Threshold Mode Structure of Square and Triangular Lattice Gain and Index Coupled Photonic Crystal Lasers

The main properties of photonic crystals stem from the existence of frequency ranges for which the propagation of electromagnetic waves in the medium is not permitted. These frequency ranges are commonly known as photonic band gaps and give the ability to modify the structure parameters, e.g., group velocity, coherence length, gain, and spontaneous emission. Photonic crystals’ properties are beneficial for both passive and active devices. This Chapter is devoted especially to the latter.


Introduction
Photonic crystals (PC) are periodic structures with variation of the refractive index in one, two or three spatial dimensions. The dynamic development of experimental and theoretical works on photonic crystals has been launched by Yablonovitch [1], [2] and John [3] publications, although the idea of periodic structures had been known since Rayleigh [4].
The main properties of photonic crystals stem from the existence of frequency ranges for which the propagation of electromagnetic waves in the medium is not permitted. These frequency ranges are commonly known as photonic band gaps and give the ability to modify the structure parameters, e.g., group velocity, coherence length, gain, and spontaneous emission. Photonic crystals' properties are beneficial for both passive and active devices. This Chapter is devoted especially to the latter.
Lasers with defects within two-dimensional photonic crystals are known for their high finesse [12] and very low threshold [13].
Photonic crystal band-edge lasers allow to obtain edge [11] and surface emission [14], [15] of coherent light from large cavity area. These devices are able to emit single mode, high-power electromagnetic radiation by utilizing the presence of band-edge in the photonic band structure [16], [17]. They also allow to control the output beam pattern by manipulation of the structure geometry [18], [19], provide low threshold [20], and beams which have small divergence angle and can be focused to a size less than the wavelength [21]. Recently, the operation of PC lasers as an on-chip dynamical control of the emitted beam direction have been demonstrated [19], [22].

Modeling of photonic crystal lasers
Laser action in photonic crystal structures has been theoretically studied and centered on the estimation of the output parameters e.g., [30], [31], and models describing light generation processes e.g., [32]- [34]. The most general semi-classical model of light generation in photonic structures is presented in [34], where the description of one-, two-, and three-dimensional structures is given.
Theoretical analysis of photonic crystal lasers based on two-dimensional plane wave expansion method (PWEM) [15], [35]- [37] and finite difference time domain method (FDTD) [35], [38]- [40] confirm experimental results. Nevertheless these methods suffer from important disadvantages, i.e., plane wave method gives a good approximation for infinite structures, whereas finite difference time domain method is suited for structures with only a few periods and consumes huge computer resources for the analysis of real photonic structures. Therefore these methods are not very convenient for design and optimization of actual photonic crystal lasers. Hence, different, less complicated methods of analysis of two-dimensional photonic crystal lasers are being developed. These methods are meant to effectively support the design process of such lasers. They are based on a coupled-wave theory (CWT) [15], [41] and focused on square and triangular lattice photonic crystals e.g., [32], [33], [42]- [48]. Most of the works e.g., [32], [42]- [46] contain a mathematical description and numerical results of the threshold analysis of two-dimensional (2-D) square and triangular lattice photonic crystal laser with TM and TE polarization. They introduce general coupled mode relations for a threshold gain, a Bragg frequency deviation and field distributions, and give calculation results for some specific values of coupling coefficients. Further, in [42] the effect of boundary reflections has been investigated, and it has been shown that the mode properties can be adjusted by changing refractive index or boundary conditions. In, [46], the achievements of these works were summarized and supplemented with the analysis for the wide range of coupling coefficient. These studies concerned structures which were infinite in the direction normal to the 2D PC plane. This approach was improved and presented in [47], where a three-dimensional (3D) couple wave model was shown. This theory addressed some key issues in a modeling of threshold operation of surface-emitting-type PC lasers, i.e. the surface emission and the inplane higher-order coupling effects. It has also been further developed to incorporate finitesize effects, and presented in [48]. Some other works such as for example [33], [44], [45], [49] present an above threshold analysis of 2D PC lasers. They illustrate gain saturation effect and describe the impact of structure parameters on the system efficiency.
In all of the cited works non have given much attention to simultaneous index and gain coupling. Thus in addition to the works already mentioned, this Chapter aims to remind crucial points of CMT and to show 2D coupled-wave analysis for structures with gain and index coupling. The study includes square and triangular lattice structures with TE and TM polarization of light.
The subsequent parts of this chapter include structure definition (Section 2), threshold analysis (Section 3), where 2D coupled-wave theory is reminded (Section 3.1), coupled-wave equations are shown (Section 3.2), and numerical analysis is performed (Section 3.3). The perspectives are sketched in Section 4, and finally conclusions are given in Section 5.

Structure definition
The Chapter describes two-dimensional photonic crystals which are characterized by the relative permittivity ε and gain α. Both parameters depend on the two-dimensional spatial structure of the medium. The cross-sections of discussed structures are schematically shown in Figure 1. (b) (a) From this point on, since photonic structures resemble the microscopic nature of crystals, a crystallography terminology will be used, see e.g., [49]. Throughout this Chapter only 2-D photonic crystals with a square, and hexagonal (also referred to as triangular) symmetry will be discussed, as it is depicted in Figure 1. The periodic pattern is created by cylinders called rods or holes. The structures in Figure 1 a) and b) are constrained in the xy plane by the square region of length L, and are assumed to be uniform and much larger than the wavelength in the z direction. The permittivity and gain of the rods and background material are represented by ε a , α a and ε b , α b, respectively. The number of periods in the xy plane is finite, but large enough to be expanded in Fourier series with small error. Schemes in Fig. 1 a) and 1 b) illustrate two spatial distributions of rods for two-dimensional photonic crystal, respectively, with square and triangular lattice.     show the reciprocal lattices corresponding, respectively, to the real square and triangular lattices. In the described case, the nodes of a two-dimensional structure can be expressed by (e.g., see [50]) where a 1 and a 2 are primitive vectors, l 1 and l 2 are arbitrary integers, x ∥ specifies the placement on the plane, x ∥ = xx + ŷ y, where x and ŷ are unit vectors along x and y axis, respectively.
In general, the reciprocal vectors can be written in the following form: where h 1 and h 2 are arbitrary integers, b 1 and b 2 are the primitive vectors of the two-dimensional reciprocal space, which are expressed by the following equations: is the j-th Cartesian component (x or y) of the a i vector (i = 1 or 2) (e.g., see [32]).
The areas of primitive cells are a c = |a 1 × a 2 | = a 2 and a c = |a 1 × a 2 | = 3a 2 / 2 in case of square and triangular lattices, respectively.
The spatial arrangement of periodic rods for the infinite two-dimensional square or triangular lattice can be expressed by the function: in terms of relative permittivity, and by in terms of gain. In Equations (6) and (7), function S specifies the location of rods in the structure, O is the area of the xy plane defined by the cross section of the rod, which symmetry axis intersects the plane at the point x ∥ = 0.
In the next section an analysis based on the coupled mode theory is shown. It is conducted in the frequency domain, thus the relative permittivity as well as gain have to be Fourier transformed to fit reciprocal space [37], [51]. Functions (6) and (7) are now, respectively, written in the following form and where f = πr 2 / a 2 -square lattice filling factor, f = ( 2π / 3 ) r 2 / a 2 -triangular lattice filling factor, rrod radius, J 1 -Bessel function of the first kind.
In next parts of this Chapter four different cases are analyzed. Two of them are dedicated to the square lattice cavities with TE and TM polarization, and two remaining to the triangular lattice structures also with TE and TM polarization. For the purpose of this work, it is assumed that there is no gain in the background material, i.e., α b = 0, but there is a gain in the rods α a ≠ 0. The structure where α a = α b will be referred to as Index Coupled, and where α a ≠ α b as Index and Gain Coupled.
The threshold analysis of the photonic crystal laser operation for the defined structures is shown in the next section.

2D Coupled-wave model for 2D PC cavity
The electromagnetic wave behavior in the two-dimensional periodic system can be described by the set of scalar wave equations. Depending on the polarization of light it is easier to choose one specific field component, since then the set of equations may be reduced to a single one.
Thus, the scalar wave equations for the electric and magnetic fields E z and H z , respectively, are written in the following forms [37], [51]: for TM polarization, and for TE polarization.
The periodic variation in the refractive index and gain is included as a small perturbation and appears in as the coupling constant κ ( G ) of the form: In the above equation, plus sign refers to TM polarization (Equation (12)), while minus sign refers to TE polarization (Equation (13)). For the simplicity, it is set that η ( G ) = -πε(G) / λε 0 1/2 and α ( G ) = α ( G ) / 2, and Equation (14) is rewritten in the following form: In the two-dimensional system which is not confined in the third direction, in the vicinity of the Bragg wavelength only some of the diffraction orders contribute in a significant way. In general, a periodic perturbation produces an infinite set of diffraction orders. Keeping this in mind, the Bragg frequency orders have to be cautiously chosen. The Bragg frequency corresponding to the Γ (e.g., see [42]) is chosen for the purpose of this work, and the most significantly contributing coupling constants are expressed as follows: In Equations (10) and (11) electric and magnetic fields for the infinite periodic structure are given by the Bloch modes, [15], [37]: and where the functions e ( G ) and h ( G ) correspond to plane wave amplitudes, and the wave vector is denoted by k. In the first Brillouin zone at the Γ point the wave vector vanishes k = 0, see e.g., [41].
In a finite two-dimensional structure, the amplitude of each plane wave is not constant, so e ( G ) and h ( G ) become functions of space. At the Γ point, only the amplitudes (e ( G ) , h ( G ) ) which are meant to be significant are considered, i.e., in most cases with | G | = β 0 s,t , except for square lattice with TE polarization where additional h ( G ) amplitudes with | G | = 2β 0 s have to be included [41]. The contributions of other waves of higher order in the Bloch mode are considered to be negligible. In general, where for example there is a three-dimensional confinement, this assumption have to be reconsidered.

Square lattice -TM polarization
For square lattice photonic crystal cavity in the case of TM polarization, it is assumed that at the center point of the Brillouin zone the most significant contribution to coupling is given by the electric waves which fulfill the condition | G | = β 0 s . Therefore in the following derivation all higher order electric wave expansion coefficients ( | G | ≥ 2β 0 s ) are neglected. Four basic waves most significantly contributing to coupling are depicted in Figure 4. Equation (17) in general describes infinite structures. It is possible to take into account the fact that the structure is finite by using the space dependent amplitudes, e.g., [42]. Thus, the electric field given by Equation (17) in the finite periodic structure can be expressed in the following way:

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In Equation (19) .4 are the four basic electric field amplitudes propagating in four directions +x, -x, +y, -y. These amplitudes correspond to e ( G ) (Equation (17)) satisfying the condition: | G | = β 0 s . In further analysis, the space dependence is omitted for the simplicity of notation.
Using derived earlier reciprocal lattice vectors and Fourier expansions of spatial dependences of the square lattice PC with circular rods, the coupling coefficients κ(G) (Equation (16)) can be written as: Combining Equations (12) and (19) with Equation (10), and assuming the slow varying electromagnetic field, one can get the set of coupled mode equations [42]: where is the Bragg frequency deviation. Coupling coefficients κ 2 and κ 3 are expressed by Equations (21) and (22). The κ 2 coefficient is responsible for orthogonal coupling (e.g., the coupling of E 1 s to E 2 s and E 4 s ), and κ 2 corresponds to backward coupling (e.g., the coupling of E 1 s to E 3 s ).
Solution of Equations (23)-(26) for the following boundary conditions: defines eigenmodes of the photonic structure. The analysis of this solution is given in Section 3.3.

Square lattice -TE polarization
In the 2D square lattice PC-like resonator with TE polarization the coupling process in the most significant way involves magnetic waves satisfying following conditions: | G | = β 0 and | G | = 2β 0 , [32]. In the presented analysis the higher order Bloch modes are neglected. Eight basic waves fulfilling the specified conditions are depicted in Figure 5. Similarly as it was stated in the case of TM polarization, the equation for magnetic field (18) describes modes for infinite structure, and the finite dimensions of the structure are introduced by spatial dependence of magnetic field amplitudes [32]. Thus, the magnetic field (18) is written in the following form:

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In Equation (29) In Equations (30)-(33), the spatial dependence of four magnetic field components H i s , i=2,4,6,8 was neglected, and it was assumed that α ≪ δ. These steps let to formulate not eight but four partial differential equations (for details see [32] or [46]). The Bragg frequency deviation δ is given by (27). The coupling coefficients κ 1 , κ 2 , and κ 3 defined by Equations (16) are expressed by [32], [43]: In contrast to TM polarization, in Equations (30)-(33), the coupling coefficient responsible for coupling in perpendicular direction κ 2 vanishes. The coupling coefficient κ 3 has the same meaning as described in the previous (TM) case, whereas the coupling coefficient κ 1 describes the coupling of e.g., waves H 1 s , H 2 s , and H 8 s . Solution of Equations (30)-(33) for the following boundary conditions: defines structure eigenmodes at lasing threshold i.e. in the linear case.

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In Equation (38), E i t , i=1..6, are the six electric field amplitudes propagating in the symmetry directions, Figure 6. Combining Equations (12), (38) with Equation (10), and assuming slowly varying amplitudes, the coupled wave equations for TM modes in triangular lattice PC are obtained: In Equations (39)- (44), like in the case of square lattice, δ is the Bragg frequency deviation (17), while κ 1 , κ 2 , and κ 3 are the coupling coefficients, which are defined by the following relations [44]: These coefficients describe strength and direction of the coupling of the waves, e.g., the coupling of E 1 1  2  2  3  3   4  5  5  6  6 , 0, , , defines structure eigenmodes at lasing threshold.

Triangular lattice -TE polarization
The simple approximation of coupling process in 2D triangular lattice PC with TE polarization includes waves satisfying the same condition as it was shown for TM polarization, i.e., | G | = β 0 , [43]. Six waves satisfying this condition are depicted in Figure 7.
In Equation (49), H i t , i=1..6, are the six magnetic field amplitudes propagating in the symmetry directions, Figure 7. Combining Equations (13), (49) and (11), and assuming slowly varying magnetic field amplitudes, the coupled wave equations for TE modes in triangular lattice PC are obtained: where the coupling coefficients κ 1 , κ 2 , and κ 3 are described by and have the same physical meaning like it was described in the TM polarization case. The boundary conditions for the square region of PC with triangular symmetry are written as:  Figure 8 shows enlarged areas of square lattice photonic crystal dispersion characteristics for the first four modes (A,B,C,D) in the vicinity of Γpoint (where the cavity finesse increases, and the active medium is used more efficiently). The dispersion curves are plotted for a) TM polarization and b) TE polarization. They have been obtained by using Plane Wave Method (PWM) [52], and they describe the infinite two-dimensional PC structures with circular holes ε b = 9.8 arranged in square lattice with background material permittivity: ε a = 12.0. The rods radius to lattice constant ratio amounts to 0.24.

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In each plot, i.e., Figure 8  In Figure 8 each of the marked points (A,B,C,D) represents a mode, which is characterized by: Bragg frequency deviationδ, threshold gainα, and threshold field distribution. These characteristic quantities were calculated by the numerical solution of Equations (23)-(26) for TM polarization and Equations (30)-(33) for TE polarization. The similar description of modes, shown in Figure 8, where no gain coupling is considered has already been presented in [46] or [49].
In order to assign appropriate points A,B,C,D to the obtained numerical values, it was necessary to use the analytic expressions for the Bragg frequency deviation. These expressions are not affected by the gain modulation, and have the following form: in case of TM polarization, and in case of TE polarization.
The numerical solution of Equations (23)- (26) and (30)-(33) for the wide range of coupling coefficient is divided into two stages. In the first phase the gain expansion coefficients α 1 , α 2 , and α 3 are neglected, and the equations and their solutions are reduced to known forms, e.g., see [46]. The second step uses the solutions obtained in first stage and iteratively solves Equations (23)- (26) and (30)  In each case (TM and TE polarization), the doubly degenerate modes are orthogonal and show saddle-shaped patterns. The slight discrepancies arise from numerical inaccuracy. All nondegenerate modes are similar and exhibit Gaussian-like pattern, and this suggests that these modes should more efficiently use the photonic cavity. These modes (A) also have lower threshold, Figure 11. increasing the value of coupling coefficient the Bragg frequency deviation increases and the threshold gain decreases. Simultaneously, the value of threshold gain saturates for all modes and eventually tends to similar values. This tendency is a consequence of growing field confinement in the cavity (for high index contrast all modes become Gaussian-like). For this reason the mode designation for higher values of coupling coefficients is difficult and only possible by the careful comparison of frequency deviation δ, and threshold gain α values. It is also worth noting that the threshold gain values for mode A are the lowest in wide range of coupling coefficient.
(a) (b) The impact of simultaneous gain and index coupling is depicted in Figure 12, where threshold gain for mode A is plotted as a function of coupling coefficient |κ 3 L| ∈ ( 0.01; 50 ) . The characteristics compare the structure with gain (solid line) and without gain coupling (dashed line). It can be easily observed that the nonuniformity of the gain in the low index contrast structures has a strong impact on the threshold gain and cannot be disregarded. Therefore, by inducing gain coupling in the index coupled structure it is possible to lower threshold gain particularly for low index contrast photonic crystals.

Triangular lattice -TM and TE polarization
By repeating all the calculations shown for square lattice structures, threshold characteristics for triangular lattice structures are obtained. In Figure 13  Bragg frequency deviation (for points marked as A,B,C,D,E,F in Figure 13) depending on coupling coefficient is analytically expressed in the following form for the TM polarization: and for TE polarization:  The values of the normalized coupling coefficients for TM and TE polarization are set as follows: |κ 1 L| = 7.0, |κ 2 L| = 3.3, |κ 3 L| = 2, and the value of the filling factor f=0. 16.
In both discussed cases, all degenerate modes are orthogonal and show similar patterns. For TM polarization, Figure 14, modes B,C are very similar to the non-degenerate mode A. This means that the coupling coefficients, for which they are plotted, have high enough values to achieve strong field confinement. Similar situation is shown for TE polarization, Figure 15, where two pairs of doubly-degenerate modes are comparable to non-degenerate mode. Likewise, it is due to relatively high values of coupling coefficients and mode confinement.
In Figure 16   decreases. Simultaneously, for larger values of coupling coefficient the threshold gain tends to alike values. This fact is due to the growing field confinement in the cavity (all modes become Gaussian-like, e.g., Figure 14 and Figure 15. The difference in the threshold gain values of degenerate modes stems from numerical inaccuracy, and the degenerate modes' threshold gain values should be averaged. structure has a significant effect on the threshold gain characteristics. They are substantially changed in the lower range of coupling coefficient values and slightly lower the threshold gain in the entire range. This shows that the index and gain coupled structures can lower the threshold gain, especially in low index contrast photonic crystals. Now, as an example of the model utilization let us consider the square lattice PC structure, which schematic cross-section is shown in Figure 1 a). For this structure following parameters are assumed: cavity length L = 50 nm, lattice constant a = 290 nm, and filling factor f = 0.16. The background material has higher permittivity than the rods ε a < ε b , and the active material is situated in the rods α a ≠ 0, α b = 0. As schematically shown in Figure 12 a) for lower values of the coupling coefficient, i.e., a low refractive index difference, (e.g., κ 3 L ∈ ( 0.01, 0.1 ) ) this structure has lower lasing threshold than it would have if the gain was uniformly distributed in the Advances in Photonic Crystals 130 medium, i.e., α a = α b . Thus for example: ε b = 12.00 and ε a = 11.94, then κ 3 L = 0.09 and the threshold gain drops by a factor of ~1.5. The supported lasing wavelength in such a cavity amounts to λ = aε 0 1/2 , where ε 0 1/2 = ε a f + ε b ( 1 -f ) , that is λ ∼ 1μm.

Perspectives
This Chapter discusses only some problems in threshold operation of 2D PC lasers. Thus, future work should be devoted to further investigation of gain coupling in photonic crystal cavities, e.g., such as comparison of solely index and solely gain coupled structures. Moreover, an above threshold analysis for gain coupled PC laser may apply as well as it did to index coupled structures, e.g., see [44], [45]. Finally, since more and more works on three-dimensional structures are published, it seems interesting to develop coupled wave models for threshold analysis of different symmetries incorporating gain and index modulation.

Conclusions
This work presents the systematic studies on the threshold operation of two-dimensional photonic crystal laser. It gives the comprehensive coupled mode description of gain and index coupled photonic crystal laser threshold operation. The calculations are conducted in the wide range of coupling coefficient for all four cases (square and triangular lattice with TM and TE polarization). It has been shown that the nonuniformity of the gain in the low index contrast structures has a strong impact on the threshold gain, by lowering it. Consequently, by inducing gain coupling in the index coupled structure it is possible to lower threshold gain particularly for low index contrast photonic crystals. This outcome helps understand the principles of PC band-edge laser operation and it may be useful in supporting the design process of PC laser structures.