arrows in (a). The data are calculated by using the same parameters as specified in the caption of Fig. 3. Note that the field intensity envelopes are plotted on hexagonal grids with a circular-shape computational domain (dashed circle). The radius of the circular domain, L = 30 μm, is discretized to span seven grid cells for which the eigenvalues converge well (see Appendix B for details).
(a) Comparison of the measured (in color) and calculated (white dashed curves) band structures. (b) Lasing spectrum measured above the lasing threshold in the direction normal to the PC plane. The threshold current (Ith) at room temperature CW operation was 25 mA. Band structure and lasing spectrum were measured at CW current levels of 0.9Ith and 1.2Ith, respectively. The frequency of the lasing peak in (b) is 0.3434 (a/λ), indicating that the lasing mode is band-edge mode C (yellow dashed line).
(a) Calculated and (b) measured FFPs of mode C. A scanning microscope image of the fabricated PC with a = 341 nm and f = 0.15 (where the r/a ratio is 0.20 and r is the air-hole radius) is shown in the left inset of (b). Ex (Ey) displayed in the right insets represents the x(y) component of the FFP. Parameters used for the calculations are the same as those shown in the caption of Fig. 4. The yellow arrows in (b) indicate the directions of the measured beam polarization. The beam divergence angle of the FFPs for both cases is around 1°, reflecting the large area of coherent oscillation.
(a) Calculated and (b) measured FFPs of mode A. A scanning microscope image of the fabricated PC with a = 341 nm and f = 0.26 (where r/a = 0.27) is shown in the left inset of (b). Ex (Ey) displayed in the right insets represents the x (y) component of the FFP. Parameters used for the calculations are specified in the caption of Table 3. The yellow arrows in (b) indicate the directions of the measured beam polarization.
(a) Location of field components (basic waves) and coupling coefficients in the vicinity of the (j,k)th hexagonal cell of the grid. Positions of the unknown field components (colored hollow dots) are staggered from the positions of the known coupling coefficients (solid dots). The colored hollow dots correspond to the points that are updated using the finite-difference scheme, while the solid points are points that are not solved. (b) Schematic of a circular computational domain (yellow shaded region) discretized on the hexagonal grids with L = 2h (L is the radius of the circular shape and h is the distance between two adjacent grid points). The green hollow dots inside the black squares indicate the boundary of R3; these are set to be zero in the calculations.
Table 1 Structural parameters of the PC-SEL device.
Table 2 Normalized threshold gain (αL) of the low-threshold modes indicated by arrows in Fig. 4(a).
Table 3 Normalized threshold gain (αL) of the low-threshold modes A–C. Parameters used for the calculations are the same as those shown in Fig. 4 except that a larger air-hole filling factor f = 0.26 is used.
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Structural parameters of the PC-SEL device.
Normalized threshold gain (αL) of the low-threshold modes indicated by arrows in Fig. 4(a).
Normalized threshold gain (αL) of the low-threshold modes A–C. Parameters used for the calculations are the same as those shown in Fig. 4 except that a larger air-hole filling factor f = 0.26 is used.