1. Introduction

Photonic crystal (PC) lasers based on the band-edge effect have attracted much attention due to the large-area coherent oscillation that they produce [1–13]. We have developed these PC lasers to realize continuous-wave operation at room temperature [7] using a square-lattice PC, and have produced a range of beam patterns including circular single-lobed and doughnut-shaped beams [8]. We have also extended coupled-wave theory [14] to include square-lattice PCs and have derived a set of coupled-wave equations [15,16] that allows the basic resonant mode properties to be analyzed. For example, electromagnetic field distributions, radiation loss (output power) from the PC cavity and thus the threshold gains will be obtained. We have used these equations to obtain analytic expressions that describe the relations between the resonant mode frequencies and the coupling constants [15]. These analytic expressions allow the feedback strength in square-lattice PC cavities to be evaluated.

In order to further develop the PC laser, it is important to investigate different crystal geometries such as the triangular lattice; this has six-fold rotational symmetry and is thus expected to have resonant modes with stronger two-dimensionality and higher controllability than the square-lattice PC with four-fold rotational symmetry. A PC laser with triangular-lattice geometry has been studied [1,2] and was found to have six resonant modes at the edge of the photonic band structure, as shown in Fig. 1. Although some of these resonant mode properties have been studied, such as the electromagnetic field [4,6], analytic expressions that allow the feedback strength in triangular-lattice PC cavities to be evaluated have not yet been obtained. In this paper, we present a coupled-wave model and derive analytic expressions that describe the relations between the resonant mode frequencies and the coupling constants for triangular-lattice PC cavities without introduced defects. In section 2 of this paper, we derive a set of coupled-wave equations for a triangular-lattice PC with a transverse electric (TE) polarization and then derive the analytic expressions that describe the relations between the resonant mode frequencies and the coupling constants. In section 3, we use these expressions to calculate the resonant mode frequencies, which we compare with those obtained using the two-dimensional (2D) plane wave expansion method (PWEM) to assess the validity of our model. The coupling constants of previously fabricated devices are then evaluated using their measured resonant mode frequencies. Using our analytic expressions, we are thus able to evaluate the feedback strength of triangular-lattice PC cavities in fabricated samples.

 

Fig. 1. Photonic bandstructure for triangular lattice photonic crystal with a TE mode. The right-hand figure shows the detailed structure in the dotted circle, which has six dispersion curves. Each curve has a band edge that is indicative of the associated resonant mode (A – D). Modes B and D are doubly degenerate.

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2. Coupled-wave model for triangular lattice

The PC structure investigated here consists of a triangular lattice of circular holes in the x-y plane with period a, as shown in Fig. 2(a). The structure is assumed to be uniform in the z-direction. The circular holes form a 2D Bravais lattice with sites given by the vectors:

Here a ₁ and a ₂ are the two primitive translation vectors of the lattice, while l ₁ and l ₂ are any two integers. The area enclosed by the primitive unit cell of this lattice is a_c=|a ₁ ×a ₂|=√3a ²/2.

 

Fig. 2. (a). Triangular lattice photonic crystal. The black arrows indicate the primitive translation vectors of the lattice. (b). The reciprocal lattice of (a). The black arrows indicate the primitive reciprocal lattice vectors. The gray arrows indicate the wavevectors considered in this paper.

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The corresponding reciprocal lattice is shown in Fig. 2(b) and the reciprocal lattice vectors G(h) are given by:

Here h ₁ and h ₂ are any two integers, denoted collectively by h, and the primitive translation vectors of this lattice are given by:

and

where a ⁽ⁱ⁾ _j is the j ^th Cartesian component, x or y, of a _i (i=1 or 2). If we express the primitive translation vectors as a ₁=(√3a/2, a/2) and a ₂=(0,a), as shown in Fig. 2(a), the primitive reciprocal lattice vectors are b ₁=(4π/√3a, 0) and b ₂=(-2π/√3a, 2π/a), as shown in Fig. 2(b).

The scalar wave equations for the magnetic field H_z in the TE mode are written in the form [17]:

Here the constant k is given by [18]:

In the above expression λ is the wavelength of the light in free space, ε _G is the Fourier coefficient of the modulated dielectric constant ε(r), ε ₀ (=ε _G=0) is the averaged dielectric constant, and α _G is the Fourier coefficient of the modulated gain constant α(r). In PC lasers, it is assumed that the gain is small over distances of the order of a wavelength, and that the modulations of the dielectric constant and gain constant are small, such that:

These assumptions allow us to express the constant k in the form:

Here, α(=α _G=0/2) is the averaged gain constant and κ _G is the coupling constant defined as:

It should be noted that we consider the resonance at the Γ-point with the smallest non-zero frequency in the photonic bandstructure [4], in which the coupling constants for |G|=4π/√3a, 4π/a, 8π/√3 a contribute significantly. We list the corresponding coupling constants as:

where β ₀=4π/√3a. In a periodic structure, the magnetic field is given by the Bloch mode [17]:

where H _G is the amplitude of each plane wave, and k is a wavevector in the first Brillouin zone that becomes zero at the Γ-point. In principle, a periodic perturbation of the medium generates an infinite set of diffraction orders. However, at the specific Γ-point discussed in this paper, only the amplitudes H _G with |G|=β ₀ are significant. All other amplitudes are small and can be neglected. Therefore, six waves with |G|=β ₀, as indicated by the gray arrows in Fig. 2(b), are considered in our model. These six waves propagate in the PC structure and interfere with each other due to diffraction by the circular holes. As a result, the amplitudes of the waves become position-dependent. We describe these waves using the complex amplitude H_i (r) (i=1 to ~6), and rewrite the expression for the magnetic field as the sum:

In view of Eq. (7), these amplitudes vary slowly and their second derivatives can thus be neglected. By substituting Eqs. (8), (9) and (14) into Eq. (5), using expressions (10)–(12), and comparing the exponential terms, we obtain six equations of the form:

The parameter δ is a normalized frequency defined by:

where n is the averaged refractive index which is equal to ε ₀ ^1/2 and c is the speed of light in free space. The parameter δ is a measure of the deviation of the oscillation frequency ω from the Bragg frequency ω ₀. Because this frequency deviation is assumed to be small, we have set β/β ₀≈1 in the above derivation.

The above set of equations expresses the coupling of waves propagating in the triangular-lattice PC structure. For example, Eq. (15-1) describes the coupling of waves H ₁ and H ₄ that travel in opposite directions; the coupling constant is κ ₃. The same equation also describes the coupling of waves that propagate in oblique directions. That is, wave H ₁ propagating along the x-axis couples to waves H ₂ and H ₆ with the coupling constant κ ₁, and it also couples to waves H ₃ and H ₅ with the coupling constant κ ₂. These oblique couplings with constants κ ₁ and κ ₂ provide 2D optical feedback, which gives rise to coherent 2D oscillation. By numerically solving the set of equations (15) under some boundary conditions, the eigenvalues α and δ provide the threshold gain and frequency of the resonant mode in triangular-lattice PC cavities, respectively. Note that, for other crystal geometries such as square-lattice PCs, different set of equations, as shown in Ref. 15, are required.

If we assume an infinite periodic structure, we can use Eq. (15) to obtain analytic expressions that describe the relations between the resonant mode frequencies and the coupling constants. Neglecting the derivatives and the threshold gain α in Eq. (15), the normalized frequency δ can be derived from the condition that the determinant of the matrix formed by the amplitude coefficients is zero. Using Eq. (16), the analytic expressions for the resonant modes (A - D) are obtained in the form:

Here, modes B and D are doubly degenerate. Given the coupling constants of the specific PC structure under consideration, one can then calculate the resonant mode frequencies using the set of equations (17).

Equation (17) can also be derived by considering the point group C₆. By using the array of group characters for an irreducible representation, we can obtain the same results, (e.g. ϕ=(1, 1, 1, 1, 1, 1) [19] leads to Eqs. (17-2)).

3. Resonant mode frequencies and coupling constants

In order to assess the validity of the analytic expressions (17), we now obtain the theoretical coupling constants and calculate the resonant mode frequencies for a PC with circular holes. We then compare the resultant mode frequencies with those obtained using the 2D PWEM. As shown in Eq. (9), the coupling constants are found using the Fourier coefficients of both the modulated dielectric constant and the gain constant. Using the dielectric constant ε_a and gain constant α_a of the circular hole, and the dielectric constant ε_b and gain constant α_b of the surrounding material, as shown in Fig. 2(a), the Fourier coefficients are obtained in the form [17]:

Here f is the filling factor (the fraction of the area occupied by the circular hole) and A is the region inside the circular hole. For a circular hole with radius R, Eqs. (18) and (19) can be expressed for G≠0 as:

Here, J ₁(x) is a Bessel function of the first kind for integer order 1 and the filling factor is given by f=(2π/√3)R ²/a ². By substituting Eqs. (20) and (21) into Eq. (9) and using λ ₀=aε ₀ ^1/2, the theoretical coupling constant for a circular hole is given by:

We calculate the coupling constant using the parameters in Ref. 4, where ε_a=10.5625, ε_b=10.6834, a=462 nm, α_a=0, and α_b=0. The calculated coupling constants are plotted as a function of the filling factor in Fig. 3. At f≈0.9, neighboring holes are in contact with each other. The coupling constant κ ₃ becomes zero at f≈0.25 and the coupling constant κ ₂ becomes zero at f≈0.33, where the feedback effect induced by each type of coupling vanishes. Even if these coupling constants vanish, the coupling constant κ ₁ still provides the 2D optical feedback. Therefore the 2D lasing oscillation can be still maintained.

 

Fig. 3. Coupling constants as a function of the filling factor for a circular hole with ε_a=10.5625, ε_b=10.6834, and a=462 nm.

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Fig. 4. Normalized resonant mode frequencies as a function of the filling factor, calculated using (a) Eq. (17), and (b) the 2D PWEM (225 waves).

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In Fig. 4(a) we present the resonant mode frequencies as a function of the filling factor, calculated using Eq. (17) and the coupling constants from Fig. 3. The corresponding frequencies calculated using the 2D PWEM are shown in Fig. 4(b). The two sets of results are in good agreement; the minor deviations are due to the different number of waves used in each method. We used 6 waves to determine the resonant mode frequencies from Eq. (17), while 225 waves were used in the 2D PWEM.

Using the set of analytic expressions (17), we can extract the coupling constants from the resonant mode frequencies:

Using the resonant mode frequencies (ω=a/λ) calculated from the lasing wavelength in Ref. 4 (ω _A=0.35906, ω _B=0.35942, ω _C=0.35965, and ω _D=0.35987), the following coupling constants are obtained: κ ₁~130 cm^-1, κ ₂~74 cm^-1, and κ ₃~39 cm^-1. Taking into account the diameter of the lasing area, L=480 µm, the strength of the coupling between waves propagating in opposite directions is found to be κ ₃L≈1.87, while the strength of the oblique couplings are found to be κ ₁L≈6.24 and κ ₂L≈3.55. According to the 1D coupled wave theory [14], if the coupling strength in the opposite direction κ ₃L is about 1~2, then the electromagnetic field intensity uniformly distributes throughout the cavity. In addition, sufficient 2D optical coupling is considered to be achieved since the oblique coupling strength κ ₁L and κ ₂L have larger values than the 1D optical coupling strength κ ₃L.

4. Conclusion

We have presented a coupled-wave model for a triangular-lattice 2D PC with a TE polarization and derived a set of coupled-wave equations. We have used these equations to obtain analytic expressions that describe the relations between the resonant mode frequencies and the coupling constants. The resonant mode frequencies were then calculated as a function of the filling factor for a PC with circular holes. Our calculated frequencies are in good agreement with those obtained using the plane wave expansion method. The coupling constants in previously fabricated devices were evaluated using our set of analytic expressions. The method presented in this paper allows the feedback strength of PC cavities in fabricated devices to be evaluated.