Coupled Mode Theory of Photonic Crystal Lasers

The main property of photonic crystal is the existence of a frequency range, for which the propagation of electromagnetic waves in the medium is not permitted. These frequency ranges are commonly known as photonic band gaps, giving the ability to modify the structure parameters, e.g. group velocity, coherence length, gain, and spontaneous emission. This type of periodic structures is used in both passive and active devices.


Introduction
Photonic crystals (PC) are structures with periodic variation of the refractive index in one, two or three spatial dimensions.The dynamic development of experimental and theoretical work on photonic crystals has been launched by Yablonovitch (1987;1993) and Sajeev John (1987) publications, although the idea of periodic structures had been known since Strutt (1887).
The main property of photonic crystal is the existence of a frequency range, for which the propagation of electromagnetic waves in the medium is not permitted.These frequency ranges are commonly known as photonic band gaps, giving the ability to modify the structure parameters, e.g. group velocity, coherence length, gain, and spontaneous emission.This type of periodic structures is used in both passive and active devices.
Lasers with defect two-dimensional photonic crystals are known for their high finesse (Monat et al. (2001)) and very low threshold (Nomura et al. (2008)).
Photonic crystal band-edge lasers allow to obtain edge (Cojocaru et al. (2005)) and surface emission (Turnbull et al. (2003); Vurgaftman & Meyer (2003)) of coherent light from large cavity area.They also allow to control the output beam pattern by manipulation of the primitive cell geometry (Iwahashi et al. (2010); Miyai et al. (2006)), provide low threshold (Susa (2001)), and beams which can be focused to a size less than the wavelength (Matsubara et al. (2008)).

Radiation generation modeling in photonic crystal lasers
Laser action in photonic crystal structures has been theoretically studied and centered on the estimation of the output parameters (Czuma & Szczepanski (2005); Lesniewska-Matys et al. (2005)) and models describing light generation processes e.g.(Florescu et al. (2002); Koba, Szczepanski & Kossek (2011); Sakai et al. (2010)).The most sophisticated and general (it describes one-, two-, and three-dimensional structures) semiclassical model of light generation in photonic structures is presented in (Florescu et al. (2002)).Theoretical analysis of photonic crystal lasers based on two-dimensional plane wave expansion method (PWEM) (Imada et al. (2002); Sakai et al. (2005)) and finite difference time domain method (FDTD) (Imada et al. (2002); Noda & Yokoyama (2005)) 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 developed.These methods are meant to effectively support the design process of such lasers.They are based on a coupled mode theory (Sakai et al. (2006); Vurgaftman & Meyer (2003)) and focused on square and triangular lattice photonic crystals (Koba & Szczepanski (2010); Koba, Szczepanski & Kossek (2011); Koba, Szczepanski & Osuch (2011); Sakai et al. (2007;2010;2008)).
The Sakai et al. (2007;2010) works contain a mathematical description and numerical results of the threshold analysis of two-dimensional (2-D) square 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.Additionally, in (Sakai et al. (2007)) 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 Sakai et al. (2008) paper, the analytical description of triangular lattice photonic crystal cavity for TE polarization has been given.In this work the analysis was focused on the coupled wave equations and the dependence of the resonant frequencies on the coupling coefficients.
In Sakai et al. (2007;2010;2008) works threshold analysis has been conducted for specific values of coupling coefficient and TM polarization for triangular has not been considered.
The equations for triangular lattice photonic crystal laser with TM polarization has been shown in Koba, Szczepanski & Kossek (2011), and the evaluation of these is shown in this chapter.
The mentioned semiclassical model, presented by Florescu et al. (2002) describing an above threshold analysis is complicated and difficult to implement.To overcome this drawback, this chapter also includes an overview of our works (Koba & Szczepanski (2010); Koba, Szczepanski & Kossek (2011); Koba, Szczepanski & Osuch (2011)), where we introduced easy to implement models for an above threshold analysis of a two-dimensional photonic crystal laser.

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Structure definition
This paper describes the two-dimensional photonic crystals which properties can be described by the complex relative electrical permittivity ε.The cross sections of these structures are shown schematically in Fig. 1  In crystallography the ideal crystal is described by the elementary cell.The shape of the cell is defined by the basic vectors which linear combination allows to specify the location of all nodes of the structure.Each node is connected to the base which may be constituted by an atom, a group of atoms, molecules, etc.The photonic structures perfectly resemble the microscopic nature of the crystal lattice in the mesoscopic scale.This allows using the terminology adopted in the solid state physics to describe the photonic crystal.
In this chapter, only 2-D photonic crystals will be discussed.In the two-dimensional space, there are five basic types of crystal lattice.This comprises a square, hexagonal, rectangular, oblique, and rhombic lattice (Kittel (1995)).The square and hexagonal (also known as triangular) lattices are the most common types of symmetry used in the practical realizations of photonic cavities.rods or holes depending on the relative difference between the refractive index of the cylinders and the surrounding material.
The structures in Fig. 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 the z direction.The permittivity of the holes and background material is ε a and ε 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.
where a 1 and a 2 are primitive vectors (Kittel (1995)), l 1 and l 2 are arbitrary integers, x specifies the placement on the plane, x = xx + ŷy, where x and ŷ are unit vectors along x and y axis, respectively.The area of primitive cell is a c = |a 1 × a 2 | = a 2 in case of square lattice, and a c = |a 1 × a 2 | = √ 3a 2 /2 in case of triangular lattice.Primitive vectors for square lattice are described by the expressions: a 1 = (a,0), a 2 = (0, a), and for the triangular lattice: In general, the reciprocal vecotrs 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 lattice: where a (i) j is the j-th cartesian component (x or y)o ft h ea i vector (i = 1 lub 2) (Sakai et al. (2010)).
Using Equation 3 and the expressions for square and triangular lattice primitive vectors the reciprocal primitive vectors are described by the following formulas: (5) The infinite square or triangular photonic crystal can be described in terms of relative permittivity by the functions: in case of TE polarization, where it is more convenient to use the inverse of relative permittivity, and for TM polarization.In previous Equations, function 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.

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Coupled Mode Theory of Photonic Crystal Lasers www.intechopen.com The functions describing the structure need to be transformed to the frequency domain in order to solve the wave equations.To do so, the crystal geometry is expressed in terms of reciprocal lattice vector by the Fourier transformation of functions 6 and 7 (M.Plihal & Maradudin (1991); M. Plihal et al. (1991)).
For TE polarization function ǫ −1 (G) is written in the following form: and for the TM polarization function ǫ(G): where f = πr 2 /a 2 -square lattice filling factor, f = 2π/ √ 3 r 2 /a 2 -triangular lattice filling factor, r -rod radius, J 1 -Bessel function of the first kind.
In further parts of this chapter four different cases have been analyzed.Two of them are dedicated to square lattice cavities with TE and TM polarization, and two remaining to triangular lattice structures also with TE and TM polarization.
In the next parts of this chapter the threshold and above threshold analysis of the photonic crystal laser operation has been shown for the defined structures.

Coupled-wave equations
In general, the scalar wave equations for the electric and magnetic fields E z and H z , respectively, are written in the following form (M. Plihal & Maradudin (1991); M. Plihal et al. (1991)): where the constant k is given by (Sakai et al. (2007))  2010)) in case of TE modes.In Equations 13 and 14 β = 2πε 1/2 0 /λ where ε 0 = ε (G = 0) is the averaged dielectric permittivity (ε 1/2 0 corresponds to averaged refractive index n), α is an averaged gain in the medium, κ(G) is the coupling constant, λ is the Bragg wavelength, and G = (mb 1 , nb 2 ) is the reciprocal lattice vector, m and n are arbitrary integers, b 1 and b 2 vary depending on the structure symmetry.Therefore, these vectors are expressed in the following forms b 1 = β s 0 ,0 and b 2 = 0, β s 0 for square lattice, and b 1 = β t 0 ,0 and b 2 = −β t 0 /2, √ 3β t 0 /2 for triangular lattice structure, where β s 0 = 2π/a and β t 0 = 4π/ √ 3a.In the derivation of Equations 13 and 14 following e.g.(Sakai et al. (2007)), we set α and α G ≪ β.In these equations the periodic variation in the refractive index is included as a small perturbation and appears in the third term through the coupling constant κ (G) of the form: In Equation 15, plus sign refers to TM polarization (Equation 13), while minus sign refers to TE polarization (Equation 14).Furthermore, we set α(G)| G =0 = 0 neglecting spatial periodicity of gain.In the vicinity of the Bragg wavelength only some of the diffraction orders contribute in a significant way, where in general, a periodic perturbation produces an infinite set of diffraction orders.Therefore the Bragg frequency orders have to be cautiously chosen.The Bragg frequency corresponding to the Γ point in the photonic band structure, e.g.(Sakai et al. (2007)) is chosen for the purpose of this paper, and the most significantly contributing coupling constants are expressed as follows: In Equations 11 and 12 electric and magnetic fields for the infinite periodic structure are given by the Bloch modes, (M.Plihal & Maradudin (1991); Vurgaftman & Meyer (2003)): 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.(Sakai et al. (2006)).For a finite structure, the amplitude of each plane wave is not constant, so e(G) and h(G) become functions of space.At the Γ point we consider only the amplitudes (e(G), h(G)) which are meant to be significant, i.e. in most cases with |G| = β s,t 0 , except for square lattice with TE polarization where additional h(G) amplitudes

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Coupled Mode Theory of Photonic Crystal Lasers www.intechopen.com with |G| = √ 2β s 0 have to be included (Sakai et al. (2006)).The contributions of other waves of higher order in the Bloch mode are considered to be negligible.

Square lattice -TM polarization
Considering square lattice photonic crystal with TM polarization, it is assumed that at the Γ point the most significant contribution to coupling is given by the electric waves which fulfill the condition |G| = β s 0 .Thus, all higher order electric wave expansion coefficients |G| ≥ √ 2β 0 are negligible.Four basic waves most significantly contributing to coupling are depicted in Fig. 4. Equation 17 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.(Sakai et al. (2007)).Thus, the electric field given by Equation 17 in the finite periodic structure can be expressed in the following way: In Equation 19E s i , i = 1..4 are the four basic electric field amplitudes propagating in four directions +x, −x, +y, y.These amplitudes correspond to e(G) in Equation 17.In the further analysis, we will drop the space dependence notation.
Knowing the reciprocal lattice vectors for the square lattice PC, the coupling coefficients κ(G) 16 can be written as: Putting Equations 13 and 19 into Equation 11, and assuming the slow varying electromagnetic field, one can get the set of coupled mode equations (Sakai et al. (2007)): is the Bragg frequency deviation, κ 2 and κ 3 are coupling coefficients expressed by Equations 21 and 22 (Sakai et al. (2007)).The κ 2 coefficient is responsible for orthogonal coupling (e.g. the coupling of E s 1 to E s 2 and E s 4 ), and κ 2 corresponds to backward coupling (e.g. the coupling of E s 1 to E s 3 ).The additional coefficient κ 0 denotes surface emission losses, and it is proportional to κ 1 (Sakai et al. (2007;2010)).Solution of Equations 23-26 for the boundary conditions: defines eigenmodes of the photonic structure.The analysis of this solution will be shown in section 3.2.

Square lattice -TE polarization
In the square lattice photonic crystal cavity with TE polarization, as mentioned before, the coupling process involves magnetic waves satisfying following conditions: (|G| = β 0 ) and |G| = √ 2β 0 , (Sakai et al. (2010)), neglecting higher order Bloch modes.Eight basic waves most significantly contributing to coupling are depicted in Fig. 5.
Similarly as in the case of TM polarization, the equation for magnetic field (Equation 18) describes modes for infinite structure.Thus, the finite dimensions of the structure are described by spatial dependence of magnetic field amplitudes (Sakai et al. (2010)), and the magnetic field 18 is written in the following form: In Equation 29H s i , i = 1..8 are the eight basic magnetic field amplitudes of waves propagating in directions schematically shown in Fig. 5.These amplitudes correspond to h(G) in Equation 18. Joining Equations 14, 29, and 12, and assuming slowly varying amplitudes, the coupled In Equations 30-33, the spatial dependence of H s i , i = 2, 4, 6, 8 amplitudes was neglected, and it was assumed that α ≪ δ.In Equations 30-33, δ is the Bragg frequency deviation, given by 27.The coupling coefficients κ 1 , κ 2 , and κ 3 , defined by Equations 16 are expressed by (Sakai et al. (2010;2008)): 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 defines structure eigenmodes at lasing threshold i.e. in the linear case.

Triangular lattice -TM polarization
In the triangular lattice photonic crystal cavity with TM polarization, the coupling process involves waves satisfying following conditions (|G| = β 0 ), neglecting higher order Bloch modes (Koba, Szczepanski & Kossek (2011); Sakai et al. (2008)).Six basic waves most significantly contributing to coupling are depicted in Fig. 6.The space dependent amplitudes for electric field e(G) (Equation 17) in triangular lattice photonic crystal cavity are written in the following form (Koba, Szczepanski & Kossek (2011)): In Equation 38, E t i , i = 1..6, are the six electric field amplitudes propagating in the symmetry directions, Fig. 6.Combining Equations 13, 38 and 11, and assuming slowly varying amplitudes, the coupled wave equations for TM modes in triangular lattice PC are obtained: 301 Coupled Mode Theory of Photonic Crystal Lasers In Equations 39-44, like in the case of square lattice, δ is the Bragg frequency deviation, given by Equation 27, while κ 1 , κ 2 , and κ 3 are the coupling coefficients, which are defined by 16 and as follows (Koba, Szczepanski & Kossek (2011)): These coefficients describe strength and direction of the coupling of the waves, e.g. the coupling of E t 1 and E t 4 is described by κ 3 , the coupling of E t 1 , E t 2 , and E t 6 by κ 1 , and the coupling of E t 1 , E t 3 , and E t 5 by κ 2 .In Equations 39-44, there is an additional coefficient κ 0 which, like in the square lattice case, is responsible for surface emission losses (Kazarinov & Henry (1985); Vurgaftman & Meyer (2003)).Solution of Equations 39-44 for the boundary conditions: defines structure eigenmodes at lasing threshold.

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Triangular lattice -TE polarization
In the triangular lattice photonic crystal cavity with TE polarization, the coupling process involves waves satisfying the same condition as it was stated in TM polarization case, i.e.
(|G| = β 0 ), (Sakai et al. (2008)), neglecting higher order Bloch modes.Six basic waves most significantly contributing to coupling are depicted in Fig. 7.The magnetic field amplitudes h(G) (Equation 18) in the triangular lattice photonic crystal cavity are written as follows (Sakai et al. (2008)): In Equation 49, H t i , i = 1..6, are the six magnetic field amplitudes propagating in the symmetry directions, Fig. 7. Combining Equations 14, 49 and 12, and assuming slowly varying magnetic field amplitudes, the coupled wave equations for TE modes in triangular lattice PC are obtained: Coupled Mode Theory of Photonic Crystal Lasers 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:  2006)): in case of TM polarization, and in case of TE polarization.These expressions were obtained from Equations 23-26 and 30-33 where no gain (α = 0), no loss (κ 0 = 0, α L = 0), and no spatial dependence of electric or magnetic field amplitude were assumed.Sets of Equations 23-26 and 30-33 were solved numerically for the wide range of coupling coefficients (κ 1 , κ 2 , κ 3 ).We grouped obtained solutions: , where κ 3i corresponds to subsequent values of coupling coefficient for different modes j = A, B, C, D; m = 1..4, s-denotes square lattice.
Assigning numerical values of δ j to analytical solutions 60 and 61 (δ A , δ B,C , δ D ), we obtained the mode structure of 2-D square lattice PC laser with TM and TE polarization.9a), D -Fig.9b), B,C -Fig.9c), d) for TM modes, and A -Fig. 10a), B -Fig. 10b), C, D -Fig.10c), d) for TE modes.The plots were made for the normalized coupling coefficients |κ 1 L| = 10.96,|κ 2 L| = 8, |κ 3 L| = 4 and filling factor f = 0.16.In each case (TM and TE polarization), the doubly degenerate modes are orthogonal and show saddle-shaped patterns.All non-degenerate modes are similar and exhibit Gaussian-like pattern, and this suggests that these modes should more efficiently use the photonic cavity.These modes also have lower threshold , Fig. 11.
In Fig. 11a) and 11b), the normalized threshold gain αL was plotted as a function of Bragg frequency deviation δL, for various values of the normalized coupling coefficient |κ 3 L| (it takes values from 0.01 to 50).mode designation is only possible by the frequency deviation δ.It is also worth noting that the threshold gain values for mode A are the lowest in wide range of coupling coefficient.These modes (A for TM and TE polarization) by having the lowest threshold and by using the active medium in the most efficient way, are favored for lasing.

Triangular lattice -TM and TE polarization
Repeating all the calculations shown for square lattice structures, we obtained threshold characteristics for triangular lattice structures.In Fig. 12 enlarged areas of triangular lattice photonic crystals dispersion curves for the first six modes (A,B,C,D,E,F) in the vicinity of Γ point are shown.Fig. 12a) coresponds to TM polarization, and Fig. 12b) refers to TE polarization.The circular holes ε b = 9.8 arranged in triangular lattice with background material ε a = 12.0 were assumed.The rods radius to lattice constant ratio was set to 0.24.
In each plot, i.e.Fig. 12a) and Fig. 12b), there can be two pairs of doubly degenerate modes observed: B,C and D,E for TM polarization, and B,C and E,F for TE polarization (they have the same frequency at the Γ point).Modes A have the lowest frequency.
Bragg frequency deviation (for points marked as A,B,C,D,E,F in Fig. 12) depending on coupling coefficient is analytically expressed in the following form for the TM polarization:

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Coupled Mode Theory of Photonic Crystal Lasers Photonic Crystals Book 1 and for TE polarization:

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Coupled Mode Theory of Photonic Crystal Lasers orthogonal and show similar patterns.For TM polarization, Fig. 13, modes B,C are very similar to the non-degenerate mode A. This means that the coupling coefficients have values for which the modes tend to converge.Similarly for TM polarization, Fig. 14, where two pairs of doubly-degenerate modes are similar to non-degenerate mode A. Likewise, it is due to high values of coupling coefficients and mode convergence.
In Fig. 15a), and 15b) the normalized threshold gain αL was plotted as a function of Bragg frequency deviation δ, for various values of the normalized coupling coefficient |κ 3 L|∈ (0.01; 50).

An above threshold analysis
The above threshold analysis of light generation in square and triangular lattice two-dimensional photonic crystal laser is based on the energy theorem, presented in e.g.(Koba & Szczepanski (2010)).The introduction of the energy theorem into previously presented coupled wave equations is straightforward but requires laborious calculations.This section presents the results of these calculations, while accurate derivations can be found in (Koba & Szczepanski (2010); Koba, Szczepanski & Kossek (2011); Koba, Szczepanski & Osuch (2011)).
At the basis of the described analysis lies a statement that the energy generated in the structure is equal to the energy leaving the structure and the energy lost in it.In general, the gain coefficient is a function of a small signal gain coefficient α 0 , saturation intensity I S , electric field intensity in the laser structure I, and the shape of gain bandwidth.In the case of a homogenous broadening and the laser action near resonance the gain coefficient is expressed in the following form: In this equation I in = ∑ i |E i | 2 denotes noncoherent component of the electric field, whereas I coh = ∑ i =j E i E * j is the coherent component, and is responsible for the spatial hole burning effect.The strength of this effect is described by the phenomenological coefficient η ∈ (0, 1).Equations presented in this section describe the relations between normalized small signal gain coefficient and the laser output power, structure losses, and structure coupling coefficient.

Square lattice -TM and TE polarization
In order to obtain the expressions describing the small signal gain coefficient in square lattice photonic crystal laser for TM and TE polarization we used the sets of coupled wave Equations 23 -26 and 30 -33, (Koba & Szczepanski (2010); Koba, Szczepanski & Osuch (2011)).We added 310 Photonic Crystals -Introduction, Applications and Theory www.intechopen.comthese sets of equations respectively with their complex conjugates and into each obtained equation we introduced the expression for the nonlinear gain Equation 64.These steps led us to the equations for small signal gain with above threshold field distributions.We replaced the above threshold distributions with the threshold field distributions which we found by numerical solutions of the sets of Equations 23-26 and 30-33.The accuracy of this threshold approximation has been discussed in (Szczepanski (1985)).The final expressions for the small signal gain coefficient of square lattice photonic crystal laser are: where and in case of TM polarization, and where and in case of TE polarization.In these equations E t i , i = 1..4 and H t i , i = 1, 3, 5, 7 are the electric and magnetic field amplitudes at the lasing threshold (Koba & Szczepanski (2010); Koba, Szczepanski & Osuch (2011)).

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Triangular lattice -TM and TE polarization
Expressions describing the small signal gain coefficients for triangular lattice photonic crystal laser are obtained in the analogical way as we have done for square lattice structure.All necessary calculations can be found in (Koba, Szczepanski & Kossek (2011); Koba, Szczepanski & Osuch (2011)).The starting points for these calculations are Equations 39-44 and 50-55 for TM and TE polarization, respectively.The small signal gain coefficient in triangular lattice photonic crystal laser with TM polarization is described as follows: where and and for the TE polarization: where and for TE polarization.In Equations 67 and 68 E t i , i = 1..6 and H t i , i = 1, 3, 5, 7 are the electric and magnetic field components at the lasing threshold, respectively.
In Equations 65-68, the distinguished factors M th , W th , and T th are associated with total power in the structure, outgoing power, and the spatial hole burning effect.Moreover, in case of TE polarization, an additional factors T TE th and M TE th are included to take into account the electric dipole interaction in terms of magnetic field.
Equations 65-68 allow us to plot the characteristics showing the behavior of small signal gain for different structure parameters.

Numerical analysis
This section is devoted to the analysis of numerical solutions of Equations 65-68.As mentioned earlier, the field distributions in Equations 65, 66, 67, and 68 are those which exist at lasing threshold.We obtained these threshold field distributions by numerically solving the sets of the coupled equations 23-26, 30-33, 39-44, and 50-55.The presented results describe above threshold operation of square and triangular lattice photonic crystal laser with TM and TE polarization.These results include nonlinear gain, structure imperfections losses, surface emission losses and spatial hole burning effect.In this section we discus modes which are marked as A in Fig. 8 and 12, section 3.In case of square lattice, we set the coupling coefficients ratios constant, and they are κ 2 /κ 3 = 2 and κ 1 /κ 3 = 2.74 (this corresponds to the filling factor f = 0.16).Whereas, for triangular lattice we set κ 1 /κ 3 = 3.49 and κ 2 /κ 3 = 1.65, which is related to the same filling factor as in square lattice case, i.e. f = 0.16.Constant ratio of the coupling coefficients corresponds to the situation in which the relative refractive indexes difference vary, but the filling factor remains the same, e.g.Equations 20-22 or 45-47.In the lossless structure with an increasing coupling strength (i.e., increasing Q-factor of the cavity), the small signal gain required to maintain given output power monotonically decreases.This tendency changes, when we introduce losses.In this situation (depicted by dashed lines in Fig. 16 and 17) plotted curves have minima within the considered values of the coupling coefficient κ 3 L.The minima are caused by nonlinear gain, i.e. the gain saturation effect.Their depth and curve shape depends on the output power P out /P S , refractive index difference, and filling factor.The minima represent the lowest value of small signal gain for considered system parameters.Thus, for each power level and given other structure parameters, there exists an optimal coupling strength that results in the minimal small signal gain required to maintain that output level.The small signal gain is related to the active medium pumping rate, thus we expect that the pumping level of the laser structure is also minimal.Therefore, we can say that for the optimal coupling strength the laser structure operates at the maximal power efficiency.Moreover, with an increasing output power level, the optimal coupling strength is shifted towards lower values (Koba & Szczepanski (2010); Koba, Szczepanski & Kossek (2011);Koba, Szczepanski & Osuch (2011)).

Perspectives
Here, we point out a interesting path for further investigation of photonic crystal lasers.In this chapter we discussed 2-D PC lasers, but since a lot of publications on three-dimensional (3-D) coupled mode theory are issued e.g.(Hamam et al. (2007)) and 3-D photonic crystal lasers are developed e.g.(Tandaechanurat et al. (2011)) it would be interesting to introduce this 3-D theory to PC lasers.This formulation would have to face some important issues, e.g. the estimation of the number of coupling waves, and increasing number of coupled equations, but it would give a crucial insight into 3-D photonic cavities.

Conclusions
In our work we have presented the systematic studies on the threshold and above threshold two-dimensional photonic crystal laser operation.We have shown the comprehensive coupled mode description of photonic crystal laser threshold operation, completing the works of Sakai et al. by presenting the threshold model for triangular lattice structure with TM polarization.Moreover, we conducted our calculations in the wide range of coupling coefficient for all four cases (square and triangular lattice with TM and TE polarization), which also has not yet been done.In addition, we have presented an approximate method of the above threshold analysis of a 2-D photonic crystal laser operation.We showed the approximate formulas for the small signal gain coefficients as a function of system parameters.Furthermore, we made necessary calculations to obtain above threshold characteristics, which depicted that it is possible to attain the optimal coupling strength providing maximal power efficiency of a given 2-D photonic laser structure.We believe that our analysis and methods could be useful in supporting the design process of a laser structure and help understand the principles of photonic crystal band-edge laser operation.
Fig. 1. a) Square and b) triangular lattice photonic structure cross section.(ε a and ε b are relative permittivities of rods and background material, respectively, a -lattice constant, Lcavity length)

Fig. 2 .
Fig. 2. The scheme of a) a square lattice photonic crystal with primitive vectors; and b) its representation in reciprocal space with reciprocal primitive vectors.

Fig. 3 .
Fig. 3.The scheme of a) a triangular lattice photonic crystal with primitive vectors; and b) its representation in reciprocal space with reciprocal primitive vectors.

Fig. 2 a
Fig. 2 a) and 3 a) show photonic crystal cross sections in xy plane with cylinders arranged in square or triangular lattice with period a, and with depicted primitive vectors a 1 and a 2 .Fig. 2 b) and 3 b) show the reciprocal lattices corresponding, respectively, to the real square and triangular lattice.In the described case, the nodes of a two-dimensional structure can be

Fig. 4 .
Fig. 4. Schematic cross section of square lattice photonic crystal laser active region, where the four basic waves involved in coupling for TM polarization are shown.
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Fig. 5 .
Fig. 5. Schematic cross section of square lattice photonic crystal laser active region, where the eight basic waves involved in coupling for TE polarization are shown.wave equations for TE modes in square lattice PC are obtained (Sakai et al. (2010)):

Fig. 6 .
Fig. 6.A schematic cross section of a triangular lattice photonic crystal laser active region, where the six basic waves involved in the coupling for TM polarization are shown.

Fig. 7 .
Fig. 7.A schematic cross section of a triangular lattice photonic crystal laser active region, where the six basic waves involved in the coupling for TE polarization are shown.

FigFig. 9 .Fig. 10 .Fig. 11 .
Fig.11a) and 11b) show that by increasing the value of coupling coefficient the Bragg frequency deviation increases and the threshold gain decreases.Simultaneously, for larger

Fig. 12 .
Fig. 12.An enlarged area of dispersion curves of photonic crystal for the first four modes in the vicinity of Γ point.Triangular lattice, a) TM polarization, and b) TE polarization.

Fig. 15
Fig.15shows similar tendency as in square lattice examples, i.e. by increasing the values of coupling coefficient the Bragg frequency deviation increases and the threshold gain decreases.Simultaneously, for larger values of coupling coefficient the threshold gain tends to similar values.This fact is due to the growing field confinement in the cavity (all modes become Gaussian-like, e.g.Fig.13 and 14).The mode designation is only possible by obtaining the frequency deviation δ values.The difference in the threshold gain values of degenerate modes stems from numerical inaccuracy, and the threshold gain values should be averaged.

Fig. 16 .Fig. 17 .
Fig. 16.Normalized small signal gain α 0 L vs. the normalized coupling constant κ 3 L with the normalized output power level P out /P S as a parameter, for two values of the normalized losses in the structure, α L L = 0 (solid line) and α L L = 0.05 (dashed line).Surface emission loss κ 0 = 0. Square lattice photonic crystal structures with a)TM, and b)TE polarization.

Fig. 16
Fig.16and 17 represent normalized small signal gain coefficient α 0 L as a function of the normalized coupling constant κ 3 L with the normalized output power level P out /P S as a parameter, for two values of the normalized losses in the structure, α L L = 0 (solid line) and α L L = 0.05 (dashed line), respectively.
Photonic Crystals -Introduction, Applications and Theory described in the previous (TM) case, whereas the coupling coefficient κ 1 describes the coupling of e.g.waves H s 1 , H s 2 , and H s 8 .Solution of Equations 30-33 for the following boundary conditions: 300 Photonic Crystals -Introduction, Applications and Theory www.intechopen.commeaning as