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Low threshold laser cavities of two-dimensional photonic crystal line defect mode
Optik 122 (2011) 1042–1045
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Low threshold laser cavities of two-dimensional photonic crystal line defect mode Xin Wang Physics Experimental Center, Nanjing University of Science & Technology, Nanjing, Jiang Su 210094, PR China
a r t i c l e
i n f o
Article history: Received 27 November 2009 Accepted 12 June 2010
Keywords: Photonic crystal Line defect mode Laser cavity Low threshold
a b s t r a c t Two-dimensional (2D) rod-type photonic crystal (PC) line defect waveguide (LDW) laser cavities based on three types of line defect modes with zero group velocity are studied by using finite-difference timedomain (FDTD) method. These laser cavities have high quality (Q) factor, better localization of light, nonuniform gain distribution and small overlap between gain medium and light field. Therefore, they have the advantages over conventional and air-bridge PC cavities with uniform gain, such as low threshold, single mode lasing and effectively avoiding thermal effect. From their comparison, one can find the mode at middle Brillouin zones (BZ) is the best one to be used as lasing mode. Its dynamic lasing process and lasing features are demonstrated by the numerical experiment where the FDTD method coupling Maxwell’s equations with the rate equations of electronic population is used. © 2010 Elsevier GmbH. All rights reserved.
1. Introduction Photonic crystals have the excellent abilities to control the properties of light emission and propagation [1–4]. Recently, there has been considerable interest on researching and improving the capability of conventional lasers by utilizing PC effects. At the edge of band of active PCs, the zero group velocity results in a large enhancement of stimulated emission [2]. Based on this band edge effect, many types of 2D PC band-edge lasers have been realized experimentally in semiconductor or organic medium [5–11]. By utilizing the localization of PC, active PC cavities with point defects exhibit a low lasing threshold [12,13]. When a line defect is introduced into an active PC structure, the band-gap guided modes (defect modes) with zero group velocity can be also used to design laser cavities with low threshold. Such a line defect waveguide (LDW) laser has better both transverse localization of light and lasing output direction along the line defect [14–16]. With the current developed fabrication technology, many LDW lasers have been realized by using semiconductor air-bridge slab structure [14–16]. However, in these 2D PC laser cavities, the most field energy distributes inside the gain medium, so the resulting high temperature inevitably degrade the laser’s capabilities such as the broadening of atomic linewidth about gain medium and uncontinuous output [17]. Furthermore, since a row of air-holes is filled in PC to form a line defect, the gain material at the line defect is spatially uniform, which easily results in multimode lasing due to the spatial hole burning effect [16]. The 2D PC LDW cavity composed of round
E-mail address: [email protected]. 0030-4026/$ – see front matter © 2010 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2010.06.042
air-rods can overcome effectively the thermal fault and the spatial hole burning effect since the light field of defect mode is mainly localized in air [18]. In this paper, by using finite-difference time-domain (FDTD) method, such 2D PC LDW cavities based on three types of line defect modes with zero group velocity are investigated firstly. These defect modes lie at the left boundary, the middle and the right boundary of Brillouin zones (BZ), respectively. The numerical results show the characteristics of these laser cavities, such as high quality (Q) factor and small spatial overlap between the field distribution and the gain medium, especially for the defect mode in middle BZ. Next, the dynamic lasing process of the defect mode in middle BZ is simulated and the lasing properties are analyzed. The results indicate that such LDW cavities of air-rods are really superior at single mode lasing due to both the special distributions of gain and light field in cavity. 2. PC LDW’s structure and band diagram The schematic structure of the LDW is shown in the inset of Fig. 1, which consists of a triangle lattice of GaAs cylindrical rods in air, with lattice constant =0.84 m and radius r = 0.12. The dielectric constant of GaAs rods ε = 11.4, in which InGaAsP multiple quantum wells are embedded. The whole width of LDW is chosen to be 8 layers of rods which is enough to confine the light propagating into the transverse (y) direction. In this paper, we focus mainly on the linear defect mode with the electronic field paralleling to the rod axis. Through tuning the width of the line defect W, various band-gap diagrams with zero group velocity at the left boundary, the middle and the right boundary of BZ can be obtained. For
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Fig. 1. The first band-gap diagram along –K (x) direction. Solid curves correspond to the LDW with W = 1.06, the dashed lines correspond to the LDW with W = 1.4. The inset shows the schematic structure of 2D PC LDW laser cavity.
example, the dashed curves in Fig. 1 illustrate the dispersion relation of W = 1.4, on which the left and right edge defect modes are marked by and K respectively. When W narrows down to 1.06, defect mode (T) with zero group velocity occurs in the middle of BZ aside from the left (1) and right (K1) edge defect modes, whose band diagram is also plotted with solid curves in Fig. 1. Here, W is chosen to be 1.06 in order to make the frequency of mode T be far away from both the edge defect modes (1 and K1) and the upper bands. All of these three types of defect modes with zero group velocity can be used as lasing modes with a low threshold. However, they have different characteristics of lasing oscillation, especially of field distribution. To compare the field distribution among them obviously, the modes 1, K1, and T will be investigated in the following.
Fig. 2. (a) The resonant transmission spectra of LDW with W = 1.06 and L = 14. (b) The resonant transmission spectra of LDW with W = 1.06. The blue solid curve corresponds to L = 14, the red dashed corresponds to L = 18. (For interpretation of the references to colors in this figure legend, the reader is referred to the web version of this article.)
3. Characteristics of cavity The above calculation of band diagram is based on infinite periodical structure. The normalized frequencies and Bloch wave vectors (kx ) of modes 1, K1, and T are 0.5171, 0.5229, 0.5436 and 0, 0.2887, 0.5 respectively. However, the length of LDW, L, is finite in practice, and the structure periodicity at its two ends weakens or disappears [16]. So the “propagating-mode” at two ends changes, which contributes to a phase change. This effect is similar as that of the interfaces of Fabry–Perot resonators cavity although the LDWs have no interfaces at its two ends. Owing to this reason, for the modes 1, K1, and T, the resonant frequencies always deviate a little from above ones and shift gradually to them with the increase of L [19], which will be demonstrated in the following calculation and figures. Since L is always equal to integral multiple lattice period, L = n, the resonant conditions for the modes 1 and K1 are satisfied automatically, kx L = m, where m and n are integers. However, for mode T, L can only be chosen approximately as L ≈ m/kx . In the following, L is taken as 14, which is expected to be equal to the resonant length, 4x = 13.86, where x is the Bloch wavelength of mode T (x = /0.2887). By using the FDTD method [20] the resonant transmission spectra of the LDWs are investigated at first. A dipolar source with a narrow modulated Gauss pulse is placed inside the LDWs to excite the resonant modes, then the resonant transmission spectra can be obtained by Fourier transform of time-evolved field (E(t)) at one end point outside the LDWs, which are plotted in Fig. 2. In Fig. 2(a), the two curves represent the resonant transmission spectra of the LDW with W = 1.06 and L = 14 in different time interval. The blue line is transformed from E(t) in the whole computing time. The peaks correspond to the available resonant modes in the cavity and it is easy to find out the left edge defect mode 1 and the middle defect mode T, corresponding to the bilateral peaks, whose normalized frequencies shift from 0.5171 and 0.5436 to 0.5174 and
0.5431, respectively. The red dashed line is transformed after the electric field attenuates for some time-step from the beginning. One can find that only three peaks remain there, which indicates that the modes attenuate slowly in the cavity and they have larger Q factors. The two lateral peaks correspond to the left edge defect mode 1 and another middle defect mode of normalized frequency ω = 0.5422 (notated by T1) close to mode T. The middle peak corresponds to the right edge defect mode K1 since its normalized frequencies 0.5247 is the closet one to 0.5229 and it also has a larger Q factor. For mode T, it can be regarded as the mode with zero group velocity, but it attenuates rapidly in cavity. This contradiction can be explained that L = 14 cannot satisfy accurately the resonant condition, so it is not a good resonant mode. In addition, in Fig. 2(b), the resonant transmission spectra of the LDW with L = 14 and 18 are plotted with the blue solid line and the red thin dashed line, respectively. The mode T1 shifts toward higher frequency and overlaps with mode T, and the modes 1 and K1 shift toward lower frequency, which coincides with the fact that the longer L is the more close to the theoretical values these resonant frequencies are. It can also be seen that the shift of normalized frequencies and the spectral narrowing as L increases. It means their Q values become larger. Next, Q factors of several resonant modes including 1, K1, T, T1 and T2 are calculated by using a wide modulated Gauss pulse whose center frequencies are equal to the resonant ones in Fig. 2, respectively. They could be precisely estimated from the relation of the angular frequency and the decay slope of logarithm energy-time, whose values are approximatively equal to 2027, 2265, 393, 1333 and 330, respectively. By comparing the Q factors, one can find that the Q factors of modes 1, K1 and T1 are larger than 1000, which is in accord with their slow attenuation or low group velocity. On the other hand, the Q factor of mode T1 is larger than that of mode T2 by far, which also demonstrates the superiority in ultralow lasing
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In addition, Fig. 3(a)–(f) also display obviously other characteristics about electric field distribution. Firstly, it can be found that, for such cavities made of air-rods, the most energy of electric field is localized near the line defect and distributes into air, especially for the middle defect modes. Therefore, such field distribution can avoid effectively thermal damage to gain medium. Secondly, the most energy flows out from the two ends of cavities along the line defect direction, which coincides with the localization of electric field in fact.
4. Simulation and analysis of lasing process
Fig. 3. (a)–(f) Electric field distributions of modes T, T1, 1 and K1. (g) Transverse distributions of electric field. The curves I–III correspond to the modes 1, T and K1, respectively.
threshold of the middle defect mode T1 apart from the conventional lateral-edge defect modes. During the course of calculating Q factors, the distributions of electric field in cavity about resonant modes T, T1, 1 and K1 can be obtained, which are shown in Fig. 3(a)–(f) respectively. From the figures, one can count out the rough number of Bloch wavelength, n(x ), in the cavity. n(x ) of mode T is between 4 and 4.5 (see Fig. 3(a)). For modes T1 and K1, one frequency corresponds to two Bloch wave vectors and the two Bloch states occur alternatively in the cavity. Fig. 3(b) and (c) are of mode T1 with n(x ) = 3.5 and 4.5. Fig. 3(e) and (f) are of mode K1 with n(x ) = 7 and 2. Owing to kx of mode 1 is near zero, i.e., the corresponding x ∼ ∞, so the value of the whole electromagnetic field along LDW direction has the same sign (see Fig. 3(d)). Furthermore, Fig. 3(a)–(f) also display different distribution of electric fields, especially different extension to y direction. The main electric fields of middle modes T and T1 extend less deeply into the transverse of the PC than lateral-edge modes 1 and K1. In Fig. 3(g), the four curves, thin solid line, dot dashed line, bold solid line and dashed line, represent the normalized transverse distribution of Fig. 3(a), (b), (d), and (e) (corresponding to modes T, T1, 1 and K1) at their maximum field intensity respectively. These curves illuminate more obviously the better localization of electric field of mode T than those of the two other modes. Therefore, we choose middle defect mode T1 as lasing mode to simulate the lasing process in the latter calculation.
In the next section, the method according to Ref. [21] is used to simulate the lasing behavior of the middle defect mode laser, which couples the time-dependent Maxwell’s equations with the rate equations of electron population based on a semi-classical theory about four-level atomic energy system [22,23]. And the amplification of line shape is Lorentzian and homogeneously broadened. The main parameters used in calculation are the total electron 4 density, N0 = N = 3.311 × 1024 m−3 , the lifetimes of the four i=1 i energy levels, 31 = 10−3 s, 32 = 10−14 s, 21 = 10−12 s, 10 = 10−14 s, the center frequency of gain medium, ω0 = 1.21 × 1015 Hz, and the full width at half maximum linewidth of atomic transition of quantum well, ω = 1.53 × 1013 Hz (about 1.2% of the center frequency). Owing to the advantage in thermal effect, it is reasonable to suppose such a narrow gain linewidth. In addition, the pumping rate, Pr, is chosen to be 1 × 107 s−1 . The electric field of one end point outside cavity is continuously recorded within a long period (1.2 × 106 time-steps), and is plotted as the function of time (E(t)) in Fig. 4(a), which gives a clear image of lasing process. In the first time interval from 1 to 5.0 × 105 time-step, the coherent oscillation builds up exponentially from spontaneous emission noise and trends to steady-state, hence, both the spontaneous and stimulated emission photons live together inside the cavity and its Fourier spectrum contains correspondingly multifrequency components, as shown in Fig. 4(b). With the growth of the laser intensity, the spontaneous emission weakens gradually and the stimulated emission predominates relatively. And only a few resonant lasing frequencies with larger gain than threshold compete in cavity. It can be shown by the amplitude profile of E(t) from 1.5 × 105 to 7.5 × 105 time-step and its Fourier spectrum is plotted in Fig. 4(c) with solid line, in which there are two lasing normalized frequencies, 0.5422 and 0.5427. If the gain medium is spatially uniform in cavity, these two adjacent modes maybe coexist by using different groups of atoms in gain medium. However, gain is not uniform but periodical in rod-type cavity. This kind of distribution of gain is adverse to multimode lasing based on the effect of spatial hole burning, while it is beneficial to single mode operation. In the competition between mode T1 and the other, gain coefficient becomes saturated by and by, and the lasing process trends gradually to stability of single mode lasing, which starts approximatively from 7 × 105 time-step. From its Fourier spectrum, the dashed line in Fig. 4(c), the only one peak of mode T1 illustrates distinctly that it lases really at single mode in the succeeding time. In addition, it can be seen from Fig. 4(a) that the lasing output is very stable and flatten although mode T1 has two Bloch states and these two states with different distributions use different spatial gain in the cavity. This case is very different from the multiple frequencies lasing owing to the effect of spatial hole burning in the cavity with uniform gain. For the cavity with uniform gain, the gain is used by different modes with different frequency at the same time, i.e., the different lasing mode coexist in cavity, therefore, the lasing output is fluctuant. However for mode T1, the two Bloch states alternatively appear (it can be found from the electri-
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overlap between gain medium and light field, and single mode oscillation due to non-uniform gain medium. The comparison of field distributions also demonstrates the line defect mode at middle BZ is the best candidate for lasing mode in such rod-type PC LDW cavity since it has the best localization of light field and smallest overlap between field and gain. In addition, the lasing features of the line defect mode at middle BZ disclose the advantage of single frequency with two Bloch states to suppress the spatial hole burning. The numerical experiment in this paper is much helpful to design such rod types of laser cavities in practice. Acknowledgments This work has been supported by NUST Research Funding (no. 2010ZYTS066) and Nanjing University of Science & Technology Grant (no. AB41928). References
Fig. 4. (a) Amplitude of electric field in units of V/m vs tj , where tj = t/t0 and t0 = 4.378 × 10−17 s. (b) The normalized Fourier spectrum of lasing within time interval, j ∈ (1, 5 × 105 ). (c) The similar spectra as (b), with solid and dashed curves corresponding to j ∈ (1.5 × 105 , 7.5 × 105 ) and j ∈ (7.5 × 105 , 1.2 × 106 ), respectively.
cal field evolution during stable lasing) and these two lasing states contribute to the same frequency. Therefore, mode T1 with two Bloch states has the advantage to suppress the spatial hole burning effect to result in single frequency lasing. 5. Conclusion In summary, the 2D rod-type PC LDW cavities based on three types of line defect modes with low group velocity are proposed and their characteristics, including resonant spectra, Q factor, field distribution, and dynamic lasing process, have been investigated by means of FDTD simulation. The numerical results demonstrate the advantages of these three types of laser cavities: low threshold due to low group velocity, avoiding the thermal effect due to small
[1] E. Yablonovitch, inhibited spontaneous emission in solid-state physics and electronics, Phys. Rev. Lett. 58 (1987) 2059–2062. [2] J.D. Joannopouls, R.D. Meade, J.N. Winn, Photonic Crystal: Molding the Flow of Light, Princeton University Press, Princeton, 1995. [3] T.F. Krauss, R.M. De La Rue, Photonic crystals in the optical regime – past, present and future, Prog. Quant. Electron. 23 (1999) 51–96. [4] H.Y. Ryu, H.G. Park, Y.H. Lee, Two-dimensional photonic crystal semiconductor lasers: computational design, fabrication, and characterization, IEEE J. Sel. Top. Quant. Electron. 8 (2002) 891–908. [5] M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, G. Sasaki, Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure, Appl. Phys. Lett. 75 (1999) 316–318. [6] M. Meier, A. Mekis, A. Dodabalapur, A. Timko, R.E. Slusher, J.D. Joannopoulos, O. Nalamasu, Laser action from two-dimensional distributed feedback in photonic crystals, Appl. Phys. Lett. 74 (1999) 7–9. [7] A. Mekis, M. Meier, A. Dodabalapur, R.E. Slusher, J.D. Joannopoulos, Lasing mechanism in two-dimensional photonic crystal lasers, Appl. Phys. A: Mater. Sci. Process. A 69 (1999) 111–114. [8] S. Riechel, C. Kalinger, U. Lemmer, J. Feldmann, A. Gombert, V. Wittwer, U. Scherf, A nearly diffraction limited surface emitting conjugated polymer laser utilizing a two-dimensional photonic band structure, Appl. Phys. Lett. 77 (2000) 2310–2312. [9] M. Notomi, H. Suzuki, T. Tamamura, Directional lasing oscillation of twodimensional organic photonic crystal lasers at several photonic band gaps, Appl. Phys. Lett. 78 (2001) 1325–1327. [10] S. Noda, M. Yokoyama, M. Imada, A. Chutinan, M. Mochizuki, Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design, Science 293 (2001) 1123–1125. [11] H.Y. Ryu, S.H. Kwon, Y.J. Lee, Y.H. Lee, J.S. Kim, Very-low-threshold photonic band-edge lasers from free-standing triangular photonic crystal slabs, Appl. Phys. Lett. 80 (2002) 3476–3478. [12] O. Painter, R.K. Lee, A. Scherer, A. Yariv, J.D. OBrien, P.D. Dpkus, I. Kim, Two-dimensional photonic band-gap defect mode laser, Science 284 (1999) 1819–1821. [13] H.G. Park, J.K. Hwang, J. Huh, H.Y. Ryu, Y.H. Lee, J.S. Kim, Nondegenerate monopole-mode two-dimensional photonic band gap laser, Appl. Phys. Lett. 79 (2001) 3032–3034. [14] A. Sugitatsu, S. Noda, Room temperature operation of 2D photonic crystal slab defect-waveguide laser with optical pump, Electron. Lett. 39 (2003) 213– 215. [15] A. Sugitatsu, T. Asano, S. Noda, Characterization of line-defect-waveguide lasers in two-dimensional photonic-crystal slabs, Appl. Phys. Lett. 84 (2004) 5395–5397. [16] K. Kiyota, T. Kise, N. Yokouchi, T. Ide, T. Baba, Various low group velocity effects in photonic crystal line defect waveguides and their demonstration by laser oscillation, Appl. Phys. Lett. 88 (2006) 201904–201906. [17] J.K. Hwang, H.Y. Ryu, D.S. Song, I.Y. Han, H.K. Park, D.H. Jang, Y.H. Lee, Continuous room-temperature operation of optically pumped two-dimensional photonic crystal lasers at 1.6 m, IEEE Photonic. Technol. Lett. 12 (2000) 1295–1297. [18] S.-Y. Lin, E. Chow, V. Hietala, P.R. Villeneuve, J.D. Joannopouls, Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal, Science 282 (1998) 274–276. [19] H.Y. Ryu, M. Notomi, Finite-difference time-domain investigation of band-edge resonant modes in finite-size two-dimensional photonic crystal slab, Phys. Rev. B 68 (2003) 045209–045217. [20] A. Taflove, Computational Electrodynamics: The Finite-Difference TimeDomain Method, Artech House, London, 1995 (see Chapters 3, 6 and 7). [21] X. Jiang, C.M. Soukoulis, Time dependent theory for random lasers, Phys. Rev. Lett. 85 (2000) 70–73. [22] A. Maitland, M.H. Dunn, Lasers Physics, NorthHolland Publishing Company, Amsterdam, 1969 (see Chapters 3, 8 and 9). [23] A.E. Siegman, Lasers, Mill Valley, CA, 1986 (see Chapters 2, 3, 6 and 13).
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Low threshold laser cavities of two-dimensional photonic crystal line defect mode Xin Wang Physics Experimental Center, Nanjing University of Science & Technology, Nanjing, Jiang Su 210094, PR China
a r t i c l e
i n f o
Article history: Received 27 November 2009 Accepted 12 June 2010
Keywords: Photonic crystal Line defect mode Laser cavity Low threshold
a b s t r a c t Two-dimensional (2D) rod-type photonic crystal (PC) line defect waveguide (LDW) laser cavities based on three types of line defect modes with zero group velocity are studied by using finite-difference timedomain (FDTD) method. These laser cavities have high quality (Q) factor, better localization of light, nonuniform gain distribution and small overlap between gain medium and light field. Therefore, they have the advantages over conventional and air-bridge PC cavities with uniform gain, such as low threshold, single mode lasing and effectively avoiding thermal effect. From their comparison, one can find the mode at middle Brillouin zones (BZ) is the best one to be used as lasing mode. Its dynamic lasing process and lasing features are demonstrated by the numerical experiment where the FDTD method coupling Maxwell’s equations with the rate equations of electronic population is used. © 2010 Elsevier GmbH. All rights reserved.
1. Introduction Photonic crystals have the excellent abilities to control the properties of light emission and propagation [1–4]. Recently, there has been considerable interest on researching and improving the capability of conventional lasers by utilizing PC effects. At the edge of band of active PCs, the zero group velocity results in a large enhancement of stimulated emission [2]. Based on this band edge effect, many types of 2D PC band-edge lasers have been realized experimentally in semiconductor or organic medium [5–11]. By utilizing the localization of PC, active PC cavities with point defects exhibit a low lasing threshold [12,13]. When a line defect is introduced into an active PC structure, the band-gap guided modes (defect modes) with zero group velocity can be also used to design laser cavities with low threshold. Such a line defect waveguide (LDW) laser has better both transverse localization of light and lasing output direction along the line defect [14–16]. With the current developed fabrication technology, many LDW lasers have been realized by using semiconductor air-bridge slab structure [14–16]. However, in these 2D PC laser cavities, the most field energy distributes inside the gain medium, so the resulting high temperature inevitably degrade the laser’s capabilities such as the broadening of atomic linewidth about gain medium and uncontinuous output [17]. Furthermore, since a row of air-holes is filled in PC to form a line defect, the gain material at the line defect is spatially uniform, which easily results in multimode lasing due to the spatial hole burning effect [16]. The 2D PC LDW cavity composed of round
E-mail address: [email protected]. 0030-4026/$ – see front matter © 2010 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2010.06.042
air-rods can overcome effectively the thermal fault and the spatial hole burning effect since the light field of defect mode is mainly localized in air [18]. In this paper, by using finite-difference time-domain (FDTD) method, such 2D PC LDW cavities based on three types of line defect modes with zero group velocity are investigated firstly. These defect modes lie at the left boundary, the middle and the right boundary of Brillouin zones (BZ), respectively. The numerical results show the characteristics of these laser cavities, such as high quality (Q) factor and small spatial overlap between the field distribution and the gain medium, especially for the defect mode in middle BZ. Next, the dynamic lasing process of the defect mode in middle BZ is simulated and the lasing properties are analyzed. The results indicate that such LDW cavities of air-rods are really superior at single mode lasing due to both the special distributions of gain and light field in cavity. 2. PC LDW’s structure and band diagram The schematic structure of the LDW is shown in the inset of Fig. 1, which consists of a triangle lattice of GaAs cylindrical rods in air, with lattice constant =0.84 m and radius r = 0.12. The dielectric constant of GaAs rods ε = 11.4, in which InGaAsP multiple quantum wells are embedded. The whole width of LDW is chosen to be 8 layers of rods which is enough to confine the light propagating into the transverse (y) direction. In this paper, we focus mainly on the linear defect mode with the electronic field paralleling to the rod axis. Through tuning the width of the line defect W, various band-gap diagrams with zero group velocity at the left boundary, the middle and the right boundary of BZ can be obtained. For
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Fig. 1. The first band-gap diagram along –K (x) direction. Solid curves correspond to the LDW with W = 1.06, the dashed lines correspond to the LDW with W = 1.4. The inset shows the schematic structure of 2D PC LDW laser cavity.
example, the dashed curves in Fig. 1 illustrate the dispersion relation of W = 1.4, on which the left and right edge defect modes are marked by and K respectively. When W narrows down to 1.06, defect mode (T) with zero group velocity occurs in the middle of BZ aside from the left (1) and right (K1) edge defect modes, whose band diagram is also plotted with solid curves in Fig. 1. Here, W is chosen to be 1.06 in order to make the frequency of mode T be far away from both the edge defect modes (1 and K1) and the upper bands. All of these three types of defect modes with zero group velocity can be used as lasing modes with a low threshold. However, they have different characteristics of lasing oscillation, especially of field distribution. To compare the field distribution among them obviously, the modes 1, K1, and T will be investigated in the following.
Fig. 2. (a) The resonant transmission spectra of LDW with W = 1.06 and L = 14. (b) The resonant transmission spectra of LDW with W = 1.06. The blue solid curve corresponds to L = 14, the red dashed corresponds to L = 18. (For interpretation of the references to colors in this figure legend, the reader is referred to the web version of this article.)
3. Characteristics of cavity The above calculation of band diagram is based on infinite periodical structure. The normalized frequencies and Bloch wave vectors (kx ) of modes 1, K1, and T are 0.5171, 0.5229, 0.5436 and 0, 0.2887, 0.5 respectively. However, the length of LDW, L, is finite in practice, and the structure periodicity at its two ends weakens or disappears [16]. So the “propagating-mode” at two ends changes, which contributes to a phase change. This effect is similar as that of the interfaces of Fabry–Perot resonators cavity although the LDWs have no interfaces at its two ends. Owing to this reason, for the modes 1, K1, and T, the resonant frequencies always deviate a little from above ones and shift gradually to them with the increase of L [19], which will be demonstrated in the following calculation and figures. Since L is always equal to integral multiple lattice period, L = n, the resonant conditions for the modes 1 and K1 are satisfied automatically, kx L = m, where m and n are integers. However, for mode T, L can only be chosen approximately as L ≈ m/kx . In the following, L is taken as 14, which is expected to be equal to the resonant length, 4x = 13.86, where x is the Bloch wavelength of mode T (x = /0.2887). By using the FDTD method [20] the resonant transmission spectra of the LDWs are investigated at first. A dipolar source with a narrow modulated Gauss pulse is placed inside the LDWs to excite the resonant modes, then the resonant transmission spectra can be obtained by Fourier transform of time-evolved field (E(t)) at one end point outside the LDWs, which are plotted in Fig. 2. In Fig. 2(a), the two curves represent the resonant transmission spectra of the LDW with W = 1.06 and L = 14 in different time interval. The blue line is transformed from E(t) in the whole computing time. The peaks correspond to the available resonant modes in the cavity and it is easy to find out the left edge defect mode 1 and the middle defect mode T, corresponding to the bilateral peaks, whose normalized frequencies shift from 0.5171 and 0.5436 to 0.5174 and
0.5431, respectively. The red dashed line is transformed after the electric field attenuates for some time-step from the beginning. One can find that only three peaks remain there, which indicates that the modes attenuate slowly in the cavity and they have larger Q factors. The two lateral peaks correspond to the left edge defect mode 1 and another middle defect mode of normalized frequency ω = 0.5422 (notated by T1) close to mode T. The middle peak corresponds to the right edge defect mode K1 since its normalized frequencies 0.5247 is the closet one to 0.5229 and it also has a larger Q factor. For mode T, it can be regarded as the mode with zero group velocity, but it attenuates rapidly in cavity. This contradiction can be explained that L = 14 cannot satisfy accurately the resonant condition, so it is not a good resonant mode. In addition, in Fig. 2(b), the resonant transmission spectra of the LDW with L = 14 and 18 are plotted with the blue solid line and the red thin dashed line, respectively. The mode T1 shifts toward higher frequency and overlaps with mode T, and the modes 1 and K1 shift toward lower frequency, which coincides with the fact that the longer L is the more close to the theoretical values these resonant frequencies are. It can also be seen that the shift of normalized frequencies and the spectral narrowing as L increases. It means their Q values become larger. Next, Q factors of several resonant modes including 1, K1, T, T1 and T2 are calculated by using a wide modulated Gauss pulse whose center frequencies are equal to the resonant ones in Fig. 2, respectively. They could be precisely estimated from the relation of the angular frequency and the decay slope of logarithm energy-time, whose values are approximatively equal to 2027, 2265, 393, 1333 and 330, respectively. By comparing the Q factors, one can find that the Q factors of modes 1, K1 and T1 are larger than 1000, which is in accord with their slow attenuation or low group velocity. On the other hand, the Q factor of mode T1 is larger than that of mode T2 by far, which also demonstrates the superiority in ultralow lasing
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In addition, Fig. 3(a)–(f) also display obviously other characteristics about electric field distribution. Firstly, it can be found that, for such cavities made of air-rods, the most energy of electric field is localized near the line defect and distributes into air, especially for the middle defect modes. Therefore, such field distribution can avoid effectively thermal damage to gain medium. Secondly, the most energy flows out from the two ends of cavities along the line defect direction, which coincides with the localization of electric field in fact.
4. Simulation and analysis of lasing process
Fig. 3. (a)–(f) Electric field distributions of modes T, T1, 1 and K1. (g) Transverse distributions of electric field. The curves I–III correspond to the modes 1, T and K1, respectively.
threshold of the middle defect mode T1 apart from the conventional lateral-edge defect modes. During the course of calculating Q factors, the distributions of electric field in cavity about resonant modes T, T1, 1 and K1 can be obtained, which are shown in Fig. 3(a)–(f) respectively. From the figures, one can count out the rough number of Bloch wavelength, n(x ), in the cavity. n(x ) of mode T is between 4 and 4.5 (see Fig. 3(a)). For modes T1 and K1, one frequency corresponds to two Bloch wave vectors and the two Bloch states occur alternatively in the cavity. Fig. 3(b) and (c) are of mode T1 with n(x ) = 3.5 and 4.5. Fig. 3(e) and (f) are of mode K1 with n(x ) = 7 and 2. Owing to kx of mode 1 is near zero, i.e., the corresponding x ∼ ∞, so the value of the whole electromagnetic field along LDW direction has the same sign (see Fig. 3(d)). Furthermore, Fig. 3(a)–(f) also display different distribution of electric fields, especially different extension to y direction. The main electric fields of middle modes T and T1 extend less deeply into the transverse of the PC than lateral-edge modes 1 and K1. In Fig. 3(g), the four curves, thin solid line, dot dashed line, bold solid line and dashed line, represent the normalized transverse distribution of Fig. 3(a), (b), (d), and (e) (corresponding to modes T, T1, 1 and K1) at their maximum field intensity respectively. These curves illuminate more obviously the better localization of electric field of mode T than those of the two other modes. Therefore, we choose middle defect mode T1 as lasing mode to simulate the lasing process in the latter calculation.
In the next section, the method according to Ref. [21] is used to simulate the lasing behavior of the middle defect mode laser, which couples the time-dependent Maxwell’s equations with the rate equations of electron population based on a semi-classical theory about four-level atomic energy system [22,23]. And the amplification of line shape is Lorentzian and homogeneously broadened. The main parameters used in calculation are the total electron 4 density, N0 = N = 3.311 × 1024 m−3 , the lifetimes of the four i=1 i energy levels, 31 = 10−3 s, 32 = 10−14 s, 21 = 10−12 s, 10 = 10−14 s, the center frequency of gain medium, ω0 = 1.21 × 1015 Hz, and the full width at half maximum linewidth of atomic transition of quantum well, ω = 1.53 × 1013 Hz (about 1.2% of the center frequency). Owing to the advantage in thermal effect, it is reasonable to suppose such a narrow gain linewidth. In addition, the pumping rate, Pr, is chosen to be 1 × 107 s−1 . The electric field of one end point outside cavity is continuously recorded within a long period (1.2 × 106 time-steps), and is plotted as the function of time (E(t)) in Fig. 4(a), which gives a clear image of lasing process. In the first time interval from 1 to 5.0 × 105 time-step, the coherent oscillation builds up exponentially from spontaneous emission noise and trends to steady-state, hence, both the spontaneous and stimulated emission photons live together inside the cavity and its Fourier spectrum contains correspondingly multifrequency components, as shown in Fig. 4(b). With the growth of the laser intensity, the spontaneous emission weakens gradually and the stimulated emission predominates relatively. And only a few resonant lasing frequencies with larger gain than threshold compete in cavity. It can be shown by the amplitude profile of E(t) from 1.5 × 105 to 7.5 × 105 time-step and its Fourier spectrum is plotted in Fig. 4(c) with solid line, in which there are two lasing normalized frequencies, 0.5422 and 0.5427. If the gain medium is spatially uniform in cavity, these two adjacent modes maybe coexist by using different groups of atoms in gain medium. However, gain is not uniform but periodical in rod-type cavity. This kind of distribution of gain is adverse to multimode lasing based on the effect of spatial hole burning, while it is beneficial to single mode operation. In the competition between mode T1 and the other, gain coefficient becomes saturated by and by, and the lasing process trends gradually to stability of single mode lasing, which starts approximatively from 7 × 105 time-step. From its Fourier spectrum, the dashed line in Fig. 4(c), the only one peak of mode T1 illustrates distinctly that it lases really at single mode in the succeeding time. In addition, it can be seen from Fig. 4(a) that the lasing output is very stable and flatten although mode T1 has two Bloch states and these two states with different distributions use different spatial gain in the cavity. This case is very different from the multiple frequencies lasing owing to the effect of spatial hole burning in the cavity with uniform gain. For the cavity with uniform gain, the gain is used by different modes with different frequency at the same time, i.e., the different lasing mode coexist in cavity, therefore, the lasing output is fluctuant. However for mode T1, the two Bloch states alternatively appear (it can be found from the electri-
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overlap between gain medium and light field, and single mode oscillation due to non-uniform gain medium. The comparison of field distributions also demonstrates the line defect mode at middle BZ is the best candidate for lasing mode in such rod-type PC LDW cavity since it has the best localization of light field and smallest overlap between field and gain. In addition, the lasing features of the line defect mode at middle BZ disclose the advantage of single frequency with two Bloch states to suppress the spatial hole burning. The numerical experiment in this paper is much helpful to design such rod types of laser cavities in practice. Acknowledgments This work has been supported by NUST Research Funding (no. 2010ZYTS066) and Nanjing University of Science & Technology Grant (no. AB41928). References
Fig. 4. (a) Amplitude of electric field in units of V/m vs tj , where tj = t/t0 and t0 = 4.378 × 10−17 s. (b) The normalized Fourier spectrum of lasing within time interval, j ∈ (1, 5 × 105 ). (c) The similar spectra as (b), with solid and dashed curves corresponding to j ∈ (1.5 × 105 , 7.5 × 105 ) and j ∈ (7.5 × 105 , 1.2 × 106 ), respectively.
cal field evolution during stable lasing) and these two lasing states contribute to the same frequency. Therefore, mode T1 with two Bloch states has the advantage to suppress the spatial hole burning effect to result in single frequency lasing. 5. Conclusion In summary, the 2D rod-type PC LDW cavities based on three types of line defect modes with low group velocity are proposed and their characteristics, including resonant spectra, Q factor, field distribution, and dynamic lasing process, have been investigated by means of FDTD simulation. The numerical results demonstrate the advantages of these three types of laser cavities: low threshold due to low group velocity, avoiding the thermal effect due to small
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