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We use the finite-difference time domain method to calculate the vertical optical confinement, which corresponds to the quality factor in the vertical direction, of two-dimensional photonic-crystal (PC) lasers as a function of the asymmetry of the shape of the air holes that form the PC. The vertical optical confinement for triangular air holes, which give the highest output power measured thus far, is decreased by two thirds when V-shaped air holes are used. In contrast, the vertical optical confinement becomes infinite for rhomboid air holes. The vertical optical confinement decreases when the air holes are deformed such that areas of opposing electric fields exist in regions of the PC with different dielectric constants. In this way, the vertical optical confinement can be controlled by changing the shape of the air holes.
©2008 Optical Society of America
1. Introduction
Two-dimensional (2D) photonic-crystal (PC) lasers [1–8] have recently attracted much attention because they are semiconductor lasers that can be operated at high power. They uses the periodic 2D PC structure as a resonator, and a 2D standing wave is formed by Bragg reflection [9,10]. This enables the realization of single-mode oscillation over a broad area. Since the output is obtained in the vertical direction, perpendicular to the PC slab, high-power operation while maintaining perfect single-mode oscillation is made possible by increasing the oscillation area. The effectiveness of the 2D PC laser have already been demonstrated through the achievement of polarization [11–13] and beam pattern [14] control, blue-violet operation [15], continuous wave (CW) operation at room temperature [16] and high power operation [17] at 60 mW [18].
Adequate control of the laser output efficiency, which depends on the required output range and is determined by the degree of vertical optical confinement, is an important issue to be addressed. Kunishi et al. have recently found that the output efficiency can be improved by changing the shape of the air holes that form the PC from circular to triangular [17,18]. This suggests that the vertical optical confinement can be controlled by the shape of the air holes, but thus far no systematic study of this phenomenon has been carried out.
In this paper, the vertical optical confinement is evaluated as a function of the asymmetry of the shape of the air holes. The model used in the calculations and the method used to quantitatively evaluate the vertical optical confinement are described in Section 2. The in-plane electric field distribution for each band-edge mode on the square-lattice PC is also determined. Section 3 focuses on the relationship between the calculated vertical optical confinement and the shape of the systematically deformed air holes. Finally, concluding remarks are given in Section 4.
2. Evaluation model and method
In general, the 2D PC laser uses an active layer in which the transverse electric (TE) mode (in which the electric field distribution is parallel to the active layer) is likely to exhibit oscillation. In order to avoid unnecessarily complicated calculations involving coupling between the TE mode and the transverse magnetic (TM) mode [19] (in which the electric field distribution is perpendicular to the active layer), we use a symmetric structure in the vertical direction in which the overlap integral of these two modes becomes zero. We consider a PC with a square lattice.
2.1. Calculation model and vertical quality factor
We used the finite-difference time domain (FDTD) method to calculate the in-plane electric field distribution and the vertical optical confinement. Figure 1 shows the model employed in the FDTD calculations, with the parameters Δx=Δy=Δz=1/20a and Δt=0.5Δx/c, where a is the lattice constant and c is the speed of light in vacuum. For simplicity, a unit cell with Bloch boundary conditions in the horizontal direction was employed, for which there is no loss in the in-plane direction [Fig. 1(a)]. In this model the active layer is situated in the center of the structure when viewed perpendicular to the vertical direction. The active layer is sandwiched by a PC layer and a cladding layer placed symmetrically on each side. The boundaries at the top and bottom of the structure consist of a perfectly matched layer (PML) [Fig. 1(b)].
Fig. 1. Schematic picture of model used in FDTD calculations. (a) Top view. A Bloch boundary was employed around the unit cell in the horizontal direction. (b) Side view. Perfectly matched layers were employed for the vertical boundaries.
In order to evaluate the vertical optical confinement quantitatively we used the vertical quality factor (Q v), which is the vertical component of the quality factor (Q). The quantity Q is generally used to evaluate the confinement characteristics of light in a resonator. We made the assumption that there is no loss in the in-plane direction and no internal loss. In this case, Q is the same as Q v, which can be expressed as:
where ω is the frequency and W is the energy in the resonator. In this study, the value of Q v was calculated from oscillation decay rates.
The in-plane electric field distribution at the active layer depends on that of the PC layers. The electric field distribution becomes symmetric with respect to the active layer. We evaluate the in-plane electric field distributions at the center of the upper PC layer.
2.2. In-plane electric field distribution on square lattice
Figure 2 shows the photonic band structure of a square-lattice PC with circular air holes, calculated using the plane-wave expansion method. The point at which a band in the photonic band structure has zero gradient is known as the band edge, and here light emitted from the active layer produces a standing wave; the group velocity of light at the band edge is zero. Oscillation of the light occurs under these standing wave conditions, giving a gain that surpasses the loss. Light can be radiated to the outside in the vertical direction through modes above the air light line indicated by the dashed line in Fig. 2.
Fig. 2. Photonic band structure of a square-lattice photonic-crystal with circular air holes, calculated using the plane-wave expansion method.
We focused on the modes at the gamma point indicated by the arrow in Fig. 2, for which light is emitted only in the vertical direction. A more detailed view of the four bands (A, B, C and D) in the vicinity of the gamma point is shown in Fig. 3(a). The in-plane electric field distribution of these four bands at the gamma point is shown in Figs. 3(b)–(e).
Fig. 3. (a) Detailed view of band structure around the gamma point. (b)–(e) Calculated in-plane electric field distributions for band-edges A, B, C and D.
The symmetry of the in-plane electric field distribution determines the nature of the modes, which can be either leaky or non-leaky. The in-plane electric field distribution at band-edges A and B is anti-symmetric with respect to the x- and y-axes, which intersect at the center of the circular air hole. The phase of the light diffracted in the vertical direction is thus anti-symmetric with respect to the x- and y-axes, canceling completely due to destructive interference. The band-edge modes A and B are hence non-leaky modes, for which light does not propagate in the vertical direction. In contrast, the band-edge modes C and D are leaky modes, because the symmetric in-plane electric field distribution does not result in cancellation of the diffracted light. Here we perform our calculations on band edges A and B since these non-leaky modes are expected to oscillate. By introducing a degree of leakage into these non-leaky modes, the vertical optical confinement can be controlled.
The above discussion considers a PC with an infinitely periodic structure in the lateral direction (Bloch boundary conditions). However, these results are equally valid for PCs with finite periodicity. Destructive interference caused by the anti-symmetric electric field still results in the band-edge modes A and B being non-leaky because of strong vertical confinement. However, this vertical confinement can be greatly affected by introducing asymmetry between the air holes and the electric fields. In fact, introducing asymmetry by changing the shape of the air holes from circles to triangles increases the output efficiency [17,18].
2.3. Shape of air holes
The improvement of the light output efficiency depending on the shape of air holes [17,18] suggests that the degree of vertical optical confinement decreases; the triangular air holes lead to a more asymmetric in-plane electric field distribution and less destructive interference. It is thus expected that the vertical optical confinement can be controlled at will by varying the shape of the air holes.
In our investigation we used an equilateral triangle as our reference shape because it gives rise to the maximum output power measured thus far. For simplicity, we varied the asymmetry of the shape along only one axis.
The shapes investigated in our study are shown in Fig. 4(a). Shapes H1 to H5 are described by the parameters shown in Fig. 4(b); the ratio of h1 to h2 is indicated in the table in Fig. 4(a). Shapes H6 and H7 are described by the parameters in Fig. 4(c). The lattice constant in all cases was a. The dielectric constants of the shaded region (outside the air hole) and the white region (inside the air hole) were ε 1 and ε 2, respectively. We assumed that ε 1 is larger than ε 2 since 2D PC lasers usually adopt this condition. We fixed the dielectric constants in the FDTD calculation to ε 1=12.7449 and ε 2=1.0. The ratio of the area of the large equilateral triangle to the unit cell (S 4/a 2) was 13.5% for shape H6 and 16.5% for shape H7. The corresponding ratio for the smaller triangle (S 5/a 2) was 1.5% for shape H6 and 4.5% for shape H7. All of the shapes exhibit mirror symmetry along a line running parallel to the x-axis and through the center. Shape H1 also displays mirror symmetry along a line parallel to the y-axis and running through the center. The degree of asymmetry with respect to this line increases from shape H1 to H7. The ratio of the shape area to the unit cell area (filling factor) is 12% for all the shapes.
Fig. 4. (a) Air hole shapes investigated in this paper. (b) Detailed representation of air holes H1 to H5. (c) Detailed representation of air holes H6 and H7.
3. Calculation results and discussion
In this section, we present the values of Q v calculated for different air hole shapes and we examine the factors that influence Q v. As mentioned in section 2.2, the value of Q v is affected by the destructive interference of light diffracted in the vertical direction. Therefore, Q v is dependent on the in-plane electric field distribution of the diffracted light. We will thus investigate how the shape of the air holes affects the in-plane electric field distribution and how Q v can be controlled by varying the shape.
Figure 5 shows the values of Q v calculated using the FDTD method for the systematic deformation of shape H1 towards H7. The value of Q v at band edge A is infinite for shape H1. It monotonically decreases as the shape becomes more asymmetric towards H7. Therefore, Q v can be continuously controlled by varying the asymmetry of the shape. For example, Q v is approximately 6500 for shape H5, which holds the record for the maximum measured output power, and is reduced to one third of this value for shape H7. This suggests that the light output efficiency can be improved by utilizing V-shaped air holes. The value of Q v for band edge B is also infinite for shape H1. However, Q v for this mode does not monotonically change with shape asymmetry. It instead shows two local minima at shapes H3 and H5.
To explain why the trend of Q v is different for the two band edges, we now focus on the changes that occur in the asymmetry of the in-plane electric field distribution.
We will first examine the asymmetry of the in-plane electric field distribution for shape H5, using band edge A as an example; this is shown in Fig. 6. The equilateral triangle indicated by the solid line is the air hole. The electric field represented by the arrows is distributed around the node of the electric field distribution indicated by the black dot, which we use to define the coordinate origin. The regions outlined by the solid and dashed circles are placed at symmetrical positions with respect to the y-axis. It is clear that the magnitude of the electric field inside these regions is different (the electric field strength inside the region in the air hole is larger). The electric field distribution is thus asymmetric with respect to the y-axis. In contrast, the electric field distribution is always anti-symmetric with respect to the x-axis, reflecting the mirror symmetry of the air hole shape. Therefore, we will only examine the asymmetry of the in-plane electric field with respect to the y-axis.
Fig. 6. In-plane electric field distribution calculated using the FDTD method for air hole H5 and band edge A. Arrows represent the electric field and solid and dotted lines are explained in the main text.
Figure 7 shows the calculated in-plane electric field distributions for band edge A at the center of the upper PC layer. The corresponding plots for band edge B are shown in Fig. 8. The different air holes are represented by solid lines. The vertical dashed lines indicate the y-axis, which runs through the node of the in-plane electric field distribution.
Fig. 7. In-plane electric field distributions calculated using the FDTD method for air holes H1 to H7 and band-edge A. The vertical dashed lines indicate the y-axis, which runs through the node of the in-plane electric field distribution.
Fig. 8. In-plane electric field distributions calculated using the FDTD method for air holes H1 to H7 and band-edge B. The vertical dashed lines indicate the y-axis, which runs through the node of the in-plane electric field distribution.
In the case of band edge A, the node of the electric field distribution shifts progressively further away from the centroid of the shape as the asymmetry of the air hole increases. As a result, the area of the air hole to the right of the dotted line becomes progressively larger with increasing asymmetry, and the electric field distribution in this area becomes progressively less compensated by the area left of the dotted line with the opposite electric field orientation.
In contrast, the node of the electric field distribution for band edge B remains at the centroid of the air hole shape, despite increasing asymmetry. It is apparent that the electric field strength is greater in the right-hand half of air holes H2 and H3 and greater in the left-hand half of air holes H5 and H6, resulting in uncompensated electric field distributions. The electric field is more evenly distributed between the right- and left-hand sides of air holes H1, H4 and H7.
For band edge A, the electric field strength inside the air holes becomes larger with increasing asymmetry, in similar fashion to the situation shown in Fig. 6. When the electric field distribution in the air hole becomes less compensated by an area of opposite field orientation, the asymmetry of the in-plane electric field distribution increases and Q v decreases.
A change in the magnitude of the electric field perpendicular to the dielectric constant boundary is thought to be the reason for the increase in magnitude of the electric field inside the air hole. More specifically, the perpendicular component of the dielectric flux on the high dielectric medium side of the boundary is retained on the low dielectric medium side. Thus, the perpendicular component of the electric field at the border becomes larger in the lower dielectric constant medium (the air hole). A clearer example is given in the appendix.
We have thus found the following design rule to control Q v. Deformation of the air holes, such that the area of the air hole containing an uncompensated electric field distribution increases, leads to a decrease in Q v. The in-plane electric field strength becomes larger in the air hole due to its smaller dielectric constant.
4. Summary
We have calculated the vertical optical confinement as a function of the in-plane asymmetry of the air hole shape. The degree of vertical optical confinement can be controlled by varying the relative area of the air hole with an uncompensated electric field distribution, which increases with shape asymmetry. For example, the use of V-shaped air holes reduces the vertical optical confinement to one third of its equilateral triangle value; triangles have thus far given the highest measured output power. Furthermore, the vertical optical confinement becomes infinite when a rhomboid shape is used. In summary, the vertical optical confinement changes continuously for band edge A with the degree of asymmetry of the air hole shape. We expect that our results will enable arbitrary control of the output efficiency of 2D PC lasers.
Appendix
Role of boundaries in determining in-plane electric field distribution
In section 3, we discussed how the asymmetric nature of the electric field distribution can be explained in terms of a change in magnitude of the electric field perpendicular to the boundary between regions with different dielectric constants. We can confirm this more clearly by examining the shapes of the air holes shown in Fig. A1.
Fig. A1 Fig. A1(a) Air hole shapes discussed in the appendix. (b) Detailed representation of air holes P1 to P5. (c) Detailed representation of air hole P6.
The air holes H1 to H7 shown in section 2.3 all have a simple structure which consists of an asymmetrical region with a dielectric constant of ε 2 and a surrounding region with a dielectric constant of ε 1. In contrast, the air holes P1 to P6 shown in Fig. A1(a) consist of three components; an asymmetrical region with a dielectric constant of ε 1, a circular region with a dielectric constant of ε 2 and the surrounding region with a dielectric constant of ε 1. The white region in Figs. A1(b) and (c), in which the dielectric constant is ε 2, acts as the air hole. The area of the asymmetrical region does not change from H1 to H6. The size of the circular boundary is defined as that necessary to contain the asymmetrical region. More specifically, the ratio of the area of the circular region to the unit cell is 35% and the corresponding ratio for the asymmetrical region is 12%. Thus, the ratio of the area with a dielectric constant of ε 2 to the unit cell is 23%. The shape corresponding to H7 in Fig. 4 is omitted because the asymmetrical region exceeds the circular region. We refer to the regions with dielectric constants of ε 1 and ε 2 as being outside and inside the air hole, respectively.
The in-plane electric field distributions at the center of the upper PC layer for band edges A and B are shown in Figs. A2(a) and (b), respectively. The arrows indicate the electric field. The shapes outlined by the polygons and circles indicate the air holes.
Fig. A2 Fig. A2(a) In-plane electric field distributions calculated using the FDTD method for air holes P1 to P6 and band-edge A. (b) In-plane electric field distributions calculated using the FDTD method for air holes P1 to P6 and band-edge B.
For band edge A, the magnitude of the electric field inside the air hole is the same as that outside the air hole. In contrast, for band edge B the electric field strength inside the air hole is larger than that outside the air hole. Thus, the magnitude of the electric field changes discontinuously. This is because the electric field is oriented perpendicular to the circular boundary. Figure A3 shows the calculated values of Q v, which are smaller for band edge B for all of the shapes. This suggests that the asymmetry of the in-plane electric field distribution is larger for band edge B.
Acknowledgments
This work was partly supported by a Grant-in-Aid and Global Center of Excellence (G-COE) program of the Ministry of Education, Culture, Sports, Science and Technology of Japan. Y. Kurosaka had been supported by a Reserch Fellowship of the Japan Society for Promotion of Science.
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